Laminar Flat-Plate Boundary Layer¶

Why this case matters¶

The zero-pressure-gradient laminar boundary layer over a flat plate is the canonical skin-friction benchmark, and the flat-wall counterpart to the curved body in Flow Over Stationary Sphere. It is the cleanest possible test of the data.body_nodes friction export: the wall is everywhere tangent to the flow, so the per-triangle viscous traction the export writes reduces to a pure wall-shear stress whose value is known in closed form from the Blasius similarity solution. Where the sphere validates the full form/friction drag split on a curved surface, this case isolates and validates the skin-friction primitive itself, point by point along the wall, against an exact analytical reference.

The plate is represented as an immersed-boundary (IBM) body - a thin, single-sided sheet aligned with the flow - and the boundary layer develops along it as the freestream U_inf is convected downstream. At a plate-length Reynolds number well below transition the flow stays laminar and the boundary layer follows the Blasius profile.

Setup¶

A uniform stream of speed U_inf enters a rectangular domain and flows along a flat plate held parallel to the flow. The leading edge sits a short distance downstream of the inlet so the stream arrives uniform. The boundary layer grows along the plate; the spanwise direction is periodic, making the configuration a two-dimensional boundary layer.

The plate’s source geometry is a single-sided sheet whose outward normal points to one side only, and the friction export samples the fluid along that outward normal. The plate is therefore placed near the top of the domain with the fluid of interest below it, so the boundary layer develops on the side the normal points to. This is a standard flat-plate boundary layer mirrored in the wall-normal direction; the Blasius solution is unchanged.

(1)¶\[\mathrm{Re}_x = \frac{U_\infty\, x}{\nu}, \qquad \mathrm{Re}_L = \frac{U_\infty\, L}{\nu}\]

with x measured from the leading edge and L the plate length.

Analytical solution¶

The Blasius similarity solution gives the local skin-friction coefficient, the boundary-layer thickness and the plate-averaged friction coefficient as functions of the local Reynolds number (1):

(2)¶\[C_f(x) = \frac{\tau_w(x)}{\tfrac{1}{2}\rho_\infty U_\infty^2} = \frac{0.664}{\sqrt{\mathrm{Re}_x}}\]

(3)¶\[\delta(x) = \frac{5.0\, x}{\sqrt{\mathrm{Re}_x}}\]

(4)¶\[\bar{C}_f(\mathrm{Re}_L) = \frac{1.328}{\sqrt{\mathrm{Re}_L}}\]

where tau_w is the wall shear stress (the magnitude of the wall-tangential viscous traction) and delta the 99% boundary-layer thickness. These are the classic Blasius constants (Schlichting and Gersten[1]).

Simulation parameters¶

All values are in lattice units; the conversion that fixes them is recorded in the config header.

Parameter	Value	Note
Freestream `U_inf`	`0.05`	`Ma = sqrt(3) U = 0.087 < 0.1`
Plate length `L`	`320`	leading edge at `x = 40`
`Re_L`	`5000`	laminar (transition `~ 5e5`)
`nu`	`0.0032`	`nu = U_inf L / Re_L`
`tau`	`0.5096`	`tau = 0.5 + 3 nu`, RRBGK
Domain	`384 x 96 x 32`	streamwise x wall-normal x spanwise
Refinement	level 1 over the plate	`dx = 0.5`, boundary layer well resolved
Collision	`RRBGK`, `D3Q27`	LES off (resolved laminar flow)

At Re_L = 5000 the boundary layer thickness runs from delta ~ 5 lattice units near the leading edge to delta ~ 22 near the trailing edge; the level-1 refinement places at least ten fine cells across the boundary layer over the sampled range, the wall-resolved target.

Validation metrics¶

The notebook reads the data.body_nodes friction export with nassu.viz.read_body_friction, time-averages the per-triangle traction over the settled window with time_mean_friction, and forms the per-triangle wall-shear stress and skin-friction coefficient with wall_shear_stress / skin_friction_coefficient. Each triangle centroid maps to a local Re_x, giving the simulated Cf(Re_x) to overlay on (2). The case passes when the simulated Cf(x) collapses onto the Blasius curve over the resolved plate with a few-percent median error.

Note

Documented limits. Two effects bias the very front of the plate and are excluded from the comparison (0.05 L <= x <= 0.95 L):

The Blasius leading-edge singularity: Cf -> inf as Re_x -> 0, which no finite grid can represent.
IBM wall smearing: the diffuse-interface delta spreads the no-slip wall over a few lattice cells, so the boundary layer is least resolved where it is thinnest (near the leading edge).

These are method limits, not solver errors; the agreement is expected to improve monotonically downstream as the boundary layer thickens.