acoustic wave propagation in thin shear layers · context and motivation many applications in...

Acoustic wave propagation in thin shear layers

Patrick JOLY

UMR CNRS-ENSTA-INRIA

Work in collaboration with

Anne-Sophie Bonnet-Ben Dhia, Marc DurufléLauris Joubert, Ricardo Weder

Seminar, Santiago de Compostela, September 2009

Aeroacoustics : sound propagation in flows

Context and motivation



Many applications in aeronautics


Specific difficulty : modelize the interaction between acoustic waves and walls



!



Effective boundary conditions




From E. J. Brambley (J. Sound Vibr. , 2009)

“ lining models (proposed in the litterature)are shown to be ill posed ”




From E. J. Brambley (J. Sound Vibr. , 2009)

“ lining models (proposed in the litterature)are shown to be ill posed ”

Objective : derive new lining models using rigorous asymptotic analysis

The mathematical models

Euler

!"

#

(!t + M ·!)v + (v ·!)M +!p = 0

(!t + M ·!)p +! · v = 0


Euler

!"

#

(!t + M ·!)v + (v ·!)M +!p = 0

(!t + M ·!)p +! · v = 0

Galbrun (!t + M ·!)2U "!(! · U) = 0


Euler

(!t + M ·!)U + (U ·!)M = v

!"

#

(!t + M ·!)v + (v ·!)M +!p = 0

(!t + M ·!)p +! · v = 0

Galbrun (!t + M ·!)2U "!(! · U) = 0

is the perturbation of Lagrangian displacementU

M!(y) ex

A preliminary analysis :Acoustic wave propagation

in a thin duct

!

!!

x

y

A preliminary analysis :Acoustic wave propagation

in a thin duct

xM(!y)

!y

!1

1

M!(y) = M(y/!)

Galbrun’s equations in a 2D thin duct

!"

#

(!t + M!!x)2 u! ! !x(!xu! + !yv!) = 0

(!t + M!!x)2 v! ! !y(!xu! + !yv!) = 0(P)!~

v!

u!


v!(x,±!, t) = 0

!"

#

(!t + M!!x)2 u! ! !x(!xu! + !yv!) = 0

(!t + M!!x)2 v! ! !y(!xu! + !yv!) = 0(P)!~


v!(x,±!, t) = 0

!"

#

(!t + M!!x)2 u! ! !x(!xu! + !yv!) = 0

(!t + M!!x)2 v! ! !y(!xu! + !yv!) = 0(P)!~

The problem is well-posed as soon as

M! ! W 1,!("1, 1)

A dimensionless model

Scaling

u!(x, y, t) = u!(x,y

!, t), v!(x, y, t) = ! v!(x,

y

!, t)


Scaled model

Scaling

u!(x, y, t) = u!(x,y

!, t), v!(x, y, t) = ! v!(x,

y

!, t)

!"

#

(!t + M!x)2u! ! !x(!xu! + !yv!) = 0

"2 (!t + M!x)2v! ! !y(!xu! + !yv!) = 0(P)!


Passage to the limit

u! ! u, v! ! v

Scaled model

!"

#

(!t + M!x)2u! ! !x(!xu! + !yv!) = 0

"2 (!t + M!x)2v! ! !y(!xu! + !yv!) = 0(P)!



u! ! u, v! ! v

(P)

Formal limit model!"

#

(!t + M!x)2u ! !x(!xu + !yv ) = 0

! !y(!xu + !yv ) = 0



u! ! u, v! ! v

(P)


#

(!t + M!x)2u ! !x(!xu + !yv ) = 0

!xu + !yv = d(x, t)



u! ! u, v! ! v

(P)


#

(!t + M!x)2u ! !xd = 0

!xu + !yv = d(x, t)

The limit model (2)

Introducing E(f)(x, t) :=12

! 1

!1f(x, y, t) dy

The limit model (2)


! 1

!1f(x, y, t) dy

!xu + !yv = d(x, t) =! d(x, t) = E!!xu

"

The limit model (2)

(!t + M!x)2 u! !xd = 0="

(!t + M!x)2 u! !2x

!E(u)

"= 0


! 1

!1f(x, y, t) dy

!xu + !yv = d(x, t) =! d(x, t) = E!!xu

"

The limit model (2)

A quasi-1D model, non local in y

(!t + M!x)2 u! !xd = 0="

(!t + M!x)2 u! !2x

!E(u)

