Minimal transient energy growth for plane Poiseuille flow . Whidborne 1 James F John McKernan 1 George Papadakis 2 1. Department of Aerospace Sciences, Cranfield University 2. Division of Engineering, King’s College London UKACC ICC2006 – p. 1/20
Transient energy growth P ∞ Consider the asymptotically stable linear time-invariant system described by the initial value problem x = Ax , ˙ x (0) = x 0 , n o with A ∈ R n × n , x 0 ∈ R n which has the continuous solution x : R + → R n , t �→ Φ ( t ) x 0 , where Φ ( t ) is the state transition matrix given by Φ ( t ) = e A t = i =0 A i t i /i ! The transient energy is defined as � x ( t ) � 2 : � x 0 � = 1 E ( t ) := max b | 2 where | Note that E ( t ) = | | | Φ ( t ) | | | |·| | | is the spectral norm The maximum transient energy growth is defined as E := max {E ( t ) : t ≥ 0 } In practice we require appropriate weights on the states � Wx (0) � =1 � Wx ( t ) � 2 , E ( t ) = max For simplicity we assume W = I UKACC ICC2006 – p. 2/20
b b Some properties, bounds, etc • E is lower bounded by unity • E is unity if and only if A + A T ≤ 0 � � � � � � � � � S − 1 � � � is the condition number of S • e 2 α ( A ) t ≤ E ( t ) ≤ κ ( S ) 2 e 2 α ( A ) t b where α ( A ) := max i ℜ ( λ i ) is the spectral abscissa of A { λ i } are the eigenvalues of A S is the right eigenvector matrix κ ( S ) := | | | S | | | • E is unity if A is normal (eigenvectors form an orthonormal set) • E ( t ) ≤ e 2 w ( A ) t where w ( A ) := λ max ( A + A T ) / 2 is the numerical abscissa of A (also known as initial growth rate or logarithmic norm) Other bounds in Vesili´ c (2003), Hinrichsen & Pritchard (2005), Plischke (2005), etc. UKACC ICC2006 – p. 3/20
b b b Lyapunov upper bound b We can use Lyapunov functions to describe an ellipsoid that bounds the trajectory to give an upper-bound on E ( t ) E u ≥ E is an upper bound on the maximum transient energy growth, E , where E u := λ max ( P ) λ max ( P − 1 ) � � � � � � where P = P T > 0 satisfies � � � P − 1 � � � = κ ( P ) , the condition number. PA + A T P ≤ 0 (1) Note that λ max ( P ) λ max ( P − 1 ) = | | | P | | | In the sequel, we will tighten the bound given by (1) to be a strict inequality. This eliminates some numerical difficulties with very little conservativeness UKACC ICC2006 – p. 4/20
Evaluating upper bound The upper bound can be obtained by solving the following LMI generalized eigenvalue problem (GEVP) (Boyd, El Ghaoui, Feron, Balakrishnan, 1994) b b min γ PA + A T P < 0 I ≤ P ≤ γ I , subject to where P > 0 is real and symmetric. The inequality I ≤ P ≤ γI ensures that γ ≥ λ max ( P ) ≥ λ min ( P ) ≥ 1 , thus λ max ( P ) /λ min ( P ) ≤ γ and so E ≤ E u ≤ γ UKACC ICC2006 – p. 5/20
Optimal state feedback controllers Now consider the linear time-invariant plant with state feedback controller K : x = Ax + Bu , ˙ x (0) = x 0 , u = Kx , Upper bound can be minimized by solving an LMI (Boyd et al 1994) Expanding Lyapunov inequality for u = Kx gives PA + A T P + PBK + K T B T P < 0 . By the change of variable, Q = P − 1 and Y = KQ the LMI AQ + QA T + BY + Y T B T < 0 is obtained. Now since λ max ( P ) λ max ( P − 1 ) = λ max ( Q ) λ max ( Q − 1 ) , a controller that minimizes the upper bound on the maximum transient energy growth can be obtained by solving the following LMI generalized eigenvalue problem (GEVP): min γ AQ + QA T + BY + Y T B T < 0 , Q = Q T subject to I ≤ Q ≤ γ I , and the upper-bound minimizing controller is K = YQ − 1 UKACC ICC2006 – p. 6/20
Fluid flow • Laminar fluid flow characterized by a smooth flow in which adjacent layers of fluid undergo shear • Turbulent flow characterized by unsteady flow-field in which fluctuations of widely varying length and time scales cause large amounts of mixing between adjacent layers of fluid • The transition of laminar fluid flow into turbulent flow results in large increases in fluid drag, hence the prevention of transition would lead to substantial savings in the energy required to sustain the flow • The process of transition from laminar to turbulent flow is thought to begin with the rapid growth of small perturbations in the laminar flow • Reynolds number Re is ratio of inertial forces to viscous forces — high Re ⇒ turbulance UKACC ICC2006 – p. 7/20
