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Formulation of linear hydrodynamic
stability problems
Contents:
1. Governing equations
2. Parallel shear flows
3. Linearization
4. Orr-Sommerfeld and Squire equations
5. Eigenvalue problem
6. Inviscid linear stability problem
7. Destabilizing action of viscosity
8. Instability in space
9. Gaster’s transformation
10. Squire’s transformation
11. Completeness of the solutions of the Orr-Sommerfeld equation
1
Governing equations

u x u y u z  




  u  v
 w u
x t y t z t  x
y
z 

 Euler equations 
du u
 
1
 2

 (u)u   p   u
dt t



(u )  0
NSE=momentum equations
The term taking into account presence of
stresses and responsible for presence for
development of shear stresses
(continuity equation=mass conservation)

+ boundary and initial conditions

u  ( u , v , w)






x y z
      
 i 
j k
x
y
z



- kinematic viscosity
Streamwise (x), wall-normal (y) and
spanwise (z) velocities
in Cartesian coordinates r=(x,y,z)
y
Free stream
direction
j

k i
x
z
2
Reynolds number
• There are many ways to derive the expression for Reynolds number
and display its significance
• Law of similarity: the flow about a body in simplest cases is
determined by a characteristic velocity U [m/s], viscosity n [m2/s], and
a characteristic size of the body L [m]. There is only one nondimensional combination of these parameters, expressing the similarity
of such flows:
Re=UL/ 
Any other non-dimensional parameter can be written as a function of
Re.
• In such a way the NS-equations can be made non-dimensional:


~
~
 ~ ~
du u
~ ~ 1 ~ 2 ~
~
 ~  (u )u  p 
 u,
~
dt t
Re
where
Some of these combinations also have names,
  u v w  ~ tU
~
u   , , , t 
,... e.g., Strouhal number: St=2pfL/U, that is
inverse of non-dimensional t.
L
U U U 
3
Nonlinear disturbance equations

• Let U  U (r ) and P(r) be distributions of velocity and pressure of a known
stationary solution of the Navier–Stokes equations of an incompressible
fluid:
1 2
(U)U  P 
U
Re
(for simplicity vector sign is removed)
(U)  0
• with natural boundary conditions U(S) = 0 at boundaries (walls) S.
U(x,y)
4
Nonlinear disturbance equations, cont.
Let us impose a disturbance u(r, t) and p(r,t), where r is a coordinate vector and t
is time, so that the resultant motion
U(r)=U(r)+u(r,t), and P(r) = P(r) + p(r, t),
also satisfy the Navier–Stokes and continuity equations as well as the boundary
conditions:
∂U/∂t+(U ∇) U =−∇P+∇2 U /Re,
(∇ U)=0.
or
∂(U+u)/∂t+((U+u)∇)(U+u)=−∇P+∇2(U+u)/Re,
(∇(U+u))=0.
removing the parenthesis and separating the values (u,p) from (U,P) yields
∂u/∂t+(U∇)u+(u∇)U+(u∇)u=−∇p+1/R∇2u,
(∇u) = 0.
5

Parallel shear flows


du u
 
1 2

 (u)u  p 
u
dt t
Re

(u )  0
NSE=momentum equations
(continuity equation=mass conservation)
+ boundary and initial conditions

The flow streamlines exhibit neither divergence


U

U
(
y
),
0
,
0
Assume:
nor convergence downstream: “parallel twodimensional (2D) flow”.
Local parallelity of the flow streamlines means that the streamlines diverge/converge
slowly comparing with the processes of interest.
6
Linearization
Assume that

u  U ( y)  u, v, w and suppose that disturbances u, v, and w are small,

so that the nonlinear (quadratic disturbance as u2, uv,
vw,…) terms in the NS-equations can be dropped.
+ boundary and initial conditions
equation for normal velocity:
evolution
equation for v

