Optimal rotary control of the cylinder wake using POD Reduced Order - PowerPoint PPT Presentation

Optimal rotary control of the cylinder wake using POD Reduced Order Model 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 (LEMTA) UMR 7563 (CNRS - INPL - UHP) ENSEM - 2, avenue de la Forˆ et de Haye BP 160 - 54504 Vandoeuvre Cedex, France Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.1/31

Outline I - Flow configuration and numerical methods II - Optimal control approach III - Proper Orthogonal Decomposition (POD) IV - Reduced Order Model of the cylinder wake (ROM) V - Optimal control formulation applied to the ROM VI - Results of POD ROM VII - Nelder-Mead Simplex method Conclusions and perspectives Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.2/31

Motivations Cylinder wake flow ? Prototype configuration of separated flow Experimental study of Tokumaru and Dimotakis (JFM 1991) Re = 15000 ◮ Unforced flow ◮ Forced flow = ⇒ 80% of drag reduction Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.3/31

I - Configuration and numerical method Two dimensional flow around a circular cylinder at Re = 200 Viscous, incompressible and Newtonian fluid ∂ u ∂t + ( u · ∇ ) u = − ∇ P + 1 ∇ · u = 0 ; Re ∆ u Cylinder oscillation with a tangential velocity γ ( t ) Γ sup y Γ e Γ c � � � γ ( t ) Γ s ✁ � ✁ � ✁ � ✁ � ✁ � ✁ � U ∞ x D Γ inf Control parameter : = R ˙ α ( t ) = γ ( t ) θ ( t ) = Tangential velocity U ∞ U ∞ Upstream velocity Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.4/31

I - Configuration and numerical method Fractional step method in time (pressure correction) Finite Element Method (FEM) in space ( P 1 , P 1 ) Numerical domain Ω = {− 10 ≤ x ≤ 20 ; − 10 ≤ y ≤ 10 } ; D = 1 Mesh : 25042 triangles, 12686 vertices ◮ Numerical code written by M.Braza (IMFT-EMT2) & D.Ruiz (ENSEEIHT) Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.5/31

I - Configuration and numerical method Iso vorticity at t = 100 . Iso pressure at t = 100 . 1.5 Authors S t C D C D 1 Braza et al. (1986) 0.2000 1.4000 C L C D , C L Henderson et al. (1997) 0.1971 1.3412 0.5 He et al. (2000) 0.1978 1.3560 0 this study 0.1983 1.3972 -0.5 Strouhal number and drag coefficient. 80 90 100 110 120 time units Aerodynamic coefficients. Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.6/31

II - Optimal control Definition Mathematical method allowing to determine without a priori knowledge a control law based on the optimization of a cost functional. State equations F ( φ, c ) = 0 ; (Navier-Stokes + I.C. + B.C.) Control variables c ; (Blowing/suction, design parameters ...) Cost functional J ( φ, c ) . (Drag, lift, target function, ...) Find a control law c and state variables φ such that the cost functional J ( φ, c ) reach an extremum under the constraint F ( φ, c ) = 0 . Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.7/31

II - Optimal control Lagrange multipliers Constrained optimization ⇒ unconstrained optimization ◮ Introduction of Lagrange multipliers ξ (adjoint variables). ◮ Lagrange functional : L ( φ, c, ξ ) = J ( φ, c ) − < F ( φ, c ) , ξ > ◮ Force L to be stationary ⇒ look for δ L = 0 : δ L = ∂ L ∂φ δφ + ∂ L ∂c δc + ∂ L ∂ξ δξ = 0 ◮ Hypothesis : φ , c and ξ assumed to be independent of each other : ∂ L ∂φ δφ = ∂ L ∂c δc = ∂ L ∂ξ δξ = 0 where ∂ L L ( x + ǫδx ) − L ( x ) ∂x = lim = 0 ∀ δx (Fréchet derivative) ǫ ǫ − → 0 Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.8/31

II - Optimal control Optimality system ◮ State equations ( ∂ L ∂ξ δξ = 0 ) : F ( φ, c ) = 0 ◮ Co-state (adjoint) equations ( ∂ L ∂φ δφ = 0 ) : � ∗ � ∗ � ∂ F � ∂ J ξ = ∂φ ∂φ ◮ Optimality condition ( ∂ L ∂c δc = 0 ) : � ∗ � ∗ � ∂ J � ∂ F = ξ ∂c ∂c ⇒ Expensive method in CPU time and storage memory for large system ! Bewley et al. (2000) : 10 8 grid points ⇒ Ensure only a local (generally not global) minimum Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.9/31

