SIMULATION OF SOLID BODY MOTION

May 12, 2014 19:5 WSPC/INSTRUCTION FILE SOLID BODY MOTION”

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International Journal of Modern Physics C c World Scientific Publishing Company ⃝

SIMULATION OF SOLID BODY MOTION IN A NEWTONIAN FLUID USING A VORTICITY-BASED PSEUDO-SPECTRAL IMMERSED BOUNDARY METHOD AUGMENTED BY THE RADIAL BASIS FUNCTIONS

FEREIDOUN SABETGHADAM Mechanical and Aerospace Engineering Department, Science and Research Branch, Azad University (IAU), Tehran, Iran [email protected] ELSHAN SOLTANI PhD. Candidate, Mechanical and Aerospace Engineering Department, Science and Research Branch, Azad University (IAU), Tehran, Iran Received Day Month Year Revised Day Month Year

The moving boundary conditions are implemented into the Fourier pseudo-spectral solution of the two–dimensional incompressible Navier–Stokes equations (NSE) in the vorticity–velocity form, using the radial basis functions (RBF). Without explicit definition of an external forcing function, the desired immersed boundary conditions are imposed by direct modification of the convection and diffusion terms. At the beginning of each time step the solenoidal velocities, satisfying the desired moving boundary conditions, along with a modified vorticity are obtained and used in modification of the convection and diffusion terms of the vorticity evolution equation. Time integration is performed by the explicit fourth-order Runge–Kutta method and the boundary conditions are set at the beginning of each sub-step. The method is applied to a couple of moving boundary problems and the accuracy and efficiency of the method is demonstrated. Keywords: Moving Bodies; Incompressible flow; Radial basis functions (RBF); Vorticity– velocity formulation; Pseudo-spectral; Immersed boundary method. PACS Nos.: 47.11.-j, 47.27.er, 47.10.ad.

1. Introduction The immersed/embedded boundary methods are among the most popular methods in studying the interaction of fluid with moving bodies. However, one of the main drawbackes of these methods has been their low convergence rates 1,2 . Particularly, for the spectral-based immersed boundary methods the accuracy of approximation and smoothness of imposition of the immersed boundary conditions play vital roles in achieving high orders of accuracy 3 . 1


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F. Sabetghadam, E. Soltani

There is a wide variety of methods in approximating the non-regular immersed boundary conditions on the regular grid points. With this regard, convolution by smooth delta functions 4,5,6 , and various kinds of interpolations 7,8,9 are in common use. In another family of the methods a local auxiliary equation is solved 10,11 in order to improve the accuracy. However, it is revealed that these methods may impose spurious oscillations which deteriorates the uniform convergence rates 1,9,11,12 . To overcome this difficulty, a few remedies are suggested in the literature; the methods like smooth extensions of the solution domain 11,13,14 , employing the numerical filters and mollifiers 15,16,17 , or various kinds of multivariate interpolation methods 3,18,19 . In the multivariate interpolation methods, the feasibility of using radial basis functions (RBF) in the spectral-like immersed boundary methods is demonstrated recently 3 . In the present work we employed the modified Shepard’s method 19,20 in a vorticity-based Fourier pseudo-spectral immersed boundary method. In particular, the Shepard’s method is chosen because of its fastness and flexibility, and since it results in locally C 2 functions suitable for our purposes. The other issue which affects drastically the order of accuracy is the algorithm of coupling of the momentum and continuity equations. With this regard, although the fractional step method (i.e. the projection method), in the discrete and continuous forcing formulations, is the most popular 4,5,7 , the vorticity-based formulations are appearing recently 9,21,22,23 . In the present work, we used a vorticity-based immersed boundary method, in which without explicit definition of a forcing function, the immersed boundaries are implemented via direct modification of convection and diffusion terms of the vorticity evolution equation. At the beginning of each time step the solenoidal velocity vectors, satisfying the desired boundary conditions, are obtained and used in construction of the initial condition of the vorticity evolution equation. In this manner, the immersed boundary conditions are imposed via the initial condition. The classical Fourier pseudo-spectral method is used in the spatial discretization, and the immersed boundary conditions are approximated on the regular grid points using modified Shepard’s method. Time integration of the vorticity evolution equation is carried out by the classical explicit fourth-order Runge–Kutta method. As it will be seen, the method is fast, accurate and easy to implement. The paper is continued by a brief formulation of the classical Fourier pseudospectral method and the detail description of the suggested algorithm for imposing the immersed boundary conditions. Because of its prominent role, the boundary condition setting method is presented in detail, in an individual sub-section. After that, the method is applied to a couple of two-dimensional moving boundary incompressible flow problems. Finally our summary and conclusions are provided.

