## 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

 (12.9-8)

where is the subgrid-scale turbulent viscosity. The isotropic part of the subgrid-scale stresses is not modeled, but added to the filtered static pressure term. is the rate-of-strain tensor for the resolved scale defined by
 (12.9-9)

For compressible flows, it is convenient to introduce the density-weighted (or Favre) filtering operator:
 (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:
 (12.9-11)

This term is split into its isotropic and deviatoric parts
 (12.9-12)

The deviatoric part of the subgrid-scale stress tensor is modeled using the compressible form of the Smagorinsky model:
 (12.9-13)

As for incompressible flows, the term involving can be added to the filtered pressure or simply neglected [ 99]. Indeed, this term can be re-written as where 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 : 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, , is modeled using s subgrid-scale turbulent Prandtl number by
 (12.9-14)

where 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

 (12.9-15)

where is the mixing length for subgrid scales and . In FLUENT, is computed using
 (12.9-16)

where is the von Kármán constant, is the distance to the closest wall, is the Smagorinsky constant, and is the volume of the computational cell.

Lilly derived a value of 0.17 for 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, is not an universal constant, which is the most serious shortcoming of this simple model. Nonetheless, 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, , 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 in advance. The details of the model implementation in FLUENT and its validation can be found in [ 181].

The obtained using the dynamic Smagorinsky-Lilly model varies in time and space over a fairly wide range. To avoid numerical instability, in FLUENT, 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

 (12.9-17)

where and in the WALE model are defined, respectively, as
 (12.9-18)

 (12.9-19)

In FLUENT, the default value of the WALE constant, , 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 ( ) 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

 (12.9-20)

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

The subgrid-scale eddy viscosity, , is computed using as
 (12.9-21)

where is the filter-size computed from . The subgrid-scale stress can then be written as
 (12.9-22)

is obtained by solving its transport equation
 (12.9-23)

In the above equations, the model constants, and , are determined dynamically [ 184]. 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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