Evaluation of various turbulence models in predicting airflow and turbulence in enclosed environments by CFD: Part 1 -- summary of prevalent turbulence models.

INTRODUCTION

Enclosed environments, such as commercial, institutional, and residential buildings; healthcare facilities; sport facilities; manufacturing plants; animal facilities; and transportation vehicles, are confined spaces with certain functionalities. It is essential to control air distribution in the enclosed environments. The parameters of air distribution include, but are not limited to, air velocity, temperature, relative humidity, enclosure surface temperature, air turbulence intensity, and concentrations of airborne gaseous, particulate, and liquid droplet contaminants in the enclosed environments. The air distribution control is to create and maintain a comfortable and healthy environment required by occupants and/or thermofluid conditions for industrial processes in the enclosed environments.

Air distribution in an enclosed environment can be driven by different forces, for instance, natural wind, mechanical fan, and/or thermal buoyancy. The combination of these flow mechanisms (forced, natural, and mixed convection) creates complex indoor airflow characteristics with impingement, separation, circulation, reattachment, vortices, buoyancy, etc., as illustrated in Figure 1. Most indoor environments have a low mean air velocity of less than 0.2 m/s, and the Reynolds number, Re, is generally very low (~[10.sup.5]). The corresponding flow regime may span from laminar to transitional to turbulent flows or a combination of all of the flow regimes under transient conditions. The complexity of indoor airflow makes experimental investigation extremely difficult and expensive.

[FIGURE 1 OMITTED]

With the rapid advance in computer capacity and speed, the computational fluid dynamics (CFD) technique has become a powerful alternative for predicting airflows in enclosed environments. By solving the conservation equations of mass, momentum, energy, and species concentrations, CFD can quantitatively calculate various air distribution parameters in an enclosed environment. It offers richer details, a higher degree of flexibility, and lower cost than experimental study. Nielsen (1974) was the first one who applied CFD for room airflow prediction. Applications of CFD for airflows in enclosed spaces have been increasing since the 1980s. The International Energy Agency Annex 20, for instance, has sponsored a research project on room airflow prediction with participants from 13 countries (Moser 1992).

CFD applications to airflow simulation for enclosed environments have achieved considerable successes, as reviewed by Whittle (1986), Nielsen (1989, 1998), Liddament (1991), Jones and Whittle (1992), Chen and Jiang (1992), Moser (1992), Lemaire et al. (1993), Ladeinde and Nearon (1997), Emmerich (1997), Spengler and Chen (2000), Chen and Zhai (2004), and Zhai (2006). These reviews concluded that CFD is a valuable tool for predicting air distribution in enclosed environments. However, there are many factors influencing the results predicted. Different users may obtain different results for the same problem even with the same computer program. The accuracy of the simulation heavily depends on a user's knowledge of fluid dynamics and experience and skill using numerical techniques. Among various CFD influential factors, proper selection of a turbulence modeling method is a key issue that will directly affect the simulation accuracy and efficiency.

Recent advances in CFD turbulence modeling methods may bring new potential for improving the accuracy and efficiency of indoor airflow modeling. Therefore, it is of great interest and value to review the progress in CFD turbulence modeling and provide solid suggestions for proper application of the models for indoor airflow simulation. The study consists of two parts: Part 1 (this paper) identifies the most popular (and/or new) turbulence models that have been (and/or have potential to be) used for modeling air distribution in enclosed environments by searching the most recent literature. Part 2 (a companion paper from Zhang et al. [2007]) systematically evaluates the identified models by comparing their prediction performance against a series of benchmark experimental results so as to recommend appropriate turbulence models for indoor environment modeling.

CFD APPROACHES

Generally, CFD predicts turbulent flows through three approaches: direct numerical simulation (DNS), large-eddy simulation (LES), and Reynolds-averaged Navier-Stokes (RANS) equation simulation with turbulence models.

DNS computes a turbulent flow by directly solving the highly reliable Navier-Stokes equation without approximations. DNS resolves the whole range of spatial and temporal scales of the turbulence, from the smallest dissipative scales (Kolmogorov scales) to the integral scale, L (case characteristic length), which is associated with the motions containing most of the kinetic energy. As a result, DNS requires a very fine grid resolution to capture the smallest eddies in the turbulent flow. According to the turbulence theory (Nieuwstadt 1990), the number of grid points, N, required to describe turbulent motions should be at least N ~ [Re.sup.9/4]. The computer systems must become rather large (memory at least [10.sup.10] words and peak performance at least [10.sup.12] flops) in order to compute the flow (Nieuwstadt et al. 1994). In other words, since the smallest eddy size is about 0.01 to 0.001 m in an enclosed environment, at least 1000 x 1000 x 1000 grids are needed to solve airflow in a room. In addition, the DNS method requires very small time steps, which makes the simulation extremely long. Neither existing nor near-future personal computers can meet these needs, so the application of DNS for indoor flows is not feasible now or in the near future.

