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

.

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

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

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

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 continuous direct LNS* direct global mode discretized direct LNS* base flow

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. 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

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. * 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) continuous direct LNS* direct global mode discretized direct LNS* continuous adjoint LNS* discretized adjoint LNS* adjoint global mode base flow DTO / AFD OTD / FDA

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 continuous direct LNS* direct global mode discretized direct LNS* base flow

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

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)

A linear global analysis – e.g. wake flows in papermaking (by O. Tammisola and F. Lundell at KTH, Stockholm) Absolute/convective instabilities of axial jet/wake flows with surface tension = M xx ddtddt 90,000 2 x = N 2 1. Discretize 2. Generate the linear evolution matrix 3. Calculate its eigenvalues and eigenvectors (eigenvalues with positive imaginary part are unstable)

A linear local analysis – e.g. wake flows in papermaking Absolute/convective instabilities of axial jet/wake flows with surface tension = M xx ddtddt 90, 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

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

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

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.

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. * 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) continuous direct LNS* direct global mode discretized direct LNS* continuous adjoint LNS* discretized adjoint LNS* adjoint global mode base flow DTO / AFD OTD / FDA

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

The adjoint global mode can also be estimated from a local stability analysis, via four different routes. 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

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. continuous direct G-L* base flow * G-L = Ginzburg-Landau equation continuous adjoint G-L* adjoint global mode direct global mode 14

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

The adjoint global mode can also be estimated from a local stability analysis, via four different routes. 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

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.

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

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.

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

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.

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.

spare slides

Reminder of the direct mode direct mode direct global mode

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

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

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 (?)

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

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

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

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

bla bla bla

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