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Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/unhb20
USE OF THE MOMENTUM INTERPOLATION METHOD FOR FLOWS WITH A LARGE BODY FORCE a
b
Seok-Ki Choi , Seong-O Kim , Chang-Ho Lee & Hoon-Ki Choi
c
a
Korea Atomic Energy Research Institute, KALIMER Technology Development Team, Taejon, Korea b
Machinery Research Laboratory, LG Cable Ltd., Kyungki-do, Korea
c
Changwon National University, Mechanical Engineering Department, Gyeongnam,Korea Version of record first published: 02 Feb 2011.
To cite this article: Seok-Ki Choi , Seong-O Kim , Chang-Ho Lee & Hoon-Ki Choi (2003): USE OF THE MOMENTUM INTERPOLATION METHOD FOR FLOWS WITH A LARGE BODY FORCE, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 43:3, 267-287 To link to this article: http://dx.doi.org/10.1080/713836204
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Numerical Heat Transfer, Part B, 43: 267–287, 2003 Copyright # 2003 Taylor & Francis 1040-7790/03 $12.00 + .00 DOI: 10.1080/10407790390121934
USE OF THE MOMENTUM INTERPOLATION METHOD FOR FLOWS WITH A LARGE BODY FORCE Seok-Ki Choi and Seong-O Kim
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Korea Atomic Energy Research Institute, KALIMER Technology Development Team, Taejon, Korea
Chang-Ho Lee Machinery Research Laboratory, LG Cable Ltd., Kyungki-do, Korea
Hoon-Ki Choi Changwon National University, Mechanical Engineering Department, Gyeongnam, Korea A numerical study of the use of the momentum interpolation method for flows with a large body force is presented. The inherent problems of the momentum interpolation method are discussed first. The origins of problems of the momentum interpolation method are the validity of linear assumptions employed for the evaluation of the cell-face velocities, the enforcement of mass conservation for the cell-centered velocities, and the specification of pressure and pressure correction at the boundary. Numerical experiments are performed for a typical flow involving a large body force. The numerical results are compared with those by the staggered grid method. The fact that the momentum interpolation method may result in physically unrealistic solutions is demonstrated. Numerical experiments changing the numerical grid have shown that a simple way of removing the physically unrealistic solution is a proper grid refinement where there is a large pressure gradient. An effective way of specifying the pressure and pressure correction at the boundary by a local mass conservation near the boundary is proposed, and it is shown that this method can effectively remove the inherent problem of the specification of pressure and pressure correction at the boundary when one uses the momentum interpolation method.
INTRODUCTION Since Rhie and Chow [1] developed a momentum interpolation method, it has been widely used because of the simplicity of its algorithm, especially when the numerical grid is nonorthogonal. Most recent developments of calculation methods are based on the Rhie and Chow scheme. Nearly all commercial codes employ the Received 7 December 2001; accepted 3 August 2002. This study was supported by the Nuclear Research and Development Program of the Ministry of Science and Technology of Korea. Address correspondence to Dr. Seok-Ki Choi, Korea Atomic Energy Research Institute, KALIMER Technology Development Team, 150 Dukjin-dong, Yusung-ku, Taejon, 305-353, Korea. E-mail: skchoi@ kaeri.re.kr 267
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NOMENCLATURE
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Anb Bu ; Bv Du ; Dv fþ e Hu ; Hv P P0 Pr Ra Sbody SP ; SC u; v x; y a Dx; Dy DV
algebraic coefficients of discretized transport equation terms defined in Eqs. (7) and (8) terms defined in Eqs. (5) and (6) geometric interpolation factor terms defined in Eqs. (3) and (4) pressure pressure correction Prandtl number Rayleigh number body force source term linearized source terms Cartesian velocity components Cartesian coordinates relaxation factor control-volume sizes volume of control volume
r Subscripts E; W; N; S; P e; w; n; s u; v
Superscripts l1 u; v
density
pertaining to E, W, N, S, and P grid points pertaining to east, west, north, and south sides of control volume pertaining to u and v velocity components
pertaining to previous iteration level pertaining to u and v velocity components