According to Kolmogorov's theory of self similarity (Kolmogorov 1941), large eddies of turbulent flows depend on the geometry, while the smaller scales are more universal. Smagorinsky (1963) and Deardorff (1970) developed LES with the hypothesis that the turbulent motion could be separated into large eddies and small eddies such that the separation between the two does not have a significant impact on the evolution of large eddies. The large eddies corresponding to the three-dimensional, time-dependent equations can be directly simulated on existing computers. Turbulent transport approximations are made for small eddies, which eliminates the need for a very fine spatial grid and small time step. The philosophy behind this approach is that the macroscopic structure is characteristic for a turbulent flow. Moreover, the large scales of motion are primarily responsible for all transport processes, such as the exchange of momentum and heat. The success of the method stems from the fact that the main contribution to turbulent transport comes from the large-eddy motion. Thus, the large-eddy simulation is clearly superior to turbulent transport closure wherein the transport terms (e.g., Reynolds stresses, turbulent heat fluxes, etc.) are treated empirically. In the last decade, rapid advances in computer capacity and speed have made it possible to use LES for some airflows related to enclosed environments. LES provides detailed information on instantaneous airflow and turbulence with the cost of still considerable computing time.

For the design and study of air distributions in enclosed environments, the mean air parameters are more useful than instantaneous turbulent-flow parameters. Thus, the interest is stronger in solving the RANS equations with turbulence models that can quickly predict air distributions. The RANS approach calculates statistically averaged (Reynolds-averaged) variables for both steady-state and dynamic flows and simulates the turbulence fluctuation effect on the mean airflow by using different turbulence models. Many turbulence models have been developed since the 1970s, but very few of them are for an enclosed environment. A few turbulence models developed for other engineering applications, such as the standard model k-[epsilon](Launder and Spalding 1974), have been adopted for indoor air modeling. Despite the challenges associated with turbulence modeling, the RANS approach has become very popular in modeling airflows in enclosed environments due to its significantly smaller requirements of computer resources and user skills.

TURBULENCE MODEL DEVELOPMENTS AND APPLICATIONS FOR ENCLOSED ENVIRONMENTS

As stated previously, the laminar to turbulent flow characteristics in enclosed environments are very complicated (Ferrey and Aupoix 2006) and impose significant challenges on turbulence models. This paper reviews the recent development and application of the major turbulence models for predicting air distribution in enclosed environments. Instead of developing an inclusive review article, this study focuses on identifying popular and/or most recently proposed turbulence models for indoor environments. The review focuses on recent applications with model validation and comparison. A brief introduction of key model evolutions is also included to make the paper complete.

The following part of this paper gives an overall review of various popular turbulence models for indoor airflow simulations, including both RANS and LES models. RANS turbulence models are divided into two primary categories: eddy-viscosity models and Reynolds-stress models. Among the turbulence models studied, some are well-known and in widespread use, while others may be undergoing development. For those popular models, this study emphasizes their applicability to various indoor flows without detailing the fundamentals. For more recent models, this investigation discusses their development and potential in predicting indoor airflows. The paper is not intended to judge or criticize the conclusions from the references without knowing the simulation details. In fact, opposite results and conclusions were observed even for similar cases, which may be attributed to factors beyond turbulence models. This paper is solely to sense the application popularity of the models for indoor environment and to identify the prevailing models in practice. The popular models identified will then be evaluated against a series of benchmark experiments and will be analyzed for their prediction performance in different indoor airflows as detailed in Part 2 of this study (Zhang et al. 2007).

RANS Eddy-Viscosity Models

Eddy-viscosity models are normally classified according to the number of transport equations used. This section will review various eddy-viscosity models from the simplest to the most complex.

Zero-Equation Eddy-Viscosity Models. The zero-equation turbulence models are the simplest eddy-viscosity models. The models have one algebra equation for turbulent viscosity and no (zero) additional partial differential transport equations (PDE) beyond the Reynolds-averaged equations for mass, momentum, energy, and species conservation. The earliest zero-equation model was developed by Prandtl (1925) with the mixing-length hypothesis. Although the mixing-length model is not theoretically sound and the mixing length needs calibrations for each specific type of flow, the model has yielded good results in predicting simple turbulent flows. Some simple zero-equation models, once calibrated, may even provide surprisingly good results for mean flow quantities of some complex flows. For instance, Nielsen's (1998) study revealed that the constant eddy-viscosity model provides results closer to the measured data than the standard model for the prediction of smoke movement in a tunnel. Nilsson (2007) also used the constant eddy- viscosity model to study the comfort conditions around a thermal manikin, which provided acceptable accuracy with significantly less computing effort.

One important development in zero-equation models for modeling airflows in enclosed environments is the zero-equation model developed by Chen and Xu (1998). By using the assumption of uniform turbulence intensity, they derived an algebraic formula to express turbulent viscosity,[v.sub.t], as a function of local mean velocity, and the distance to the nearest wall, L:

[v.sub.t] = 0.03874 UL (1)

The equation has an empirical constant of 0.03874 for different flows. The validations conducted by Chen and Xu (1998), Srebric et al. (1999), and Morrison (2000) have demonstrated the feasibility of this model in predicting general room airflows. The model has been widely used for simulating airflows in different indoor environments with acceptable accuracy and significant reduction in computing time (Kameel and Khalil 2003; Chen et al. 2005). Li et al. (2005) further applied this zero-equation model for outdoor thermal environment simulations, which also provided reasonable predictions when compared with the measured data. Airpak (Fluent 2002), commercial CFD software for HVAC applications, has adopted this model as its default. This model is the most popular zero-equation model for enclosed environments.

