Understanding and Controlling Turbulent Shear Flows

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1 Understanding and Controlling Turbulent Shear Flows Bassam Bamieh Department of Mechanical Engineering University of California, Santa Barbara bamieh 23rd Benelux Meeting on Systems and Controls, March 04

2 The phenomenon of turbulence Example: Flow behind a cylinder at increasing velocities Re = 0.16 Re = 13.1 Re = 2,000 Re = 10,000 Flow direction = In nature: low altitude cloud formation behind an island Von Karman vortex street ( organized turbulence) 1

3 The phenomenon of turbulence (Cont.) Technologically important flows: Flows past streamlined bodies wings channels pipes Wall-bounded shear flows Friction with the walls drives the flows 2

4 The phenomenon of turbulence (Cont.) Boundary layers form in flow past any surface Idealization: flow on a flat plate Flow direction Viewed sideways 3

5 Boundary layer turbulence and skin-friction drag A laminar BL causes less drag than a turbulent BL (for same free-stream velocity) This skin-friction drag is 40-50% of total drag on typical airliner 4

layers been turbulent Shark skin has small grooves (riblets)

6 Shark and Dolphin skins Some sharks and dolphins swim much faster than their muscle mass would allow had their boundary layers been turbulent Shark skin has small grooves (riblets) Moin & Kim, 98 One can buy swim suites that try to mimic this 5

7 Shark and Dolphin skins (Cont.) Dolphins have a different mechanisms Springs Flow Plate Dolphin skin is a compliant skin Rigid base Skin is an elastic membrane that interacts with flow and suppresses the production of turbulence Difficulties: Basic mechanisms of how these skins suppress turbulence is not well understood Therefore, difficult to scale designs to airplanes and ships 6

8 Phenomena and mathematical theories The natural phenomena As parameters change (e.g. velocity) fluid flows transition from simple (laminar) to very complex (turbulent) Depending on the flow geometry transition can occur abrubtly, gradually, or in several stages The mathematical models Transition from one stage to the next dynamical instability successful in many cases, e.g. Benard convection, Taylor-Coutte flow, etc.. In important cases, e.g. shear flows in streamlined geometries... instability is too narrow a concept to capture the notion of transition Transition in general involves questions of robustness, sensitivity and stability 7

9 Hydrodynamic Stability (mathematical formulation) The incompressible Navier-Stokes equations t u = u u grad p + 1 R u 0 = div u u u(x, y, z, t) v(x, y, z, t) w(x, y, z, t) p(x, y, z, t) 3D velocity field pressure field The central question: Given a laminar flow, is it stable? laminar flow := a stationary solution of the Navier-Stokes equations Decompose the fields as u = ū + ũ laminar fluctuations 8

10 Fluctuation dynamics: t ũ = ūũ ũū grad p + ũ ũũ 0 = div ũ In linear hydrodynamic stability, ũũ is ignored 9

11 Examples: Poiseuille Flow U(x,y)=(1 y2). y x Couette Flow U(x,y)=y. y x Pipe Flow U(r)=(1 r 2 ) Blasius boundary layer y x Others: Benard Convection (Between two horizontal plates) Taylor-Couette (Flow between two concentric rotating cylinders) 10

12 The Evolution Model Can rewrite linearized fluctuation equations using only two fields v := wall-normal velocity ω := wall-normal vorticity := u z w x. 1. y 1 z x... v w u Equivalent equations: t [ v ω t [ v ω ] ] = = [ ( 1 U x + 1 U x + 1 R 1 2) ][ ( 0 v ( U z ) U x + 1 R ) ω [ ][ ] [ ] L 0 v v =: A C S ω ω ] Abstractly, t Ψ = A Ψ 11

13 Classical linear hydrodynamic stability: Transition instability A has spectrum in right half plane existence of exponentially growing normal modes a modal instability 12

