Dynamics and control of wall-bounded shear flows

Transcription

1 Dynamics and control of wall-bounded shear flows Mihailo Jovanović mihailo Center for Turbulence Research, Stanford University; July 13, 2012

2 M. JOVANOVIĆ 1 Flow control technology: shear-stress sensors; surface-deformation actuators application: challenge: turbulence suppression; skin-friction drag reduction distributed controller design for complex flow dynamics

3 M. JOVANOVIĆ 2 Outline ❶ DYNAMICS AND CONTROL OF WALL-BOUNDED SHEAR FLOWS The early stages of transition initiated by high flow sensitivity Controlling the onset of turbulence simulation-free design for reducing sensitivity ❷ CASE STUDIES Key issue: high flow sensitivity Sensor-free flow control streamwise traveling waves Feedback flow control design of optimal estimators and controllers ❸ SUMMARY AND OUTLOOK

4 M. JOVANOVIĆ 3 Transition to turbulence successful in: Benard Convection, Taylor-Couette flow, etc. fails in: wall-bounded shear flows (channels, pipes, boundary layers) DIFFICULTY 1 Inability to predict: Reynolds number for the onset of turbulence (Re c ) LINEAR HYDRODYNAMIC STABILITY: unstable normal modes Experimental onset of turbulence: { much before instability no sharp value for Re c DIFFICULTY 2 Inability to predict: flow structures observed at transition (except in carefully controlled experiments)

5 linear stability analysis ded shear flows M. JOVANOVIĆ 4 LINEAR STABILITY: r instability For Re Re c exp. growing normal modes } corresponding e-functions l prediction Experiment = exp. growing flow structures (TS-waves) disturbances ~ 1000 ~ 350 ~ z x NOISY EXPERIMENTS: streaky boundary layers and turbulent spots on to more complex flow is ise streaks am turbulence z x Flow z x Matsubara & Alfredsson, J. Fluid Mech. 01

6 large transient responses large noise amplification small stability margins TO COUNTER THIS SENSITIVITY: must account for modeling imperfections M. JOVANOVIĆ 5 FAILURE OF LINEAR HYDRODYNAMIC STABILITY caused by high flow sensitivity TRANSITION STABILITY + RECEPTIVITY + ROBUSTNESS flow disturbances unmodeled dynamics Farrell, Ioannou, Schmid, Trefethen, Henningson, Gustavsson, Reddy, Bamieh, etc.

7 M. JOVANOVIĆ 6 Tools for quantifying sensitivity INPUT-OUTPUT ANALYSIS: spatio-temporal frequency responses d Free-stream turbulence Surface roughness Neglected nonlinearities Linearized Dynamics d 1 d 2 d 3 }{{} d Fluctuating velocity field amplification } v u v w }{{} v IMPLICATIONS FOR: transition: insight into mechanisms control: control-oriented modeling

8 M. JOVANOVIĆ 7 Transient growth analysis Re < 5772 kinetic energy: STUDY TRANSIENT BEHAVIOR OF FLUCTUATIONS ENERGY ψ t = A ψ time E-values: misleading measure of transient response

9 M. JOVANOVIĆ 8 ψ1(t), ψ2(t) [ ] ψ 1 ψ 2 A toy example = [ 1 0 k 2 ] [ ψ1 ψ 2 ]

10 M. JOVANOVIĆ 9 Response to stochastic forcing Re = 2000 forcing: white harmonic d(x, y, z, t) = ˆd(k x, y, k z, t) e i(k xx + k z z) in t and y in x and z kinetic energy

11 M. JOVANOVIĆ 10 Ensemble average energy density channel flow with Re = 2000: Dominance of streamwise elongated structures streamwise streaks! Farrell & Ioannou, Phys. Fluids A 93 Jovanović & Bamieh, J. Fluid Mech. 05 Schmid, Annu. Rev. Fluid Mech. 07 Gayme et al., J. Fluid Mech. 10

