Advances in Boundary Element & Meshless ...

Advances in Boundary Element & Meshless Techniques XVII 150

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Edited by Münevver Tezer-Sezgin Bülent Karasözen Ferri M.H. Aliabadi

The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques. Previous conferences devoted to Boundary Element and Meshless Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012), Paris, France (2013), Florence, Italy (2014), Valencia, Spain (2015).

ISBN 978-0-9576731-3-7

Advances in Boundary Element & Meshless Techniques XVII

150

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50

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Edited by Münevver Tezer-Sezgin Bülent Karasözen Ferri M.H. Aliabadi

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Advances In Boundary Element & Meshless Techniques XVII

Advances In Boundary Element & Meshless Techniques XVII

Edited by Münevver Tezer-Sezgin Bülent Karasözen Ferri M.H. Aliabadi

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Published by EC, Ltd, UK Copyright 2015, Published by EC Ltd, P.O.Box 425, Eastleigh, SO53 4YQ, England Phone (+44) 2380 260334

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ISBN 978-0-9576731-3-7

The text of the papers in this book was set individually by the authors or under their supervision. No responsibility is assumed by the editors or the publishers for any injury and/or damage to person or property as a matter of product liability, negligence or otherwise, or from any used or operation of any method, instructions or ideas contained in the material herein.

International Conference on Boundary Element and Meshless Techniques XVII 11-13 July 2016, Ankara, Turkey Organising Committee: Prof. M¨ unevver Tezer-Sezgin Prof. B¨ ulent Karas¨ ozen Institute of Applied Mathematics & Department of Mathematics Middle East Technical University 06800 Ankara, Turkey [email protected], [email protected] Prof. Ferri M. H. Aliabadi Department of Aeronautics Imperial College, South Kensington Campus London SW7 2AZ UK [email protected] International Scientific Advisory Committee: Abe, K (Japan) Baker, G (USA) Benedetti, I (Italy) Beskos, D (Greece) Bl´ azquez (Spain) Chen, Weiqiu (China) Chen, Wen (China) Cisilino, A (Argentina) Darrigrand, E (France) De Araujo, F C (Brazil) Denda, M (USA) Dong, C (China) Dumont, N (Brazil) Estorff, O.v (Germany) Gao, X.W. (China) Garca-S´ anchez (Spain) Hartmann, F (Germany) Hematiyan, M.R. (Iran)

Hirose, S (Japan) Kinnas, S (USA) Liu, G-R (Singapore) Mallardo, V (Italy) Mansur, W. J (Brazil) Mantiˆc, V (Spain) Marin, L (Romania) Matsumoto, T (Japan) Mesquita, E (Brazil) Millazo, A (Italy) Minutolo, V (Italy) Ochiai, Y (Japan) Panzeca, T (Italy) Pérez Gavilán, J J (Mexico) Pineda, E (Mexico) Procházka, P (Czech Republic) Qin, Q (Australia) Sáez, A (Spain) Sapountzakis, E.J. (Greece) Sellier, A (France) Semblat, J-F (France) Seok Soon Lee (Korea) Shiah, Y (Taiwan) Sládek, J (Slovakia) Sládek, V (Slovakia) Sollero, P. (Brazil) Stephan, E P (Germany) Taigbenu, A (South Africa) Tan, C L (Canada) Telles, J C F (Brazil) Wen, P H (UK) Wrobel, L C (UK) Yao, Z (China) Zhang, Ch (Germany)

PREFACE The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques in both boundary element methods (BEM) and boundary integral equation methods (BIEM). Previous successful conferences devoted to Boundary Element Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012), Paris, France (2013), Florence, Italy (2014) and Valencia, Spain (2015). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the Cultural and Convention Center, Middle East Technical University, Ankara, Turkey during 11-13th July 2016. Research papers received from 7 countries formed the basis for the Technical Program. The themes considered for the technical program included solid mechanics, fluid mechanics, potential theory, composite materials, fracture mechanics, damage mechanics, contact and wear, optimization, heat transfer, dynamics and vibrations, acoustics and geomechanics. The conference organizers would also like to express their appreciation to the International Scientific Advisory Board for their assistance in supporting and promoting the objectives of the meeting and for their assistance in the form of reviews of the submitted papers. Editors

CONTENT DRBEM for Solving Natural Convection in a Nanofluid-Filled Enclosure Nagehan Alsoy-Akg¨ un

1

Particle-particle interactions in 2D MHD viscous flow Sel¸cuk Han Aydın, Antoine Sellier

9

Providing Fundamental Solutions in Time Domain using Gene Expression Programming Mazda Behnia, Danial Behnia, Kamran Goshtasbi Goharizi

15

Meshless reconstruction of the support of a source B. Bin-Mohsin, D. Lesnic

23

Numerical solution to MHD pipe flow in annular-like domains Canan Bozkaya, M¨ unevver Tezer-Sezgin

29

The Role of the Second Pair of Wings in Insect Flight A simple vortex approach to complex multi-wing unsteady applying problems in 2-D Mitch Denda

35

Performance Prediction of Wings Moving Above Free Surface Ali Dogrul, S ¸ akir Bal

43

A Fast-Multipole Implementation of the Simplified Hybrid Boundary Element Method Ney A. Dumont, Hélvio F. C. Peixoto

51

Linear analysis of heterogeneous microstructures by the Boundary Element Method Guilherme A. Ohland, Jordana F. Vieira, Matheus C. dos Santos, Gabriela R. Fernandes

59

Effects of Slip Boundary Conditions on Mixed Convection flow of Nanofluids in a Lid-Driven Cavity Sevin G¨ umg¨ um MHD Stokes flow in a smoothly constricted rectangular enclosure Merve G¨ urb¨ uz, M¨ unevver Tezer-Sezgin Reducing the condition number of the coefficient matrix in the method of fundamental solutions for 2D Laplace equation M.R. Hematiyan, M. Arezou, A. Haghighi Performance Prediction of 2D Foils Moving Above and Close to Free Surface ¨ Omer Kemal Kınacı, S¸akir Bal

67

73

79

85

2D Boundary element formulation for deformable particle flow in a microchannel Cem Kurt, Barbaros C ¸ etin, and Besim Barano˘glu Dual Reciprocity Boundary Element Solution of a System Modeling Acid-Mediated Tumor Cell Invasion G¨ ulnihal Meral

93

99

DRBEM solution to ferrofluid flow and heat transfer in a semi-annulus enclosure in the presence of magnetic field F.Sidre Oglakkaya, Canan Bozkaya

107

RBF-PS Solution of the Brinkman-Forchheimer-extended Darcy model in a porous medium ¨ urk, Bengisen Pekmen Yasemin Ozt¨

113

Fast numerical convolution with the Sparse Cardinal Sine Decomposition Francois Alouges, Matthieu Aussal, Emile Parolin

119

Fundamental three-dimensional MHD creeping flow bounded by a plane solid and motionless wall Antoine Sellier

125

A boundary formulation for the axisymmetric MHD slow viscous flow about a sphere translating parallel with a uniform ambient magnetic Antoine Sellier, Sel¸cuk H. Aydın

131

Flow in a Square Cavity with an Obstacle under the Influence of a Non-uniform Magnetic Field Pelin S ¸ enel, M¨ unevver Tezer-Sezgin

139

A Solenoidal-Galerkin Approach for the Numerical Simulation of Flow Past a Circular Cylinder Hakan I. Tarman

147

Boundary integral solution of MHD pipe flow M. Tezer-Sezgin, Canan Bozkaya

155

POD Analysis of Drag Reduction in Turbulent Pipe Flow Ozan Tu˘gluk, Hakan I. Tarman

161

Numerical simulation of MHD duct flow problems using BEM and DGFEM approaches Hamdullah Y¨ ucel, Canan Bozkaya, M¨ unevver Tezer-Sezgin

167

DRBEM for Solving Natural Convection in a Nanofluid-Filled Enclosure Nagehan Alsoy-Akgün

Keywords: DRBEM, Natural convection flow, Nanofluid.

Abstract. A numerical investigation of unsteady, two-dimensional natural convection flow of nanofluids in a square cavity with a heat source placed at the bottom wall. Dual reciprocity boundary element method (DRBEM) is applied to solve governing equations for the problem in the form of stream function, vorticity and temperature. In the solution procedure, two modified Helmholtz equations are obtained for the vorticity transport and energy equations using forward difference approximation with relaxation parameters for the time derivatives. Therefore, the need of another time integration scheme is eliminated and stability problems are diminished. Also, DRBEM is a boundary only solution method. So, this provides the computations easier and less expensive than the other domain discretization methods. In this study, DRBEM is carried with the fundamental solution of Laplace equation and modified Helmholtz equation for stream function and vorticitytemperature equations, respectively. Unknown vorticity boundary conditions and nonlinear terms are treated using the coordinate matrices F which are constructed using the coordinate functions f = 1 + r for stream function and f = r2 ln r for both vorticity and energy equations. Results are given for copper-water based nanofluid to show that the effect of the problem parameters. It is observed that, the length and location of the heat source effects cooling performance of the fluids.

Introduction Natural convection heat transfer is an important phenomenon for many engineering areas and different numerical methods are used to analyze it. Heat exchanger devises must have higher performance and also they must be small and light. Conventional heat transfer fluids such as water and engine oil are used to improve their performance but they have low thermal conductivity. Using nanoparticles together with these fluids, their heat transfer capacities can be enhanced which is introduced by Choi [7]. In the literature, the are many studies about nanofluids which are used different numerical techniques. In this study, natural convection flow of nanofluids in a square cavity is solved by using DRBEM. Solution procedure is carried for the stream function-vorticity form of the governing equations after transforming the vorticity and temperature equations to the modified Helmholtz equations. By using this idea, the need of another time integration scheme is eliminated. Stream function Poisson equation and resulting modified Helmholtz equations are solved by DRBEM with the fundamental solutions of ln(x) and fundamental solution of K0 (x), respectively. The basic idea of the DRBEM is to approximate the forcing term by a series of radial basis functions f j which are related to a series of particular solutions by ∇2 uj = f j . To obtain a particular solution analytically for the Laplace operator L = ∇2 and the biharmonic operator L = ∇4 can be done by repeated integrations [5]. For that reason most of the differential equations were restricted to the form ∇2 = f (x, y, u, ux , uy ) when the DRBEM was used [5]. For the Helmholtz-type operator this technique can not be used directly. To obtain the particular solution of the modified Helmholtz equation different methods were experienced and thin plate splines was used as f j in [4].

Governing Equations Let us consider the continuity, momentum and energy equations for a Newtonian, incompressible and laminar fluid, which are governed by unsteady Navier-Stokes equations. In stream function (ψ), vorticity (w) and temperature (T ) the non-dimensional governing equations are

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Eds: M. Tezer-Sezgin, B. Karasözen, M H Aliabadi

∇2 ψ = −w (ρβ )n f ∂ T µn f 2 ∂w ∂w ∂w ∇ w = +u +v − RaPr ρn f α f ∂t ∂x ∂y ρn f β f ∂ x αn f 2 ∂T ∂T ∂T ∇ T = +u +v αf ∂t ∂x ∂y

(1)

with the relations u = ∂∂ψy , v = − ∂∂ψx , and w = ∂∂ xv − ∂∂ uy where u and v are the non-dimensional velocity components. Ra is Rayleigh number and Pr is Prandtl number. The parameters µn f , ρn f and αn f are viscosity, effective density and thermal diffusivity of the nanofluid defined by [6] µn f =

µf , (1 − ϕ)2.5

ρn f = (1 − ϕ)ρ f + ϕρs ,

αn f =

kn f (ρCp )n f

where the subscripts ’s,’ ’ f ’ and ’n f ’ refer to solid, fluid and nanofluid, respectively. Also, µ f is the dynamic viscosity of the fluid, ϕ is solid volume fraction, ρ is density and C p is the specific heat at constant pressure. Here kn f is the thermal conductivity of nanofluid and it is approximated by using the Maxwell-Garnett’s model for spherical nanoparticles as in [6, 2] (ks + 2k f ) − 2ϕ(k f − ks ) . kn f = k f (ks + 2k f ) + ϕ(k f − ks ) Also, (ρCp ) is the heat capacity of nanofluid and (ρβ )n f is the thermal expansion coefficient of nanofluid which is located in the Boussinesq term and they are defined as (ρC p )n f = (1 − ϕ)(ρCp ) f + ϕ(ρCp )s ,

(ρβ )n f = (1 − ϕ)(ρβ ) f + ϕ(ρβ )s ,

respectively. The time derivatives of vorticity and temperature in the governing equations are approximated (n+1) (n) (n+1) (n) as ∂∂tw = w ∆t−w and ∂∂tT = T ∆t−T , where w(n) = w(x, y,tn ), T (n) = T (x, y,tn ), tn = n∆t and ∆t is the time increment. The unknowns in the Laplace terms are relaxed as w(n+1) = θw w(n+1) + (1 − θw )w(n) and T (n+1) = θT T (n+1) + (1 − θT )T (n) with 0 < θw , θT < 1. Inserting these approximations into the (1) and rewriting one can obtain the following iterative equations for the natural convection flow of water-based nanofluid ∇2 ψ (n+1) = −w(n) ρn f α f ∂ ψ (n+1) ∂ w(n) ∂ ψ (n+1) ∂ w(n) (θw − 1) 2 (n) ∇2 w(n+1) − λw2 w(n+1) = ∇ w − λw2 w(n) + − θw µn f θw ∂y ∂x ∂x ∂y (ρβ )n f α f ∂ T (n) β f µn f θw ∂ x αf ∂ ψ (n+1) ∂ T (n) ∂ ψ (n+1) ∂ T (n) (θT − 1) 2 (n) − ∇ T − λT2 T (n) + θT αn f θT ∂y ∂x ∂x ∂y

−RaPr ∇2 T (n+1) − λT2 T (n+1) = ρ α

(2)

α

f f f and λT2 = αn f ∆tθ , and n indicates iteration number. These coupled equations are ready to where λw2 = µn nf ∆tθ w T solve with DRBEM. In this study, Cu is considered as a nanoparticle with uniform shape and size and the thermophysical properties of nanoparticle and base fluid (water) are given in Table (1). The dimensionless local Nusselt number on the

Table 1: Pure water Cu

Thermophysical properties of base fluid nanoparticles [6].

ρ(kgm−3 ) 997.1 8933

surface of heat source is given as [6] Nus (X) =

Cp (Jkg−1 K −1 ) 4179 385 1 Ts (X)

k(W m−1 K −1 ) 0.613 401

β × 10−5 (−1 K) 21 1.67

where Ts is local dimensionless heat source temperature.

Advances in Boundary Element and Meshless Techniques XVII

0.2

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13 12

0.18

φ=0 φ=0.05 φ=0.1 φ=0.15 φ=0.2

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Nus

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0.12

0.08

0.06

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7 6

0.55

0.6

5

0.65

0.35

0.4

0.45

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Figure 1: Profile of local temperature (left) and local Nusselt number (right) of Cu-based nanofluid for several solid volume fractions when Ra = 105 , B = 0.4 and D = 0.5.

Method of Solution In the iterative form of the non-dimensional stream function, vorticity and temperature equations, we have one Poisson and two modified Helmholtz equations. So, DRBEM solution of governing equations are obtained from ∇2 ψ = b1 ∇2 w − λw2 w = b2 (3) ∇2 T − λT2 T = b3 where b1 , b2 and b3 are previously known right hand sides of corresponding equations in (2). In the DRBEM procedure, the governing equations in (2) are transformed into boundary integral equations. For this aim, first, 1 ln x) and modified Helmholtz they are weighted by using the fundamental solutions of Laplace equation (u∗ = 2π 1 ∗ equation (u = 2π K0 x) as in [1]. The domain integrals caused by the nonhomegeneties are transformed to the boundary integral equation by approximating the right hand side functions, given by b ≈

K+L

∑ α j f j where K and

j=1

L are the numbers of boundary and interior points, respectively, as in [5]. f j ’s are radial basis functions where f j = 1 + r j is used for b1 and f j = r2j ln r j is used for b2 and b3 . Here, r j denotes the distance between the source and field points and α j are initially unknown coefficients. After discretization of the boundary using K constant elements, we obtain the matrix-vector equations as (n+1)

Hψ (n+1) − Gqψ

(n+1) H w(n+1) + G qw (n+1) (n+1) HT + G qT

(n)

− GQ ψ )F −1 b = (H Ψ 1 w )F −1 b(n) + G Q = (H W 2

T )F −1 b(n) = (H T + G Q 3

(4)

where ψ, w, T and qψ , qw , qT are vectors containing discretized values of stream function, vorticity, temperature coefficient matrices and their entriesare given and their normal derivatives. H, G, H and G are the DRBEM 1 1 ∂ ∂ 1 1 1 ln dΓ j , Hi, j = dΓ j in [5, 3, 4] as Hi, j = ln (K0 (λ ri )) dΓ j , Gi, j = 2π Γ j ∂ n ri 2π Γ j ∂ n 2π Γ j ri 1 Gi, j = K0 (λ ri )dΓ j where Γ j is the j-th boundary element and λ refers to λw and λT for vorticity transport 2π Γ j and energy equations, respectively. F and F are the coordinate matrices containing f j = 1 + r j and f j = r2j ln r j Q ψ , W w and T, Q T are matrices , Q as columns, respectively where ri j as the distance from the point i to j. Ψ, j = 1 + r j, which are constructed using particular solutions and their normal derivatives for the equations ∇2 ψ j − λw2 w j = r2j ln r j and ∇2 Tj − λT2 Tj = r2j ln r j . Using the initial values of vorticity and temperature, the ∇2 w

4

Figure 2:


Streamlines, vorticity and temperature contours (from top to bottom) of Cu-based nanofluid for several Rayleigh numbers when ϕ = 0.1,

B = 0.4 and D = 0.5.

steady-state solutions are obtained by solving equations (4), iteratively. Solution procedure starts with the solution of stream function equation. The new stream function values are used to compute the velocity vectors and vorticity boundary conditions. Then, the vorticity transport and energy equations are solved using new stream function values and the values of temperature and vorticity which belong to previous time level.

Numerical Results In this work, we solve the unsteady natural convection flow of water based nanofluid in a square cavity Ω = [(0, 1) × (0, 1)] with a heat source at the bottom wall. No slip boundary conditions for velocity are assumed on all the walls. So stream function boundary conditions are taken zero. All the wall of the cavity except the bottom wall are cooled (T = 0). The adiabatic boundary conditions are imposed on the bottom wall where kf ∂T ∂T ∂ y = 0 at 0 ≤ x < (D − 0.5B) and (D + 0.5B) < x ≤ 1, and ∂ y = − kn f at (D − 0.5B) ≤ x ≤ (D + 0.5B).

Vorticity boundary conditions are obtained using the definition w = ∂∂ xv − ∂∂ uy by using DRBEM coordinate matrix. In the computations, to obtain smooth graphs, the highest number of boundary elements (K = 124) and interior nodes (L = 225) are used for all values of parameters. The stopping criteria to achieved steady state results is ε = 10−5 for all unknowns. The relaxation parameters θw and θT are taken 0.9 which accelerate the convergence to steady state using more contribution from newly obtained results. It is noticed that as Ra and B increases we need smaller ∆t which varies between 0.1 and 0.005. The profiles of local temperature and local Nusselt number along the heat source for different values of volume fractions are given in Figure 1. From the figures they are not uniform and, the surface temperature takes the maximum value and corresponding Nusselt number takes minimum value at the middle of the heater.


Figure 3:

5

Streamlines, vorticity and temperature contours (from top to bottom) of Cu-based nanofluid for different heat source length when ϕ = 0.1,

Ra = 105 and D = 0.5.

The effect of Rayleigh number on Cu−based nanofluid is presented in Figure 2. The results are given for Ra = 103 − 106 with B = 0.4, D = 0.5 and φ = 0.1. In this analysis, the heater is placed in the middle of the bottom wall, so the symmetrical behaviors are observed from all graphs. There are two circular fluid cells occur for streamlines where they take the maximum and minimum values at the center of the left and right cells for all values of Rayleigh number. The vorticity contours show the similar behavior as forming two circular cells at the left and right side of the cavity. The temperature contours occur near the heater and due to the adiabatic boundary conditions all the contours are perpendicular to the bottom wall. When the Rayleigh number increases the shape of streamlines cells do not change but the intensity of convection increases. The center of the vorticity cells move towards the heater and the boundary layer occur near the vertical walls and heater. The temperature contours show different behavior for higher values of Rayleigh number where the boundary layer be formed near vertical and top walls. These results are good agreement with the results in [6]. Figures 3 and 4 show the effect of heat source length and location on the streamlines, vorticity and temperature, respectively. The behaviors of the contours are similar with the previous case. But, in this case, when the heater length increases the heat generation rates increases. Thus, the intensity of the convection increases and the higher temperature pattern occurs in the cavity. The effect of the heater location can be seen very clear from the Figure 4. When the heater is located near the left wall, a new circular pattern with different size occur in the streamlines. When the heater moves towards to the middle of the bottom wall, it becomes grow until they have equal size and intensity. The same behaviors take place at the vorticity contours. The temperature contours follow the heater and when the heater arrive the middle of the bottom wall, they take the symmetrical form again. Also, the maximum flow temperature decreases near the left wall. All these behaviors are observed in [6].

6

Figure 4:


Streamlines, vorticity and temperature contours (from top to bottom) of Cu-based nanofluid for different location of heat source when

ϕ = 0.1, Ra = 105 and B = 0.4.

Conclusion In this work, DRBEM solution of natural convection flow of water based nanofluid in a square cavity with a heat source at the bottom wall is presented. The numerical results show that present method is capable of solving the problem without difficulties and it has advantages of giving good accuracy with considerably small number of boundary elements.

References [1] C. A. Brebbia, The Boundary Element Method for Engineers, Pentech Press, London (1984). [2] H.F. Oztop and E. Abu-Nada, Numerical study of natural convection in a partially heated rectangular enclosures filled with nanofluids, International Journal of Heat and Fluid Flow, 29, 1326-1336 (2008). [3] M. Tezer-Sezgin, Boundary Element Method Solution of MHD Flow in a Rectangular Duct, Int. J. for Num. Methods in Fluids, 18, 937-952 (1994). [4] N. Alsoy-Akgün and M. Tezer-Sezgin, DRBEM and DQM Solution of Natural Convection Flow in Cavity Under a Magnetic Field, Progress in Computational Fluid Dynamics, 13, 270-284 (2013). [5] P. W. Partridge, C. A. Brebbia, and L. C. Wrobel, The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton (1992).


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[6] S.M. Aminossadati and B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, European Journal of Mechanics B/Fluids, 28, 630-640 (2009). [7] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of Non-Newtonian Flows, FED-vol 231,66:99–105, 1995.

Particle-particle interactions in 2D MHD viscous flow 1

S. H. Aydin1 and A. Sellier2 Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey 2 LadHyx. Ecole polytechnique, 91128 Palaiseau Cedex, ´ France

Keywords: 2D MHD Stokes flow, Boundary-integral equation, particle-particle interactions, asymptotic analysis.

Abstract. Particle-particle interactions in 2D MHD Stokes flow are asymptotically investigated for two widely-separated solid disks experiencing prescribed rigid-boby motions in a conducting Newtonian liquid subject to a uniform magnetic field. These interactions are characterized by the so-called 6 × 6 resistance matrix having entries calculated from the knowledge of the traction, per unit length, arising on each disk contour. The Cartesian components of this traction are governed, whatever the particles location (i. e. for close or distant ones), by coupled Fredholm boundary-integral equations of the first kind recently obtained in [9] and here asymptotically inverted for two distant particles. The behaviour of the entries of the resistance matrix is then found to deeply depend upon the addressed entry and the cluster orientation with respect to the uniform ambient magnetic field. Introduction Determining the flow, with velocity u and pressure p, about a moving solid body immersed in a conducting Newtonian liquid subject to a uniform ambient magnetic field B is a very challenging task. Indeed, one has to simultaneously obtain not only the so-called resulting MHD flow (u, p) but also the magnetic field B and electric field E induced in the liquid domain. More precisely, (u, p) is driven by the Lorentz body force fL = σ(E + u ∧ B ) ∧ B , with σ > 0 the liquid uniform conductivity, and one must then solve coupled unsteady Maxwell and non-linear incompressible Navier-Stokes equations [1-2]. Fortunately, in some basic circumstances the problem becomes more tractable [3]. One interesting case, handled in the present work, is for a uniform ambient magnetic field B = Be1 (with B > 0) and a plane solid body P both being normal to the unit vector e3 and the body translating at a velocity U such that U.e3 = 0 and/or rotating at the angular velocity Ωe3 . Indeed, it is then found [4] that E = 0 (i. e. there is no induced electric field in the liquid!) and one ends up with a 2D MHD flow (u, p) in the (O, x1 , x2 ) plane, to which the plane solid P belongs to, having pressure p = p(x1 , x2 ) and velocity u = u(x1 , x2 ) normal to e3 and with typical magnitude V > 0. For a body with length scale a and a liquid with uniform density ρ, viscosity µ and magnetic permeability µm it is useful to define the magnetic Reynolds number Rem , the Reynolds number Re and the Hartmann number M as follows (1) Rem = µm σV a, Re = ρV a/µ, M = a/d = aB/ µ/σ where d = ( µ/σ)/B is the so-called Hartmann layer thickness [5]. For a sufficiently small body or body migration one gets Re 1 and can ignore the non-linear convective term in the Navier-Stokes momentum equation therefore ending with the linear (unsteady) Stokes equations with Lorentz body force fL = σ(u ∧ B ) ∧ B (recall that E = 0). For most applications it turns out that Rem Re. Hence, one also gets Rem 1 and this property implies that the ambient magnetic field B is not disturbed by the body [1], i. e. that B = B in the entire liquid domain. Assuming moreover a quasisteady 2D MHD flow, (u, p) is then obtained by solving the, much more pleasant, steady linear Stokes equations with non-uniform body force fL = σ(u ∧ B) ∧ B. This steady flow (u, p) depends not only upon the plane body shape and rigid-body motion (characterized by the translational velocity U and angular velocity Ωe3 ) but also upon the Hartmann number M = a/d comparing the body length scale a with the Hartmann layer thickness d. The problem has been solved for a translating disk in [3,6] by expanding a stream function as an infinite serie of fundamental solutions involving modified Bessel functions. Recently, [7] proposed a new boundary approach to efficiently deal with the challenging case of a plane body of arbitrary shape experiencing either a translation or a rotation. The advocated

10

Eds: M. Tezer-Sezgin, B. Karasözen, M H Aliabadi

Figure 1: Two plane solid interacting particles P1 and P2 embedded in the x3 = 0 plane in a quiescent and unbounded conducting Newtonian liquid subject to the ambient uniform magnetic field B = Be1 with B > 0. (a) Arbitrary cluster. (b) 2-disk cluster. boundary technique appeals to a new fundamental MHD Stokes flow induced by a point force and obtained in [8]. An extension of this boundary approach has been also developed in [9] for two interacting and arbitrary-shaped particles. The flow about one solid particle has been found to exbibit far away from a particle a decay which deeply depends upon both the Hartmann number M and the direction (it decays either slowly or quicky in a direction normal or parallel to the ambient uniform field B, respectively). As a consequence, for a 2-particle cluster the particle-particle interactions are expected to strongly depend upon the cluster orientation with respect to B. The aim of this work is to investigate this issue for two distant disks by asymptotically solving the boundary-integral equations proposed in [9]. Governing problem and resistance matrix This section presents the governing equations for a 2-particle cluster and also introduces the associated resistance matrix which entirely characterizes the particle-particle interactions. Governing equations As sketched in Fig. 1(a), we consider two solid and plane particles P1 and P2 located in the x3 = 0 plane, with origin O and Cartesian coordinates (O, x1 , x2 ), immersed in a Newtonian conducting liquid with uniform density ρ, viscosity µ and conductivity σ > 0. Far from the cluster the liquid is quiescent with constant (zero) pressure and also subject to a prescribed uniform magnetic field B = Be1 with B > 0. Morevover, each particle Pn , with length scale an and attached point On , experiences a prescribed rigid-body motion described by its angular velocity Ω(n) e3 and translational velocity U(n) (here the velocity of point On ) normal to e3 . The resulting two-dimensional MHD liquid flow about the two-particle cluster with typical length scale a = M ax(a1 , a2 ), has pressure p(x1 , x2 ) and velocity u = u1 (x1 , x2 )e1 + u2 (x1 , x2 )e2 , with typical magnitude V > 0. As discussed in the introduction, there is no induced electric field in the liquid domain D. Moreover, assuming that Re = ρV a/µ 1 implies in practice that the magnetic field is B in the entire liquid domain and that (u, p) obey a Stokes MHD problem with lorentz body force fL = σ(u ∧ B) ∧ B. More precisely, if Pn has smooth (closed) contour Cn the flow (u, p) obeys the following problem µ∇2 u + σB 2 (u ∧ e1 ) ∧ e1 = ∇p and ∇.u = 0

in D,

(u, p) → (0, 0) as r = |OM| → ∞, u = ud on C = C1 ∪ C2

(2) (3)

with velocity ud given on the cluster boundary by ud = U(n) + Ω(n) e3 ∧ On M on Cn (n=1,2).

(4)

Resistance matrix The flow (u, p) has stress tensor σ and thus exerts on each smooth body contour Cn , with (see Fig. 1(a)) unit normal n pointing into the liquid, a force F(n) and a torque Γ(n) with respect to the attached point On . Those vectors satisfy (n) (n) σ.ndl = F1 e1 + F2 e2 , Γ(n) = On M ∧ σ.ndl = Γ(n) e3 . (5) F(n) = Cn

Cn

For convenience we henceforth use the tensor summation convention for repeated indices i, j, m, n (n) in {1, 2}. For instance, u = ui ei and U(n) = Uj ej . From (3)-(4) the flow (u, p) linearly depends

Advances in Boundary Element and Meshless Techniques XVII (1)

(1)

11 (2)

(2)

upon the generalized velocity vector X = (U1 , U2 , Ω(1) , U1 , U2 , Ω(2) ). Accordingly, there exists (n),(m) (n),(m) (n),(m) ei ⊗ ej , vectors B(n),(m) = Bi ei , C(n),(m) = Ci ei second-rank tensors A(n),(m) = Aij (n),(m) and also constants D such that F(n) = −µ{A(n),(m) .U(m) + B(n),(m) Ω(m) }, Γ(n) = −µ{C(n),(m) .U(m) + D(n),(m) Ω(m) }. (1)

(1)

(2)

(6)

(2)

Introducing the generalized force-torque vector F T = (F1 , F2 , Γ(1) , F1 , F2 , Γ(2) ) and designating by t Y the transposed vector of Y, the previous relations (6) receive the condensed form t T F = −µR.t X where R is the so-called 6 × 6 cluster resistance square matrix R defined as   (n),(m) (n),(m) (n),(m) (1),(1) A12 B1 A11 (1),(2) R R  (n),(m) (n),(m)  , R(n),(m) = A(n),(m) (7) R= . A22 B2 21 R(2),(1) R(2),(2) (n),(m) (n),(m) (n),(m) C1 C2 D

Clearly, the resistance matrix R fully characterizes the way the generalized force-torque vector F T and velocity vector X are related for a given cluster. It solely depends upon both the cluster geometry and the Hartmann number M (recall (1)). It also shows how the different motions (translation or rotation) of the particules interact. Finally, for arbitrary 2-particles clusters such as the one depicted Fig. 1(a) the following remarks hold: (i) The real-valued resistance matrix R is positive-definite and symmetric. Those basic properties, not proved here, imply that R is non-singular and therefore admits an inverse: the so-called mobility matrix M. (ii) Because R is symmetric it turns out that matrices R(1),(2) and R(2),(1) are transposed while each matrix R(n),(n) is symmetric. Therefore, R solely admits at the most 21 unknown entries (of course, whenever symmetries prevail this number decreases). The determination of those entries is achieved by successively considering some specific rigid-motions of the particles. For instance, the determination of the first column of R is readily obtained by keeping P2 at rest while P1 is translating (without rotating) at the unit velocity e1 . (iii) For particles ignoring each other (for instance very distant ones) R(n),(m) vanishes for n = m (n),(n) obtained when Pn is single in while R(n),(n) tends to the 3 × 3 symmetric resistance matrix Rs (n),(n) the liquid. For instance, for a single disk, with center On and radius an , the matrix Rs,disk admits the following simple diagonal form   (n) 0 0 A11   (n),(n) (n) Rs,disk =  0 (8) 0  A22 (n) 0 0 D (n)

(n)

with coefficients A11 , A22 and D(n) solely depending upon the disk radius an and the Hartmann num(n) (n) ber Mn = an /d. Note that [4] gives for Mn ≤ 1 the coefficients A11 and A22 but not the coefficient (n) D . Boundary approach and resulting asymptotic analysis for distant particles This section first recalls the boundary-integral equations obtained in [9] which govern the Cartesian components of the traction by unit length taking place on the cluster boundary. It then illustrates for one cluster motion how to asymptotically invert these equations for two distant disks.

