Optimal Control of Plane Poiseuille Flow

Optimal Control of Plane Poiseuille Flow Dr G.Papadakis Division of Engineering, King s College London george.papadakis@kcl.ac.uk AIM Workshop on Flow Control King s College London, 17/18 Sept 29 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 1/62

Contents Background and motivation Basic equations and spectral model Linear state-space model (plant) Initial conditions Synthesis of state feedback controllers, state estimators and output feedback controllers Controller implementation in full FV Navier-Stokes solver Recapitulation and future work AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 2/62

Background and motivation Ability to alter flow can have tremendous benefits Reduction of flow unsteadiness can benefit: Ocean ships (consumption 2.1bn barrels of oil pa) Aircrafts (consumption 1.5bn barrels of jet oil pa) Cars, trucks, trains, pipelines etc Increase of flow unsteadiness can benefit: Combustion efficiency Product quality that depends on mixing AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 3/62

Active control with feedback (within a linear framework) Application in Poiseuille Flow Wall Plane Poiseuille Flow Wall + Flow Disturbance Wall Normal, y Streamwise, x Actuation Sensing (Controller) AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 4/62

Control of Poiseuille flow Simple base flow, geometry, boundary conditions Linearly unstable R>5772, Transition R>1 Steps: Selection of state variables, actuation and sensing Transformation to state-space form Synthesis of linear optimal controllers to minimise Transient Energy Growth Implementation and testing of controllers in a NS solver AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 5/62

Perturbation Equations around a base flow Perturbation form of Navier-Stokes and Continuity eq. ( ) u + Ub u + ( u ) u +( u ) U b = p + 1 R 2 u u = Non-linear equations Divergence free velocity field Homogeneous (zero) wall boundary conditions on u Base flow U b parabolic AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 6/62

Linearised perturbation equations General form ( ) u + Ub u +( u ) U b = p + 1 R 2 u u = Parallel flows only: Wall-normal Velocity-vorticity v-η form η = u z ( 2 v) t η t w x ( 2 v) + U b x + U b y v z + U b 2 U b y 2 v x 1 R 2 ( 2 v)= η x 1 R 2 η = Boundary conditions at walls: v =, v/ y =, η = AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 7/62

Discretisation of v-η form Basis for state-space form: Spectral discretisation Fourier in streamwise and spanwise directions x, z One wavenumber pair (α, β) selected ( U b k 2 2 U b y 2 = j ( 3ṽ α y 2 t (k 2 = α 2 + β 2 ) v = R (ṽ(y, t)e j(αx+βz)) ) k4 ṽ + jrα ) ṽ k2 (jαu b + k2 R t ) η 1 R ) (U b + 2k2 2ṽ jrα y 2 2 η y + jβṽ U b 2 y = η t ( 1 ) 4ṽ jrα y 4 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 8/62

Discretisation of v-η form (cont) Chebyshev polynomials Γ n (y) in wall-normal direction y ṽ(y, t) = N Γ n (y)a v,n (t) Evaluation at collocation pts y =cos(nπ/n),n= N ṽ(y,t) Γ (1)... Γ N (1) a v,. =........ ṽ(y N,t) Similarly for η } Γ ( 1)... {{ Γ N ( 1) } D a v,n AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 9/62

Discretisation of v-η form (cont) Calculation of first order derivatives: ṽ (y, t) = N n= a v,n (t)γ n(y) ṽ (y,t). ṽ (y N,t) = Γ (1)... Γ N (1)....... } Γ ( 1)... {{ Γ N ( 1) } D1 a v,. a v,n Similarly we define matrices D2, D3, D4 for higher order derivatives. AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 1/62

Discretisation of v-η form (cont) Substituting to the v-η equations: L 11 L 12 ȧ = A 11 A 12 a L 21 L 22 A 21 A 22 A 11 = a =(a v,...a v,n a η,...a η,n ) T ) ( αūk 2 αū Ik4 jr A 12 = [], A 21 = βū D, A 22 = D + ) (αū + 2Ik2 ) (αū + Ik2 jr jr D D2 jr L 11 = j ( k 2 D + D2 ), L 12 =[], L 21 =[], L 22 = jd D2 D4 jr AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 11/62