2. Mathematical formulations Mathematical framework of the method is presented in this section. We start by the formulation of the classical pseudo-spectral method, and then the modifica-


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Simulation of Solid Body Motion in a Newtonian Fluid . . .

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tions for imposing the immersed boundary conditions are suggested. After that the interpolation method is presented, which followed by a summary of the algorithm. 2.1.

The problem set up

The flow configuration is illustrated in Fig. 1. The regular solution domain D contains the fluid domain Ωf and the solid obstacle Ωs , fixed or moving. The immersed velocity boundary conditions are given on the physical immersed boundary Γs at each time instant, and the solution domain D is covered by a N-point uniform grid G(N ). 2.2. Fourier pseudo-spectral formulation According to Fig. 1, for a two-dimensional velocity vector u = (u1 , u2 ), defined ¯ = {D + ΓD }, the dynamics of the vorticity vector on the regular closure D ω = (0, 0, ωz = ωˆ ez = ∇ × u) is obtainable from time integration of the vorticity transport equation { ∂t ω + (u · ∇)ω = ν∇2 ω in D × (0, τ ], (1) ω(x, t = 0) = ω0 (x) for x ∈ D, while the velocity vector u satisfies the following Poisson’s problem with Dirichlet boundary conditions: { 2 ˆz × ∇ω ∇ u=e in D, (2) u(ΓD ) = uΓD . As is well-known, irrespective to the history of the flow, and for any arbitrary distribution ω ∈ H1 (D), a physical (divergence-free) velocity vector is obtainable from solution of Eq. (2), provided that the appropriate boundary conditions are imposed (see Refs. 9, 24 for more discussions). In fact, one of the main advantages of the above vorticity-velocity formulation, in comparison to some primitive variable formulations, is that the time is not appeared explicitly in the elliptic part of equations. Therefore, Eq. (2) can be manipulated (almost) independent of Eq. (1). In this way, our suggestion is construction of a solenoidal velocity field, satisfying approximately the immersed boundary conditions, and then imposing it into the solution as the initial condition. For the periodic solutions, the Fourier pseudo-spectral solvers are advantageous in many circumstances. In the Fourier space the vorticity transport equation (1) recasts   ˆ = − ν|k|2 ω ˆ − (u\ · ∇)ω ,  dt ω | {z } | {z } (3) ˆ ˆ L N   ω ˆ (k, t = 0) = ω ˆ0,


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Fig. 1. The flow domain Ωf and fixed or moving obstacle Ωs are embedded in the regular solution domain D with its boundary ΓD . The immersed velocity boundary conditions are given on Γs , and periodic boundary cnditions are assumed on ΓD .

and the Poisson’s problem (2) simplifies to ˆ = −i u

k⊥ ω ˆ, |k|2

(4)

ˆ stands for the quantities in the Fourier space, k = (k1 , k2 ) is the wave where (·) number vector with the magnitude |k|2 = k12 + k22 ; while k⊥ = (−k2 , k1 ) is the ˆ and the perpendicular wave number vector, and i2 = −1. The diffusion term L ˆ are named for the future references. In the finite dimensional non-linear term N ˆ is constructed (and de-aliased) in the physical calculations, the nonlinear term N space— the algorithm known as the pseudo-spectral solution. More or less similar formulations have been used widely in numerous efficient and accurate direct solvers of the periodic flow in regular domains, which their computational efforts are scaled by (N log N ). In the sequel, we will modify equations (3) and (4), and suggest an algorithm for imposing the desired immersed velocity boundary conditions into the solution. 2.3. Imposing the immersed boundary conditions via the initial condition The above vorticity-velocity formulation gives an opportunity in imposing the immersed boundaries without explicit addition of a forcing function in the right hand side of Eq. (3). In this method the immersed surfaces are introduced via direct ˆ and L. ˆ The key idea is: at the beginning of each time step modifications in N the solenoidal velocity vector, satisfying the desired immersed velocity boundary conditions approximately, and its corresponding vorticity are introduced into the


