Spalart-Allmaras One-Equation turbulence model

1. Introduction

The Spalart-Allmaras model [1] is a one-equation turbulence model designed for aerodynamic applications involving wall-bounded flows, which solves a transport equation for the undamped turbulent kinematic viscosity. It is a low-Reynolds number model (i.e. it does not utilise wall functions), although it can be adapted for use with wall functions. Both of these options are available in PHOENICS as a high-Reynolds and low-Reynolds version.

The model has largely been used for wall-bounded aerodynamic and turbomachinery applications. It is not suited for free shear flows and decaying turbulence.

The Spalart-Allmaras model is considered to be more robust than the low-Reynolds number k-ε model, and requires much less computer time than the 2-equation models.

In the original model as proposed by Spalart and Allmaras [1], there are two trip terms included in the model. The "standard" version of the model most often employed does not use the first trip term. This first trip term is used to "start-up" the model in cases where no turbulence is present in the free-stream. PHOENICS uses this standard version of the model also without the second trip term included (see for example [2]). This second trip term is not required provided the appropriate boundary condition in the far field is employed.

2. Model Equations

The kinematic turbulent eddy viscosity,

ν_{t}

, is given by,

ν_{t} = \tilde{ν} f_{v 1}

where

f_{v 1} = \frac{χ^{3}}{χ^{3} + c_{v 1}^{3}}, χ \equiv \frac{\tilde{ν}}{ν}

and $ν$ is the molecular kinematic viscosity.

We solve for $\tilde{ν}$ , the undamped turbulent kinematic viscosity. PHOENICS uses the standard version of the Spalart-Allmaras model without the trip term [2], which obeys the following transport equation:

\frac{\partial (ρ \tilde{ν})}{\partial t} + \nabla \cdot (ρ 𝐮 \tilde{ν}) = P_{ν} - D_{ν} + \frac{1}{σ} \nabla \cdot (ρ (ν + \tilde{ν}) \nabla \tilde{ν}) + \frac{c_{b 2}}{σ} ρ \nabla \tilde{ν} \cdot \nabla \tilde{ν}

where $P_{ν}$ is the production of $\tilde{ν}$ and $D_{ν}$ is the destruction of $\tilde{ν}$ . The last two terms of the transport equation (2.1.3) are the diffusion and diffusive source terms respectively.

The production term,

P_{ν}

, is

P_{ν} = c_{b 1} \tilde{S} ρ \tilde{ν}

where

\tilde{S} = m a x (Ω + C_{r o t} m i n (0, S - Ω) + \frac{\tilde{ν}}{κ^{2} d^{2}} f_{v 2}, 0.3 Ω)

[3,4], and

f_{v 2} = 1 - \frac{χ}{1 + χ f_{v 1}} .

Here, $κ$ is von Karman's constant, and

S = \sqrt{2 S_{i j} S_{i j}} Ω = \sqrt{2 Ω_{i j} Ω_{i j}}

where $S_{ij}, Ω_{ij}$ are the mean strain rate and mean rotation rate tensors respectively:

S_{i j} = \frac{1}{2} (\nabla 𝐮 + \nabla 𝐮^{T}) Ω_{i j} = \frac{1}{2} (\nabla 𝐮 - \nabla 𝐮^{T})

In this implementation of the production term, $\tilde{S}$ (2.1.5) uses a rotation correction [3], with limit of $0.3Ω$ imposed on $\tilde{S}$ as stated in [4].

The destruction term, $D_{ν}$ , is

D_{ν} = c_{w 1} ρ f_{w} {(\frac{\tilde{ν}}{d})}^{2}

where

d

is the distance to the nearest wall,

c_{w 1} = \frac{c_{b 1}}{κ^{2}} + \frac{1 + c_{b 2}}{σ}

and

f_{w} = g {(\frac{1 + c_{w 3}^{6}}{g^{6} + c_{w 3}^{6}})}^{1 / 6}, g = r + c_{w 2} (r^{6} - r), r = m i n (\frac{\tilde{ν}}{\tilde{S} κ^{2} d^{2}}, 10)

The parameter $r$ is truncated to 10 for large values as recommended in [1].

