# On a hybrid RANS/LES approach for indoor airflow modeling (RP-1271).

INTRODUCTION

Airflow in enclosed spaces can be complicated due to complex flow features such as flow transition and lack of stability. Many indoor airflows are transitional when the Reynolds number based on the supply air grille is in the region of 2000 < Re < 3500. Turbulence is generated at the grille where the fluctuating component of velocity is a fraction of the mean velocity. As the air travels further into the room, the fluctuating component may decay gradually due to the decrease in the mean flow gradient and damping effect of the solid surfaces. Therefore, relaminization may occur within the occupied space, and the flow is transitional. In addition to flow transition, in many indoor environments, airflow with relatively high air change rates (between 5 to 20 ach) can be unstable under the transitional Reynolds number. One reason for this is that the transitional phenomenon makes the flow unstable. In the transitional region, the inertial force is approximately balanced by the viscous force. A random small impact from the main stream can break down this balance and lead to transitional flow. This mechanism results in instability of the main flow.

Another reason for unstable flow is the interaction between different flow features. Many enclosed environments are mechanically ventilated. However, the air can also be driven by buoyancy in the occupied zone, where the thermal plume can be as strong as the flow from the mechanical ventilation system. The two flows interact, and result in instability.

The complex geometry of the indoor environment can also lead to unstable flow. Furniture in an indoor environment can generate flow separation, which is usually unstable. As discussed above, flow transition and unstable behaviors cause complicated indoor airflow phenomena, making indoor airflow difficult to model.

To model indoor airflow, one approach is to apply computational fluid dynamics (CFD) to solve the flow equations. Airflow in a room is governed by the Navier-Stokes (N-S) equation. The most straightforward way to solve this equation is called direct numerical simulation (DNS), which directly solves the N-S equation together with initial and boundary conditions and produces a realization of the flow. The DNS does not need a model and can provide the most accurate and detailed flow motion. However, the DNS resolves a wide range of spatial and temporal scales, from the smallest Kolmogorov scale (dissipative eddies) to the integral scale (large flow motion), which requires a very fine meshes and time steps (Wilcox 2006). Even for steady-state flow, DNS needs to perform unsteady calculation, and calculate a long period of time to obtain the mean flow. As a result, the cost of DNS can be extremely high. For example, to simulate a small office under mechanical ventilation with Re = 5000, grid spacing should be on the order of 0.039 in. ([10.sup.-3] m) (Kolmogorov length scale). The number of cells for the room is on the order of [10.sup.10]. The size of the time step should be on the order of [10.sup.-3] s (Kolmogorov time scale). To obtain a sufficient statistical sample, flow should be calculated for several minutes; therefore, the number of time steps should be in the order of [10.sup.6]. However, it is not feasible to perform this simulation on a personal computer or a moderate computer cluster in the near future.

A more realistic and widely used approach for indoor airflow simulations is by Reynolds-Averaged Navier-Stokes (RANS) equation modeling. The RANS simulation solves the time-averaged N-S equation and models the additional Reynolds stresses. This modeling approach can significantly reduce the grid resolution requirement, and can be performed as steady-state. Many studies have used different RANS models for indoor airflow simulation (Yuan et al. 1999; Holmes et al. 2000; Gadgil et al. 2003; Hsieh and Lien 2004; Zhang et al. 2007) and found mixed results of the model performance. Zhang et al. (2007) tested eight popular turbulence models for different indoor airflows and concluded that no model was superior to the others. Wang and Chen (2009) further tested these eight models using a set of experimental cases with gradually added flow features and found that the RANS models failed to predict flow separation. This conclusion was supported by Costa et al. (1999) to some extent, who found that the low-Reynolds-number model suffered from a problem, leading to the prediction of an unrealistic local minimum of the wall heat transfer near points of flow reattachment. The failure of the RANS models to predict a separated flow could have resulted from the time-averaging approach. In the separation region, flow is very unstable, and the velocity magnitude and direction change rapidly. Therefore, it may not be meaningful to have a time-averaged solution for this case since the information contained by the mean value of flow variables is too limited to describe the rapidly varying unstable flow. Thus, the RANS models may not be capable of correctly predicting unstable airflow features in enclosed spaces, such as the separation caused by furniture, impingement flow, and unsteady plumes.

With the advancement of computing power, large eddy simulation (LES) is becoming increasingly popular for engineering applications due to its ability to solve unstable separated flow. LES solves the filtered N-S equation for the energy-containing eddies (large eddies) and models the subgrid-scale flow motions (small eddies). The turbulence model for LES is not as important as that for a RANS model (Wang and Chen 2009). In other words, LES relies more on fundamental flow physics than on modeling assumptions. As a result, LES is more accurate and informative than RANS models for airflow modeling, especially for a separated and unstable flow. Since LES solves only the large-scale eddies, the computing cost is much lower than that of DNS. The cell size required by LES could be much larger than the Kolmogorov length scale. Therefore, LES uses significantly less computational time than DNS. For example, to calculate a flow over a backward-facing step by DNS (Le et al. 1997) and by LES (Akselvoll and Moin 1993) at Reynolds number of 5000, LES requires only 3% of the cells and 2% of the computing time needed by DNS, while having equally good results compared with the experimental data (Wilcox 2006). However, it is difficult to apply LES for the near-wall region of indoor airflow modeling, where the size of the energy-bearing eddies (large eddies) is comparable to that of the dissipative eddies. If LES is to resolve most energy-bearing eddies, the computing cost could be the same as that of the DNS. For airflow in enclosed spaces, the flow is bounded by walls, furniture, and occupants. All of these solid surfaces could significantly increase the computing cost by LES.

To overcome the disadvantages of LES and RANS, the concept of the RANS/LES hybrid model has emerged in the past decade. The hybrid model applies LES in the separated flow region to capture the three-dimensional, time-dependent flow, and applies a RANS model in the attached boundary layers to avoid the excessively fine meshes required by LES. This approach was first introduced by Spalart et al. (1997), adopting the one-equation Spalart model. Some studies used Smagorinsky (1963) zero-equation LES model with zero-equation RANS model such as Cebeci-Smith and Baldwin-Lomax models (Georgiadis et al. 2003; Kawai and Fujii 2005). Many other studies adopted multi-equation RANS models with LES, such as k-[omega] model (Strelets 2001; Davidson and Peng 2003), k-[epsilon] model (Hamba 2001, 2003), and k-l model (Tucker and Davidson 2004). After reviewing the RANS/LES hybrid models, our effort was to develop a new hybrid model, the semi-v2f/LES model, for modeling airflows in an enclosed environment. The new model used transport equations for k and [epsilon], and an algebraic equation for the normal stress near a wall to model turbulence viscosity in the RANS region and subgrid turbulence viscosity for the LES region. The new model was tested by applying it to a mixed convection flow in a model room, and to a strong buoyancy-driven flow with a high temperature gradient in a room.

MODEL DEVELOPMENT

Hybrid RANS/LES Simulation for Indoor Airflow

Indoor airflow is governed by the N-S equation together with the energy equation for the fluid phase (FLUENT 2003). Since it is not feasible to use DNS for solving such a flow, the N-S equation should be approximated in order to make it solvable with the present capacity of computers. By using Reynolds-averaged approach, flow variables can be written as

[empty set] = [PHI] + [empty set]' (1)

where [PHI] is the mean value of flow variables, and [empty set]' is the fluctuating part of flow variables. On the other hand, LES uses a filter to obtain large-scale flow variables:

[bar.[empty set]]([bar.x], t) = [integral][[integral] V][integral][empty set]([bar.x], t)G([bar.x][bar.x'], t)d[bar.x'] (2)

where the overbar denotes filtered variables, and G is a filter function.

