COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O˜ nate and E. Dvorkin (Eds.) c
CIMNE, Barcelona, Spain 1998
A SEGREGATED FINITE ELEMENT APPROACH TO THE SOLUTION OF THE NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLOW C.G. du Toit School for Mechanical and Materials Engineering
Potchefstroom University for CHE Private Bag X6001, Potchefstroom 2520, South Africa Email:
[email protected] Key Words: Segregated algorithm, SIMPLER, SIMPLEST, SUPG formulation, NavierStokes Abstract. A segregated finite element algorithm for the solution of the SUPG formulation of the incompressible steady-state Navier-Stokes equations for non-isothermal flow is presented in this paper. The method features equal order interpolation for all the flow variables. The SIMPLER and SIMPLEST algorithms are employed and the sets of nonsymmetric linear equations are solved by means of the preconditioned conjugate gradient squared solver, whilst the preconditioned conjugate gradient solver is used to solve the sets of symmetric linear equations. The effect of the value of the Euclidian norm to test for the convergence of the solution is investigated. It is shown that the Euclidian norm is grid dependent. Also the effect of applying the SUPG weighting functions to all the terms in the momentum and energy equations is compared with the effect of applying the functions to the convective terms only. Three cases are investigated, i.e. the flow in a square cavity, the flow between parallel plates and natural convection in a square cavity. In the case of the driven cavity flow the full SUPG formulation results in a grid independent solution being reached sooner. However, in the case of the flow between parallel plates this is not necessarily the case. The problem is attributed to the weighting of the pressure terms in the momentum equations. It is concluded that this needs thorough investigation.
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1
INTRODUCTION
Based on the success of the finite difference and finite volume methods, a number of finite element segregated solution schemes have been presented.1–7 These vary from schemes that solve for pressure corrections to schemes that solve directly for pressure, and schemes that employ mixed-order interpolation to schemes that employ equal-order interpolation for all variables. The pressure is obtained either via a pressure Poisson formulation which is formed by differentiation of the momentum equations, or via the continuity equation by incorporating an appropriate velocity-pressure relation in the equation. This study is based on the scheme by Rice et al.4 which solves directly for pressure, a´nd employs equal order interpolation for all variables. They employed a stream-line upwinding scheme8 which may be cumbersome to extend to three-dimensional flows. Van Zijl et al.9 adapted the scheme by implementing the SUPG weighting functions of Brooks et al.10 Van Zijl et al.11 later implemented a variation of the SIMPLEST algorithm. This resulted in the symmetization of the coefficient matrices associated with the momentum equations. Du Toit12 replaced the frontal solver used by Van Zijl et al.11 with the conjugate gradient solver and showed that the implementation of an iterative solver can result in a substantial reduction in the storage requirements and execution time. Du Toit13 investigated the implementation of the conservative form of the Navier-Stokes equations. This study investigates the effect that the value, which is specified for the Euclidian norm, to test for convergence, can have on the results. The consistent and inconsistent SUPG formulations along with the SIMPLER and SIMPLEST algorithms are also compared. Three cases are investigated, i.e. the flow in a square cavity, the flow between parallel plates and natural convection in a square cavity. 2 2.1
THEORY Governing equations
The governing equations are the non-conservative form of the two-dimensional NavierStokes equations for steady, incompressible, non-isothermal flow. The equations expressed in tensor notation14 are : ∂uj ∂xj ∂ui ρuj ∂xj ∂T ρcp uj ∂xj
= 0
(1)
