Comparison of two equations closure turbulence models for the prediction of heat and mass transfer in a mechanically ventilated enclosure

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1 Proceedings of 4 th ICCHMT May 17-0, 005, Paris-Cachan, FRANCE 381 Comparison of two equations closure turbulence models for the prediction of heat and mass transfer in a mechanically ventilated enclosure Kuznik Frédéric a *; Rusaouën Gilles a ; Hohotă Raluca b a Thermal Sciences Center, National Institute of Applied Sciences, Lyon Bât. Freyssinet, 0 av. A. Einstein Villeurbanne Cedex b Technical University of Civil Engineering of Bucharest *Correspondence author: Fax: kuznik@etb.insa-lyon.fr ABSTRACT In this paper, different turbulence models are tested for the prediction of the airflow in a mechanically ventilated room. The is carried out in a full scale test room with a hot and a cold air jet. The computational fluid dynamics is used to predict the air flow by testing various two equations closure turbulence models: a k ε realizable model, a standard k ω model and a Shear-Stress Transport (SST) k ω model. Different velocity magnitude and temperature profiles are used for the comparisons. The results show that, among the turbulence models tested, the k ε realizable model is in better agreement with al results for the hot case. Concerning the cold case, none of the models used is reliable to predict the airflow. NOMENCLATURE D flow rate (m 3 h 1 ) T in air inlet temperature ( C) T m room mean air temperature ( C) k turbulent kinetic energy (m s ) u mean velocity magnitude (ms 1 ) ε dissipation rate of k (m s 3 ) ω specific dissipation rate (s 1 ) INTRODUCTION As we spend most of our time in enclosed spaces, environmental control of buildings is becoming very important. Among the thermal comfort indices proposed in the literature, the Fanger s empirical one, PMV (Predicted Mean Vote) has become a standard approach to asses the indoor thermal comfort, but evaluating PMV is only possible if the velocity and thermal fields are known. A detailed determination of this fields is possible with codes of the type CFD (Computational Fluid Dynamics). Therefore, the CFD technique was increasingly employed to predict the airflow into rooms during the last decade. Nowadays, the turbulence modelling is the main stake in airflow modelling. In one hand, the k ε model of Launder and Splading [1] have been widely used for the prediction of airflow in rooms. Many modifications have been made to this first model: low Reynolds number k ε model, two layers k ε model, two-scale k ε model and renormalization group (RNG) k ε model, k ε realizable model...as a general rule, the problem of the k ε model family is the near wall treatment. However, among the k ε models, the realizable one [] have shown his capability to predict the behavior of mechanically ventilated room configurations [3]. On the other hand, Wilcox [4] developed a k ω model applicable to wall-bounded flows and free-shear flows. The shear-stress transport (SST) k ω model was developed by Menter [5] to effectively blend a robust and accurate formulation of the k ω model by combining good near-wall behavior of the Wilcox s first k ω model of 1988,and the k ε model away from the walls. The objective of this study is to compare the realizable k ε model, the standard k ω model and the SST k ω model for the prediction of airflow and temperature field of a mechanically ventilated enclosure. Velocity and temperature fields are obtained by in our full scale test room. Numerical approach was performed using computational fluid dynamics (CFD) Fluent code. EXPERIMENTAL SET-UP The al full scale test room MINIBAT (CETHIL, France) consists of an enclosure which dimensions are 3.10m, 3.10m,.50m according to (x, y, z) (see figure 1); the air inlet, with a round

2 4th International Conference on Computational Heat and Mass Transfer opening of diameter 0.1m, creates an axisymmetric jet in the room. A thermal guard allows us to control the temperatures of the cell faces. The ventilation system regulates the inlet and outlet flow rates which are measured by two flow-meters. The jet temperature is imposed by the means of an air treatment system. median plan x=1.55m inlet boundary conditions cell air inlet.5 tions description and the detailed dynamic and thermal field measurements to compare with numerical data based on our models. AIR FLOW MODELLING This part is devoted to a description of the combined heat and fluid flow modelling. The mean air flow modelling principles are as follow: turbulence modelling, treatment of air flow near wall, boundary conditions, computational domain discretisation and numerical solution. Basically, our numerical model is based on a commercial CFD code, FLUENT. This general purpose code is a finite volume, Navier Stokes solver. 1.5 z [m] 0.5 y=0.6m y=1m y=m 0 1 y [m ] 3 3 cell air outlet 0 1 x [m] Turbulence modelling: The three turbulence models are based on the Reynolds averaging of the Navier-Stokes and energy equations. The fluid is incompressible with density computed via the ideal gas law considered as varying only with temperature. The closure problem is solved by using two additional equations. outlet boundary conditions The k ε realizable model: This model using the Figure 1 Experimental test cell The test room have been equipped with thermocouples to measure the wall surfaces temperature with a resolution of ±0.4 C, each face having nine thermocouples. The air temperature is measured with two thermocouples with a resolution of ±0.4 C. The mean air velocity is measured by an omnidirectional velocity probe with temperature compensation which resolution is the worst of ±0.5ms 1 and ±3% of the measurement value for a temperature contained between 0 C and 6 C adding 0.5%/ C outside this range. The velocity measurement rate is 0 samples per second, the mean velocity is measured over 1000 samples. A mobile three axis arm allows to move the temperature and air sensor in the room. The was realized under steady state conditions which are given in table 1. hot case cold case Tin ( C) 31.0 Tm ( C) The standard k ω model: The two equation k ω models are based on model transport equations for the turbulent kinetic energy and the specific dissipation rate which can also be thought of as the ration of ε to k. The model tested is the one of Wilcox [4] including a low-reynolds correction. The Shear-Stress Transport (SST) k ω model: Table 1 Experimental conditions Re,d transport equations of k and ε to compute the turbulent viscosity. We employ in this study a revised k ε turbulence model called the realizable k ε model of Shih and al. []. Compared with the other k ε models, the realizable one satisfies certain mathematical constraints on the Reynolds stress tensor, consistent with the physics of turbulent flows (for example the normal Reynolds stress terms must always be positive). Moreover, a new model for the dissipation rate is taking into account to predict the spread of both round and plan jets. D(m3 h 1 ) In conclusion, the al methodology has permitted us to obtain a complete boundary condi- The SST k ω model is so named because the definition of the turbulent viscosity is modified to account for the transport of the principal turbulent shear stress. The model tested here has been developed by Menter [5] and include a low-reynolds correction. Near wall treatment: Correct calculation of a wall bounded flow and its associated transport phenomena

