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Orr-Sommerfeld - Foundations of Fluid Mechanics II - Handout, Exercises 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 of this lecture are: Orr-Sommerfeld, Stability of Nearly Parallel Flows, Solutions of the Orr-Sommerfeld, Qualitative Example Problems, Method of Normal Modes, Wavenumber Vec

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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()
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).
pf3
pf4
pf5

Partial preview of the text

Download Orr-Sommerfeld - Foundations of Fluid Mechanics II - Handout and more Exercises Fluid Mechanics in PDF only on Docsity!

M E 522 Spring 2008 Professor John M. Cimbala Lecture 16 02/20/

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

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 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 :

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 ˆ ( p y ).

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 2 4

Re

yy yy yyyy yy

U c k U k k

ik

(7), the Orr-Sommerfeld equation.

  • Step 6. Examine stability: Finally, we examine solutions of the Orr-Sommerfeld equation (to be done in class).