Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface Yiu-Chung Hon a,n, Božidar Šarler b, Dong-fang Yun a a b
Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China Laboratory for Multiphase Process, University of Nova Gorica, Nova Gorica, Slovenia
art ic l e i nf o
a b s t r a c t
Article history: Received 30 April 2014 Received in revised form 18 October 2014 Accepted 6 November 2014
This paper explores the application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) for the solution of coupled heat transfer and fluid flow problems with a free surface. The method employs the representation of temperature, velocity and pressure fields on overlapping fivenoded sub-domains through collocation by using Radial Basis Functions (RBFs). This simple representation is then used to compute the first and second derivatives of the fields from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time integration scheme. For numerical efficiency, the Artificial Compressibility Method (ACM) with Characteristic Based Split (CBS) technique is firstly adopted to solve the pressure–velocity coupled equations. The performance of the method is assessed based on solving the classical two-dimensional De Vahl Davis steady natural convection benchmark problem with an upper free surface for Rayleigh number ranged from 103 to 105 and Prandtl number equals to 0.71. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Free surface flow natural convection Buoyancy driven cavity Meshfree and meshless methods Local radial basis function collocation method Artificial compressibility method
1. Introduction The development of approximate methods for the numerical solutions of multiphasic problems has attracted the attention of engineers, physicists and mathematicians in the last several decades, the finite element method (FEM) has been well established for the modelling of complex problems in applied mechanics and related fields. It is well-known that the FEM relies on mesh discretization, which leads to complications for certain classes of realistic problems with complex three dimensional geometrical shape. The loss of accuracy is observed when the elements in the mesh become extremely skewed or distorted. The meshing (polygonization) process is often the most time consuming part and this process usually cannot be automated realized. The boundary element method (BEM) has become an efficient and popular alternative to the FEM, especially for stress concentration problems, or for boundary value problems wherein a part of the boundary extends to infinity. In spite of the great success of the finite and boundary element methods as effective numerical tools for the solution of boundary value problems in complex domains, there has been a growing interest in the so-called meshless methods over the past decade
n
Corresponding author. E-mail addresses:
[email protected] (Y.-C. Hon),
[email protected] (B. Šarler),
[email protected] (D.-f. Yun).
[1–6]. The solution is represented on an arbitrary distributed set of nodes which have no topological relations among them. This makes meshfree methods easy to be implemented in high dimensional space. A number of meshless methods have been proposed such as the smooth particle hydrodynamics (SPH) method, the diffuse element method (DEM), the element free Galerkin (EFG) method, the reproducing kernel particle method (RKPM), the moving leastsquares reproducing kernel (MLSRK) method, the partition of unity finite element method (PUFEM) the finite point method, boundary node method (BNM), and the local boundary integral equation (LBIE) method, see [7–10] for reference. The term meshless or meshfree stems from the ability of an approximation or interpolation scheme to be constructed entirely from a set of nodes [11]. During the solving process by meshless method, there is no need to create meshes neither in the domain nor on the boundary. There are basically two types of meshless methods: the weakform-based approach and the collocation-based type. Both collocation and weak-form approaches are effective for discretizing the partial differential equations. Numerical computations indicate that these meshless methods are ideal for solving complex physical problems on irregular domains. The radial basis function collocation method (RBFCM) [12,13], which is the simplest meshless method that has been vastly studied [14–19]. The computational cost of collocation-based type is generally lower than the weak-form approach since no numerical quadrature is needed. On the other hand, the collocation-based approach requires a higher