7
Linearization, cont.
To describe the complete three-dimensional flow field, a second equation is needed (the
third one is provided by the continuity equation). This is most conveniently the equation of
normal vorticity, which describes the “horizontal” flow
y
Free stream
direction
x

evolution
equation for 

+ initial conditions
This pair of these ordinary differential equations equations provides a complete
description of the evolution of an arbitrary linear disturbance.
8
Orr-Sommerfeld and Squire equations
The classical approach to the solution of such stability problems is the method of normal
modes, consisting of a reduction of the linear initial-boundary-value problem to an
eigenvalue problem. Let us suppose that the full solution can be expressed as a sum of
elementary solutions (modes), which have the form
Arbitrary ‘small’ complex amplitudes
  streamwise (generally complex) wave number
i(x  z -t )
~
v( x, y, z, t )  Av ( y )e
  spanwise (generally complex) wave number
i(x  z -t )
~
 ( x, y, z, t )  B ( y )e
  circular (generally complex) frequency
Then, the evolution equations for the normal velocity and normal vorticity are reduced to
the Orr-Sorrmerfeld
equation
the Squire equation
d
with boundary conditions v~  Dv~  ~  0 at solid walls and in free stream, D 
dy
dU
d 2U
U' 
,U ' ' 
.
dy
dy 2
The OS-equation with homogeneous b.c. constitutes an eigenvalue problem for normal
velocity disturbances. In the Squire equation the right-hand term depending on the normal
velocity is a ‘driving’ force for the disturbances of normal vorticity.
9
Examples of the eigenoscillations
duct solid boundary
c

duct solid boundary
A duct with acoustic waves or electromagnetic microwave

“liquid boundary”
U

a boundary layer with an OS mode
vr
vi
solid boundary
10
Eigenvalue problem
To remind you the meaning of the eigenvalue problem, let us consider a model algebraic problem
ax  by  0  a b  x   0 
 or      , a  0.
cx  dy  0   c d  y   0 
It has nontrivial solutions only if
ad  bc  0.
This solvabilit y relation is sometimes called characteri stic or dispersion relation.I f it holds,
the solution is given by the vector
 -b / a 

  k , k  C ,
1


which is called eigenvecto r. In our case, supposing U and Re are given, the role of the
coefficien ts (eigenvalu es) play  ,  and . From the solvabilit y relation it is clear that given,
for example, b, c and d , it is always possible to determine a (eigenvalu e). The same with the OS and Squire
equations. Given, e.g.,  ,  allow us to find , or given ,  allow us to find  .
11
Solution of the Orr-Sommerfeld
equation
OSE
12
Solution of the Squire equation
imaginary part
real part
13
Interpretation of modal results
(some definitions)
k

g

14
Analysis and Synthesis
original momentum
equations for disturbances
linearization
reduction to eigenvalue problem
(spectral analysis)
complete set of linearlyindependent modes (waves)
solution of initial value problem
expansion of any smallamplitude disturbance to the
set of the modes
15
Spectral formulation of stability
F=0
16
Spectral formulation of stability, cont.
Re=hU/
17
Inviscid linear stability problem
The instability waves related to the solutions of the Rayleigh equation are called
Rayleigh waves.
Note that Rayleigh equation in the final form is unchanged when  is replaced by -.
Thus, it is customary to consider >0. Also if v~ is an eigenfunction with c for some 
then so is v* with eigenvalue c* for the same . Thus, to each unstable mode, there exists
a corresponding stable mode (* means here complex conjugation).
Considering the inviscid stability problem in time for two-dimensional (direct) waves,
Rayleigh proved some important general theorems.
18
Rayleigh’s inflection point criterion
(first Rayleigh theorem)
The inflection point is a point ys such that
 2U  y s 
U 
0
2
y
That is if the growth rate ci or i≠0, then 2nd derivative of flow mean velocity U”
''
s
profile changes sign between flow boundaries.
Formal multiplication of the Rayleigh
equation by its complex-conjugate
solution v~ * and integration by parts
over y.
The limits are taken [-1;+1], but this
does not lead to a lack of generality.
The procedure can be repeated for any
other limits.
19
Rayleigh’s inflection point criterion,
cont.
on
20
Second Rayleigh theorem
and the critical layer
Rayleigh eqn. (R.E.)
U ' ' ~
 2
2
D  
v  0
U c

