ANSYS FLUENT 12.0 Theory Guide - 4.6.2 Transport Equations for the

4.6.2 Transport Equations for the - - $\omega$ Model

The - - $\omega$ model is considered to be a three-equation eddy-viscosity type, which includes transport equations for turbulent kinetic energy ( ), laminar kinetic energy ( ), and the inverse turbulent time scale ( $\omega$ )

$\frac{Dk_T}{Dt} = P_{K_T}+R+R_{NAT}-\omega k_T - D_T + \frac... ...a_T}{\alpha_k}\right) \frac{\partial k_T}{\partial x_j}\right]$

(4.6-1)

$\frac{Dk_L}{Dt} = P_{K_L}-R-R_{NAT}-D_L + \frac{\partial}{\partial x_j}\left[ \nu \frac{\partial k_L}{\partial x_j}\right]$

(4.6-2)

$\frac{D_{\omega}}{D_t} = C_{\omega 1} \frac{\omega}{k_T}P_{k... ..._{\omega}}\right) \frac{\partial \omega}{\partial x_j} \right]$

(4.6-3)

The inclusion of the turbulent and laminar fluctuations on the mean flow and energy equations via the eddy viscosity and total thermal diffusivity is as follows:

$\overline{-u_i u_j} = \nu_{TOT}\left(\frac{\partial U_i}{\p... ...artial U_j}{\partial x_i}\right)-\frac{2}{3}k_{TOT}\delta_{ij}$

(4.6-4)

$\overline{-u_i \theta}=\alpha_{\theta,TOT}\frac{\partial \theta}{\partial x_i}$

(4.6-5)

The effective length is defined as

$\lambda_{eff}=MIN(C_{\lambda} d,\lambda_T)$

(4.6-6)

where $\lambda_T$ is the turbulent length scale and is defined by

$\lambda_T = \frac{\sqrt{k}}{\omega}$

(4.6-7)

and the small scale energy is defined by

$k_{T,s}=f_{ss}f_W k_T$

(4.6-8)

$f_W = \frac{\lambda_{eff}}{\lambda_T}$

(4.6-9)

$f_{ss} = exp\left[ -\left(\frac{C_{ss}\nu \Omega}{k_T}\right) ^2\right]$

(4.6-10)

The large scale energy is given by

$k_{T,l}=k_T - k_{T,s}$

(4.6-11)

Note that the sum of Equations 4.6-8 and 4.6-11 yields the turbulent kinetic energy .

The turbulence production term generated by turbulent fluctuations is given by

$P_{k_T} = \nu_{T,s} S^2$

(4.6-12)

where the small-scale turbulent viscosity is $\nu_{T,s}$

$\nu_{T,s} = f_{\nu} f_{INT} C_{\mu}\sqrt{k_{T,s}}\lambda_{eff}$

(4.6-13)

and

$C_{\mu}=\frac{1}{A_0+A_s(S/\omega)}$

(4.6-14)

$f_{\nu}=1-exp\left(-\frac{\sqrt{Re_{T,s}}}{A_{\nu}}\right)$

(4.6-15)

A damping function defining the turbulent production due to intermittency is given by

$f_{INT} = MIN\left(\frac{k_L}{C_{INT}k_{TOT}},1\right)$

(4.6-16)

$Re_{T,s} = \frac{f^2_W k_T}{\nu \omega}$

(4.6-17)

In Equation 4.6-2, $P_{k_L}$ is the production of laminar kinetic energy by large scale turbulent fluctuations, such that

$P_{k_L} = \nu_{T,l}S^2$

(4.6-18)

The large-scale turbulent viscosity $\nu_{T,1}$ is modeled as

$\nu_{T,1} = MIN\left\lbrace \nu^*_{T,1},\frac{0.5(k_L+k_{T,1})}{S}\right\rbrace$

(4.6-19)

where

$\nu^*_{T,1}=f_{\tau,1}C_{11}\left(\frac{\Omega \lambda^2_{e... ...sqrt{k_{T,1}}\lambda_{eff}+\beta_{TS}C_{12}\phi_{NAT}d^2\Omega$

(4.6-20)

The limit in Equation 4.6-19 binds the realizability such that it is not violated in the two-dimensional developing boundary layer. The time-scale-based damping function $f_{\tau,1}$ is

$f_{\tau,1}=1-exp\left[ -C_{\tau,1}\frac{k_{T,1}}{\lambda^2_{eff}\Omega^2}\right]$