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vorticity evolution equation (3). Then this modified vorticity evolution equation is allowed to evolve in time. As it will be seen, by establishing the solution, the immersed boundaries will be implemented appropriately. The procedure of finding the solenoidal velocities will be discussed in the next section. Given these velocities and the corresponding vorticity, the modified vorticity evolution equation reads { dt ω ˆ = −ν|k|2 ω ˆ BC − (uSol\ · ∇)ω BC , (5) BC ω ˆ (k, t = 0) = ω ˆ . Time integration of the above equation yields ω ˆ in the next time, and the solution will be continued. 2.4. Solenoidal velocities satisfying immersed boundary conditions For any arbitrary distribution ω ∈ H1 (D) the physical (solenoidal) velocity vector is achievable from solution of Eq. (2), subjected to the periodic boundary conditions (see Refs. 9, 24 for more discussions). In fact, given an ω ∈ H1 (D), solution of the Poisson’s equations ∇2 u = ez × ∇ω,

on

¯ D;

(6)

implies ∇2 (∇ · u) = 0,

on D,

(7)

which means although ∇ · u is not necessarily zero, it is a harmonic function, and therefore, attains its maximum values on the boundary ΓD 21,25,26,? . Now, for the periodic boundary conditions on ΓD (which means a no-boundary problem), achieving a solenoidal velocity vector is anticipated via solution of Eq. (6) in the Fourier space. With regard to the above issues, the suggested procedure can be summarized in the following steps: ¯ is found on i) Given the velocity vector u(Ωf ), the extended velocity uBC (D) BC ¯ ¯ ¯ D via addition of the velocities of D \ Ωf region, such that u (D) satisfies the desired immersed boundary conditions. This procedure is explained in the next section. ii) The extended vorticity ω BC (D) = ∇ × uBC (D) is obtained on D. iii) The final solenoidal velocity vector uSol (D) is obtained from solution of the Poisson’s equations ˆz × ∇ω BC ∇2 uSol = e

x∈D

(8)

subjected to the periodic boundary conditions. iv) If needed, the solenoidal velocity vector in the fluid region ( i.e. uSol (Ωf )) ¯ \ Ωf region. In our may be obtained by discarding the velocities on the D pseudo-spectral solution merely uSol (D) is needed in the next steps.


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Fig. 2. Finding the solenoidal velocities ( i.e. the SOL procedure). The initial velocity u(Ωf ) is extended and modified to satisfy the desired immersed boundary conditions, and to be defined on D. The extended vorticity ω(D) is obtained, and then the solenoidal velocity vector uSol (D) is calculated by solution of the Poisson’s equations. The solenoidal fluid velocities uSol (Ωf ) can be extracted from uSol (D), if it is needed.

The above procedure is illustrated in Fig. 2, and will be referred as the SOL algorithm in the next sections. The most time consuming step in the above algorithm is solution of the Poisson’s problems solving with the fast Poisson solvers. Therefore, the computational costs of the algorithm, as a whole, is scaled by O(N log N ). 2.5. High-order smooth interpolations In the majority of the immersed boundary methods, including the present method, evaluation of the immersed boundary conditions on the regular grid requires interpolations on the scattered data 7,28 . With this regard, a wide variety of interpolation methods is suggested in the literature. Among these methods one can refer to the polynomial interpolations 29 , using the radial basis functions 30,31 , the Wiener interpolation 32,33 , and the Shepard distance-weighted interpolation 20,34 . In the present work, since high global convergence rates are desired, the modified Shepard method 19 is used, which results in locally C 2 fields. In this method, as a weighted least square method, the objective function M ∑ i=1,i̸=k

2

Φik [Qk (x(i) ) − fi ] ,

(9)


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Fig. 3. The modified Shepard’s interpolation is used in evaluation of the immersed boundary conditions on the regular grid points. The physical boundary Γs is divided into ∆s ≈ ∆x; the coefficients of the interpolation polynomials are found using the velocity vector in ∆1 ∈ (Γs ∪ Ωf ) and ∆3 ∈ Ωs ; and finally the velocities in ∆2 are calculated using these polynomials.

is minimized for [ (Rw − dik )+ ]2 , Rw dik

Φik (x, y) =

(10)

where M is the number of involved points. In this way, the coefficients of the cubic function Qk (x, y) are found such that Qk (x, y) = fk + a1k (x − xk )3 + a2k (x − xk )2 (y − yk ) + a3k (x − xk )(y − yk )2 +a4k (y − yk )3 + a5k (x − xk )2 + a6k (x − xk )(y − yk ) +a7k (y − yk )2 + a8k (x − xk ) + a9k (y − yk ).

