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Formulation of linear hydrodynamic stability problems

Contents:

1. Governing equations2. Parallel shear flows3. Linearization4. Orr-Sommerfeld and Squire equations5. Eigenvalue problem6. Inviscid linear stability problem7. Destabilizing action of viscosity8. Instability in space9. Gaster’s transformation10. Squire’s transformation11. Completeness of the solutions of the Orr-Sommerfeld equation

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Governing equations

upuut

u

t

u

21)(

d

d

0)( u

+ boundary and initial conditions

- kinematic viscosity

(continuity equation=mass conservation)

NSE=momentum equations

),,( wvuu

zyx

in Cartesian coordinates r=(x,y,z)

kz

jy

ix

Euler equations

Streamwise (x), wall-normal (y) and spanwise (z) velocities

y

x

z

Free streamdirection

uz

wy

vx

ut

z

z

u

t

y

y

u

t

x

x

u

The term taking into account presence of stresses and responsible for presence for development of shear stresses

ij

k

3

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 non-dimensional 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:

,...~,,,~

where

,~~Re

1~~~)~~(~

~~d

~d 2

L

tUt

U

w

U

v

U

uu

upuut

u

t

u

Some of these combinations also have names, e.g., Strouhal number: St=2fL/U, that is inverse of non-dimensional t.

4

Nonlinear disturbance equations• Let and P(r) be distributions of velocity and pressure of a known

stationary solution of the Navier–Stokes equations of an incompressible fluid:

• with natural boundary conditions U(S) = 0 at boundaries (walls) S.

)(rUU

0)(Re

1)( 2

U

UPUU(for simplicity vector sign is removed)

U(x,y)

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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.

Nonlinear disturbance equations, cont.

6

Parallel shear flows

upuut

u

t

u

2

Re

1)(

d

d

0)( u

+ boundary and initial conditions

(continuity equation=mass conservation)

NSE=momentum equations

Assume: 0,0),( yUU The flow streamlines exhibit neither divergence

nor convergence downstream: “parallel two-dimensional (2D) flow”.

Local parallelity of the flow streamlines means that the streamlines diverge/converge slowly comparing with the processes of interest.

7

LinearizationAssume that wvuyUu ,,)(

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

evolution

equation for v

equation for normal velocity:

8

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

xFree streamdirection

+ initial conditions

This pair of these ordinary differential equations equations provides a complete description of the evolution of an arbitrary linear disturbance.

evolutionequation for

9

Orr-Sommerfeld and Squire equations

)-i(

)-i(

e)(~),,,(

e)(~),,,(tzx

tzx

yBtzyx

yvAtzyxv

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

Then, the evolution equations for the normal velocity and normal vorticity are reduced to

frequency complex) (generallycircular

number wavecomplex) (generally spanwise

number wavecomplex) (generally streamwise

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.

Arbitrary ‘small’ complex amplitudes

0~~~ vDv

the Orr-Sorrmerfeldequation

the Squire equationwith boundary conditions at solid walls and in free stream,

dy

dD

.'','2

2

dy

UdU

dy

dUU

10

Examples of the eigenoscillations

A duct with acoustic waves or electromagnetic microwave

duct solid boundary

vr

vi

duct solid boundary

solid boundary

“liquid boundary”

a boundary layer with an OS mode

c

U

11

Eigenvalue problem

. find tousallow , givenor , find tousallow , e.g., Given, equations.

Squire and OS the withsame The e).(eigenvalu determine topossible always isit , and ,example,for

given, clear that isit relationy solvabilit the From. and ,play es)(eigenvalu tscoefficien

theof role thegiven, are Reand supposing case,our In r.eigenvecto called is which

,,1

/

vectorby the given is solution the

holds,it frelation.I dispersionor sticcharacteri called sometimes is relationy solvabilit This

.0

ifonly solutions nontrivial hasIt

0. ,0

0or

0

0

problem algebraic modela consider uslet problem, eigenvalue theof meaning the youremind To

adcb

U

Ckka-b

bcad

ay

x

c d

a b

dycx

byax

12

Solution of the Orr-Sommerfeld equation

OSE

13

Solution of the Squire equation

imaginary part

real part

14

Interpretation of modal results (some definitions)

15

Analysis and Synthesis

original momentum equations for disturbances

linearization

reduction to eigenvalue problem(spectral analysis)

complete set of linearly-independent modes (waves)

expansion of any small-amplitude disturbance to the

set of the modes

solution of initial value problem

16

Spectral formulation of stability

F=0

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Spectral formulation of stability, cont.

Re=hU/

18

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 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.

v~

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Rayleigh’s inflection point criterion(first Rayleigh theorem)

The inflection point is a point ys such that

That is if the growth rate ci or i≠0, then 2nd derivative of flow mean velocity U” profile changes sign between flow boundaries.

0

2

2''

y

yUU s

s

Formal multiplication of the Rayleigh equation by its complex-conjugate solution 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.