∂ui ∂p ∂ ∂uj µ + ρgi = − + +µ ∂xi ∂xj ∂xj ∂xi ∂ui ∂uj ∂ui ∂ ∂T = + k + qT + µ ∂xj ∂xj ∂xj ∂xi ∂xj
(2) (3)
with i, j = 1,2 and where ρ, µ, uj , p, gi , cp , k and ST are the density, dynamic viscosity, velocity components, pressure, gravitational acceleration, specific heat, conductivity and heat source respectively. 2
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2.2 2.2.1
Finite element formulation Weak formulation
The weak statement of the governing equations can be written as: Z Z ∂w uj dΩ = wuj nj dΓ (4) ∂xj Ω Γ Z Z Z ∂ui ∂w ∂ui ∂p ∂w ∂uj wρuj dΩ = − w wρgi − µ dΩ (5) +µ dΩ + ∂xj ∂xj ∂xj ∂xi ∂xj ∂xi Ω Ω Ω Z ∂ui ∂uj + wµ nj dΓ + ∂xj ∂xi Γ Z Z ∂ui ∂uj ∂ui ∂T ∂w ∂T +k w qT + µ + wρcp uj dΩ = dΩ (6) ∂xj ∂xj ∂xj ∂xj ∂xi ∂xj Ω Ω Z ∂T nj dΓ + wk ∂xj Γ where w is an appropriate weighting function and Ω and Γ denote the interior and the boundary of the domain of interest respectively. nj are the direction cosines of the outward pointing vector normal to the boundary. 2.2.2
Discretisation
The Galerkin method of weighted residuals is employed to solve equations (4)–(6). The domain is discretised into isoparametric quadilateral elements and bilinear interpolation is used to approximate both the velocities a´nd pressure: φh =
4 X
φek Nk
(7)
k=1
where φ = u1 ,u2 , T and p and Nk are the Lagrange interpolation functions. The Streamline Upwind Petrov-Galerkin (SUPG) weighting functions: wi = Ni + pi
(8)
proposed by Brooks et al.10 are used. The streamline upwind contribution pi is defined as: ∂Ni k˜ uj (9) pi = 2 u1 + u22 ∂xj The artificial diffusion k˜ is a function of the directional element Peclet numbers P eξ and P eη . The critical approximation is used, thus pi = 0 when P eξ and P eη ≤ 2. Two 3
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weighting strategies are considered, i.e. the consistent formulation as proposed by Brooks et al.10 where all the terms in the momentum equations are weighted with the SUPG functions, and an inconsistent approach where only the convective terms are weighted with the SUPG functions. The source terms in equation (5) are linearized according to the first-order Taylor expansion as outlined by Du Toit.13 2.2.3
Momentum and energy equations
Substituting the approximations and weighting functions described in section 2.2.2 into equations (5) and (6) and assembling the equations in the conventional manner, leads to the global equations (in matrix notation; superscripts and subscripts i, j = 1, 2 refer to the particular velocity component and subscripts k, l, m = 1, . . . , N to the discretised values and associated coefficients; asterisk indicate the last known value): Z ∂Nl i i i i i i 1 ∗ ([C ] + [D ] − [S2 ]){U } = {S1 } − [S2 ]{U } − wk pl dΩ (10) ∂xi Ω ([C T ] + [DT ]){T } = {S T } (11) with cikl
j∗ ∂Nl dΩ wk ρ Nm um ∂xj Ω Z ∂Nk ∂Nl dΩ µ ∂xj ∂xj Ω Z Z ∂Nk ∂Nm j∗ µ dΩ + wk ρgi dΩ − u ∂xj ∂xj m Ω Ω Z ∂ui ∂uj + wk µ nj dΓ + ∂xj ∂xi Γ Z ∂Nk ∂Nk µ dΩ − ∂xi ∂xi Ω 0 for k 6= l Z j∗ ∂Nl dΩ wk ρcp Nm um ∂xj Ω Z ∂Nk ∂Nl dΩ k ∂xj ∂xj Ω Z Z ∂ui ∂uj ∂ui ∂T dΩ + wk wk qT + µ + nj dΓ ∂xj ∂xi ∂xj ∂xj Ω Γ Z
=
dikl = i S1k =
i S2kk = i S2kl =
cTkl = dTkl = SkT =
(12) (13) (14)
(15) (16) (17) (18) (19)
The resulting global velocity coefficient matrices are nonsymmetric due to the convective terms in equations (12) and (17). In the SIMPLEST formulation these coefficient matrices 4
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are symmetrized by taking the non-diagonal coefficient contributions from the convective terms to the right hand side of equations (10) and (11). The equations then become: Z ∂Nl ∗ i i i i i i i i ([CD ] + [D ] − [S2 ]){U } = {S1 } − ([S2 ] + [CO ]){U } − wk pl dΩ (20) ∂xi Ω T ] + [DT ]){T } = {S T } − [C T ]{T }∗ ([CD (21) O
i and C T are the diagonals of the convective matrices and C i and C T the Where CD D O O off-diagonal terms of the convective matrices. The velocity and temperature solutions are constrained by prescribed velocities and temperatures at inlets, wall boundaries, or any nodes in the flow domain where the values are known, as well as by velocity and temperature gradients at the outlet boundaries. 2.2.4
Pressure equation
