VALIDATION OF TURBULENT NATURAL CONVECTION IN A SQUARE CAVITY FOR APPLICATION OF CFD MODELLING TO HEAT TRANSFER AND FLUID FLOW IN ATRIA GEOMETRIES C.A. Rundle and M.F. Lightstone Department of Mechanical Engineering, McMaster University Hamilton, ON, L8S 4L7, Canada Email: lightsm@mcmaster.ca ABSTRACT Atria are a common design feature in modern buildings. They prove many benefits if properly designed. However there is a lack of appropriate tools for designers to use. There is a potential role for Computational Fluid Dynamics (CFD) to play in this design process since CFD can afford a designer the opportunity to evaluate the performance of the structure before it is constructed. As such, use of CFD could provide an important aide to designers interested in including an atrium in their structures. However, in order for this to be effective, the ability of the code to accurately model turbulent natural convection, a very important parameter in atria performance, must be validated and the available turbulence models evaluated. In order to validate the ANSYS CFX 5.7 code for turbulent natural convection, predictions using three different turbulence models will be assessed. In addition, the buoyancy turbulence production term will be evaluated within each turbulence model. These numerical results will be compared to experimental benchmarks for turbulent natural convection in a square cavity. INTRODUCTION Atria are spaces encompassing large glazed areas, used for natural ventilation and lighting in many large buildings. They can have a significant effect on the thermal performance of the building. If incorrectly designed they can be a major source of unwanted heat loss and heat gains, depending on the time of day (Voeltzel et. al., 2001). As such it is important that atria be designed to minimize these problems. There are computer programs that are capable of aiding in atria design (Voeltzel et. al., 2001, Laouadi & Atif, 1999), however, they have shortcomings that could be improved upon by CFD. In particular, the velocity field is rarely fully calculated and there is a reliance on estimating variables that requires extensive experience and testing thus reducing applicability to new designs. In some cases the velocity field must be inputted from experiments in the atrium (Laouadi & Atif, 1999). Other programs use equations that model the heat transfer using a special adaptation of heat transfer coefficients specific to each atrium (Voeltzel et. al., 2001). If ANSYS CFX can be validated for the appropriate physical phenomena, it can form the basis of a program that can accurately model the heat transfer and fluid flow in atria geometries without requiring specialist knowledge. One such relevant phenomenon is turbulent natural convection. A validation of ANSYS CFX for this phenomenon is the focus of the current paper. Turbulent Natural Convection Validation There are numerous examples of validations of turbulent natural convection in academic literature to use as reference material. In, particular the work of Henkes and Hoogendorn, Sharif and Liu and Wen et al. were considered (Henkes & Hoogendoorn, 1995, Sharif & Liu, 2003, Wen et. al., 2000). A wide variety of grids were used in these validations however, they were all expanding, two dimensional grids, with anywhere from forty to one hundred and sixty nodes per side. In terms of grid independence there was some variation in the findings. Wen et al. found that grid independent solutions could be achieved with a 80x80 expanded grid (Wen et. al., 2000). However, in 2003 Sharif and Liu required a 100x100 grid to achieve grid independence (Sharif & Liu, 2003). In the reviewed literature convergence was also an issue that required the application of under relaxation techniques to achieve convergence. This was especially true for Henkes and Hoogendoorn where all of the results required under relaxation (Henkes & Hoogendoorn, 1995).