12 ANALYSIS OF LINEAR DYNAMICAL SYSTEMS M. JOVANOVIĆ 11 Schmid, Annu. Rev. Fluid Mech. 07

13 state equation: output equation: Solution to state equation ψ(t) = e At ψ(0) + ψ(t) = A ψ(t) + B d(t) φ(t) = C ψ(t) t 0 e A(t τ) B d(τ) dτ M. JOVANOVIĆ 12 State-space representation unforced response forced response

14 M. JOVANOVIĆ 13 Transform techniques ψ(t) = A ψ(t) + B d(t) Laplace transform ψ(t) = e At ψ(0) + t 0 s ˆψ(s) ψ(0) = A ˆψ(s) + B ˆd(s) e A(t τ) B d(τ) dτ ˆψ(s) = (si A) 1 ψ(0) + (si A) 1 B ˆd(s)

15 M. JOVANOVIĆ 14 Natural and forced responses Unforced response Forced response matrix exponential ψ(t) = e A t ψ(0) impulse response H(t) = C e A t B resolvent ˆψ(s) = (si A) 1 ψ(0) transfer function H(s) = C (si A) 1 B Response to arbitrary inputs φ(t) = t 0 H(t τ) d(τ) dτ Laplace transform ˆφ(s) = H(s) ˆd(s)

16 UNFORCED RESPONSES M. JOVANOVIĆ 15

17 M. JOVANOVIĆ 16 Systems with non-normal A ψ(t) = A ψ(t) Non-normal operator: doesn t commute with its adjoint E-value decomposition of A A A A A

18 M. JOVANOVIĆ 17 Let A have a full set of linearly independent e-vectors normal A: unitarily diagonalizable A = V Λ V

19 choose w i such that w i v j = δ ij M. JOVANOVIĆ 18 E-value decomposition of A

20 M. JOVANOVIĆ 19 Use V and W to diagonalize A solution to ψ(t) = A ψ(t) ψ(t) = e A t ψ(0) = n e λit v i w i, ψ(0) i = 1

21 identify initial conditions with simple responses ψ(t) = n i = 1 e λ it v i w i, ψ(0) ψ(0) = v k ψ(t) = e λ kt v k M. JOVANOVIĆ 20 Right e-vectors

22 { v 1 = { w 1 = [ k 2 k [ 1 + k 2 0 ], v 2 = ], w 2 = [ 0 1 [ k 1 solution to ψ(t) = A ψ(t): ]} ]} ψ(t) = ( e t v 1 w 1 + e 2t v 2 w 2) ψ(0) M. JOVANOVIĆ 21 E-value decomposition of A = [ 1 0 k 2 ] [ ψ1 (t) ψ 2 (t) ] = [ e t ψ 1 (0) k ( e t e 2t) ψ 1 (0) + e 2t ψ 2 (0) ] E-values: misleading measures of transient response

23 FORCED RESPONSES M. JOVANOVIĆ 22

24 M. JOVANOVIĆ 23 Amplification of disturbances Harmonic forcing d(t) = ˆd(ω) e iωt Frequency response steady-state response ˆφ(ω) = C (iωi A) 1 B }{{} H(ω) φ(t) = ˆφ(ω) e iωt ˆd(ω) example: 3 inputs, 2 outputs [ ˆφ1 (ω) ˆφ 2 (ω) ] = [ H11 (ω) H 12 (ω) H 13 (ω) H 21 (ω) H 22 (ω) H 23 (ω) ] ˆd1 (ω) ˆd 2 (ω) ˆd 3 (ω) H ij (ω) response from jth input to ith output

25 M. JOVANOVIĆ 24 Input-output gains Determined by singular values of H(ω) left and right singular vectors: H(ω)H (ω) u i (ω) = σ 2 i (ω) u i(ω) H (ω)h(ω) v i (ω) = σ 2 i (ω) v i(ω) {u i } orthonormal basis of output space {v i } orthonormal basis of input space

