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Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models.

Michel Bergmann Laurent Cordier & 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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Reduced Order Model (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 A simple configuration, a rich dynamical behavior

Stf = 0, 1 CD = 4, 25. Stf = 0, 2 CD = 2, 24. Stf = 0, 3 CD = 1, 57. Stf = 0, 4 CD = 1, 25. Stf = 0, 5 CD = 1, 09. Stf = 0, 6 CD = 1, 02. Stf = 0, 7 CD = 1, 03. Stf = 0, 8 CD = 1, 07. Stf = 0, 9 CD = 1, 13. Stf = 1, 0 CD = 1, 18.

• Fig. : Iso-values of the vorticity fields ωz for A = 3

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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.

Control of the cylinder wake in the laminar regimeby Trust-Region methods and POD Reduced Order Models. – p.11/21

Surrogate drag function and model objective function Generalities

◮ 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

Control of the cylinder wake in the laminar regimeby Trust-Region methods and POD Reduced Order Models. – p.17/21

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 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 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 Optimal control of the channel flow at Reτ = 180 Test other reduced basis method than classical POD Centroidal Voronoi Tessellations (Gunzburguer, 2004) : "intelligent" sampling in the control parameter space Balanced POD (Rowley, 2004) Model-based POD (Willcox, CDC-ECC 2005) : modify the definitions of the POD modes

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