1. Computing Internal and External Flows Undergoing Instability and Bifurcations V. V. S. N. Vijay, Neelu Singh and T. K. Sengupta High Performance Computing Laboratory Department of Aerospace Engineering I. I. T. Kanpur, Kanpur 208016 INDIA

2. Organization of The Presentation  Instability, Bifurcation and Multi-Modal Dynamics.  Hopf- bifurcation and Landau-Stuart-Eckhaus (LSE) Equation.  Numerical Methods: Aliasing Error, Dispersion Error.  Evidence of Multiple Hopf-Bifurcations for Flow Past a Cylinder.  Multiple Bifurcations in Lid-Driven Cavity Flow.  Dynamical System Approach - Proper Orthogonal Decomposition (POD).  Free Stream Turbulence (FST) : Effects and Its Modeling.  Conclusions.

3. Bifurcation and Instabilities Flow Past a Circular Cylinder, Re = 60Flow Inside a Square Cavity, Re = 8500

4. Landau Equation and Hopf-Bifurcation  If disturbance velocity is given by (1)‏  And if its amplitude is given by (2)‏ then, linearized evolution equation for amplitude is (3)‏  Landau (1944) proposed a corresponding nonlinear evolution equation (4)‏

5. Landau Equation and Hopf-Bifurcation  Above and near Re cr, (5)‏  For Eq. (4), an equilibrium exists (using Eq. (5)) that is given by (6)‏  Variation of |A e | beyond Re cr is parabolic and is symptomatic of Hopf- bifurcation.

6. Analysis of Numerical Methods Non-dimensional effectiveness of convection terms Dissipation Discretization

7. Error Dynamics for Convection Discretization

8. Evidences of Multiple Modes in Flow Past a Cylinder Present Computations, Re = 60Expt. by Strykowski (1986), Re = 49 Presence of multiple modes is evident in both the cases.

9. Multiple Hopf-Bifurcations and Its Modeling in Flow past a Cylinder 84.615-0.217.6-9.5825.7133 -250 63.868-7.2105-5124.480 -133 51.934136-67492.67.6951.93 – 80.0 Re cr k 4 x 10 9 k 3 x 10 8 k 2 x 10 6 k 1 x 10 4 Re Strykowski’s Data (1986)‏

10. Further Evidences of Multiple Bifurcations in Flow Past a Cylinder  Vortex shedding began at Re = 65 in Homann’s experiment.  The first bifurcation was suppressed by use of highly viscous oil as working media.  Provansal et. al (1987) also have reported different critical Reynolds number in the same tunnel.  Results due to Homann (1936)‏ 6911 cm10.0 630.8 cm12.5 580.6 cm16.7 Re cr DL/D

11. Multiple Bifurcations in Lid-Driven Cavity Flow

12. Proper Orthogonal Decomposition (POD) of DNS Data  A spatio-temporal fluid dynamical system can be analyzed by POD. energy/ enstrophy  We project the space-time dependent field into Fourier space with definitive energy/ enstrophy content.  Disturbance vorticity is represented as an ensemble of M snapshots  The covariance matrix as given below  The eigenvalues and eigenvectors of [R] provide the enstrophy content and the POD modes, respectively.

13. Flow Past a Circular Cylinder : Cumulative Enstrophy Distribution for Different Time Range

14. Flow in LDC : Cumulative Enstrophy Distribution for Different Time Range

15. Leading Eigenvalues For Flow Past a Cylinder Obtained by POD of Enstrophy for Re = 60 -- 0.0026040.021043 0.0026040.021151 0.0367050.04038 0.0371150.04224 0.2809160.17497 0.2832520.17554 --0.48416 2.4149351.61711 2.4975581.66274 Eigenvalues (t = 351-430)‏Eigenvalues ( t = 200-430)‏

16. Leading Eigenvalues for Lid-Driven Cavity Flow Obtained by POD of Enstrophy for Re = 8500 -- 0.00004470.0002115 0.00005480.0002225 0.0002899-- 0.00032810.0003878 0.0012810.0020369 0.0193140.013615 0.0235610.016495 Eigenvalues (t = 300-500)‏Eigenvalues (t = 200-500)‏

