High resolution of 2D natural convection in a cavity by the DQ method

Transcription

1 Journal of Computational and Applied Mathematics 7) High resolution of D natural convection in a cavit b the DQ method D.C. Lo a,, D.L. Young b, C.C. Tsai c a Department of Animation Design and Game Programming, Toko Universit, Chia-Yi Count, 66, Taiwan b Department of Civil Engineering and Hdrotech Research Institute, National Taiwan Universit, Taipei 67, Taiwan c Department of Information Technolog, Toko Universit, Chia-Yi Count, 66, Taiwan Received 6 November 5; received in revised form 5 Februar 6 Abstract In this paper, the differential quadrature method DQ) is applied to solve the benchmark problem of D natural convection in a cavit b utilizing the velocit vorticit form of the Navier Stokes equations, which is governed b the velocit Poisson equation, continuit equation and vorticit transport equation as well as energ equation. The coupled equations are simultaneousl solved b imposing the vorticit definition at boundar without an iterative procedure. The present model is properl utilized to obtain results in the range of Raleigh number 7 ) and H/L aspect ratios varing from to. sselt numbers computed for Ra 7 in a cavit show excellent agreement with the results available in the literature. Additionall, the detailed features of flow phenomena such as velocit, temperature, vorticit, and streamline plots are also delineated in this work. Thus, it is convinced that the DQ method is capable of solving coupled differential equations efficientl and accuratel. 6 Elsevier B.V. All rights reserved. Kewords: Natural convection; Differential quadrature; Velocit vorticit form; Aspect ratios; sselt numbers. Introduction Computational fluid dnamics CFD) finds wide range of applications in sciences and engineering. Computation of incompressible Navier Stokes equations is the most significant area in CFD. The velocit vorticit form of the Navier Stokes equations has been developed in [5] as an alternative scheme without involving the pressure term. The main advantage of the velocit vorticit formulation is that its numerical formulation is independent of whether or not the reference frame is inertial. To be precise, the vorticit is one of the field variables, thus velocit vorticit formulation can be exploited to stud vortex dominated flows no matter D or D flow is considered. Furthermore, the extension of D flow is also straightforward while the velocit vorticit formulation is used. When the incompressible Navier Stokes equations are solved in the velocit vorticit form, the numerical algorithm varies depending upon how the vorticit term is treated with respect to the vorticit transport equation especiall for the vorticit values at the boundar wall. In man practical problems vorticit values at the boundar wall are one of the variables to be considered for design purpose. Thereb, the velocit vorticit form of the Navier Stokes equations is an alternative Corresponding author. Tel.: address: loderg@ntu.edu.tw D.C. Lo). 77-7/$ - see front matter 6 Elsevier B.V. All rights reserved. doi:.6/j.cam.6..