Boundary conditions (without control) Lower wall ṽ = ṽ/ y = Similarly for upper wall Similarly for η N n= N n= Γ n ( 1)a v,n (t) = Γ n( 1)a v,n (t) = These b.c. replace the first, second, last, next-to-last rows. Leads to spurious eigenvalues (easily mapped to an arbitrary location in the complex plane) AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 12/62

Introduction of wall transpiration We aim to control the flow via wall transpiration. New b.c. Lower wall: ṽ( 1,t)= q l (t) Upper wall: ṽ(1,t)= q u (t) q l (t), q u (t) are unknown functions (to be determined) For both walls: ṽ/ y =,η = AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 13/62

Transformation of variables Introduction of variable ṽ h (y, t) that has homogenous b.c. ṽ(y, t) =ṽ h (y, t)+f u (y) q u (t)+f l (y) q l (t) We want ṽ h (±1,t)=ṽ h (±1,t)= This can be achieved if f u (1) = f l ( 1) = 1,f u ( 1) = f l (1) = f u(±1) = f l (±1) = Suitable functions (3rd order polynomials) f u (y) = y3 +3y +2,f l (y) = y3 3y +2 4 4 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 14/62

Novel Basis functions To eliminate spurious modes, we introduce novel basis functions Γ DN n (y) and Γ D n We require them to satisfy the conditions: Γ D n (y = ±1) = Γ DN n (y = ±1) = Γ DN n (y = ±1) = ṽ(y, t) = η(y, t) = N n= Γ DN n (y)a v,n (t)+f u (y)ṽ u (t)+f l (y)ṽ l (t) N Γ D n (y)a η,n (t) n= AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 15/62

Form of new basis functions Γ D n,odd = Γ n Γ 1 Γ D n,even =Γ n Γ 2 Γ DN n,odd = (Γ n+4 Γ 1 ) (n +3) 2 (Γ n+2 Γ 1 )/(n +1) 2 Γ DN n,even = (Γ n+4 Γ 2 ) ((n +3) 2 1) (Γ n+2 Γ 2 )/ [ (n +1) 2 1 ] 1.5 f u f l Γ D last y.5 1 Γ DN 1 Γ DN Γ D Γ DN last Γ D 1 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 16/62

State-space form L 11 L 12 ȧ = A 11 A 12 a + B 11 B 12 q E 11 E 12 q L 21 L 22 A 21 A 22 B 21 B 22 E 21 E 22 }{{} effect of transpiration where q =( q u, q l ) T and (N +1) 1 vectors are B 11 = B 12 = ) ( αūk 2 αū Ik4 jr ) ( αūk 2 αū Ik4 jr f u + f l + ( ) αū + 2k2 I jr ( ) αū + 2k2 I jr B 21 = βū f u, B 22 = βū f l E 11 = j ( k 2 f u + f ) u, E12 = j ( k 2 f l + f ) l E 21 = (), E 22 =() f u f l AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 17/62

State-space form (cont) Invert L ȧ = L 1 Aa + L 1 Bq L 1 E q Input vectors q and q are not independent. Recast system so that q is part of the state variable vector and control input is q. ȧ q = L 1 A L 1 B a q + L 1 E q I New system matrix has the same eigenvalues as L 1 A plus two eigenvalues equal to. AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 18/62

Final form of State-Space Model General form required for controller synthesis;- X = AX + BU, Y = CX For our problem: ȧ q }{{} X = ỹ = L 1 A L 1 B }{{} A ( ) C D a }{{} q C }{{} X a q }{{} X + L 1 E I } {{ } B q }{{} U AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 19/62

Output in physical space Measurement of shear stresses in x,z directions in the 2 walls. τ yx = 1 ( u R y + v ) x τ yz = 1 ( w R y + v ) z Since v is known it can be subtracted out So the measurement vector can be defined as: y = 1 ( ) T u u w w y y y y R y=+1 y= 1 y=+1 y= 1 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 2/62

Output in Fourier space ỹ = 1 R ũ y ũ y w y w y y=+1 y= 1 y=+1 y= 1 = 1 R j α 2 + β 2 α 2 ṽ y 2 α 2 ṽ y 2 β 2 ṽ y 2 β 2 ṽ y 2 β η y β η y + α η y + α η y y=+1 y= 1 y=+1 y= 1 or ỹ = j α 2 + β 2 C 11 C 12 C 21 C 22 a + j α 2 + β 2 D 11 D 21 q ỹ = ( C D ) a q AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 21/62