momentum interpolation method, and its use in the structured and unstructured grid situations is well established. However, the original Rhie and Chow scheme had some minor problems. It did not take into account the presence of underrelaxation factors in the discretized momentum equations, and Majumdar [2] found that the resulting converged solution is slightly relaxation factor-dependent and proposed a remedy for this problem. Choi [3] also modified the original Rhie and Chow scheme to obtain a converged solution that is independent of the size of the time step in the unsteady flow calculations. A recent study by Yu et al. [4] showed that the modifications by Majumdar [2] and Choi [3] also prevent the checkerboard pressure oscillation when one uses a small velocity underrelaxation factor and small time step size. Two important findings of drawbacks of the momentum interpolation method are due to Miller and Schmidt [5] and Gu [6]. Miller and Schmidt [5] found that the momentum interpolation method may result in physically unrealistic solutions where the cell-face velocities do not lie between the neighboring cell-centered velocities. Miller and Schmidt [5] observed that such nonphysical velocities occur in the region of a rapidly varying pressure gradient. Gu [6] also found that the momentum interpolation method may result in physically unrealistic solutions when the body force term is large. During one of the present authors’ analysis of thermal stratification in a curved pipe [7], the nonphysical behavior of velocities was observed near the pipe wall. The fact that the momentum interpolation method may result in a nonphysical solution is a problem that should be resolved. The present study concerns this problem of the momentum interpolation method for solution of flows with a large body force. The objective of the present study is to investigate the origin and remedy of such a physically unrealistic solution when one uses the momentum interpolation method for flows involving a large body force. First, the inherent problems of the momentum interpolation method are discussed, and a simple remedy for such problems is explained. Second, detailed numerical experiments are performed to demonstrate how such problems occur and are remedied.
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MOMENTUM INTERPOLATION METHOD Mathematical Formulation
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In the Rhie and Chow scheme the momentum equations are solved at the cell-centered locations. For the sake of simplicity, we assume the flow is two-dimensional, steady, and incompressible. Consider the Cartesian grid shown in Figure 1. The discretized momentum equations for the cell-centered velocity components can be written as follows, with the underrelaxation factors and body force terms expressed explicitly: uP ¼ ðHu ÞP þ ðDu ÞP ðPw Pe ÞP þ ðBu ÞP þ ð1 au Þul1 P
ð1Þ
vP ¼ ðHv ÞP þ ðDv ÞP ðPs Pn ÞP þ ðBv ÞP þ ð1 av Þvl1 P
ð2Þ
where Hu ¼
Hv ¼
au ½
P
av ½
P
Aunb unb þ ðSuc DVÞ AuP
ð3Þ
Avnb vnb þ ðSvc DVÞ AvP
ð4Þ
Du ¼
au Dy AuP
Figure 1. A typical control volume.
ð5Þ
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av Dx AvP
ð6Þ
Bu ¼
au Sbody DV u u AP
ð7Þ
Bv ¼
av Sbody DV v v AP
ð8Þ
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Dv ¼
AuP ¼ AvP ¼
X X
Aunb SuP DV
ð9Þ
Avnb SvP DV
ð10Þ
It is noted that the source terms of momentum equations (Suc ; SuP ; Svc ; SvP ) are source terms without the body force term (Sbody ; Sbody ), and the body force term is written u v explicitly as a term in the discretized momentum equations. The discretized form of momentum equation for the cell-face velocity components can be written as follows: ue ¼ ðHu Þe þ ðDu Þe ðPP PE Þ þ ðBu Þe þ ð1 au Þul1 e
ð11Þ
vn ¼ ðHv Þn þ ðDv Þn ðPP PN Þ þ ðBv Þn þ ð1 av Þvl1 n
ð12Þ
In the Rhie and Chow scheme these cell-face velocity components are obtained explicitly through the interpolation of momentum equations for neighboring cellcentered velocity components. The following assumptions are commonly introduced to evaluate these cell-face velocity components, for example, the cell-face velocity component at the east side, ðHu Þe feþ ðHu ÞE þ ð1 feþ ÞðHu ÞP
ð13Þ
1 fþ ð1 feþ Þ eu þ u ðAP Þe ðAP ÞE ðAuP ÞP
ð14Þ
where feþ is the geometric interpolation factor defined in terms of distances between nodal points. Similar assumptions can be introduced for the evaluation of the cellface velocity component at the north face, vn . Using the above assumptions, Eq. (11) can be written as follows: ue ¼ ½ feþ uE þ ð1 feþ ÞuP þ ðDu Þe ðPP PE Þ feþ ðDu ÞE ðPw Pe ÞE ð1 feþ ÞðDu ÞP ðPw Pe ÞP þ ½ðBu Þe feþ ðBu ÞE ð1 feþ ÞðBu ÞP þ l1 þ l1 þ ð1 au Þ½ul1 e fe uE ð1 fe ÞuP
ð15Þ
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The term in the first set of brackets on the right-hand side of Eq. (15) is the original Rhie and Chow scheme. The term in the second set of brackets originated from the existence of a body force source term, and the term in the last set of brackets is due to Majumdar [2], to obtain a converged solution that is independent of relaxation factors.