One-Equation Eddy-Viscosity Models. The turbulent viscosity correlations of zero-equation models may sometimes fail due to the inherent physical deficiencies, such as not considering nonlocal and flow-history effects on turbulent eddy viscosity. One-equation turbulence models use additional turbulence variables (such as the turbulent kinetic energy, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) to calculate eddy viscosity, [v.sub.t], such as follows:

[v.sub.t] = C[k.sup.[1/2]]l (2)

where k is obtained by solving a transport equation, l is a turbulence length scale, and C is a constant coefficient. The one-equation models need to prescribe the length scale, l, in a similar manner as that for the zero-equation models.

Most one-equation models solve the transport equation for turbulent kinetic energy, k. Some one-equation models derive transport equations for other turbulent variables, such as the turbulent Reynolds number (Baldwin and Barth 1990). Spalart and Allmaras (1992) proposed to directly solve a transport equation for eddy viscosity (the S-A model). Unlike most other one-equation models, the S-A model is local so that the solution at one point is independent of the solutions at neighboring cells and thus compatible with grids of any structure. This model is most accurate for free-shear and boundary-layer flows. The literature review shows that the S-A model, among very few one-equation models used for indoor environment simulation, is a relatively popular and reliable one-equation model at present. Torano et al. (2006) simulated ventilation in tunnels and galleries with the constant turbulent eddy viscosity model, the k-[epsilon] model, and the S-A model. The comparison of simulation results with detailed experimental data shows great performance of the k-[epsilon] and the S-A models. In addition, the S-A model has been incorporated by one of the newest turbulence modeling methods--detached eddy simulation (DES)--discussed below.

Two-Equation Eddy-Viscosity Models. In addition to the k-equation, two-equation eddy-viscosity models solve a second partial differential transport equation for z(z = [k.sup.[alpha]][l.sup.[beta]])to represent more turbulence physics. Different [alpha] and [beta] values form various kinds of two-equation models. Two-equation models are generally superior to zero- and one-equation models because they do not need prior knowledge of turbulence structure. The eddy viscosity can be calculated from the k and the length scale, l. Table 1 lists some typical two-equation models.

Table 1. Typical Forms of z Variable in Two-Equation Eddy-Viscosity
Models

z =        [k.sup.1/2]/l  [k.sup.3/2]/l  k/[l.sup.2]    k/l

Symbol     [omega]        [epsilon]      W            kl

Reference  Kolmogorov     Chou (1945)    Spalding     Rodi and
           (1942)                        (1972)       Spalding (1984)

k-[epsilon] two-equation eddy-viscosity model. The k-[epsilon] model family is the most popular turbulence model and has the largest number of variants. The "standard" k-[epsilon] model developed by Launder and Spalding (1974) is one of the most prevalent models for indoor airflow simulation due to its simple format, robust performance, and wide validations. The turbulent eddy viscosity, [v.sub.t], is calculated in the k-[epsilon] model as follows (Equation 3):

[v.sub.t] = [C.sub.[mu]][[k.sup.2]/[epsilon]] (3)

where k is the turbulence kinetic energy, [epsilon] is the dissipation rate of turbulence energy, and [C.sub.[mu]] = 0.09 is an empirical constant. The standard k-[epsilon] model was developed for high Reynolds number flows. To apply the model for low Reynolds number flows, such as near-wall flows, wall functions (Launder and Spalding 1974) are usually used to connect the outer-wall free stream and the near-wall flow. The use of wall functions avoids modeling the rapid changes of flow and turbulence near the walls with a fine grid and, thus, saves computing time.

The standard k-[epsilon] k-[epsilon] model with wall functions can predict the airflow and turbulence in enclosed environments reasonably well. For example, Holmes et al. (2000) simulated two ideal rooms with several one- and two-equation k-[epsilon] models and found that the standard k-[epsilon] model provides a reasonably good prediction. Gadgil et al. (2003) also verified that the k-[epsilon] model predicted the indoor-pollutant mixing time in an isothermal closed room fairly well. Nahor et al. (2005) simulated a complex airflow case in an empty and a loaded cool store with agricultural product by using the standard k-[epsilon] model. They concluded that the model was capable of predicting both the air and product temperature with reasonable accuracy. Zhang and Chen (2006) successfully applied the standard k-[epsilon] model to predict the airflow and particle distribution in a room with an underfloor air distribution system.

Meanwhile, the renormalization group (RNG) k-[epsilon] model (Yakhot and Orszag 1986) has also been widely used for predicting indoor airflows with many successes. For instance, Yuan et al. (1999a, 1999b) simulated airflow, temperature, and gas concentration distribution in a room with displacement ventilation and obtained good agreement with the experimental data. Sekhar and Willem (2004) successfully used the RNG k-[epsilon] model to study flow in a large office area. Craven and Settles (2006) modeled the thermal plume from a highly simplified human model with the RNG k-[epsilon] model, and the results agreed quite well with the particle image velocimetry data. Zhang et al. (2005) conducted a comprehensive validation of the RNG k-[epsilon] model for air distributions in an individual office, a cubicle office, and a quarter of a classroom with displacement ventilation. The study found that the computed air temperature and velocity agreed reasonably well with the measured data.

