Low Reynolds Number Model
Low Reynolds Number: Theoretical Foundations
Overview
Teacher! Are low-Reynolds number models the turbulence models that don't use wall functions?
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.
What kind of damping functions are used?
A representative one is the Launder-Sharma (1974) k-ε model, which multiplies the eddy viscosity by a damping function $f_\mu$.
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
Is the ε equation also modified?
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
What kind of mesh is needed to use a Low-Re model?
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 $y^+ = 1$:
Example: For a flat plate with $U = 10$ m/s, $Re_L = 10^6$, $\Delta y_1 \approx 2 \times 10^{-5}$ m.
That requires a very fine mesh. I'm worried about the computational cost.
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.
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 (5
Computational Methods for Low Reynolds Number
Key Points in Numerical Implementation
What points require special attention in the numerical implementation of Low-Re models?
Because the mesh near the wall becomes extremely fine, several numerical difficulties arise.
Wall Boundary Condition for ε
We impose $\tilde{\varepsilon} = 0$ at the wall, right?
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
Do problems ever occur with damping functions?
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
How are Low-Re models used in commercial solvers?
| 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 |
So in Fluent, it's provided as Enhanced Wall Treatment.
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 |
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.
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