Hegedus Aerodynamics

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SA-noft2

The Spalart-Allmaras turbulence model without the $f_{t2}$ term comes from References (1)(2)(3). $$\frac{\partial \widehat{\nu }}{\partial t}+u_j\frac{\partial \widehat{\nu }}{\partial x_j}=c_{b1}\left(1-f_{t2}\right)\widehat{S}\widehat{\nu }-\left[c_{w1}{f}_w-\frac{c_{b1}}{{\kappa }^2}{f}_{t2}\right]{\left(\frac{\widehat{\nu }}{d}\right)}^2+\frac{1}{\sigma }\left[\frac{\partial }{\partial x_j}\left(\left(\nu +\widehat{\nu }\right)\frac{\partial \widehat{\nu }}{\partial x_j}\right)+c_{b2}\frac{\partial \widehat{\nu }}{\partial x_j}\frac{\partial \widehat{\nu }}{\partial x_j}\right]$$ $${\mu }_t=\rho \widehat{\nu }{f}_{v1}$$ $$f_{v1}=\frac{{\chi }^3}{{\chi }^3+c_{v1}^3}$$ $$\chi =\frac{\widehat{\nu }}{\nu }$$ $$f_{t2}=0$$ $$\widehat{S}=\Omega +\frac{\widehat{\nu }}{{\kappa }^2{d}^2}{f}_{v2}$$ $$f_{v2}=1-\frac{\chi }{1+\chi f_{v1}}$$ $$f_w=g{\left[\frac{1+c_{w3}^6}{g^6+c_{w3}^6}\right]}^{\frac{1}{6}}$$ $$g=r+c_{w2}\left(r^6-r\right)$$ $$r=\min \left[\frac{\widehat{\nu }}{\widehat{S}{\kappa }^2{d}^2}\text{,}10\right]$$ Constants:
$$\sigma =2/3$$ $$\kappa =0.41$$ $$c_{b1}=0.1355$$ $$c_{b2}=0.622$$ $$c_{w1}=\frac{c_{b1}}{{\kappa }^2}+\frac{1+c_{b2}}{\sigma }$$ $$c_{w2}=0.3$$ $$c_{w3}=2$$ $$c_{v1}=7.1$$ Boundary conditions:
$$\frac{{\widehat{\nu }}_{\mathrm{wall}}}{{\nu }_{\mathrm{wall}}}=0$$ $$\frac{{\widehat{\nu }}_{\infty }}{{\nu }_{\infty }}=3$$

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Discretization Form

Before the Spalart-Allmaras turbulence model is discretized, it will be placed into another form to make things easier.  The modification involves the $\partial \widehat{\nu }\partial \widehat{\nu }$ term.

Starting with the chain rule:

$$\partial \left(\widehat{\nu }\partial \widehat{\nu }\right)=\partial \widehat{\nu }\partial \widehat{\nu }+\widehat{\nu }{\partial }^2\widehat{\nu }$$ and rearranging terms, results in:

$$\partial \widehat{\nu }\partial \widehat{\nu }=\partial \left(\widehat{\nu }\partial \widehat{\nu }\right)-\widehat{\nu }{\partial }^2\widehat{\nu }$$ Substituting this into the Spalart-Allmaras turbulence model results in:

$$\frac{\partial \widehat{\nu }}{\partial t}+u_j\frac{\partial \widehat{\nu }}{\partial x_j}=c_{b1}\left(1-f_{t2}\right)\widehat{S}\widehat{\nu }-\left[c_{w1}{f}_w-\frac{c_{b1}}{{\kappa }^2}{f}_{t2}\right]{\left(\frac{\widehat{\nu }}{d}\right)}^2+\frac{1}{\sigma }\frac{\partial }{\partial x_j}\left[\left(\nu +\left(1+c_{b2}\right)\widehat{\nu }\right)\frac{\partial \widehat{\nu }}{\partial x_j}\right]-\frac{c_{b2}}{\sigma }\widehat{\nu }\frac{{\partial }^2\widehat{\nu }}{\partial x_j^2}$$ The form shown above is different than the one given in Reference (1) but is that same as that given in Reference (2).

