An Internet Book on Fluid Dynamics
Stability of Planar Parallel Flows
In this section we examine some of the results obtained from the linear stability analyses described in
section (Bkc). It is convenient to begin with the data for boundary layers presented in Figures 1 to 4.
The equations governing the perturbations in section (Bkc) suggest that it is convenient and appropriate
to non-dimensionalize the coordinate xby using a surrogate parameter in the form of the displacement
thickness, δD, and to non-dimensionalize this in the form of δDU/ν which is a Reynolds number based
on the displacement thickness and the velocity, U, at the edge of the boundary layer. Thus δDU/ν will
represent the distance, x, in the data that follows. Moreover, a convenient non-dimensional frequency is
ωRν/U2.
Figure 1: Neutral stability boundary (kI= 0) for the Blasius boundary layer velocity profile in a graph of the perturbation
frequency, ωRν/U2, plotted against the surrogate distance parameter, δDU/ν.
As a first example of the results of a stability calculation, Figure 1 presents the neutral stability contour
(kI= 0) for the Blasius boundary layer velocity profile in a graph of the potential perturbation frequency,
ωRν/U2, plotted against the surrogate distance parameter, δDU/ν. The regions of stable behavior surround
a region of instability and the neutral stability line demonstrates that the boundary layer, which is stable
when δDis small, first becomes unstable when δDU/ν ≈600 as we can see by constructing a vertical
tangent to the neutral stability curve. Since we have seen in section (Bjd) that δD=1.72(νx/U)1
2for the
Blasius boundary layer it follows that this will first become unstable when the distance xfrom the leading
edge increases to a value given by
Ux
ν=(Rex)=600
1.722
=1.22 ×105(Bkd1)
where Rexdenotes the Reynolds number based on Uand x. It also follows that the first frequency which
becomes unstable is given by the point where the vertical tangent touches the neutral stability curve and
from Figure 1 that frequency is given by
ωR≈1.6×10−4U2
ν(Bkd2)
Since it is that frequency which is first amplified, it is usually that frequency that is observed experimentally
during the first stages of transition. This assumes that the inherent noise available for amplification is at
least broadband if not white.