26 Right singular vectors ˆφ(ω) = H(ω) ˆd(ω) = r i = 1 identify input directions with simple responses ˆφ(ω) = σ 1 (ω) σ 2 (ω) > 0 r i = 1 σ i (ω) u i (ω) v i (ω), ˆd(ω) σ i (ω) u i (ω) v i (ω), ˆd(ω) ˆd(ω) = vk (ω) M. JOVANOVIĆ 25 Action of H(ω) on ˆd(ω) ˆφ(ω) = σ k (ω) u k (ω) σ 1 (ω): the largest amplification at any frequency

27 G = H 2 σ 2 1 (H(ω)) = max output energy input energy = max ω σ2 1 (H(ω)) M. JOVANOVIĆ 26 Worst case amplification H norm: an induced L 2 gain (of a system)

28 M. JOVANOVIĆ 27 Robustness interpretation ˆd Nominal Linearized Dynamics Γ modeling uncertainty (can be nonlinear or time-varying) ˆφ Closely related to pseudospectra of linear operators ψ(t) = (A + B Γ C) ψ(t) LARGE worst case amplification small stability margins

29 M. JOVANOVIĆ 28 ROBUSTNESS [ ] ψ 1 ψ 2 d 1 s + λ 1 Back to a toy example = [ ] [ ] λ1 0 ψ1 k λ 2 ψ 2 G = max ω H(iω) 2 = ψ 1 k γ modeling uncertainty 1 s + λ 2 k 2 + (λ 1 λ 2 ) 2 ψ 2 [ 1 0 ] d stability γ < λ 1 λ 2 /k det ([ s 0 0 s ] [ ]) λ1 γ k λ 2 = s 2 + (λ 1 + λ 2 ) s + (λ 1 λ 2 γ k) }{{} > 0

30 M. JOVANOVIĆ 29 Response to stochastic forcing White-in-time forcing Hilbert-Schmidt norm E (d(t 1 ) d (t 2 )) = I δ(t 1 t 2 ) power spectral density: H(ω) 2 HS = trace (H(ω) H (ω)) = r i = 1 σ 2 i (ω) H 2 norm variance amplification: H 2 2 = 1 2 π H(ω) 2 HS dω = 0 H(t) 2 HS dt

31 M. JOVANOVIĆ 30 Computation of H 2 and H norms H 2 norm Lyapunov equation E (d(t 1 ) d (t 2 )) = W δ(t 1 t 2 ) ψ(t) = A ψ(t) + B d(t) φ(t) = C ψ(t) { A P + P A H 2 2 = trace (C P C ) = B W B H norm E-value decomposition of Hamiltonian in conjunction with bisection H γ [ 1 A γ B B 1 γ C C A ] has at least one imaginary e-value

32 BACK TO FLUIDS M. JOVANOVIĆ 31

33 M. JOVANOVIĆ 32 Frequency response: channel flow harmonic forcing: d(x, y, z, t) = ˆd(k x, y, k z, ω) e i(k xx + k z z + ωt) steady-state response v(x, y, z, t) = ˆv(k x, y, k z, ω) e i(k xx + k z z + ωt) Frequency response: operator in y ˆd(k x, y, k z, ω) H(k x, k z, ω) ˆv(k x, y, k z, ω) componentwise amplification u v w = H u1 H u2 H u3 H v1 H v2 H v3 H w1 H w2 H w3 d 1 d 2 d 3 Jovanović & Bamieh, J. Fluid Mech. 05

34 kx kx M. JOVANOVIĆ 33 kx k z k z k z

35 M. JOVANOVIĆ 34 Amplification mechanism in flows with high Re HIGHEST AMPLIFICATION: (d 2, d 3 ) u normal/spanwise forcing glorified diffusion (si 1 2 ) 1 vortex tilting ReAcp LINEARIZED DYNAMICS OF NORMAL VORTICITY η viscous dissipation (si ) 1 streamwise velocity η = η + Re A cp v source

