Acoustic wave propagation in thin shear layers · Context and motivation Many applications in...
Transcript of Acoustic wave propagation in thin shear layers · Context and motivation Many applications in...
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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
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Aeroacoustics : sound propagation in flows
Context and motivation
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Aeroacoustics : sound propagation in flows
Context and motivation
Many applications in aeronautics
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
!
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
!
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Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
Effective boundary conditions
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Aeroacoustics : sound propagation in flows
Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
From E. J. Brambley (J. Sound Vibr. , 2009)
“ lining models (proposed in the litterature)are shown to be ill posed ”
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Aeroacoustics : sound propagation in flows
Context and motivation
Specific difficulty : modelize the interaction between acoustic waves and walls
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
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The mathematical models
Euler
!"
#
(!t + M ·!)v + (v ·!)M +!p = 0
(!t + M ·!)p +! · v = 0
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The mathematical models
Euler
!"
#
(!t + M ·!)v + (v ·!)M +!p = 0
(!t + M ·!)p +! · v = 0
Galbrun (!t + M ·!)2U "!(! · U) = 0
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The mathematical models
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
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The mathematical models
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
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M!(y) ex
A preliminary analysis :Acoustic wave propagation
in a thin duct
!
!!
x
y
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A preliminary analysis :Acoustic wave propagation
in a thin duct
xM(!y)
!y
!1
1
M!(y) = M(y/!)
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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!
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Galbrun’s equations in a 2D thin duct
v!(x,±!, t) = 0
!"
#
(!t + M!!x)2 u! ! !x(!xu! + !yv!) = 0
(!t + M!!x)2 v! ! !y(!xu! + !yv!) = 0(P)!~
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Galbrun’s equations in a 2D thin duct
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)
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A dimensionless model
Scaling
u!(x, y, t) = u!(x,y
!, t), v!(x, y, t) = ! v!(x,
y
!, t)
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A dimensionless model
Scaling
u!(x, y, t) = u!(x,y
!, t), v!(x, y, t) = ! v!(x,
y
!, t)
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A dimensionless model
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)!
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A dimensionless model
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)!
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A dimensionless model
Passage to the limit
u! ! u, v! ! v
(P)
Formal limit model!"
#
(!t + M!x)2u ! !x(!xu + !yv ) = 0
! !y(!xu + !yv ) = 0
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A dimensionless model
Passage to the limit
u! ! u, v! ! v
(P)
Formal limit model!"
#
(!t + M!x)2u ! !x(!xu + !yv ) = 0
!xu + !yv = d(x, t)
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A dimensionless model
Passage to the limit
u! ! u, v! ! v
(P)
Formal limit model!"
#
(!t + M!x)2u ! !xd = 0
!xu + !yv = d(x, t)
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The limit model (2)
Introducing E(f)(x, t) :=12
! 1
!1f(x, y, t) dy
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The limit model (2)
Introducing E(f)(x, t) :=12
! 1
!1f(x, y, t) dy
!xu + !yv = d(x, t) =! d(x, t) = E!!xu
"
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The limit model (2)
(!t + M!x)2 u! !xd = 0="
(!t + M!x)2 u! !2x
!E(u)
"= 0
Introducing E(f)(x, t) :=12
! 1
!1f(x, y, t) dy
!xu + !yv = d(x, t) =! d(x, t) = E!!xu
"
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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
Introducing E(f)(x, t) :=12
! 1
!1f(x, y, t) dy
!xu + !yv = d(x, t) =! d(x, t) = E!!xu
"
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The quasi 1D model
(P) (!t + M!x)2 u! !2x
!E(u)
"= 0
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The quasi 1D model
When is constant, and commute :M M E
(P) (!t + M!x)2 u! !2x
!E(u)
"= 0
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The quasi 1D model
• One advected 1D wave equation
When is constant, and commute :M M E
(P) (!t + M!x)2 u! !2x
!E(u)
"= 0
(!t + M!x)2!E(u)
"! !2
x
!E(u)
"= 0
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The quasi 1D model
• One advected 1D wave equation
When is constant, and commute :M M E
• 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
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Main questions relative to this model
For a general Mach profile, is the evolution problem well-posed ?(P)
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Main questions relative to this model
If not, what are the conditions on the Mach profile for the problem to be well-posed ?
For a general Mach profile, is the evolution problem well-posed ?(P)
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Main questions relative to this model
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 ?