The two methods can transform the N-S equation into a single form:

[[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.i]] = 0 (3)

[[partial derivative][[bar.u].sub.i]/[partial derivative]t] + [[bar.u].sub.j][[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.j]] = - [1/[rho]][[partial derivative][bar.P]/[partial derivative][x.sub.i]] + v[[[[partial derivative].sup.2][[bar.u].sub.i]]/[[partial derivative][x.sub.j][partial derivative][x.sub.j]]] + [[partial derivative][[tau].sub.ij]/[partial derivative][x.sub.j]] (4)

where the overbar denotes mean variables for RANS, and filtered variables for LES. The term [[tau].sub.ij] in the equation is the Reynolds stress for RANS:

[[tau].sub.ij] = -[bar.[u.sub.i]'[u.sub.j]'] (5)

and the subgrid-scale stress for LES:

[[tau].sub.ij] = [[bar.u].sub.i][[bar.u].sub.j] - [bar.[u.sub.i][u.sub.j]] (6)

Many turbulence models use a Boussinesq approximation to simplify the second-order symmetrical tensor by correlating it to a scalar [v.sub.t], and transform Equation 4 into

[[partial derivative][[bar.u].sub.i]/[partial derivative]t] + [[bar.u].sub.j][[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.j]] = - [1/[rho]][[partial derivative][bar.P]/[partial derivative][x.sub.i]] + [[partial derivative]/[partial derivative][x.sub.j]][(v + [v.sub.t])[[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.j]]] (7)

where [v.sub.t] is the turbulence viscosity for the RANS model and the subgrid-scale turbulence viscosity for LES.

Spalart et al. (1997) introduced the first hybrid RANS/LES method: the detached eddy simulation Spalart-Allmaras (DES-SA) model. This method used the Spalart-Allmaras one-equation model to calculate the turbulence viscosity for the RANS part and modified the length scale in the equation to calculate the subgrid-scale turbulence viscosity for the LES part. The transport equation for v is written as

[[partial derivative][~.v]/[partial derivative]t] + [[bar.u].sub.j][[partial derivative][~.v]/[partial derivative][x.sub.j]] = [c.sub.b1]S[~.v] - [c.sub.w1][[florin].sub.w][([[~.v]/[~.l]]).sup.2] + [1/[sigma]][[partial derivative]/][partial derivative][x.sub.k]]][(v + [~.v])[[partial derivative][~.v]/[[partial derivative][x.sub.k]]]] + [[c.sub.b2]/[sigma]][[partial derivative][~.v]/[[partial derivative][x.sub.k]]][[partial derivative][~.v]/[[partial derivative][x.sub.k]]] (8)

where [~.v] is the modified turbulence viscosity used as a working variable, [c.sub.b1], [c.sub.b2], [c.sub.w1], and [sigma] are constants, and

[~.l] = min([d.sub.w], [C.sub.DES][DELTA]) [DELTA] = max([DELTA]x, [DELTA]y, [DELTA]z) (9)

[~.l] is the DES-blending function. In Equation 9, [d.sub.w] is the distance to the nearest wall and also the RANS length scale used by the S-A model, [C.sub.DES] is a constant, and [DELTA] is the largest cell spacing. [C.sub.DES][DELTA] represents the grid length scale. [~.l] compares the RANS length scale with the grid length scale. If the cell is close to the wall (i.e., the wall distance is smaller than the grid size), the DES-blending function is equal to the wall distance, and the transport equation has the same form as the Spalart-Allmaras model. Otherwise, the equation becomes a Smagorinsky-like subgrid-scale turbulence viscosity based on the local grid size.

Many hybrid approaches with more advanced RANS models have been developed based on the idea of DES-SA (Hamba 2001; Strelet 2001; Tucker and Davidson 2004). One can explicitly replace the [d.sub.w] in the DES-SA model with the turbulence length scale, or implicitly involve other RANS models. Strelet (2001) developed a RANS/LES hybrid model based on the shear stress transport (SST) k-[omega] model from Menter et al. (2003). The major modification was on the dissipation term in the SST k-[omega] model:

[D.sub.RANS.sup.k] = [rho][k.sup.[3/2]]/[~.l] (10)

where

[~.l] = min([l.sub.[k-[omega]]], [C.sub.DES][DELTA]) (11)

Here, [C.sub.DES][DELTA] was the same as that in the DES-SA model. Note that the RANS scale used in this model is the turbulence length scale, which means that switching between RANS and LES is based on the local eddy size and local grid size.

Hybrid RANS/LES models are usually classified based on the RANS model they use. The performance of a hybrid RANS/LES model also depends on its RANS model. For indoor airflow simulation, a few preliminary studies in the literature show that the available RANS/LES hybrid models did not perform as well as LES or even as well as some RANS models. This is because the RANS models used were not good (Zhang et al. 2007; Wang and Chen 2009). Therefore, our effort was to identify an appropriate RANS model for indoor airflow simulations and to develop a new hybrid model with the RANS model.

Identification of a Suitable RANS Model for Hybrid Simulation

As discussed above, the accuracy of the RANS model is essential to the performance of a hybrid model. Some studies have evaluated different RANS turbulence models for indoor airflow, which could be helpful for choosing a suitable RANS model. Zhai et al. (2007) identified 17 models that may be appropriate for calculating indoor airflows. Wang and Chen (2009) tested six of them that are most promising for indoor airflows and concluded that the v2f, RNG k-[epsilon], and Reynolds stress models were the best. This conclusion was consistent with that from Zhang et al. (2007) for forced, natural, and mixed convection flows.

The RNG k-[epsilon] model uses a wall function near solid boundaries, which makes it unsuitable for a hybrid model. The Reynolds stress model can resolve the near-wall region, but it has six scalar transport equations that are too computationally expensive. The v2f model can be used for the hybrid model, since it can resolve the near-wall viscous region at a reasonable computational cost (three scalar transport equations) and accounts for anisotropic behavior near the wall. However, studies from the literature (Durbin 1991; Davidson et al. 2003) and our preliminary research show that the transport equation for the wall normal stress [v'.sup.2] and the elliptic equation for the relaxation function [florin] make the model numerically unstable. Therefore, it is essential to simplify the v2f model before implementing it in a hybrid simulation.

The Semi-v2f/LES Model

Our effort to develop the v2f model further was to introduce an algebraic equation for the normal stress [v'.sup.2]:

[[bar.v'].sup.2] = [florin](k, [epsilon], [upsilon], y ...) (12)

to replace the transport equation for [bar.[v'.sup.2]] and the elliptic equation for [florin], which have been problematic. The procedure was first to derive new equations for [v'.sup.2] and turbulence viscosity. The constant in the equations was then determined through curve fitting of the flow data obtained for indoor airflow. The detailed procedure was as follows.

* [bar.[v'.sup.2]] Equation

Starting from the Boussinesq approximation for incompressible flow:

-[[tau].sub.ij] = -2[v.sub.t][S.sub.ij] + [2/3] k * [[delta].sub.ij] (13)

considering the diagonal entries at wall normal direction (i = j = 2), and substituting the [v.sub.t] formulation of v2f model, it is easy to obtain a correlation:

[[[bar.v'].sup.2]/k] = [C.sub.1](1 - [C.sub.2] [[[bar.v'].sup.2]/[epsilon]] [[partial derivative]V/[partial derivative]y]) (14)

where [C.sub.1] and [C.sub.2] are constants. Through dimensional analysis of Equation (14), there are three nondimensional variables: [v'.sup.2]/k, y* = (y[C.sub.[mu].sup.[1/4]] [k.sup.[1/2]])/v, and l* = [k.sup.[3/2]]/([epsilon]y). Therefore, Equation 14 can be written as a dimensionless form:

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - g(y*, l*)] (15)

where g(y*, l*) is the function to be determined. If the y* value becomes large (far from the wall), the turbulence should be homogeneous:

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - g(y*, l*)] [right arrow] [C.sub.1] as y* [right arrow] [infinity] (16)

Near a wall, the normal kinetic energy goes to zero:

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - g(y*, l*)] [right arrow] 0 as y* [right arrow] 0 (17)

Assuming the wall damping effect is only related to the geometry, the influence of l* (the turbulence eddy scale) could be neglected. Thus, Equations 16 and 17 could be a good approximation of Equation 14. Namely,

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - exp(-[y*/[y*.sub.0]])] (18)

where [C.sub.1] and [y*.sub.0] are constants.

