Adjoints

MEC651 [email protected]

Adjoints

1

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

2

Governing equations Incompressible Navier-Stokes equations: 𝜕𝑡 𝑢 + 𝑢𝜕𝑥 𝑢 + 𝑣𝜕𝑦 𝑢 = −𝜕𝑥 𝑝 + 𝜈 𝜕𝑥𝑥 𝑢 + 𝜕𝑦𝑦 𝑢 + 𝑓 𝜕𝑡 𝑣 + 𝑢𝜕𝑥 𝑣 + 𝑣𝜕𝑦 𝑣 = −𝜕𝑦 𝑝 + 𝜈 𝜕𝑥𝑥 𝑣 + 𝜕𝑦𝑦 𝑣 + 𝑔 −𝜕𝑥 𝑢 − 𝜕𝑦 𝑣 = 0 Can be recast into: 1 ℬ𝜕𝑡 𝑤+ 𝒩 𝑤, 𝑤 + ℒ𝑤 = 𝑓 2 where: 𝑢 𝑓 𝑤= 𝑝 𝑓= 0 1 0 ℬ= , 0 0 𝑢 ⋅ 𝛻𝑢2 + 𝑢2 ⋅ 𝛻𝑢1 𝒩 𝑤1 , 𝑤2 = 1 0 −𝜈Δ() 𝛻() ℒ= −𝛻 ⋅ () 0 Boundary conditions: Dirichlet, Neumann, Mixed MEC651 [email protected]

Adjoints

3

Some properties a) 𝒩 𝑤1 , 𝑤2 = 𝒩 𝑤2 , 𝑤1 b)

1 𝒩 2

𝑤0 + 𝜖𝛿𝑤, 𝑤0 + 𝜖𝛿𝑤 =

c) 𝒩𝑤0 𝛿𝑤 = 𝒩 𝑤0 , 𝛿𝑤 =

1 𝒩 2

𝑤0 , 𝑤0 + 𝜖 𝒩 𝑤0 , 𝛿𝑤 + Jacobian

𝛿𝑢 ⋅ 𝛻𝑢0 + 𝑢0 ⋅ 𝛻𝛿𝑢 0

𝜖2 𝒩 2

𝛿𝑤, 𝛿𝑤 + ⋯ Hessian

=𝒩𝑤0 𝛿𝑤

𝑢 𝑢 d) ℬ𝑤 = ℬ 𝑝 = 0 e) 𝜕𝑡 𝑢 + 𝑢 ⋅ 𝛻𝑢 = −𝛻𝑝 + 𝜈𝛻 2 𝑢 ⇒ −𝛻 2 𝑝 = 𝛻 ⋅ 𝑢 ⋅ 𝛻𝑢 , 𝜕𝑛 𝑝 = 𝜈𝛻 2 𝑢 ⋅ 𝑛 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: < 𝑤1 , 𝑤2 > = ∬ 𝑢1∗ 𝑢2 + 𝑣1∗ 𝑣2 𝑑𝑥𝑑𝑦 = ∬ (𝑤1 ⋅ ℬ𝑤2 )𝑑𝑥𝑑𝑦 so that < 𝑤, 𝑤 > is the energy.

MEC651 [email protected]

Adjoints

4

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

5

Asymptotic development Solution: 𝑤 𝑡 = 𝑤0 + 𝜖𝑤1 𝑡 + ⋯ with ϵ ≪ 1 Governing equations:

1 ℬ𝜕𝑡 𝑤+ 𝒩 𝑤, 𝑤 + ℒ𝑤 = 𝑓 2 Introduce solution into governing eq:: 1 ℬ𝜕𝑡 (𝑤0 +𝜖𝑤1 + ⋯ )+ 𝒩 𝑤0 + 𝜖𝑤1 + ⋯ , 𝑤0 + 𝜖𝑤1 + ⋯ + ℒ(𝑤0 +𝜖𝑤1 + ⋯ ) = 𝑓 2 1 𝒩 𝑤0 , 𝑤0 + ℒ𝑤0 = 𝑓 at order 𝑂(1) 2 1 ℬ𝜕𝑡 𝑤1 + [𝒩 𝑤1 , 𝑤0 + 𝒩 𝑤0 , 𝑤1 ] + ℒ𝑤1 = 0 at order 𝑂(𝜖) ⇒ 2 𝒩𝑤0 𝑤1

1 ℬ𝜕𝑡 𝑤2 +𝒩𝑤0 𝑤2 + ℒ𝑤2 = − 𝒩 𝑤1 , 𝑤1 at order 𝑂(𝜖 2 ) 2

MEC651 [email protected]

Adjoints

6

Oder

0 𝜖 :

Base-flow

Definition: 𝑤 𝑡 = 𝑤0 + 𝜖𝑤1 (𝑡) + ⋯

1 𝐹 𝑤 = 𝒩 𝑤, 𝑤 + ℒ𝑤 − 𝑓 2

Non-linear equilibrium point : 1 𝒩 𝑤0 , 𝑤0 + ℒ𝑤0 = 𝑓 2

𝑤

How to compute a base-flow ? 𝑤0 Newton iteration: 1 𝒩 𝑤0 + 𝛿𝑤0 , 𝑤0 + 𝛿𝑤0 + ℒ(𝑤0 +𝛿𝑤0 ) = 𝑓 2 Linearization: 1 𝒩 𝑤0 , 𝛿𝑤0 + ℒ𝛿𝑤0 = 𝑓 − 𝒩 𝑤0 , 𝑤0 − ℒ𝑤0 2 1 −1 ⇒ 𝛿𝑤0 = 𝒩𝑤0 + ℒ 𝑓 − 𝒩 𝑤0 , 𝑤0 − ℒ𝑤0 2 MEC651 [email protected]

Adjoints

7

0

Oder 𝜖 : Base-flow The case of cylinder flow

𝑅𝑒 = 47 Streamwise velocity field of base-flow.

