INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 24,89-99 (1 987)

NUMERICAL STUDIES OF BUOYANCY-DRIVEN TURBULENT FLOW IN A RECTANGULAR CAVITY

C. P. THOMPSON,* N. s. WILKES+ AND I. P. JONES*

A E R E Harwell, Didcot, Oxon OX11 ORA, U . K .

SUMMARY

In this paper some numerical predictions are presented for the buoyancy-driven turbulent flow and heat transfer in a large cavity using both a prescribed eddy viscosity model and a k-E model. The results from these models are in good agreement, although the calculations with the prescribed eddy viscosity model are considerably cheaper. The results also show the correct qualitative behaviour, but there are some differences between the predictions and the available experimental data. Possible reasons for these are discussed.

INTRODUCTION

Laminar buoyancy-driven flows are reasonably well understood and can usually be predicted accurately using numerical mcthods. For turhulcnt buoyancy-driven flows, however, the situation is quite different. The principal turbulence models in use were developed for high Reynolds number forced convective flows. Furthermore, the boundary conditions usually adopted for these models are obtained lrom semi-empirical laws of the wall derived from one-dimensional forced flows and are not appropriate for strongly buoyant flows. As well as thc modelling uncertaintics several authors have reported numerical difficulties in attempting to tackle the problem of predicting buoyant turbulent flows.

This paper summarizes part of the theoretical work that has been carried out at AERE Harwell as part of a wider programme to obtain an understanding of buoyant turbulent flows and to develop tcchniques for their prediction. In particular, the paper considers the turbulent double glazing problem, that is the flow in a two-dimensional rectangular cavity of height L and width D with the horizontal walls adiabatic, one vertical wall heated to a constant temperature TI and the other cooled to a uniform temperature To. This configuration is sketched in Figure 1.

Only limited experimental information is available for this p r ~ b l e m l - ~ . These works indicate that, at high Rayleigh numbers, the flow takes place in very narrow boundary layers and that the bulk of the cavity is stably stratified and virtually stagnant. These features, as will be seen later, pose great difficulties for the effective numerical solution of these problems.

The first part of the paper describes the turbulencc models that have been used. These are a prescribed eddy viscosity model (PEV) and the well-known k--E model. Artificial boundary conditions for these arc also considered. The results froin these models, and their variants, are

*Computer Science and Systems Divison ' Engineering Sciences JXvision

0029-5981/87/010089-11$05.50 0 1987, UKAEA

90 C. P. THOMPSON, N. S. WILKES AND 1. P. JONES

Figure 1. Configuration

then discussed for a standard case for which very limited experirncntal data are available. The predictions from the different models are found to be in good agreement, although the computations with the PEV model are considerably cheaper. The results show the correct qualitativc behaviour: the narrow boundary layers, revcrsc flow in the boundary layers and large stagnant regions in the centre of the cavity, The detailed comparisons indicate, however, that the predicted velocities are too high. Possible reasons for this are discussed.

PRESCRIBED EDDY VISCOSITY TURBULENCE MODEL

Basic model

Following a suggestion by Leslie: a prescribed eddy viscosity model has been developed and used successfully for a variety of flows. This model was based upon the work of George and Capp5 who carried out a theoretical and experimental investigation into the boundary layer flow near a single heated wall. By using scaling arguments based upon the wall heat flux, George and Capp postulated the existence of three subregions within the boundary layer. These regions are

1. The conductive sublayer adjacent to the vertical wall. In this region the kinematic boundary conditions require the turbulence to be small and that conduction is the major heat transfer mechanism.

2. The buoyant sublayer, a ‘buffer-zone’ between the visco-conductive sublayer and the outer flow. In this region (molecular) viscous and conduction effects are negligible, since it is sufficiently far from the wall for these effects to be dominated by turbulence. Moreover, the region is sufficiently close to the wall for mean flow convection to be unimportant. As a result the heat flux is constant across the layer. In such a region the only governing parameters are the heat flux and the buoyancy gfl, where g is the acceleration due to gravity and B the coefficient of volumetric expansion.

3. The outer flow, where both turbulence and convection contribute.

BUOYANCY-DRIVEN TURBULENT FLOW 91

Experimental verification of this analysis appears to bc far more difficult than in the constant shear-layer case because of the extreme narrowncss of the boundary layers. Some doubt has been cast on these basic assumptions when the boundary layer thickness is increasing.6 However, one-dimensional studies by Betts et d7 suggest that, for cavities, there is reasonable agreement with the derived forms of the equations.