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Inherent Problems of the Momentum Interpolation Method In the conventional staggered-grid method the velocity components are stored at the cell-face locations. The momentum equations are solved at the cell-face locations using Eqs. (11) and (12). After solving the momentum equations, the velocity-correction equations are introduced in terms of pressure correction in the SIMPLE algorithm [8] as follows: ue ¼ ue þ ðDu Þe ðPP0 P0E Þ
ð16Þ
vn ¼ vn þ ðDv Þn ðP0P P0N Þ
ð17Þ
where the starred velocities, ue ; vn , are the velocity components which satisfy only the momentum equations. If the velocity correction equations are inserted into the continuity equation, the pressure-correction equation is obtained. After solving the pressure-correction equation, the velocity components are corrected to satisfy the continuity equation using Eqs. (16) and (17). Thus, the velocity components in the staggered-grid method satisfy both the momentum equations and the continuity equation. Unlike the momentum interpolation method, the pressure at the boundary is not needed in the whole solution process in the staggered-grid method. Therefore, there is no room for producing physically unrealistic solutions in the staggered-grid method. The staggered-grid method may produce an inaccurate solution if the numerical grids are not fine enough, but it does not produce a physically unrealistic solution. In the momentum interpolation method there exist two kinds of velocities, the cell-centered velocities and the cell-face velocities. The momentum equations are solved at the cell-centered locations using Eqs. (1) and (2). After solving the pressurecorrection equation in the SIMPLE algorithm, the cell-centered velocities are corrected by the following equations: uP ¼ uP þ ðDu ÞP ðP0w P0e Þ
ð18Þ
vP ¼ vP þ ðDv ÞP ðP0s P0n Þ
ð19Þ
However, the above velocity-correction equations for the cell-centered velocities are not constrained by the continuity equation, and the pressure-correction terms in the above equation are usually obtained by the linear interpolation of neighboring pressure correction stored at the cell-centered locations. Thus, the cell-centered velocities satisfy the continuity equation on the assumption of linearly varying pressure correction. There is room for producing velocity components that do not satisfy the continuity equation in the region of a rapidly varying pressure gradient when the numerical grids are not fine enough. Another problem in solving the
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cell-centered velocity components is that the pressure and pressure correction at the boundary should be specified to solve the momentum equations or to correct the velocity components near the boundary. The pressure and pressure correction at the boundary are usually specified by a linear extrapolation of interior nodal point values. This linear extrapolation practice may result in an erroneous solution near the boundary if the pressure or pressure correction do not vary linearly in the rapidly varying pressure gradient region due to improper grid refinement. The cell-face velocities in the momentum interpolation method satisfy the continuity equation, but they satisfy the momentum equations based on the assumptions given in Eqs. (13) and (14). Thus, the inherent problems of the momentum interpolation method are: 1. Mass conservation of the cell-centered velocity components 2. Momentum conservation of the cell-face velocity components 3. Specification of pressure and pressure correction at the boundary In the current momentum interpolation method, the above inherent problems are resolved using linear assumptions, such as Eqs. (13) and (14) and the linear variation of pressure and the pressure-correction field. If the numerical grids are properly refined so that theses assumptions are valid, there is no reason for the momentum interpolation method to produce physically unrealistic solutions. However, the flow involving a large body force is usually highly nonlinear, so if the numerical grids are not properly refined, there exists much room for producing physically unrealistic solutions. Specification of Boundary Pressure and Pressure Correction by Local Mass Conservation As explained in the previous section, the specification of pressure and pressure correction at the boundary by a linear extrapolation may cause a problem when the numerical grids are not properly refined near the boundary. We propose a simple way of specifying the pressure and pressure correction at the boundary that satisfies the local mass conservation near the boundary. Consider half of a control volume near the boundary as shown in Figure 2. The equation of mass conservation in this half control volume can be written as follows: rb ub Dy rP uP Dy þ 0:5rn vn Dx 0:5rs vs Dx ¼ 0
ð20Þ
The velocity-correction equation for uP can be written as follows: uP ¼ uP þ ðDu ÞP ðPP0 Pb0 Þ
ð21Þ
In Eq. (21), we use the forward differencing scheme for pressure correction to relate the pressure correction at the cell center and at the boundary. If we insert Eq. (21) into Eq. (20), we obtain the pressure correction and pressure at the boundary as follows: Pb0 ¼ PP0 þ
ðrP uP Dy rb ub Dy þ 0:5rs vs Dx 0:5rn vn DxÞ rP ðDu ÞP Dy
ð22Þ
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Figure 2. Half of a control volume near the boundary.
