Boundary layers

Understand it in 30 seconds

Viscosity makes fluid touching a stationary wall come to rest. Moving away from the wall, speed increases until it approaches the free-stream speed $U_\infty$ [1,2].

The wall-adjacent region containing this velocity change is the boundary layer. Viscous effects may be small outside it, but skin friction and separation are determined by the velocity profile within it.

What happens near the wall

Consider a uniform stream meeting a flat plate. Fluid touching the leading edge decelerates sharply. Viscosity transmits this retardation to adjacent layers, so the wall's influence reaches farther from the plate downstream.

Each curve in the figure is a velocity profile. Velocity rises continuously from $u=0$ at the plate to $U_\infty$ outside, and the height required to reach the free stream increases downstream.

Defining boundary-layer thickness

Velocity approaches $U_\infty$ gradually, so there is no exact material line at the boundary-layer edge. An engineering convention defines thickness $\delta$ as the height where local speed $\boldsymbol{u}$ reaches 99% of $U_\infty$ [1].

The thickness $\delta$ describes the flow state; it is not a material boundary. Fluid particles can move into and out of the layer.

Equation (1)No-slip condition

u(x,0)=0

Equation (2)Boundary-layer thickness

u\!\left(x,\delta\right)=0.99U_\infty

Reynolds number from the leading edge

For a flat-plate boundary layer, use distance $x$ from the leading edge as the characteristic length. The local Reynolds number $\mathrm{Re}_x$ compares inertia with viscosity and increases downstream at fixed fluid properties and free-stream speed.

Even on a smooth plate, transition depends on free-stream turbulence, roughness, pressure gradient and vibration. $\mathrm{Re}_x\approx5\times10^5$ is a representative guide, not a universal threshold [1,3].

Equation (3)Local flat-plate Reynolds number

\mathrm{Re}_x=\frac{U_\infty x}{\nu}=\frac{\rho U_\infty x}{\mu}

Symbols and units

Reynolds number is dimensionless. Keep physical quantities separate from SI units and use fraction notation for velocity and kinematic viscosity [4].

Symbols and SI units for a flat-plate boundary layer
Symbol	Meaning	SI unit
$x$	Distance from the leading edge	$\mathrm{m}$
$y$	Distance normal to the wall	$\mathrm{m}$
$\boldsymbol{u}$	Local velocity in the boundary layer	$\mathrm{m/s}$
$U_\infty$	Free-stream speed outside the layer	$\mathrm{m/s}$
$\delta$	Boundary-layer thickness	$\mathrm{m}$
$\mathrm{Re}_x$	Local Reynolds number based on x	$\text{—}$
$\rho$	Density	$\mathrm{kg/m^3}$
$\mu$	Dynamic viscosity	$\mathrm{Pa{\cdot}s}$
$\nu$	Kinematic viscosity	$\mathrm{m^2/s}$
$\tau_{\mathrm w}$	Wall shear stress	$\mathrm{Pa}$

Laminar flat-plate estimates

For steady, incompressible Newtonian flow over a smooth plate with zero pressure gradient, the Blasius solution gives useful approximations for thickness and skin friction [1].

Equation (4) shows that the layer grows downstream while its relative thickness $\delta/x$ decreases as $\mathrm{Re}_x$ increases.

Equation (4)Laminar boundary-layer thickness

\delta(x)\approx\frac{5x}{\sqrt{\mathrm{Re}_x}}

Steady, incompressible, Newtonian, smooth flat plate, zero pressure gradient, laminar

Equation (5)Local skin-friction coefficient

C_{f,x}=\frac{0.664}{\sqrt{\mathrm{Re}_x}}

Same conditions as Equation (4)

Worked example: thickness over a flat plate

Air at 20 °C flows at $U_\infty=10\ \mathrm{m/s}$ over a smooth flat plate. Use $\nu=1.5\times10^{-5}\ \mathrm{m^2/s}$ and find the boundary-layer thickness at $x=0.50\ \mathrm{m}$ from the leading edge.

Equation (3) gives $\mathrm{Re}_x\approx3.33\times10^5$ , below the representative transition guide, so use the laminar estimate. Equation (4) then gives $\delta\approx0.0043\ \mathrm{m}=4.3\ \mathrm{mm}$ .

Equation (6)Local Reynolds number

\mathrm{Re}_x=\frac{10\times0.50}{1.5\times10^{-5}}\approx3.33\times10^5

Equation (7)Boundary-layer thickness

\delta\approx\frac{5\times0.50}{\sqrt{3.33\times10^5}}\approx4.3\ \mathrm{mm}

Pressure rise and separation

In an adverse pressure gradient, pressure increases downstream and slow fluid near the wall loses forward momentum. Wall velocity gradient and shear fall; if reverse flow develops near the surface, the boundary layer separates.

Separation creates a large wake and pressure drag and appears in wing stall, diffuser loss and bends. Smooth shaping often aims to avoid an abrupt pressure rise and delay separation.

Conditions and limits

Equations (4) and (5) apply to a laminar layer on a smooth flat plate at zero pressure gradient. Do not apply them unchanged to curved surfaces, strong pressure gradients, suction, blowing, rough walls or turbulent layers.

High-speed flow may require density, temperature and viscosity variations. Also check thin-layer assumptions when boundary-layer thickness is not small relative to the body.

Common mistakes

Before using a boundary-layer correlation, identify the flow state, characteristic length, pressure gradient and surface condition.

Viscosity is zero outside: Viscosity remains a fluid property; small velocity gradients merely make viscous terms negligible there.
δ is a material surface: It is a conventional scale, commonly defined where $\boldsymbol{u}$ reaches $0.99U_\infty$ .
Use the pipe Re_D threshold: A plate uses $\mathrm{Re}_x$ and has a different transition mechanism.
Use δ alone to locate separation: Separation is identified from wall velocity gradient, shear and pressure gradient.

Applications and related knowledge

Boundary layers underpin drag on wings and vehicles, pipe entrances, turbine blades, heat exchangers, coating flows and convective transport. A thermal boundary layer similarly connects wall and free-stream temperatures.

Review Reynolds number and laminar versus turbulent flow first. Continue with compressibility and Mach number or internal flow to connect boundary-layer ideas to external- and internal-flow design.

Prerequisites: Reynolds number, laminar and turbulent flow, viscosity, no slip
Continue with: Compressibility and Mach number, internal flow, friction loss, convective heat transfer

References

Sources accessed 14 August 2026. Independent expert review has not yet been completed.