Anisotropic Linear Forcing (ALF)

The Anisotropic Linear Forcing (ALF) is as an efficient way of generating turbulence fluctuations in a volume for scale-resolving simulations.

The ALF method proposed by De Laage de Meux et al. [369] is a good alternative to other synthetic turbulence generation methods, such as digital-filtering-based methods [372] or synthetic-eddy methods [370], [371]. The principal of the ALF is to force the mean (statistical) properties of a scale-resolving simulation towards target values for mean velocity and mean Reynolds-stresses. The target values can be obtained from RANS simulations of the same flow field.

In the following, the superscript ~ denotes a filtered quantity and the brackets <> denote averaging.

The volumetric forcing source term $f_{b, ALF}$ is added to the LES filtered momentum equation given by Eqn. (1384) as:

Figure 1. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ \tilde{v}) + \nabla\cdot (ρ \tilde{v} \otimes \tilde{v}) = - \nabla\cdot \tilde{p} I + \nabla\cdot (T + T_{SGS}) + ρ f_{b, ALF}

(1470)

De Laage de Meux et al. showed that all volumetric forcing formulations can be written using a general tensorial linear function of the following form:

Figure 2. EQUATION_DISPLAY

f_{b, ALF} = A \tilde{v} + b

(1471)

where $A$ and $b$ depend on the volumetric forcing approach.

In order to derive adequate formulation for $A$ and $b$ , De Laage de Meux et al. use the following properties:

The contribution of the forcing term to the averaged momentum equation is equal to:
Figure 3. EQUATION_DISPLAY

$〈 f_{b, ALF} 〉 = A 〈 \tilde{v} 〉 + b$
(1472)
The contribution of the forcing to the budget of the resolved stresses is equal to:
Figure 4. EQUATION_DISPLAY

$P^{f} = A T_{res} + T_{res} A$
(1473)

where $T_{res} = 〈 \tilde{v}' \otimes \tilde{v}' 〉$ is the resolved mean Reynolds-stress tensor of the LES solution.

The mean velocity and Reynolds-stresses of the scale resolving simulation can thus be driven towards target values using the following constraints:

The mean restoring force to drive the mean velocity towards the target velocity $v_{target}$ can be imposed as:
Figure 5. EQUATION_DISPLAY

$〈 f_{b, ALF} 〉 = \frac{1}{τ_{v}} (v_{target} - 〈 \tilde{v} 〉)$
(1474)

where $τ_{v}$ is a relaxation time scale.
A similar constraint can be imposed for the Reynolds-stresses as:
Figure 6. EQUATION_DISPLAY

$P^{f} = \frac{1}{τ_{r}} (T_{target} - T_{res})$
(1475)

where:
- $τ_{r}$ is a second relaxation time scale.
- $T_{target} = {〈 v' \otimes v' 〉}_{target}$ is the target Reynolds stress tensor.

De Laage de Meux et al. showed that both properties and both constraints above lead to the following definition for $A$ and $b$ :

Figure 7. EQUATION_DISPLAY

A = \frac{1}{2 {III}_{r} (I_{r} {II}_{r} - {III}_{r})} [I_{r} T_{res}^{2} H T_{res}^{2} - I_{r}^{2} (T_{res}^{2} H T_{res} - T_{res} H T_{res}^{2}) + (I_{r} {II}_{r} - {III}_{r}) (T_{res}^{2} H + H T_{res}^{2}) + (I_{r}^{3} + {III}_{r}) T_{res} H T_{res} - I_{r}^{2} {II}_{r} (T_{res} H + H T_{res}) + (I_{r}^{2} {III}_{r} + {II}_{r} (I_{r} {II}_{r} - {III}_{r})) H]

(1476)

Figure 8. EQUATION_DISPLAY

b = \frac{1}{τ_{v}} (v_{target} - 〈 \tilde{v} 〉) - A 〈 \tilde{v} 〉

(1477)

where:

$I_{r}$ , $I I_{r}$ , and ${III}_{r}$ are the first, second, and third invariant of $T_{res}$ .
$H = P^{f}$ , where $P^{f}$ is given by Eqn. (1475)

The two time-scales ( $τ_{v}$ related to the velocity forcing term and $τ_{r}$ related to the Reynolds-stress term ) are defined as:

Figure 9. EQUATION_DISPLAY

τ_{v} = \frac{C_{v}}{S_{target}}

(1478)

Figure 10. EQUATION_DISPLAY

τ_{r} = \max [2 Δt, \max (\frac{C_{r}}{0.09} \frac{k_{target}}{ε_{target}}, \sqrt{\frac{ν}{ε_{target}}})]

(1479)

where:

$C_{v}$ and $C_{r}$ are Model Coefficients.
$S_{target}$ is the modulus of the strain rate tensor given by Eqn. (1129) and computed from the target velocity field $v_{target}$ .
$k_{target}$ and $ε_{target}$ are the target turbulent kinetic energy and dissipation rate, respectively.
$ν$ is the kinematic viscosity.

The mean velocity $〈 \tilde{v} 〉$ and the mean Reynolds-stress components $T_{res}$ are computed on the fly using an exponentially-weighted averaging (EWA) method over a specified number of time-steps $n_{EWA}$ .

Simcenter STAR-CCM+ applies a limiter to the resulting forcing term $f_{b, ALF}$ , which ensures that the Courant number associated with the forced momentum equation stays below a specified maximum value ${CFL}_{\max}$ in each cell.

$n_{EWA}$ and ${CFL}_{\max}$ are available as Model Coefficients.

Model Coefficients


$C_{r}$	$C_{v}$	${CFL}_{\max}$	$n_{EWA}$
1	0.1	0.1	750