MEC651 [email protected]

Adjoints

8

1

Order 𝜖 : Global modes Definition 𝑤 𝑡 = 𝑤0 + 𝜖𝑤1 (𝑡) + ⋯ Linear governing equation: ℬ𝜕𝑡 𝑤1 + 𝒩𝑤0 𝑤1 + ℒ𝑤1 = 0 Solution 𝑤1 under the form: 𝑤1 = 𝑒 𝜆𝑡 𝑤 + c.c This leads to : 𝜆ℬ𝑤 + 𝒩𝑤0 + ℒ 𝑤 = 0 Eigenvalue: 𝜆 = 𝜎 + 𝑖𝜔 Eigenvector: Real solution:

𝑤 = 𝑤r + iw𝑖 𝑤1 = 𝑒 𝜆𝑡 𝑤 + c.c = 2𝑒 𝜎𝑡 (cos 𝜔𝑡 𝑤𝑟 − sin 𝜔𝑡 𝑤𝑖 )

MEC651 [email protected]

Adjoints

9

1

Order 𝜖 : Global modes How to compute global modes ? Eigenvalue problem solved with shift-invert strategy: - Power method, easy to find largest magnitude eigenvalues of 𝐴𝑥 = 𝜆𝑥. For this, evaluate 𝐴𝑛 𝑥0 - To find eigenvalues of 𝐴 closest to zero, search largest magnitude eigenvalues of 𝐴−1 : 𝐴−1 𝑥 = 𝜆−1 𝑥. For this, evaluate 𝐴−1 𝑛 𝑥0 - To find eigenvalues of 𝐴 closest to 𝑠, search largest magnitude eigenvalues of 𝑛 𝐴 − 𝑠𝐼 −1 : 𝐴 − 𝑠𝐼 −1 𝑥 = 𝜆 − 𝑠 −1 𝑥. For this, evaluate 𝐴 − 𝑠𝐼 −1 𝑥0 - Instead of power-method, use Krylov subspaces -> Arnoldi technique - Cost of algorithm = cost of several complex matrix inversions

MEC651 [email protected]

Adjoints

10

1

Order 𝜖 : Global modes Case of cylinder flow

Spectrum 𝑅𝑒 = 47

MEC651 [email protected]

Real part of cross-stream velocity field Marginal eigenmode

Adjoints

11

The Ginzburg-Landau eq. We consider the linear Ginzburg-Landau equation 𝜕𝑡 𝑤1 + ℒ𝑤1 = 0 where

𝑥2 ℒ = 𝑈𝜕𝑥 − 𝜇 𝑥 − 𝛾𝜕𝑥𝑥 , 𝜇 𝑥 = 𝑖𝜔0 + 𝜇0 − 𝜇2 . 2 Here 𝑈, 𝛾, 𝜔0 , 𝜇0 and 𝜇2 are positive real constants. The state 𝑤(𝑥, 𝑡) is a complex variable on −∞ < 𝑥 < +∞ such that |𝑤| → 0 as 𝑥 → ∞. In the following, +∞

𝑤𝑎 𝑥 ∗ 𝑤𝑏 𝑥 𝑑𝑥 .

𝑤𝑎 , 𝑤𝑏 = −∞

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

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Adjoints

12

The Ginzburg-Landau eq. 2/ Show that 𝑤(𝑥) = 𝜁𝑒

𝑈 𝜒2 𝑥2 𝑥− 2𝛾 2

1

with 𝜒 =

𝜇2 4 2𝛾

and 𝜁 =

𝜒 2 1 1 𝑈 2 2 𝜋4 𝑒 8𝛾 𝜒

verifies 𝜆𝑤 + ℒ𝑤 = 0.

What is the eigenvalue 𝜆 associated to this eigenvector? The constant 𝜁 has been selected so that 𝑤, 𝑤 = 1. 3/ Show that the flow is unstable if the constant 𝜇0 is chosen such that: 𝜇0 > 𝜇𝑐 , where 𝜇𝑐 =

Nota:

𝑈2 4𝛾

+

𝛾𝜇2 . 2

𝜆𝑛 = i𝜔0 + 𝜇0 −

𝑈2 4𝛾

− 2𝑛 + 1

𝛾𝜇2 , 𝑤𝑛 2

= 𝜁𝑛 𝐻𝑛 𝜒𝑥 𝑒

𝑈 𝜒2 𝑥2 𝑥− 2𝛾 2

are

all

the

eigenvalues/eigenvectors of ℒ, 𝐻𝑛 being Hermite polynomials.