The discretised momentum equations (10) and (20) can be rearranged to give : i i i ∂p uk = u˜k − Kp ∂xi k where u˜ik are the pseudo-velocities:
u˜ik =
(22)
1 i − kkl ul +fki i |{z} kkk
(23)
k6=l
i and where Kpk are the pressure (gradient)-velocity coupling coefficients: Z 1 i Kpk = i wk dΩ kkk Ω
(24)
The pressure gradients (∂p/∂xi )k are assumed to be known in the above derivation, or it can be seen as lumped least squares fits of the pressure gradients. Applying the Galerkin method of weighted residuals to the continuity equation (4), substituting (22) into the result and expanding the pressure gradients, gives: [K p ]{P } = {F p } with p kkl
fkp
∂Nk j ∂Nl = Nm Kpm dΩ ∂xj ∂xj Ω Z Z ∂Nk j = Nm u˜m dΩ − Nk Nm ujm nj dΓ ∂xj Ω Γ
(25)
Z
5
(26) (27)
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The resulting pressure coefficient matrix K p is symmetric and positive-definite. The pressure field is constrained explicitly by prescribed nodal pressures at points of known pressure, e.g. outlets, but implicitly by prescribed velocities through the velocitypressure equations (22). At nodes where velocities are prescribed, the pseudo-velocities (23) are decoupled from the pressure. To enforce this constraint, the pseudo-velocities are set equal to the prescribed nodal velocities, whilst the nodal velocity-pressure coefficients are set to zero. At in- and outlet boundaries the boundary integral in the pressure equation, as given by (27), is evaluated. 2.2.5
Velocity updating
To ensure continuity during each iteration, the velocity components are updated, using the pressure field obtained, through the solution of equation (25), in the velocity correction equations : Z 1 ∂Nl i i uk = u˜k − i wk pl dΩ (28) kkk Ω ∂xi 3
COMPUTATIONAL ASPECTS
The segregated solution algorithm as illustrated by Rice et al.15 is employed in this study. Because the matrix-vector multiplication in the solvers is row-oriented, the global coefficient matrices are stored according to a slightly modified version of the compressed row storage scheme.16 It uses up to 15.6% less memory per variable than the element by element method17 for two-dimensional problems and up to 36.7% less in the case of threedimensional problems. The non-symmetric linearised algebraic momentum equations are solved by means of the preconditioned conjugate squared method and the solution of the equations is considered to be converged when the residual norm ||r||: P |rk+1| ||r|| = P (29) |r0 | is less than εr = 10−2 or when 100 iterations are completed, where εr is a user specified tolerance. The symmetric linearised algebraic momentum and linear algebraic pressure equations are solved by means of the preconditioned conjugate gradient method. The solution of the pressure equations is considered to be converged when ||r|| ≤ 10−3 or when 1000 iterations are completed. The primitive variable field, i.e. the outer iterations, is taken to be converged when the euclidian norms ||eφ || of the variables for successive iterations satisfy the test P |φn |2 − P φn−1 2 i i ||eφ || = (30) < εe P n2 |φi | where φ = ui, T and p and εe is a user specified convergence tolerance. 6
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In the segregated algorithm the coupled sets of nonlinear equations are solved sequentially in an iterative scheme. In such iterative solution schemes it is necessary to under-relax the variables to ensure stability and to prevent the solution from diverging. The velocities and temperatures are under-relaxed using the false or pseudo time step formulation and linear under-relaxation, whilst linear under-relaxation is applied in the case op the pressures.13 4
NUMERICAL EVALUATION
Some aspects of the performance of the segregated finite element algorithm is assessed by investigating the flow in a square cavity, the flow between parallel plates and natural convection in a square cavity. In particular, attention is paid to the effect of the convergence tolerance, the consistent and inconsistent SUPG formulations and the SIMPLER and SIMPLEST algorithms on the results. 4.1
Flow in a square cavity
Figure 1: Pressure distribution for εe = 0.001 for various grids.
