Explicit nonlinear constitutive relation
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(New page: {{Template: Turbulence modeling}} <math> \begin{align} - \frac{\mathbf{u u}}{k} + \frac{2}{3} \mathbf{I} & = \beta_1 \tilde{\mathbf{S}} \\ & + \beta_2 \left( \tilde{\mathbf{S}...) |
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{{Template: Turbulence modeling}} | {{Template: Turbulence modeling}} | ||
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+ | == General Concept == | ||
+ | |||
+ | An explicit nonlinear constitutive relation for the Reynolds stresses represents an explicitly-postulated expansion over the [[Linear eddy viscosity models|linear Boussinesq hypothesis]]. | ||
+ | |||
+ | One of such ''explicit and nonlinear'' expansion over the Boussinesq hypothesis, as proposed by [[#References|[Wallin & Johansson (2000)]]], is given by | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
- | - \frac{\mathbf{u u}}{k} + \frac{2}{3} \mathbf{I} | + | - \frac{\mathbf{u u}}{k} & + \frac{2}{3} \mathbf{I} = \beta_1 \tilde{\mathbf{S}} |
\\ | \\ | ||
& + \beta_2 \left( \tilde{\mathbf{S}}^2 - \frac{II_S}{3} \mathbf{I} \right) | & + \beta_2 \left( \tilde{\mathbf{S}}^2 - \frac{II_S}{3} \mathbf{I} \right) | ||
+ \beta_3 \left( \tilde{\mathbf{\Omega}}^2 - \frac{II_\Omega}{3} \mathbf{I} \right) | + \beta_3 \left( \tilde{\mathbf{\Omega}}^2 - \frac{II_\Omega}{3} \mathbf{I} \right) | ||
- | |||
\\ | \\ | ||
- | & + \beta_5 \left( \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 \right) | + | & + \beta_4 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \right) |
- | + | + \beta_5 \left( \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 \right) | |
- | + | \\ | |
+ | & + \beta_6 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}} - \frac{2}{3} IV \mathbf{I} \right) | ||
+ | \\ | ||
+ | & + \beta_7 \left( \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}}^2 - \frac{2}{3} V \mathbf{I} \right) | ||
\\ | \\ | ||
& + \beta_8 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 + \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \right) | & + \beta_8 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 + \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \right) | ||
+ \beta_9 \left( \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} \right) | + \beta_9 \left( \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} \right) | ||
- | + | \\ | |
+ | & + \beta_{10} \left( \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}}^2 + \tilde{\mathbf{\Omega}}^2 \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} \right) | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
+ | |||
+ | Note that the terms in the first line are exactly the linear relation as expressed by the Boussinesq hypothesis. | ||
+ | |||
+ | == Reference == | ||
+ | |||
+ | * {{reference-paper|author=Wallin, S., and Johansson, A. V.|year=2000|title=An Explicit Algebraic Reynolds Stress Model for Incompressible and Compressible Turbulent Flows|rest=Journal of Fluid Mechanics, Vol. 403, Jan. 2000, pp. 89–132}} | ||
{{stub}} | {{stub}} |
Latest revision as of 20:15, 4 November 2009
General Concept
An explicit nonlinear constitutive relation for the Reynolds stresses represents an explicitly-postulated expansion over the linear Boussinesq hypothesis.
One of such explicit and nonlinear expansion over the Boussinesq hypothesis, as proposed by [Wallin & Johansson (2000)], is given by
Note that the terms in the first line are exactly the linear relation as expressed by the Boussinesq hypothesis.
Reference
- Wallin, S., and Johansson, A. V. (2000), "An Explicit Algebraic Reynolds Stress Model for Incompressible and Compressible Turbulent Flows", Journal of Fluid Mechanics, Vol. 403, Jan. 2000, pp. 89–132.