MEC651 [email protected]

Adjoints

13

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

14

Bi-orthogonal basis and adjoint global modes (1/3) In finite dimension Global modes: The eigenvectors 𝑤𝑖 form a basis:

𝐴𝑤𝑖 = 𝜆𝑖 𝑤𝑖 𝑤=

𝛼𝑖 wi 𝑖

Definition of adjoint global modes: with as a given scalar-product (say < 𝑤1 , 𝑤2 > = 𝑤1∗ 𝑤2 ), there exists for each 𝛼𝑖 a unique 𝑤𝑖 such that 𝛼𝑖 =< 𝑤𝑖 , 𝑤 > for all 𝑤. The adjoint global modes are the structures 𝑤𝑖 . In the following: 𝑤𝑖 , 𝑤𝑖 = 1. Properties:  𝑤𝑘 and wj are bi-orthogonal bases: they verify 𝑤𝑗 = < 𝑤𝑘 , wj > = 𝛿𝑘𝑗 (in matrix notations 𝑊 ∗ 𝑊 = 𝐼)

𝑖

1 2

< 𝑤𝑖 , 𝑤𝑗 > wi and so 1 2

 Cauchy-Lifschitz: 1 = < 𝑤𝑖 , wi > ≤< 𝑤𝑖 , 𝑤𝑖 > < 𝑤𝑖 , 𝑤𝑖 > 1 2

Hence: < 𝑤𝑖 , 𝑤𝑖 > ≥ 1 and cos angle 𝑤𝑖 , 𝑤𝑖 = MEC651 [email protected]

Adjoints

1

1

2 15

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

𝑤 = (𝑤1 ⋅ 𝑤)𝑤1 + (𝑤2 ⋅ 𝑤)𝑤2

Def of 𝑤1 : 𝑤1 ⋅ 𝑤1 = 1 𝑤1 ⋅ 𝑤2 = 0

𝑤2 𝑤2

𝑤1

Def of 𝑤2 : 𝑤2 ⋅ 𝑤2 = 1 𝑤2 ⋅ 𝑤1 = 0

𝑤1 Method 1 : 𝑊 = W ∗−1

𝑊 ∗𝑊 = 𝐼

Method 2 : 𝑊 = 𝑊 𝑋 ⇒ 𝑋 ∗ 𝑊 ∗ 𝑊 = 𝐼 ⇒ 𝑋 = 𝑊 ∗ 𝑊 Method 3 : adjoint global modes MEC651 [email protected]

Adjoints

−1

⇒ 𝑊 = 𝑊 𝑊 ∗𝑊

−1

16

Bi-orthogonal basis and adjoint global modes (3/3) Global modes: 𝜆𝑖 ℬ𝑤𝑖 + 𝒩𝑤0 + ℒ 𝑤𝑖 = 0 The eigenvectors 𝑤𝑖 form a basis: 𝑤=

𝛼𝑖 wi 𝑖

Definition of adjoint global modes: with as a given scalar-product, there exists for each 𝛼𝑖 a unique 𝑤𝑖 such that 𝛼𝑖 =< 𝑤𝑖 , ℬ𝑤 > for all 𝑤. The adjoint global modes are the structures 𝑤𝑖 . In the following: < 𝑤𝑖 , ℬ𝑤𝑖 > = 1. Properties:  𝑤𝑘 and wj are bi-orthogonal bases: they verify 𝑤𝑗 = < 𝑤𝑘 , ℬwj > = 𝛿𝑘𝑗

1 2

𝑖

< 𝑤𝑖 , ℬ𝑤𝑗 > wi and so 1 2

 Cauchy-Lifschitz: 1 = < 𝑤𝑖 , ℬwi > ≤< 𝑤𝑖 , ℬ𝑤𝑖 > < 𝑤𝑖 , ℬ𝑤𝑖 > . Hence: 1 2

< 𝑤𝑖 , ℬ𝑤𝑖 > ≥ 1 and cos angle 𝑤𝑖 , 𝑤𝑖 = MEC651 [email protected]

Adjoints

1

1 2

17

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

18

Optimal initial condition (1/3) Definition of optimal initial condition Initial-value problem: ℬ𝜕𝑡 𝑤1 + 𝒩𝑤0 + ℒ 𝑤1 = 0,

𝑤1 𝑡 = 0 = 𝑤 𝐼

Solution: < 𝑤𝑖 , ℬ𝑤 𝐼 > 𝑒 𝜆𝑖 𝑡 𝑤𝑖

𝑤1 𝑡 = 𝑖

If (𝑤1 , 𝜆1 ) is the global mode which displays largest growth rate, at large times: 𝑤1 𝑡 ≈< 𝑤1 , ℬ𝑤 𝐼 > 𝑒 𝜆1𝑡 𝑤1 We look for unit-norm 𝑤 𝐼 (< 𝑤 𝐼 , ℬw I >= 1) which maximizes the amplitude of the response at large times. 𝑤𝐼 is the optimal initial condition.

