Understanding and Controlling Turbulent Shear Flows

Bassam Bamieh

Department of Mechanical EngineeringUniversity of California, Santa Barbara

http://www.engineering.ucsb.edu/bamieh

23rd Benelux Meeting on Systems and Controls, March ’04

The phenomenon of turbulence

Example: Flow behind a cylinder at increasing velocities

Re = 0.16Re = 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

The phenomenon of turbulence (Cont.)

Technologically important flows: Flows past streamlined bodies

pipes

channels

wings

Wall-bounded shear flows Friction with the walls drives the flows

2

The phenomenon of turbulence (Cont.)

Boundary layers form in flow past any surface

Idealization: flow on a flat plate

Flow direction

Viewed sideways

3

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 dragis 40-50% of total drag ontypical airliner

4

Shark and Dolphin skins

Some sharks and dolphins swim much faster than their muscle mass would allowhad their boundary layers been turbulent

Shark skin has small “grooves” (riblets)

Moin & Kim, ’98

One can buy swim suitesthat try to mimic this

5

Shark and Dolphin skins (Cont.)

Dolphins have a different mechanisms

Dolphin skin is a compliant skin Rigid base

SpringsFlow Plate

Skin is an elastic membrane that interacts with flow and “suppresses” the productionof turbulence

Difficulties:

• Basic mechanisms of how these skins suppress turbulence is not well understood

• Therefore, difficult to scale designs to airplanes and ships

6

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 geometrytransition can occur abrubtly, gradually, or in several stages

The mathematical models

• Transition from one stage to the next ↔ dynamical instabilitysuccessful 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

Hydrodynamic Stability (mathematical formulation)

The incompressible Navier-Stokes equations

∂tu = −∇uu − grad p + 1R∆u

0 = div u

u ↔⎡⎣ u(x, y, z, t)

v(x, y, z, t)w(x, y, z, t)

⎤⎦ 3D velocity field

p(x, y, z, t) 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 = u + u↑ ↑

laminar fluctuations

8

• Fluctuation dynamics:

∂tu = −∇uu −∇uu − grad p + ∆u − ∇uu

0 = div u

In linear hydrodynamic stability, − ∇uu is ignored

9

Examples:

• Poiseuille Flow�

����

�����

.........................................................................................................................................................................................................

�

��

U(x,y)=(1−y2)y

x

• Couette Flow��

���

�����

��

��

��

��

��

��.........................................................................................................................................................................................................

�

�

yxU(x,y)=y

• Pipe Flow��

���

�����

�U(r)=(1−r2)

• Blasius boundary layer

� � ����������

� � �

��

�

�

yx

• Others:

– Benard Convection (Between two horizontal plates)– Taylor-Couette (Flow between two concentric rotating cylinders)

10

The Evolution Model

����������

����������

............................................................

............................................................

............................................................

.............................................

............................................................

............................................................

............................................................

.............................................

�

�

���� �����

..

�

.........................................................................................................................................................................................................

�����

�����

�

.........................................................................................................................................................................................................

�����

�����

x

1

−1uwv

y

zCan rewrite linearized fluctuation equationsusing only two fields

v := wall-normal velocity

ω := wall-normal vorticity :=∂u

∂z− ∂w

∂x.

Equivalent equations:

∂t

[vω

]=

[ (−∆−1U∂x∆ + ∆−1U ′′∂x + 1R∆−1∆2

)0

(−U ′∂z)(−U∂x + 1

R∆) ] [

vω

]

∂t

[vω

]=

[ L 0C S

] [vω

]=: A

[vω

]

Abstractly,

∂tΨ = A Ψ

11

Classical linear hydrodynamic stability:

Transition ←→ instability ≡ A has spectrum in right half plane

existence of exponentially growing normal modes a modal instability

12

The Eigenvalue Problem

∂

∂t

[vω

]=

[ (−∆−1U ∂∂x∆ + ∆−1U ′′ ∂

∂x + 1R∆−1∆2

)0(−U ′ ∂

∂z

) (−U ∂∂x + 1

R∆) ] [

vω

]∂

∂t

[vω

]=

[ L 0C S

] [vω

]= A

[vω

]Remark: A translation invariant in x, z (but not in y!). Fourier transform in x and z:

∂

∂t

[vω

]=

[ (−ikx∆−1U∆ + ikx∆−1U ′′ + 1R∆−1∆2

)0

(−ikzU′)

(−ikxU + 1R∆

) ] [vω

]kx, kz: spatial frequencies in x, z directions (wave-numbers).

