Thèse de doctorat présentée pour obtenir le grade de
Docteur de l’École Polytechnique
par Elena Vyazmina
Bifurcations in a swirling flow*
* Bifurcations d’un écoulement tournant
Directeurs de thèse: Jean-Marc Chomaz et Peter Schmid
13 juillet 2010
1
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.
2
Vortex breakdown: definition
Introduction
Rotating cylinder, fixed lid: S. Harris
Free jet: Gallaire (2002)
→ Swirling flow
→ Vortex breakdown
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
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.
•
•
•
Main Features:
core of vorticity and axial
velocity
stagnation point
reverse flow or “recirculation
bubble”
3
Applications
Introduction
Combustion burner
Tornado
→ Swirling flow
→ Vortex breakdown
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Aeronautics
Active open-loop
control
Summary and
perspectives
4
Vortex breakdown: classification
Introduction
Bubble or axisymmetric form
Double helix form
→ Swirling flow
→ Vortex breakdown
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
Faler & Leibovich (1977)
Faler & Leibovich (1977)
3D vortex breakdown
Active open-loop
control
Spiral form
Cone form
Summary and
perspectives
Faler & Leibovich (1977)
Billant et al. (1998)
5
Problematic
Pipe
Introduction
→ Swirling flow
→ Vortex breakdown
Experiments: Sarpkaya (1971), Faler & Leibovich (1978), Leibovich
(1978,1983), Althaus (1990), Escudier & Zehnder (1982)…
→ Applications
→ Classification
→ Problematic
Numerical method
2D vortex breakdown
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)…
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…
6
Problematic: open flow, “no” lateral confinement
Introduction
→ Swirling flow
Boundary
condition allowing
entrainment!
→ Vortex breakdown
→ Applications
→ Classification
→Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
Governing parameters
Re 
rcore
u x 0 rcore

, S
u 0 (rcore )
ux 0
- the radius of the vortex core;
ux0
- the inlet axial velocity;
u 0
- the azimuthal velocity;
7
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
8
Introduction
Numerical method
Flow configuration
→DNS
→RPM
→Arc-length
continuation
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
Numerical method
•
Flow configuration
•
Direct numerical simulations (DNS)
•
Recursive projection method (RPM)
•
Arc-length continuation
9
Flow configuration
Introduction
Numerical method
→Flow configuration
→DNS
→RPM
→Arc-length
continuation
R  10rcore
2D vortex breakdown
x0  20rcore
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
The numerical simulations are based on the incompressible
time-dependent axisymmetric Navier-Stokes equations in
cylindrical coordinates (x,r,)
[ L]  rcore , [  ]  1,
[U ]  u x 0 , [T ] 
rcore
.
ux0
10
Flow configuration
R  10rcore
x0  20rcore
Grabowski profile (matches experiments of Mager (1972))
u x 0 (0, r )  1,
uniform flow
ur 0 (0, r )  0,
u 0 (0, 0  r  1)  Sr (2  r 2 ),
u 0 (0,1  r )  S / r.
Grabowski & Berger (1976)
Flow configuration: open lateral boundary
R  10rcore
x0  20rcore
Traction-free
 n  0
Boersma et al. (1998)
Ruith et al. (2003)
ur
( x, R)  0,
r
u x
ur
( x, R ) 
( x, R)  0,
x
r
u
u
( x, R)   ( x, R)  0.
r
r
Flow configuration: open outlet boundary
R  10rcore
x0  20rcore
Convective outlet conditions
(steady state)
Ruith et al. (2003)
u x
u
( x0 , r )  C x ( x0 , r )  0,
t
x
ur
ur
( x0 , r )  C
( x0 , r )  0,
t
x
u
u
( x0 , r )  C  ( x0 , r )  0.
t
x
u x
( x0 , r )  0,
x
ur
( x0 , r )  0,
x
u
( x0 , r )  0.
x
Direct Numerical Simulation (DNS)
Introduction
Code adapted from the code developed by Nichols, Nichols et al. (2007)
Numerical method
→Flow configuration
→DNS
Mesh:
• clustered around centreline in radial direction Hanifi et al. (1996)
→RPM
→Arc-length
continuation
2D vortex breakdown
Discretization:
• sixth-order compact-difference scheme in space
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
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
14
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”
F determines the asymptotic rate of the
The dominant eigenvalue of the Jacobian J 
u
convergence of the fixed point iteration
|| us  u n 1 |||  | max || us  u n ||
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 (!)
Recursive Projection Method (RPM)
Reconstruct solution:
un+1 = p+q=PN(p,q)+QF
n  n +1
Initial state un
• Treats timestepping routine
as a “black-box”
DNS evaluates
un+1=F(un)
Picard
iterations
Subspace
Q =I-P
Subspace P
of few slow &
unstable
eigenmodes
un+1 =F(un)
• Recursively identifies subspace
of slow eigenmodes, P
F(un)
• Substitutes pure Picard
iteration with
DNS
Newton
iterations
Newton method in P
no
Convergence?
yes
Steady state us
Shroff et al. (1993)
Picard 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
Arc-length continuation
Continuation of a branch of steady solution with respect to the parameter :
•
S
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
 F  u, c ) 
det 
0
u


