Combined Local and Global Stability Analyses (work in progress)
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Transcript of Combined Local and Global Stability Analyses (work in progress)
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Combined Local and Global Stability Analyses
(work in progress)
Matthew Juniper, Ubaid Qadri, Dhiren Mistry, Benoît Pier, Outi Tammisola,Fredrik Lundell
Department of Engineering
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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.)
continuousdirect LNS*
direct global mode
discretizeddirect LNS*
continuousadjoint LNS*
discretizedadjoint LNS*
adjoint global mode
base flow
* LNS = Linearized Navier-Stokes equations
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Local stability analyses use the WKBJ approximation to reduce the large 2D eigenvalue problem into a series of small 1D eigenvalue problems.
continuousdirect LNS*
continuousdirect O-S**
discretizeddirect O-S**
base flow
* LNS = Linearized Navier-Stokes equations** O-S = Orr-Sommerfeld equation
continuousadjoint LNS*
continuousadjoint O-S**
discretizedadjoint O-S**
adjoint global mode
direct global mode
1 2 3 4
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We have compared global and local analyses for simple wake flows(with O. Tammisola and F. Lundell at KTH, Stockholm)
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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
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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 globalanalyses for the flow behind a cylinder
Juniper, Tammisola, Lundell (2011) , comparison of local and global analyses for co-flow wakes
Re
Re = 100
local
localglobal
global
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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.
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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.)
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local analysis
global analysis
global modegrowth rate
The combined local and global analysis explains why confinement destabilizes these wake flows at Re ~ 100.
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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), structuralsensitivity of the flow behind a cylinder(global analysis)
structural sensitivity of a co-flow wake(local analysis)
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Recently, we have looked at swirling jet/wake flows
Ruith, Chen, Meiburg & Maxworthy (2003) JFM 486Gallaire, Ruith, Meiburg, Chomaz & Huerre (2006) JFM 549
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At entry (left boundary) the flow has uniform axial velocity, zero radial velocity and varying swirl.
(base flow)
(base flow)
(base flow)
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(base flow)
(base flow)
(base flow)
(absolute growth rate)
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(absolute growth rate, local analysis)
(spatial growth rate at global mode frequency from local analysis)
centre of global mode
wavemaker region
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(first direct eigenmode)
(absolute growth rate, local analysis)
(first direct eigenmode)
(first direct eigenmode)
centre of global mode
(global analysis)
(global analysis)
(global analysis)
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(first adjoint eigenmode)
(first adjoint eigenmode)
(first adjoint eigenmode)
(absolute growth rate)
(global analysis)
(global analysis)
(global analysis)
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(absolute growth rate)
(global analysis)
(global analysis)
(global analysis)
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(global analysis)
Axial momentum
Radial momentum
Azimuthal momentum
Sensitivity of growth rate
Sensitivity of frequency
max sensitivity
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spare slides
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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 analysis Current study, local analysis
receptivity to spatially-localized feedback receptivity to spatially-localized feedback
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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
adjoint mode direct mode
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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.
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The adjoint global mode can also be estimated from a local stability analysis.
continuousdirect LNS*
continuousdirect O-S**
discretizeddirect O-S**
base flow
* LNS = Linearized Navier-Stokes equations** O-S = Orr-Sommerfeld equation
continuousadjoint LNS*
continuousadjoint O-S**
discretizedadjoint O-S**
adjoint global mode
direct global mode
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The adjoint global mode is formed from the k+ branch (red) upstream of the wavemaker and the k- branch (green) downstream
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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.
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Reminder of the direct mode
direct mode
direct global mode
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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
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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 (?)
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The direct global mode can also be estimated with a local stability analysis. This relies on the parallel flow assumption.
continuousdirect LNS*
continuousdirect O-S**
discretizeddirect O-S**
base flow
* LNS = Linearized Navier-Stokes equations** O-S = Orr-Sommerfeld equation
direct global mode
WKBJ
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Preliminary results indicate a good match between the local analysis and the global analysis
u,u_adj overlap fromlocal analysis(Juniper)
u,u_adj overlap fromglobal analysis(Tammisola & Lundell)
0 10
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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.
continuousdirect LNS*
continuousdirect O-S**
discretizeddirect O-S**
base flow
direct global mode
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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
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For the direct global mode, the local analysis agrees very well with the global analysis.
Giannetti & Luchini, JFM (2007), global analysis Current study, local analysis
direct global mode direct global mode
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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 analysis Current study, local analysis
adjoint global mode adjoint global mode
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global modegrowth rate(no slip case)
local analysis
global analysis
local analysis
global analysis
global modegrowth rate(perfectslip case)
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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