* Turbulence Viscosity

In the v2f model, the turbulence viscosity is modeled as

[v.sub.t] = [C.sub.[mu]] [[bar.v'].sup.2][k/[epsilon]] (19)

where [C.sub.[mu]] = 0.22 is a constant. By substituting Equation 18 into Equation 19, the turbulence viscosity can be determined by

[v.sub.t] = A * [1 - exp(-[y*/[y*.sub.0]])] * [[k.sup.2]/[epsilon]] (20)

where A and [y*.sub.0] are constants to be determined.

* Determination of Constant

The model derived above was to mimic the behavior of the v2f model. Therefore, this study simulated several room airflows using the v2f model, extracted the data from different locations, and performed a curve fitting on Equation 18 to determine the constants. The curve-fitting results led to the new algebraic [v'.sub.2] equation:

[[bar.v'].sup.2]/k [approximately equal to] 0.308 [1 - exp(-[y*/50.836])] (21)

and the turbulence viscosity:

[v.sub.t] = 0.07 [1 / exp([ - y*/50.836])] * [[k.sup.2]/[epsilon]] (22)

Figure 1 compares the [bar.[v'.sub.2]]/k near the wall by the algebraic equation with the DNS data (Del Alamo et al. 2004). The agreement is reasonably good.

[FIGURE 1 OMITTED]

* New Semi-v2f/LES Model

The new semi-v2f/LES model used the same k and [epsilon] equations as those in the DES realizable k-[epsilon] model, but has a new algebraic equation for [v'.sup.2] and a new eddy-viscosity correlation. The new model does not modify the formulation of the turbulence heat flux, but uses the same energy equation as that used by the DES realizable k-[epsilon] model. The appendix provides a more detailed formulation.

PERFORMANCE OF THE SEMI-V2F/LES MODEL

This study evaluated the performance of the semi-v2f/LES model by applying it to two benchmark cases: a mixed convection flow in a model room with moderate buoyancy (Wang and Chen 2009) and a strong buoyancy-driven flow in a model fire room (Murakami et al. 1995). The first case had a simple geometry but most features of indoor airflow. The second case was selected to test the robustness of the new model in an extreme scenario.

Mixed Convection Flow

Figure 2a shows the case schematic. Air velocity distributions were measured streamwise and at the cross sections in the 8 x 8 x 8 ft (2.44 x 2.44 x 2.44 m) cubic room. Air was supplied from a slot diffuser located at the upper left corner of the room, and exhausted from a slot at the lower right corner. The Reynolds number, based on inlet height, was about 2500, which was in the transitional region. A 4 x 4 x 4 ft (1.22 x 1.22 x 1.22 m) heated box was placed in the center with 700 W of power, which produced a thermal plume and caused flow separation. This case represents a typical indoor airflow scenario with the most important flow features. For more information about this case, see Wang and Chen (2009).

[FIGURE 2 OMITTED]

The semi-v2f/LES model was implemented in FLUENT (2003) by user-defined functions. According to Wang and Chen (2009), two grid resolutions were used for the new model: one was a 44 x 44 x 44 grid (coarse); the other was a 110 x 77 x 101 grid (fine). The v2f model modified by Davidson (2003) (v2f-Dav), the LES dynamic Smagorinsky-Lilly (LES-DSL) model, and the DES realizable k-[epsilon] model were also included in this study to evaluate the performance of the new model. The LES-DSL and DES realizable k-[epsilon] models used a 110 x 77 x 101 grid, and the v2f-Dav model used a 44 x 44 x 44 grid. The comparison was based on the velocity, temperature, and turbulence kinetic energy at 10 positions, as shown in Figure 2b. The results shown in this paper were at the positions where turbulence models had the best, worst, and average performances. For definition of the three categories, please see Wang and Chen (2009).

Figure 3 depicts the four models' air velocity predictions. In general, all the models predicted very similar, acceptable results. The new model slightly underpredicted the velocity at position (6) with a fine grid. The realizable k-[epsilon] had a similar prediction at this position. However, the discrepancy was very small. At position (5), where the flow was separated, all the models underpredicted the velocity. However, the new model with the fine grids and the DES realizable k-[epsilon] model did better than the other models. The v2f-Dav model failed primarily because Reynolds averaging may not be appropriate for separated flow. The LES-DSL model failed to provide good results because it requires a finer grids than DES. This also reflects the advantage of the DES models. At position (3), all the models gave similar results, though with small discrepancies.

[FIGURE 3 OMITTED]

Figure 4 shows the temperature prediction by the models. At position (8), the new model with coarse grids surprisingly predicted better results compared with the other models. The new model with fine grids also predicted slightly better results than the realizable k-[epsilon] and v2f-Dav models. The results of the LES-DSL model were comparable with those of the new model with fine grids. At position (4), the results of the new model with the two grid distributions were better than those from the other models. At position (5), the new model with the fine grids predicted one of the best profiles. The new model with coarse grids predicted an incorrect peak at Y/L = 0.6. The grid distribution of 44 x 44 x 44 was not fine enough for the new model to capture the separated flow.

[FIGURE 4 OMITTED]

Figure 5 shows the turbulence kinetic energy profiles predicted by the four models. The new model with fine grids predicted the best result at position (1). The DES realizable k-[epsilon] model performed similarly. The LES-DSL model significantly overpredicted the turbulence level, while the v2f-Dav model underpredicted it. At position (5), none of the four models predicted the turbulence kinetic energy well; it is indeed difficult to capture fluctuating, highly separated flow. At position (10), where most models showed average performance, the LES-DSL model was slightly better.

[FIGURE 5 OMITTED]

Strong Buoyancy-Driven Flow

The second case tested was a strong buoyancy-driven flow in a fire room. This case, chosen to test the new model's robustness, was designed by Murakami et al. (1995), who measured detailed fluctuating velocity using two-component laser doppler velocimetry (LDV), and the temperature using thermocouples. The test room was 5.9 ft (1.8 m) long, 3.9 ft (1.2 m) wide, and 3.9 ft (1.2 m) high, as shown in Figure 6. A 8.625 Btu/s (9.1 kW) heat source was placed at the corner of the room. The surface temperature at the heat source reached more than 932[degrees]F (500[degrees]C). An opening of size 1.3 x 3.0 ft (0.4 x 0.9 m) connected the air between the test room and the outside chamber. The fluctuating velocity and temperature were measured at 12 lines at two sections.

[FIGURE 6 OMITTED]

The numerical simulations calculated the air inside the test room as well as the outer enclosure (a very large laboratory). This helped to avoid instability in using a pressure boundary condition for the opening. The measured surface temperature was used as boundary conditions for the simulations. Due to the high temperature difference between the heat source surface and ambient air, the Boussinesq approximation is no longer valid. Instead, the ideal gas law was used for determining the air density:

[rho] = [[P.sub.op]/RT] (23)

where [P.sub.op] is the pressure at the opening, R is the gas constant, and T is the air temperature. Using variable density would change the form of equations presented in section 2. Since typical indoor airflow has a small temperature difference, it is much better to use the equations given in the Model Development section. This case is exceptional, so this article omits the differences on the governing equations.

Although this case was primarily used to test the robustness of the new model, the results from the DES Spalart-Allmaras (DES-SA) and LES-DSL models are included for comparison. The test was based on the comparison of the air velocity, air temperature, and root-mean-square (RMS) velocity predicted. The RMS reflects the turbulence predicted by the models. The results at all 12 measurement lines were compared, but only those at lines 2, 4, and 6 are shown here due to limited space.