Final form of State-Space Model: a recap States X : Coefficients of basis a n and Wall-normal velocities at walls q u, q l Input U: rate of change of wall-normal velocities at walls q u, q l Output Y: Wall-shear stress measurements τ u, τ l A has two integrators, associated with transpiration from either wall, free of spurious eigenvalues We are ready to synthesize controllers now! AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 22/62

Initial Conditions Worst initial conditions lead to non-normal growth Non-orthogonal system matrix A (Trefethen 1993) Variational method used (Butler and Farrell 1992, Bewley 1998) States X (t) =Ξe Λt χ, Energy E = X (t) T QX (t) Transient Energy Growth θ = E(τ)/E() from θ ( Ξ T QΞ ) ) χ = (e ΛT τ Ξ QΞe Λτ χ Hence X () = Ξχ u() AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 23/62

Test Cases Two cases representing early transition 1) α =1,β =,R = 1 Tollmien Schlichting waves One Unstable eigenvalue 2) α =,β =2.44,R = 5 Streamwise vortices Stable, but the largest transient energy growth Both are 2-D cases AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 24/62

case 1: Comparison with Orszag(1971) 1 λ i λ i (Orszag) / λ i (Orszag) 1 2 1 4 1 6 1 8 1 1 5 1 15 2 25 3 mode i AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 25/62

case 1: Poles and zeros.1 Poles Zeros I.2.3.4.5.6.7.8.9 1 1.8.6.4.2 R AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 26/62

case 1: Unstable ṽ eigenvector Comparison with Thomas(1953) 1.2 1 R I Thomas R Thomas I.8.6 v.4.2.2 1.5.5 1 y AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 27/62

case 1: ṽ eigenvectors Real Part (solid) and Imaginary Part (dot-dashed) 1 y 1 1 2 3 4 5 6 7 8 9 1 11 12 13 y 1 R I 1 14 15 16 17 18 19 2 Mode i AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 28/62

case 1: ũ eigenvectors Real Part (solid) and Imaginary Part (dot-dashed) 1 (Scaled by 1/25 c.f. ṽ eigenvectors) y 1 1 2 3 4 5 6 7 8 9 1 11 12 13 y 1 R I 1 14 15 16 17 18 19 2 Mode i AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 29/62

Transpiration mode (m=2) Re=5 u contours y=1.5.5.5.5.5.5.5 1.5 y= 1 x= v contours x=6.23 y=1.8.6.4.2.6.8.6.5 1.4.2.5.4.6.8.4.2 y= 1 x= x=6.23.2 Results from linear simulations and solution of NS (in-house code) are identical. AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 3/62

Case 1: Observability 1 8 1 6 Current Model Bewley s Data 1 4 κ o 1 2 1 1 2 1 2 3 4 5 6 Mode i AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 31/62

Open loop, case 1, Transient energy From Initial Conditions X worst Scaled to E =1 1 9 8 7 N=1 N=2 N=3 N=4 N=5 6 E 5 4 3 2 1 5 1 15 t AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 32/62

Case 2: Poles and zeros.5.4 Poles Zeros.3.2.1 I.1.2.3.4.5 1.8.6.4.2 R AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 33/62

Case 2: ṽ eigenvectors (imaginary) 1 y 1 1 1 2 3 4 5 6 7 8 9 1 11 12 13 I y 1 14 15 16 17 18 19 2 Mode i AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 34/62

Case 2: Observability 1 8 Current Model Bewley s Data 1 6 κ o 1 4 1 2 1 5 1 15 2 Mode i AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 35/62

Open loop, case 2, Transient energy From Initial Conditions X worst Scaled to E =1 6 5 4 N=1 N=2 N=3 N=4 N=5 E 3 2 1 5 1 15 2 25 t AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 36/62

Optimal State Feedback (LQR controller) AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 37/62

Optimal State Feedback (LQR controller) Given the real system;- X = AX + BU, Y = CX State feedback control signal to minimize;- ( X (t) T QX (t)+u(t) T RU(t) ) dt Given by U = KX where K = R 1 B T X X = X T from ARE A T X + XA XBR 1 B T X + Q = AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 38/62