R.E.
(for ci≠0).
critical
layer
large velocity
gradient
effects of viscosity
at the wall
21
Criterion of the maximum vorticity
on
Here Us is the velocity at inflection point.
:
U " (U  cr ) ~ 2
| v | dy    | Dv~ |2  2 | v~ |
2
|U  c |
1
1
1
1

 dy.
2 2
if we add
cr  U s  | Dv~ |2  2 | v~ |2 2dy  0
1
1
to the left hand side, we obtain
U " (U  U s ) ~ 2
~ |2  2 | v~ |2 2 dy  0

|
v
|
dy


|
D
v
2
 |U  c |

1
1
1
from which follows that
U " (U  U s ) ~ 2
| v | dy  0.
| U  c |2
1
1

1
Thus, only the velocity profiles with the
inflection points associated with the
maximum shear are unstable; i.e. the
inequality U”(U-Us)<0 should be
satisfied over a certain range of y.
Taking into account the first Rayleigh
theorem, this is equivalent to the
requirement of a relative maximum of the
absolute value of the vorticity at ys for the
instability to occur. It follows in particular
from the criterion that the Couette flow is
stable in the inviscid approach.
Analysis of stability of flows through the stability criteria: a stable, U’’<0; b stable, U’’>0, c stable, U’’=0, but
U’’(U-Us)≥0; d probably unstable, U’’s=0 and U’(U-Us)≤0.
22
Semicircle theorem
The complex wa ve speed c for unstable mode lies inside the semicircle in the
upper half c - plane which has the range of U as its diameter along the real axis.
If we denote
Q | v~' |2  2 | v~ |2  0,
then the real and imaginary parts of the equation used to proof the 1st Raylei gh theorem
becomes :
 U  c 
1
2
r

1
 c Qdz  0,
2
i
1
2ci  U  cr Qdz  0.
1
Now using the 2nd Rayleigh theorem we have
1
 (U-U
min
)(U-U max )Qdz  0, but this can be written as
Umin and Umax are minimum and maximum
velocity in the flow
-1
 (c
2
r
stable

1
 c )-(U max  U max )cr  U max U max Qdz  0.
2
i
-1
unstable
It follows that
(cr2  ci2 )-(U max  U max )cr  U max U max  0,
which on rearrangem ent gives
2
1


1

2
c

(
U

U
)

c

(
U

U
)
r
max
max
i
max
max


 2

2
2
23
Behavior of u and v of 2D wave
at finite Re in the critical layer
| v~ |
| u~ |
U”
U
Falkner-Skan basic velocity profile
with backflow, H=‒0.1
=0.06
inflection
point
The streamwise velocity tends to
infinity in the inviscid limit like:
u
backflow
U " ( ys )
ln( y  y s )
U ' ( ys )
unless U''=0 in the critical layer.
24
Destabilizing action of viscosity
As the Reyleigh inviscid instability acts at the Reynolds number Re→∞, it is clear
that viscosity and, as a consequence, the viscous instability can destabilize the flow
only at a finite Reynolds number.
The physical mechanism of the viscous instability can be seen from the equation of
energy balance:
Considering two-dimensional disturbances in a parallel flow, the equation can be
simplified to the form:
dE
 v u 
1
V
This term is zero
without the shift
dt
  vu*  uv * U ' dy 
  dxdy
Re   x y 
re is a
25
Examples
Results of viscous computations
Different asymptotic behavior at Re→∞.
That is:
• The inviscid fluid may be unstable and the viscous fluid stable. The effect
of viscosity is then purely stabilizing.
• The inviscid fluid may be stable and the viscous fluid unstable. In this
case viscosity would be the cause of the instability.
26
Instability in space
The problem of initial conditions or stability in time. If the initial
disturbance decays in time at each fixed point of space (or, at least,
does not monotonically grow), the system is called stable to these
disturbances. Otherwise, if the initial disturbance monotonically
grows in time at a fixed point of space, the system is called
absolutely unstable. In physiscs such systems are denoted
sometimes as “generators”.

