A boundary layer does not have one perfectly sharp outer edge, so a single geometric thickness cannot capture all of its effects. Integral thicknesses summarize the velocity deficit in ways tied directly to mass flow and momentum.

Let $U_e(x)$ be the outer-flow speed and $u(x,y)$ the tangential velocity inside a steady incompressible boundary layer.

Conventional boundary-layer thickness

A common visual thickness is the 99% thickness $\delta_{99}$, defined by

$$\boxed{u(x,\delta_{99})=0.99,U_e(x)}.$$

This is useful for visualizing how far the velocity disturbance extends, but the numerical level $99%$ is conventional rather than fundamental.

Displacement thickness

The boundary layer carries less volume flow near the wall than a hypothetical uniform stream of speed $U_e$ occupying the same region. The displacement thickness is

$$\boxed{ \delta^*(x) =\int_0^\infty \left(1-\frac{u}{U_e}\right)dy }.$$

Multiply by $U_e$:

$$U_e\delta^* =\int_0^\infty(U_e-u),dy.$$

The right side is the volume-flow deficit per unit span relative to the uniform outer flow. Thus $\delta^*$ is the distance by which an ideal inviscid boundary would need to be displaced outward to reproduce the same missing volume flow.

This is why boundary-layer growth can alter the outer flow even when the physical wall itself has not moved.

Momentum thickness

The momentum thickness is

$$\boxed{ \theta(x) =\int_0^\infty \frac{u}{U_e} \left(1-\frac{u}{U_e}\right)dy }.$$

Multiplying by $\rho U_e^2$ gives

$$\rho U_e^2\theta =\int_0^\infty \rho u(U_e-u),dy.$$

This quantity measures the streamwise momentum deficit associated with the boundary layer. It is therefore the natural thickness that appears in an integrated momentum balance.

Shape factor

Define the boundary-layer shape factor

$$\boxed{H=\frac{\delta^*}{\theta}}.$$

$H$ contains information about the shape of the normalized velocity profile, not merely its absolute thickness. Two boundary layers can have the same $\delta_{99}$ but different $\delta^*$, $\theta$, and $H$ because their velocity deficits are distributed differently across $y$.

Shape factor is particularly useful when comparing laminar, turbulent, and strongly decelerated boundary layers.

Worked example: a simple polynomial profile

Suppose an approximate boundary layer has thickness $\delta$ and profile

$$\frac{u}{U_e} =\frac32\eta-\frac12\eta^3, \qquad \eta=\frac y\delta,$$

for

$$0\le\eta\le1,$$

with

$$u=U_e$$

for $y\ge\delta$.

The profile satisfies

$$u(0)=0,$$

$$u(\delta)=U_e,$$

and has zero velocity gradient at the outer edge.

The displacement thickness is

$$\delta^* =\delta\int_0^1 \left[1-\left(\frac32\eta-\frac12\eta^3\right)\right]d\eta.$$

Evaluating,

$$\boxed{\delta^*=\frac38\delta}.$$

The momentum thickness is

$$\theta =\delta\int_0^1 \left(\frac32\eta-\frac12\eta^3\right) \left[1-\left(\frac32\eta-\frac12\eta^3\right)\right]d\eta,$$

which gives

$$\boxed{\theta=\frac{39}{280}\delta}.$$

Therefore

$$H =\frac{3/8}{39/280} =\boxed{\frac{35}{13}\approx2.69}.$$

The three thickness measures answer different questions:

• $\delta$ or $\delta_{99}$ describes the visible extent of the layer;
• $\delta^*$ measures missing volume flow;
• $\theta$ measures missing momentum;
• $H$ characterizes the normalized profile shape.

Integral thicknesses are valuable because they allow boundary-layer forces and growth to be analyzed without knowing every detail of the velocity field point by point.