NOMENCLATURE For the purposes of this paper and all related discussion the variables and parameters are defined in Tables 1 and 2. Table 1. Variable Definitions and Values Sym. Definition Value Unit A Area of cavity wall 0.75 m 2 g Gravity acceleration -9.81 m/s 2 k Thermal conductivity 2.61e-2 W/mûK L Cavity side length 0.75 m Q Cavity heat flow W T Temperature ûk T Cold wall temperature 283.15 ûk C T Hot wall temperature 321.57 ûk h v Vertical velocity m/s α Thermal diffusivity 2.19e-5 m 2 /s β Thermal expansivity 3.36e-3 1/ûK υ Kinematic viscosity 1.54e-5 m 2 /s Variable Dimensionless Temperature Table 2. Dimensionless Number Definitions Dimensionless Velocity Average Nusselt Number Rayleigh Number Equation T v ND ND T T = T T = Nu = Ra = h C C v gβl( T T ) C h QL Ak( T h TC ) gβ ( Th TC ) L υα 3 SIMULATION Turbulence Models There are many different turbulence models available through the code. The correct selection for use in atria is dependent on a combination of accuracy and acceptable computational time. For the purpose of this paper three different turbulence models will be considered. Reynolds stress models have been excluded from this study due to their long computational times and problems experienced with stability. The k-ε Model The k-ε model has an extensive and long documented history of use. For this paper the standard k-ε model will be used. Henkes and Hoogendoorn believe that more modern variations, such as k-ε model with low Reynolds number (LRN) terms may give better physical accuracy (Henkes & Hoogendoorn, 1995). However, more recent work by Walsh and Leong found that the extremely low computational requirements of the standard k-ε model make it preferable to more complicated k-ε models (Walsh & Leong, 2004). In addition good results have previously been obtained from the standard k ε model at McMaster (Dworkin et. al., 2003). Therefore it was determined that the standard k-ε model should be validated. The Wilcox k-ω Model The Wilcox k-ω model is the standard k-ω model offered by ANSYS CFX. It is included in the validation for two reasons. It is a common twoequation model and forms the basis for the shear stress transport model that is initially believed to be the most promising choice. The Shear Stress Transport Model (SST) The SST model is a combination of the previous k-ε and k-ω models. A blending function, which is dependent on the local turbulent variables, is used to switch between models. This model was recommended by the ANSYS CFX documentation as the best two-equation model for flows near a wall. Buoyancy Turbulence The code offers an optional production term for buoyant flows. The production term model accounts for the additional production of turbulent kinetic energy arising from density gradients. This term can be
included in only the turbulent kinetic equation or both the turbulent kinetic energy equation and either the dissipation or the frequency equation, depending on the type of model selected. DISCUSSION AND RESULTS ANALYSIS GRID AND GRID CONVERGENCE Physical Setup The physical domain is shown in Figure 1. The top and bottom walls are insulated and hot and cold uniform temperatures are specified on the left and right vertical walls, respectively. The dimensions and temperature difference are chosen to yield the Rayleigh number of interest. Figure 1. Fluid Domain Validation The numerical results for ANSYS CFX will be validated against the experimental data provided by Ampofo and Karayiannis (Ampofo & Karayiannis, 2003). This benchmark is at a Rayleigh number of 1.58 x10 9. Unless otherwise noted all velocity and temperature profiles are taken at half of the cavity height or y/l=0.5. Table 3. Experimental Uncertainty Parameter Uncertainty Wall Temperature 0.15 K Air Temperature 0.10 K Air Velocity 0.07 % Rayleigh Number 0.62 % Nusselt Number 0.25-1.13% (Ampofo and Karayiannis, 2003) Figure 2. 80x80 Expanding Grid The grid shown in figure 2 is the standard grid used in these validations. It was based on a similar grid used by Davidson in the paper presented by Henkes and Hoogendoorn in 1995 (Henkes & Hoogendoorn, 1995). An expanding 140x140 grid was also used for confirming grid independence. Uniform grids of 20x20 to 320x320 were used however they never converged. In order to ensure that the results obtained were independent of the grid selected a second expanding grid of 140x140 was created. Figure 3 and Figure 4 below shows a comparison of the velocity and temperature profiles for the standard Wilcox k-ω model generated on each of the grids. Figure 3. Temperature profile of standard k-ω model on various grids
Figure 4 Velocity profile of standard k-ω model on various grids Both figures 3 and 4 clearly show that both grids produce very similar results. The peak velocities differed by approximately 0.5% and the average temperature of the cavity differed by less than 0.01%. Therefore, it was deemed the solution for the 80x80 grid was effectively grid independent. Convergence Issues Typically a convergence of the root mean square residual of 1E-4 was sufficient for significant physical results. As figure 5 and figure 6 below shows at higher iterations a convergence level of 1E-4 is sufficient for convergence and no significant change in result occur when compared to a convergence level of 1E-6 for the Wilcox k-ω model. The average change per node for the temperature was 0.22%. These figures are specific to the k-ω model but similar trends were present in the SST model as well. The k-ε model proved something of a different matter, which will be discussed in the k-ε section. Figure 6. Velocity profile of the k-ω model at different convergence criteria. In all cases under-relaxation for the mass and time step loops was used to enable convergence. The values used ranged from 0.7 to 0.95. There was not discernable pattern to the appropriate relaxation values. Without relaxation convergence was impossible. Convergence was also impossible with a constant time step. In order to converge, the automatic time step function was used. This created problems in the k-ε model that were overcome by limiting the maximum value of allowable time step. The k-ε Model Convergence was never reached with the k-ε model. A variety of methods to aide convergence including relaxation of time step loop, relaxation of mass equations and variation of grid, were attempted. The outer loop relaxations number which is the relaxation factor for the time step was attempted at 0.05 intervals from 0.25 to 1.00. The k-ε models did demonstrate a false convergence in the initial iterations. The root mean square residuals would typically all drop below 1E-4. This level was sufficient for the Wilcox k-ω models as previously mentioned. However, the results at this level of convergence were not physically realistic and with further iterations the solution would diverge until a convergence of 1e-3 or higher was reached. Figure 5.Temperature profile of the k-ω model at different convergence criteria.