(11)

Given the interpolation coefficients, the desired interpolations are performed using the weighted interpolant ∑M Wk (x, y)Qk (x, y) F (x, y) = k=1 , (12) ∑M k=1 Wk (x, y) in which Wk (x, y) = where

{ (Rw − dk )+ =

[ (Rw − dk )+ ]3 , Rw dk

Rw − dk if dk < Rw , 0 if dk ⩾ Rw .

(13)

(14)

The computational costs of the method depends on the distribution of the data points. The maximum performance, in which the computational costs scales by


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Fig. 4. The main steps of the proposed algorithm. The SOL box is illustrated in Fig. 2, and the time integration box contains a classical Fourier pseudo-spectral solver (i.e. calculation of the right hand side of Eq. (3), solution of Eq. (4), de-aliasing and so forth).

O(M ), obtains on the structured grids, while for a general scattered data O(M 2 ) operations are needed at worst. In the present work, the standard ACM790 Fortran code is used 19 . According to the aforementioned SOL procedure, given the fluid velocity vector u(Ωf ) and the solid velocity vector u(Ωs ∪ Γs ) at the time instant t0 , it is aimed to find uBC such that the desired immersed boundary conditions be satisfied, and uBC be as smooth as possible. In this context, according to Fig. 3, three regions are defined in the vicinity of the physical boundary Γs : a fluid region ∆1 ∈ Ωf ∪ Γs , a solid region ∆3 ∈ Ωs , and an overlap region ∆2 . The velocities in ∆1 and ∆3 are given, and the velocities in ∆2 are desired. At first, the coefficients of the cubic functions Qk (x, y) are found using the data of ∆1 and ∆3 , and then the un-known values in ∆2 are found using the interpolation function F (x, y). 2.6. Summary of the solution procedure The main steps of the method and their computational costs are summarized in this section. The procedure is also illustrated in Fig. 4, in which the SOL block was explained previously in §2.4 and Fig. 2.


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1. Given the vorticity field ω ˆ n from the initial condition or the last time step, ¯ is obtained from Eq. (4) and two inverse FFTs the velocity vector u(D) with the cost of O(N log N ). 2. The velocity vector u is modified to satisfy the immersed boundary conditions, using the Shepard’s interpolations. This conditioned velocity is called uBC and is not necessarily solenoidal. The computational cost of this step is O(M 2 ) at worst, in which M ≪ N is the number of points in ∆1 ∪ ∆3 and therefore is negligible in comparison with the other costs. 3. The uBC is fed into the SOL algorithm, which results in the condition vorticity ω BC and solenoidal velocity vector uSol . The computational cost of this step is O(N log N ) as explained in § 2.4. 4. The conditioned vorticity ω BC , along with the solenoidal velocity uSol are fed into a classical pseudo-spectral solver, where Eq. (3) is time integrated and ω ˆ n+1 is obtained. This step is a classical pseudo-spectral solution with the computational cost of O(N log N ). An attractive feature of the above procedure is that it can be added easily into a classical pseudo-spectral solver without any change in its internal structure. In fact, the time integration box contains all steps of a classical pseudo-spectral solution (as it was formulated in §2.2). In our numerical experiments that are presented in this paper, the fourth-order Runge–Kutta method is used for time integration, and the boundary conditions are set at the beginning of each sub-step.

3. Numerical experiments To assess the capability of the method in facing with irregular immersed boundaries, a couple of moving boundary problems are considered in this section. Since the uniform grids are employed, the symbol G(N ) is used in referring to an N -point grid for the sake of simplicity.