Figure 7 depicts the calculated and measured velocity profiles in the x direction. All the models were stable in this extreme case. The semi-v2f/LES model predicted comparable results as the other models.

[FIGURE 7 OMITTED]

Figure 8 compares the predicted and measured air temperature profiles. Because of their correct velocity predictions, it is not surprising that all the models predicted reasonable results. However, all the models underpredicted the temperature near the ceiling. The semi-v2f/LES model performed relatively better than the other models due to its better turbulence viscosity formulation in the near wall region.

[FIGURE 8 OMITTED]

Figure 9 shows the RMS velocity profiles in the x direction. This flow quantity was obtained statistically using the solution at each time step. Although it is more difficult to get good agreement for the second-order momentum than the first-order one, the three models used were able to predict the correct trend of the profile. The DES-SA model predicted poorer results than the other two models because the model solved only one turbulence transport quantity (the modified turbulence viscosity) and could not calculate the turbulence length scale related to the local shear layer thickness. Therefore, the model may not be accurate for a complicated shear flow. The new semi-v2f/LES and LES-DSL models had comparable performances.

[FIGURE 9 OMITTED]

The computational cost of the new model was very similar to that of the DES realizable model, since they both solved two additional equations for turbulence. The DES-SA model took slightly less computational time, since it only solved one additional equation for turbulence. The LES-DSL model had a simpler formulation than the new model, but required much a finer grid near the wall for full LES simulation. Thus, it could need much more computational time.

CONCLUSION

This investigation developed a new RANS/LES hybrid model, the semi-v2f/LES model, for indoor airflow simulations. The development simplified the v2f model by replacing the partial differential equation for [v'.sup.2] and [florin] with an algebraic equation for [v'.sup.2].

The study tested the performance of the semi-v2f/LES model by applying it to predict mean and turbulence quantities in two indoor airflows. The first case was a mixed convection flow in a model room with typical indoor airflow features. The new model predicted the best turbulence kinetic energy results and comparable velocity and temperature results among the several models tested. The second case was a strong buoyancy-driven flow in a room. The new model performed well in this case. It gave the best temperature prediction and was one of the best models for turbulence predictions.

NOMENCLATURE

Acronyms

ach = air changes per hour

CFD = computational fluid dynamics

DES = detached eddy simulation

DES-SA = detached eddy simulation with Spalart-Allmaras model

DNS = direct numerical simulation

LDV = laser doppler velocimetry

LES = large eddy simulation

LES-DSL = large eddy simulation with dynamic Smagorinsky-Lilly model

N-S = Navier-Stokes

RANS = Reynolds-averaged Navier-Stokes equations

Re = Reynolds number

RMS = root mean square

S-A = Spalart-Allmaras

SST = shear stress transport

v2f = v2f model

v2f-Dav = v2f model modified by Davidson Top

Symbols

A = model constant

C = model constant

[D.sub.RANS.sup.k] = dissipation term in the SST k-[omega] model

[d.sub.w] = distance to nearest wall

G = filter function for LES

k = turbulence kinetic energy

L = size of test room

l = turbulence length scale

[~.l] = DES blending function

l* = distance to wall, nondimensional

R = gas constant

P = pressure

[S.sub.ij] = strain rate tensor

T = temperature

u = velocity

[bar.[v'.sub.2]] = wall normal stress

x, y, z = Cartesian coordinates

y* = y[C.sub.[mu].sup.[1/4]] [k.sup.[1/2]]/v

Greek Symbols

[DELTA] = grids spacing

[[delta].sub.ij] = Kronecker delta

[epsilon] = turbulence dissipation rate

v = kinematic viscosity

[~.v] = modified turbulence viscosity

[rho] = density

[sigma] = model constant

[[tau].sub.ij] = for RANS: -[bar.[u.sub.i][u.sub.j]]; for LES: [[bar.u].sub.i][[bar.u].sub.j] - [bar.[u.sub.i][u.sub.j]]

[PHI] = mean flow variables

[empty set] = flow variables

[omega] = specific turbulence dissipation rate

Subscripts

DES = detached eddy simulation

i = i th direction

j = j th direction

LES = large eddy simulation

max = maximum value

min = minimum value

op = at opening

RANS = RANS model

RKE = RANS k-[epsilon] model

t = turbulence quantities

Overbars

Overbar = mean and filtered flow variables for RANS and LES, respectively

Tilde = variables for DES-SA model

REFERENCES

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APPENDIX

This appendix provides all the equations for the semi-v2f/LES model. The transport equation for k is

[[partial derivative]/[partial derivative]t]([rho]k) + [[partial derivative]/[partial derivative][x.sub.j]]([rho][ku.sub.j]) = [[partial derivative]/[partial derivative][x.sub.j]][([mu] + [[[mu].sub.t]/[[sigma].sub.k]])[[partial derivative]k/[partial derivative][x.sub.j]]] + [G.sub.b] - [rho][epsilon] - [Y.sub.M] + [S.sub.k]

The transport equation for [epsilon] is

[[partial derivative]/[partial derivative]t]([rho][epsilon]) + [[partial derivative]/[partial derivative][x.sub.j]]([rho][epsilon][u.sub.j]) = [[partial derivative]/[partial derivative][x.sub.j]][([mu] + [[[mu].sub.t]/[[sigma].sub.[epsilon]]])[[partial derivative][epsilon]/[partial derivative][x.sub.j]]] + [rho][C.sub.1][S.sub.[epsilon]] - [rho][C.sub.2][[[epsilon].sup.2]/[k + [square root of v[epsilon]]]] + [C.sub.1[epsilon]][[epsilon]/k][C.sub.3[epsilon]][G.sub.b] + [S.sub.[epsilon]]

The algebraic equation for [[bar.v].sup.'2] is

[[bar.v'].sup.2] = 0.308[1 - exp(-[y*/50.836])]k

Turbulence viscosity is determined by

[v.sub.t] = min{0.07 [[k.sup.2]/[epsilon]], 0.22[[bar.v'].sup.2T]} = 0.07 min{[[k.sup.2]/[epsilon]][1 - exp(-[y*/50.836])]} * T

where

T = min {2.0 [L/[square root of k]], max[[k/[epsilon]], 6[([[[bar.v'].sup.2]/[epsilon]]).sup.[1/2]]]}

and

L = [C.sub.L] min{[[k.sup.[3/2]]/[epsilon]], [V.sub.CELL.sup.[1/3]]}

The terms in k and [epsilon] equations are

[G.sub.k] = [[mu].sub.t][S.sup.2]

where

S = [square root of 2[S.sub.ij][S.sub.ij]]

[G.sub.b] = -[g.sub.i][[[mu].sub.t]/[rho][Pr.sub.t]][[partial derivative][rho]/[partial derivative][x.sub.i]]

[Y.sub.k] = [[rho][k.sup.[3/2]]/[l.sub.DES]]

where

[l.sub.DES] = min([l.sub.rke], [l.sub.LES])

[l.sub.rke] = [[k.sup.[3/2]]/[epsilon]]

[l.sub.LES] = [C.sub.DES][DELTA]

[C.sub.1] = max[0.43, [[eta]/[eta] + 5]], [eta] = S[k/[epsilon]]

The model constant is

[C.sub.1[epsilon]] = 1.44, [C.sub.2] = 1.9, [[sigma].sub.k] = 1.0, [[sigma].sub.[epsilon]] = 1.2, [C.sub.DES] = 0.61

Received January 5, 2010; accepted March 11, 2010

This paper is based on findings resulting from ASHRAE Research Project RP-1271.

Miao Wang is a doctoral candidate and Qingyan (Yan) Chen is a professor at the Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN.

