Adjoints MEC651 denis.sipp@onera.fr Adjoints 1 Outline - - - PowerPoint PPT Presentation

β–Ά
adjoints
SMART_READER_LITE
LIVE PREVIEW

Adjoints MEC651 denis.sipp@onera.fr Adjoints 1 Outline - - - PowerPoint PPT Presentation

Mar 20, 2024 381 likes β€’874 views

Adjoints MEC651 denis.sipp@onera.fr Adjoints 1 Outline - Governing equations - Asymptotic development Order 0 : Base-flow Order 1 : Global modes - Bi-orthogonal basis and adjoint global modes Definition of adjoint

slide-1
SLIDE 1

Adjoints

MEC651 denis.sipp@onera.fr Adjoints 1

slide-2
SLIDE 2

MEC651 denis.sipp@onera.fr Adjoints 2

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-3
SLIDE 3

Incompressible Navier-Stokes equations: πœ–π‘’π‘£ + π‘£πœ–π‘¦π‘£ + π‘€πœ–π‘§π‘£ = βˆ’πœ–π‘¦π‘ž + πœ‰ πœ–π‘¦π‘¦π‘£ + πœ–π‘§π‘§π‘£ + 𝑔 πœ–π‘’π‘€ + π‘£πœ–π‘¦π‘€ + π‘€πœ–π‘§π‘€ = βˆ’πœ–π‘§π‘ž + πœ‰ πœ–π‘¦π‘¦π‘€ + πœ–π‘§π‘§π‘€ + 𝑕 βˆ’πœ–π‘¦π‘£ βˆ’ πœ–π‘§π‘€ = 0 Can be recast into: β„¬πœ–π‘’π‘₯+ 1 2 π’ͺ π‘₯, π‘₯ + β„’π‘₯ = 𝑔 where: π‘₯ = 𝑣 π‘ž 𝑔 = 𝑔 ℬ = 1 0 , π’ͺ π‘₯1, π‘₯2 = 𝑣1 β‹… 𝛼𝑣2 + 𝑣2 β‹… 𝛼𝑣1 β„’ = βˆ’πœ‰Ξ”() 𝛼() βˆ’π›Ό β‹… () Boundary conditions: Dirichlet, Neumann, Mixed

Governing equations

Adjoints 3 MEC651 denis.sipp@onera.fr

slide-4
SLIDE 4

a) π’ͺ π‘₯1, π‘₯2 = π’ͺ π‘₯2, π‘₯1 b) 1

2 π’ͺ π‘₯0 + πœ—πœ€π‘₯, π‘₯0 + πœ—πœ€π‘₯ = 1 2 π’ͺ π‘₯0, π‘₯0 + πœ— π’ͺ π‘₯0, πœ€π‘₯

Jacobian

=π’ͺ

π‘₯0πœ€π‘₯

+

πœ—2 2 π’ͺ πœ€π‘₯, πœ€π‘₯

Hessian + β‹― c) π’ͺ

π‘₯0πœ€π‘₯ = π’ͺ π‘₯0, πœ€π‘₯ = πœ€π‘£ β‹… 𝛼𝑣0 + 𝑣0 β‹… π›Όπœ€π‘£

d) ℬπ‘₯ = ℬ 𝑣 π‘ž = 𝑣 e) πœ–π‘’π‘£ + 𝑣 β‹… 𝛼𝑣 = βˆ’π›Όπ‘ž + πœ‰π›Ό2𝑣 β‡’ βˆ’π›Ό2π‘ž = 𝛼 β‹… 𝑣 β‹… 𝛼𝑣 , πœ–π‘œπ‘ž = πœ‰π›Ό2𝑣 β‹… π‘œ on solid

  • walls. Hence, p is a function of u and should not be considered as a degree of

freedom of the flow. f) Scalar-product: < π‘₯1, π‘₯2 > = ∬ 𝑣1

βˆ—π‘£2 + 𝑀1 βˆ—π‘€2 𝑒𝑦𝑒𝑧 = ∬ (π‘₯1 β‹… ℬπ‘₯2)𝑒𝑦𝑒𝑧 so

that < π‘₯, π‘₯ > is the energy.

Some properties

Adjoints 4 MEC651 denis.sipp@onera.fr

slide-5
SLIDE 5

MEC651 denis.sipp@onera.fr Adjoints 5

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-6
SLIDE 6

Adjoints 6

Solution: π‘₯ 𝑒 = π‘₯0 + πœ—π‘₯1 𝑒 + β‹― with Ο΅ β‰ͺ 1 Governing equations: β„¬πœ–π‘’π‘₯+ 1 2 π’ͺ π‘₯, π‘₯ + β„’π‘₯ = 𝑔 Introduce solution into governing eq:: β„¬πœ–π‘’(π‘₯0+πœ—π‘₯1 + β‹― )+ 1 2 π’ͺ π‘₯0 + πœ—π‘₯1 + β‹― , π‘₯0 + πœ—π‘₯1 + β‹― + β„’(π‘₯0+πœ—π‘₯1 + β‹― ) = 𝑔 β‡’ 1 2 π’ͺ π‘₯0, π‘₯0 + β„’π‘₯0 = 𝑔 at order 𝑃(1) β„¬πœ–π‘’π‘₯1+ 1 2 [π’ͺ π‘₯1, π‘₯0 + π’ͺ π‘₯0, π‘₯1 ]

π’ͺ

π‘₯0π‘₯1

+ β„’π‘₯1 = 0 at order 𝑃(πœ—) β„¬πœ–π‘’π‘₯2+π’ͺ

π‘₯0π‘₯2 + β„’π‘₯2 = βˆ’ 1

2 π’ͺ π‘₯1, π‘₯1 at order 𝑃(πœ—2)

Asymptotic development

MEC651 denis.sipp@onera.fr

slide-7
SLIDE 7

Oder πœ—0: Base-flow

Adjoints 7

Definition: π‘₯ 𝑒 = π‘₯0 + πœ—π‘₯1(𝑒) + β‹― Non-linear equilibrium point : 1 2 π’ͺ π‘₯0, π‘₯0 + β„’π‘₯0 = 𝑔 How to compute a base-flow ? Newton iteration: 1 2 π’ͺ π‘₯0 + πœ€π‘₯0, π‘₯0 + πœ€π‘₯0 + β„’(π‘₯0+πœ€π‘₯0) = 𝑔 Linearization: π’ͺ π‘₯0, πœ€π‘₯0 + β„’πœ€π‘₯0 = 𝑔 βˆ’ 1 2 π’ͺ π‘₯0, π‘₯0 βˆ’ β„’π‘₯0 β‡’ πœ€π‘₯0 = π’ͺ