3.1. Couette flow between concentric cylinders As our first test case, the steady Couette flow between two concenteric cylinders is considered. Presence of curved rotating solid surfaces and availability of an exact solution, make this problem suitable for evaluation of the accuracy of the immersed boundary methods 6,22,35 . The problem setup is illustrated in Fig. 5. We consider two concentric cylinders with radii R1 and R2 and the corresponding angular velocities ω1 and ω2 . The outer cylinder is assumed to be stationary (ω2 = 0), while the inner one is rotating. Since a steady laminal flow is desired, the Taylor number is set as Ta =

(R + R ) ω2 1 2 (R2 − R1 )3 21 = 12.59 < Tac ≈ 1700 2 ν

(15)


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Fig. 5. The problem setup for the Couette flow between two concentric cylinders. The fluid ˜ f are used for calculation of different convergence domain is Ωf and different fluid sub-domains Ω rates (see Fig. 8).

The detail of geometric and physical parameters is given in Tab. 1. The solutions were begun from zero velocities and were proceeded until acheiving the steady flow within the machine accuracy. There is an analytical solution ua = ua (r) = (uar , uaθ ) for this steady flow, which is used in estimation of the numerical errors uar (r) = 0; uaθ (r) =

(

)

ω1 − ω2 R12 R22 1 ω2 R22 − ω1 R12 r + . R22 − R12 R22 − R12 r

(16) (17)

Moreover, taking curl of the above velocity vector yields an essentially constant vorticity distribution in the annular slot ω a = ∇ × ua =

1 ∂(ruθ ) ∂ur ω2 R22 − ω1 R12 . − =2 r ∂r ∂θ R22 − R12

(18)

In order to find the rate of convergence, the problem was solved on the D = {2π 2 } square domain with different grid resolutions. A constant time step ∆t = 5×10−5 sec was chosen which yielded a stable solution on the finest grid. Note that in contrast to the analytical solution (which is steady and thus, is independent of the viscosity ν), our numerical solution is unsteady. Therefore, the stability of the solution depends on ν. (The same issue was reported by other authors 22 .) Consequently, the viscosity is set to meet the stability criterion on the finest grid (see Tab. 1). The regions ∆i , where i = 1, 2, 3 are chosen to have at least two grid points in each direction, in each region. As an example, the details of ∆i are given in Tab. 2 for G(128). In Fig. 6 the horizontal velocity profile at x = π plane for a rather coarse grid G(32) is compared with the analytical solution. A very good agreement is observable with the analytical solution. Achieving this accuracy on a


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R1 (m) 0.3π

R2 (m) 0.9π

ν(m2 /sec.) 2

∆t(sec.) 5 × 10−5

ω1 (rad/sec.) 2

11

ω2 (rad/sec.) 0

Fig. 6. Comparison of the horizontal velocity profiles of the Couette flow problem on x = π plane. The solution on G(32) is compared with the analytical solution and the staircase solution. Accuracy of the solution on this coarse grid is interesting.

fairly coarse grid is interesting. In order to stress the rule of accuracy of evaluation of the immersed boundary conditions on the global accuracy, the velocity profile of a staircase solution is added in the figure as well. A significant difference is observable between the staircase solution and the analytical solution, especially close to the moving boundary. The vorticity profiles at x = π plane are given in Fig. 7, for different grid resolutions. As one can see, there is not a systematic error even on the coarsest grid; and the numerical results is close to the exact solution (i.e, the constant value ω a = 0.5 according to Eq. (18)), especially far from the immersed boundaries. Moreover, by increasing the grid resolution the results converge to the analytical solution very fast.

G(128)

Inner cylinder Outer cylinder

∆1

∆2

∆3

0.1R1 0.04R2

0.16R1 0.0125R2

0.1R1 0.0125R2


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Fig. 7. Vorticity profiles at x = π plane on different grid resolutions compared with the analytical solution (i.e a constant value ω a = 0.5 according to Eq. (18)). The vorticity converges to the analytical solution by increasing the grid resolution.

In order to evaluate the rates of convergence, the velocity errors are calculated between two cylinders εG (Ωf ) =∥ uG (Ωf ) − ua (Ωf ) ∥2 .

