Domain-free discretization method for doubly connected ... - CiteSeerX

Comput. Methods Appl. Mech. Engrg. 191 (2002) 1827–1841 www.elsevier.com/locate/cma

Domain-free discretization method for doubly connected domain and its application to simulate natural convection in eccentric annuli C. Shu *, Y.L. Wu Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 23 October 2000; received in revised form 29 March 2001; accepted 5 October 2001

Abstract In this paper, an efficient numerical approach, the domain-free discretization (DFD) method, was presented to solve partial differential equations (PDEs) on a doubly connected domain. For any doubly connected domain, the numerical discretization is always based on the cylindrical coordinate system. No coordinate transformation is involved in the present approach. The mesh points are only distributed along the radial line. So, overall, there is no structure for the mesh point distribution. The proposed method was applied to simulate the natural convection in eccentric annuli. The vorticity-stream function formulation in the cylindrical coordinate system is taken as the governing equation, and the pressure single-value condition is used to update the stream function value on the inner cylinder wall. It was found that the numerical results obtained by the DFD method agree well with available data in the literature. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Domain-free discretization; Interpolation; Extrapolation; Natural convection; Eccentric annuli

1. Introduction It is well known that a partial differential equation (PDE) can be solved by analytical method or numerical method. For the analytical method, the closed form solution is usually obtained by two steps. In the first step, a general solution is pursued which is only based on the given PDE. Then in the second step, the expression of the general solution is substituted into the boundary conditions to determine the unknown coefficients in the general solution. Clearly, the first step does not involve the solution domain. The solution domain (geometry of the problem) is only involved in the second step when the boundary condition is implemented. So, the analytical method can be well applied to both regular and irregular domain problems.

*

Corresponding author. Tel.: +65-874-6476; fax: +65-779-1459. E-mail address: [email protected] (C. Shu).

0045-7825/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 0 1 ) 0 0 3 5 5 - 3

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In contrast, the numerical method solves the PDE by directly coupling it with the boundary condition. In other words, the numerical solution is obtained in just one step. In this step, the PDE is discretized on the solution domain with proper implementation of the boundary condition. We can see clearly that the numerical discretization of the PDE in a numerical method is problem-dependent. Due to this feature, some numerical methods can only be applied to regular domain problems. Examples are the finite difference schemes [1–3] and the global method of differential quadrature (DQ) [4–7]. When these methods are applied to solve irregular domain problems, the coordinate transformation is a must. In general, the process of coordinate transformation is very complicated, and problem-dependent. In addition, it may bring additional errors into the numerical computation. To overcome the drawbacks of conventional numerical methods which strongly couple the PDE with the solution domain, a domain-free discretization (DFD) method was proposed by Shu and Fan [8] to solve PDEs on a singly connected domain. In the work of Shu and Fan [8], for any irregular singly connected domain, the numerical discretization is always discretized in the Cartesian coordinate system without any coordinate transformation. The basic idea of the DFD method is that the discrete form of the given differential equation is irrelevant of solution domain. In other words, the discrete form can involve some points outside the solution domain. Obviously, the key process in the DFD method is how to evaluate the functional values at the points outside the solution domain. In the work of Shu and Fan [8], the functional values at the points outside the solution domain are computed by an approximate form of the solution along a vertical or horizontal line. This paper further extends the work of Shu and Fan [8] to solve a PDE on any doubly connected domain. In this work, the numerical discretization is always made in the cylindrical coordinate system for any doubly connected domain. We should indicate that the essence of the DFD method is kept in this application. In other words, the discrete form of given PDEs is irrelevant of the solution domain, which may involve functional values at points outside the solution domain. In the paper, the proposed method is introduced in detail, and then applied to simulate natural convection in an annulus between two eccentric cylinders.