2 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 Nomenclature weighting coefficients for the lth order derivatives of the function with respect to x weighting coefficients for the mth order derivatives of the function with respect to L number of grid points in the x direction M number of grid points in the direction avg the average sselt number on the boundar of the cavit at x = max the maximum value of local sselt number on the boundar of the cavit at x = min the minimum value of local sselt number on the boundar of the cavit at x = A l) i,k B m) j,k Pr Prandtl number Ra Raleigh number t dimensionless time T dimensionless temperature u, v dimensionless velocities in the x and directions x, dimensionless Cartesian coordinates ω dimensionless vorticit in the z direction H/L aspect ratios of height and length Γ problem boundar Ω computational domain scheme in man CFD applications. However, the number of field variables increases from three to six for the cases of D Navier Stokes equations, which ma be a disadvantage. In order to circumvent the pitfalls associated with the vorticit values at the boundar, a global DQ method is emploed to compute vorticit boundar values implicitl without using an iterative procedure. As pointed out in [] the satisfaction of the continuit equation reduces to enforcing the vorticit definition at the boundar in terms of curl of the velocit field. The Navier Stokes equations in velocit vorticit form indicate that vorticit is created at the boundar b satisfing the velocit boundar conditions []. The definition of vorticit is given b the curl of the velocit field. Consequentl, in order to correctl satisf the vorticit definition at the boundar, computation of accurate velocit gradients or the vorticit values at the boundar are ver crucial. As indicated b the investigators [7,7,9], in order to enforce the solenoidalit of vorticit in the computation of flow problems, the vorticit transport equations have to be solved with the vorticit values at the boundar. It is well known that the curl of vorticit definition results in the kinematic velocit Poisson equations for an incompressible fluid flow. Additionall, the curl of the momentum equations in primitive variable form gives rise to the dnamic vorticit transport equations. A detailed discussion was briefl presented in [7]. It should be noted that the solenoidalit of velocit field can be assured onl b coupling the kinematic and dnamic equations. In the present work, the velocit u in the x direction is solved b the velocit Poisson equation and the velocit v in the direction is obtained from the differentiated form of the continuit equation, hence assuring a divergence-free velocit field. For the variables of the vorticit and temperature field, the vorticit transport equation and energ equation are used for solving the vorticit and temperature variables of the flow field. When higher-order approximation is discretized to compute all flow variables, also a numerical coupling between the velocit, vorticit and temperature is required to satisf the solenoidalit constraint of the velocit field. Therefore the motivation for the present work originated from the fact that how to enforce the vorticit definition accuratel as the curl of velocit both at the interior and at the boundaries of the computational domain. B coupling the entire field variables the vorticit boundar values can be computed implicitl from the velocit fields and enforced at the boundar nodes. In recent ears, the DQ method is becoming a powerful numerical scheme to solve partial differential equations with higher-order accurac. The DQ method was first developed in [] to approximate the derivative of a smooth function using a higher-order polnomial. The application of the DQ method has been successfull implemented for solving engineering problems b man researchers [,,]. Shu and Xue [] applied generalized DQ method to solve the natural convection in a square cavit. Two boundar conditions Dirichlet and Neumann tpe) were discussed therein for the streamfunction at each boundar. Recentl, a number of meshless methods have been developed to solve natural convection problems. Shu et al. [] developed a local radial basis function-based differential quadrature method and its

3 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 application to the simulation of natural convection in a square cavit. Additionall, the local radial interpolation method was developed in [9] for solving D natural convection problems with enclosed domain of different geometries. Ding et al. [] also presented a meshless method based on the use of a weighted least-square approximation to solve D natural convection in a square cavit. Natural convection in a differentiall heated enclosure is ver popular problem as a test case to demonstrate the capabilit of an proposed numerical algorithm. The reason for selecting this problem is to prove the efficienc and accurac of the proposed coupled numerical scheme to solve governing equations for vortex dominated flow. Since the vorticit transport equation is coupled to the energ equation through the buoanc force, we choose this problem to test the proposed coupled numerical algorithm. The four field variables, two velocities, one vorticit, and one temperature are solved at the same time using a coupled numerical technique. The proposed numerical scheme is applied to determine the velocit, vorticit and temperature variations for a natural convection problem in a differentiall heated enclosure for Raleigh number range of 7 and H/L aspect ratios varing from to. Comparisons of the present results with those obtained b other numerical schemes are presented.. Differential quadrature method The DQ method replaces a given spatial partial derivative of a function fx)b a linear weighted sum of the function values at the discrete sample points considered along the coordinate direction. As a result, the given partial differential equation reduces to a set of algebraic equations. Thereb the numerical solution of partial differential equations can be solved b the DQ method. This method has been briefl presented in Refs. [,,]. For a function of two variables fx,) the lth order derivatives and mth order derivatives of the function with respect to x and can be obtained as [,] f l) x x i, j ) = f m) x i, j ) = L A l) i,k fx k, j ), l =,,...,L, a) B m) j,k fx i, k ), m =,,...,M for i =,,...,L, j =,,...,M, b) where L, M are the numbers of grid points in the x, direction, respectivel, A l) i,k,bm) j,k are the weighting coefficients. For the first-order derivatives, the weighting coefficients A ) i,k,b) j,k can be obtained as follows: A ) i,j = L ) x i ) x i x j )L ), i,j =,,...,L, but j = i, a) x j ) B ) i,j = M ) i ) i j )M ), i,j =,,...,M, but j = i, b) j ) where L ) x i ) = L j=,j =i x i x j ), M ) i ) = M j=,j =i i j ). ) Similarl the weighting coefficients for the lth order derivatives and mth order derivatives of the function with respect to x and can be obtained as ) A l) i,j = l A l ) i,i A ) i,j Al ) i,j for i, j =,,...,L, but j = i, l =,,...,L, a) x i x j B m) i,j = m B m ) i,i B ) i,j Bm ) i,j i j ) for i, j =,,...,M, but j = i, m =,,...,M. b)

