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

Adjoints MEC651 denis.sipp@onera.fr Adjoints 1

Outline - Governing equations - Asymptotic development  Order 𝜗 0 : Base-flow  Order 𝜗 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 denis.sipp@onera.fr Adjoints 2

Governing equations Incompressible Navier-Stokes equations: 𝜖 𝑢 𝑣 + 𝑣𝜖 𝑦 𝑣 + 𝑤𝜖 𝑧 𝑣 = −𝜖 𝑦 𝑞 + 𝜉 𝜖 𝑦𝑦 𝑣 + 𝜖 𝑧𝑧 𝑣 + 𝑔 𝜖 𝑢 𝑤 + 𝑣𝜖 𝑦 𝑤 + 𝑤𝜖 𝑧 𝑤 = −𝜖 𝑧 𝑞 + 𝜉 𝜖 𝑦𝑦 𝑤 + 𝜖 𝑧𝑧 𝑤 + 𝑕 −𝜖 𝑦 𝑣 − 𝜖 𝑧 𝑤 = 0 Can be recast into: ℬ𝜖 𝑢 𝑥+ 1 2 𝒪 𝑥, 𝑥 + ℒ𝑥 = 𝑔 where: 𝑥 = 𝑣 𝑞 𝑔 = 𝑔 0 ℬ = 1 0 0 , 0 𝒪 𝑥 1 , 𝑥 2 = 𝑣 1 ⋅ 𝛼𝑣 2 + 𝑣 2 ⋅ 𝛼𝑣 1 0 −𝜉Δ() 𝛼() ℒ = −𝛼 ⋅ () 0 Boundary conditions: Dirichlet, Neumann, Mixed MEC651 denis.sipp@onera.fr Adjoints 3

Some properties a) 𝒪 𝑥 1 , 𝑥 2 = 𝒪 𝑥 2 , 𝑥 1 𝜗 2 b) 1 1 2 𝒪 𝑥 0 + 𝜗𝜀𝑥, 𝑥 0 + 𝜗𝜀𝑥 = 2 𝒪 𝑥 0 , 𝑥 0 + 𝜗 𝒪 𝑥 0 , 𝜀𝑥 + 2 𝒪 𝜀𝑥, 𝜀𝑥 + ⋯ Hessian Jacobian =𝒪 𝑥0 𝜀𝑥 𝑥 0 𝜀𝑥 = 𝒪 𝑥 0 , 𝜀𝑥 = 𝜀𝑣 ⋅ 𝛼𝑣 0 + 𝑣 0 ⋅ 𝛼𝜀𝑣 c) 𝒪 0 d) ℬ𝑥 = ℬ 𝑣 𝑞 = 𝑣 0 e) 𝜖 𝑢 𝑣 + 𝑣 ⋅ 𝛼𝑣 = −𝛼𝑞 + 𝜉𝛼 2 𝑣 ⇒ −𝛼 2 𝑞 = 𝛼 ⋅ 𝑣 ⋅ 𝛼𝑣 , 𝜖 𝑜 𝑞 = 𝜉𝛼 2 𝑣 ⋅ 𝑜 on solid walls . Hence, p is a function of u and should not be considered as a degree of freedom of the flow. ∗ 𝑣 2 + 𝑤 1 ∗ 𝑤 2 𝑒𝑦𝑒𝑧 = ∬ (𝑥 1 ⋅ ℬ𝑥 2 )𝑒𝑦𝑒𝑧 so f) Scalar-product: < 𝑥 1 , 𝑥 2 > = ∬ 𝑣 1 that < 𝑥, 𝑥 > is the energy. MEC651 denis.sipp@onera.fr Adjoints 4

Outline - Governing equations - Asymptotic development  Order 𝜗 0 : Base-flow  Order 𝜗 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 denis.sipp@onera.fr Adjoints 5

Asymptotic development Solution: 𝑥 𝑢 = 𝑥 0 + 𝜗𝑥 1 𝑢 + ⋯ with ϵ ≪ 1 Governing equations: ℬ𝜖 𝑢 𝑥+ 1 2 𝒪 𝑥, 𝑥 + ℒ𝑥 = 𝑔 Introduce solution into governing eq:: ℬ𝜖 𝑢 (𝑥 0 +𝜗𝑥 1 + ⋯ )+ 1 2 𝒪 𝑥 0 + 𝜗𝑥 1 + ⋯ , 𝑥 0 + 𝜗𝑥 1 + ⋯ + ℒ(𝑥 0 +𝜗𝑥 1 + ⋯ ) = 𝑔 1 2 𝒪 𝑥 0 , 𝑥 0 + ℒ𝑥 0 = 𝑔 at order 𝑃(1) ℬ𝜖 𝑢 𝑥 1 + 1 2 [𝒪 𝑥 1 , 𝑥 0 + 𝒪 𝑥 0 , 𝑥 1 ] + ℒ𝑥 1 = 0 at order 𝑃(𝜗) ⇒ 𝒪 𝑥0 𝑥 1 𝑥 0 𝑥 2 + ℒ𝑥 2 = − 1 2 𝒪 𝑥 1 , 𝑥 1 at order 𝑃(𝜗 2 ) ℬ𝜖 𝑢 𝑥 2 +𝒪 MEC651 denis.sipp@onera.fr Adjoints 6

