Temporal Stability of Locally Parallel Flow – The Orr-Sommerfeld Equation
1. Problem Setup:
x
y
U(y)
λ
cr
Consider an incompressible, locally parallel flow with some known steady
basic state U(y). We examine this problem using linear stability analysis. Note:
A boundary layer type of flow is sketched, but the procedure applies to any
kind of parallel or nearly parallelflow.
2. Summary of Linear Stability Analysis:
The in-class analysis follows Kundu, Section 12.8 closely, filling in some of the details. Start with the normalized incompressible
Navier-Stokes equations for total flow variables ( ):
q
0
i
i
u
x
∂=
∂
(1) and
2
1
Re
ii
ji
j
ij
uu u
p
u
txx x
∂∂ ∂
∂
+=−+
∂∂∂ ∂∂
j
x
(2) where 0
Re UL
ν
≡ and U0 and L are a characteristic velocity and a
characteristic length, respectively. This represents 4 equations and 4 unknowns, nonlinear p.d.e.s.
• Step 1. Start with the basic state (Q): Ui = (U(y), 0, 0). Continuity yields 00
=
(1b). The y and z momentum equations show
that P = P(x) only, and the x-momentum reduces to 2
1
0Re
dP U
dx
=− + ∇ (2b).
• Step 2. Add disturbances (uU , ,u=+
0vv=+
0ww
=
+
,
p
Pp
=
+
), and plug them into (1) & (2): This generates the
total equations (1t) and (2t).
• Step 3. Subtract the basic state equations from the total equations: This generates the disturbance equations (1d) and (2d).
• 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
(,,,) ()ikx mz kct
uxyzt uye +−
=, ()
ˆ
(,,,) ()ikx mz kct
vxyzt vye +−
=, ()
ˆ
(,,,) ()ikx mz kct
wxyzt wye +−
=, and
(
ˆ
(,,,) ()i kx mz kct
pxyzt pye +−
=)
, where variables with hats are complex amplitudes. k and m are the x and z components,
respectively, of wavenumber vector . 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:
K
G
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
() Re yy
ik U c w imp w k m w
ˆ
⎡
⎤
−=−+ −+
⎣
⎦ (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, , vy, and .
ˆ()uy ˆ() ˆ()py
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:
()
2
1
() 2
Re
yy yy yyyy yy
Uc k U k k
ik
24
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).
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