∂

∂t

[v(t, kz, kx)ω(t, kz, kx)

]= A(kx, kz)

[v(t, kz, kx)ω(t, kz, kx)

], v(t, kz, kx), ω(t, kz, kx) ∈ L2[−1, 1]

Essentially:

spectrum (A) =⋃

kx,kz

spectrum(A(kx, kz)

)

13

Tollmien-Schlichting Instability

Example: Poiseuille flow at R = 6000, kx = 1, kz = 0 has the following eigenvalues:

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

real part

imag

inar

y pa

rt

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, 1940.

14

Problems with the theory (1st difficulty)

Rc: The critical Reynolds number at which instability occursR ≥ Rc ⇒ ∃ unstable eigenvalues of Acorresponding 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 caseShear flows: (e.g. flows in channels, pipes, and boundary layers)

Flow type Classical linear theory Rc Experimental Rc

Poiseuille 5772 ≈ 1000Couette ∞ ≈ 350

Pipe ∞ ≈ 2200

Experimental Rc is highly dependent on conditions such as wall roughness,external disturbances, etc...

15

Problems with the theory (2nd difficulty)

Second failure: Incorrect prediction of “natural transition” flow structures

• Classical theory predicts Tollmien-Schlichting waves in Poiseuille and boundarylayer flows:

• Except in very noise-free and controlled experiments, flow structures in transitionare more like turbulent spots and streaky boundary layers:

16

Boundary layer with free-stream disturbances

(a) (b)

(d )(c)

From Matsubara & Alfredsson, 2001

17

Transition Scenarios

1.Infinitesimaldisturbances

=⇒ TS-wave =⇒3D

Secondaryinstabilities

=⇒ Transition

Demonstrable in “clean” experiments

2.Infinitesimaldisturbances

=⇒ Stream-wise vorticesand streaks

=⇒ Transition

Occurs when small amounts of noise is present

Possible explanation:

noisy environments −→ big disturbances −→ “non-linear effects” dominate

18

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

The Evolution Model

∂

∂t

[vω

]=

[ (−∆−1U ∂∂x∆ + ∆−1U ′′ ∂

∂x + 1R∆−1∆2

)0(−U ′ ∂

∂z

) (−U ∂∂x + 1

R∆) ] [

vω

]∂

∂t

[vω

]=

[ L 0C S

] [vω

]= A

[vω

]

����������

����������

............................................................

............................................................

............................................................

.............................................

............................................................

............................................................

............................................................

.............................................

�

�

���� �����

..

�

.........................................................................................................................................................................................................

�����

�����

�

.........................................................................................................................................................................................................

�����

�����

x

1

−1uwv

y

z

After Fourier transform:

∂

∂t

[vω

]=

[ (−ikx∆−1U∆ + ikx∆−1U ′′ + 1R∆−1∆2

)0

(−ikzU′)

(−ikxU + 1R∆

) ] [vω

]

kx, kz: spatial frequencies in x, z directions (wave-numbers).

20

Transient energy growth of perturbed flow fields

The energy density of a perturbation for a given kx, kz is

E =kxkz

16π2

∫ 1

−1

∫ 2π/kx

0

∫ 2π/kz

0

(u2 + v2 + w2) dz dx dy

which can be rewritten as a quadratic form on the normal velocity and vorticity fieldsas:

E =

⟨[vω

],

[I − 1

k2x+k2

z

∂2

∂y2 00 1

k2x+k2

zI

] [vω

]⟩=: 〈Ψ, Q Ψ〉 .

