Bifurcations in a swirling flow*

1

Bifurcations in a swirling flow*

Thèse de doctorat présentée pour obtenir le grade de

Docteur de l’École Polytechnique

par Elena Vyazmina

* Bifurcations d’un écoulement tournant

13 juillet 2010

Directeurs de thèse: Jean-Marc Chomaz et Peter Schmid

2

Swirling flow

Introduction

→ Swirling flow

→ Vortex breakdown

→ Applications

→ Classification

→ Problematic

Numerical method

2D vortex breakdown

3D vortex breakdown

Active open-loop control

Summary and perspectives

A flow is said to be ’swirling’ when its mean direction is aligned with its rotation axis, implying helical particle trajectories.

3

Vortex breakdown: definition

Main Features:

• core of vorticity and axial velocity

• stagnation point

• reverse flow or “recirculation bubble”

Vortex breakdown is defined as a dramatic change in the structure of the flow core, with the appearance of stagnation points followed by regions of reversed flow referred to as the vortex breakdown bubble.

Free jet: Gallaire (2002) Rotating cylinder, fixed lid: S. Harris Introduction

→ Swirling flow


→ Applications

→ Classification

→ Problematic

Numerical method

2D vortex breakdown

3D vortex breakdown



4

Applications

Combustion burner

Aeronautics

TornadoIntroduction

→ Swirling flow


→ Applications

→ Classification

→ Problematic

Numerical method

2D vortex breakdown

3D vortex breakdown



5

Vortex breakdown: classification

Bubble or axisymmetric form

Faler & Leibovich (1977)


Spiral form

Billant et al. (1998)

Cone form


Double helix formIntroduction

→ Swirling flow


→ Applications

→ Classification

→ Problematic

Numerical method

2D vortex breakdown

3D vortex breakdown



6

ProblematicPipe

Experiments: Sarpkaya (1971), Faler & Leibovich (1978), Leibovich (1978,1983), Althaus (1990), Escudier & Zehnder (1982)…

Theoretical and numerical investigations: Squire (1960), Benjamin (1962,1965,1967), Batchelor (1967), Escudier & Keller (1983), Keller et al. (1985), Beran (1989), Beran & Culick (1992), Lopez (1994), Wang & Rusak and coll. (1996, 1997, 1998, 2000, 2001, 2004), Buntine & Saffman (1995), Derzho & Grimshaw (2002), Herrada & Fernandez-Feria (2006)…

Introduction

→ Swirling flow


→ Applications

→ Classification

→ Problematic

Numerical method

2D vortex breakdown

3D vortex breakdown



Open flow

Experiments: Billant (1998)

Numerical investigations: Ruith et al. (2003) – 2D; Ruith et al. (2002, 2003, 2004), Gallaire & Chomaz (2003), Gallaire et al. (2006) – 3D

Theoretical investigations: not so many…

7

Problematic: open flow, “no” lateral confinement

Governing parameters

0 0

0

( )Re ,x core core

x

u r u rS

u

0

0

core

x

r

u

u

- the inlet axial velocity;

- the azimuthal velocity;

- the radius of the vortex core;

Boundary condition allowing

entrainment!

Introduction

→ Swirling flow


→ Applications

→ Classification

→Problematic

Numerical method

2D vortex breakdown

3D vortex breakdown



8

Overview

• Introduction

• Numerical method

• 2D (axisymmetric) vortex breakdown

• 3D vortex breakdown

• Active open-loop control: effect of an external axial pressure gradient on 2D vortex breakdown

• Summary and perspectives

9

Numerical method

Introduction

Numerical method

Flow configuration

→DNS

→RPM

→Arc-length continuation

2D vortex breakdown

3D vortex breakdown



•Flow configuration

•Direct numerical simulations (DNS)

•Recursive projection method (RPM)

•Arc-length continuation

10

Flow configuration

The numerical simulations are based on the incompressible time-dependent axisymmetric Navier-Stokes equations in

cylindrical coordinates (x,r,)