MEC651 [email protected]

Adjoints

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Optimal initial condition (2/3) If direct global mode as initial condition: 𝑤 𝐼 = 𝑤1 In this case, at large time:

𝑤1 𝑡 ≈ 𝑒 𝜆1𝑡 𝑤1

If adjoint global mode as initial condition: 𝑤𝐼

𝑤1

=
2

> ≤
2
2

1

MEC651 [email protected]

Adjoints

20

Optimal initial condition (3/3) Estimation of gain: From Causchy-Lifschitz: < Amplitude gain:
2 ≥

1 𝑤1 , ℬ𝑤1 >2 =

1

1 cos angle 𝑤1 , 𝑤1

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In finite dimension 𝑤1

𝑤2 ≈ −1.2𝑤1 + 1.8𝑤2 𝑤2 = 1

𝑤2

𝑤2 = 0𝑤1 + 1𝑤2 𝑤2 = 1

𝑤2

𝑤1

𝑤1 𝑤2

𝑤1 ≈ 1.9𝑤1 − 1.3𝑤2 𝑤1 = 1 𝑤1 = 1𝑤1 + 0𝑤2 𝑤1 = 1

MEC651 [email protected]

Adjoints

22

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

23

Optimal forcing in stable flow (1/2) Problem:

1 ℬ𝜕𝑡 𝑤+ 𝒩 𝑤, 𝑤 + ℒ𝑤 = 𝜖ℬ𝑓1 2 𝑤 = 𝑤0 + 𝜖𝑤1

At first order: ℬ𝜕𝑡 𝑤1 + 𝒩𝑤0 + ℒ 𝑤1 = 𝑓1

In frequency domain: 𝑤1 = 𝑒 𝑖𝜔𝑡 𝑤 and 𝑓1 = 𝑒 𝑖𝜔𝑡 𝑓 Governing equation: 𝑖𝜔ℬ𝑤 + 𝒩𝑤0 + ℒ 𝑤 = ℬf Where to force (𝑓) and at which frequency (𝜔) to obtain strongest response (𝑤)?

MEC651 [email protected]

Adjoints

24

Optimal forcing in stable flow (1/2) Introducing global mode basis: 𝑤 =

𝑖

< 𝑤𝑖 , ℬ𝑤 > 𝑤𝑖 and ℬ𝑓 =

𝑖𝜔 < 𝑤𝑖 , ℬ𝑤 > ℬ𝑤𝑖 − 𝜆𝑖 < 𝑤𝑖 , ℬ𝑤 > ℬ𝑤𝑖 = 𝑖

𝑖

< 𝑤𝑖 , ℬ𝑓 > ℬ 𝑤𝑖 :

< 𝑤𝑖 , ℬ𝑓 > ℬ𝑤𝑖 𝑖

Scalar-product with 𝑤𝑗 and using bi-orthogonality:< 𝑤𝑗 , ℬ𝑤𝑖 > = 𝛿𝑖𝑗 < 𝑤𝑗 , ℬ𝑤 > 𝑖𝜔 − 𝜆𝑗 =< 𝑤𝑗 , ℬf > < 𝑤𝑗 , ℬf > < 𝑤𝑗 , ℬ𝑤 >= 𝑖𝜔 − 𝜆𝑗 Solution:

𝑤1 𝑡 =

𝑒 𝑖𝜔𝑡 𝑖

< 𝑤𝑖 , ℬf > ℬ𝑤𝑖 𝑖𝜔 − 𝜆𝑖

To maximize response: a/ force at frequencies i𝜔 closest to 𝜆𝑖 b/ force with f =

𝑤i

1

2

MEC651 [email protected]

Adjoints

25

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

26

Adjoint operator Definition Definition of adjoint operator: Let ⟨𝑤1 , 𝑤2 ⟩ be a scalar product and 𝒜 a linear operator. The adjoint operator of 𝒜 verifies ⟨𝑤1 , 𝒜𝑤2 ⟩ = ⟨𝒜 𝑤1 , 𝑤2 ⟩ whatever 𝑤1 and 𝑤2 .

MEC651 [email protected]

Adjoints

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Adjoint operator Example in finite dimension Space:

𝑤 ∈ ℂ𝑁

Scalar-product:

< 𝑤1 , 𝑤2 > = 𝑤1∗ 𝑄𝑤2 with 𝑄 a Hermitian matrix 𝑄∗ = 𝑄. Linear operator: 𝒜 matrix. Adjoint operator: < 𝑤1 , 𝒜𝑤2 > = 𝑤1∗ 𝑄𝒜𝑤2 = 𝑤1∗ 𝑄𝒜𝒬 −1 𝒬𝑤2 = 𝒬 −1 𝒜∗ 𝒬𝑤1 ∗ 𝒬𝑤2 =< 𝒜 𝑤1 , 𝑤2 > with 𝒜 = 𝒬 −1 𝒜∗ 𝒬 If 𝒬 = 𝐼, then 𝒜 = 𝒜∗

MEC651 [email protected]

Adjoints

28

The Ginzburg-Landau eq. (cont’d) 4/ Determine the operator ℒ adjoint to ℒ, considering the scalar product ⋅,⋅ .

