understanding and controlling turbulent shear flowsbamieh/pubs/benelux... · 2004-03-27 ·...
TRANSCRIPT
![Page 1: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/1.jpg)
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
![Page 2: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/2.jpg)
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
![Page 3: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/3.jpg)
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
![Page 4: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/4.jpg)
The phenomenon of turbulence (Cont.)
Boundary layers form in flow past any surface
Idealization: flow on a flat plate
Flow direction
Viewed sideways
3
![Page 5: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/5.jpg)
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
![Page 6: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/6.jpg)
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
![Page 7: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/7.jpg)
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
![Page 8: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/8.jpg)
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
![Page 9: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/9.jpg)
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
![Page 10: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/10.jpg)
• Fluctuation dynamics:
∂tu = −∇uu −∇uu − grad p + ∆u − ∇uu
0 = div u
In linear hydrodynamic stability, − ∇uu is ignored
9
![Page 11: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/11.jpg)
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
![Page 12: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/12.jpg)
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
![Page 13: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/13.jpg)
Classical linear hydrodynamic stability:
Transition ←→ instability ≡ A has spectrum in right half plane
existence of exponentially growing normal modes a modal instability
12
![Page 14: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/14.jpg)
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
![Page 15: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/15.jpg)
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
![Page 16: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/16.jpg)
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
![Page 17: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/17.jpg)
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
![Page 18: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/18.jpg)
Boundary layer with free-stream disturbances
(a) (b)
(d )(c)
From Matsubara & Alfredsson, 2001
17
![Page 19: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/19.jpg)
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
![Page 20: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/20.jpg)
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
![Page 21: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/21.jpg)
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
![Page 22: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/22.jpg)
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
![Page 23: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/23.jpg)
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
![Page 24: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/24.jpg)
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
![Page 25: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/25.jpg)
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
![Page 26: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/26.jpg)
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
![Page 27: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/27.jpg)
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
![Page 28: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/28.jpg)
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
![Page 29: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/29.jpg)
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
![Page 30: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/30.jpg)
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
![Page 31: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/31.jpg)
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
![Page 32: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/32.jpg)
Stream-wise vortices and streaks
31
![Page 33: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/33.jpg)
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
![Page 34: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/34.jpg)
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
![Page 35: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/35.jpg)
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
![Page 36: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/36.jpg)
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
![Page 37: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/37.jpg)
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
![Page 38: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/38.jpg)
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
![Page 39: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/39.jpg)
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
![Page 40: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/40.jpg)
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
![Page 41: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/41.jpg)
The nature of turbulence
Fluid dynamics are described by deterministic equations
Why does fluid flow “look random” at high Reynolds numbers??
40
![Page 42: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/42.jpg)
The nature of turbulence (Cont.)
Common view of turbulence
41
![Page 43: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/43.jpg)
The nature of turbulence (cont.)
Common view of turbulence
Intuitive reasoningComplex, statistical looking behavior ←→ System with chaotic dynamics
42
![Page 44: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/44.jpg)
The nature of turbulence (cont.)
43
![Page 45: Understanding and Controlling Turbulent Shear Flowsbamieh/pubs/benelux... · 2004-03-27 · transition can occur abrubtly, gradually, or in several stages The mathematical models](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f96691a9c795f3a6a6c75f8/html5/thumbnails/45.jpg)
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