Relevant boundary-integral equations As highlighted in the previous subsection, the determination of the 2-particle cluster resistance matrix appeals to the knowledge of the traction f = σ.n = fi ei exerted, by unit length, on each particle closed contour Cn for a few prescribed rigid-body motions of the particles. As mentioned in [9], it is possible to show that the required Cartesian components fi satisfy, for given rigid-body

12


motions u = U(n) +Ω(n) e3 ∧On M of the particles, four coupled Fredholm boundary-integral equations of the first kind on the cluster boundary C = C1 ∪ C2 . Setting n = nk ek , those equations read σB 2 1 − Gji (x, y)fi (y)dl(y) = uj (x) + [(u(x) ∧ e1 ) ∧ e1 ].y(1) Gji (x, y)ni (y)dl(y) 4πµ C 4πµ C1 1 1 ui (y)Tijk (y, x)nk (y)dl(y) − [ui (y) − ui (x)]Tijk (y, x)nk (y)dl(y) for x on C1 , (9) − 4π C2 4π C1 σB 2 1 Gji (x, y)fi (y)dl(y) = uj (x) + [(u(x) ∧ e1 ) ∧ e1 ].y(2) Gji (x, y)ni (y)dl(y) − 4πµ C 4πµ C2 1 1 ui (y)Tijk (y, x)nk (y)dl(y) − [ui (y) − ui (x)]Tijk (y, x)nk (y)dl(y) for x on C2 (10) − 4π C1 4π C2 where y(n) = y − OOn and the quantities Gji (x, y) and Tijk (y, x) have been obtained in [8]. Here we content ourselves with giving Gji (x, y) below while directing the reader to [8] for Tijk (y, x). Denoting by K0 and K1 = −K0 the usual modified Bessel function of order zero and order one, it has been established in [8] that x ˆ1 ˆ1 ˆ2 x ˆ1 rˆ rˆ x x ˆ1 rˆ x )K0 ( ) + sinh( )K1 ( ) , G12 (x, y) = sinh( )K1 ( ) , 2d 2d 2d 2d rˆ 2d 2d rˆ x ˆ1 ˆ1 x ˆ1 rˆ rˆ x G21 (x, y) = G12 (x, y), G22 (x, y) = cosh( )K0 ( ) − sinh( )K1 ( ) 2d 2d 2d 2d rˆ

G11 (x, y) = cosh(

(11) (12)

ˆ = x − y, x ˆ .ei and rˆ = |ˆ with the notations x î = x x|. Clearly, Gij (x, y) = Gji (y, x) = Gij (y, x). Asymptotic approximation of the resistance matrix We illustrate below the aymptotic procedure used to approximate the resistance matrix of a cluster made of two distant particles by considering, as shown in Fig. 1(b), a 2-disk cluster. More precisely, (n),(n) the disk Pn has center On and radius an and also if alone the 3 resistance matrix Rs,disk given by (8). We also set L = |O1 O2 | and introduce the angle θ in [0, π/2] such that O1 O2 = L(cosθe1 + sinθe2 ). The θ = 0 and θ = π/2 cases correspond to in-line and side-by-side disks (with respect to the magnetic field Be1 ). We show below how to asymptotically estimate some entries of the 2-disk cluster 6 × 6 resistance matrix R = (Rαβ ) (with α and β running in {1, ..., 6}) for distant disks by considering the cases for which P2 is at rest while P1 translates at a given velocity U = Uj ej . Upon introducing on C1 the new traction s(y) = f (y) + σB 2 {[(U ∧ e1 ) ∧ e1 ].y(1) }n(y), one can then cast (9)-(10) into the simple coupled boundary-integral equations 1 1 − Gji (x, y)si (y)dl(y) − Gji (x, y)fi (y)dl(y) = δn1 Uj for x on Cn (13) 4πµ C1 4πµ C2 where δ designates the usual Kronecker Delta. Inspecting (11)-(12) reveals that we can speak of distant disks when u = L/(2d) = M L/(2a) 1 where we set a= M ax(a1 , a2 ) and M = a/d is the Hartmann number. From [10] it turns out that K0 (u) ∼ K1 (u) ∼ π/(2u)e−u for large u. Accordingly, for x and y belonging to two different closed contours one gets the approximations Gji (x, y) ∼ Gji (u, θ) as u → ∞ with = u−1/2 e−u(1−cos θ) 1.

(14)

Curtailing the details, it is found from (11)-(12) that the O(1) constant quantities Gji read π π {1 + cos θ + (1 − cos θ)e−2u cos θ) }, G12 (u, θ) = sin θ{1 − e−2u cos θ }, (15) G11 (u, θ) = 8 8 π {1 − cos θ + (1 + cos θ)e−2u cos θ }. G12 (u, θ) = G21 (u, θ), G22 (u, θ) = (16) 8 Let us now select U = e1 in order to determine the first column of the cluster resistance matrix R, i .e the entries Rα1 . From (13)-(14) it clearly appears that, at the leading order, one gets on the contour


13

(1),1

C1 the approximation s ∼ fs which is the traction obtained on C1 when the disk P1 translates at the velocity e1 as single, i. e. in absence of the second disk. Recalling (8) note that we have the relations (1) (17) fs(0),1 (y).ei dl(y) = −µA11 δ1i . C1

Using again (13)-(14), one thus also obtains, at the leading order, for the traction on the second disk P2 the boundary-integral equation 1 (1) − (18) Gji (x, y)fi (y)dl(y) = − Gj1 A11 4πµ C2 4π which has the solution f =−

(1) [A ]{G11 fs(2),1 + G21 (fs(2),2 + σB 2 [(e2 ∧ e1 ) ∧ e1 ].y(2) n)} 4π 11

(19)

(2),i

designates the traction obtained on C2 when the disk P2 translates at the velocity ei in where fs absence of the first disk. After some manipulations one then obtains the following estimates R41 ∼ R41 , R41 = −[

G11 (1) (2) G12 (1) (2) a2 ]A A , R51 ∼ R51 , R51 = −[ ]A [A + M 2 π( )2 ]. 4π 11 11 4π 11 22 a

(20)

In a similar fashion, using the previous solution (19) for the traction on the contour C2 and exploiting (13) yields for the traction on the contour C1 the following expansion f = fs(1),1 −

2 {G1i R3+i1 fs(1),1 + G2i R3+i1 [fs(1),2 − σB 2 y(1) .e2 n]} 4π

(21)

(1),i

where of course fs designates the traction obtained on C1 when the disk P1 translates at the velocity ei in absence of the second disk. As the reader may easily check, one then arrives at the following additional asymptotic estimates (1)

R11 ∼ A11 {1 − R21 ∼ −

2 [G11 R41 + G12 R51 ]}, 4π

2 a1 (1) [G21 R41 + G22 R51 ][A22 + M 2 π( )2 ]. 4π a

(22) (23)

Inspecting (20) and (22)-(23) clearly reveals that different entries Rα1 exhibit different scalings for distant disks. Moreover, it turns out (recall (15)-(16)) that both G21 and G12 vanish for in-line (θ = 0) and side-by-side (θ = π/2) disks. For such cases it then appears that R51 = o() while R21 = o(2 ). Conclusions The boundary approach recently proposed elsewhere (see [9]), and consiting of 4 coupled boundaryintegral equations, has been used to asymptotically approximate the resistance matrix entries of a a cluster made of two plane and distant solid particles immersed in a conducting Newtonian liquid subject to a uniform ambient magnetic field. The procedure is illustrated for a 2-disk cluster by deriving asymptotic estimates of a few coefficients of the resistance matrix associated with a translation parallel to the magnetic field of one disk the second one being at rest. The case of other coefficients together with convincing comparisons against the results obtained by numerically inverting the boundary-integral equations will be both given and discussed at the oral presentation. References [1] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970).

14


[2] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [3] K. Gotoh Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 15 (4), 696-705 (1960). [4] H. Yosinobu and T. Kakutani Two-dimensional Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 14 (10), 1433-1444 (1959). [5] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [6] A. B. Tsinober and A. G. Shtern Stokes flow around a circular cylinder in a transverse magnetic field. MagnetoHydrodynamics, 3 (4), 146-147 (1967). [7] A. Sellier, M. Tezer-Sezgin and S. H. Aydin A new boundary approach for the 2D slow viscous MHD flow of a conducting liquid about a solid particle. In Advances in Boundary Element & Meshless Techniques XV. Editors V. Mantic, A. Saez and M. H. Aliabadi. 329-334 (2014). [8] A. Sellier, S. H. Aydin and M. Tezer-Sezgin Free-Space Fundamental Solution of a 2D Steady Slow Viscous MHD Flow. CMES, 102 (5), 393-406 (2014). [9] A. Sellier, S. H. Aydin and M. Tezer-Sezgin A new BEM approach for the slow 2D MHD flow about interacting soid particles. In Advances in Boundary Element & Meshless Techniques XVI. Editors V. Mantic, A. Saez and M. H. Aliabadi. 67-72 (2015). [10] M. Abramowitz and I. A. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York. (1965).

Providing Fundamental Solutions in Time Domain using Gene Expression Programming Mazda Behnia1, Danial Behnia2, Kamran Goshtasbi Goharizi3 1- Department of civil Engineering, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran, Email: [email protected] 2- Department of mining Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran, Email: [email protected] 3- Department of Mining Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran, Email: [email protected] Keyword: Boundary elements methods, GEP, Inverse Laplace transform, Fundamental solutions, Poroelasticity.

Abstract. The inverse Laplace transform of fundamental solution’s that have been found for some dynamic problems in the transformed domain could not be found analytically. So, numerical methods of inversion are required to evaluate the functions and ensued tractions which could be a time consuming procedure. In this paper an easy, versatile method is introduced to find a proper function to emulate the results of numerical inversion. Consequently the time consuming steps of numerical inversion could be avoided. A sample problem of saturated poroelasticity is mentioned and the method is implemented, however the procedure for any other cases could be the same. The present research attempts to prepare a surrogate function instead of the fundamental solutions in time domain using gene expression programming (GEP). With respect to the accuracy of the GEP method, they may be recommended in solutions of dynamic problems by the boundary elements methods in the time domain for future purposes. Introduction The Boundary Element Method (BEM) in the time domain is especially important to treat wave propagation problems in infinite and semi-infinite domains. Certainly this is not the only advantage of a time domain BEM, but very often the main motivation as, e.g., in earthquake engineering. The use of the powerful features of the boundary element method in reduction of mesh size and dimension of matrixes is another expedient to use of this method. These salient features are owed to the fundamental solutions as a main part of this method. The fundamental solutions of the governing equations represent the field generated by a concentrated unit charge acting at a predefined point [1]. The Kupradze method and the Laplace transform have been creating a framework to find the fundamental solutions for dynamic problems in the Laplace transform domain. However, in many cases the result could not be found in the time domain, when the integrals in the reverse Laplace transform could not be found analytically. It is not a real barrier to use the method and different ways are devised to pass it. Using the diverse numerical methods is the first and the usual choice. However, this solution raises some new problems such as time consuming and instability [2]. The instability problem could be overcome by the convolution quadrature method (“CQM”) [3] which uses the Laplace domain fundamental solutions and removes the problem of finding the inverse Laplace transform, but the repetitious numerical calculation remains unsolved. This piece of the method could be getting a cumbersome feature, especially when for a time depending problem it is needed for many times.

16


In this paper, we are presenting a method to address this problem. The main idea is to find functions that could emulate the integrals in the inverse Laplace transform which could not be written in terms of ordinary functions and should be evaluated using an iterative numerical method. However, the proposed procedure could be done in the case of the quadrature method too. In order to show the suggested procedure, a known dynamic problem of poroelasticity is selected. The fundamental solutions have been found later in both Laplace transform and time domain later. However, the time domain functions include some integrals which could not be written in terms of the ordinary functions. The values of the integrals have been found numerically using a wide range of data which could be expected in practice. This produced data have been divided to test and train parts and entered to the GEP procedure which resulted in some ordinary functions which could be gotten as the results of the integrals in the domain that defines by input data. In this very introductory step, the singular or nearly singular points do not enter into the procedure, however, they could be found in other similar procedure. The convolution quadrature method is not probed here, but the same procedure could be applied to avoid the integration procedure in this method too. The Sample problem In order to present the method, a sample problem is required. So evaluating the time domain, 3D fundamental solutions for saturated porous media with incompressible components is selected. It is a simplified version of the saturated porous media problem introduced by Biot [4]. The u-p model and incompressibility of the constituents make a simpler problem. The fundamental solutions for the general case have been found by Chen and the fundamental solutions for the simplified problem have been found by Kamalian et al [5]. The governing equations of dynamic poroelasticity were first derived by Biot [4] using the concept of variational formulation. However, other theories result in the same equations [6]. Zienkiewicz and Shiomi outlined some approximations for the problem. Among them is the “u-p” approximation which ignores the relative acceleration of the fluid phase and is advisable for quasi-static and consolidation type problems. Moreover, it is valid for most problems of the earthquake engineering and other problems with slower frequencies [7]. After these assumptions, one can write the equations of the conservation of total momentum, the constitutive equation of the solid skeleton, the flow conservation for the fluid phase and the generalized Darcy’s law respectively. They make a framework for analyzing such a medium. As an approximation, the fluid phase could be taken as a solid which makes the model introduced by Gatmiri and Nguyen [8] or both grains and fluid phases could be gotten as solid materials. This viewpoint is exploited by Kamalian et al to have simpler equations [5]. Needless to say, this new assumption restricts the domain for frequencies, but still could be useful. The governing equations and their derivation could be found in [5] but they are repeated here for the sake of clarity: u i , jj       u i , ji  ui   p ,i  f i  0

(1)

kp,ii  ui ,i  Fi ,i    0

(2) Where u i represents the displacement field of the solid skeleton, p denotes the excessive fluid pore pressure, λ and μ are the Lame’s constants of the drained skeleton, k stands for the permeability coefficient of the solid skeleton, ρ represents the density of solid–fluid mixture and α is a material parameter that relates to compressibility of phase and as a result of solidity of phases here is equal to 1. Finally, f i and Fi denote the body forces on the fluid and the solid phases; respectively. In addition, γ represents the rate of fluid injection into the media.


17

The Laplace transform provides a powerful and experienced way to solve such problems. After applying the transformation and the known procedures, the boundary integral equations of the problem in the Laplace transform domain and consequently, in time domain could be found:

 t i *uˆi  tî *ui d     p * qî  pˆ * q i d     f i *uˆi  fi *u i  d     i *uˆi  î *u i  d  ˆ













(3)

The star stands for the convolution operator. The required fundamental solutions have been found in the Laplace transform domain by Kamalian et al [5]: Gîj  (

A ij

1

 s 2  s  2 



B ij

1

v d s s  s  2  



C ij 1

v d 2 s

) exp(

r vd

s  s  2  )

 A 1 B 1  A ij  1 C ij r 1 1 1 ij   1     ij     D ij  exp( s ) 3 2 2 2     s 3 v s s 2  sv s 2 s v  s 2 s s s 2 2 2     d            r,i  1  r,i r,i  1 1  1 1   exp( r s  s  2   )  Gˆ 4i          2  4 r  s s  2   2 rv d s  s  2    vd 4 r 2  s s  2    

(4)



1 Gî 4  Gˆ 4i s 1 4 kr

Gˆ 44

Where 

 Aij

(6)  1 1    s s 2  

 r  exp(  vd 

s  s  2  ) 

1   s  2 

(7)

2

(8)

2 k

  2 

vd 

(5)

(9)

1 3x i x j  ij ( 5  3) 4 r r

(10)

1 3x i x j  ij ( 4  2) 4 r r 1 xi x j ( ) C ij  4 r 3 1 D ij   ij 4 r 

(11)

B ij 

(12) (13)

Then they returned the Laplace transform to the real time domain and the time domain fundamental solutions represented as: G ij  (

 A Aij C ij   ij t     (1  exp(2  (t   )))  2   2  vd2    2   vd 

t

 r

  B  g 1 ( )  ij g 2 ( ) d  )H (t  r )   vd v d  

 A ij 1 B ij C ij 1 1 (1  exp(2  t )  2  t  2  t )))   * 3 * 2  D ij  2 2  v s 2   s s s v  s  2   s  Aij

2 2

 r  exp( s ) v d 

(14)

18


r,i    r2 4 

G 4i 

r,i 4 r 2



t

r /v d

 1  exp(2 t      g

1

  d 

  2 rrv ,i

d

 r g 2    H (t  )k v d 

1  exp(2 t    

 r,i  t   r,i   1 1  G i 4   exp(2 t     g 1   d      t    rv d 2 2 2  4 r 2  r /v d    r,i   1 1 exp(2  t       t  4 r 2  2 2 

G44



1   4k  r 

t

r /v d

t

r

vd

 r g 2  d   H (t  ) vd 

1 exp 2 t   g   d  H (t vr )  exp(2 t  

Where ݃ଵ and ݃ଶ are:



1

d

  r /v  2 d g 1 ( )  exp(2  t )  I 1   t 2   r / v d     (t  r / v d )    t 2  r /v d      2    2 g 2 ( )  exp(2 t )  I 0   t   r / v d      



(15)

(16)

(17)

(18) (19)

In the above equations I 0 and I 1 stand for the Bessel functions of order 0 and 1 respectively and  

 2 . It 2 k

could be seen that the integrals in (14) to (17) cannot be calculated analytically and numerical methods are the usual choice to find out their values despite their costs. In the succeeding sections, a new method based on the genetic expression method will be introduced to found a function to emulate the values of the integrals so, the time consuming step of numerical integration could be omitted. Gene Expression Programming Ferreira suggested a new algorithm according to a genetic algorithm and genetic programming. This was called as “Gene Expression Programming or GEP”, a new evolved algorithm being used to overcome on most restrictions of GA and GP [9]. A genetic algorithm is a model of machine learning that its behavior has been inspired from nature, evolution mechanism [10]. This method is implemented by creating a population, which its individuals are considered as a chromosome which is designated for evolutionary processes. GP is a specific genetic algorithm in which modified genetic operators change the size of chromosomes and is a sub-field of genetic algorithms (GAs). The main difference between GA and GP is that program evolution in GP is in the form of parse trees, while this program is as binary fibers with constant length in GA [11]. GEP is a new sub-field of GP which rendering solution is the difference between these two methods (12). The solutions have the treelike structure that is called “Expression Tree or ET” which a sample could be seen in Fig. 1. This algorithm randomly builds an initial chromosome, which indicates a mathematical function. Then, it is converted into an expression tree (ET). This method includes two major parts known as chromosomes and expression trees (ET) [9]. GEP has four operators, including selection, mutation, transposition and crossover. The selection operator chooses each chromosome, according to roulette wheel method and elitism. In coding sequence, the chromosome is selected and changed by the mutation operator; however, its structure does not vary. Transposition operator randomly copies a part of chromosome and locates it in another position. Then, coding of chromosomes, which have been randomly selected, changes by crossover. These are the principles of GEP [13]. The algorithm could be seen at a glance in Fig. 2 [14].


19

Figure 1- the expression tree for ඥሺܽ ൅ ܾሻሺܿ െ ݀ሻ [13]

Figure 2- The Algorithm of GEP [14] Dimensionless Version of the Integrals The train and the test step of GEP need for a big amount of data to execute. Our goal in this paper, to have a function as the values of unsolved integrals that remain in (14) to (18), requires their numerical values. These numerical values should swipe a suitable range of variables. The dimensionless variables and equations could help to decrease the number of the variants and to find a suitable domain for them, here a practical one. Hence, in this section, two dimensionless variables are introduced and then the integrals are changed accordingly to use them. The required integrals to probe are: t

M 1   r g 1   d  vd

t

M 2   r  g 1   d  vd

M 3



t r

vd

exp  2 t     g 1   d 

(20) (21) (22)

20

Eds: M. Tezer-Sezgin, B. Karasözen, M H Aliabadi t

N   r g 2   d 

(23)

vd

We introduce two new dimensionless variables which decrease the number of variables:



 r

vd t 0  r vd r  vd Rewrite the integrals of Equations (20) to (23) in terms of the new variables:

M 1  exp(





0 exp(  ) r r ) I 1   2  1 d   exp( )  M 1( , ) 2 1 vd v  1 d

M 2 tM 1  ( 





r 2 0  exp( ) r2 ) I 1   2  1 d  tM 1  (  2 )M 2 ( ,  ) 2 1 vd vd  2 1

M 3   exp(20 ) 

0

1

exp( )

 1 2







I 1   2  1 d   exp(2 t 



r 0 r N ( )  exp( )I 0   2  1 d  ( )N ( ,  )  vd 1 vd

r r ) M 3 ( , )  exp(2 t  ) vd vd

(24) (25) (26)

(27) (28)

(29)

(30)

Data and results It could be seen that the primed functions in (27) to (30) should be found to have the explicit expressions for desired functions. The required inputs could be produced by numerical integration methods when the dimensionless variants swipe a useful domain. These dimensionless variables are introduced in (24) and (26). The first one, θ, is the ratio of the current time to the interval needed to the wave be received at the desired point and of course should be greater than one. It is because of the interpretation of the fundamental solutions, besides, the Heaviside’s function in (14) to (17) highlights this issue. For the upper limit of θ, we considered the vanishing property of the fundamental solutions, which makes them unimportant after a proper period of time. Precisely, a hundred time of the period, which takes the wave to receive a point the first time has been taken as a proper time to ignore the effects of the excitement. The other dimensionless variable  has been gotten between

103 and 102 based on the practical values for the soil hydraulic permeability and the specific mass of water. The required data have been produced on the domain which has been defined in recent lines; however the nearly singular values have been omitted in this very introductory use of this method in this area. In a standard procedure of GEP, the inputs have been divided into two groups to train and test phases. The different ratio for the number of data in these groups are common, here the more restrict one, half of data for each phase has been gotten. It is needed to examine each guessed functions, “Chromosomes” in order to find the fitness value to find the best one. Different statistical criteria are available, among them are root mean square error (RMSE) and


21

correlation coefficient (R), respectively, given by the Equations (31) and (32) (9). In addition, evaluation of the models by test datasets assures that the training process is performed optimally. In other words, it prevents over-training or over-fitting and informs under-training. 1 n ( )(ai  pi )2 n i1

RMSE  n

R

( p i 1

n

( p i 1

i

(31)

 p )( ai  a ) (32)

n

 p ) 2  ( ai  a ) 2

i

i 1

Then the GEP has given the following results for the dimensionless function defined in (27):

 tan  L n 

M1  ,   

1

(33)

1.25   L n 

  3  2  2L n    

  4 

(34)

This function emulates the dimensionless integral,��, very well. The fitness criteria for both test and phases are RMSE=0.010 and R=0.999. The same results for ��, are:

 M2  ,

 1 Ln    2      2 tanh    sin 1   3Ln e    2  10 





   

(35)

The fitness criteria for both test and phases are RMSE=0.279 and R=0.997. Accordingly for ��, :

M3  ,   tan 1

  Ln      4 tan  tan      4     1   e 2  2     2  Ln       

 

(36)

Where RMSE=0.002 and R=0.973. Finally for � , :

 tanh  Ln      Ln  sinh     N  ,   12      10   cos 2   





(37)

Where RMSE=0.273 and R=0.999. Summary and Conclusions

One of the novel intelligent methods, GEP, is employed to overcome the problem of finding the time domain fundamental solutions of a sample problem in terms of ordinary functions which helps to avoid the time consuming numerical integration for time domain dynamic problem of poroelasticity. The dimensionless parameters and equations have been introduced to reduce the number of variables and define a suitable domain for validity of the proposed substitute functions. Two numerous set of data have been provided for training and testing (divided equally) by numerical integration. The proposed functions are very favorable, so the time

22


consuming procedure of numerical integration could be avoided and cost of the calculation will be significantly reduced. The same procedure is useful for many cases like CQM and other variations of the time domain BEM. However we ignored the important nearly singular domain, the ability of GEP to use more complicate functions will remove this shortage easily. 1. References [1] Brebbia, C.A. and Dominguez, I. Boundary Elements:An Introductory Course. Southampton , WIT Press, 1993. [2] Peirce, A. and Siebrits, Stability analysis and design of time-stepping schemes for general elastodynamic boundary element models. , Int. J. Numer. Methods. Engrg., 1997, Vol. 40. [3] Schanz, Martin, On a reformulated convolution quadrature based boundary element method : Computer Modeling in Engineering & Sciences(CMES), 2010, Vol. 58. [4] Biot, M.A, Theory of propagation of elastic waves in a fluid-saturated porous solid:I- Low frequency range., Journal of the Acoustical Society of America, 1956, Vol. 28. [5] Kamalian, M. and Gatmiri, B. and Jiryaei Sharahi, Time domain 3D fundamental solutions for saturated poroelastic media with incompressible constituents. , Communications in Numerical Methods in Engineering, 2008, Vol. 24. [6] Coussy, Olivier and Detournay., Emmanuel and Dormieux, Luc., From mixture theory to Biot’s approach for porous media. 34, s.l. : International Journal of Solids and Structures, 1998, Vol. 35. [7] Zienkiewicz, O. C and Shiomi, Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution., International journal for numerical and analytical methods in Geomechanics, 1984, Vol. 8. [8] Gatmiri, B. and Nguyen, K.V, Time 2D fundamental solution for saturated porous media with incompressible fluid., Commun. Numer. Meth. Engng, 2004, Vol. 21. [9] Kayadelen, C., Soil liquefaction modeling by Genetic Expression Programming and Neuro-Fuzzy., Expert Systems with Applications, 2011, Vol. 38. [10] Sivanandam, SN and Deepa, SN., Introduction to genetic algorithms., Springer Science & Business Media, 2007. [11] Mollahasani, Ali and Alavi, Amir Hossein and Gandomi, Amir Hossein, Empirical modeling of plate load test moduli of soil via gene expression programming. , Computers and Geotechnics, 2011, Vol. 38. [12] Ferrira, C., Gene expression programming: a new adaptive algorithm for solving problems., Complex System, 2001, Vol. 13. [13] Ferrira, C. Gene expression programming: mathematical modeling by an artificial intelligence. : Springer, 2006. [14] Teodorescu, Liliana and Sherwood, Daniel, High energy physics event selection with gene expression programming., Computer Physics Communications, 2008, Vol. 178.

Meshless reconstruction of the support of a source B. Bin-Mohsin1 and D. Lesnic2 1

Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia, [email protected] 2 Department of Applied Mathematics, University of Leeds, UK, [email protected]

Keywords: Inverse source problem; Method of fundamental solutions; Nonlinear optimization. Abstract. The meshless reconstruction of the support of a three-dimensional volumetric source from a single pair of exterior boundary Cauchy data is investigated. The underlying potential satisfying the Laplace equation is sought as a discretised single-layer boundary integral representation but with sources relocated outside the solution domain, as in the method of fundamental solutions (MFS). The unknown source domain is parametrised by the radial coordinate, as a function of the spherical angles. The resulting least-squares functional estimating the gap between the measured and the computed data is minimized using the lsqnonlin toolbox routine in Matlab. Numerical results are presented and discussed for both exact and noisy data.

Introduction In this paper, the aim is to reconstruct numerically in a stable and accurate manner the support of a volumetric source by employing a combined meshless technique with nonlinear optimization which have recently been developed by the authors, [1], in two-dimensions. We consider the inverse problem of determining the support Ω2 ⊂ Ω ⊂ R3 of an unknown volumetric source of unit intensity in the Laplace equation ∇2 u = χ(Ω2 )

in Ω,

(1)

where χ(Ω2 ) denotes the characteristic function of the domain Ω2 and u is the potential. We assume that the domains Ω and Ω2 are bounded with smooth boundaries and that Ω1 := Ω\Ω2 is connected. By defining u1 in Ω1 , u =: (2) u2 in Ω2 , equation (1) can be rewritten as the following transmission problem: ∇2 u1 = 0

u 1 = u2 ,

in Ω1 ,

(3)

∇2 u2 = 1 in Ω2 , ∂u2 ∂u1 = on ∂Ω2 , ∂n ∂n

(4) (5)

where n denotes the outward unit normal to the boundary. We also prescribe one pair of Cauchy boundary data on ∂Ω, namely, u1 = f,

∂u1 =g ∂n

on ∂Ω.

We have the following uniqueness theorem in the class of star-shaped domains, [2].

(6)

24


Theorem 1. The inverse problem (3)-(6) has at most one solution Ω2 in the class of star-shaped domains. The next task is to reconstruct the source domain Ω2 numerically. Recently, the MFS has proved, [1,3], easy to use in detecting cavities, rigid inclusions, as well as inhomogeneities in inverse geometric problems. In this paper, we investigate yet another application of the MFS to reconstruct the source domain Ω2 from the Cauchy data (6). In the next section we describe the MFS, as well as the nonlinear minimization proposed for reconstructing the star-shape support of the unknown source.

The method of fundamental solutions (MFS) Prior to applying the MFS, we need to move the right-hand side inhomogeneity in (4) to the boundary conditions (5) and (6). For this, we decompose u2 = uh2 +

|x|2 , 6

(7)

in Ω2 .

(8)

where the homogeneous part uh2 satisfies ∇2 uh2 = 0

With the superposition (7), the transmission interface conditions (5) become u1 = uh2 +

|x|2 , 6

∂u1 ∂uh2 x·n = + ∂n ∂n 3

on ∂Ω2 .

(9)

On applying the MFS we approximate the solutions u1 and uh2 of the Laplace equations (3) and (8) by finite linear combinations of fundamental solutions of the form, [4], u1,2N M (x) =

2 M N

aSi,j G(x, ξ i,j ), S

S=1 i=1 j=1

uh2,N M (x) =

M N

bi,j G(x, ξ i,j ), 3

i=1 j=1

x ∈ Ω1 ,

(10)

x ∈ Ω2 ,

(11)

where a = (aSi,j )i=1,N ,j=1,M ,S=1,2 and b = (bi,j )i=1,N ,j=1,M are unknown coefficients to be determined, ) are source points located outside the annular domain Ω1 , (ξ i,j S i=1,N ,j=1,M ,S=1,2 ) are source points located outside the domain Ω2 , and G is the fundamental solution of the (ξ i,j 3 i=1,N ,j=1,M three-dimensional Laplace equation given by, G(x, ξ) =

1 . 4π|x − ξ|

(12)

The source points (ξ i,j ) ∈ Ω are placed on a (fixed) dilated pseudo-boundary ∂Ω of similar 1 i=1,N ,j=1,M ) ∈ Ω2 and (ξ i,j ) ∈ Ω2 are placed on shape as ∂Ω. The remaining source points (ξ i,j 2 i=1,N ,j=1,M 3 i=1,N ,j=1,M contraction and dilation (moving) pseudo-boundaries ∂Ω2 and ∂Ω2 similar to ∂Ω2 at a distance δ > 0 in the inward and outward directions, respectively. Without loss of generality, we may assume that the domain Ω is the unit sphere B(0; 1). We also assume that the unknown support Ω2 is star-shaped with respect to the origin, i.e. ∂Ω2 = r(θ, φ) cos(θ) sin(φ), sin(θ) sin(φ), cos(φ) | θ ∈ [0, 2π), φ ∈ [0, π) , (13)


25

where r is a smooth function with values in (0, 1). In this setup of particular domains Ω and Ω2 , the collocation and source points are uniformly distributed, as follows:

X 2i,j

X 1i,j = (cos(θi ) sin(φj ), sin(θi ) sin(φj ), cos(φj )), = r(θi , φj ) cos(θi ) sin(φj ), sin(θi ) sin(φj ), cos(φj ) , i = 1, N , j = 1, M , ξ k, = R cos(θ˜k ) sin(φ˜ ), sin(θ˜k ) sin(φ˜ ), cos(φ˜ ) , ξ k, = (1 − δ)X 2 , 1

2

= (1 + δ)X 2k, , ξ k, 3

(14)

k,

k = 1, N , = 1, M ,

(15)

where θi = 2πi/N, θ˜k = 2πk/N,

i = 1, N ,

φj = πj/M,

j = 1, M ,

φ˜ = π/M,

= 1, M ,

k = 1, N ,

ri,j := r(θi , φj ) for i = 1, N , j = 1, M , R > 1 and δ ∈ (0, 1). The unknown radii vector r = (ri,j )i=1,N ,j=1,M , characterising the star-shaped support Ω2 , together with the unknown MFS coefficients a and b, giving the approximations of the solutions u1 and u2 , are simulateneously determined by imposing the transmission conditions (9) and the Cauchy data (6) at the collocating points (14) in a least-squares sense. This results into minimizing the following (regularized) least-squares nonlinear objective function: 2 2 2 u |x|2 1 T (a, b, r) := u1 − f − g + u1 − uh2 − + 2 2 ∂n 6 2 L (∂Ω) L (∂Ω2 ) L (∂Ω) 2 ∂u h x · n 1 ∂u2 (16) − − + + λ a2 + b2 + rθ 2 + rφ 2 , ∂n ∂n 3 2 L (∂Ω2 )

where λ ≥ 0 is a regularization parameter to be prescribed. In (16), g is a noisy perturbation of the exact data g given by g (X 1i,j ) = (1 + ρi,j p)g(X 1i,j ),

i = 1, N , j = 1, M ,

(17)

where p represents the percentage of noise and ρi,j is a pseudo-random noisy variable drawn from a uniform c command -1+2 rand(1,NM), and distribution in [–1, 1] using the MATLAB * rθ 2 =

N M ri,j − ri−1,j 2 , 2π/N i=2 j=1

rφ 2 =

N M ri,j − ri,j−1 2 . π/M

(18)

i=1 j=2

The minimization of the functional (16) is performed using the Matlab toolbox routine lsqnonlin which does not require the user to provide the gradient and, in addition, it offers the option of imposing lower and upper bounds on the vector of unknowns (a, b, r).

Numerical results and discussion In all numerical experiments, the initial guess for the unknown vectors a and b are 0, and the initial guess for Ω2 is a sphere centred at the origin of radius 0.7. The Matlab toolbox routine lsqnonlin was run iteratively until a user-specified tolerance of XT OL = 10−6 was achieved, or until when a user-specified maximum number of iterations M AXCAL = 1000 × 4M N was reached. We have also set the simple bounds on the variable (a, b, r) as the box [−1010 , 1010 ]2N M × [−1010 , 1010 ]N M × (0, 1)N M . The choices of the regularization parameter λ in (16) was based on trial and error. We consider retrieving a sphere centred at the origin of radius R0 = 0.5. That is, we seek the star-shape approximation (13) for the spherical radius function r(θ, φ) ≡ R0 = 0.5,

θ ∈ [0, 2π), φ ∈ [0, π).

(19)

26


We then take the analytical solutions of the equations (3)-(5) and (8) to be given by R02 R03 − 6 3r 2 R02 r u2 (r, θ) = − , 6 3 2 R uh2 (r, θ, φ) = − 0 , 3

u1 (r, θ, φ) =

(r, θ, φ) ∈ (R0 , 1) × [0, 2π) × [0, π),

(20)

(r, θ, φ) ∈ (R0 , 1) × [0, 2π) × [0, π),

(21)

(r, θ, φ) ∈ (R0 , 1) × [0, 2π) × [0, π).

(22)

Based on (20), the input Cauchy data (6) are given by R02 R03 − , 6 3 3 ∂u1 R (1, θ, φ) = g(θ, φ) = 0 , ∂n 3

u1 (1, θ, φ) = f (θ, φ) =

θ ∈ [0, 2π), φ ∈ [0, π),

(23)

θ ∈ [0, 2π), φ ∈ [0, π),

(24)

and the transmission interface conditions (9) become r2 (θ, φ) , u1 r(θ, φ), θ, φ = uh2 (r(θ, φ), θ, φ) + 6 ∂u1 ∂uh2 r(θ, φ) r(θ, φ), θ, φ = (r(θ, φ), θ, φ) + , ∂n ∂n 3

θ ∈ [0, 2π), φ ∈ [0, π),

(25)

θ ∈ [0, 2π), φ ∈ [0, π).

