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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:

${\delta^*}= \int_0^\delta {(1-{u\over u_0})dy}$

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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

$\theta = \int_0^\infty {{u(z)\over u_o} {\left[1 - {u(z)\over u_o}\right]}} dz$

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

$\rho \int_0^\infty {u(z) [u_o - u(z)]} dz$

(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

$\theta = 0.664 \sqrt{{\nu x}\over u_o}$

(2.3)

Other length scales describing viscous boundary layers include the boundary-layer thickness, $δ$ , displacement thickness , $δ *$ , and energy thickness, $δ 3$ .

Category: Boundary layers