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Minimal transient energy growth for plane Poiseuille flow

James F . Whidborne1 John McKernan1 George Papadakis2

• 1. Department of Aerospace Sciences, Cranfield University
• 2. Division of Engineering, King’s College London

UKACC ICC2006 – p. 1/20

Transient energy growth

Consider the asymptotically stable linear time-invariant system described by the initial value problem ˙ x = Ax, x(0) = x0, with A ∈ Rn×n, x0 ∈ Rn which has the continuous solution x : R+ → Rn, t → Φ(t)x0, where Φ(t) is the state transition matrix given by Φ(t) = eAt =

i=0 Aiti/i!

The transient energy is defined as E(t) := max

x(t)2 : x0 = 1

• Note that E(t) = |

| |Φ(t)| | |2 where | | |·| | | 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 E(t) = max

Wx(0)=1 Wx(t)2 ,

For simplicity we assume W = I

UKACC ICC2006 – p. 2/20

Some properties, bounds, etc

• b

E is lower bounded by unity

• b

E is unity if and only if A + AT ≤ 0

• e2α(A)t ≤ E(t) ≤ κ(S)2e2α(A)t

where α(A) := maxi ℜ(λi) is the spectral abscissa of A {λi} are the eigenvalues of A S is the right eigenvector matrix κ(S) := | | |S| | |

• S−1
• is the condition number of S
• b

E is unity if A is normal (eigenvectors form an orthonormal set)

• E(t) ≤ e2w(A)t

where w(A) := λmax(A + AT )/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

Lyapunov upper bound

We can use Lyapunov functions to describe an ellipsoid that bounds the trajectory to give an upper-bound on E(t)

Eu ≥

E is an upper bound on the maximum transient energy growth,

E, where

Eu := λmax(P)λmax(P−1) where P = PT > 0 satisfies PA + AT P ≤ 0

(1)

Note that λmax(P)λmax(P−1) = | | |P| | |

• P−1
• = κ(P), the condition number.

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) min γ subject to I ≤ P ≤ γI, PA + AT P < 0 where P > 0 is real and symmetric. The inequality I ≤ P ≤ γI ensures that γ ≥ λmax(P) ≥ λmin(P) ≥ 1, thus λmax(P)/λmin(P) ≤ γ and so

E ≤

Eu ≤ 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) = x0, u = Kx, Upper bound can be minimized by solving an LMI (Boyd et al 1994) Expanding Lyapunov inequality for u = Kx gives PA + AT P + PBK + KT BT P < 0. By the change of variable, Q = P−1 and Y = KQ the LMI AQ + QAT + BY + YT BT < 0 is obtained. Now since λmax(P)λmax(P−1) = λmax(Q)λmax(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 subject to I ≤ Q ≤ γI, AQ + QAT + BY + YT BT < 0, Q = QT 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

• f 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
• ccurrence
• f

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) = xT (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.

500 1000 1500 2000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

t E(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

(Controller) Plane Poiseuille Flow Flow Disturbance Lower Wall Upper Wall Actuation Sensing Spanwise, z Wall−normal, y Streamwise, x

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 +
• U · ∇
• U = − 1

ρ∇P + µ ρ ∇2 U ∇ · U = 0

• U is velocity, P is pressure, ρ is density, µ is viscosity
• Laminar flow has a parabolic stream-wise velocity profile

Ub = ((1 − y2)Ucl, 0, 0), Pb 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

˙

• u +
• Ub · ∇
• u + (

u · ∇)

• u +

Ub

• = −∇p + 1

R ∇2 u ∇ · u = 0 where R := ρUclh/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(ei(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

• State variables x are wall-normal ve-

locity ˜ v and vorticity, ˜ η := ∂u/∂z − ∂w/∂x, perturbation Chebyshev coeffi- cients, plus the upper and lower wall v transpiration velocities

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 ←ΓDN

0 (8)

←ΓDN

1 (4.6)

←ΓDN

last(2.1)

←ΓD

0(2)

←ΓD

1(1.5)

←ΓD

last(2)

←fu

lmi(1)

←fl

lmi(1)

y

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
• This

test case is linearly stable but has the largest linear transient en- ergy growth

• ver

all unit initial con- ditions, time and wave-number, and represents the very earliest stages

• f

the transition to turbulence

500 1000 1500 2000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

t E(t)

UKACC ICC2006 – p. 13/20

Minimal upper bound control

Solving LMI gives upper bound

Eu = 1722 Actual maximum transient energy growth is

E = 883 Open loop

E = 4941

500 1000 1500 100 200 300 400 500 600 700 800 900

t E(t)

UKACC ICC2006 – p. 14/20

Worst case control signal

• The worst case wall control uwc = Kxwc where xwc 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 xwc being symmetrical with respect

to y

• The control signals are impractically large

500 1000 1500 −1 1 2 3 4 5 6 x 10

4

t uwc(t)

UKACC ICC2006 – p. 15/20

Constraint on control signal

A constraint max

t≥0 u(t)2 ≤ µ2

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 ≤ γI AQ + QAT + BY + YT BT < 0

Q YT Y µ2I

≥ 0

UKACC ICC2006 – p. 16/20

Dependence on control constraint µ

• The GEVP with the control constraint was solved over a range of µ
• Minimal

E does not coincide with largest µ (minimal

Eu)

• Minimal

E occurs for µ = 20 with upper bound

Eu = 1722, and

E = 825 — but control gains quite large with the largest control gain element, max |K| = 13.648

10

−2

10

−1

10 10

1

10

2

10

3

10

4

10

3

10

4

µ

• E,

Eu

UKACC ICC2006 – p. 17/20

Minimal upper bound control with µ = 0.8

Control effort can be substantially reduced without a significant performance deterioration Solving GEVP with µ = 0.8 constraint gives upper bound

Eu = 1722 Actual maximum transient energy growth is

E = 842 Open loop

E = 4941 Control signal magnitudes and the control gains are small with the largest control gain element, max |K| = 0.558.

500 1000 1500 100 200 300 400 500 600 700 800 900

t E(t)

UKACC ICC2006 – p. 18/20

Worst case control signal

• The control signal magnitudes are relatively modest
• Control gains are small with the largest control gain element, max |K| = 0.5588

100 200 300 400 500 −0.05 0.05 0.1 0.15 0.2

t uwc(t)

UKACC ICC2006 – p. 19/20

Conclusions and Comments

Limitations:

• Upper bound minimized — actual maximum energy can be minimized (Whidborne et al,

2005), but very computationally intensive

• State feedback required
• Periodicity assumption on the flow velocity field — control for a single wave number

(non-periodic plane Poiseuille flow has been tackled by Baramov et al (2004))

• Transient energy growth has no explicit robustness

Other comments:

• Maximum transient energy growth is particularly useful in fluid flow control because it is

easier to verify through experiment and CFD simulation than H∞-norm measures

• Validation of results by means of CFD simulation has been performed — will be

presented at IMechE Symposium in October

UKACC ICC2006 – p. 20/20