The following text was automatically extracted from the image on this page using optical character recognition software:
NACA TM 1265 3
report, therefore, deals first, with the distribution of the amplitude
of the disturbance over the flow section, that is, the calculation of
the characteristic functions and second, with the study of the energy
distribution and energy balance of the disturbance motion. The investi-
gations are based upon the disturbances of the laminar flow past a flat
plate which are situated exactly at the boundary between amplification
and damping (neutral oscillations).
Chapter I
AMPLITUDE DISTRIBUTION
2. Discussion of the differential equation of disturbance.
Let U(y) be the velocity distribution of the -undamental flow
(fig. 1) and 4 the flow function of the superposed disturbanQe motion,
which is assumed as a wave motion moving in the x direction (direction
of primary flow), whose amplitude 9 is solely dependent on y, hence
(x,y,t) -= (y)ei(Ca-t) = q(y)eia(x-ct)
a is real and X = 2~a/ is the wave length of the disturbance;
P = Br + ii and c = [/a are, in general, complex; T = 21/Pr is
the period of oscillation; [i indicates the amplification or the damping,
depending upon whether positive or negative; cr = [r/m is the phase
velocity of the disturbance. For the disturbance amplitude cp, after
introduction of dimensionless variables from the Navier-Stokes differential
equations, it results in a linear-differential equation of the fourth
order, the differential equation of the disturbance
(U - c)(p" - aq) - 'p = - (" -2a2 + 9) (1)-
(R -- Um /V = Reynolds number, Um = constant velocity outside of the
boundary layer, 5 = characteristic length of the boundary-layer
profile = boundary-layer thickness, v = kinematic viscosity.) The
general solution cp of the disturbance equation is built up from four
particular solutions 9~1, *I , o4
9 = C1' + - + + 0393 + C09G (2)