Another high Reynolds number k-[epsilon]-model family is realizable k-[epsilon] models. Realizable k-[epsilon] models usually provide much improved results for swirling flows and flows involving separation when compared to the standard k-[epsilon] model. For example, Van Maele and Merci (2006) indicated that the realizable k-[epsilon] model (Shih et al. 1995) performs better than the standard k-[epsilon] model for predicting various buoyancy plumes. It was observed that the model developed by Shih et al. (1995) is the most-used realizable model for indoor environment.

The high Reynolds number models may usually fail when the near-wall region is of great concern (Chen 1995) due to the equilibrium assumption of turbulence production and dissipation in wall functions. One method to remedy the near-wall problem is to use two-layer or even three-layer turbulence models. The two-layer models divide the wall vicinity into a viscosity-affected near-wall region resolved with a one-equation model and an outer region simulated with the standard k-[epsilon] model. Another approach for handling near-wall flows is to use a low Reynolds number (LRN) turbulence model to solve the governing equations all the way down to the solid surfaces. LRN models request a very fine grid near the walls so that the computing time is much longer. Many LRN models have been proposed since the 1970s, with most having a similar form. The observation of the applications of LRN models for indoor simulation reveals that the LRN model may only improve model accuracy for specific cases and lack wide applicability. For instance, Bosbach et al. (2006) simulated airflows in a generic airplane cabin with a group of high and low Reynolds k-[epsilon] turbulence models and two-layer k-[epsilon] models. Comparison with particle image velocimetry measurements showed that, for reliable prediction of isothermal cabin flow, LRN turbulence models had to be used. However, Costa et al. (1999) tested eight LRN k-[epsilon] models to simulate the mixed convection airflow generated by two nonisothermal plane wall jets and found that some LRN models may be able to provide good overall performance but suffer from singular defects occurring near separation/reattachment points of the flow. Hsieh and Lien (2004) also indicated that most LRN models tend to relaminarize the core-region low-turbulence flow and, as a consequence, significantly underpredict the near-wall turbulence intensities and boundary-layer thickness when modeling buoyancy-driven turbulent flows in enclosures. It was observed that the LRN models developed by Jones and Launder (1973) and Launder and Sharma (1974) are the most popular and commonly used models, upon which a few variation models were developed (Radmehr and Patankar 2001) but used less.

Comparison studies of these models can be found in literature. Chen (1995) compared five k-[epsilon]-based turbulence models in predicting various convective airflows and an impinging flow. The results showed that the RNG k-[epsilon] model had the best overall performance in terms of accuracy, numerical stability, and computing time, while the standard k-[epsilon] model had competitive performance. Rouaud and Havet (2002) confirmed that both the standard k-[epsilon] and the RNG k-[epsilon] models well predict the main features of the flow in cleanrooms. Kameel and Khalil (2003) used the standard k-[epsilon] model, the RNG k-[epsilon] model, and the zero-equation model (Chen and Xu 1998) to calculate airflows in a surgical operating room and found both the k-[epsilon] models are superior in predicting flow characteristics in near-wall and steep-gradient zones. Gebremedhin and Wu (2003) used five RANS models to simulate a ventilated animal facility and concluded that the RNG k-[epsilon] model is most appropriate for characterizing the flow field and is computationally stable. Posner et al. (2003) evaluated several k-[epsilon] models by simulating the airflow in a model room. They found that the simulation results with the laminar flow and the RNG k-[epsilon] models agreed better with the experimental data than those with the standard k-[epsilon] model. Yang (2004) investigated the mean ventilation flow rates through a naturally ventilated building with the standard k-[epsilon] model and the RNG k-[epsilon] model and found that the mean ventilation rates predicted by both models agreed well with the measurements. Walsh and Leong (2004) assessed the performance of several commonly used turbulence models, including the standard k-[epsilon], the RNG k-[epsilon], and a Reynolds stress model (RSM) in predicting heat transfer due to natural convection inside an air-filled cubic cavity. The study found that the standard k-[epsilon] model was the most effective model and the RSM did not improve any results. It is clearly observed from the literature search that both the standard and the RNG k-[epsilon] models have been widely used for indoor environment simulation, and the majority of comparison studies indicate that the RNG k-[epsilon] model is slightly better than the standard k-[epsilon] model in terms of the overall simulation performance.

k-[omega] two-equation eddy-viscosity model. The [omega] two-equation eddy-viscosity models (Wilcox 1988; Menter 1994) have also received increasing attention in many industrial applications in the last decade. In the k-[omega] models, [omega] is the ratio of [epsilon] over k. Compared to the k-[epsilon] models, the k-[omega] models are superior in predicting equilibrium adverse pressure flows (Wilcox 1988; Huang et al. 1992), while less robust in wake region and free-shear flows (Menter 1992). This led to the development of an integrated model that takes advantage of both models, a fairly successful model named shear stress transport (SST) k-[omega] model developed by Menter (1994). The SST k-[omega] model is essentially a k-[omega] model near wall boundaries and is equivalent to a Transformed k-[epsilon] model in regions far from walls. The switch between the k-[omega] and k-[epsilon] formulations is controlled by blending functions.