17. Relating LSE Equation with POD Modes  The LSE modes are related to the POD modes by,  For example, two leading LSE modes are governed by,  These two constitute stiff-differential equations that can be reformulated as where

18. Amplitude Functions of POD Modes for Flow Past a Cylinder for Re = 60

19. Amplitude Functions of POD Modes for Flow in a Square-Cavity for Re = 8500

20. Leading Eigenfunctions of POD Modes for Flow Past a Cylinder for Re = 60 and t : 200-430

21. Leading Eigenfunctions of POD Modes for Flow Past a Cylinder for Re = 60 and t : 200-430 (contd.) ‏

22. Multi-Modal Feature in the Wake  Vorticity variation and FFT of vorticity data shown at different streamline positions ( 1.5D, 3D, 6D and 10D )‏  Note the presence of additional modes seen in FFT of data at 6D and 10D.

23. Vortex Topology Inside a Lid-Driven Square Cavity

24. Leading Eigenfunctions of POD Modes for Flow in a Square-Cavity for Re = 8500

25. Higher Eigenfunctions for LDC Flow for Re = 8500

26. Different Critical Reynolds Number Reported in Different Facilities for Flow Past a Circular Cylinder  Different Re cr have been reported : Batchelor (1988) – 40 Landau & Lifshitz (1959) - 34 Reported byCritical Reynolds Number (Re cr )‏ Kovasznay (1949)‏40 Strykowski, Sreenivasan & Olinger (1990)‏ Between 45 - 46 Schumm, Berger & Monkewitz (1994)‏ 46.7 Roshko (1954)‏50 Kiya et al. (1982)‏52 Tordella & Cancelli (1991)‏53 Homann (1936)‏65.3

27. Extrinsic or Intrinsic Dynamics ? D = 5mm, U  = 17cm/s, Re = 53D = 1.8mm, U  = 46.9cm/s, Re = 53 Note: The input disturbance at the Strouhal number on the left frame is one order of magnitude larger as compared to the case on the right.

28. Modeling Free Stream Turbulence (FST) ‏  Higher order statistics are modeled by the 1st-order moving-average time series model (Fuller,1978) with the low frequency noise added separately.  FST is modeled by the following equation: where the first two terms are given by Gaussian distribution and the last term represents low frequency component of noise and u’ is the stream-wise disturbance component of velocity.

29. FFT of Experimental and Synthetic FST Data Experimental Disturbance VelocitySynthetic Disturbance Velocity

30. Effect of FST on Results Variation of lift and drag coefficient with time with and without FST for Re = 100 Comparison of saturation amplitude of lift coefficient (peak-to-peak) for the cases of without and with FST (Tu = 0.06%)‏ Note: The noise causes early onset of asymmetry and very high frequency fluctuations in drag data.

31. Effect of Different Levels of FST  Dynamic range increases directly with Tu.  Higher than the shown levels, could display completely different dynamics that is also indicated by the early onset time implying separation bubbles not allowed to form as in no-FST case.

32. Streamfunction Contours for Re = 50, With and Without FST  At this sub-critical Re, symmetric wake bubble is formed for no-FST case.  In the presence of FST, asymmetry builds up slowly from the tip of the bubble. Kelvin-Helmholtz instability  For higher FST levels, interface of the separation bubble can suffer Kelvin-Helmholtz instability interrupting vortex street formation.

33. Conclusions  Multiple Hopf-bifurcations have been shown computationally as seen in Ae vs Re plots for both the cases.  For the cylinder, we have identified the second bifurcation at Re = 63.86, that is closer to value reported by Homann (1936) at Re = 65.3.  Multi-modal behaviour is explained with the help of POD analysis for both the flows. POD analysis is also used to explain the Landau-Stuart-Eckhaus Equation. do not satisfy  We have identified anomalous modes during the instability phase for both the flows from the POD amplitude functions. These modes do not satisfy the LSE Equations and need to be modeled differently.  For LDC flow, the higher modes which do not account for significant contribution to enstrophy are seen as localized spatio-temporal structures, e.g. the triangular vortex seen in the core. extrinsic dynamics  With the help of results with FST and collective results in the literature, for flow past a cylinder we conclude that the onset of instability at first bifurcations is governed by extrinsic dynamics.

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