Newton
iterations

 u 

u

  s  0
 
s
 s 
T
•
Full system
F (u,  )  0

 u 

u

  s  0
 
s
 s 
T
RPM procedure:
–
Picard iteration in Q
– Newton in other
Introduction
Numerical method
Axisymmetric vortex
breakdown
→Transcritical
bifurcation (inviscid)
2D (axisymmetric) vortex breakdown
→Viscous effect
→Resolution test
•
Transcritical bifurcation (inviscid)
3D vortex breakdown
•
Viscous effects
Active open-loop
control
•
Resolution test
Summary and
perspectives
J. Kostas
18
Axisymmetric vortex breakdown: review
Pipe flow
•
Non uniqueness of the solution
u (r )
on the parameter
S   0 core
ux0
•
Hysteretic behavior
•
Theory of Wang and Rusak for a finite
domain
Critical swirl
Stability of the inviscid solution
Viscous effect
Beran & Culick (1992)
Open flow
?
Transcritical bifurcation (inviscid) open flow
Base flow : Grabowski inlet profile q0(r)=(ux0(r),ur0(r),u0(r))
Small disturbance analysis q(x,r)=q0(r) +eq1(x,r)+…, q1(x,r)=(ux1(x,r),ur0(x,r),u0(x,r))
of Euler equations  equation for the radial velocity ur1:
Analytical solution:
separation of variables ur1(x,r)=sinpx/2x0)F(r)
ODE for F=F(r) and W=S2
2u 0 d  ru 0 ) 
d  1 d  rF )   p 2

 2 W 2
 F  0,
dr  r dr   4 x0
r u x 0 dr 
F (0)  0,
dF
( R)  0
dr
Eigen value problem on W
W1=S12- the “critical swirl” .
Solution q1 determined up to a
multiplicative constant q1= Aq’1
Vyazmina et al. (2009)
Viscous effects: asymptotics of an open flow
Introduction
Wang & Rusak (1997) showed in a pipe: regular expansion is invalid near W1S12
Vyazmina et al. (2009): non-homogeneous expansion for open flow
Numerical method
Axisymmetric vortex
breakdown
WW1+eW’, e2’,
with W’=O(1), ’=O(1)
→Transcritical
bifurcation (inviscid)
Linearization of
Navier-Stokes
→Viscous effect
→Resolution test
Three-dimensional
vortex breakdown
Active open-loop
control
Summary and
perspectives
q(x,r)=q0(r)+ e q1(x,r)+ e 2 q2(x,r) + …
q1= Aq’1
Fredholm alternative
e : L ur1=0
e
2:
ur†1 | s  0
L ur2=s(q1,q0),
Amplitude equation:
A2M1+AW’M2+’ W1M3=0,
with
M 1  ur†1 | s 1 , M 2  ur†1 | s 2 , M 3  ur†1 | s 3
21
Viscous effect: asymptotics of an open flow
Introduction
A2M1+AW’M2+’ W1M3=0,
Numerical method
Axisymmetric vortex
breakdown
A
W ' M 2 
→Transcritical
bifurcation (inviscid)
 W ' M 2 )
2M1
2
 4 ' M 1M 3
,
| W ' | 2
M 1M 3
| M2 |
'
→Viscous effect
→Resolution test
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
Sc2 1  W1  2
Sc2 2  W1  2
M 1M 3
| M2 |
M 1M 3
| M2 |
,