0 Pb ¼ Pl1 b þ aP Pb
ð23Þ
where the velocity components vn ; vs are the corrected (continuity satisfying) velocities and Pl1 is the pressure at the boundary at the previous iteration level. After b solving the pressure-correction equation, the cell-face velocities are corrected first, the pressure correction and pressure at the boundary are updated by the above equations, and then the cell-centered velocities are corrected. In our experience, this way of specifying the pressure and pressure correction at the boundary did not cause any convergence or stability problems. RESULTS AND DISCUSSION Before going into the detailed analysis of momentum interpolation method for flows with a large body force, we first validate the local mass conservation method for specifying the pressure and pressure correction at the boundary explained above. The test problems considered in the present study are lid-driven cavity flow [9] and natural convection in a cavity [10], which are commonly used for the validation of numerical methods. The Reynolds number of lid-driven cavity flow is 1,000, the Rayleigh number of natural convection problem is 106 , and the Prandtl number is 0.7. The lid-driven cavity flow is solved using 42642 nonuniform grids and the natural-convection problem is solved using 82682 nonuniform grids. The nonuniform grids used in the present study are the same as those used by Hortmann et al. [10]. Figure 3–7 show the comparisons of the predicted results and convergence histories between two different treatments of pressure and pressure correction at the boundary, one by a linear extrapolation (MIM-1) and the other by a local mass conservation method (MIM-2). These figures show that two methods with different specification of pressure and pressure correction at the boundary result in nearly the same solutions and convergence behaviors. However, the specification of the pressure and pressure correction at the boundary by a linear extrapolation causes problems in certain problems, as will be shown later. These facts show that the specification of the pressure and pressure correction at the boundary by a local mass
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Figure 3. Centerline velocity distributions of lid-driven cavity flow.
conservation method is a good alternative for specifying the pressure and pressure correction at the boundary when one uses the momentum interpolation method, although this method requires some extra programming. In order to demonstrate the physically unrealistic solutions for flows with a large body force and the remedy for such problems when one uses the momentum interpolation method, test calculations for a typical two-dimensional naturalconvection problem in a cavity, shown in Figure 8, are performed. The Rayleigh number of this problem is 106 and the Prandtl number is 1. The computer code based on the algorithm by Patankar [8] has been modified for the present purpose. The SIMPLE algorithm is used and the convection term is treated by the power-law scheme [8]. Calculations are performed changing the numerical grids. Convergence is declared when the sum of absolute mass residuals of the pressure correction equation is less than 1014 . Calculations are performed first employing the staggered-grid method (STG), and two calculations with different treatments of specification of pressure and pressure correction at boundaries (MIM-1, MIM-2) are performed employing the momentum interpolation method. Four different numerical grids (42642 uniform grid, 42642 nonuniform grid, 82682 uniform grid, and 82682 nonuniform grid) are used in the present investigations. Figures 9–12 show the numerical results of velocity vector by the staggeredgrid method (STG). We can see that the staggered-grid method does not produce any
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Figure 4. Convergence histories of lid-driven cavity flow.