http://dx.doi.org/10.1016/j.enganabound.2014.11.006 0955-7997/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Hon Y-C, et al. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Boundary Elem. (2015), http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
Y.-C. Hon et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
smoothness requirement to satisfy the derivative condition in solving problems with Neumann boundary condition. The use of higher order basis functions in the collocation-based approach makes the method more sensitive and unstable, and in the worst case, provides unsatisfactory approximation to the solution [20]. The thermo-driven fluid-flow problem arises as one of the realistic industrial problems. The use of RBFCM suffers from the well-known ill-conditioning problem due to the resultant full matrix. In this paper, we apply the recently developed local radial basis function collocation method (LRBFCM) to simulate the solution for the thermo-driven fluid-flow problem [23–27]. The LRBFCM enjoys also the advantages of RBFCM on truly meshless, highly accurate, and easy implementation of the ill-conditioning problem and hence is ideal for solving large scale problems. Furthermore, the LRBFCM is readily parallelized for solving high dimensional problems. In particular, the well-known sensitivity problem in choosing the shape parameter for the RBFCM no longer exists. This local variant has been applied successfully to solve some problems [15,16,19,21,22]. In the present paper, the LRBFCM will be used to obtain numerical solution of coupled mass, momentum and energy equations. In compared with the classical benchmark problems [28–30], we pose a Neumann condition on the upper fixed boundary for velocity, which imposes an extra difficulty in seeking numerical approximation. For stability, the Artificial Compressibility Method (ACM) [31] is combined with the Characteristic Based Split (CBS) [32–35] to deal with the pressure velocity coupling system. The LRBFCM is tested on natural convection problems for Rayleigh numbers ranged within at three different node arrangements, respectively varying from 1681 to 16641 nodes, i.e. 41 41, 81 81, and 101 101. Results are presented in terms of stream functions, isotherm, and mid-plane velocities. This paper is organized as follows: In Section 2, the governing equations for thermo-driven fluid-flow problems are given and the solving procedure is listed. The algorithm of LRBFCM is discussed in Section 3. Numerical benchmark tests to solve the system of equations with prescribed boundary conditions are presented in
Section 4. In the final section, conclusions are made on this paper and future work.
2. Governing equations and solution procedure The transient natural convection problem in a fixed domain Ω is described by three coupled mass, momentum, and energy conservation equations as follows: ∇ v ¼ 0;
ð1Þ
∂ ðρvÞ þ ∇ ðρvvÞ ¼ ∇P þ ∇ ðμ∇vÞ þ f; ∂t
ð2Þ
∂ ðρcp TÞ þ ∇ ðρcp TvÞ ¼ ∇ ðλ∇TÞ; ∂t
ð3Þ
where t; v; P; T; λ; cp ; g; ρ; βB ; T ref ; μ and f represents time, velocity, pressure, temperature, thermal conductivity, specific heat, gravitational acceleration, density, coefficient of thermal expansion, reference temperature for Boussinesq approximation, viscosity and body force, respectively. The Boussinesq approximation for the body force is given by f ¼ ρ½1 βB ðT T ref Þg;
ð4Þ
which is the source of the thermo-driven force for the cavity flow of fluid inside the domain. The problem is to be solved on the fixed domain Ω with boundary Γ which includes a free surface. Dirichlet boundary conditions for temperature on the fixed boundaries and Dirichlet boundary condition for velocity on the free surface are imposed to model this thermo-driven fluid-flow problem. In the present paper, the meshless LRBFCM will be employed to solve the above problem (1)–(3) in which the spatial derivatives will be approximated by the radial basis functions. For the time derivative, we apply the explicit finite difference quotient formulae and the Artificial Compressibility Method (ACM) with Characteristic Based
Calculate new pressure P T v P
Solve momentum equation (5) for intermediate velocity
by solving (7) (8) Correct intermediate velocity
by(6) to get new velocity v at time t + t
v
Solve Energy e quation (9) for new temperature
Check steady state criteria (10)
T
Fig. 1. Schematic diagram on the solution procedure for solving the thermo-driven fluid-flow problem.
Problem domain
Collocation
Boundary
Corner
Subdomain Interior
Fig. 2. Local collocation scheme (left) and collocation sub-domain selection strategy (right).
