Test case calculated with CASTEM2000 v.99 developed by CEA.
9.4.1. Introduction
The test case chosen is the computation of the steady-state, natural convection flow of a compressible fluid (air) in a square cavity, with differentially heated vertical walls. It is customary to assume incom-pressible flow (the local Mach numbers are extremely low), constant flow properties and to model buoyancy forces with the so-called Boussinesq approximation (Gray and Giorgini, [1976]). Under this
hypothesis, density is assumed constant in the flow equations except in the gravity force term, where small variations due to temperature differences are modelled in the form g(ρ-ρ0 )/ρ0 ≈ g∆T/T0 . Most natural convection solvers used in the industry are indeed based on this model, but these assumptions are only valid for small temperature differences, and may be the source of substantial errors in heat transfer coefficients. For large temperature differences, compressibility effects come into play even if the local Mach numbers remain extremely small so that it becomes necessary to use a compressible model to compute the flow.
This test case will illustrate the following guidelines:
• the effect of the flow model on the heat transfer results (modelling errors);
• the grid-sensitivity of results (numerical errors).
9.4.2. Geometry and boundary conditions
The geometry is a square cavity, of length L, with thermally insulated horizontal walls, and differentially heated vertical walls. The right wall is maintained at a “cold” temperature Tc, the left wall at a “hot”
temperature Th. No slip boundary conditions are assumed on all solid walls.
9.4.3. Grids
The grids used consist of a series of structured quadrilateral meshes, slightly clustered near the solid boundaries where stronger gradients occur. In order to conduct a grid-convergence study, the follow-ing meshes were used: 20x20, 40x40, 80x80, and 160x160.
9.4.4. Features of the simulation
The cavity is filled with air, assumed to behave as a perfect gas, with Prandtl number Pr=0.71 and the ratio of specific heats γ equal to 1.4. The viscosity µ and the thermal conductivity λ are assumed to depend only on temperature, according to Sutherland’s law,
S Ra=106, for which the flow is known to be laminar. We recall that the Rayleigh number is defined as:
2
Uniform initial conditions are assumed, with air at atmospheric pressure Po=101325 Pa, and a tem-perature To= (Th + Tc)/2 = 600 K. Two temperature ratios ε = (Th - Tc) / (Th + Tc) will be considered, ε = 0.01 for which the Boussinesq flow model is valid, and ε = 0.2 for which a compressible flow model is required. With these definitions, expressions for the hot and cold temperatures Th and Tc are respec-tively Th = To (1 + ε) and Tc = To (1 - ε).
The numerical scheme used for the computations is an elliptic finite element solver based on an as-ymptotic approximation of the Navier-Stokes equations at low Mach numbers in which the pressure is split into an average time-dependent thermo-dynamic pressure which obeys the perfect gas law, and a hydrodynamic fluctuation pressure which obeys a Poisson pressure equation just as for incompressi-ble flow. The discretisation is based on bilinear quadrilateral elements (Galerkin method), and the so-lutions are obtained by time-marching to steady state using a time-accurate scheme.
9.4.5. Numerical results
Small temperature difference caseThe main objective of the simulation is to compute the heat transfer to the walls. Hence, the figures below show distribution plots of the local Nusselt number along the vertical walls. The Nusselt number is a non-dimensional number defined as:
c wall
In this first computation, we perform a grid convergence study for the Ra = 106 and ε = 0.01 case.
Table 1 shows the evolution of the average, maximum and minimum Nusselt number computed on the right wall as the mesh is progressively refined. In all cases, steady-state was obtained with a drop of the momentum residual of more than 10 orders of magnitude. A reference solution correspond to grid-converged Boussinesq solutions, which assume incompressible flow and constant flow properties. In that case, a symmetric solution is obtained so that identical heat transfer distributions are obtained on the walls.
Table 1: Heat transfer results for ε = 0.01 case; grid convergence study and comparison with refer-ence solutions; the figures in brackets represent the relative error with respect to Le Quéré et al.
[1985].
The grid convergence study shows that the solution varies sensibly up to the 80x80 mesh, and very lit-tle when the mesh is refined further (see Fig. 1). Thus, one can assume that the solution on the 160x160 mesh is nearly grid converged. For that solution, the results are in close agreement (relative error in heat transfer of the order of 0.2%) with reference solutions found in the literature, which corre-spond to incompressible flow models with the Boussinesq approximation. Thus, discrepancies be-tween the solutions computed on the coarse meshes and the reference solutions can be attributed to numerical errors, stemming from the use of too coarse meshes or excessively dissipative schemes.