Miao Wang

Student Member ASHRAE

Qingyan (Yan) Chen, PhD

Fellow ASHRAE

Airflow in enclosed spaces can be complicated due to complex flow features such as flow transition and lack of stability. Many indoor airflows are transitional when the Reynolds number based on the supply air grille is in the region of 2000 < Re < 3500. Turbulence is generated at the grille where the fluctuating component of velocity is a fraction of the mean velocity. As the air travels further into the room, the fluctuating component may decay gradually due to the decrease in the mean flow gradient and damping effect of the solid surfaces. Therefore, relaminization may occur within the occupied space, and the flow is transitional. In addition to flow transition, in many indoor environments, airflow with relatively high air change rates (between 5 to 20 ach) can be unstable under the transitional Reynolds number. One reason for this is that the transitional phenomenon makes the flow unstable. In the transitional region, the inertial force is approximately balanced by the viscous force. A random small impact from the main stream can break down this balance and lead to transitional flow. This mechanism results in instability of the main flow.

Another reason for unstable flow is the interaction between different flow features. Many enclosed environments are mechanically ventilated. However, the air can also be driven by buoyancy in the occupied zone, where the thermal plume can be as strong as the flow from the mechanical ventilation system. The two flows interact, and result in instability.

The complex geometry of the indoor environment can also lead to unstable flow. Furniture in an indoor environment can generate flow separation, which is usually unstable. As discussed above, flow transition and unstable behaviors cause complicated indoor airflow phenomena, making indoor airflow difficult to model.

To model indoor airflow, one approach is to apply computational fluid dynamics (CFD) to solve the flow equations. Airflow in a room is governed by the Navier-Stokes (N-S) equation. The most straightforward way to solve this equation is called direct numerical simulation (DNS), which directly solves the N-S equation together with initial and boundary conditions and produces a realization of the flow. The DNS does not need a model and can provide the most accurate and detailed flow motion. However, the DNS resolves a wide range of spatial and temporal scales, from the smallest Kolmogorov scale (dissipative eddies) to the integral scale (large flow motion), which requires a very fine meshes and time steps (Wilcox 2006). Even for steady-state flow, DNS needs to perform unsteady calculation, and calculate a long period of time to obtain the mean flow. As a result, the cost of DNS can be extremely high. For example, to simulate a small office under mechanical ventilation with Re = 5000, grid spacing should be on the order of 0.039 in. ([10.sup.-3] m) (Kolmogorov length scale). The number of cells for the room is on the order of [10.sup.10]. The size of the time step should be on the order of [10.sup.-3] s (Kolmogorov time scale). To obtain a sufficient statistical sample, flow should be calculated for several minutes; therefore, the number of time steps should be in the order of [10.sup.6]. However, it is not feasible to perform this simulation on a personal computer or a moderate computer cluster in the near future.

A more realistic and widely used approach for indoor airflow simulations is by Reynolds-Averaged Navier-Stokes (RANS) equation modeling. The RANS simulation solves the time-averaged N-S equation and models the additional Reynolds stresses. This modeling approach can significantly reduce the grid resolution requirement, and can be performed as steady-state. Many studies have used different RANS models for indoor airflow simulation (Yuan et al. 1999; Holmes et al. 2000; Gadgil et al. 2003; Hsieh and Lien 2004; Zhang et al. 2007) and found mixed results of the model performance. Zhang et al. (2007) tested eight popular turbulence models for different indoor airflows and concluded that no model was superior to the others. Wang and Chen (2009) further tested these eight models using a set of experimental cases with gradually added flow features and found that the RANS models failed to predict flow separation. This conclusion was supported by Costa et al. (1999) to some extent, who found that the low-Reynolds-number model suffered from a problem, leading to the prediction of an unrealistic local minimum of the wall heat transfer near points of flow reattachment. The failure of the RANS models to predict a separated flow could have resulted from the time-averaging approach. In the separation region, flow is very unstable, and the velocity magnitude and direction change rapidly. Therefore, it may not be meaningful to have a time-averaged solution for this case since the information contained by the mean value of flow variables is too limited to describe the rapidly varying unstable flow. Thus, the RANS models may not be capable of correctly predicting unstable airflow features in enclosed spaces, such as the separation caused by furniture, impingement flow, and unsteady plumes.

With the advancement of computing power, large eddy simulation (LES) is becoming increasingly popular for engineering applications due to its ability to solve unstable separated flow. LES solves the filtered N-S equation for the energy-containing eddies (large eddies) and models the subgrid-scale flow motions (small eddies). The turbulence model for LES is not as important as that for a RANS model (Wang and Chen 2009). In other words, LES relies more on fundamental flow physics than on modeling assumptions. As a result, LES is more accurate and informative than RANS models for airflow modeling, especially for a separated and unstable flow. Since LES solves only the large-scale eddies, the computing cost is much lower than that of DNS. The cell size required by LES could be much larger than the Kolmogorov length scale. Therefore, LES uses significantly less computational time than DNS. For example, to calculate a flow over a backward-facing step by DNS (Le et al. 1997) and by LES (Akselvoll and Moin 1993) at Reynolds number of 5000, LES requires only 3% of the cells and 2% of the computing time needed by DNS, while having equally good results compared with the experimental data (Wilcox 2006). However, it is difficult to apply LES for the near-wall region of indoor airflow modeling, where the size of the energy-bearing eddies (large eddies) is comparable to that of the dissipative eddies. If LES is to resolve most energy-bearing eddies, the computing cost could be the same as that of the DNS. For airflow in enclosed spaces, the flow is bounded by walls, furniture, and occupants. All of these solid surfaces could significantly increase the computing cost by LES.

To overcome the disadvantages of LES and RANS, the concept of the RANS/LES hybrid model has emerged in the past decade. The hybrid model applies LES in the separated flow region to capture the three-dimensional, time-dependent flow, and applies a RANS model in the attached boundary layers to avoid the excessively fine meshes required by LES. This approach was first introduced by Spalart et al. (1997), adopting the one-equation Spalart model. Some studies used Smagorinsky (1963) zero-equation LES model with zero-equation RANS model such as Cebeci-Smith and Baldwin-Lomax models (Georgiadis et al. 2003; Kawai and Fujii 2005). Many other studies adopted multi-equation RANS models with LES, such as k-[omega] model (Strelets 2001; Davidson and Peng 2003), k-[epsilon] model (Hamba 2001, 2003), and k-l model (Tucker and Davidson 2004). After reviewing the RANS/LES hybrid models, our effort was to develop a new hybrid model, the semi-v2f/LES model, for modeling airflows in an enclosed environment. The new model used transport equations for k and [epsilon], and an algebraic equation for the normal stress near a wall to model turbulence viscosity in the RANS region and subgrid turbulence viscosity for the LES region. The new model was tested by applying it to a mixed convection flow in a model room, and to a strong buoyancy-driven flow with a high temperature gradient in a room.

MODEL DEVELOPMENT

Hybrid RANS/LES Simulation for Indoor Airflow

Indoor airflow is governed by the N-S equation together with the energy equation for the fluid phase (FLUENT 2003). Since it is not feasible to use DNS for solving such a flow, the N-S equation should be approximated in order to make it solvable with the present capacity of computers. By using Reynolds-averaged approach, flow variables can be written as

[empty set] = [PHI] + [empty set]' (1)

where [PHI] is the mean value of flow variables, and [empty set]' is the fluctuating part of flow variables. On the other hand, LES uses a filter to obtain large-scale flow variables:

[bar.[empty set]]([bar.x], t) = [integral][[integral] V][integral][empty set]([bar.x], t)G([bar.x][bar.x'], t)d[bar.x'] (2)

where the overbar denotes filtered variables, and G is a filter function.