’
Disturbance source

The problem of boundary conditions or amplification in space. If an
external signal at the entrance to the system decays whilst propagating
in it, it is said that the spatial attenuation (non-transmission of a
signal) takes place. Otherwise, there is a spatial amplification, and the
system is called convectively unstable. In physics such systems are
Disturbance source
called sometimes “amplifiers”.
27
Difference in the evolution equations

That is, we consider the
development in time for
the wave with given 
and .
evolution
equation for v
28
Difference in the evolution
equations, cont.
They provide quasi-linear parabolic partial differential equations in time:
+
 
 1
2
2 ~
~( y , t )



i

U

D

k

(
y
,
t
)


i

U
'
v


 t

 Re


Parabolic, has first order derivative in time (initial-value problem)
29
Difference in the evolution
equations, cont.
s:
Development along x and z. For twodimensional flows, the symmetry
prescribe  real, so we can consider
the growth only along x.
1 2 ~

~( y , t )



i


i

U



(
y
,
t
)


i

U
'
v

Re 
Elliptic, absence of the time derivative (b.v. problem)
A formulation of the later problem as the initial value problem is ill-posed, as the
solution depends on the conditions at the downstream boundary and there are
solutions propagating upstream.
One has to regularize the problem by imposing additional constraints to the initial
data. It was proposed to exclude all solutions propagating upstream. A possible
physical explanation of this is a fast ‘loss’ of the effect of the downstream boundary
conditions in the bulk of the boundary layer or channel flow due to a quick decay of
the upstream propagating disturbances. We will show this by considering the structure
of the spectrum of the Orr-Sommerfeld and Squire equations.
30
Eigenvalue problem
OS
SQ
Note that negative i correspond to unstable disturbances, because of the factor:
ei(x+y-t)=e-iei(rx+y-t).
The OS-equation constitutes the 4th order polynomial (sometimes called “nonlinear”)
eigenvalue problem in .
31
Gaster’s transformation
Gaster (1962) proved that ther e are simple mathematic al relations between
parameters of instabilit ies in time and in space for two - dimensiona l waves.
In particular , he showed that
 rs   t ,
 s  rt ,
it
r


  cg ,
s
i
 r
where cg is group velocity and superscrip ts s and t denotes spatial and temporal
solutions, respective ly.
The transform ation is valid, only if i and  i are rather small (almost neutral
disturbanc es).
32
Gaster’s transformation, cont.
values at thу neutral curve
The prove is centered in considering the total differential of the general form of the
disturbance relation about a neutral disturbance in complex plane.
Using it we obtain:
relates small changes in a to small
changes in w through the group velocity
For flows with small dispersion , i.e. when
cg  c p 
r
,
r
the transform ation becomes particular y simple.
For strongly unstable flows, the transformation is not accurate, as it is based
on a linear expansion about a neutral value.
33
Inviscid instability in space
• The Rayleigh theorems is impossible to
prove for complex .
• However, the results for neutral
disturbances are equally applicable for
both cases.
• For non-neutral cases (slightly stable or
unstable) the Gaster’s transformation is
applicable.
34
Squire’s transformation
It is clear that both equations are equivalent, if put
  2   2 Re 2 D   Re
35
Completeness of the solutions of the
Orr-Sommerfeld equation
For the OS and Squire equations a proof is required for the completeness of the
solutions.
36
Structure of the solutions
of the Orr-Sommerfeld equation
37
Structure of the solutions
of the Orr-Sommerfeld equation
(pressure waves)
vr
vi
38
Structure of the solutions
of the Orr-Sommerfeld equation
(vorticity waves)
downstream
upstream of the source
vi
vr
39
Structure of the solutions
of the Orr-Sommerfeld equation
(discrete waves)
40
Further reading
• Betchov R. and Criminale W. O. (1967) Stability of parallel flows, NY:
Academic.
• Drazin P. G. and Reid W. H. (1981) Hydrodynamic Stability, Cambridge
University Press
• Gaster M. (1962) A note on the relation between temporally-increasing and
spatially-increasing disturbances in hydrodynamic stability, J. Fluid Mech.,
Vol. 14, pp. 222‒224.
• Schmid P.J., Henningson D.S. (2000) Stability and transition in shear flows,
Springer, p. 1‒60.
41