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Orr Sommerfeld Equation - Foundations of Fluid Mechanics II - Lecture Notes, Study notes of Fluid Mechanics

This is the second course of a two-semester fluid mechanics sequence for graduate students in the thermal sciences. This course includes topics like fully turbulent flows, turbulent boundary layers and free shear flows, turbulence modeling, laminar boundary layers including axisymmetric and 3-D boundary layers. Key points in this lecture are: Orr Sommerfeld Equation, Temporal Stability, Locally Parallel Flow, Linear Stability Analysis, Navier-Stokes Equations, Method of Normal Modes, Wavenumber

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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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Temporal Stability of Locally Parallel Flow – The Orr-Sommerfeld Equation

1. Problem Setup:

x

y

U ( y )

λ

c

r

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 ):

i

i

u

x

(1) and

2

Re

i i

j

i

j i j

u u p u

u

t x x x

j

x

(2) where

0

Re

U L

ν

and U 0

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 ): U

i

= ( U ( y ), 0, 0). Continuity yields 0 = 0 (1b). The y and z momentum equations show

that P = P ( x ) only, and the x -momentum reduces to

2

Re

dP

U

dx

= − + ∇ (2b).

  • Step 2. Add disturbances ( u  = U + u , v  = 0 + v , w  = 0 + w , p  = P + p ), 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

Re

u u dU p

U v u

t x dy x

,

2

Re

v v p

U v

t x y

,

2

Re

w w p

U w

t x z

(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

( )

i kx mz kct

u x y z t u y e

( )

i kx mz kct

v x y z t v y e

( )

i kx mz kct

w x y z t w y e

= , and

(

i kx mz kct

p x y z t p y e

)

, 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

y

iku + v + imw =

(1n), ( )

2 2

Re

y yy

ik U c u vU ikp u k m u

,

2 2

Re

y yy

ik U c v p v k m v

, and

2 2

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: m = 0 &

w = 0. The normal mode equations simplify:

y

iku + v = (4),

2

Re

y yy

ik U c u vU ikp u k u

(5), and

2

Re

y yy

ik U c v p v k v

(6).

We are now down to 3 o.d.e.s and 3 unknowns, u y ˆ( ), v y ˆ( ) , and p y ˆ ( ).

Orr-Sommerfeld Equation : Define a disturbance stream function,

(

ik x ct

ψ x y t φ y e

)

. Note : φ ( ) y 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

Re

yy yy yyyy yy

U c k U k k

ik

2 4

− φ − φ − φ= ⎡φ − φ + φ⎤

(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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