4.11.3 Subgrid-Scale Models

The subgrid-scale stresses resulting from the filtering operation are unknown, and require modeling. The subgrid-scale turbulence models in ANSYS FLUENT employ the Boussinesq hypothesis [ 130] as in the RANS models, computing subgrid-scale turbulent stresses from

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

For compressible flows, it is convenient to introduce the density-weighted (or Favre) filtering operator:

The Favre Filtered Navier-Stokes equation takes the same form as Equation  4.11-5. The compressible form of the subgrid stress tensor is defined as:

This term is split into its isotropic and deviatoric parts

The deviatoric part of the subgrid-scale stress tensor is modeled using the compressible form of the Smagorinsky model:

As for incompressible flows, the term involving $\tau_{kk}$ can be added to the filtered pressure or simply neglected [ 89]. Indeed, this term can be re-written as $\tau_{kk} = \gamma {M^2}_{sgs} \overline{p}$ 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.

ANSYS 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.

The subgrid-scale turbulent flux of a scalar, $\phi$, is modeled using s subgrid-scale turbulent Prandtl number by

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 [ 106] to the subgrid-scale flux.

Smagorinsky-Lilly Model

This simple model was first proposed by Smagorinsky [ 320]. In the Smagorinsky-Lilly model, the eddy-viscosity is modeled by

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 ANSYS FLUENT, $L_s$ is computed using

where $\kappa$ is the von Kármán constant, $d$ is the distance to the closest wall, $C_s$ is the Smagorinsky constant, and $\Delta$ is the local grid scale. In ANSYS FLUENT, $\Delta$ is computed according to the volume of the computational cell using

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 a universal constant, which is the most serious shortcoming of this simple model. Nonetheless, a $C_s$ 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 ANSYS FLUENT.

Dynamic Smagorinsky-Lilly Model

Germano et al. [ 106] and subsequently Lilly [ 197] 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 concept of the dynamic procedure is to apply a second filter (called the test filter) to the equations of motion. The new filter width $\hat {\Delta}$ is equal to twice the grid filter width $\Delta$. Both filters produce a resolved flow field. The difference between the two resolved fields is the contribution of the small scales whose size is in between the grid filter and the test filter. The information related to these scales is used to compute the model constant. In ANSYS FLUENT, the variable density formulation of the model is considered as explained below.

At the test filtered field level, the SGS stress tensor can be expressed as:

Both $T_{ij}$ and $\tau_{ij}$ are modeled in the same way with the Smagorinsky-Lilly model, assuming scale similarity:

In Equation  4.11-19 and Equation  4.11-20, the coefficient C is asumed to be the same and independent of the filtering process (note that per Equation  4.11-25, $C=C_s^2$). The grid filtered SGS and the test-filtered SGS are related by the Germano identity [ 106] such that:

Where $L_{ij}$ is computable from the resolved large eddy field. Substituting the grid-filter Smagorinsky-Lilly model and Equation  4.11-20 into Equation  4.11-21, the following expressions can be derived to solve for C with the contraction obtained from the least square analysis of Lilly (1992).

With

More details of the model implementation in ANSYS FLUENT and its validation can be found in [ 165].

The $C_s=\sqrt{C}$ obtained using the dynamic Smagorinsky-Lilly model varies in time and space over a fairly wide range. To avoid numerical instability, both the numerator and the denominator in Equation  4.11-22 are locally averaged (or filtered) using the test-filter. In ANSYS FLUENT, $C_s$ is also clipped at zero and 0.23 by default.

Wall-Adapting Local Eddy-Viscosity (WALE) Model

In the WALE model [ 248], the eddy viscosity is modeled by

where $L_s$ and $S_{ij}^{d}$ in the WALE model are defined, respectively, as

In ANSYS 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 ANSYS FLUENT replicates the model proposed by Kim and Menon [ 168].

The subgrid-scale kinetic energy is defined as

which is obtained by contracting the subgrid-scale stress in Equation  4.11-7.

The subgrid-scale eddy viscosity, $\mu_t$, is computed using $k_{\rm sgs}$ as

where $\Delta_f$ is the filter-size computed from $\Delta_f \equiv V^{1/3}$.

The subgrid-scale stress can then be written as

$k_{\rm sgs}$ is obtained by solving its transport equation

In the above equations, the model constants, $C_k$ and $C_{\varepsilon}$, are determined dynamically [ 168]. $\sigma_k$ is hardwired to 1.0. The details of the implementation of this model in ANSYS FLUENT and its validation is given by Kim [ 165].