For a general Mach profile, is the evolution problem well-posed ?(P)
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Main questions relative to this model
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 ?
Can we solve numerically this problem ?
For a general Mach profile, is the evolution problem well-posed ?(P)
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Main questions relative to this model
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 ?
Can we solve numerically this problem ?
For a general Mach profile, is the evolution problem well-posed ?(P)
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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
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where is the operator in A(M) L2(!1, 1)2
First order evolution problem:
A(M) =
!
"M I
E M
#
$
Towards the well-posedness analysis
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
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Towards the well-posedness analysis
A(M)
!U(k, t) = e!ikA(M)t !U0(k)
As the operator is bounded, we can write
.
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Towards the well-posedness analysis
A(M)
!U(k, t) = e!ikA(M)t !U0(k)
As the operator is bounded, we can write
The problem is to get uniform bounds in k .
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Towards the well-posedness analysis
A(M)
!U(k, t) = e!ikA(M)t !U0(k)
As the operator is bounded, we can write
The problem is to get uniform bounds in k
As A(M) is non normal, general theorems from semi-group theory do not apply.
.
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Towards the well-posedness analysis
A(M)
!U(k, t) = e!ikA(M)t !U0(k)
As the operator is bounded, we can write
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}
".
.
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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.
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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
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Re z
Im z
Structure of the spectrum of A(M)
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M(y)
Re z
Im z
y
Structure of the spectrum of A(M)
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Im z
Re z
Structure of the spectrum of A(M)
y
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y
Im z
Re z
Structure of the spectrum of A(M)
M(y)
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Eigenvalues of A(M)
With an explicit computation, one establishes that
(1)
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Eigenvalues of A(M)
With an explicit computation, one establishes that
! ! 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
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Eigenvalues of A(M)
With an explicit computation, one establishes that
! ! 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
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A(M) =
!
"M I
E M
#
$
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A(M) =
!
"M I
E M
#
$
A(M) U = λ U
!M u + v = λ u
E(u) + M v = ! v!"
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A(M) =
!
"M I
E M
#
$
A(M) U = λ U
!
E(u) + M v = ! v!"
u = v / (!!M)
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A(M) =
!
"M I
E M
#
$
A(M) U = λ U
!
v = E(u) / (!!M)!"
u = v / (!!M)
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A(M) =
!
"M I
E M
#
$
A(M) U = λ U
!
v = E(u) / (!!M)!"
u =E(u)
(!!M)2=!
u = v / (!!M)
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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)
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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)
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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!"
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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
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FM (!) :=! 1
!1
dy"!!M(y)
#2
!
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FM (!) :=! 1
!1
dy"!!M(y)
#2
M+M! !
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FM (!) :=! 1
!1
dy"!!M(y)
#2
M+M! !
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FM (!) :=! 1
!1
dy"!!M(y)
#2
M+M! !
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FM (!) :=! 1
!1
dy"!!M(y)
#2
M+M! !
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FM (!) :=! 1
!1
dy"!!M(y)
#2
2
M+M! !
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FM (!) :=! 1
!1
dy"!!M(y)
#2
2
!! !+M+M! !
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Im z
Re z
Back to the spectrum of A(M)
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Im z
Re z
Back to the spectrum of A(M)
!! !+
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Im z
Re z
Back to the spectrum of A(M)
!! !+
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Definition of a stable profile
is unstable if Definition : a Mach profile
( E ) has non real solutions
and stable if not.
M
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Definition of a stable profile
is unstable if Definition : a Mach profile
( E ) has non real solutions
and stable if not.
M
Theorem : if M is unstable, is strongly ill-posed(P)
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Definition of a stable profile
is unstable if Definition : a Mach profile
( E ) has non real solutions
and stable if not.
M
Theorem : if M is unstable, is strongly ill-posed(P)
Conjecture : if M is stable, is well-posed(P)
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Definition of a stable profile
is unstable if Definition : a Mach profile
( E ) has non real solutions
and stable if not.
M
Theorem : if M is unstable, is strongly ill-posed(P)
Conjecture : if M is stable, is well-posed(P) (*)
(*) has been proven in some cases (see later)
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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!
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A by-product : hydrodynamic instabilities
!
(P)!
These are new results for hydrodynamic instabilities in compressible fluids, proven by perturbation theory
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!
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A by-product : hydrodynamic instabilities
!
(P)!