The values for the model constants are:

c_{b 1} = 0.1355, c_{b 2} = 0.622, c_{v 1} = 7.1, σ = 2 / 3

c_{w 2} = 0.3, c_{w 3} = 2.0, κ = 0.41, C_{r o t} = 2.0

3. Boundary Conditions

3.1. Wall

For the high Reynolds-number model, the condition at the wall is

\tilde{ν} = κ δ_{w} u_{τ}

where $κ$ is von Karman's constant (=0.41), $δ_{w}$ is the distance between the wall and the first grid node, and $u_{τ}$ is the friction velocity, calculated from the wall shear-stress as

u_{τ} = \sqrt{\frac{τ_{w}}{ρ}}

.

The relation (3.1.1) is valid throughout the log layer.

For the low Reynolds-number model, the boundary condition at the wall is $\tilde{ν} = 0 .$ This is due to there being no fluctuations at the wall.

In both cases, the wall boundary conditions are automatically set when selecting the model in PHOENICS.

3.2. Inlet

At the inlet, $ν_{t}$ should be specified, then

\tilde{ν} = \frac{ν_{t}}{ν}

Alternatively, as in the free-stream (section 3.3), $\tilde{ν}$ can be set as a ratio of $ν$ , i.e. $\tilde{ν} = ratio * ν$ .

In PHOENICS, the value of $\tilde{ν}$ at the inlet can be specified in the VR Viewer in a number of alternate ways, by choosing the appropriate option in the Attributes menu of the inlet object:

Intensity+auto length scale: The user inputs the desired turbulent intensity and the length scale is automatically calculated within PHOENICS, with the inlet value of $\tilde{ν}$ calculated from:

$\tilde{ν} = {(C_{μ} C_{d})}^{1 / 4} (U I) l_{m}$

where $C_{μ} = 0.5478$ , $C_{d} = 0.1643$ such that $C_{μ} C_{d} = 0.09$ , $U$ is the mean flow velocity, $I$ is the turbulent intensity and $l_{m}$ is the turbulent mixing length. The turbulent mixing length is taken to be the hydraulic radius of the inlet. This is half the hydraulic diameter, which is calculated as 4*area/perimeter. The area and perimeter are always based on the bounding box of the object, so may not be accurate for a non-cuboid shape. If desired, the values of $C_{μ}$ and $C_{d}$ can be specified by the user from the Main Menu, under Models, Turbulence models, Settings.
Viscosity ratio: The user specifies a multiplier of the kinematic viscosity, i.e. $\tilde{ν} = viscosity ratio * ν .$
Intensity+length scale: This is the same as option 1, with the value at the inlet calculated using equation (3.2.2), but the value of the mixing length is supplied by the user in addition to the intensity.
User-set: The user can set a desired inlet value for $\tilde{ν}$ .

3.3. Free-Stream

Turbulence in the free-stream can be estimated from the turbulent Reynolds number

{Re}_{t} = \frac{ν_{t}}{ν}

with $\tilde{ν}$ in the free-stream set as

\tilde{ν} = {Re}_{t} ν_{∞}

where $ν_{∞}$ is the free-stream molecular kinematic viscosity.

It is recommended that $\tilde{ν}$ is set to $3 ν_{∞}$ or greater. With this, the model will provide fully turbulent results, i.e. it becomes turbulent in regions containing shear.