(26)

We solve numerically the inverse problem given by equations (3), (8), (23)-(26) to retrieve the triplet solution (r(θ, φ), u1 (r, θ, φ), uh2 (r, θ, φ)) to compare with the analytical solutions given by equations (19), (20) and (22). Also, once uh2 has been obtained, equation (7) yields u2 . Initially, we have performed several numerical runs with various values of the input MFS parameters and, for illustrative purposes, we have decided to show results only for a typical set of values δ = 0.5, R = 2 and N = M = 10. We consider first the case of exact data, i.e. p = 0 in equation (17). In Figure 1, we present the numerically reconstructed domain for various numbers of iterations for no noise and no regularization, as well as the exact sphere. From this figure, it can be seen that even if the input data is exact, as the number of iterations increases the numerical solution becomes more inaccurate. This is to be expected because no regularization has been imposed yet and the inverse problem under investigation is ill-posed. In order to restore stability regularization should be employed with a positive regularization parameter λ in (16).


0.5

27

0.5

0

0

-0.5

-0.5

0.5 0 -0.5

0.5

0

-0.5

0.5 0 -0.5

(a) Exact

-0.5

0

0.5

(b) iter=10

0.5

0.5

0

0

-0.5

-0.5

0.5 0 -0.5

-0.5

(c) iter=100

0

0.5

0.5 0 -0.5

-0.5

0

0.5

(d) iter=200

Figure 1: The reconstructed source for various numbers of iterations for no noise and no regularization.

Figure 2 shows the higher accuracy and stabilising effect that the regularization has on the retrieved shapes for values of λ between 10−3 and 10−1 . We also perturb by a large amount of p = 10% noise the flux g, as in (17), in order to investigate the stability of the numerical solution. The numerically obtained results with various values of the regularization parameter λ after 200 iterations are shown in Figure 3. From this figure, we observe that overall λ between 10−3 and 10−2 yields accurate and stable results.

Conclusions In this paper, an inverse geometric problem which consists of reconstructing the unknown support of a volumetric source in the three-dimensional Poisson equation from a single pair of exterior boundary Cauchy data has been investigated using the MFS. The numerical results show satisfactory reconstructions for the unknown support with reasonable stability against inverting noisy data.

References

[1] B. Bin-Mohsin and D. Lesnic, Reconstruction of a source domain from boundary measurements, Applied Mathematical Modelling, submitted. [2] V. Isakov, S. Leung and J. Qian, A fast local level set method for inverse gravimetry, Communications in Computational Physics, 10, 1044-1070 (2011). [3] B. Bin-Mohsin and D. Lesnic, Determination of inner boundaries in modified Helmholtz inverse geometric problems using the method of fundamental solutions, Mathematics and Computers in Simulation, 82, 14451458 (2012). [4] A. Karageorghis, D. Lesnic and L. Marin, A moving pseudo-boundary MFS for three-dimensional void detection, Advances in Applied Mathematics and Mechanics, 5, 510-527 (2013).

28


0.5

0.5

0

0

-0.5

-0.5

0.5

0

-0.5

0.5

0

-0.5

0.5

0

(a) Exact

-0.5

-0.5

0

0.5

(b) λ = 0

0.5

0.5 0

0

-0.5

-0.5

0.5 0 -0.5

-0.5

0

0.5

0.5

(c) λ = 10−3

0

-0.5

-0.5

0

0.5

(d) λ = 10−1

Figure 2: The reconstructed source after 200 iterations for no noise and with regularization.

0.5

0.5

0

0

-0.5

-0.5

0.5

0

-0.5

-0.5

0

0.5

0.5

-0.5

-0.5

0

0.5

(b) λ = 10−4

(a) λ = 0

0.5

0.5

0

0

-0.5

-0.5

0.5

0

0

-0.5

-0.5

0

(c) λ = 10−3

0.5

0.5

0

-0.5

-0.5

0

0.5

(d) λ = 10−2

Figure 3: The reconstructed source after 200 iterations for p = 10% noise and with regularization.

Numerical solution to MHD pipe flow in annular-like domains Canan Bozkaya1 and M. Tezer-Sezgin2 Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: 1 [email protected], 2 [email protected]

Keywords: MHD flow, EDEM, BEM, annular domain.

Abstract. In the present study, we consider the magnetohydrodynamic (MHD) pipe flow in annular-like domains with electrically conducting walls. The coupled convection-diffusion type equations governing the MHD flow are decoupled into two modified Helmholtz equations through appropriate transformations. The resulting elliptic boundary value problems are solved numerically by using both the extended-domain-eigenfunction method (EDEM) and the well-known boundary element method (BEM). The extended-domain-eigenfunction method aims to reformulate the original problem on an extended domain which possesses symmetry. The new domain is obtained by transforming the inner pipe wall to a smaller circle towards the center of the pipe. Due to the symmetry of the new domain, an eigenfunction solution can be obtained for the problem theoretically. By collocating only the inner circular wall the theoretical solution values are transformed back to the original inner wall. This solution now can be regarded as the semi-theoretical solution of the problem in the annular-like domain. On the other hand, the boundary element method is a boundary only nature technique which transforms the differential equations into boundary integrals by using the fundamental solution of the problem under consideration. The outer and inner boundaries of the original problem are discretized by using constant boundary elements and taking into account only the directions of the normals. The numerical calculation are carried out for increasing values of Hartmann number (M) in the annular-like domain using ellipse as an inner boundary with constant conductivity k. The results of the EDEM are compared with those of the BEM for several values of M and k. Although the results obtained from EDEM and BEM are very compatible for small values of M, the EDEM is computationally less expensive and faster in convergence compared to BEM. On the other hand, for higher values of M the BEM is able to give more accurate results than the EDEM due to the large numerical errors occurred particularly close to the inner boundary.

Introduction The analytical solutions to elliptic boundary value problems (BVP) on domains with complex geometries are very rarely obtainable. Thus, these type of problems are preferably solved by some traditional numerical techniques such as the finite difference method (FDM), the finite element method (FEM) and the boundary element method. However, researchers recently have considered some alternate semi-analytic approaches to solve elliptic boundary value problems. One of these numerical implementations is the Trefftz method [1] which utilizes eigenfunctions of the differential operator to construct a finite sum approximation to the elliptic BVP. Another semi-analytic method, which is called extended-domain-eigenfunction method, has been introduced by Aarão et.al. [2] for solving elliptic boundary value problems on annular-like domains. In this technique, the original domain of the problem is embedded into an extended or larger one with simple boundaries where an eigenfunction solution like in the Trefftz method can be generated using standard techniques such as separation of variables. When the solution in the larger domain is restricted to the original domain, one can obtain the solution of the original problem. In the work [2], the Laplace’s equation in an annular-like domain is solved by EDEM. Later, the EDEM is also employed to solve the modified Helmholtz BVPs in annular-like domains by Aarão et.al. [3]. In this paper, we aim to extend the implementation of the extended-domain-eigenfunction method to solve the magnetohydrodynamic flow in annular-like domains, for the first time to the best of author’s knowledge. MHD is the study of the interaction of electrically conducting fluids and electromagnetic forces, which has many applications in cooling systems, MHD generators, nuclear reactors, flow-meters and etc. The flow of an incompressible, viscous, electrically conducting fluid inside an annular pipe gives rise to coupled convection-

30


diffusion type equations in velocity and induced magnetic field. Due to this coupling, the analytical solutions for the MHD flow equations are available only for some simple geometries subject to simple boundary conditions. Therefore, some numerical techniques have been used for the solution of MHD flow problems, e.g. a direct BEM approach is employed by Bozkaya and Tezer-Sezgin [4, 5] both in finite and infinite problem domains. Moreover, in the present study we will also apply the BEM solution of the MHD flow in annular-like domains and not only compare but also validate the results obtained by the semi-theoretical EDEM, since there are no analytical solution for MHD flow in annular-like domains.

Basic equations The equations governing the steady, laminar, fully developed flow of an incompressible, viscous, electrically conducting fluid in an annular-like pipe subject to a constant and uniform horizontally applied magnetic field of intensity H0 , are the same as those MHD duct flow equations [6] and are given in non-dimensional form as ∂B ∂x ∂V 2 ∇ B+M ∂x

∇2V + M

= −1

in Ω

(1)

= 0

where Ω is the annular-like domain (the cross-section of the pipe) between the inner boundary Γ1 and the outer boundary Γ2 as shown in Figure 1(a). V (x, y) and B(x, y) are respectively the dimensionless velocity and induced magnetic field in the z-direction (axis of the annular pipe). Hartmann number M is defined by √ √ M = H0 L0 σ / µ, where L0 is the characteristic length, σ and µ are the electrical conductivity and the coefficient of viscosity of the fluid, respectively.

H0

ΩE

Ω

r=b

Γ0

Γ1

(a)

Γ2

Γ2

(b)

Figure 1: Cross-section of the annular-like pipe (a) original domain Ω, (b) extended domain ΩE between Γ0 and Γ2 (circle with radius r = b) to which the EDEM is applied. The corresponding boundary conditions are given as V = 0 on Γ1 and Γ2 , B = k on Γ1 , B = −

x M

on Γ2

(2)

where k is a constant. With the change of variables U1 = V + B, U2 = V − B, the governing equations (1) are decoupled and the corresponding boundary conditions (2) become as follows: ∂U1 ∂x ∂U2 2 ∇ U2 − M ∂x

∇2U1 + M

= −1 = −1

in Ω,

U1 = k, U2 = −k x x U1 = − , U2 = M M

on Γ1 on Γ2 .

(3)

The problem is further simplified by the change of variables W1 = U1 + x/M and W2 = U2 − x/M, which give the homogeneous equations ∇2W1 + M

∂W1 = 0, ∂x

∇2W2 − M

∂W2 = 0, ∂x

in Ω .

(4)


31

Then by the transformations u1 = W1 eνx and u2 = W2 e−νx with ν = M/2, finally one obtains the following homogeneous modified Helmholtz equations with the corresponding boundary conditions ∇2 u1 − ν 2 u1 = 0 2

2

∇ u2 − ν u2 = 0

in Ω,

x x νx )e , u2 = −(k + )e−νx M M u1 = u2 = 0, u1 = (k +

on Γ1 on Γ2 .

(5)

Once the problem (5) is solved for u1 and u2 , the solution in terms of the original variables velocity V and induced magnetic field B can be obtained by the formulas 1 1 −νx x V = (e−νx u1 + eνx u2 ), B= e u1 + eνx u2 − 2 . (6) 2 2 M

Application of EDEM In this section, the extended-domain-eigenfunction method is briefly presented for the solution of MHD flow in annular-like pipe. The EDEM based on the methodology introduced in [2, 3], is employed for the elliptic BVP involving the modified Helmholtz equation  2  ∇ u − ν 2 u = 0 in Ω Original u|Γ1 = f1 A: (7)  Problem u|Γ2 = 0

where f1 ∈ L2 (Γ1 ) is a given function and Ω is the annular-like domain which is enclosed by the curves Γ1 : r = t1 (θ ) and Γ2 : r = t2 (θ ) = b (see Figure 1(a)) in polar coordinates. Both Γ1 and Γ2 are centered at the origin. Here, u can be considered as either u1 with f1 = (k + x/M)eνx or u2 with f1 = −(k + x/M)e−νx as given in equation (5). The basic idea of EDEM is to extend the original domain Ω to a larger annular domain, denoted by ΩE , such that the new extended domain contains greater symmetry. This extension is done by choosing two concentric circles Γ2 and Γ0 : r = t0 (θ ) = a ≤ min (t1 (θ )) (that is, the curve Γ0 is enclosed by Γ1 ) (see Figure 1(b)). Then, a related BVP is formulated as  2  ∇ w − ν 2 w = 0 in ΩE Extended w|Γ0 = f0 AE : (8)  Problem w|Γ2 = 0 in which f0 is initially unknown. Due to the symmetry of the extended domain, an eigenfunction solution to problem AE can be obtained by using separation of variables technique. The most general solution to the modified Helmholtz equation satisfying homogeneous Dirichlet condition on the outer boundary Γ2 is [7] ∞ Km (νb)Im (νr) (Am cos (mθ ) + Bm sin (mθ )), (9) w(r, θ ) = ∑ Km (νr) − Im (νb) m=0

where Im and Km are m-th order modified Bessel functions of the first and the second kind, respectively. The unknown coefficients {Am , Bm }∞ m=0 are determined through the application of the boundary condition at Γ0 as in the Fourier series representations. However, since the boundary condition f0 is not yet specified, as an alternative the boundary condition on the original inner boundary Γ1 is used to determine the coefficients {Am , Bm }. This is accomplished by defining an invertible mapping F of f1 on Γ1 to f0 on Γ0 , such that the solution w of problem AE , when restricted to Ω, is the solution of problem A, [2]. Thus, an f0 is required such that w|Ω = u and w|Γ1 = f1 . Now, to determine the coefficients {Am , Bm }, first the expansion (9) is truncated to a finite sum L Km (νb)Im (νr) (Am cos (mθ ) + Bm sin (mθ )) w(r, θ ) = ∑ Km (νr) − (10) Im (νb) m=0 then, we consider a finite number (2L + 1) of points on the inner boundary Γ0 . This identifies 2L + 1 unique points, {x j = (θ j ,t1 (θ j ))}2L+1 j=1 , on the original inner boundary Γ1 corresponding to those points chosen on

32


Γ0 . Imposing the boundary conditions w|Γ1 = f1 in the system (10) for those 2L + 1 points results in 2L + 1 equations in terms of the unknown coefficients {Am , Bm }Lm=0 (where B0 ≡ 0), which can be written as the matrix equation Az = C, [3], where 

  A= 

α01 α02 .. .

α0,2L+1

α11 α12 .. .

··· ···

α1,2L+1

αM1 αL2 .. .

··· · · · αL,2L+1

β11 β12 .. .

β1,2L+1

··· ···

βL1 βL2 .. .

··· · · · βL,2L+1



  , 



     z=     

A0 A1 .. .



        AL  , C =    B1  ..  .  BL



f1 (r(θ1 ), θ1 ) f1 (r(θ2 ), θ2 ) .. .

f1 (r(θ2L+1 ), θ2L+1 )

   (11) 

with αi j = gi (r(θ j )) cos (iθ j ), βi j = gi (r(θ j )) sin (iθ j ), and for non-zero ν,

gi (r(θ j )) = Ki (νr(θ j )) − Ki (νb)Ii (νr(θ j ))/Ii (νb), i = 1, · · · , L . The unknown coefficients {Am , Bm }Lm=0 are found by solving the system (11)for z. Then, the solution w within the original domain Ω, i.e. u = w|Ω , is obtained by direct application of equation (10). Thus, the solutions u1 and u2 given in problem (5) are obtained by taking f1 = (k + x/M)eνx and f1 = −(k + x/M)e−νx , respectively, in the problem A given in (7). Once u1 and u2 are obtained, the original unknowns, velocity V and the induced magnetic field B, can be determined back through relationships in (6).

Application of BEM A direct BEM with constant elements is also applied to the MHD flow problem in annular-like pipes for comparison with EDEM. BEM is a boundary only nature technique which transforms the domain integrals into boundary integrals by using the fundamental solution of the governing equations of the problem under consideration. The application of BEM to problem A in (7) by using the fundamental solution of the modified (νr) Helmholtz equation, (u∗ = K02π , r is the magnitude of the distance vector between the source and field point), results in [8] (12) −cS u(S) + Hu + Gq = 0 where H and G are the BEM matrices with entries Hi j =

ν 2π

Γj

K1 (νr)

∂r 1 dΓ j and Gi j = ∂n 2π

Γj

K0 (νr)dΓ j ,

∂u . Here, cS is 1/2 or 1 when the source point S is on the boundary or inside, respectively. respectively; and q = ∂n After the imposition of the corresponding boundary conditions given in equation (5) for the unknowns u1 and u2 in (12), one can obtain solutions u1 , u2 and their normal derivatives on the boundaries Γ1 and Γ2 . The interior values for u1 and u2 can be obtained by taking cS = 1 in equation (12). The original unknowns, velocity V and the induced magnetic field B, are also determined through the equation (6).

Results and discussions The two-dimensional MHD flow subject to an external horizontally applied magnetic field is considered in an annular-like domain Ω bounded by the ellipse Γ1 : x2 + 4y2 = 1 and the circle Γ2 : r = b = 3. The elliptical inner boundary andthe circular outer boundary can be written in polar coordinates (r, θ ), respectively, as

Γ1 : r = t1 (θ ) = 1/ 1 + 3 sin2 (θ ), Γ2 : r = t2 (θ ) = 3, for θ ∈ [0, 2π). The numerical simulations are carried out for various values of Hartmann number (0.2 ≤ M ≤ 300) and the inner boundary is taken as insulated (k = 0) and with conductivity k = 1. The results are presented in terms of equivelocity and current lines including a comparison of the EDEM and BEM methods. The EDEM solutions are obtained by using maximum 2L + 1 = 79 collocation points for the discretization of the inner boundary Γ1 and N = 250 constant boundary elements for the discretization of Γ = Γ1 ∪ Γ2 . The comparison of the EDEM and BEM solutions under the effect of increasing Hartmann number (M = 0.2, 8, 16) when the inner wall is insulated is displayed in Figure 2. The results show good agreement of


33

the solutions obtained by both methods. It is also observed that for low values of M the EDEM with coarse discretization produces as accurate results as does the BEM. On the other hand, for higher values of M > 16 V

BEM

EDEM

B

BEM

M = 16

M=8

M = 0.2

EDEM

Figure 2: The effect of M(= 0.2, 8, 16) on V and B when k = 0 comparing EDEM with BEM. the rate of change in the obtained results for V and B close to the inner boundary increases and the EDEM solution suffers from large numerical errors particularly close to Γ1 due to the formation of boundary layer, and increasing the number of points in EDEM does not improve the solution significantly. However, BEM is able to give quite accurate results by using an adequate number of boundary elements for higher values of Hartmann number as shown in Figure 3. The boundary layer formation close to inner boundary for velocity is wellobserved for high M, and the induced magnetic field current lines distributed symmetrically about the vertical centerline x = 0. Moreover, both the velocity and induced magnetic field values decrease with an increase in M, indicating the retarding effect of the Hartmann number on the velocity field. As M increases, the velocity also becomes stagnant along the line x = 0 and velocity contours are concentrated along y = 0 in the direction of applied magnetic field. V V B B

M = 50

M = 300

Figure 3: V and B by BEM when M(= 50, 300) and k = 0. Figure 4 displays the velocity and induced magnetic field contours obtained by EDEM for the case when the inner boundary has the conductivity k = 1 and M = 0.2, 8, 16. It is observed that the symmetry about the vertical centerline x = 0 in both velocity and induced magnetic field is destroyed by increasing k from 0 to 1 for each values of M (see Figure 2). The fluid action concentrates around the inner pipe in the direction of applied magnetic field as M increases as in the insulated wall case. However, the induced magnetic field profile

34

Eds: M. Tezer-Sezgin, B. Karasözen, M H Aliabadi M = 0.2

M=8

M = 16

V

B

Figure 4: V and B by EDEM when M(= 0.2, 8, 16) and k = 1. is significantly changed compared to the case of k = 0 for increasing M. That is, the current lines circulate and form a thick boundary layer around the inner wall in the direction of applied magnetic field as M increases.

Conclusion The MHD flow in an annular-like pipe under the effect of an external applied magnetic field is solved numerically by EDEM and BEM. The effects of the Hartmann number and the conductivity of the inner boundary on the velocity and induced magnetic field are investigated. It is observed that the EDEM is computationally less expensive and faster than BEM for small values of M, while BEM is more effective and accurate than EDEM to obtain the solution for higher values of M. The well-known behavior of MHD flow is captured in both methods.

References [1] I. Herrera. Trefftz method: A general theory. Numer Meth Part D E, 16(6):561–580, 2000. [2] J. Aarão, B.H. Bradshaw-Hajek, S.J. Miklavcic, and D.A. Ward. The extended-domain-eigenfunction method for solving elliptic boundary value problems with annular domains. J Phys A-Math Theor, 43(18):185202, 2010. [3] J. Aarão, B.H. Bradshaw-Hajek, S.J. Miklavcic, and D.A. Ward. Numerical implementation of the EDEM for modified helmholtz BVPs on annular domains. J Comput Appl Math, 235(5):1342 – 1353, 2011. [4] C. Bozkaya and M. Tezer-Sezgin. Fundamental solution for coupled magnetohydrodynamic flow equations. J Comput Appl Math, 203:125–144, 2007. [5] M. Tezer-Sezgin and C. Bozkaya. The boundary element solution of magnetohydrodynamic flow in an infinite region. J Comput Appl Math, 225:510–521, 2009. [6] L. Dragos¸. Magnetofluid Dynamics. Abacus Press, 1975. [7] A.D. Polyanin. Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall/CRC, 2002. [8] M. Tezer-Sezgin and S. Dost. Boundary element method for mhd channel flow with arbitrary wall conductivity. Appl Math Model, 18:429–436, 1994.

The Role of the Second Pair of Wings in Insect Flight A Simple Vortex Approach to Complex Multi-wing Unsteady Flapping Problems in 2-D M. Denda Mechanical and Aerospace Engineering Department, Rutgers University, 98 Brett Road, Piscataway, NJ 08854-8058, USA e-mail: [email protected]

Keywords: Discrete Vortex Method, 2-D Unsteady Insect Flight, Two Pairs of Wing Flapping.

Abstract. This paper is inspired by Wigglesworth [1] who discusses the potential disadvantage of the second pair of insect wings in flapping flight working in the region of turbulence created by the first pair. While the Orthoptera and lower Neuroptera species suffer, the dragonflies overcome this difficulty. In an attempt to reveal the secrets of this two-pairs of wing flapping problem, we have developed a simple but highly reliable approach based on vortices that capture all essential unsteady aerodynamics phenomena involved. Since the species considered have a long wing span, the problem can be reduced to 2-D for which the flow field in the span direction is constant. Introduction While birds have two wings, flying insects have (in general) four wings. According to Wigglesworth [1], in Orthoptera (grasshopper, cricket, katydid, and locust.), Neuroptera (lacewing, mantidfly, and antlion), Isoptera (termite), and Odonata (dragonfly and damselfly), the fore- and hind-wings move independently. In Hymenoptera, Tricoptera, Lepidoptera, and Hemiptera, ”fore- and hind-wings are united by various mechanisms to make a functional unit.” In Coreoptera, the fore-wings (called elytra) are held ”aloft” (sometimes kept folded, as in Cetonia) during the flight. The hind-wings of Diptera and fore-wings of male Strepsiptera have been reduced to a tiny club-shaped structure, the haltres, that regulate the flight. With the exception of the most primitive insects, Orthoptera, Neuroptera, Isoptera, and Odonata, most insect wings are modified to form a single functional unit in order to [1] ”avoid the disadvantage of the second pair of wings working in a region of turbulence produced by the first pair.” The primary goal of this paper is to investigate numerically the independently moving fore- and hind-wings using observation of the flight of these primitive insects. The vortex based method used in this paper has been verified as simple but highly reliable for the study of unsteady flapping aerodynamics [2]. The method, originally developed for a single wing, is extended to two wings in 2-D. Coordinate Systems For insects with a long wing span length the flow approximately remains constant in the span direction, justifying the two-dimensional (2-D) modeling. Two original infinitely long wings are represented by their intersections with the 2-D plane as thin rigid chords. Each wing undergoes two translational (lunge and heave) and one rotational (pitch) motions. For each wing, its position is described by the wing-fixed system O − ξη, locally, and the space˜ ξ˜η˜, globally, as shown in Figure 1. The third coordinate system (the wing-translating fixed system O− ˆ placed system) has its axes ξˆ and ηˆ parallel to the global ξ˜ and η˜ axes, respectively, with its origin O at the center of rotation of the wing, which is located at a distance a (the rotational offset) along the negative ξ axis. While the wing-fixed system rotates and translates with the wing, the wing-translating

36

Eds: M. Tezer-Sezgin, B. Karasözen, M H Aliabadi ^F η

ηF

Top ^H η ^ ξF

aF ^ OF αF OF

Top

^ ηC

βF

dF

^ C

^ ξC

δ

aH

dH

bH

^ BF

^ ξH

^ OH

ξF

bF

βH ^ BH

ηH

~ η

αH OH

βF − δ βH − δ Bottom

~ ξ

ξH

~ O

Bottom

˜ − ξ˜η˜) , wing-fixed (OF − ξF ηF , OH − ξH ηH ), wing-translating (O ˆ F − ξˆF ηˆF , Figure 1: Space-fixed (O ˆ ˆ ˆ OH − ξH ηˆH ) , and body (C − xC zC ) coordinate systems. Subscripts F and H indicate the fore- and hind-wings, respectively. system only translates. For each wing, the trajectory of the translational wing motion, or the trace of ˆ of the wing-translating system, is described by a line called the stroke line. the origin O In addition, we introduce the body-fixed system Cˆ − ξˆC ηˆC whose origin is located at the mass center Cˆ with vertical and horizontal axes. In addition, introduce a straight axis, body axis, aligned with the body of the insect. The angle of the body axis and the stroke line angle with respect to the body axis are given by δ and β, respectively, such that the stroke line angle in the space-fixed system is β − δ. Figure 1 shows two sets of wing-fixed and wing-translating systems along with the common ˜ − ξ˜η˜ and the body-fixed system Cˆ − ξˆC ηˆC . space-fixed system O 2-D Motion of Flapping Insect Wings Wing Position. While the rolling of a single finite length insect wing, either right or left, in 3-D is described by the rotation around the body axis, it is represented, in 2-D, by a translational motion of an infinitely long wing along the stroke plane. The projection of the stroke plane onto the twodimensional plane is a straight inclined line (stroke line) with the slope, β, measured positive counterclockwise from the body axis. This angle is called the stroke angle. The translational motion is decomposed into the horizontal (heave), H, and vertical (lunge), L, components. The wing undergoes ˆ of the wing-translating system, which is called the pitch another kind of rotation around the origin O and denoted by α. The forward pitch, in the down-stroke direction with α < 0, is called the pronation and the backward pitch, in the direction of the up-stroke with α > 0, is called the supination. The distance a between the origins of the wing-fixed and wing-translating systems can be varied. Consider a 3-D rolling motion of an insect wing of the span length l over a stroke plane, which is the trajectory of the center line of the wing span. Let the upper and lower extents of the rolling motion be given by the upper and lower stroke angles, φT and φB . These three quantities, taken from the actual insect flight motions, are used to specify the extent of the stroke line by the trajectory of the mid point, 0.5l, of the span into 2-D plane. Since the linear travel of points along the wing span increases from zero, at the base, to the maximum, at the wing tip, its projection is taken at the mid point, giving the average. The time variation of the lunge and heave is expressed using the sinusoidal function. The pitch motion occurs as a sudden rotation, described by the step function, of the wing at the extremes of each stroke (symmetric pitch); pronation in the downward direction and the supination at the bottom in the upward direction at the top and the bottom of the stroke, respectively . The


37

η

H-V

•

^ η

• • α

L-U ^ O

a

^ ξ

α O

P (ξ, η)

ξ

Figure 2: Translational and rotational velocities of the wing-translating and wing-fixed coordinate systems. Only one wing is shown. pitch motions that occur before and after reaching the top or bottom are called advanced and delayed pitches, respectively. The most insects do not have the capability to pitch instantaneously and the rotation is smoothed out throughout the stroke, for which a smoothed out step function is used [2]. The effect of the insect flight speed, (U, V ), is incorporated by moving the body itself in the direction opposite to the air velocity under zero ambient velocity. The non-zero ambient velocity can be superposed on top of this. For a constant air velocity the total translational motion of the wing is obtained by superposing the contributions, L and H, from the flapping motion and the air velocity to give (L − U t, H − V t), where t is the time.

Wing Velocity. The origin of the wing-translating system moves with the velocity (L˙ − U, H˙ − V ) while the origin of the wing-fixed system moves with a combined translational (L˙ − U, H˙ − V ) and ˆ of the wing-translating system is the rotational (α) ˙ velocities as shown in Figure 2. The origin O center of rotation. The velocity of an arbitrary point P = (ξ, η) on the wing is given, in terms of these wing velocity parameters, by Vξ˜ = L˙ − U + α(η ˙ cos(α) − (a + ξ) sin(α)), ˙ Vη˜ = H − V − α((a ˙ + ξ) cos(α) + η sin(α)).

(1)

in the space-fixed coordinate system [2]. Consider the wing with the camber specified by a function η = f (ξ) in the wing-fixed system, then the complex valued unit normal vector is given by −df (ξ)/dξ + i −iα n ˜ = nξ˜ + inη˜ = n ˆ e−iα = e 1 + (df (ξ)/dξ)2

(2)

in the space-fixed system, where i is the imaginary number. The normal velocity of the wing is given, from (1) and (2), by V n˜ = (V¯ζñ ˜ ) = Vξñξ˜ + Vηñη˜, (3) where () is the real part of a complex variable.

Motion of two wings. The motion of the second wing is described exactly in the same manner as the first wing mentioned above. When two wings are present, parameters for each wing are distinguished

38


from those of the other wing using subscripts, F and H, for the forward and hind wings as can be seen in Figure 1. Vortex Equations Single Vortex. Vortices used in this paper are conveniently described in terms of complex variables [2]. Consider a line vortex with the circulation Γ (positive counter-clockwise) located at ζ˜0 = ξ˜0 +iη˜0 . Its complex potential function is given by ˜ =− ω(ζ)

iΓ log(ζ˜ − ζ˜0 ), 2π

(4)

where ζ˜ = ξ˜ + i˜ η and ζ˜0 are complex position vectors in the space-fixed system. The corresponding conjugate complex velocity is given by ˜ = v(ζ)

iΓ 1 dω =− , 2π ζ˜ − ζ˜0 dζ˜

(5)

¯ indicates the complex conjugate. ˜ = v ˜ + ivη˜ and an overbar () where v(ζ) ξ In order to alleviate an excessively high value of the velocity, when it is evaluated in the neighborfood of the source point, we introduce the Rankine vortex [2] for which formula (5) is used outside a core radius only. Within a core distance from the source the velocity changes linearly from zero to a value, given by the formula (5), on the radius of the core. Discretization of the Wing. For each wing introduce m vortices, Γj at ζ˜0j (j = 1, 2..., m) and m − 1 collocation points ζ˜i (i = 1, 2..., m − 1) on the wing. The collocation points are placed at the midpoints of vortex points. The number of vortices for two wings, mF and mH , may differ (notice the subscripting convention, F and H, used for two wing system). It is important to place vortices at the leading and trailing edges and in between. The spacing of the vortex points could be equidistant at the middle and gradually narrowed toward the two end points, which are mathematically known singular points [2]. Influence Coefficients. For the two wing system, there are two types of influence coefficients between vortices located on two wings: self- and cross- influence coefficients. The complex conjugate velocity at the fore-wing collocation point ζ˜F i due to a vortex ΓF j at ζ˜F 0j on the fore-wing is given by v¯F iF j = −

iΓF j 1 . 2π ζ˜F i − ζ˜F 0j

(6)

The normal component of this velocity at the fore-wing collocation point is given, from Eq. (3), by vF iF j n ˜F i) = vFn˜ FiF j = (¯

n ˜F i ΓF j F ( ) ≡ VFn˜iF j ΓF j , ˜ 2π ζF i − ζ˜F 0j

(7)

where n ˜ F i is given by Eq. (2) at the fore-wing collocation point, is the imaginary part of a complex variable, and n ˜F i 1 F VFn˜iF ( ), (8) j = 2π ζ˜F i − ζ˜F 0j

is the self-influence coefficient for the fore-wing. Another self-influence coefficient for the hind-wing, n ˜H VHiHj , is obtained from (8) by replacing the subscript F by H. Two more cross-influence coefficients n ˜H F and VHiF that describe the interaction between the fore-and hind-wings, VFn˜iHj j , are obtained similarly. For example, the former is the cross-influence coefficient for the normal velocity on a fore-wing collocation point F i due to a hind-wing vortex ΓHj .


39

System of Equations for Discrete Vortices on the Two Wings Contribution from the Bound Vortices on the Two Wings. For the fore-wing add contributions from the entire discrete bound vortices on its own and other wings to get the normal velocity component at the fore-wing collocation point ζ˜F i , vFn˜ Fi =

mF

F VFn˜iF j ΓF j +

F j=1

mH

F VFn˜iHj ΓHj ,

(9)

Hj=1

n ˜F F where VFn˜iF j is given by Eq.(8) and VF iHj is one of cross-influence coefficients between two wings. The non-penetration condition requires that this normal velocity must be equal to the normal velocity of the fore-wing, VFn˜iF , at each collocation point, mF

mH

F VFn˜iF j ΓF j +

F j=1

F VFn˜iHj ΓHj = VFn˜iF ,

(10)

Hj=1

for collocation points F i = 1, 2, ..., mF − 1. An additional equation, required to match the number of mF unknowns, ΓF i , for the fore-wing is give by the conservation of the vortices, mF

ΓF j = 0.

(11)

F j=1

Similarly, system of equations for the hind-wing is obtained by swapping subscripts F and H in (10) and (11) with the result mF

F j=1

mH

n ˜H VHiF j ΓF j +

n ˜H n ˜H VHiHj ΓHj = VHi ,

(12)

Hj=1

for collocation points Hi = 1, 2, ..., mH − 1 and the hind-wing vortex conservation equation mH

ΓHj = 0.

