Reynolds Stress Model (RSM)
Theory and Physics
Overview
Professor, what's the difference between the Reynolds Stress Model (RSM) and other RANS models?
The biggest difference is that it does not use the eddy viscosity hypothesis (Boussinesq assumption). Models like k-epsilon or k-omega relate the Reynolds stress $\overline{u_i'u_j'}$ to the mean strain rate via the eddy viscosity $\mu_t$. RSM does not employ this assumption and instead directly solves transport equations for each of the six components of the Reynolds stress tensor.
Six components? Does that mean six more equations?
The Reynolds stress tensor is symmetric, so it has six independent components. Adding the equation for $\varepsilon$ (Dissipation Rate) gives a total of seven additional equations to solve. Compared to two-equation models (like k-omega), the computational cost increases by 2 to 3 times.
Governing Equations
Could you show me the specific equations?
The transport equation for the Reynolds stress $R_{ij} = \overline{u_i'u_j'}$ is as follows.
Let's summarize the meaning of each term.
| Term | Expression | Physical Meaning |
|---|---|---|
| Production $P_{ij}$ | $-R_{ik}\frac{\partial U_j}{\partial x_k} - R_{jk}\frac{\partial U_i}{\partial x_k}$ | Production by mean velocity gradient (can be calculated exactly) |
| Pressure-Strain $\Pi_{ij}$ | $\overline{p'\left(\frac{\partial u_i'}{\partial x_j}+\frac{\partial u_j'}{\partial x_i}\right)}$ | Redistribution of energy among components (requires modeling) |
| Turbulent Diffusion $D_{ij}^T$ | $-\frac{\partial}{\partial x_k}\overline{u_i'u_j'u_k'}$ | Transport by turbulence (requires modeling) |
| Viscous Diffusion $D_{ij}^\nu$ | $\nu\nabla^2 R_{ij}$ | Diffusion by molecular viscosity (exact) |
| Dissipation $\varepsilon_{ij}$ | $2\nu\overline{\frac{\partial u_i'}{\partial x_k}\frac{\partial u_j'}{\partial x_k}}$ | Viscous dissipation (requires modeling) |
So not everything can be calculated exactly.
Exactly. The production term $P_{ij}$ and viscous diffusion can be calculated in closed form, but the pressure-strain term, turbulent diffusion term, and dissipation tensor require modeling. The accuracy of RSM heavily depends on the quality of these models.
Pressure-Strain Term Models
What kinds of models are there for the pressure-strain term?
Let me list some representative models.
| Model | Proposer | Features |
|---|---|---|
| LRR (Linear Return to Isotropy) | Launder, Reece, Rodi (1975) | Linear model. Standard for industrial applications. |
| SSG (Speziale-Sarkar-Gatski) | Speziale et al. (1991) | Quadratic nonlinear model. Improved accuracy. |
| GL (Gibson-Launder) | Gibson, Launder (1978) | Accounts for wall reflection effects. |
In the LRR model, $\Pi_{ij}$ is decomposed as follows.
$\Pi_{ij,1}$ is the slow return term (effect of returning anisotropy to isotropy), $\Pi_{ij,2}$ is the rapid term (redistribution by mean strain), and $\Pi_{ij,w}$ is the wall reflection term.
The Luxury of Solving 7 Equations — Why RSM is the "Pinnacle of Theory"
The Reynolds Stress Model (RSM) solves individual transport equations for the six stress components $\overline{u_i u_j}$ and the dissipation rate ε, making it the most rigorous model within RANS. By not using the eddy viscosity hypothesis, it can naturally represent turbulence anisotropy caused by swirling flows, curvature, and buoyancy. The trade-off is computational cost: it requires 2-3 times more memory and computation time than two-equation models, and convergence is difficult. While there are real-world scenarios like secondary flows in turbomachinery or cyclone separator design where "RSM is absolutely necessary," it also serves as a prime example that "a more accurate model is not necessarily the model you should always use."
