Transcript global analysis

Department of Engineering
Combined Local and Global Stability Analyses
(work in progress)
Matthew Juniper, Ubaid Qadri, Dhiren Mistry, Benoît Pier, Outi Tammisola,
Fredrik Lundell
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*
discretized
direct LNS*
direct global mode
base flow
continuous
adjoint LNS*
discretized
adjoint LNS*
adjoint global mode
* LNS = Linearized Navier-Stokes equations
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**
direct global mode
base flow
1
2
3
continuous
adjoint LNS*
continuous
adjoint O-S**
discretized
adjoint O-S**
4
adjoint global mode
* 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 overpredicts the global growth rate. This highlights the weakness of the parallel flow
assumption.
Re = 100
Re
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
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
wavemaker position
absolute growth rate
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)
(first direct eigenmode)
(global analysis)
(first direct eigenmode)
(global analysis)
(first direct eigenmode)
(global analysis)
centre of global mode
(absolute growth rate)
(first adjoint eigenmode)
(global analysis)
(first adjoint eigenmode)
(global analysis)
(first adjoint eigenmode)
(global analysis)
(absolute growth rate)
(global analysis)
(global analysis)
(global analysis)
Axial momentum
Radial momentum
Azimuthal momentum
(global analysis)
Sensitivity of growth rate
Sensitivity of frequency
max sensitivity
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
Giannetti & Luchini, JFM (2007), global analysis
receptivity to spatially-localized feedback
Current study, local analysis
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
direct mode
adjoint 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.
The adjoint global mode can also be estimated from a local stability analysis.
continuous
direct LNS*
continuous
direct O-S**
discretized
direct O-S**
direct global mode
base flow
continuous
adjoint LNS*
continuous
adjoint O-S**
discretized
adjoint O-S**
adjoint global mode
* 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.
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
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.
WKBJ
continuous
direct LNS*
continuous
direct O-S**
discretized
direct O-S**
direct global mode
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)
0
10
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.
continuous
direct LNS*
continuous
direct O-S**
discretized
direct O-S**
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
Giannetti & Luchini, JFM (2007), global analysis
direct global mode
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
Giannetti & Luchini, JFM (2007), global analysis
adjoint global mode
Current study, local analysis
global mode
growth rate
(perfect
slip case)
local analysis
global analysis
global mode
growth rate
(no slip case)
local analysis
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
wavemaker position
absolute growth rate