Hydrodynamic Stability
Lecture 22
Stability of density stratified shear flows.
Similar to
K – H
Competition between shear resulting in instability, that does work against stable density
stratification
Assumptions – Inviscid, nondiffusive, incompressible, Boussinesq approximation, statically
stable
( )
( )
( )
( )
( )
0
0
Basic State
,0,0
0
z
z
u
U
z
z
p
p z
p
g
z dz
ρ
ρ
ρ
ρ
∂
≤
∂
=
=
′
=
=
−
∫
±
Horizontal, rectilinear flow whose density varies with height.
Heuristic argument:
Source of instability comes from KE in basic state, can be converted to perturbation in K.E., has to
do work against density (a potentional energy mechanism).
If KE liberated by the instability is less than the work required to liberate it, then expect the flow
to be stable.
1
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(
)
1
1
2
2
2
2
1
2
1
particle at
, has
,
particle at
, has
,
Part at
:
z
U
z
U
z
z
z
z
1
2
ρ
ρ
ρ
ρ
ρ
∂
=
+
−
∂
Using Taylor series expansion, (they’re close enough together).
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 Fall '08
 Staff
 Fluid Dynamics, Ri, Boussinesq, Boussinesq approximation, stable density stratification

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