Flow in a square cavity driven by the motion of the lid of the cavity is analysed. The results are compared with the numerical results obtained from the finite element analysis done by Winters et al.18 They used the finite element package WIFE, employing triangular elements with quadratic approximation for the velocity and linear approximation for the pressure. They investigated several meshes and believed their results on a grid consisting of (57 × 57,4) nodes to be converged or grid independent. (The 4 represent the number 7
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Figure 2: Pressure distribution for the inconsistent SUPG formulation.
Figure 3: Averaged Euclidian norm as a function of number of iterations.
of additional interior grid points near the singularities at the top corners.) In this study four meshes, employing 529, 1369, 4225 and 14641 nodes respectively, are investigated. The grids were generated in the same way as that shown by Du Toit.13 Three sets of results are discussed. The first two sets are for a Reynolds number of 400. The first set is concerned with the effect of the value of the convergence tolerance on
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Figure 4: Pressure distribution for the consisten SUPG formulation.
the results. Du Toit13 suggested that the pressure distribution on the horizontal centre line of the cavity could be used as measure to determine whether convergence or grid independence has been reached. Alternatively the pressure distribution on the lid could also be considered. Fig. 1 shows the scaled pressure distribution for a convergence tolerance εe = 0.001 for the various grids. The pressure values shown in the figure (and subsequent figures) have been scaled by Re(p − pr ), where pr is the reference pressure at the origin of the vertical centre line. Following Winters et al.18 the reference pressure was taken as pr = 0. It can be seen that initially the solution apparently improves as the grid is refined and then deteriorates significantly. This is even more clear when the results are compared with those for a convergence tolerance of εe = 0.00001 shown in fig. 2. It, therefore, appears that the fixed Euclidian norm (εe = 0.001) is not necessarily a consistent test for convergence on various grids. The effect of grid refinement on the averaged Euclidian norm can be seen in fig. 3. When very small relaxation parameters are used, or on fine grids this may terminate the calculation prematurely before a converged solution has been obtained. The flattening out of the curves is due to the fact that the coarse components of the solution take much longer to resolve on a finer grid and suggests the need for a multigrid approach. A thorough investigation should be performed to find a more consistent test for convergence. In the case of the Euclidian norm it may be that every fifth or tenth iteration should be compared instead of every successive iteration. A residual norm and an alternative formulation for the Euclidian norm are currently under investigation. The second set of results show that there is a marked difference between the results 9
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for the inconsistent SUPG formulation, fig. 2, and those for the consistent SUPG formulation, fig. 4. The results for the consistent SUPG formulation seem to approach a grid independent solution much quicker. It can be seen that the convective SUPG results for the grid consisting of 4225 (64 × 64 elements) nodes are in very good agreement with the (57 × 57,4) of Winters et al. This seems to suggest that the inconsistent SUPG formulation is better than the consistent SUPG formulation. However, a close study of the results of Winters et al. revealed that the consistent SUPG results seem to be in better agreement with the (41 × 41,12) results of Winters et al. To test for grid independence calculations were performed for both the inconsistent and consistent SUPG formulations on a grid consisting of 14641 nodes. The pressure distributions for this case (120 × 120 elements) are also shown in figs. 2 and 4. As the grid is refined and the element Peclet numbers therefore decrease, the streamline upwind contributions to the SUPG weighting functions (see eqn. (9)) also decrease and the inconsistent and consistent SUPG formulations should therefore approach one another. This can be seen to be the case. The results also suggest that the values for the grid of 4225 nodes for the full SUPG formulation can already be considered to be grid independent. Winters et al. found that grid refinement near the singularities had a marked effect on the converged solution and it may well be that the grid refinement in the case of the (57 × 57,4) grid was not sufficient to obtain grid independence. The third set of results compares the number of iterations needed to obtain a converged solution on the grid of 1369 nodes for the inconsistent and consistent formulations along with the SIMPLER and SIMPLEST algorithms. The results are tabulated in tab. 1. In each column the solution for the previous Reynolds number was used to initialize
Re 100 200 400 800 1600 3200
CONSISTENT SIMPLER SIMPLEST 589 588 395 394 257 255 199 198 847 391 — —
INCONSISTENT SIMPLER SIMPLEST 587 588 378 379 211 212 373 374 438 438 493 494
Table 1: Number of iterations needed for convergence for various formulations.
