6.4 Boundary layers

Boundary layers4 are regions of fluid formed along solid boundaries in which the velocity eqn varies: from zero at the boundary (no-slip condition, Sec. 4.4 ); to a value largely unaffected by the proximity of the boundary, determined by the flow conditions.

PICT\relax \special {t4ht=

The figure above shows a boundary layer for flow in the eqn-direction at speed eqn, along a flat solid boundary oriented in the eqn-normal direction. At the boundary surface, the vorticity eqn is significant.

Vorticity can be shown over a planar section of the boundary layer of width eqn and height eqn (above, right). Applying Stokes’s theorem, Eq. (2.39 ), the integral = eqn along the upper line (with zero along the wall and the verticals sides). The average vorticity over plane area eqn is therefore eqn.

Boundary layers are the main source of vorticity for turbulence. Turbulence occurs when instabilities, e.g. induced by roughness of the boundary surface, cause the vorticity to become chaotic, sustained by a sufficiently high eqn.

The growth of boundary layers is related to vorticity transport. For flow over a flat plate, vorticity generated at the leading edge is advected by the flow, while diffusing away from the plate.  

Vorticity propagates by diffusion by a distance eqn in time eqn, see Sec. 2.22 . In that time, it is advected a distance eqn, where eqn is the freestream flow speed. Comparing the distances over the same eqn, the boundary layer thickness is
 r -- --- r -1- --/ -----= ---: x u1 x Re \relax \special {t4ht=
The relation eqn is suitable for laminar boundary layers, with coefficient eqn depending on the definition of eqn. Data and analysis, including e.g. the Blasius solution5, indicate eqn in the case of the “99% thickness”, i.e. the distance from the wall where velocity reaches 99% of its asymptotic value.

In turbulent boundary layers, the diffusion front advances more rapidly due to mixing, see Sec. 6.11 . As a result, eqn is relatively insensitive to eqn, e.g. the analytical solution eqn , based on a one-seventh (eqn) power law for the velocity profile.

4Ludwig Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, 1904.
5Paul Richard Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, 1908.

Notes on CFD: General Principles - 6.4 Boundary layers