4 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 When j = i, the weighting coefficients are written as A l) i,i = B m) i,i = L j=,j =i j=,j =i A l) i,j, i =,,...,L, l=,,...,l, 5a) B m) i,j, i =,,...,M, m=,,...,m. 5b) It is quite important to note that the weighting coefficients of the second- and higher-order derivatives can be computed from the first-order derivatives themselves.. Governing equations and boundar conditions In the case of natural convection problems, the Navier Stokes equations include the buoanc force generated as a result of fluid densit difference caused b the temperature difference. The buoanc term is computed based on the Boussinesq approximation. B taking the curl of the primitive variable form of the Navier Stokes equations for natural convection and making use of the vorticit definition, the velocit vorticit form of the natural convection equations can be derived as follows: Velocit equations u = ω, 6) u x + v =. 7) We use the continuit equation Eq. 7)) instead of the velocit Poisson equation v = ω/ x) to solve velocit component v in order to reduce the CPU time for the flow computation as recommendation in [8]. Vorticit transport equation ω t Energ equation T t + u ω x + v ω = Pr ω + Ra Pr T x. 8) + u T x + v T = T, 9) where L, α/l, α/l, L /α and T = θ θ c )/θ h θ c ) are used as scale factors for length, velocit, vorticit, time, and temperature in the above non-dimensional form of the governing equations. The non-dimensional parameters are defined as Ra = gβδtl /αυ and Pr = υ/α. The velocit Poisson and continuit equations 6), 7) remain unaffected as the represent the kinematic condition of the flow field. Eqs. 8), 9) with ω as vorticit and T as the scalar temperature field as well as velocit field Eqs. 6), 7)) are the final form of the governing equations to be solved in the domain Ω with the specified boundar conditions for velocit, vorticit and temperature on the solid boundar Γ. In Eqs. 6), 7), u and v are the velocities in x and directions, respectivel, in a computational domain Ω surrounded b a boundar Γ. We seek a solution in the domain Ω, which enforces the Dirichlet boundar conditions of velocit given b u = u, v) = u b ) and the corresponding vorticit values on the boundar can be obtained using the definition given as ω = v x u. )

5 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 The Dirichlet and Neumann boundar conditions of present cases for temperature are as follows: T = T b, T =. a) b). Formulations b DQ method The velocit equations 6), 7) can be approximated b the DQ method as follows: L L A ) i,k u k,j + A ) i,k k= k= B ) j,k u i,k + B ) j,k u k,k + B ) j,k ω i,k =, ) B ) j,k v i,k =. ) For the vorticit transport equation, the non-linear term is treated explicitl, the viscous term is treated implicitl, as well as the time derivative is discretized b a second-order time stepping of finite difference tpe see Refs. [6,6])b the global method of DQ approximation. The vorticit transport equation can be discretized as the following formula: ω n+ ω n + ω n Δt + u ω x + v ω ) n u ω x + v ω ) n ) T n+ Pr ω n+ Ra Pr =. x 5) B using the DQ approximation the above equation can be expressed as ) ω n+ L i,j Pr A ) i,k Δt ω k,j + = Δt + ) ω n i,j Δt L u i,j ) ω n i,j A ) i,k ω k,j + v i,j B ) j,k ω i,k) n+ Ra Pr L u i,j A ) i,k ω k,j + v i,j Similarl, the energ equation is written as ) L n+ Ti,j n+ A ) i,k Δt T k,j + B ) j,k i,k) T = Δt L A ) i,k T k,j) n+ B ) j,k ω i,k B ) j,k ω i,k) n. 6) + ) n ) ) Ti,j n Ti,j n Δt L u i,j L u i,j A ) i,k T k,j + v i,j A ) i,k T k,j + v i,j B ) j,k T i,k ) n B ) j,k T i,k) n. 7) The algebraic equations ) ), 6) 7) represent the DQ form of the governing equations for the velocit, vorticit and temperature. B combining the coefficient matrices of all the field variables, the following global matrix form is

