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Structural sensitivity calculated with a local stability analysis

Department of Engineering. LMFA, CNRS - École Centrale de Lyon. Structural sensitivity calculated with a local stability analysis. Matthew Juniper and Beno î t Pier.

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Structural sensitivity calculated with a local stability analysis

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  1. Department of Engineering LMFA, CNRS - École Centrale de Lyon Structural sensitivity calculated with a local stability analysis Matthew Juniper and Benoît Pier

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  3. The steady flow around a cylinder at Re = 50 is unstable. The linear global mode frequency and growth rate can be calculated with a 2D eigenvalue analysis. Giannetti & Luchini, JFM (2007), base flow, Re = 50 D. C. Hill (1992) NASA technical memorandum 103858

  4. The structural sensitivity can also be calculated - Giannetti and Luchini (JFM 2007): “where in space a modification in the structure of the problem is able to produce the greatest drift of the eigenvalue”. Giannetti & Luchini, JFM (2007), Receptivity to spatially localized feedback

  5. overlap The receptivity to spatially localized feedback is found by overlappingthe direct global mode and the adjoint global mode. Direct global mode Adjoint global mode Receptivity to spatially localized feedback Giannetti & Luchini, JFM (2007)

  6. continuous direct LNS* discretized direct LNS* direct global mode base flow The direct global mode is calculated by linearizing the Navier-Stokes equations around a steady base flow, then discretizing and solving a 2D eigenvalue problem. * LNS = Linearized Navier-Stokes equations

  7. continuous direct LNS* discretized direct LNS* base flow adjoint global mode The adjoint global mode is found in a similar way. The discretized adjoint LNS equations can be derived either from the continuous adjoint LNS equations or the discretized direct LNS equations. direct global mode continuous adjoint LNS* discretized adjoint LNS* * LNS = Linearized Navier-Stokes equations

  8. continuous direct LNS* discretized direct LNS* DTO / AFD base flow OTD / FDA adjoint global mode The adjoint global mode is found in a similar way. The discretized adjoint LNS equations can be derived either from the continuous adjoint LNS equations or the discretized direct LNS equations. direct global mode continuous adjoint LNS* discretized adjoint LNS* * LNS = Linearized Navier-Stokes equations DTO / AFD = Discretize then Optimize (Bewley 2001) / Adjoint of Finite Difference (Sirkes & Tziperman 1997) OTD / FDA = Optimize then Discretize (Bewley 2001) / Finite Difference of Adjoint (Sirkes & Tziperman 1997)

  9. continuous direct LNS* discretized direct LNS* direct global mode base flow The direct global mode is calculated by linearizing the Navier-Stokes equations around a steady base flow, then discretizing and solving a 2D eigenvalue problem. * LNS = Linearized Navier-Stokes equations

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

  11. A linear global analysis – e.g. wake flows in papermaking(by O. Tammisola and F. Lundell at KTH, Stockholm) 1. Discretize 2. Generate the linear evolution matrix 3. Calculate its eigenvalues and eigenvectors (eigenvalues with positive imaginary part are unstable)

  12. A linear global analysis – e.g. wake flows in papermaking(by O. Tammisola and F. Lundell at KTH, Stockholm) d dt M = x x 90,0002 1. Discretize x = 2. Generate the linear evolution matrix 3. Calculate its eigenvalues and eigenvectors (eigenvalues with positive imaginary part are unstable) N 2 Absolute/convective instabilities of axial jet/wake flows with surface tension

  13. A linear local analysis – e.g. wake flows in papermaking d dt M = x x 90,0002 1. Slice the flow 2. Calculate the absolute growth rate of each slice 3. Work out the global complex frequency 4. Calculate the response of each slice at that frequency 5. Stitch the slices back together again Absolute/convective instabilities of axial jet/wake flows with surface tension

  14. A linear local analysis – e.g. wake flows in papermakingAt Re = 400, the local analysis gives almost exactly the same result as the global analysis Base Flow Absolute growth rate global analysis local analysis

  15. 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. local global local Re = 100 global Re Juniper, Tammisola, Lundell (2011) , comparison of local and global analyses for co-flow wakes Giannetti & Luchini, JFM (2007), comparison of local and global analyses for the flow behind a cylinder

  16. 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. global analysis local analysis

  17. continuous direct LNS* discretized direct LNS* DTO / AFD base flow OTD / FDA adjoint global mode The adjoint global mode is found in a similar way. The discretized adjoint LNS equations can be derived either from the continuous adjoint LNS equations or the discretized direct LNS equations. direct global mode continuous adjoint LNS* discretized adjoint LNS* * LNS = Linearized Navier-Stokes equations DTO / AFD = Discretize then Optimize (Bewley 2001) / Adjoint of Finite Difference (Sirkes & Tziperman 1997) OTD / FDA = Optimize then Discretize (Bewley 2001) / Finite Difference of Adjoint (Sirkes & Tziperman 1997)

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

  19. continuous direct LNS* continuous direct O-S** discretized direct O-S** direct global mode adjoint global mode The adjoint global mode can also be estimated from a local stability analysis, via four different routes. 1 2 3 4 base flow continuous adjoint LNS* continuous adjoint O-S** discretized adjoint O-S** * LNS = Linearized Navier-Stokes equations ** O-S = Orr-Sommerfeld equation

  20. continuous direct G-L* We compared routes 1 and 4 rigorously with the Ginzburg-Landau equation, from which we derived simple relationships between the local properties of the direct and adjoint modes. These carry over to the Navier-Stokes equations. direct global mode base flow 1 4 continuous adjoint G-L* adjoint global mode * G-L = Ginzburg-Landau equation

  21. adjoint mode direct mode 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. adjoint mode direct mode

  22. continuous direct LNS* continuous direct O-S** discretized direct O-S** direct global mode adjoint global mode The adjoint global mode can also be estimated from a local stability analysis, via four different routes. 1 2 3 4 base flow continuous adjoint LNS* continuous adjoint O-S** discretized adjoint O-S** * LNS = Linearized Navier-Stokes equations ** O-S = Orr-Sommerfeld equation

  23. 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.

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

  25. By overlapping the direct and adjoint modes, we can get the sensitivities. This is equivalent to the calculation of Giannetti & Luchini (2007) but takes much less time.

  26. 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) 10 0

  27. 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.

  28. This shows that the wavemaker region defined by Pier, Chomaz etc. from the local analysis is equivalent to that defined by Giannetti & Luchini from the global analysis.

  29. spare slides

  30. direct mode Reminder of the direct mode direct global mode

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

  32. 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. receptivity to spatially-localized feedback receptivity to spatially-localized feedback Giannetti & Luchini, JFM (2007), global analysis Current study, local analysis

  33. direct mode 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. • leads to • quick structural sensitivity calculations for slowly-varying flows • quasi-3D structural sensitivity (?)

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

  35. continuous direct LNS* continuous direct O-S** discretized direct O-S** The absolute growth rate (ω0) is calculated as a function of streamwise distance. The linear global mode frequency (ωg) is estimated. The wavenumber response, k+/k-, of each slice at ωg is calculated. The direct global mode follows from this. direct global mode base flow

  36. The absolute growth rate (ω0) is calculated as a function of streamwise distance. The linear global mode frequency (ωg) is estimated. The wavenumber response, k+/k-, of each slice at ωg is calculated. The direct global mode follows from this. direct global mode

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

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  44. 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. adjoint global mode adjoint global mode Giannetti & Luchini, JFM (2007), global analysis Current study, local analysis

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