2. Domain-free discretization method in the cylindrical coordinate system As described in Section 1, the idea of the DFD method is different from the conventional numerical techniques. That is, after numerical discretization, the discrete form of PDE may involve some points outside the given physical domain. Obviously, the key procedure in the DFD method is to evaluate the functional values at those points. As shown in the work of Shu and Fan [8], the functional values at those points can be computed from the expression of analytical solution. Nevertheless, we have to indicate that this process cannot be implemented in practice. As we know, the numerical solution of a PDE is pursued only when its analytical solution is difficult to be given. If the analytical solution of a problem can be easily obtained, there is no need to pursue the numerical solution, which is just an approximate solution. On the other hand, we notice that although the analytical expression of the solution in the given physical domain is difficult to be given, we may be able to find its approximate form in part of the given domain. An example is to find the approximate form of the solution along a line. Suppose that the solution of a problem is expressed by wðh; rÞ in the cylindrical coordinate system. Along a radial line of h ¼ hi , the solution wðhi ; rÞ is only the function of the variable r. If the functional values at certain mesh points along this line are given, then wðhi ; rÞ can be approximated by spline function (local approximation) or Lagrange interpolated polynomial (global approximation). The approximate form of the solution along a line is then used to compute the functional values at any point along the line no matter what the point is inside or outside the given physical domain. This is the key step in the DFD method.

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To apply the DFD method, the doubly connected domain is distributed by a set of radial lines through the coordinates hi ; i ¼ 1; 2; . . . ; N . Then all the derivatives with respect to h in the given PDE are discretized by a numerical method such as the second order finite difference scheme. After numerical discretization in the h direction, the given PDE will be reduced to a set of ordinary differential equations (ODEs). Further, we can distribute mesh points along each line, in which the numerical solution is defined. Note that in general, the number of mesh points at different lines may be different, and the r coordinates of relevant mesh points on adjacent lines may not be the same. So, overall, there is no structure for the mesh point distribution. This is an important feature of the DFD method. After the mesh points are distributed along each radial line, we can further discretize the derivatives with respect to r in the resultant ODEs. As a consequence, a set of algebraic equations will be obtained. It should be indicated that the point selected to further discretize the ODE along a radial line is an interior point of the physical domain. However, since all the ODEs are coupled, the resulting algebraic equation may involve some points at neighbouring lines, which are not the mesh points, and can be inside the physical domain or outside the physical domain. We compute the functional values at these points by using approximate form of the solution along relevant lines. The process of computing the functional value at a point inside the physical domain is called interpolation, while the process of computing the functional value at a point outside the physical domain is termed extrapolation. So, basically, the DFD method involves two aspects. One is the derivative discretization by some numerical methods. The other is the computation of functional values at a point by using interpolation/extrapolation technique. We will take the two-dimensional Poisson equation as an example to illustrate the procedure of the DFD method. Poisson equation in the cylindrical coordinate system is given as follows: o2 w 1 ow 1 o2 w þ þ ¼ f ðh; rÞ: or2 r or r2 oh2

ð1Þ

At first, we distribute N radial lines in the physical domain with the coordinate hi , i ¼ 1; 2 . . . ; N . Along each radial line, we further distribute different mesh points. For numerical discretization, Eq. (1) is first discretized in the h direction by a numerical method. As a result, Eq. (1) is reduced to a set of ODEs in terms of the independent variable r. The resultant ODEs are further discretized in the r direction at internal mesh points by a numerical method. In this work, the second order central difference scheme is used to do numerical discretization in the h direction, and the DQ method [4–7] is applied in the r direction. Since the DQ method is based on the Lagrange interpolated polynomial, the use of DQ method in the r direction is consistent with Lagrange interpolation in the DFD method. With the second order central difference scheme and the DQ method, Eq. (1) can be discretized at a mesh point ðhi ; rj Þ as Mi Mi 1 wi1 ðrj Þ  2wi ðrj Þ þ wiþ1 ðrj Þ X 1 X þ brj;k wi ðrk Þ þ arj;k wi ðrk Þ ¼ f ðhi ; rj Þ 2 2 rj rj k¼1 Dh k¼1

for i ¼ 2; 3; . . . ; N  1; j ¼ 2; 3; . . . ; Mi  1;

ð2Þ

where Mi is the number of mesh points along the line of h ¼ hi , and brj;k and arj;k are the DQ weighting coefficients of the second and first order derivatives in the r direction. Note that the weighting coefficients brj;k arj;k on different radial lines may be different. Eq. (2) is a set of algebraic equations. As shown in Fig. 1, this equation involves the functional values wi1 ðrj Þ, wiþ1 ðrj Þ at two points on two neighbouring lines of h ¼ hi1 and h ¼ hiþ1 . For the general case, these two points may not be the mesh points. In other words, their functional values are not defined. On the other hand, we can see that when the approximate forms of the solution on the two neighbouring lines are found, then the functional values at the two points can be easily calculated from the approximate forms of the solution. Along each radial line, Lagrange interpolated polynomial is a natural choice for the approximate form of the solution. On a radial line of h ¼ hk , Lagrange interpolated polynomial can be written as

