Momentum thickness
= Momentum thickness
{title2=$\theta$}
The <momentum thickness> weights the streamwise <velocity> deficit by the local <velocity> carrying it:
$$
\theta=\int_0^\infty\frac{u}{U}\left(1-\frac{u}{U}\right)\,dy,\qquad \rho U^2\theta=\rho\int_0^\infty u(U-u)\,dy.
$$
Thus $\rho U^2\theta$ measures a <momentum flux> deficit per unit span relative to transporting the same local <mass flow rate> at $U$. It is not $\rho\int(U^2-u^2)\,dy$, which compares a different reference flux. A <Blasius boundary layer> has $\theta\simeq0.664115\sqrt{\nu x/U}$; the ratio of <displacement thickness> to <momentum thickness> is about $2.591$.<Modules on High Reynolds Number Flows/Boundary-layer Thickness, Skin friction, and Energy dissipation>{ref}
For the <steady flow> over an impermeable flat plate at constant $U$, integrating the <boundary-layer equation> and using <mass conservation> gives
$$
\frac{d\theta}{dx}=\frac{\tau_w}{\rho U^2}.
$$
This connects growth of the <momentum thickness> to local <wall shear stress>. Consequently the <drag> per unit span on one face up to $x=L$ is $D'=\rho U^2\theta(L)$ in the ideal <Blasius boundary layer> with $\theta(0)=0$. Measured <velocity profiles> can therefore estimate integrated <drag> without differentiating noisy near-wall readings. The identity assumes no wall suction and zero outer <pressure gradient>.<Modules on High Reynolds Number Flows/Momentum Integral>{ref}