Please cite this article as: Hon Y-C, et al. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Boundary Elem. (2015), http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
Y.-C. Hon et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Split (CBS) technique to iteratively solve the pressure velocity coupling equation (2). The solution procedure is briefly outlined as follows: In the first time-step, the intermediate velocity is obtained from solving the discretized transient form of Eq. (2) without the pressure term: Δt v ¼ v0 þ ∇ μ∇v0 þ f 0 ∇ ρv0 v0 ; ρ n
ð5Þ
where Δt stands for the length of the time-step; vn denotes the intermediate velocity at time t 0 þ Δt; v0 and f 0 denote the velocity and force at initial time t0. We note here that the calculated velocity vn is merely the intermediate velocity which does not in general satisfy the mass continuity equation (1). In the second time-step, the velocity at next time-step t 0 þ Δt are corrected from the computed intermediate velocity by using the pressure gradient: v1 ¼ v n
Δt ∇P ρ 1
In the third time-step, the discretized form of transient energy equation (3) is solved by T1 ¼ T0 þ
Δt ∇ λ∇T 0 ∇ ρcp T 0 v0 ; ρcp
ð9Þ
where T0 and T1 denote respectively the temperatures at time t0 and t 0 þ Δt. After these three steps, the updated velocity v1 , pressure P1 and temperature T1 at time t 0 þ Δt are obtained. The iteration is continued until the following steady-state conditions are achieved: jT 1 Tjo ϵT ;
jv1 vj o ϵV ;
jP 1 Pjo ϵP ;
ð10Þ
where ϵT , ϵV and ϵP are constants prescribing error tolerances for respectively the temperature, velocity and pressure. The solution procedure is described by the schematic diagram given in Fig. 1.
ð6Þ
where the pressure field P1 is obtained from the initial pressure field P0 by ð7Þ P 1 ¼ P 0 βΔt ρ∇ vn Δt∇2 P 0 satisfying the Neumann boundary condition ∂P 1 ¼0 ∂n
3
on Γ ;
ð8Þ
where n stands for the outer unit normal vector. We remark here that Δt∇2 P 0 in (7) serves as a stabilization term and the constant β is called the compressibility coefficient. The selection of optimal compressibility coefficient is problem dependent.
3. Local radial basis function collocation method for the thermo-driven fluid-flow problem In the Local Radial Basis Function Collocation Method (LRBFCM), the pressure, velocity and temperature fields are interpolated on a set of neighbouring nodes that can be uniformly or non-uniformly distributed in the domain and on the boundary. Suppose this set consists of N nodes, including domain and boundary nodes. Suppose also the domain is divided into Nlocal overlapping subdomains, each of which consists of l N (in general) equally or non-equally distributed nodes, denoted as l P n , where l ¼ 1; 2; …; Nlocal and n ¼ 1; 2; …; l N is the number of nodes in the lth subdomain. In each subdomain, define a function θ as lN
T y
0,
u 0, v y
θðpÞ ∑ l αk l Λk ðpÞ
0
ð11Þ
k¼1
T =T u = 0, v 0
T =T u = 0, v 0
where p represents a position vector in the lth subdomain, l Λk are the collection of RBFs centred at points l pk , and fl αk g denote the expansion coefficients. The LRBFCM in the present paper is implemented with Hardy's scaled multiquadric (MQ), which is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 ðpÞ l k ; l r 2k ðpÞ ¼ ðp l pk Þ ðp l pk Þ; ð12Þ l Λk ðpÞ ¼ r 2 þ c2 l 0 where the constant c called shape parameter. The scaling parameter l r 20 is set to the maximum nodal distance in the lth subdomain. In order to obtain coefficients fl αk g, we only need the collocation lN
θðl pn Þ ¼ l θn ¼ ∑ l αk l Λk ðl pn Þ; k¼1
T y
0, u
0, v
n ¼ 1; 2; …; l N ;
ð13Þ
which gives the following linear system of l N equations from Eq. (11)
0
Fig. 3. Natural convection with an upper free surface on a square cavity.
