Combined Local and Global Stability Analyses

(work in progress)

Matthew Juniper, Ubaid Qadri, Dhiren Mistry, Benoît Pier, Outi Tammisola,Fredrik Lundell

Department of Engineering

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

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

We have compared global and local analyses for simple wake flows(with O. Tammisola and F. Lundell at KTH, Stockholm)

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

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

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.

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.)

local analysis

global analysis

global modegrowth rate

The combined local and global analysis explains why confinement destabilizes these wake flows at Re ~ 100.

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)

Recently, we have looked at swirling jet/wake flows

Ruith, Chen, Meiburg & Maxworthy (2003) JFM 486Gallaire, 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

(first direct eigenmode)

(absolute growth rate, local analysis)

(first direct eigenmode)

(first direct eigenmode)

centre of global mode

(global analysis)

(global analysis)

(global analysis)

(first adjoint eigenmode)

(first adjoint eigenmode)

(first adjoint eigenmode)

(absolute growth rate)

(global analysis)

(global analysis)

(global analysis)

(absolute growth rate)

(global analysis)

(global analysis)

(global analysis)

(global analysis)

Axial momentum

Radial momentum

Azimuthal momentum

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.

Giannetti & Luchini, JFM (2007), global analysis Current study, local analysis

receptivity to spatially-localized feedback receptivity to spatially-localized feedback

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

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.

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

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.

continuousdirect LNS*

continuousdirect O-S**

discretizeddirect O-S**

base flow

* LNS = Linearized Navier-Stokes equations** O-S = Orr-Sommerfeld equation

direct global mode

WKBJ

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

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

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.

Giannetti & Luchini, JFM (2007), global analysis Current study, local analysis

direct global mode direct global mode

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

global modegrowth rate(no slip case)

local analysis

global analysis

local analysis

global analysis

global modegrowth rate(perfectslip case)

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