14 The Eigenvalue Problem t [ v ω t [ v ω ] ] = = [ ( 1 U x + 1 ( U ) x + 1 R 1 2) ( 0 U z U x + 1 R [ ][ ] [ ] ) L 0 v v = A C S ω ω ][ v ω Remark: A translation invariant in x, z (but not in y!). Fourier transform in x and z: [ ] [ ( ikx = t ˆvˆω 1 U +ik x 1 U + 1 R 1 2) ][ ] ( 0 ( ik z U ) ikx U + 1 R ˆvˆω ) k x,k z : spatial frequencies in x, z directions (wave-numbers). [ ] [ ] ˆv(t, kz,k x ) ˆv(t, = t ˆω(t, k z,k x ) Â(k kz,k x,k z ) x ), ˆv(t, k ˆω(t, k z,k x ) z,k x ), ˆω(t, k z,k x ) L 2 [ 1, 1] ] Essentially: spectrum (A) = spectrum (Â(kx,k z )) k x,k z 13

15 Tollmien-Schlichting Instability Example: Poiseuille flow at R =6000, k x =1, k z =0has the following eigenvalues: imaginary part real part Typical stability regions in K, R space: (Poiseuille, Blasius boundary layer flows) Unstable eigenvalue corresponds to a slowly growing travelling wave: the Tollmien-Schlichting wave Tollmien 29, Schlichting 33. (also occurs in boundary layer flows) First seen in experiments by Skramstad & Schubauer,

16 Problems with the theory (1st difficulty) R c : The critical Reynolds number at which instability occurs R R c unstable eigenvalues of A corresponding eigenfunctions are flow structures that grow exponentialy at transition Classical linear hydrodynamic stability is successful in many problems, e.g. Benard Convection Taylor-Couette Flow (Flow between two rotating concentric cylinders) Classical linear hydrodynamic stability fails badly in a very important special case Shear flows: (e.g. flows in channels, pipes, and boundary layers) Flow type Classical linear theory R c Experimental R c Poiseuille Couette 350 Pipe 2200 Experimental R c is highly dependent on conditions such as wall roughness, external disturbances, etc... 15

Problems with the theory (2nd difficulty) Second failure:

Classical theory predicts Tollmien-Schlichting waves in

noise-free and controlled experiments, flow structures in

17 Problems with the theory (2nd difficulty) Second failure: Incorrect prediction of natural transition flow structures Classical theory predicts Tollmien-Schlichting waves in Poiseuille and boundary layer flows: Except in very noise-free and controlled experiments, flow structures in transition are more like turbulent spots and streaky boundary layers: 16

18 Boundary layer with free-stream disturbances (a) (b) (c) (d) From Matsubara & Alfredsson,

19 Transition Scenarios 1. Infinitesimal disturbances = TS-wave = 3D Secondary instabilities = Transition Demonstrable in clean experiments 2. Infinitesimal disturbances = Occurs when small amounts of noise is present Stream-wise vortices and streaks = Transition Possible explanation: noisy environments big disturbances non-linear effects dominate 18

20 The emerging new theory Since 89, a new mathematical approach to linear hydrodynamic stability emerged (Farrell, Ioannou, Butler, Henningson, Reddy, Trefethen, Driscoll, Gustavsson,...) Recent book: Schmidt & Henningson Transient growth Pseudo-spectrum Noise amplification Amazing similarity to notions from Robust Control Even though systems are stable (subcritical), they have Large transient growth Large input-output norms Small stability margins BASICALLY: Use robustness analysis, rather than eigenvalues to quantify transition Transition is no longer on/off Significant implications for control-oriented modelling 19

21 The Evolution Model t [ v ω t [ v ω ] ] = = [ ( 1 U x + 1 ( U ) x + 1 R 1 2) ( 0 U z U x + 1 R [ ][ ] [ ] ) L 0 v v. = A C S ω ω y 1. z. x 1. ][ ] v ω v w u After Fourier transform: [ ] [ ( ikx = t ˆvˆω 1 U +ik x 1 U + 1 R 1 2) ][ ( 0 ( ik z U ) ikx U + 1 R ˆvˆω ) ] k x,k z : spatial frequencies in x, z directions (wave-numbers). 20

22 Transient energy growth of perturbed flow fields The energy density of a perturbation for a given k x,k z is E = k xk z 16π π/kx 0 2π/kz 0 (u 2 + v 2 + w 2 ) dz dx dy which can be rewritten as a quadratic form on the normal velocity and vorticity fields as: E = [ ˆv ˆω ], [ I 1 k 2 x +k2 z 2 y k 2 x +k2 zi ] [ ˆv ˆω ] =: Ψ, Q Ψ. 21