l
Λlα ¼ lθ;
ð14Þ
Table 1 Scales of dimensionless, time steps Δt, number of iterations M, compressibility coefficient β and criteria ϵabs for different Ra numbers and grids size. Ra
3
10 104 105
ϵabs
5
10 10 5 10 5
β
10 10 10
41 41
81 81
101 101
Scale
Δt
M
Scale
Δt
M
Scale
Δt
M
10 10 20
0.001 0.001 0.001
7672 6161 23130
50 50 50
0.001 0.001 0.001
14 455 11 061 13 105
50 50 50
0.001 0.001 0.001
12 567 10 629 13 961
Please cite this article as: Hon Y-C, et al. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Boundary Elem. (2015), http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
Y.-C. Hon et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
0 l
Λ11
l
Λ12 l Λ22
⋯
⋮
⋱
Λl N2
⋯
l
B B l Λ21 B B ⋮ @
Λl N1
l
1
Λ1l N 0 l α1 1 CB α C C l Λ2l N CB l 2 C l
⋯
⋮
1
Λl N l N
0
θ 1 B θ C Bl 2 C l 1
ð15Þ
C¼B C; CB A@ ⋮ A @ ⋮ A l
αl N
l
lN lN lN ∂2 ∂2 θðpÞ ∑ 2 l Λk ðpÞ ∑ l Λij 1 l θj ¼ ∑ 2 ∂y j¼1 j¼1 k ¼ 1 ∂y
0.9 0.8 0.7
2
0.6
y
ð17Þ
where pσ ¼ x; y stands for Cartesian coordinates. All necessary derivatives to construct the involved divergence, gradient and Laplace operators can be calculated through Eqs. (16) and (17). Furthermore, the integral of function θ over pσ (used in the present context in the calculation of the stream function) can be evaluated as well lN
θðpÞ dpσ ¼ ∑ l αk k¼1
Z l
Λk ðpÞ dpσ :
0.3 0.2 0.1
ð18Þ
lN ∂ ∂ θ ð p Þ ¼ ∑ l αk l Λk ðl pi Þ; ∂pσ l i ∂p σ k¼1
0
α ¼ lΛ
l
lN
k¼1
j¼1
j¼1
lN
1
∑ l Λk ðpÞl Λij
k¼1
θ
l j:
lN
∂ 1 l Λk ðpÞl Λij k ¼ 1 ∂y ∑
!
0.8
1
0.8
1
Ra=104, 101x101
0.5 0.4 0.3 0.2 0.1 0
0
0.2
0.4
0.6
x Ra=105, 101x101
1 0.9 0.8
The first partial spatial derivatives of θðpÞ can be expressed as ! lN lN lN lN ∂ ∂ ∂ 1 1 θðpÞ ∑ Λ ðpÞ ∑ Λ θ ¼ ∑ ∑ Λ ðpÞ Λ l ij l j l k l k l ij l θj ; ∂x j¼1 j ¼ 1 k ¼ 1 ∂x k ¼ 1 ∂x
lN lN lN ∂ ∂ 1 θðpÞ ∑ l Λk ðpÞ ∑ l Λij l θ j ¼ ∑ ∂y j¼1 j¼1 k ¼ 1 ∂y
1
0.6
ð21Þ
0.7 0.6
y
ð22Þ
0.8
0.7
y
lN
0.6
0.8
By taking into account the expressions for the calculation of the coefficients, the collocation representation of θðpÞ can be expressed as ! lN
0.4
0.9
ð20Þ
θðpÞ ∑ l Λk ðpÞ ∑ l Λij 1 l θj ¼ ∑
0.2
1
ð19Þ
θ:
0
x
where the index i stands for the ith node in the subdomain where derivative values are given. The reason that all matrix elements l Λij and coefficients l αk are only need to be calculated once is due to the use of the following technique to deal with the interpolation matrix l Λ ¼ ðl Λij Þl N l N . The coefficients l α can be computed by inverting the system (19) l
0.5 0.4
All matrix elements l Λij and coefficients l αk need to be evaluated only once before time stepping begins. In the present paper, only the simplest five-noded overlapping subdomains are used as shown in Fig. 2. In every five node subdomain, the first and the second spatial derivatives in the central node are approximated. If the derivatives instead of the function values are given to nodes on the boundary where Neumann boundary condition is set, Eq. (13) should be replaced by
1
θ
l j;
Ra=103, 101x101
1
ð16Þ
∂ ∂ θðpÞ ∑ l αk 2 l Λk ðpÞ; ∂pσ ∂p2σ k¼1
Z
!
