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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

• Re = 190: Subcritical Mode A inst. (λd ≈ 4d)

• Re = 240: Mode B instability (λd ≈ 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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are needed to see this picture.

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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…

AtmosphericShear

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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 Asubcritical

Mode Bsupercritical

U

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Subcritical (hysteretic transition)

First 3D cylinder wake transition (Mode A, Re=190)

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Supercritical transition

Mode B (3D cylinder wake at Re=260)

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Shear Layer Instability

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

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture. Periodic inflow/outflow

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Jet instability

U(y) = sech2(y) - Symmetric jet

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Again periodic boundaries

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Cylinder wake results

Shear Layer Instability in a Cylinder Wake

Re > 1000-2000

Transition point fromConvective to AbsoluteInstability

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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

ExperimentsRayleigh equationDNS

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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)

Absolutely unstableConvectively unstable

Saturated state

Either - pre-shedding or time-mean wake

Velocity profiles on vertical lines used for analysis

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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 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.

» 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!

x

Real x

Saddle point

Complex x

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Accuracy of saddle point prediction

Prediction of preferred frequency is:

• 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

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…

Frequency variation Based on

Landau equation

Supercritical transition

Wake saturating….

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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

ExperimentsRayleigh equationDNS

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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.

• Predictions of growth rate

as a function of Reynolds number for the steady cylinder wake.

Predicted wake frequency

Absolute instability

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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.

DNS

Downstream A-->C transition(Koch)

Experiments

IRC criterion(= nonlinear prediction)

(Monkewitz and Nguyen)

From mean flow(saddle point criterion)

Saddle point onSteady flow

Max amplication (Pierrehumbert)

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Global stability analysis

Prediction based on Global instability analysis of time-mean wake. (Barkley 2006).

Match with experiments & DNSFor wake frequency

Predicted mode is neutrally stable…