Oder 𝜗 0 : Base-flow Definition: 𝐺 𝑥 = 1 2 𝒪 𝑥, 𝑥 + ℒ𝑥 − 𝑔 𝑥 𝑢 = 𝑥 0 + 𝜗𝑥 1 (𝑢) + ⋯ Non-linear equilibrium point : 1 2 𝒪 𝑥 0 , 𝑥 0 + ℒ𝑥 0 = 𝑔 𝑥 𝑥 0 How to compute a base-flow ? Newton iteration: 1 2 𝒪 𝑥 0 + 𝜀𝑥 0 , 𝑥 0 + 𝜀𝑥 0 + ℒ(𝑥 0 +𝜀𝑥 0 ) = 𝑔 Linearization: 𝒪 𝑥 0 , 𝜀𝑥 0 + ℒ𝜀𝑥 0 = 𝑔 − 1 2 𝒪 𝑥 0 , 𝑥 0 − ℒ𝑥 0 −1 𝑔 − 1 ⇒ 𝜀𝑥 0 = 𝒪 𝑥 0 + ℒ 2 𝒪 𝑥 0 , 𝑥 0 − ℒ𝑥 0 MEC651 denis.sipp@onera.fr Adjoints 7

Oder 𝜗 0 : Base-flow The case of cylinder flow 𝑆𝑓 = 47 Streamwise velocity field of base-flow. MEC651 denis.sipp@onera.fr Adjoints 8

Order 𝜗 1 : Global modes Definition 𝑥 𝑢 = 𝑥 0 + 𝜗𝑥 1 (𝑢) + ⋯ Linear governing equation: ℬ𝜖 𝑢 𝑥 1 + 𝒪 𝑥 0 𝑥 1 + ℒ𝑥 1 = 0 Solution 𝑥 1 under the form: 𝑥 1 = 𝑓 𝜇𝑢 𝑥 + c.c This leads to : 𝜇ℬ𝑥 + 𝒪 𝑥 0 + ℒ 𝑥 = 0 Eigenvalue: 𝜇 = 𝜏 + 𝑗𝜕 Eigenvector: = 𝑥 𝑥 r + iw 𝑗 Real solution: 𝑥 1 = 𝑓 𝜇𝑢 𝑥 + c.c = 2𝑓 𝜏𝑢 (cos 𝜕𝑢 𝑥 𝑠 − sin 𝜕𝑢 𝑥 𝑗 ) MEC651 denis.sipp@onera.fr Adjoints 9

Order 𝜗 1 : Global modes How to compute global modes ? Eigenvalue problem solved with shift-invert strategy: - Power method, easy to find largest magnitude eigenvalues of 𝐵𝑦 = 𝜇𝑦 . For this, evaluate 𝐵 𝑜 𝑦 0 To find eigenvalues of 𝐵 closest to zero, search largest magnitude eigenvalues of - 𝐵 −1 : 𝐵 −1 𝑦 = 𝜇 −1 𝑦 . For this, evaluate 𝐵 −1 𝑜 𝑦 0 To find eigenvalues of 𝐵 closest to 𝑡 , search largest magnitude eigenvalues of - 𝐵 − 𝑡𝐽 −1 𝑜 𝑦 0 𝐵 − 𝑡𝐽 −1 : 𝐵 − 𝑡𝐽 −1 𝑦 = 𝜇 − 𝑡 −1 𝑦 . For this, evaluate - Instead of power-method, use Krylov subspaces -> Arnoldi technique - Cost of algorithm = cost of several complex matrix inversions MEC651 denis.sipp@onera.fr Adjoints 10

Order 𝜗 1 : Global modes Case of cylinder flow Real part of cross-stream velocity field Spectrum 𝑆𝑓 = 47 Marginal eigenmode MEC651 denis.sipp@onera.fr Adjoints 11

The Ginzburg-Landau eq. We consider the linear Ginzburg-Landau equation 𝜖 𝑢 𝑥 1 + ℒ𝑥 1 = 0 where 𝑦 2 ℒ = 𝑉𝜖 𝑦 − 𝜈 𝑦 − 𝛿𝜖 𝑦𝑦 , 𝜈 𝑦 = 𝑗𝜕 0 + 𝜈 0 − 𝜈 2 2 . Here 𝑉, 𝛿, 𝜕 0 , 𝜈 0 and 𝜈 2 are positive real constants. The state 𝑥(𝑦, 𝑢) is a complex variable on −∞ < 𝑦 < +∞ such that |𝑥| → 0 as 𝑦 → ∞ . In the following, +∞ 𝑥 𝑏 , 𝑥 𝑐 = 𝑥 𝑏 𝑦 ∗ 𝑥 𝑐 𝑦 𝑒𝑦 . −∞ 1/ What do the different terms in the Ginzburg Landau equation represent? MEC651 denis.sipp@onera.fr Adjoints 12