Weighting Matrix Q Choose Q to form energy density E; E = X T QX = 1 V E = 1 8k 2 1 1 1 y= 1 2π/α x= 2π/β z= (k 2 ṽ T ṽ + ṽ T ṽ y y + ηt η)dy (u 2 + v 2 + w 2 ) dz dx dy 2 Curtis-Clenshaw quadrature for integration E = 1 8k 2 X T T T Q Q T +( T/ y) T Q ( T/ y) X X T QX Q contains quadrature weights T converts states to ṽ, η at collocation points AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 39/62

Weighting Matrix R Choose R = r 2 I Max energy vs r plotted from linear simulations Large r leads to small control effort (and larger energy) Small r leads to larger control effort (and smaller energy) r =.25 for Case 1 and r = 128 for Case 2. AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 4/62

Case 1: Open loop v.s. LQR control 1 9 8 7 N=1 N=2 N=3 N=4 N=5 14 12 1 N=3 N=4 N=5 N=7 N=1 E 6 5 4 E 8 6 3 4 2 1 2 5 1 15 t 2 4 6 8 1 t without control with LQR control AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 41/62

Case 2: Open loop v.s. LQR control 6 5 4 N=1 N=2 N=3 N=4 N=5 1 9 8 7 6 N=3 N=4 N=5 N=7 N=1 E 3 2 1 5 1 15 2 25 t without control E 5 4 3 2 1 5 1 15 t with LQR control AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 42/62

How does the controller affect the flow? u(y, z, t) =u v U b t (Ellingsen and Palm (1975)) (a) t = (b) t = τ/8 (c) t = τ (d) t =4τ Figure from Bewley and Liu (1998) (no control) v component (dashed) magnified by 25 as compared to η (solid) Solid contours: u>, Dashed contours: u< t = τ corresponds to maximum transient energy growth AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 43/62

How does the controller affect the flow? (cont) u(y, z, t) =u v U b t (Ellingsen and Palm (1975)) (a) t = (b) t = τ/8 (c) t = τ (d) t =4τ Figure from Bewley and Liu (1998) (with control) v component (dashed) magnified by 25 as compared to η (solid) Solid contours: u>, Dashed contours: u< t = τ corresponds to maximum transient energy growth AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 44/62

Optimal State Estimation AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 45/62

Optimal State Estimation Given the real estimator;- X est = AX est + BU + L (Y CX est ) The optimal L to minimize;- E { } [X X est ] T [X X est ] Is given by L = Y CV 1 where Y = Y T from ARE AY + Y A T Y C T V 1 CY + W = AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 46/62

Weighting Matrices W, V W represents covariance of process noise Transform system states to values at collocation points Choose W =(1 y 1 ) 2 (1 y 2 ) 2, where y 1,y 2 are collocation points Disturbances near the centreline more variable/larger Better estimation than W = I V represents covariance of measurement noise Choose V = αi, α by inspection AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 47/62

Case 1: Performance of tuned estimator (open loop) 1 9 8 7 6 E 5 4 3 2 1 E, N=1 E Est, N=3 E Est, N=4 E Est, N=5 E Est, N=7 E Est, N=1 5 1 15 t AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 48/62

Case 2: Performance of estimator (open loop) 5 45 4 35 3 E, N=1 E Est, N=3 E Est, N=4 E Est, N=5 E Est, N=7 E Est, N=1 E 25 2 15 1 5 5 1 15 2 25 t AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 49/62

Optimal Output feedback AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 5/62

Effect of control: A summary Linear Simulation Case OL Time LQR Time LQG Time 1 unstable - 12.64 18.3 29.42 19.2 2 4896.94 379.5 848.8 187.5 934 196.3 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 51/62

Implementation of Controller into a Navier-Stokes solver Off-line: Generate matrices A, B, C,Q Synthesize controller matrix K and estimator matrix L (for output feedback) Store K and A, B, C, L (for output feedback) On start-up: Read matrices in NS solver and store Read initial velocity field On each time-step of the solver: For state feedback Calculate X from velocity field Actuation U = KX For output feedback: FFT on τ u,τ l to get Y =( τ u, τ l ) T Run estimator X est = AX est + BU + L (Y CX est ) Actuation U = KX est Integration of U to get q u and q l Inverse FFT to get q u,q l Use q u,q l to set boundary conditions for next time step AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 52/62

