Dynamic Smagorinsky Model
Theory and Physics
Overview
Professor, what aspect of the Smagorinsky model did the dynamic Smagorinsky model improve?
The biggest problem with the Smagorinsky model is that it uses a fixed value for the model constant $C_s$, even though it depends on the flow. The dynamic Smagorinsky model (Germano et al., 1991; Lilly, 1992) is a method to determine $C_s$ locally and dynamically during the computation. It utilizes a mathematical identity called the Germano Identity.
Germano Identity
What is the Germano Identity?
It is an identity concerning two-stage filtering: the grid filter $\bar{\phantom{u}}$ (width $\Delta$) and the test filter $\hat{\phantom{u}}$ (width $\hat{\Delta} = 2\Delta$).
This $L_{ij}$ (Leonard stress tensor) can be calculated directly from known resolved-scale quantities. On the other hand, from the consistency condition with the model expression for the SGS stress,
Dynamic Calculation of $C_s^2$
How is $C_s^2$ determined?
Use Lilly's (1992) least squares method.
$\langle \cdot \rangle$ represents averaging in spatial directions (homogeneous directions) or Lagrangian averaging. Without this averaging, $C_s^2$ can become locally negative or oscillate violently, making the computation unstable.
What is the physical meaning of $C_s^2$ becoming negative?
A negative $C_s^2$ means backscatter, i.e., reverse energy transport from the SGS scale to the resolved scale. This is a physically possible phenomenon, but numerically it can cause instability. In implementation, the lower limit of $C_s^2$ is clipped to zero, or negative values are suppressed using Lagrangian averaging.
Advantages of the Dynamic Model
What are the specific advantages of the dynamic model?
| Advantage | Explanation |
|---|---|
| $C_s \to 0$ at walls | Automatically gives correct wall behavior without Van Driest damping |
| Reproduction of transitional flow | $C_s \to 0$ in laminar regions, avoiding excessive dissipation |
| Adaptation to different Re numbers | No need for prior tuning of constants |
| Partial reproduction of backscatter | Physically allows $C_s^2 < 0$ |
The Revolutionary Nature of Germano's Idea: "Let the Flow Itself Determine the Constant"
The revolutionary aspect of the dynamic procedure proposed by Massimo Germano in 1991 lies in the idea that "the SGS constant does not need to be determined by humans." By solving the Germano identity using a test filter and determining the locally optimal $C_s$ from the flow field itself—this idea fundamentally changed the conventional thinking of "calibrating model parameters through experiments." However, the initial paper had the problem of the constant becoming negative and causing divergence in the computation. The following year, Lilly (1992) stabilized it using averaging via the least squares method, giving it a practical form.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, 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 due to heartbeats, or the fluctuation of flow 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 this term is set to zero. Since computational cost is significantly reduced, 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 current, right? This is "convection"—the effect where fluid motion transports objects. The warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term includes "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 things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever added milk to coffee and left it? Even without stirring, after a while, it naturally mixes, right? That is molecular diffusion. Now, next question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, 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 overwhelms, and diffusion plays a supporting 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 becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are densely packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD often refers to gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, it might be due to confusion between absolute/gauge pressure.
- Source term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed upward 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 the source term. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result, like turning on a heater in a winter room but the warm air doesn't rise.
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 numbers above 0.3, 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 flow (requires shock capturing), free surface flow (requires VOF/Level Set, etc.)
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
Test Filter Implementation
How exactly is the test filter implemented?
For unstructured grids, the average (volume-weighted) of cell center values from neighboring cells is often used as the test filter. For structured grids, top-hat filters or Gaussian filters can be used.
| Filter Type | Implementation Method | Accuracy |
|---|---|---|
| Top-hat (Box) | Simple average of neighboring cells | Low (problematic on non-uniform grids) |
| Volume-weighted average | $\hat{\phi}_P = \sum_f V_f \phi_f / \sum_f V_f$ | Medium |
| Gaussian | Gaussian weights based on distance | High (suited for structured grids) |
Stabilization Techniques
How do you deal with the problem of $C_s^2$ becoming negative and causing instability?
There are mainly three approaches.
1. Clipping: Clip to zero when $C_s^2 < 0$. Simplest but loses physical backscatter.
2. Spatial averaging: Average $\langle L_{ij}M_{ij}\rangle$ in homogeneous directions (e.g., spanwise direction). Suitable for channel flow.
3. Lagrangian averaging (Meneveau et al. 1996): Time average along fluid particle paths. Applicable to non-homogeneous flows.
The Lagrangian dynamic model is the most versatile and is implemented in OpenFOAM as dynLagrangian. It is also available in Fluent as the Dynamic Smagorinsky-Lilly model.
Solver Settings
Please tell me how to set it up in each solver.
The dynamic Smagorinsky is theoretically the most elegant method among LES SGS models, and its major strength is that it requires no constant tuning. However, the trade-off is implementation complexity and cost (test filter computation).
The Problem: Is "Twice" the Right Size for the Test Filter Width?
The test filter width $\hat{\Delta}$ in the dynamic procedure is conventionally set to $\hat{\Delta} = 2\Delta$, but many people are at a loss for a clear answer when asked why it's twice. Theoretically, it's "the ratio that yields the maximum information within the range where inertial subrange scale similarity holds," but in reality, it's a rather empirical choice. Studies have tried $\hat{\Delta} = 3\Delta$ or $4\Delta$, and cases of grid dependence have been reported. Instead of "just setting it to twice," we recommend conducting sensitivity tests.
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 > 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 multi-physics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when the residuals for each of the continuity equation, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2~0.3, Velocity: 0.5~0.7. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5~20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it reflects the physics that upstream information determines downstream. Accuracy is first-order, but it is highly stable because it correctly captures the flow direction.
Practical Guide
Applicability
In what situations should the dynamic Smagorinsky model be used?
| Suitable Applications | Reason |
|---|---|
| Transitional flow (flow including laminar→turbulent transition) | $C_s$ automatically becomes zero in laminar regions |
| Industrial LES with complex geometry | No tuning required, versatile |
| LES resolving near-wall regions | Correct wall behavior without Van Driest damping |
| Systematic studies at different Re numbers | Constants adjust automatically |
How much does the computational cost increase compared to the standard Smagorinsky model?
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