Curvature Correction

Rotation and by association streamline curvature have a strong influence on the evolution of turbulent quantities. Such sensitivity is absent in one- and two-equation eddy viscosity models, for pure derivation equations. Different methods have been proposed to account for this deficiency. Three different curvature correction (CC) formulations are used in Simcenter STAR-CCM+, depending on the turbulence model being considered.

CC for Spalart-Allmaras models

Shur and others [300] proposed a correction term for system-rotation and streamline curvature effects. The model acts directly on the eddy viscosity by including a rotation function $f_{r 1}$ in the production term $P_{\tilde{ν}}$ :

Figure 1. EQUATION_DISPLAY

f_{r 1} = (1 + c_{r 1}) \frac{2 r *}{1 + r *} [1 - c_{r 3} {tan}^{- 1} (c_{r 2} \tilde{r})] - c_{r 1}

(1286)

where:

$c_{r 1}$ , $c_{r 2}$ , and $c_{r 3}$ are Model Coefficients.
$r * = | S | / | \tilde{W} |$
$r = 2 (\tilde{W} S) : (\frac{D S}{D t}) \frac{1}{D^{4}}$
$D^{2} = 0.5 ({| S |}^{2} + {| \tilde{W} |}^{2})$

$S$ and $\tilde{W}$ are given by Eqn. (1130) and Eqn. (1133), respectively.

Model Coefficients


$c_{r 1}$	$c_{r 2}$	$c_{r 3}$
$1$	$12$	$1$

CC for K-Epsilon and K-Omega models

The transport equation for the turbulent kinetic energy is insensitive, by construction, to stabilizing and destabilizing effects usually associated with strong (streamline) curvature and frame-rotation. These effects can be incorporated by using a curvature correction factor, which alters the turbulent kinetic energy production term according to the local rotation and vorticity rates.

The curvature correction factor $f_{c}$ [320] is calculated as:

Figure 2. EQUATION_DISPLAY

f_{c} = min (C_{max}, \frac{1}{C_{r_{1}} (| η | - η) + \sqrt{1 - min (C_{r_{2}}, 0.99)}})

(1287)

where:

$C_{max}$ , $C_{r_{1}}$ , and $C_{r_{2}}$ are Model Coefficients.
$η = T^{2} (S : S - W : W)$

The time-scale $T$ is limited in order to have the correct near-wall asymptotic behavior:

T = \begin{matrix} max (T_{1}, T_{3}) \end{matrix}

where:


Model	$T_{1}$	$T_{2}$	$T_{3}$
K-Epsilon models	$\frac{ε}{k}$	$6 \sqrt{\frac{ν}{ε}}$	$\begin{matrix} {({T_{1}}^{1.625} T_{2})}^{\frac{1}{1.625 + 1}} \end{matrix}$
K-Omega models	$\begin{matrix} \frac{1}{β^{*} ω} \end{matrix}$	$6 \sqrt{\frac{ν}{β^{*} k ω}}$

The strain-rate tensor $S$ is as defined in Eqn. (1130), and the absolute rotation-rate tensor $W$ is calculated as:

Figure 3. EQUATION_DISPLAY

W = W^{l} + W^{f} + (C_{ct} - 1) W^{S}

(1288)

where $C_{c t}$ is a Model Coefficient and:

Figure 4. EQUATION_DISPLAY

W^{l} = \begin{matrix} \frac{1}{2} (\nabla \bar{v} - {\nabla \bar{v}}^{T}) \end{matrix}

(1289)

Figure 5. EQUATION_DISPLAY

W^{f} = E \cdot ω^{f}

(1290)

Figure 6. EQUATION_DISPLAY

\begin{matrix} W^{S} = W^{f} - E \cdot (A^{- 1} \cdot ω) \end{matrix}

(1291)

$W^{l}$ , $W^{f}$ , and $W^{S}$ are the contributions due to the vorticity computed in the local frame of reference, the rotating frame of reference (defined by the frame rotation vector $ω^{f}$ ), and the streamline curvature, respectively.

$E$ is the Levi-Civita tensor (also know as the three-dimensional permutation tensor).

For the last contribution, two additional terms are needed:

Figure 7. EQUATION_DISPLAY

A = I - \frac{3 S^{2}}{2S : S} and ω = E \cdot \frac{S \cdot (D_{t} S) - (D_{t} S) \cdot S}{2S : S}

(1292)

where $D_{t} S$ is the strain-rate total derivative tensor. For more details on the derivation of this method, see [320].

If curvature correction is not activated, $f_{c}$ is set to unity.

Model Coefficients


$c_{r 1}$	$c_{r 2}$	$C_{c t}$	$C_{\max}$ (Upper Limit)
0.04645	0.25	2	1.25