physically unrealistic solutions. The accuracy of solutions is improved as the numerical grids are refined. Figures 13–16 show the numerical results of the momentum interpolation method when the boundary pressure and pressure correction are obtained by the linear extrapolation of values at the interior nodal points (MIM-1). We can observe that very strange and physically unrealistic velocities exist near the bottom and top boundaries when the numerical grids are coarse (42642 uniform grid). It is noted that the plotted velocities are the cell-centered velocities. The nonphysical behavior of velocity vectors near the top and bottom boundaries is relieved when the numerical grids are refined near the boundaries (42642 nonuniform grid). Nonphysical velocities still exist very near the top and bottom boundaries (42642 nonuniform grid). The refinement of the numerical grid near the boundaries with the same number of grids results in relatively coarse grids in the center region. We observe nonphysical velocity vectors in the upper center region, which are not observed in the results by the staggered-grid method and in the solutions by the momentum interpolation using 42642 uniform grids. The existence of nonphysical velocities in these regions is due to the invalidity of linear assumptions employed in the momentum interpolation method when the numerical grids are coarse and the flow is highly nonlinear. We can observe that solution behaviors are very sensitive to
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Figure 5. Streamlines for natural convection in a cavity (solid, MIM-1; dashed, MIM-2).
the numerical grid when the momentum interpolation method is used. Even the numerical results employing the 82682 uniform grid exhibit nonphysical velocity vectors very near the top and bottom boundaries when the momentum interpolation method is used. However, the results by the 82682 nonuniform grids, given in Figure 16, show that the momentum interpolation method results in nearly the same results as the staggered-grid method when the numerical grids are properly refined. The nonphysical velocity vectors observed in Figures 13–15 disappear completely. These figures clearly show the importance of grid refinement when the momentum interpolation method is used for the numerical solution of flow involving a large body force. It is worthwhile to mention that we could not obtain nearly the same solution as the staggered-grid method without the term in the second set of brackets of Eq. (15) even though we use the 82682 nonuniform grid. Figures 17–20 show the numerical results of the momentum interpolation method when the pressure and pressure correction at the boundaries are obtained by a local mass conservation near the boundaries (MIM-2). The nonphysical velocity vectors near the top and bottom boundaries observed in Figure 9 disappear completely. This shows clearly that the nonphysical vectors observed at the top and bottom boundaries in Figure 13 are due to the dissatisfaction of mass conservation for the cell-centered velocities in these regions. It also shows that the specification of pressure and pressure correction by local mass conservation is a very effective method to remove such nonphysical vectors near the boundary due to improper
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Figure 6. Isotherms for natural convection in a cavity (solid, MIM-1; dashed, MIM-2).
specification of pressure and pressure correction by a linear extrapolation. We notice that this method also results in a physically unrealistic solution in the upper center region when the numerical grids are relatively coarse in this region (42642 nonuniform grid). Thus, the nonphysical solution in the upper center region for the 42642 nonuniform grid is due to the invalidity of linear assumptions employed in the momentum interpolation method and is already observed in Figure 14. We can observe that the results by this method employing 82682 uniform and nonuniform grids are nearly the same as the results by the staggered-grid method. It is worthwhile to investigate how the nonphysical velocity vectors shown in Figure 13 affect the solution behavior of other variables such as the temperature field in the natural-convection problem. Figures 21–22 show the predicted local Nusselt number distribution at the upper plane and the isotherms in the whole domain by the three methods (MIM-1, MIM-2, STG). It is surprising to observe that the three methods result in nearly identical solutions for the temperature field even though there exist nonphysical velocities near the upper and bottom boundaries in the solution by the MIM-1 method. In order to understand this phenomenon, the cellface velocities predicted by the MIM-1 method are plotted, and are shown in Figure 23. The plotted cell-face velocities do not show any strange nonphysical behavior. At present, it is not clearly understood by the present authors why the cell-face velocities do not exhibit nonphysical behavior in the region where the cell-centered velocities
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Figure 7. Convergence histories of natural convection in a cavity.
exhibit strange nonphysical behavior. One should note that the temperature field is governed by the cell-face velocities (usually the mass fluxes are stored at the cell faces), not the cell-centered velocities. Thus the temperature field is properly predicted if the continuity satisfying cell-face velocities are properly predicted. During one of the present authors’ analysis of thermal stratification in a curved pipe [7], nonphysical behavior of the cell-centered velocities was observed near the pipe wall, but the temperature field was properly predicted. This is one reason why people ignore or do not observe the problem. However, it is definitely not a desirable feature that nonphysical behavior of the cell-centered velocities can occur in the computation of fluid flow and heat transfer involving a large body force when one uses the momentum interpolation method. The present work shows the origin of and remedy for such a problem. CONCLUSIONS A numerical study of the use of the momentum interpolation method for flows with a large body force is performed. The inherent problems of the nonstaggered momentum interpolation method are discussed, and it is shown that a simple way of avoiding a physically unrealistic solution for flows with a large body force is a proper
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Figure 8. Test problem of natural convection in a cavity.