(4.6-21)

where $\beta_{TS}$ from Equation 4.6-20 is

$\beta_{TS}=1-exp\left(-\frac{MAX(\phi_{NAT} - C_{TS,crit},0)^2}{A_{TS}}\right)$

(4.6-22)

$\phi_{NAT}=\frac{d^2\Omega}{\nu}$

(4.6-23)

Near-wall dissipation is given by

$D_T=2\nu \frac{\partial \sqrt{k_T}}{\partial x_j}\frac{\partial \sqrt{k_T}}{\partial x_j}$

(4.6-24)

$D_L=2\nu \frac{\partial \sqrt{k_L}}{\partial x_j}\frac{\partial \sqrt{k_L}}{\partial x_j}$

(4.6-25)

In Equation 4.6-1 - 4.6-3, represents the averaged effect of the breakdown of streamwise fluctuations into turbulence during bypass transition:

$R=C_R\beta_{BP}k_L\omega/f_W$

(4.6-26)

$\beta_{BP}$ , which is the threshold function controls the bypass transition process:

$\beta_{BP}=1-exp\left(-\frac{\phi_{BP}}{A_{BP}}\right)$

(4.6-27)

$\phi_{BP}=MAX\left[ \left(\frac{k_T}{\nu \Omega}-C_{BP,crit}\right),0\right]$

(4.6-28)

The breakdown to turbulence due to instabilities is considered to be a natural transition production term, given by

$R_{NAT}=C_{R,NAT}\beta_{NAT}k_L\Omega$

(4.6-29)

$\beta_{NAT}=1-exp\left[ -\frac{MAX(\phi_{NAT}-C_{NAT,crit}/f_{NAT,crit},0)}{A_{NAT}}\right\rbrace$

(4.6-30)

$f_{NAT,crit}=1-exp\left(C_{NC} \frac{\sqrt{k_L}d}{\nu}\right)$

(4.6-31)

The use of $\omega$ as the scale-determining variable can lead to a reduced intermittency effect in the outer region of a turbulent boundary layer, and consequently an elimination of the wake region in the velocity profile. From Equation 4.6-3, the following damping is defined as

$f_{\omega} = 1-exp\left[ -0.41 \left(\frac{\lambda_{eff}}{\lambda_T}\right)^4\right]$

(4.6-32)

The total eddy viscosity and eddy diffusivity included in Equations 4.6-4 and 4.6-5 are given by

$\nu_{TOT} = \nu_{T,s} + \nu_{T,l}$

(4.6-33)

$\alpha_{\theta,TOT}=f_W\left(\frac{k_T}{k_{TOT}}\right)\fra... ...Pr_\theta}+(1-f_W)C_{\alpha,\theta} \sqrt{k_{T}}\lambda_{eff}$

(4.6-34)

The turbulent scalar diffusivity in Equations 4.6-1 - 4.6-3 is defined as

$\alpha_T=f_\nu C_{\mu,std} \sqrt{k_{T,s}}\lambda_{eff}$

(4.6-35)

$k_{TOT}=k_T + k_L$

(4.6-36)

Model Constants

The model constants for the - - $\omega$ transition model are listed below [ 364]

$A_0 = 4.04, \;\; A_s = 2.12, \;\; A_{\nu} = 6.75, \;\; A_{BP}= 0.6$

$A_{NAT} = 200, \;\; A_{TS} = 200, \;\; C_{BP,crit} = 1.2, \;\; C_{NC}= 0.1$

$C_{NAT,crit} = 1250, \;\; C_{INT} = 0.75, \;\; C_{TS,crit} = 1000, \;\; C_{R,NAT}= 0.02$

$C_{11} = 3.4 \times 10^{-6}, \;\; C_{12} = 1.0 \times 10^{-10}, \;\; C_R = 0.12, \;\; C_{\alpha,\theta}= 0.035$

$C_{SS} = 1.5, \;\; C_{\tau,1} = 4360, \;\; C_{\omega 1} = 0.44, \;\; C_{\omega 2} = 0.92$

$C_{\omega 3} = 0.3, \;\; C_{\omega R} = 1.5, \;\; C_{\lambda} = 2.495, \;\; C_{\mu,std} = 0.09$

$Pr_{\theta} = 0.85, \;\; \sigma_k = 1, \;\; \sigma_{\omega} = 1.17$

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4.6.2 Transport Equations for the - - Model

4.6.2 Transport Equations for the - - $\omega$ Model