Spalart-Allmaras Model
Theory and Physics
Overview
Professor, I heard that the Spalart-Allmaras model is often used in the aerospace field. What kind of model is it?
The Spalart-Allmaras (SA) model is a one-equation model that describes turbulence with just a single transport equation for the modified eddy viscosity $\tilde{\nu}$. It was published by Philippe Spalart and Steven Allmaras in 1992. It was specifically developed for flows around aircraft wings and has become one of NASA's standard turbulence models.
One equation means it's lighter than the two-equation k-epsilon model?
Exactly. With one less transport equation, it has lower computational cost and faster convergence. It has good behavior near walls and demonstrates high accuracy for attached boundary layers and flows under weak adverse pressure gradients.
Governing Equations
Could you please tell me the specific equations?
The transport equation for the modified eddy viscosity $\tilde{\nu}$ is as follows.
The physical meaning of each term is as follows.
| Term | Equation | Physical Meaning |
|---|---|---|
| Production Term | $c_{b1}\tilde{S}\tilde{\nu}$ | Turbulence production due to mean velocity gradient |
| Destruction Term | $c_{w1}f_w(\tilde{\nu}/d)^2$ | Destruction dependent on wall distance $d$ |
| Diffusion Term | $\frac{1}{\sigma}[\nabla\cdot((\nu+\tilde{\nu})\nabla\tilde{\nu}) + c_{b2}(\nabla\tilde{\nu})^2]$ | Molecular diffusion + turbulent diffusion |
Eddy Viscosity Calculation
How do you obtain the eddy viscosity $\nu_t$ from $\tilde{\nu}$?
It is calculated using the damping function $f_{v1}$ as follows.
Near the wall ($\chi \ll 1$), $f_{v1} \to 0$, causing $\nu_t$ to automatically decay. This is a major feature of the SA model, eliminating the need to add a separate Low-Re damping function.
Model Constants
How many constants are there?
The standard constants are as follows.
| Constant | Value | Constant | Value |
|---|---|---|---|
| $c_{b1}$ | 0.1355 | $c_{b2}$ | 0.622 |
| $\sigma$ | 2/3 | $\kappa$ | 0.41 |
| $c_{w1}$ | $c_{b1}/\kappa^2 + (1+c_{b2})/\sigma$ | $c_{w2}$ | 0.3 |
| $c_{w3}$ | 2.0 | $c_{v1}$ | 7.1 |
It looks like there are many constants, but $c_{w1}$ is derived from other constants. So essentially there are about 6 independent constants, right?
Exactly. Moreover, the wall distance $d$ is the only geometric input, and there's no need to solve a separate equation for the turbulent length scale. This is the simplicity of a one-equation model.
"Solving Wings with One Equation" — The Bet Between Philip and Amir
The SA model published by Philip Spalart and Steven Allmaras in 1992 was designed with the concept of "solving just one transport equation for the eddy viscosity itself." At the time, the CFD community had the preconception that "turbulence cannot be computed without solving two equations," and the one-equation approach was met with skepticism. However, its computational accuracy for wing boundary layers was not only comparable but sometimes superior to two-equation models, leading to its rapid adoption in the aerospace industry. The reason it has been recommended for years in NASA's external aerodynamics CFD guidelines is precisely this combination of lightness and robustness from the single equation.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It's looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, starting with a steady-state solution is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. The warm air from a heater reaching the other side of the room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, the liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference creates the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget a source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number 0.3 and above, consider compressibility effects
- Boussinesq Approximation (Natural Convection): Consider density changes only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flows (shock capturing required), free surface flows (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge pressure and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Discretization in FVM
How do you discretize the SA model in a CFD solver?
Discretization using the Finite Volume Method (FVM) follows the same framework as other scalar transport equations. Convection and diffusion terms are converted into face fluxes and discretized.
However, there are specific points to note for the SA model. The nonlinear diffusion term $c_{b2}(\nabla\tilde{\nu})^2$ does not fit the standard diffusion term form, so it needs to be handled as a source term or rewritten into a conservative form with some ingenuity.
Wall Distance $d$ Calculation
How do you calculate the wall distance? You need the distance from every cell to the nearest wall, right?