MEC651 [email protected]

Adjoints

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Adjoint operator Example with linear PDE and B.C. (1/2) Space: Functions 𝑥 ∈ 0,1 → ℂ such that 𝑢 0 = 𝜕𝑥 𝑢 1 = 0. Scalar-product: 1

< 𝑢1 , 𝑢2 > = Linear operator 𝒜:

0

𝑢1∗ 𝑢2 𝑑𝑥

𝒜𝑢 = 𝑈𝜕𝑥 𝑢 − 𝛼𝑢 − 𝜈𝜕𝑥𝑥 𝑢 Adjoint operator: 1 < 𝑢1 , 𝒜𝑢2 > = 0 𝑢1∗ 𝑈𝜕𝑥 𝑢2 − 𝛼𝑢2 − 𝜈𝜕𝑥𝑥 𝑢2 𝑑𝑥 =

1 ∗ 𝑢1 𝑈𝜕𝑥 𝑢2 − 𝛼𝑢1∗ 𝑢2 − 𝜈𝑢1∗ 𝜕𝑥𝑥 𝑢2 𝑑𝑥 = 0 1 𝑢1∗ 𝑈𝑢2 − 𝜈𝑢1∗ 𝜕𝑥 𝑢2 10 + 0 −𝜕𝑥 𝑢1∗ 𝑈 𝑢2 − 𝛼𝑢1∗ 𝑢2 + 𝜈𝜕𝑥 𝑢1∗ 𝜕𝑥 𝑢2 𝑑𝑥 = 1 𝑢1∗ 𝑈𝑢2 − 𝜈𝑢1∗ 𝜕𝑥 𝑢2 + 𝜈(𝜕𝑥 𝑢1∗ )𝑢2 10 + 0 −𝜕𝑥 𝑈𝑢1 − 𝛼𝑢1 − 𝜈𝜕𝑥𝑥 𝑢1 ∗ 𝑢2 𝑑𝑥

=

< 𝒜 𝑢1 , 𝑢2 > Hence: 𝒜 𝑢 = −𝜕𝑥 𝑈𝑢 − 𝛼𝑢 − 𝜈𝜕𝑥𝑥 𝑢 = −𝑈𝜕𝑥 𝑢 −𝑢𝜕𝑥 𝑈 − 𝛼𝑢 − 𝜈𝜕𝑥𝑥 𝑢 MEC651 [email protected]

Adjoints

30

Adjoint operator Example with linear PDE and B.C. (2/2) Boundary integral term: 𝑢1∗ 𝑈𝑢2 − 𝜈𝑢1∗ 𝜕𝑥 𝑢2 + 𝜈(𝜕𝑥 𝑢1∗ )𝑢2

1 0

=0

At 𝑥 = 0: 𝑢2 = 0 and 𝜕𝑥 𝑢2 ≠ 0, so that 𝑢1 = 0 At 𝑥 = 1: 𝜕𝑥 𝑢2 = 0 and 𝑢2 ≠ 0, so that 𝑢1∗ 𝑈 + 𝜈(𝜕𝑥 𝑢1∗ ) = 0, or 𝑢1 𝑈 + 𝜈𝜕𝑥 𝑢1 = 0 𝑢1 should be in the following space: Functions 𝑥 ∈ 0,1 → ℂ such that 𝑢 0 = 𝑢1 (1)𝑈 + 𝜈𝜕𝑥 𝑢 1 = 0.

MEC651 [email protected]

Adjoints

31

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

32

Adjoint global modes and biorthogonality (1/4) In finite dimension Theorem: Let wi , 𝜆𝑖 be eigenvalues/eigenvectors of 𝐴wi = 𝜆𝑖 wi . Then there exists 𝑤𝑖 , 𝜆∗𝑖 solution of the adjoint eigenproblem 𝐴∗ 𝑤i = 𝜆∗𝑖 𝑤i . These structures are the adjoint global modes and may be scaled such that 𝑤𝑖∗ wj = 𝛿𝑖𝑗 . The vectors 𝑤i are biorthogonal with respect to the vectors wj .

MEC651 [email protected]

Adjoints

33

Adjoint global modes and biorthogonality (2/4) In finite dimension Proof:

𝜆𝑖 wi = 𝐴wi 𝜆𝑗∗ 𝑤𝑗 = 𝐴∗ 𝑤𝑗 ∗

∗

𝜆𝑖 𝑤𝑗∗ wi = 𝑤𝑗∗ 𝐴wi = 𝐴∗ 𝑤𝑗 wi = 𝜆𝑗∗ 𝑤𝑗 wi = 𝜆𝑗 𝑤𝑗∗ wi 𝜆𝑖 − 𝜆𝑗 𝑤𝑗∗ wi = 0 If 𝜆𝑖 ≠ 𝜆𝑗 , then 𝑤𝑗∗ wi = 0 If 𝑤𝑗∗ wi ≠ 0, then 𝜆𝑖 = 𝜆𝑗 . Conclusion: 𝑤𝑗 can be chosen such that

MEC651 [email protected]

𝑤𝑗∗ wi = 𝛿𝑗𝑖

Adjoints

34

Adjoint global modes and biorthogonality (3/4) Theorem: Let wi , 𝜆𝑖 be eigenvalues/eigenvectors of 𝜆𝑖 ℬwi + 𝒩𝑤0 + ℒ wi = 0. Then there exists 𝑤𝑖 , 𝜆∗𝑖 solution of the adjoint eigenproblem 𝜆∗𝑖 ℬ𝑤𝑖 + 𝒩𝑤0 + ℒ 𝑤𝑖 = 0. These structures are the adjoint global modes and may be scaled such that < 𝑤𝑖 , ℬwj >= 𝛿𝑖𝑗 . The vectors 𝑤i are bi-orthogonal with respect to the vectors wj .

