Résonance et contrôle en cavité ouverte Jérôme Hœpffner KTH, Sweden Avec Espen Åkervik, Uwe Ehrenstein, Dan Henningson
Outline The flow case Investigation tools resonance Reduced dynamic model for feedback control Control performance
Boundary layer with cavity y x 2D flow over a smooth cavity Inflow: Blasius profile Reynolds number : 320
Cavity: Shear layer mode (compressible) From Rowley et al, JFM 2002 Self sustained cycle: perturbation growth pressure wave new perturbation
Cavity: Wake mode From Rowley et al, JFM 2002 Subcritical bifurcation to oscillating state Low frequency ejection of large vortices. (for large aspect ratio)
Sound generation in organ pipes Interaction of the jet and the edge generates sound. from http://www.fluid.tue.nl/gdy/acous/
Cavity: linear instability of the incompressible flow? Stable boundary layer, or convectively unstable convectively unstable shear layer Questions: Can we have an globally unstable cavity flow? What role does the pressure play in the incompressible case? Can we control the cavity flow?
Investigation tools DNS to compute the base flow: Chebyshev in wall normal, finite difference in streamwise. Stability analysis by computation of 2D eigenmodes: Chebyshev/Chebyshev and Arnoldi Optimal growth by optimization over initial conditions : Singular value decomposition, using the reduced model Control optimization by solution of two Riccati equations : Using the reduced order model
The eigensolver 2D Navier-Stokes + continuity iωû = (U )û (û )U ˆp x + 1/Re 2 û iωˆv = (U )ˆv (û )V ˆp y + 1/Re 2ˆv 0 = u Generalized eigenproblem: Bωu = Au To be rewritten A 1 Bu = 1 ω u Solved by Arnoldi iterations. Matrix formulation: iωû...... x iωˆv =...... y 0 C x y Additional constraints C û ˆv p
DNS: EIG: Grids & resolution The resolution are: nx=2048 finite difference, ny=97 Chebyshev, Lx=409, Ly=80 nx=250 Chebyshev, ny= 50 Chebyshev, Lx=270. Ly=15. DNS grid vs eigenmode grid y 70 60 50 40 30 20 10 0 0 50 100 150 200 250 300 350 400 x
Recirculating zone inside the cavity The base flow Streamwise velocity profiles u: 6 4 2 y 0 2 4 50 100 150 200 250 x Boundary layers (before and after the cavity) Shear-layer over the cavity Flow is composed of :
The base flow Globally unstable flow Base flow obtained from time averaging
Eigenvalues Spectra 0 0.02 (m5) (m1) (m4) (m2) (m3) λ c 0.04 (m6) 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25 λ r
Spectra: unstable shear layer mode, (m1)
Spectra: unstable shear layer mode, (m2)
Spectra: higher frequency mode, (m3)
Spectra: more damped, (m4)
Spectra: propagative mode, (m5)
Spectra: propagative mode free-stream, (m6)
eigenmodes and their adjoint, Integrated 0.5 Eigenmodes integrated in y integrated in y 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 3.5 x 106 3 Adjoint eigenmodes integrated in y x integrated in y 2.5 2 1.5 1 0.5 0 0 50 100 150 200 250 Where are the modes localised and where are they sensitive? x
Optimal transient energy growth from initial conditions System x(t) = Ax, ẋ(0) = x 0, with solution x(t) = e At x 0 Find the initial condition x 0 maximizing G(t) = max x0 < x(t), x(t) > < x 0, x 0 >, adjoint: < Ax 1, x 2 >=< x 1, A + x 2 > x 1, x 2 leads to G(t) = max < eat x 0, e At x 0 > < x 0, x 0 > = max < ea+t e At x 0, x 0 > < x 0, x 0 > Maximum growth at time t: eigenvalue of e A+t e At.
Optimal growth in the cavity Global instability Potentiality of strong energy growth Low frequency cycle 10 8 Energy envelopes, number of modes from 1 to 260 10 6 max E 10 4 10 2 10 0 0 100 200 300 400 500 600 700 800 900 1000
10 8 6 Trajectories from the worst initial conditions x 106 Several energy trajectories out of the envelope E 4 2 0 0 100 200 300 400 500 600 700 800 900 1000 time
The most dangerous initial condition A wave packet at the beginning of the shear layer.