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Fig. 1. Configuration of a doubly connected domain with mesh, interpolation and extrapolation points.

wk ðrÞ ¼

Mk X

Lðhk ; rl ; rÞwk;l ;

ð3Þ

l¼1

where Lðhk ; rj ; rÞ ¼

Mk Y ll¼1;l6¼ll

r  rll : rl  rll

Note that Eq. (3) involves all the mesh points on the line of h ¼ hk . For any point on this line, its functional value can be easily computed by Eq. (3). Substituting Eq. (3) into Eq. (2) gives " # Miþ1 Mi1 Mi Mi X X X 1 1 X  2w þ Lðh ; r r Þw þ Lðh ; r ; r Þw br w þ arj;k wi;k ¼ f ðhi ; rj Þ: i1 l; j iþ1 l j j;k i1;l i;j iþ1;l i;k rj k¼1 rj2 ðDhÞ2 l¼1 l¼1 k¼1 ð4Þ

Note that equation system (4) only involves the functional values at mesh points, which can be solved by iterative methods such as SOR approach. It is indicated that no coordinate transformation was introduced in deriving equation system (4). Equation system (4) can be applied to any doubly connected domain. It should be indicated that equation system (4) can only be applied at interior mesh points. However, it involves the functional values at boundary points, which have to be given by the boundary condition. When the Dirichlet condition (function value is given) is imposed, the implementation of boundary condition is very easy and straightforward. One just needs to substitute the given functional values into equation system (4). When the Neumann condition (normal derivative of the function is given) is considered, we need to discretize the derivative first by using some numerical methods such as the one-side finite difference scheme, and then derive an expression to update the functional value at the boundary point. Note that the expression may involve some interior points. As shown in the work of Shu and Fan [8], the expression for updating the functional value at the boundary cannot include the points outside the physical domain because the boundary condition is to define the specific problem. The points outside the physical domain are only appeared in the discrete form of a PDE. Eq. (3) is an approximate form of the solution on a radial line. As shown in Fig. 1, it can be used to do interpolation for a point inside the physical domain or extrapolation for a point outside the physical domain. It was found that the Lagrange interpolation can give very accurate results, but the Lagrange extrapolation may cause a large numerical error due to the following fact. Lagrange extrapolation coefficients Lðhk ; rl ; rj Þ are very large, especially for the case in which the order of Lagrange interpolated polynomial is

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high, and the point for extrapolation is far away from the physical domain. The large extrapolation coefficients may introduce a large round-off error, and the large errors can eventually lead the computation to diverge. To seek more accurate extrapolation method, we can take local extrapolation instead of the global extrapolation. In this work, we use the three local points to construct a second order polynomial to do the extrapolation. The respective formulations are given below wk ðrÞ ¼

ðr  rk;2 Þðr  rk;3 Þ ðr  rk;3 Þðr  rk;1 Þ ðr  rk;1 Þðr  rk;2 Þ w þ w þ w ; ðrk;1  rk;2 Þðrk;1  rk;3 Þ k;1 ðrk;2  rk;3 Þðrk;2  rk;1 Þ k;2 ðrk;3  rk;1 Þðrk;3  rk;2 Þ k;3

ð5Þ

when r < ri ðhk Þ, wk ðrÞ ¼

ðr  rk;Mk 1 Þðr  rk;Mk 2 Þwk;Mk ðr  rk;Mk 1 Þðr  rk;Mk Þwk;Mk 2 þ ðrk;Mk  rk;Mk 1 Þðrk;Mk  rk;Mk 2 Þ ðrk;Mk 2  rk;Mk 1 Þðrk;Mk 2  rk;Mk Þ þ

ðr  rk;Mk 2 Þðr  rk;Mk Þwk;Mk 1 ; ðrk;Mk 1  rk;Mk 2 Þðrk;Mk 1  rk;Mk Þ

ð6Þ

when r > ro ðhk Þ. As shown in Fig. 1, ri ðhÞ; ro ðhÞ represent, respectively, the inner and outer surface of the doubly connected domain.