MEC651 [email protected]

Adjoints

35

Adjoint global modes and biorthogonality (4/4) Proof: 𝜆𝑖 ℬwi + 𝒩𝑤0 + ℒ wi = 0 𝜆𝑗∗ ℬ𝑤𝑗 + 𝒩𝑤0 + ℒ 𝑤𝑗 = 0 < 𝑤𝑗 , 𝒩𝑤0 + ℒ wi > = −𝜆𝑖 < 𝑤𝑗 , ℬwi > < 𝑤𝑗 , 𝒩𝑤0 + ℒ wi > =< 𝒩𝑤0 + ℒ 𝑤𝑗 , wi > =< −𝜆𝑗∗ ℬ𝑤𝑗 , wi > = −𝜆𝑗 < 𝑤𝑗 , ℬwi > 𝜆𝑖 − 𝜆𝑗 < 𝑤𝑗 , ℬwi >= 0 If 𝜆𝑖 ≠ 𝜆𝑗 , then < 𝑤𝑗 , ℬwi > = 0 If < 𝑤𝑗 , ℬwi >≠ 0, then 𝜆𝑖 = 𝜆𝑗 . Conclusion: 𝑤𝑗 can be chosen such that < 𝑤𝑗 , ℬwi > = 𝛿𝑗𝑖

MEC651 [email protected]

Adjoints

36

The Ginzburg-Landau eq. (cont’d) 𝜒2 𝑥2

𝑈

1

− 𝑥− 2 𝜉𝑒 2𝛾

−4

5/ Show that: 𝑤(𝑥) = with 𝜉 = 𝜒𝜋 is solution of 𝜆∗ 𝑤 + ℒ 𝑤 = 0. Note that the normalization constant 𝜉 has been chosen so that: 𝑤, 𝑤 = 1. Can you qualitatively represent 𝑤(𝑥) and 𝑤 𝑥 ? 6/ Noting that: 1

𝑈2

2 2 3 12 𝛾2 𝜇2

𝑤, 𝑤 = 𝑒 , what does 𝑤, 𝑤 represent? What is the effect of the advection velocity 𝑈 and viscosity 𝛾 on this coefficient? Nota: 𝑤𝑛 (𝑥) = 𝜉𝑛 𝐻𝑛

𝑈

𝜒2 𝑥2

− 𝑥− 2 (𝜒𝑥)𝑒 2𝛾

MEC651 [email protected]

are all the adjoint eigenvectors.

Adjoints

37

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

38

Adjoint of linearized advection operator (1/4) Theorem:

𝑢 ⋅ 𝛻𝑢0 + 𝑢0 ⋅ 𝛻𝑢 be an operator acting on w = (𝑢, 𝑣, 𝑝) such 0 that 𝑢 = 𝑣 = 0 on boundaries. If < 𝑤1 , 𝑤2 > = ∬ 𝑢1∗ 𝑢2 + 𝑣1∗ 𝑣2 + 𝑝1∗ 𝑝2 𝑑𝑥𝑑𝑦, the adjoint operator of 𝒩𝑤0 is ∗ 𝜕𝑥 u0 𝜕𝑦 𝑢0 0 −𝑢0∗ 𝜕𝑥 − 𝑣0∗ 𝜕𝑦 0 0 𝒩𝑤0 = 𝜕𝑥 𝑣0 𝜕𝑦 𝑣0 0 + 0 −𝑢0∗ 𝜕𝑥 − 𝑣0∗ 𝜕𝑦 0 0 0 0 0 0 0 Let 𝒩𝑤0 𝑤 =

MEC651 [email protected]

Adjoints

39

Adjoint of linearized advection operator (2/4) < 𝑤1 , 𝒩𝑤0 𝑤2 >=< 𝒩𝑤0 𝑤1 , 𝑤2 > 𝑢1∗ 𝑢0 𝜕𝑥 𝑢2 + 𝑣0 𝜕𝑦 𝑢2 + 𝑢2 𝜕𝑥 𝑢0 + 𝑣2 𝜕𝑦 𝑢0 +𝑣1∗ 𝑢0 𝜕𝑥 𝑣2 + 𝑣0 𝜕𝑦 𝑣2 + 𝑢2 𝜕𝑥 𝑣0 + 𝑣2 𝜕𝑦 𝑣0

𝑢1∗ 𝑢0 𝜕𝑥 𝑢2 + 𝑢1∗ 𝑣0 𝜕𝑦 𝑢2 + 𝑣1∗ 𝑢0 𝜕𝑥 𝑣2 + 𝑣1∗ 𝑣0 𝜕𝑦 𝑣2

=

=

𝑑𝑥𝑑𝑦 𝑑𝑥𝑑𝑦

+ 𝑢1∗ 𝜕𝑥 𝑢0 𝑢2 + 𝑢1∗ 𝜕𝑦 𝑢0 𝑣2 + 𝑣1∗ 𝜕𝑥 𝑣0 𝑢2 + 𝑣1∗ 𝜕𝑦 𝑣0 𝑣2

𝑢1∗ 𝑢0 𝜕𝑥 𝑢2 + 𝑢1∗ 𝑣0 𝜕𝑦 𝑢2 + 𝑣1∗ 𝑢0 𝜕𝑥 𝑣2 + 𝑣1∗ 𝑣0 𝜕𝑦 𝑣2 𝑑𝑥𝑑𝑦 ∗

+ MEC651 [email protected]