These assumptions lead to a temperature profile within the constant heat flux layer adjacent to a cold wall of the form

T* = - [A,(Pr) + K , x * - ~ ' ~ ] ( 1 )

In this expression T* is the non-dimensional temperature, x* is the non-dimensional distance from the vertical wall, A, is a function of the Prandtl number ( P r = v/ti) alone, K , is a constant, v is the kinematic viscosity and ti the thermal diffusivity. The non-dimensionalization in these expressions has been carried out using the wall heat flux and the buoyancy. That is, the length scale is

(&) the velocity scale is

(SBF, w4 and the temperature scale is

F , is a scaled wall flux and is given by

The Boussinesq approximations for turbulence arc that

and

We use these, equation ( 1 ) and an assumption of constant eddy Prandtl number to derive a turbulent eddy viscosity which is consistent with the constant heat flux layer. In order to be useful for cavity flows, this profile has to be extended to two-dimensional flows. The prescribed eddy viscosity model that has been used in this work has the form

where vT is the turbulent eddy diffusivity, Gr, the Grashof number based on height, yB(T, - T0)L3/v2, A the aspect ratio LID, N u the Nusselt number based upon width and a length scale. For this model the Huleev length scale' has been used throughout. Near to a single wall this length scale is similar to the distance from that wall, and has the advantage of being smooth everywhere, unlike the distance to the nearest wall. The constant K , was given the value 5.6 that George and Capp obtained from their analyses of experimental results.

Note that this model requires an estimate of the Nusselt number, which is not known a priori .

92 C. P. THOMPSON, N. S. WlLKES AND I. P. JONES

The calculations presented here used an estimate based upon experimental results:'

Nu = 0.043 Ra"' ( 3 )

although it would be relatively easy to iterate to find a suitable value. Numerical experiments haw shown that the calculations are not particularly sensitive to the value used.

The strong boundary layer effects found in these flows make it essential to use very strongly stretched grids in order both to resolve the boundary layers and yet have economical computing. As a result investigations have bcen performed into discretization techniques for problems such as these, where large variations of the solution take place over small intervals. These techniques, based on co-ordinate transformations, enabled grids which gave adequate resolution of the boundary layers to be used. This work has been described by Jones and Thompsong and Jones," and the techniques have been successfully used in the calculations presented here.

The prescribed eddy viscosity model was implemented into a stream-function-vorticity code IOTA4 and predictions obtained for the flows in large cavities and also other configurations. It is worth noting that both upwind and central differencing of the advective terms in the governing equations have been used, with little difference to the results. Some of these calculations will be discussed later.

One of the main advantages of this model is its simplicity, but it does rely upon a suitable prescription of the structure of the turbulent flow. The choice that has been discussed so far does not allow for laminar values of the eddy viscosity and diffusivity in the central core region. It is possible, however, to extend the model to rcmove some of its deficiencies.

Attenuation of the viscosity

One of the main drawbacks of thc prescription given in equation (2) is that the eddy viscosity is a maximum in the centre of the cavity. This is clearly not physically reasonable since the flow there is stagnant, and hence laminar. This does not matter for the calculations for the velocity field since the high levels of the viscosity ensure that the core flow is zero. However, because of the assumption that the eddy Prandtl number is a constant, the thermal diffusivity is also far too high in the cavity. Experimental evidence indicates that there are fairly high levels of vertical temperature stratification in the stagnant core. Because of the high thermal diffusivity present in the model, however, the predictions have significantly lower levels of vertical stratification. This drawback may be overcome by attenuating the prescription for the eddy viscosity given by equation (2). Two forms of attenuation have bcen tried. The first form used was a Van Driest attenuation. This, however, only modifies the eddy-viscosity in the outer edges of the boundary layers and leaves the turbulence field unchanged in the interior of the cavity. Therefore this made very little difference to the overall form of the results.

A second form used was to multiply the turbulent viscosity in equation (2) by an udhoc function

0.511 +- tanh[- 103(d-b)])

where d is the distance of the point from the nearest boundary and b an estimate of the boundary layer thickness. This modification reduces the value of thc turbulent viscosity to zero outside the boundary layers, so that laminar values of the viscosity and thermal diffusivity prevail in the centre of the cavity. The implementation of this modification gave rise to numerical difficulties for the iterative method in the finite difference code used initially for the predictions. The reasons for these problems arc discusscd latcr in this paper.