"= 0


! 1

!1f(x, y, t) dy

!xu + !yv = d(x, t) =! d(x, t) = E!!xu

"

The quasi 1D model

(P) (!t + M!x)2 u! !2x

!E(u)

"= 0

The quasi 1D model

When is constant, and commute :M M E

(P) (!t + M!x)2 u! !2x

!E(u)

"= 0

The quasi 1D model

• One advected 1D wave equation


(P) (!t + M!x)2 u! !2x

!E(u)

"= 0

(!t + M!x)2!E(u)

"! !2

x

!E(u)

"= 0

The quasi 1D model

• One advected 1D wave equation


• Decoupled 1D transport equations

(P) (!t + M!x)2 u! !2x

!E(u)

"= 0

(!t + M!x)2 u = !2x

!E(u)

"

(!t + M!x)2!E(u)

"! !2

x

!E(u)

"= 0

Main questions relative to this model

For a general Mach profile, is the evolution problem well-posed ?(P)


If not, what are the conditions on the Mach profile for the problem to be well-posed ?




Is this model helpful for building effective lining conditions ?




Is this model helpful for building effective lining conditions ?

Can we solve numerically this problem ?


Towards the well-posedness analysis

u(x, y, t) !" u(k, y, t)Fx

U(k,y,t) =!

u(k,y,t),"(!t + ikM)u

#(k, y, t)

$t

where is the operator in A(M) L2(!1, 1)2

First order evolution problem:

A(M) =

!

"M I

E M

#

$


U̇ + ikA(M)U = 0

u(x, y, t) !" u(k, y, t)Fx

U(k,y,t) =!

u(k,y,t),"(!t + ikM)u

#(k, y, t)

$t


A(M)

!U(k, t) = e!ikA(M)t !U0(k)

As the operator is bounded, we can write

.


A(M)



The problem is to get uniform bounds in k .


A(M)



The problem is to get uniform bounds in k

As A(M) is non normal, general theorems from semi-group theory do not apply.

.


A(M)



The problem is to get uniform bounds in k

As A(M) is non normal, general theorems from semi-group theory do not apply.

Intuitively, one expects well-posedness if and only if

!!A(M)

"! C!

!C! := {Im z " 0}

".

.

General properties of A(M)

S(u, v) = (v, u)Let , then one has

A(M)! = S ! A(M) ! S

The spectrum of A(M) is symmetric w.r.t. the real axis.

General properties of A(M)

S(u, v) = (v, u)Let , then one has

A(M)! = S ! A(M) ! S

The spectrum of A(M) is symmetric w.r.t. the real axis.

A0(M) =

!

"M I

0 M

#

$

is a compact perturbation of A(M)The operator

Re z

Im z

Structure of the spectrum of A(M)

M(y)

Re z

Im z

y


Im z

Re z


y

y

Im z

Re z


M(y)

Eigenvalues of A(M)

With an explicit computation, one establishes that

(1)

Eigenvalues of A(M)


! ! C \ Im MLemma : A number is an eigenvalue of if and only if:A(M)

(1)

( E ) FM (!) = 2, where FM (!) :=! 1

!1

dy"!!M(y)

#2

Eigenvalues of A(M)


! ! C \ Im MLemma : A number is an eigenvalue of if and only if:A(M)

This eigenvalue is simple associated with

(u!, u̇!) =! 1

(!!M)2,

1(!!M)

"

(1)

( E ) FM (!) = 2, where FM (!) :=! 1

!1

dy"!!M(y)

#2

A(M) =

!

"M I

E M

#

$

A(M) =

!

"M I

E M

#

$

A(M) U = λ U

!M u + v = λ u

E(u) + M v = ! v!"

A(M) =

!

"M I

E M

#

$

A(M) U = λ U

!

E(u) + M v = ! v!"

u = v / (!!M)

A(M) =

!

"M I

E M

#

$

A(M) U = λ U

!

v = E(u) / (!!M)!"

u = v / (!!M)

A(M) =

!

"M I

E M

#

$

A(M) U = λ U

!

v = E(u) / (!!M)!"

u =E(u)

(!!M)2=!

u = v / (!!M)

A(M) =

!

"M I

E M

#

$

A(M) U = λ U

!

v = E(u) / (!!M)!"

u =E(u)

(!!M)2=!

=! E(u)!E

" 1(!!M)2

#! 1

$= 0

u = v / (!!M)

A(M) =

!

