In physics and fluid dynamics, the Spalart–Allmaras model is a popular mathematical model used in computational fluid dynamics (CFD) to simulate the effects of turbulence. It is a one-equation model, meaning it solves a single transport equation to calculate a variable ν ~ {\displaystyle {\tilde {\nu }}} which is related to the turbulent viscosity. Its main advantages are its relative simplicity and low computational cost, making it widely used for practical engineering problems. The Spalart–Allmaras model was designed specifically for aerospace applications involving airflow over surfaces (known as wall-bounded flows), and it gives good results for flows subject to slowing pressure, known as adverse pressure gradients. Because of its robustness, it is also gaining popularity in turbomachinery applications. However, it is less accurate for simulations of free-flowing turbulence, such as jets, and cannot predict the natural decay of turbulence in the absence of a surface.
Original model The turbulent eddy viscosity is given by
ν t = ν ~ f v 1 , f v 1 = χ 3 χ 3 + C v 1 3 , χ := ν ~ ν {\displaystyle \nu _{t}={\tilde {\nu }}f_{v1},\quad f_{v1}={\frac {\chi ^{3}}{\chi ^{3}+C_{v1}^{3}}},\quad \chi :={\frac {\tilde {\nu }}{\nu }}}
∂ ν ~ ∂ t + u j ∂ ν ~ ∂ x j = C b 1 [ 1 − f t 2 ] S ~ ν ~ + 1 σ { ∇ ⋅ [ ( ν + ν ~ ) ∇ ν ~ ] + C b 2 | ∇ ν ~ | 2 } − [ C w 1 f w − C b 1 κ 2 f t 2 ] ( ν ~ d ) 2 + f t 1 Δ U 2 {\displaystyle {\frac {\partial {\tilde {\nu }}}{\partial t}}+u_{j}{\frac {\partial {\tilde {\nu }}}{\partial x_{j}}}=C_{b1}[1-f_{t2}]{\tilde {S}}{\tilde {\nu }}+{\frac {1}{\sigma }}\{\nabla \cdot [(\nu +{\tilde {\nu }})\nabla {\tilde {\nu }}]+C_{b2}|\nabla {\tilde {\nu }}|^{2}\}-\left[C_{w1}f_{w}-{\frac {C_{b1}}{\kappa ^{2}}}f_{t2}\right]\left({\frac {\tilde {\nu }}{d}}\right)^{2}+f_{t1}\Delta U^{2}}
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