ð25Þ
lN ∂ ∂ θðpÞ ∑ l αk Λ ðpÞ; ∂pσ ∂pσ l k k¼1
lN
∂2 ∑ Λ ðpÞl Λij 1 2 l k k ¼ 1 ∂y
θl N
where l Λij ¼ l Λj ðpi Þ, i; j ¼ 1; 2; …; l N. The solution of the above linear system provides the collocation coefficients fl αk g. Moreover, spatial derivatives of the function θ can be easily obtained through derivation of Eq. (13) due to the linearity of RBFs expansion:
2
lN
0.5 0.4
l θj ;
ð23Þ and the second partial spatial derivatives of θðpÞ can be expressed as ! lN lN lN lN ∂2 ∂2 ∂2 1 1 θ ðpÞ ∑ Λ ðpÞ ∑ Λ θ ¼ ∑ ∑ Λ ðpÞ Λ l ij l θj ; l ij l j 2 l k 2 l k ∂x2 j¼1 j ¼ 1 k ¼ 1 ∂x k ¼ 1 ∂x ð24Þ
0.3 0.2 0.1 0
0
0.2
0.4
x
0.6
Fig. 4. Isolines of stream function and temperature contours.
Please cite this article as: Hon Y-C, et al. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Boundary Elem. (2015), http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
Y.-C. Hon et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ lN lN ∂2 ∂2 1 θðpÞ ∑ l Λk ðpÞ ∑ l Λij l θ j ∂x∂y j¼1 k ¼ 1 ∂x∂y ! lN lN ∂2 1 ¼ ∑ ∑ l Λk ðpÞl Λij l θj : j ¼ 1 k ¼ 1 ∂x∂y
where ΔT stands for maximum temperature difference and L stands for enclosure length. The LRBFCM described in the last section is performed on 41 41 (with N max ¼ 1677), 81 81 (with N max ¼ 6557) and 101 101 (with Nmax ¼ 10 197) nodes under different Ra number, where Nmax stands for the total number of nodes. Results are presented in Table 1 in terms of compressibility parameter β, time-step Δt and the number of iterations M. We set steady state criteria ϵT ; ϵV ; ϵP to be the same and denoted by ϵabs. All cases are calculated using multiquadric RBF with shape parameter c¼32. We remark here that the choice of optimal shape parameter in using meshless collocation method with global multiquadric radial basis function is well known to be an open problem. Although there is still no theoretical justification, we adapt the experimental sensitivity result in the shape parameter c using LRBFCM for the problem (1)–(4) by Gregor and Šarler in [23] to choose c¼32 in our computation. Moreover, the numerical experiments given in [36] have verified that the choice of the shape parameter c is no longer sensitive due to the use of small number of local nodes in LRBFCM. Results in terms of isolines of stream functions and temperature contours are plotted with solid line (—) representing stream lines and dotted line (⋯) representing isolines of temperature in Fig. 4. The stream function Ψ is calculated through integration of the velocity component
ð26Þ
For each subdomain, from the function values l θ ¼ fl θj g at each five nodes, the spatial derivatives at the central node can be approximated by multiplying vectors lN
∂ 1 l Λk ðpÞl Λij ; ∂x k¼1 ∑
lN
∂ 1 l Λk ðpÞl Λij ; ∂y k¼1 ∑
lN
lN
∂2 Λ ðpÞl Λij 1 ; 2 l k ∂y k¼1 ∑
lN
∂2 Λ ðpÞl Λij 1 ; 2 l k k ¼ 1 ∂y
∂2 1 l Λk ðpÞl Λij : k ¼ 1 ∂x∂y ∑
∑
ð27Þ
Finally, by substituting the function values at these nodes for Eqs. (22)–(26) and using the Neumann condition, the value at these nodes for Eq. (21) can be derived.
