Transcript Stability of Parallel Flows Betchov and Criminale (1967)

Stability of Parallel Flows
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Analysis by LINEAR STABILITY ANALYSIS.

Transitions as Re increases
• 0 < Re < 47: Steady 2D wake
• Re = 47: Supercritical Hopf bifurcation
• 47 < Re < 180: Periodic 2D vortex street
B 190:
instability
in the Mode A inst. (λ ≈ 4d)
•Mode
Re =
Subcritical
d
QuickTime™ and a
wake behind a circular
YUV420
codec decompressor
• Re = 240: Mode B instability (λd ≈ 1d)
are
needed
to see this picture.
cylinder at Re = 250
• Re increasing: spatio-temporal chaos, rapid transition to
Thompson (1994)
turbulence.
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Atmospheric Shear Instability

Examples: • Kelvin-Helmholtz instability
» Velocity gradient in a continuous fluid or
» Velocity difference between layers of fluid
May also involve density differences, magnetic fields…


Atmospheric
Shear
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Cylinder Wake - High Re

Bloor-Gerrard Instability (cylinder shear layer instability)
Karman shedding
Shear layer
instability
Prasad and Williamson JFM 1997
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Transition Types

Instability Types:
• Convective versus Absolute instability
»
A convective instability is convected away downstream - it grows as
it does so, but at a fixed location, the perturbation eventually dies out.
Example: KH instability

»
Absolute instability means at a fixed location a perturbation will
grow exponentially. Even without upstream noise - the instability will
develop
Example: Karman wake

2D vortex street behind a circular
cylinder at Re = 140
Photograph: S. Taneda (Van
Dyke 1982)
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Transition Types

Supercritical versus Subcritical transition
• A supercritical transition occurs at a fixed value of the control parameter
» Example: Initiation of vortex shedding from a circular cylinder at
Re=46. Mode B for a cylinder wake, Shedding from a sphere.
• A subcritical transition occurs over a range of the control parameter
depending on noise level. There is an upper limit above which transition
must occur.
» Example: Mode A instability - first three-dimensional mode of a
cylinder wake.
Mode A
subcritical
U
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Mode B
supercritical
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Subcritical (hysteretic transition)

First 3D cylinder wake transition (Mode A, Re=190)
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Supercritical transition
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Mode B (3D cylinder wake at Re=260)
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Shear Layer Instability

U(y) = tanh(y) - Symmetric Shear Layer
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
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Periodic inflow/outflow
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Jet instability

U(y) = sech2(y) - Symmetric jet
Again periodic boundaries
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
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Cylinder wake results
Shear Layer Instability in a Cylinder Wake
Re > 1000-2000
Transition point from
Convective to Absolute
Instability
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Frequency Prediction for a Cylinder Wake

Numerical Stability Analysis based on Time-Mean Flow
• Extract velocity profiles across wake
• Analyze using parallel stability analysis to predict Strouhal number
DNS
Rayleigh equation
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Experiments
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Interesting Recent Work

Barkley (2006 EuroPhys L)
• Time mean wake is neutrally stable - preferred frequency corresponds to
observed Strouhal number to within 1%

Chomaz, Huerre, Monkewitz… Extension to non-parallel wakes…

Pier (2002, JFM)
• Non-linear stability modes to predict observed shedding frequency of a
cylinder wake

Hammond and Redekopp (JFM 1997)
• Analysis of time-mean flow of a flat plate.
• Also of interest: Non-normal mode analysis/optimal growth theory….to
predict transition in Poiseiulle flow.
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Basic Stability Theory 2: Absolute & Convective Instability

Background
• Generally, part of a wake may be convectively unstable and part may be
absolutely unstable
» Recall
Convective instability means a disturbance will die out locally but
will grow in amplitude as it convects downstream.
» Think of shear layer vortices
Absolute instability means that a disturbance will grow in
amplitude locally (where it was generated)
» Think of the Karman wake.


2D vortex street behind a circular
cylinder at Re = 140
Photograph: S. Taneda (Van
Dyke 1982)
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Absolute & Convective unstable zones
2D vortex street behind a circular
cylinder at Re = 140
Photograph: S. Taneda (Van
Dyke 1982)
Saturated state
Velocity profiles on vertical lines
used for analysis
Convectively unstable
Absolutely unstable
Either - pre-shedding or time-mean wake
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Selection of the wake frequency
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Problem: Wake absolutely unstable over a finite spatial range.
• Prediction of frequency at any point in this range.
• So what is the selected frequency?

