This preview shows pages 1–3. Sign up to view the full content.
Hydrodynamic Stability
Lecture 11
RayleighBenard solution continued.
Take
(momentum equation)
∇
i
0
u
∇=
i
±
Divergence of gradient gives Laplacian.
2
pg
z
ρ
∂
−
∂
A Poisson equation for
p
in terms of
.
Take
of
z
component of momentum equation.
2
∇
22
2
2
2
2
1
00
0
0
0
11
gg
vw
p
g
tz
z
z
g
ρρ
∂∂
∂
∂
−∇ ∇
=−
∇ −
∇
−
−
∇
∇
∂
∂
Horizontal Laplacian operator.
2
1
2
2 2
1
0
Take
of previous equation.
xy
K
t
g
Kv
w
K
tt
t
2
+
∂
−∇
∂
∂
−∇∇ =
− ∇
∂
R.H.S is identical to LHS of energy equation.
()
2
10
0
2
1
So one equation in one unknown:
g
w
w
g
ραβ
αβ
∇
−
∇
2
w
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentNondimensionalize
Length scale:
Lh
=
Time scale:
2
h
K
=
T
For diffusion of heat (we’ve dropped temperature).
Velocity scale:
K
h
=
V
ˆ
ˆ
;
ˆ
tx
Th
u
u
V
==
=
±
±
±
±
Plug into the equation.
()
223
ˆ
, etc.
multiply result by
and drop hats again
k
uu
h
hhh
kvk
=
±
²
So equation is for nondimensional quantities.
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 Staff

Click to edit the document details