Displacement Thickness and Momentum Thickness Calculator

Displacement thickness (delta*) is the distance the outer inviscid flow is effectively pushed outward, away from the wall, because the slower-moving fluid inside the boundary layer carries less mass than an equivalent layer of free-stream flow. For the Blasius solution, delta* = 1.7208x / sqrt(Re_x).

Momentum thickness (theta) measures the momentum deficit in the boundary layer compared to free-stream flow, and it is directly related to skin friction drag on a flat plate. For the Blasius solution, theta = 0.664x / sqrt(Re_x), the same 0.664 that appears in the local skin friction coefficient formula.

The shape factor H = delta*/theta describes the general shape of the boundary layer velocity profile. For a Blasius laminar boundary layer H is always 2.5916, a fixed invariant. In real flows, H rising well above 2.6 signals the boundary layer is approaching separation under an adverse pressure gradient, while a falling H is typical of the transition to turbulence.

The 99% thickness delta marks where velocity reaches 99% of free-stream U, while displacement thickness delta* is a mass-flow-deficit integral over the whole profile. For a Blasius boundary layer delta* is only about 1.7208/5.0, roughly 34%, of delta, since most of the mass deficit sits close to the wall.

Momentum thickness weights the velocity deficit by the local velocity itself, so its integral is concentrated in the thin region very near the wall where both the deficit and the local velocity are significant. This makes theta smaller than displacement thickness delta*, which in turn is smaller than the 99% thickness delta.

Re_x = rho U x / mu, where rho is fluid density, U is free-stream velocity, x is distance from the leading edge, and mu is dynamic viscosity. It is called 'local' because it changes with position x along the plate, and it is the single number that sets delta, delta*, theta, and H at that station.

The Blasius formulas for delta*, theta, and H assume laminar flow, which holds until the local Reynolds number Re_x reaches roughly 5x10^5, the classic transition value for a flat plate. Past that point the boundary layer becomes turbulent and different, higher-value correlations for delta* and theta apply instead.

This calculator takes fluid density (rho) and dynamic viscosity (mu) as separate inputs, matching how most textbooks and material property tables present the data. Kinematic viscosity, if you have it instead, is nu = mu / rho, so multiply nu by rho to recover mu before entering it.

Momentum thickness theta is directly proportional to the local skin friction drag on a flat plate through the momentum integral equation, so it is the preferred thickness measure for computing total viscous drag. Displacement thickness delta* is instead used to correct the effective shape seen by the outer inviscid flow, for example when sizing wind tunnel test sections or duct inlets.

For air at 10 m/s flowing 0.3 m from the leading edge of a flat plate, the Blasius displacement thickness delta* is about 1.146 mm and the momentum thickness theta is about 0.442 mm, both a fraction of a millimetre, which shows how thin these boundary layer measures are at everyday lab-scale speeds and distances.

No. Because H = delta*/theta = 1.7208/0.664 comes directly from dividing two constants in the Blasius similarity solution, it always equals 2.5916 for laminar flow regardless of the fluid density, viscosity, velocity, or position x. H only changes once the flow leaves the laminar Blasius regime.