𝑢1 𝜕𝑥 𝑢0∗ + 𝑣1 𝜕𝑥 𝑣0∗ Adjoints

∗

∗

𝑢2 + 𝑢1 𝜕𝑦 𝑢0∗ + 𝑣1 𝜕𝑦 𝑣0∗ 𝑣2 𝑑𝑥𝑑𝑦 40

Adjoint of linearized advection operator (3/4) 0

∗ =

𝑢1∗ 𝑢0 𝑛𝑥 𝑢2 + 𝑢1∗ 𝑣0 𝑛𝑦 𝑢2 + 𝑣1∗ 𝑢0 𝑛𝑥 𝑣2 + 𝑣1∗ 𝑣0 𝑛𝑦 𝑣2 𝑑𝑠 −

𝜕𝑥 𝑢1∗ 𝑢0 𝑢2 + 𝜕𝑦 𝑢1∗ 𝑣0 𝑢2 + 𝜕𝑥 𝑣1∗ 𝑢0 𝑣2 + 𝜕𝑦 𝑣1∗ 𝑣0 𝑣2 𝑑𝑥𝑑𝑦

=−

𝜕𝑥 𝑢1 𝑢0∗ + 𝜕𝑦 𝑢1 𝑣0∗

∗

𝑢2 + 𝜕𝑥 𝑣1 𝑢0∗ + 𝜕𝑦 𝑣1 𝑣0∗

∗

𝑣2 𝑑𝑥𝑑𝑦

−𝜕𝑥 𝑢1 𝑢0∗ − 𝜕𝑦 𝑢1 𝑣0∗ 𝑢1 𝜕𝑥 𝑢0∗ + 𝑣1 𝜕𝑥 𝑣0∗ 𝒩𝑤0 𝑤1 = 𝑢1 𝜕𝑦 𝑢0∗ + 𝑣1 𝜕𝑦 𝑣0∗ + −𝜕𝑥 𝑣1 𝑢0∗ − 𝜕𝑦 𝑣1 𝑣0∗ 0 0 −𝑢0∗ 𝜕𝑥 𝑢1 −𝑣0∗ 𝜕𝑦 𝑢1 −𝑢0∗ 𝜕𝑥 𝑣1 −𝑣0∗ 𝜕𝑦 𝑣1 0 MEC651 [email protected]

Adjoints

(using 𝜕𝑥 𝑢0 + 𝜕𝑦 𝑣0 = 0 )

41

Adjoint of linearized advection operator (4/4) Conclusion:

∗

𝜕𝑥 u0 𝜕𝑦 𝑢0 0 𝑢1 −𝑢0∗ 𝜕𝑥 − 𝑣0∗ 𝜕𝑦 𝒩𝑤0 𝑤1 = 𝜕𝑥 𝑣0 𝜕𝑦 𝑣0 0 𝑣1 + 0 0 0 0 𝑝1 0 𝜕𝑥 u0 𝜕𝑦 𝑢0 0 𝑢2 𝑢0 𝜕𝑥 + 𝑣0 𝜕𝑦 𝒩𝑤0 𝑤2 = 𝜕𝑥 𝑣0 𝜕𝑦 𝑣0 0 𝑣2 + 0 0 0 0 𝑝2 0

0 0 𝑢1 −𝑢0∗ 𝜕𝑥 − 𝑣0∗ 𝜕𝑦 0 𝑣1 0 0 𝑝1 0 0 𝑢2 𝑢0 𝜕𝑥 + 𝑣0 𝜕𝑦 0 𝑣2 0 0 𝑝2

𝒩𝑤0 ≠ 𝒩𝑤0 because of: - component-type non-normality => 𝑣 → 𝑢 becomes 𝑢 → 𝑣 - convective-type non-normality => upstream convection

MEC651 [email protected]

Adjoints

42

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

43

Adjoint of Stokes operator (1/4) Theorem:

−𝜈Δ() 𝛻() be an operator acting on w = (𝑢, 𝑣, 𝑝) such that −𝛻 ⋅ () 0 𝑢 = 𝑣 = 0 on boundaries. If 𝑤1 , 𝑤2 = ∬ 𝑢1∗ 𝑢2 + 𝑣1 ∗ 𝑣2 + 𝑝1∗ 𝑝2 𝑑𝑥𝑑𝑦, the operator ℒ is self-afjoint : ℒ = ℒ. Let ℒ =

MEC651 [email protected]