The Ginzburg-Landau eq. 1 2𝛿 𝑦− 𝜓2𝑦2 𝑉 𝜈 2 𝜓 4 and 𝜂 = 2/ Show that 𝑥 (𝑦) = 𝜂𝑓 2 with 𝜓 = verifies 𝜇𝑥 + ℒ𝑥 = 0 . 𝑉2 2𝛿 1 1 𝛿2𝜓2 8 𝜌 4 𝑓 What is the eigenvalue 𝜇 associated to this eigenvector? The constant 𝜂 has been selected so that 𝑥 , 𝑥 = 1 . 3/ Show that the flow is unstable if the constant 𝜈 0 is chosen such that: 𝜈 0 > 𝜈 𝑑 , where 𝑉 2 𝛿𝜈 2 𝜈 𝑑 = 4𝛿 + 2 . 2𝛿 𝑦− 𝜓2𝑦2 𝑉 𝜇 𝑜 = i𝜕 0 + 𝜈 0 − 𝑉 2 𝛿𝜈 2 4𝛿 − 2𝑜 + 1 2 , 𝑥 𝑜 = 𝜂 𝑜 𝐼 𝑜 𝜓𝑦 𝑓 2 Nota: are all the eigenvalues/eigenvectors of ℒ , 𝐼 𝑜 being Hermite polynomials. MEC651 denis.sipp@onera.fr Adjoints 13

Outline - Governing equations - Asymptotic development  Order 𝜗 0 : Base-flow  Order 𝜗 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 denis.sipp@onera.fr Adjoints 14

Bi-orthogonal basis and adjoint global modes (1/3) In finite dimension Global mod odes: 𝐵𝑥 𝑗 = 𝜇 𝑗 𝑥 𝑗 The eigenvectors 𝑥 𝑗 form a basis: 𝑥 = 𝛽 𝑗 w i 𝑗 Definition of adjoint global modes: with <> as a given scalar-product (say ∗ 𝑥 2 ), there exists for each 𝛽 𝑗 a unique 𝑥 < 𝑥 1 , 𝑥 2 > = 𝑥 1 𝑗 such that 𝛽 𝑗 =< 𝑥 𝑗 , 𝑥 > for all 𝑥 . The adjoint global modes are the structures 𝑥 𝑗 . In the following: 𝑥 𝑗 , 𝑥 𝑗 = 1 . Properties:  𝑥 𝑙 and w j are bi-orthogonal bases: they verify 𝑥 𝑘 = < 𝑥 𝑗 , 𝑥 𝑘 > w i and so 𝑗 ∗ 𝑋 = 𝐽 ) < 𝑥 𝑙 , w j > = 𝜀 𝑙𝑘 (in matrix notations 𝑋 1 1  Cauchy-Lifschitz: 1 = < 𝑥 𝑗 , w i > ≤< 𝑥 𝑗 , 𝑥 𝑗 > 2 < 𝑥 𝑗 , 𝑥 𝑗 > 2 1 1 Hence: < 𝑥 𝑗 , 𝑥 𝑗 > 2 ≥ 1 and cos angle 𝑥 𝑗 , 𝑥 𝑗 = 1 <𝑥 𝑗 ,𝑥 𝑗 > 2 MEC651 denis.sipp@onera.fr Adjoints 15

Bi-orthogonal basis and adjoint global modes (2/3) In finite dimension 𝑥 = (𝑥 1 ⋅ 𝑥)𝑥 1 + (𝑥 2 ⋅ 𝑥)𝑥 2 Def of 𝑥 1 : 𝑥 2 𝑥 1 ⋅ 𝑥 1 = 1 𝑥 1 ⋅ 𝑥 2 = 0 𝑥 2 Def of 𝑥 2 : 𝑥 2 ⋅ 𝑥 2 = 1 𝑥 2 ⋅ 𝑥 1 = 0 𝑥 1 𝑥 1 ∗ 𝑋 = 𝐽 𝑋 = W ∗−1 Method 1 : 𝑋 −1 ⇒ 𝑋 −1 = 𝑋 𝑌 ⇒ 𝑌 ∗ 𝑋 ∗ 𝑋 = 𝐽 ⇒ 𝑌 = 𝑋 ∗ 𝑋 = 𝑋 𝑋 ∗ 𝑋 Method 2 : 𝑋 Method 3 : adjoint global modes MEC651 denis.sipp@onera.fr Adjoints 16