Momentum thickness

In aeronautics and viscous fluid theory, the boundary layer thickness ($\{delta\}$) is the distance from a fixed boundary wall where zero flow is considered to occur, and beyond $\{delta\}$ the fluid is considered to move at a constant velocity. This distance is calculated based on the total momentum of the fluid, rather than the total mass, as in the case of displacement thickness ($\{delta^*\}$). The region of moving fluid contains a percentage (typically 97%) of the fluid's momentum, leading to the definition (from incompressible fluid theory and the continuity equation) mathematically, of:

:$\{delta^*\}=\; int\_0^delta\; \{(1-\{uover\; u\_0\})dy\}$

The momentum thickness, θ, is a theoretical length scale to quantify the effects of fluid viscosity in the vicinity of a physical boundary. Mathematically it is defined as

:$heta\; =\; int\_0^infty$u(z)over u_o} {left [1 - {u(z)over u_o} ight] dz

(2.1)

where the vertical coordinate, z, is increasing upward from the boundary and u_o is the velocity in the ideal flow of the free stream. The velocity in a frictional boundary layer is subject to the no-slip boundary condition at the surface (z = 0) and asymptotically approaches the free stream value (u_o). Compared to potential flow, this would be the distance that the surface would be displaced for the flow to have the same momentum. The influence of fluid viscosity creates a wall shear stress, $au\_w$, which extracts energy from the mean flow. The boundary layer can be considered to possess a total momentum flux deficit,

:$ho\; int\_0^infty\; \{u(z)\; [u\_o\; -\; u(z)]\; \}\; dz$

(2.2)

due to the frictional dissipation. It may also be designated as δ₂ as in Schlicting, H. (1979) Boundary-Layer Theory McGraw Hill, New York, U.S.A. 817 pp.

For a flat plate at no angle of attack with a laminar boundary layer, the Blasius solution gives

:$heta\; =\; 0.664\; sqrt$ u x}over u_o}

(2.3)

Other length scales describing viscous boundary layers include the boundary-layer thickness, $delta$, displacement thickness , $delta^*$, and energy thickness, $delta\_3$.