"M I

E M

#

$

A(M) U = λ U

!

v = E(u) / (!!M)!"

=! E(u)!E

" 1(!!M)2

#! 1

$= 0

= 0

u = v / (!!M)

A(M) =

!

"M I

E M

#

$

A(M) U = λ U

!

v = E(u) / (!!M)!"

=! E(u)!E

" 1(!!M)2

#! 1

$= 0

= 0

u = v / (!!M)

FM (!) = 2!"

Eigenvalues of A(M) (2)

Lemma : The operator A(M) has exactly two real eigenvalues outside the interval [M!, M+]

!! < M! < M+ < !+

The study of real eigenvalues is easier because FM (!)is real-valued along the real axis

FM (!) :=! 1

!1

dy"!!M(y)

#2

!

FM (!) :=! 1

!1

dy"!!M(y)

#2

M+M! !

FM (!) :=! 1

!1

dy"!!M(y)

#2

2

M+M! !

FM (!) :=! 1

!1

dy"!!M(y)

#2

2

!! !+M+M! !

Im z

Re z

Back to the spectrum of A(M)

Im z

Re z

Back to the spectrum of A(M)

!! !+

Definition of a stable profile

is unstable if Definition : a Mach profile

( E ) has non real solutions

and stable if not.

M




and stable if not.

M

Theorem : if M is unstable, is strongly ill-posed(P)




and stable if not.

M


Conjecture : if M is stable, is well-posed(P)




and stable if not.

M


Conjecture : if M is stable, is well-posed(P) (*)

(*) has been proven in some cases (see later)

A by-product : hydrodynamic instabilities

!

(P)!

Theorem : if M is unstable, is unstable, i. e.(P)!"

#

(!t + M!x)2u! ! !x(!xu! + !yv!) = 0

"2 (!t + M!x)2v! ! !y(!xu! + !yv!) = 0

!u!!L2x(L2

y) + !v!!L2x(L2

y) " C(u0, u1) e" t!


!

(P)!

These are new results for hydrodynamic instabilities in compressible fluids, proven by perturbation theory


#

(!t + M!x)2u! ! !x(!xu! + !yv!) = 0

"2 (!t + M!x)2v! ! !y(!xu! + !yv!) = 0

!u!!L2x(L2

y) + !v!!L2x(L2

y) " C(u0, u1) e" t!


!

(P)!

Most known results concern the incompressible case: Rayleigh, Fjortoft, Drazin, Schmid-Henningson...


#

(!t + M!x)2u! ! !x(!xu! + !yv!) = 0

"2 (!t + M!x)2v! ! !y(!xu! + !yv!) = 0

!u!!L2x(L2

y) + !v!!L2x(L2

y) " C(u0, u1) e" t!


!

(P)!

Most known results concern the incompressible case: Rayleigh, Fjortoft, Drazin, Schmid-Henningson...


#

(!t + M!x)2u! ! !x(!xu! + !yv!) = 0

"2 (!t + M!x)2v! ! !y(!xu! + !yv!) = 0

!u!!L2x(L2

y) + !v!!L2x(L2

y) " C(u0, u1) e" t!

This is a low freq. approach in opposition to the high freq. approach of O. Laffite & al for Rayleigh -Taylor instability

Stability results

They have been obtained with the following process

Stability results


1. The profile M is approximated by a piecewise linear continuous profile Mh such that

!Mh "M!L! # 0, h # 0

Stability results



2. One analyzes the equation ( E ) for Mh

( the function FMh(!) is a rational fraction )

!Mh "M!L! # 0, h # 0

Stability results



2. One analyzes the equation ( E ) for Mh

( the function FMh(!) is a rational fraction )

3. One concludes using perturbation theory for eigenvalue problems (Kato)

!Mh "M!L! # 0, h # 0

Stability results

Theorem : the profile M is stable in the following 3 cases

1. M is convex or concave in [-1,1]

Stability results


1. M is convex or concave in [-1,1]2. M is decreasing and convex - concave

Stability results


1. M is convex or concave in [-1,1]2. M is decreasing and convex - concave

3. M is increasing and concave - convex

Instability results

It is more difficult to establish general instability results

Instability results


However, it is possible to obtain several results in the caseof odd profiles, increasing and convex - concave.

Instability results


However, it is possible to obtain several results in the caseof odd profiles, increasing and convex - concave.

