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Spalart - Allmaras model

Spalart-Allmaras model (SA) presented by Spalart and Allmaras [11] is an one-equational model written in terms of modified eddy viscosity. The model uses empiricism and arguments of dimensional analysis, it is independent on $ y^+$, but requires the distance to the nearest wall $ d_w$. The turbulent eddy viscosity is developed with the help of model’s transport equation:
\begin{multline} \frac{\partial \tilde{\nu}}{\partial t} + u_j \frac{\partial \tilde{\nu}}{\partial x_j} = \underbrace{ C_{b1} [1 - f_{t2}] \tilde{S} \tilde{\nu} }_{\text{production term}} + \underbrace{ \frac{1}{\sigma} \{ \frac{\partial }{\partial x_j} [(\nu + \tilde{\nu}) \frac{\partial \tilde{\nu}}{\partial x_j}] + C_{b2} \frac{\partial \tilde{\nu}}{\partial x_j} \frac{\partial \tilde{\nu}}{\partial x_j} \} }_{\text{diffusion term}} - \\ - \underbrace{ \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right] \left( \frac{\tilde{\nu}}{d} \right)^2 }_{\text{destruction term}} + f_{t1} \Delta U^2 \end{multline}
where the production term is developed with the help of norm of vorticity $ \vert\Omega\vert$. The diffusion terms are naturally connected with spatial derivatives of $ \tilde{\nu}$. The destruction term arose from dimensional analysis. $ f_{t1}$ and $ f_{t2}$ are transition functions, that provide control over the laminar and turbulent regions. All the details are step by step discussed in the original article [11]. The turbulent eddy viscosity is then:
$\displaystyle \nu_T = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi= \frac{\tilde{\nu}}{\nu}$ (27.29)
where $ \chi$ $ =$ $ Re_T$ is Reynolds turbulent number. Model is completed by following formulas:
$\displaystyle \tilde{S} \equiv \vert\Omega\vert + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}$ (27.30)
$\displaystyle f_w = g \left[ \frac{ 1 + C_{w3}^6 }{ g^6 + C_{w3}^6 } \right]^{1/6}, \quad g = r + C_{w2}(r^6 - r), \quad r \equiv \frac{\tilde{\nu} }{ \tilde{S} \kappa^2 d^2 }$ (27.31)
$\displaystyle f_{t1} = C_{t1} g_t \exp\left( -C_{t2} \frac{\omega_t^2}{\Delta U^2} [ d^2 + g^2_t d^2_t] \right)$ (27.32)
$\displaystyle f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)$ (27.33)
The following table gives the model constants present in the formulas above:
$ \sigma$ $ C_{b1}$ $ C_{b2}$ $ \kappa$ $ C_{w1}$ $ C_{w2}$ $ C_{w3}$ $ C_{v1}$ $ C_{t1}$ $ C_{t2}$ $ C_{t3}$ $ C_{t4}$
$ \frac{2}{3}$ 0.1355 0.622 0.41 $ C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma$ 0.3 2.0 7.1 1.0 2.0 1.1 2.0
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Previous: Performance, applicability and limitations Up: Appendix:         RANS Theory Next: Finite volume method applied

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