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Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition.

Laurent CORDIER Michel BERGMANN & Jean-Pierre BRANCHER

Laurent.Cordier@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

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Introduction

Configuration and numerical method

Two dimensional flow around a circular cylinder at Re = 200 Viscous, incompressible and Newtonian fluid Cylinder oscillation with a tangential sinusoidal velocity γ(t) γ(t) = VT U∞ = A sin(2πStft)

• θ

x y Γe Γsup Γs Γinf Γc Ω U∞ D VT (t)

Find the control parameters c = (A, Stf)T such that the mean drag coefficient is minimized

CDT = 1 T

2 p nx R dθ dt − 1 T

2 Re

∂x nx + ∂u ∂y ny

• R dθ dt,

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Introduction

Parametric study

0.9929 6.9876 1.3928 1.3878 1.3928 1.3878 1.3905 1.1728 1.0320 3.6306 2.5516 1.0529 1.3526 1.2327 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

Amplitude Strouhal

Variation of the mean drag coefficient with A and Stf. Numerical minimum

• Amin, Stfmin
• = (4.3, 0.74).

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Introduction

Mean drag coefficient & steady unstable base flow

25 50 75 100 125 150 175 200 0.75 1 1.25 1.5 1.75 2 2.25 2.5

CDT Re CDT Cbase

D

C0

D

Natural flow Base flow

• Fig. : Variation with the Reynolds number of the mean drag coefficient. Contributions and

corresponding flow patterns of the base flow and unsteady flow.

Protas, B. et Wesfreid, J.E. (2002) : Drag force in the open-loop control of the cylinder wake in the laminar

• regime. Phys. Fluids, 14(2), pp. 810-826.

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Reduced Order Model (ROM) and optimization problems

x ∆ Initialization Optimization Optimization on simplified model a(x), grad a(x) f(x), grad f(x) Recourse to detailed model (TRPOD) High−fidelity model Approximation model

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ROM

Proper Orthogonal Decomposition (POD)

◮ Introduced in turbulence by Lumley (1967). ◮ Method of information compression ◮ Look for a realization Φ(X) which is clo- ser, in an average sense, to realizations u(X). (X = (x, t) ∈ D = Ω × R+) ◮ Φ(X) solution of the problem : max Φ |(u, Φ)|2 s.t. Φ2 = 1. ◮ Snapshots method, Sirovich (1987) :

T

C(t, t′)a(n)(t′) dt′ = λ(n)a(n)(t). ◮ Optimal convergence in L2 norm (energy)of Φ(X) ⇒ Dynamical order reduction is possible.

Coordinate axis Coordinate axis

Φ1 Φ2

Original point cloud Point cloud, mean shifted (centered around origin)

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ROM

Parameter sampling in an optimization setting

General configuration. Ideal sampling. Unsuitable sampling. Unsuitable sampling. Discussion of parameter sampling in an optimization setting (from Gunzburger, 2004). − − − − path to optimizer using full system, initial values, optimal values, and • parameter values used for snapshot generation.

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ROM

Non-equilibrium modes (Noack et al. 2004)

◮ Necessity for a given reference flow to introduce new modes : either new

• perating conditions or shift-modes
• Controlled space

Dynamics I Dynamics II φI φI

2

φII

1

φII

2

φI

1

φII φI→II

neq

• Fig. : Schematic representation of a dynamical transition with a non-equilibrium mode

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ROM

A robust POD surrogate for the drag coefficient

◮ POD approximations consistent with our approach : U(x, t) = (u, v, p)T =

N

i=0

ai(t) φi(x)

Galerkin modes +

N+M

i=N+1

ai(t) φi(x)

non-equilibrium modes + γ(c, t) Uc(x)

control function Physical aspects Modes Dynamical aspects actuation mode Uc predetermined dynamics mean flow mode Um, i = 0 a0 = 1 Galerkin modes Dynamics of the reference flow I i = 1 POD ROM Temporal dynamics of the modes (eventually, the mode i = 0 is solved then a0 ≡ a0(t)) i = 2 · · · i = N non-equilibrium modes Inclusion of new operating conditions II, III, IV, · · · i = N + 1 · · · i = N + M

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ROM

Galerkin projection

◮ Galerkin projection of NSE onto the POD basis :

• φi, ∂u

∂t + ∇ · (u ⊗ u)

• =
• φi, −∇p + 1

Re∆u

• .

◮ Reduced order dynamical system where only (N + M + 1) (≪ NP OD) modes are retained (state equations) : d ai(t) d t =

N+M

j=0

Bij aj(t) +

N+M

j=0 N+M

k=0

Cijk aj(t)ak(t) + Di d γ d t + Ei +

N+M

j=0

Fij aj(t)

γ(c, t) + Giγ2(c, t), ai(0) = (U(x, 0), φi(x)). Bij, Cijk, Di, Ei, Fij and Gi depend on φi, U c and Re.

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Surrogate drag function and model objective function

◮ Drag operator : CD : R3 → R u → 2

• u3nx − 1

Re ∂u1 ∂x nx − 1 Re ∂u1 ∂y ny

• R dθ,

(1)

◮ Surrogate drag function :

CD(t) = a0(t)N0 +

N+M

i=N+1

ai(t)Ni

evolution of the mean drag +

N

i=1

ai(t)Ni

fluctuations C′

D(t)

with Ni = CD(φi). ◮ Model objective function : m =

CD(t)T = 1 T

a0(t)N0 +

N+M

i=N+1

ai(t)Ni

dt.

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Surrogate drag function

Test case A = 2 and St = 0.5

◮ Comparison of real drag coefficient CD (symbols) and model function

CD (lines) at the design parameters.

