Compressible Turbulence Modeling
Theory and Physics
Fundamental Theory of Compressible Turbulence
Professor, what's the difference between compressible turbulence and incompressible turbulence? Can't we just use models like k-epsilon as-is?
Good question. In compressible turbulence, density fluctuations can no longer be ignored. For incompressible flows, we could directly use Reynolds decomposition $u_i = \bar{u}_i + u_i'$, but for compressible flows, we need to introduce Favre averaging (density-weighted averaging).
Favre averaging is defined as follows.
Using this, the mass conservation equation takes the same form as in the incompressible case, making it easier to handle.
I see, so you incorporate the density fluctuations into the averaging. But what specific additional terms appear?
RANS Equations via Favre Averaging
The Favre-averaged momentum equation takes the following form.
Here, $u_i'' = u_i - \tilde{u}_i$ is the Favre fluctuation component. The last term corresponds to the Reynolds stress tensor.
The Reynolds stress part looks the same as in the incompressible case. So what about the turbulent kinetic energy equation?
The transport equation for the Favre-averaged turbulent kinetic energy $\tilde{k} = \widetilde{u_i''u_i''}/2$ contains terms specific to compressibility that are not present in the incompressible case.
The important terms here are the pressure-dilation correlation $\Pi_d = \overline{p'\frac{\partial u_k''}{\partial x_k}}$ and the compressible dissipation $\varepsilon_c$ (dilatational dissipation).
So $\Pi_d$ and $\varepsilon_c$ are the terms unique to compressibility. Under what conditions do these become non-negligible?
Compressibility Correction Models
These terms become significant when the turbulent Mach number $M_t = \sqrt{2k}/a$ (where $a$ is the speed of sound) becomes large. Specifically, their influence becomes noticeable from around $M_t > 0.3$. Let me introduce some representative compressibility correction models.
| Model | Compressible Dissipation | Application Range |
|---|---|---|
| Sarkar (1992) | $\varepsilon_c = \alpha_1 \bar{\rho} \varepsilon M_t^2$ | Free shear flows, mixing layers |
| Zeman (1990) | $\varepsilon_c = \alpha_2 \bar{\rho} \varepsilon f(M_t)$ | Flows including shock waves |
| Wilcox (1992) | Compressibility modification for $k$-$\omega$ | General compressible flows |
| SST Compressibility Correction | Modification via $F(M_t)$ function | Addition to Menter SST model |
In the Sarkar model, $\alpha_1 \approx 1.0$ is used, and it can well reproduce the reduction in spreading rate in high Mach number mixing layers. Experiments show that as $M_c$ (convective Mach number) increases, the growth rate of the mixing layer decreases significantly. To capture this correctly, compressibility corrections are essential.
I'll remember to be cautious when the turbulent Mach number exceeds 0.3. Professor, does this relate to things like supersonic jet noise prediction?
Exactly. In supersonic jet noise prediction, accurate modeling of compressible turbulence directly affects the estimation accuracy of source strength. This field is actively researched at institutions like NASA as well.
"Density Fluctuations" in Compressible Turbulence—They Actually Consume Turbulent Energy
In subsonic turbulence, density is almost constant, so "you only need to worry about velocity fluctuations." However, when the Mach number exceeds about 0.3, density fluctuations can no longer be ignored. According to Morkovin's hypothesis, the influence of density fluctuations is small at low to moderate Mach numbers, but the story changes in the hypersonic region above Mach 5. As shown by Sarkar et al. in the 1990s, an additional energy dissipation pathway arises from the "interaction between pressure and dilatation" besides viscous dissipation. Ignoring this term leads to overestimation of turbulent kinetic energy and significant errors in heat flux prediction—a term that can lead to fatal mistakes in the thermal design of high-speed vehicles.
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, splashing 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 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 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, solving first in steady-state 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 "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 things" → 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, they naturally mix. 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 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, and 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 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: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange 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 source terms. What happens if you forget a 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 room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular 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 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 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
Details of Numerical Methods
When solving compressible turbulence with CFD, do the numerical schemes change compared to incompressible flows?
They change significantly. In compressible flows, discontinuous surfaces like shock waves appear, so upwind schemes become essential for discretizing the convection term. Let's organize some representative ones.
| Scheme | Characteristics | Accuracy | Notes for Compressible Turbulence |
|---|---|---|---|
| Roe | Approximate Riemann solver | 2nd order (with MUSCL) | Excessive dissipation at low Mach numbers |
| AUSM+ | Mass flux splitting type | 2nd order or higher | Handles both low and high Mach numbers |
| HLLC | 3-wave approximate Riemann | 2nd order | Good resolution of contact discontinuities |
| Central Difference + Artificial Viscosity | Jameson type | 2nd order | Suitable for LES, dissipation control is key |
What about when solving compressible turbulence with LES? Does the choice of scheme change?
