What mesh is required to use a Low-Re model?

The first cell centroid from the wall must be within the viscous sublayer, ideally at $y^+ \approx 1$. The required first cell height to achieve $y^+ = 1$ is: $$ \Delta y_1 = \frac{y^+ \mu}{\rho u_\tau} = \frac{\mu}{\rho \sqrt{\tau_w/\rho}} $$ Example: For a flat plate with $U = 10$ m/s and $Re_L = 10^6$, $\Delta y_1 \approx 2 \times 10^{-5}$ m.

Low Reynolds Number Model

Q: Professor! Are low-Reynolds number models the turbulence models that do not use wall functions?

That's correct. Low-Reynolds number (Low-Re) models are designed to resolve the viscous sublayer near the wall (where $y^+ < 5$), so they do not rely on wall functions.

Q: What kind of damping functions are used?

A representative example is the Launder-Sharma (1974) k-ε model, which applies a damping function $f_\mu$ to the eddy viscosity. $$ \mu_t = C_\mu f_\mu \rho \frac{k^2}{\tilde{\varepsilon}} $$ $$ f_\mu = \exp\left(-\frac{3.4}{(1 + Re_t/50)^2}\right), \quad Re_t = \frac{\rho k^2}{\mu \tilde{\varepsilon}} $$ Near the wall, $Re_t \to 0$, so $f_\mu \to \exp(-3.4) \approx 0.033$, making the eddy viscosity nearly zero. In the far field, where $Re_t \gg 50$, $f_\mu \to 1$.

Q: Is the ε equation also modified?

In the Launder-Sharma model, a modified variable $\tilde{\varepsilon} = \varepsilon - 2\nu(\partial\sqrt{k}/\partial y)^2$ is used, imposing the boundary condition $\tilde{\varepsilon} = 0$ at the wall. Damping functions $f_1$ and $f_2$ are also added to the ε equation. Other notable Low-Re models: | Model | Year | Key Feature | | :--- | :--- | :--- | | Jones-Launder | 1972 | First Low-Re k-ε model | | Launder-Sharma | 1974 | Most widely used | | Lam-Bremhorst | 1981 | $Re_y$-based damping functions | | Abe-Kondoh-Nagano | 1994 | Uses Kolmogorov velocity scale | | Yang-Shih | 1993 | Improved near-wall asymptotic behavior |

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

CAE visualization for low re model theory - technical simulation diagram

低Reynolds数モデル

Theory and Physics

Overview

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Teacher! Are low-Reynolds number models the turbulence models that don't use wall functions?

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Exactly. Low-Reynolds number (Low-Re) models are turbulence models with damping functions designed to directly resolve the near-wall region, including the viscous sublayer ($y^+ < 5$). They faithfully represent the physics of turbulence at the wall without using approximations like wall functions.

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What kind of damping functions are used?

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A representative one is the Launder-Sharma (1974) k-ε model, which multiplies the eddy viscosity by a damping function $f_\mu$.

$$ \mu_t = C_\mu f_\mu \rho \frac{k^2}{\tilde{\varepsilon}} $$

$$ f_\mu = \exp\left(-\frac{3.4}{(1 + Re_t/50)^2}\right), \quad Re_t = \frac{\rho k^2}{\mu \tilde{\varepsilon}} $$

At the wall, $Re_t \to 0$, so $f_\mu \to \exp(-3.4) \approx 0.033$, making the eddy viscosity nearly zero. Far away, where $Re_t \gg 50$, $f_\mu \to 1$.

Governing Equations

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Is the ε equation also modified?

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In the Launder-Sharma model, a variable $\tilde{\varepsilon} = \varepsilon - 2\nu(\partial\sqrt{k}/\partial y)^2$ is used, and the boundary condition $\tilde{\varepsilon} = 0$ is imposed at the wall. Damping functions $f_1$, $f_2$ are also added to the ε equation.

Other famous Low-Re models:

Model	Year	Damping Function Feature
Jones-Launder	1972	First Low-Re k-ε
Launder-Sharma	1974	Most widely used
Lam-Bremhorst	1981	$Re_y$ based damping function
Abe-Kondoh-Nagano	1994	Uses Kolmogorov scale
Yang-Shih	1993	Improved near-wall asymptotic behavior

Mesh Requirements

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What kind of mesh is needed to use a Low-Re model?

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A first cell $y^+ < 1$ at the wall is a mandatory condition. At least 5-10 cells within the viscous sublayer ($y^+ < 5$) and a total of 15-30 prism layers up to the log-law layer ($y^+ = 30$-$300$) are required.

First cell height to achieve $y^+ = 1$:

$$ \Delta y_1 = \frac{y^+ \mu}{\rho u_\tau} = \frac{\mu}{\rho \sqrt{\tau_w/\rho}} $$

Example: For a flat plate with $U = 10$ m/s, $Re_L = 10^6$, $\Delta y_1 \approx 2 \times 10^{-5}$ m.

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That requires a very fine mesh. I'm worried about the computational cost.