(13)

Hj=1

Notice that the vortex conservation condition is applied for each wing independently in order to provide two additional equations to secure the sufficient number of equations that matches the total number of bound vortices, mF + mH . Contribution from the Wake Vortices. At each time step a pair of fore-wing vortices, from the leading (F j = mF ) and trailing (F j = 1) edges, are shed from the fore-wing. This results in, during the p-th time period, 2(p − 1) fore-wing wake vortices, [k]

located at

[k]

Γ F 1 , ΓF m ,

(14)

[p] ˜[k] [p] ˜[k] ζF 1 , ζF m ,

(15)

where the pre-superscript [p] () indicates the current step [p] and the post-superscript ()[k] ((k = 1, 2, ..., p − 1) indicates the originating time step. Since the values of ΓF ’s remain constant once the vortices are shed into the flow, they do not have the pre-superscript. Following the procedure used

40


earlier for the calculation of bound-vortex induced velocity, we can calculate [2] the normal compon ˜H nents, [p] vFn˜ FiF p and [p] vHiF p , of fore-wing wake induced velocity at the fore-and hind-wing collocation ˜ ˜ points ζF i and ζHi , respectively. The effects of the wake vortices shed from the hind-wing can be treated similarly to obtain the n ˜H normal components, [p] vFn˜ FiHp and [p] vHiHp , of hind-wing wake induced velocity at the fore-and hind˜ ˜ wing collocation points ζF i and ζHi , respectively. Eqs. (10) and (12) are now modified to give mF

and

F VFn˜iF j ΓF j +

mH

F j=1

Hj=1

mF

mH

n ˜H VHiF j ΓF j +

F j=1

F VFn˜iHj ΓHj + [p] vFn˜ FiF p + [p] vFn˜ FiHp = VFn˜iF ,

(16)

n ˜H n ˜H ˜H n ˜H [p] n VHiHj ΓHj + [p] vHiF vHiHp = VHi . p+

(17)

Hj=1

Notice that for the first step the wake is absent. Convection of Wake Vortices, Shedding of Bound Edge Vortices and the Kutta Condition In our proposed method the Kutta condition is not enforced at the leading (LE) and trailing (TE) edges of two wings [2]. Instead the bound vortices at these points are shed regularly at each time step. Right after shedding, the two edges of each wing lose vortices to effectively satisfy the Kutta condition momentarily until the new bound vortices are built up in the next time step. In addition, all wake vortices are convected using the velocity calculated at the wake vortex site. The velocity contributions come from the bound and wake vortices of the two wings. Time Marching Solution Procedure In the time marching solution procedure, for each time step, specify the wing position and velocity of the two wings. Determine the magnitudes of the bound vortices using the non-penetration condition, Eqs. (16), (17), and vortex conservation equations, Eqs. (11), (13), for two wings. Calculate the induced velocity at the trailing and leading edges of two wings and wake vortex sites. Shed the edge vortices and convect the wake vortices. Repeat the whole process for the subsequent time steps. For a detailed description of the time marching solution procedure for the single wing, see [2]. Also see [2] for the selection of the time increment ∆t that is consistent with the spacial resolution, determined by the spacing of bound vortices on the wings. Impulses and Forces/Moment Space-fixed System. It is shown [2] that the linear and angular impulses of a line vortex located at ζ˜ = ξ˜ + i˜ η with the circulation Γ are given by ˜ −iρΓζ, and

(18)

1 ˜ 2, − ρΓ|ζ| (19) 2 where Γ is a signed circulation with positive counter-clockwise. The time derivatives of the linear and angular impulses will provide the force and moment exerted by the vortex onto the air. The force and moment acting on the wing are obtained by reversing the signs of these obtained for the air mass.


41

Wing-translating System. The linear and angular impulses must be calculated in the space-fixed system. However, the angular momentum around the origin of the space-fixed system is undesirable in practical applications. Rather, it should be calculated about the origin Cˆ of the body-fixed system (see Figure 1). The problem however is that its origin is moving. To resolve this issue, we introduce another space-fixed system that has the same origin as the body-fixed system and calculate the angular impulse in this coordinate system. Although this system needs to be updated as the body-fixed system moves on, each one of them in time history is a space-fixed system and is the legitimate system for the calculation of the impulses. ˜ − ξ˜η˜ and the body-fixed system Cˆ − ξˆc ηˆc . The Consider, first, the original space-fixed system O transformation between the two systems is given by ζ˜ = r + ζˆC ,

(20)

where ζˆC = ξˆC + iˆ ηC and r = −U t − iV t. Substitute this relation into Eq. (18) and (19) and simplify to get ˆ ˆ IÃ = IÂ + (¯ rI), (21) I˜ = I, where

ˆ IÂ = − 1 ρΓ|ζ| ˆ 2. Iˆ = −iρΓζ, 2 Now take the time derivative in Eq.(21) to get the force and moment, ˆ˙ M ˜ = I˜˙ A = Iˆ˙ A + (r¯˙ I) ˆ + (¯ ˆ˙ F˜ = I˜˙ = I, rI)

(22)

(23)

At this point we switch from the original space-fixed system to the space-fixed system that is placed on top of the body-fixed system, for which r = 0, giving the updated relations ˆ˙ M ˜ = I˜˙ A = Iˆ˙ A + (r¯˙ I), ˆ F˜ = I˜˙ = I,

(24)

where r¯˙ = −U + iV . Finally, the force and moment acting on the wing is obtained by reversing the signs. In 2-D, the force and moment are calculated per unit depth. Consider an insect with the span length l. In 3-D, the wing undergoes a radial flapping motion about the wing base. In 2-D adaptation of this flapping motion we determine its stroke line length d by projecting the 3-D flapping motion of the mid-point at 0.5l rather than the wing tip, at l of the wing, in order to represent the average (over the span) stroke line length of the 3-D flapping wing. Note that, the program inputs the half-span length 0.5l and obtains the force and moment per unit span length. The total force and moment are calculated by multiplying the span length l. Counting on the right and left wings, we further double the total force and moment. Evaluation of numerical performance Denda et al. [2] has evaluated the numerical performance of the proposed vortex method for the single wing problem in 2-D in comparison with the solution obtained by a Navier-Stokes solver, OpenFOAM. The spacial resolution is determined by the number of bound vortices, m, which determines the spacing between bound vortices. The first candidate for the time increment, ∆td , depends on the lunge motion and is determined in terms of m and the stroke line length d, such that the distance covered by the lunge motion in a time increment is equal to the spacing between the bound vortices. Another candidate, ∆tp , is determined by the pitch speed p, such that the pitch event is fully contained within the time increment. The smaller of the two is used for the actual time increment. Notice that m is the only non-physical parameter that can be selected independently from all other physical parameters, including d and p, that define the problem.

42


Our surprising discovery is that, although the results using a small number of m is quantitatively inferior, they still preserve the essence of the solution, obtained using a much higher number of m qualitatively. We have found that for the value of m bigger than 35 the computational time increases exponentially but the numerical results deteriorate as compared to the viscous solution obtained by the Navier-Stokes solver. The best use of the proposed method is as a quick solver for the unsteady flapping problems. Using a small number of m, such as m = 5 and m = 10, we can explore the unknown unsteady behavior of the flapping flight. Once we discover interesting phenomena or bahaviors, we can increase the number of m (up to m = 35) for more accurate results. If the phenomena need to be further scrutinized, then we switch on to one of the Navier-Stokes solvers. We can remain confident that use of this method with a small number of m does not miss the essential behavior and is quick to calculate. Summary The numerical study of the flapping flight with two wings performed in this paper is based on the preliminary work done with the single wing. The method developed in this paper inherits all the characteristics of the technique developed for the single wing. These include the modeling of the boundary layers (or the viscous effects) by the arrays of discrete bound vortices, flow separation from the edges of the wings, and the convection of wake vortices. This method is applicable regardless of the value of the Reynolds number; for large Reynolds numbers, the number of vortices increases and the problem of turbulence is reduced to the many-wake-vortices problem in the inviscid fluid. The remarkable simplicity of the method makes it even more attractive tool to investigate the two-wings flapping problems. During the conference numerical results will be shown to shed light to issues related to the twowings flapping. When adding the second pair of wings behind the first pair, where should it be placed (separation effect)? What is the best phase shift for the second pair relative to the first. It is also demonstrated how simple it is to use the proposed method compared to the existing Navier-Stokes solvers.

References [1] V. B. Wigglesworth The Principles of Insect Physiology, 7th ed., Chapman and Hall (1972). [2] M. Denda, P. K. Jujjavarapu and B. C. Jones A Vortex Approach to Unsteady Insect Flight Analysis in 2-D, European J. Compu. Mech, to be published.

Performance Prediction of Wings Moving Above Free Surface Ali Dogrul1, Sakir Bal2 1

2

Department of Naval Architecture and Marine Engineering, Yildiz Technical University, Istanbul, Turkey, [email protected]

Department of Naval Architecture and Marine Engineering, Istanbul Technical University, Istanbul, Turkey, [email protected]

Keywords: Free water surface, Iterative Boundary Element Method (IBEM), Computational Fluid Dynamics (CFD), Wing-in-ground (WIG) Abstract. The performance of 3D (three-dimensional) wings moving with a constant speed above free water surface has been investigated by an Iterative Boundary Element Method (IBEM). The Computational Fluid Dynamics (CFD) technique has also been applied for validation. The CFD technique has first been applied to compare the IBEM results in case of unbounded flow domain for a 3D rectangular wing which has constant NACA0012 foil sections along its span-wise direction. The CFD code is based on finite volume method (FVM). Inviscid analyses have been performed by solving Euler equations. On the other hand, the IBEM originally developed for 3D hydrofoils moving under a free surface has been modified and applied to 3D wings moving with a constant speed above free water surface. In the IBEM applied here, the integral equation based on Green's theorem can be divided into two parts: (1) the wing part, (2) the ground or free surface part. These two problems are solved separately, with the effects of one on the other being accounted for in an iterative manner. The wing part including wake surface has been modeled with constant strength dipole and source panels. The free surface part has also been modeled with constant strength dipole and source panels. Source strengths on the free surface can be expressed by the linearized free surface conditions. In order to prevent upstream waves the first and second derivatives of the perturbation potential with respect to horizontal axis have been enforced to be equal to zero. No radiation condition has been enforced at the downstream boundary and transverse boundaries. The potential induced by wing on the free surface and the potential induced by free surface on wing surface can be considered on the right hand sides of each corresponding integral equations. The effects of aspect ratio and Froude number on the wing performance have been discussed. It has been found that the free surface can affect the wing performance significantly. Introduction Marine vehicles operating near the ground or above free surface can utilize wings in order to reach high speeds. WIG (wing-in-ground) effect craft and some racing boats including catamarans with hydrofoils can take advantage of air lifting surfaces to support completely or partially the vehicle weight. Higher economical speeds mean faster transportation which is crucial in shipping. With the help of wings, air is compressed under the vehicle and an air cushion is created above the free water surface. This air cushion helps to reduce the drag of the vehicle and only air induced drag occurs. Also an accurate distance between WIG and the free surface may lead to an additional lift force which means additional payload, operating range or speed. Many WIG designs have been made in recent years. Detailed information about design principles of WIG crafts can be found in [1, 2]. The advantages of WIG crafts have inspired the researchers for decades. Several studies have been made for the aerodynamics and hydrodynamics of the WIG effect crafts. WIG crafts have been studied in a theoretical manner in [3]. The flow induced by a 2-D body near a plane wall has been analyzed by considering the irrotational flow except the vortex sheet which is representing the wake. The body has been positioned near the wall that the distance of the wall to the body is very small in order to investigate the extreme ground effect unsteadily. In another study, an airfoil has been observed in close proximity to the ground using flap-like appendage [4]. The effect of the appendage angle on flow pattern behind the trailing edge and the stability of the wing have been presented. The flow around a 3D rectangular wing in ground effect has been investigated numerically in [5]. The numerical method based on a commercial CFD code has been validated with the available experimental data for one ground clearance. Then the results have been compared with experiments and vortex lattice method (VLM). It has been shown that CFD method is applicable for the aerodynamic optimization of WIG vehicles. Another numerical investigation of a wing in ground effect with endplates has been made by

44


taking the free surface deformations into account in [6]. The numerical analyses have been performed for 2-D (two-dimensional) and 3-D (three-dimensional) cases with and without endplates. The numerical results have shown that the deformation on the free surface is caused by wing tip vortices rather than the pressure distribution under the wing. It has also been shown that the endplates in the wing tips have dramatically reduced the free surface deformation. The lifting surface theory has been used in order to solve the flow around a 3-D wing for the inviscid model in [7]. Ground effect and free surface are taken into account and the wing has been represented by horseshoe vortices using finite number panels. Numerical results have been compared with the experimental ones and the effects of ground clearance on the lift force have been shown for different Mach numbers. In a similar study in [8], Prandtl’s lifting line theory has been used to calculate the lift force of a wing in the case of ground effect and in the vicinity of a free surface. Linearized free surface boundary condition has been employed using horseshoe vortices for the representation of the wing. The numerical results have been compared with the experimental data. An experimental study has also been made very recently in [9]. This experimental study involves the aerodynamic investigation of a NACA6409 foil with ground effect in a wind tunnel. Lift and drag forces, pitching moment and center of pressure have been measured and compared via various angles of attacks, aspect ratios, endplate types and ground clearances. Also smoke tests have been conducted in order to visualize the flow around the wing. The experimental study in [10] focuses on the investigation of a NACA4412 wing with tip sails. At a constant ground clearance, the lift and drag coefficients have been observed for different angles of attacks taking the tip sails into account. On the other hand, various numerical methods have been developed to treat the cavitating or non-cavitating flows around hydrofoils moving under free surface. Important studies by using the boundary element methods for the flow analysis of 2-D and 3-D cavitating or non-cavitating hydrofoils and propellers can be found in [11, 12]. Specifically, the Boundary Element Methods (BEMs) have also been found to be computationally efficient and robust tools for the inviscid analysis of cavitating or non-cavitating flows around arbitrary geometries both in two- and three-dimensions, including the effects of free surface. For instance, Kelvin and Rankine types of singularities have modeled the flow around cavitating or non-cavitating hydrofoil under a free surface in [13] and [14], respectively. The linearized free surface condition was used in both methods. An IBEM (iterative boundary element method) for the solution of cavitating or non-cavitating hydrofoil moving under a free surface was described in detail in [14, 15]. The integral equation obtained by applying Green's theorem on the surfaces of the problem was divided into two parts; the hydrofoil part and the free surface part. The hydrofoil influence on the free surface and vice versa was considered via their potential values. This iterative method was modified and extended to apply to the surface piercing bodies inside a numerical towing tank or without a numerical towing tank, the method has been validated with those of others and experiments and extensive numerical results of the method have been presented in [16, 17, 18, 19, and 20]. In the present study, the performance of a NACA0012 wing moving with a constant speed above free surface has been investigated by an Iterative Boundary Element Method (IBEM). The same wing geometry in unbounded flow domain has also been investigated with both IBEM and Computational Fluid Dynamics (CFD) technique based on finite volume method (FVM). The effects of the aspect ratio and Froude number on the aerodynamic performance of the wing have been discussed. Iterative Boundary Element Method It is considered that the wing above free surface is subjected to a uniform inflow, U. The x-axis is positive in the direction of uniform inflow, the z-axis is positive upwards and the y-axis completes the right-handed system as shown in Fig. 1. The wing above undisturbed free surface is located at z = h. It is assumed that the fluid is inviscid and incompressible and the flow field is irrotational. The perturbation potential, and the total potential, should satisfy the Laplace’s equation in the fluid domain,  2    2  0 (1)


45

z

Wake Panels x

y

U Wing Panels

Fig. 1 Definition of coordinate system.

The following boundary conditions should also be satisfied by perturbation potential: i) Kinematic boundary condition: The flow should be tangent to the wing surface,

    U  n n

(2)

 where n is the unit normal vector to the wing surface directed into the fluid domain. ii) Kutta condition and boundary condition on the wake surface: The velocity at the trailing edge of the wing should be finite,   finite; at the trailing edge (3) An iterative pressure Kutta condition is forced at the trailing edge of the wing. The force-free condition is also satisfied on the wake surface. The trailing wake surface is assumed to be constant on z=0. The dipole strength in the wake however is convected along the assumed wake model in order to ensure that the pressure jump in the wake is equal to zero. In other words, in order for wake surface to be force-free, the pressure across the wake surface must be continuous, p+=p-=p on wake surface (4)  If t is a unit vector in the direction of the mean velocity, equation (4) implies that the streamwise velocity must be continuous across the surface,

    V  t  V  t

(5)

Equation (5) may be written in terms of ,

Φ  Φ   t t







  Φ  Φ  0 t

(6)

which means that the jump of the potential across the wake remains constant in the streamwise direction,

Φ   Φ         constant in the t direction

(7)

Equation (7) can be shown to reduce iterative Morino’s Kutta condition [12], T+

T-

      T   T   Δ w

(8)

where and are the values of the potential at the upper and lower sides of the hydrofoil trailing edge, respectively. Refer to [12, 20] for details. iii) Kinematic free surface condition: The fluid particles should follow the free surface (x,y),

DF (x, y, z)  0 on z  ζ(x, y) Dt

(9)

46


where F(x,y,z)=z-(x,y). iv) Dynamic free surface condition: The pressure on the free surface should be equal to the atmospheric pressure (patm). Applying Bernoulli’s equation, the following equation can be given as,





1 ( ) 2  U 2  gζ  0 on z  ζ(x, y) 2

(10)

 2   k0  0 on z  0 z x 2

(11)

where g is the gravitational acceleration. If we linearize (omit the second-order terms) and combine Eq. (9) and Eq. (10), the following linearized free surface equation can be obtained as,

Here, k0=g/U2 is the wave number. The corresponding wave elevation in linearized form can be obtained as,

ζ

U  g x

(12)

v) Radiation condition: No upstream waves should occur. In order to prevent upstream waves, both the firstderivative and the second-derivative of the perturbation potential with respect to x is forced to be equal to zero for the upstream region on the free surface [20],

 2    0 as x   x 2 x

(13)

According to the Green’s third identity the perturbation potential on the wing surface surface) and on the free surface can be expressed as,

2 

G  G    G dS   W  dS  n n  n SW S H  S FS 



(14)

where SH, SW and SFS are the boundaries of the wing surface, wake surface and the free surface, respectively. G is the Green function (G=1/r), (r is the distance between the singularity point and field point). W is the potential jump across the wake surfaces, and n+ is the unit vector normal to the wake surface pointing upwards. The iterative method described in [14, 15] is applied to solve Eq. (14). The iterative method here in general is composed of two parts: (1) the wing part and (2) the free surface part. On the other hand, the potential in the fluid domain due to the influence of the wing, H, can be given as,

 G

2 H 





  n  n G dS   

W

SH

SW

G dS n 

(15)

The potential in the fluid domain due to the influence of the free surface, FS however, can be given as,

2 FS 



 G



  n  n G dS

(16)

S FS

By substituting Eq. (16) into Eq. (14), the following integral equation for the flow on the wing surface can be written as,

2 

 G





  n  n G dS   

SH

W

SW

G dS  4 FS n 

(17)

and by substituting Eq. (15) into Eq. (14) similarly, the following integral equation for the flow on the free surface can be written as,

2 

 G





  n  n G dS  4

S FS

H

(18)

After applying the kinematic condition on the wing surface and linearized free surface condition, Eqs. (17) and (18) can be reduced to,

Advances in Boundary Element and Meshless Techniques XVII 2 

 G



  n  Un G dS    x

SH

2 

W

SW

47 G dS  4 FS n 

 G  2 G    n  x 2 k 0 dS  4 H S FS 

(19) (20)

Here, nx is the x component of normal vector on the wing surface. Integral Equations (19) and (20) can be solved iteratively by a low-order panel method with the potentials H and FS being updated during the iterative process. Here, the wing surface and the free surface communicate each other via potential. The wing surface and the free surface are discretized into panels with constant strength source and dipole distributions. The discretized integral equations provide two matrix equations with respect to the unknown potential values and can be solved by any matrix solver. Computational Fluid Dynamics Technique The governing equations are the continuity equation and the well-known RANS equations for the steady, threedimensional, incompressible flow. The continuity can be given as;

U i 0 x i

(21)

While the momentum equations are expressed as [21];

  Ui U j  x j

1 P     ρ x i x j

  U U j    u i' u 'j i         x j xi   x j

(22)

In inviscid case, only first term is taken into account in the right hand side of the equation which refers as Euler equation. P expresses the mean pressure, ρ the density.

  Ui U j  x j

 

1 P ρ x i

(23)

In order to simulate the flow accurately, unstructured hexahedral mesh type has been employed in CFD analyses. Four different computational domains have been created for the analyses. In addition, some volumetric control blocks have been created for mesh refinement in upstream and downstream of the wing. Total mesh number is 2.5 million. In CFD technique, inviscid analyses have been performed by solving Euler equations. Numerical Results and Discussion The rectangular 3-D wing which has NACA0012 sections along span-wise direction has been modeled with aspect ratios of 4 and 6. The chord length and span of the wing are represented as c and s, respectively. The flow around the wing has been first simulated for the infinite case (no free surface effect, unbounded flow domain). The analyses have been performed for the angle of attack, =5°. The total number of panels on the wing is 50x40=2000, (the number of panels along chord-wise direction and span-wise direction are 50 and 40, respectively) as shown half of the wing due to symmetry with respect to span in Fig. 1. On the other hand the total number of panels on the free surface is selected as 100x20=2000, (the number of panels along x direction and y direction are 100 and 20, respectively). Cp distribution calculated by CFD analyses have an oscillating trend especially on the back surface of the wing which may be caused by the grid structure. The oscillations have been smoothed by applying least square method based on sixth order parabola. In Fig. 2, the non-dimensional pressure coefficients ( Cp 

p0  p , p0: pressure far filed from wing, p: pressure on the wing, and : density of air) on the 2 1 2 ρU

mid-section of the 3-D wing by CFD technique are shown as compared with those of IBEM. The agreement between them is satisfactory.

48


Fig. 2 Comparison of BEM and FVM results.

Fig. 3 Effect of Froude number on lift and drag coefficients for two different clearances.

L D , L: lift , CD  1 2 ρAU 2 ρAU 2 U force, D: drag force, A: planform area of the wing) with Froude number ( Fn  ) is presented for two gc The variation of lift and drag (induced drag + wave drag) coefficients ( C L 

1 2

different h/c ratios (=0.5 and 1.0) in Fig. 3. The aspect ratio (AR=s2/A) is chosen as 4. The lift and drag coefficients (only induced) in the case of unbounded flow domain (no free surface effect) have been computed as CL=0.3283 and CD=0.0095, respectively. Therefore for the selected range of Froude numbers the free surface caused an increase in both lift and drag coefficients. In Fig. 4, the wave contours on the free surface by IBEM have been shown for Fn=0.7 and h/c=0.5. The Kelvin wave pattern can be seen clearly here.

Fig. 4 Kelvin wave contours on free surface.

Fig.5 Non-dimensional pressure distribution.

The pressure coefficient on the mid-section of the wing for Fn=0.7 and h/c=0.5 are shown as compared with those of infinite case in Fig. 5. Free surface caused an increase in pressure values which is consistent with the results given in Fig. 3. Later the effects of aspect ratio on the results have been investigated. In Fig. 6, the non-dimensional pressure distribution on the mid-section of the wing with AR=4 in unbounded flow domain has been compared with those of AR=6. An increase in aspect ratio caused an increase in pressure distribution slightly. The same is true for the case of Fn=0.7 and h/c=1.0 (e.g. with free surface effect) as given in Fig. 7.


49

1.6 1.4

AR=4 (Face) AR=4 (Back) AR=6 (Face) AR=6 (BacK)

1.2 1 0.8

No Free Surface Effect 0 =5

-Cp

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

0.25

0.5

x/c

0.75

1

Fig. 6 Effect of aspect ratio on pressure distribution.

Fig. 7 Effect of aspect ratio on mid-section pressure distribution in case of free surface.

Conclusions In the present paper, an iterative boundary element method and finite volume method have been employed in order to predict the aerodynamic performance of 3-D wing moving steadily over a free water surface. The followings have been found: 1-) the free surface caused an increase in loading (lift and drag coefficients and pressure distribution) of the wing related to those of infinite domain (the case of unbounded flow domain, no free surface effect). 2-) If the clearance is small, free surface effect on wing can become significant. 3-) The Kelvin wave pattern has also been occurred on the free water surface. 4-) An increase in aspect ratio also causes an increase in the loading of the wing. Unsteady and/or viscous effects will also be studied in near future. References [1] K. V. Rozhdestvensky, “Wing-in-ground effect vehicles,” Prog. Aerosp. Sci., vol. 42, no. 3, pp. 211–283, May 2006. [2] K. V. Rozhdestvensky, Aerodynamics of a Lifting System in Extreme Ground Effect. Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 2010. [3] E. O. Tuck, “A nonlinear unsteady one-dimensional theory for wings in extreme ground effect,” J. Fluid Mech., vol. 98, no. 01, pp. 33–47, May 1980. [4] E. O. Tuck, “Steady flow and static stability of airfoils in extreme ground effect,” J. Eng. Math., vol. 15, no. 2, pp. 89–102, Apr. 1981. [5] M. S. Seif and M. T. Dakhrabadi, “A practical method for aerodynamic investigation of WIG,” Aircr. Eng. Aerosp. Technol., vol. 88, no. 1, pp. 73–81, Jan. 2016. [6] T. J. Barber, “A Study of Water Surface Deformation Due to Tip Vortices of a Wing-in-Ground Effect,” J. Ship Res., vol. 51, no. 2, pp. 182–186, Jun. 2007. [7] H. Liang and Z. Zong, “A subsonic lifting surface theory for wing-in-ground effect,” Acta Mech., vol. 219, no. 3–4, pp. 203–217, Feb. 2011. [8] Z. Zong, H. Liang, and L. Zhou, “Lifting line theory for wing-in-ground effect in proximity to a free surface,” J. Eng. Math., vol. 74, no. 1, pp. 143–158, Aug. 2011.

50


[9] K. H. Jung, H. H. Chun, and H. J. Kim, “Experimental investigation of wing-in-ground effect with a NACA6409 section,” J. Mar. Sci. Technol., vol. 13, no. 4, pp. 317–327, Aug. 2008. [10] C. Sun and C. Dai, “Experimental Study on Ground Effect of a Wing with Tip Sails,” Procedia Eng., vol. 126, pp. 559–563, 2015. [11] N. E. Fine and S. A. Kinnas, "A boundary element method for the analysis of the flow around 3-D cavitating hydrofoils," J Ship Res., vol. 37, pp. 213-224, 1993. [12] S. A. Kinnas and C. Y. Hsin, "A boundary element method for the analysis of the unsteady flow around extreme propeller geometries," AIAA J., vol. 30, pp. 688-696, 1992. [13] C. S. Lee, C. W. Lew and Y. G. Kim, "Analysis of a two-dimensional partially or supercavitating hydrofoil advancing under a free surface with a finite Froude number," 19th Symposium on Naval Hydrodynamics, Korea, pp. 605-618, 1992. [14] S. Bal and S.A. Kinnas, "A BEM for the prediction of free surface effect on cavitating hydrofoils," Computational Mechanics, vol. 28, pp. 260-274, 2002. [15] S. Bal, S. A. Kinnas and H. Lee, "Numerical analysis of 2-D and 3-D cavitating hydrofoils under a free surface," J Ship Res., vol. 45, no. 1, p. 34-49, 2001. [16] S. Bal, "A numerical method for the prediction of wave pattern of surface piercing cavitating hydrofoils," Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Sciences, vol. 221, p. 1623-1633, 2007. [17] S. Bal, "High-speed submerged and surface piercing cavitating hydrofoils, including tandem case," Ocean Engineering, vol. 34, p. 1935-1946, 2007. [18] S. Bal, "Prediction of wave pattern and wave resistance of surface piercing bodies by a boundary element method," International Journal for Numerical Methods in Fluids, vol. 56, pp. 305-329, 2008. [19] S. Bal, " Performance prediction of surface piercing bodies in numerical towing tank," International Journal of Offshore and Polar Engineering, vol. 18, p. 106-111, 2008. [20] S. Bal, "The effect of finite depth on 2-D and 3-D cavitating hydrofoils," Journal of Marine Science and Technology, vol. 16, no. 2, p. 129-142, 2011. [21] D. C. Wilcox, “Formulation of the k-w Turbulence Model Revisited,” AIAA J., vol. 46, no. 11, pp. 2823– 2838, 2008.

A Fast-Multipole Implementation of the Simplified Hybrid Boundary Element Method Ney A. Dumont* and Hélvio F. C. Peixoto Civil Engineering Department, PUC-Rio, Rio de Janeiro, Brazil *

Corresponding author, email: [email protected]

Keywords: Hybrid boundary elements, fast multipole, variational methods. Abstract. The ultimate subject of this work is the implementation and testing of a novel numerical tool that can simulate on a personal computer and only in a few minutes a problem with many millions of degrees of freedom. The authors have already successfully developed and tested a technique that turned out to be a modified, reverse fast-multipole implementation for the conventional BEM. On the other hand, the variationally-based hybrid BEM already leads to a computationally less intensive formulation than in the conventional BEM for large-scale 2D and 3D problems of potential and elasticity. This formulation is especially advantageous for problems of complicated geometry and topology or requiring complicated fundamental solutions. The proposed implementation of the fast multipole method (FMM) for the simplified, hybrid BEM deals with the transpose of the double-layer potential matrix as well as with the nodal matrix expression of the potential fundamental solution (as an advantageous alternative to using the single-layer potential matrix – in a slightly different conceptual environment and adequately dealing with the ensuing singularities). The natural order of outlining the proposed implementation of the FMM for the simplified, hybrid BEM is by firstly introducing the basic aspects of the FMM for the conventional BEM. This takes most part of the present paper, which ends up with some validating numerical results. The FMM outline for the simplified hybrid BEM is shown in a separate section, as its numerical implementation is still in progress.

Introduction The present research work is part of the studies carried out by the second author [1] together with Novelino [2] to develop a robust and efficient fast multipole code applicable to problems with generally curved boundaries, in a framework that is almost completely independent from the underlying fundamental solution [3, 4]. The basic concept of the (FMM), with the expansion of the fundamental solution about successive layers of source and field poles, is described in a compact algorithm that is more straightforward to lay out and promises to be more efficient than the ones available in the technical literature [5-7]. In the proposed FMM implementation, a hierarchical tree of poles is built upon a topological concept of superelements inside superelements, which in part circumvents the need of evaluating geometrical distances between nodes as well as the need of concepts such as quadtrees or octrees for 2D or 3D problems. This FMM – which differs from the formulations classically presented in the literature not only because it follows a reverse strategy – has been already assessed for a variety of patch and cut-out tests for 2D potential problems and is being presently implemented for elasticity and 3D problems. It has not been inserted into an iterative solver yet, since our goal has consisted in first to validate and assess the isolated FMM algorithm for accuracy, computational effort and storage allocation. The code is written in C++ and can automatically deal with elements of any order – although only linear and quadratic elements have actually been tested. (A separate code for constant elements is also implemented.)

Proposed FM algorithm for a general, complex function   

The following basic definitions are used in the present developments: z  z0 = difference between the source point z0 and the field point z . zc = point about which the fundamental solution will be expanded for the field point z . z L = point about which the fundamental solution will be expanded for the source point z0 . Expansions about successively farther poles zck , k  1, 2,  nc (where, by definition, zc0  zc ) and z Ll ,

 l 1, 2,  nL (where, by definition, z L0  z L ) are also undertaken.

52


A generic fundamental solution for 2D problems is expanded as

 f ( z  z0 )

n 1

1

1

n 1

 (i  1)! P ( z  z ) ( j  1)! P ( z

i c nk i 1 j 1

for truncation order given by max

 (z  z

c nk

) / ( z  z0 )

n 1

j

Lnl

 z0 )Qi  j 1 ( zcnk  z Lnl )

, ( z Lnl  z0 ) / ( z  z0 )

n 1

(1)

 and with P( z) and

Q( z ) defined as P ( z )  1 z Q( z ) [ f ( z ) f ( z ) / z

z2

f 2 ( z ) / z 2

z 3  z n 1 

(2)

f 3 ( z ) / z 3

 f n 1 ( z ) / z n 1 ]

(3)

In general, the higher derivatives of the fundamental solution tend rapidly to zero when evaluated for large arguments. Equation (1) is the starting point for a procedure that leads to a computationally fast and economical evaluation of a given fundamental solution f ( z  z0 ) for a very large number of source points z0 and of field points z by means of a sufficiently approximate expression. As shown, f ( z  z0 ) is expanded in terms of successive arrays of source poles z Ll as well as field poles zc k . The expansion ends up with a series of products of binomials Pi ( z  zcnk ) and Pi ( z0  z Lnl ) , which are independent from the complexity of the

function f ( z  z0 ) , multiplied by functions Qi ( z Lnl  zcnk ) that are given as f ( z Lnl  zcnk ) and its 2n first

derivatives, for the expansion indicated in eq (1). Although these latter functions may be computationally intensive to evaluate, they are only needed for the multiplication of the arrays of poles represented by z Ll and

zck . Then, the evaluation of Qi ( z Lnl  zcnk ) may end up orders of magnitude less intensive than the direct

evaluation of f ( z  z0 ) for all source and field points.

Expansion of Pi ( z  zck ) and Pj ( z Lnl  z0 ) about successive poles

In eq (1), the superscripts nk and nl are the highest levels of field and source poles used in a given

expansion. The binomials Pi ( z  zcl ) defined in eq (2) are expressed for a lower-level pole zcl 1 exactly as

 Pi (zzcl )

i

C j 1

j ,i 1 j

Pj (zzcl 1 ) Pi 1 j (zcl1 zcl )

(4)

where Cij  1 if i  1 or j  1 , otherwise, Cij  Ci 1, j  Ci , j 1 . In eq (4), Pi 1 j ( zcl 1  zcl ) is defined as in eq

(2) since the argument refers to two consecutive levels. On the other hand, Pj ( z  zcl 1 ) is recursively



evaluated according to eq (4) until the lowest level Pj z  zc0

 is obtained. With this recursive approach, the

binomials Pi ( z  zcnk ) and Pj ( z Lnl  z0 ) always end up expressed in terms of arguments given as differences of poles in two consecutive levels.

