Compressible Turbulence Modeling
Compressible Turbulence Modeling: Theoretical Foundations
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.
Computational Methods for Compressible Turbulence Modeling
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.
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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.
Experience the theory firsthand with the interactive simulator for this field
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