# Explicit nonlinear constitutive relation

(Difference between revisions)
 Revision as of 20:14, 4 November 2009 (view source)← Older edit Latest revision as of 20:15, 4 November 2009 (view source) 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]]. An explicit nonlinear constitutive relation for the Reynolds stresses represents an explicitly-postulated expansion over the [[Linear eddy viscosity models|linear Boussinesq hypothesis]]. Line 28: Line 30: Note that the terms in the first line are exactly the linear relation as expressed by the Boussinesq hypothesis. Note that the terms in the first line are exactly the linear relation as expressed by the Boussinesq hypothesis. - === Reference === + == 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}} * {{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.