4. Activation of the model

The one-equation Spalart-Allmaras model can be activated from the VR menu, or alternatively by inserting the PIL command TURMOD(SPALART-ALLMARAS) in the Q1 file, which is equivalent to the following PIL commands:

   TURMOD(SPALART-ALLMARAS); SOLVE(ENTI)
   ENUT=GRND2;IENUTA=21
   DISWAL;GENK=T
   STORE(VOR1);STORE(GEN1)
   STORE(DUDY,DUDZ,DVDX,DVDZ,DWDX,DWDY)
   PRT(ENTI)=2./3.
   PRNDTL(ENTI)=2./3.
   SOLUTN(ENTI,Y,Y,Y,N,N,Y)      
   
   PATCH(SASOPRO,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
   COVAL(SASOPRO,ENTI,FIXFLU,GRND4)

   PATCH(SASODIS,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
   COVAL(SASODIS,ENTI,GRND4,0.0)

   PATCH(SASODIF,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
   COVAL(SASODIF,ENTI,FIXFLU,GRND4)

This activates the high Reynolds version of the Spalart-Allmaras model with wall functions, which has wall boundary conditions:

   COVAL(FIXVAL, 0.41*SQRT(STRS)*WDIS)

where STRS is the wall-shear stress divided by density, $\frac{τ_{w}}{ρ}$ , and WDIS is the minimum distance to the wall.

The DISWAL command activates the solution of a scalar variable LTLS, from which WDIS is deduced.

The low-Reynolds version can be activated from the VR menu, or by inserting the PIL command TURMOD(SPALART-ALLMARAS-LOWRE). This is equivalent to the high-Reynolds, but with IENUTA=22, and wall boundary conditions:

	
	COVAL(1.,0.)

5. Sources of further information

5.1. Library Cases

The following cases demonstrating the use of the Spalart-Allmaras model are available from the PHOENICS library:

One-dimensional:
- 1D Fully-Developed Pipe Flow Low-Re: t901
- 1D Fully-Developed Pipe Flow High-Re: t902
Two-dimensional:
- 2D Elliptic Pipe Flow Low-Re: t903
- 2D Elliptic Pipe Flow High-Re: t904
- 2D Channel Flow Low-Re X-Y: t905
- 2D Channel Flow Low-Re Z-Y: t906
- 2D Flow Past a Flat Plate Low-Re X-Y: t907
- 2D Flow Past a Flat Plate Low-Re X-Z t908
- 2D Flow Past a Flat Plate High-Re X-Y: t909
- 2D Flow Past a Flat Plate High-Re X-Z: t910
- Backward-Facing Step High-Re: t911
- Backward-Facing Step Low-Re: t912
- 2D Flow Across a Blunt Plate High-Re: t914
- 2D Flow Past a Square Rib High-Re: t915
- 2D Flow Across a Blunt Plate Low-Re: t916
- 2D Flow Past a Square Rib Low-Re: t917
- 2D Flow in A Turnaround Duct High-Re (BFC): t918
- 2D Flow in A Turnaround Duct Low-Re (BFC): t919
- 2D NACA0012 Aerofoil: t921
- 2D Model-A Aerofoil: t922
Three-dimensional:
- 3D Flow past a Cube in a Channel: t913
- 3D Flow in an Elbow Meter (BFC): t920
- 3D NACA2414 Aerofoil: t923

5.2. References

1. Spalart, P.R. & Allmaras, S.R. (1992). "A One-Equation Turbulence Model for Aerodynamic Flows" AIAA-92-0439

2. Aupoix, B. & Spalart, P. R. Extensions of the Spalart-Allmaras Turbulence Model to Account for Wall Roughness. International Journal of Heat and Fluid Flow. 2003; 24(4):454-462, https://doi.org/10.1016/S0142-727X(03)00043-2.

3. Dacles-Mariani, J., Zilliac, G. G., Chow, J. S., and Bradshaw, P. Numerical/Experimental Study of a Wingtip Vortex in the Near FieldAIAA Journal 1995; 33(9):1561-1568, https://doi.org/10.2514/3.12826

4. NASA. The Spalart-Allmaras Turbulence Model. https://turbmodels.larc.nasa.gov/spalart.html. [Accessed 21st March 2025]

3.2.2 Spalart-Allmaras One-Equation turbulence model

Contents

1. Introduction

2. Model Equations

3. Boundary Conditions

3.1. Wall

3.2. Inlet

3.3. Free-Stream

4. Activation of the model

5. Sources of further information

5.1. Library Cases

5.2. References