36 wall-normal direction M. JOVANOVIĆ 35 AMPLIFICATION MECHANISM: vortex tilting or lift-up spanwise direction

37 Has much of the phenomenology of boundary AMPLIFICATION: STABILITY: la v = H d singular values of H ψ = A ψ Transitional (bypass) and fully turbulent cohe e-values of A very elongated in streamwise direction Simpler than full 3D models M. JOVANOVIĆ 36 Linear analyses: Input-output vs. Stability typical structures cross sectional dynamics sta typical structures cross-sectional dynamics 2D models

38 Objective FLOW CONTROL controlling the onset of turbulence Transition initiated by M. JOVANOVIĆ 37 high flow sensitivity Control strategy reduce flow sensitivity

39 riblets surface roughness super-hydrophobic surfaces BODY FORCES temporally/spatially oscillatory forces traveling waves M. JOVANOVIĆ 38 Sensor-free flow control GEOMETRY MODIFICATIONS WALL OSCILLATIONS transverse wall oscillations common theme: PDEs with spatially or temporally periodic coefficients

40 NORMAL VELOCITY: V (y = ±1) = α cos ( ω x (x ct) ) TRAVELING WAVE PARAMETERS: M. JOVANOVIĆ 39 Blowing and suction along the walls spatial frequency: speed: amplitude: ω x c α { c > 0 downstream c < 0 upstream INVESTIGATE THE EFFECTS OF c, ω x, α ON: base flow cost of control onset of turbulence

41 M. JOVANOVIĆ 40 Min, Kang, Speyer, Kim, J. Fluid Mech. 06 drag SUSTAINED SUB-LAMINAR DRAG THIS TALK: cost of control onset of turbulence time CHALLENGE: selection of wave parameters

42 M. JOVANOVIĆ 41 Effects of blowing and suction? bulk flux net efficiency fluctuations energy TRAVELING WAVE induces a bulk flux (pumping) PUMPING DIRECTION DESIRED EFFECTS OF CONTROL: opposite to a traveling wave direction

43 M. JOVANOVIĆ 42 Nominal velocity V (y = ±1) = α cos (ω x (x ct)) = α cos (ω x x) SMALL AMPLITUDE BLOWING/SUCTION weakly-nonlinear analysis ū = (U( x, y), V ( x, y), 0) steady in a traveling wave frame periodic in x U( x, y) = parabola {}}{ mean {}} drift { U 0 (y) + α 2 U 2,0 (y) + α oscillatory: no mean drift {}}{ ( U1, 1 (y) e iω x x + U 1,1 (y) e iω x x ) + α 2 ( U 2, 2 (y) e 2iω x x + U 2,2 (y) e 2iω x x ) + O(α 3 )

44 M. JOVANOVIĆ 43 Best-case scenario for net efficiency ASSUME: { no control: laminar with control: laminar { no control: turbulent ASSUME: with control: laminar

45 M. JOVANOVIĆ 44 Velocity fluctuations: DNS preview Lieu, Moarref, Jovanović, J. Fluid Mech. 10

46 M. JOVANOVIĆ 45 Ensemble average energy density: controlled flow EVOLUTION MODEL: linearization around (U( x, y), V ( x, y), 0) periodic coefficients in x = x ct ψ t = A ψ + B d v = C ψ } d = d( x, y, z, t) stochastic body forcing v = (u, v, w) velocity fluctuations ψ = (v, η) normal velocity/vorticity

47 Simulation-free approach to determining energy density effect of small wave amplitude: Moarref & Jovanović, J. Fluid Mech. 10 energy density = energy density + }{{} α 2 E 2 (θ, k z ; Re; ω x, c) + O(α 4 ) small with control w/o control M. JOVANOVIĆ 46 (θ, k z ) spatial wavenumbers

