I. Verification

3. Programs (Codes)

3.1. FRECON

A modified version of the three-dimensional numerical code FREVON3V (FRE) has been used to obtain reference solutions and to perform several tests of the model. This finite difference false transient solver, developed at the University of New South Wales,  uses vorticity-vector potential formulation of the Navier-Stokes and energy equations for steady, laminar flow of a viscous, incompressible fluid. Solutions were obtained for Cartesian coordinates on uniform meshes. Reliability and robustness of the code has been tested over many years and has been reported in numerous papers [4]. It is also the fastest solver from among all other tested in this paper.

The two-dimensional solutions were obtained using the code FRE with only 5 grid points for the channel depth, with slip kinematic boundary condition and adiabatic thermal boundary conditions for the side walls to eliminate the flow components in the third direction. These 2D results, generated for the sequence of mesh resolutions from 21x21 to 301x301 are given in the Table 2. The maximum and minimum of the two velocity components and the averaged Nusselt number for the cold wall are displayed as basic monitors of the code convergence performance. Additionally we present max./min. values of the velocity components on horizontal and vertical mid-line. The solutions were assumed to converge when all residues of the equations imbalance were less than 10^-9.  A typical CPU time necessary to reach converged solution shows nearly cubic growth with a grid resolution and varies from 180sec for FRE1 to 3.6^.10⁵sec for FRE7 case (all CPU times are scaled to PentiumHT 3GHz processor with 2GB memory).

Table 2. FRECON3V mesh dependence test: 

3.2. FLUENT

A steady state two-dimensional solutions for the problem defined above were obtained using commercial finite volume code Fluent 6.2 [5]. Several uniform structural grids were tested, results obtained for three of them are displayed in Table 3. Fluent gives possibility to select large spectrum of different solvers and solving strategies using the user friendly interface. Selection of the appropriate model is often crucial both in terms of speed as well as accuracy of the result. Following experience gained during several test runs the implicit false transient method was used to reach efficiently a steady state. Spatial derivatives were approximated using QUICK scheme, which is based on a weighted average of second-order-upwind and central interpolation of the variable. Pressure-velocity coupling was done using SIMPLE algorithm. The non-linear density variation given by (2.5) was implemented in the solver. Fluent uses internally dimensional variables, hence results of the simulations were scaled  using relations (2.6). The convergence criteria was given by the residua of the solution less than 10^-6. It turned out that solutions obtained using single precision solver were different by more than 5 % for extremes of velocity values and their location, even for the finest mesh 380x380. Therefore, all the results reported in this work were performed using double precision solver. A typical CPU time necessary to reach converged solution for the coarse mesh case (FLU0) is 2^.10⁴s, which is much slower in comparison with the FRECON solver. It is worth noting, however, that solution obtained for this relative coarse mesh is much closer to the fine mesh solution (FRE7) then similar solution obtained with the FRECON (FRE2).

3.3. FIDAP

3.4. SOLVSTR

3.5. SOLVMEF

In above h stands for the distance between two neighbouring points (see Figure 2), and F_P, F_N, F_S, F_W, F_E, F_NW, F_NE, F_SW, F_SE denote the values of a function being approximated in the vicinity of arbitrary point P, w₁,w₂ denote the value of a weighting function. In the present calculations the following weighting function is used:

Figure 2. Computational molecule used in the diffuse approximation method

The resulting algebraic equations result in a sparse matrix with at most nine non-zero elements in each row of a matrix, and are solved making use of Gauss – Seidel algorithm. The number of non-zero elements in the matrix is closely related to the number of neighbouring points taken into account during application of diffuse approximation. Each computational molecule consists of nine nodes (see Figure 2). The use of Gauss-Seidel method to solve approximated equations makes the algorithm much slower in comparison with classical approach, for instance the one applied in SOLVSTR. The application of much more sophisticated solvers will be considered in future work as well as the use of a preconditioner to improve performance of the algorithm. A CPU time necessary to reach converged solution (error less than 10^-6) for 100² mesh is fifteen times slower than corresponding case performed by SOLVSTR algorithm. This made more extensive discretization dependence study out of reach.