Adjoints

44

Adjoint of Stokes operator (2/4) < 𝑤1 , ℒ𝑤2 >=< ℒ 𝑤1 , 𝑤2 >

ℒ=

−𝜈Δ() 𝛻() −𝛻 ⋅ () 0

𝑢1∗ −𝜈𝜕𝑥𝑥 𝑢2 − 𝜈𝜕𝑦𝑦 𝑢2 + 𝜕𝑥 𝑝2 + 𝑣1∗ −𝜈𝜕𝑥𝑥 𝑣2 − 𝜈𝜕𝑦𝑦 𝑣2 + 𝜕𝑦 𝑝2

+ 𝑝1∗ −𝜕𝑥 𝑢2 − 𝜕𝑦 𝑣2 𝑑𝑥𝑑𝑦 =

−𝜈𝑢1∗ 𝜕𝑥𝑥 𝑢2 − 𝜈𝑢1∗ 𝜕𝑦𝑦 𝑢2 + 𝑢1∗ 𝜕𝑥 𝑝2 − 𝜈𝑣1∗ 𝜕𝑥𝑥 𝑣2 − 𝜈𝑣1∗ 𝜕𝑦𝑦 𝑣2

+ 𝑣1∗ 𝜕𝑦 𝑝2 − 𝑝1∗ 𝜕𝑥 𝑢2 − 𝑝1∗ 𝜕𝑦 𝑣2 𝑑𝑥𝑑𝑦 =

−𝜈𝑢1∗ 𝑛𝑥 𝜕𝑥 𝑢2 − 𝜈𝑢1∗ 𝑛𝑦 𝜕𝑦 𝑢2 + 𝑢1∗ 𝑛𝑥 𝑝2 − 𝜈𝑣1∗ 𝑛𝑥 𝜕𝑥 𝑣2 − 𝜈𝑣1∗ 𝑛𝑦 𝜕𝑦 𝑣2 + 𝑣1∗ 𝑛𝑦 𝑝2 − 𝑝1∗ 𝑛𝑥 𝑢2 − 𝑝1∗ 𝑛𝑦 𝑣2 𝑑𝑠

−

−𝜈𝜕𝑥 𝑢1∗ 𝜕𝑥 𝑢2 − 𝜈𝜕𝑦 𝑢1∗ 𝜕𝑦 𝑢2 + 𝜕𝑥 𝑢1∗ 𝑝2 − 𝜈𝜕𝑥 𝑣1∗ 𝜕𝑥 𝑣2 − 𝜈𝜕𝑦 𝑣1∗ 𝜕𝑦 𝑣2

+ 𝜕𝑦 𝑣1∗ 𝑝2 − 𝜕𝑥 𝑝1∗ 𝑢2 − 𝜕𝑦 𝑝1∗ 𝑣2 𝑑𝑥𝑑𝑦 MEC651 [email protected]

Adjoints

𝑢 = 𝑣 = 0 on boundaries 45

Adjoint of Stokes operator (3/4) 𝜕𝑥 𝑢1∗ 𝑝2 + 𝜕𝑦 𝑣1∗ 𝑝2 − 𝜕𝑥 𝑝1∗ 𝑢2 − 𝜕𝑦 𝑝1∗ 𝑣2 𝑑𝑥𝑑𝑦

=−

+

− −𝜈𝜕𝑥 𝑢1∗ 𝜕𝑥 𝑢2 − 𝜈𝜕𝑦 𝑢1∗ 𝜕𝑦 𝑢2 − 𝜈𝜕𝑥 𝑣1∗ 𝜕𝑥 𝑣2 − 𝜈𝜕𝑦 𝑣1∗ 𝜕𝑦 𝑣2 𝑑𝑥𝑑𝑦 (∗)

∗ =

− −𝜈𝜕𝑥 𝑢1∗ 𝑛𝑥 𝑢2 − 𝜈𝜕𝑦 𝑢1∗ 𝑛𝑦 𝑢2 − 𝜈𝜕𝑥 𝑣1∗ 𝑛𝑥 𝑣2 − 𝜈𝜕𝑦 𝑣1∗ 𝑛𝑦 𝑣2 𝑑𝑠

+

−𝜈𝜕𝑥𝑥 𝑢1∗ 𝑢2 − 𝜈𝜕𝑦𝑦 𝑢1∗ 𝑢2 − 𝜈𝜕𝑥𝑥 𝑣1∗ 𝑣2 − 𝜈𝜕𝑦𝑦 𝑣1∗ 𝑣2 𝑑𝑥𝑑𝑦

−𝜈𝜕𝑥𝑥 − 𝜈𝜕𝑦𝑦 0 ℒ 𝑤1 = −𝜕𝑥 MEC651 [email protected]

0 −𝜈𝜕𝑥𝑥 − 𝜈𝜕𝑦𝑦 −𝜕𝑦 Adjoints

𝜕𝑥 𝜕𝑦 0

𝑢1 𝑣1 𝑝1 46

Adjoint of Stokes operator (4/4) −𝜈𝜕𝑥𝑥 − 𝜈𝜕𝑦𝑦 0 ℒ 𝑤1 = −𝜕𝑥

0 −𝜈𝜕𝑥𝑥 − 𝜈𝜕𝑦𝑦 −𝜕𝑦

𝜕𝑥 𝜕𝑦 0

𝑢1 𝑣1 𝑝1

−𝜈𝜕𝑥𝑥 − 𝜈𝜕𝑦𝑦 0 ℒ𝑤2 = −𝜕𝑥

0 −𝜈𝜕𝑥𝑥 − 𝜈𝜕𝑦𝑦 −𝜕𝑦

𝜕𝑥 𝜕𝑦 0

𝑢2 𝑣2 𝑝2

ℒ=ℒ

MEC651 [email protected]

Adjoints

47

Outline - Governing equations - Asymptotic development  Order 𝜖 0 : Base-flow  Order 𝜖 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

MEC651 [email protected]

Adjoints

48

The adjoint global mode of cylinder flow

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

MEC651 [email protected]

Adjoints

49