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Low-Re k-epsilon models

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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

Contents

Introduction

Good reviews of classical Low-Re k-epsilon models can be found in [Patel (1995)] and [Rodi (1993)]. There are hundreds of different low-Re k-epsilon models in the literature. This article tries to summarize and describe the most common and classical models. Feel free to add more models here, but please only add models that have gained a widespread use in the CFD community.

Overview of models

The models presented presented here are:

Model Reference Description
Chien Model [Chien (1982}] A very common model in turbomachinery applications. Has nice numerical properties.
Launder-Sharma Model [Launder (1974)] An old classical model which has attracted some attention for its ability to in model cases predict by-pass transition.
Nagano-Tagawa Model [Nagano (1990)] A model originally developed for heat-transfer applications.

Governing equations

These models can be written in a general form like:

\frac{\partial}{\partial t} \left( \rho k \right) + \frac{\partial}{\partial x_j} \left[ \rho k u_j - \left( \mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j} \right] = P - \rho \epsilon - \rho D
\frac{\partial}{\partial t} \left( \rho \epsilon \right) + \frac{\partial}{\partial x_j} \left[ \rho \epsilon u_j - \left( \mu + \frac{\mu_t}{\sigma_\epsilon} \right) \frac{\partial \epsilon}{\partial x_j} \right] = \left( C_{\epsilon_1} f_1 P - C_{\epsilon_2} f_2 \rho \epsilon \right) \frac{\epsilon}{k} + \rho E
\mu_t = C_\mu f_\mu \rho \frac{k^2}{\epsilon}
P = \tau_{ij}^{turb} \frac{\partial u_i}{\partial x_j}

Where C_{\epsilon_1}, C_{\epsilon_2}, C_\mu, \sigma_k and \sigma_\epsilon are model constants. The damping functions f_\mu, f_1 and f_2 and the extra source terms D and E are only active close to solid walls and makes it possible to solve k and \epsilon down to the viscous sublayer. Table below summarizes the constants, damping functions and boundary conditions for all k-epsilon models presented here.

Model constants, daming functions and boundary conditions

Chien Launder-Sharma Nagano-Tagawa
c_\mu 0.09 0.09 0.09
\sigma_k 1 1 1.4
\sigma_\epsilon 1.3 1.3 1.3
D 2\, \nu \, \frac{k}{y^2} 2 \, \nu \, \left( \frac{\partial \sqrt{k}}{\partial y} \right)^2 0
E -\frac{2 \nu \epsilon}{y^2} \exp{-0.5y^+} 2 \, \nu \, \nu_t \, \left(\frac{\partial^2 u}{\partial y^2}\right)^2 0
\epsilon_{wall} 0 0 \nu \, \left( \frac{\partial \sqrt{k}}{\partial y} \right)^2
C_{\epsilon_1} 1.35 1.44 1.45
C_{\epsilon_2} 1.8 1.92 1.9
f_\mu 1-\exp{-0.0115 y^+} \exp{\frac{-3.4}{\left( 1 + R_t/50 \right)^2}} A \left(1-\exp{\frac{-y^+}{26}}\right)^2 \left(1+\frac{4.1}{Re_t^{3/4}}\right)
f_1 1 1 1
f_2 1-0.22 \exp{-\left(\frac{Re_t}{6}\right)^2} 1-0.3 \exp{-Re_t^2} \left(1\!-\!0.3\exp{-\left(\frac{Re_t}{6.5}\right)^2}\right)\!\! \left(1\!-\!\exp{\frac{-y^+}{6}}\right)^2

Where Re_t \equiv \frac{k^2}{\nu \epsilon}, y^+ \equiv \frac{u^*y}{\nu} and k_{wall} = 0.

A According to P. Gardin, M.Brunet, J.F.Domgin and K. Pericleous, "An experimental and numerical CFD study of turbulence in a tundish container", 2nd International Conference on CFD, CSIRO, 1999. There are three options for f_\mu


D1

\min \left( 1.0 , \frac{R_t^{1/3}}{45} \right)


D2

1 - \exp{\frac{-R_t}{ 30.1862 }}


D3

\exp{\frac{-3.4}{\left( 1 + R_t/50 \right)^2}}

Performance, applicability and limitations

Not written yet

Implementation issues

Not written yet

References

Chien, K.-Y. (1982), "Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds Number Turbulence Model", AIAA Journal, Vol. 20, No. 1, pp. 33-38.

Nagano, Y. and Tagawa, M. (1990), "An Improved k-epsilon Model for Boundary Layer Flows", Journal of Fluids Engineering, Vol. 112, pp. 33-39.

Patel, V. C. and Rodi, W. and Scheuerer, G. (1985), "Turbulence Models for Near-Wall and Low Reynolds Number Flows: A Review", AIAA Journal, Vol. 23, No. 9, pp. 1308-1319.

Rodi, W. and Mansour, N. N. (1993), "Low Reynolds Number k-epsilon Modeling with the Aid of Direct Simulation Data", Journal of Fluid Mechanics, Vol. 250, pp. 509-529.

Launder, B. E. and Sharma, B. I. (1974), "Application of the Energy-Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc", Letters in Heat and Mass Transfer, Vol. 1, No. 2, pp. 131-138.

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