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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. Directeurs de thèse: Jean-Marc Chomaz et Peter Schmid. 13 juillet 2010. Swirling flow. Introduction - PowerPoint PPT Presentation
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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
→ Vortex breakdown
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
4
Applications
Combustion burner
Aeronautics
TornadoIntroduction
→ Swirling flow
→ Vortex breakdown
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
5
Vortex breakdown: classification
Bubble or axisymmetric form
Faler & Leibovich (1977)
Faler & Leibovich (1977)
Spiral form
Billant et al. (1998)
Cone form
Faler & Leibovich (1977)
Double helix formIntroduction
→ Swirling flow
→ Vortex breakdown
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
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
→ Vortex breakdown
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
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
→ Vortex breakdown
→ Applications
→ Classification
→Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
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
Active open-loop control
Summary and perspectives
•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
→Arc-length continuation
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
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
→Arc-length continuation
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
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
Active open-loop control
Summary and perspectives
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
Axisymmetric vortex breakdown
→Transcritical bifurcation (inviscid)
→Viscous effect
→Resolution test
Three-dimensional vortex breakdown
Active open-loop control
Summary and perspectives
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
Axisymmetric vortex breakdown
→Transcritical bifurcation (inviscid)
→Viscous effect
→Resolution test
3D vortex breakdown
Active open-loop control
Summary and perspectives
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
Axisymmetric vortex breakdown
→Transcritical bifurcation (inviscid)
→Viscous effect
→Resolution test
3D vortex breakdown
Active open-loop control
Summary and perspectives
27
Introduction
Numerical method
2D vortex breakdown
Three-dimensional vortex breakdown
→Mathematical formulation
→Spiral vortex breakdown
Active open-loop control
Summary and perspectives
Three-dimensional 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
Three-dimensional vortex breakdown
→Mathematical formulation
→Spiral vortex breakdown
Active open-loop control
Summary and perspectives
29
3D vortex breakdown: mathematical formulation
2D axisymmetric state is stable to axisymmetric perturbations
3D perturbations?
Introduction
Numerical method
2D vortex breakdown
Three-dimensional vortex breakdown
→Mathematical formulation
→Spiral vortex breakdown
Active open-loop control
Summary and perspectives
( , ), ( , ), ( , )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
Active open-loop control
→Theoretical expectations
→Numerical results
Summary and perspectives
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
Active open-loop control
→Theoretical expectations
→Numerical results
Summary and perspectives
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
Active open-loop control
→Theoretical expectations
→Numerical results
Summary and perspectives
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
Active open-loop control
Summary and perspectives
→Summary
→Perspectives
Summary and 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
Active open-loop control
Summary and perspectives
→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
Active open-loop control
Summary and perspectives
→Summary
→Perspectives
Hopf bifurcation and period doublings perspectives
Chaotic dynamics ?
42
Merci pour votre attention!