The two methods can transform the N-S equation into a single form:

[[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.i]] = 0 (3)

[[partial derivative][[bar.u].sub.i]/[partial derivative]t] + [[bar.u].sub.j][[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.j]] = - [1/[rho]][[partial derivative][bar.P]/[partial derivative][x.sub.i]] + v[[[[partial derivative].sup.2][[bar.u].sub.i]]/[[partial derivative][x.sub.j][partial derivative][x.sub.j]]] + [[partial derivative][[tau].sub.ij]/[partial derivative][x.sub.j]] (4)

where the overbar denotes mean variables for RANS, and filtered variables for LES. The term [[tau].sub.ij] in the equation is the Reynolds stress for RANS:

[[tau].sub.ij] = -[bar.[u.sub.i]'[u.sub.j]'] (5)

and the subgrid-scale stress for LES:

[[tau].sub.ij] = [[bar.u].sub.i][[bar.u].sub.j] - [bar.[u.sub.i][u.sub.j]] (6)

Many turbulence models use a Boussinesq approximation to simplify the second-order symmetrical tensor by correlating it to a scalar [v.sub.t], and transform Equation 4 into

[[partial derivative][[bar.u].sub.i]/[partial derivative]t] + [[bar.u].sub.j][[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.j]] = - [1/[rho]][[partial derivative][bar.P]/[partial derivative][x.sub.i]] + [[partial derivative]/[partial derivative][x.sub.j]][(v + [v.sub.t])[[partial derivative][[bar.u].sub.i]/[partial derivative][x.sub.j]]] (7)

where [v.sub.t] is the turbulence viscosity for the RANS model and the subgrid-scale turbulence viscosity for LES.

Spalart et al. (1997) introduced the first hybrid RANS/LES method: the detached eddy simulation Spalart-Allmaras (DES-SA) model. This method used the Spalart-Allmaras one-equation model to calculate the turbulence viscosity for the RANS part and modified the length scale in the equation to calculate the subgrid-scale turbulence viscosity for the LES part. The transport equation for v is written as

[[partial derivative][~.v]/[partial derivative]t] + [[bar.u].sub.j][[partial derivative][~.v]/[partial derivative][x.sub.j]] = [c.sub.b1]S[~.v] - [c.sub.w1][[florin].sub.w][([[~.v]/[~.l]]).sup.2] + [1/[sigma]][[partial derivative]/][partial derivative][x.sub.k]]][(v + [~.v])[[partial derivative][~.v]/[[partial derivative][x.sub.k]]]] + [[c.sub.b2]/[sigma]][[partial derivative][~.v]/[[partial derivative][x.sub.k]]][[partial derivative][~.v]/[[partial derivative][x.sub.k]]] (8)

where [~.v] is the modified turbulence viscosity used as a working variable, [c.sub.b1], [c.sub.b2], [c.sub.w1], and [sigma] are constants, and

[~.l] = min([d.sub.w], [C.sub.DES][DELTA]) [DELTA] = max([DELTA]x, [DELTA]y, [DELTA]z) (9)

[~.l] is the DES-blending function. In Equation 9, [d.sub.w] is the distance to the nearest wall and also the RANS length scale used by the S-A model, [C.sub.DES] is a constant, and [DELTA] is the largest cell spacing. [C.sub.DES][DELTA] represents the grid length scale. [~.l] compares the RANS length scale with the grid length scale. If the cell is close to the wall (i.e., the wall distance is smaller than the grid size), the DES-blending function is equal to the wall distance, and the transport equation has the same form as the Spalart-Allmaras model. Otherwise, the equation becomes a Smagorinsky-like subgrid-scale turbulence viscosity based on the local grid size.

Many hybrid approaches with more advanced RANS models have been developed based on the idea of DES-SA (Hamba 2001; Strelet 2001; Tucker and Davidson 2004). One can explicitly replace the [d.sub.w] in the DES-SA model with the turbulence length scale, or implicitly involve other RANS models. Strelet (2001) developed a RANS/LES hybrid model based on the shear stress transport (SST) k-[omega] model from Menter et al. (2003). The major modification was on the dissipation term in the SST k-[omega] model:

[D.sub.RANS.sup.k] = [rho][k.sup.[3/2]]/[~.l] (10)

where

[~.l] = min([l.sub.[k-[omega]]], [C.sub.DES][DELTA]) (11)

Here, [C.sub.DES][DELTA] was the same as that in the DES-SA model. Note that the RANS scale used in this model is the turbulence length scale, which means that switching between RANS and LES is based on the local eddy size and local grid size.

Hybrid RANS/LES models are usually classified based on the RANS model they use. The performance of a hybrid RANS/LES model also depends on its RANS model. For indoor airflow simulation, a few preliminary studies in the literature show that the available RANS/LES hybrid models did not perform as well as LES or even as well as some RANS models. This is because the RANS models used were not good (Zhang et al. 2007; Wang and Chen 2009). Therefore, our effort was to identify an appropriate RANS model for indoor airflow simulations and to develop a new hybrid model with the RANS model.

Identification of a Suitable RANS Model for Hybrid Simulation

As discussed above, the accuracy of the RANS model is essential to the performance of a hybrid model. Some studies have evaluated different RANS turbulence models for indoor airflow, which could be helpful for choosing a suitable RANS model. Zhai et al. (2007) identified 17 models that may be appropriate for calculating indoor airflows. Wang and Chen (2009) tested six of them that are most promising for indoor airflows and concluded that the v2f, RNG k-[epsilon], and Reynolds stress models were the best. This conclusion was consistent with that from Zhang et al. (2007) for forced, natural, and mixed convection flows.

The RNG k-[epsilon] model uses a wall function near solid boundaries, which makes it unsuitable for a hybrid model. The Reynolds stress model can resolve the near-wall region, but it has six scalar transport equations that are too computationally expensive. The v2f model can be used for the hybrid model, since it can resolve the near-wall viscous region at a reasonable computational cost (three scalar transport equations) and accounts for anisotropic behavior near the wall. However, studies from the literature (Durbin 1991; Davidson et al. 2003) and our preliminary research show that the transport equation for the wall normal stress [v'.sup.2] and the elliptic equation for the relaxation function [florin] make the model numerically unstable. Therefore, it is essential to simplify the v2f model before implementing it in a hybrid simulation.

The Semi-v2f/LES Model

Our effort to develop the v2f model further was to introduce an algebraic equation for the normal stress [v'.sup.2]:

[[bar.v'].sup.2] = [florin](k, [epsilon], [upsilon], y ...) (12)

to replace the transport equation for [bar.[v'.sup.2]] and the elliptic equation for [florin], which have been problematic. The procedure was first to derive new equations for [v'.sup.2] and turbulence viscosity. The constant in the equations was then determined through curve fitting of the flow data obtained for indoor airflow. The detailed procedure was as follows.

* [bar.[v'.sup.2]] Equation

Starting from the Boussinesq approximation for incompressible flow:

-[[tau].sub.ij] = -2[v.sub.t][S.sub.ij] + [2/3] k * [[delta].sub.ij] (13)

considering the diagonal entries at wall normal direction (i = j = 2), and substituting the [v.sub.t] formulation of v2f model, it is easy to obtain a correlation:

[[[bar.v'].sup.2]/k] = [C.sub.1](1 - [C.sub.2] [[[bar.v'].sup.2]/[epsilon]] [[partial derivative]V/[partial derivative]y]) (14)

where [C.sub.1] and [C.sub.2] are constants. Through dimensional analysis of Equation (14), there are three nondimensional variables: [v'.sup.2]/k, y* = (y[C.sub.[mu].sup.[1/4]] [k.sup.[1/2]])/v, and l* = [k.sup.[3/2]]/([epsilon]y). Therefore, Equation 14 can be written as a dimensionless form:

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - g(y*, l*)] (15)

where g(y*, l*) is the function to be determined. If the y* value becomes large (far from the wall), the turbulence should be homogeneous:

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - g(y*, l*)] [right arrow] [C.sub.1] as y* [right arrow] [infinity] (16)

Near a wall, the normal kinetic energy goes to zero:

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - g(y*, l*)] [right arrow] 0 as y* [right arrow] 0 (17)

Assuming the wall damping effect is only related to the geometry, the influence of l* (the turbulence eddy scale) could be neglected. Thus, Equations 16 and 17 could be a good approximation of Equation 14. Namely,

[[[bar.v'].sup.2]/k] = [C.sub.1][1 - exp(-[y*/[y*.sub.0]])] (18)

where [C.sub.1] and [y*.sub.0] are constants.