21

The evolution of the disturbance’s energy

‖Ψ(t)‖2 = ‖eAtΨ(0)‖2 =⟨Ψ(0) ,

{eA

∗tQeAt}

Ψ(0)⟩

In the fluids literature, the following is emphasized:

• If A is stable and normal (w.r.t. Q), then ‖eAtΨ(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)

22

Input-output analysis of Linearized Navier-Stokes

Add forcing terms to the NS equations

∂tu = −∇uu −∇uu − ∇p +1R

∆u + d

0 = ∇ · u

Possible sources of d

• NEGLECTED NONLINEAR TERMS (small gain analysis)

• NON-LAMINAR FLOW PROFILES (conservative)

• BODY FORCES

• NON-SMOOTH GEOMETRIES

23

Input-output analysis of Linearized Navier-Stokes (Cont.)

In standard form

∂tψ = A ψ + B d

∂tψ =

[−∆−1U∂x∆ + ∆−1U ′′∂x + 1

R∆−1∆2 0

−U ′∂z −U∂x + 1R∆

]︸ ︷︷ ︸

A

ψ +

[−∆−1∂xy ∆−1(∂xx + ∂zz) −∆−1∂yz

∂z 0 − ∂x

]︸ ︷︷ ︸

B

⎡⎣ dx

dy

dz

⎤⎦

Studied in M. Jovanovic & B. Bamieh, ’02

24

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 theleast damped modes of A

(as spatio-temporal patterns)

25

Input-Output vs. Modal Analysis

A simple example: a finite-dimensional Single-Input Single-Output system

x = Ax + Buy = Cx

⇐⇒ H(s) = C(sI − A)−1B

Theorem: Let z1, . . . , zn be any locations in the left half of the complex plane.Any stable frequency response function in H2 can be arbitrarily closely approximatedby a transfer function of the following form:

H(s) =N1∑i=1

α1,i

(s − z1)i+ · · · +

Nn∑i=1

αn,i

(s − zn)i

by choosing any of the Nk’s large enough

i.e.: No connection between underdamped modes of A and peaks of frequencyresponse H(jω)

Re(s)

Im(s)

|H(s)|

X X

X

26

Spatio-temporal Frequency Response

∂

∂tΨ(t, kx, kz) = A(kx, kz) Ψ(kx, kz) + B(kx, kz) d(t, kx, kz)

Fourier transform in time

Ψ(ω, kx, kz) =((jωI − A(kx, kz))

−1B)

d(ω, kx, kz) =: H(ω, kx, kz) d(ω, kx, kz)

The operator-valued spatio-temporal frequency response

H(ω, kx, kz)

is a MIMO (in y) frequency response of several frequency variables

Not allways straight forward to visualize, has lots of information

27

Dominance of Stream-wise Constant Flow Structures

λmax

(∫ ∞

−∞H(ω, kx, kz)H∗(ω, kx, kz) dω

)= a scalar function of (kx, kz)

averaging out time, and taking σmax in y direction

05

1015

20

−20

2

0

5

10

15

20

25

kz

kx

σm

ax(k

x, kz)

Poiseuille flow at R=2000.

28

Dominance of Stream-wise Constant Flow Structures (cont.)

sup−∞<ω<∞

λmax (H(ω, kx, kz)H∗(ω, kx, kz))

05

1015

-2

0

2

0

10

20

30

kz

kx

[||H

|| ∞](

k x,kz)

10-2

100

102

100

-60

-40

-20

0

20

40

log10

(kz)log

10(k

x)

20 lo

g 10([

||H|| ∞

](k x,k

z))

A log-log plot

Poiseuille flow at R=2000.

29

Input-output analysis of Linearized Navier-Stokes (Cont.)

��������

��������

................................................................................

................................................................................

.................................................................

................................................................................

................................................................................

.................................................................

�

�

��� �����..

�

.........................................................................................................................................................................................................

�� ���

�� ���

�

.........................................................................................................................................................................................................