4. Numerical example The classical De Vahl Davis [28] natural convection problem is considered for benchmarking purposes. The domain of the problem is a closed air (Prandtl number is 0.71) filled square cavity with differentially heated vertical walls (ΔT ¼ T H T C ) and insulated horizontal walls (Fig. 3). Different from previous related works, we give an upper Neumann boundary condition for velocity ∂u ¼ 0; ∂n
v ¼ 0;
v ¼ ðu; vÞ;
Ψ ðx; yÞ ¼
y y¼ ; L
u¼
uLρcp
λ
;
ð28Þ
v¼
vLρcp
λ
;
Ψ¼
uLρcp
λ
;
τ¼t
λ : ρc p L 2 ð29Þ
where x; y stand for the dimensionless coordinates, u; v stand for the dimensionless horizontal and vertical velocity components respectively, Ψ stands for the dimensionless temperature and τ is denoted as the dimensionless time. Prandtl and Rayleigh numbers are related to other parameters as Pr ¼
Ra ¼
μcp ; λ
ð30Þ
g βB ΔTL3 ρ2 cp
λμ
;
Z ð32Þ
uðx; yÞ dy:
We draw a contour plot for stream functions with 10 contour levels, and a contour plot for temperature with 20 contour levels for Ra ¼ 103 ; 104 ; 105 . Table 2 gives us comparisons on maximum and minimum values of velocities and their locations for different Ra numbers and nodes distributions. Moreover, the values of velocities on mid-plane are also presented in Fig. 5 respectively with values in Table 3 as well. In the above computations, the fixed compressibility coefficient is chosen as β ¼ 10. In fact, it has been proven that the numerical approximation obtained by using the ACM method will converge to the true solution of the incompressible flow problem as 1=β tends to 0, i.e., the value of the compressibility coefficient β in theory should be very large. In practical computation, however, the convergence rates of the numerical solutions are no good when the values of β are too small or too large. Hence, we chose a moderate value of β ¼ 10 as suggested in [32]. In Table 4, we indicate that the numerical solution is insensitive to small change in the value of β in which the table lists the maximum error differences in temperature, u and v between the numerical results obtained under different values of β and the results when β ¼ 10. In this comparison, the computation is implemented when Ra ¼ 103 with 41 41 nodes.
which arises new difficulties for finding steady state solutions. The steady-state is achieved through a time transient from the initial temperature, pressure and velocity. All variables are stated in Cartesian coordinates and standard dimensionless form [30] x x¼ ; L
5
ð31Þ
Table 2 Comparison of maximum and minimum values of velocities for different Ra and nodes distributions. vmax
Ra
umax
y
103
3.961 3.954 3.950
0.200 0.200 0.200
104
16.19 16.16 16.14
0.200 0.1875 0.190
22.25 22.72 22.79
1.000 1.000 1.000
21.44 21.77 21.76
0.1250 0.1125 0.110
105
31.91 33.61 33.88
0.150 0.150 0.150
64.01 61.20 60.71
1.000 1.000 1.000
71.93 71.91 71.59
0.075 0.070 0.060
umin 5.435 5.517 5.564
y 1.000 1.000 1.000
4.399 4.484 4.533
x 0.175 0.175 0.180
vmin
x
Nodes distribution
0.825 0.8125 0.810
41 41 81 81 101 101
20.74 20.72 20.73
0.875 0.875 0.870
41 41 81 81 101 101
63.62 67.99 68.70
0.925 0.925 0.930
41 41 81 81 101 101
4.369 4.326 4.299
Please cite this article as: Hon Y-C, et al. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Boundary Elem. (2015), http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
Y.-C. Hon et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
u(0.5,y)
40
Table 3 Values of velocities on mid-plane for different Ra numbers.
3
Ra=10 , Pr=0.71 4
30
Ra=10 , Pr=0.71 5
Ra=10 , Pr=0.71
20
u(0.5,y)
10 0 −10 −20 −30 −40 −50 −60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y u(x,0.5)
80
Ra=103, Pr=0.71 Ra=104, Pr=0.71
60
Ra=105, Pr=0.71
Ra
103
104
105
x=y
u
v
u
v
u
v
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.000 1.951 3.140 3.756 3.950 3.840 3.510 3.023 2.423 1.740 0.997 0.213 0.598 1.420 2.239 3.035 3.786 4.463 5.024 5.415 5.564
0.000 2.474 3.877 4.470 4.478 4.084 3.428 2.614 1.711 0.768 0.184 1.120 2.015 2.836 3.536 4.051 4.293 4.152 3.495 2.169 0.000
0.000 8.235 13.230 15.632 16.117 15.287 13.607 11.411 8.924 6.284 3.560 0.761 2.139 5.172 8.339 11.590 14.805 17.803 20.340 22.119 22.791
0.000 16.014 21.548 20.702 17.033 12.849 9.192 6.284 3.975 2.009 0.138 1.857 4.168 6.951 10.290 14.083 17.841 20.430 19.946 13.958 0.000
0.000 19.117 30.542 33.879 30.973 24.813 18.049 12.310 8.217 5.706 4.359 3.582 2.648 0.706 3.177 9.840 19.692 32.212 45.511 56.326 60.712
0.000 69.153 61.082 33.599 13.551 3.684 0.054 0.715 0.394 0.302 1.141 2.025 2.773 2.904 1.330 4.023 16.196 37.332 61.850 64.466 0.000
40
Table 4
v(x,0.5)
20
Numerical comparison with respect to compressibility coefficient when Ra ¼ 103 over 41 41 nodes.