Most known results concern the incompressible case: Rayleigh, Fjortoft, Drazin, Schmid-Henningson...
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!
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A by-product : hydrodynamic instabilities
!
(P)!
Most known results concern the incompressible case: Rayleigh, Fjortoft, Drazin, Schmid-Henningson...
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!
This is a low freq. approach in opposition to the high freq. approach of O. Laffite & al for Rayleigh -Taylor instability
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Stability results
They have been obtained with the following process
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Stability results
They have been obtained with the following process
1. The profile M is approximated by a piecewise linear continuous profile Mh such that
!Mh "M!L! # 0, h # 0
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Stability results
They have been obtained with the following process
1. The profile M is approximated by a piecewise linear continuous profile Mh such that
2. One analyzes the equation ( E ) for Mh
( the function FMh(!) is a rational fraction )
!Mh "M!L! # 0, h # 0
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Stability results
They have been obtained with the following process
1. The profile M is approximated by a piecewise linear continuous profile Mh such that
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
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Stability results
Theorem : the profile M is stable in the following 3 cases
1. M is convex or concave in [-1,1]
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Stability results
Theorem : the profile M is stable in the following 3 cases
1. M is convex or concave in [-1,1]
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Stability results
Theorem : the profile M is stable in the following 3 cases
1. M is convex or concave in [-1,1]2. M is decreasing and convex - concave
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Stability results
Theorem : the profile M is stable in the following 3 cases
1. M is convex or concave in [-1,1]2. M is decreasing and convex - concave
3. M is increasing and concave - convex
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Instability results
It is more difficult to establish general instability results
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Instability results
It is more difficult to establish general instability results
However, it is possible to obtain several results in the caseof odd profiles, increasing and convex - concave.
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Instability results
It is more difficult to establish general instability results
However, it is possible to obtain several results in the caseof odd profiles, increasing and convex - concave.
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Instability results
It is more difficult to establish general 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
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Instability results (1)
Theorem : Assume that M is odd,M is unstable
of class C2
if and only if,increasing and convex - concave
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Instability results (1)
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
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Instability results (1)
Application : M(y) = a tanh(! y), a > 0, ! > 0.
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Instability results (1)
Application : M(y) = a tanh(! y), a > 0, ! > 0.
!1 1 2a
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Instability results (1)
Application : M(y) = a tanh(! y), a > 0, ! > 0.
!1 1
greater a
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Instability results (1)
Application : M(y) = a tanh(! y), a > 0, ! > 0.
!1 1
greater !
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Instability results (1)
Application : M(y) = a tanh(! y), a > 0, ! > 0.
Let !! the unique solution of
! tanh! = 1 (!! ! 1.1996)
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Instability results (1)
Application : M(y) = a tanh(! y), a > 0, ! > 0.
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
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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)
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!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
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!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
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!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
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!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
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!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
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!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
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!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
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!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
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!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
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Instability results (2)
M be continuous and Let {yj} be a regular
mesh of [!1, 1] of stepsize h > 0.
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Instability results (2)
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]
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Instability results (2)
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]
Then, for small enough, is unstable.Mhh
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Instability results (2)
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]
Then, for small enough, is unstable.Mhh
This points out how delicate is the numerical approximation of the problem
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A well-posedness result
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A well-posedness result
(A) ( E ) only has real solutions. is stableM !"! !
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A well-posedness result
(B) M ! C3,!("1, 1), M ! #= 0, M !! #= 0 in [!1, 1]
(A) ( E ) only has real solutions. is stableM !"! !
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A well-posedness result
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]
(A) ( E ) only has real solutions. is stableM !"! !
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A well-posedness result
!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]
(A) ( E ) only has real solutions. is stableM !"! !
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Proof of the theorem
(!t + M!x)2 u = !2x
!E(u)
"Since , it suffices to study
U(x, t) :=!E(u)
"(x, t)
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Proof of the theorem
Using the Fourier-Laplace transform in
(!t + M!x)2 u = !2x
!E(u)
"Since , it suffices to study
U(x, t) :=!E(u)
"(x, t)
U(x, t) !" !U(k, !), k # R, ! # C
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Proof of the theorem
Using the Fourier-Laplace transform in
(!t + M!x)2 u = !2x
!E(u)
"Since , it suffices to study
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
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Proof of the theorem
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(·, ")
#
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Proof of the theorem
With inverse Laplace transform in time:
with k !I > 0 .