6 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 obtained for the coupled solution algorithm: A A u f u B B v C C ω = f v f ω. 8) D T f T The submatrices and the load vectors can be written as [A ]= [B ]= L L A ) i,k + M A ) i,k k= k= ) L [C ]= Pr Δt B ) j,k, [A ]= B ) j,k, [B ]= A ) ) L [D ]= A ) Δt [f u ]=, [f v ]=, [f ω ]= Δt + [f T ]= Δt + i,k + M i,k + M B ) j,k B ) j,k ), ) ) ω n i,j ω n i,j Δt L u i,j A ) i,k ω k,j + v i,j ) ) Ti,j n Ti,j n Δt L u i,j A ) i,k T k,j + v i,j B ) j,k, B ) j,k, ) L, [C ]= Ra Pr L u i,j B ) j,k ω i,k) n, L u i,j B ) j,k T i,k) n, A ) i,k ω k,j + v i,j A ) i,k T k,j + v i,j A ) i,k, B ) j,k ω i,k B ) j,k T i,k where [A ], [A ], [B ], [B ], [C ], [C ] and [D ] are the submatrices of Eq. 8) which represent the various differential operators that appear in the DQ approximation. The unknown field variables are the velocities u, v), vorticit ω) and temperature T ). The load vectors are the [f u ], [f v ], [f ω ] and [f T ]. The velocit boundar condition can be computed as u,j =, u L,j =, u i, =, u i,m =, v,j =, v L,j =, v i, =, v i,m = for i =,...,L and j =,...,M. 9) The vorticit values at boundar wall expressed in terms of velocit gradients b Eq. ) can also be approximated b the DQ method as follows: ω i,j L A ) i,k v k,j + B ) j,k u i,k = for i =,...,L, and j = orj = M for j =,...,M, and i = ori = L. ) ) n ) n

7 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) The Dirichlet temperature boundar conditions at the left and the right walls of the square cavit can be represented as T,j =, T L,j =. ) The adiabatic boundar conditions on the top and bottom sides of D natural convection in a cavit can be derived in the DQ form as [] M B ), B) M,M B) M, B),M )T i, B ),M B) M,k B) M,M B),k )T i,k =, a) k= M B ),M B) M, B) M,M B), )T i,m B ) M,k B), B),k B) M, )T i,k = k= for i =,...,L, j=,,...,m. b) It should be noted that Eqs. a), b) also involve the implicit scheme for the Neumann boundar conditions. The simultaneous equations resulting from the global matrix sstem of Eq. 8) are solved using a bi-conjugate iterative equation solver []. Since the coefficient values are sparse not onl for submatrices but also for the global matrix sstem as expressed in Eq. 8), onl the non-zero entries are stored in a column storage format. The numerical solution procedures used here can be briefl described as follows:. Get initial conditions and boundar conditions.. Solve the submatrices of Eq. 8).. Compute all the non-zero entries in a column storage format in order to solve Eq. 8) efficientl.. As far as boundar conditions are concerned, we used the velocit boundar condition of Eq. 9), the vorticit definition of Eq. ) and temperature boundar conditions of Eqs. ), ) to satisf the sstem of Eq. 8). 5. Eq. 8) is advanced in time to solve the unknown velocit, vorticit and temperature. 6. Check for the convergence of the velocit, vorticit and temperature components in the present stud. We used the velocit, vorticit and temperature components at the previous time step as the initial guess for the next iteration. The computations are carried out until stead state conditions are reached. The convergence criteria used in the time loop to achieve stead state conditions are u n+ u n )/u n 6, v n+ v n )/v n 6, ω n+ ω n )/ω n 6, T n+ T n )/T n 6. ) For the DQ method, the mesh point distribution in the two spatial coordinates is expressed as [] cos[π/l)] cos[i )π/l)] x i = cos[π/l)] cos[l )π/l)] L x, i =,,...,L cos[π/m)] cos[j )π/m)] j = cos[π/m)] cos[m )π/m)] L, j =,,...,M ) where L, M are the numbers of grid points in the x, directions, respectivel. L x is the length in the x direction, and L is the width in the direction, respectivel. 5. Results and discussion A numerical model was developed to validate the accurac for the solutions of D natural convection in a cavit as shown in Fig.. The no-slip boundar conditions of the velocities at boundar walls are assumed. Temperature equal to and are assumed at the left and the right walls, respectivel. Adiabatic conditions are assumed on the top and bottom sides of the cavit. The aspect ratios of height and length is defined as H/L, and H/L varing from to was undertaken in the present work. In order to establish benchmark solution, the obtained numerical results are compared