23 The evolution of the disturbance s energy Ψ(t) 2 = e At Ψ(0) 2 = Ψ(0), { e A t Qe At} Ψ(0) In the fluids literature, the following is emphasized: If A is stable and normal (w.r.t. Q), then e At Ψ(0) decays monotonically for t>0. If A is non-normal, then large transient energy growth is possible. (Farrell, Butler, Trefethen, Driscoll, Henningson, Reddy, et.al present) 22

24 Input-output analysis of Linearized Navier-Stokes Add forcing terms to the NS equations t ũ = ūũ ũū p + 1 R ũ + d 0 = ũ Possible sources of d NEGLECTED NONLINEAR TERMS (small gain analysis) NON-LAMINAR FLOW PROFILES (conservative) BODY FORCES NON-SMOOTH GEOMETRIES 23

25 Input-output analysis of Linearized Navier-Stokes (Cont.) In standard form t ψ = A ψ + B d t ψ = [ ] 1 U x + 1 U x + 1 R U z U x + 1 R }{{ ψ + } A [ ] 1 xy 1 ( xx + zz ) 1 yz d x d y z 0 x }{{} d z B Studied in M. Jovanovic & B. Bamieh, 02 24

26 The Input-Output View Question: Investigate the system mapping d Ψ Surprises: Even when A is stable, the mapping d Ψ has large norms (and scales badly with R) The input-output resonances are very different from the least damped modes of A (as spatio-temporal patterns) 25

27 Input-Output vs. Modal Analysis A simple example: a finite-dimensional Single-Input Single-Output system ẋ = Ax + Bu y = Cx H(s) = C(sI A) 1 B Theorem: Let z 1,..., z n be any locations in the left half of the complex plane. Any stable frequency response function in H 2 can be arbitrarily closely approximated by a transfer function of the following form: H(s) = N 1 i=1 α 1,i (s z 1 ) i + + N n i=1 α n,i (s z n ) i Im(s) by choosing any of the N k s large enough H(s) X X X Re(s) i.e.: No connection between underdamped modes of A and peaks of frequency response H(jω) 26

28 Spatio-temporal Frequency Response t Ψ(t, k x,k z ) = A(k x,k z )Ψ(k x,k z ) + B(k x,k z ) d(t, k x,k z ) Fourier transform in time ( ) Ψ(ω, k x,k z ) = (jωi A(k x,k z )) 1 B d(ω, k x,k z ) =: H(ω, k x,k z ) d(ω, k x,k z ) The operator-valued spatio-temporal frequency response H(ω, k x,k z ) is a MIMO (in y) frequency response of several frequency variables Not allways straight forward to visualize, has lots of information 27

29 λ max ( Dominance of Stream-wise Constant Flow Structures ) H(ω, k x,k z )H (ω, k x,k z ) dω averaging out time, and taking σ max in y direction = a scalar function of (k x,k z ) σ max (k x, k z ) k x k z Poiseuille flow at R=

30 Dominance of Stream-wise Constant Flow Structures (cont.) sup λ max (H(ω, k x,k z )H (ω, k x,k z )) <ω< 30 [ H ](k x,k z ) k x k z log 10 ([ H ](k x,k z )) log 10 (k x ) log 10 (k z ) 10 2 A log-log plot Poiseuille flow at R=

31 Input-output analysis of Linearized Navier-Stokes (Cont.) Most resonant flow structures are stream-wise vortices and streaks 1 y z 1. x... v u w Cross sectional view 30

32 Stream-wise vortices and streaks 31

33 Mechanism of Generation of Stream-wise Vortices and Streaks t [ ˆvˆω ] = [ ( 1 R 1 2) ][ ( 0 ( ik z U ) 1 R ˆvˆω ) ] + [ dv d ω ] =: [ 1 R L 0 C 1 R S ][ ˆvˆω ] + [ d d d w d v (si 1 R L) 1 ik z U (si 1 ω R S) x Kz Kz Kz

Figure 3: Streamwise velocity perturbation

High speed streaks are represented by red

34 Figure 3: Streamwise velocity perturbation development for largest singular value (first row) and second largest singular value (second row) of operator Ĥ at {k x =0.1, k z =2.12, ω= 0.066}, in Poiseuille flow with R = High speed streaks are represented by red color, and low speed streaks are represented by green color. Isosurfaces are taken at ±

