Stability of Parallel Flows

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Stability of Parallel Flows
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• Analysis by LINEAR STABILITY ANALYSIS.

• Transitions as Re increases
• 0 lt Re lt 47 Steady 2D wake
• Re 47 Supercritical Hopf bifurcation
• 47 lt Re lt 180 Periodic 2D vortex street
• Re 190 Subcritical Mode A inst. (?d 4d)
• Re 240 Mode B instability (?d 1d)
• Re increasing spatio-temporal chaos, rapid
  transition to turbulence.

Mode B instability in the wake behind a circular
cylinder at Re 250 Thompson (1994)
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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

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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 Re46. 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
Mode B supercritical
U
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Subcritical (hysteretic transition)

• First 3D cylinder wake transition (Mode A, Re190)

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Supercritical transition

• Mode B (3D cylinder wake at Re260)

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Shear Layer Instability

• U(y) tanh(y) - Symmetric Shear Layer

Periodic inflow/outflow
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Jet instability

• U(y) sech2(y) - Symmetric jet

Again periodic boundaries
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Cylinder wake results

• Shear Layer Instability in a Cylinder Wake

Re gt 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

Experiments
Rayleigh equation
DNS
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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.

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Absolute Convective unstable zones
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

• 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

• Is the wake parallel?
• This indicates how parallel the wake is at Re160

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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.
• Can use complex Taylor series Cauchy-Riemann
  equations to project off the real axis (the only
  place where you know anything).

Here, both omega and x are complex!
Complex x
Saddle point
x
Real x
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Accuracy of saddle point prediction

• Prediction of preferred frequency is
• Parallel inviscid theory at Re160 gives 0.1006
• Numerical simulation of (saturated) shedding at
  Re160 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

• 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

Experiments
Rayleigh equation
DNS
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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--gtC 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