Combined Local and Global Stability Analyses (work in progress)

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Department of Engineering. Combined Local and Global Stability Analyses (work in progress). Matthew Juniper, Ubaid Qadri, Dhiren Mistry, Beno î t Pier, Outi Tammisola, Fredrik Lundell. continuous direct LNS*. discretized direct LNS*. base flow. adjoint global mode.

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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

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

discretized

* LNS = Linearized Navier-Stokes equations

continuous

direct LNS*

continuous

direct O-S**

discretized

direct O-S**

direct 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

continuous

discretized

* 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)

(global analysis)

(global analysis)

(absolute growth rate)

(global analysis)

(global analysis)

(global analysis)

Axial momentum

Azimuthal momentum

Sensitivity of growth rate

Sensitivity of frequency

max sensitivity

(global analysis)

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

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

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

The adjoint global mode can also be estimated from a local stability analysis.

base flow

continuous

continuous

discretized

* 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

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

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.

• 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

local analysis

(Juniper)

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.

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