48 M. JOVANOVIĆ 47 Energy amplification: controlled flow with Re = 2000 explicit formula: energy density with control energy density w/o control (θ = 0, k z = 1.78): most energy w/o control g 2, downstream: 1 + α 2 g 2 (θ, k z ; ω x, c) g 2, upstream:

49 M. JOVANOVIĆ 48 Recap facts revealed by perturbation analysis: Blowing/Suction Type Nominal flow analysis Energy amplification analysis Downstream reduce bulk flux reduce amplification Upstream increase bulk flux promote amplification Moarref & Jovanović, J. Fluid Mech. 10

50 M. JOVANOVIĆ 49 DNS results: avoidance/promotion of turbulence DOWNSTREAM: NO TURBULENCE small initial energy (flow with no control stays laminar) UPSTREAM: PROMOTES TURBULENCE

51 DOWNSTREAM moderate initial energy M. JOVANOVIĆ 50 NO TURBULENCE:

52 UPSTREAM moderate initial energy M. JOVANOVIĆ 51 TURBULENCE:

53 OPTIMAL CONTROL AND ESTIMATION M. JOVANOVIĆ 52

54 M. JOVANOVIĆ 53 Linear Quadratic Regulator (LQR) Minimize quadratic objective subject to linear dynamic constraint minimize J(ψ, u) = 1 2 T subject to A ψ(t) + B u(t) ψ(t) = 0 ψ(0) = ψ 0, t [0, T ] optimization variable is a function 0 ( ψ(τ), Q ψ(τ) + u(τ), R u(τ) ) dτ ψ(t ), Q T ψ(t ) u: [0, T ] H u state and control weights { Q, QT self-adjoint, non-negative R self-adjoint, positive infinite number of constraints

55 Introduce Lagrangian L(ψ, u, λ) = J(ψ, u) + form variations wrt ψ, u, λ T L(ψ, u + ũ, λ) L(ψ, u, λ) = 0 λ(τ), A ψ(τ) + B u(τ) ψ(τ) dτ T 0 R u(τ) + B λ(τ), ũ(τ) dτ = 0 has to hold for all ũ M. JOVANOVIĆ 54 u(t) = R 1 B λ(t), t [0, T ] necessary conditions for optimality: wrt λ ψ(t) = A ψ(t) + B u(t), ψ(0) = ψ 0 wrt ψ λ(t) = Q ψ(t) A λ(t), λ(t ) = Q T ψ(t ) wrt u u(t) = R 1 B λ(t), t [0, T ]

56 M. JOVANOVIĆ 55 Solution to finite horizon LQR [ ψ(t) λ(t) [ ψ0 0 Differential Riccati Equation ] ] two-point boundary value problem: [ ] [ ] A B R = 1 B ψ(t) Q A λ(t) [ ] [ ] [ ] [ I 0 ψ(0) 0 0 ψ(t ) = λ(0) I λ(t ) u(t) = R 1 B λ(t) can show: Q T λ(t) = X(t) ψ(t) ] Ẋ(t) = A X(t) + X(t) A + Q X(t) B R 1 B X(t) X(T ) = Q T optimal controller: determined by state-feedback u(t) = K(t) ψ(t) K(t) = R 1 B X(t)

57 minimize J = 1 2 subject to Optimal controller: 0 ( ψ(τ), Q ψ(τ) + u(τ), R u(τ) ) dτ ψ(t) = A ψ(t) + B u(t) { u(t) = K ψ(t) K = R 1 B X M. JOVANOVIĆ 56 Infinite horizon LQR X = X non-negative solution to Algebraic Riccati Equation (ARE) A X + X A + Q X B R 1 B X = 0 (A, B) (A, Q) stabilizable detectable } stability of ψ(t) = (A B K) ψ(t)

58 M. JOVANOVIĆ 57 Scalar example Optimal controller k lqr = a + J = 1 2 a 2 + q r 0 ( q ψ 2 (τ) + r u 2 (τ) ) dτ ( ψ(t) = exp a 2 + qr ) t ψ(0) ψ = a ψ + u tradeoff: large q/r small q/r convergence rate fast slow control effort large low