0
β
Max err in temperature
Max err in u
Max err in v
5 15 20
2:98E 05 8:40E 07 3:26E 06
3:18E 03 2:43E 04 1:65E 04
1:23E 03 2:39E 04 2:69E 04
−20 −40 −60 −80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x Fig. 5. Mid-plane velocities.
The numerical implementation is done by using Matlab (64 bits). All calculations are done on a laptop computer HP DV43011tx, with dual core i5-2410 and 8 GB of RAM. For the node distribution of 41 41, the CPU time was about 400 s for error bound of magnitude 10 5. For benchmark purpose, we choose only rectangular domain with uniform distributed nodes. The LRBFCM method, however, is in general applicable to irregular domain with non-uniform nodes.
5. Conclusions The LRBFCM was firstly applied to solve the coupled heat transfer and fluid flow benchmark problem with upper free boundary condition for velocity, and combined with the ACM with CBS technique to successfully simulate stable solution. From the plots and values of temperature contours, streamlines and velocities on middle line, it was observed that the numerical isolines of temperature and stream function are not symmetric. This is due to the Neumann boundary condition on the upper free surface and hence is different from the symmetric results given in [23]. Instead of using the pressure correction technique for solving pressure–velocity coupling, the use of ACM with CBS in this paper provides a stable numerical simulation to this thermo-driven fluid-flow problem in square cavity with free surface. Our further
work will be placed on the efficiency and accuracy of this LRBFCM in dealing with higher Rayleigh numbers and non-uniform nodes distribution.
Acknowledgement The work described in this paper was partially supported by a grant from Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101112). References [1] Atluri SN, Shen S. The meshless method. Encino: Tech Science Press; 2002. [2] Atluri SN. The meshless method (MLPG) for domain and BIE discretization. Forsyth: Tech Science Press; 2004. [3] Kansa EJ. Multiquadrics—a scattered data approximation scheme with application to computational fluid dynamics, Part I. Comput Math Appl 1990;19:127–45. [4] Kansa EJ. Multiquadrics—a scattered data approximation scheme with application to computational fluid dynamics, Part II. Comput Math Appl 1990;19:147–61. [5] Liu GR. Mesh free methods. Boca Raton: CRC Press; 2003. [6] Liu GR, Gu YT. An introduction to meshfree methods and their programming. Dordrecht: Springer; 2005. [7] Atluri SN, Shen S. The meshless local Petrov–Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods. Comput Model Eng Sci 2002;3:11–52. [8] Lin H, Atluri SN. Meshless local Petrov Galerkin method (MLPG) for convection–diffusion problems. Comput Model Eng Sci 2001;1:45–60. [9] Šarler B, Kuhn G. Primitive variable dual reciprocity boundary element method solution of incompressible Navier–Stokes equations. Eng Anal Bound Elem 1999;23:443–55. [10] Šarler B. From global to local radial basis function collocation method for transport phenomena. Berlin: Springer; 2007 pp. 257–82.