Animation of flow cycle
Flow cycle, u and v
Flow cycle, the pressure Generation of global pressure change when the wave-packet impacts on the downstream lip Regeneration of disturbances when the pressure hits the upstream lip
Flow cycle, the pressure
Control
Control Seek to minimize the energy growth 10 Actuation Disturbance 5 Sensing 0 50 100 150 200 250 One actuator upstream One sensor downstream Oscillating disturbance in the shear layer
Feedback control Using a dynamic model of the system: { ẋ = Ax + Bu r = Cx One can optimize for the feedback u = G(r) The model in 2D is too big for optimization reduced model. For reduction: project the dynamics on the least stable eigenmodes. Finally, couple the reduced controller and the flow system
Physical space: { ẋ = Ax + Bu Model reduction Galerkin projection on least stable eigenmodes: Eigenmode space: P ẋ }{{} k = P AP 1 }{{} A M P x }{{} k r = Cx r = CP }{{ 1 }}{{} P x C M k Projection on eigenmodes biorthogonal set of vectors: Eigenmodes: q i, Adjoint operator: A + / < Ax 1, x 2 >=< x 1, A + x 2 >, x 1, x 2 + }{{} P B u B M Adjoint eigenmodes: q i +, Biorthogonality: δ ij =< q i, q j + >, Projection: k i =< x, q i + >
Control terminology Estimation: From sensor information, recover the instantaneous flow field. Full information control: From full knowledge of the flow state, apply control. Compensation: Model reduction: Control penalty: Close the loop by using the estimated flow state for control. Project the dynamics on a set of selected basis vectors. Penalisation of the actuation amplitude. sensor noise: Disturbances: Uncertainty in the measured signal. External forcing exciting the flow. Objective function: Function of the flow state to be minimized.
Central elements of the design 1) From the sensors, estimate the flow state: Sensor location Sensor noise Disturbance model (here perturbations at the inflow) 2) Using the flow state information, apply control : Actuator location Control penalty Objective function Optimization is done by solving two Riccati equations
Testing procedure 1. Decide penalties, sensor noise, locations 2. Reduce the model by projection 3. Optimize for the feedback 4. Couple flow system and controller The reduced controller (20 states) is applied on the full system (20,000 states) 5. Compute energy of controlled flow
Compensation performance 0.1 0.08 Flow, compensated flow Flow energy, Global cycle + exponential growth E 0.06 0.04 Compensated flow 0.02 0 0 200 400 600 800 1000 1200 time Good control performance from the second cycle Reduced order model:20 states
Flow/compensated flow animation
Flow/compensated flow, x/t diagram
Flow/compensated flow, x/t diagram
Actuation signal actuation 0.05 0 Actuator signal Flow signal, V in the vicinity 0.05 0 200 400 600 800 1000 1200 time Signal, V, downstream lip 40 Compensated flow signal Signal 20 0 20 Flow signal 40 0 200 400 600 800 1000 1200
Starting the compensator at later times 2 x 105 Flow 1.5 Starting at 150 1 Starting at 0 Starting at 200 0.5 Starting at 300 0 0 100 200 300 400 500 600 700 800 900 1000 Compensator cannot affect the disturbance propagation but can affect the disturbance generation
Dynamic distortion blue :flow Red :compensated flow Spectra with and without compensation 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0.15 0.2 0.25
Control gain Function used to extract the actuation signal from the flow
Estimation gain Function used to force the estimator flow
Compensator impulse response Compensator: input (sensor signal, r ) output (Control signal, u ) linear system The input-output relation is described by convolution u(t) = τ=0 G(τ)r(t τ) x 10 5 Compensator Impulse response 2 G 0 2 0 50 100 150 200 250 300 350 400 450 500 τ
Conclusion Found supercritical Hopf bifurcation for long cavity Incompressible cavity can have global cycle due to pressure. Global eigenmodes can be used for analysis and model reduction. Model reduction allows optimal feedback design for large systems. Non-parallel effects/global instabilities can be treated.
Extra slides
Base flow: 15 10 5 0 5 Normal velocity: 20 40 60 80 100 120 140 0.5 Quiver: 0 0.5 1 1.5 2 2.5 3 5 4 3 2 1 0 1 2 3 4 3.5 4 4.5 30 40 50 60 70 80 90 Re500/Blasius: 5 0 50 100 150 200
10 5 0 5 Re 300/Re 500: 20 40 60 80 100 120 140 160 0.08 0.06 0.04 0.02 0 Local stability: 150 100 50 0 0.5 1 1.5
Boundary conditions 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 Re 300 20 0.2 0.1 40 60 80 100 120 140 and re 500: 20 0.2 0.1 40 60 80 100 120 140 fringe 20 1 0.6 0.5 0.9 0.8 0.7 0.4 0.6 0.3 0.5 0.4 0.2 0.3 0.1 20 40 60 80 100 120 140 low resolution: 20 0.2 0.1 40 60 80 100 120 140 long and strong 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 fringe: 20 0.2 0.1 40 60 80 100 120 140 160 no fringe 20 0.2 0.1 40 60 80 100 120 140