8 6 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 u = ; v = ; T/ = u = ; v = ; T = H u = ; v = ; T = L u = ; v = ; T/ = Fig.. Laout of the problem. Table Grid-independence stud results of natural convection in a square cavit for Ra = and Ra Method DQ DQ DQ Bench. [6] DQ DQ DQ Bench. [6] Mesh size avg Error %) with the results due to Vahl Davis [5] who used a grid size of 8 8 in a square cavit. The significant parameter for natural convection is to compute the sselt number as follows:. The sselt number on the vertical boundar at x =, evaluated as H T = x d.. The average sselt number throughout the cavit, evaluated as avg = H/L) L 5.. Grid-independence stud dx. For the case of a natural convection in a square cavit, the average sselt number throughout the cavit is considered as important parameters for comparison purposes. Tables and depict the comparisons of the above parameters obtained in the present work for Ra 6 with the benchmark solution obtained using streamfunction vorticit formulation in [5]. The present stud produces almost the same results as obtained in [5]. However, it can be observed that the present scheme used three grid sizes to produce the same results compared to the benchmark solution. From the results obtained, one can observe the computational grid was refined successivel from to 5 5 and, in which ver accurate results are obtained in all the three grids sstem. Thereb, the present coupled numerical algorithm

9 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) Table Grid-independence stud results of natural convection in a square cavit for Ra = 5 and 6 Ra 5 6 Method DQ DQ DQ Bench. [6] DQ DQ DQ Bench. [6] Mesh size avg Error %) Table Comparison of CPU time s) for different Raleigh number Mesh size Δt CPU time s) Ra = Ra = Ra = Ra = based on the velocit vorticit form of the Navier Stokes equations and the DQ method computes the average sselt number throughout the cavit consistentl with grid refinement. 5.. Results on flow and hdrothermal characteristics As far as the CPU time for different Raleigh number is concerned, Table summarizes the total CPU time of different Raleigh number for three grids sstem. From the above computed tabulation, the CPU time increases as Raleigh number increased. Also, the computation of CPU time is increasing from the different numbers of grid nodes 5 and.intable, we present comparisons of an excellent agreement with the results obtained in [5] for Ra =,, 5, and 6. For the case of higher Raleigh number equal to 7, the evaluations of the maximum sselt number and minimum sselt number as well as average sselt number are presented in Table, respectivel. The obtained results indicate that present work is capable of handling high Raleigh number without difficulties. The present coupled numerical scheme is further demonstrated b plotting the velocit, temperature, vorticit and streamline contours as shown in Fig. b using a grid of points for Ra =,, 5, 6 and 7, respectivel. After successfull validating the present algorithm for natural convection in a square cavit, we further analze the different H/L aspect ratios of and. For the case of the different ratios in a cavit, Ismail and Scalon [8] developed a finite element method for numerical solutions of free convection in a cavit with side walls maintained at constant but different temperatures over a wide range of Raleigh number Ra 6 ) and different L/H aspect ratios. Table 5 displas the comparison of the sselt number predicated in a cavit for various H/L aspect ratios. The increase in the values of average sselt number with increase in Raleigh number has been accuratel solved as expected trend. From the tabulated results it can be noted that the present model obtained the benchmark solution with a coarser grid for all five values of Raleigh number and different H/Laspect ratios varing from to. Additionall, the distributions of velocit, temperature and vorticit fields are plotted in Figs. 5 for Ra = 5, 6 and 7 in the case of H/L= /, respectivel.