Obtain solution q1= Aq’1
22
Viscous effects: numerical simulations Re=1000
Importance of the resolution for high Re
Resolution 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 %
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
26
Introduction
Numerical method
2D vortex breakdown
Three-dimensional
vortex breakdown
→Mathematical
formulation
→Spiral vortex
breakdown
Three-dimensional vortex breakdown
•
Mathematical formulation
•
Spiral vortex breakdown
Active open-loop
control
Summary and
perspectives
Lim & Cui (2005)
27
3D vortex breakdown: short review
Introduction
Numerical method
2D vortex breakdown
Three-dimensional
vortex breakdown
→Mathematical
formulation
→Spiral vortex
breakdown
Active open-loop
control
Summary and
perspectives
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)
28
3D vortex breakdown: mathematical formulation
Introduction
Numerical method
2D axisymmetric state U  U x ( x, r ),U r ( x, r ),U ( x, r ) ) is stable to
axisymmetric perturbations
2D vortex breakdown
Three-dimensional
vortex breakdown
3D perturbations?
→Mathematical
formulation
→Spiral vortex
breakdown
•
Base flow is axisymmetric and stable to 2D perturbations
Active open-loop
control
•
Since the base flow is independent of time and azimuthal angle, the
perturbations are
Summary and
perspectives
u  u( x, r )eim iwt ,
p  p( x, r )eim iwt ,
where m – azimuthal wavenumber, w - complex frequency;
the growth rate sRe(-i w )
the frequency
Re(-i w )
29
Spiral vortex breakdown: non-axisymmetric mode
m=-1
S=1.3 growth rate vs Re
Ruith et al. (2003) solved
fully nonlinear 3D equations
Re=150, S=1.3, m=-1
Introduction
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
Effect of the external pressure gradient
→Theoretical
expectations
•
Theoretical expectations
→Numerical results
•
Numerical results
Summary and
perspectives
31
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
Rusak et al. (1997)
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
Leclaire (2010)
32
Pressure gradient: Theoretical expectations
Carrying out the similar non-homogeneous asymptotic analysis with two competitive
small parameters:  and  using dominant balance (e2’,  =e2  ’) we obtain the
amplitude equation in the form
A2M1-AW’M2+’ W1M3- ’ M4=0,
M4 did not calculated, since there is not analytical solution for the adjoint problem.
A
W ' M 2 
2
 4M1  ' M 3   ' M 4 )
2M1
Sc2 1  W1  2
Sc2 2  W1  2
c  
 W ' M 2 )
M3
M4
M 1  M 3   M 4 )
| M2 |
M 1  M 3   M 4 )
| M2 |
,
| W ' | 2
M 1  ' M 3   ' M 4 )
| M2 |
Schematic bifurcation surface
,
,
Pressure gradient: bridging the gap
Introduction
Numerical method
Schematic bifurcation surface
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
→Theoretical
expectations
→Numerical results
Summary and
perspectives
35
Pressure gradient: numerical results Re=1000
N3
N1
N2
N1
Does the steady solution
exist down to  =0?
No, in the case Re=1000
N3
N2
N3
N3
N3
N3
N2
N3
Favorable pressure gradient
delays vortex breakdown
Introduction
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
Summary and perspectives
→Summary
→Perspectives
37
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
Perspectives
Introduction
•
Numerical method
to find vortex breakdown-free state at S >Sc 2
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
 1 

W


Computations at higher Reynolds numbers Re ~ O 
•
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
→Summary
→Perspectives
Investigation of the stability of the solution
39
Perspectives: Supercritical Hopf bifurcation
Introduction
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop
control
Summary and
perspectives
→Summary
→Perspectives
40
Hopf bifurcation and period doublings  perspectives
 Chaotic dynamics ?
Merci pour votre attention!
42