M(y)2 !M !(0)2 y2

Instability results (1)

Theorem : Assume that M is odd,M is unstable

of class C2

if and only if,increasing and convex - concave


Theorem : Assume that M is odd,M is unstable

of class C2

if and only if,

! 1

!1

M "(0)2 y2 !M(y)2

y2 M(y)2dy < 1 + M "(0)2

increasing and convex - concave


Application : M(y) = a tanh(! y), a > 0, ! > 0.



!1 1 2a



!1 1

greater a



!1 1

greater !



Let !! the unique solution of

! tanh! = 1 (!! ! 1.1996)



Let !! the unique solution of

! tanh! = 1 (!! ! 1.1996)

The profile M is unstable if and only if

! > !! =! ! tanh! < 1.(*)

(*)

a <!1! ! tanh!

" 12! > !! and

Computation of discrete spectra

A(M)With finite dimensional approximation spaces one

constructs discrete approximations Ah(M) of

One computes the spectrum of Ah(M)

!1.5 !1 !0.5 0 0.5 1 1.5!0.03

!0.02

!0.01

0

0.01

0.02

0.03

real(!)

imag(!)

N=400

N=200

N=100

Im z

Re z

The case of a linear profile M(y) = y

!1.5 !1 !0.5 0 0.5 1 1.5

!0.02

!0.01

0

0.01

0.02

Re(!)

Im(!)

The case of a stable tanh profileIm z

Re z

M(y) = a tanh(! y)

N=200

N=400

N=100

!1.5 !1 !0.5 0 0.5 1 1.5!0.08

!0.06

!0.04

!0.02

0

0.02

0.04

0.06

0.08

real(!)

imag(!

)

N = 400

N = 200

N = 100

The case of an unstable tanh profile

Im z

Re z


M be continuous and Let {yj} be a regular

mesh of [!1, 1] of stepsize h > 0.




MhLet be the piecewise constant profile given by

Mh(y) =1h

! yj+1

yj

M(y) dy, y ! [yj , yj+1]





Mh(y) =1h

! yj+1

yj

M(y) dy, y ! [yj , yj+1]

Then, for small enough, is unstable.Mhh





Mh(y) =1h

! yj+1

yj

M(y) dy, y ! [yj , yj+1]

Then, for small enough, is unstable.Mhh

This points out how delicate is the numerical approximation of the problem

A well-posedness result


(A) ( E ) only has real solutions. is stableM !"! !


(B) M ! C3,!("1, 1), M ! #= 0, M !! #= 0 in [!1, 1]



Theorem : Under assumptions (A) and (B), (P) is

well-posed : if

there exists a unique solution

u ! C0!R+;H1

x(L2y)

"" C1

!R+;L2

x(L2y)

"

(u0, u1) ! H4x(L2

y)"H3x(L2

y),weakly

(B) M ! C3,!("1, 1), M ! #= 0, M !! #= 0 in [!1, 1]



!u(·, t)!H1x(L2

y) " C(M) (1 + t3)!!u0!H4

x(L2y) + !u1!H3

x(L2y)

"

Theorem : Under assumptions (A) and (B), (P) is

well-posed : if

there exists a unique solution

u ! C0!R+;H1

x(L2y)

"" C1

!R+;L2

x(L2y)

"

(u0, u1) ! H4x(L2

y)"H3x(L2

y),weakly

(B) M ! C3,!("1, 1), M ! #= 0, M !! #= 0 in [!1, 1]


Proof of the theorem

(!t + M!x)2 u = !2x

!E(u)

"Since , it suffices to study

U(x, t) :=!E(u)

"(x, t)


Using the Fourier-Laplace transform in

(!t + M!x)2 u = !2x

!E(u)


U(x, t) :=!E(u)

"(x, t)

U(x, t) !" !U(k, !), k # R, ! # C


Using the Fourier-Laplace transform in

(!t + M!x)2 u = !2x

!E(u)


U(x, t) :=!E(u)

"(x, t)

one obtains an expression of the form :

where ! is known explicitly from (u0, u1)

U(x, t) !" !U(k, !), k # R, ! # C

!U(k,!) ="(#, k)

2! FM (#), ! = # k


f0(y,!) = iM(y)

!!!M(y)

"2 ! i1

!!M(y)

f1(y,!) =1

!!!M(y)

"2

is singular along [M!, M+]! !" "(!, k)

!(", k) = E!f0(·, ") "u0(·, ")

#+ E

!f1(·, ") "u1(·, ")

#


With inverse Laplace transform in time:

with k !I > 0 .