3. Simulation of natural convection in concentric and eccentric annuli by the DFD method 3.1. Governing equations and numerical discretization Natural convective heat transfer in enclosed spaces has been extensively studied due to its wide applications in engineering. The horizontal concentric and eccentric annular geometries are most commonly encountered in practical uses. There are various experimental [9,10] and numerical [11–15] investigations available. Among the numerical works, some numerical methods, such as the finite difference method (FD) and the DQ method, always require the computational domain to be regular or a combination of regular sub-domains. When a problem with complex geometry like the eccentric case here is concerned, transformation has to be conducted to map an irregular physical domain into a regular computational domain. As a result, a simple governing equation is often transformed into a lengthy and complicated one, which will involve laborious coding and additional computer storage. As will be shown later on, these difficulties can be removed by the DFD method. The governing equations of natural convection in an annulus between two circular cylinders can be written in the cylindrical coordinate system as o2 w 1 ow 1 o2 w þ þ ¼ x; or2 r or r2 oh2    2  ox ox v ox o x 1 ox 1 o2 x oT 1 oT þu þ ¼ Pr þ þ cos h ; þ  Pr  Ra sin h ot or r oh or2 r or r2 oh2 or r oh oT oT v oT o2 T 1 oT 1 o2 T þu þ ¼ 2þ þ 2 2; ot or r oh or r or r oh where u¼

1 ow ; r oh

v¼

ow : or

ð7Þ

ð8Þ

ð9Þ

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The dimensionless parameters appeared in the above equations are the Prandtl number, Pr ¼ lc=k and the Rayleigh number, Ra ¼ qgbL3 ðTi  To Þ=la, Ti and To are the temperature on the inner and outer cylinders, respectively. The inner cylinder is assumed to be heated. The length of the cylinders is assumed to be infinite, thus the flow and heat transfer in the annular are regarded as two-dimensional. The boundary conditions on two impermeable isothermal walls are given by w ¼ u ¼ v ¼ 0;

x¼

o2 w ; T ¼ 1; or2

ð10Þ

on the inner cylinder and w ¼ u ¼ v ¼ 0; x ¼

o2 w ; T ¼ 0; or2

ð11Þ

on the outer cylinder. The periodic condition is imposed in the h direction. As shown in the previous section, Eqs. (7)–(9) can be discretized by the DFD method in the cylindrical coordinate system for any doubly connected domain. Applying the second order central difference scheme in the h direction and the DQ method [5] in the r direction, Eqs. (7)–(9) can be discretized at a mesh point ðhi ; ri;j Þ as Mi X

brðj; k; iÞwi;k þ

k¼1

Mi wiþ1 ðri;j Þ  2wi;j þ wi1 ðri;j Þ 1 X arðj; k; iÞwi;k þ ¼ xi;j ; 2 ri;j k¼1 ri;j Dh2

ð12Þ

Mi X dxi;j vi;j xiþ1 ðri;j Þ  xi1 ðri;j Þ þ ui;j arðj; k; iÞxi;k þ 2Dh dt ri;j k¼1

" ¼ Pr 

Mi X k¼1

Mi 1 X xiþ1 ðri;j Þ  2xi;j þ xi1 ðri;j Þ brðj; k; iÞxi;k þ arðj; k; iÞxi;k þ 2 ri;j k¼1 ri;j Dh2

"  Pr  Ra sin hi

Mi X k¼1

# 1 Tiþ1 ðri;j Þ  Ti1 ðri;j Þ arðj; k; iÞTi;k þ cos hi ; ri;j 2Dh

#

ð13Þ

Mi X dTi;j vi;j Tiþ1 ðri;j Þ  Ti1 ðri;j Þ þ ui;j arðj; k; iÞTi;k þ 2Dh dt ri;j k¼1

¼

Mi X

brðj; k; iÞTi;k þ

k¼1

Mi 1 X Tiþ1 ðri;j Þ  2Ti;j þ Ti1 ðri;j Þ arðj; k; iÞTi;k þ ; 2 ri;j k¼1 ri;j Dh2

ð14Þ

where arðj; k; iÞ and brðj; k; iÞ are the DQ weighting coefficients of the first and second order derivatives along a radial line of h ¼ hi . Following the work of Shu and Richards [5], these weighting coefficients can be computed by arðj; k; iÞ ¼