π‘₯0 + β„’ βˆ’1 𝑔 βˆ’ 1

2 π’ͺ π‘₯0, π‘₯0 βˆ’ β„’π‘₯0 π‘₯ 𝐺 π‘₯ = 1 2 π’ͺ π‘₯, π‘₯ + β„’π‘₯ βˆ’ 𝑔

MEC651 denis.sipp@onera.fr

π‘₯0

slide-8
SLIDE 8

Oder πœ—0: Base-flow The case of cylinder flow

Adjoints 8 MEC651 denis.sipp@onera.fr

𝑆𝑓 = 47 Streamwise velocity field of base-flow.

slide-9
SLIDE 9

Order πœ—1: Global modes Definition

Adjoints 9

π‘₯ 𝑒 = π‘₯0 + πœ—π‘₯1(𝑒) + β‹― Linear governing equation: β„¬πœ–π‘’π‘₯1 + π’ͺ

π‘₯0π‘₯1 + β„’π‘₯1 = 0

Solution π‘₯1 under the form: π‘₯1 = π‘“πœ‡π‘’π‘₯ + c.c This leads to :

MEC651 denis.sipp@onera.fr

Eigenvalue: πœ‡ = 𝜏 + π‘—πœ• Eigenvector: π‘₯ = π‘₯ r + iw 𝑗 Real solution: π‘₯1 = π‘“πœ‡π‘’π‘₯ + c.c = 2π‘“πœπ‘’(cos πœ•π‘’ π‘₯ 𝑠 βˆ’ sin πœ•π‘’ π‘₯ 𝑗) πœ‡β„¬π‘₯ + π’ͺ

π‘₯0 + β„’ π‘₯

= 0

slide-10
SLIDE 10

Order πœ—1: Global modes How to compute global modes ?

Adjoints 10 MEC651 denis.sipp@onera.fr

Eigenvalue problem solved with shift-invert strategy:

  • Power method, easy to find largest magnitude eigenvalues
  • f 𝐡𝑦 = πœ‡π‘¦. For this, evaluate π΅π‘œπ‘¦0
  • To find eigenvalues of 𝐡 closest to zero, search largest magnitude eigenvalues of

π΅βˆ’1: π΅βˆ’1𝑦 = πœ‡βˆ’1𝑦. For this, evaluate π΅βˆ’1 π‘œπ‘¦0

  • To find eigenvalues of 𝐡 closest to 𝑑, search largest magnitude eigenvalues of

𝐡 βˆ’ 𝑑𝐽 βˆ’1: 𝐡 βˆ’ 𝑑𝐽 βˆ’1𝑦 = πœ‡ βˆ’ 𝑑 βˆ’1𝑦. For this, evaluate 𝐡 βˆ’ 𝑑𝐽 βˆ’1 π‘œπ‘¦0

  • Instead of power-method, use Krylov subspaces -> Arnoldi technique
  • Cost of algorithm = cost of several complex matrix inversions
slide-11
SLIDE 11

Order πœ—1: Global modes Case of cylinder flow

Adjoints 11 MEC651 denis.sipp@onera.fr

Spectrum 𝑆𝑓 = 47 Real part of cross-stream velocity field Marginal eigenmode

slide-12
SLIDE 12

The Ginzburg-Landau eq.

MEC651 denis.sipp@onera.fr Adjoints 12

We consider the linear Ginzburg-Landau equation πœ–π‘’π‘₯1 + β„’π‘₯1 = 0 where β„’ = π‘‰πœ–π‘¦ βˆ’ 𝜈 𝑦 βˆ’ π›Ώπœ–π‘¦π‘¦, 𝜈 𝑦 = π‘—πœ•0 + 𝜈0 βˆ’ 𝜈2 𝑦2 2 . Here 𝑉, 𝛿, πœ•0, 𝜈0 and 𝜈2 are positive real constants. The state π‘₯(𝑦, 𝑒) is a complex variable on βˆ’βˆž < 𝑦 < +∞ such that |π‘₯| β†’ 0 as 𝑦 β†’ ∞. In the following, π‘₯𝑏, π‘₯𝑐 = π‘₯𝑏 𝑦 βˆ—π‘₯𝑐 𝑦 𝑒𝑦

+∞ βˆ’βˆž

. 1/ What do the different terms in the Ginzburg Landau equation represent?

slide-13
SLIDE 13

The Ginzburg-Landau eq.

MEC651 denis.sipp@onera.fr Adjoints 13

2/ Show that π‘₯ (𝑦) = πœ‚π‘“

𝑉 2π›Ώπ‘¦βˆ’πœ“2𝑦2 2 with πœ“ =

𝜈2 2𝛿

1 4 and πœ‚ =

πœ“ 𝜌

1 4𝑓 1 8 𝑉2 𝛿2πœ“2

verifies πœ‡π‘₯ + β„’π‘₯ = 0. What is the eigenvalue πœ‡ associated to this eigenvector? The constant πœ‚ has been selected so that π‘₯ , π‘₯ = 1. 3/ Show that the flow is unstable if the constant 𝜈0 is chosen such that: 𝜈0 > πœˆπ‘‘, where πœˆπ‘‘ =

𝑉2 4𝛿 + π›Ώπœˆ2 2 .