The k-[omega] models have been recently used for a few indoor airflow simulations. Liu and Moser (2003) indicated that the SST k-[omega] model can predict the transient turbulent flow and heat transfer of forced ventilated fire in enclosures if the transient conjugate heat transfer and thermal radiation are properly modeled. Sharif and Liu (2003) used the LRN k-[omega] model from Wilcox (1994) and the LRN k-[omega] model from Lam and Bermhorst (1981) to simulate the buoyancy-driven flow in a two-dimensional square cavity. The performance of the k-[omega] model was found to be better in capturing the flow physics, such as the strong streamline curvature in the corner regions. However, both models failed to predict the boundary-layer transition from laminar to turbulent. Arun and Tulapurkara (2005) computed the turbulent flow inside an enclosure with central partition with three advanced turbulence models: the RNG k-[epsilon] model, a Reynolds stress model, and the SST k-[omega] model. They found that the SST k-[omega] model can capture complex flow features such as the movement of vortices downstream of the partition, flow in reverse direction in the top portion of the enclosure, and exit of flow with swirl. Hu et al. (2005) simulated cross-ventilation by using the standard k-[epsilon], RNG k-[epsilon], standard k-[omega], and SST k-[omega] models, as well as LES, and also concluded that the SST k-[omega] model can depict the flow features satisfactorily. Stamou and Katsiris (2006) used the SST k-[omega] model, the standard model, the RNG model, and the laminar flow model to predict air velocity and temperature distributions in a model office room with a task ventilation system. By comparing the simulation and experimental results, the study concluded all three turbulent models satisfactorily predict the main qualitative features of the flow with slightly better performance from the SST model. Kuznik et al. (2007) also evaluated the realizable model (Shih et al. 1995), the RNG model, the standard model (Wilcox 1988), and the SST model with experimental measurements of air temperature and velocity for a mechanically ventilated room with a strong jet inflow. The research found that all of the models can accurately predict the global occupied zone temperature and velocity for the isothermal and hot cases, but none of the models is good and reliable for the cold cases. The model appears most reliable and can simulate the expansion rates in the highly anisotropic cold case at the same magnitude order as but is not a match with the measurements.

The k-[omega] models undoubtedly present new potential for modeling indoor environment with good accuracy and numerical stability. Many existing studies indicate that the SST k-[omega] model has a better overall performance than the standard k-[epsilon] model and the RNG k-[epsilon] model. Recently, one of the commercial CFD tools, CFX (ANSYS 2006), placed its emphasis on [omega]-equation-based turbulence models due to its multiple advantages, such as simple and robust formulation, accurate and robust wall treatment (low-Re formulation), high quality for heat transfer predictions, and easy combination with other models. However, a systematic model evaluation must be performed in order to reach a solid conclusion, especially for modeling indoor environment airflows.

Multiple-Equation Eddy-Viscosity Models. Another noticed development in eddy-viscosity models is multiple-equation eddy-viscosity models. A multiple-equation eddy-viscosity model is often developed and used for near-wall flows. Durbin (1991) suggested that the wall blocking effect, i.e., zero normal velocity at walls, is much more crucial than the viscous effect on near-wall flows. Instead of using the turbulent kinetic energy to calculate near-wall turbulence eddy viscosity, he suggested the use of a more proper quantity, the fluctuation of normal velocity [bar.[v'.sup.2], as the velocity scale in the near-wall eddy viscosity calculation. Durbin introduced a transport equation of [bar[.v'.sup.2]] and a corresponding damping function,, for the [bar.[v'.sup.2]] equation, which created a three-equation eddy-viscosity model (v2f model) including k,[epsilon] and [bar.[v'.sup.2]] transport equations. The model received continuous improvement and modification afterward (Durbin 1995; Lien and Durbin 1996; Davidson et al. 2003; and Laurence et al. 2004).

The v2f model, as one of the most recently developed eddy-viscosity models, has a more solid theoretical ground than LRN models but is less stable for segregated solvers. Choi et al. (2004) tested the accuracy and numerical stability of the original v2f model (Durbin 1995) and a modified v2f model (Lien and Kalitzin 2001) along with a two-layer model (Chen and Patel 1988) for natural convection in a rectangular cavity. The study found the original v2f model with the algebraic heat-flux model best predicted the mean velocity, velocity fluctuation, Reynolds shear stress, turbulent heat flux, local Nusselt number, and wall shear stress. The predicted results agreed with the measurements fairly well. However, this model exhibits the numerical stiffness problem in a segregate solution procedure, such as the SIMPLE algorithm, which requires remedy. Davidson et al. (2003) discovered that the v2f model could overpredict model could [bar.[v'.sup.2]] in regions far away from walls. They analyzed the f equation in isotropic condition and postulated a simple but effective way to limit [[bar].v'.sup.2] in nearly isotropic flow regions. With this restriction function, the v2f model can improve the accuracy in regions far away from walls. The v2f model brings more turbulence physics, especially for low-speed near-wall flows, which are critical in enclosed environments. However, the model has not been well tested and evaluated for indoor environment modeling under different flow conditions. A comprehensive and quantitative evaluation is inevitable before the model can be recommended.

Other than the v2f models, some other multiple-equation eddy-viscosity models can be found in literature. For instance, Hanjalic et al. (1996) proposed a new three-equation eddy-viscosity model by introducing a transport equation for RMS temperature fluctuation, [[bar.[[theta]'.sup.2]], for high Raleigh number flows. However, all these models become more complicated and have not been well accepted and applied for predicting air distributions in enclosed environments.