0

10

20core

core

R r

x r

Introduction

Numerical method

→Flow configuration

→DNS

→RPM


2D vortex breakdown

3D vortex breakdown



00

[ ] , [ ] 1,

[ ] , [ ] .

core

corex

x

L r

rU u T

u

Flow configuration

0

10

20core

core

R r

x r

Grabowski profile (matches experiments of Mager (1972))

0

0

20

0

(0, ) 1,

(0, ) 0,

(0,0 1) (2 ),

(0,1 ) / .

x

r

u r

u r

u r Sr r

u r S r

Grabowski & Berger (1976)

uniform flow

Flow configuration: open lateral boundary

0

10

20core

core

R r

x r

( , ) 0,

( , ) ( , ) 0,

( , ) ( , ) 0.

r

xr

ux R

ruu

x R x Rx ru u

x R x Rr r

0 nBoersma et al. (1998)

Ruith et al. (2003)

Traction-free

Flow configuration: open outlet boundary

Convective outlet conditions

0 0

0 0

0 0

( , ) ( , ) 0,

( , ) ( , ) 0,

( , ) ( , ) 0.

x x

r r

u ux r C x r

t xu u

x r C x rt x

u ux r C x r

t x

(steady state)

0

0

0

( , ) 0,

( , ) 0,

( , ) 0.

x

r

ux r

xu

x rxu

x rx

Ruith et al. (2003)

0

10

20core

core

R r

x r

14

Direct Numerical Simulation (DNS)

Code adapted from the code developed by Nichols, Nichols et al. (2007)

Mesh:

• clustered around centreline in radial direction Hanifi et al. (1996)

Discretization:

• sixth-order compact-difference scheme in space

Timestepping method:

• fourth-order Runge-Kutta scheme in time

• computation of the predicted velocity

• computation of pressure from the Poisson equation

• correction of the new velocity

Introduction

Numerical method

→Flow configuration

→DNS

→RPM


2D vortex breakdown

3D vortex breakdown



Recursive Projection Method (RPM)

Steady solutions with b.c.can be found by the iterative procedure: un+1=F(un),

where F(un) is the “Runge-Kutta integrator over one time-step”

The dominant eigenvalue of the Jacobian determines the asymptotic rate of the

convergence of the fixed point iteration

RPM: method implemented around existing

DNS alternative to Newton!

• Identifies the low-dimensional unstable

subspace of a few “slow” eigenvalues

• Stabilizes (and speeds-up) convergence of

DNS even onto unstable steady-states.

• Efficient bifurcation analysis by computing

only the few eigenvalues of the small subspace.

Even when the Jacobian matrix is not explicitly available (!)

FJ

u

1max|| || | | || ||n n

s su u u u

Recursive Projection Method (RPM)

Newton

iterations

Initial state un

DNSun+1 =F(un)

Convergence?

Subspace P of few slow &

unstable eigenmodes

Subspace Q =I-P

Reconstruct solution:un+1 = p+q=PN(p,q)+QF

Steady state us

Picarditerations

no

yes

n n +1

F(un)

• Treats timestepping routine

as a “black-box”

DNS evaluates

un+1=F(un)

• Recursively identifies subspace of slow eigenmodes, P

• Substitutes pure Picard iteration with

Newton method in PPicard iteration in

Q = I-P• Reconstructs solution u from

sum of the projectors P and Q onto subspace P and its orthogonal complement Q, respectively:

u = PN(p,q) + QF Shroff et al. (1993)

Arc-length continuation

Continuation of a branch of steady solution with respect to the parameter :

• F(u=0, where in our case

• We assume that the solution curve u( is a multi-valued

function of

• At c

• Pseudo – arc length condition

• Full system

,det 0cF u

u

Newton

iterations

0T

uu s

s s

( , ) 0

0T

F u

uu s

s s

RPM procedure:– Picard iteration in Q

– Newton in other

S

18

Introduction

Numerical method

Axisymmetric vortex breakdown

→Transcritical bifurcation (inviscid)

→Viscous effect

→Resolution test

3D vortex breakdown



2D (axisymmetric) vortex breakdown

•Transcritical bifurcation (inviscid)

•Viscous effects

•Resolution test

J. Kostas

Axisymmetric vortex breakdown: review

Pipe flow• Non uniqueness of the solution

on the parameter

• Hysteretic behavior

• Theory of Wang and Rusak for a finite domain

Critical swirl

Stability of the inviscid solution

Viscous effectBeran & Culick (1992)

Open flow

?