The model was also implemented in a finite element code based upon the TGSL finite element subroutine library.' A standard Galerkin formulation with four-noded quadrilateral elements

BUOYANCY-DRIVEN TURBULENT FLOW 93

was used with bilinear interpolation for velocities and temperature, and piecewise constant interpolation for the pressure. The numerical difficulties with the attenuated eddy viscosity were not encountered here because a different solution procedure, a Newton method with direct solution of the resulting linear equations, was used. Because of this it was also easier to use more strongly stretched mesh distributions. As will be seen later, the results were little different to those from the finite difference code, apart from an increased level of stratification in the core regions.

This choice of' elements was somewhat fortuitous. Later work, by Cliffe et a1.,12 showed that the conventional choice of pressure interpolation i s unable to represent this type of flow on coarse grids. This is because, in the stagnant core, the pressure gradient balances the buoyancy forces and linear pressure elements (coupled with quadratic variation for velocities and temperatures) do not have sufficicnt degrees of freedom to satisfy this degenerate form of the equations.

Artificial boundary conditions, prescribed eddy viscosity

The boundary conditions on the solid surfaces create difficulties for the predictio,, of these flows because of the very fine scales present in the flow. For example, it is important to have grid points present in the visco-conductive sublayer in order to impose properly the no-slip velocity conditions and the prescribed temperature conditions, i.e. the 'natural' boundary conditions. For forced convective flows it is usual to apply the semi-empirica1 logarithmic law of the wall and to match the solutions to these profiles. However, this procedure takes no account of buoyant action, and the analysis of Gcorgc and Capp indicates that the behaviour of the variables near a vertical wall is unlike that of the log law. A detailed analysis of the flow in the buffer layer has been performed using boundary layer assumptions and the results used provide artificial boundary conditions for use in computer codes. This analysis is quite complex, particularly for the stream-function-vorticity formulation, and is not given here. The analysis also indicates that the expression given by George and Capp for the velocity profile appears to be inconsistent with the behaviour of the temperature in equation (1). However, this depends upon various other assumptions which must be made regarding the turbulence modelling. Full details may be found in Reference 13. With this approach the first grid point need only be in the buffer layer and not in the conductive sublayer. This point is however still between the velocity peak in the boundary layer and the wall and considerable grid stretching is still required.

These conditions also require a value for the empirical function A,(Pr) in equation (1). This was chosen by matching the integral of the heat flux through the vertical walls to the value given by the experimental correlation for the Nusselt number, equation (3) .

k--6 MODEL

Description of the model

One of the drawbacks of the prescribed eddy viscosity model is that the turbulence levels have to be defined ci priori. For simple flows this is reasonably straightforward, but it is not practicable for more complex situations. Predictions have also been obtained using a fairly standard k--E turbulence model. Details of this model may be found, for example, in Reference 14. Note that terms involving the destruction of turbulence were included in the equations for k and E. These have the form recommended by Goussebaile and Vi01let.l~ The standard k-c model was initially developed for high Reynolds number forced flows and it is known to be inadequate for low Reynolds number near-wall flows such as we effectively have here. N o attempt, however, was made to use a low Reynolds number model, apart from including the molecular diffusivities

94 C. P. THOMPSON, N. S. WILKES AND I. P. JONES

in the effective turbulent diffusivitics. A constant eddy Prandtl number of unity was also used. This model was implemented into the Harwell finite difference code TUFC,16 which is itself a

modified version of the well-known TEACH program.

Numerical details

The previous section has highlighted the importance of thc ncar wall regions, particularly for the heat transfer. There are difficulties, however, with the implementation of the type of artificial boundary condition described earlier in this paper, since the behaviour of the k--E model is inconsistent with the assumptions of the prescribed eddy viscosity model. For this reason the usual logarithmic boundary conditions were imposed, although care was taken to ensure that the grid resolved the bulk of the turbulcnt flow, and several points wcrc between the velocity peak in the boundary layer and the wall. In fact, for the calculation reported here, the first grid point for the velocity parallel to the walls lay in the laminar sublayer and not the logarithmic layer.