Nota: πœ‡π‘œ = iπœ•0 + 𝜈0 βˆ’ 𝑉2

4𝛿 βˆ’ 2π‘œ + 1 π›Ώπœˆ2 2 , π‘₯

π‘œ = πœ‚π‘œπΌπ‘œ πœ“π‘¦ 𝑓

𝑉 2π›Ώπ‘¦βˆ’πœ“2𝑦2 2

are all the eigenvalues/eigenvectors of β„’, πΌπ‘œ being Hermite polynomials.

slide-14
SLIDE 14

MEC651 denis.sipp@onera.fr Adjoints 14

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-15
SLIDE 15

Bi-orthogonal basis and adjoint global modes (1/3)

In finite dimension

Adjoints 15

Global mod

  • des:

𝐡π‘₯ 𝑗 = πœ‡π‘—π‘₯ 𝑗 The eigenvectors π‘₯ 𝑗 form a basis: π‘₯ = 𝛽𝑗w i

𝑗

Definition of adjoint global modes: with <> as a given scalar-product (say < π‘₯1, π‘₯2 > = π‘₯1

βˆ—π‘₯2), there exists for each 𝛽𝑗 a unique π‘₯

𝑗 such that 𝛽𝑗 =< π‘₯ 𝑗, π‘₯ > for all π‘₯. The adjoint global modes are the structures π‘₯ 𝑗. In the following: π‘₯ 𝑗, π‘₯ 𝑗 = 1. Properties:

  • π‘₯

𝑙 and w j are bi-orthogonal bases: they verify π‘₯ π‘˜ = < π‘₯ 𝑗, π‘₯ π‘˜ > w i

𝑗

and so < π‘₯ 𝑙, w j > = πœ€π‘™π‘˜ (in matrix notations 𝑋 βˆ—π‘‹ = 𝐽)

  • Cauchy-Lifschitz: 1 = < π‘₯

𝑗, w i > ≀< π‘₯ 𝑗, π‘₯ 𝑗 >

1 2< π‘₯

𝑗, π‘₯ 𝑗 >

1 2

Hence: < π‘₯ 𝑗, π‘₯ 𝑗 >

1 2β‰₯ 1 and cos angle π‘₯

𝑗, π‘₯ 𝑗 =

1 <π‘₯ 𝑗,π‘₯ 𝑗>

1 2

MEC651 denis.sipp@onera.fr

slide-16
SLIDE 16

Bi-orthogonal basis and adjoint global modes (2/3)

In finite dimension

Adjoints 16

π‘₯ 1 π‘₯ 2 π‘₯ 1 π‘₯ 2 Def of π‘₯ 1: π‘₯ 1 β‹… π‘₯ 1 = 1 π‘₯ 1 β‹… π‘₯ 2 = 0 Def of π‘₯ 2: π‘₯ 2 β‹… π‘₯ 2 = 1 π‘₯ 2 β‹… π‘₯ 1 = 0 𝑋 βˆ—π‘‹ = 𝐽 Method 1 : 𝑋 = W βˆ—βˆ’1 Method 2 : 𝑋 = 𝑋 π‘Œ β‡’ π‘Œβˆ—π‘‹ βˆ—π‘‹ = 𝐽 β‡’ π‘Œ = 𝑋 βˆ—π‘‹

βˆ’1 β‡’ 𝑋

= 𝑋 𝑋 βˆ—π‘‹

βˆ’1

Method 3 : adjoint global modes π‘₯ = (π‘₯ 1β‹… π‘₯)π‘₯ 1 + (π‘₯ 2β‹… π‘₯)π‘₯ 2

MEC651 denis.sipp@onera.fr

slide-17
SLIDE 17

Bi-orthogonal basis and adjoint global modes (3/3)

Adjoints 17

Global mod

  • des:

πœ‡π‘—β„¬π‘₯ 𝑗 + π’ͺ

π‘₯0 + β„’ π‘₯

𝑗 = 0 The eigenvectors π‘₯ 𝑗 form a basis: π‘₯ = 𝛽𝑗w i

𝑗

Definition of adjoint global modes: with <> as a given scalar-product, there exists for each 𝛽𝑗 a unique π‘₯ 𝑗 such that 𝛽𝑗 =< π‘₯ 𝑗, ℬπ‘₯ > for all π‘₯. The adjoint global modes are the structures π‘₯ 𝑗. In the following: < π‘₯ 𝑗, ℬπ‘₯ 𝑗 > = 1. Properties:

  • π‘₯

𝑙 and w j are bi-orthogonal bases: they verify π‘₯ π‘˜ = < π‘₯ 𝑗, ℬπ‘₯ π‘˜ > w i

𝑗

and so < π‘₯ 𝑙, ℬw j > = πœ€π‘™π‘˜

  • Cauchy-Lifschitz: 1 = < π‘₯

𝑗, ℬw i > ≀< π‘₯ 𝑗, ℬπ‘₯ 𝑗 >

1 2< π‘₯

𝑗, ℬπ‘₯ 𝑗 >

1

  • 2. Hence:

< π‘₯ 𝑗, ℬπ‘₯ 𝑗 >

1 2β‰₯ 1 and cos angle π‘₯

𝑗, π‘₯ 𝑗 =

1 <π‘₯ 𝑗,ℬπ‘₯ 𝑗>

1 2

MEC651 denis.sipp@onera.fr

slide-18
SLIDE 18

MEC651 denis.sipp@onera.fr Adjoints 18

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint global modes of cylinder flow

Outline

slide-19
SLIDE 19

Optimal initial condition (1/3)

Adjoints 19

Definition of optimal initial condition Initial-value problem: β„¬πœ–π‘’π‘₯1 + π’ͺ

π‘₯0 + β„’ π‘₯1 = 0,

π‘₯1 𝑒 = 0 = π‘₯𝐽 Solution: π‘₯1 𝑒 = < π‘₯ 𝑗, ℬπ‘₯𝐽 > π‘“πœ‡π‘—π‘’π‘₯ 𝑗

𝑗

If (π‘₯ 1, πœ‡1) is the global mode which displays largest growth rate, at large times: π‘₯1 𝑒 β‰ˆ< π‘₯ 1, ℬπ‘₯𝐽 > π‘“πœ‡1𝑒π‘₯ 1 We look for unit-norm π‘₯𝐽 (< π‘₯𝐽, ℬwI >= 1) which maximizes the amplitude of the response at large times. π‘₯𝐽 is the optimal initial condition.

