Turbulent Time, Length, and Velocity Scales

Turbulent time, length, and velocity scales are approximate scales that are used to define model parameters. The Kolmogorov and Taylor scales provide information about the local limits for LES/DES meshes.

Turbulence models, for example, use certain definitions of turbulent time and length scales to define boundary conditions (see RANS Boundary, Region, and Initial Conditions ).

Aeroacoustics, combustion, Eulerian, and Lagrangian models require different scale definitions that are suitable for certain parameters of these models. The following table lists these scale definitions depending on the respective turbulence model:


Turbulence Model	Turbulent Time Scale $τ$	Turbulent Length Scale $l$	Kolmogorov Length Scale $η$	Kolmogorov Time Scale $τ_{η}$	Taylor Micro Scale $λ$
Spalart-Allmaras	Figure 1. EQUATION_DISPLAY $(1 - f_{v 2}) \min (\frac{2}{\tilde{S}}, \frac{ρ κ d^{2}}{μ_{t}})$ (1480)	Figure 2. EQUATION_DISPLAY $\frac{1}{2} (1 - f_{v 2}) \sqrt{τ \frac{μ_{t}}{{ρ C}_{μ}}}$ (1481)	-	-	-
K-Epsilon	Figure 3. EQUATION_DISPLAY $\frac{k}{ε}$ (1482)	Figure 4. EQUATION_DISPLAY $τ k^{1 / 2}$ (1483)	Figure 5. EQUATION_DISPLAY ${(\frac{ν^{3}}{ε})}^{1 / 4}$ (1484)	Figure 6. EQUATION_DISPLAY ${(\frac{ν}{ε})}^{1 / 2}$ (1485)	Figure 7. EQUATION_DISPLAY $\sqrt{10 ν \frac{k}{ε}}$ (1486)
K-Omega	Figure 8. EQUATION_DISPLAY ${(β^{*} ω)}^{- 1}$ (1487)	Figure 9. EQUATION_DISPLAY $τ k^{1 / 2}$ (1488)	Figure 10. EQUATION_DISPLAY ${(\frac{ν^{3}}{β^{*} ω k})}^{1 / 4}$ (1489)	Figure 11. EQUATION_DISPLAY ${(\frac{ν}{β^{*} ω k})}^{1 / 2}$ (1490)	Figure 12. EQUATION_DISPLAY $\sqrt{10 ν \frac{1}{β^{*} ω}}$ (1491)
RST	-	-	Figure 13. EQUATION_DISPLAY ${(\frac{ν^{3}}{ε})}^{1 / 4}$ (1492)	Figure 14. EQUATION_DISPLAY ${(\frac{ν}{ε})}^{1 / 2}$ (1493)	Figure 15. EQUATION_DISPLAY $\sqrt{10 ν \frac{k}{ε}}$ (1494)
LES	Figure 16. EQUATION_DISPLAY $\frac{C_{t}}{S}$ (1495)	Figure 17. EQUATION_DISPLAY $τ \sqrt{C_{t} \frac{μ_{t}}{ρ} S}$ (1496)	-	-	-
DES	As for the respective RANS model.		-	-	-

where:

$f_{v 2}$ is a Spalart-Allmaras Damping Function.
$\tilde{S}$ is given by Eqn. (1155).
$ρ$ is the density.
$κ$ is the von Karman constant. See Spalart-Allmaras Model—Model Coefficients.
$d$ is the distance to the wall.
$μ_{t}$ is the turbulent eddy viscosity.
$C_{μ}$ is a Model Coefficient.
$k$ is the turbulent kinetic energy.
$ε$ is the turbulent dissipation rate.
$ν$ is the kinematic viscosity.
$β^{*}$ is given in K-Omega Model—Model Coefficients.
$ω$ is the specific dissipation rate.
$C_{t}$ is a Model Coefficient and must be greater than $(\sqrt{3}) / 2$ to satisfy realizability constraints [358].
$S$ is given by Eqn. (1129) and computed from the resolved velocity field $\tilde{v}$ .

For LES, the relation for $τ$ is obtained from the ratio of the subgrid scale turbulent kinetic energy $k_{S G S}$ and its dissipation rate $ε_{S G S}$ . When computing this ratio, the subgrid scale turbulent kinetic energy is modeled as:

Figure 18. EQUATION_DISPLAY

k_{S G S} = C_{t} \frac{μ_{t}}{ρ} S

(1497)

The dissipation rate $ε_{S G S}$ is modeled from equilibrium considerations as:

Figure 19. EQUATION_DISPLAY

ε_{S G S} = \frac{μ_{t}}{ρ} S^{2}

(1498)

The turbulent length scale satisfies $l = \sqrt{k_{S G S}} τ$ .

For all turbulence models, the turbulent velocity scale is calculated from the respective turbulent length scale and turbulent time scale as:

Figure 20. EQUATION_DISPLAY

v = \frac{l}{τ}

(1499)

Model Coefficients


$C_{t}$	$C_{μ}$
3.5	0.09