### 2.5.1. Subgrid-Scale Models

#### 2.5.1.1. Smagorinsky Model

The Smagorinsky model [34] is an algebraic model for the SGS viscosity . Based on dimensional analysis, the SGS viscosity can be expressed as:

| (2–165)

where is the length scale of the unresolved motion (usually the mesh size ) and is the velocity of the unresolved motion.

Based on an analogy to the Prandtl mixing length model, the velocity scale is related to the gradients of the filtered velocity:

| (2–166)

This yields the Smagorinsky model [34] for the subgrid-scale (SGS) viscosity:

| (2–167)

with the Smagorinsky constant. The value of the Smagorinsky constant for isotropic turbulence with inertial range spectrum:

| (2–168)

is:

| (2–169)

For practical calculations, the value of is changed depending on the type of flow and mesh resolution. Its value is found to vary between a value of 0.065 (channel flows) and 0.25. Often a value of 0.1 is used and has been found to yield the best results for a wide range of flows. This is also the default value of .

The coefficient is, therefore, not a universal constant and this is the most serious shortcoming of this model. Furthermore, damping functions are needed close to walls.

##### 2.5.1.1.1. Wall Damping

Close to walls, the turbulent viscosity can be damped using a combination of a mixing length minimum function, and a viscosity damping function :

| (2–170)

with:

| (2–171)

C S and are constants which you can set; their default values are 0.1 and 0.4 respectively.

By default, the damping function is 1.0. A Van Driest and a Piomelli like damping can be specified by the user. For the Van Driest case, the damping function is:

| (2–172)

with A = 25. For the Piomelli case it is:

| (2–173)

with A = 25. The normalized wall distance:

| (2–174)

is defined as a function of the calculated wall distance , kinematic viscosity , and local velocity scale .

The Van Driest or Piomelli wall damping can be switched on when the LES turbulence model is selected. The damping factor A is defaulted to 25.0.

#### 2.5.1.2. WALE Model

The wall-adapted local eddy-viscosity model (Nicoud and Ducros [200]) is formulated locally and uses the following equation to compute the eddy-viscosity:

| (2–175)

where denotes the traceless symmetric part of the square of the velocity gradient tensor:

| (2–176)

and where , and is the Kronecker symbol. The tensor can be rewritten in terms of the strain-rate and vorticity tensors in the following way:

| (2–177)

where the vorticity tensor is given by:

| (2–178)

The main advantages of the WALE model are the capability to reproduce the laminar to turbulent transition and the design of the model to return the correct wall-asymptotic -variation of the SGS viscosity. It offers therefore the same advantages as the Dynamic Smagorinsky-Lilly model, and at the same time does not require an explicit (secondary) filtering. The constant has been calibrated using freely decaying isotropic homogeneous turbulence: the default value is 0.5 and is available in CFX-Pre.

#### 2.5.1.3. Dynamic Smagorinsky-Lilly Model

Germano et al. [198] and subsequently Lilly [199] introduced a method for evaluating subgrid scale model coefficients using information contained in the resolved turbulent velocity field, in order to overcome the deficiencies of the Smagorinsky model. The model coefficient is no longer a constant value and adjusts automatically to the flow type. The basic idea behind the model is an algebraic identity, which relates subgrid scale stresses at two different filter widths. The filtering on the smaller filter width is given implicitly by the mesh size, whereas the filtering on the larger filter width requires the use of an explicit filtering procedure. Because it is required that the filter works on unstructured meshes, a volume-weighted averaging of the variables from neighboring element centers to the corresponding vertex is used.

The so-called Germano identity reads:

| (2–179)

where represents the subgrid-scale (SGS) stress at scale and the SGS stress at scale {}:

| (2–180)

| (2–181)

and {…} denotes secondary filtering of a quantity with . The stress formulated by turbulent motions of scale intermediate between and {} reads:

| (2–182)

Although the Germano procedure can be applied with any SGS model, the Smagorinsky model has been used to compute the SGS stresses at the different filtering levels:

| (2–183)

| (2–184)

Using the Germano identity one obtains:

| (2–185)

The resulting system of Equation 2–185 is over-determined and the coefficient appears inside the secondary filter operation. In order to solve this system of equations, the error associated with the Smagorinsky model is defined in the following way:

| (2–186)

Lilly [199] applied a least square approach to minimize the error. The coefficient is taken out of the filtering procedure. This leads to

| (2–187)

| (2–188)

Using the coefficient , the eddy viscosity is obtained by:

| (2–189)

The model coefficient obtained using the dynamic Smagorinsky-Lilly model varies in time and space over a fairly wide range. To avoid numerical instability, a relaxation of in time is applied and an upper and lower limit on the coefficient is imposed in the following way:

1. Lower bound in order to avoid negative viscosity:

   | (2–190)

2. Upper bound:

   | (2–191)

3. Relaxation in time:

   | (2–192)

Freely decaying isotropic homogeneous turbulence has again been used to calibrate . The value of should be in the range of 0.04 to 0.09. The default value is 0.04 and can be specified in CFX-Pre.

Note: Density fluctuations are not accounted for in the current subgrid turbulence formulation as they scale with the subgrid Mach number, which is small. For flows with strong density variations, buoyancy effects are captured by the resolved large scales and not modeled for the subgrid scales.

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