5 10 15 1.045 1.05 1.055 1.06 1.065 1.07 1.075 1.08 1.085 1.09 1.095 1.1

t CD &

CD

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Robustness of the model objective function

Test case A = 2 and St = 0.5

0.4 0.5 0.6 1.5 2 2.5 1.293 1.275 1.257 1.239 1.221 1.203 1.185 1.167 1.149 1.131 1.113 1.095 1.077 1.059 1.041

A St (a) Real objective function J

0.4 0.5 0.6 1.5 2 2.5 1.221 1.209 1.197 1.185 1.173 1.161 1.149 1.137 1.125 1.113 1.101 1.089 1.077 1.065 1.053

A St (b) Model objective function m

• Fig. : Comparison of the real and the model objective functions associated to the mean drag coefficient.

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Trust-Region Proper Orthogonal Decomposition

Generalities

Range of validity of the POD ROM restricted to a vicinity of the design parameters Objective : Use ROMs to solve large- scale optimization problems with assu- rance of :

• 1. Automatic

restriction

• f

the range of validity

• 2. Global convergence
• Solution

◮ Embed the POD technique into the concept of trust-region methods : Trust-Region Proper Orthogonal Decomposition (Fahl, 2000)

Conn, A.R., Gould, N.I.M. et Toint, P .L. (2000) : Trust-region methods. SIAM, Philadelphia.

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Trust-Region Proper Orthogonal Decomposition (TRPOD)

Algorithm

Initialization : c0, Navier-Stokes resolution, J0. k = 0. ∆0 Construction of the POD ROM and evaluation of the model

• bjective function mk

Solve the optimality system based on the POD ROM under the constraints ∆k ck+1 and mk+1 Solve the Navier-Stokes equations and estimate a new POD basis Jk+1 Evaluation of the performance (Jk+1 − Jk)/(mk+1 − mk)

poor medium good

∆k+1 < ∆k ∆k+1 ∆k ∆k+1 > ∆k k = k + 1 k = k + 1 c0 ck ck+1 ck+1

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TRPOD

Numerical results

Initial control parameters : A = 1.0 et St = 0.2

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

A St

5 10 15 0.9 1 1.1 1.2 1.3 1.4 1.5

Jk Iteration number k Optimal control parameters : A = 4.25 et St = 0.74 Mean drag coefficient : J = 0.993 8 resolutions of the Navier-Stokes equations

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TRPOD

Numerical results

Initial control parameters : A = 6.0 et St = 0.2

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

A St

5 10 15 0.5 1 1.5 2 2.5

Jk Iteration number k Optimal control parameters : A = 4.25 and St = 0.74 Mean drag coefficient : J = 0.993 6 resolutions of the Navier-Stokes equations

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TRPOD

Numerical results

Initial control parameters : A = 6.0 et St = 1.0

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

A St

5 10 15 0.975 1 1.025 1.05

Jk Iteration number k Optimal control parameters : A = 4.25 and St = 0.74 Mean drag coefficient : J = 0.993 4 resolutions of the Navier-Stokes equations

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TRPOD

Numerical results

Initial control parameters : A = 1.0 et St = 1.0

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

A St

5 10 15 0.9 1 1.1 1.2 1.3 1.4

Jk Iteration number k Optimal control parameters : A = 4.25 and St = 0.74 Mean drag coefficient : J = 0.993 5 resolutions of the Navier-Stokes equations

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TRPOD

Drag coefficient

◮ Optimal control law : γopt(t) = A sin(2πSt t) avec A = 4.25 et St = 0.74

25 50 75 100 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

γ = 0 γ(t) = γopt(t) Base flow

CD Time units ◮ Relative drag reduction of 30% (J0 = 1, 4 ⇒ Jopt = 0, 99)

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TRPOD

Vorticity contour plots

Uncontrolled flow, γ = 0. Controlled flow, γ = γopt.

• Fig. : Iso-values of vorticity ωz.

Controlled flow : near wake strongly unsteady, far wake (after 5 diameters) steady and symmetric → steady unstable base flow

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Numerical costs

Discussion

Optimal control of NSE by He et al. (2000) : ⇒ 30% drag reduction for A = 3 and St = 0.75. Optimal control POD ROM by Bergmann et al. (2005) with no reactualization

• f the POD ROM :

⇒ 25% drag reduction for A = 2.2 and St = 0.53. Reduction costs compared to NSE : CPU time : 100 Memory storage : 600 but no mathematical proof concerning the Navier-Stokes optimality. TRPOD (this study) : ⇒ More than 30% of drag reduction for A = 4.25 and St = 0.738. Reduction costs compared to NSE : CPU time : 4 Memory storage : 400 but global convergence. ֒ → "Optimal" control of 3D flows becomes possible !

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Conclusions and perspectives

Conclusions on TRPOD Important relative drag reduction : more than 30% of relative drag reduction Global convergence : mathematical assurance that the solution is identical to the one of the high-fidelity model TRPOD compared to NSE ⇒ important reduction of numerical costs : ֒ → Reduction factor of the CPU time : 4 ֒ → Reduction factor of the memory storage : 400 "OPTIMAL" CONTROL OF 3D FLOWS POSSIBLE BY POD ROM Perspectives Test mode interpolations for controlled flows (Morzynski and Tadmor’s talks, GAMM 2006) Test other reduced basis method than classical POD Centroidal Voronoi Tessellations (Gunzburguer, 2004) : "intelligent" sampling in the control parameter space Model-based POD (Willcox, 2005) : modify the definitions of the POD modes Optimal control of the channel flow at Reτ = 180

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