Combined Local and Global Stability Analyses (work in progress)

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

continuous direct LNS* discretized direct LNS* base flow adjoint global mode 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.) direct global mode continuous adjoint LNS* discretized adjoint LNS* * LNS = Linearized Navier-Stokes equations

continuous direct LNS* continuous direct O-S** discretized direct O-S** direct global mode adjoint global mode Local stability analyses use the WKBJ approximation to reduce the large 2D eigenvalue problem into a series of small 1D eigenvalue problems. 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

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

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

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

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.) absolutely unstable region absolute growth rate wavemaker position

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

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)

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

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

(base flow) (base flow) (base flow) (absolute growth rate)

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

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

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

(absolute growth rate) (global analysis) (global analysis) (global analysis)

Axial momentum Radial momentum Azimuthal momentum Sensitivity of growth rate Sensitivity of frequency max sensitivity (global analysis)

spare slides

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

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

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.

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

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

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.

direct mode Reminder of the direct mode direct global mode

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

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 (?)

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

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

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

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

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

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

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

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 absolute growth rate wavemaker position

Combined Local and Global Stability Analyses (work in progress)