59 M. JOVANOVIĆ 58 State-feedback H 2 controller minimize subject to Minimum variance controller lim E( ψ(t), Q ψ(t) + u(t), R u(t) ) t ψ(t) = A ψ(t) + B d d(t) + B u u(t) E (d(t 1 ) d (t 2 )) = W d δ(t 1 t 2 ) state-feedback controller: u(t) = K ψ(t) K = R 1 B u X 0 = A X + X A + Q X B u R 1 B u X

60 M. JOVANOVIĆ 59 State estimation state equation: measured output: Estimator (observer) ψ(t) = A ψ(t) + B d d(t) + B u u(t) ϕ(t) = C ψ(t) + n(t) d(t) process disturbance; n(t) measurement noise copy of the system + linear injection term ˆψ(t) = A ˆψ(t) + 0 d(t) + B u u(t) + L (ϕ(t) ˆϕ(t)) ˆϕ(t) = C ˆψ(t) + 0 n(t) estimation error: ψ(t) = ψ(t) ˆψ(t) ψ(t) = (A L C) ψ(t) + [ B d ϕ(t) = C ψ(t) + n(t) L ] [ d(t) n(t) ] (A, C): detectable can design L to provide stability of the error dynamics

61 M. JOVANOVIĆ 60 Kalman filter ϕ(t) = C ψ(t) + n(t) E (d(t 1 ) d (t 2 )) = W d δ(t 1 t 2 ); E (n(t 1 ) n (t 2 )) = W n δ(t 1 t 2 ) Kalman filter: optimal estimator minimizes steady-state variance of ψ(t) = ψ(t) ˆψ(t) ψ(t) = A ψ(t) + B d d(t) + B u u(t) L = Y C W 1 n Kalman gain: 0 = A Y + Y A + B d W d B d Y C W 1 n C Y

62 M. JOVANOVIĆ 61 Output-feedback H 2 controller minimize subject to lim E( ψ(t), Q ψ(t) + u(t), R u(t) ) t ψ(t) = A ψ(t) + B d d(t) + B u u(t) ϕ(t) = C ψ(t) + n(t) E (d(t 1 ) d (t 2 )) = W d δ(t 1 t 2 ); E (n(t 1 ) n (t 2 )) = W n δ(t 1 t 2 ) Minimum variance controller observer-based controller: ˆψ(t) = (A L C) ˆψ(t) + B u u(t) + L ϕ(t) u(t) = K ˆψ(t) feedback and observer gains: { K LQR gain L Kalman gain

63 M. JOVANOVIĆ 62 H controller finite energy inputs Modeling Uncertainty Nominal Plant performance outputs BLENDS CLASSICAL WITH OPTIMAL CONTROL control inputs measured outputs Controller

64 M. JOVANOVIĆ 63 Boundary actuation φ t (y, t) = φ yy (y, t) + d(y, t) φ( 1, t) = u(t) φ(+1, t) = 0 Problem: control doesn t enter additively into the equation Coordinate transformation Example: heat equation Choose f(y) to obtain ψ(±1, t) = 0 Many possible choices Conditions for selection of f: ψ(y, t) = φ(y, t) f(y) u(t) {f( 1) = 1, f(1) = 0} simple option f(y) = 1 y 2

65 In new coordinates: φ t (y, t) = φ yy (y, t) + d(y, t) φ( 1, t) = u(t) φ(+1, t) = 0 φ(y,t) = ψ(y,t) + f(y) u(t) ψ t (y, t) + f(y) u(t) = ψ yy (y, t) + f (y) u(t) + d(y, t) ψ(±1, t) = 0 M. JOVANOVIĆ 64 New input: v(t) = u(t) d dt [ ] [ ] [ ψ(t) A0 f = ψ(t) u(t) 0 0 u(t) φ(t) = [ I f ] [ ] ψ(t) u(t) ] + [ I 0 ] d(t) + [ f I ] v(t) A 0 = d2 dy 2 with Dirichlet BCs