Since the model adopted allows for the destruction of turbulcncc in the strongly stratified regions of the flow, severe numerical difficulties of the type discussed earlier were encountered. To overcome these a timc-stepping strategy was adopted, with the time steps chosen so that

where y is the vertical co-ordinate. Using this strategy, the calculational procedure converged to a steady state. For larger time steps the procedure did not converge, and instead the numerical perturbations were converted into gravity wavcs which did not die away. Thc underlying reasons for this, and a simple convergence analysis, are given by Jones.17 It should be noted that time stepping amounts to severe undcr-relaxation. Since the conduction lime-scale for the strongly stratified regions is considerably greater than that given by equation (4) many time steps are required before convergencc is obtained. The calculations reported here were obtained with a 30 x 30 non-uniform grid and continued till the maximum time derivatives of all the variables were less than 5 x This required many thousands of times steps and hence was comparatively expensive. Plausible-looking results were obtained in this case after only a few steps. The temperature field, however, altered very slowly because the time steps were much smaller than the conduction time-scale. There were, however, significant changes in the temperature field, particularly in the level of vertical stratification, over many time steps. In their study of a similar problem Markatos and Pericleous18 used very small time steps in order to ‘accelerate convergence’, although they used far fewer sweeps than was found to be necessary in the current case.

This configuration has also been studied by Ozoe et al.,” at slightly lower Rayleigh numbers. They report ‘oscillating but non-diverging’ solutions which may be symptomatic of the problems discussed above.

At present, research into a multigrid method based upon the block solution of all variables associated with a particular control volume is being performed. Initial results with laminar flows suggest that good rates of convergence can be obtained for reasonable levels of stratification.*’

The scaling of the problem gives rise to difficulties in another way. The vertical momentum equation is approximately

BUOYANCY-DRIVEN TURBULENT PLOW 95

In the large, stagnant core we have U = V = 0, so this degenerates to

It is possible to decompose the pressure into two components. The first is analogous to the hydrostatic pressure (related to the buoyancy). This is a reasonably large fixed quantity. The other component is to balance velocity fluctuations. Thesc are very small in the stagnant region (effectively tending to zero). Therefore, the first component swamps the contributions from the second and there is a serious loss of accuracy due to the round-off error. This was overcome by adjusting the arbitrary reference temperature To by a quantity determined iteratively to neutralize this contribution. In fact, we put To = T(D/2 , y). This technique improved the accuracy of the calculations considerably.

RESULTS

Results have been successfully obtained using these codes and models for a variety of different configurations. Very few numerical predictions have been presented in the literature,18’21*22 fo r the problem studied here. For this reason, the results in this paper concentrate on a case for which very limited experimental evidence is available. This is a water-filled cavity of aspect ratio 2 for Rayleigh numbers in the range lo’*- lot3. The first set of results consists of a comparison of the different models for the same case at a Rayleigh number of It should be noted that all the codes used the Boussinesq approximation for the variation of the density with the temperature and that the physical properties were taken as constant.

Figure 2 shows a comparison between the vertical velocity profilcs half-way up the vertical heated face for thc various cases considered here. The different calculations from the prescribed eddy viscosity model fall within a fairly narrow range. Because of this only the upper and lower bounds ofthe band are indicated. The results show that the velocity rises very sharply from the wall with the velocity peak lying very close to the wall. On the other sidc of the peak the velocity falls much more slowly and eventually there is a weak reverse flow. This is not shown on the scale of the figure. The predictions from the k--E model show the same features. In particular, the velocity peak is at the same distance from the wall, although the velocity is slightly lower in magnitudc and the

t I I 0

1~ox10-2 2.0x10-2 X / /

Figure 2. Comparison of PEV and k--8 models; large scale water rig at Ra = k--8 results

96 C. P. THOMPSON, N. S. WILKES AND 1. P. JONES

peak is less broad. All of the results also show a stagnant core in the centre of the cavity but they do indicate different levels of vertical stratification. This is illustrated by the figures in Table 1, which give the levels of vertical stratification,

dT L S = - 8Y TI - To

It is difficult to assess the accuracy of calculations which are as expensive as these since it is not possible to perform proper mesh refinement. However, as can be seen from Table I, different types of differencing schemes have been employed (upwind and central) for some of the calculations. Comparison of these indicates that the important features of the flow are represented reasonably accurately.

The table, and other results not presented here, show that the stratification is mainly influenced by the nature of the eddy viscosity in the central core. In particular, it is important to use a realistic attenuation of the prescribed eddy viscosity. It is not sensitive, however, to the distance of the first grid point from the wall.

The Nusselt number is another important parameter. Table I also contains this information from the cases considered here. In the prescribed eddy viscosity model it is, in fact, one of the parameters in the prescription. The correlation of equation (3) used in the prescription gives the value 926 for these parameter values. It should be emphasized, however, that the correlation is being used outside the range of parameters examined by Cowan et a1.' The predictions from the PEV model are quite close to this value, particularly when artificial boundary conditions are used. The k-E results are also reasonably close, considering the completely different approximations used. There is, however, considerable sensitivity in these values to the distance of the first grid node from the boundary, particularly when there is insufficient resolution in this region and natural boundary conditions are used.

