Control of the cylinder wake in the laminar regime by Trust-Region - PowerPoint PPT Presentation

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 Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.1/20

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 ) = V T U ∞ = A sin(2 πSt f t ) Γ sup � ∂u � Z T Z 2 π Z T Z 2 π y Ω Γ e V T ( t ) θ ��� ��� Γ s ��� ��� 0 U ∞ x Γ c D Γ inf Find the control parameters c = ( A, St f ) T such that the mean drag coefficient is minimized � C D � T = 1 2 p n x R dθ dt − 1 2 ∂x n x + ∂u ∂y n y R dθ dt, T T Re 0 0 0 0 Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.2/20

Introduction Parametric study 1.3928 6 3.6306 1.0320 5 1.3878 0.9929 4 6.9876 Amplitude � � 3 1.3928 1.1728 1.0529 2 2.5516 1.3905 1.3878 1 1.3526 1.2327 0 0 0.2 0.4 0.6 0.8 1 Strouhal Variation of the mean drag coefficient with A and St f . Numerical minimum = (4 . 3 , 0 . 74) . A min , St f min Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.3/20

Introduction Mean drag coefficient & steady unstable base flow 2.5 Natural flow 2.25 Base flow 2 1.75 � C D � T � C D � T 1.5 1.25 C 0 D 1 C base D 0.75 25 50 75 100 125 150 175 200 Re 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. Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.4/20

Reduced Order Model (ROM) and optimization problems 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 Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.5/20

ROM Proper Orthogonal Decomposition (POD) ◮ Introduced in turbulence by Lumley (1967). ◮ Method of information compression Z Φ 2 ◮ Look for a realization Φ ( X ) which is clo- ser, in an average sense, to realizations u ( X ) . Coordinate axis Φ 1 ( X = ( x , t ) ∈ D = Ω × R + ) ◮ Φ ( X ) solution of the problem : � Φ � 2 = 1 . �| ( u , Φ ) | 2 � max s.t. Φ Original point cloud ◮ Snapshots method, Sirovich (1987) : Coordinate axis C ( t, t ′ ) a ( n ) ( t ′ ) dt ′ = λ ( n ) a ( n ) ( t ) . Point cloud, mean shifted (centered around origin) T ◮ Optimal convergence in L 2 norm (energy)of Φ ( X ) ⇒ Dynamical order reduction is possible. Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.6/20

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. Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.7/20

ROM Non-equilibrium modes (Noack et al. 2004) ◮ Necessity for a given reference flow to introduce new modes : either new operating conditions or shift-modes Dynamics I φ I 2 ���������� ���������� ���������� ���������� φ I ���������� ���������� 1 ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� ���������� Dynamics II ���������� ���������� φ I φ I → II 0 φ II neq 2 φ II ����������� ����������� 1 ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� 0 ����������� ����������� ����������� ����������� φ II ����������� ����������� ����������� ����������� 0 ����������� ����������� ����������� ����������� ����������� ����������� Controlled space Fig. : Schematic representation of a dynamical transition with a non-equilibrium mode Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.8/20

X X | {z } | {z } | {z } ROM A robust POD surrogate for the drag coefficient ◮ POD approximations consistent with our approach : N N + M U ( x , t ) = ( u, v, p ) T = a i ( t ) φ i ( x ) + a i ( t ) φ i ( x ) + γ ( c , t ) U c ( x ) i =0 i = N +1 control function Galerkin modes non-equilibrium modes Physical aspects Dynamical aspects Modes actuation mode predetermined dynamics U c a 0 = 1 mean flow mode U m , i = 0 i = 1 Galerkin modes i = 2 POD ROM Dynamics of the reference · · · Temporal dynamics of the flow I i = N modes (eventually, the mode i = 0 is solved then non-equilibrium modes i = N + 1 a 0 ≡ a 0 ( t ) ) Inclusion of new operating · · · conditions II, III, IV, · · · i = N + M Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.9/20

� � � � ROM Galerkin projection X X X ◮ Galerkin projection of NSE onto the POD basis : ! X φ i , ∂ u φ i , − ∇ p + 1 ∂t + ∇ · ( u ⊗ u ) = Re ∆ u . ◮ Reduced order dynamical system where only ( N + M + 1) ( ≪ N P OD ) modes are retained (state equations) : N + M N + M N + M d a i ( t ) = B ij a j ( t ) + C ijk a j ( t ) a k ( t ) d t j =0 j =0 k =0 N + M + D i d γ γ ( c , t ) + G i γ 2 ( c , t ) , d t + E i + F ij a j ( t ) j =0 a i (0) = ( U ( x , 0) , φ i ( x )) . B ij , C ijk , D i , E i , F ij and G i depend on φ i , U c and Re . Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.10/20

Z 2 π � � Surrogate drag function and model objective function X X f ◮ Drag operator : | {z } | {z } C D : R 3 → R (1) u 3 n x − 1 ∂u 1 ∂x n x − 1 ∂u 1 u �→ 2 ∂y n y R dθ, Re Re 0 ! Z T X f ◮ Surrogate drag function : N + M N C D ( t ) = a 0 ( t ) N 0 + a i ( t ) N i + a i ( t ) N i with N i = C D ( φ i ) . i = N +1 i =1 fluctuations C ′ evolution of the mean drag D ( t ) ◮ Model objective function : N + M C D ( t ) � T = 1 m = � a 0 ( t ) N 0 + a i ( t ) N i dt. T 0 i = N +1 Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.11/20

f Surrogate drag function Test case A = 2 and St = 0 . 5 ◮ Comparison of real drag coefficient C D (symbols) and model function C D (lines) at the design parameters. 1.1 1.095 1.09 1.085 1.08 C D f 1.075 C D & 1.07 1.065 1.06 1.055 1.05 1.045 0 5 10 15 t Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.12/20

Robustness of the model objective function Test case A = 2 and St = 0 . 5 2.5 2.5 1.293 1.221 1.275 1.209 1.257 1.197 1.239 1.185 1.221 1.173 1.203 1.161 1.185 1.149 A A 2 2 1.167 1.137 1.149 1.125 1.131 1.113 1.113 1.101 1.095 1.089 1.077 1.077 1.059 1.065 1.041 1.053 1.5 1.5 0.4 0.5 0.6 0.4 0.5 0.6 St St (a) Real objective function J (b) Model objective function m Fig. : Comparison of the real and the model objective functions associated to the mean drag coefficient. Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition. – p.13/20