Laminar Flat Plate

Overview

A uniform stream flows over a sharp flat plate aligned with the flow, forming a laminar boundary layer. With zero pressure gradient and constant properties, the boundary-layer equations admit the Blasius similarity solution, which provides analytic references for the skin-friction distribution, the boundary-layer growth, and, with heat transfer, the local Nusselt number. The solver result is compared against this known analytic answer.

Problem Setup

The plate is 0.3048 m long. The freestream is air at low Mach number (≈ 0.2) with velocity 69.1687 m/s and temperature 297.62 K, giving a length-based Reynolds number $R{e}_L$ of about 1.3 × 10⁶, within the laminar regime over the plate. The plate is held isothermal at 197.62 K, so a thermal boundary layer ($\Delta T=100$ K) develops alongside the velocity boundary layer; the energy equation is solved with one-way coupling at constant density. The leading edge sits at the origin, with a symmetry plane upstream to avoid a singular leading-edge boundary layer.

Quantities of Interest

• Velocity profile $u/V_{\infty }=f^{\prime }(\eta )$ across the boundary layer at $x=0.15$ m, plotted against the Blasius similarity variable $\eta =y\sqrt{V_{\infty }/(\nu x)}$, normalised by the boundary-layer edge velocity at the station (which equals the free-stream $V_{\infty }$ here).
• Skin-friction coefficient $$C_f=\frac{{\tau }_w}{\frac{1}{2}\rho V_{\infty }^2}\approx 0.664R{e}_x^{-1/2},$$ compared pointwise along the plate, plus the integrated friction-drag coefficient $C_D=1.328R{e}_L^{-1/2}$. The computed $C_D$ is 0.0011639, within 0.03% of the Blasius value 0.0011642.
• Local Nusselt number $$N{u}_x=\frac{\dot{q}x}{\kappa \Delta T}\approx 0.33R{e}_x^{1/2}P{r}^{1/3},$$ from the wall heat flux, comparing the thermal boundary layer to the Blasius correlation.

These analytic curves provide references for the solver's viscous flux discretization, wall-shear-stress evaluation, and wall heat-transfer for laminar boundary layers.