Figure 9. A velocity vector plot of natural convection in a cavity by the staggered-grid method (42642 uniform grid).
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Figure 10. A velocity vector plot of natural convection in a cavity by the staggered-grid method (42642 nonuniform grid).
Figure 11. A velocity vector plot of natural convection in a cavity by the staggered-grid method (82682 uniform grid).
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Figure 12. A velocity vector plot of natural convection in a cavity by the staggered-grid method (82682 nonuniform grid).
Figure 13. A velocity vector plot of natural convection in a cavity by the momentum interpolation method (42642 uniform grid).
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Figure 14. A velocity vector plot of natural convection in a cavity by the momentum interpolation method (42642 nonuniform grid).
Figure 15. A velocity vector plot of natural convection in a cavity by the momentum interpolation method (82682 uniform grid).
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Figure 16. A velocity vector plot of natural convection in a cavity by the momentum interpolation method (82682 nonuniform grid).
Figure 17. A velocity vector plot of natural convection in a cavity by the momentum interpolation method with the local mass conservation method (42642 uniform grid).
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Figure 18. A velocity vector plot of natural convection in a cavity by the momentum interpolation method with the local mass conservation method (42642 nonuniform grid).
Figure 19. A velocity vector plot of natural convection in a cavity by the momentum interpolation method with the local mass conservation method (82682 uniform grid).
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Figure 20. A velocity vector plot of natural convection in a cavity by the momentum interpolation method with the local mass conservation method (82682 nonuniform grid).
Figure 21. Nusselt number distributions along the upper wall predicted using the 42642 uniform grid.
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Figure 22. Isotherms predicted using the 42642 uniform grid (solid, STG; dash-dot, MIM-1; dashed, MIM-2).
Figure 23. A vector plot of the cell-face velocities for natural convection in a cavity by the momentum interpolation method (42642 uniform grid).
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grid refinement. An effective way of specifying the pressure and pressure correction at the boundaries by a local mass conservation near the boundaries is proposed, and it is shown that this method effectively removes the nonphysical solution behavior near the boundaries due to the improper specification of pressure and pressure correction at the boundaries. As a final remark, one should be very careful about the grid refinement when one solves highly nonlinear flows involving a large body force by the momentum interpolation method. Otherwise one could obtain physically unrealistic solutions for the cell-centered velocities.
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REFERENCES 1. C. M. Rhie and W. L. Chow, Numerical Study of the Turbulent Flow past an Airfoil with Trailing Edge Separation, AIAA J., vol. 21, no. 11, pp. 1525–1532, 1983. 2. M. Majumdar, Role of Under-relaxation in Momentum Interpolation for Calculation of Flow with Non-staggered Grids, Numer. Heat Transfer, vol. 13, pp. 125–132, 1988. 3. S. K. Choi, Note on the Use of Momentum Interpolation Method for Unsteady Flows, Numer. Heat Transfer A, vol. 36, pp. 545–550, 1999. 4. B. Yu, Y. Kawaguchi, W.-Q. Tao, and H. Ozoe, Checkerboard Pressure Predictions due to the Underrelaxation Factor and Time Step Size for a Nonstaggered Grid with Momentum Interpolation Method, Numer. Heat Transfer B, vol. 41, pp. 85–94, 2002. 5. T. F. Miller and F. W. Schmidt, Use of a Pressure-Weighted Interpolation Method for the Solution of the Incompressible Navier-Stokes Equations on a Nonstaggered Grid System, Numer. Heat Transfer, vol. 14, pp. 213–233, 1988. 6. C. Y. Gu, Computation of Flows with Large Body Forces, Proc. Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, Swansea, UK, 1991. 7. J. C. Jo and S. K. Choi, Numerical Analysis of Initial Formation of Thermal Stratification in a Curved Pipe, Proc. Korean Nuclear Society Spring Meeting, Kori, Korea, 2000. 8. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980. 9. U. Ghia, K. N. Ghia, and C. T. Shin, High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Muligrid Method, J. Comput. Phys., vol. 48, pp. 387–411, 1982. 10. M. Hortmann, M. Peric, and G. Scheuerer, Finite Volume Multigrid Prediction of Laminar Natural Convection: Bench-Mark Solutions, Int. J. Numer. Meth. Fluids, vol. 11, pp. 189–207, 1990.