MEC651 denis.sipp@onera.fr

slide-20
SLIDE 20

Optimal initial condition (2/3)

Adjoints 20 MEC651 denis.sipp@onera.fr

If direct global mode as initial condition: π‘₯𝐽 = π‘₯ 1 In this case, at large time: π‘₯1 𝑒 β‰ˆ π‘“πœ‡1𝑒π‘₯ 1 If adjoint global mode as initial condition: π‘₯𝐽 = π‘₯ 1 < π‘₯ 1, ℬπ‘₯ 1 >

1 2

Then, at large time: π‘₯1 𝑒 β‰ˆ< π‘₯ 1, ℬπ‘₯ 1 >

1 2 π‘“πœ‡1𝑒π‘₯

1 This is optimal since: < π‘₯ 1, ℬπ‘₯𝐽 > ≀< π‘₯ 1, ℬπ‘₯ 1 >

1 2< π‘₯𝐽, ℬπ‘₯𝐽 > 1 2 1

slide-21
SLIDE 21

Optimal initial condition (3/3)

Adjoints 21

Estimation of gain: From Causchy-Lifschitz: < π‘₯ 1, ℬπ‘₯ 1 >

1 2β‰₯ 1

Amplitude gain: < π‘₯ 1, ℬπ‘₯ 1 >

1 2=

1 cos angle π‘₯ 1, π‘₯ 1

MEC651 denis.sipp@onera.fr

slide-22
SLIDE 22

In finite dimension

Adjoints 22

π‘₯1

MEC651 denis.sipp@onera.fr

π‘₯ 1 π‘₯ 2 π‘₯ 2 π‘₯ 1 π‘₯2 π‘₯2 β‰ˆ βˆ’1.2π‘₯ 1 + 1.8π‘₯ 2 π‘₯2 = 1 π‘₯1 β‰ˆ 1.9π‘₯ 1 βˆ’ 1.3π‘₯ 2 π‘₯1 = 1 π‘₯ 2 = 0π‘₯ 1 + 1π‘₯ 2 π‘₯ 2 = 1 π‘₯ 1 = 1π‘₯ 1 + 0π‘₯ 2 π‘₯ 1 = 1

slide-23
SLIDE 23

MEC651 denis.sipp@onera.fr Adjoints 23

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-24
SLIDE 24

Optimal forcing in stable flow (1/2)

Adjoints 24

Problem: β„¬πœ–π‘’π‘₯+ 1 2 π’ͺ π‘₯, π‘₯ + β„’π‘₯ = πœ—β„¬π‘”

1

π‘₯ = π‘₯0 + πœ—π‘₯1 At first order: β„¬πœ–π‘’π‘₯1 + π’ͺ

π‘₯0 + β„’ π‘₯1 = 𝑔 1

In frequency domain: π‘₯1 = π‘“π‘—πœ•π‘’π‘₯ and 𝑔

1 = π‘“π‘—πœ•π‘’π‘”

Governing equation: π‘—πœ•β„¬π‘₯ + π’ͺ

π‘₯0 + β„’ π‘₯

= ℬf Where to force (𝑔 ) and at which frequency (πœ•) to obtain strongest response (π‘₯ )?

MEC651 denis.sipp@onera.fr

slide-25
SLIDE 25

Optimal forcing in stable flow (1/2)

Adjoints 25

Introducing global mode basis: π‘₯ = < π‘₯ 𝑗, ℬπ‘₯ >

𝑗

π‘₯ 𝑗 and ℬ𝑔 = < π‘₯ 𝑗, ℬ𝑔 > ℬ

𝑗

π‘₯ 𝑗: π‘—πœ• < π‘₯ 𝑗, ℬπ‘₯ > ℬπ‘₯ 𝑗 βˆ’ πœ‡π‘— < π‘₯ 𝑗, ℬπ‘₯ > ℬπ‘₯ 𝑗 = < π‘₯ 𝑗, ℬ𝑔 > ℬπ‘₯ 𝑗

𝑗 𝑗

Scalar-product with π‘₯ π‘˜ and using bi-orthogonality:< π‘₯ π‘˜, ℬπ‘₯ 𝑗 > = πœ€π‘—π‘˜ < π‘₯ π‘˜, ℬπ‘₯ > π‘—πœ• βˆ’ πœ‡π‘˜ =< π‘₯ π‘˜, ℬf > < π‘₯ π‘˜, ℬπ‘₯ >= < π‘₯ π‘˜, ℬf > π‘—πœ• βˆ’ πœ‡π‘˜ Solution: π‘₯1 𝑒 = π‘“π‘—πœ•π‘’ < π‘₯ 𝑗, ℬf > π‘—πœ• βˆ’ πœ‡π‘— ℬπ‘₯ 𝑗

𝑗

To maximize response: a/ force at frequencies iπœ• closest to πœ‡π‘— b/ force with f =

π‘₯ i <π‘₯ 𝑗,ℬπ‘₯ 𝑗>

1 2

MEC651 denis.sipp@onera.fr

slide-26
SLIDE 26

MEC651 denis.sipp@onera.fr Adjoints 26

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-27
SLIDE 27

Adjoint operator Definition

Adjoints 27

Definition of adjoint operator: Let ⟨π‘₯1, π‘₯2⟩ be a scalar product and 𝒝 a linear operator. The adjoint operator of 𝒝 verifies ⟨π‘₯1, 𝒝π‘₯2⟩ = βŸ¨π’ π‘₯1, π‘₯2⟩ whatever π‘₯1 and π‘₯2.

MEC651 denis.sipp@onera.fr

slide-28
SLIDE 28

Adjoint operator Example in finite dimension

Adjoints 28

Space: π‘₯ ∈ ℂ𝑂 Scalar-product: < π‘₯1, π‘₯2 > = π‘₯1

βˆ—π‘…π‘₯2

with 𝑅 a Hermitian matrix π‘…βˆ— = 𝑅. Linear operator: 𝒝 matrix. Adjoint operator: < π‘₯1, 𝒝π‘₯2 > = π‘₯1

βˆ—π‘…π’π‘₯2 = π‘₯1 βˆ—π‘…π’π’­βˆ’1𝒭π‘₯2 = π’­βˆ’1π’βˆ—π’­π‘₯1 βˆ—π’­π‘₯2 =< 𝒝

π‘₯1, π‘₯2 > with 𝒝 = π’­βˆ’1π’βˆ—π’­ If 𝒭 = 𝐽, then 𝒝 = π’βˆ—

MEC651 denis.sipp@onera.fr

slide-29
SLIDE 29

The Ginzburg-Landau eq. (cont’d)

MEC651 denis.sipp@onera.fr Adjoints 29

4/ Determine the operator β„’ adjoint to β„’, considering the scalar product β‹…,β‹… .

slide-30
SLIDE 30

Adjoint operator Example with linear PDE and B.C. (1/2)