10 8 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 Table merical results of natural convection in a square cavit for different average sselt number max min avg Ra = Bench. [6] Position) =.9) = ) Present DQ) Error.66%.5% % Position) =.) = ) Ra = Bench. [6] Position) =.) = Present DQ) Error.85%.7%.5% Position) =.6) = Ra = 5 Bench. [6] Position) =.8) = ) Present DQ) Error.7%.7%.% Position) =.95) = ) Ra = 6 Bench. [6] Position) =.78) = ) Present DQ) Error.7%.78%.6% Position) =.) = ) Ra = 7 Present DQ) Position) =.) = ) Further, when H/L= / for the distributions of velocit vector, temperature and vorticit fields are shown in Figs. 6 8, respectivel. As the Raleigh number increases, one can observe the rapid changes occurring in the velocit vectors indicating an increase of the convection participation in the heat transfer process for all the cases. The increase of vorticit values near the isothermal walls indicate that a near-stagnant interior core occurs along with the distinct boundar laers near the end walls with increase in the values of the Raleigh number. As the gap height increases, it means that the aspect ratios H/L increase, for all the cases of high Raleigh number the heat transfer b conduction is reduced while the natural convection effect starts to get stronger. As the convection spreads to the entire cavit, we observe the formation of circulating cells adjacent to the corners of the cavit. The increase in the fluid convection process with increase in Raleigh number results in the thinning of the boundar laers as observed in above figures. In the present formulation, the vortex which was originall in circular pattern for Ra = now changes its shape to elliptical forms with increase in the value of the Raleigh number. From the results obtained, we can note the onset of vortex domination throughout the entire cavit, thus affecting the flow field. The above figures also indicate the increase in the value of the constant vorticit lines is observed as the value of the Raleigh number increases. In Fig. 9, we exhibit comparisons of the variations of u and x v velocit profile plots for Ra 7 and H/L aspect ratios varing from to, respectivel. Since all no-slip velocit boundar conditions are enforced at boundar walls for the natural convection problem, a smmetric velocit distribution is expected in the above figure. For the cases of the Ra = at different aspect ratios varing from to, there is not enough convective motion of the fluid within the cavit due to the viscous force domination, which is delineated b a line which is almost vertical. As the Raleigh number increases, the buoanc force increases giving rise to an effective fluid convection, which will lead to a circulator gre pattern. The circulator gre pattern is indicated b the positive and negative u-velocit values along the smmetric vertical plane as observed in the above figure. The velocit value reaches its maximum for Ra = 7

11 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) Velocit Temperature Vorticit Streamline b) b) b) b) c) c) c) c) d) d) d) d) a) a) a) a) e) e) e) e) Fig.. Distributions of velocit, temperature, vorticit and streamline for a D square cavit at different Ra number for Ra = a a), Ra = b b), Ra = 5 c c), Ra = 6 d d), Ra = 7 e e), H/L= /.