the solution for the following Reynolds number. No clear pattern can be distinguished. However, it will appear as if the inconsistent formulation might be more stable. A much more thorough investigation should be performed employing various grid sizes and distributions. The SIMPLEST algorithm, along with the inconsistent formulation, is also more stable than the fixed-point method employed by Robichaud et al.19
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4.2
Flow between parallel plates
Developing laminar flow between parallel plates is also considered. In this case only the results of the consistent and inconsistent SUPG formulations employing the SIMPLER algorithm are investigated. Due to symmetry it is only necessary to consider one half of the flow field. The domain is 500mm long and 5mm wide. Four regular grids consisting of 20 × 2, 40 × 2, 80 × 4 and 160 × 8 elements are investigated. Plug flow with a magnitude of unity was prescribed at the inlet and the Reynolds numbers based on twice the distance between the plates varied between 8 and 800. The maximum element Reynolds numbers, therefore, varied between approximately 1500 for the coarsest grid to 1.88 for the finest grid. This means that on the finest grid no upwinding takes place in the case of the smallest Reynolds number. The velocity on the centreline at the outlet, for
Grid 20 × 2 40 × 2 80 × 4 160 × 8
Reynolds number 8 80 800 1.600 1.600 1.600 1.600 1.600 1.600 1.524 1.524 1.523 1.506 1.507 1.500
Table 2: Velocity on centreline at outlet for inconsistent SUPG formulation.
the inconsistent SUPG formulation, is tabulated in tab. 2 and for the consistent SUPG formulation in tab. 3, for the various cases. This should be compared with the theoretical value of 1.5. It can be seen that the inconsistent SUPG formulation converges apparently far quicker to the grid independent (analytical) solution. This is just the opposite of what had been found for the driven cavity case. The consistent formulation is seems to be more grid sensitive, particularly for the lower Reynolds numbers. As expected the two formulations give the same result for the finest grid and the smallest Reynolds number. Reynolds number Grid 8 80 800 20 × 2 1.885 1.703 1.610 40 × 2 1.819 1.656 1.604 80 × 4 1.599 1.548 1.527 160 × 8 1.506 1.516 1.501 Table 3: Velocity on centreline at outlet for consistent SUPG formulation.
The difference between the driven cavity flow and the Poiseulle flow is that in the case of the driven cavity the flow is dominated by the action of the lid and the recirculation 11
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(mixing) which takes place, whilst in the case of the parallel plates the flow is driven by the pressure gradient and dominated by convection. It will appear as if it is the SUPG weighting of the pressure gradients which might be the cause of the difference in the especially in the latter case. A close look at the implementation of the SUPG functions by Brooks et al.10 shows that because of the (reduced order pressure) formulation they employed, the SUPG functions were never applied to the pressure terms. It therefore seems as if it might be necessary to revisit the SUPG formulation by Brooks et al. and closely examine the implemetation of the method. 4.3
Natural convection in a square cavity
Natural convection in a square cavity is also analysed. The results are compared with the numerical results obtained by Rice et al.15 and the bench mark data published by De Vahl Davis et al.20 and De Vahl Davis.21 In this study three meshes consisting of 529, 1369 and 4225 nodes respectively were employed. The grids are the same as those used for the driven cavity. The boundary conditions are identical to those given by De Vahl Davis et al.,20 whilst the ideal gas law was used to calculate densities where the reference pressure was assumed to be 101.325 kPa and the reference temperature 273.15 K. Only the results for the second grid will be presented. Three sets of results will be discussed. The first set is concerned with the temperature distribution. Fig. 5(a) shows
(a) Figure 5: Temperature distribution for (a) Ra = 103 and (b) Ra = 106 .