35 Figure 8: Streamwise vorticity perturbation development for largest singular value of operator Ĥ at {k x = 0.1, k z =2.12, ω= 0.066}, in Poiseuille flow with R = High vorticity regions are represented by yellow color, and low vorticity regions are represented by blue color. Isosurfaces are taken at ±

36 Linearized NS (LNS) Equations With stochastic forcing H 2 norm of LNS scales like R 3 Amplification mechanism is 3D, streak spacing mathematically related to dynamical coupling (Bamieh & Dahleh, 99) Spatio-temporal Impulse response of LNS has features of Turbulent spots (Jovanovic & Bamieh, 01) Wall blowing/suction control in simulations of channel flow Cortelezi, Speyer, Kim, et.al. (UCLA) Bewley, Hogberg, et.al. (UCSD) Using H 2 as a problem formulation Achieved re-laminarization at low Reynolds numbers Fluid dynamics community gradually warming up to model-based control Matching channel flow statistics by input noise shaping in LNS (Jovanovic & Bamieh, 01) Corrugated surfaces (Riblets) effect on drag reduction (current work) 35

37 Basic Issue Uncertainty and robustness in the Navier-Stokes (NS) equations Streamlined geometries = Extreme sensitivity of the NS equations Transition instability (linear or non-linear). Transition (instability, fragility, sensitivity). 36

38 Uncertainty in a Dynamical System Stability analysis deals with uncertainty in initial conditions ψ(0) If x(0) isknowntobeprecisely x e, then x(t) =x e, t 0 We introduce the concept of stability because we can never be infinitely certain about the initial condition Shortcomings of stability analysis Perturbs only initial conditions Cares mostly about asymptotic behavior 37

39 Uncertainty in a Dynamical System (cont.) Stability ẋ = f(x) uncertain initial conditions investigate lim x(t) t investigate transients e.g. sup t 0 x(t) dynamical uncertainty ẋ = F (x)+ (x) exogenous disturbances ẋ(t) =F (x(t),d(t)) combinations ẋ(t) =F (x(t),d(t))+ (x(t),d(t)) More Uncertainty Linearized version: transient growth eigenvalue stability ẋ = Ax umodelled dynamics ẋ =(A + )x Psuedo-spectrum exogenous disturbances ẋ(t) =Ax(t)+Bd(t) input-output analysis combinations ẋ(t) =(A + B C)x(t)+ (F + G H)d(t) 38

40 Uncertainty in a Dynamical System (cont.) Stability ẋ = f(x) uncertain initial conditions investigate lim x(t) t investigate transients e.g. sup t 0 x(t) dynamical uncertainty ẋ = F (x)+ (x) exogenous disturbances ẋ(t) =F (x(t),d(t)) combinations ẋ(t) =F (x(t),d(t))+ (x(t),d(t)) More Uncertainty Linearized version: transient growth eigenvalue stability ẋ = Ax umodelled dynamics ẋ =(A + )x Psuedo-spectrum exogenous disturbances ẋ(t) =Ax(t)+Bd(t) input-output analysis combinations ẋ(t) =(A + B C)x(t)+ (F + G H)d(t) 39

41 The nature of turbulence Fluid dynamics are described by deterministic equations Why does fluid flow look random at high Reynolds numbers?? 40

42 The nature of turbulence (Cont.) Common view of turbulence 41

43 The nature of turbulence (cont.) Common view of turbulence Intuitive reasoning Complex, statistical looking behavior System with chaotic dynamics 42

44 The nature of turbulence (cont.) 43

45 Amplification vs. instability Most turbulent flows probably have both instabilities (leading to spatio-temporal patterns) high noise amplification The statistical nature of turbulence may be due to ambient uncertainty amplification? as opposed to chaos 44

Systems Theory and Shear Flow Turbulence Linear Multivariable Systems Theory is alive and well

Systems Theory and Shear Flow Turbulence Linear Multivariable Systems Theory is alive and well Dedicated to J. Boyd Pearson Bassam Bamieh Mechanical & Environmental Engineering University of California