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 @ Santa Barbara joint work with Mohammed Dahleh Mihailo Jovanovic

Motivation and Outline Two basic problems Understanding transition from laminar to turbulent flow Hydrodynamic Stability Understanding the structure of turbulent flow (statistics, coherent structures, averaged quantities) Considerations of uncertainty and robustness in the Navier-Stokes equations Extreme sensitivity of the Navier-Stokes equations in streamlined geometries Using the tools of Robust Control Theory to analyze this sensitivty Linearized Navier-Stokes equations with signal and system uncertainties Exact models vs. uncertain models: Robust Linear Multivariable Systems Theory seems to provide the missing link in shear flow turbulence.

Key Points 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, in several stages or at once The mathematical models Transition from one stage to the next is often studied as a problem of the dynamical instability of the solutions of the NS equations successful in many cases, e.g. Benard convection, Taylor-Coutte flow, curved plates, etc.. In important cases, e.g. shear flows in streamlined geometries instability is too narrow a concept to capture the notion of transition Example of a system with Highly Optimized Tolerance (HOT)

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) 3D velocity fields The central question: Given a laminar flow, is it stable? p(x, y, z, t) pressure field laminar flow := a stationary solution of the Navier-Stokes equations Decompose the fields as u = ū + ũ laminar fluctuations Fluctuation dynamics: t ũ = ūũ ũū grad p + ũ ũũ 0 = div ũ In linear hydrodynamic stability, the quadratic term is ignored

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) 5

The Evolution Model Can rewrite linearized fluctuation equations using only two fields v := wall-normal velocity ω := wall-normal vorticity := u z w x. Equivalent equations: y 1. 1 z x... v w u 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, the evolution equation is written as t Ψ = A Ψ where A generates the evolution operator e ta Classical linear hydrodynamic stability: Transition instability A has eigenvalues in right half plane existence of exponentially growing normal modes 6

Problems with the theory R c : The critical Reynolds number at which instability occurs For R R c there are unstable eigenvalues of A with corresponding eigenfunctions the corresponding eigenfunctions are the flow structures that grow exponentialy at transition Classical linear hydrodynamic stability is enormously 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 5772 1000 Couette 350 Pipe 2200 Experimental R c is highly dependent on conditions such as wall roughness, external disturbances, etc... 7

Problems with the theory (cont.) The second faliure of classical linear hdrodynamic stability theory: does not predict the commonly experimentally observed flow structures at transtion Classical theory predicts Tollmien-Schlichting waves in Poiseuille and boundary layer flows: Except in very noise-free and controlled experiments, flow structures are transition are more like turbulent spots and streaky boundary layers: 8

The emerging new theory (Farrell, Ioannou, Butler, Henningson, Reddy, Trefethen, Driscoll, Gustavsson, 89-present) Even though the systems are stable (subcritical) Large transients (large H 2 norms) Large frequency singular value plots (large H norms) Small stability margins with respect to unmodeled dynamics (Essentially a small gain analysis, termed pseudo-spectra ) Small stability margins with respect to perturbations in boundary values Large amplification of disturbance variances (large H 2 norms) Key idea Use robust measures like norms, stability margins, etc. rather than eigenvalues Connections with Robust Control Theory: In the past two decades, Robust Control Theory has been concerned with quantifying uncertainty and the Robustness of systems. 9

The Evolution Model [ v t 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 ] Observation: A is translation invariant in x, z (but not in y!). Fourier transform in x and z: [ ] [ ( ˆv ikx = 1 U +ik x 1 U + 1 R 1 2) ][ ] ( 0 t ˆω ( 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, kz,k = A(k t ˆω(t, k z,k x ) 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] Fact: A generates a stable evolution if and only if for all k x,k z, A(k x,k z ) is stable. 10

Transient energy growth of perturbed flow fields The energy density of a perturbation for a given k x,k z is 1 2π/kx 2π/kz E = k xk z (u 2 + v 2 + w 2 ) dz dx dy 16π 2 1 0 0 which can be rewritten as a quadratic form on the normal velocity and vorticity fields as: [ ] [ ] ˆv I 1 2 [ ] 0 k E =, x 2 ˆω +k2 z y 2 ˆv 1 0 k I =: Ψ, Q Ψ. ˆω x 2+k2 z The evolution of the disturbance s energy } Ψ(t) 2 = e At Ψ(0) 2 = Ψ(0), {e A t Qe At Ψ(0) Recently, research in this field has been motivated by the following distinction: 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... 89-present) 12

Transient energy growth of perturbed flow fields (Cont.) It turns out: Convection flows (linearized) have normal generators (typical when body forces are dominant) Thus, eigenvalue locations give a reasonable measure of stability margins. Strongly sheared flows have highly non-normal generators Thus, in non-normal problems, conclusions based on eigenvalue locations can be very misleading Hydrodynamic stability without eigenvalues, (Trefethen et.al., Science 93) Extreme sensitivity to external forcing and/or unmodelled dynamics Must explicitly treat uncertainty 13