Standard Models Figure 7. Velocity profile of standard models. Figure 8. Velocity profiles in the boundary layer As seen in figure 7 both standard models correctly predict the vertical velocity field outside of the boundary layer. However, figure 8 shows a marked difference between the models in the boundary layer. Within the boundary layer the Wilcox k-ω model s velocity profile has a similar width and peak velocity to the benchmark. The peak velocity difference is smaller than the experimental uncertainty, less than 0.01%. The SST model s boundary layer is significantly thinner and has a peak velocity that is 19% larger than the benchmark. Therefore the standard Wilcox k-ω model more accurately predicts the velocity profile. Figure 10. Temperature profile of standard models in the boundary layer. Figure 9 and 10 show both models produce similar temperature profiles, both inside and outside of the boundary layer. A comparison of the difference between the models shows that the average difference in temperature predicted is 0.27ûK. Note that the k-ω model the profile in the boundary layer is slightly closer to the benchmark values for a small section. Compared to the benchmark both models under predict the temperature outside of the boundary layer by less than one degree Kelvin. Therefore there is no definite advantage to either model in terms of modeling temperature distribution. Table 4 Average Nusselt number for standard models Wilcox k-ω SST Model Nu 61.92 60.87 % Error 1.32% 3.00% Table 4 clearly shows that the Wilcox k-ω model is superior in predictions of heat transfer, with a percentage error of less than half that of the SST model. Also the estimated error on the benchmark value is 0.25%-1.13% (Ampofo & Karayiannis, 2003). This means that the error in the k-ω model is very close to experimental error. Figure 9. Temperature profile of standard models.
The Wilcox k-ω Model with Buoyancy Turbulence Figure 11. Velocity Profiles with the Buoyancy Turbulence term for the k-ω model. Figure 13 Temperature Profiles with the Buoyancy Turbulence term for the k-ω model. Figure 12. Velocity Profiles in the Boundary Layer with the Buoyancy Turbulence term for the k-ω model. Figure 11 shows the velocity profiles across the cavity and it is clear that all of the models have similar zero profiles in the middle of the cavity. In order to determine the differences between the models the magnified view the boundary layer provided in figure 12 is required. The production run and production and dissipation run produce nearly identical results with the velocity profiles nearly overlapping perfectly. The average difference between the buoyancy turbulence production term velocity profiles shown in figure 11 was 0.025% compared to the peak velocity. The buoyancy turbulence production term created a narrower and more intense boundary layer than observed in either the experimental results or the standard model. The peak velocity was 14.2% larger than the benchmark value. Since this is significantly greater than the previously mentioned 0.01% error in the standard model the standard model was the most accurate in terms of peak magnitude and location. Figure 14 Temperature Profiles in boundary layer with the Buoyancy Turbulence term for the k-ω model. Figures 13 and 14 above show that there is no significant difference between the temperature profiles created by the production run and the production and dissipation run. The average difference was less than 0.03%. There is an average difference of 2.0% between the standard model and the models that include the buoyancy turbulence production term. That difference translates into an average temperature difference of 0.21ûK. Therefore it is reasonable to claim that the buoyancy turbulence production term had a minimal effect on the predicted temperature distribution. Table 5 Average Nusselt Number for the k-ω model Standard and Dissipation Nu 61.92 61.50 61.49 % Error 1.32 1.99 2.01
Table 5 shows that all of the buoyancy turbulence production models produced similar average Nusselt numbers as the standard model. The change was approximately 0.6% of the standards model, however this change resulted in an increase in error from 1.32% to approximately 2.00% when compared to the benchmark. Therefore the addition of the buoyancy turbulence production terms in the Wilcox k-ω model did not improve the accuracy of the predictions. Both the velocity profiles, particularly in the boundary layer, and the average Nusselt number showed a decrease in accuracy and the temperature distribution had no noticeable change. difference of 1.2% between the standard model and the buoyancy turbulence production models. Comparing the peak velocities shown in figure 16 the production run and production and dissipation run have identical values to the third