Constitutive Relations

Constitutive relations describe the relation between the stress tensor and the mean strain rate used in the Boussinesq approximation.

By default, the Boussinesq approximation implies a linear constitutive relation. Non-linear constitutive relations [311] account for anisotropy of turbulence by adding non-linear functions of the strain and rotation tensors. Two non-linear constitutive relations are available for the Standard K-Epsilon model variants (SKE, SKE 2L, and SKE LRe) in Simcenter STAR-CCM+:


Constitutive Relation	Formulation
Quadratic	Figure 1. EQUATION_DISPLAY $\begin{matrix} T_{RANS, N L} = - 4 μ_{t} \frac{k}{ε} {C_{1} [S ∙ S - \frac{1}{3} I (S : S)] + C_{2} (W ∙ S + S ∙ W^{T}) \\ \begin{matrix} + C_{3} [W ∙ W^{T} - \frac{1}{3} I (W : W^{T})] \end{matrix}} \end{matrix}$ (1203) where: $C_{1} = \begin{matrix} \frac{C_{N L 1}}{(C_{N L 6} + C_{N L 7} {\bar{S}}^{3}) C_{μ}} \end{matrix}$ $C_{2} = \frac{C_{N L 2}}{(C_{N L 6} + C_{N L 7} {\bar{S}}^{3}) C_{μ}}$ $C_{3} = \frac{C_{N L 3}}{(C_{N L 6} + C_{N L 7} {\bar{S}}^{3}) C_{μ}}$
Cubic	Figure 2. EQUATION_DISPLAY $T_{RANS, N L} = T_{RANS, q u a d} - 8 μ_{t} \frac{k^{2}}{ε^{2}} {C_{4} [(S ∙ S) ∙ W + W^{T} ∙ (S ∙ S)] + C_{5} (S : S - W : W^{T}) [S - \frac{1}{3} t r (S) I]}$ (1204) where $T_{RANS, q u a d}$ is given by Eqn. (1203) and: ${C_{4} = C}_{N L 4} {C_{μ}}^{2}$ $C_{5} = C_{N L 5} {C_{μ}}^{2}$

where:

$S$ and $t r (S)$ are given by Eqn. (1130) and Eqn. (5215), respectively.
$I$ is the identity tensor.
$W$ is given by Eqn. (1132).
$C_{N L 1}$ , $C_{N L 2}$ , $C_{N L 3}$ , $C_{N L 4}$ , $C_{N L 5}$ , $C_{N L 6}$ , and $C_{N L 7}$ are Model Coefficients.

The coefficient $C_{μ}$ is given by:

Figure 3. EQUATION_DISPLAY

\begin{matrix} C_{μ} = \frac{C_{a 0}}{C_{a 1} + C_{a 2} \bar{S} + C_{a 3} \bar{W}} \end{matrix}

(1205)

where:

$C_{a 0}$ , $C_{a 1}$ , $C_{a 2}$ , and $C_{a 3}$ are Model Coefficients.
$\begin{matrix} \bar{S} = \frac{k}{ε} \sqrt{2 S : S} \end{matrix}$
$\begin{matrix} \bar{W} = \frac{k}{ε} \sqrt{2 W : W} \end{matrix}$

When non-linear constitutive relations are used, the variable coefficient $C_{μ}$ (Eqn. (1205)) replaces the constant value of $C_{μ}$ in the relation for the turbulent viscosity $μ_{t}$ , see Eqn. (1163).

For the momentum equation Eqn. (665), the stress tensor $T_{RANS}$ is defined as:

Figure 4. EQUATION_DISPLAY

T_{RANS} = T_{RANS, L} + T_{RANS, N L}

(1206)

where $T_{RANS, L}$ is given by Eqn. (1147).

Model Coefficients


$C_{N L 1}$	$C_{N L 2}$	$C_{N L 3}$	$C_{N L 4}$	$C_{N L 5}$	$C_{N L 6}$	$C_{N L 7}$
0.75	3.75	4.75	-10	-2	1000	1


$C_{a 0}$	$C_{a 1}$	$C_{a 2}$	$C_{a 3}$
0.667	1.25	1	0.9