!U(k, t) ="

Im!=!I

!(", k)2! FM (")

e!ik!t d"


This integral is studied using complex variable techniques.


with k !I > 0 .

!U(k, t) ="

Im!=!I

!(", k)2! FM (")

e!ik!t d"




with k !I > 0 .

!U(k, t) ="

Im!=!I

!(", k)2! FM (")

e!ik!t d"

is analytic outside ! !" "(!, k) [M!, M+]



We have to use the analyticity properties of FM (!)


with k !I > 0 .

!U(k, t) ="

Im!=!I

!(", k)2! FM (")

e!ik!t d"

.

is analytic outside ! !" "(!, k) [M!, M+]

Inverting the Laplace transform in time

Im !

Re !


Branch cut of FM (!)

Im !

Re !


Branch cut of FM (!)

Im !

Re !

Poles of !FM (!)! 2

"!1


Im !

Re !


Im !

Re !

Im ! = !I


Im !

Re !

!!


Im !

Re !


Im !

Re !!+ !!

Residue Residue

A principal value integral

!lim!!0

"g(!± ı")!± ı"

d! = P.V.

"g±(!)

!d!! ı#g±(0)

#

U(x, t) = Up(x, t) + U c(x, t)

U(x, t) = Up(x, t) + U c(x, t)

Up is a solution of the generalized wave equation

[(!t ! "+ !x)(!t ! "! !x)

]Up = 0

U(x, t) = Up(x, t) + U c(x, t)

Up is a solution of the generalized wave equation

U c is a continuous superposition on ! of solutionsof squared transport equations

[(!t ! "+ !x)(!t ! "! !x)

]Up = 0

(!t ! " !x)2 U c,! = 0U c =! M+

M!

U c,! d!

A numerical illustration


We get a quasi-analytic (through mutiple integrals)of the solution which is complicated but can beexploited for numerical computations



M(y) = M0 y

We present a numerical result for a linear profile

M(!y)

M0

!M0



M(y) = M0 y

and for the following initial conditions

We present a numerical result for a linear profile

where is a gaussian profile.g

u0(x, y) = g(x), u0(x, y) = 0.

The red arrows move at velocities !+ !! = !!+and

The function U(x, t), M0 = 0.4

The function u(x, y, t), M0 = 0.4

The function U(x, t), M0 = 1


The function u(x, y, t), M0 = 1

The function U(x, t), M(y) = M0 tan!y/ tan!


u(x, y, t), M(y) = M0 tan!y/ tan!

The function U(x, t), M(y) = M0 (1! y2)

The red arrows move at velocities !+ and !!

The function u(x, y, t), M(y) = M0 (1! y2)

Effective boundary conditions (in progress)


!M!(y)


v! ! !2x

!TMv!

"= "

#!xu! + !yv!

$


TM : !(x, t) !"!TM!(x, t)

":= E

!u(!)

"

v! ! !2x

!TMv!

"= "

#!xu! + !yv!

$


(!t + M!x)2 u(")! !2x

!E

"u(")

#$= "

u(!) = "tu(!) = 0 at t = 0.

TM : !(x, t) !"!TM!(x, t)

":= E

!u(!)

"

v! ! !2x

!TMv!

"= "

#!xu! + !yv!

$


(!t + M!x)2 u(")! !2x

!E

"u(")

#$= "

u(!) = "tu(!) = 0 at t = 0.

TM : !(x, t) !"!TM!(x, t)

":= E

!u(!)

"

v! ! !2x

!TMv!

"= "

#!xu! + !yv!

$

The well-posedness of the initial boundary problemin the half-space has been proven (Kreiss method)


(!t + M!x)2 u(")! !2x

!E

"u(")

#$= "

u(!) = "tu(!) = 0 at t = 0.

TM : !(x, t) !"!TM!(x, t)

":= E

!u(!)

"

v! ! !2x

!TMv!

"= "

#!xu! + !yv!

$

Describe the reflection of wavesInvestigate the existence of surface waves

Questions (1)


(!t + M!x)2 u(")! !2x

!E

"u(")

#$= "

u(!) = "tu(!) = 0 at t = 0.

TM : !(x, t) !"!TM!(x, t)

":= E

!u(!)

"

v! ! !2x

!TMv!

"= "

#!xu! + !yv!

$

Find an efficient numerical method

Questions (2)

acoustic wave propagation in thin shear layers · context and motivation many applications in...

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