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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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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
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.
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
dx
= − + ∇ (2b).
total equations (1t) and (2t).
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 ).
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 :
y
iku + v + imw =
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
−
)
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.