1. 6/29/20151 Stability of Parallel Flows

2. 6/29/20152 Analysis by LINEAR STABILITY ANALYSIS. l 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 ≈ 4d) Re = 240: Mode B instability (λ 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)

3. 6/29/20153 Atmospheric Shear Instability l Examples: - Kelvin-Helmholtz instability » Velocity gradient in a continuous fluid or » Velocity difference between layers of fluid l May also involve density differences, magnetic fields… l Atmospheric Shear

4. 6/29/20154 Cylinder Wake - High Re l Bloor-Gerrard Instability (cylinder shear layer instability) Karman shedding Shear layer instability Prasad and Williamson JFM 1997

5. 6/29/20155 Transition Types l 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. l Example: KH instability » Absolute instability means at a fixed location a perturbation will grow exponentially. Even without upstream noise - the instability will develop l Example: Karman wake 2D vortex street behind a circular cylinder at Re = 140 Photograph: S. Taneda (Van Dyke 1982)

6. 6/29/20156 Transition Types l 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

7. 6/29/20157 Subcritical (hysteretic transition) l First 3D cylinder wake transition (Mode A, Re=190)

8. 6/29/20158 Supercritical transition l Mode B (3D cylinder wake at Re=260)

9. 6/29/20159 Shear Layer Instability l U(y) = tanh(y) - Symmetric Shear Layer Periodic inflow/outflow

10. 6/29/201510 Jet instability l U(y) = sech 2 (y) - Symmetric jet Again periodic boundaries

11. 6/29/201511 Cylinder wake results Shear Layer Instability in a Cylinder Wake Re > 1000-2000 Transition point from Convective to Absolute Instability

12. 6/29/201512 Frequency Prediction for a Cylinder Wake l Numerical Stability Analysis based on Time-Mean Flow Extract velocity profiles across wake Analyze using parallel stability analysis to predict Strouhal number ExperimentsRayleigh equationDNS

13. 6/29/201513 Interesting Recent Work l Barkley (2006 EuroPhys L) Time mean wake is neutrally stable - preferred frequency corresponds to observed Strouhal number to within 1% l Chomaz, Huerre, Monkewitz… Extension to non-parallel wakes… l Pier (2002, JFM) Non-linear stability modes to predict observed shedding frequency of a cylinder wake l 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.

14. 6/29/201514 Basic Stability Theory 2: Absolute & Convective Instability l Background Generally, part of a wake may be convectively unstable and part may be absolutely unstable » Recall l Convective instability means a disturbance will die out locally but will grow in amplitude as it convects downstream. »Think of shear layer vortices l 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)

15. 6/29/201515 Absolute & Convective unstable zones 2D vortex street behind a circular cylinder at Re = 140 Photograph: S. Taneda (Van Dyke 1982) Absolutely unstable Convectively unstable Saturated state Either - pre-shedding or time-mean wake Velocity profiles on vertical lines used for analysis

16. 6/29/201516 Selection of the wake frequency l Problem: Wake absolutely unstable over a finite spatial range. Prediction of frequency at any point in this range. So what is the selected frequency? l 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.

17. 6/29/201517 Selection of wake frequency - Saddle Point Criterion l 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.

18. 6/29/201518 Test Case - Flow over trailing edge forming a wake l Hammond and Redekopp (JFM 1997) Considered the general case below, but » Focus on symmetric wake without base suction.

19. 6/29/201519 Linear theory assumptions l Is the wake parallel? This indicates how parallel the wake is at Re=160

20. 6/29/201520 Frequency prediction with downstream distance l 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

21. 6/29/201521 Saddle point prediction l 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

22. 6/29/201522 Accuracy of saddle point prediction l 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 l 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.

23. 6/29/201523 Linear theory - inviscid and viscous l Predictions from Hammond and Redekopp (1997) Inviscid = Rayleigh equation on downstream profiles Viscous = Orr-Sommerfeld equation on downstream profiles » Re = 160.

24. 6/29/201524 Saturation of wake (Landau Model) l Further points: Wake frequency varies as the wake saturates… Frequency variation Based on Landau equation Supercritical transition Wake saturating….

25. 6/29/201525 Frequency Prediction for a Circular Cylinder Wake l Numerical Stability Analysis based on Time-Mean Flow Extract velocity profiles across wake Analyze using parallel stability analysis to predict Strouhal number ExperimentsRayleigh equationDNS

26. 6/29/201526 Inadequacy of theory? l 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… l 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

27. 6/29/201527 Non-linear theory l 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

28. 6/29/201528 Frequency predictions based on near-parallel, inviscid assumption l 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 on Steady flow Max amplication (Pierrehumbert)

29. 6/29/201529 Global stability analysis l Prediction based on Global instability analysis of time-mean wake. (Barkley 2006). Match with experiments & DNS For wake frequency Predicted mode is neutrally stable…