LES/DES for Compressible Turbulence
In LES (Large Eddy Simulation), we solve the Favre-filtered Navier-Stokes equations, where grid filtering is performed with density weighting. SGS (Sub-Grid Scale) models also need to be compatible with compressibility.
$$ \bar{\rho} \frac{\partial \tilde{u}_i}{\partial t} + \bar{\rho} \tilde{u}_j \frac{\partial \tilde{u}_i}{\partial x_j} = -\frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}(\bar{\tau}_{ij} - \bar{\rho}\tau_{ij}^{sgs}) $$
For SGS models, compressible extensions of the Smagorinsky model or the WALE (Wall-Adapting Local Eddy-viscosity) model are used. Also, in DES (Detached Eddy Simulation), we switch between a compressibility-corrected SST model in the RANS region and an SGS model in the LES region.
The balance between numerical dissipation and physical dissipation seems important. Are there guidelines for grid resolution?
Near walls, $y^+ < 1$ is ideal, but in compressible flows, mesh density near shock waves is also important. Resolving down to the shock wave thickness (on the order of mean free path) is unnecessary, but we want at least 5-10 cells across the shock front. Managing the Courant number is also crucial; for explicit methods, $\text{CFL} < 1$ must be strictly observed.
Time Integration Methods
Are there special time integration methods for compressible flows?
For RANS where a steady solution is desired, local time stepping or implicit LU-SGS methods are efficient. For unsteady LES/DES, dual time stepping or explicit Runge-Kutta methods (3-stage or 4-stage) are standard.
$$ \frac{\partial \mathbf{U}}{\partial \tau} + \frac{3\mathbf{U}^{n+1} - 4\mathbf{U}^n + \mathbf{U}^{n-1}}{2\Delta t} + \mathbf{R}(\mathbf{U}^{n+1}) = 0 $$
This is the equation for dual time stepping, where $\tau$ is pseudo-time and $\mathbf{R}$ is the spatial residual. It maintains second-order accuracy in physical time while converging implicitly in pseudo-time.
I see, so we need to choose appropriately between steady and unsteady. Are there any implementation pitfalls?
The most common mistake is forgetting to enable compressibility corrections. For example, in Fluent's k-epsilon model, the "Compressibility Effects" checkbox is off by default. Forgetting this in a high Mach number mixing layer calculation leads to a significant overprediction of the spreading rate.
Coffee Break Trivia
Morkovin Hypothesis—Why Turbulence Models "Transform" Near the Speed of Sound
In 1962, Mark Morkovin proposed a bold hypothesis: "If density fluctuations are small, even in compressible turbulence, incompressible turbulence models should be usable as-is." This is the Morkovin hypothesis. In practice, it works fairly well up to about Mach 5, but beyond Mach 5, density fluctuations become non-negligible and the hypothesis breaks down. In the field, you hear anecdotes like "for some reason, only the SST model converged," but that might be less about the model's inherent superiority and more because the calculation happened to be in a speed range where the Morkovin hypothesis coincidentally holds.
Upwind Differencing (Upwind)
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 (Central Differencing)
2nd-order accurate, but numerical oscillations occur for Peclet number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy 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 time step.
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In LES (Large Eddy Simulation), we solve the Favre-filtered Navier-Stokes equations, where grid filtering is performed with density weighting. SGS (Sub-Grid Scale) models also need to be compatible with compressibility.
For SGS models, compressible extensions of the Smagorinsky model or the WALE (Wall-Adapting Local Eddy-viscosity) model are used. Also, in DES (Detached Eddy Simulation), we switch between a compressibility-corrected SST model in the RANS region and an SGS model in the LES region.
The balance between numerical dissipation and physical dissipation seems important. Are there guidelines for grid resolution?
Near walls, $y^+ < 1$ is ideal, but in compressible flows, mesh density near shock waves is also important. Resolving down to the shock wave thickness (on the order of mean free path) is unnecessary, but we want at least 5-10 cells across the shock front. Managing the Courant number is also crucial; for explicit methods, $\text{CFL} < 1$ must be strictly observed.
Are there special time integration methods for compressible flows?
For RANS where a steady solution is desired, local time stepping or implicit LU-SGS methods are efficient. For unsteady LES/DES, dual time stepping or explicit Runge-Kutta methods (3-stage or 4-stage) are standard.
This is the equation for dual time stepping, where $\tau$ is pseudo-time and $\mathbf{R}$ is the spatial residual. It maintains second-order accuracy in physical time while converging implicitly in pseudo-time.
I see, so we need to choose appropriately between steady and unsteady. Are there any implementation pitfalls?
The most common mistake is forgetting to enable compressibility corrections. For example, in Fluent's k-epsilon model, the "Compressibility Effects" checkbox is off by default. Forgetting this in a high Mach number mixing layer calculation leads to a significant overprediction of the spreading rate.
Morkovin Hypothesis—Why Turbulence Models "Transform" Near the Speed of Sound
In 1962, Mark Morkovin proposed a bold hypothesis: "If density fluctuations are small, even in compressible turbulence, incompressible turbulence models should be usable as-is." This is the Morkovin hypothesis. In practice, it works fairly well up to about Mach 5, but beyond Mach 5, density fluctuations become non-negligible and the hypothesis breaks down. In the field, you hear anecdotes like "for some reason, only the SST model converged," but that might be less about the model's inherent superiority and more because the calculation happened to be in a speed range where the Morkovin hypothesis coincidentally holds.
Upwind Differencing (Upwind)
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 (Central Differencing)
2nd-order accurate, but numerical oscillations occur for Peclet number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy 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 time step.
Related Topics
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