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Standard k-epsilon model

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Turbulence modeling
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      1. Prandtl's one-equation model
      2. Baldwin-Barth model
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    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
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        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
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      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
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        4. Realisability and Schwarz' inequality
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    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
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Turbulence near-wall modeling
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  1. Turbulence intensity
  2. Turbulence length scale

Contents

Transport equations for standard k-epsilon model

For turbulent kinetic energy k

\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k


For dissipation \epsilon

\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon}

Modeling turbulent viscosity

Turbulent viscosity is modelled as:

\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}

Production of k

P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i}


P_k = \mu_t S^2

Where S is the modulus of the mean rate-of-strain tensor, defined as :

S \equiv \sqrt{2S_{ij} S_{ij}}

Effect of buoyancy

P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i}



where Prt is the turbulent Prandtl number for energy and gi is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Prt is 0.85.

The coefficient of thermal expansion, \beta , is defined as

\beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p

Model constants

C_{1 \epsilon} = 1.44, \;\;\; C_{2 \epsilon} = 1.92,\;\;\; C_{3 \epsilon} = -0.33, \;\; \; C_{\mu} = 0.09, \;\;\; \sigma_k = 1.0, \;\;\; \sigma_{\epsilon} = 1.3
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