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Explicit nonlinear constitutive relation

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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]].
An explicit nonlinear constitutive relation for the Reynolds stresses represents an explicitly-postulated expansion over the [[Linear eddy viscosity models|linear Boussinesq hypothesis]].
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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.
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=== Reference ===
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== 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}}
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Latest revision as of 20:15, 4 November 2009

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

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.


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