Navier-Stokes solver Finite-volume discretisation method Second order in space (CDS), implicit 2 nd order in time PISO algorithm, Collocated grid (approx 1 1) Boundary Conditions: Streamwise - cyclic at α Walls - transpiration set by controller Spanwise - cyclic at β Code modified to solve for the non-linear perturbation u about base flow U b AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 53/62

Small Perturbations: Cases 1 and 2, LQR Perturbation energy at t =corresponds to ṽ max =1 4 U cl E C1 =8.23 1 8 (Case 1) E C2 =2.26 1 9 (Case 2) 1.2 x 1 6 1 Non linear E Linear E 2 x 1 6 1.8 1.6 Non linear E Linear E.8 1.4 1.2 E.6 E 1.4.2 2 4 6 8 1 t.8.6.4.2 1 2 3 4 5 6 t Case 1 (LQR control) Case 2 (LQR control) AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 54/62

Large Perturbations that trigger non-linearities Perturbation energy at t =corresponds to ṽ max =1 2 U cl 1 4 E C1 (Case 1) 1 4 E C2 (Case 2) which is around 1 times the transition threshold of Reddy et al (1998).8.7.6 Non linear E Non linear E Est Linear E Linear E Est.12.1 Non linear E Non linear E Est Linear E Linear E Est.5.8 E.4 E.6.3.4.2.1.2 5 1 15 t 2 4 6 8 1 t Case 1 (Open loop) Case 2 (Open loop) AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 55/62

Control of large Perturbations (LQR).12.1 Non linear E Linear E.2.18.16 Non linear E Linear E.8.14.12 E.6 E.1.4.2 5 1 15 t.8.6.4.2 2 4 6 8 1 t Case 1 (LQR control) Case 2 (LQR control) AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 56/62

Multi-wave number control: Motivation Case 2 x 1 5.25 E max =.23951 at t=123.9 E max /E(t=)=159.8.2 E Σ i=:2 E i E E 1 E 2 1.2 1 E max =1.929e 5 at t=379.5 E max /E(t=)=4835.6 E Σ i=:2 E i E E 1 E 2.15.8 E E.6.1.4.5.2 5 1 15 2 25 3 t 5 1 15 2 25 3 t Open loop, E() = 1 4 E C2 Open loop, E() = E C2 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 57/62

Nominal vs Multi-wave number control Case 2, E() = 1 4 E C2 x 1 3.1 E max =.92747 at t=87.9 E max /E(t=)=41.39 E Σ i=:2 E i E 9 8 E max =.87311 at t=86.1 E max /E(t=)=386.33 E Σ i=:2 E i E.8 E 1 E 2 7 E 1 E 2 E.6 E 6 5.4 4 3.2 2 1 5 1 15 2 25 3 t 5 1 15 2 25 3 t Nominal control Multi-wave number control AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 58/62

Comparison of nominal vs Multi-wave number control Energy Growth: A summary max t (E(t))/E(t =) Linear-Sized Small Non-Linear Large Non-linear E C2 1 4 E C2 1 6 E C2 Uncontrolled 4835.6 159.8 21.1 One wavenumber Control 837.5 41.4 21.8 Multiple wavenumber Control 838.6 386.3 12.7 AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 59/62

Control of oblique vortices (3D simulations) α =1, β =1, ṽ max =1 3 U cl 1.5 x 1 4 n. d. E vs n. d. t 1 n. d. E.5 1 2 3 4 5 6 n. d. t Open loop (red line) vs nominal control (green line) and multi-wave number control (blue line). AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 6/62

Recapitulation Development of a linear state-space model for plane Poiseuille flow Synthesis of optimal controllers at (α, β) Synthesis of multi-wave number controllers Implementation of controllers in FV Navier-Stokes code. Small Perturbations Spectral linear and FV CFD results very close. Large Perturbations Trigger non-linearities and generate multi wave numbers. Multi-wave number controllers show benefit over single-wave number controllers AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 61/62

The End Any Questions? AIM workshop on Flow Control, King s College London, 17/18 Sept 29 p. 62/62