2.5 Boundary Layer Flows
41
commonly used boundary layer thickness is b where the local velocity u(z) has
risen to 99% of the free-stream velocity, uo, i. e.:
u(z) = 0.99uo for z = b.
(2.46)
The boundary layer thickness is a convenient, single number characterizing
the boundary layer. However, thickness, b, provides information about flow at
one selected point only, and no information is given about the shape of the
boundary layer along the solid surface.
The assumption of the slip condition, when du/dz = 0 and no force is imposed
on the surface, implies that the fluid is non-viscous (perfect). However, when
the no-slip condition is assumed, the velocity u(z) depends on the regime of
flow, laminar or turbulent, within the boundary layer. We now consider various
forms of the boundary layer.
2.5.3 Laminar Boundary Layer
Consider a thin, flat plate oriented parallel to uniform flow. The plate retards
the oncoming stream at the exact level of the plate. At points located further
downstream from the leading edge of the plate, more space is available for
boundary layer generation and the boundary layer becomes thicker (Fig. 2.14).
The thickness of the boundary layer for a steady, uniform flow over a flat plate
increases as a function of distance from plate edge according to the formula
derived by von Karman (Schlichting, 1960):
f5
5.5x
b(x) ~ 5.5x -
=
~,
UoX
v Rex
(2.4 7)
in which Rex = uox/v is called the local Reynolds number. In the von Karman
solution the velocity profile within the boundary layer has a parabolic shape:
( 2z z2)
u(x, z) ~ Uo b(x) - b2(x) , 0::; z ::; b(x),
(2.48)
and for very small distance from the plate velocity, this profile becomes approximately linear, i. e.:
2uoz
u(x, z) ~ b(x)'
(2.49)
In general, a boundary layer is considered 'thin' when the ratio b / x is less
than about 0.1. Thus, in Eq. (2.47), when b(x)/x = 5.5/..jRe x = 0.1, local
Reynolds number becomes ~ 3000. For Rex having value less than 3000, and
b(x)/x > 0.1, the thick layer has a significant effect on the outer inviscid flow.
41
commonly used boundary layer thickness is b where the local velocity u(z) has
risen to 99% of the free-stream velocity, uo, i. e.:
u(z) = 0.99uo for z = b.
(2.46)
The boundary layer thickness is a convenient, single number characterizing
the boundary layer. However, thickness, b, provides information about flow at
one selected point only, and no information is given about the shape of the
boundary layer along the solid surface.
The assumption of the slip condition, when du/dz = 0 and no force is imposed
on the surface, implies that the fluid is non-viscous (perfect). However, when
the no-slip condition is assumed, the velocity u(z) depends on the regime of
flow, laminar or turbulent, within the boundary layer. We now consider various
forms of the boundary layer.
2.5.3 Laminar Boundary Layer
Consider a thin, flat plate oriented parallel to uniform flow. The plate retards
the oncoming stream at the exact level of the plate. At points located further
downstream from the leading edge of the plate, more space is available for
boundary layer generation and the boundary layer becomes thicker (Fig. 2.14).
The thickness of the boundary layer for a steady, uniform flow over a flat plate
increases as a function of distance from plate edge according to the formula
derived by von Karman (Schlichting, 1960):
f5
5.5x
b(x) ~ 5.5x -
=
~,
UoX
v Rex
(2.4 7)
in which Rex = uox/v is called the local Reynolds number. In the von Karman
solution the velocity profile within the boundary layer has a parabolic shape:
( 2z z2)
u(x, z) ~ Uo b(x) - b2(x) , 0::; z ::; b(x),
(2.48)
and for very small distance from the plate velocity, this profile becomes approximately linear, i. e.:
2uoz
u(x, z) ~ b(x)'
(2.49)
In general, a boundary layer is considered 'thin' when the ratio b / x is less
than about 0.1. Thus, in Eq. (2.47), when b(x)/x = 5.5/..jRe x = 0.1, local
Reynolds number becomes ~ 3000. For Rex having value less than 3000, and
b(x)/x > 0.1, the thick layer has a significant effect on the outer inviscid flow.