Adjoints 30

Space: Functions 𝑦 ∈ 0,1 β†’ β„‚ such that 𝑣 0 = πœ–π‘¦π‘£ 1 = 0. Scalar-product: < 𝑣1, 𝑣2 > = 𝑣1

βˆ—π‘£2𝑒𝑦 1

Linear operator 𝒝: 𝒝𝑣 = π‘‰πœ–π‘¦π‘£ βˆ’ 𝛽𝑣 βˆ’ πœ‰πœ–π‘¦π‘¦π‘£ Adjoint operator: < 𝑣1, 𝒝𝑣2 > = 𝑣1

βˆ— π‘‰πœ–π‘¦π‘£2 βˆ’ 𝛽𝑣2 βˆ’ πœ‰πœ–π‘¦π‘¦π‘£2 𝑒𝑦 1

= 𝑣1

βˆ—π‘‰πœ–π‘¦π‘£2 βˆ’ 𝛽𝑣1 βˆ—π‘£2 βˆ’ πœ‰π‘£1 βˆ—πœ–π‘¦π‘¦π‘£2 𝑒𝑦 1

= 𝑣1

βˆ—π‘‰π‘£2 βˆ’ πœ‰π‘£1 βˆ—πœ–π‘¦π‘£2 0 1 +

βˆ’πœ–π‘¦ 𝑣1

βˆ—π‘‰ 𝑣2 βˆ’ 𝛽𝑣1 βˆ—π‘£2 + πœ‰πœ–π‘¦π‘£1 βˆ—πœ–π‘¦π‘£2 𝑒𝑦 1

= 𝑣1

βˆ—π‘‰π‘£2 βˆ’ πœ‰π‘£1 βˆ—πœ–π‘¦π‘£2 + πœ‰(πœ–π‘¦π‘£1 βˆ—)𝑣2 0 1 +

βˆ’πœ–π‘¦ 𝑉𝑣1 βˆ’ 𝛽𝑣1 βˆ’ πœ‰πœ–π‘¦π‘¦π‘£1 βˆ—π‘£2𝑒𝑦

1

= < 𝒝 𝑣1, 𝑣2 > Hence: 𝒝 𝑣 = βˆ’πœ–π‘¦ 𝑉𝑣 βˆ’ 𝛽𝑣 βˆ’ πœ‰πœ–π‘¦π‘¦π‘£ = βˆ’π‘‰πœ–π‘¦π‘£ βˆ’π‘£πœ–π‘¦ 𝑉 βˆ’ 𝛽𝑣 βˆ’ πœ‰πœ–π‘¦π‘¦π‘£

MEC651 denis.sipp@onera.fr

slide-31
SLIDE 31

Adjoint operator Example with linear PDE and B.C. (2/2)

Adjoints 31

Boundary integral term: 𝑣1

βˆ—π‘‰π‘£2 βˆ’ πœ‰π‘£1 βˆ—πœ–π‘¦π‘£2 + πœ‰(πœ–π‘¦π‘£1 βˆ—)𝑣2 0 1 = 0

At 𝑦 = 0: 𝑣2= 0 and πœ–π‘¦π‘£2 β‰  0, so that 𝑣1 = 0 At 𝑦 = 1: πœ–π‘¦π‘£2 = 0 and 𝑣2 β‰  0, so that 𝑣1

βˆ—π‘‰ + πœ‰(πœ–π‘¦π‘£1 βˆ—) = 0, or 𝑣1𝑉 + πœ‰πœ–π‘¦π‘£1 = 0

𝑣1 should be in the following space: Functions 𝑦 ∈ 0,1 β†’ β„‚ such that 𝑣 0 = 𝑣1(1)𝑉 + πœ‰πœ–π‘¦π‘£ 1 = 0.

MEC651 denis.sipp@onera.fr

slide-32
SLIDE 32

MEC651 denis.sipp@onera.fr Adjoints 32

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-33
SLIDE 33

Theorem: Let w i, πœ‡π‘— be eigenvalues/eigenvectors of 𝐡w i = πœ‡π‘—w

  • i. Then there exists π‘₯

𝑗, πœ‡π‘—

βˆ—

solution of the adjoint eigenproblem π΅βˆ—π‘₯ i = πœ‡π‘—

βˆ—π‘₯

  • i. These structures are the adjoint

global modes and may be scaled such that π‘₯ 𝑗

βˆ—w

j = πœ€π‘—π‘˜. The vectors π‘₯ i are bi-

  • rthogonal with respect to the vectors w

j.

Adjoint global modes and biorthogonality (1/4)

In finite dimension

Adjoints 33 MEC651 denis.sipp@onera.fr

slide-34
SLIDE 34

Adjoint global modes and biorthogonality (2/4)

In finite dimension

Adjoints 34

Proof: πœ‡π‘—w i = 𝐡w i πœ‡π‘˜

βˆ—π‘₯

π‘˜ = π΅βˆ—π‘₯ π‘˜ πœ‡π‘—π‘₯ π‘˜

βˆ—w

i = π‘₯ π‘˜

βˆ—π΅w

i = π΅βˆ—π‘₯ π‘˜

βˆ—w

i = πœ‡π‘˜

βˆ—π‘₯

π‘˜

βˆ—w

i = πœ‡π‘˜π‘₯ π‘˜

βˆ—w

i πœ‡π‘— βˆ’ πœ‡π‘˜ π‘₯ π‘˜

βˆ—w

i = 0 If πœ‡π‘— β‰  πœ‡π‘˜, then π‘₯ π‘˜

βˆ—w

i = 0 If π‘₯ π‘˜

βˆ—w

i β‰  0, then πœ‡π‘— = πœ‡π‘˜. Conclusion: π‘₯ π‘˜ can be chosen such that π‘₯ π‘˜

βˆ—w

i = πœ€

π‘˜π‘—

MEC651 denis.sipp@onera.fr

slide-35
SLIDE 35

Theorem: Let w i, πœ‡π‘— be eigenvalues/eigenvectors of πœ‡π‘—β„¬w i + π’ͺ

π‘₯0 + β„’ w

i = 0. Then there exists π‘₯ 𝑗, πœ‡π‘—

βˆ— solution of the adjoint eigenproblem πœ‡π‘— βˆ—β„¬π‘₯

𝑗 + π’ͺ π‘₯0 + β„’ π‘₯ 𝑗 =

  • 0. These structures are the adjoint global modes and may be scaled such that

< π‘₯ 𝑗, ℬw j >= πœ€π‘—π‘˜. The vectors π‘₯ i are bi-orthogonal with respect to the vectors w j.