12 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 Table 5 sselt number predicated in a cavit for various H/L aspect ratios Mesh size max min avg Ra = DQ H/L = /) Position) =.) = DQ H/L = /) Position) =.9) = ) DQ H/L = /) Position) =.) = ) Ra = DQ H/L = /) Position) =.6) = ) DQ H/L = /) Position) =.9) = ) DQ H/L = /) Position) =.) = ) Ra = 5 DQ H/L = /) Position) =.95) = ) DQ H/L = /) Position) =.86) = ) DQ H/L = /) Position) =.9) = ) Ra = 6 DQ H/L = /) Position) =.) = ) DQ H/L = /) Position) =.) = ) DQ H/L = /) Position) =.) = ) Ra = 7 DQ H/L = /) Position) =.) = ) DQ H/L = /) Position).) = ) DQ H/L = /) Position) =.) = ) with steep velocit gradients at the middle of the vertical central plane. This is because at higher values of the Raleigh number the non-linear characteristic of fluid convection becomes more dominant in the flow field. A similar trend is observed with the v-velocit variation seen from Fig. 9. The sselt number is an important non-dimensional parameter in convective heat transfer. Thus, the sselt number for several H/L aspect ratios at different Raleigh number is briefl sketched in Fig., respectivel. It is interesting to indicate that the maximum values of the sselt number along the direction is not at bottom wall, but close to the bottom wall. An initial look at the range of the sselt number values shown in these figures clearl indicates that the sselt number increases with increase in the value of the Raleigh number as expected. When convective heat transport reaches minimum at lower value of Raleigh number up to, the value of the sselt number is smaller compared to higher Raleigh number. The pronounced convective heat transfer due to the increase in value of the Raleigh number to 7 is vividl indicated in Fig.. The present coupled numerical algorithm has accuratel predicted the convective heat transport process inside the cavit. Thus, it is convinced that the present scheme is an effective and robust method for predicating the heat transfer problems when the governing equations in velocit vorticit form are used as a full coupled sstem of equations.

13 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) a) b) c) Fig.. Distributions of a) velocit, b) temperature and c) vorticit for Ra = 5, H/L= / a) b) c) Fig.. Distributions of a) velocit, b) temperature and c) vorticit for Ra = 6, H/L= /.

14 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) a) b) c) Fig. 5. Distribution of a) velocit, b) temperature and c) vorticit for Ra = 7, H/L= / a) b) c) Fig. 6. Distributions of a) velocit, b) temperature and c) vorticit for Ra = 5, H/L= /.

15 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) a) b) c).5 Fig. 7. Distributions of a) velocit, b) temperature and c) vorticit for Ra = 6, H/L= / a) b) c) Fig. 8. Distributions of a) velocit, b) temperature and c) vorticit for Ra = 7, H/L= /.

16 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) u v - - Ra= Ra= Ra= 5 Ra= 6 Ra= Ra= Ra= Ra= 5 Ra= 6 Ra= x 8 u v - - Ra= Ra= Ra= 5 Ra= 6 Ra= Ra= Ra= Ra= 5 Ra= 6 Ra= x 5 u - - Ra= Ra= Ra= 5 Ra= 6 Ra= 7 v -5 - Ra= Ra= Ra= 5 Ra= 6 Ra= x Fig. 9. Centerline velocit profiles of the smmetr for Ra Conclusions The D Navier Stokes equations in velocit vorticit form are numericall solved using the DQ method and the numerical algorithm developed has been successfull implemented to stud natural convection in a differentiall heated cavit. Higher-order polnomial approximation used for discretizing the partial differential derivatives in the DQ method has been effectivel used to obtain accurate numerical results while solving the velocit vorticit form of the Navier Stokes equations. More importantl the higher-order polnomial approximation used in the DQ method enabled the computation of the vorticit values at boundar more accuratel. The coupled solution procedure followed in the present scheme completel eliminated the explicit specification of the vorticit boundar values without involving

17 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) Ra =, H/L = /) Ra =, H/L = /) Ra =, H/L = /) Ra = 5, H/L = /) Ra = 5, H/L = /) Ra = 5, H/L = /) Ra = 6, H/L = /) Ra = 6, H/L = /) Ra = 6, H/L = /) Ra = 7, H/L = /) Ra = 7, H/L = /) Ra = 7, H/L = /) Fig.. sselt number for several H/L aspect ratios at different Ra numbers.