Plane Poisieuille flow Consider plane channel (or Poisieuille) flow (unidirectional flow between two infinite parallel planes) • A simple flow that is prone to transition • Experiments show flow undergoes transition to turbulence for Reynolds number as low as 1000 • But flow is known to be linearly stable for Reynolds numbers below 5772 • The occurrence of transition in the lin- early stable regime is thought to be due to large transient growth causing non-linear effects UKACC ICC2006 – p. 8/20
Transient energy growth in Poiseuille flow From an initial state x (0) with unit kinetic energy density E (0) = x T (0) Wx (0) = 1 , large transient growth in the kinetic energy density E ( t ) of the state occurs before an eventual exponential decay of energy at the rate of the least-stable constituent eigenmode. 5000 4500 4000 3500 3000 E ( t ) 2500 2000 1500 1000 500 0 0 500 1000 1500 2000 t UKACC ICC2006 – p. 9/20
Feedback control of plane channel flow Objective is to maintain laminar flow by measuring the shear at the wall and using the controller to actively modify the boundary conditions by blowing/suction at the walls Upper Wall Wall−normal, y Streamwise, x Plane Poiseuille Spanwise, z Flow Disturbance Flow Lower Wall Actuation (Controller) Sensing UKACC ICC2006 – p. 10/20
� � Feedback control of plane channel flow • Incompressible fluid flow is described by the Navier-Stokes and the continuity equations U = − 1 ρ ∇ P + µ ˙ ρ ∇ 2 � � � � U + U · ∇ U ∇ · � U = 0 � U is velocity, P is pressure, ρ is density, µ is viscosity • Laminar flow has a parabolic stream-wise velocity profile � U b = ((1 − y 2 ) U cl , 0 , 0) , P b with no slip occurring at the bounding parallel planes • It undergoes transition to turbulence when small disturbances � � � � � u = ( u, v, w ) , p about the steady base profile, grow spatially and temporally to form a self-sustaining turbulent flow • Non-dimensionalizing the perturbation equations gives = −∇ p + 1 ˙ � u + � R ∇ 2 � u + � U b · ∇ u + ( � � u · ∇ ) � U b u ∇ · � u = 0 where R := ρU cl h/µ is the Reynolds number UKACC ICC2006 – p. 11/20
Linearized model • For control by wall transpiration, no-slip wall boundary conditions at y = ± 1 are replaced by prescribed wall transpiration velocities, ( u ( ± 1) = 0 , v ( ± 1) � = 0 , w ( ± 1) = 0) • Variations in span-wise and stream-wise directions are assumed to be periodic ℜ ( e i ( αx + βz ) ) , and flow disturbances grow in time, but not in space • Boundary control at wave numbers α and β respectively can be represented as linear state-space system in the standard form ˙ x = Ax + Bu where linearized Navier-Stokes equations are evaluated at N locations in the wall-normal direction y 1 lmi (1) ← f l 0.8 0.6 ← f u lmi (1) • State variables x are wall-normal ve- 0.4 ←Γ D last (2) locity ˜ v and vorticity, ˜ η := ∂u/∂z − 0.2 ←Γ D 1 (1.5) ∂w/∂x , perturbation Chebyshev coeffi- 0 y cients, plus the upper and lower wall v ←Γ D 0 (2) −0.2 transpiration velocities ←Γ DN last (2.1) −0.4 ←Γ DN 1 (4.6) −0.6 ←Γ DN 0 (8) −0.8 −1 0 0 0 0 0 0 0 0 UKACC ICC2006 – p. 12/20
Feedback control of plane channel flow • For this study, we assume system state can be accurately measured. • Control inputs u are the rates of change of transpiration velocity on the upper and lower walls • The test case considered here is α = 0 , β = 2 . 044 , R = 5000 • The model is discretized in the wall-normal direction with N = 20 5000 • This test case is 4500 linearly stable but 4000 has the largest 3500 linear transient en- ergy growth over 3000 all unit initial con- E ( t ) 2500 ditions, time and wave-number, and 2000 represents the very 1500 earliest stages of the transition to 1000 turbulence 500 0 0 500 1000 1500 2000 t UKACC ICC2006 – p. 13/20
b b b Minimal upper bound control Solving LMI gives upper bound E u = 1722 E = 883 Actual maximum transient energy growth is Open loop E = 4941 900 800 700 600 500 E ( t ) 400 300 200 100 0 0 500 1000 1500 t UKACC ICC2006 – p. 14/20
Worst case control signal • The worst case wall control u wc = Kx wc where x wc is the transient x ( t ) that results in the largest energy gain is shown below • Note that the second control signal is zero due to x wc being symmetrical with respect to y • The control signals are impractically large 4 x 10 6 5 4 3 u wc ( t ) 2 1 0 −1 0 500 1000 1500 t UKACC ICC2006 – p. 15/20
Constraint on control signal A constraint t ≥ 0 � u ( t ) � 2 ≤ µ 2 max can be can be included in the LMI’s. The GEVP then can be formulated as (McKernan et al 2005) min γ " # subject to I ≤ Q ≤ γ I AQ + QA T + BY + Y T B T < 0 Y T Q ≥ 0 µ 2 I Y UKACC ICC2006 – p. 16/20
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