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

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Structural sensitivity calculated with a local stability analysis

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


overlap Giannetti and Luchini (JFM 2007): “

The receptivity to spatially localized feedback is found by overlappingthe direct global mode and the adjoint global mode.

Direct global mode

Adjoint global mode

Receptivity to spatially localized feedback

Giannetti & Luchini, JFM (2007)


continuous Giannetti and Luchini (JFM 2007): “

direct LNS*

discretized

direct LNS*

direct global mode

base flow

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 Giannetti and Luchini (JFM 2007): “

direct LNS*

discretized

direct LNS*

base flow

adjoint global mode

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.

direct global mode

continuous

adjoint LNS*

discretized

adjoint LNS*

* LNS = Linearized Navier-Stokes equations


continuous Giannetti and Luchini (JFM 2007): “

direct LNS*

discretized

direct LNS*

DTO / AFD

base flow

OTD / FDA

adjoint global mode

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.

direct global mode

continuous

adjoint LNS*

discretized

adjoint LNS*

* 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 Giannetti and Luchini (JFM 2007): “

direct LNS*

discretized

direct LNS*

direct global mode

base flow

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 Giannetti and Luchini (JFM 2007): “

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


A linear global analysis – e.g. wake flows in papermaking Giannetti and Luchini (JFM 2007): “(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 Giannetti and Luchini (JFM 2007): “(by O. Tammisola and F. Lundell at KTH, Stockholm)

d

dt

M

=

x

x

90,0002

1. Discretize

x =

2. Generate the linear evolution matrix

3. Calculate its eigenvalues and eigenvectors

(eigenvalues with positive imaginary part are unstable)

N 2

Absolute/convective instabilities of axial jet/wake flows with surface tension


A linear local analysis e g wake flows in papermaking
A linear local analysis – e.g. wake flows in papermaking Giannetti and Luchini (JFM 2007): “

d

dt

M

=

x

x

90,0002

1. 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

Absolute/convective instabilities of axial jet/wake flows with surface tension


A linear local analysis – e.g. wake flows in papermaking Giannetti and Luchini (JFM 2007): “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


continuous complex frequency from the global analysis, we get exactly the same result.

direct LNS*

discretized

direct LNS*

DTO / AFD

base flow

OTD / FDA

adjoint global mode

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.

direct global mode

continuous

adjoint LNS*

discretized

adjoint LNS*

* 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 complex frequency from the global analysis, we get exactly the same result.

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


continuous complex frequency from the global analysis, we get exactly the same result.

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, via four different routes.

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


continuous complex frequency from the global analysis, we get exactly the same result.

direct G-L*

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.

direct global mode

base flow

1

4

continuous

adjoint G-L*

adjoint global mode

* G-L = Ginzburg-Landau equation


adjoint mode complex frequency from the global analysis, we get exactly the same result.

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


continuous complex frequency from the global analysis, we get exactly the same result.

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, via four different routes.

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


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 strong co-flow). The direct 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
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


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 etc. from the local analysis is equivalent to that defined by Giannetti & Luchini from the global analysis.


Reminder of the direct mode

direct mode etc. from the local analysis is equivalent to that defined by Giannetti & Luchini from the global analysis.

Reminder of the direct mode

direct global mode


So once the direct mode has been calculated the adjoint mode can be calculated at no extra cost

adjoint mode etc. from the local analysis is equivalent to that defined by Giannetti & Luchini from the global analysis.

direct mode

So, once the direct mode has been calculated, the adjoint mode can be calculated at no extra cost.

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

receptivity to spatially-localized feedback

Giannetti & Luchini, JFM (2007), global analysis

Current study, local analysis


direct mode feedback, the local analysis agrees reasonably well with the global analysis in the regions that are nearly locally parallel.

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 feedback, the local analysis agrees reasonably well with the global analysis in the regions that are nearly locally parallel.

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


continuous feedback, the local analysis agrees reasonably well with the global analysis in the regions that are nearly locally parallel.

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 ( feedback, the local analysis agrees reasonably well with the global analysis in the regions that are nearly locally parallel. 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
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


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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

adjoint global mode

Giannetti & Luchini, JFM (2007), global analysis

Current study, local analysis


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