Lecture 3
Reynolds Supposition
Assume
steady solution to N. S. equations and B. C.
()
ux
±±
Examine Behavior
() ( )
,
ux u xt
′
+
±
Analytical Methods
1. Linear theory – assume small
u
′
, linearize equations
eigenvalue problem.
→
Gives growth rate of disturbances.
2. Weakly nonlinear theory – extends linear theory, perturbation methods
3. Energy methods – examine overall energetics, see if total energy of system will grow or
decay.
Don’t generally need to linearize, but not too useful.
Variational methods
(calculus or variations) are used.
Minimization/ maximization principles.
4. Direct Simulations – no linearization (fully nonlinear).
Experimental Methods
Also experimental approaches generally set up a “clean” laminar flow – examine instability that
develops.
1. Natural
2. Forced
Often linear theory gives sufficient condition for instability
.
If linear theory predicts instability, small amplitude perturbations will grow.
Large amplitude
perturbations will grow or decay, if it decays, linear theory will come in.
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 Fall '08
 Staff
 Fluid Dynamics, Fundamental physics concepts, B. C., linear theory

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