Constitutive Relations

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

By default, the Boussinesq approximation implies a linear constitutive relation. Non-linear constitutive relations account for anisotropy of turbulence by adding non-linear functions of the strain and vorticity tensors. For the SST (Menter) K-Omega model in Simcenter STAR-CCM+, quadratic and cubic constitutive relations are available. The cubic constitutive relation is derived from a Reynolds stress model and thus represents an Explicit Algebraic Reynolds Stress Model (EARSM):


Constitutive Relation	Formulation
Quadratic [325]	Figure 1. EQUATION_DISPLAY $T_{RANS, N L} = - 2 μ_{t} 0.04645 (O ∙ S - S ∙ O)$ (1241)
Cubic [322], [327]	Figure 2. EQUATION_DISPLAY $T_{RANS, N L} = - ρ k [β_{3} (W^{} ∙ W^{} - \frac{1}{3} {I I}_{Ω} I) + β_{4} (S^{} ∙ W^{} - W^{} ∙ S^{}) + β_{6} (S^{} ∙ W^{} ∙ W^{} + W^{} ∙ W^{} ∙ S^{} - {I I}_{Ω} S^{*} - \frac{2}{3} I V I)]$ (1242)

where:

$ρ$ is the density.
$I$ is the identity tensor.
$β_{3}$ , $β_{4}$ , and $β_{6}$ are Model Coefficients.

S

is the strain rate tensor given by Eqn. (1130) and

S^{*}

is given by:

Figure 3. EQUATION_DISPLAY

S^{*} = \frac{1}{β^{*} ω} S

(1243)

where $β^{*}$ is given in K-Omega Model—Model Coefficients.

W

is the vorticity tensor given by Eqn. (1132),

W^{*}

is given by:

Figure 4. EQUATION_DISPLAY

W^{*} = \frac{1}{β^{*} ω} W

(1244)

and:

Figure 5. EQUATION_DISPLAY

O = \frac{W}{\sqrt{(S - W) (S - W)}}

(1245)

Figure 6. EQUATION_DISPLAY

{I I}_{Ω} = W^{*} {: W^{*}}^{T}

(1246)

Figure 7. EQUATION_DISPLAY

I V = S^{*} ∙ W^{*} ∙ W^{*}

(1247)

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

Figure 8. EQUATION_DISPLAY

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

(1248)

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

Relation for Turbulent Viscosity

When applying the linear or the quadratic constitutive relation, the turbulent viscosity for the SST (Menter) K-Omega model is given by Eqn. (1207). For the cubic constitutive relation, $μ_{t}$ is computed as:

Figure 9. EQUATION_DISPLAY

μ_{t} = \frac{C_{μ}}{β^{*}} ρ k T

(1249)

The coefficient $C_{μ}$ is obtained from:

Figure 10. EQUATION_DISPLAY

C_{μ} = - \frac{1}{2} (β_{1} + {I I}_{Ω} β_{6})

(1250)

where $β_{1}$ and $β_{6}$ are Model Coefficients.

Model Coefficients

The coefficients $β_{i}$ are given by:

Figure 11. EQUATION_DISPLAY

β_{1} = - \frac{N (2 N^{2} - 7 {I I}_{Ω})}{Q}

(1251)

Figure 12. EQUATION_DISPLAY

β_{3} = - \frac{12 I V}{N Q}

(1252)

Figure 13. EQUATION_DISPLAY

β_{4} = - \frac{2 (N^{2} - 2 {I I}_{Ω})}{Q}

(1253)

Figure 14. EQUATION_DISPLAY

β_{6} = - \frac{6 N}{Q}

(1254)

Figure 15. EQUATION_DISPLAY

Q = \frac{5}{6} (N^{2} - 2 {I I}_{Ω}) (2 N^{2} - {I I}_{Ω})

(1255)

where $N$ is given by:

Figure 16. EQUATION_DISPLAY

N = {\begin{matrix} \frac{A_{3}^{'}}{3} + \sqrt[3]{P_{1} + \sqrt{P_{2}}} + s i g n (P_{1} - \sqrt{P_{2}}) \sqrt[3]{| P_{1} - \sqrt{P_{2}} |} & f o r P_{2} \geq 0 \\ \frac{A_{3}^{'}}{3} + 2 \sqrt[6]{P_{1}^{2} - P_{2}} \cos [\frac{1}{3} \cos^{-1} (\frac{P_{1}}{\sqrt{P_{1}^{2} - P_{2}}})] & f o r P_{2} < 0 \end{matrix}

(1256)

where:

Figure 17. EQUATION_DISPLAY

P_{1} = A_{3}^{'} (\frac{{A_{3}^{'}}^{2}}{27} + \frac{9}{20} {I I}_{S} - \frac{2}{3} {I I}_{Ω})

(1257)

Figure 18. EQUATION_DISPLAY

P_{2} = P_{1}^{2} - {(\frac{{A_{3}^{'}}^{2}}{9} + \frac{9}{10} {I I}_{S} - \frac{2}{3} {I I}_{Ω})}^{3}

(1258)

Figure 19. EQUATION_DISPLAY

A_{3}^{'} = \frac{9}{5} + \frac{9}{4} 2.2 \max (1 + β_{1}^{e q} {I I}_{S}, 0)

(1259)

Figure 20. EQUATION_DISPLAY

β_{1}^{e q} = - \frac{6}{5} (\frac{\frac{81}{20}}{{(\frac{81}{20})}^{2} - 2 {I I}_{Ω}})

(1260)

Figure 21. EQUATION_DISPLAY

{I I}_{S} = S^{*} {: S^{*}}^{T}

(1261)