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Stability of Parallel Flows. 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. ( λ d ≈ 4 d )

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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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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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2D vortex street behind a circular cylinder at Re = 140

Photograph: S. Taneda (Van Dyke 1982)

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 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

Mode B

supercritical

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

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 > 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

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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2D vortex street behind a circular cylinder at Re = 140

Photograph: S. Taneda (Van Dyke 1982)

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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2D vortex street behind a circular cylinder at Re = 140

Photograph: S. Taneda (Van Dyke 1982)

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 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!

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 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…

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

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 onlinear 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 assumption

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

Match with experiments & DNS

For wake frequency

Predicted mode is neutrally stable…