A very brief outline of the conventional BEM The basic equation of the conventional BEM, as applied to the Laplace equation  2  0 , is

Hd  Gt

(5)

In complex notation, as adequate for a FMM implementation, * H  H sf    ,*j ( z  zs ) j ( z )  f ( z )d( z ), G  Gs    ( z  zs )q ( z )d( z ) 



(6)

are the double-layer and single-layer potential matrices, d are nodal potentials and t are nodal flux attributes. In eq (6),  ,*j ( z  zs )  q*js and  * ( z  zs )   s* are the flux and potential fundamental solutions of the potential problem – which have global support – and ( z ) is the integration boundary. The subscript s refers to a given source node (at which the unit point source of the singular fundamental solution is applied) and the


53

subscripts f (which stands for field) and  (also a field reference) indicate the nodes to which the potentialinterpolation function  f ( z ( )) and the flux-interpolation function q ( z ( )) are referred. In the above equation and in the following, repeated indices mean summation.  j ( z ) are the Cartesian components of the outward unity vector to ( z ) , and  f ( z ) formally comes from the piecewise (with local support) interpolation of potentials  ( z ) along the boundary:  ( z )   f ( z )d f , where d f are the nodal potentials. In a usual isoparametric representation, potential and geometry are represented identically for each one of their Cartesian components, so that  f ( z )   f ( z ( )) are actually replaced with shape functions N k ( ) , for a

given boundary segment, with   (0,1) or   (1,1) , depending on the preferred parametric representation. For z  z0 , strong and weak singularities must be taken care of in the evaluation of H and G . However, this issue can be disregarded in the present developments, which are actually only concerned with what occurs when z and z0 are very far from each other. The Jacobian used in the definition of  j ( z ) cancels out with

J ( z ) d , in terms of the parametric variable  . Then, expanding q*js according to the Jacobian of d( z )  eq (1) ends up with the evaluation of a polynomial integral corresponding to the first of eq (6). For the singlelayer potential matrix G , it is proposed that the usual interpolation polynomials q of normal fluxes in eq (6) be replaced with q  q J (at  ) / J , (7) where J

(at  )

is the value of the Jacobian at the point characterized by the subscript  [8, 9]. Nothing changes

formally in the developments of the BEM, except that the numerical integration of the matrix G becomes much easier and actually more consistent as compared to proposed implementations given in the literature [8]. In fact, J cancels out in the product q d in eq (6) for q defined as suggested, and the integrand of G , in the frame of a fast multipole implementation, also becomes a polynomial, independently from the assumed kernel  s* . In a practical implementation, the functions q ( z ( )) are replaced with the same shape functions

N k ( ) used to represent displacements, although the context differs conceptually, as G , among other

features, is in general a rectangular matrix (  in general spans a larger number of nodes than f ).

Implemented FM algorithm This section describes a compact version of the implemented fast-multipole algorithm, as applied to potential problems in the conventional BEM and described above. The number of code lines is actually very small. However, as the algorithm calls a recursive routine (PoleExpansion) inside another recursive routine (Adjacencies), this makes it fairly convoluted and difficult to explain verbally, although the flowcharts can be easily translated into a code. The basic version presented below gives an overview of the algorithm's four major routines: Main, Adjacencies, Source and PoleExpansion. The procedure Main (Fig. 1) loads the input data, generates the hierarchical mesh according to the concepts briefly discussed in the Introduction and evaluates the kernel expansions according to eq (3). Then it executes a small loop over all elements of the first level ( k  0 ) of the hierarchical mesh in order to create the adjacency structure for each macro-element ( ie ), carrying out, at the same time, all the possible field evaluations for the child elements of element ie . The routine Adjacencies (Fig. 2), which is actually preceded by an initializing routine Adjacencies0 also called by the procedure Main, assembles the adjacency structure, and when it reaches the most refined level ( k  nv ), calls the routine Source. This routine (Fig. 3) handles integrations in terms of the conventional BEM matrix-vector products (routine BEMAdj, not shown) for the adjacent elements, as well as in terms of FM expansions (routine BEMFM, also not shown). The analytical integrations carried out in the frame of the routine BEMFM refer to the closest field pole and are successively stored for use with far-field elements in the routine PoleExpansion. The routine Source also leads to the successive expansion of the FM integration terms,

54


thus delivering data information to the upper refinement levels, if this is the case, by calling the routine PoleExpansion. Finally, the recursive routine PoleExpansion (Fig. 4) delivers the FM-integrated data to the source poles that are considered sufficiently far by calling the routine Qvector to evaluate the Q vectors defined in eq (3) and then evaluating the expansion series for the source point, eq (1). It also checks if the level k  kexp has been reached, as expansions stop at this level, indicating that the results obtained so far are directly delivered to the remaining source points. If k  kexp has not been reached, the routine calls routine Pvector (not shown) to convey the obtained data, according to eq (4), to the upper pole levels and, when all elements of level k have been processed, calls itself (thus recursively) to proceed to the immediately upper refinement level of the hierarchical structure.

Numerical results Figure 5 shows two irregularly-shaped domains that will undergo the hierarchical mesh refinement discussed in the Introduction and then will be submitted to a given potential field [10]. This potential field is an in principal arbitrary analytical solution of the Laplace equation for the open-field domain, for which a vector d of potential values and a vector t of normal gradient values are obtained along a boundary drawn in the open field, and applied to the body in which is called a cut-out test, as the accuracy of the BEM eq (5) will be assessed for different mesh refinements. The code is implemented in the language C++ and runs on a desktop computer (i7-4770 CPU 3.4GHz, 16GB RAM in Windows 7). The errors presented on the right of Figs. 6 and 7 represent the Euclidean error norm   Hd  Gt / Gt (8) For the domain on the left of Fig. 5, a quadratic field x 2  y 2 was applied as the test analytical solution of the Laplace equation. The boundary is discretized with constant, linear and quadratic elements with up to 224  16, 777, 216 degrees of freedom as represented in the horizontal axis of both graphs in Fig. 6. Figure 6 shows on the left the time required for running simulations with different degrees of freedom (horizontal axes) and different numbers n of terms in the series, according to eq (1), for each element type. It may be noticed that, for a given element type, increasing the number of terms in the series does not lead to a considerable increase in the execution time. On the other hand, the graph on the right of Fig. 6 shows that the number n of expansion terms considerably affects the numerical accuracy. The full circles characterize in both graphs of Fig. 6 results obtained by evaluating the matrix-vector products Hd and Gt as in a conventional BEM implementation. Since the applied analytical open-field is quadratic, the conventional BEM solution for quadratic elements (dashed lines) is as accurate as the numerical integration and round-off errors allow. However, when evaluated via the FMM, there is an intrinsic error due to the series expansions. This error poses an accuracy threshold to simulations with the lower-order elements. For the domain on the right of Fig. 5, only quadratic elements are used in order to have a cleaner display of results. This structure is discretized with up to 5  222  20,971,520 degrees of freedom and is submitted to a

zs 12.5  15i is the logarithmic field ln z  z s , where z x  iy is the field point of the domain, and  source point, represented by () in Fig. 5. The execution time and error results are given in Fig. 7 as outlined for the first numerical example. As already observed, one sees on the left of Fig. 7 that the computational effort increases only slightly as the number n of expansion terms increases. This graph also displays the curves proportional to N (dotted line), N log N (dashed line), and N 2 (dash-dot line), which shows that, while the implementation of the matrix-vector product in terms of the conventional BEM requires a computational time proportional to N 2 , the present FMM implementation performs close to N , as already suggested by Liu [7] as an achievable goal. The error assessment on the right of Fig. 7 goes only up to 5  218 degrees of freedom, as the error threshold for the FMM expansions is arrived at already for 5  210 degrees of freedom, with the same convergence behavior observed in the first example.


55

Start Main

Recursive routine Adjacencies(ie,k)

Input data

For ie pr  1..nc ie p  ie  (nc  1)  ie pr

End

Set adjacencies to element iep

Initialize some constants and control parameters Routines  Hierarchical mesh  Fundamental solutions

If k+1=nv

T

F

Recursive routine Adjacencies(iep,k+1)

For ie=1.. nek[0]

Routine Source(iep)

Routine Adjacencies0(ie) Recursive routine Adjacencies(ie,0)

Figure 2. Procedure Adjacencies.

Routine Source(iep)

Add discontinuous part of H

Routine BEMAdj(iep) End Routine BEMFM(iep)

Figure 1. Procedure Main.

IfF ie p multiple of nc T

Figure 3. Procedure Source.

Recursive Routine PoleExpansion(nv-1) End

The expedite boundary element method The expedite boundary element method, a simplified, variationally-based formulation, has been well explained in Reference [9], for example. For the simplest case of a potential problem, it relies on the assumption that the potential  and its gradients  ,l inside a domain can be described in terms of a series of point source parameters p s applied along the boundary plus some arbitrary particular solution  p ,

  s, j ps  , pj  s  Cs  ps   p , , j 

(9)

where  s is a fundamental solution of the corresponding differential equation of the problem. In the present case,  2 s  0 except for the point of application of p s , when  s becomes undetermined. Moreover,  s is

56


obtained except for a constant Cs [8, 9]. This belongs to the basic theory of the conventional boundary element method only bearing in mind that in a variational formulation  s is used as a numerical approximation of the actual problem and not just as a weighting function (references given in [9]). Assigning subscripts D and N to subvectors of nodal potentials d and equivalent nodal gradients p to characterize whether the boundary conditions are of Dirichlet or Neumann type, the final matrix equation system of the expedite boundary element method is expressed as  HTN    p N  p Np   UN   d N  d Np  (10) ,   T  p    p  p  p  pD  pD  d D  d D  HD  UD  Recursive routine PoleExpansion(k) F

child=elsplit(k)

T

For all elements ia p that are adjacent to child 's parent at level k  1 F

If iap's child iac is adjacent to child at level k T

Routine Qvector(child,iac,k)

T

If k  kexp

 For els 1..nek  k  k c  F

End

If els adjacent to child at level k  kc T

For ic 1..nc

sc=(els – 1)×nc+ic Routine Qvector(child,sc,k) parent=elsplit(k-1) Routine Pvector(k,child,parent)

Figure 4. Procedure PoleExpansion. If child multiple of nc

F

T

Recursive routine PoleExpansion(k-1)

End

Figure 5. Domains used in the numerical assessments. Left: stepwise linear boundary submitted to a quadratic field. Right: irregularly-shaped domain submitted to a logarithmic field with source applied at  [10].


57

Figure 6. Execution times (left) for the evaluation of eq (5) for the domain on the left of Fig. 5 using constant, linear and quadratic elements, and accuracy results (right) for different numbers n of expansion terms [10].

Figure 7. Execution times (left) for the evaluation of eq (5) for the domain on the right of Fig. 5 using quadratic elements, and accuracy results (right) for different numbers n of expansion terms [10]. where the quantities with superscript p stand for nodal potentials or gradients belonging to the assumed, arbitrary particular solution of the problem. H is the double-layer potential matrix and U represents the fundamental solutions  s evaluated at the boundary nodal points, that is, U   s* ( z f  z0 ) . The equations above can be firstly solved for p  in terms of the known nodal quantities, provided that the problem is well posed, with the subsequent evaluation of the remaining boundary potentials and gradients. Results at internal points are obtained directly by using eq (9). Results close to or at nodal points can also be obtained [9]. The implemented 2D boundary element code works with linear, quadratic or cubic elements. In the proposed implementation, H can be approximated for far source and field nodal points as [9]

H  TT L, with  L  Lf

 

f

( z )q ( z )d( z )

(11)

The actually small block matrix L [9] comes from the transformation between normal flux gradients and equivalent nodal flux gradients: p f  Lf t , which links the quantities introduced in eqs (5) and (10). The

58


nodal values of the fundamental solution U  ,*j ( z  zs ) j U fs  * ( z f  zs ) and T  Ts 

(at  )

can be

obtained also when source and field points coincide by simply assigning them a patch test interpretation [9]. The evaluation of the matrix-vector products given in eq (10) has already been shown feasible in the frame of the proposed FMM. Accuracy of results and computational costs are being currently assessed.

Summary This paper presents a novel, kernel-independent fast multipole formulation to be used with the BEM. The formulation relies on a hierarchical mesh refinement strategy for generally curved boundary elements, which is also used in the evaluation of element adjacencies and is key to the proposed algorithm. A compact version of the implemented algorithm is presented, and its application is illustrated for two irregularly-shaped domains with up to N  20,971,520 degrees of freedom. The numerical assessments show that the proposed algorithm is seamlessly applicable to generally curved elements of any order. The simulation of extremely convoluted shapes including multiply-connected domains seems to present no difficulties. The computational cost for all examples run so far has shown to be proportional to O( N ) , as opposed to a conventional BEM implementation, which requires operations of order O ( N 2 ) . As a matter of fact, the proposed FMM implementation is superior to a conventional BEM implementation in terms of computational costs even for a very small number of degrees of freedom, as observed in the graphs on the left of Figs. 6 and 7. The implementation for the variationally-based, simplified hybrid boundary element method is in progress. Preliminary tests have shown that applications to problems with very complicated fundamental solutions, such as in fracture mechanics, may become advantageous regardless of problem size.

Acknowledgments This work was supported by the Brazilian agencies CAPES, CNPq and FAPERJ.

References [1] H.F.C. Peixoto. Application of the Hybrid Boundary Element Method to Large-Scale Problems Using Fast Multipole Techniques. Msc. thesis, PUC-Rio (2014). [2] L.S. Novelino. A Novel Fast Multipole Technique in the Boundary Element Methods. Msc. thesis, PUCRio (2015). [3] H.F.C. Peixoto, L.S. Novelino and N.A. Dumont. Basics of a fast-multipole unified technique for the analysis of several classes of continuum mechanics problems with the boundary element method. In Boundary Elements and Other Mesh Reduction Methods XXXVIII 47-59, WITPress Southampton (2015). [4] H.F.C. Peixoto, L.S. Novelino and N.A. Dumont. A fast-multipole unified technique for the analysis of continuum mechanics problems with the boundary element methods. In CILAMCE – XXXVI Iberian LatinAmerican Congress on Computational Methods in Engineering, 12 pp, Rio de Janeiro, Brazil (2015). [5] N. Nishimura. Fast multipole accelerated boundary integral equation methods. Applied Mechanics Reviews 55(4) 299-324 (2002). [6] Y. Liu and N. Nishimura. The fast multipole boundary element method for potential problems: A tutorial. Engineering Analysis with Boundary Elements 30(5) 371–381 (2006). [7] Y. Liu. Fast Multipole Boundary Element Method: Theory and Applications in Engineering, Cambridge University Press (2009). [8] N.A. Dumont. The boundary element method revisited. In Boundary Elements and Other Mesh Reduction Methods XXXII 227-238, WITPress, Southampton (2010). [9] N.A. Dumont and C.A. Aguilar. The best of two worlds: The expedite boundary element method. Engineering Structures 43 235–244 (2012). [10] H.F.C. Peixoto and N.A. Dumont. A kernel-independent fast multipole technique for the analysis of problems with the BEM. In XII Simpósio de Mecânica Computacional, 8 pp, Diamantina, Brazil (2016).

Linear analysis of heterogeneous microstructures by the Boundary Element Method Guilherme A. Ohland1, Jordana F. Vieira2, Matheus C. dos Santos2 and Gabriela R. Fernandes2 1

FEIS- Faculdade de Engenharia de Ilha Solteira CEP: 15385-000, Ilha Solteira, SP, Brazil

2

Civil Engineering Department, Federal University of Goiás (UFG) CAC – Regional Catalão, Av. Dr. Lamartine Pinto de Avelar, 1120, Setor Universitário- CEP 75700-000 Catalão – GO Brazil, email [email protected]

Keywords: boundary elements, stretching problem, zoned plates.

Abstract. A formulation of the boundary element method (BEM) to perform linear analysis of heterogeneous microstructures is presented. Inside the matrix can be defined voids or inclusions with different elastic properties. The microstructure is modelled by a zoned plate, where different values of Poisson’s ratio and Young’s modulus can be defined for each sub-region. To solve the domain integrals written in terms of in-plane displacements, the matrix and inclusions domains are discretized into cells where the displacements have to be approximated. Thus, in this model, besides the boundary values for inplane displacements and tractions, nodal values of in-plane displacements are defined in the domain. Then, adopting homogenization techniques, the homogenized values for stress and constitutive tensor are computed. In a future work this formulation will be extended to solve the non-linear problem of the microstructure or RVE (representative volume element) in terms of displacement fluctuations. Coupling this extended formulation to the BEM model adopted for the macro-continuum, the multi-scale analysis could be performed using only the Boundary Element Method. Some numerical examples are presented and compared to a finite element code to show the accuracy of the proposed model. 1. Introduction The boundary element method (BEM) has already proved to be a suitable numerical tool to deal with plate problems. The method is particularly recommended to evaluate internal force concentrations due to loads distributed over small regions that very often appear in practical problems. Several works have already considered BEM formulations to treat different kinds of problems [1-5]. In general, the materials, even the metallic, are heterogeneous at the micro and grain scale. Besides, the material microstructure can be also appropriately manipulated by adding certain constituents to a matrix phase, in order to change the material properties to attend specific applications, as the MMCs (metal matrix composites). As any heterogeneity of the material as well as the microcracking initiation and propagation in the micro-scale affect directly the macro-continuum response, modelling heterogeneous material in different scales is very important to better represent the behaviour of such complex materials [6-8]. In multi-scale analysis, the RVE (representative volume element) represents the microstructure, at grain level, of the macro-continuum at the infinitesimal material neighbourhood of a point (see [9]-[11]). In this paper a BEM formulation to perform linear analysis of RVEs is proposed, where it is considered as a zoned plate, whose sub-regions define the different materials of the microstructure. 2. Basic Equations The domain Ω of a heterogeneous microstructure is assumed to consist in general of a solid part, a void part

 s and

v , being   v  s , where the solid part can be made of distinct materials (or phases),

each one defined by a sub-domain, whose material can have different elastics properties. Without loss of generality, let us consider the microstructure depicted in Fig. 1 represented by a zoned plate, where subregion Ω1 represents the matrix whose external boundary is Γ1, sub-region Ω2 is an inclusion and Ω3 a void. Besides, in Fig. (1) Γjk represents the interface between the adjacent sub-regions Ωj and Ωk and the Cartesian system of co-ordinates (axes x1 and x2) is defined on the plate surface.

60

Eds: M. Tezer-Sezgin, B. Karasözen, M H Aliabadi Ω1 Ω2 ᴦ 21 ᴦ12

X2

Ω3 ᴦ13

ᴦ1

X1

Figure 1- Heterogeneous microstructure represented by a zoned plate For a point placed at any of those plate sub-regions, the following in-plane equilibrium equation can be defined:

N ij , j bi  0

i, j=1, 2

(1)

where bi are body forces distributed over the plate middle surface and Nij is the membrane internal force, which, for plane stress conditions, can be written in terms of the in-plane deformations  ij as follows:

N ij 

E 1     2

kk

 ij  1   ij 

(2)

where E is the Young’s modulus,  the Poisson’s ratio and  ij the Kronecker delta. The problem definition is then completed by assuming the following boundary conditions over Γ:

U i  U i on Γu (in-plane displacements) and Pi  P i on Γp (in-plane tractions), where  u   p   . 3. Integral Representation for Displacement As described in [5], from Betti’s theorem, the following equation can be obtained for any sub-region  m :



m* kij

N ijm d 

m

N

 d

m* m kij ij

i, j, k=1, 2

(3)

m

m* m* and N kij are fundamental solutions. where  kij

Eq. (3) can be now modified by writing the fundamental strains of sub-region  m in terms of the values



* kij

and E referred to the sub-region where the load point s is placed. This simplifies the formulation

because allows to eliminate the tractions along the interfaces. Thus, the following relation can be defined:

 kijm *   kij* E / E m

(4)

where E m  E m t m , being Em the Young’s modulus in the sub-region  m . m* * can be also written in terms of  and N kij referred to the Considering Eq. (4) the membrane force N kij

sub-region where the load point is placed as follows: *( m )  N kij

1   1   2

2 m

m

*  N kij

E  m  * 1   kij 1  m2   





(5)

Replacing Eq. (4) and (5) into (3) and considering all sub-regions, one obtains the following relation for the whole plate:


NS

Em 

*   kij N ij d    m 1





m

 E 

m

ij

61

    * * N kij d  E m 1  m    ij  kij d m    m  

where Ns is the sub-regions number; E m 

i, j, k=1, 2

(6)

Em

1   . m

2

Integrating eq (6) by parts we obtain the following representation of in-plane displacements:

Ck 1u1 s   Ck 2u2 s      N inc  E m m  E 1 1    



E

m 1



N voids



m 1

E1 1 E

E1 1 E

 u P  p s, P   u P  p s, P d  * k1

1

* k2

2

1

 u P  p s, P   u P  p s, P d 1

* k1

* k2

2

m1

m1

 u P  p s, P   u P  p s, P d 1

* k1

2

* k2

1m

1 m



 u s, P  p P   u s, P  p P d  * k1

1

* k2

2

1

s       *  E1 1  1   u1 P  k*1 s, P   u2 P  k*2 s, P  d   E m 1  m   ui  p  kij , j s, p d m    m   1 m 1



N



N inc           E m 1  m   E1 1  1   u1 P  k*1 s, P   u2 P  k*2 s, P  dm1       m1 m 1 





k, i, j=1,2 (7)

where k is the fundamental load direction, Nint is the interfaces number; Nvoids is the voids number; 1 represents the matrix domain and 1 its external boundary; m1 represents an interface between the matrix and inclusion m and 1m the interface between the matrix and a void m; the free terms values Ck1 and Ck2 depend on the position of the collocation point s. 4. Algebraic Equations The integral representation Eq. (7) is transformed into algebraic expression after discretizing the external boundary and interfaces into elements and the domain into cells. Geometrically linear elements have been adopted, where linear shape functions have been assumed to approximate the variables. Moreover, triangular cells have been used to discretize the sub-regions domain, where the displacements u1 and u2 are approximated by continuous linear shape functions, being their cell nodal values new independent values. Along the external boundary the nodal values are: two in-plane displacements (u1 and u2) and two inplane tractions (p1 and p2). As two of these values must be given as boundary conditions, two equations must be written for each boundary node. Besides, two unknowns (u1 and u2) are also defined at interface and cell nodes, being two more integral equations required at all these nodes. Therefore, after writing two algebraic equations of displacements at all nodes, one can get the following set of equations:

H BB  H  iB 

H Bi  U B  G BB  P  H ii   U i   G iB  B

(8)

In Eq. (8) the subscript B is related to the external boundary while i is referred to interface and internal nodes; U and P are displacement and traction vectors, respectively; H  is obtained by integrating all boundary and interfaces elements as well as the cells defined in the matrix and inclusions domain; G  is computed by integrating all boundary elements.

62


In a multi-scale analysis, the macro-strain has to be imposed to the RVE, leading to prescribe linear displacements over the external boundary as boundary conditions. Thus, in this case the unknowns

PB  . U i 

vector{X} computed from eq (8) is given by: X   

To solve the RVE equilibrium problem discussed in next section, the normal forces have also to be computed for each cell, where they are adopted constant.

5. Homogenised values for Stress and Constitutive tensor In the multi-scale analysis, the stress as well as the constitutive tensor at a point x of the macrocontinuum is evaluated from their respective fields over the RVE by using homogenization techniques. For that, the RVE is assumed sufficiently large in order to be considered as a continuum and the concept of stress to be valid at the microscopic scale. Then, it is also assumed that the strain tensor ε and the stress tensor σ at a point x of the macro-continuum are the volume average of their respective microscopic field ( or ) over the RVE associated with x (see [9]-[11]), i. e.:

 x    x  

1 V



1 V



    y dV

(9)

    y dV

(10)

where  and  are, respectively, the macroscopic or homogenised strain and stress. The homogenised constitutive tangent modulus Cep can be also evaluated by applying the homogenization process, as follows:

 x   C ep x     x 

1 V

    y dV



S

 x 



1 V

 f    y dV y



S

 x 

(11)

Note that in a non-linear analysis fy would be the constitutive functional defined by the adopted criterion. As in this work no dissipative phenomenon is considered, fy is given by the elastic tensor. The final expressions for homogenised stress and constitutive tensor are obtained according to the formulation proposed in [9]. For that, let us now split the microscopic displacement field u  in the following sum:

u   y   u L  y   u~  y 

(12)

In Eq. (12) u L  y  represents a linear displacement field, being u L  y    ( x) y , where  (x) is the

macroscopic strain and y represents the coordinates of an arbitrary point of the RVE; u~ is the displacement

fluctuation which represents the strain variation in the RVE. As described in [9-11] after discretising the RVE into cells and elements (or finite elements for the FEM formulation developed in [9-11]), the following microscopic equilibrium equation must hold for a discretisation h:

Rh 

B

 h

T





N cel

f y   Bu~ dV   BeT N e Ae  0 e 1

(13)


63

h

where   denotes the discretised RVE domain, Ncel is the number of cells used to discretize the RVE, Be the cell strain-displacement matrix, Ae is the cell area and Ne the normal force vector in the cell, which is considered constant over the cell. After imposing the macroscopic strain tensor ε to the RVE boundary, the microscopic equilibrium problem consists of finding the displacement fluctuation field

~ being u 

i 1

u~i1  u~i  u~i1 that satisfies equation (13),

the fluctuations corrections to be imposed in iteration i+1. Thus, by applying the Newton-

~ i 1 is computed by the following expression: Raphson Method, u  F i  K iu~i 1  0

(14)

where F is the traction vector and K the rigidity matrix, being defined as:

Fi 





N cel

T i T e(i )  B f y  n 1  Bu~ dV   Be N Ae

(15)

e 1

 h

N cel R K i   ~F   BeT DNe Be Ae u e 1

(16)

where DN is the microscopic constitutive tangent (in the proposed formulation is given by the elastic tensor), relating forces and strains. The RVE formulation is completed with the choice of kinematical constraints to be imposed on the RVE that leads to different classes of multi-scale models and therefore, to different numerical results. Three different boundary conditions can be imposed to the RVE in terms of displacement fluctuations: (i) linear boundary displacements, (ii) periodic fluctuations and (iii) uniform tractions (see more details in [9]-[11]). According to the formulation developed in [9], the homogenised stress tensor (equation 10) can be written as:

 where  

Nb

 F y i 1

b i

T i

1   T 2V





(17)

, being Nb the number of external boundary nodes and the forces Fb are obtained

from the forces Pb defined in Eq. (8), multiplying Pb by the influence length of the respective node. It is important to point out that the forces Pb defined in Eq. (8) are computed considering the nodal displacements along the external boundary U b obtained after solving the RVE equilibrium problem, i. e.,

U b is given by adding the displacement fluctuation field u~ to the linear displacement field u L .

As discussed in [9], the homogenised constitutive tangent modulus Cep (eq. 11) can be split into two parts, as follows:

~ Cep  Cep(Taylor)  C ep

(18)

where C ep (Taylor ) is denoted the Taylor model tangent operator (obtained by assuming

~ ep

computed by the volume average of the microscopic constitutive tangent; C

S u~  0 ) and

represents the influence of

the displacement fluctuation into the homogenised tangent modulus; they are given by (see [9]):

64


C ep (Taylor ) 

~ C ep 

1 V

1 V





   

 g  y

S

dV 

1 V



u~ dV

 h





Np

Vp

 D dV   V D

p

(19)

p 1



1 T 1 GR K R GR V

(20)

where Dμ is the microscopic constitutive tangent (in the proposed formulation is given by the elastic tensor) and Np the number of phases defined in the RVE; G 

Ncel

 D B V ; K e 1

e

e e

R

and GR are, respectively, reduced

forms of K (Eq. 16) and G, being they defined according to the adopted multi-scale model (see [9]-[11]). 6. Numerical Application In the numerical example is considered the RVE depicted in Fig. (2), where five inclusions are defined. For the matrix the following elastic properties are adopted: Poisson’s ratio =0,3; Young’s modulus E=70GPa, and for the inclusions it is assumed: =0.2; E= 200GPa. In the discretization (see Fig. 2) 520 elements and 293 nodes have been defined in the RVE. Note that in order to show the robustness of the proposed modelling, the inclusion distribution adopted for the RVE is random and it has been chosen in order to introduce strong heterogeneities due to different sizes and locations of the inclusions. The macro strain imposed to the RVE boundary is given by:    1  2 12   0,00054146 0,0020585 0. Three different boundary conditions have been imposed to the RVE in terms of displacements fluctuations: (i) linear boundary displacements, (ii) periodic fluctuations and (iii) uniform tractions, being the results compared to the ones obtained with the RVE formulation developed in [9]-[11], using the Finite Element Method (see tables 1 and 2). As can be observed, the results for constitutive tensor are the same and the ones for stress are very similar. Note that in Table 1 the stress vector is defined as:     1  2  12  .

Figure 2- Discretization of a RVE with inclusions.

Linear displacements Model

Stress Vector(MPa)

Periodic fluctuations Stress Vector (MPa)

Uniform tractions Stress Vector (MPa)

BEM

0.1056

205.70 0.867

0.303

204.88 0.689

3.88

200.83 .72

FEM

- 0.619

206.97 0.90

- 0.37

206.00 0.697

3.41

201.71 0.76

Table 1: Homogenised Values for stress


Linear displacements

Periodic fluctuations

65

Uniform tractions

Model

Constitutive Tensor(GPa)



BEM

 107.22 27.90 - .00478  27.90 107.42 .436   - .00478 .436 38.72 

 106.85 27.93 - 0.0144  27.93 107.42 0.335    - 0.0144 0.335 37.69 

104.91 29.25 - 0.051  29.25 105.68 0.356    - 0.051 0.356 37.44 

FEM

 107.22 27.90  27.90 107.89  - .00478 .436

 106.85 27.93 - 0.0144  27.93 107.42 0.335    - 0.0144 0.335 37.69 

104.91 29.25 - 0.051  29.25 105.68 0.356    - 0.051 0.356 37.44 

- .00478 .436  38.72 

Table 1: Homogenised Values for constitutive tensor 7. Conclusions A BEM formulation to analyze micro-structures of heterogeneous materials has been presented, where the RVE (Representative Volume Element) is modeled as a zoned plate. The different phases of the RVE are assumed to have elastic behavior, where different elastic properties can be defined in order to represent the material heterogeneity. The homogenized values for stress and constitutive tensor compared very well to the FEM solution, showing the accuracy of the presented formulation. Acknowledgements: The author wish to thank FAPEG (Fundação de Apoio à Pesquisa do Estado de Goiás), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and CAPES (Foundation, Ministry of Education of Brazil) for the financial support.

References [1] Aliabadi MH. The Boundary Element Method: applications in solids and structures, v.2, Wiley, 577pp., 2002. [2] Beskos D.E., Boundary element analysis of plates and shells. Springer Verlag, Berlin, 1991. [3] Brebbia CA, Telles JCF, Wrobel LC. Boundary element techniques. Theory and applications in engineering, Springer-Verlag, Berlin and New York, 1984. [4] Fernandes GR and de Souza Neto EA. Self-consistent linearization of non-linear BEM formulations with quadratic convergence. Computational Mechanics., v.52, p.1125-1139, 2013. [5] Fernandes G. R., Rosa Neto J. Analysis of stiffened plates composed by different materials by the boundary element method. Structural Engineering and Mechanics, an International Journal. v.56 No.4 p.605-623, 2015. [6] Gal, E. and Kryvoruk, R, Fiber reinforced concrete properties – a multiscale approach, Comput Concrete, 8 (5), (2011), 525-539. [7] Nguyen, V.P. , Lloberas Valls, O., Stroeven, M. and Sluys, L.J., Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks, Comput. Methods Appl. Mech. Engrg., 200 (9–12), (2011), 1220–1236. [8] S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier Science, Amsterdam, 1999. [9] de Souza Neto, E.A. & Feijóo, R.A. Variational foundations of multi-scale constitutive models of solid: Small and large strain kinematical formulation. National Laboratory for Scientific Computing (LNCC/MCT), Brazil, Internal Research & Development Report No. 16, 2006. [10] Fernandes G. R., Pituba J. J. C. & de Souza Neto E. A. Multi-Scale Modelling For Bending Analysis of Heteregeneous Plates by Coupling BEM AND FEM. Engineering Analysis with Boundary Elements. v. 51 p.1-13, 2015. [11] Fernandes G. R., Pituba J. J. C. & de Souza Neto E. A. FEM/BEM formulation for multi-scale analysis of stretched plates. Engineering Analysis with Boundary Elements.V. 54 p.47-59, 2015.

Effects of Slip Boundary Conditions on Mixed Convection flow of Nanofluids in a Lid-Driven Cavity S. Gümgüm

Keywords: Nanofluids, Knudsen number, DRBEM, Mixed convection

Abstract. In this study, effects of slip boundary conditions on mixed convection flow nanofluids in a lid-driven cavity is investigated by DRBEM. The cavity is filled with water (Pr = 6.2) and alumina (Al2 O3 ) nanoparticles. The unknown vorticity boundary conditions, all space derivatives of the unknowns and the slip boundary conditions are calculated by using the coordinate matrix in DRBEM. A relaxation parameter is used for the vorticity between two time level results to accelerate the convergence. Computations are performed for Richardson number Ri = 100, Grashof number Gr = 104 , solid volume fraction values ϕ = 0 − 0.05, and Knudsen number 0.001 ≤ Kn ≤ 0.1. It is observed that increasing Knudsen number yields a decrease in the temperature gradients and also reduces the average Nusselt number, on the other hand increasing solid volume fraction increases the average Nusselt number.

Introduction Flows in micro-devices is an important application field of fluid dynamics such as medicine, fuel cells, biomedical reaction chambers, heat exchangers for electronic cooling. It has been demonstrated that at nano and micro scales, the mechanical properties at the fluid-solid interface cannot be understood by extrapolating known properties of the bulk fluid [1]. Variation in slip length arises from the fact that, during a collision with a solid surface, a fluid molecule will transfer some of its tangential momentum to the solid. The collision frequency is not high enough to ensure thermodynamic equilibrium, and a certain degree of slip tangential velocity must be allowed [2]. Therefore it is important to investigate slip flows in order to simulate convective heat transfer in micro-devices. The fluids which are generally used for heat transfer applications such as water, engine oil and ethylene glycol have low heat transfer performance due to their low thermal conductivity. In order to improve the thermal conductivity of these fluids, nanoparticles are suspended in the base fluid and the resulting mixture is called nanofluid. These mixtures have high thermal conductivity and thus, enhances overall heat transfer capability and have many usages such as industrial cooling applications, nuclear reactors, transportation industry, microelectromechanical systems, electronics and instrumentation and biomedical. Talebi et. al. [3] studied laminar mixed convection flows through a copper-water nanofluid in a square lid-driven cavity. They observed that the solid concentration affects the flow pattern and thermal behavior for a higher Rayleigh number and fixed Reynolds number. Mizzi et. al. [4] investigated the differences between Navier-Stokes-Fourier slip/jump boundary conditions and direct simulation Monte-Carlo computations for a micro lid-driven cavity problem. They found that for complex flows non-equilibrium effects are more appreciable and their onset occurs at lower Knudsen numbers than expected. Perumal et. al. [5] studied micro flows using Lattice Boltzmann Method and simulated the pressure driven micro-channel flows and micro lid-driven cavity flows. Kuo et. al. [6] studied the thermal convection under several slip boundary conditions. They observed that the slip boundary conditions of vertical side walls and horizontal plates affect the pattern selections of the flow and temperature fields. Rahmati et. al. [7] investigated the effects of slip boundaries on mixed convection of Al2 O3 -water nanofluid in microcavity. They showed that increasing the Knudsen and Richardson numbers decreases the average Nusselt number. Mixed convection flow of nanofluids is studied by Gümgüm and Tezer [8] by DRBEM. It is disclosed that the average Nusselt number increases with the increase in volume fraction, and decreases with an increase in both the Richardson number and heat source length.