0

0

( )core

x

u rS

u

Transcritical bifurcation (inviscid) open flow

Base flow : Grabowski inlet profile q0(r)=(ux0(r),ur0(r),u0(r))

Small disturbance analysis q(x,r)=q0(r) +q1(x,r)+…, q1(x,r)=(ux1(x,r),ur0(x,r),u0(x,r))

of Euler equations equation for the radial velocity ur1:

Analytical solution:separation of variables ur1(x,r)=sinx/2x0(r)

ODE for =(r) and =S2

Eigen value problem on

=S12- the “critical swirl” .

Solution q1 determined up to a

multiplicative constant q1= Aq’1

200

2 20 0

210,

4

(0) 0, ( ) 0

x

d r d ruud

dr r dr x r u dr

dR

dr

Vyazmina et al. (2009)

21

Viscous effects: asymptotics of an open flow

Wang & Rusak (1997) showed in a pipe: regular expansion is invalid near 1S12

Vyazmina et al. (2009): non-homogeneous expansion for open flowIntroduction

Numerical method



→Viscous effect

→Resolution test

Three-dimensional vortex breakdown



1+’, 2’, with ’=O(1), ’=O(1)

q(x,r)=q0(r)+ q1(x,r)+ 2 q2(x,r) + …q1= Aq’1

: L ur1=0

2: L ur2=(q1,q0),

Fredholm alternative

†1 | 0ru

Amplitude equation:

A21+A’2+’ 13=0,

with

† † †1 1 1 2 1 2 3 1 3| , | , |r r rM u M u M u

Linearization of

Navier-Stokes

22

Viscous effect: asymptotics of an open flow

A21+A’2+’ 13=0, Introduction

Numerical method



→Viscous effect

→Resolution test

3D vortex breakdown



22 2 1 3 1 3

1 2

' ' 4 ', | ' | 2 '

2 | |

M M M M M MA

M M

Obtain solution q1= Aq’1

1 321 1

2

1 322 1

2

2 ,| |

2| |

c

c

M MS

M

M MS

M

Viscous effects: numerical simulations Re=1000

Importance of the resolution for high ReResolution N1:

NR =127; Nx =257

Other resolutions:

N2=2N1; N3=3N1; N4=4N1

Point C: comparison N1 and N4

?

Point A:

• N1 error 4 %

• N2 error 0.7 %

• N3 error 0.2 %

Point B:

• N1 error 2.5 %

• N2 error 0.4 %

• N3 error 0.1 %

Point C:

• N1 error 8 %

• N2 error 1 %

• N3 error 0.2 %

26

Viscous effect, Re=1000: second bifurcation ?

Introduction

Numerical method



→Viscous effect

→Resolution test

3D vortex breakdown



27

Introduction

Numerical method

2D vortex breakdown


→Mathematical formulation

→Spiral vortex breakdown




•Mathematical formulation

•Spiral vortex breakdown

Lim & Cui (2005)

28

3D vortex breakdown: short review

Spiral vortex breakdown has been observed

• Experimentally: Sarpkaya (1971), Faler & Leibovich (1977), Escudier & Zehnder (1982), Lambourne & Bryer (1967)

• DNS: Ruith et al. (2002, 2003)

Transition to helical breakdown:

sufficiently large pocket of absolute instability in the wake of the bubble, giving rise to a self-excited global mode Gallaire et al. (2003, 2006)

Introduction

Numerical method

2D vortex breakdown






29

3D vortex breakdown: mathematical formulation

2D axisymmetric state is stable to axisymmetric perturbations

3D perturbations?