Please cite this article as: Hon Y-C, et al. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Boundary Elem. (2015), http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
Y.-C. Hon et al. / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [11] Babuška I, Banerjee U, Osborn John E. Survey of meshless and generalized finite element methods: a unified approach. Acta Numer 2003;12:1–125. [12] Chen W. New RBF collocation schemes and kernel RBFs with applications. Lecture notes in computational science and engineering, vol. 26, 2002. p. 75–86. [13] Hon YC, Cheung KF, Mao XZ, Kansa EJ. Multiquadric solution for Shallow Water equations. ASCE J Hydraul Eng 1999;125:524–33. [14] Wong SM, Hon YC, Li TS. A meshless multi-layer model for a coastal system by radial basis functions. Comput Math Appl 2002;43:585–605. [15] Sadat H, Couturier S. Performance and accuracy of a meshless method for laminar natural convection. Numer Heat Transf Part B 2000;37:455–67. [16] Šarler B. A radial basis function collocation approach in computational fluid dynamics. Comput Model Eng Sci 2005;7:185–93. [17] Zhou X, Hon YC, Sun S, Mak AFT. Numerical simulation of the steady state deformation of a smart hydrogel under an external electric field. Smart Mater Struct 2002;11:459–67. [18] Wen PH, Hon YC, Wang W. Dynamic responses of shear flows over a deformable porous surface layer in a cylindrical tube. Appl Math Model 2009;33:423–36. [19] Šarler B, Perko J, Chen CS. Radial basis function collocation method solution of natural convection in porous media. Int J Numer Methods Heat Fluid Flow 2004;14:187–212. [20] Buhmann MD. Radial basis functions. Cambridge: Cambridge University Press; 2000. [21] Divo E, Kassab AJ. An efficient localized RBF meshless method for fluid flow and conjugate heat transfer. ASME J Heat Transf 2007;129:124–36. [22] Šarler B. Towards a mesh-free computation of transport phenomena. Eng Anal Bound Elem 2002;26:731–8. [23] Gregor K, Šarler B. Solution of thermo-fluid problems by collocation with local pressure correction. Int J Numer Methods Heat Fluid Flow 2008;18:868–82. [24] Manzari MT. An explicit finite element algorithm for convection heat transfer problems. Int J Numer Methods Heat Fluid Flow 1999;9:860–77.
7
[25] Šarler B, Vertnik R. Meshfree explicit local radial basis function collocation method for diffusion problems. Comput Math Appl 2006;51:1269–82. [26] Vertnik R, Šarler B. Meshless local radial basis function collocation method for convective–diffusive solid–liquid phase change problems. Int J Numer Methods Heat Fluid Flow 2006;16:617–40. [27] Vertnik R, Založnik M, Šarler B. Solution of transient direct-chill aluminium billet casting problem with simultaneous material and interphase moving boundaries by a meshless method. Eng Anal Bound Elem 2006;30:847–55. [28] De Vahl Davis G. Natural convection of air in a square cavity: a bench mark numerical solution. Int J Numer Methods Fluids 1983;3:249–64. [29] Hortmann M, Perić M, Scheuerer G. Finite volume multigrid prediction of laminar natural convection—bench-mark solutions. Int J Numer Methods Fluids 1990;11:189–207. [30] Wan DC, Patnaik BSV, Wei GW. A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numer Heat Transf Part B 2001;40:199–228. [31] Chorin A. A numerical method for solving incompressible viscous flow problems. J Comput Phys 1967;2:12–26. [32] Mramor K, Vertnik R, Šarler B. Low and intermediate Re solution of lid driven cavity problem by local radial basis function collocation method. Comput Mater Continua 2013;36(1):1–21. [33] Massarotti N, Nithiarasu P, Zienkiewicz OC. Characteristic-based-split (CBS) algorithm for incompressible flow problems with heat transfer. Int J Numer Methods Heat Fluid Flow 1998;8:969–90. [34] Zienkiewicz O, Codina R. A general algorithm for compressible and incompressible flow—Part I. The split, characteristic-based scheme. Int J Numer Methods Fluids 1995;20:869–85. [35] Zienkiewicz O, Morgan K, Codina BVK, Vasquez R. A general algorithm for compressible and incompressible flow—Part II. Tests on the explicit form. Int J Numer Methods Fluids 1995;3:887–913. [36] Li M, Chen W, Chen CS. The localized RBFs collocation methods for solving high dimensional PDEs. Eng Anal Bound Elem 2013;37:1300–4.
Please cite this article as: Hon Y-C, et al. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Boundary Elem. (2015), http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