On the other hand, on fine enough meshes, the incompressible flow models and the more general low Mach number compressible flow models yield extremely close results (and within the band of refer-ence solutions). Thus, for small temperature differrefer-ences, the modelling errors (errors due to the choice of flow model) are negligible compared to numerical errors.
“Large” temperature difference case
In this case, we consider a larger temperature difference, corresponding to ε = 0.2. Note that this pa-rameter is still quite small compared to 1, and that it would be tempting to use an incompressible flow model with the Boussinesq approximation. However, we shall see that even for this moderately large temperature difference, and in spite of the extremely low Mach numbers, a compressible flow formula-tion is required to solve the problem.
Table 2 shows the grid converged heat transfer results obtained on the fine mesh (160x160) (validated against other compressible solvers by Paillère et al., [1999]), alongside the reference incompressible flow results. Note that the average Nusselt number has remained almost constant (as was observed in Chenoweth et al., [1986], Le Quéré et al., [1992] and Paillère et al., [1999]). However, the minimum and maximum Nusselt numbers are substantially different, as may be observed from the distribution plots in Fig. 2, which display a noticeable non-symmetric character attributed to compressibility effects.
The maximum local Mach number in the flow was found to be of the order of 3.4 x10-4 . Relative errors in heat transfer prediction between the compressible and the incompressible flow models are of the order of 5%.
Thus, for large temperature differences, the incompressible flow model with Boussinesq approximation and the low Mach number compressible flow model yield substantially different results. The difference can be attributed to modelling errors, as it is known (but unfortunately often forgotten) that the former model is only valid for small temperature differences in Gray et al. [1976].
ε=0.2 160x160
Ref. Incomp.
Sol.
<Nu> 8.817 8.826 (0.1%)
Nu_min 0.930 0.979
(5%)
Nu_max 16.869 17.536
(4%)
Table 2: Heat transfer results for ε = 0.2 case ; the figures in brackets represent the relative error be-tween the reference incompressible solution (Le Quéré et al., [1985]) and the ‘grid converged’ com-pressible solution
9.4.6. Conclusions
This example illustrates two sources of error that can affect a CFD simulation, numerical error due to insufficient grid refinement and modelling error due to inappropriate choice of flow model. When ana-lysing flow with heat transfer and very low Mach numbers, care must be used in the choice of flow model (i.e extremely low Mach does not imply incompressible flow), and grid-sensitivity assessed by comparing results on grids of different size.
9.4.7. References
Chenoweth, D. and Paolucci, S. (1986), “Natural Convection in an Enclosed Vertical Layer with Large Temperature Differences”, J. Fluid Mech., Vol. 169, pp. 173-210.
Gray, D.D. and Giorgini, A., (1976) “The Validity of the Boussinesq Approximation for Liquids and Gases”, Int. J. Heat Mass Transfer, Vol. 15, pp. 545-551, 1976
Le Quéré, P., Masson, R. and Perrot, P. (1992), “A Chebyshev Collocation Algorithm for 2D Non-Boussinesq Convection”, J. Comput. Phys., 57, pp. 320-335.
Le Quéré, P. and Alziary de Roquefort, T. (1985), “Computation of Natural Convection in Two-dimensional Cavities with Chebyshev Polynomials”, J. Comput. Phys., Vol. 57, pp.210-228.
Paillère, H. and Magnaud, J.P. (1998), “A Finite Element Elliptic Flow Solver for Low Mach Number Compressible Flows”, pp. 419-424, Proc. 10th Int. Conf. on Finite Elements in Fluids, Tucson, Arizona, January 5-8, 1998.
Paillère, H., Viozat, C., Kumbaro, A. and Toumi, I. (1999), “Comparison of Low Mach Number Models for Natural Convection Problems”, EUROTHERM Seminar No. 63 ‘Single and Two-Phase Natural Circulation`, Genoa, Italy, 6-8 September 1999
0 0.25 0.5 0.75 1
y/L
0 5 10 15 20
Nu(y)
right wall 20x20 right wall 40x40 right wall 80x80 right wall 160x160
Figure 1: Grid convergence study for ε = 0.01 case.
0 0.25 0.5 0.75 1
y/L
0 5 10 15 20
Nu(y)
epsi=0.2, right wall epsi=0.2, left wall epsi=0.01, right wall epsi=0.01, left wall
Figure 2: Effect of temperature difference on heat transfer results