Could you provide the specific equations?

The two additional transport equations are as follows: Transport equation for intermittency $\gamma$: $$ \frac{\partial(\rho\gamma)}{\partial t} + \nabla\cdot(\rho\mathbf{u}\gamma) = P_\gamma - E_\gamma + \nabla\cdot\left[(\mu + \frac{\mu_t}{\sigma_\gamma})\nabla\gamma\right] $$ Transport equation for the transition momentum thickness Reynolds number $\widetilde{Re}_{\theta t}$: $$ \frac{\partial(\rho\widetilde{Re}_{\theta t})}{\partial t} + \nabla\cdot(\rho\mathbf{u}\widetilde{Re}_{\theta t}) = P_{\theta t} + \nabla\cdot\left[\sigma_{\theta t}(\mu + \mu_t)\nabla\widetilde{Re}_{\theta t}\right] $$

Transition Model (γ-Reθ)

Q: Professor, how is the transition from laminar to turbulent flow handled in CFD? Models like k-epsilon and SST k-omega assume fully turbulent flow everywhere, don't they?

That's correct. Standard RANS models assume fully turbulent flow, which leads to an overprediction of drag in laminar regions and an inability to capture the transition location. To address this, the $\gamma$-$Re_\theta$ transition model (Langtry-Menter, 2009) was developed. It adds two transport equations—one for the intermittency $\gamma$ and one for the transition momentum thickness Reynolds number $\widetilde{Re}_{\theta t}$—to the SST k-omega model.

Q: What is intermittency?

$\gamma$ takes a value between 0 and 1, where $\gamma = 0$ represents fully laminar flow and $\gamma = 1$ represents fully turbulent flow. In the transition region, $0 < \gamma < 1$.

Q: How does $\gamma$ affect the turbulence model?

$\gamma$ modifies the production term in the $k$ equation of the SST k-omega model. $$ P_k^{\text{eff}} = \gamma_{\text{eff}} \cdot P_k $$ In laminar regions ($\gamma = 0$), turbulence production becomes zero, keeping $k$ at a low value. This suppresses the eddy viscosity, resulting in the reproduction of a laminar-like velocity profile.

Category: 流体解析（CFD） | Integrated 2026-04-06

CAE visualization for transition model theory - technical simulation diagram

遷移モデル（γ-Reθ）

Theory and Physics

Overview

🧑‍🎓

Professor, how is the transition from laminar to turbulent flow handled in CFD? With k-epsilon or SST k-omega, it's fully turbulent everywhere, right?

🎓

Exactly. Standard RANS models assume fully turbulent flow, so they overpredict drag in laminar regions and cannot reproduce the transition location. That's why the $\gamma$-$Re_\theta$ transition model (Langtry-Menter, 2009) was developed. It adds two transport equations for the intermittency $\gamma$ and the transition momentum thickness Reynolds number $\widetilde{Re}_{\theta t}$ to the SST k-omega model.

🧑‍🎓

What is intermittency?

🎓

$\gamma$ takes a value from 0 to 1, where $\gamma = 0$ represents fully laminar flow and $\gamma = 1$ represents fully turbulent flow. In the transition region, $0 < \gamma < 1$, representing a state where laminar and turbulent flows are locally mixed.

Governing Equations

🧑‍🎓

Please tell me the specific equations.

🎓

The two additional transport equations are as follows.

Transport equation for intermittency $\gamma$:

$$ \frac{\partial(\rho\gamma)}{\partial t} + \nabla\cdot(\rho\mathbf{u}\gamma) = P_\gamma - E_\gamma + \nabla\cdot\left[(\mu + \frac{\mu_t}{\sigma_\gamma})\nabla\gamma\right] $$

Transport equation for transition momentum thickness Reynolds number $\widetilde{Re}_{\theta t}$:

$$ \frac{\partial(\rho\widetilde{Re}_{\theta t})}{\partial t} + \nabla\cdot(\rho\mathbf{u}\widetilde{Re}_{\theta t}) = P_{\theta t} + \nabla\cdot\left[\sigma_{\theta t}(\mu + \mu_t)\nabla\widetilde{Re}_{\theta t}\right] $$

🧑‍🎓

How does $\gamma$ affect the turbulence model?

🎓

$\gamma$ modifies the production term in the $k$ equation of the SST k-omega model.

$$ P_k^{\text{eff}} = \gamma_{\text{eff}} \cdot P_k $$

In laminar regions ($\gamma = 0$), turbulence production becomes zero, keeping $k$ at a low value. This suppresses eddy viscosity, resulting in the reproduction of a laminar-like velocity profile.

Physical Mechanisms of Transition

🧑‍🎓

What specific types of transition are there?

🎓

The main transition mechanisms are as follows.

Transition Type	Physical Mechanism	Typical Scenario
Natural Transition	Growth of Tollmien-Schlichting waves	Airfoil surfaces with low turbulence intensity
Bypass Transition	High freestream turbulence directly disturbs the boundary layer	Gas turbine blade rows
Separation-Induced Transition	Transition occurs within a laminar separation bubble	Low-Re airfoils, compressor blades
Crossflow Instability Transition	Instability of the crossflow velocity profile in a 3D boundary layer	Leading edge of swept wings

🎓

The $\gamma$-$Re_\theta$ model mainly covers natural and bypass transitions. It also handles separation-induced transition to some extent, but additional correlation functions are needed for crossflow instability transition.