The Input-Output View Allow for external excitation into the dynamics: [ ] [ ( v = 1 U x + 1 ( U ) x + 1 R 1 2) ( 0 t ω U z U x + 1 R ) ][ v ω ] + B [ dv d ω ] Sources of external disturbance d: Disturbance body forces (free stream disturbances, thermal fluctuations, etc.) Non-smooth geometries (rough walls, uncertain laminar flow profiles) Neglected nonlinearity Unmodeled dynamics (a small gain -type robust stability analysis) 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) 14

Input-Output vs. Modal Analysis A simple example: given a finite-dimensional Single-Input Single-Output system ẋ = Ax + Bu H(s) = C(sI A) 1 B y = Cx 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 by choosing any of the N k s large enough i=1 α n,i (s z n ) i Im(s) H(s) X X X Re(s) i.e.: No connection between underdamped modes of A and peaks of frequency response H(jω)

Spatio-temporal frequency response t Ψ(k x,k z ) = A(k x,k z )Ψ(k x,k z ) + B(k x,k z ) d(k x,k z ) Fourier transform in time Ψ(ω, k x,k z ) = H(ω, k x,k z ) d(k x,k z ) The operator-valued spatio-temporal frequency response H(ω, k x,k z ) amimo(iny) frequency response of several frequency variables not allways straight forward to visualize, has lots of information

Dominance of Stream-wise Contstant Flow Structures λ max ( ) H(ω, k x,k z )H (ω, k x,k z ) dω averaging out time, and taking σ max in y direction Poiseuille flow at R=2000. 17

Stochastically Forced Linearized N-S (Farrell & Ioannou 93, 96) At a fixed k x,k z,consider t Ψ = A Ψ + d, where d is a white (spatially and temporally) stationary random field. The correlation operator of Ψ is V := E{Ψ(t) Ψ(t)} = eāteā t dt, 0 where V solves: AV + V A = I. Note: The quantity {trace(v )} is the H 2 norm of the system from d to Ψ. For 3D shear flows this quantity is found to be surprisingly large. e.g. for Couette and Poiseuille flow @ R=2000, trace(v )iso(1000) trace(v ) increases with Reynolds number The above implies that random disturbances in the flow are amplified by orders of magnitude by the system s dynamics Basic mechanism: energy transfer from mean flow to perturbed flow 19

Stochastically Forced Linearized N-S (cont.) trace(v )isreferredtoastheensemble Average Energy of flow perturbations Farrell & Ioannou 93, observed through extensive numerical computations (for both Poiseuille and Couette flows): Ensemble Average Energy growth vs. Reynolds number (R) For small R, trace(v ) R For large R, trace(v ) R 3 (True only for 3D disturbances) Note: This is in stark contrast to 2D disturbances where R 3 growth is not possible! Worst case growth occurs at k x =0,andk z 0 i.e. Dominant flow structures are elongated in the stream-wise direction! (stream-wise vortices and streaks) Theorem: (Bamieh & Dahleh, 98) For parallel shear flows, stream-wise constant disturbances achieve O(R 3 ) energy growth. More precisely, at k x =0: trace(v ) = f 1 (k z ) R + f 2 (k z ) R 3 In the case of Couette flow, f 1, f 2 can be calculated analytically. 20

The Impulse Response of the Linearized Model t Ψ = A Ψ + d, with d of the form d(x, y, z, t) = δ(x, z, t)f(y), f(y) :an approximate impulse localized near lower wall approximates a body force impulse near lower wall y 1. 1 z x... v w u Response should approximate the evolution of so-called turbulent spots 21

The Impulse Response of the Linearized Model (cont.) isosurface plot of stream-wise velocity showing high speed (red) and low speed (green) streaks 23

Remarks The impulse response of the linearized NS equations has many of the qualitative features of real turbulent spots

Remarks The impulse response of the linearized NS equations has many of the qualitative features of real turbulent spots Need to develop correspondences between multi-dimensional spatio-temporal frequency response plots and system properties (like intuition from bode plots in ODEs) An infinite dimensional linear system can have very rich behavior! Complex behavior Non-linearity That s because we know of low-order non-linear models with complex behavior

Turbulent boundary layers Forced linearized model seems to be the proper model for the near-wall viscous sublayer in a turbulent boundary layer This models contains the streamwise vortices and streaks found in turbulent boundary layers Can compute the second order statistics for the forced linearized equations (from solution of operator Lyapunov equation) These correllations can be used in closure schemes for averaged equations of fully turbulent flows (with Marsden & Shkoller) using these correlation operators in the α-averaged equations (they seem to contain the right kind of non-isotropy)

Active and passive control ofboundary layers The forced linearized NS equations seem to provide an excellent control-oriented model for boundary layer turbulence suppression. Passive control with regular wall geometries (Ribletts) Passive control with moving walls (compliant walls), and flutter Two groups working on laminarization of turbulent 3D channel flow with wall transpirations (blowing/suction) (simulations) Kang, Cortelezzi, Speyer, Kim Bewley et. al. using techniques of LQG, H 2 and H. More complex systems (MEMS devices)