Introduction

Numerical method

2D vortex breakdown






( , ), ( , ), ( , )x rU x r U x r U x rU

• Base flow is axisymmetric and stable to 2D perturbations

• Since the base flow is independent of time and azimuthal angle, the perturbations are

where m – azimuthal wavenumber, - complex frequency;

the growth rate Re(-i )

the frequency Re(-i )

( , ) , ( , ) ,im i t im i tx r e p p x r e u u

Spiral vortex breakdown: non-axisymmetric mode m=-1

S=1.3 growth rate vs Re

Re=150, S=1.3, m=-1

Ruith et al. (2003) solved fully nonlinear 3D

equations

31

Introduction

Numerical method

2D vortex breakdown

3D vortex breakdown


→Theoretical expectations

→Numerical results


Effect of the external pressure gradient

•Theoretical expectations

•Numerical results

32

An imposed pressure gradient: review for a pipe• Batchelor (1967): in a diverging pipe solution families have a fold

as the swirl increased.

• Numerically Buntine & Saffman (1995) showed the existence of bifurcation where two equilibrium solutions exist in a certain range of swirl below this limit level.

• Asymptotic analysis of Rusak et al. (1997) of inviscid flow due to the pipe convergence or divergence.

• Converging tube Leclaire (2006)

Introduction

Numerical method

2D vortex breakdown

3D vortex breakdown





Rusak et al. (1997)

Leclaire (2010)

Pressure gradient: Theoretical expectations

Carrying out the similar non-homogeneous asymptotic analysis with two competitive small parameters: and using dominant balance (2’, =2 ’) we obtain the amplitude equation in the form

A21-A’2+’ 13-’ 4=0,

4 did not calculated, since there is not analytical solution for the adjoint problem.

4

2

2 2 1 3 3 41

1 2

' ' 4 ' ', | ' | 2

2 | |

' 'M M M M M MA

M M

M M

1 321 1

2

1 322 1

4

2

3

4

4

2 ,| |

2 ,| |

c

c

c

M MS

M

M MS

M

M

M

M

M

Schematic bifurcation surface

35

Pressure gradient: bridging the gap

Introduction

Numerical method

2D vortex breakdown

3D vortex breakdown





Schematic bifurcation surface

Pressure gradient: numerical results Re=1000

N1

N1

N2

N2

N2

N3

N3

N3

N3

N3

Does the steady solution exist down to =0?

No, in the case Re=1000

Favorable pressure gradient delays vortex breakdown

N3

N3

37

Introduction

Numerical method

2D vortex breakdown

3D vortex breakdown



→Summary

→Perspectives


Summary

• 2D: Bifurcation due the viscosity: numerical and theoretical analysis.

• 3D: 2D stable solution is unstable to 3D perturbations. Spiral vortex breakdown, m = -1.

• 2D: external negative pressure gradient can delay or even prevent vortex breakdown;

– Bifurcation with respect to S and is more complex than a double fold

39

Perspectives

• Computations at higher Reynolds numbers

to find vortex breakdown-free state at S >Sc2

• Asymptotic analysis with two competitive parameters and , determine the adjoint mode numerically

• Compute 3D global modes of the adjoint Navier-Stokes linearized around the axisymmetric vortex breakdown state. Proceed sensitivity analysis

• The slow convergence along the vortex breakdown branch

Introduction

Numerical method

2D vortex breakdown

3D vortex breakdown



→Summary

→Perspectives

1Re ~ O

Investigation of the stability of the solution

40

Perspectives: Supercritical Hopf bifurcation

Introduction

Numerical method

2D vortex breakdown

3D vortex breakdown



→Summary

→Perspectives

Hopf bifurcation and period doublings perspectives

Chaotic dynamics ?

42

Merci pour votre attention!

Date post:	30-Dec-2015
Category:	Documents
Upload:	lucy-burt
View:	43 times
Download:	3 times

Bifurcations in a swirling flow*

Documents