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12.9.3 Subgrid-Scale Models

The subgrid-scale stresses resulting from the filtering operation are unknown, and require modeling. The subgrid-scale turbulence models in FLUENT employ the Boussinesq hypothesis [ 142] as in the RANS models, computing subgrid-scale turbulent stresses from

 \tau_{ij} - \frac{1}{3} \tau_{kk} \delta_{ij} =-2\mu_{t}\overline{S}_{ij} (12.9-8)

where $\mu_t$ is the subgrid-scale turbulent viscosity. The isotropic part of the subgrid-scale stresses $\tau_{kk}$ is not modeled, but added to the filtered static pressure term. $\overline{S}_{ij}$ is the rate-of-strain tensor for the resolved scale defined by
 \overline{S}_{ij} \equiv \frac{1}{2}\left(\frac{\partial \o... ...x_j} + \frac{\partial \overline{u}_j}{\partial x_i}\right) (12.9-9)

For compressible flows, it is convenient to introduce the density-weighted (or Favre) filtering operator:
 \phi = \frac{\overline{\rho \phi}}{\overline{\rho}} (12.9-10)

The Favre Filtered Navier-Stokes equation takes the same form as Equation  12.9-5. The compressible form of the subgrid stress tensor is defined as:
 T_{ij} = -\rho u_i u_j + \overline{\rho} u_i u_j (12.9-11)

This term is split into its isotropic and deviatoric parts
 T_{ij} = \underbrace{T_{ij} - \frac{1}{3} T_{ll} \delta_{ij... ...underbrace{\frac{1}{3} T_{ll} \delta_{ij}}_{\mbox{isotropic}} (12.9-12)

The deviatoric part of the subgrid-scale stress tensor is modeled using the compressible form of the Smagorinsky model:
 T_{ij} - \frac{1}{3} T_{ll} \delta_{ij} = 2 \mu_t (\delta_{ij} - \frac{1}{3} \delta_{ii} \delta_{ij}) (12.9-13)

As for incompressible flows, the term involving $T_{ll}$ can be added to the filtered pressure or simply neglected [ 99]. Indeed, this term can be re-written as $T_{ll} = \gamma {M^2}_{sgs} \overline{p}$ where $M_{sgs}$ is the subgrid Mach number. This subgrid Mach number can be expected to be small when the turbulent Mach number of the flow is small.

FLUENT offers four models for $\mu_t$: the Smagorinsky-Lilly model, the dynamic Smagorinsky-Lilly model, the WALE model, and the dynamic kinetic energy subgrid-scale model.

Subgrid-scale turbulent flux of a scalar, $\phi$, is modeled using s subgrid-scale turbulent Prandtl number by
 q_j = -\frac{\mu_t}{\sigma_t}\frac{\partial \phi}{\partial x_j} (12.9-14)

where $q_j$ is the subgrid-scale flux.

In the dynamic models, the subgrid-scale turbulent Prandtl number or Schmidt number is obtained by applying the dynamic procedure originally proposed by Germano [ 116] to the subgrid-scale flux.



Smagorinsky-Lilly Model


This simple model was first proposed by Smagorinsky [ 337]. In the Smagorinsky-Lilly model, the eddy-viscosity is modeled by

 \mu_{t} = \rho L_s^2 \left\vert \overline{S} \right\vert (12.9-15)

where $L_s$ is the mixing length for subgrid scales and $\left\vert \overline{S} \right\vert \equiv \sqrt{2 {\overline{S}_{ij}} {\overline{S}_{ij}}}$. In FLUENT, $L_s$ is computed using
 L_s = {\rm min} \left(\kappa d, C_s V^{1/3}\right) (12.9-16)

where $\kappa$ is the von Kármán constant, $d$ is the distance to the closest wall, $C_s$ is the Smagorinsky constant, and $V$ is the volume of the computational cell.

