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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 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
Typology: Exercises
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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 K
. 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
−
potential function, but simply the magnitude of the disturbance stream function. Plugging this into Eqs. (4) to (6) yields one
2 2 4
Re
yy yy yyyy yy
U c k U k k
ik
(7), the Orr-Sommerfeld equation.