Abstract
The linearized equation of motion (2·10) and the equation of mean motion ρ
u̅v̅
=
Py
+ μ
dU
/
dy
are solved simultaneously to find periodic solutions of finite amplitude. The expression
U
of the mean flow and the integrals ψ are interdependent; ψ, however, depends mainly on the value of
U
"
0
/
U
'
0
at the critical point (defined in §3). Accordingly, the linearized equation is solved for a set of values of
U
"
0
/
U
'
0
; for each case the critical point and corresponding critical Reynolds number are determined. These integrals, which are functions of
U
"
0
/
U
'
0
, are inserted into the equation of mean motion; this leads to an equation between the amplitude of the vibration and the Reynolds number.
Results
. To each Reynolds number there corresponds a definite value of the amplitude
a
. For infinitesimal values of the amplitude the critical
R
is about 5000; and
R
is reduced with increase of the amplitude, reaching its minimum value of about 2900. The maximum value of
u
is about 11.5 % of
U
(0), and the mean value of |u| across the channel is about 8.4 % of the mean value of the main flow. The actual profile differs slightly from the parabola; the second derivative, however, is considerably changed.
Cited by
101 articles.
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