Spalart–Allmaras turbulence model

Short description: One-equation turbulence model for computational fluid dynamics

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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~ representing the turbulent viscosity.[original research?] 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.[original research?] 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.

## Contents

* 1 Original model
* 2 Modifications to original model
* 3 Model for compressible flows
* 4 Boundary conditions
* 5 References
* 6 External links

## Original model

The turbulent eddy viscosity is given by

νt=ν~fv1,fv1=χ3χ3+Cv13,χ:=ν~ν

∂ν~∂t+uj∂ν~∂xj=Cb1[1−ft2]S~ν~+1σ{∇⋅[(ν+ν~)∇ν~]+Cb2|∇ν~|2}−[Cw1fw−Cb1κ2ft2](ν~d)2+ft1ΔU2

S~≡S+ν~κ2d2fv2,fv2=1−χ1+χfv1

fw=g[1+Cw36g6+Cw36]1/6,g=r+Cw2(r6−r),r≡ν~S~κ2d2

ft1=Ct1gtexp⁡(−Ct2ωt2ΔU2[d2+gt2dt2])

ft2=Ct3exp⁡(−Ct4χ2)

S=2ΩijΩij

The rotation tensor is given by

Ωij=12(∂ui/∂xj−∂uj/∂xi)

where d is the distance from the closest surface and ΔU2 is the norm of the difference between the velocity at the trip (usually zero) and that at the field point we are considering.

The constants are

σ=2/3Cb1=0.1355Cb2=0.622κ=0.41Cw1=Cb1/κ2+(1+Cb2)/σCw2=0.3Cw3=2Cv1=7.1Ct1=1Ct2=2Ct3=1.1Ct4=2

## Modifications to original model

According to Spalart it is safer to use the following values for the last two constants:

Ct3=1.2Ct4=0.5

Other models related to the S-A model:

DES (1999) [1]

DDES (2006)

## Model for compressible flows

There are several approaches to adapting the model for compressible flows.

In all cases, the turbulent dynamic viscosity is computed from

μt=ρν~fv1

where ρ is the local density.

The first approach applies the original equation for ν~.

In the second approach, the convective terms in the equation for ν~ are modified to

∂ν~∂t+∂∂xj(ν~uj)=RHS

The third approach involves inserting the density inside some of the derivatives on the LHS and RHS.

The second and third approaches are not recommended by the original authors, but they are found in several solvers.

## Boundary conditions

Walls: ν~=0

Freestream:

Ideally ν~=0, but some solvers can have problems with a zero value, in which case ν~≤ν2 can be used.

This is if the trip term is used to "start up" the model. A convenient option is to set ν~=5ν in the freestream. The model then provides "Fully Turbulent" behavior, i.e., it becomes turbulent in any region that contains shear.

Outlet: convective outlet.

## References

* Spalart, Philippe R.; Allmaras, Steven R. (1992). "A One-Equation Turbulence Model for Aerodynamic Flows". AIAA Paper (92–0439). doi:10.2514/6.1992-439.
* Javaherchi, Teymour (2010). Review of Spalart-Allmaras turbulence model and its modifications (PDF) (MS Thesis).
* Allmaras, Steven R.; Johnson, Forrester T. (2012). "Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model". 7th International Conference on Computational Fluid Dynamics (ICCFD7). Big Island, Hawaii. Paper ICCFD7-1902. https://iccfd.org/iccfd7/assets/pdf/papers/ICCFD7-1902_paper.pdf.

## External links

* This article was based on the Spalart-Allmaras model article in CFD-Wiki
* What Are the Spalart-Allmaras Turbulence Models? from kxcad.net
* The Spalart-Allmaras Turbulence Model at NASA's Langley Research Center Turbulence Modelling Resource site

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