RANS Reynolds Stress Models

Most eddy viscosity models assume isotropic turbulence structures, which could fail for flows with strong anisotropic behaviors, such as swirling flows and flows with strong curvatures. RSMs, instead of calculating turbulence eddy viscosity, explicitly solve the transport equations of Reynolds stresses and fluxes. However, the derivation of the Reynolds stresses transport equations leads to higher-order unsolved turbulence correlations, such as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which need be modeled to close the equations.

The development and application of Reynolds stress models can be traced back to the 1970s. Studies directed toward three-dimensional flows, however, began to appear in the 1990s. Early applications of the RSM in room airflow computation include those by Murakami et al. (1990) and Renz and Terhaag (1990). They computed airflow patterns in a room with jets. The results showed that the Reynolds stress model is superior to the standard k-[epsilon] model because anisotropic effects of turbulence are taken into account. The same conclusions were reached recently by Moureh and Flick (2003), who investigated the characteristic of airflow generated by a wall jet within a long and empty slot-ventilated enclosure. Dol and Hanjalic (2001) predicted the turbulent natural convection in a side-heated near-cubic enclosure. They found that the secondmoment closure is better at capturing thermal three-dimensional effects and strong streamline curvature in the corners, while the k-[epsilon] model still provides reasonable predictions of the first moments away from the corners.

Chen (1996) compared three RSMs with the standard k-[epsilon] model for natural convection, forced convection, mixed convection, and impinging jet in a room. He concluded that the RSMs are only slightly better than the k-[epsilon] model and have a severe penalty in computing time. Based on a large number of applications for engineering flows, Leschziner (1990) concluded that RSMs are appropriate and beneficial when the flow is dominated by a recirculation zone driven by a shear layer. Among various RSMs, the model developed by Gibson and Launder (1978) and the one by Gatski and Speziale (1993) are often used in practice. The models, however, still have some weaknesses that need to be addressed. Tornstrom and Moshfegh (2006), for instance, found that the RSM with linear pressure-strain approximation overpredicted the lateral spreading rate and the turbulent quantities of three-dimensional cold wall jets. The RSMs are still receiving continuous study and improvement, mostly related to the fundamental research of turbulence mechanism. The models will need significant justification of application advantages before they can soundly be accepted and used for room airflow prediction. Most existing room airflow studies indicated that the marginal improvement on prediction quality of RSMs is not well justified by the high computational costs.

To reduce the computing time of the RSMs, algebraic Reynolds stress models (ASMs) were developed accordingly (Rodi 1976). The ASM derives algebraic equations for all Reynolds stress and fluxes from the differential stress models in which each Reynolds stress correlates with others and the derivatives of velocity and temperature. Jouvray et al. (2007) tested several ASMs against the standard k-[epsilon] and k-l models (Wolfshtein 1969) for two rooms with displacement and mixing ventilation. The results showed that the nonlinear ASMs give marginally better agreement with the measured data than do others. The application of the nonlinear models may not be well justified due to the strong case-dependent stability performance and the high additional computational costs.

Large-Eddy Simulation Models

Due to the rapid development of computer speed and power, LES models have recently received increased attention for modeling engineering flows. LES is an intermediate modeling technique between DNS and RANS. LES solves filtered (transformed) Navier-Stokes equations for large-scale eddies while modeling small-scale (also known as subgrid scale) eddies. Filtering of various variables in the Navier-Stokes equations is similar to the process of Reynolds averaging, and the resulting equations for incompressible flow can be written in a form similar to the RANS equations. Smagorinsky (1963) proposed the first subgrid model correlating eddy viscosity to the strain rate, which can be written in the form of eddy viscosity as follows:

[[tau].sub.ij] = [1/3][[tau].sub.kk][[delta].sub.ij]-2[[upsilon].sub.t][[bar.[S.sub.ij]] (4)

where [[bar].S.sub.ij] is the strain-rate tensor based on the filtered velocity field, and the isotropic part, [[tau].sub.kk], is an unknown scalar and is usually combined with [bar].p. The eddy viscosity is expressed as follows:

[[upsilon].sub.t] = [([c.sub.s][DELTA]).sup.2]|[bar.S]| (5)

where |[[bar].S| = [square root of 2[bar.S.sub.ij] [bar.S.sub.ij], [DELTA] is the filter width, and [C.sub.s] is the Smagorinsky constant. Different Smagorinsky constants were proposed by various researchers. Lilly (1996) suggested a value of 0.17 for [C.sub.s] in homogeneous isotropic turbulent flow. Many variants of the Smagorinsky model were proposed thereafter. In physics, the [C.sub.s] may not be a constant. Thus, the dynamic Smagorinsky-Lilly model based on the Germano identity (Germano et al. 1991; Lilly 1992) calculates the [C.sub.s] with the information from resolved scales of motion as follows:

[([C.sub.s]).sup.2] = [<[L.sub.ij][M.sub.ij]>/<[M.sub.ij][M.sub.ij]>] (6)

where [L.sub.ij] and [M.sub.ij] are the resolved stress tensors and < > is an average operation on a homogeneous region. Without the average, the dynamic model has been found to yield a highly variable eddy-viscosity field with negative values, which causes the numerical instability. However, the average operation is difficult to implement when the flow field does not have statistical homogeneous direction. Meneveau et al. (1996) proposed the Lagrangian dynamic model in which a Lagrangian time average was applied to Equation 6. Zhang and Chen (2000) proposed the application of an additional filter to Equation 6, which improved the simulation of indoor airflows. Other, more complex models have been proposed to improve accuracy, such as the dynamic models as reviewed by Meneveau and Katz (2000).

