Explicit nonlinear constitutive relation

(Difference between revisions)
 Revision as of 12:00, 4 November 2009 (view source) (New page: {{Template: Turbulence modeling}} \begin{align} - \frac{\mathbf{u u}}{k} + \frac{2}{3} \mathbf{I} & = \beta_1 \tilde{\mathbf{S}} \\ & + \beta_2 \left( \tilde{\mathbf{S}...)← Older edit Latest revision as of 20:15, 4 November 2009 (view source) (2 intermediate revisions not shown) Line 1: Line 1: {{Template: Turbulence modeling}} {{Template: Turbulence modeling}} + + == 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 [itex] [itex] \begin{align} \begin{align} - - \frac{\mathbf{u u}}{k} + \frac{2}{3} \mathbf{I} & = \beta_1 \tilde{\mathbf{S}} + - \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_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_4 \left( \tilde{\mathbf{S}} \tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}} \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_5 \left( \tilde{\mathbf{S}}^2 \tilde{\mathbf{\Omega}} - \tilde{\mathbf{\Omega}} \tilde{\mathbf{S}}^2 \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_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) + \\ + & + \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} [/itex] + + 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

 Turbulence RANS-based turbulence models Linear eddy viscosity models Nonlinear eddy viscosity models Explicit nonlinear constitutive relation v2-f models $\overline{\upsilon^2}-f$ model $\zeta-f$ model Reynolds stress model (RSM) Large eddy simulation (LES) Detached eddy simulation (DES) Direct numerical simulation (DNS) Turbulence near-wall modeling Turbulence free-stream boundary conditions

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

\begin{align} - \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_3 \left( \tilde{\mathbf{\Omega}}^2 - \frac{II_\Omega}{3} \mathbf{I} \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_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}

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.