There were three completing theories:
• Monkewitz and Nguyen (1987) proposed the Initial Resonance Condition
» The frequency selected corresponds to the predicted frequency at the
point where the initial transition from convective to absolute instability
occurs.
• Koch (1985) proposed the downstream resonance condition.
» This states that it is the downstream transition from absolute to
convective instability that determines the selected frequency.
• Pierrehumbert (1984) proposed that the selection is determined by the
point in the absolute instability range with the maximum amplification rate.
• These theories are largely ad-hoc.
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Selection of wake frequency - Saddle Point Criterion

Since then
• Chomaz, Huerre, Redekopp (1991) & Monkewitz in various papers
have shown that the global frequency selection for (near) parallel flows is
determined by the complex frequency of the saddle point in complex
space, which can be determined by analytic continuation from the
behaviour on the real axis.
• This was demonstrated quite nicely by the work of Hammond and
Redekopp (1997), who examined the frequency prediction for the wake
from a square trailing edge cylinder.
•
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Test Case - Flow over trailing edge forming a wake

Hammond and Redekopp (JFM 1997)
• Considered the general case below, but
» Focus on symmetric wake without base suction.
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Linear theory assumptions
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Is the wake parallel?
• This indicates how parallel the wake is at Re=160
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Frequency prediction with downstream distance

The real and imaginary components of the complex frequency is
determined using both Orr-Sommerfeld (viscous) and Rayleigh
(inviscid) solvers from velocity profiles across the wake.
• These are used to construct the two plots below:
Predicted
oscillation frequency
Predicted
Growth rate
Downstream distance
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Saddle point prediction

Prediction of selected frequency:
• First note that the downstream point at which the minimum frequency
occurs does not correspond with the point at which the maximum growth
rate occurs.
» This means that the saddle point occurs in complex space!!!!
» This is the complex point at which the frequency and growth rate
reach extrema together.
Here, both omega and x are complex!
»
Can use complex Taylor series + Cauchy-Riemann equations to
project off the real axis (…the only place where you know anything).
Complex x
Saddle point
x
Real x
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Accuracy of saddle point prediction

Prediction of preferred frequency is:
•
•
•
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Parallel inviscid theory at Re=160 gives 0.1006
Numerical simulation of (saturated) shedding at Re=160 gives 0.1000.
» Better than 1% accuracy!
Saddle point at
Things to note:
•
•
•
•
•
Spatial selection point is within 1D of the trailing edge.
Amazing accuracy.
Generally, imaginary component of saddle point position is small.
The predicted frequency (on the real axis) may not vary all that much anyway over
the absolute instability region, and may not vary much from the position of
maximum growth rate. Hence all previous adhoc conditions are generally close.
Note prediction is based on time-mean wake not the steady (pre-shedding) wake.
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Linear theory - inviscid and viscous
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Predictions from Hammond and Redekopp (1997)
• Inviscid = Rayleigh equation on downstream profiles
• Viscous = Orr-Sommerfeld equation on downstream profiles
» Re = 160.
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Saturation of wake (Landau Model)

Further points:
• Wake frequency varies as the wake saturates…
Wake
saturating….
Frequency variation
Based on
Landau equation
Supercritical
transition
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Frequency Prediction for a Circular Cylinder Wake

Numerical Stability Analysis based on Time-Mean Flow
• Extract velocity profiles across wake
• Analyze using parallel stability analysis to predict Strouhal number
DNS
Rayleigh equation
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Experiments
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Inadequacy of theory?

We need to know the time-mean flow (either by numerical simulation
or running experiments) to computed the preferred wake frequency!!!
• This is not very satisfying…

Other option is to undertake a non-linear stability analysis on the
steady base flow (when the wake is still steady - prior to shedding).
• This was done by Pier (JFM 2002).
Vorticity field - cylinder wake
Re = 100
Unstable steady wake
Re = 100
Time-mean wake
Re = 100
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Non-linear theory

Pier (JFM 2002) & Pier and Huerre (2001).
• Frequency selection based on the (imposed) steady cylinder wake using
non-linear theory.
Absolute instability
•
Predictions of growth rate
as a function of Reynolds number
for the steady cylinder wake.
Predicted wake frequency
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Frequency predictions based on near-parallel, inviscid assumption

Nonlinear theory indicates that the saturated wake frequency
corresponds to the frequency predicted from the Initial Resonance
Criterion (IRC) of Monkewitz and Nguyen (1987) based on linear
analysis.
IRC criterion
(= nonlinear prediction)
(Monkewitz and Nguyen)
DNS
Experiments
From mean flow
(saddle point criterion)
Downstream A-->C transition
(Koch)
Max amplication
(Pierrehumbert)
Saddle point on
Steady flow
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Global stability analysis

Prediction based on Global instability analysis of time-mean wake.
(Barkley 2006).
Match with experiments & DNS
For wake frequency
Predicted mode is neutrally stable…
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