Coffee Break Yomoyama Talk

"The Moment It Becomes Turbulent" — A 100-Year Question in Transition Research

The "Transition" from laminar to turbulent flow is one of the oldest research themes among the unsolved problems in fluid mechanics. Starting from Reynolds' experiments in the 19th century, followed by the discovery of Tollmien–Schlichting waves (1929) and the recognition of bypass transition (1960s), there is still no universal model to predict "under what conditions and from where turbulence begins." The γ-Reθ model is one of the most robust engineering solutions within these constraints, a practical compromise that "forces transition into the framework of turbulence models" using four equations.

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, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow due to 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 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 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" → 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 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. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick, sluggish" 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? 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 densely 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 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 source terms. What happens if you forget a source term? In natural convection analysis, forgetting to include 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 (mean free path ≪ characteristic length)
Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (viscosity model required for non-Newtonian fluids)
Incompressibility assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered.
Boussinesq approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used 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	$Re = \rho u L / \mu$. Criterion for laminar/turbulent transition.
CFL number	Dimensionless	$CFL = u \Delta t / \Delta x$. Directly related to time step stability.

Numerical Methods and Implementation

Transition Correlation Functions

🧑‍🎓

How is it determined when transition begins?

🎓

The core of the $\gamma$-$Re_\theta$ model is the empirical correlation function. It determines the critical Reynolds number $Re_{\theta c}$ from the freestream turbulence intensity $Tu$ and the pressure gradient parameter $\lambda_\theta$.

$$ Re_{\theta c} = f(Tu, \lambda_\theta) $$

🎓

This correlation is based on experimental correlations like Abu-Ghannam-Shaw (1980) and Mayle (1991). The specific functional form is published in Menter et al.'s papers.

Parameter	Effect
Freestream Turbulence Intensity $Tu$	Higher values move transition upstream ($Re_{\theta c}$ decreases)
Pressure Gradient $\lambda_\theta$	Favorable pressure gradient delays transition, adverse pressure gradient promotes it

Role of the $\widetilde{Re}_{\theta t}$ Equation

🧑‍🎓

What is the purpose of the $\widetilde{Re}_{\theta t}$ equation? Isn't $\gamma$ alone enough?

🎓

Good question. The transition correlation function requires the freestream $Tu$ and $\lambda_\theta$ as inputs. However, in CFD, it is technically difficult to refer to freestream values near the wall (non-local information is needed).

🎓

The transport equation for $\widetilde{Re}_{\theta t}$ is a mechanism to solve this non-locality problem. It calculates the correlation function value in the freestream and diffuses it to the near-wall region via the transport equation. This allows access to freestream Tu information even near the wall.

🧑‍🎓

So $\widetilde{Re}_{\theta t}$ is like a "postman" delivering freestream information to the wall?

🎓

Exactly that image.

Coupling with SST k-omega

🧑‍🎓

In terms of implementation, is it added to the SST k-omega model?

🎓

Exactly. Transition SST ($\gamma$-$Re_\theta$ SST) solves the following four equations simultaneously.

1. $k$ equation (SST k-omega, but production term modified by $\gamma$)

2. $\omega$ equation (SST k-omega, unchanged)

3. $\gamma$ equation (intermittency)

4. $\widetilde{Re}_{\theta t}$ equation (transition Re number)

Mesh Requirements

🧑‍🎓

Does the transition model require a finer mesh than standard RANS?

🎓

Higher resolution than standard RANS is needed to correctly capture the transition region.

Parameter	SST k-omega (Wall Function)	Transition SST
Wall $y^+$	30〜100	1 or less
Layers within boundary layer	8〜15	20〜30
Streamwise (chordwise)	100 cells/chord	200〜300 cells/chord
Spanwise	--	Finer if solving 3D structure of transition front

🧑‍🎓

$y^+ \approx 1$ is mandatory because transition is a subtle phenomenon near the wall, right?

Coffee Break Yomoyama Talk

γ-Rθ Transition Model—"RANS Prediction of Transition" Made Practical by Menter (2006)

The Menter-Langtry γ-Reθ transition model (2006, 2009), which predicts the "transition" from laminar to turbulent flow using RANS, is the most widely used transition model in engineering. It adds two transport equations for the transition onset momentum thickness Reynolds number Reθt and the intermittency coefficient γ to the k-ω SST, handling three modes—natural transition, bypass transition, and separation-induced transition—within a single framework. It is implemented as standard in Fluent and OpenFOAM and is used for transition prediction in aircraft wings, gas turbine blades, and wind turbine blades. However, it has high sensitivity to the settings of turbulence intensity (Tu) and length scale (Λ), and the handling of inlet conditions greatly affects accuracy.

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 number > 2. Suitable for low Reynolds number diffusion-dominated flows.

TVD Scheme (MUSCL, QUICK, etc.)

Maintains high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks or 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 method: CFL ≤ 1 is the stability condition. Implicit method: 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.

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

Pressure: 0.2〜0.3, Velocity: 0.5〜0.7 are typical initial values. 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,