1. Combined Local and Global Stability Analyses (work in progress) Matthew Juniper, Ubaid Qadri, Dhiren Mistry, Benoît Pier, Outi Tammisola, Fredrik Lundell Department of Engineering

2. Global stability analyses linearize around a 2D base flow, discretize and solve a 2D matrix eigenvalue problem. (This technique would also apply to 3D flows.) continuous direct LNS* direct global mode discretized direct LNS* continuous adjoint LNS* discretized adjoint LNS* adjoint global mode base flow * LNS = Linearized Navier-Stokes equations

3. Local stability analyses use the WKBJ approximation to reduce the large 2D eigenvalue problem into a series of small 1D eigenvalue problems. continuous direct LNS* continuous direct O-S** discretized direct O-S** base flow * LNS = Linearized Navier-Stokes equations ** O-S = Orr-Sommerfeld equation continuous adjoint LNS* continuous adjoint O-S** discretized adjoint O-S** adjoint global mode direct global mode 1234

4. We have compared global and local analyses for simple wake flows (with O. Tammisola and F. Lundell at KTH, Stockholm)

5. Base Flow Absolute growth rate global analysis local analysis At Re = 400, the local analysis gives almost exactly the same result as the global analysis

6. The weak point in this analysis is that the local analysis consistently over- predicts the global growth rate. This highlights the weakness of the parallel flow assumption. Giannetti & Luchini, JFM (2007), comparison of local and global analyses for the flow behind a cylinder Juniper, Tammisola, Lundell (2011), comparison of local and global analyses for co-flow wakes Re Re = 100 local global

7. global analysis local analysis If we re-do the final stage of the local analysis taking the complex frequency from the global analysis, we get exactly the same result.

8. absolutely unstable region wavemaker position absolute growth rate The local analysis gives useful qualitative information, which we can use to explain the results seen in the global analysis. (Here, the confinement increases as you go down the figure.)

9. local analysis global analysis global mode growth rate The combined local and global analysis explains why confinement destabilizes these wake flows at Re ~ 100.

10. By overlapping the direct and adjoint modes, we can get the structural sensitivity with a local analysis. This is equivalent to the global calculation of Giannetti & Luchini (2007) but takes much less time. Giannetti & Luchini, JFM (2007), structural sensitivity of the flow behind a cylinder (global analysis) structural sensitivity of a co-flow wake (local analysis)

11. Recently, we have looked at swirling jet/wake flows Ruith, Chen, Meiburg & Maxworthy (2003) JFM 486 Gallaire, Ruith, Meiburg, Chomaz & Huerre (2006) JFM 549

12. At entry (left boundary) the flow has uniform axial velocity, zero radial velocity and varying swirl. (base flow)

13. (absolute growth rate)

14. (absolute growth rate, local analysis) (spatial growth rate at global mode frequency from local analysis) centre of global mode wavemaker region

15. (first direct eigenmode) (absolute growth rate, local analysis) (first direct eigenmode) centre of global mode (global analysis)

16. (first adjoint eigenmode) (absolute growth rate) (global analysis)

17. (absolute growth rate) (global analysis)

18. Axial momentum Radial momentum Azimuthal momentum Sensitivity of growth rate Sensitivity of frequency max sensitivity

19. spare slides

20. Similarly, for the receptivity to spatially-localized feedback, the local analysis agrees reasonably well with the global analysis in the regions that are nearly locally parallel. Giannetti & Luchini, JFM (2007), global analysisCurrent study, local analysis receptivity to spatially-localized feedback

21. The adjoint mode is formed from a k - branch upstream and a k + branch downstream. We show that the adjoint k - branch is the complex conjugate of the direct k + branch and that the adjoint k + is the c.c. of the direct k - branch. direct mode adjoint mode direct mode

22. Here is the direct mode for a co-flow wake at Re = 400 (with strong co-flow). The direct global mode is formed from the k- branch (green) upstream of the wavemaker and the k+ branch (red) downstream.

23. The adjoint global mode can also be estimated from a local stability analysis. continuous direct LNS* continuous direct O-S** discretized direct O-S** base flow * LNS = Linearized Navier-Stokes equations ** O-S = Orr-Sommerfeld equation continuous adjoint LNS* continuous adjoint O-S** discretized adjoint O-S** adjoint global mode direct global mode

24. The adjoint global mode is formed from the k+ branch (red) upstream of the wavemaker and the k- branch (green) downstream

25. This shows that the ‘core’ of the instability (Giannetti and Luchini 2007) is equivalent to the position of the branch cut that emanates from the saddle points in the complex X-plane.

26. Reminder of the direct mode direct mode direct global mode

27. So, once the direct mode has been calculated, the adjoint mode can be calculated at no extra cost. direct mode adjoint mode adjoint global mode

28. In conclusion, the direct mode is formed from the k -- branch upstream and the k + branch downstream, while the adjoint mode is formed from the k + branch upstream and the k -- branch downstream. direct mode leads to quick structural sensitivity calculations for slowly-varying flows quasi-3D structural sensitivity (?)

29. The direct global mode can also be estimated with a local stability analysis. This relies on the parallel flow assumption. continuous direct LNS* continuous direct O-S** discretized direct O-S** base flow * LNS = Linearized Navier-Stokes equations ** O-S = Orr-Sommerfeld equation direct global mode WKBJ

30. Preliminary results indicate a good match between the local analysis and the global analysis u,u_adj overlap from local analysis (Juniper) u,u_adj overlap from global analysis (Tammisola & Lundell) 0 10

31. The absolute growth rate (ω 0 ) is calculated as a function of streamwise distance. The linear global mode frequency (ω g ) is estimated. The wavenumber response, k + /k -, of each slice at ω g is calculated. The direct global mode follows from this. continuous direct LNS* continuous direct O-S** discretized direct O-S** base flow direct global mode

32. The absolute growth rate (ω 0 ) is calculated as a function of streamwise distance. The linear global mode frequency (ω g ) is estimated. The wavenumber response, k + /k -, of each slice at ω g is calculated. The direct global mode follows from this. direct global mode

33. For the direct global mode, the local analysis agrees very well with the global analysis. Giannetti & Luchini, JFM (2007), global analysisCurrent study, local analysis direct global mode

36. For the adjoint global mode, the local analysis predicts some features of the global analysis but does not correctly predict the position of the maximum. This is probably because the flow is not locally parallel here. Giannetti & Luchini, JFM (2007), global analysisCurrent study, local analysis adjoint global mode

37. global mode growth rate (no slip case) local analysis global analysis local analysis global analysis global mode growth rate (perfect slip case)

38. The local analysis gives useful qualitative information, which we can use to explain the results seen in the global analysis. (Here, the central speed reduces as you go down the figure.) absolutely unstable region wavemaker position absolute growth rate