3 is not possible without and adequate description of the flow in the near wall region. Near wall treatment for the k ε model: In FLUENT, the near wall treatment combines a twolayer model with enhanced wall functions. In one hand, the first cell values of temperature and velocity are given by enhanced wall functions applicable in the entire near-wall region, according to the method of Kader [6]. On the other hand, the viscosity affected region is resolved by the two-layer model of Wolfstein [7]: the demarcation of this region and the fully turbulent region (where the k ε realizable equations are used) is determined by a wall-distancebased turbulent Reynolds number. The k equation is solved in all cases in the whole domain, using for the walls k n = 0 where n is the local coordinate normal to the wall. Near wall treatment for the k ω models: The near wall treatment for the k ω and k ω SST models is computed following the same logic as for the k ε realizable model. The first cells values are given by the same equations as in the previous section. However, there is no need for a special treatment for the viscosity affected region because of the low Reynolds correction in the k ω and k ω SST models. Boundary conditions: The numerical solution precision deeply depends on the boundary conditions accuracy and the way that these conditions are integrated within the numerical model. In our case, there are three kinds of boundary conditions: air inlet conditions, air outlet conditions and wall boundary conditions. In order to avoid errors due to the lack of knowledge about the exact physical parameters fields like temperature, velocity and pressure, we choose to model the air supply at the inlet. The inlet conditions are imposed far from the inlet diffuser, at a fully developed flow section (see figure 1). The velocity and temperature values are given as known values using the al data. Concerning the turbulence quantities, they are imposed by means of two parameters, the turbulence intensity and the hydraulic diameter, assuming a fully developed duct flow upstream. In the same way as the inlet boundary conditions, the outlet boundary conditions are imposed at a fully developed flow section. The outlet velocity is computed from mass balance, the gradients normal to flow direction of other variables are also set to zero at the exit section. Finally, we need to provide boundary conditions of wall surfaces. Therefore, the classical no-slip boundary conditions are assured to the walls. We imposed either fixed values of temperatures using measured values or zero thermal fluxes at internal surfaces. Discretisation: The discretisation of computational domain is achieved by means of an unstructured mesh. The grid contains tetrahedral elements obtained due to a mesh generation algorithm based on the Delaunay criterion [8]. There are three reasons that justify our option regarding an unstructured grid: first of all, the unstructured grid is ideally suited for the discretisation of complicated geometrical domains. In our case, this allows us an exact description of supply inlet. Besides, our flow is a complex one, therefore there is no advantage to using a hexahedral (structured) mesh since the flow is not aligned with the mesh (the numerical diffusion is minimized if the flow tracks the shape of grid elements). Finally, another advantage of the unstructured mesh is that it allows us to refine without difficulty the grid based on geometric or numerical solution data. This property is very useful for the regions of the domain where strong flow gradients occur (boundary layers, plumes or jets). The grid in these zones can be refined without adding unnecessary cells in the other parts of the domain as classically happens in the structured grid approach. Our final mesh is composed of finite volumes. Numerical scheme: The solution method is based on the following main hypothesis: the diffusion terms are second-order central-differenced and the second order upwind scheme for convective terms is used to reduce the numerical diffusion. The velocitypressure coupling method is the simple algorithm. The multigrid scheme allows to accelerate the convergence as our model contains a very large number of control volumes RESULTS AND DISCUSSION This section presents the numerical and al results for the hot and cold cases. In the ventilated enclosure, the dynamic of the flow is quite simple because there are two principal zones: the jet zone and the weakly moving fluid zone where the fluid velocity is less than 0.05m/s and then not measurable by our means. In order to evaluate the accuracy of our models, we then compare the profiles, in the jet zone, of the mean velocity and temperature along three straight lines in the median plan (see fig-