�� ���

�� ���

x1−1

uwvy

z

“Most resonant flow structures”are stream-wise vortices and streaks

Cross sectional view

30

Stream-wise vortices and streaks

31

Mechanism of Generation of Stream-wise Vortices and Streaks

∂

∂t

[vω

]=

[ (1R∆−1∆2

)0

(−ikzU′)

(1R∆

) ] [vω

]+

[dv

dω

]=:

[1RL 0C 1

RS] [

vω

]+

[dd

����

(sI − 1RL)−1 −ikzU

′ (sI − 1RS)−1� � � ��

�

+dv

dw

ω

0 2 4 6

0.1

0.15

0.2

0.25

0.3

0.35

Kz0 2 4 6

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

Kz

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4x 10

−4

Kz10

−110

−7

10−6

10−5

10−4

10−3

32

Figure 3: Streamwise velocity perturbation development for largest singular value (first row) and secondlargest singular value (second row) of operator H at {kx = 0.1, kz = 2.12, ω = −0.066}, in Poiseuille flowwith R = 2000. High speed streaks are represented by red color, and low speed streaks are represented bygreen color. Isosurfaces are taken at ±0.5.

33

Figure 8: Streamwise vorticity perturbation development for largest singular value of operator H at {kx =0.1, kz = 2.12, ω = −0.066}, in Poiseuille flow with R = 2000. High vorticity regions are represented byyellow color, and low vorticity regions are represented by blue color. Isosurfaces are taken at ±0.4.

34

Linearized NS (LNS) Equations

• With stochastic forcing

– H2 norm of LNS scales like R3

– Amplification mechanism is 3D, streak spacing mathematically related todynamical 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 H2 as a problem formulationAchieved re-laminarization at low Reynolds numbersFluid 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

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

Uncertainty in a Dynamical System

Stability analysis deals with uncertainty in initial conditions

ψ(0)

If x(0) is known to be precisely xe, then x(t) = xe, t ≥ 0

We introduce the concept of stability because we can never be infinitely certain aboutthe initial condition

Shortcomings of stability analysis

⎧⎨⎩

• Perturbs only initial conditions

• Cares mostly about asymptotic behavior

37

Uncertainty in a Dynamical System (cont.)

Stabilityx = f(x)

uncertain initialconditions

investigate limt→∞

x(t)

investigate transientse.g. supt≥0 ‖x(t)‖

dynamical uncertaintyx = F (x) + ∆(x)

exogenous disturbancesx(t) = F (x(t), d(t))

combinationsx(t) = F (x(t), d(t))+

∆(x(t), d(t))

More Uncertainty

Linearized version:

eigenvalue stabilityx = Ax

transient growth

umodelled dynamicsx = (A + ∆)x

Psuedo-spectrum

exogenous disturbancesx(t) = Ax(t) + Bd(t)

input-output analysis

combinationsx(t) = (A + B∆C)x(t)+

(F + G∆H)d(t)

38

Uncertainty in a Dynamical System (cont.)

Stabilityx = f(x)

uncertain initialconditions

investigate limt→∞

x(t)

investigate transientse.g. supt≥0 ‖x(t)‖

dynamical uncertaintyx = F (x) + ∆(x)

exogenous disturbancesx(t) = F (x(t), d(t))

combinationsx(t) = F (x(t), d(t))+

∆(x(t), d(t))

More Uncertainty

Linearized version:

eigenvalue stabilityx = Ax

transient growth

umodelled dynamicsx = (A + ∆)x

Psuedo-spectrum

exogenous disturbancesx(t) = Ax(t) + Bd(t)

input-output analysis

combinationsx(t) = (A + B∆C)x(t)+

(F + G∆H)d(t)

39

The nature of turbulence

Fluid dynamics are described by deterministic equations

Why does fluid flow “look random” at high Reynolds numbers??

40

The nature of turbulence (Cont.)

Common view of turbulence

41

The nature of turbulence (cont.)

Common view of turbulence

Intuitive reasoningComplex, statistical looking behavior ←→ System with chaotic dynamics

42

The nature of turbulence (cont.)

43

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 uncertaintyamplification?as opposed to “chaos”

44