significant digit. These values are 3.04% higher, when compared to the benchmark, than the standard model making them less accurate. Therefore there is no significant improvement to the velocity profile from adding the buoyancy turbulence production term. The Shear Stress Transport Model with Buoyancy Turbulence Figure 17 Temperature Profiles with the Buoyancy Turbulence term for the SST model. Figure 15. Velocity Profiles with the buoyancy Turbulence Term for SST model Figure 16. Velocity Profiles in the Boundary layer with the Buoyancy Turbulence term for SST model Figure 15 and 16 show a similar velocity profile for all of the SST models. For the velocity profile across the width of the cavity there is an average difference of 0.05% compared to the velocity peak magnitude between the two models with the buoyancy turbulence production term. There is also an average Figure 18 Temperature Profiles in boundary layer with the Buoyancy Turbulence term for the SST model. Figures 17 and 18, shown above, depict the nearly identical temperature profiles for the SST model when run with the buoyancy turbulence production term included to various degrees. The average difference between the production run and production and dissipation run is 0.03%. This means that the two curves are virtually identical. When compared to the standard model s temperature profile the models with buoyancy turbulence production s temperature profiles there is a difference of 0.61% or 0.2ûK. Therefore the inclusion of the buoyancy turbulence production term has no significant affect on the predicted temperature profile.
Table 6. Average Nusselt number for SST model Standard and Dissipation Nu 60.87 60.93 60.96 % Error 3.00 2.90 2.85 Table 6 shows the first evidence that the buoyancy turbulence production term can improve the accuracy of the models. The percentage error for the average Nusselt number decreases when the buoyancy turbulence production term is added. CONCLUSION Based on the data from the simulations the following conclusions were obtained. 1. While it cannot be definitely stated the k-ε model will not converge the effort to does so is out of proportion to its worth. Especially given the accuracy of the Wilcox k-ω model. 2. The Wilcox k-ω model is the preferred turbulence model for calculating natural convection flows given it superior accuracy. 3. The buoyancy turbulence production terms do not increase the accuracy of the simulation and should not be included. There is some indication that inclusion of the buoyancy turbulence production term may have slightly improved the accuracy of the SST model. This improvement was very small and insignificant compared to the amount of error in the SST model. However it should be noted that this might not be true at higher Rayleigh numbers. Future work will confirm or refute this finding 4. Under-relaxation of the time step loop and mass equations be used to aid in convergence specifically starting with a factor 0.7 and increasing in units of 0.05 until either convergence or a factor 1.0 is reached. Should it be the latter restart at 0.7 and subtract 0.05 ACKNOWLEDGEMENTS This work was funded in part by the Solar Buildings Research Network under the Strategic Network Grants Program of the Natural Sciences and Engineering Research Council of Canada. This work would also not be possible without the support supplied by ANSYS-CFX through the use of their code. REFERENCES Ampofo, F., Karayiannis, T.G. Experimental benchmark data for turbulent natural convection in an air filled square cavity. International Journal of Heat and Mass Transfer, 46:3551-3572, 2003 Dworkin, S.B., Woods, J. and Lightstone, M.F.. Analysis of insulating panels for hot water radiator heating systems. Report for Natural Resources Canada, 2003 Henkes, R.A.W.M., Hoogendoorn, C.J. Comparison exercise for computations of turbulent natural convection in enclosures. Numerical Heat Transfer, Part B, 28:59-78, 1995 Laouadi, A., Atif, M.R. Comparison between computed and field measured thermal parameters in an atrium building. Building and Environment, 34:129-138, 1999 Sharif, M.A.R and Liu, W. Numerical Study of Turbulent Natural Convection in a Side-heated Square Cavity at Various Angles of Inclination. Numerical Heat Transfer, Part A,43:693-716, 2003 Walsh, P.C., Leong, W.H. Effectiveness of several turbulence models in natural convection. International Journal of Numerical Methods for Heat & Fluid Flow, 14(5):633-648, 2004 Wen J.X., Liu, F. and Lo, S. Performance Comparison of a Buoyancy-Modified Turbulence Model with Three LRN Turbulence Models for a Square Cavity. Numerical Heat Transfer Part B- Fundamentals, 39:257-276, 2000 Voeltzel, A., Carrie, F.R., Guarracino, G. Thermal and ventilation modelling of large highly-glazed spaces. Energy and Buildings, 33:121-132, 2001