In the last decade, LES has been increasingly applied to model airflows in enclosed environments due to its rich dynamic details as compared to RANS models. Some representative applications include forced convection flow in a room (Davidson and Nielsen 1996; Emmerich and McGrattan 1998) or an airliner cabin (Lin et al. 2005), fire-driven air and smoke flows (McGrattan et al. 2000), natural ventilation flow in buildings (Jiang and Chen 2001), particle dispersion in buildings (Jiang and Chen 2002; Beghein et al. 2005; Chang et al. 2006; Zhang and Chen 2006).

Chow and Yin (2004) indicated that, due to the short computing time and less knowledge demand of users, the k-[epsilon] turbulence model is still a practical approach (the first choice) for simulating fire-induced airflows; although, the LES approach would give more detailed information that is important for understanding dynamic fire and smoke behaviors. Tian et al. (2006) compared the predictions of indoor particle dispersion and contaminant concentration distribution in a model room with LES and the standard k-[epsilon] model and the RNG k-[epsilon] model. Their study showed that all three of the turbulence model predictions were in good agreement with the experimental data, while the LES model yielded the best agreement. Their paper thus concluded that the LES prediction can be effectively employed to validate various k-[epsilon] models that are widely applied in building simulations. Musser and McGrattan (2002) evaluated LES for indoor-air-quality modeling and smoldering fires and indicated that LES can, in general, predict the experimental data reasonably well; however, care must be taken in defining convection from heated surfaces and grid resolution. As indicated by most studies, the LES model provides more detailed and accurate prediction of air distributions in enclosed environments, which could be important for understanding the flow mechanism; however, the high demand on computing time and user knowledge makes LES still viable mainly for research and RANS model development purposes.

Detached-Eddy Simulation Models

The DES method presents the most recent development in turbulence modeling, which couples the RANS and LES models to solve problems where RANS is not sufficiently accurate and LES is not affordable. The earliest DES work includes Spalart et al. (1997) and Shur et al. (1999), in which the one-equation eddy-viscosity model (Spalart and Allmaras 1992) was used for the attached boundary layer flow, while LES was used for free-shear flows away from the walls. Since the formation of eddy viscosity in RANS and LES models is similar, the S-A and LES models can be coupled using this similarity. In the near-wall region, the wall distance, d, of a cell is normally much smaller than the stream-wise and span-wise grid size. In the regions far away from the wall, the wall distance is usually much larger than the cubic root of the cell volume, . Hence, the switch between the S-A and LES models can be determined by comparing d and [DELTA]. When d is much larger than [DELTA], LES is performed; otherwise, the RANS (S-A) model is executed.

In practice, the switch between the RANS and LES models requires more programming and computing effort than simply changing the calculation of the length scale. In fact, many implementations of the DES approach allow for regions to be explicitly designated as RANS or LES regions, overruling the distance-function calculation. Squires (2004) reviewed and summarized the current status and perspectives of DES for aerospace applications. Keating and Piomelli (2006) combined a RANS near-wall layer with an LES outerflow with a dynamic stochastic forcing method, which can provide more accurate predictions of the mean velocity and velocity fluctuations.

Some comparison studies of DES, LES, and RANS can be found in the recent literature (Roy et al. 2003; Jouvray and Tucker 2005; Jouvray et al. 2007). These recent studies indicate that DES appears to be a promising model, giving the best velocity agreement and overall good agreement with measured Reynolds stresses. However, they also mention that the encouraging DES results could be fortuitous because the method has the potential for LES zones to occur downstream of RANS zones, resulting in poor LES boundary conditions. In addition, the eddy resolving approaches (LES and DES) demand extremely high computational costs and computer powers. As an emerging technology, DES still needs more studies before it can be applied for predictions of air distributions in enclosed environments.

CONCLUSION

This paper reviews the primary turbulence models that have been used for CFD prediction of air distributions in enclosed environments. The selective literature review shows that a large collection of turbulence models can be (and have been) applied for diverse indoor-air simulations. Each turbulence model has its own pros and cons. There are no universally preferable turbulence models for indoor-airflow simulation. The selection of a suitable model depends mainly on accuracy needed and computing time afforded.

Table 2 presents the selected prevalent turbulence models for predicting airflows in enclosed environments, ranging from RANS to LES. The models are organized into eight subcategories. Based on the perceived model popularity, we have identified one prevailing turbulence model from each of the eight subcategories. The prediction performance of these models for indoor airflows has been further evaluated and analyzed by modeling a series of benchmark test cases. This evaluation is presented in the companion paper (Zhang et al. 2007).