Adjoint global modes and biorthogonality (3/4)

Adjoints 35 MEC651 denis.sipp@onera.fr

slide-36
SLIDE 36

Adjoint global modes and biorthogonality (4/4)

Adjoints 36

Proof: πœ‡π‘—β„¬w i + π’ͺ

π‘₯0 + β„’ w

i = 0 πœ‡π‘˜

βˆ—β„¬π‘₯

π‘˜ + π’ͺ π‘₯0 + β„’ π‘₯ π‘˜ = 0 < π‘₯ π‘˜, π’ͺ

π‘₯0 + β„’ w

i > = βˆ’πœ‡π‘— < π‘₯ π‘˜, ℬw i> < π‘₯ π‘˜, π’ͺ

π‘₯0 + β„’ w

i > =< π’ͺ π‘₯0 + β„’ π‘₯ π‘˜, w i > =< βˆ’πœ‡π‘˜

βˆ—β„¬π‘₯

π‘˜, w i > = βˆ’πœ‡π‘˜ < π‘₯ π‘˜, ℬw i > πœ‡π‘— βˆ’ πœ‡π‘˜ < π‘₯ π‘˜, ℬw i >= 0 If πœ‡π‘— β‰  πœ‡π‘˜, then < π‘₯ π‘˜, ℬw i > = 0 If < π‘₯ π‘˜, ℬw i >β‰  0, then πœ‡π‘— = πœ‡π‘˜. Conclusion: π‘₯ π‘˜ can be chosen such that < π‘₯ π‘˜, ℬw i > = πœ€

π‘˜π‘—

MEC651 denis.sipp@onera.fr

slide-37
SLIDE 37

The Ginzburg-Landau eq. (cont’d)

MEC651 denis.sipp@onera.fr Adjoints 37

5/ Show that: π‘₯ (𝑦) = πœŠπ‘“

βˆ’ 𝑉

2π›Ώπ‘¦βˆ’πœ“2𝑦2 2 with 𝜊 =

πœ“πœŒβˆ’1

4 is solution of πœ‡βˆ—π‘₯

+ β„’ π‘₯ = 0. Note that the normalization constant 𝜊 has been chosen so that: π‘₯ , π‘₯ = 1. Can you qualitatively represent π‘₯ (𝑦) and π‘₯ 𝑦 ? 6/ Noting that: π‘₯ , π‘₯ = 𝑓

1 2 2 𝑉2 𝛿

3 2𝜈2 1 2,

what does π‘₯ , π‘₯ represent? What is the effect of the advection velocity 𝑉 and viscosity 𝛿

  • n this coefficient?

Nota: π‘₯ π‘œ(𝑦) = πœŠπ‘œπΌπ‘œ(πœ“π‘¦)𝑓

βˆ’ 𝑉

2π›Ώπ‘¦βˆ’πœ“2𝑦2 2 are all the adjoint eigenvectors.

slide-38
SLIDE 38

MEC651 denis.sipp@onera.fr Adjoints 38

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-39
SLIDE 39

Theorem: Let π’ͺ

π‘₯0π‘₯ = 𝑣 β‹… 𝛼𝑣0 + 𝑣0 β‹… 𝛼𝑣

be an operator acting on w = (𝑣, 𝑀, π‘ž) such that 𝑣 = 𝑀 = 0 on boundaries. If < π‘₯1, π‘₯2 > = ∬ 𝑣1

βˆ—π‘£2 + 𝑀1 βˆ—π‘€2 + π‘ž1 βˆ—π‘ž2 𝑒𝑦𝑒𝑧, the adjoint operator of π’ͺ π‘₯0 is

π’ͺ π‘₯0 = πœ–π‘¦u0 πœ–π‘§π‘£0 πœ–π‘¦π‘€0 πœ–π‘§π‘€0

βˆ—

+ βˆ’π‘£0

βˆ—πœ–π‘¦ βˆ’ 𝑀0 βˆ—πœ–π‘§

βˆ’π‘£0

βˆ—πœ–π‘¦ βˆ’ 𝑀0 βˆ—πœ–π‘§

Adjoint of linearized advection operator (1/4)

Adjoints 39 MEC651 denis.sipp@onera.fr

slide-40
SLIDE 40

< π‘₯1, π’ͺ

π‘₯0π‘₯2 >=< π’ͺ

π‘₯0π‘₯1, π‘₯2 > 𝑣1

βˆ— 𝑣0πœ–π‘¦π‘£2 + 𝑀0πœ–π‘§π‘£2 + 𝑣2πœ–π‘¦π‘£0 + 𝑀2πœ–π‘§π‘£0

+𝑀1

βˆ— 𝑣0πœ–π‘¦π‘€2 + 𝑀0πœ–π‘§π‘€2 + 𝑣2πœ–π‘¦π‘€0 + 𝑀2πœ–π‘§π‘€0

𝑒𝑦𝑒𝑧 = 𝑣1

βˆ—π‘£0πœ–π‘¦π‘£2 + 𝑣1 βˆ—π‘€0πœ–π‘§π‘£2 + 𝑀1 βˆ—π‘£0πœ–π‘¦π‘€2 + 𝑀1 βˆ—π‘€0πœ–π‘§π‘€2

+ 𝑣1

βˆ—πœ–π‘¦π‘£0𝑣2 + 𝑣1 βˆ—πœ–π‘§π‘£0𝑀2 + 𝑀1 βˆ—πœ–π‘¦π‘€0𝑣2 + 𝑀1 βˆ—πœ–π‘§π‘€0𝑀2

𝑒𝑦𝑒𝑧 = 𝑣1

βˆ—π‘£0πœ–π‘¦π‘£2 + 𝑣1 βˆ—π‘€0πœ–π‘§π‘£2 + 𝑀1 βˆ—π‘£0πœ–π‘¦π‘€2 + 𝑀1 βˆ—π‘€0πœ–π‘§π‘€2 𝑒𝑦𝑒𝑧 βˆ—