the temperature distribution for a Rayleigh number of 103 whilst fig. 5(b) shows the temperature distribution for a Rayleigh number of 106 . These results were obtained with the inconsistent SUPG formulation and are in very good agreement with those presented by De Vahl Davis21 and Rice et al.15 12
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3
(u1 )max x2 (u2 )max x1
10 3.685 0.821 3.781 0.179
Rayleigh number 104 105 106 15.394 38.727 63.972 0.821 0.857 0.857 19.140 69.221 220.767 0.107 0.054 0.036
Table 4: Velocities for inconsistent SUPG formulation.
The second set of results is concerned with the maximum horizontal velocity on the vertical centreline and the maximum vertical velocity on the horizontal centreline for various Rayleigh numbers. The non-dimensional velocities, for the grid of 1369 nodes, for the inconsistent SUPG formulation are tabulated in tab. 4, whilst those for the consistent SUPG formulation are tabulated in tab. 5. These are not interpolated values, but those
(u1 )max x2 (u2 )max x1
Rayleigh number 103 104 105 106 3.685 15.394 39.185 65.537 0.821 0.821 0.857 0.821 3.781 19.140 71.640 222.785 0.179 0.107 0.071 0.036
Table 5: Velocities for consistent SUPG formulation.
obtained directly from the results. The values are in good agreement with those tabulated by De Vahl Davis et al.20 with the maximum discrepancy being 11%. It is suggested that a much more thorough investigation should be performed employing various grid sizes and distributions. The last set of results compares the number of iterations needed to obtain a converged solution on the grid of 1369 nodes for the inconsistent and consistent formulations along with the SIMPLER and SIMPLEST algorithms. The results are tabulated in tab. 6. In each column the solution for the previous Rayleigh number was used to initialize the solution for the following Rayleigh number. No real differences can be distinguished. This can probably be attributed to the fact that the actual velocities are very small, particularly for the lower Rayleigh numbers and that no or very little upwinding is taking place. It is suggested that in this case a much more thorough investigation should be performed employing various grid sizes, distributions and Rayleigh numbers. However, it can nevertheless be concluded that the current method is performing very well.
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Ra 103 104 105 106
CONSISTENT SIMPLER SIMPLEST 955 955 514 512 498 498 580 585
INCONSISTENT SIMPLER SIMPLEST 955 955 514 512 500 500 577 581
Table 6: Number of iterations needed for convergence for various formulations.
5
CONCLUSIONS
A segregated finite element algorithm which features equal order interpolation for all the flow variables for the non-conservative formulation of the Navier-Stokes equations for steady-state, incompressible, non-isothermal flow was discussed in this paper. The SIMPLER and SIMPLEST algorithms along with the preconditioned conjugate gradient squared solver for the sets of non-symmetric linear equations and the preconditioned conjugate gradient solver for the sets of symmetric linear equations were used to solve the non-linear problem. Three cases were investigated, i.e. the flow in a square cavity, the flow between parallel plates and natural convection in a square cavity to assess the method. The effect of a fixed value of εe = 0.001 for the Euclidian norm to test for the convergence of the solution was investigated. It was shown that the Euclidian norm is grid dependent and that it might not necessarily be a consistent test for convergence. It is recommended that a thorough investigation should be performed to find a more consistent test for convergence. The effect of applying the SUPG weighting functions to all the terms in the momentum equations was also compared with the effect of applying the functions to the convective terms only. In the case of the driven cavity flow the consistent (full) SUPG formulation results in a grid independent solution being reached sooner. However, in the case of the flow between parallel plates the opposite was apparently found. The problem was attributed to the weighting of