# Hegedus Aerodynamics

Spalart-Allmaras Turbulence Model for Aero Troll CFD

Note: the equations below require MathML to be displayed.

SA-noft2

The Spalart-Allmaras turbulence model without the ${f}_{t2}$ term comes from References (1)(2)(3). $∂ ν ^ ∂ t + u j ∂ ν ^ ∂ x j = c b 1 1 - f t 2 S ^ ν ^ - c w 1 f w - c b 1 κ 2 f t 2 ν ^ d 2 + 1 σ ∂ ∂ x j ν + ν ^ ∂ ν ^ ∂ x j + c b 2 ∂ ν ^ ∂ x j ∂ ν ^ ∂ x j$ $μ t = ρ ν ^ f v 1$ $f v 1 = χ 3 χ 3 + c v 1 3$ $χ = ν ^ ν$ $f t 2 = 0$ $S ^ = Ω + ν ^ κ 2 d 2 f v 2$ $f v 2 = 1 - χ 1 + χ f v 1$ $f w = g 1 + c w 3 6 g 6 + c w 3 6 1 6$ $g = r + c w 2 r 6 - r$ $r = min⁡ ν ^ S ^ κ 2 d 2 , 10$ Constants:
$σ = 2 / 3$ $κ = 0.41$ $c b 1 = 0.1355$ $c b 2 = 0.622$ $c w 1 = c b 1 κ 2 + 1 + c b 2 σ$ $c w 2 = 0.3$ $c w 3 = 2$ $c v 1 = 7.1$ Boundary conditions:
$ν ^ wall ν wall = 0$ $ν ^ ∞ ν ∞ = 3$

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 $\partial \stackrel{^}{\nu }\partial \stackrel{^}{\nu }$ term.

Starting with the chain rule:

$∂ ν ^ ∂ ν ^ = ∂ ν ^ ∂ ν ^ + ν ^ ∂ 2 ν ^$ and rearranging terms, results in:

$∂ ν ^ ∂ ν ^ = ∂ ν ^ ∂ ν ^ - ν ^ ∂ 2 ν ^$ Substituting this into the Spalart-Allmaras turbulence model results in:

$∂ ν ^ ∂ t + u j ∂ ν ^ ∂ x j = c b 1 1 - f t 2 S ^ ν ^ - c w 1 f w - c b 1 κ 2 f t 2 ν ^ d 2 + 1 σ ∂ ∂ x j ν + 1 + c b 2 ν ^ ∂ ν ^ ∂ x j - c b 2 σ ν ^ ∂ 2 ν ^ ∂ x j 2$ The form shown above is different than the one given in Reference (1) but is that same as that given in Reference (2).

Spatial Discretization

The following is substituted in for the diffusion terms:

$∂ ∂ x j f⁡ ν ^ ∂ ν ^ ∂ x j ≈ k j → ∂ ∂ k j k j → f⁡ ν ^ ∂ ∂ k j$ Resulting in:

$∂ ν ^ ∂ t + k → j · V → ∂ ν ^ ∂ k j = c b 1 1 - f t 2 S ^ ν ^ - c w 1 f w - c b 1 κ 2 f t 2 ν ^ d 2 + k j → σ ∂ ∂ k j k j → ν + 1 + c b 2 ν ^ ∂ ν ^ ∂ k j - c b 2 σ ν ^ k j → ∂ ∂ k j k j → ∂ ν ^ ∂ k j$ 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).

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).

$D + h L ξ o D - 1 D + h L η o D - 1 D + h L ζ o Δ ν ^ k = - I + h L ξ + h L η + h L ζ n ν ^ k - ν ^ n - h r n$ $L ξ = L ξ d + L ξ o , L η = L η d + L η o , L ζ = L ζ d + L ζ o$ $D = I + h L ξ d + h L η d + h L ζ d - h ∂ ν ^ f n$ $Δ ν ^ k = ν ^ k + 1 - ν ^ k$ Huang's subiteration DDADI method is as follows:

${ D + h L ξ o Δ ν 1 k + 1 = - h r n k = 0 D + h L ξ o Δ ν ^ 1 k + 1 - Δ ν ^ 1 k = D Δ ν ^ 3 k - 1 - Δ ν ^ 1 k k ≠ 0$   $D + h L η o Δ ν ^ 2 k + 1 - Δ ν ^ 2 k = D Δ ν ^ 1 k + 1 - Δ ν ^ 2 k$ $D + h L ζ o Δ ν ^ 3 k + 1 - Δ ν ^ 3 k = D Δ ν ^ 2 k + 1 - Δ ν ^ 3 k$

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.