Department of Engineering
LMFA, CNRS - École Centrale de Lyon
Structural sensitivity calculated with a
local stability analysis
Matthew Juniper and Benoît Pier
.
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
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.
Direct global mode
Adjoint global mode
overlap
Receptivity to spatially localized feedback
Giannetti & Luchini, JFM (2007)
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.
continuous
direct LNS*
discretized
direct LNS*
direct 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.
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
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*
discretized
direct LNS*
direct global mode
DTO / AFD
base flow
OTD / FDA
continuous
adjoint LNS*
discretized
adjoint LNS*
adjoint global mode
* 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)
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.
continuous
direct LNS*
discretized
direct LNS*
direct global mode
base flow
* LNS = Linearized Navier-Stokes equations
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
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)
1. Discretize
x=
2. Generate the linear evolution matrix
d
x =
dt
3. Calculate its eigenvalues and eigenvectors
(eigenvalues with positive imaginary part are unstable)
Absolute/convective instabilities of axial
jet/wake flows with surface tension
M
x
90,0002
N2
A linear local analysis – e.g. wake flows in papermaking
1. Slice the flow
2. Calculate the absolute growth rate of each slice
d
x =
dt
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
M
90,0002
x
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
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 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*
discretized
direct LNS*
direct global mode
DTO / AFD
base flow
OTD / FDA
continuous
adjoint LNS*
discretized
adjoint LNS*
adjoint global mode
* 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)
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 can also be estimated from a local stability analysis, via
four different routes.
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 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*
direct global mode
base flow
1
4
continuous
adjoint G-L*
adjoint global mode
* G-L = Ginzburg-Landau equation
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
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**
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
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.
receptivity to spatially-localized feedback
Giannetti & Luchini, JFM (2007), global analysis
receptivity to spatially-localized feedback
Current study, local analysis
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
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
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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.
adjoint global mode
Giannetti & Luchini, JFM (2007), global analysis
adjoint global mode
Current study, local analysis