* Turbulence Viscosity

In the v2f model, the turbulence viscosity is modeled as

[v.sub.t] = [C.sub.[mu]] [[bar.v'].sup.2][k/[epsilon]] (19)

where [C.sub.[mu]] = 0.22 is a constant. By substituting Equation 18 into Equation 19, the turbulence viscosity can be determined by

[v.sub.t] = A * [1 - exp(-[y*/[y*.sub.0]])] * [[k.sup.2]/[epsilon]] (20)

where A and [y*.sub.0] are constants to be determined.

* Determination of Constant

The model derived above was to mimic the behavior of the v2f model. Therefore, this study simulated several room airflows using the v2f model, extracted the data from different locations, and performed a curve fitting on Equation 18 to determine the constants. The curve-fitting results led to the new algebraic [v'.sub.2] equation:

[[bar.v'].sup.2]/k [approximately equal to] 0.308 [1 - exp(-[y*/50.836])] (21)

and the turbulence viscosity:

[v.sub.t] = 0.07 [1 / exp([ - y*/50.836])] * [[k.sup.2]/[epsilon]] (22)

Figure 1 compares the [bar.[v'.sub.2]]/k near the wall by the algebraic equation with the DNS data (Del Alamo et al. 2004). The agreement is reasonably good.

[FIGURE 1 OMITTED]

* New Semi-v2f/LES Model

The new semi-v2f/LES model used the same k and [epsilon] equations as those in the DES realizable k-[epsilon] model, but has a new algebraic equation for [v'.sup.2] and a new eddy-viscosity correlation. The new model does not modify the formulation of the turbulence heat flux, but uses the same energy equation as that used by the DES realizable k-[epsilon] model. The appendix provides a more detailed formulation.

PERFORMANCE OF THE SEMI-V2F/LES MODEL

This study evaluated the performance of the semi-v2f/LES model by applying it to two benchmark cases: a mixed convection flow in a model room with moderate buoyancy (Wang and Chen 2009) and a strong buoyancy-driven flow in a model fire room (Murakami et al. 1995). The first case had a simple geometry but most features of indoor airflow. The second case was selected to test the robustness of the new model in an extreme scenario.

Mixed Convection Flow

Figure 2a shows the case schematic. Air velocity distributions were measured streamwise and at the cross sections in the 8 x 8 x 8 ft (2.44 x 2.44 x 2.44 m) cubic room. Air was supplied from a slot diffuser located at the upper left corner of the room, and exhausted from a slot at the lower right corner. The Reynolds number, based on inlet height, was about 2500, which was in the transitional region. A 4 x 4 x 4 ft (1.22 x 1.22 x 1.22 m) heated box was placed in the center with 700 W of power, which produced a thermal plume and caused flow separation. This case represents a typical indoor airflow scenario with the most important flow features. For more information about this case, see Wang and Chen (2009).

[FIGURE 2 OMITTED]

The semi-v2f/LES model was implemented in FLUENT (2003) by user-defined functions. According to Wang and Chen (2009), two grid resolutions were used for the new model: one was a 44 x 44 x 44 grid (coarse); the other was a 110 x 77 x 101 grid (fine). The v2f model modified by Davidson (2003) (v2f-Dav), the LES dynamic Smagorinsky-Lilly (LES-DSL) model, and the DES realizable k-[epsilon] model were also included in this study to evaluate the performance of the new model. The LES-DSL and DES realizable k-[epsilon] models used a 110 x 77 x 101 grid, and the v2f-Dav model used a 44 x 44 x 44 grid. The comparison was based on the velocity, temperature, and turbulence kinetic energy at 10 positions, as shown in Figure 2b. The results shown in this paper were at the positions where turbulence models had the best, worst, and average performances. For definition of the three categories, please see Wang and Chen (2009).

Figure 3 depicts the four models' air velocity predictions. In general, all the models predicted very similar, acceptable results. The new model slightly underpredicted the velocity at position (6) with a fine grid. The realizable k-[epsilon] had a similar prediction at this position. However, the discrepancy was very small. At position (5), where the flow was separated, all the models underpredicted the velocity. However, the new model with the fine grids and the DES realizable k-[epsilon] model did better than the other models. The v2f-Dav model failed primarily because Reynolds averaging may not be appropriate for separated flow. The LES-DSL model failed to provide good results because it requires a finer grids than DES. This also reflects the advantage of the DES models. At position (3), all the models gave similar results, though with small discrepancies.

[FIGURE 3 OMITTED]

Figure 4 shows the temperature prediction by the models. At position (8), the new model with coarse grids surprisingly predicted better results compared with the other models. The new model with fine grids also predicted slightly better results than the realizable k-[epsilon] and v2f-Dav models. The results of the LES-DSL model were comparable with those of the new model with fine grids. At position (4), the results of the new model with the two grid distributions were better than those from the other models. At position (5), the new model with the fine grids predicted one of the best profiles. The new model with coarse grids predicted an incorrect peak at Y/L = 0.6. The grid distribution of 44 x 44 x 44 was not fine enough for the new model to capture the separated flow.

[FIGURE 4 OMITTED]

Figure 5 shows the turbulence kinetic energy profiles predicted by the four models. The new model with fine grids predicted the best result at position (1). The DES realizable k-[epsilon] model performed similarly. The LES-DSL model significantly overpredicted the turbulence level, while the v2f-Dav model underpredicted it. At position (5), none of the four models predicted the turbulence kinetic energy well; it is indeed difficult to capture fluctuating, highly separated flow. At position (10), where most models showed average performance, the LES-DSL model was slightly better.

[FIGURE 5 OMITTED]

Strong Buoyancy-Driven Flow

The second case tested was a strong buoyancy-driven flow in a fire room. This case, chosen to test the new model's robustness, was designed by Murakami et al. (1995), who measured detailed fluctuating velocity using two-component laser doppler velocimetry (LDV), and the temperature using thermocouples. The test room was 5.9 ft (1.8 m) long, 3.9 ft (1.2 m) wide, and 3.9 ft (1.2 m) high, as shown in Figure 6. A 8.625 Btu/s (9.1 kW) heat source was placed at the corner of the room. The surface temperature at the heat source reached more than 932[degrees]F (500[degrees]C). An opening of size 1.3 x 3.0 ft (0.4 x 0.9 m) connected the air between the test room and the outside chamber. The fluctuating velocity and temperature were measured at 12 lines at two sections.

[FIGURE 6 OMITTED]

The numerical simulations calculated the air inside the test room as well as the outer enclosure (a very large laboratory). This helped to avoid instability in using a pressure boundary condition for the opening. The measured surface temperature was used as boundary conditions for the simulations. Due to the high temperature difference between the heat source surface and ambient air, the Boussinesq approximation is no longer valid. Instead, the ideal gas law was used for determining the air density:

[rho] = [[P.sub.op]/RT] (23)

where [P.sub.op] is the pressure at the opening, R is the gas constant, and T is the air temperature. Using variable density would change the form of equations presented in section 2. Since typical indoor airflow has a small temperature difference, it is much better to use the equations given in the Model Development section. This case is exceptional, so this article omits the differences on the governing equations.

Although this case was primarily used to test the robustness of the new model, the results from the DES Spalart-Allmaras (DES-SA) and LES-DSL models are included for comparison. The test was based on the comparison of the air velocity, air temperature, and root-mean-square (RMS) velocity predicted. The RMS reflects the turbulence predicted by the models. The results at all 12 measurement lines were compared, but only those at lines 2, 4, and 6 are shown here due to limited space.