(19)

The convergence rates are given in Fig. 8. In this figure, to observe the effects of the solid walls, the rates of convergence are found in different distances away from the ˜ f inside the flow boundaries. With this regard, we define different sub-domains Ω domain Ωf (see Fig. 5) and their fraction of grid points as Fr =

˜f) Np (Ω , Np (Ωf )

(20)

in which Np (·) stands for the number of regular grid points in the region. Now the convergence rates are found for different fraction numbers (see Fig. 8). As one can see, the minimum rate O(2.15) is obtained for the whole flow domain (i.e., Fr = 1); and it is increased by moving away from the solid walls. The convergence rates are calculated up to Fr = 0.3, resulted in the maximum rate of convergence O(2.85). In order to observe the effects of the interpolations accuracy, the results of staircase solution is shown in Fig. 8 as well. As one can see, the staircase solution showed an O(1.4) convergence rate in average, which is substantially lower than the present method. 3.2. Impulsively started circular cylinder The impulsively started circular cylinder problem has been studied several times experimentally as well as numerically 36,37,38,40,41 . These studies showed a steady


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Fig. 8. Velocity convergence rates for the Couette flow problem. The rates (obtained for different fraction numbers Fr ), are compared with the staircase solution. The lowest convergence rate of the present method is higher than the staircase solution.

wake behind the cylinder for the low Reynolds numbers (say Re< 60), while for the higher Reynolds numbers the wake was revealed to be unsteady 37,38 . In the present work, to cover both steady and unsteady wakes two better documented Reynolds numbers (i.e. Re= 20, 550) are considered. 3.2.1. Flow at Re = 20 Consider a circular cylinder of diameter d moving with velocity u = (U0 , 0) in an incompressible Newtonian quiescent fluid with kinematic viscosity ν, which is confined in a square box l1 × l2 = 2π × 2π. The Reynolds number is set as Re = U0 d/ν = 20, and the ratio λ = d/l1 = 0.07 is chosen for the sake of comparisons 37 . The detail of the physical and geometric parameters are given in Tab. 3. Our numerical experiments have shown that the grid convergence (based on the changes in the wake length and the horizontal velocity profiles in the symmetric axis behind the cylinder) occurs with G(256), in which 15 regular grid points are across the cylinder diameter in average. On this grid the fourth-order Runge–Kutta method was used with a constant time step ∆t = 10−4 sec. Time history of the horizontal velocity profile on the symmetry axis behind the cylinder is compared with the experimental data in Fig. 9, in which the non-

Re 20 550

ν(m2 /sec.) 10−3 3.63 × 10−5

d(m) 0.4 0.4

U0 (m/sec.) 0.05 0.05

G 256 512

λ 0.07 0.03


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dimensional time is defined as t∗ = Ud0 t . Good agreements can be seen between the numerical results and the experimental data. Furthermore, a stable wake was observed with length L/d = 0.75, which shows about 2.66% difference with the experimental data 37 (see Fig. 9 for definition of the wake length L). It is worth mentioning that after stabilizing the closed wake region the velocity field is still changing, both within and outside the standing eddies. It is also in agreement with the experimental observations 37 . The spatial rate of convergence of the velocity field is calculated at t∗ = 0.5 as an example. The solution on G(1024) is chosen as the reference solution, and the errors are found in Ωf according to Eq. (19). The results are presented in Fig. 10, which shows a rate O(2.5) in average.

Fig. 9. Simulation of the impulsively started circular cylinder at Re = 20. Left: Time evolution of the horizontal velocity profile at the symmetry axis compared with the experimental data 37 . Lines show the results of the present method, while symbols show the experimental data. Right: A steady wake is developed with length L = 0.75d.

3.2.2. Flow at Re = 550 The experimental observations and the numerical simulations have revealed substantial differences between the wake structures in the low and moderate Reynolds numbers 38 . In this section Re = 550 is considered as a moderate Reynolds number. The flow in this regim is characterized by formaton of an unsteady wake, which is two-dimensional at least in the early stages. Near the cylinder wall the so-called bulge phenomenon is observable which follows by a secondary vortex in the next times 38,39,41 . The problem is solved using the proposed method on G(512), and the ratio λ = 0.03. The detail of the numerical and physical parameters of the solution is given in Tab. 3.


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Fig. 10. The spacial rate of convergence of impulsively started circular cylinder at Re = 20. An average rate O(2.5) is observable.

Fig. 11. Time evolution of the horizontal velocity profile of impulsively started circular cylinder at Re = 550 compared with the experimental data 38 . Lines show the results of the present method, while symbols show the experimental data citeCoutanceau3.