+ 𝑣1πœ–π‘¦π‘£0

βˆ— + 𝑀1πœ–π‘¦π‘€0 βˆ— βˆ— 𝑣2 + 𝑣1πœ–π‘§π‘£0 βˆ— + 𝑀1πœ–π‘§π‘€0 βˆ— βˆ—π‘€2 𝑒𝑦𝑒𝑧

Adjoint of linearized advection operator (2/4)

Adjoints 40 MEC651 denis.sipp@onera.fr

slide-41
SLIDE 41

βˆ— = 𝑣1

βˆ—π‘£0π‘œπ‘¦π‘£2 + 𝑣1 βˆ—π‘€0π‘œπ‘§π‘£2 + 𝑀1 βˆ—π‘£0π‘œπ‘¦π‘€2 + 𝑀1 βˆ—π‘€0π‘œπ‘§π‘€2 𝑒𝑑

βˆ’ πœ–π‘¦ 𝑣1

βˆ—π‘£0 𝑣2 + πœ–π‘§ 𝑣1 βˆ—π‘€0 𝑣2 + πœ–π‘¦ 𝑀1 βˆ—π‘£0 𝑀2 + πœ–π‘§ 𝑀1 βˆ—π‘€0 𝑀2 𝑒𝑦𝑒𝑧

= βˆ’ πœ–π‘¦ 𝑣1𝑣0

βˆ— + πœ–π‘§ 𝑣1𝑀0 βˆ— βˆ— 𝑣2 + πœ–π‘¦ 𝑀1𝑣0 βˆ— + πœ–π‘§ 𝑀1𝑀0 βˆ— βˆ— 𝑀2 𝑒𝑦𝑒𝑧

π’ͺ π‘₯0π‘₯1 = 𝑣1πœ–π‘¦π‘£0

βˆ— + 𝑀1 πœ–π‘¦π‘€0 βˆ—

𝑣1πœ–π‘§π‘£0

βˆ— + 𝑀1πœ–π‘§π‘€0 βˆ—

+ βˆ’πœ–π‘¦ 𝑣1𝑣0

βˆ— βˆ’ πœ–π‘§ 𝑣1𝑀0 βˆ—

βˆ’πœ–π‘¦ 𝑀1𝑣0

βˆ— βˆ’ πœ–π‘§ 𝑀1𝑀0 βˆ— βˆ’π‘£0

βˆ—πœ–π‘¦π‘£1βˆ’π‘€0 βˆ—πœ–π‘§π‘£1

βˆ’π‘£0

βˆ—πœ–π‘¦π‘€1βˆ’π‘€0 βˆ—πœ–π‘§π‘€1

(using πœ–π‘¦π‘£0 + πœ–π‘§π‘€0 = 0 )

Adjoint of linearized advection operator (3/4)

Adjoints 41 MEC651 denis.sipp@onera.fr

slide-42
SLIDE 42

Conclusion: π’ͺ π‘₯0π‘₯1 = πœ–π‘¦u0 πœ–π‘§π‘£0 πœ–π‘¦π‘€0 πœ–π‘§π‘€0

βˆ— 𝑣1

𝑀1 π‘ž1 + βˆ’π‘£0

βˆ—πœ–π‘¦ βˆ’ 𝑀0 βˆ—πœ–π‘§

βˆ’π‘£0

βˆ—πœ–π‘¦ βˆ’ 𝑀0 βˆ—πœ–π‘§

𝑣1 𝑀1 π‘ž1 π’ͺ

π‘₯0π‘₯2 =

πœ–π‘¦u0 πœ–π‘§π‘£0 πœ–π‘¦π‘€0 πœ–π‘§π‘€0 𝑣2 𝑀2 π‘ž2 + 𝑣0πœ–π‘¦ + 𝑀0πœ–π‘§ 𝑣0πœ–π‘¦ + 𝑀0πœ–π‘§ 𝑣2 𝑀2 π‘ž2 π’ͺ

π‘₯0 β‰  π’ͺ

π‘₯0 because of:

  • component-type non-normality => 𝑀 β†’ 𝑣 becomes 𝑣 β†’ 𝑀
  • convective-type non-normality => upstream convection

Adjoint of linearized advection operator (4/4)

Adjoints 42 MEC651 denis.sipp@onera.fr

slide-43
SLIDE 43

MEC651 denis.sipp@onera.fr Adjoints 43

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-44
SLIDE 44

Theorem: Let β„’ = βˆ’πœ‰Ξ”() 𝛼() βˆ’π›Ό β‹… () be an operator acting on w = (𝑣, 𝑀, π‘ž) such that 𝑣 = 𝑀 = 0 on boundaries. If π‘₯1, π‘₯2 = ∬ 𝑣1

βˆ—π‘£2 + 𝑀1 βˆ—π‘€2 + π‘ž1 βˆ—π‘ž2 𝑒𝑦𝑒𝑧, the

  • perator β„’ is self-afjoint : β„’

= β„’.

Adjoint of Stokes operator (1/4)

Adjoints 44 MEC651 denis.sipp@onera.fr

slide-45
SLIDE 45

< π‘₯1, β„’π‘₯2 >=< β„’ π‘₯1, π‘₯2 > 𝑣1

βˆ— βˆ’πœ‰πœ–π‘¦π‘¦π‘£2 βˆ’ πœ‰πœ–π‘§π‘§π‘£2 + πœ–π‘¦π‘ž2 + 𝑀1 βˆ— βˆ’πœ‰πœ–π‘¦π‘¦π‘€2 βˆ’ πœ‰πœ–π‘§π‘§π‘€2 + πœ–π‘§π‘ž2

+ π‘ž1

βˆ— βˆ’πœ–π‘¦π‘£2 βˆ’ πœ–π‘§π‘€2 𝑒𝑦𝑒𝑧

= βˆ’πœ‰π‘£1

βˆ—πœ–π‘¦π‘¦π‘£2 βˆ’ πœ‰π‘£1 βˆ—πœ–π‘§π‘§π‘£2 + 𝑣1 βˆ—πœ–π‘¦π‘ž2 βˆ’ πœ‰π‘€1 βˆ—πœ–π‘¦π‘¦π‘€2 βˆ’ πœ‰π‘€1 βˆ—πœ–π‘§π‘§π‘€2