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The aim of this study is to investigate the effects of the slip boundary conditions on mixed convection flow of Al2 O3 -water based nanofluid in a lid-driven cavity by using DRBEM. The coordinate matrix in DRBEM is used to calculate the slip boundary conditions.

Governing Equations The non-dimensional, steady equations of motion and energy can be written as follows ∇2 u = − ∇2 v =

∂ω ∂y

∂ω ∂x

µf (ρβ )n f ∂ T ρf 2 ∂ω ∂ω +v − Ri ∇ ω = u Re(1 − ϕ)2.5 ρn f ∂x ∂y ρn f β f ∂ x 1 αn f 2 ∇ T Pr Re α f

= u

(1)

∂T ∂T +v ∂x ∂y

where u and v are the velocity components, T is the fluid temperature. Re and Pr are the Reynolds and Prandtl number, respectively. The subscripts ‘nf ’ and ‘f ’ refer to nanofluid and fluid. The density, ρn f , the heat capacity, (ρCp )n f , the thermal expansion coefficient, (ρβ )n f of the nanofluid are calculated as follows ρn f = (1 − ϕ)ρ f + ϕρ p (ρCp )n f = (1 − ϕ)(ρCp ) f + ϕ(ρCp ) p (ρβ )n f = (1 − ϕ)(ρβ ) f + ϕ(ρβ ) p

where ϕ is the nanoparticle volume fraction and C p is the specific heat at constant pressure. The subscript ‘p’ refers to particle. The thermal conductivity, kn f , and the effective dynamic viscosity, µn f , of the nanofluid is taken as in [3] k p + 2k f + 2ϕ(k f − k p ) kn f = kf k p + 2k f − ϕ(k f − k p ) µn f =

µf (1 − ϕ)2.5

The left and right walls as well as the top wall of the cavity is cooled while the bottom wall is heated. The bottom wall moves to the left with a constant speed, the other walls are stationary. The slip and temperature jump boundary conditions are taken as [7] Uslip −Uwall = Tslip − Twall

2 − σν ∂Us Kn σν ∂n

2 − σT 2γ Kn ∂ T = σT γ + 1 Pr ∂ n

(2)

where σν and σT are the tangential momentum and energy accommodation coefficients taken as 1. Kn is the Knudsen number which represents the classification of the flow in micro-devices. When Kn < 0.001, the flow is modeled by the classical no-slip boundary conditions. For 0.001 ≤ Kn ≤ 0.1, the flow is modeled by the slip boundary conditions [7].


69

The vorticity boundary conditions are derived from the definition of vorticity, ω = average Nusselt numbers are calculated at the heated wall.

∂v ∂u − . The local and ∂x ∂y

Numerical Method The right hand side functions in Eq. (1) is approximated by, [9] b

N+L

∑ αj fj

(3)

j=1

where N is the number of boundary nodes, L is the number of internal nodes, α j are unknown coefficients and f j are radial basis functions which are linked with the particular solution uˆ j by ∇2 uˆ j = f j . When these approximations are substituted back in Eq. (1), Laplace operator appears on both sides of the equations. Then 1 1 ln ) and integrated these equations are multiplied by the fundamental solution of Laplace equation (u∗ = 2π r over the domain. Then, Green’s second identity is applied to the Laplacian terms on the left and right hand side for each source point and domain integrals are transformed to boundary integrals. Then, the obtained equation is discretized by linear elements and integrated over each boundary element. The final equations can be expressed in the matrix-vector form as ˆ (4) Hu − Gq = (HUˆ − GQ)α where G and H are (N + L) × (N + L) matrices defined by, [9] 1 1 ∂ ln dΓ j , Hi j = ci δi j + 2π Γ j ∂ n r

Gi j =

1 2π

Γj

ln

1 r

dΓ j .

with Γ j being the boundary of the j-th element. The matrices Uˆ and Qˆ are constructed by taking the corresponding particular solution uˆ j and its normal derivative uˆq j respectively as columns for i, j = 1, ..., N + L. By taking the value of b at (N + L) different points Eq. (3) can be expressed in matrix form as b = Fα

(5)

and α can be obtained by inverting coordinate matrix F containing coordinate functions f j ’s as columns evaluated at N + L points. Eq. (4) can be rewritten as ˆ −1 b. Hu − Gq = (HUˆ − GQ)F

(6)

where b includes the the derivatives of u with respect to x and y which are approximated by a similar idea, [9] u = Fβ .

(7)

Differentiating the above equation with respect to x and y yields ∂u ∂F = β, ∂x ∂x

∂u ∂F = β. ∂y ∂y

(8)

∂ u ∂ F −1 = F u. ∂y ∂y

(9)

Inverting Eq. (7) and substituting back in (8) gives ∂ u ∂ F −1 = F u, ∂x ∂x

Hence, the nodal values of the derivatives are expressed as the product of two unknown matrices and known nodal values of the problem variable. The unknown vorticity boundary conditions, all convective terms and slip boundary conditions are calculated by the above DRBEM idea.

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Results and Discussion The results are presented for Grashof number Gr = 104 , Prandtl number Pr = 6.84, Richardson number Ri = 100, Knudsen number 0.001 ≤ Kn ≤ 0.1 and solid volume fraction ϕ = 0 − 0.05. Quadratic radial basis functions are used. The boundary is discretized using 96 boundary elements. Fig. 1 shows the average Nusselt number of the hot wall for several solid volume fraction values when Knudsen number is fixed at 0.001. It is observed that when the other parameters are fixed and the solid volume fraction is increased, average Nusselt number increases. This is the expected effect of adding nanoparticles in the base fluid. On the other hand, from Fig. 2 it can be seen that the average Nusselt number decreases when Knudsen number increases. 10.5 10.4 10.3

Average Nusselt Number

10.2 10.1 10 9.9 9.8 9.7 9.6 9.5

0

0.01

0.02 0.03 Solid Volume Fraction

0.04

0.05

Figure 1: Average Nusselt number values for several volume fraction, Ri = 100, Kn = 0.001

10.5 10.45 10.4

Average Nusselt Number

10.35 10.3 10.25 10.2 10.15 10.1 10.05 10 0,001

0,01

Knudsen Number

0,1

Figure 2: Average Nusselt number values for several Knudsen number, Ri = 100, ϕ = 0.05

Conclusion Mixed convection flow of nanofluids in a lid-driven cavity is analyzed by DRBEM for various values of problem parameters. Natural convection effect is obtained by setting the bottom wall of the cavity to a higher temperature than those of the other walls which are kept at the same temperature, The forced convection effect is obtained by moving the bottom wall of the cavity. Results showed that using velocity slip and temperature jump boundary conditions affect the heat transfer within the cavity. It is observed that increasing the nanofluid volume fraction


71

increases the average Nusselt number but increasing Knudsen number results in a decrease in the average Nusselt number.

References [1] S. Granick Motions and relaxations of confined liquids. Science, 253, 1374-1379, (1991). [2] H. Power, J. Soavi, P. Kantachuvesiri and C. Nieto The effect of Thompson and Troian’s nonlinear slip condition on Couette flows between concentric rotating cylinders. Z. Angew Math. Phys., 66, 2703-2718 (2015). [3] F. Talebi, A.H. Mahmoudi and M. Shahi Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluids, International Communications in Heat and Mass Transfer, 37, 79-90 (2010). [4] S. Mizzi, D.R. Emerson, S. Stefanov, R.W. Barbery and J.M. Reese Micro-scale cavities in the slip and transition flow regimes, European Conference on Computational Fluid Dynamics, Netherlands, (2006). [5] D.A. Perumal, V. Krishna, V. Sarvesh and A.K. Das Numerical simulation of gaseous microflows by lattice boltzmann method, Int. J. of Recent Trends in Eng., 1(5) (2009). [6] L.S. Kuo, W.P. Chou and P.H. Chen Effects of slip boundaries on thermal convection in 2D box using lattice Boltzmann method, Int. J. Heat Mass Trans., 54, 1340-1343 (2009). [7] A.R. Rahmati, T. Azizi, S.H. Mousavi and A. Zarareh Effects of slip boundaries on mixed convection of Al2 O3 -water nanofluid in microcavity , Int. J. Advanced Design and Manufacturing Technology, 8(2), 47-54 (2015). [8] S. Gümgüm and M. Tezer-Sezgin DRBEM solution of mixed convection flow of nanofluids in enclosures with moving walls, JCAM, 259 Part B, 730-740 (2014). [9] P.W. Partridge, C.A. Brebbia L.C. and Wrobel The Dual Reciprocity Boundary Element Method. Comp. Mech. Pub., Southampton and Elsevier Sci, London, UK (1992).

MHD Stokes Flow in a Smoothly Constricted Rectangular Enclosure M. Gürbüz1 and M. Tezer-Sezgin2 1 2

Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: [email protected] Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: [email protected]

Keywords: MHD Stokes flow, constricted channel.

Abstract. In this paper, the two-dimensional steady slow flow of a viscous, incompressible and electrically conducting fluid is considered in a constricted channel in the presence of a uniform vertically applied magnetic field. In the momentum equations viscous forces dominate over the inertia forces leading to the Stokes flow and the Lorentz force terms are present. We solve these MHD Stokes flow equations iteratively in terms of velocity components, pressure, stream function and vorticity by using radial basis function (RBF) approximation. Particular solution itself which is approximated by RBFs to satisfy both differential equation and boundary conditions becomes the solution of the differential equation. The unknown boundary conditions for pressure and vorticity are obtained from the momentum equations and the vorticity definition, respectively. Stream function equation is discretized by using finite difference including interior values for computing vorticity boundary values. Pressure boundary conditions are derived by using coordinate matrix for the space derivatives and finite difference for pressure gradients. The RBF approximation procedure is tested on the rectangular cavity problem whose middle section is smoothly constricted. No-slip boundary conditions are imposed except on the left wall which is moving upwards with a velocity v = 1. The numerical results are obtained for Hartmann number (M) values in the range 0-50 and the constriction ratio (CR) is taken up to 75% (0% < CR < 75%). As Hartmann number increases flow is flattened and boundary layers are developed close to the moving left wall. Pressure is concentrated near the corners of the right wall due to the increase in the constriction ratio. The solution is obtained in a considerably low computational cost through the use of radial basis functions in the approximation of problem unknowns.

Introduction Magnetohydrodynamics (MHD) is the branch of fluid mechanics that focuses on the flow of electrically conducting fluids under the effect of a magnetic field. It has many industrial applications such as purification of molten metals from non-metallic inclusions, liquid metal plasma physics, metal working process and geothermal energy extractions. The Navier-Stokes equations of fluid motion coupled with Maxwell’s equations of electromagnetics through Ohm’s law constitute the governing MHD equations. The incompressible, viscous flow in slow motion which is called Stokes flow results in neglecting convection terms due to small values of Reynolds number (Re 0 deisgnates the liquid uniform conductivity. One has then to solve coupled Maxwell and non-linear incompressible Navier-Stokes equations [1-2] to get (B, E, u). For a body with typical length scale a and a MHD flow (u, p) with velocity scale V > 0 one introduces the Reynolds magnetic number Rem = µm σVa with µm > 0 the fluid magnetic permeability. Assuming that Rem 1 (a very good assumption in practice) and that the liquid domain boundary (made of the body surface and eventually solid boundaries) has the same permeability as the fluid then yields B = B0 in the entire liquid. In other words, we shall consider that the magnetic fields B is uniform and prescribed in the liquid therefore ending up with three unknown fields: p, u and E. Those fields deeply depend upon the flow Reynolds number Re = ρVa/µ and Hartmann number M = a/d where ρ and µ denotethe liquid uniform density and viscosity and d is the so-called Hartmann layer thickness [3] given by d = ( µ/σ )/|B|. For most of the applications Rem Re so that assuming Re 1 ensures a uniform magnetic field B in the liquid and also a quasi-steady MHD flow governed by the steady Stokes equations with body force j ∧ B. Within this pleasant Low-Reynolds-Number framework the case of a sphere with radius a translating parallel with B has been handled for small [4] and large [5] Hartmann number M = a/d. For such an axisymmetric MHD flow without swirl it turns out that E = 0 [6] a property which facilitates the problem treatment. Recently, [7] obtained two basic axisymmetric MHD Stokes flows without swirl, produced by a ring distribution of axial or radial forces, which are currently used get the axisymmetric MHD Stokes flow around a solid arbitrary axisymmetric body translating parallel to both its axis of revolution and the ambient magnetic field (i. e. to extend by a quite different approach [4,5]). Note that the determination of those two fundamental axisymmetric flows appeals to the fundamental general (i. e. not axisymmetric and fully three-dimensional) MHD Stokes flow produced by a point force with arbitrary strength nicely derived in [8]. The calculation of the fully three-dimensional MHD Stokes flow (u, p) about a solid and arbitray-shaped particle experiencing a rigid-body motion (made in general of a translation and a rotation) remains a tremendously-involved challenge since one has this time to simultaneously determine (u, p) and the occurring electric field E. Because ∇ ∧ E = 0 it turns out that E = −∇φ with φ the unknown electric potential satisying, from the charge conservation ∇.j = 0, the equation ∆φ = ∇.(u ∧ B). The difficult problem of getting (u, p, φ ) has been solved in [8] for the MHD Stokes flow produced in absence of body by a point force. In contrast and to the author’s best knowledge, it has not yet been solved for the MHD Stokes flow

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due to a solid and arbitrary-shaped body experiencing a general rigid-body motion in a quiescent unbounded conducting liquid. Since boundaries are also encountered for applications it is worth examining to which extent a general (creeping or not) MHD flow may be affected when bounded. This issue has been investigated in [9] for different Low-Reynolds-Number steady axisymmetric MHD flow bounded by a plane solid wall Σ normal to the uniform magnetic field B prevailing in the entire liquid domain. One should note that [9] handles different boundary conditions at the wall both for u (not necessarily a no-slip condition) and when needed φ . Moreover, it considers two types of axisymmetric flows: flows without swirl (for which E = 0, as mentioned above) and also the case of swirling Stokes flow (with velocity having only a swirl component) for wich E = 0 (and is found to have no swirl component) and boundary conditions for φ are also prescribed on Σ. In a second paper [10] the Green function for a Stokes axisymmetric MHD flow without swirl produced near a plane motionless wall by a point force normal to the wall has been handled. In the present paper we look at the general three-dimensional MHD Stokes flow (u, p, φ ) produced by a point force with arbitrary unit strentgh e near a plane motionless and no-slip wall Σ.

Addressed fundamental MHD Stokes flow and resulting problem This section introduces the considered three-dimensional fundamental MHD problem and shows how to reduce its solution to the treatment of another equivalent problem. Governing problem for the fundamental MHD Stokes flow As illustrated in Fig. 1, we consider a concentrated force with arbitrary unit strength e located at point x0 in a conducting liquid domain with uniform viscosity µ and conductivity σ > 0 occupying the z > 0 domain D which is bounded by the solid and motionless z = 0 plane wall Σ. i h

g

c

k d

e

b

j f

a m

l

Figure 1: A concentrated force with unit strength e located at point x0 in the liquid domain z > 0 liquid domain bounded by the plane motionless z = 0 wall Σ. The point x0 is the symmetric of x0 with respect to the wall. We assume vanishing Reynolds and magnetic numbers (see the introduction) and the uniform magnetic field B in the liquid to be normal to the wall, i. e. we henceforth take B = Bez . The point force produces a threedimensional MHD Stokes flow with velocity field u (with respect to the motionless wall Σ), pressure field p and electrostatic potential field φ to be simultaneously determined. The flow is driven by the Lorentz body force fL = j ∧ B with current density j following Ohm’s law and thus given by j = σ (−∇φ + u ∧ B). Adding the charge conservation ∇.j = 0 in the liquid, the flow (u, p, φ ) then obeys in the z > 0 liquid domain D the coupled equations µ∇2 u = ∇p + σ B∇φ ∧ ez − σ B2 (u ∧ ez ) ∧ ez − δ (x − x0 )e for x = x0 ∈ D,

∇.u = 0 and ∆φ = B∇.(u ∧ ez ) for x = x0 ∈ D

(1) (2)


127

with ∆ and δ the three-dimensional Laplacian operator and Dirac delta pseudo-function, respectively. Of course, (1)-(2) must be supplemented with proper boundary conditions far from the source x0 and on the wall Σ. Although other more general requirements for φ on the motionless wall Σ may be also handled, were impose the conditions (u, ∇φ , p) → (0, 0, 0) far from x0 , u = 0

on Σ(z = 0), φ = 0 or ∇φ .ez = 0 on Σ(z = 0).

(3)

Clearly the condition φ = 0 or ∇φ .ez = 0 on Σ is used for a perfectly conducting (j ∧ n = 0) or insulating (j.n = 0) wall, respectively.

Free-space analytical solution In absence of wall one only keeps the far-field condition in (3) and it is possible to analytically get the resulting fundamental MHD Stokes with velocity field v, pressure field q and electrostatic potential ψ. More precisely, upon introducing [3] the so-called Hartmann layer thickness d = ( µ/σ )/|B|, it is shown in [7,8] that (v, ψ) solution to (1)-(2) with the far-field behaviour (3) takes the following form (for the present work it is no use giving the associated pressure q, also displayed in terms of He in [7]) 1 B v(x0 , x) = {∇ ∧ (∇ ∧ [He])}, ψ(x0 , x) = ∇.[H(ez ∧ e)] (4) µ µ where the key auxiliary function H = H(x0 , x) satisfies the equation 1 ∂ 2H = δ (x − x0 ) for x = x0 . (5) d 2 ∂ z2 The function H(x0 , x) is given in [7] together with the velocity v(x0 , x) but not the associated electrostatic potential ψ. Setting R = |x − x0 | and taking Cartesian coordinates (x0 , y0 , z0 ) for the source point x0 and (x, y, z) for the observation point x, one actually gets the basic results: (i) Case 1: e = ez : ∆(∆H) −

vx (x0 , x) = sinh(

2d x − x0 e−R/(2d) z − z0 )[1 + ][ ] , 2d R R 8π µR

z − z0 2d y − y0 e−R/(2d) )[1 + ][ ] , vy (x0 , x) = sinh( 2d R R 8π µR z − z0 z − z0 2d z − z0 e−R/(2d) ) + sinh( )[1 + ][ ] , ψ(x0 , x) = 0. vz (x0 , x) = cosh( 2d 2d R R 8π µR

(ii) Case 2 e = ex : vx (x0 , x) =

z − z0 e−R/(2d) 1 2 cosh( ) + d[T1 − (x − x0 )2 T2 ] , 8π µ 2d R

d(x − x0 )(y − y0 ) z − z0 2d x − x0 e−R/(2d) ]T2 , vz (x0 , x) = sinh( )[1 + ][ ] , 8π µ 2d R R 8π µR Bd 2 (y − y0 ) ψ(x0 , x) = − 8π µ (x − x0 )2 + (y − y0 )2 e−|x−x0 |/(2d) e(z−z0 )/(2d) e−(z−z0 )/(2d) − ] , [ + |x − x0 | |x − x0 | − (z − z0 ) |x − x0 | + z − z0 vy (x0 , x) = −[

∂ψ 2d y − y0 e−R/(2d) z − z0 (x0 , x) = B sinh( )[1 + ][ ] . ∂z 2d R R 8π µR

(6) (7) (8)

(9) (10)

(11) (12)

with occurring functions T1 (x0 , x) and T2 (x0 , x) given in [7] and too long to be reproduced here. For symmetry reasons, it is no use handling the last case e = ey since its solution will be easily deduced from the one to above Case 2. As previously mentioned, the pressure field q has also been obtained in closed form in both Case 1 and Case 2 (see [7]).

128


Advocated flow decomposition and resulting problem The idea consists in decomposing the fundamental MHD flow (u, p, φ ) governed by (1)-(3) as below u(x0 , x) = v(x0 , x) − v(x0 , x) + U(x0 , x), p(x0 , x) = q(x0 , x) − q(x0 , x) + P(x0 , x),

φ (x0 , x) = ψ(x0 , x) − ψ(x0 , x) + Φ(x0 , x)

(13) (14)

where, as shown in Fig. 1, the point x0 is the symmetric of the source point x0 with respect to the z = 0 plane wall Σ. Clearly, the new MHD flow (U, P, Φ) must satisfy (1) in which one has to omit the term δ (x − x0 )e and the equations (2). The far-field behaviour and boundary conditions on the wall (3) result in conditions for the new flow (U, P, Φ). Setting R = |x − x0 | it appears that on the wall R = R and also for previous Case 2 (e = ex ) that L(x0 , x) = L(x0 , x) for L = φ , T1 , T2 . From (6)-(12) the required conditions thus read (U, ∇Φ, P) → (0, 0, 0) far from x0 in both Case 1 and Case 2,

Ux (x0 , x) = −2vx (x0 , x), Uy (x0 , x) = −2vy (x0 , x), Uz (x0 , x) = 0 on Σ(z = 0) in Case 1, Φ(x0 , x) = 0

or

∂Φ ∂ z (x0 , x)

= 0 on Σ(z = 0) in Case 1,

Ux (x0 , x) = Uy (x0 , x) = 0, Uz (x0 , x) = −2vz (x0 , x) on Σ(z = 0) in Case 2, Φ(x0 , x) = 0

or

∂Φ ∂ z (x0 , x)

=

−2 ∂∂ψz (x0 , x)

on Σ(z = 0) in Case 2.

(15) (16) (17) (18) (19)

In summary, the problem has been reduced to the determination of the auxiliary MHD Stokes flow (U, P, Φ) governed by (1)-(2) (taking e = 0) and the conditions (15)-(19).

Solution for the unit force normal to the wall In this section we solve the Case 1 of the unit force e = ez while the solution of Case 2 will be given at the oral presentation. Adopted form of the auxilary MHD flow The point MO is such that x0 = OM0 . In that case we expect the auxiliary flow (U, P, Φ) to be axisymmetric about the axis (M0 , ez ) and to be without swirl. In view of the boundary condition (17) we take Φ = 0 and seek (U, P) in the form took by the solution in free-space. Since e = ez we thus write from (4) µwx (x0 , x) = [

∂ 2F ∂ 2F ∂ 2F ∂ 2F ](x0 , x), µwy (x0 , x) = [ ](x0 , x), µwz (x0 , x) = −[ 2 + 2 ](x0 , x) ∂ x∂ z ∂ y∂ z ∂ x ∂ y

(20)

with unknown function F(x0 , x) regular in the entire liquid domain and solution to ∆(∆F) −

1 ∂ 2F = 0 for z = x.ez > 0. d 2 ∂ z2

(21)

The Cartesian components of the free-space velocity v(x0 , x) are related to the free-space function H(x0 , x) by relations similar to the ones given in (20). Invoking (15)-(16), one thus also requires that ∂ 2F ∂ 2F ∂ 2F ∂ 2F , , + 2 )(x0 , x) → (0, 0, 0) far from x0 , ∂ x∂ z ∂ y∂ z ∂ 2 x ∂ y 2 ∂ F ∂ 2H ∂ 2F ∂ 2H [ ](x0 , x) = −2[ ](x0 , x) and [ ](x0 , x) = −2[ ](x0 , x) on Σ(z = 0), ∂ x∂ z ∂ x∂ z ∂ y∂ z ∂ y∂ z 2 2 ∂ F ∂ F [ 2 + 2 ](x0 , x) = 0 on Σ(z = 0). ∂ x ∂ y (

Consequently, the task reduces to the determination of function F governed by (21)-(24).

(22) (23) (24)


129

Solution by a two-dimensional Fourier transform We have h = x0 .ez > 0 (see Fig. 1) and clearly F(x0 , x) = Fh (X1 , X2 , z) where X1 = (x − x0 ).ez and X2 = (x − x0 ).ey . In a similar fashion, for the free-space we have H(x0 , x) = Hh (X1 , X2 , z). The linearity of the new problem (21)-(24) for Fh suggests solving it using a Fourier transform. Here we resort to the two-dimensional Fourier tranform gˆ of a function g(X1 , X2 , z) defined as g(λ ˆ ; z) =

1 2π

∞ ∞

−∞ −∞

g(X1 , X2 , z)ei(λ1 X1 +λ2 X2 ) dX1 dX2

(25)

where i denotes the usual complex number such that i2 = −1 and λ = λ1 ex + λ2 ey is the vector in the Fourier ˆ space with magnitude λ = {λ12 + λ22 }1/2 . Denoting further the Fourier transform of Fh by Fˆ and of Hh by H, the problem (21)-(24) becomes in Fourier space 1 ∂ 2 Fˆ ∂ 4 Fˆ ˆ ; z) → 0 as z → +∞, ) + 4 = 0, F(λ d2 ∂ 2z ∂z ∂ Fˆ ∂ Hˆ ˆ ; 0) = 0. (λ ; 0) = −2 (λ ; 0) and F(λ ∂z ∂z

λ 4 Fˆ − (2λ 2 +

(26) (27)

As shown in [9], the solution to (26) admits the general form ˆ ; z) = A1 (λ )eα1 z + A2 (λ )eα2 z , α1 = − 1 − (λ 2 + 1 )1/2 < α2 = 1 − (λ 2 + 1 )1/2 < 0. F(λ 2d 4d 2 2d 4d 2

(28)

ˆ ; z) = H(λ ˆ ; z). Enforcing the conditions (27) then finally gives the solution It also turns out that H(λ

ˆ ˆ ; z) = F(λ ˆ ; z) = −4d sinh( z )[ ∂ H (λ ; 0)]e− F(λ 2d ∂ z

λ 2+

1 4d 2

.

Setting ρ = {X12 + X22 }1/2 and recalling that h = x0 .ez is can be shown from [7] that √ 2 2 ∞ −√t 2 +h2 /(2d) e d h e− ρ +h /(2d) ∂ Hˆ d h ∂H , t[ √ ]J0 (λt)dt (x, y, 0) = sinh( ) (λ ; 0) = sinh( ) 2 2 ∂z 4π 2d ∂ z 4π 2d 0 t 2 + h2 ρ +h

(29)

(30)

where J0 designates the usual Bessel function (of the first kind) of order zero. We now take the two-dimensional ˆ ; z) to get the required function F(X1 , X2 , z). It is given by inverse transform of F(λ F(X1 , X2 , z) =

1 2π

∞ ∞

−∞ −∞

ˆ 1 ; z)e−i(λ1 R1 +λ2 R2 ) dλ1 dλ2 . F(λ

(31)

ˆ 1 ; z) the result is ˆ 1 ; z) = F(λ Since F(λ F(x0 , x) = F(ρ, z; h) =

∞ 0

ˆ ; z)J0 (ρλ )dλ λ F(λ

(32)

ˆ ; z) with its value gained from (29)-(30). where one has to replace F(λ

Conclusions A method has been proposed to build the fundamental 3D MHD Stokes flow induced by a source point with strength e located at point M0 in a conducting Newtonian liquid bounded by a motionless solid plane wall and subject to a uniform magnetic field B normal to the wall. Assuming the plane wall to be either perfectly conducting or insualting and using the solution in absence of wall, the task is reduced to the determination of a three-dimensional auxiliary MHD Stokes flow regular in the entire liquid domain. For a unit force e normal to the wall this flow, which is axisymmetric with respect to the (M0 , B) axis and has no swirl, is determined. The solution for a unit force e parallel with the wall requires more efforts and will be given at the oral presentation.

130


References [1] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970). [2] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [3] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [4] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech.”, vol 3, 304-308 (1957). [5] W. Chester The effect of a magnetic field on the flow of a conducting fluid past a body of revolution. J. Fluid Mech.”, vol 10, 459-465 (1961). [6] H. Yosinobu and T. Kakutani Two-dimensional Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 14 (10), 1433-1444 (1959). [7] A. Sellier and S. H. Aydin Fundamental free-space solutions of a steady axisymmetric MHD viscous flow. To appear in The European Journal of Computational Mechanics. [8] J. Priede Fundamental solutions of MHD Stokes flow arXiv: 1309.3886v1. Physics. fluid. Dynamics, (2013). [9] A. B. Tsinober Axisymmetric MagnetoHydrodynamic Stokes flow in a half-space. MagnetoHydrodynamics, 4, 450-461 (1973). [10] A. B. Tsinober Green’s function for axisymmetric MHD Stokes flow in a half-space. MagnetoHydrodynamics, 4, 559-562 (1973).

A Boundary Formulation for the Axisymmetric MHD Slow Viscous Flow about a Sphere Translating Parallel with a Uniform Ambient Magnetic A. Sellier1 and S.H. Aydin2 1

2

LadHyx. Ecole polytechnique, 91128 Palaiseau Cédex, France e-mail: [email protected] Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey e-mail: [email protected]

Keywords: MagnetoHydroDynamics, Axisymmetric MHD flow, Boundary-integral equation, spherical body.

Abstract. A new boundary formulation is proposed to compute the axisymmetric creeping MHD flow about a solid and axisymmetric body translating in a conducting Newtonian liquid parallel with both its axis of revolution and a uniform ambient magnetic field. The procedure rests on integral representations for the liquid velocity and pressure fields obtained by appealing to two axisymmetric fundamental MHD flows recently obtained in [7]. The task then reduces to the treatment of a key Fredholm boundary-integral equation of the first kind for the unknown surface traction exerted on the body surface. The numerical implementation perfectly matches at small Hartmann number M with the asymptotic prediction derived, by a quite different approach, in [5] for the drag experienced by a translating sphere.

Introduction Determining the MagnetoHydrodynamic flow about a solid particle experiencing a prescribed rigid-body migration in a conducting and unbounded Newtonian liquid subject to a given ambient uniform magnetic field B is a very challenging task since one has to simultaneously solve the unsteady coupled Maxwell and NavierStokes equations [1,2] which govern the magnetic field B , the electric field E and the flow velocity field u and pressure field p prevailing in the liquid. The flow (u, p) is driven by the Lorentz body force f = j ∧ B with current density j = σ (E + u ∧ B ) (usual Ohm’s law) where σ > 0 designates the uniform liquid conductivity. For convenience one usually introduces the body length scale a, the flow velocity magnitude V > 0 and the liquid uniform density ρ, viscosity µ and magnetic permeability µm > 0. Setting B = |B| > 0, it is then possible to define three key dimensionless numbers: the magnetic Reynolds number Rem , the Reynolds number Re and the Hartmann number M. These numbers are defined as (1) Rem = µm σVa, Re = ρVa/µ, M = a/d = aB/ µ/σ where the length d = ( µ/σ )/B is the so-called Hartmann layer thickness [3]. It then turns out that (u, p, E , B ) not only depends upon the particle shape and motion but also upon (Re, Rem , M). If the liquid and the body have the same magnetic permeability and Rem 1 it appears [1,2] that B = B, i. e. that the ambient magnetic field B is not disturbed by the body. One however then still ends up with three unknown unsteady fields: E , u and p. Assuming further a quasi-steady and axisymmetric problem yields E = 0 (see, for details, [1,4]). This is the case for a solid axisymmetric body translating parallel to both its axis of revolution and the uniform ambient magnetic field B. Assuming moreover that Re 1, one in that case faces the more tractable determination of a MHD axisymmetric Stokes flow about the translating body of revolution. Nevertheless, getting this later MHD flow whatever the Hartmann number M is still involved and therefore the available literature confines the investigations to a spherical body for either small [5] or large [6] Hartmann numbers M. The aim of the present work is to fill the gap, i. e. to present a new boundary approach able to deal with arbitrary axisymmetric body and Hartmann number.

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Addressed axisymmetric MHD Stokes flows This section presents the governing problem for the steady MHD axisymmetric Stokes flow about a solid axisymmetric body translating parallel, in an unbounded conducting Newtonian liquid, to both its axis of revolution and a prescribed uniform magnetic field. It also introduces a fundamental axisymmetric MHD flow obtained in [7] and of importance in establishing the proposed new boundary formulation.