Table I . Vertical stratification and Nusselt numbers. Ra =

Boundary Grid Model conditions size Code s N u

PEV natural 32 x 32 Finite difference, upwind differences for the 0.16 740 advection tcrms

PEV George and 32 x 32 Finite difference, upwind differences for the 0.06 965 CaPP advcction terms

PEV George and 32x 32 Finite difference, central differences for the 0.10 970 CaPP advection terms

PEV natural 32 x 32 Finite element, inadequate resolution near 0.22 492 the wall, no attenuation of viscosity

PEV natural 32 x 32 Finite elemcnt, realistic attenuated eddy 0.42 1172 viscosity, points very close to wall

PEV natural 32 x 32 Finite element, realistic attenuated eddy 0.42 815 viscosity, points slightly further from wall than the previous case

k - E log layers 30 x 30 Finite difference 0.34 1292

BUOYANCY-DRIVEN TURBULENT FLOW 97

-\ - - - - _ _ _ Distance from hot wall

Figure 3. Comparison of vertical velocity profiles for the large-scale water rig half-way up the hot race: --- 30 x 30 k--c model with Rayleigh number 4.3 x ~ -experimental rcsults

Experimental results, particularly for velocity distributions, are extremely limited, especially at these very high Rayleigh numbers. There are also considerable experimental difficulties in obtaining accurate measurements. Limited experimental results have been obtained at Harwell using LDA.23 Figure 3 shows a comparison between the predictions of the k-s code at Ra = 4.3 x 10” and the velocity measuremcnts.

Since the PEV results are in fairly close agreement with the k--E predictions they are not shown here, The comparison shows that the velocity pcak is predicted to be at the corrcct location, although the magnitude is too high by about a factor 2. The predicted boundary layer also appears to be too broad and the flow reversal is nearer the centre of the cavity.

It is difficult, however, to ascribe an exact Rayleigh number for the experiments since the physical properties of water vary significantly with temperature. Furthermore, it was difficult to maintain the vertical surfaces at a constant temperature and there was a considerable temperature rise on these surfaces, particularly the hot wall. This increased stratification is likely to reduce the velocity in the boundary layer.

CONCLUSIONS

Numerical results have been presented for the buoyancy-driven turbulent flow and heat transfer in a large vertical cavity using both a prescribed cddy viscosity model and the k-c model. Results havc also been obtained for the PEV model with artificial boundary conditions designed to model better the heat transfer near to the vertical surfaces. Numerical difficulties associated with the prediction of these flows appear to have been largely overcome, although they may require many iterations or time steps if a realistic representation of the stagnant core of the fluid is required, particularly for the lc-c model. The results from the different models are in relatively good agreement, particularly for the velocity distributions with in the boundary layer. The prescribed eddy viscosity model is thcrefore considerably cheaper than the k--E model and is very useful for parametric surveys and providing initial approximations for more sophisticated models. In particular, because it is more closely based upon the physics of vertical boundary layers, it can provide cheap and good estimates of the boundary layer behaviour, and hence provide information on the gridding needed for more sophisticated models.

The results indicate that the models predict the qualitative features of the flows; narrow

98 C. P. THOMPSON, N. S. WILKES AND I. P. JONES

boundary layers, a stagnant core and high Nusselt numbers, although the predicted velocities appear to be too high within the boundary layer and the boundary layer itself too wide. Some of this may well be due to inadequate numerical resolution, not only immediately adjacent to the vertical walls but also at the edge of the boundary layers. This is more likely to affect thc heat transfer than the velocity profiles.

I t is difficult to make further comparisons between the results of the turbulence models without more detailed velocity and temperature measurements at these high Rayleigh numbcrs. The codes should also attempt to model the experimental situations more closely, both the actual thermal boundary conditions imposed and the variation of physical properties with temperature. This should give a better indication of the weaknesses of the models. It is already clear that the behaviour of the flow close to the wall is of crucial importance in determining the heat transfer, and there is a need for greater understanding of this regions. In particular, more sophisticated artificial boundary conditions are needed for use in cases where it is not practicable to obtain the necessary degree of resolution within the boundary layers.

ACKNOWLEDGEMENTS

This work was partially funded by the U. K. National Nuclear Corporation. The authors would like to thank them for the permission to publish this paper.

The authors would also like to thank D. C . Leslie, G. L. Quarini and K . A. Cliffe for their many suggestions and contributions to this work.

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BUOYANCY-DRIVEN TURBULENT FLOW 99

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