[PPT] - Optimal control of the cylinder wake flow using Proper Orthogonal PowerPoint Presentation

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Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD)

Michel Bergmann, Laurent Cordier & Jean-Pierre Brancher

Michel.Bergmann@ensem.inpl-nancy.fr

Laboratoire d’ ´ Energ´ etique et de M´ ecanique Th´ eorique et Appliqu´ ee UMR 7563 (CNRS - INPL - UHP) ENSEM - 2, avenue de la Forˆ et de Haye BP 160 - 54504 Vandoeuvre Cedex, France

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 1/16

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Outline

I - Configuration and numerical method II - Proper Orthogonal Decomposition (POD) III - Reduced Order Model of the cylinder wake (ROM) IV - Optimal control formulation based on the reduced order model V - Closed loop results Conclusions and perspectives

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 2/16

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I - Configuration and numerical method

Two dimensional flow around a circular cylinder at Re = 200 Viscous, incompressible and Newtonian fluid Cylinder oscillation with a tangential velocity γ(t)

Γ1 Γ4 Γ2 Γ3 Γc ρ U

D y x

γ

Fractional steps method in time Finite Elements Method (FEM) in space

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 3/16

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I - Configuration and numerical method

Iso pressure at t = 100.

80 90 100 110 120

0.5

0.5 1 1.5

time units CD, CL CD CL

Aerodynamic coefficients. Iso vorticity at t = 100. Authors St CD Braza et al. (1986) 0.2000 1.4000 Henderson et al. (1997) 0.1971 1.3412 He et al. (2000) 0.1978 1.3560 this study 0.1983 1.3972 Strouhal number and drag coefficient.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 4/16

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II - Proper Orthogonal Decomposition (POD)

◮ Introduced in fluid mechanics (turbulence context) by Lumley (1967). ◮ Look for a realization φ(X) which is closer, in an average sense, to the realizations u(X). (X = (x, t) ∈ D = Ω × R+) ◮ φ(X) solution of the problem : max φ |(u, φ)|2 φ2 . ◮ Snapshots method, Sirovich (1987) :

T

C(t, t′)a(n)(t′) dt′ = λ(n)a(n)(t). ◮ Optimal convergence L2 norm (energy) of φ(X) ⇒ Dynamical order reduction is possible. ◮ Decomposition of the velocity field : u(x, t) =

NP OD

i=1

a(i)(t)φ(i)(x).

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 5/16

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III - Reduced Order Model of the cylinder wake (ROM)

◮ Galerkin projection of NSE on the POD basis :

φ(i), ∂u

∂t + (u · ∇)u

=
φ(i), −∇p + 1

Re∆u

.

◮ Integration by parts (Green’s formula) leads :

φ(i), ∂u

∂t + (u · ∇)u

=
p, ∇ · φ(i)

− 1 Re

(∇ ⊗ φ(i))T , ∇ ⊗ u
− [p φ(i)] + 1

Re[(∇ ⊗ u)φ(i)]. with [a] =

Γ

a · n dΓ and (A, B) =

Ω

A : B dΩ =

i, j
Ω

AijBji dΩ.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 6/16

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III - Reduced Order Model of the cylinder wake (ROM)

◮ Velocity decomposition with NP OD modes : u(x, t) = um(x) + γ(t) uc(x) +

NP OD

k=1

a(k)(t)φ(k)(x). ◮ Reduced order dynamical system where only Ngal (≪ NP OD) modes are retained (state equations) :                    d a(i)(t) d t =Ai +

Ngal

j=1

Bij a(j)(t) +

Ngal

j=1

Ngal

k=1

Cijk a(j)(t)a(k)(t) + Di d γ d t +  Ei +

Ngal

j=1

Fij a(j)(t)   γ + Giγ2 a(i)(0) = (u(x, 0), φ(i)(x)). Ai, Bij, Cijk, Di, Ei, Fij and Gi depend on φ, um, uc and Re.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 7/16

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III - Reduced Order Model of the cylinder wake (ROM) Stabilization

Integration and (optimal) stabilization of the reduced order dynamical system with γ = A sin(2πStt), A = 2 and St = 0, 5.

2 4 6 8 10

1
0.5

0.5 1

a(n)

time units

Temporal evolution of the 6 first POD modes.

1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

|a(n)|

index

Average amplitudes of POD modes.

− − − − projection (DNS) − − prediction before stabilization (low order model) · · · prediction after stabilization (low order model).