Axisymmetric MHD Stokes flow about a translating solid body of revolution As illustrated in Fig. 1, we consider a solid axisymmetric body with smooth boundary S immersed in a conducting and unbounded Newtonian liquid with uniform viscosity µ and conductivity σ > 0. x z y a

h

t b

u

nn c

g

f

e

d

i

Figure 1: A solid axisymmetric body translating in a conducting Newtonian liquid at the velocity U = Uez parallel to both its axis of revolution and a prescribed uniform ambient magnetic field B = Bez . The body, with attached point O and axis of revolution (O, ez ), translates at the velocity U = Uez parallel with the imposed uniform ambient magnetic field B = Bez with B > 0. The resulting MHD flow about the body has pressure p and velocity u with typical magnitude V = |U| while the body has typical length scale a and the same uniform magnetic permeability µm as the liquid. Assuming that Rem = µm σVa 1 shows that the magnetic field is the ambient one, B, in the entire liquid domain D. As seen in the introduction, the problem symmetries also imply that there is no induced electric field in the liquid. Assuming that Re = ρVa/µ 1 one can further neglect all inertial effects. The axisymmetric MHD Stokes liquid flow (u, p) then obeys µ∇2 u = ∇p − σ B2 (u ∧ ez ) ∧ ez and ∇.u = 0 in D ,

(u, p) → (0, 0) as |x| → ∞ , u = Uez on S

(2) (3)

where |x| = (x12 + x22 + x32 )1/2 if one uses Cartesian coordinates (O, x1 , x2 , x3 ), with associated unit vectors (e1 , e2 , e3 ), so that x = x1 e1 + x2 e2 + x3 e3 . The MHD flow (u, p) has stress tensor σ . It exerts on the body surface S, with unit normal n directed into the liquid (see Fig. 1), a traction f = σ .n. For symmetry reasons, the resulting torque about the point O is zero while the force F exerted on the body reads F=

S

fdS = −6π µaλUez

(4)

with λ > 0 the so-called drag coefficient. In dimensionless “coordinates” x = x/a the surface S becomes immediately shows that u = Uu and p = µU p/a with the dimensionless body surface S. Inspecting (2)-(3) (u, p) solely depending upon (x; S, M) where M = aB σ /µ is the Hartmann number. Note that, accordingly, λ = λ (S, M). Since the problem (2)-(3) is axisymmetric it is convenient to rather locate x by its usual cylindrical coordinates (r, z, θ ) taking θ ∈ [0, 2π], z = x.ez and r = {|x|2 − z2 }1/2 ≥ 0. Then, x = rer + zez with unit vector


133

er = er (θ ). At point x the axisymmetric MHD flow (u, p) has pressure p(x) = p(r, z) and velocity u(x) = ur (r, z)er + uz (r, z)ez with radial and axial velocity components ur and uz . In a similar fashion, the traction f (recall (4)) reads, for x located on the axisymmetric body surface, f(x) = fr (r, z)er + fz (r, z)ez where fr and fz are the radial and axial components, respectively. As shown below it is actually possible to compute the key traction f without necessarily obtaining at the same time the flow (u, p) in the liquid domain.

Relevant axisymmetric fundamental MHD Stokes flow For further purposes it is necessary to determine, at x(r, z, θ ), the fundamental MHD axisymmetric flow, with velocity v(x) = vr (r, z)er + vz (r, z)ez and pressure q(x), produced by distributing a force with strength Fr er + Fz ez , taking Fr and Fz constant, on the circular ring with radius r0 > 0 located in the z = z0 plane. This basic flow (v, q) has been recently obtained in [7] at x(r, z, θ ) arbitrarily-located off the ring. Introducing points M(r, z) (see also Fig. 1) and M0 (r0 , z0 ) located in the half θ = 0 plane, taking indices α and β in {r, z} and employing henceforth the usual tensor summation convention over repeated indices, it has been found that vα (x) = [

1 1 ]G (M, M0 )Fβ and q(x) = [ ]Pβ (M, M0 )Fβ 8π µ αβ 8π

for M = M0

(5)

where Gαβ (M, M0 ) and Pβ (M, M0 ) are the so-called Green tensor velocity components and Green pressure vector components. For a sake of conciseness, the rather-intricated formulae expressing in [7] these quantities, solely in terms of (z − z0 , r, r0 ) and of the Hartmann layer thickness [3] d = ( µ/σ )/B, are not reproduced here. The associated stress tensor σ (v,q) takes at point x(r, z, θ ) located off the ring, with associated point M(r, z) in the θ = 0 half plane, the following form (taking also γ in {r, z}) σ (v,q) (x) = σαγ (M)eα ⊗ eγ + [τθ θ − q](M)eθ ⊗ eθ , σαγ (M) = [ταγ − δαγ q](M), eθ = ez ∧ er ∂ vz ∂ vr ∂ vz vr ∂ vr τrr (M) = 2µ , τzz (M) = 2µ , τrz (M) = τzr (M) = µ( + ), τθ θ (M) = 2µ ∂r ∂z ∂z ∂r r

(6) (7)

with δ the usual Kronecker delta. In contrast to ez , both unit vectors er and eθ depend upon x(r, z, θ ) with, actually, er = er (θ ) and eθ = e theta (θ ). Substituting (5) into (6)-(7) yields the relation σαγ (M) = [

∂ Gαβ ∂ Gγβ 1 ]T (M, M0 )Fβ , Tαβ γ (M, M0 ) = [ + − δαγ Pβ ](M, M0 ). 8π αβ γ ∂γ ∂α

(8)

Using (8) permits one to determine each component σαγ from the Green quantities Gαβ (M, M0 ) and Pβ (M, M0 ) obtained in [7]. Such a task has however not been achieved in [7].

Advocated boundary approach This section derives a boundary formulation suitable to efficiently and accurately compute the required MHD axisymmetric Stokes flow (u, p) about the translating solid body of revolution.

Basic velocity and pressure boundary integral representations We start with a basic extension of the usual reciprocal identity usually encountered for Stokes flows (see, for instance [8,9]) for a non-conducting liquid (case σ = 0, i. e. M = 0). First, consider two MHD flows (u, p) and (u , p ), with stress tensors σ and σ , obeying (2) in a bounded domain Ω with smooth boundary ∂ Ω on which the unit normal n points into the conducting liquid. In addition, let us define the linear operator L applied to a vector field a by L[a] = σ B2 (a ∧ ez ) ∧ ez . Exploiting (2) then easily yields the relation

∂Ω

[u.σ .n − u .σ .n]dS =

Ω

[u.L(u ) − u .L(u)]dΩ.

(9)

134


From the definition of L[a] the domain integral on the right-hand side of (9) vanishes and so does the boundary integral in (9). The same property holds for an unbounded domain Ω (assuming also this time the far-field behaviour (3) for both flows). In summary, whatever the liquid domain (bounded or unbounded) one gets the basic reciprocal relation ∂Ω

u.σ .ndS =

∂Ω

u .σ .ndΩ.

(10)

Let us consider a torus Tε , immersed in our liquid domain D, having circular cross section with radius ε > 0, closed surface ∂ Tε and mean curve with equation r = r0 > 0 and z = z0 . In other words, as ε vanishes Tε collapses to the circular ring located in the half θ = 0 plane by the point M0 (r0 , z0 ). We then apply (10) for Ω = D \ Tε taking for (u, p) the MHD flow governed by (2)-(3) and for (u , p ) the fundamental flow (v, q) with stress tensor σ (v,q) (recall (5)-(8)). By virtue of the boundary condition (3), one gets

∂ Tε

[u.σ (v,q) .n − v.σ .n](x)dS =

S

[v.σ .n](x)dS −U

S

ez .[σ (v,q) .n](x)dS.

(11)

Since M0 (r0 , z0 ) is located in the liquid the fundamental flow (v, q) fulfills also the MHD equations (2) inside the solid body. This is also true for the flow with pressure Q = 0 and uniform velocity w = Uez having a zero stress tensor. Applying (10) to these flows for the domain inside the body P yields

S

ez .[σ (v,q) .n](x)dS = 0.

(12)

On the axisymmetric boundary S ∪ ∂ Tε we have dS = rdldθ and also n(x) = [nγ eγ ](M), [σ .n](x) = [ fα eα ](M) and, using (6), [σ (v,q) .n](x) = [σαγ nγ eα ](M). In the θ = 0 half plane the trace of ∂ Tε is the circle with center M0 (r0 , z0 ) and radius ε while the trace of S is a contour C (since O belongs to the body, C is not a closed contour). Combining (12) with (11) and performing the integration over θ in [0, 2π] gives, using the relations (5) and (8), the identities (since Fr and Fz are constant and arbitray) µ

Cε

uα (M)Tαβ γ (M, M0 )nγ (M)rdl − =

C

Cε

Gαβ (M, M0 ) fα (M)rdl

Gαβ (M, M0 ) fα (M)rdl for β = r, z.

(13)

The next step consists in letting ε tend to zero in (13). Setting ρ = MM0 = {(r − r0 )2 + (z − z0 )2 }1/2 it appears that ρ = ε for M belonging to Cε . Accordingly, we need to approximate both Gαβ (M, M0 ) and Tαβ γ (M, M0 ) as M tends to M0 . Appealing to [7] first gives the following behaviours 2 log ρ as M → M0 , r0 2(z − z0 )(r − r0 ) as M → M0 . Grz (M, M0 ) ∼ Gzr (M, M0 ) ∼ r0 ρ 2

Grr (M, M0 ) ∼ Gzz (M, M0 ) ∼ −

(14) (15)

On Cε one has ρ = ε, r − r0 = ε sin ϕ and z − z0 = ε cos ϕ with angle ϕ in [0, 2π]. From (14)-(15) it follows that the second integral on the right-hand side of (13) vanishes with ε. One should note that the leading expansions (14)-(15) do not depend upon the Hartmann layer d. As noticed in [7], this is because the behaviour of the quantities Gαβ (M, M0 ) and Pβ (M, M0 ) as M → M0 is the one obtained for d → ∞, i. e. for the Stokes fundamental flow (zero Hartmann number M = a/d). In approximating, at the leading order, Tαβ γ (M, M0 ) for M close to M0 we can therefore expand the form taken by Tαβ γ (M, M0 ) for a Stokes flow and derived in [9]. Curtailing the cumbursome details, one arrives at the key behaviors 8(z − z0 )3 8(r − r0 )3 and rTrrr (M, M0 ) ∼ − as M → M0 , ρ4 ρ4 8(r − r0 )(z − z0 )2 as M → M0 , rTzzr (M, M0 ) ∼ rTrzz (M, M0 ) ∼ rTzrz (M, M0 ) ∼ − ρ4 8(z − z0 )(r − r0 )2 as M → M0 . rTzrr (M, M0 ) ∼ rTrrz (M, M0 ) ∼ rTrzr (M, M0 ) ∼ − ρ4 rTzzz (M, M0 ) ∼ −

(16) (17) (18)


135

In addition, on Cε note that nr (M) = (r − r0 )/ε and nz (M) = (z − z0 )/ε. Exploiting (16)-(18) then easily shows that the first integral on the right-hand side of (13) tends to −8πuβ (x0 ) as ε vanishes. Moreover, from [7] it appears that Gαβ (M, M0 ) = Gβ α (M0 , M). Switching the roles played by M0 and M and also by α and β , one then arrives at the following key single-layer integral representation for each velocity component ur or uz uα (x) = −

1 8π µ

C

Gαβ (M, P) fβ (P)r(P)dl(P) for x ∈ D and α = r, z.

(19)

The key results (19) suggests the following comments or remarks: (i) Of course, in (19) the point P located in the θ = 0 half plane on the contour C has coordinates z(P) and r(P). (ii) For symmetry reasons one must have ur (x) = 0 for M on the (O, ez ) axis (i. e. when r = 0). This requirement is here ensured by (19) because (see [7]) Gαβ (M, P) = 0 as soon as r = 0 whatever the point P. (iii) As it stands, (19) shows that the flow velocity u is obtained inside the liquid domain (and also, see the next subsection, on the solid boundary S) by spreading there a surface distribution of rings with surface density f(P) = [ fα eα ](P). From (iii) and (5) it is clear that the flow pressure p admits the integral representation p(x) = −

1 8π

C

Pβ (M, P) fβ (P)r(P)dl(P) for x ∈ D.

(20)

Resulting relevant coupled boundary-integral equation and strategy It turns out that the integral representation (19) also holds on S because of the previous asymptotic behaviour of Gαβ (M, P) as M tends to P (recall (14)-(15)). Consequently, the unknown surface traction f = fr (r, z)er + fz (r, z)ez on the body surface S is gained by inverting the following coupled boundary-integral equations of the first kind (using the boundary condition (3))

C C

Grr (M, P) fr (P)r(P)dl(P) + Gzr (M, P) fr (P)r(P)dl(P) +

C C

Grz (M, P) fz (P)r(P)dl(P) = 0 for M on C ,

(21)

Gzz (M, P) fz (P)r(P)dl(P) = −8π µU for M on C .

(22)

For symmetry reasons for P on the (O, ez ) axis, i. e. for r(P) = 0, one knows that fr (P) = 0. One in practice thus only imposes (22) at M such that r = 0. The proposed strategy to efficiently solve the problem (2)-(3) then consists of the two following steps: (i) Get the tractions f on the body surface S by inverting (21)-(22) for its unknown components fr and fz . (ii) Compute in the liquid domain D the MHD Stokes flow (u, p) about the translating body by resorting to the integral representations (19)-(20).

Numerical implementation and preliminary results for a translating sphere This section briefly describes the employed numerical strategy. It also gives for a translating sphere comparisons against [5] for the drag coefficient at small Hartmann number M.

Numerical method The coupled boundary-integral equations of the first kind (21)-(22) have been numerically solved, for a spherical body with center O and radius a, by calculating each influence coefficient Gαβ (M, M0 ) as proposed in [7]. The half-circle contour C is splitted into 16 curved 3-node quadratic boundary elements with equal length. The discretized coupled boundary-integral equations (21)-(22) are numerically enforced at the resulting 31 nodal

136


points located off the sphere axis of revolution (O, ez ) while only (22) is imposed at the two remaining nodal points located on the sphere axis of revolution. Adopting Ne = 16 curved boundary elements has been found sufficient to ensure a five-digit accuracy for the drag coefficient λ defined by (4). For instance, if M = 30 the results are λ = 11.837020 for Ne = 16 and λ = 11.837016 for Ne = 32.

Benchmark test at small M for the drag coefficient Using a procedure quite different from the proposed boundary approach, Chester [5] built, at small Hartmann number M, the following asymptotic approximation of the shere drag coefficient λ ∼ λa = 1 +

3M 7M 2 43M 3 + − . 8 960 7680

(23)

As shown in Table 1, our numerical results very well agree with the prediction (23) for M in the range [0, 1]. Table 1: Computed λ and approximated λa (as given by (23)) drag coefficients for M ≤ 1. M λ λa

0.01 1.00381 1.00375

0.1 1.03763 1.03757

0.3 1.11310 1.11301

0.5 1.18886 1.18862

0.7 1.26479 1.26415

1 1.37884 1.37669

Conclusions A new boundary formulation is proposed to accurately and efficiently compute the MHD axisymmetric Stokes flow about a solid body of revolution translating, in a conducting Newtonian liquid, parallel to both its axis of symmetry and an imposed uniform magnetic field. So doing, the task finally solely reduces to the treatment of a boundary-integral equation governing the surface traction prevailing on the body surface. For a sphere the achieved numerical implementation yields for the drag at small Hartmann number M results in perfect agreement with the asymptotic predictions obtained in [5]. As will be shown at the oral presentation, it also permits us to compute the flow patterns about the translating sphere for different values of M.

References [1] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970). [2] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [3] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [4] K. Gotoh Magnetohydrodynamic flow past a sphere. Journal of the Physical Society of Japan, 15 (1), 189196 (1960). [5] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech.”, vol 3, 304-308 (1957).


137

[6] W. Chester The effect of a magnetic field on the flow of a conducting fluid past a body of revolution. J. Fluid Mech.”, vol 10, 459-465 (1961). [7] A. Sellier and S. H. Aydin Fundamental free-space solutions of a steady axisymmetric MHD viscous flow. To appear in The European Journal of Computational Mechanics. [8] S. Kim and S. J. Karrila Microhydrodynamics: principles and selected applications (Butterworth, 1991). [9] C. Pozrikidis Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, (1992).

Flow in a Square Cavity with an Obstacle under the Influence of a Non-uniform Magnetic Field. Pelin Senel1 and M. Tezer-Sezgin2 Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey, e-mail: 1 [email protected], 2 [email protected]

Keywords: DRBEM, Navier-Stokes equations, point magnetic source, concentric pipes.

Abstract. In this paper, we investigate fully developed, laminar, steady flow of an electrically non-conducting, viscous, incompressible, magnetizable fluid in concentric pipes of square cross-sections under the effect of a non-uniform magnetic field. Magnetic field is generated by an electric wire placed below the bottom of the outer pipe lying along the pipes-axis. The problem is reduced to the flow of a magnetizable fluid in a square cavity with a square obstacle in the two-dimensional cross-section of the pipes. Governing equations are solved in terms of the velocity-stream function-pressure, and also the axial velocity. The effect of the position and the size of the obstacle on the solution is investigated. The Dual Reciprocity Boundary Element Method (DRBEM) is applied iteratively in obtaining numerical results. In DRBEM all the terms other than Laplacian are taken as inhomogeneity and the fundamental solution of the Laplace equation is used in order to convert domain integrals to boundary integrals through Divergence theorem. The boundary is discretized with constant elements and sufficient number of interior points are taken. Pressure boundary conditions are approximated by using finite difference for the spatial derivatives of the pressure and the DRBEM coordinate matrix F for the other terms in the momentum equations. The numerical results reveal that when the obstacle is in the middle of the cavity, magnetic effect divides the flow into four vortices. Two eddies are attached to the obstacle horizontally, squeezing the flow caused by the point magnetic source through the vertical walls of the cavity and enlarging as magnetic field intensity increases. An increase in the size of the obstacle squeezes the eddies through the boundaries of the cavity. Pressure is highly concentrated around the magnetic source. As magnetic field intensity increases the axial velocity decelerates around the magnetic point source.

Introduction Flow in concentric pipes under magnetic effect has numerous industrial applications, such as power transformer cooling and audio speakers. If there is temperature difference between the walls heat exchangers can also be added to the list. Laminar mixed convection flow and double diffusive natural convection flow in a square cavity with a heated square blockage inside is investigated by Islam et al. [1] and Nazari et al. [2] by using the finite volume method and Lattice Boltzmann method, respectively. Selimefendigil and Oztop [3] studied natural convection flow in a nano-fluid filled cavity having different shaped obstacles installed under the influence of a uniform heat generation and magnetic field by using Galerkin weighted residual finite element formulation. Stokes flow in a rectangular cavity with moving lids and a (rotating) cylinder in the center is investigated by Galaktionov et al. [4]. They implement a general analytical method of superposition. Tzirtzilakis et al. [5] studied the flow of a non-conducting biomagnetic fluid in a 3D rectangular duct under the influence of the magnetic field by using pressure-linked pseudotransient method on a common grid. In this paper, we investigate the effect of a point magnetic source on a fully developed, steady flow of viscous, electrically non-conducting, magnetizable fluid in a square cavity with a square obstacle inside. The governing equations in velocity-stream function-pressure form are solved iteratively by DRBEM with constant elements. The equation for the axial velocity is also solved. Unknown pressure boundary conditions are approximated through x− and y−components of the momentum equations. Numerical results are given for various magnetic numbers, and the effect of the size and the position of the obstacle is also investigated. The DRBEM gives the solution at a small expense due to its boundary only nature.

140


Governing equations We consider the two-dimensional, fully developed, steady and laminar flow of an electrically non-conducting, incompressible, viscous, magnetizable fluid in concentric pipes with square cross-sections. The fluid is influenced by a magnetic field generated by an electric current going through a thin wire placed c units below the outer pipe parallel to the axis of the pipes. There is no electric current or electric charge in the case of nonconducting fluid but the flow undergoes a force created by the magnetization of the fluid. The flow in the axial direction is generated by a given constant pressure along the z−axis. Since the flow is fully developed, we are able to split the pressure as in [5, 6] P(x, y, z) = p(x, y) + p(z) ˜

(1)

and

∂ P ∂ p˜ = = Pz = constant . (2) ∂z ∂z Splitting the pressure enables us to solve the problem in the cross-section of the pipes (square cavities). Thus, the inner cavity serves as an obstacle inside the outer cavity enforcing the fluid to flow on the transverse plane and the electric wire resembles to a point magnetic source. The governing equations are similar to the case of Ferrohydrodynamics [5, 7]. Continuity and momentum equations defining the two-dimensional flow in terms of non-dimensional velocity (u, v, w) and pressure p are ∂u ∂v + =0 ∂x ∂y

(3)

∂u ∂u ∂H ∂ 2u ∂ 2u ∂ p + u + v − MnH + = ∂ x2 ∂ y2 ∂x ∂x ∂y ∂x

(4)

∂v ∂v ∂H ∂ 2v ∂ 2v ∂ p + u + v − MnH + = ∂ x 2 ∂ y2 ∂y ∂x ∂y ∂y

(5)

∂w ∂w ∂ 2w ∂ 2w +v + 2 = Pz + u ∂ x2 ∂y ∂x ∂y

(6)

where Mn is the Magnetic number defined as Mn =

µ0 KH02 h2 . ν 2ρ

(7)

Here, ρ is the fluid density, ν is the kinematic viscosity, µ0 is the magnetic permeability, K is the magnetic susceptibility of the fluid and H0 is the magnetic field strength at the point (h/2, 0) (h denotes the dimension of the outer pipe). Magnetic field strength is defined by, H(x, y) =

|b|/h

(x − a/h)2 + (y − b/h)2

(8)

where (a, b) = (h/2, −c) is the place of the magnetic source. The magnetization force appears in the x− and y−momentum equations as additional terms and it is absent in the z−direction since ∂ H/∂ z = 0. The equation for pressure p(x, y) is obtained by taking the derivatives of eqs. (4), (5) with respect to x and y, adding them and using the continuity equation as 2 2 ∂H 2 ∂H 2 ∂ 2H ∂ 2H ∂u ∂v ∂v ∂u ∂2p ∂2p . + = Mn( + + H( + )) − − −2 ∂ x2 ∂ y2 ∂x ∂y ∂ x2 ∂ y2 ∂x ∂y ∂x ∂y The two-dimensional stream function Ψ is defined as u =

∂Ψ ∂y ,

v = − ∂∂Ψx satisfying the continuity equation

(9)


141

which results in the stream function equation ∇2 Ψ =

∂u ∂v − . ∂y ∂x

(10)

Thus, the velocity components in the non-linear terms of x− and y− momentum equations can be interchanged with their stream functions equivalences ∂ 2u ∂ 2u ∂ p ∂ Ψ ∂ u ∂ Ψ ∂ u ∂H + − − MnH + = ∂ x 2 ∂ y2 ∂x ∂y ∂x ∂x ∂y ∂x

(11)

∂ 2v ∂ 2v ∂ p ∂ Ψ ∂ v ∂ Ψ ∂ v ∂H + = + − − MnH . (12) ∂ x 2 ∂ y2 ∂y ∂y ∂x ∂x ∂y ∂y The problem configuration for the domain Ω between the squares and the boundary conditions are given in Figure 1.

d

1

point magnetic source

u=v=w=Ψ=0

u=v=w=Ψ=0 obstacle

u=v=w=Ψ=0

u=v=w=Ψ=0

u=v=w=Ψ=0

Figure 1: The problem geometry and the boundary conditions

Application of DRBEM We use the dual reciprocity boundary element idea to transform equations (6), (9),(10), (11) and (12) into the boundary integral equations using the fundamental solution of the Laplace equation (u∗ = (1/2π)ln(1/r)) [8]. We take all the terms other than the Laplacian as inhomogeneity and approximate the right-hand sides of the equations by the radial basis function f j (r) = 1 + r j which is connected to the particular solution as ∇2 uˆ j = f j , [8]. Then, eqs. (6), (9),(10), (11) and (12) take the form ∇2 u =

N+L

∑ α j ∇2 uˆ j ,

∇2 v =

j=1

N+L

∑ β j ∇2 uˆ j ,

j=1

∇2 w =

N+L

∑ γ j ∇2 uˆ j ,

j=1

∇2 p =

N+L

∑ ζ j ∇2 uˆ j ,

∇2 Ψ =

j=1

N+L

∑ λ j ∇2 uˆ j

(13)

j=1

where N and L are the number of boundary and interior nodes, respectively, and α j , β j , γ j ζ j and λ j are the undetermined coefficients. Weighting these equations by the fundamental solution u∗ , applying Green’s second identity two times and discretizing the boundary with constant elements we obtain (N + L) × (N + L) DRBEM discretized matrix-vector equations. Hu − G

∂u ˆ −1 ( ∂ p + ∂ Ψ ∂ u − ∂ Ψ ∂ u − MnH ∂ H ) ˆ − GQ)F = (HU ∂n ∂x ∂y ∂x ∂x ∂y ∂x

∂v ˆ −1 ( ∂ p + ∂ Ψ ∂ v − ∂ Ψ ∂ v − MnH ∂ H ) ˆ − GQ)F = (HU ∂n ∂y ∂y ∂x ∂x ∂y ∂y ∂ w ∂ w ∂w ˆ −1 (Pz + u ˆ − GQ)F = (HU +v ) Hw − G ∂n ∂x ∂y 2 2 2 2 ∂p ˆ −1 (Mn( ∂ H + ∂ H + H∇2 H) − ∂ u − ∂ v − 2 ∂ v ∂ u ) ˆ − GQ)F Hp − G = (HU ∂n ∂x ∂y ∂x ∂y ∂x ∂y Hv − G

(14) (15) (16) (17)

142


HΨ − G Here, Hi j = ci δi j +

Γj

∂Ψ ˆ −1 ( ∂ u − ∂ v ) ˆ − GQ)F = (HU ∂n ∂y ∂x

q∗ dΓ j ,

Gi j =

Γj

u∗ dΓ j ,

Gii =

l 2 (ln( ) + 1) 2π l

(18)

(19)

ˆ and F are constructed by ˆ Q where δi j is the Kronecker delta and l is the length of the element. The matrices U, taking each vector uˆ j , qˆ j = ∂ uˆ j /∂ n and f j as columns, respectively. All the space derivatives of the unknowns on the right hand sides of (14)-(18) are approximated by the DRBEM coordinate matrix F ∂A ∂ 2A ∂ F −1 ∂ F −1 ∂ F −1 (20) = F A, = F F A ∂η ∂η ∂ξ∂η ∂η ∂ξ with A being u, v or p , and ξ and η denote x or y. We use an iterative process for solving the system of equations numerically. The iteration starts with initial velocities u, v for solving the stream function equation and then the velocity components u and v using equations (11)-(12) with initial pressure gradient. Equations (4) and (5) are used to generate the boundary conditions of pressure. We approximate the spatial derivatives of the velocities by DRBEM coordinate matrix F, and the spatial derivatives of pressure by a finite difference approach using inner pressure values. Then, pressure equation (9) and the axial-velocity equation (6) are solved by using newly obtained u− and v− velocities. The iteration continues until a preassigned tolerance is reached. The stopping criteria for the iteration is ||z(n+1) − z(n) ||∞ < tolerance ||z(n) ||∞

(21)

where z denotes u, v, w, p or Ψ and n is the iteration number. The tolerance is taken as 10−3 in the computations.

Numerical results Numerical results are obtained with axial pressure gradient Pz = −8000 and the point magnetic source is placed at (0.5, −0.05) below the outer cavity. The magnetic source effect on the flow is investigated by taking a medium obstacle with size d = 1/3 placed at the center of the cavity. We present flow profiles, streamlines and the pressure profile for Mn = 0, 30, 500, 14000 at Fig. 2. The numerical results show that when there is no magnetic source (Mn = 0) variations of the flow and pressure on the transverse plane are zero. Fluid and its pressure are almost symmetrically spread around the obstacle. The axial velocity collapses at the center of the cavity due to the obstacle and it shows a parabola like profile between the corners of the cavity and the obstacle. Magnetic effect creates a flow on the transverse plane and divides streamlines into two symmetrical domains. Boundary layers are developed on the left and right vertical walls of the obstacle and on the bottom of the cavity. Two eddies are attached to the obstacle horizontally, squeezing the main flow through the vertical walls of the cavity and enlarging as magnetic field intensity increases. Velocity in the x− direction is decomposed into 4 vortices as Mn increases mainly emanating from the corners of the obstacle. The vortices attached to the top corners are weaker in magnitude than the ones attached to the bottom corners due to the placement of the magnetic source below the bottom side of the cavity. The velocity in y−direction is divided into 5 vortices with two small eddies appearing at the bottom corners and enlarging with increasing value of magnetic number Mn. Pressure is highly concentrated with large magnitude around the magnetic source. The profiles are similar to the ones obtained for a square cavity without an obstacle [9] as entering and then leaving the hole. Under the magnetic effect the axial velocity develops a peak around the magnetic source. Increasing magnetic field intensity accelerates the flow on the transverse plane. For Mn values greater than 14000 no significant change is observed in the flow behavior, only planar velocities and the pressure increase, and the axial velocity retardation around x = 0.5 is more pronounced. This is the well-known behavior of the flow under the magnetic effect. Fig. 3 displays the streamlines, pressure and velocity profiles for Mn = 5000 with different size and place of the obstacle. When the dimension of the obstacle is increased (Fig. 3 (a)-(c)) both the eddies around the vertical sides of the obstacle and the ones close to the bottom wall are squeezed through the vertical and bottom walls


143

between cavities, respectively. Meantime new flows appear above the obstacle through the upper wall of the cavity. Pressure due to the magnetic source is increasing with the enlargement of the obstacle and shows a profile as if emanating from the obstacle corners. Horizontal velocity is divided into new symmetric vortices between the upper wall of the cavity and the obstacle. The vortices of the vertical velocity at the bottom corners of the cavity enlarge separating the three main flows. Since the shear stress caused by the obstacle is more pronounced the axial velocity decreases. If the obstacle with size d = 1/5 is shifted 1/5 units above (Fig. 3 (d)-(e)), the profiles in planar velocities and the streamlines stay the same with enlarged vortices below the obstacle. Pressure is also moved to the obstacle area. When it is shifted 1/5 units to the right the symmetry in the profiles is destroyed as expected. The eddy attached to the left vertical wall of the obstacle strengthens and meets with the right bottom vortex while the eddy close to the right vertical wall is weakening. Similarly, in the u− and v− velocities the vortices on the right part of the obstacle shrink through the right vertical wall of the cavity. The left main vortex in the vertical velocity covers almost all parts of the cavity. Pressure around the point magnetic source extends to the top of the cavity.

Conclusion Steady, laminar, fully developed flow of viscous, incompressible, electrically non-conducting, magnetizable fluid in a square cavity with an obstacle under the influence of a point source magnetic field is investigated by using DRBEM approach. The formation of the vortices and their sizes depend on the magnetic number Mn as well as the position and the dimensions of the obstacle. The presence of the magnetic field generates two main vortices on the transverse plane meantime the obstacle disturbs the flow, and two additional eddies are attached to it horizontally. Pressure is highly concentrated with increasing magnitude and the axial velocity flattens around the magnetic source as the intensity of the source is increasing. Enlarging the size of the obstacle shrinks the eddies through the walls of the cavities. Displacement of the obstacle on the symmetry axis through the top of the cavity does not destroy the planar flow behavior due to the place of the magnetic point source.

References [1] A.W. Islam, M.A.R. Sharif, E.S. Carlson Mixed convection in a lid driven square cavity with an isothermally heated square blockage inside, International Journal of Heat and Mass Transfer 55 5244-5255 (2012). [2] M. Nazari, L. Louhghalam, M.H. Kayhani Lattice Boltzman simulation of double diffusive natural convection in a square cavity with a hot square obstacle, Chinese Journal of Chemical Engineering 23 22-30 (2015). [3] F. Selimefendigil, H.F. Oztop Natural convection and entropy generation of nanofluid filled cavity having different shaped obstacles under the influence of magnetic field and internal heat generation, Journal of the Taiwan Institute of Chemical Engineers 56 42-56 (2015). [4] O.S. Galaktionov, V.V. Meleshko, G.W.M. Peters, H.E.H. Meijer Stokes flow in a rectangular cavity with a cylinder Fluid Dynamics Research 24 81-102 (1999) [5] E.E. Tzirtzilakis, V.D. Sakalis, N.G. Kafoussias, P.M. Hatzikonstantinou Biomagnetic Fluid Flow in a 3D Rectangular Duct, International Journal for Numerical Methods in Fluids, 44, 1279-1298 (2004). [6] C.A.J. Fletcher Computational Techniques for Fluid Dynamics 2. (2nd eddition) Springer, Berlin. (1991) [7] R.E. Rosensweig Ferrohydrodynamics. Dover Publications, Mineola, New York. (2014) [8] P.W. Partridge, C.A. Brebbia, L.C. Wrobel The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, (1992). [9] P. Senel, M. Tezer-Sezgin Flow in a Rectangular Duct Under the Influence of The Magnetic Field, Proceedings of the 10th UK Conference on Boundary Integral Methods, Brighton, UK, (13-14 July 2015).


0.2

0.4

0.1

0.6

0.8

1

0

0.2

0.4

w

0.6

0.8

647

83

0

0.2

0.4

0.6

0.8

0.2 0.1 0

1

200 150

100

100

100

100

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y

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083 31.5

737.9

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w

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−0.8 199 5

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25.6634

w

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p

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25.6634

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0.4

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2976

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1

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97

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p

1 0.9

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0.2

58

1

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7.3796 17.219

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796

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5

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1

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376

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0.3

0.07

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0.15032

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−10.6

−7.3

0.2

0.4

0.3

0

0.4

0.1

594

594

0.3

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03

77

0.4

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0.4

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0.7

0.3

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0.4

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0.8

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0.64516

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0.2

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0.4

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0.8

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v

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0.4

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0.6

0.2

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0.4

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59 .038

14

0.2

8599

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0

0

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0.6

0.2

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0.2

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−7.6329e−020

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0

−7.6329e−020

0.3

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0.4

6

0.7

0.2

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1

0.01286

0.8

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0.5

0

1

48

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0.7 0.6

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1

e−0

0.8

0.3

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0.8

0.6

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0.9

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0.4

0.4

0.81

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0.2

0.5

0.2 0.1

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1 0.9

0

0.6

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u

0.0025964

3

0

1

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−0.0 −0.0025964

2

78

0.8

0.3

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986

0.1

0.4

0.7

64

0.6

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0.5

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7 −0.1445

0.8

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0.01704

.0 −0

0.4

8

0.2

94

0

0

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7.

0.0

0.2

− 5e

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1 02

7.7105e−021

0.1

0.3

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5e

72

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0.2

0

2 −0

−0

0.4

0.3

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−0.017 0.017048

0.6

0.00099862

0.6 0.5

0.4

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0.7

03

0

0.8

0.0005

0.7

0.0

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0.5

9917

59917

0.9

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28e

0.6

0

−0.000

0.8

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Ψ

1

0.9

859

10

.7

−7

0.7

Ψ

1

0.9

0.0

0.8

Ψ

1

−7.7105e−021

0.03

1

02

− 5e

54.9708

Ψ

1 0.9

0.81921

144

x

0 1

1

0.5

y

0.5 0 0

x

Figure 2: Medium obstacle (d = 1/3) at the center of the cavity. Streamlines Ψ, horizontal velocity u, vertical velocity v, pressure p and axial velocity w profiles for Mn = 0, 30, 500, 14000 from left to right using N = 160, 160, 160, 320, respectively.


