The subgrid-scale stresses resulting from the filtering operation are unknown,
and require modeling. The subgrid-scale turbulence models in
FLUENT employ the Boussinesq
142] as in the RANS models, computing subgrid-scale turbulent stresses from
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
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
12.9-5. 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
can be added to the filtered pressure or simply neglected [
99]. Indeed, this term can be re-written as
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
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 [
to the subgrid-scale flux.
This simple model was first proposed by Smagorinsky [
In the Smagorinsky-Lilly model, the eddy-viscosity is modeled by
is the mixing length for subgrid scales and
is computed using
is the von Kármán constant,
is the distance to the
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.
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
Dynamic Smagorinsky-Lilly Model
Germano et al. [
116] and subsequently Lilly [
conceived a procedure in which the Smagorinsky model constant,
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
FLUENT and its validation can be found in [
obtained using the dynamic Smagorinsky-Lilly model varies in time and space over a fairly wide range.
To avoid numerical instability, in
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
in the WALE model are defined, respectively, as
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
) 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 [
The subgrid-scale kinetic energy is defined as
which is obtained by contracting the subgrid-scale stress in Equation
The subgrid-scale eddy viscosity,
, is computed using
is the filter-size computed from
The subgrid-scale stress can then be written as
is obtained by solving its transport equation
In the above equations, the model constants,
determined dynamically [
is hardwired to 1.0. The details
of the implementation of this model in
FLUENT and its validation is given by Kim [