+ 𝑀1

βˆ—πœ–π‘§π‘ž2 βˆ’ π‘ž1 βˆ—πœ–π‘¦π‘£2 βˆ’ π‘ž1 βˆ—πœ–π‘§π‘€2 𝑒𝑦𝑒𝑧

β„’ = βˆ’πœ‰Ξ”() 𝛼() βˆ’π›Ό β‹… () = βˆ’πœ‰π‘£1

βˆ—π‘œπ‘¦πœ–π‘¦π‘£2 βˆ’ πœ‰π‘£1 βˆ—π‘œπ‘§πœ–π‘§π‘£2 + 𝑣1 βˆ—π‘œπ‘¦π‘ž2 βˆ’ πœ‰π‘€1 βˆ—π‘œπ‘¦πœ–π‘¦π‘€2 βˆ’ πœ‰π‘€1 βˆ—π‘œπ‘§πœ–π‘§π‘€2 + 𝑀1 βˆ—π‘œπ‘§π‘ž2

βˆ’ π‘ž1

βˆ—π‘œπ‘¦π‘£2 βˆ’ π‘ž1 βˆ—π‘œπ‘§π‘€2 𝑒𝑑

βˆ’ βˆ’πœ‰πœ–π‘¦π‘£1

βˆ—πœ–π‘¦π‘£2 βˆ’ πœ‰πœ–π‘§π‘£1 βˆ—πœ–π‘§π‘£2 + πœ–π‘¦π‘£1 βˆ—π‘ž2 βˆ’ πœ‰πœ–π‘¦π‘€1 βˆ—πœ–π‘¦π‘€2 βˆ’ πœ‰πœ–π‘§π‘€1 βˆ—πœ–π‘§π‘€2

+ πœ–π‘§π‘€1

βˆ—π‘ž2 βˆ’ πœ–π‘¦π‘ž1 βˆ—π‘£2 βˆ’ πœ–π‘§π‘ž1 βˆ—π‘€2 𝑒𝑦𝑒𝑧

𝑣 = 𝑀 = 0 on boundaries

Adjoint of Stokes operator (2/4)

Adjoints 45 MEC651 denis.sipp@onera.fr

slide-46
SLIDE 46

= βˆ’ πœ–π‘¦π‘£1

βˆ—π‘ž2 + πœ–π‘§π‘€1 βˆ—π‘ž2 βˆ’ πœ–π‘¦π‘ž1 βˆ—π‘£2 βˆ’ πœ–π‘§π‘ž1 βˆ—π‘€2 𝑒𝑦𝑒𝑧

+ βˆ’ βˆ’πœ‰πœ–π‘¦π‘£1

βˆ—πœ–π‘¦π‘£2 βˆ’ πœ‰πœ–π‘§π‘£1 βˆ—πœ–π‘§π‘£2 βˆ’ πœ‰πœ–π‘¦π‘€1 βˆ—πœ–π‘¦π‘€2 βˆ’ πœ‰πœ–π‘§π‘€1 βˆ—πœ–π‘§π‘€2 𝑒𝑦𝑒𝑧 (βˆ—)

βˆ— = βˆ’ βˆ’πœ‰πœ–π‘¦π‘£1

βˆ—π‘œπ‘¦π‘£2 βˆ’ πœ‰πœ–π‘§π‘£1 βˆ—π‘œπ‘§π‘£2 βˆ’ πœ‰πœ–π‘¦π‘€1 βˆ—π‘œπ‘¦π‘€2 βˆ’ πœ‰πœ–π‘§π‘€1 βˆ—π‘œπ‘§π‘€2 𝑒𝑑

+ βˆ’πœ‰πœ–π‘¦π‘¦π‘£1

βˆ—π‘£2 βˆ’ πœ‰πœ–π‘§π‘§π‘£1 βˆ—π‘£2 βˆ’ πœ‰πœ–π‘¦π‘¦π‘€1 βˆ—π‘€2 βˆ’ πœ‰πœ–π‘§π‘§π‘€1 βˆ—π‘€2 𝑒𝑦𝑒𝑧

β„’ π‘₯1 = βˆ’πœ‰πœ–π‘¦π‘¦ βˆ’ πœ‰πœ–π‘§π‘§ πœ–π‘¦ βˆ’πœ‰πœ–π‘¦π‘¦ βˆ’ πœ‰πœ–π‘§π‘§ πœ–π‘§ βˆ’πœ–π‘¦ βˆ’πœ–π‘§ 𝑣1 𝑀1 π‘ž1

Adjoint of Stokes operator (3/4)

Adjoints 46 MEC651 denis.sipp@onera.fr

slide-47
SLIDE 47

β„’ π‘₯1 = βˆ’πœ‰πœ–π‘¦π‘¦ βˆ’ πœ‰πœ–π‘§π‘§ πœ–π‘¦ βˆ’πœ‰πœ–π‘¦π‘¦ βˆ’ πœ‰πœ–π‘§π‘§ πœ–π‘§ βˆ’πœ–π‘¦ βˆ’πœ–π‘§ 𝑣1 𝑀1 π‘ž1 β„’π‘₯2 = βˆ’πœ‰πœ–π‘¦π‘¦ βˆ’ πœ‰πœ–π‘§π‘§ πœ–π‘¦ βˆ’πœ‰πœ–π‘¦π‘¦ βˆ’ πœ‰πœ–π‘§π‘§ πœ–π‘§ βˆ’πœ–π‘¦ βˆ’πœ–π‘§ 𝑣2 𝑀2 π‘ž2 β„’ = β„’

Adjoint of Stokes operator (4/4)

Adjoints 47 MEC651 denis.sipp@onera.fr

slide-48
SLIDE 48

MEC651 denis.sipp@onera.fr Adjoints 48

  • Governing equations
  • Asymptotic development
  • Order πœ—0 : Base-flow
  • Order πœ—1 : Global modes
  • Bi-orthogonal basis and adjoint global modes
  • Definition of adjoint global modes
  • Optimal initial condition
  • Optimal forcing in stable flow
  • Adjoint operator
  • Definition
  • Adjoint global modes as solutions of adjoint eigen-problem
  • Adjoint linearized Navier-Stokes operator
  • Adjoint of linearized advection operator
  • Adjoint of Stokes operator
  • Adjoint global modes of cylinder flow

Outline

slide-49
SLIDE 49

The adjoint global mode

  • f cylinder flow

Adjoints 49 MEC651 denis.sipp@onera.fr

Real part of cross-stream velocity field. Marginal adjoint global mode