## 6.4Boundary layers

Boundary layers4 are regions of ﬂuid formed along solid boundaries in which the velocity 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.

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

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

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 .

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 in time , see Sec. 2.22 . In that time, it is advected a distance , where is the freestream ﬂow speed. Comparing the distances over the same , the boundary layer thickness is
 (6.6)
The relation is suitable for laminar boundary layers, with coeﬃcient depending on the deﬁnition of . Data and analysis, including e.g. the Blasius solution5, indicate 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, is relatively insensitive to , e.g. the analytical solution , based on a one-seventh () 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