# UFR 4-10 Best Practice Advice

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# Natural convection in simple closed cavity

## Best Practice Advice for the UFR

### 7.1 KEY PHYSICS

The case consists of a turbulent flow in a tall cavity driven by buoyancy forces and hence the key physical aspects which must be modelled adequately are:

• Hydrodynamics of turbulent shear flow;
• Turbulent scalar transport;
• Density dependence upon temperature;
• Buoyancy forces;
• Interaction of turbulence and buoyancy forces.

### 7.2 NUMERICAL MODELLING

Probably the most important numerical aspect is the grid resolution across the cavity width. Based on the calculations of Ince and Launder [38], using a low-Re k-ε model, it is recommended that at least 60 nodes are employed, with strong symmetric clustering towards the walls. The case is much less sensitive to grid resolution in the vertical direction and indeed can be solved for the mid-height distributions quite adequately using a one-dimensional model [19, 38].

The application of low-Re turbulence models in particular necessitates the use of a very fine mesh normal to the heated walls. In general the near wall cell should have a thickness normal to the wall of not greater than ν/uτ where uτ is the local friction velocity (i.e. y+ ≤ 1) however this depends to some degree on the specific model employed.

As buoyant flows in cavities typically display various forms of recirculation it is important, as with any such recirculating flow, that a suitably-accurate (i.e. at least second-order) convection discretization is employed, in particular for the momentum and energy equations.

### 7.3 physical modelling

#### Thermodynamic effects

The first consideration in modelling buoyant flows in both the laminar and turbulent regimes is the use of an appropriate modelling assumption for density. The Boussinesq approximation is valid in many applications of engineering interest and can lead to computational savings but care should be taken to assess the size of density variations relative to the average density. In the present case, the Boussinesq approximation is quite valid as the maximum non-dimensional temperature difference, θ, is only 0.064.

In the non-Boussinesq case care should be taken to ensure that mass is appropriately conserved, especially for closed cavities for which it is clear that the enclosed mass of fluid must remain constant. This care is particularly necessary in the case of steady flows for which many solution procedures (including those of many commercial CFD codes) do not conserve the total enclosed mass between iterations. One answer is to specify an initial condition with the appropriate mass and then solve the problem using a physical time-marching procedure, ensuring good convergence at every time step.

#### Turbulence effects — momentum equations

As reviewed above, there exist a wide variety of turbulence modelling approaches which have been applied to turbulent buoyant flows in cavities using both RANS and LES. Whichever level of modelling is employed, there is general agreement in the literature that it is necessary to integrate right up to the wall using some form of “low-Reynolds-number” treatment and that a wall-function approach is not sufficient. This applies both to fully turbulent cases and to transitional flows where a turbulent wall function clearly cannot be assumed.

Simple two-dimensional flows in tall cavities, such as the present UFR case, are in fact relatively straightforward shear flows from the hydrodynamic perspective and as such can be modelled quite effectively using a simple eddy-viscosity assumption. In more complex cavity flows, however, a more sophisticated anisotropic treatment may be required using, for example, an algebraic-stress or non-linear two-equation model or even a full Reynolds-stress transport model.

Depending upon the level of accuracy required, and indeed the quantities of interest, probably the most basic acceptable approach would be an isotropic two-equation model with low-Reynolds-number wall treatment. The approach of Ince and Launder [38], combining the Launder-Sharma k-ε model with the Yap correction (assuming a constant turbulent Prandtl number), has been shown to give reasonable predictions of mean heat transfer rate, temperature and velocity though its predictions of turbulence quantities is much less reliable. It is important to note however that the Launder-Sharma model alone without the Yap correction can lead to a strong overprediction of the heat-transfer rate across a tall cavity and hence great care must be taken to validate the chosen model. The use of standard buoyancy modifications to the chosen turbulence model is also important, for example the additional terms in the k-ε model described in [28].

#### Turbulence effects — energy equation

As with any flow with heat transfer, the effects of turbulence not only on the momentum equations but also the energy equation must be considered. Frequently these are represented using an isotropic assumption and constant turbulent Prandtl number (Prt) however the turbulent heat fluxes are not generally isotropic and the turbulent Prt is not constant.

At least five levels of treatment for the turbulent heat fluxes can be identified:

1. Isotropic with constant Prandtl number.

2. Isotropic with one/two equation model for eddy diffusivity: αt=fn(

3. Generalised gradient diffusion hypothesis (GGDH).

4. Anisotropic with algebraic flux model (based on one/two equation model for and ε).

5. Anisotropic with full heat flux transport model.

Hanjalić [31] recommends the use of at least level 4 for flows of this type, on the basis that, in general, some representation of counter-gradient diffusion is essential but that level 5 is probably impractical in most industrial applications. It should be noted however that such models are not in general available in standard commercial CFD packages.

For the specific case considered here, it is possible to predict the mean temperature profile quite accurately using an appropriate eddy-viscosity model (discussed above) combined with level 1 above, i.e. a simple constant Prandtl number assumption. It should be noted that this will not be sufficient if more detailed data are required or for more complex buoyant cavity flows.

### 7.4 Application uncertainties

In comparing simulation results with experimental data, great care should be taken to ensure that the thermal boundary conditions employed in the simulation genuinely represent the conditions prevailing in the experiment. If a 2D simulation is being carried out then it should be checked that the flow in the experimental cross-section considered is indeed two-dimensional in nature.

### 7.5 RECOMMENDATIONS FOR FUTURE WORK

There still remains much scope for further studies concerning this test case  principally in relation to the appropriate level of turbulence modelling required, finding a suitable compromise between accuracy and computational expense. There is also not full agreement between the DNS and experimental data for this case, a point which is worthy of further investigation.