Boundary-layer thickness

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In a fluid with velocity flowing over a surface or a body moving through a fluid, viscous effects are limited to regions with fluid shear. Consider a stationary body with flow over it such that the flow is steady. In situations with a solid boundary, the fluid satisfies the no-slip boundary condition in the boundary-layer and asymptotically approaches the free stream mean velocity, u_o. Therefore it is impossible to define a clear thickness of the boundary layer. Several quantities with dimensions of length are used to quantify the thickness of boundary layers.

[edit] 99% Velocity Thickness

The boundary-layer thickness, δ, is used for a thickness beyond which the velocity is essentially the free-stream velocity $u o$ . This is customarily defined as the distance from the wall to the point where

u (y) = 0.99 u o

For laminar boundary-layers over a flat plate, the Blasius solution gives

$\delta \approx 4.5 \sqrt{ {\nu x}\over u_o}$

[edit] Displacement Thickness

The displacement thickness, δ^* or δ₁ is the distance by which a surface would have to be moved parallel to itself towards the reference plane in an inviscid fluid stream of velocity $u 0$ to give the same volumetric flow as occurs between the surface and the reference plane in a real fluid.

In practical aerodynamics, the displacement thickness essentially modifies the shape of a body immersed in a fluid to allow an inviscid solution. It is commonly used in aerodynamics to overcome the difficulty inherent in the fact that the fluid velocity in the boundary layer approaches asymptotically to the free stream value as distance from the wall increases at any given location. The mathematical definition of the displacement thickness for incompressible flow is given by

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

and for compressible flow, by

${\delta^*}= \int_0^\infty {\left(1-{\rho u(y)\over \rho_0 u_0}\right)dy}$

where $ρ 0$ and $u 0$ refer to the density and velocity outside the boundary layer.

[edit] Momentum Thickness

The momentum thickness, θ or δ₂, is the distance by which a surface would have to be moved parallel to itself towards the reference plane in an inviscid fluid stream of velocity $u 0$ to give the same total momentum as exists between the surface and the reference plane in a real fluid.

Mathematically it is defined as

$\theta = \int_0^\infty {{u(y)\over u_o} {\left(1 - {u(y)\over u_o}\right)}} dy$

where the vertical coordinate, y, 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 (y = 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 possess a total momentum flux deficit,

$\rho \int_0^\infty {u(y) \left(u_o - u(y)\right)} dy$

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

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