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Exactly. Low-Re models require 2-5 times more mesh cells compared to using wall functions, and the computational cost scales proportionally. Nowadays, the SST k-ω model can handle both wall-resolved and wall-function approaches, so the use of classical Low-Re k-ε is limited.

Coffee Break Yomoyama Talk

Low-Reynolds Number Turbulence Models—The Need to Solve the "Viscous Sublayer" Near the Wall

The boundary layer near the wall has a three-layer structure: "viscous sublayer (y+<5)", "buffer layer (530)". Standard k-ε accurately solves only the log-law layer and uses a "wall function" to approximate the near-wall region. On the other hand, to precisely predict wall heat transfer, transition, and separation, a "low-Reynolds number turbulence model (Low-Re model)" that resolves down to the viscous sublayer is necessary. Jones-Launder (1972) proposed the first Low-Re k-ε model, explicitly accounting for viscous effects using wall damping functions f_μ, f₁, f₂. Modern k-ω model families (SST, BSL) incorporate low-Re wall treatment, so a separate Low-Re model is often not needed.

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 spluttering and unstable, but after a while, the flow becomes steady, right? This term describes the "state of change." The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens/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. This significantly reduces computational cost, so starting with a steady-state solution 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 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'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. That's molecular diffusion. Now a question—honey and water, which flows easier? 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 flows, convection overwhelmingly dominates, 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 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 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 (lower density) than its surroundings, so it's pushed up 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 heated winter room.

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): Density is treated as constant. For Mach number ≥ 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 gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (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: ~1.225 kg/m³ @20°C, Water: ~998 kg/m³ @20°C
Viscosity coefficient $\mu$	Pa·s	Note confusion 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

Key Points in Numerical Implementation

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What points require special attention in the numerical implementation of Low-Re models?

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Because the mesh near the wall becomes extremely fine, several numerical difficulties arise.

Wall Boundary Condition for ε

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We impose $\tilde{\varepsilon} = 0$ at the wall, right?

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In Launder-Sharma type, $\tilde{\varepsilon} = 0$ (Dirichlet condition). On the other hand, in Jones-Launder type, $\varepsilon_{wall} = 2\nu (\partial\sqrt{k}/\partial y)^2_{wall}$ is set as a finite value. The latter can be easier to handle numerically in some cases.

In OpenFOAM, instead of using epsilonWallFunction, you choose the appropriate one: fixedValue or zeroGradient.

Numerical Stability of Damping Functions

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Do problems ever occur with damping functions?

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Since the calculation of $f_\mu$ involves $Re_t = \rho k^2/(\mu\varepsilon)$, there is a risk of division by zero in regions where $\varepsilon \to 0$. Lower limit clipping ($\varepsilon_{min} > 0$, $k_{min} > 0$) is mandatory.

Also, if the mesh expansion ratio is large in the region $y^+ = 5$-$30$ where the damping function changes rapidly, convergence can deteriorate due to gradient discontinuity. The mesh expansion ratio should be kept to 1.1-1.2.

Settings in Various Solvers

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How are Low-Re models used in commercial solvers?

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Solver	Setting Method	Notes
Fluent	Viscous → k-epsilon → Realizable or RNG + Enhanced Wall Treatment	Automatically handles Low-Re treatment with a two-layer model
CFX	No direct Low-Re k-ε. Use SST + Automatic Wall Treatment as alternative
STAR-CCM+	Select k-epsilon Low-Re	Launder-Sharma, Abe-Kondoh-Nagano, etc.
OpenFOAM	`LaunderSharmaKE`	Wall BCs must be set manually

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So in Fluent, it's provided as Enhanced Wall Treatment.

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Fluent's Enhanced Wall Treatment is not strictly a Low-Re model; it's a blend of a two-layer model (switching to a one-equation model near the wall) and an All $y^+$ wall function. It effectively provides Low-Re-like resolution but differs from pure Launder-Sharma.

Computational Cost Comparison

Model	Mesh Amount (Relative)	Compute Time (Relative)
k-ε + Wall Function	1.0	1.0
Low-Re k-ε	2-5x	3-8x
SST ($y^+=1$)	2-3x	2-4x
SST + Wall Function	1.0-1.5x	1.0-1.5x

Coffee Break Yomoyama Talk

Numerical Implementation of Low-Reynolds Number k-ε Models—Selection of Wall Damping Functions

The wall damping functions f_μ (and corrections to the ε equation production term) for Low-Re k-ε models have many variations depending on the model. Launder-Sharma (1974), Lam-Bremhorst (1981), Chien (1982), Myong-Kasagi (1990) are representative, each with different Reynolds number-dependent functional forms. An implementation note is the definition of "distance from the wall y," which requires geodesic distance calculation for complex geometries. Fluent automatically calculates wall distance functions, but in OpenFOAM, the `wallDist` utility must be explicitly executed. In Low-Re regions, mesh resolution of y+≦1 is required, making grid costs 5~10 times higher than wall function methods.

Upwind Scheme (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 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 shock waves and steep gradients.

Finite Volume Method vs Finite Element Method

FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex geometries 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 recommended. Physical meaning: Information should not travel more than one cell per time step.

Residual Monitoring

Convergence is judged when residuals for the continuity equation, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.