Figure 7 depicts the calculated and measured velocity profiles in the x direction. All the models were stable in this extreme case. The semi-v2f/LES model predicted comparable results as the other models.

[FIGURE 7 OMITTED]

Figure 8 compares the predicted and measured air temperature profiles. Because of their correct velocity predictions, it is not surprising that all the models predicted reasonable results. However, all the models underpredicted the temperature near the ceiling. The semi-v2f/LES model performed relatively better than the other models due to its better turbulence viscosity formulation in the near wall region.

[FIGURE 8 OMITTED]

Figure 9 shows the RMS velocity profiles in the x direction. This flow quantity was obtained statistically using the solution at each time step. Although it is more difficult to get good agreement for the second-order momentum than the first-order one, the three models used were able to predict the correct trend of the profile. The DES-SA model predicted poorer results than the other two models because the model solved only one turbulence transport quantity (the modified turbulence viscosity) and could not calculate the turbulence length scale related to the local shear layer thickness. Therefore, the model may not be accurate for a complicated shear flow. The new semi-v2f/LES and LES-DSL models had comparable performances.

[FIGURE 9 OMITTED]

The computational cost of the new model was very similar to that of the DES realizable model, since they both solved two additional equations for turbulence. The DES-SA model took slightly less computational time, since it only solved one additional equation for turbulence. The LES-DSL model had a simpler formulation than the new model, but required much a finer grid near the wall for full LES simulation. Thus, it could need much more computational time.

CONCLUSION

This investigation developed a new RANS/LES hybrid model, the semi-v2f/LES model, for indoor airflow simulations. The development simplified the v2f model by replacing the partial differential equation for [v'.sup.2] and [florin] with an algebraic equation for [v'.sup.2].

The study tested the performance of the semi-v2f/LES model by applying it to predict mean and turbulence quantities in two indoor airflows. The first case was a mixed convection flow in a model room with typical indoor airflow features. The new model predicted the best turbulence kinetic energy results and comparable velocity and temperature results among the several models tested. The second case was a strong buoyancy-driven flow in a room. The new model performed well in this case. It gave the best temperature prediction and was one of the best models for turbulence predictions.

NOMENCLATURE

Acronyms

ach = air changes per hour

CFD = computational fluid dynamics

DES = detached eddy simulation

DES-SA = detached eddy simulation with Spalart-Allmaras model

DNS = direct numerical simulation

LDV = laser doppler velocimetry

LES = large eddy simulation

LES-DSL = large eddy simulation with dynamic Smagorinsky-Lilly model

N-S = Navier-Stokes

RANS = Reynolds-averaged Navier-Stokes equations

Re = Reynolds number

RMS = root mean square

S-A = Spalart-Allmaras

SST = shear stress transport

v2f = v2f model

v2f-Dav = v2f model modified by Davidson Top

Symbols

A = model constant

C = model constant

[D.sub.RANS.sup.k] = dissipation term in the SST k-[omega] model

[d.sub.w] = distance to nearest wall

G = filter function for LES

k = turbulence kinetic energy

L = size of test room

l = turbulence length scale

[~.l] = DES blending function

l* = distance to wall, nondimensional

R = gas constant

P = pressure

[S.sub.ij] = strain rate tensor

T = temperature

u = velocity

[bar.[v'.sub.2]] = wall normal stress

x, y, z = Cartesian coordinates

y* = y[C.sub.[mu].sup.[1/4]] [k.sup.[1/2]]/v

Greek Symbols

[DELTA] = grids spacing

[[delta].sub.ij] = Kronecker delta

[epsilon] = turbulence dissipation rate

v = kinematic viscosity

[~.v] = modified turbulence viscosity

[rho] = density

[sigma] = model constant

[[tau].sub.ij] = for RANS: -[bar.[u.sub.i][u.sub.j]]; for LES: [[bar.u].sub.i][[bar.u].sub.j] - [bar.[u.sub.i][u.sub.j]]

[PHI] = mean flow variables

[empty set] = flow variables

[omega] = specific turbulence dissipation rate

Subscripts

DES = detached eddy simulation

i = i th direction

j = j th direction

LES = large eddy simulation

max = maximum value

min = minimum value

op = at opening

RANS = RANS model

RKE = RANS k-[epsilon] model

t = turbulence quantities

Overbars

Overbar = mean and filtered flow variables for RANS and LES, respectively

Tilde = variables for DES-SA model

REFERENCES

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APPENDIX

This appendix provides all the equations for the semi-v2f/LES model. The transport equation for k is

[[partial derivative]/[partial derivative]t]([rho]k) + [[partial derivative]/[partial derivative][x.sub.j]]([rho][ku.sub.j]) = [[partial derivative]/[partial derivative][x.sub.j]][([mu] + [[[mu].sub.t]/[[sigma].sub.k]])[[partial derivative]k/[partial derivative][x.sub.j]]] + [G.sub.b] - [rho][epsilon] - [Y.sub.M] + [S.sub.k]

The transport equation for [epsilon] is

[[partial derivative]/[partial derivative]t]([rho][epsilon]) + [[partial derivative]/[partial derivative][x.sub.j]]([rho][epsilon][u.sub.j]) = [[partial derivative]/[partial derivative][x.sub.j]][([mu] + [[[mu].sub.t]/[[sigma].sub.[epsilon]]])[[partial derivative][epsilon]/[partial derivative][x.sub.j]]] + [rho][C.sub.1][S.sub.[epsilon]] - [rho][C.sub.2][[[epsilon].sup.2]/[k + [square root of v[epsilon]]]] + [C.sub.1[epsilon]][[epsilon]/k][C.sub.3[epsilon]][G.sub.b] + [S.sub.[epsilon]]

The algebraic equation for [[bar.v].sup.'2] is

[[bar.v'].sup.2] = 0.308[1 - exp(-[y*/50.836])]k

Turbulence viscosity is determined by

[v.sub.t] = min{0.07 [[k.sup.2]/[epsilon]], 0.22[[bar.v'].sup.2T]} = 0.07 min{[[k.sup.2]/[epsilon]][1 - exp(-[y*/50.836])]} * T

where

T = min {2.0 [L/[square root of k]], max[[k/[epsilon]], 6[([[[bar.v'].sup.2]/[epsilon]]).sup.[1/2]]]}

and

L = [C.sub.L] min{[[k.sup.[3/2]]/[epsilon]], [V.sub.CELL.sup.[1/3]]}

The terms in k and [epsilon] equations are

[G.sub.k] = [[mu].sub.t][S.sup.2]

where

S = [square root of 2[S.sub.ij][S.sub.ij]]

[G.sub.b] = -[g.sub.i][[[mu].sub.t]/[rho][Pr.sub.t]][[partial derivative][rho]/[partial derivative][x.sub.i]]

[Y.sub.k] = [[rho][k.sup.[3/2]]/[l.sub.DES]]

where

[l.sub.DES] = min([l.sub.rke], [l.sub.LES])

[l.sub.rke] = [[k.sup.[3/2]]/[epsilon]]

[l.sub.LES] = [C.sub.DES][DELTA]

[C.sub.1] = max[0.43, [[eta]/[eta] + 5]], [eta] = S[k/[epsilon]]

The model constant is

[C.sub.1[epsilon]] = 1.44, [C.sub.2] = 1.9, [[sigma].sub.k] = 1.0, [[sigma].sub.[epsilon]] = 1.2, [C.sub.DES] = 0.61

Received January 5, 2010; accepted March 11, 2010

This paper is based on findings resulting from ASHRAE Research Project RP-1271.

Miao Wang is a doctoral candidate and Qingyan (Yan) Chen is a professor at the Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN.

Miao Wang

Student Member ASHRAE

Qingyan (Yan) Chen, PhD

Fellow ASHRAE

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Author: | Wang, Miao; Chen, Qingyan (Yan) |
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Publication: | HVAC & R Research |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Nov 1, 2010 |

Words: | 6190 |

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