Lilly derived a value of 0.17 for $C_s$ for homogeneous isotropic turbulence in the inertial subrange. However, this value was found to cause excessive damping of large-scale fluctuations in the presence of mean shear and in transitional flows as near solid boundary, and has to be reduced in such regions. In short, $C_s$ is not an universal constant, which is the most serious shortcoming of this simple model. Nonetheless, $C_s$ value of around 0.1 has been found to yield the best results for a wide range of flows, and is the default value in FLUENT.



Dynamic Smagorinsky-Lilly Model


Germano et al. [ 116] and subsequently Lilly [ 211] conceived a procedure in which the Smagorinsky model constant, $C_{s}$, is dynamically computed based on the information provided by the resolved scales of motion. The dynamic procedure thus obviates the need for users to specify the model constant $C_s$ in advance. The details of the model implementation in FLUENT and its validation can be found in [ 181].

The $C_s$ obtained using the dynamic Smagorinsky-Lilly model varies in time and space over a fairly wide range. To avoid numerical instability, in FLUENT, $C_s$ is clipped at zero and 0.23 by default.



Wall-Adapting Local Eddy-Viscosity (WALE) Model


In the WALE model [ 262], the eddy viscosity is modeled by

 \mu_{t} = \rho L_s^2 \frac{(S_{ij}^{d} S_{ij}^{d})^{3/2}}{(... ... \overline{S}_{ij})^{5/2} + (S_{ij}^{d} S_{ij}^{d})^{5/4}} (12.9-17)

where $L_s$ and $S_{ij}^{d}$ in the WALE model are defined, respectively, as
 L_s = {\rm min} \left(\kappa d, C_w V^{1/3}\right) (12.9-18)


 S_{ij}^{d} = \frac{1}{2} \left( \overline{g}_{ij}^{2} + \ov... ...e{g}_{ij} = \frac{\partial \overline{u_i}}{\partial x_{j}} (12.9-19)

In FLUENT, the default value of the WALE constant, $C_w$, is 0.325 and has been found to yield satisfactory results for a wide range of flow. The rest of the notation is the same as for the Smagorinsky-Lilly model. With this spatial operator, the WALE model is designed to return the correct wall asymptotic ( $y^{3}$) behavior for wall bounded flows.



Dynamic Kinetic Energy Subgrid-Scale Model


The original and dynamic Smagorinsky-Lilly models, discussed previously, are essentially algebraic models in which subgrid-scale stresses are parameterized using the resolved velocity scales. The underlying assumption is the local equilibrium between the transferred energy through the grid-filter scale and the dissipation of kinetic energy at small subgrid scales. The subgrid-scale turbulence can be better modeled by accounting for the transport of the subgrid-scale turbulence kinetic energy.

The dynamic subgrid-scale kinetic energy model in FLUENT replicates the model proposed by Kim and Menon [ 184].

The subgrid-scale kinetic energy is defined as

 k_{\rm sgs} = \frac{1}{2}\left(\overline{u_k^2} - \overline{u}_k^2 \right) (12.9-20)

which is obtained by contracting the subgrid-scale stress in Equation  12.9-7.

The subgrid-scale eddy viscosity, $\mu_t$, is computed using $k_{\rm sgs}$ as
 \mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta_f (12.9-21)

where $\Delta_f$ is the filter-size computed from $\Delta_f \equiv V^{1/3}$. The subgrid-scale stress can then be written as
 \tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta_f \overline{S}_{ij} (12.9-22)

$k_{\rm sgs}$ is obtained by solving its transport equation
 \frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\... ...\sigma_k} \frac{\partial k_{\rm sgs}}{\partial x_{j}} \right) (12.9-23)

In the above equations, the model constants, $C_k$ and $C_{\varepsilon}$, are determined dynamically [ 184]. $\sigma_k$ is hardwired to 1.0. The details of the implementation of this model in FLUENT and its validation is given by Kim [ 181].


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