P ð1Þ ðri;j Þ ; ðri;j  ri;k Þ  P ð1Þ ðri;k Þ

arðj; j; iÞ ¼ 

Mi X k¼1;k6¼j

arðj; k; iÞ;

when j 6¼ k;

ð15aÞ

ð15bÞ

C. Shu, Y.L. Wu / Comput. Methods Appl. Mech. Engrg. 191 (2002) 1827–1841

 brðj; k; iÞ ¼ 2arðj; k; iÞ  arðj; j; iÞ 

brðj; j; iÞ ¼ 

Mi X

 1 ; ri;j  ri;k

when j 6¼ k;

1833

ð16aÞ

ð16bÞ

brðj; k; iÞ;

k¼1;k6¼j QMi k¼1;k6¼j

where P ð1Þ ðri;j Þ ¼ ðri;j  ri;k Þ. It is noted that N is the number of radial lines. Along each radial line, the number of mesh points, Mi , may be different. Thus, the radial coordinate of a mesh point is represented by two indexes i and j, and the DQ weighting coefficients may be different along different radial lines. Eqs. (12)–(14) involve evaluation of functional values wi1 ðri;j Þ, xi1 ðri;j Þ, Ti1 ðri;j Þ, wiþ1 ðri;j Þ, xiþ1 ðri;j Þ, Tiþ1 ðri;j Þ at two points ðhi1 ; ri;j Þ, ðhiþ1 ; ri;j Þ on two radial lines of h ¼ hi1 and h ¼ hiþ1 . It is indicated that these two points may not be the mesh points. We need to use the interpolation or extrapolation formulation to compute the functional values at these two points for the general case. The implementation of boundary conditions for w and T is very simple. One just needs to substitute their values into Eqs. (12)–(14) directly. The vorticity value at the boundary involves the second order derivative of w, which is approximated by the DQ method. The time derivatives in Eqs. (12)–(14) can be discretized by Euler implicit scheme. Then the resultant algebraic equations are solved by SOR method. Eqs. (12)–(14) can be applied to simulate natural convection in an arbitrarily eccentric annulus without introducing coordinate transformation. A schematic configuration of the eccentric annulus is shown in o  R  i , where R  i and R  o are the diFig. 2. In the present study, the reference length L is taken as L ¼ R mensional radii of the inner and outer cylinders. As shown in Fig. 2, the positions of the inner and outer cylinders can be represented by the eccentricity e and the angular position u0 ð0° 6 u0 6 360°Þ. As required by the DFD method, the non-dimensional functions ri ðhÞ and ro ðhÞ for the inner and outer surfaces of a doubly connected domain should be given. For an eccentric annulus, the origin is put at the center of the  o =R i, inner cylinder. So, ri ðhÞ is a constant, but ro ðhÞ is a function of h. If the radius ratio is defined as rr ¼ R then the non-dimensional radii of the inner and outer cylinders are Ri ¼ 1=ðrr  1Þ and Ro ¼ rr=ðrr  1Þ. Therefore, ri ðhÞ and ro ðhÞ are given by ri ðhÞ ¼ Ri ¼ 1=ðrr  1Þ;

ð17aÞ

Fig. 2. Configuration of an eccentric annulus.

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ro ðhÞ ¼ e  cosðh  u0 Þ þ ½R2o  e2 sin2 ðh  u0 Þ

:

ð17bÞ

3.2. Single-value pressure condition For the concentric annulus, the flow and thermal fields are symmetric to the vertical center line. So, there is no global circulation around the inner cylinder, and the stream function values on the inner and outer cylinder walls can be simply set to zero. However, for the eccentric annulus, the intracenter line, which is the line to connect the centers of the inner and outer cylinders, is inclined with respect to the gravity vector, that would cause a global circulation around the inner cylinder. As a result, the stream function values on the two cylinder walls may not be the same. In general, the stream function value is set to zero on the outer cylinder wall. Thus, we need to take the stream function value on the inner cylinder wall as an unknown, which can be determined from the pressure single-value condition. H According to the pressure single-value condition, we have l rp  dl ¼ 0, where l can be an arbitrary circumferential line in the flow field. For simplicity, we take l as the inner cylinder boundary along anticlockwise direction. Along the inner cylinder boundary, we have Z 2p op dh ¼ 0: ð18Þ oh 0 On the other hand, from the momentum equation in the cylindrical coordinate system, we have   2 1 op ov 1 ov ¼ Pr þ Pr  Ra  sin h þ ri ðhÞ oh or2 ri ðhÞ or on the inner cylinder wall. Substituting Eq. (19) into Eq. (18) and using v ¼ ow=or, we obtain  Z 2p