II - Optimal control Iterative method ◮ c (0) given ; 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 c ( n ) ; → state variables φ ( n ) ֒ 2. From t = T to t = 0 solve the co-state equations with φ ( n ) ; → co-state variables ξ ( n ) ֒ 3. Solve the optimality condition with φ ( n ) and ξ ( n ) ; → objective gradient δc ( n ) ֒ → c ( n +1) = c ( n ) + ω ( n ) δc ( n ) 4. New control law ֒ ◮ End do. Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.10/31

II - Optimal control Reduced Order Model (ROM) "without an inexpensive method for reducing the cost of flow computation, it is unlikely that the solution of optimization problems involving the three dimensional unsteady Navier-Stokes system will become routine" M. Gunzburger, 2000 Initialization Recourse to detailed model (TRPOD) High−fidelity model f(x), grad f(x) a(x), grad a(x) Approximation model Optimization ∆ x Optimization on simplified model Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.11/31

III - 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 � 2 = 1 . �| ( u , φ | 2 � ◮ φ ( X ) solution of the problem : max s.t. φ ◮ Snapshots method, Sirovich (1987) : � C ( t, t ′ ) a ( n ) ( t ′ ) dt ′ = λ ( n ) a ( n ) ( t ) . T ◮ Optimal convergence L 2 norm (energy) of φ ( X ) ⇒ Dynamical order reduction is possible. ◮ Decomposition of the velocity field : N P OD � a ( i ) ( t ) φ ( i ) ( x ) . u ( x , t ) = i =1 Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.12/31

III - POD POD modes : uncontrolled flow ( γ = 0 ) First POD mode. Second POD mode. Third POD mode. Fourth POD mode. Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.13/31

III - Reduced Order Model of the cylinder wake (ROM) ◮ Galerkin projection of NSE on the POD basis : � � � � φ ( i ) , ∂ u φ ( i ) , − ∇ p + 1 ∂t + ( u · ∇ ) u = Re ∆ u . ◮ Integration by parts (Green’s formula) leads : � � φ ( i ) , ∂ u − 1 � p, ∇ · φ ( i ) � � ( ∇ ⊗ φ ( i ) ) T , ∇ ⊗ u � ∂t + ( u · ∇ ) u = Re − [ p φ ( i ) ] + 1 Re [( ∇ ⊗ u ) φ ( i ) ] . � � � � with [ a ] = a · n d Γ and ( A, B ) = A : B d Ω = A ij B ji d Ω . Γ Ω Ω i, j Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.14/31

III - Reduced Order Model of the cylinder wake (ROM) ◮ Velocity decomposition with N P OD modes : N P OD � a ( k ) ( t ) φ ( k ) ( x ) . u ( x , t ) = u m ( x ) + γ ( t ) u c ( x ) + k =1 ◮ Reduced order dynamical system where only N gal ( ≪ N P OD ) modes are retained (state equations) : N gal N gal N gal d a ( i ) ( t ) � � � B ij a ( j ) ( t ) + C ijk a ( j ) ( t ) a ( k ) ( t ) = A i + d t j =1 j =1 k =1   N gal d γ � F ij a ( j ) ( t )  γ + G i γ 2 + D i d t +  E i + j =1 a ( i ) (0) = ( u ( x , 0) , φ ( i ) ( x )) . A i , B ij , C ijk , D i , E i , F ij and G i depend on φ , u m , u c and Re . Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.15/31

IV - Reduced Order Model of the cylinder wake Stabilization Integration and "optimal" stabilization of the POD ROM for γ = A sin(2 πS t t ) , A = 2 and S t = 0 . 5 . POD reconstruction errors ⇒ temporal modes amplification 1.5 ◮ Reasons : 1 Extraction by POD only of the large energetic eddies 0.5 a ( n ) Dissipation takes place in small 0 eddies -0.5 ◮ Solution : -1 Addition of an optimal artificial -1.5 0 5 10 time units viscosity on each POD mode Temporal evolution of the first 6 POD temporal modes. projection (Navier-Stokes) prediction before stabilization (POD ROM) prediction after stabilization (POD ROM). Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.16/31

IV - Reduced Order Model of the cylinder wake Stabilization 0 10 0.007 0.006 proj |�| |�| a ( n ) |� − �| a ( n ) 0.005 �| a ( n ) |� 0.004 -1 10 0.003 0.002 0.001 -2 10 0 0 2 4 6 8 10 12 14 5 10 POD modes index POD modes index Comparison of energetic spectrum. Comparison of absolute errors. ◮ Good agreements between POD ROM spectrum and DNS spectrum ◮ Reduction of the reconstruction error between predicted (POD ROM) and projected (DNS) modes ⇒ Validation of the POD ROM Optimal rotary control of the cylinder wake using POD Reduced Order Model – p.17/31