!U(k, t) ="
Im!=!I
!(", k)2! FM (")
e!ik!t d"
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Proof of the theorem
With inverse Laplace transform in time:
with k !I > 0 .
!U(k, t) ="
Im!=!I
!(", k)2! FM (")
e!ik!t d"
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Proof of the theorem
This integral is studied using complex variable techniques.
With inverse Laplace transform in time:
with k !I > 0 .
!U(k, t) ="
Im!=!I
!(", k)2! FM (")
e!ik!t d"
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Proof of the theorem
This integral is studied using complex variable techniques.
With inverse Laplace transform in time:
with k !I > 0 .
!U(k, t) ="
Im!=!I
!(", k)2! FM (")
e!ik!t d"
is analytic outside ! !" "(!, k) [M!, M+]
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Proof of the theorem
This integral is studied using complex variable techniques.
We have to use the analyticity properties of FM (!)
With inverse Laplace transform in time:
with k !I > 0 .
!U(k, t) ="
Im!=!I
!(", k)2! FM (")
e!ik!t d"
.
is analytic outside ! !" "(!, k) [M!, M+]
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Inverting the Laplace transform in time
Im !
Re !
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Inverting the Laplace transform in time
Branch cut of FM (!)
Im !
Re !
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Inverting the Laplace transform in time
Branch cut of FM (!)
Im !
Re !
Poles of !FM (!)! 2
"!1
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Inverting the Laplace transform in time
Im !
Re !
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Inverting the Laplace transform in time
Im !
Re !
Im ! = !I
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Inverting the Laplace transform in time
Im !
Re !
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Inverting the Laplace transform in time
Im !
Re !
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Inverting the Laplace transform in time
Im !
Re !
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Inverting the Laplace transform in time
Im !
Re !
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Inverting the Laplace transform in time
Im !
Re !
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!!
Inverting the Laplace transform in time
Im !
Re !
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Inverting the Laplace transform in time
Im !
Re !!+ !!
Residue Residue
A principal value integral
!lim!!0
"g(!± ı")!± ı"
d! = P.V.
"g±(!)
!d!! ı#g±(0)
#
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U(x, t) = Up(x, t) + U c(x, t)
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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
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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!
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A numerical illustration
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A numerical illustration
We get a quasi-analytic (through mutiple integrals)of the solution which is complicated but can beexploited for numerical computations
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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
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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
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.
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The red arrows move at velocities !+ !! = !!+and
The function U(x, t), M0 = 0.4
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The function u(x, y, t), M0 = 0.4
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The function U(x, t), M0 = 1
The red arrows move at velocities !+ !! = !!+and
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The function U(x, t), M0 = 1
The red arrows move at velocities !+ !! = !!+and
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The function u(x, y, t), M0 = 1
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The function u(x, y, t), M0 = 1
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The function U(x, t), M(y) = M0 tan!y/ tan!
The red arrows move at velocities !+ !! = !!+and
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The function U(x, t), M(y) = M0 tan!y/ tan!
The red arrows move at velocities !+ !! = !!+and
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u(x, y, t), M(y) = M0 tan!y/ tan!
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The function U(x, t), M(y) = M0 (1! y2)
The red arrows move at velocities !+ and !!
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The function U(x, t), M(y) = M0 (1! y2)
The red arrows move at velocities !+ and !!
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The function u(x, y, t), M(y) = M0 (1! y2)
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Effective boundary conditions (in progress)
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Effective boundary conditions (in progress)
!M!(y)
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Effective boundary conditions (in progress)
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Effective boundary conditions (in progress)
v! ! !2x
!TMv!
"= "
#!xu! + !yv!
$
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Effective boundary conditions (in progress)
TM : !(x, t) !"!TM!(x, t)
":= E
!u(!)
"
v! ! !2x
!TMv!
"= "
#!xu! + !yv!
$
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Effective boundary conditions (in progress)
(!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!
$
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Effective boundary conditions (in progress)
(!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)
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Effective boundary conditions (in progress)
(!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)
![Page 175: Acoustic wave propagation in thin shear layers · Context and motivation Many applications in aeronautics. Context and motivation Specific difficulty : modelize the interaction](https://reader036.fdocuments.us/reader036/viewer/2022071212/60259169f39dd43ae2049944/html5/thumbnails/175.jpg)
Effective boundary conditions (in progress)
(!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)