22 General Criteria for Instability

A distinctive feature of the velocity profiles shown in Fig. 4.1 is that one has an inflexion point but the other does not. The importance of this fact and its bearing on the stability or instability of the flow was first recognized by Rayleigh (1880), who proved

• Rayleigh s inflexion-point theorem. A necessary condition for instability is that the basic velocity profile should have an inflexion point.

To prove this theorem first rewrite Rayleigh s equation in the form

and suppose that c _i>0 so that the equation is non-singular. On multiplying this equation by *, integrating from z ₁ to z ₂, and then integrating the first term by parts, we obtain

The imaginary part of this equation is

from which it follows that U" must change sign at least once in the open interval ( z ₁, z ₂).

A stronger form of this condition was obtained later by Fj rtoft (1950) who proved

• Fj rtoft s theorem. A necessary condition for instability is that U ?( U ? U _s)<0 somewhere in the field of flow, where z _s is a point at which U ?=0 and U _s= U( z _s).

To prove this theorem consider the real part of equation (22.2):

If we now add

to the left-hand side of equation (22.4) we obtain

from which the result follows. Thus, if U(z) is a monotone function with...