That's correct. There are mainly two methods for calculating wall distance.
| Method | Computational Cost | Accuracy | Parallelization |
|---|---|---|---|
| Geometric Search (Brute Force) | $O(N_{cell} \times N_{wall})$ | Accurate | High communication cost |
| Poisson Equation Method | $O(N_{cell})$ | Approximate | Easy |
The Poisson equation method solves $\nabla^2 \phi = -1$ and estimates $d \approx |\nabla\phi| + \sqrt{|\nabla\phi|^2 + 2\phi}$. Fluent uses this by default. It is more efficient than geometric search for large-scale parallel computation.
Boundary Conditions
What boundary conditions are set at the wall and far field?
Boundary Condition Remarks Wall $\tilde{\nu} = 0$ Full no-slip condition Inlet $\tilde{\nu}_{\text{in}} = 3\nu$ 〜 $5\nu$ External flow with low turbulence intensity Far Field $\tilde{\nu}_{\infty} / \nu = 3$ 〜 $5$ NASA recommended values
The inlet $\tilde{\nu}$ setting affects results. NASA Turbulence Modeling Resource recommends $\tilde{\nu}/\nu = 3$, but adjustment based on turbulence intensity is necessary for comparison with wind tunnel experiments.
Numerical Stability Techniques
Can calculations become unstable with the SA model?
If $\tilde{\nu}$ becomes negative, it is physically meaningless. The following countermeasures are used.
- Negativity clipping: Clip to zero when $\tilde{\nu} < 0$
- SA-neg variant: Negative-value tolerant version proposed by Allmaras et al. (2012). Modifies production and destruction terms to allow stable computation even when $\tilde{\nu} < 0$
- Implicit linearization of source term: Decompose into the form $S_{\tilde{\nu}} = S_c + S_p \cdot \tilde{\nu}$ and incorporate the $S_p < 0$ part into the coefficient matrix
Is the SA-neg variant implemented in OpenFOAM as well?
In OpenFOAM v2306 and later, the SpalartAllmaras class includes negative value handling. Fluent also supports SA-neg.
Coffee Break Yomoyama Talk
SA Model's Wall Distance—The Big Impact of a Small Input
The Spalart-Allmaras model uses the wall distance $d$ as its only geometric input, and the accuracy of this $d$ determines the overall accuracy of the model. For complex shapes, such as gaps between wings and flaps or junctions between fuselage and engine, calculating wall distance becomes difficult. Particularly, if an internal wall (like the inner surface of a cavity) is mistakenly recognized as an "external wall," the wall distance becomes overestimated, leading to locally overestimated eddy viscosity. When using the SA model, it's an ironclad rule in the field to always visualize and check the wall distance field to safeguard calculation accuracy.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy using limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Related Topics
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What boundary conditions are set at the wall and far field?
| Boundary | Condition | Remarks |
|---|---|---|
| Wall | $\tilde{\nu} = 0$ | Full no-slip condition |
| Inlet | $\tilde{\nu}_{\text{in}} = 3\nu$ 〜 $5\nu$ | External flow with low turbulence intensity |
| Far Field | $\tilde{\nu}_{\infty} / \nu = 3$ 〜 $5$ | NASA recommended values |
The inlet $\tilde{\nu}$ setting affects results. NASA Turbulence Modeling Resource recommends $\tilde{\nu}/\nu = 3$, but adjustment based on turbulence intensity is necessary for comparison with wind tunnel experiments.
Can calculations become unstable with the SA model?
If $\tilde{\nu}$ becomes negative, it is physically meaningless. The following countermeasures are used.
Is the SA-neg variant implemented in OpenFOAM as well?
In OpenFOAM v2306 and later, the SpalartAllmaras class includes negative value handling. Fluent also supports SA-neg.
SA Model's Wall Distance—The Big Impact of a Small Input
The Spalart-Allmaras model uses the wall distance $d$ as its only geometric input, and the accuracy of this $d$ determines the overall accuracy of the model. For complex shapes, such as gaps between wings and flaps or junctions between fuselage and engine, calculating wall distance becomes difficult. Particularly, if an internal wall (like the inner surface of a cavity) is mistakenly recognized as an "external wall," the wall distance becomes overestimated, leading to locally overestimated eddy viscosity. When using the SA model, it's an ironclad rule in the field to always visualize and check the wall distance field to safeguard calculation accuracy.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy using limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Related Topics
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