CFD Solution Verification · Turbulence Modelling · interactive

Turbulent Flat-Plate Boundary Layer: k–ω SST vs the Law of the Wall

OpenFOAM simpleFoam (v2312) · steady incompressible RANS, k–ω SST · wall-resolved y⁺≈0.13 · digitalmodel CFD capability (issue #1168, epic #1161)

VERIFIED — law of the wall captured; C_f within ~9% of correlations

SolversimpleFoam + k–ω SST

Reynolds No.Re_L = 10⁶

Mesh50,400 cells

Wall resolutiony⁺ = 0.12–0.31

C_f vs correlation~9% mean

TurbulenceRANS (first turbulent case)

1 Objective

Verify the digitalmodel CFD pipeline for turbulent flow — the regime of essentially all real engineering CFD, and the first turbulent case in the validation suite (the flat-plate Blasius and cylinder Re=100 cases are laminar). The test is the zero-pressure-gradient turbulent boundary layer, whose near-wall structure obeys the universal law of the wall and whose skin friction follows well-established empirical correlations.

viscous sublayer u⁺ = y⁺ log law u⁺ = (1/κ) ln y⁺ + B, κ≈0.41, B≈5.0

skin friction C_f(Re_x) ≈ 0.0592 Re_x^−1/5 (Prandtl) or (2 log₁₀ Re_x − 0.65)^−2.3 (Schlichting)

What this gates in real work

Real marine flows are turbulent — this case is the full-scale friction gate. A wall-resolved k-ω SST boundary layer that lands on the universal law of the wall qualifies:

Ship & FPSO hull friction — full-scale Cf prediction and the ITTC/Schoenherr line comparisons behind every powering estimate.
Appendage and rudder RANS — the same turbulence model, wall treatment (y⁺ ≈ 0.1–0.3, no wall functions) and near-wall meshing used for lifting surfaces (see the NACA0012 case).
Current loads on hulls and jackets — drag coefficients used in mooring and DP capability studies inherit their credibility from exactly this near-wall physics.

2 Primary result — the law of the wall

Velocity profile at Re_x=10⁶ in wall units, normalised by the computed wall friction velocity. The solution follows u⁺=y⁺ through the viscous sublayer, bends through the buffer layer, and lies on the log law across the logarithmic region before lifting into the wake.

This is the definitive turbulence check. With the near-wall mesh resolved to y⁺≈0.13 (viscous sublayer, no wall functions bridging the gap), the k–ω SST solution reproduces all three canonical layers — sublayer, log law (κ≈0.41, B≈5.0), and wake. A model that got the turbulent wall physics wrong could not collapse onto the universal profile.

3 Skin friction

Computed C_f(Re_x) against the Prandtl and Schlichting turbulent correlations. The comparison window (shaded) is the developed turbulent region Re_x=5×10⁵–10⁶.

Metric	Value
C_f level at Re_x=10⁶	0.00342 (vs 0.00375 correlation; ~5× the laminar Blasius value — unambiguously turbulent)
Mean \|error\| vs Schlichting (Re_x 5×10⁵–10⁶)	9.3% (max 9.9%)
Trend	correct C_f ∝ Re_x^−1/5 decay

On the ~9% offset. The empirical turbulent correlations themselves scatter by 5–10% (Prandtl, Schlichting, White, Kármán–Schoenherr), and they assume a boundary layer turbulent from the leading edge; the computed layer has a short resolved transition region near the plate nose (visible as the larger deficit at low Re_x, shrinking downstream). The self-consistent law-of-the-wall collapse above — using the solution's own friction velocity — is the stronger and more fundamental verification of the turbulence model; the skin-friction level agrees with the classical correlations to within their own spread.

4 Numerical model

Item	Specification
Solver / model	`simpleFoam` + `kOmegaSST` (RANS), steady, SIMPLEC
Domain / mesh	Two-block: 0.2 m symmetry run-in + 1.0 m plate, H=0.3 m; 50,400 cells; wall-normal grading to y⁺=0.12–0.31 (sublayer-resolved, low-Re wall treatment)
Flow	U∞=1, ν=10⁻⁶ → Re_L=10⁶
Wall BCs	k `kLowReWallFunction`, ω `omegaWallFunction`, ν_t `nutLowReWallFunction`
Verification	y⁺ via the `yPlus` function object; C_f from `wallShearStress`; u⁺ profile normalised by the local computed u_τ

5 Conclusion

Verified. The digitalmodel CFD pipeline reproduces a wall-resolved turbulent boundary layer with the k–ω SST model: the velocity profile collapses onto the universal law of the wall (sublayer + log law + wake) at y⁺≈0.13, and the skin friction matches the classical turbulent correlations to within ~9% (their own mutual scatter). This extends the verified capability from laminar (flat-plate Blasius, cylinder Re=100) to turbulent RANS — the regime of practical engineering CFD.

References

Schlichting, H. & Gersten, K. (2017). Boundary-Layer Theory, 9th ed., Springer.
White, F. M. (2006). Viscous Fluid Flow, 3rd ed., McGraw-Hill.
Menter, F. R. (1994). "Two-equation eddy-viscosity turbulence models for engineering applications." AIAA J. 32(8), 1598–1605.
NASA Turbulence Modeling Resource — 2D zero-pressure-gradient flat plate.