18 6 D.C. Lo et al. / Journal of Computational and Applied Mathematics 7) 9 6 an iterative procedure. Test results for natural convection in a differentiall heated cavit indicates that: Velocit vector contours and temperature contours plotted and the sselt number variations for Ra 7 in a square cavit are in quantitative agreement with the results available in the literature. Vorticit contours plotted show the increase in vorticit values with increase in the Raleigh number in terms of the variation of the aspect ratios H/L of the cavit. Since the computer memor and computational effort are directl proportional to the number of grid points used, the present numerical scheme indicates that a significant saving in computational time can be achieved. The present stud proposed a new numerical scheme to solve two-dimensional Navier Stokes equations b efficientl exploiting the advantages of both the velocit vorticit form of the Navier Stokes equations and the DQ method. Acknowledgments The support under Grants NSC 9-8-E-6-, 9-6-E--7 and 9--E-6- b the National Science Council of Taiwan are gratefull acknowledged. References [] G.K. Batchelor, An Introduction to Fluid Mechanics, Cambridge Universit Press, Cambridge, UK, 967. [] R.E. Bellman, B.G. Kashef, J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phs. 97) 5. [] O. Daube, Resolution of the D Navier Stokes equations in velocit vorticit form b means of an influence matrix technique, J. Comput. Phs. 99). [] H. Ding, C. Shu, K.S. Yeo, D. Xu, Development of least-square-based two-dimensional finite-difference schemes and their application to simulate natural convection in a cavit, Comput. & Fluids ) 7 5. [5] H. Fasel, Investigation of the stabilit of boundar laers b a finite-difference model of the Navier Stokes equations, J. Fluid Mech ) [6] P.M. Gresho, R.L. Sani, Advection diffusion and isothermal laminar flow, Incompressible Flow and the Finite Element Method, vol.. Wile, New York,. [7] G. Guj, F. Stella, A vorticit velocit method for the numerical solution of D incompressible flows, J. Comput. Phs. 6 99) [8] K.A.R. Ismail, V.L. Scalon, A finite element free convection model for the side wall heated cavit, Internat. J. Heat Mass Transfer ) [9] D.C. Lo, K. Murugesan, D.L. Young, merical solution of three-dimensional velocit vorticit Navier Stokes equations b finite difference method, Internat. J. mer. Methods Fluids 7 5) [] W.H. Press, S.A. Teukolsk, W.T. Vetterling, B.P. Flanner, merical Recipes in Fortran 9, second ed., Cambridge Universit Press, New York, 996. [] C. Shu, Differential Quadrature and Its Application in Engineering, Springer, London,. [] C. Shu, H. Ding, K.S. Yeo, Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier Stokes equations, Comput. Methods Appl. Mech. Eng. 9 ) [] C. Shu, B.E. Richards, Application of generalized differential quadrature to solve -dimensional incompressible Navier Stokes equations, Internat. J. mer. Methods Fluids 5 99) [] C. Shu, H. Xue, Comparison of two approaches for implementing stream function boundar conditions in DQ simulation of natural convection in a square cavit, Internat. J. Heat Fluid Flow 9 998) [5] G. de Vahl Davis, Natural convection of air in a square cavit, a bench mark numerical solution, Internat. J. mer. Methods Fluids 98) 9 6. [6] J.M. Vanel, R. Peret, P. Bontoux P, in: K.W. Morton, M.J. Baines Eds.), merical Methods for Fluid Dnamics, vol. II, Clarendon Press, Oxford, 986, pp [7] K.L. Wong, A.J. Baker, A D incompressible Navier Stokes velocit vorticit weak form finite element algorithm, Internat. J. mer. Methods Fluids 8 ) 99. [8] X.H. Wu, J.Z. Wu, J.M. Wu, Effective vorticit velocit formulations for three-dimensional incompressible viscous flows, J. Comput. Phs. 995) [9] Y.L. Wu, G.R. Liu, A meshfree formulation of local radial point interpolation method LRPIM) for incompressible flow simulation, Comput. Mech. )