4 ure 1): at y = 0.6m (3cm from the room air inlet), at y = 1m and at y = m Figure Velocity profiles at y = 0.6m, y = 1m and y = m - hot case Model relevancy: In order to validate our mesh, we created a mesh with finite volumes (the original mesh is composed of volumes) by grid adaptation on the jet zone. The differences between the two meshes concerning the velocity magnitude don t exceed 0.5%. This comparison validates our choices for the mesh. Profiles analysis: The hot case: Figure 3 Temperature profiles at y = 0.6m, y = 1m and y = m - hot case The figure shows the velocity profiles at three locations. First of all, close to the ventilation inlet, the al and numerical velocity profiles are in good agreement which validates the model used for the ventilation inlet. Table Maximum values of k (m s ) in the profiles y = 0.6m, y = 1m and y = m - hot case k ε k ω k ω SST k k k

5 The more we go away from the inlet, the more the numerical models present behaviors different than the. In fact, the k ε realizable and the k ω SST models overestimate the maximum velocity when the k ω model underestimates the last one. In the meantime, the k ε realizable model predicts best the velocity profiles Figure 4 Velocity profiles at y = 0.6m, y = 1m and y = m - cold case The figure 3 presents the temperature profiles for the hot case. Near the inlet, the temperature profiles are all in good agreement. But, as for the velocity profiles, the temperature profiles are different far from the inlet. The and the k ω SST models overestimate the maximum temperature when the k ω and the k ε realizable models seem in good agreement with the al data. The table gives the maximum values of k in the three profiles. The turbulent kinetic energy is the higher for the k ω model and then explain that it overestimates the jet expansion and underestimate the maximum velocity and temperature values. The k ω SST model products turbulent values lower than the two other models and then shows its inability to predict the expansion of the jet. The k ε, which fits best the al data, gives k values contained between those of the two others models. The cold case: Figure 5 Temperature profiles at y = 0.6m, y = 1m and y = m - cold case

6 The cold case is more specific because, contrary to the hot case, the cold air trend to go down and then the effect of the wall is less important. The figure 4 shows the velocity profiles for the cold case. At y = 0.6m, the al simulations and the al data are in good agreement. But the more we go away from the inlet, the more the numerical profiles differ from the : it seems that the numerical profiles still attached to the wall instead of falling like the al jet. Table 3 Maximum values of k (m s ) in the profiles y = 0.6m, y = 1m and y = m - cold case k ε k ω k ω SST k k k The temperature profiles of the cold case are represented figure 5. The conclusions are the same as for the velocity profile, excepted that the maximum values are quite well evaluated by the k ω model. The table 3 gives the maximum values of k in the three profiles. The values of k yield to conclusions similar to those made for the hot case, even if none of the models predicts the good behavior of the jet. CONCLUSIONS In this article, al data concerning a mechanically ventilated enclosure have been performed in a full scale test room. Three numerical modelling with different two equations closure turbulence models have been tested in order to predict the ventilated enclosure airflow: a k ε realizable model, a k ω model and a k ω SST model. Two cases were tested: a hot air jet and a cold air jet. Concerning the hot case, the k ε realizable model fits better the al velocity and temperature data. The k ω model overestimate the spreading of the jet and then underestimate the maximum velocity values. On the contrary, the SST model underestimate the jet spreading and then overestimate the maximum velocity values. These conclusions are confirmed by the maximum turbulent kinetic energy values in the jet zone. For the cold case, none of the turbulence models predict the good behavior of the jet. Even if the velocity and temperature profiles are underestimated and overestimated by the models, the jet dynamic is not well predict, especially the falling of the jet. It seems that the numerical shear layers behaviors are quite different compared with. Further investigations are needed to better comprehend the behavior of such jets: turbulence measurements will help us to better understand the jet behavior, especially in the shear layer zone. KEYWORDS CFD, k ω models; k ε model; Building room; Ventilation REFERENCES 1. Launder, B.E., Splading, D.B., 1974, The numerical computation of turbulent flow, Computer Methods in Applied Mechanics and Energy, 3, pp Shih T., Liou W.W., Shabbir A., Yang Z., Zhu J., 1995, A new k ε Eddy Viscosity Model for High Reynolds Turbulent Flows. J. Computer Fluids, 4(3), pp Teodosiu C., 001, Modelisation des Systèmes Techniques dans le Domaine des Equipements des Bâtiments à l aide des Codes CFD, PhD thesis presented at the INSA of Lyon. 4. Wilcox D.C., 1998, Turbulence modelling for CFD,1998, nd Edition, DCW Industries. 5. Menter F.R., 1994, Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, 3(8), pp Kader B., 1993, Temperature and Concentration Profiles in Fully Turbulent Boundary Layers, Int. J. Heat Mass Transfer,4(9), pp Wolfstein M., 1969, The Velocity and Temperature Distribution of One Dimensionnal Flow with Turbulence Augmentation and Pressure Gradient, Int. J. Heat and Mass Transfer, 1,pp Thompson J.F., Soni B.K., Weatherill N.P., 1999, Handbook of Grid Generation, London: CRC Press.