*~v

20

Rayleigh’s inflection point criterion, cont.

on

21

Second Rayleigh theoremand the critical layer

Rayleigh eqn. (R.E.)

R.E.

criticallayer

effects of viscosityat the wall

0~''22

v

cU

UD

(for ci≠0).

large velocitygradient

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Criterion of the maximum vorticity

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.

1

1

22

1

1

22221

1

22

1

1

2222

1

1

22221

1

22

.0|~|||

)("

thatfollows whichfrom

0|~||~||~|||

)("

obtain weside, handleft theto

0|~||~|

add weif

.|~||~||~|||

)("

dyvcU

UUU

dyvvDdyvcU

UUU

dyvvDUc

dyvvDdyvcU

cUU

s

s

sr

r

Here Us is the velocity at inflection point.

Thus, only the velocity profiles with the inflection points associated with the maximum shear are unstable; i.e. theinequality 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.

:

on

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Semicircle theorem

stable

unstable

2

maxmax2

2

maxmax

maxmaxmaxmax22

1

1-

maxmaxmaxmax22

max

1

1-

min

1

1

i

1

1

22

222

)(2

1)(

2

1

givesent rearrangem on which

,0)()(

thatfollowsIt

.0)()(

as writtenbe can but this ,0)()(

have we theorem Rayleigh2nd theusingNow

.02c ,0

:becomes

theoremgh1st Raylei theproof toused equation theof partsimaginary and real thethen

,0|~||'~|

denote weIf

axis. real thealongdiameter its as of range thehas whichplane- halfupper

thein semicircle theinside lies mode unstablefor speed vecomplex wa The

UUcUUc

UUcUU-cc

QdzUUcUU-cc

QdzU-UU-U

QdzcUQdzccU

vvQ

Uc

c

ir

rir

rir

rir

Umin and Umax are minimum and maximum velocity in the flow

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Behavior of u and v of 2D waveat finite Re in the critical layer

|~| u

|~| v

UU”

inflection point

Falkner-Skan basic velocity profilewith backflow, H=‒0.1

backflow

=0.06

)ln()('

)("s

s

s yyyU

yUu

The streamwise velocity tends toinfinity in the inviscid limit like:

unless U''=0 in the critical layer.

25

Destabilizing action of viscosity

dxdyy

u

x

vdyUuvvu

dt

dEV

Re

1' **

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:

re is a

This term is zerowithout the shift

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Examples

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.

Results of viscous computations

27

Disturbance source

’

Disturbance source

Instability in space

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 called sometimes “amplifiers”.

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”.

28

evolutionequation for v

Difference in the evolution equations

That is, we consider thedevelopment in time for the wave with given and .

29

Difference in the evolution equations, cont.

They provide quasi-linear parabolic partial differential equations in time:

+

),(~'i),(~Re

1i 22 tyvUtykDU

t

Parabolic, has first order derivative in time (initial-value problem)

30

Difference in the evolution equations, cont.

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.

s:Development along x and z. For two-dimensional flows, the symmetry prescribe real, so we can consider the growth only along x.

),(~'i),(~Re

1ii 2 tyvUtyU

Elliptic, absence of the time derivative (b.v. problem)

31

Eigenvalue problem

The OS-equation constitutes the 4th order polynomial (sometimes called “nonlinear”) eigenvalue problem in .

OS

SQ

Note that negative i correspond to unstable disturbances, because of the factor:ei(x+y-t)=e-iei(rx+y-t).

32

Gaster’s transformation

es).disturbanc

neutral(almost smallrather are and ifonly valid,is ation transformThe

ly.respective solutions,

temporaland spatial denotes and tssuperscrip and velocity group is where

,

,

,

thatshowed he ,particular In

waves.ldimensiona-for two space in and timein iesinstabilit of parameters

between relations almathematic simple are e that therproved (1962)Gaster

i

i

g

gr

rsi

ti

tr

s

tsr

tsc

c

33

Gaster’s transformation, cont.

simple.y particular becomes ation transformthe

,

wheni.e. ,dispersion small withflowsFor

r

rpg cc

The prove is centered in considering the total differential of the general form of the disturbance relation about a neutral disturbance in complex plane.

For strongly unstable flows, the transformation is not accurate, as it is based on a linear expansion about a neutral value.

valu

es a

t thу

neu

tral

cur

ve Using it we obtain:

relates small changes in a to small changes in w through the group velocity

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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.

35

Squire’s transformation

It is clear that both equations are equivalent, if put

ReRe222 D

36

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.

37

Structure of the solutionsof the Orr-Sommerfeld equation

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Structure of the solutionsof the Orr-Sommerfeld equation

(pressure waves)

vr

vi

39

Structure of the solutionsof the Orr-Sommerfeld equation

(vorticity waves)

vr

vi

downstream upstream of the source

40

Structure of the solutionsof the Orr-Sommerfeld equation

(discrete waves)

41

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.