Spalart-Allmaras Model

This section describes the theory behind the Spalart-Allmaras model. Information is presented in the following sections:

4.2.1. Overview

4.2.2. Transport Equation for the Spalart-Allmaras Model 4.2.3. Modeling the Turbulent Viscosity

4.2.4. Modeling the Turbulent Production 4.2.5. Modeling the Turbulent Destruction 4.2.6. Model Constants

4.2.7. Wall Boundary Conditions

4.2.8. Convective Heat and Mass Transfer Modeling

For details about using the model in ANSYS Fluent, see Modeling Turbulence and Setting Up the Spalart-Allmaras Model in the User's Guide.

4.2.1. Overview

The Spalart-Allmaras model [445] (p. 783) is a one-equation model that solves a modeled transport

equation for the kinematic eddy (turbulent) viscosity. The Spalart-Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity in turbomachinery applications.

In its original form, the Spalart-Allmaras model is effectively a low-Reynolds number model, requiring the viscosity-affected region of the boundary layer to be properly resolved ( meshes). In ANSYS Fluent, the Spalart-Allmaras model has been extended with a -insensitive wall treatment, which allows the application of the model independent of the near-wall resolution. The formulation blends auto-matically from a viscous sublayer formulation to a logarithmic formulation based on . On intermediate grids, , the formulation maintains its integrity and provides consistent wall shear stress and Turbulence

heat transfer coefficients. While the sensitivity is removed, it still should be ensured that the boundary layer is resolved with a minimum resolution of 10-15 cells.

The Spalart-Allmaras model was developed for aerodynamic flows. It is not calibrated for general indus-trial flows, and does produce relatively larger errors for some free shear flows, especially plane and round jet flows. In addition, it cannot be relied on to predict the decay of homogeneous, isotropic tur-bulence.

4.2.2. Transport Equation for the Spalart-Allmaras Model

The transported variable in the Spalart-Allmaras model, , is identical to the turbulent kinematic viscosity except in the near-wall (viscosity-affected) region. The transport equation for the modified turbulent viscosity is

(4.15)

where is the production of turbulent viscosity, and is the destruction of turbulent viscosity that occurs in the near-wall region due to wall blocking and viscous damping. and are the constants and is the molecular kinematic viscosity. is a user-defined source term. Note that since the turbulence kinetic energy, , is not calculated in the Spalart-Allmaras model, the last term in Equation 4.14 (p. 41) is ignored when estimating the Reynolds stresses.

4.2.3. Modeling the Turbulent Viscosity

The turbulent viscosity, , is computed from

(4.16) where the viscous damping function, , is given by

(4.17)

and

(4.18)

4.2.4. Modeling the Turbulent Production

The production term, , is modeled as

(4.19) where

(4.20) and

(4.21) Spalart-Allmaras Model

and are constants, is the distance from the wall, and is a scalar measure of the deformation tensor. By default in ANSYS Fluent, as in the original model proposed by Spalart and Allmaras, is based on the magnitude of the vorticity:

(4.22) where is the mean rate-of-rotation tensor and is defined by

(4.23)

The justification for the default expression for is that, for shear flows, vorticity and strain rate are identical. Vorticity has the advantage of being zero in inviscid flow regions like stagnation lines, where turbulence production due to strain rate can be unphysical. However, an alternative formulation has been proposed [89] (p. 764) and incorporated into ANSYS Fluent.

This modification combines the measures of both vorticity and the strain tensors in the definition of : (4.24) where

with the mean strain rate, , defined as

(4.25)

Including both the rotation and strain tensors reduces the production of eddy viscosity and consequently reduces the eddy viscosity itself in regions where the measure of vorticity exceeds that of strain rate.

One such example can be found in vortical flows, that is, flow near the core of a vortex subjected to a pure rotation where turbulence is known to be suppressed. Including both the rotation and strain tensors more accurately accounts for the effects of rotation on turbulence. The default option (including the rotation tensor only) tends to overpredict the production of eddy viscosity and hence over-predicts the eddy viscosity itself inside vortices.

You can select the modified form for calculating production in the Viscous Model Dialog Box.

4.2.5. Modeling the Turbulent Destruction

The destruction term is modeled as

(4.26)

where

(4.27) (4.28) (4.29) Turbulence

, , and are constants, and is given by Equation 4.20 (p. 43). Note that the modification

described above to include the effects of mean strain on will also affect the value of used to compute .

4.2.6. Model Constants

The model constants , and have the following default values [445] (p. 783):

4.2.7. Wall Boundary Conditions

The Spalart-Allmaras model has been extended within ANSYS Fluent with a -insensitive wall treatment, which automatically blends all solution variables from their viscous sublayer formulation

(4.30) to the corresponding logarithmic layer values depending on .

(4.31) where is the velocity parallel to the wall, is the friction velocity, is the distance from the wall,

is the von Kármán constant (0.4187), and .

The blending is calibrated to also cover intermediate values in the buffer layer .

4.2.8. Convective Heat and Mass Transfer Modeling

In ANSYS Fluent, turbulent heat transport is modeled using the concept of the Reynolds analogy to turbulent momentum transfer. The “modeled” energy equation is as follows:

(4.32)

where , in this case, is the thermal conductivity, is the total energy, and is the deviatoric stress tensor, defined as