o 3 w o2 w ri ðhÞ 3 þ 2 dh ¼ 0: or or 0

ð19Þ

ð20Þ

We will show that the boundary conditions can be directly substituted into Eq. (20). From Eqs. (10) and (11), we have Mi X ow vi;1 ¼ arð1; k; iÞwi;k ¼ 0; ð21aÞ r¼ri ðhÞ ¼ or k¼1 vi;Mi ¼

Mi X ow arðMi ; k; iÞwi;k ¼ 0: r¼ro ðhÞ ¼ or k¼1

ð21bÞ

Coupling of Eqs. (21a) and (21b) gives Mi X

wi;2 ¼ 

pi;k wi;k ;

ð22aÞ

qi;k wi;k ;

ð22bÞ

k¼1;k6¼2;Mi 1

wi;Mi 1 ¼

Mi X k¼1;k6¼2;Mi 1

where pi;k ¼ farðMi ; Mi  1; iÞarð1; k; iÞ  arð1; Mi  1; iÞarðMi ; k; iÞg=D;

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qi;k ¼ farðMi ; 2; iÞarð1; k; iÞ  arð1; 2; iÞarðMi ; k; iÞg=D; D ¼ arð1; 2; iÞarðMi ; Mi  1; iÞ  arðMi ; 2; iÞarð1; Mi  1; iÞ: The derivatives in Eq. (20) can be discretized by the DQ method. After DQ discretization, Eq. (20) can be reduced to ) I ( X Mi Mi X ri;1 crð1; k; iÞwi;k þ brð1; k; iÞwi;k dh ¼ 0; ð23Þ k¼1

k¼1

where crð1; k; iÞ are the DQ weighting coefficients of the third order derivative. Eq. (23) can be further simplified by using Eqs. (22a) and (22b) I Mi X ½pq3ði; kÞ þ pq2ði; kÞ wi;k dh ¼ 0; ð24Þ k¼1;k6¼2;Mi 1

where pq3ði; kÞ ¼ ri;1 crð1; k; iÞ  ri;1 crð1; 2; iÞpi;k þ ri;1 crð1; Mi  1; iÞqi;k ; pq2ði; kÞ ¼ brð1; k; iÞ  brð1; 2; iÞpi;k þ brð1; Mi  1; iÞqi;k : Since wi;Mi ¼ 0, Eq. (24) leads to H PMi 2 k¼3 ½pq3ði; kÞ þ pq2ði; kÞ wi;k dh H : wwall ¼ wi;1 ¼  ½pq3ði; 1Þ þ pq2ði; 1Þ dh

ð25Þ

Eq. (25) indicates that the stream function value on the inner cylinder wall can be updated by the stream function values at the interior points, which are computed from the governing equations. 3.3. Natural convection in a concentric annulus The simulation of natural convection in a concentric annulus is first selected to validate the DFD method. For this special case, when the number of grid points is taken the same at each radial line, the DFD method is actually reduced to the combination of the second order FD scheme in the h direction and the DQ method in the r direction. Neither interpolation nor extrapolation is involved in the computation. In addition, there is no need to use Eq. (25) since the flow and thermal fields are symmetric, and the stream function value on the inner cylinder wall can be set to zero. The computed values of average equivalent conductivities are used to compare the present results with available data in the literature. The average equivalent conductivity is defined as [14] Z 2p oT keqi ¼  lnðrrÞ  dh ð26aÞ 2pðrr  1Þ 0 or for the inner cylinder, and Z 2p oT keqo ¼  rr  lnðrrÞ  dh 2pðrr  1Þ 0 or

ð26bÞ

for the outer cylinder. The computed keqi and keqo by the present method for the case of Pr ¼ 0:71 and rr ¼ 2:6 are compared with the results of Kuehn and Goldstein [9] and Shu [14]. The results of Shu [14] were obtained by using the DQ method. They are from the grid-independent study, and can thus be considered as the benchmark solution. In the present computation, a uniform mesh is used in the h direction while the

