Momentum thickness

In aeronautics and viscous fluid theory, the boundary layer thickness (δ) is the distance from a fixed boundary wall where zero flow is considered to occur, and beyond δ 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 (δ ^* ). 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:

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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

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, τ_w, which extracts energy from the mean flow. The boundary layer can be considered to posses a total momentum flux deficit,

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

Other length scales describing viscous boundary layers include the boundary-layer thickness, δ, displacement thickness , δ ^* , and energy thickness, δ₃.