the pressure terms in the momentum equations. It is recommended that this needs thorough investigation. In the case of the natural convection in the square cavity, very good agreement was obtained with the bench mark data. Although it is suggested that the investigation should be extended to include a wider range of grid sizes, grid distributions, Rayleigh and Reynolds numbers, it can nevertheless be concluded that the current method is performing very well. REFERENCES [1] G.E. Schneider, G.D. Raithby and M.M. Yovanovich, “Finite element solution procedures for solving the incompressible, Navier-Stokes equations using equal order vari14
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able interpolation”, Num. Heat Transf., 1, 433-451 (1978). [2] C. Prakash and S.V. Patankar, “A control volume-based finite-element method for solving the Navier-Stokes equations using equal-order velocity-pressure interpolation”, Num. Heat Transf., 8, 259-280 (1985). [3] A.C. Benim and W. Zinser, “A segregated formulation of Navier-Stokes equations with finite elements”, Comp. Meth. Appl. Mech. Engrg., 57, 223-237 (1986). [4] J.G. Rice and R.J. Schnipke, “An equal-order velocity-pressure formulation that does not exhibit spurious pressure modes”, Comp. Meth. Appl. Mech. Engrg., 58, 135-149 (1986). [5] C.T. Shaw, “Using a segregated finite element scheme to solve the incompressible Navier-Stokes equations”, Int. J. Num. Meth. Fluids, 12, 81-92 (1991). [6] H. Araseki, “Finite element method for thermal hydraulic analysis using equal-order interpolation”, Num. Heat Transf., Part B , 19, 153-174 (1991). [7] V. Haroutunian, M.S. Engelman and I. Hasbani, “Segregated finite element algorithms for the numerical solution of large-scale incompressible problems”, Int. J. Num. Meth. Fluids, 17, 323-348 (1993). [8] J.G. Rice and R.J. Schnipke, “A monotone streamline upwind finite element method for convection-dominated flows”, Comp. Meth. Appl. Mech. Engrg., 48, 313-327 (1985). [9] G.P.A.G. van Zijl and C.G. du Toit, “A SIMPLER finite element solution of the incompressible Navier-Stokes equations”, In: Proc. 2nd Nat. Symp. Comp. Fluid Dynamics (Vereeniging, SA, 1991). [10] A.N. Brooks and T.J.R. Hughes, “Streamline Upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations”, Comp. Meth. Appl. Mech. Engrg., 32, 199-259 (1982). [11] G.P.A.G. van Zijl and C.G. du Toit, “A SIMPLEST finite element solution of the incompressible Navier-Stokes equations”, In: Proc. Symp. Finite Element Meth. in South Africa 92 (Cape Town, SA, 1992). [12] C.G. du Toit, “A SIMPLEST iterative solution of the Navier-Stokes equations”, In: Proc. Symp. Finite Element Meth. in South Africa 95 (Stellenbosch, SA, 1995). [13] C.G. du Toit, “Finite element solution of the Navier-Stokes equations for incompressible flow using a segregated algorithm”, Int. J. Comp. Meth. Appl. Mech. Engrg., 151, 131-141 (1998). [14] R. Peyret and T.D. Taylor, Computational methods for fluid flow , Springer Verlag, (1983). [15] R.J. Schnipke and J.G. Rice, “A finite element method for free and forced convection heat transfer”, Int. J. Num. Meth. Engrng., 24, 117-128 (1987). [16] G.F. Carey and J.T. Oden, Finite elements : Computational aspects, Prentice-Hall, Vol. III, (1984). [17] M.P. Reddy and J.N. Reddy, “Penalty finite element analysis of incompressible flows using element by element solution algorithms”, Comp. Meth. Appl. Mech. Engrng., 15
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100, 169-205 (1992). [18] K.H. Winters and K.A.Cliffe, A finite element study of driven laminar flow in a square cavity, AERE Harwell Report HL79/1212 (C.7) (1979). [19] M.P. Robichaud, P.A. Tanguy and M. Fortin, “An iterative implementations of the Uzawa algorithm for 3-D fluid flow problems”, Int. J. Num. Meth. Fluids, 10, 429-442 (1990). [20] G. De Vahl Davis and I.P. Jones, “Natural convection in a square cavity: A comparison exercise”, Int. J. Num. Meth. Fluids, 3, 227-248 (1983). [21] G. De Vahl Davis, “Natural convection of air in a square cavity: A bench mark numerical solution”, Int. J. Num. Meth. Fluids, 3, 249-264 (1983).
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