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Spatial Discretization

The following is substituted in for the diffusion terms:

$$\frac{\partial }{\partial x_j}\left[f\left(\widehat{\nu }\right)\frac{\partial \widehat{\nu }}{\partial x_j}\right]\approx \overrightarrow{k_j}\frac{\partial }{\partial k_j}\left[\overrightarrow{k_j}f\left(\widehat{\nu }\right)\frac{\partial }{\partial k_j}\right]$$ Resulting in:

$$\frac{\partial \widehat{\nu }}{\partial t}+\left({\overrightarrow{k}}_j·\overrightarrow{V}\right)\frac{\partial \widehat{\nu }}{\partial k_j}=c_{b1}\left(1-f_{t2}\right)\widehat{S}\widehat{\nu }-\left[c_{w1}{f}_w-\frac{c_{b1}}{{\kappa }^2}{f}_{t2}\right]{\left(\frac{\widehat{\nu }}{d}\right)}^2+\frac{\overrightarrow{k_j}}{\sigma }\frac{\partial }{\partial k_j}\left[\overrightarrow{k_j}\left(\nu +\left(1+c_{b2}\right)\widehat{\nu }\right)\frac{\partial \widehat{\nu }}{\partial k_j}\right]-\frac{c_{b2}}{\sigma }\widehat{\nu }\overrightarrow{k_j}\frac{\partial }{\partial k_j}\left(\overrightarrow{k_j}\frac{\partial \widehat{\nu }}{\partial k_j}\right)$$ The advection terms are discretized using first order accurate upwinding and the diffusion terms are discretized using second order accurate central differencing, as in Reference (1).

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Iterative Diagonal Dominant Alternating Direction Implicit (DDADI) Method

The Diagonal Dominant Alternating Direction Implicit (DDADI) method with Huang subiterations is used to invert the implicit matrix, Reference (4) and (5).

$$\left[D+{hL}_{\xi }^o\right]D^{-1}\left[D+h{L}_{\eta }^o\right]D^{-1}\left[D+h{L}_{\zeta }^o\right]\Delta {\widehat{\nu }}^k=-{\left[I+h{L}_{\xi }+h{L}_{\eta }+h{L}_{\zeta }\right]}^n\left({\widehat{\nu }}^k-{\widehat{\nu }}^n\right)-h{r}_n$$ $$L_{\xi }=L_{\xi }^d+L_{\xi }^o\text{,}{L}_{\eta }=L_{\eta }^d+L_{\eta }^o\text{,}{L}_{\zeta }=L_{\zeta }^d+L_{\zeta }^o$$ $$D=\left[I+h{L}_{\xi }^d+h{L}_{\eta }^d+h{L}_{\zeta }^d-h{\partial }_{\widehat{\nu }}fn\right]$$ $$\Delta {\widehat{\nu }}^k={\widehat{\nu }}^{k+1}-{\widehat{\nu }}^k$$ Huang's subiteration DDADI method is as follows:

$$\{\begin{array}{cc}\left[D+h{L}_{\xi }^o\right]\Delta {\nu }_1^{k+1}=-h{r}^n& k=0\\ \left[D+h{L}_{\xi }^o\right]\left(\Delta {\widehat{\nu }}_1^{k+1}-\Delta {\widehat{\nu }}_1^k\right)=D\left(\Delta {\widehat{\nu }}_3^{k-1}-\Delta {\widehat{\nu }}_1^k\right)& k\ne 0\end{array}$$ $$$$ $$$$ $$\left[D+h{L}_{\eta }^o\right]\left(\Delta {\widehat{\nu }}_2^{k+1}-\Delta {\widehat{\nu }}_2^k\right)=D\left(\Delta {\widehat{\nu }}_1^{k+1}-\Delta {\widehat{\nu }}_2^k\right)$$ $$\left[D+h{L}_{\zeta }^o\right]\left(\Delta {\widehat{\nu }}_3^{k+1}-\Delta {\widehat{\nu }}_3^k\right)=D\left(\Delta {\widehat{\nu }}_2^{k+1}-\Delta {\widehat{\nu }}_3^k\right)$$

References

1) Spalart, P. R. and Allmaras, S. R., "A One-Equation Turbulence Model for Aerodynamic Flows," AIAA-92-0439, 1992.
2) Krist, S. L., Biedron, R. T., and Rumsey, C. L., "CFL3D User's Manual (Version 5.0)," NASA TM 1998-208444, June 1998.
3) http://turbmodels.larc.nasa.gov/spalart.html
4) Walsh, P. C., and Pulliam T., "The Effect of Turbulence Model Solution On Viscous Flow Problems," AIAA-2001-1018.
5) Klopfer, G. H., Van der Wijngaart, R. F., Hung, C. M, and Onufer, J. T., "A Diagonalized Diagonal Dominant Alternating Direction Implicit (D3ADI) Scheme and Subiteration Correction," AIAA Paper 98-2824, 1998.