Time evolution of the horizontal velocity profiles on the symmetry axis behind the cylinder are compared with the experimental data in Fig. 11. The figure shows a very good agreement between the results. Moreover, note that for t∗ > 2 velocities more than U0 are observable in places in the wake region. This is the main reason of causing bulge in the streamlines near the cylinder wall, and formation of the secondary vortex in the next times (see Ref. 38 for a detailed discussion). To show that


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Fig. 12. The impulsively started circular cylinder problem at Re = 550. Left: Time evolution of the maximum non-dimensional horizontal velocity at the symmetry axis. The horizontal velocity exceeds U0 after t∗ > 2. Right: Time evolution of the wake length. Displacement of the cylinder center is shown by the dashed line and the wake length (the solid line) is compared with the experimental data 38 . The growth rate of the wake length is less than the cylinder center displacement.

the simulation was able to capture the phenomenon correctly, time history of the maximum non-dimensional velocity −umax /U0 is compared with the experimental data in the left panel of Fig. 12. As one can see, the maximum velocity is increased in time monotonically, and is exceeds U0 after t∗ > 2. In the right panel of Fig. 12, time history of the non-dimensional wake length L/d is compared with the experimental data 38 . Again a good agreement is observable. Furthermore, the nearly linear growth in the wake length is noticeable, which means an approximately constant growth rate in the wake length. However, this growth rate has been less than the velocity of the cylinder, which shows by the dashed line in the figure. Particularly this means that the cylinder motion was followed by the fluid particles behind the cylinder. In order to provide a better insight of the flow structure, the streamlines of the numerical simulations on different grids are compared with the experimental realization at t∗ = 2.5 (see Fig. 13). As one can see, the main structure of the secondary vortex is captured correctly in a position between the separation and stagnation points behind the cylinder, even for a fairly coarse grid G(256). The spatial rate of convergence at t∗ = 1 is presented in Fig. 14. The errors are calculated using relation (19). Obviously the convergence rate O(2.3) is achieved in average. 3.3. Flow arround an insect-like flapping wing In order to show the ability of the method in handling fairly complex geometries and body motions, a combination of translating and rotating motion of an insectlike flapping wing is studied here. Understanding the unsteady characteristics of a


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Fig. 13. Comparison of the flow patterns for Re = 550 at t∗ = 2.5. Up: experimental observation 38 . Down: The results of the present simulations; left to right, G(256), G(512) and G(1024). Even for a rather coarse grid G(256) the structure of the flow is captured correctly.

Fig. 14. Velocity convergence rate for simulation of the Re = 550 regime at t∗ = 1. The errors are decreased by an average rate of O(2.3).

flapping wing at high frequencies is one of the attractive problems in the field of aerodynamics of natural and man-made flyers 42 . The problem has been studied by many authors experimentally, analytically and numerically 43,44,45 . Here we study a simplified model for the kinematics of an insect-like flapping wing proposed by Wang 46 . In this model the wing motion comprises two down-


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β π/6

A0 (m) 1.25

c(m) 0.5

e 0.25

T (sec.) 0.0125

α0 π/4

Re 157

stroke and up-stroke translational phases, in addition to two rotational phases which change the angle of attack during the translational motions. Therefore, the flow is under the influence of the stroke amplitude, the Reynolds number and the rotational and translational speeds, in addition to the initial angle of attack 46 . In the present work, the better documented regime Re = 157 is considered 47 , where the Reynolds number will be defined later. The geometric parameters are illustrated in Fig. 15. The regular domain l1 × l2 = 2π × 2π is chosen, and the results are presented on G(512). The fourth-order Runge–Kutta method is used in time integration with a constant time step ∆t = 10−5 sec., which resulted in a stable solution. In this model an ellipse is chosen as the wing with chord c and thickness ratio e. The wing center position A(t) changes in the β direction as A(t) =

A0 2πt [cos( ) + 1], 2 T

(21)

while the angle of attack α(t) changes as α(t) = α0 [1 − sin(

2πt )]. T

(22)

In these relations T is the flappling period, and α0 and A0 are the initial angle of attack, and the stroke amplitude respectively. Using the above definitions, the 0 Reynolds number is defined as Re = πcA νT . The detail of the physical and geometric parameters are given in Tab. 4.

