= Blasius boundary layer
{c}
{wiki}
= Flat-plate laminar boundary layer
{synonym}
A <Blasius boundary layer> is the <steady flow>, two-dimensional <laminar flow> along a smooth stationary flat plate with uniform outer <velocity> $U>0$. Let $x>0$ measure distance from its leading edge and $y\geq0$ distance normal to its upper face. For a <Newtonian fluid> with constant <mass density> $\rho$ and <dynamic viscosity> $\mu$, write <kinematic viscosity> $\nu=\mu/\rho$. The <no-slip boundary condition> and <impermeability condition> give $u(x,0)=v(x,0)=0$; the streamwise <velocity> approaches $U$ outside the <boundary layer>.
\Image[/flat-plate-laminar-velocity-profiles.png]
{title=Computed laminar velocity profiles and thickness measures along a flat plate}
{description=The <Blasius equation> supplies all curves. Normal distances are in millimetres and are magnified relative to downstream distances. The dashed <boundary layer thickness> curve marks $u/U=0.99$, rather than a material interface. The shaded normalized <velocity profile> shows the <displacement thickness> integrand. Parameters and distances are illustrative model inputs, not experimental measurements.}
For example, choose $U=10\,\mathrm{m\,s^{-1}}$ and $\nu=1.5\times10^{-5}\,\mathrm{m^2\,s^{-1}}$. At $x=0.30\,\mathrm m$, the local <Reynolds number> is $\mathrm{Re}_x=Ux/\nu=2.0\times10^5$ and the <boundary layer thickness> is about $3.29\,\mathrm{mm}$. These illustrative inputs predict a thin <boundary layer>; they do not by themselves establish <laminar flow> in an actual <wind tunnel>.
The <boundary-layer scaling> follows by comparing <advection>, of size $U^2/x$, with normal viscous transport, of size $\nu U/\delta^2$:
$$
\delta\sim\sqrt{\frac{\nu x}{U}},\qquad \frac{\delta}{x}\sim\mathrm{Re}_x^{-1/2},\qquad \mathrm{Re}_x=\frac{Ux}{\nu}.
$$
Equivalently, the available travel time $x/U$ equals the <viscous diffusion time> $\delta^2/\nu$ in order of magnitude. Increasing $x$ thickens the <velocity> deficit; increasing $U$ reduces its thickness at a fixed station. This <boundary-layer scaling> requires $\mathrm{Re}_x\gg1$. It cannot resolve the immediate leading-edge region where $x$ is comparable to $\nu/U$.<Modules on High Reynolds Number Flows/Boundary Layer Governing Equations>{ref}
The <boundary-layer equation> assumes a thin, attached <laminar flow> with streamwise changes over distances much larger than $\delta$, negligible spanwise changes, <incompressible flow>, and constant material properties. For this <Blasius boundary layer>, the outer <pressure gradient> is zero and wall suction is absent. <Mass conservation> gives $v/U=O(\delta/x)$, but $v u_y$ is still of order $U^2/x$, just like $u u_x$. Thus the <Prandtl boundary-layer equation> retains both <advection> terms:
$$
u_x+v_y=0,\qquad uu_x+vu_y=\nu u_{yy},\qquad p_y\simeq0.
$$
The discarded streamwise viscous term $\nu u_{xx}$ is smaller than $\nu u_{yy}$ by $O((\delta/x)^2)$. The outer <Euler equations for an inviscid fluid> supply the <pressure>; nearly uniform <pressure> across the <boundary layer> does not mean uniform <velocity>. <Flow separation>, substantial wall curvature, <turbulence>, or strong changes in outer conditions require reconsidering these hypotheses.
Introduce the <stream function> and a dimensionless normal coordinate:
$$
\eta=y\sqrt{\frac{U}{\nu x}},\qquad \psi=\sqrt{\nu Ux}\,f(\eta),\qquad u=\psi_y=Uf'(\eta),\qquad v=-\psi_x=\frac12\sqrt{\frac{\nu U}{x}}\,[\eta f'(\eta)-f(\eta)].
$$
Substitution gives the <Blasius equation> $f'''+ff''/2=0$, with $f(0)=f'(0)=0$ and $f'(\infty)=1$. Its numerical <boundary value problem> gives $f''(0)\simeq0.332057$. All streamwise <velocity profiles> therefore collapse onto the same $f'(\eta)$ after rescaling $y$ by $\sqrt{\nu x/U}$. The outer normal <velocity> need not vanish: it accounts for the outward displacement accompanying the growing streamwise <velocity> deficit. The MIT <Modules on High Reynolds Number Flows/Similarity Solution> uses the alternative coordinate $\eta_{\mathrm{MIT}}=\eta/\sqrt2$ and obtains $f_{\mathrm{MIT}}''(0)\simeq0.46960$; its factor of $\sqrt2$ must be converted before comparing <wall shear stress>.
The historical <Motion of fluids with very little viscosity by L. Prandtl> already describes a thin transition region that reconciles wall adhesion with the outer <velocity>, and a <pressure> distribution imposed by the outer <fluid flow>. Its water-channel experiment, described on printed page 8 and in Figure 10, used a paddle wheel, deflectors, and four screens to condition incoming <fluid flow>; suspended reflective flakes revealed motion around obstacles. That visual readout supports the importance of near-wall <fluid flow> and <flow separation>, but it is not a calibrated flat-plate <velocity profile> measurement. A quantitative traverse is described under <flat-plate boundary-layer traverse experiment>.
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