Ψ

1

0

1

0.8

1

.03

0.8

51

8.4227

66 −0.973

−1.2307

0.4

0.33412

−8.4707

0.1

−0.28536 0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

0.6

0.8

0

1106 .137 1

10

10

357 3.7 106 454

32 .88

77

497.869

1

0

0

0.2

0.4

1

0.8

4

83

1.0

0.6

272.46

06

0.5

0

0.2

183.9824

537.8

953

0.2

−1.3471

6

0.1 0.4

0.6

1

0.4 0.3

369

0.8

0.7

15.6663

−0.5

0.6

p

0.9

7.5647

0.1 1

70

338 25.5 0.4

v

0.2

09 3

0.2

0.1

0.2

0.3

6.0

0.2

827

0.5

0.4 0.3

0.3

0

0.6

0.5

0.2 0.1

0.8

1.1

0.1

0.4

63

0.5

0.3

−1.929

−1.929

0.7

0.6

27

0.6

217.8784

0.5 0.4

0.9

0.7

0.7

0.6

3.6215

1

4.4004

−2

0

−202.1074 −132.1097

84 217.87

26

−0.21653

0.6

0.7

95.5042

0.26529

0.4

0.8

2.2339

1

7.0

0.8

0.2

0.9

0.4

u

0.9

0.9

0

0

0.15244

0.8

p

1

4.3154

0.6

9

1

0.4

38

0.8

551

2370.4

0.2

0.9

0.6

6.6

65 .30 −9 36 .03 15 159

−0.0025

0.49684

0.4

0

36

0.2

−20

7 47

12.1

0.29147

−0.79114

0.24144

0

0

357.8737

0

1

1654.5938

0.1

454

0.1

357

0.2

0.1

0.8

357.8737

0.2

4

8.63

0.6

v 4.3154

0.5

−8.634

938.7326

32

0.3

0.2

994

0.4

1

366.0436

938.7326

.88

0.3

−0.33

0.2

35

0.4

0.3

0.33994

0

0.6

0.4

366.0436

0.4

77

97

8

.4

−4

0.5

0.8

0.5

−2.5

.25

84

0.6

0.6

p

0.2

−2.5

0.7

4.48

0.4

0.3

0.8

0.7

97

9

53

11

−0.496

0.026149

−0.026149

0.8

0.2

1

0.6

0.9

−3.799

0

0.9

. 23

1

3.799

0

3.7

1

0.5

0.1

45

0.1

u

0.9

v 0.6

0.13

57

1663.73

0.1 1

6.6

0.8

0.8

1263.7357

−20

0.6

0.6

106

0.4

1

0.13075

0.8

(e)

0.2

57

1654.5938

Ψ

0

0.4

9

1

0.2

.53

0.8

0

0.2

0

1

463.73

7

0.2

−0.97345

14

0.6

−0.13075

0.6

0.3

0

5

39

0.4

1

0.7

0.3

463.735

0.3

−0.9734

.5

0.2

0.9

0.4

0.8

0.4

0.7

0.4

70

0.5

0.8

0.5

0.6

0.6

7

26

.4

25

7

0.7

0.5

0.4

p

0.8

0.8

0.6

9

0

−7.9016

0.6

0.1

−0.34874

6

−0

0

0.2

1

26

1

.115

4

487

0.3

.4

0.9

0.1

−12

0

25

0.2

1

0

0.9

0.4

0.1

0.8

8

0.7

0.6

0.6

7.9016

0.2

0.1

1

6.3599

1.9599

23

0.5097

0.2

6.3599

0.8

u

0.7 −0.24144

0.3

0.4

.16

0.8

144

−0.5097

0.4

0.2

1

0.6 0.5

0

0.9

0.24

0.8

.76

0.0025

4144

−0.2

0.7

0

0.6

0

.875

1388

0.1

v

10

Ψ

0.8

77

1

0.4

0.2

−3.8071

0.3

0.1 0.8

0.2

0.5

6

0.6

0

0.9

5.0112

33

803

.45

−0

0.4

1

0.7

2 −5.011

0.4

5.0

803

0.2

1

27

7.00

−3.807

1

−1

0.45 0

0.8

0.5

0.2

1

(d)

0.6

0.3

0.9

(c)

0.7

0.2

0

0.6

u

0.8

0.3

0.1

0.4

.14

0.4

0.2

065 9.3

0.5

2

0.79114

0.6

0

1

−0.29147

0.7

0

0.9

0.12492

249

0.8

0

−2

−0.1

0.1

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1

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5

1

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.25 3

0.8

24

0.1

682.0292

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15.6524

116.552

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71

0.6

Ψ

0.6

1106.13

0.4

0.7

0.6

11

78 85

78

−0.46695

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(b)

.65

0

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5.4584

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−15

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0

−

0.4

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0.46695

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85

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961

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8

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8.547

8.547

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338 25.5

0.3631

0.7

18

.9 −0

(a)

−0.363

0.6

2

v

1

−1.6192

19

−1.619

0.8

1.6

1.6192

0.8 0.7

u

1 0.9

5.94

Ψ

1 0.9

145

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0

0

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0.8

1

Figure 3: Flow and pressure profiles for Mn = 5000. Inner cavity positioned at the center, (a) d = 1/5, N = 168 (b) d = 1/3, N = 160 (c) d = 1/2, N = 168. Inner cavity positions d = 1/5, N = 288 (d) near the upper wall, left bottom corner placed at (2/5, 3/5) (e) near the right wall, left bottom corner placed at (3/5, 2/5).

A Solenoidal-Galerkin Approach for the Numerical Simulation of Flow Past a Circular Cylinder Hakan I. Tarman Department of Engineering Sciences, Middle East Technical University, Ankara, Turkey e-mail: [email protected] Keywords: Two Dimensional Flow past Circular Cylinder, Solenoidal Bases, Galerkin Projection.

Abstract. Flow past a circular cylinder embodies many interesting features of fluid dynamics as a challenging fluid phenomena. In this preliminary study, flow past a cylinder is simulated numerically using a Galerkin procedure based on solenoidal bases. The advantages of using solenoidal bases are two folds: first, the incompressibility condition is exactly satisfied due to the expansion of the flow field in terms of the solenoidal bases and second, the pressure term is eliminated in the process of Galerkin projection onto solenoidal dual bases. The formulation is carried out using nodal Fourier expansion in the angular variable while modal polynomial expansion is used in the radial variable. A variational approach to recover the pressure variable is also presented. Some numerical tests are performed. Introduction In the numerical modeling of the incompressible flow phenomena, the incompressibility flow condition, namely the divergence-free condition or continuity condition stands as an important source of difficulty. Yet, another related issue is the numerical handling of the pressure variable appearing in the governing partial differential Navier-Stokes equations that usually comes without any boundary conditions. There are schemes developed solely to satisfy the continuity equation such as the fractional step scheme in [1] and the influencematrix method in [2]. However, this can only be achieved to a limited degree of accuracy. In this work, a solenoidal spectral representation for the flow field is used in a Galerkin approach. As a consequence, the incompressibility and boundary conditions on the flow field are strictly enforced. The pressure term is then eliminated in the Galerkin projection procedure onto a solenoidal dual space. There have been various works utilizing solenoidal spectral expansions. Moser et al. [3] presented a spectral method to automatically satisfy the continuity equation and boundary conditions and tested their method on the channel flow and the flow between concentric cylinders. They expanded the vertical and horizontal extents with Chebyshev polynomials and Fourier series, respectively. Kessler [4] studied steady and oscillatory regimes of Rayleigh-Benard convection with explicitly constructed solenoidal bases based on poloidal-toroidal decomposition. Trigonometric polynomials and the beam functions were used in the construction of the solenoidal bases satisfying the boundary conditions in a rectangular container. Clever and Busse [5] used toroidal-poloidal expansion in their numerical approach satisfying the solenoidal condition exactly; however, the procedure for eliminating the pressure leads to higher order derivatives. Most recently Meseguer and Trefethen [6] proposed a spectral Petrov-Galerkin formulation based on divergence-free bases in terms of Chebyshev polynomials to study stability of pipe flow. Another recent study utilizing solenoidal bases is also the study of pipe flow in [7]. A mathematical analysis of the solenoidalGalerkin approach for the Stokes problem can be found in [8]. In this study, uniform flow past a circular cylinder is numerically simulated by using a solenoidal-galerkin procedure. The flow configuration consists of a simple geometry of cylinder with simple no-slip boundary

148


conditions on the cylinder. However, the resulting flow field shows strong inhomogeneities in the wake region developing behind the cylinder with many different mechanisms at play. It is also physically relevant that this type of flow is an everyday phenomena that can be observed when objects move in a fluid medium such as air and water. There is a vast literature on this subject since it is one of the classical problems of fluid mechanics. Two volumes text on the flow around circular cylinders may be a good starting point in [9,10]. A related low dimensional Galerkin approach to study of three dimensional flow around a circular cylinder is presented in [11]. In their approach, the solenoidal bases are constructed using Laguerre functions in the radial direction modified to resolve the boundary layer on the cylinder. Periodic bases are used in the spanwise direction with a physical period L and in the angular direction with natural period 2 . Gram-Schmidt process is used for the orthogonalization of the bases. In this work, nodal Fourier representation is used in the angular direction while modal polynomial representation based on Legendre polynomials is used in the radial direction to ensure resolution of the boundary layer around the cylinder [12]. The use of Legendre polynomials with their natural interval of definition [1,1] , of course, requires the truncation of the computational domain. This is an issue that we would like to explore further in a future study. A variational approach to recover the pressure that is vanished in the process of Galerkin projection onto dual solenidal basis is also presented. In this work, our main focus has been the formulation and the implementation of the numerical procedure. Thus, we opted to work with linearized and steady form of the governing Navier-Stokes equations and more realistic settings are postponed to a future study. Governing Equations

The steady Navier-Stokes equations in polar coordinates v  0

(1a),

1 p 1  2 2 v 1  v v r  v2     vr    vr  2 r r Re  r  r 2  1 1 p 1  2 2 v r 1  v v  v r v     v    v  2 r r  Re  r  r 2 

(1b), (1c),

where Re is the Reynolds number based on the cylinder radius and the free-stream velocity U  that drives the flow v  (v r , v ) , are linearized in u  (u, v) that is the flow field superimposed over the basic mode u 0 , i.e.

 v u 0  u , subject to the boundary conditions,

u(r  1, )  0, lim u(r, )  0, r 

u(r,  2 )  u(r, ) . The basic mode u 0 is given by  0   1  0 u 0 ( x )    0 e z   , , 0  r  r  

(2)

with



1 

 r 1 

 0 ( x)   r   1  exp     sin   r    

(3)

Advances in Boundary Element and Meshless Techniques XVII where   

149

Re and   4 following [11]. Here x stands for (r,  ) . It can be shown the basic mode satisfies

the free-stream flow condition

lim u 0 (r, ) r 

 cos ,  sin  

at infinity. Numerical Formulation

This system is approximated using a Solenoidal-Galerkin approach. The Solenoidal-Galerkin projection procedure starts with the expansion of the flow field u in terms of solenoidal expansion functions Uq (x) which are generated from a scalar field  q ( x) as follows:  q   1  q U q ( x)    qe z   , , 0  r  r  

(4)

with the scalar field in the form

  ( x)

ˆ  (x) ˆ  q

q

q

q

q

Pm (r)SN (   n )

(5)

for the index vector q  (m, n) and satisfy   Uq 

 1 1  Uq  Uq  Vq  0 r r r 

(6)

where U q  (U q , Vq ) and the boundary conditions

Uq (r  1, )  0, lim Uq (r, )  0, r 

Uq (r,  2 )  Uq (r, ) . In the expansion in eq(5), Pm (r) represents a polynomial with m associated with its degree and 1 sin( N21 (   n )) SN (   n )  N  1 sin( 12 (   n ))

(7)

is the cardinal function in the Fourier space of 2 -periodic functions (see Fig.1). The projection procedure proceeds with the definition of a weighted inner product

  Up , Uq 





2

0



d  Up  Uq  (r) r dr 1

onto the dual space spanned by U p . It can be shown that the pressure term in eq(1) vanishes under projection

U

p

, p   0 

provided that dual expansion functions U p satisfy

  ( U p )  0 with the boundary conditions U p (r  1, )  0,

lim r (r) U p (r, )  0 . r 

(8)

150


In this study, the weight is chosen as unity,  (r)  1 , with the obvious benefits and the dual expansion functions

U p are selected to be the same as the solenoidal expansion functions Uq . Numerical Procedure Before the numerical implementation of the formulation, two issues are to be resolved: First, the infinite domain is truncated at r  R and boundary condition at infinity is approximated by

Uq (r  R, )  0 . This, in turn, dictates the selection Pm (r)  (x 2  1) 2 L m (x)

(9)

in the expansion, eq(5), where zeros of Uq (x) at r  1 and r  R are double zeros of  q ( x) in the variable r (see Fig.1). Here, L m (x) denotes Legendre polynomial of order m in x(r) 

2 R 1

(r  1)  1 for 1  x  1 .

Fig.1: The first five of the radial expansion functions Pm (r) with increasing zero crossings indicating increasing resolving capacity on the left. The angular expansion function SN (   n ) is the cardinal function in the Fourier space of 2 -periodic functions on the right. The second is the numerical evaluation of the inner product integrals in the projection procedure. Here, the numerical integration is performed using quadrature integration. It is known that Legendre-Gauss quadrature formula



1

1

f (x)dx   k 0 kf (x k ) K

is exact for all polynomials of degree  2K  1 over the quadrature nodes x k that are the roots of the polynomial L K 1 (x) [12]. It is also known that the quadrature formula 2

1 1 N  g()d  g(n )  2 0 N  1 n 0

2n (N  1) [12]. Thus, the inner product is exact for any f () eij with | j | N over the quadrature nodes n  integrals

 U , U  p

q

2

0

R

d  Up  Uq r dr 1

 

N K R 1 2  n 0 k 0

 k Up (rk , n )  Uq (rk , n ) rk

Advances in Boundary Element and Meshless Techniques XVII where  rk r(x k )

R 1 2

151

(x k  1)  1 , can be evaluated exactly for the functions U p and Uq associated with the

expansion in eq(5) provided that the degree m in eq(9) satisfies m  K  4 and due to the representation of the cardinal function 1 N 1



N 2 j  N 2

exp(ij(   n ))  SN (   n ) .

It should be noted that the cardinal function satisfies the cardinality property SN ( j   n )   jn where  jn is the kronecker delta and that spatial distribution of the radial nodes rk associated with Legendre-Gauss quadrature nodes x k are denser closer to the boundaries r  1 and r  R thus providing added spatial resolution where it is needed as shown in quadrature node distribution over the domain in Fig. 2.

Fig. 2: The quadrature node distribution for N  32 and K  16 with R  10 .

Pressure Calculation The pressure field, where it vanished in the Galerkin projection operation onto solenoidal bases in the previous section, can be recovered by using the velocity field v  (v r , v ) computed above. In order construct a variational formulation, the pressure field p(r, ) is expanded in the form

 p(r, )

 pˆ q

q

Lm (x)SN (   n )

(10)

with q  (m, n) where L m (x) denotes Legendre polynomial of order m in x(r) 

2 R 1

(r  1)  1 for 1  x  1 . The

linearized equations are then projected

p , V   N( v, v)  Re1 L( v ) , V   

(11),

onto the space V  V  V  spanned by V span{Pk (r)SN (   n )} with Pk (r)  (x 2  1)L k (x) . Here, k,n

 v v r  1r v2  N ( v, v )    1  v v  r v r v 

and

 

  2 v r  22  v  12 v r r r L( v )     2 v  22  v r  12 v r r 

are linearized in u that is superimposed over the basic mode u 0 with v  u 0  u .

  

(12),

152


Results Some numerical experiments are performed for Re  1,10 and R  10 with the resulting flow field shown below:

Fig. 3: The flow field at Re  1 for a resolution of N  32 and K  16 is shown on the left with its close-up on the right. The flow follows the contour of the cylinder up to a distance from the cylinder surface.

Fig. 4: The flow field at Re  10 for a resolution of N  32 and K  16 is shown with its close-up on the right. The wake behind the cylinder is forming. In this preliminary work, our objective is to formulate and implement the numerical procedure presented. For this purpose the linearized and steady form of the governing Navier-Stokes equations are used. This limits the numerical experiments to small Re values at higher values of which nonlinearity becomes important. The future work is planned to include nonlinear terms under unsteady flow conditions for higher Re flow regimes.


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References [1] S. A. Orzag and L. C. Kells, “Transition to turbulence in plane poiseuille and plane couette flow,” Journal of Fluid Mechanics, vol. 96, no. 1, pp. 159–205, 1980. [2] L. Kleiser and U. Schumann, “Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows,” in Proceedings of the 3rd Conference on Numerical Methods in Fluid Mechanics, E. H. Hirschel, Ed., pp. 165–173, 1980. [3] R. D. Moser, P. Moin, and A. Leonard, “A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow,” Journal of Computational Physics, vol. 52, no. 3, pp. 524–544, 1983. [4] R. Kessler, “Nonlinear transition in three-dimensional convection,” Journal of Fluid Mechanics, vol. 174, pp. 357–379, 1987. [5] R. M. Clever and F. H. Busse, “Nonlinear Oscillatory Convection”, Journal of Fluid Mechanics, vol. 176, pp. 403–417, 1987. [6] A. Meseguer and L. N. Trefethen, “Linearized pipe flow to Reynolds number 107,” Journal of Computational Physics, vol. 186, no. 1, pp. 178–197, 2003. [7] Ozan Tuğluk and Hakan I. Tarman, “Direct numerical simulation of pipe flow using a solenoidal spectral method”, Acta Mechanica, vol. 223, no. 5, pp. 923-935, 2012. [8] A. F. Pasquarelli, A. Quarteroni and G. Sacchi-Landriani, “Spectral approximations of the Stokes problem by divergence-free functions”, Journal of Scientific Computing, vol. 2, no. 3, pp. 195-226, 1987 [9] M. M. Zdravkovich (1997), Flow Around Circular Cylinders 1: Fundamentals, Oxford Science Publications. [10] M. M. Zdravkovich (2003), Flow Around Circular Cylinders 2: Applications, Oxford Science Publications. [11] B. R. Noack and H. Eckelmann, A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder, Phys. Fluids 6 (1), Jan. 1994. [12] J. S. Hesthaven and D. Gottlieb (2007), Spectral Methods for Time-Dependent Problems, Cambridge University Press.

Boundary integral solution of MHD pipe flow M. Tezer-Sezgin1 and Canan Bozkaya2 1

2

Keywords: MHD pipe flow, boundary integrals, exterior Neumann problem.

Abstract. A mathematical model is given for magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross-section of the pipe, coupled with an outer Neumann problem. Pipe is under the influence of a transverse magnetic field, and both the fluid and the external medium including the pipe wall are electrically conducting. Inner Dirichlet problem is given as the coupled convection-diffusion equations for the velocity and the induced magnetic field of the fluid. Neumann problem is defined with the Laplace equation for the induced magnetic filed of the exterior. The two problems are also coupled with the continuity requirements of the induced magnetic fields on the pipe wall. Unique solution of Dirichlet problem is obtained reducing it to boundary integral equations defined on the pipe wall by using the fundamental solution of convectiondiffusion equation. Exterior solution is given on the pipe wall with Poisson’s integral formula with an additive constant which is found through coupled boundary conditions. The theoretical part of the solution results in solving these three coupled boundary integral equations simultaneously including the solvability condition of Neumann problem. The collocation method is made use of to obtain linear system of equations for discrete values of the solution on the pipe wall. Then, the solution is extended to the interior and exterior regions with the calculated numerical values on the boundary. The velocity of the fluid and the induced magnetic fields of the fluid and exterior medium are simulated for several values of problem parameters as Reynolds number Re, magnetic pressure Rh and magnetic Reynolds numbers Rm1 and Rm2 of the fluid and outside medium, respectively. The well-known MHD characteristics are observed inside the pipe for increasing values of Rm1 , and the continuity of induced magnetic fields is maintained accordingly with the ratio Rm1 /Rm2 .

Introduction MHD flow through pipes has many practical applications in the design of cooling systems with liquid metals for nuclear reactors, MHD generators, electromagnetic pumps, MHD flow-meters measuring blood pressure, biomedical instruments using magnetic sources. The flow of an incompressible, viscous, electrically conducting fluid inside the pipe gives rise to an induced magnetic field with the interaction of transverse magnetic field. When the outside medium has also electrical conductivity, although small, the inside and outside induced magnetic fields (induced currents) continue on the pipe wall. Some analytical solutions are available when external region is insulated [1]. Numerous numerical methods have been used for MHD pipe flow in insulating mediums with mixed wall conditions, and Sheu and Lin [2], Neslitürk and Tezer-Sezgin [3], Tezer-Sezgin and Bozkaya [4], Dehghan and Mirzai [5], Loukopoulos et.al. [6], Tezer-Sezgin and Han Aydın [7] and Cai et.al. [8] are some of them. In this paper, a theoretical solution is provided for MHD pipe flow in an electrically conducting exterior medium, reducing the inner Dirichlet and outer Neumann problems to coupled boundary integral equations on the pipe wall. Unique solution of Dirichlet problem and the solution of the Poisson’s integral equation (exists with an additive constant) for Neumann problem are combined. Solvability condition is added to boundary integral equations and the additive constant is obtained through the continuity requirements of induced currents on the pipe wall. The effects of external magnetic filed on the flow and induced currents are visualized for several values of problem parameters.

156


Mathematical formulation MHD flow in a circular pipe in an electrically conducting external medium under the effect of a transverse magnetic field is given with the coupled equations and coupled boundary conditions [9] ∇2V + ReRh

∂ BΩ = −1 ∂y

in Ω

∂V ∇ BΩ + Rm1 =0 ∂y

(1)

2

∇2 BΩ = 0

in Ω

V =0 BΩ = BΩ

on Γ

1 ∂ BΩ 1 ∂ BΩ = . Rm1 ∂ n Rm2 ∂ n

(2)

Ω Γ n Ω n y B0 x Figure 1: Cross-section of the pipe in 2D By taking B1 =

√ ReRh BΩ and B2 = BΩ where M = ReRhRm1 , and the new variables M u1 = V + B1 +

1 y, M

u2 = V − B1 −

1 y M

equations (1) are decoupled as inner and outer problems ∇2 u1 + M

∂ u1 =0 ∂y

∂ u2 ∇ u2 − M =0 ∂y

in Ω

(3)

2

∇ 2 B2 = 0

in Ω

(4)

with the boundary conditions still coupled u1 = −u2 ∂ u2 ∂ u1 M ∂ B2 2 ∂y = +2 − ∂n ∂n Rm2 ∂ n M ∂n u1 =

M y B2 + Rm1 M

on Γ

(5)

Advances in Boundary Element and Meshless Techniques XVII since B1 =

M B2 , Rm1

1 ∂ B2 1 ∂ B1 =− M ∂n Rm2 ∂ n

157

on Γ .

(6)

We consider convection-diffusion equations (3) with the assumption that u1 and u2 are known on Γ as inner ∂ B2 Dirichlet problem, and Laplace equation (4) assuming is given on Γ as outer Neumann problem. ∂n Inner Dirichlet problem has a unique solution and it is reduced to two integral equations on Γ ∂ g1 ∂ u1 g1 − u1 dΓ + M g1 u1 ny dΓ c p u1 (ξ , η) = ∂n ∂n Γ Γ (7) ∂ g2 ∂ u2 g2 c p u2 (ξ , η) = − u2 dΓ − M g2 u2 ny dΓ ∂n ∂n Γ Γ α(P) , P = (ξ , η) is the source point and α(P) is the internal angle at P. with cP = 2π The first two boundary conditions in (5) rearrange the boundary integral equations (7) as 1 ∂ g1 ∂ u1 dΓ + M g1 u1 ny dΓ g1 = u1 (P) − u1 2 ∂n ∂n Γ Γ 1 − u1 (P) = 2 for the three unknowns u1 ,

g2

Γ

2M ∂ B2 2 ∂y ∂ g2 ∂ u1 + u dΓ + M u1 g2 ny dΓ + − 1 ∂ n Rm2 ∂ n M ∂n ∂n Γ

(8)

∂ B2 ∂ u1 and on Γ. ∂n ∂n

∂ B2 There exists solution to Neumann problem ∇2 B2 = 0 in Ω with the assumption is known on Γ and is ∂ n given in terms of a Poisson’s integral formula [10] R 2π ∂ B2 r 2 − 2Rr cos (φ − ψ) + R2 dψ +C (9) ln B2 (r , φ ) = − 2π 0 ∂ r r 2 with an additive constant C. Here R is the radius of the pipeand r = x2 + y2 for a point Q(x, y) on Ω ∪ Γ. Rm 1 1 u1 (R, φ ) − R sin φ from the third condition in Thus, on the pipe wall Γ, r = R and B2 (R, φ ) = M M (5), and then Poisson’s integral becomes u1 (R, φ ) =

1 MR R sin φ − M 2πRm1

2π ∂ B2 0

∂n

(R, ψ) ln [2(1 − cos (φ − ψ))]dψ +C

(10)

containing u1 and ∂∂Bn2 unknowns on Γ and the normal n is in the r direction. Boundary integral equations (8) and (10) are solved simultaneously including the solvability condition of Neumann problem which is 2π ∂ B2 ∇2 B2 dΩ = dφ = 0 . ∂ n 0 Ω

Numerical calculations The integral over the boundary Γ is divided into N subintervals and solvability condition is discretized as 2π ∂ B2 0

∂n

(R, φ )dφ =

N

∂ B2 |Γ j (φ j+1 − φ j ) = 0 , j=1 ∂ n

∑

158


φ j+1 and φ j are the angles at the end points of Γ j with the x-axis. Taking the source (ξ , η) and the field (x, y) points as the mid points of Γ j subintervals and discretizing the boundary integrals (8), (10), we arrive at 3N × 3N linear system of equations (taking C = 0 first) H 1 u1 + G1

∂ u1 =0 ∂n

2 ∂ u1 ∂ B2 ∂y + K2 = G2 ∂n ∂n M ∂n ∂ B2 R = sin φ Iu1 + (∆φ I + K 3 ) ∂n M

H 2 u1 + G2

where u1 , ∂∂un1 and ∂∂Bn2 are the values at the midpoints of N subintervals and considered as constant over each subinterval. ∆φ = φ j+1 − φ j for Γ j , ∂∂ ny and sin φ are vectors evaluated at discretized N boundary points. The entries of the matrices are 1 ∂ g1 1 ∂ g2 Hi1j = + dΓ, Hi2j = − + dΓ −Mg1 ny + −Mg2 ny − 2 ∂n 2 ∂n Γj Γj G1i j = − Ki2j = −

Γj

2M g2 dΓ, Rm2

Γj

G2i j = −

g1 dΓ, Ki3j =

Γj

Γj

g2 dΓ

MR ln[2(1 − cos (φ − ψ))]dψ . 2πRm1

Solution can be extended to the interior and exterior of the pipe wall by using obtained values of u1 ,

∂ u1 , ∂n

∂ B2 on Γ and taking c p = 1 in (8) for interior u1 and r ∈ Ω in (9) for exterior B2 . Constant C is computed ∂n M (B2 +C) since B2 on Γ is obtained by taking C = 0 first. The velocity and through the requirement B1 = Rm1 the induced magnetic field of the fluid are obtained from V=

u1 + u2 , 2

B1 =

u1 − u2 − M2 2

since u2 = −u1 .

Numerical results The results are simulated in terms of equi-velocity and current lines for several values of the problem physical parameters. In Figure 2, the velocity and induced magnetic fields are displayed when Rm1 = Rm2 = 1, Re = 1, V

B

Figure 2: Velocity and induced magnetic field when Rm1 = Rm2 = Re = 1 and Rh = 10.


159

Rh = 10. Flow attains its maximum value at the center of the pipe and reduces to V (x, y) = 0 on the pipe wall. Inner and outer induced magnetic fields continue on the pipe wall obeying to the boundary conditions (6). Figure 3 shows the effect of the magnetic Reynolds number Rm1 on the inner and outer induced magnetic fields at fixed values of Rm2 = Re = 1 and Rh = 10. As Rm1 increases from Rm1 = 1 to Rm1 = 50, the inner induced magnetic field behaves as if it is the current in a pipe with insulated wall. Both inner and outer induced magnetic fields smoothly connect on the pipe wall according to the wall conditions (6). Rm1 = 1

Rm1 = 10

Rm1 = 50

Figure 3: Effect of Rm1 (= 1, 10, 50) on the induced magnetic field when Rm2 = Re = 1 and Rh = 10. Re = 1

Re = 10

Re = 50

Figure 4: Effect of Re(= 1, 10, 50) on the induced magnetic field when Rm1 = Rh = 10 and Rm2 = 1. Rh = 5

Rh = 10

Rh = 20

Figure 5: Effect of Rh(= 5, 10, 20) on the induced magnetic field when Rm1 = Re = 10 and Rm2 = 1.

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References [1] L. Drago¸s. Magnetofluid Dynamics. Abacus Press, London, 1975. [2] Tony W. H. Sheu and R. K. Lin. Development of a convection-diffusion-reaction magnetohydrodynamic solver on non-staggered grids. International Journal for Numerical Methods in Fluids, 45(11):1209–1233, 2004. [3] A.I. Nesliturk and M. Tezer-Sezgin. Finite element method solution of electrically driven magnetohydrodynamic flow. Journal of Computational and Applied Mathematics, 192(2):339 – 352, 2006. [4] M. Tezer-Sezgin and C. Bozkaya. Boundary element method solution of magnetohydrodynamic flow in a rectangular duct with conducting walls parallel to applied magnetic field. Computational Mechanics, 41(6):769–775, 2007. [5] Mehdi Dehghan and Davoud Mirzaei. Meshless local boundary integral equation (lbie) method for the unsteady magnetohydrodynamic (mhd) flow in rectangular and circular pipes. Computer Physics Communications, 180(9):1458 – 1466, 2009. [6] V. C. Loukopoulos, G. C. Bourantas, E. D. Skouras, and G. C. Nikiforidis. Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions. Computational Mechanics, 47(2):137–159, 2010. [7] M. Tezer-Sezgin and S. Han Aydın. Bem solution of mhd flow in a pipe coupled with magnetic induction of exterior region. Computing, 95(1):751–770, 2013. [8] Xinghui Cai, G.H. Su, and Suizheng Qiu. Local radial point interpolation method for the fully developed magnetohydrodynamic flow. Applied Mathematics and Computation, 217(9):4529 – 4539, 2011. [9] Adrian Carabineanu, Adrian Dinu, and Iuliana Oprea. The application of the boundary element method to the magnetohydrodynamic duct flow. Zeitschrift für angewandte Mathematik und Physik ZAMP, 46(6):971–981, 1995. [10] D.A. Polyanin. Handbook of Linear PDE for Engineers and Scientists. Chapman and Hall/CRC, 2002.

POD Analysis of Drag Reduction in Turbulent Pipe Flow O. Tugluk1 and H.I. Tarman2 1

METU Center For Wind Energy (METUWIND), Ankara, Turkey e-mail: [email protected] 2 Department of Engineering Sciences, METU, Ankara, Turkey e-mail: [email protected]

Abstract. In this study, proper orthogonal decomposition (POD) based analysis is employed to investigate the dynamics of drag reduction in turbulent pipe flow via phase randomization. The flow in a circular cylindrical pipe is simulated numerically employing solenoidal spectral basis functions, hence the continuity equation is automatically satisfied. These simulations are performed for the case of constant mass flux, at a Reynolds number (Re) of 4900. Following the simulation, POD (also known as Karhunen-Loève decomposition) is performed. Differences between controlled and baseline flow are observed.

Introduction Turbulent pipe flow is an ubiquitous case of wall bounded flows. Countless industrial applications as well as numerous academic studies are available [10, 5, 13]. Reduction of turbulent drag in flow problems, including wall bounded flows, is an active area of research. Reducing skin friction, which is directly related to drag, has immediate implications in practical applications. At its simplest, drag reduction would mean less pumping is necessary to achieve a given flow rate. Reduction in drag has been observed in wall bounded flows via different methods. To name a few, these methods are wall oscillations [3, 5, 7, 9, 10], particle addition [4], and phase randomization [6, 11, 14]. In the case of pipe flow a net drag reduction on the order of 20% was observed via phase randomization [14]. In this preliminary study the effects of phase randomization are investigated using proper orthogonal decomposition (POD). POD is also known as Karhunen-Loève (KL) or empirical eigenfunction decomposition, and is an effective tool in analyzing and storing flow-fields obtained from numerical studies [1, 8]. Decomposition of the flow-field into its KL-mode components, allows easier identification of underlying structures in the flow. Subsequently, the obtained KL modes can be utilized in order to construct a nonlinear ODE system which enables a reduced order dynamical study of the underlying flow.

Numerical Method The turbulent pipe flow is generally mathematically modeled with the incompressible Navier-Stokes (NS) equation. In its dimensionless form this equations reads, ∂t u + (u · ∇)u = G(t) ez − ∇p + ∇·u =0

u(1, θ , z,t) = 0,

1 2 ∇ u Re

(1)

u(r, θ , z, 0) = u0 .

Here z is the direction along the pipe axis, u represents the velocity vector, p is the pressure variable, G the pressure gradient present between the inlet and the outlet of the pipe. Re is a dimensionless parameter. No-slip boundary condition is applied at the pipe wall and the flow is assumed to be periodic along the axial direction. Equation 1 is solved using a spectral solenoidal method, where the velocity is expressed as a sum of precomputed divergence-free flowlets, for details the reader is referred to [12, 13]. Pipe flow is simulated for

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constant mass-flux at bulk Reynolds number 4900. Simulations were performed for flow without control and phase randomization at regular intervals [14]. As previously mentioned spatial discretization is handled by a solenoidal-spectral method, and time dependence is handled by a 3rd order semi implicit (IMEX) scheme. In Fourier space the phase randomization operation is exactly as the name implies, i.e., at equispaced time intervals, the Fourier coefficients alnm corresponding to the velocity field are transformed as, alnm → eiφnm alnm .

(2)

However, if the phases of all modes are randomized this results in a drag increase. Hence, a subset of modes is selected for randomization, kl2 + kn2 ≤ kmax /6, 1

Advances in Boundary Element & Meshless ...

Advances in Boundary Element & Meshless ...

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