M E 522 Spring 2008 Professor John M. Cimbala Lecture 16 02/20/2008
Today, we will:
•
Continue to discuss the stability of nearly parallel flows – the Orr-Sommerfeld Eq.
•
Look at some qualitative example problems – Solutions of the Orr-Sommerfeld Eq.
We left off here last time …
• Step 4. Linearize the disturbance equations to generate the linearized disturbance equations
0
i
i
u
x
∂=
∂
(1l) and
2
1
Re
uudUp
Uv u
txdyx
∂∂ ∂
++=−+∇
∂∂ ∂
,
2
1
Re
vvp
Uv
txy
∂∂∂
+=−+∇
∂∂∂
,
2
1
Re
wwp
Uw
txz
∂
∂∂
+
=− + ∇
∂
∂∂
(2l).
This still represents 4 equations and 4 unknowns, but the equations are now linear. (Note: the disturbance variables are now
the unknowns since the basic state is known.) These are still p.d.e.s since u, v, w, and p are functions of (x, y, z, t).
• Step 5. Solve the linearized disturbance equations (1l) and (2l): We use the method of normal modes.
Method of Normal Modes: Assume disturbances that are periodic in x and z, but not growing or decaying in x or z, and may
be periodic and may be growing or decaying in t. (temporal instability.) Specifically, let the disturbances be of the form
()
ˆ
(,,,) ()
ikxmz kct
uxyzt uye
+−
=
,
()
ˆ
(,,,) ()
i kx mz kct
vxyzt vye
+−
=
,
()
ˆ
(,,,) ()
ikx mz kct
wxyzt wye
+−
=
, and
()
ˆ
(,,,) ()
ikx mz kct
pxyzt pye
+−
=
, where variables with hats are complex amplitudes. k and m are the x and z components,
respectively, of wavenumber vector
K
G
. For temporal stability analysis, both k and m must be real, while complex wave
speed c can be complex. (Otherwise spatial instability would also be possible.) Plug these disturbances into Eqs. (1l) and
(2l) to get the normal mode equations:
ˆˆ ˆ0
y
iku v imw++ =
(1n),
()
22
1
ˆˆ ˆ ˆ ˆ
() Re
yyy
ik U c u vU ikp u k m u
⎡
⎤
−+ =−+ −+
⎣
⎦
,
()
22
1
ˆˆ ˆ ˆ
() Re
yyy
ik U c v p v k m v
⎡⎤
−=−+ −+
⎣⎦
, and
()
22
1
ˆˆ ˆ ˆ
() Re
yy
ik U c w imp w k m w
⎡
⎤
−=−+ −+
⎣
⎦
(2n).
Note: For convenience in Eqs. (1n) and (2n), subscript y denotes differentiation with respect to y. We are now down to 4
o.d.e.s and 4 unknowns since U(y) is known, along with its derivatives.
Squire’s Theorem: In 2-D parallel flow, for each unstable 3-D disturbance, there corresponds a more unstable 2-D
disturbance. In other words, the most unstable case is the 2-D one:
0m
=
&
ˆ0w
=
. The normal mode equations simplify:
ˆˆ 0
y
iku v+=
(4),
2
1
ˆˆ ˆ ˆ ˆ
() Re
yyy
ik U c u vU ikp u k u
⎡
⎤
−+ =−+ −
⎣
⎦
(5), and
2
1
ˆˆ ˆ ˆ
() Re
yyy
ik U c v p v k v
⎡⎤
−=−+ −
⎣⎦
(6).
We are now down to 3 o.d.e.s and 3 unknowns,
ˆ()uy
,
ˆ()vy
, and
ˆ()
p
y
.
Orr-Sommerfeld Equation: Define a disturbance stream function,
()
(,,) ()
ik x ct
xyt ye
ψφ
−
=
. Note:
()y
φ
is not a velocity
potential function, but simply the magnitude of the disturbance stream function. Plugging this into Eqs. (4) to (6) yields one
o.d.e. and one unknown:
()
224
1
() 2
Re
yy yy yyyy yy
Uc k U k k
ik
φ
φφ φ φφ
⎡
⎤
−−−= −+
⎣
⎦
(7), the Orr-Sommerfeld equation.
• Step 6. Examine stability: Finally, we examine solutions of the Orr-Sommerfeld equation (to be done in class).