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine turning 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—these are all 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 drops significantly, trying a steady-state solution first is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? 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 speed increases, this term rapidly strengthens, making it difficult to control. 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 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 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 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 $S_\phi$: Heated air rises—why? Because it becomes lighter (less dense) 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 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, forgetting to include buoyancy means the fluid won't move at all—a physically impossible result where warm air doesn't rise in a heated winter room.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path of molecules ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (viscosity model needed for non-Newtonian fluids)
- 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 variation 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, be careful with 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 $\nu = \mu/\rho$ [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 time step stability. |
Numerical Methods and Implementation
Discretization and Solution in FVM
How do you solve the seven equations of RSM?
In a segregated solver, the six Reynolds stress equations and one $\varepsilon$ equation are solved iteratively in sequence. An RSM update step is inserted between the momentum and pressure equations.
1. Solve the momentum equations (using the stress tensor directly from $R_{ij}$, not via eddy viscosity)
2. Solve the pressure correction equation (SIMPLE/PISO)
3. Solve the six Reynolds stress equations
4. Solve the $\varepsilon$ equation
5. Convergence Check, iterate
What do you mean by using Reynolds stress directly in the momentum equations?
In the k-epsilon model, the eddy viscosity hypothesis $-\overline{u_i'u_j'} = \mu_t(\partial U_i/\partial x_j + \partial U_j/\partial x_i) - (2/3)k\delta_{ij}$ treats it as a diffusion-like term. In RSM, $\overline{u_i'u_j'}$ is inserted directly as a source term into the momentum equations. This allows anisotropic effects to be accurately reflected.
Numerical Stability Challenges
I've heard RSM is difficult to converge.
Because it solves seven strongly coupled nonlinear equations, convergence is significantly more difficult compared to k-epsilon. Let me list the main problems and countermeasures.
| Problem | Cause | Countermeasure |
|---|---|---|
| Reynolds Stress Realizability Violation | $R_{ij}$ ceases to be positive definite symmetric | Apply Realizability constraint |
| Overestimation of $\varepsilon$ | Excessive dissipation near walls | Use $\omega$-based RSM (BSL-RSM) |
| Slow Convergence | Strong coupling between equations | Set Under-relaxation to 0.3–0.5 |
| Sensitivity to Initial Values | Initial $R_{ij}$ is non-physical | Initialize with k-epsilon results |
As a practical TIP, when starting an RSM calculation, the golden rule is to first obtain a converged solution with k-epsilon or SST k-omega, then use that result to initialize the RSM.
Omega-based RSM
Are there RSM variants that use $\omega$ instead of $\varepsilon$?
Yes. The BSL-RSM (Baseline RSM) proposed by Menter uses the $\omega$ equation from SST k-omega instead of the $\varepsilon$ equation. This improves near-wall behavior and works well with wall functions.
| RSM Variant | Dissipation Equation | Wall Treatment | Solver Support |
|---|---|---|---|
| LRR-RSM | $\varepsilon$ | Low-Re or Wall Function | Fluent, CFX, OpenFOAM |
| SSG-RSM | $\varepsilon$ | Low-Re | Fluent, CFX, STAR-CCM+ |
| BSL-RSM ($\omega$-based) | $\omega$ | Automatic WT | CFX, Fluent |
| LRR-RSM-w | $\omega$ | Wall Function Compatible | OpenFOAM |
Is BSL-RSM recommended in CFX?
In CFX, BSL-RSM is the default RSM. It can use the same wall treatment as SST k-omega (Automatic Wall Treatment), making it robust with respect to mesh $y^+$.
Scenes Where RSM Achieves a Come-from-Behind Victory in Cyclone Separator Design
In the analysis of cyclone separators (devices that separate particles or droplets via centrifugal force), it is known that standard k-ε or SST models can significantly mispredict separation efficiency. This is because strong swirling flows cause severe anisotropy in turbulent stresses, breaking down the eddy viscosity hypothesis. When RSM is applied to this problem, many cases report that swirl velocity distribution and pressure drop align much more closely with experimental values. This is why some companies in chemical plant cyclone design use RSM as standard. "For specific flows, only RSM works"—this reality justifies the high cost.
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
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 or 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. SPH and other mesh-fr
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