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Table 1 Comparison of average equivalent heat conductivity for a concentric case keqi (Inner cylinder) Ra 2

10 103 104 5 104

keqo (Outer cylinder)

Present

Shu [14]

Kuehn et al. [9]

Present

Shu [14]

Kuehn et al. [9]

1.001 1.082 1.976 2.953

1.001 1.082 1.979 2.958

1.000 1.081 2.010 3.024

1.001 1.082 1.976 2.952

1.001 1.082 1.979 2.958

1.002 1.084 2.005 2.973

non-uniform mesh as shown in [5] is used in the r direction. The use of non-uniform mesh in the r direction is due to requirement of the DQ method [5]. Table 1 shows the comparison of keqi and keqo for Rayleigh numbers of 102 , 103 , 104 , 5 104 . The mesh sizes used in the present computation are 61 9, 61 11, 61 19, 71 19, respectively for Ra ¼ 102 , 103 , 104 , 5 104 . Note that the number of mesh points used in the h direction is much larger than that used in the r direction. This is because the DQ method is applied in the r direction. Since DQ is a global method, it can obtain very accurate results by using a considerably small number of grid points. In contrast, the difference scheme used in the h direction is just second order. So, to achieve the same order of accuracy as the DQ method, a much larger number of grid points is needed in the h direction. As Rayleigh number increases, the boundary layer becomes thinner and thinner. So, to capture the thin boundary layer, more and more mesh points are needed with increase of Ra. It can be obviously observed from Table 1 that the present results agree very well with the benchmark solution of Shu [14] and the results of Kuehn and Goldstein [9]. It should be indicated that the computed keqi and keqo by the present approach are very close. This confirms the theoretical analysis. Since there is no energy loss in the whole system, the theoretical average equivalent conductivities for the inner and outer cylinders should be the same. 3.4. Natural convection in an arbitrarily eccentric annulus Eccentric annulus is an irregular, doubly connected domain. We will show that for this problem, the DFD method is able to obtain very accurate numerical results without introducing any coordinate transformation. Actually, the program used for this case is the same as that used for the concentric case except that the two statements for ri ðhÞ and ro ðhÞ are changed and Eq. (25) is used to update the stream function value on the inner cylinder wall. It should be indicated that for the eccentric case, interpolation or extrapolation might be involved, which can be automatically considered in the program. At first, we consider the natural convection in a horizontal eccentric annulus. This case has been studied by Guj and Stella [13] using the finite difference scheme, and Shu et al. [15] using the DQ method. Thus, the results of Guj and Stella [13] and Shu et al. [15] are used to validate the DFD results. A comparison of wmax , wwall and Nui is shown in Table 2. Note that for the present results in Table 2, the mesh size of 41 21, Pr ¼ 0:71, rr ¼ 2:36 and u0 ¼ 90° were used. It can be seen from the table that the computed wmax and Nui by the DFD method generally agree well with those of Guj and Stella [13] and Shu et al. [15]. However, there are some deviations for wwall . This is because wwall is a very sensitive parameter as compared to other parameters. As shown in [15], the Nui value given by Guj and Stella [13] for the case of e ¼ 0:75 ð3:174Þ may be too large. As mentioned by Guj and Stella [13], the overall Nusselt number has been demonstrated to be weakly sensitive to both vertical and horizontal eccentricities. Kuehn and Goldstein [9] pointed out that the overall heat transfer coefficient for the eccentric case are all found to be within 10% of that for the concentric case. With the confidence of the DFD method for horizontal eccentric annuli, we further apply the DFD method to simulate the natural convection in an arbitrarily annulus for Ra ¼ 104 , Pr ¼ 0:71, rr ¼ 2:6. The

C. Shu, Y.L. Wu / Comput. Methods Appl. Mech. Engrg. 191 (2002) 1827–1841

1837

Table 2 Comparison of numerical results for a horizontal eccentric annulus ðRa ¼ 5300Þ Results

References

e 0.0

0.25

0.50

0.75

12.47 12.259 12.323

14.27 13.773 13.931

15.35 14.549 14.884

wmax

Guj et al. [13] Shu et al. [15] Present

9.23 9.3604 9.374

wwall

Guj et al. [13] Shu et al. [15] Present

0