The Smagorinsky Model -- Theory and Governing Equations
Theory and Physics
Overview of the Smagorinsky Model
Professor, the Smagorinsky model is the most basic SGS model, right?
That's correct. It was proposed by Joseph Smagorinsky in 1963 for atmospheric circulation models and can be considered the origin of SGS models in LES. The concept is simple: it calculates the SGS eddy viscosity from the magnitude of the local strain rate and the filter width.
Derivation of the Basic Equations
What are the specific equations?
In the Smagorinsky model, the SGS eddy viscosity $\nu_{sgs}$ is defined as follows.
Here, $C_s$ is the Smagorinsky constant, $\Delta$ is the filter width, and $|\bar{S}|$ is the magnitude of the filtered strain rate tensor.
It makes sense dimensionally as well. $\nu_{sgs}$ has dimensions of $[length^2/time]$, $(C_s \Delta)^2$ is $[length^2]$, and $|\bar{S}|$ is $[1/time]$.
Good observation. In fact, this model can also be interpreted as the LES version of the mixing length theory. It has the same structure as Prandtl's mixing length theory in RANS, $\nu_t = l_m^2 |S|$, where the mixing length $l_m$ is replaced by the filter width $C_s \Delta$.
Value of the Smagorinsky Constant $C_s$
How is the value of $C_s$ determined?
According to Lilly's theoretical analysis (1967), assuming isotropic turbulence and aligning it with Kolmogorov's energy spectrum $E(k) = C_K \varepsilon^{2/3} k^{-5/3}$, we get:
(where $C_K \approx 1.6$). However, in practice, values in the range $C_s = 0.1$ to $0.2$ are often used.
So the optimal value changes depending on the flow?
Exactly. While $C_s \approx 0.17$ is appropriate for isotropic turbulence, in shear flows it needs to be around $C_s \approx 0.1$ to avoid excessive dissipation. Near walls, an even smaller value is needed, and it's common to use the Van Driest damping function to make $C_s \to 0$ at the wall.
Here, $A^+ \approx 25$. This drawback of "needing to adjust $C_s$ according to the flow" was the motivation for developing the dynamic Smagorinsky model.
Strengths and Weaknesses of the Smagorinsky Model
Could you summarize the strengths and weaknesses?
| Strengths | Weaknesses | ||
|---|---|---|---|
| Extremely simple implementation | $C_s$ is a problem-dependent parameter | ||
| Low computational cost | Excessive dissipation near walls (requires Van Driest damping) | ||
| Numerically stable (always $\nu_{sgs} \geq 0$) | Generates dissipation even in laminar/transition regions (if $ | \bar{S} | \neq 0$ then $\nu_{sgs} > 0$) |
| Long history and abundant validation cases | Cannot represent inverse energy cascade (backscatter) |
The Reason the Smagorinsky Model Was Born—The Computational Cost Problem in Weather Forecasting
The paper published by Joseph Smagorinsky in 1963 was originally research for numerical weather prediction. At that time, it was computationally impossible to resolve all turbulence scales smaller than the grid of atmospheric general circulation models. Hence, the idea emerged: "Let's represent the effects of small scales with a local eddy viscosity." The Smagorinsky constant $C_s \approx 0.17$ was calibrated from atmospheric boundary layer data, so applying it to engines or building wind environments is somewhat of a stretch.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, the flow becomes steady, right? This "period of change" is described by the temporal term. The pulsation of blood flow due to heartbeats, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks 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. Warm air from a heater reaching the far corner of a room is also because the air, as a "carrier," 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, it naturally mixes after a while. 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. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re number flows, convection overwhelms and diffusion plays a minor role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, 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 provides 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 occurs 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: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from 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 the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (viscosity model needed for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach number ≥ 0.3, consider compressibility effects
- Boussinesq Approximation (Natural Convection): Consider density variation only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (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 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 | Note confusion with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to timestep stability |
Numerical Methods and Implementation
Implementation Details
When implementing the Smagorinsky model in code, what are the specific steps?
For each timestep and each cell, calculate the following.
1. Calculate the filtered velocity gradient $\partial \bar{u}_i / \partial x_j$
2. Calculate the strain rate tensor $\bar{S}_{ij}$
3. Calculate $|\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}}$
4. Calculate the filter width $\Delta$ (e.g., cube root of cell volume)
5. Compute the SGS eddy viscosity $\nu_{sgs} = (C_s \Delta)^2 |\bar{S}|$
6. Reflect it in the diffusion term of the momentum equation as the effective viscosity $\nu_{eff} = \nu + \nu_{sgs}$
No additional transport equations for the SGS model are required, so the implementation is very simple.
Filter Width Calculation Methods
There are several types of filter width $\Delta$ calculation methods, right?
Correct. Representative filter width definitions are as follows.
| Definition | Formula | Characteristics |
|---|---|---|
| Volume-based | $\Delta = V_{cell}^{1/3}$ | Most common, suitable for unstructured grids |
| Maximum edge length | $\Delta = \max(\Delta x, \Delta y, \Delta z)$ | Conservative, overestimates for high aspect ratios |
| Geometric mean | $\Delta = (\Delta x \cdot \Delta y \cdot \Delta z)^{1/3}$ | Used for structured grids |
For cells with high aspect ratios (e.g., prism layers near walls), the filter width definition significantly affects the results.
Implementation in OpenFOAM
How do you set up the Smagorinsky model in OpenFOAM?
Write the following in constant/turbulenceProperties.
```
simulationType LES;
{
LESModel Smagorinsky;
turbulence on;
printCoeffs on;
delta cubeRootVol;
SmagorinskyCoeffs
{
Ck 0.094;
Ce 1.048;
}
}
```
In OpenFOAM, it's defined with $C_k$ and $C_e$ instead of $C_s$, with the relation $C_s = C_k^{3/4}/\pi$.
Setting in Ansys Fluent
What about in Fluent?
In Fluent, select LES in the Viscous Model dialog and choose the Smagorinsky-Lilly Model as the SGS model. The Smagorinsky constant is set to $C_s = 0.1$ by default and can be changed as needed. Van Driest damping near walls is applied automatically, so no special settings are required.
Considerations for Numerical Stability
Are there any numerical considerations when using the Smagorinsky model?
The Smagorinsky model guarantees $\nu_{sgs} \geq 0$ at all times, making it very stable numerically. This is a major advantage, especially for beginners starting with LES, as it can be used with confidence. However, conversely, this also means it cannot represent inverse energy cascade (energy transfer from small to large scales) at all, which is a physical limitation.
So there's a trade-off between stability and physical accuracy.
Exactly. When using the Smagorinsky model in practice, if you carefully choose the value of $C_s$ and properly set wall damping, you can obtain sufficient accuracy for many industrial problems.
SGS Model Implementation in "Just 5 Lines"
The implementation of the Smagorinsky model is surprisingly simple. Calculate the strain rate tensor $\bar{S}_{ij}$, find its second invariant $|\bar{S}|$, and add $(C_s \Delta)^2 |\bar{S}|$ as eddy viscosity to the diffusion term. Looking at OpenFOAM's source code, it indeed completes in just a few dozen lines. Yet, despite criticism for being "too simple," it has remained in active use for over 60 years because of its overwhelming practicality in terms of low computational cost and ease of implementation. Simplicity is strength.
Upwind Scheme
1st-order upwind: Large numerical diffusion but stable. 2nd-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
2nd-order accurate, but numerical oscillations occur for Pe > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via 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)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is judged when residuals for the Continuity Equation, momentum, and energy each drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
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