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 8/16

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IV - Optimal control formulation based on reduced order model

◮ Objective functional : J (a, γ(t)) = T J(a, γ(t)) dt = T  

Ngal

i=1

a(i)2 + α 2 γ(t)2   dt. α : regularization parameter (penalization). ◮ Adjoint equations :        d ξ(i)(t) dt = −

Ngal

j=1

 Bji + γ Fji +

Ngal

k=1

(Cjik + Cjki) a(k)   ξ(j)(t) − 2a(i) ξ(i)(T) = 0. ◮ Optimality condition (gradient) : δγ(t) = −

Ngal

i=1

Di dξ(i) dt +

Ngal

i=1

 Ei +

Ngal

j=1

Fija(j) + 2Giγ(t)   ξ(i) + αγ.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 9/16

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IV - Optimal control formulation based on reduced order model

◮ γ(0)(t) done ; for n = 0, 1, 2, ... and while a convergence criterium is not satisfied, do :

1. From t = 0 to t = T solve the state equations with γ(n)(t) ;

֒ → state variables a(n)(t)

2. From t = T to t = 0 solve the adjoint equations with a(n)(t) ;

֒ → adjoint variables ξ(n)(t)

3. Solve the optimality condition with a(n)(t) and ξ(n)(t) ;

֒ → objective gradient δγ(n)(t)

4. New control law ֒

→ γ(n+1)(t) = γ(n)(t) + ω(n) δγ(n)(t) ◮ End do.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 10/16

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V - Closed loop results Generalities

◮ No reactualization of the POD basis. ◮ The energetic representativity is a priori different to the dynamical

ne :

֒ → possible inconvenient for control, ֒ → a POD dynamical system represents a priori only the dynamics (and its vicinity) used to build the low dynamical model. ◮ Construction of a POD basis representative of a large range of dynamics : ֒ →excitation of a great number of degrees of freedom scanning γ(t) in amplitudes and frequencies.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 11/16

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V - Closed loop results Excitation

10 20 30 40 50

5
4
3
2
1

1 2 3 4 5

γe

time units

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500

St

◮ γ = 0 : ֒ → 2 modes out of 100 are sufficient to represent 97% of the kinetic energy. ◮ γ = γe : ֒ → 30 modes out of 100 are then necessary to represent 97%

f the kinetic energy.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 12/16

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V - Closed loop results Optimal control

10 20 30

3
2
1

1 2 3

γopt

time units

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 2500 3000

St

◮ Reduction of the wake instationarity. γopt ≃ A sin(2πStt) with A = 2.2 and St = 0.53 J (γe) = 9.81 = ⇒ J (γopt) = 5.63. ◮ The control is optimal for the reduced order model based on POD. ◮ Is it also optimal for the Navier Stokes model ?

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 13/16

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V - Closed loop results Comparison of wakes’ organization

◮ No mathematical proof concerning the Navier Stokes optimality. a) no control γ = 0 b) optimal control γ = γopt Isocontours of vorticity ωz. ◮ no control : γ = 0 ⇒ Asymmetrical flow. ֒ → Large and energetic eddies. ◮ optimal control : γ = γopt ⇒ Symmetrization of the (near) wake. ֒ → Smaller and lower energetic eddies.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 14/16

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V - Closed loop results Aerodynamic coefficients

10 20 30 40 50 60 70 1 1.1 1.2 1.3 1.4 1.5

CD

time units

10 20 30 40 50 60 70

0.5

0.5

CL

time units

◮ Very consequent drag reduction : CD = 1.40 for γ = 0 et CD = 1.06 for γ = γopt (more than 25%). ◮ Decrease of the lift amplitude : CL = 0.68 for γ = 0 et CL = 0.13 for γ = γopt.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 15/16

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Conclusions and perspectives

◮ Conclusions Significative drag reduction minimizing the wake instationnarity of the ROM. Numerical costs (CPU and memory) negligible. ◮ Perspectives Improve the representativity of the low order model. ֒ → "Optimize" the temporal excitation γe, ֒ → Mix snapshots corresponding to several differents dynamics (temporal excitations). Look for harmonic control γ(t) = A sin(2π St t) with POD basis reactualization. Couple this optimal system with trust region methods (TRPOD) = ⇒ proof of convergence. Couple pressure with the POD dynamical system. Optimal control of the Navier Stokes equations.

Optimal control of the cylinder wake flow using Proper Orthogonal Decomposition (POD) – p. 16/16