66 v(x, ±1, z, t) = K ± v (x ξ, y, z ζ) v(ξ, y, ζ, t) dy dξ dζ + K ± η (x ξ, y, z ζ) η(ξ, y, ζ, t) dy dξ dζ Optimal controller: exponentially decaying convolution kernels K v (0 ξ, y, 0 ζ): M. JOVANOVIĆ 65 Blowing and suction along the walls Högberg, Bewley, Henningson, J. Fluid Mech. 03

67 DNS verification M. JOVANOVIĆ 66 Optimal localized control Blowing and suction along the discrete lattice Moarref, Lieu, Jovanović, CTR Summer Program 2010

68 M. JOVANOVIĆ 67 Sparsity-promoting optimal control Strike balance between quadratic performance and sparsity of K minimize J(K) + γ card (K) variance amplification card (K) number of non-zero elements of K sparsity-promoting penalty function K = γ > 0 quadratic performance vs. sparsity tradeoff card (K) = 8 Lin, Fardad, Jovanović, IEEE TAC 11 (conditionally accepted; arxiv: v1)

69 SUMMARY AND OUTLOOK M. JOVANOVIĆ 68

70 M. JOVANOVIĆ 69 Summary: transition INPUT-OUTPUT ANALYSIS quantifies flow sensitivity reveals distinct mechanisms for subcritical transition streamwise streaks, oblique waves, TS-waves exemplifies the importance of streamwise elongated flow structures LATER STAGES OF TRANSITION Self-Sustaining Process (SSP) Jovanović & Bamieh, J. Fluid Mech. 05 challenge: relative roles of flow sensitivity and nonlinearity advection of mean shear Streaks O(1) instability of U(y,z) Streamwise Rolls O(1/R) nonlinear self interaction Streak wave mode (3D) O(1/R) Waleffe, Phys. Fluids 97 Farrell & Ioannou, CTR Summer Program 12 WKH 1993, HKW 1995, W 1995, 1997

71 M. JOVANOVIĆ 70 Summary: sensor-free flow control CONTROLLING THE ONSET OF TURBULENCE facts revealed by perturbation analysis: Blowing/Suction Type Nominal flow analysis Energy amplification analysis Downstream reduce bulk flux reduce amplification Upstream increase bulk flux promote amplification POWERFUL SIMULATION-FREE APPROACH TO PREDICTING FULL-SCALE RESULTS DNS verification Moarref & Jovanović, J. Fluid Mech. 10 Lieu, Moarref, Jovanović, J. Fluid Mech. 10

72 M. JOVANOVIĆ 71 Outlook: model-based sensor-free flow control GEOMETRY MODIFICATIONS WALL OSCILLATIONS BODY FORCES riblets super-hydrophobic surfaces transverse oscillations oscillatory forces traveling waves USE DEVELOPED THEORY TO DESIGN GEOMETRIES AND WAVEFORMS FOR control of transition/skin-friction drag reduction CHALLENGES control-oriented modeling of turbulent flows optimal design of periodic waveforms Flow disturbances Spatially Invariant PDE Fluctuations energy + Cost of control Spatially Periodic Multiplication

73 CONTROL OF TURBULENT FLOWS model-based control design control-oriented modeling flow structures M. JOVANOVIĆ 72 Moarref & Jovanović, J. Fluid Mech. 12 (in press; arxiv: )

74 M. JOVANOVIĆ 73 Acknowledgments TEAM: SUPPORT: Rashad Moarref Binh Lieu Armin Zare (Caltech) (U of M) (U of M) NSF CAREER Award CMMI NSF Award CMMI U of M IREE Early Career Award CTR Summer Programs 06, 10, 12 COMPUTING RESOURCES: Minnesota Supercomputing Institute SPECIAL THANKS: Prof. Moin