Fig. 15.

The geometric parameters of the insect-like flapping wing problem.


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Fig. 16. Vorticity snapshots of the insect-like flapping wing in the Re = 157 regime at different time instances. Left: the results of the present method. Right: the results of Xu et al. 47 .

In Fig. 16 vorticity snapshots of one flapping period are compared with the results of the second-order immersed boundary solution of Xu et al. 47 . Obviousely, there is a good agreement between the captured vortical structures. During the down-stroke phase, a pair of counter-rotating vortices are generated and grow (see parts (a) and (b) of Fig. 16). In the (d) part of Fig. 16, the wing in the middle of the upstroke phase is shown. As one can see, the wing vortices separate from the wing such that the wing is no longer in the influence of these vortices at the end of this phase. 4. conclusion A high-order vorticity-based immersed boundary method is proposed and used in solution of the two-dimensional incompressible NSE with moving boundaries. The NSE in the vorticity-velocity form are descretized using the Fourier pseudo-spectral


20

”SIMULATION OF


method and the no-slip conditions are imposed by direct modification of the diffusion and convection terms of the vorticity evolution equation at the beginning of each sub-step. The immersed boundary conditions are evaluated on the regular grid points using the modified Shepard’s distance-weighted interpolation, which results in locally C 2 velocity fields. The explicit fourth-order Runge–Kutta method is employed in time integration of the vorticity evolution equation, and the no-slip conditions are set once at the beginning of each time step. Since the pseudo-spectral method is used, the total computational costs is scaled by N log N . The accuracy and efficiency of the method is demonstrated through solution of two-dimensional rotating and translating moving boundary problems. Depending on the smoothness of the original problem, different rates of convergence were obtained as O(hm ), where 2 < m ≤ 2.85. In order to emphasize the influence of the RBF interpolations the results were compared with the staircase solutions and definite effectiveness of smooth imposition of the immersed boundary conditions on the accuracy of the solutions, especially near the solid moving boundaries, were observed. Since the velocity–vorticity formulation is employed, the method can be extended easily to the three-dimensional flows. Addition of a mean velocity for simulation of the inflow/outflow problems, and coupling with an elastic structure solver can be seen as the other extension lines of the present method. References 1. B. E. Griffith and C. S. Peskin, J. Comput. Phys. 208, 75 (2005) . 2. X. Zeng and C. Farhat, J. Comput. Phys. 233, 2892 (2012). 3. J. Fang, M. Diebold, C. Higgins and M. B. Parlange, J. Comput. Phys. 230, 8179 (2011). 4. C. S. Peskin, Acta Num. 11, 479 (2002). 5. E. A. Fadlun, R. Verzicco, Y. P. Orlandi and J. Mohd-Yusof, J. Comput. Phys. 161, 35 (2000). 6. K. Taira T. Colonius, J. Comput. Phys. 225, 2118 (2007). 7. M. Vanella, P. Rabenold and E. Balaras, J. Comput. Phys. 229, 6427 (2007). 8. B. Sj¨ ogreen and N. A. Petersson, Commun. Comput. Phys.. 2, 1199 (2007). 9. F. Sabetghadam, M. Badri, S. Sharafatmandjoor, and H. Kor, under review (Comput. Fluid), also available from arXiv (arXiv:1110.5984). 10. D. Russell, Z. J. Wang, J. Comput. Phys. 191, 177 (2003). 11. F. Sabetghadam, S. Sharafatmandjoor and F. Norouzi, J. Comput. Phys. 228, 55 (2009). 12. L. Lee and R.J. Leveque, SIAM J. sci. comput. 25, 832 (2003). 13. A. Liang, X. Jing and X. Sun, J. Comput. Phys. 227, 8341 (2008). 14. J. P. Boyd, J. Comput. Phys. 178, 118 (2002). 15. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM, 1977), p. 35. 16. E. Tadmor, Act. Numer. 16, 305 (2007). 17. G. H. Keetels, U. Dortona, W. Kramer, H. J. H. Clercx, K. Schneider and G. J. F. Van Heijst, J. Comput. Phys. 227, 919 (2007). 18. Q. Wang, P. Moin and G. Iaccarino, J. Comput. Phys. 229, 6343 (2009).


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