Table 2. List of Popular and Prevalent Turbulence Models for
Predicting Airflows in Enclosed Environments

                Model                        Primary      Prevalent
            Classification                  Turbulence      Models
                                         Models Used in   Identified
                                           Indoor-Air
                                           Simulations

RANS            EVM  Zero-equation      Zero-equation    Indoor
                                        (Chen and Xu     zero-equation
                                        1998)

                     Two-equation       Standard         RNG
                                        k-[epsilon]      k-[epsilon]
                                        (Launder and
                                        Spalding
                                        1974)
                                        RNG
                                        k-[epsilon]
                                        (Yakhot and
                                        Orszag 1986)
                                        Realizable
                                        k-[epsilon]
                                        (Shih et al.
                                        1995)

                                        LRN-LS           LRN-LS
                                        (Launder and
                                        Sharma 1974)
                                        LRN-JL (Jones
                                        and Launder
                                        1973)
                                        LRN-LB (Lam
                                        and Bremhorst
                                        1981)

                                        LRN k-[omega]    SST k-[omega]
                                        (Wilcox 1994)
                                        SST k-[omega]
                                        (Menter 1994)

                     Multiple-equation  v2f-dav          v2f-dav
                                        (Davidson et
                                        al. 2003)
                                        v2f-lau
                                        (Laurence et
                                        al. 2004)

                RSM                     RSM-IP (Gibson   RSM-IP
                                        and Launder
                                        1978)
                                        RSM-EASM
                                        (Gatski and
                                        Speziale
                                        1993)

LES                                     LES-Sm           LES-Dyn
                                        (Smagorinsky
                                        1963)
                                        LES-Dyn
                                        (Germano et
                                        al. 1991;
                                        Lilly 1992)
                                        LES-Filter
                                        (Zhang and
                                        Chen 2000,
                                        2005)

DES                                     DES (S-A)        DES-SA
                                        (Shur et al.
                                        1999)
                                        DES (ASM)
                                        (Batten et al.
                                        2002)

Note that Table 2 is not necessarily a comprehensive representation of all previously developed turbulence models. Rather, this paper only reviews the turbulence models that consider the turbulence at single time and length scale at a point for simplicity needed in practice. Increasing knowledge of turbulence modeling has made it challenging to conduct a systematic classification and review of existing turbulence models.

As observed from the literature, the conclusions from past studies are not always consistent. Opposite observations can be attributed to the differences in cases simulated, numerical factors (e.g., scheme, grid, and program), judging criteria, and user skills. Without knowledge of all details of the simulations and cases studied, it is difficult to pass judgment on the merits of each turbulence model based solely on its presentation in the literature. This study has instead identified turbulence models that are either popularly used or which have been proposed recently and have some potential for indoor airflow applications. Part 2 of this study evaluates the selected models by comparing them to published experimental data. Despite the disparities among the studies in the literature, some general remarks for turbulence modeling of air distributions in enclosed environments can be made:

1. The standard k-[epsilon] model with wall functions (Launder and Spalding 1974) is still widely used and provides acceptable results (especially for global flow and temperature patterns) with good computational economy. The model may have difficulty dealing with special room situations (e.g., high buoyancy effect and/or large temperature gradient).

2. The RNG k-[epsilon] model (Yakhot and Orszag 1986) provides similar (or slightly better) results as the standard model and is also widely used for airflow simulations in enclosed environments.

3. The zero- and one-equation models with specially tuned coefficients are appropriate (sometimes even better than significantly more detailed models) for cases with similar flow characteristics as those used to develop the models.

4. Most LRN k-[epsilon] models and nonlinear RANS models provide no or marginal improvements on prediction accuracy but suffer from strong case-dependent stability problems and have long computing times.

5. The Reynolds stress models can capture some flow details that cannot be modeled by the eddy viscosity models. The marginal improvements on the mean variables, however, are not well justified by the severe penalty on computing time.

6. The k-[omega] model (Wilcox 1988) presents a new potential to model airflows in enclosed environments with good accuracy and numerical stability. Most existing studies indicate that the SST k-[omega] model (Menter 1994) has a better overall performance than the standard k-[epsilon] and RNG k-[omega] models, but a systematic evaluation (especially for modeling indoor airflows) is needed before a solid conclusion can be reached.

7. The v2f model (Durbin 1995) looks very promising for indoor environment simulations but needs to resolve some inherent numerical problems and undergo a comprehensive evaluation.

8. The LES model provides more detailed and, perhaps, more accurate predictions for indoor airflows, which could be important for understanding the flow mechanism. However, the high demand on computing time and user knowledge still makes LES a tool mainly for research and RANS model development.

9. The DES model can be a valuable intermediate modeling approach but needs significant further study, improvement, and validation before it can be used for room airflow predictions.

ACKNOWLEDGMENTS

Z. Zhang and Q. Chen would like to acknowledge the financial support to the study presented in this paper by the U.S. Federal Aviation Administration (FAA) Office of Aerospace Medicine through the Air Transportation Center of Excellence for Airliner Cabin Environment Research under cooperative agreement 04-C-ACE-PU. Although the FAA has sponsored this study, it neither endorses nor rejects the findings of this research. The presentation of this information is in the interest of invoking technical community comments on the results and conclusions of the research.

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Zhiqiang (John) Zhai, PhD

Member ASHRAE

Wei Zhang, PhD

Member ASHRAE

Zhao Zhang

Student Member ASHRAE

Qingyan (Yan) Chen, PhD

Fellow ASHRAE

Zhiqiang (John) Zhai is an assistant professor in the Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder. Zhao Zhang is a graduate research assistant, Wei Zhang is an affiliate, and Qingyan (Yan) Chen is a professor of mechanical engineering in the School of Mechanical Engineering, Purdue University, West Lafayette, IN. No. 6, November 2007. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHR

Received April 16, 2007; accepted August 29, 2007