Page 254 - Computational Fluid Dynamics for Engineers

244 8. Stability and Transition

{
- tL- f—\ — - — (8.1.3)
a \ a* J uo uo
The solution of the Orr-Sommerfeld equation and its boundary conditions
may be obtained by temporal or spatial amplification theories. The former takes
uo to be complex (= uj r + iuii) so that the amplitude of the disturbance varies
with time as exp(c<;^), in contrast to the spatial amplification theory which takes
u to be real and a to be complex [Eq. (2.5.13)], or u to be real and c to be
complex [Eq. (8.1.2)], so that the amplitude varies with x as exp(—otix). Note
that for LOi — 0 and given a r, a complex value of c implies values of the real
frequency, u; r, and the spatial amplification rate (—c^). Also, if a and uo are
both real, then the disturbance propagates through the parallel mean flow with
constant amplitude | </>(£/) |; if ot and UJ are complex, the disturbance amplitude
will vary in both time and space.
While both procedures have advantages, it is more convenient to use the
spatial amplification theory since the amplitude change of disturbance with
distance can be measured in a steady mean flow. The amplitude at a fixed
point is independent of time and spatial theory gives the amplitude change in
a more direct manner than does the temporal theory.
The solution of the system given by Eqs. (2.5.13) and (8.1.1), with the real
parts of £i, £2 strictly positive, exists only for certain combinations of Reynolds
number R and the parameters of the disturbance a and u since all the boundary
conditions are homogeneous. Thus the problem is an eigenvalue problem in
which values of i?, a and uo are the eigenvalues and the corresponding amplitude
functions are eigenfunctions. Hence, in general, no nontrivial solution of this
system exists. Only if a, cj, R satisfy one or more relations of the form

F{a,u),R) = 0 (8.1.4)

can such a solution be found.
The eigenvalues of the OS equation for the spatial amplification case are
often presented in (a, R) and (a;, R) diagrams that describe the three states
of a disturbance at a given Reynolds number R as damped, neutral or ampli-
fied. For two-dimensional flows the locus cii = 0, or q = 0, called the curve of
neutral stability, separates the damped (stable) region from the amplified (un-
stable) region. The point on this curve at which R has its smallest value is of
special interest, because at values of R less than this value, all disturbances
are stable. This smallest Reynolds number is known as the critical Reynolds
number, R cr. The neutral curve is the same in both temporal and spatial am-
plification theories. It is defined by a characteristic length L, velocity uo and
kinematic viscosity v, that is