8  2D Laminar Flat-Plate Boundary Layer — FR Blasius Verification

Verification of the p2 Flux Reconstruction solver against the Blasius velocity profile and skin-friction solution

8.1 Run Provenance and Artifacts

Item Value
Wake CFD baseline version 0.22.0
Solver FR
Compute device GPU (NVIDIA RTX A2000 12GB)
Parallel execution 1 MPI rank
Arithmetic precision FP64
Run date 2026-09-23
Note

Results depend on the solver build that produced them. Read every figure in this report together with the version and date above — re-running the same input on a later build may give different numbers.

8.1.1 Downloadable Files

Not available: standalone input decks and solver result files have not yet been published for download.

8.2 Verification Objective

This case simulates the developing laminar boundary layer over a two-dimensional flat plate with the Flux Reconstruction (FR) solver. The final numerical solution is compared with the Blasius similarity solution to verify:

  • the no-slip viscous-wall treatment in the FR solver
  • third-order p2 solution accuracy on a stretched quadrilateral mesh
  • collapse of velocity profiles at several streamwise stations in similarity coordinates
  • local skin-friction evaluation from FR wall flux points
  • dual-time integration of a low-Mach viscous flow

The geometry and freestream condition match the DBNS laminar flat-plate case, so the result also adds an independently implemented solver track to the common flat-plate verification family.

8.3 Reference Solution and Data

8.3.1 Analytical Solution

For a zero-pressure-gradient laminar boundary layer, let x denote distance from the plate leading edge, y the wall-normal distance, U_inf the freestream velocity, and nu the kinematic viscosity. The similarity coordinate and local Reynolds number are

\[ \eta = y\sqrt{\frac{U_\infty}{\nu x}}, \qquad Re_x = \frac{U_\infty x}{\nu}. \]

The Blasius solution gives u/U_inf = f'(eta) and

\[ \delta_{99}\sqrt{\frac{U_\infty}{\nu x}} \approx 4.91, \qquad C_f(x)=\frac{0.664}{\sqrt{Re_x}}. \]

Velocity profiles taken at different x/L stations should therefore collapse onto one curve when plotted against eta.

8.4 Simulation Conditions

Item Setting
Dimension 2D
Domain x=[0, 0.762] m, y=[0, 0.1524] m
Flat plate x=[0.1524, 0.4572] m, L=0.3048 m
Analysis Unsteady, compressible laminar Navier–Stokes, evaluated at t=12 s
Mach number 0.1
Reynolds number Re_L=100,270
Freestream P_inf=101325 Pa, T_inf=297.62 K, rho_inf=1.1827 kg/m³, U_inf=34.63 m/s
Viscosity Constant, mu=1.245×10^-4 Pa·s

The viscosity is selected to obtain Re_L≈10^5 at M=0.1; this is a numerical verification condition rather than a specific ambient-air experiment. Although the wall is isothermal, this report does not make a heat-transfer claim because the retained run output does not contain the required integrated heat-flux monitor.

8.5 Mesh

The structured mesh has 80×40=3,200 linear quadrilateral cells. It is clustered at the plate leading and trailing edges and stretched in the wall-normal direction, with a first wall-cell height of 1.5×10^-4 m. With an FR p2 basis, each cell contains nine solution points, giving 28,800 solution points and 144,000 scalar conservative-variable degrees of freedom.

Structured quadrilateral mesh with streamwise edge clustering and wall-normal stretching

8.6 Boundary and Initial Conditions

Boundary Type Setting
Left Farfield inlet M=0.1, P=101325 Pa, T=297.62 K
Right Pressure outlet P=101325 Pa
Top Farfield Freestream condition
Bottom outside plate Free-slip wall Zero normal velocity
Plate No-slip isothermal wall T_wall=327.38 K

Initial field: Not reported.

8.7 Numerical Methods

Item Setting
Spatial discretization FR p2 (third order), quadrilateral elements
Convective interface flux Rusanov
Viscous interface parameters LDG beta=0.5, tau=0.3
Physical-time integration Dual-time BDF1, dt=5×10^-3 s, 2,400 steps
Pseudo-time integration RK45 with local-PI control, CFL=0.3
Inner iterations 10 minimum, 300 maximum per physical step

The run advances from a uniform freestream state to t=12 s. Each physical step reached the configured 300-inner-iteration cap, and no inner residual stopping tolerance was enabled. The comparison below therefore assesses the final time-marched solution; it is not evidence of formal pseudo-time or grid convergence.

8.8 Results

The final velocity profiles at x/L=0.25, 0.50, and 0.75 follow the Blasius curve closely. RMS errors in normalized velocity remain below 1% at all three stations. The boundary-layer thickness is slightly underpredicted, with the largest difference at the downstream station.

x/L Re_x RMS error in u/U_inf Computed delta_99(eta) Difference from 4.91
0.25 25,068 0.53% 4.673 -4.8%
0.50 50,135 0.66% 4.621 -5.9%
0.75 75,203 0.97% 4.517 -8.0%

Over 0.2≤x/L≤0.9, the median absolute deviation of local Cf from the Blasius solution is 2.0%; the worst sampled deviation is 10.8%. The median signed ratio is Cf_FR/Cf_Blasius=1.021, indicating a small overall overprediction of wall shear in the evaluated interval.

FR velocity profiles and local skin friction compared with the Blasius solution

The velocity contour shows the boundary layer growing downstream from the plate leading edge. The plotted region is limited to the near-wall field so that the thin low-speed layer remains visible.

Velocity-magnitude contour near the flat-plate leading edge

8.9 Conclusions and Applicable Scope

  • The FR p2 solution reproduces the Blasius normalized velocity profiles with less than 1% RMS error at all three sampled stations.
  • Local skin friction agrees with the analytical solution to a 2.0% median absolute deviation over 0.2≤x/L≤0.9; the largest sampled deviation is 10.8%.
  • The boundary-layer thickness is underpredicted by 4.8–8.0%, with the discrepancy increasing downstream.
  • This result verifies the combined FR viscous-wall, wall-flux, and dual-time path for one p2 mesh at M=0.1 and Re_L≈10^5.
  • The run does not establish mesh, polynomial-order, time-step, or pseudo-time convergence. Those variables should be varied independently before broader accuracy claims are made.
  • No Pohlhausen heat-transfer result is reported because the retained history file lacks the integrated heat-flux quantities required for that comparison.

This case establishes a baseline for laminar flat-plate calculations with the FR solver. It does not cover turbulent flow, pressure gradients, separation, transition, curved walls, or other Reynolds and Mach numbers.

8.10 References

  • Blasius, H., “Grenzschichten in Flüssigkeiten mit kleiner Reibung,” Zeitschrift für Mathematik und Physik, 56, 1908, pp. 1–37.
  • White, F. M., Viscous Fluid Flow, 3rd ed., McGraw-Hill, 2006, Chapter 4.