1. Bifurcations in a swirling flow*
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

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
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
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.

4. Applications Tornado Combustion burner Aeronautics Introduction
Swirling flow
Vortex breakdown
Applications
Classification
Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
Tornado
Combustion burner
Aeronautics

5. Vortex breakdown: classification
Bubble or axisymmetric form
Double helix form
Introduction
Swirling flow
Vortex breakdown
Applications
Classification
Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
Faler & Leibovich (1977)
Faler & Leibovich (1977)
Spiral form
Cone form
Faler & Leibovich (1977)
Billant et al. (1998)

6. Problematic Pipe Open flow
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
Introduction
Swirling flow
Vortex breakdown
Applications
Classification
Problematic
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
Boundary condition allowing entrainment!
Governing parameters
- the radius of the vortex core;
- the inlet axial velocity;
- the azimuthal velocity;

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 Flow configuration Direct numerical simulations (DNS)
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

10. Flow configuration
Introduction
Numerical method
Flow configuration
DNS
RPM
Arc-length continuation
2D vortex breakdown
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,q)

11. Grabowski profile (matches experiments of Mager (1972))
Flow configuration
Grabowski profile (matches experiments of Mager (1972))
uniform flow
Grabowski & Berger (1976)

12. Flow configuration: open lateral boundary
Traction-free
Boersma et al. (1998)
Ruith et al. (2003)

13. Flow configuration: open outlet boundary
Convective outlet conditions
(steady state)
Ruith et al. (2003)

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

15. 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 (!)

16. Recursive Projection Method (RPM)
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 P
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
Reconstruct solution:
un+1 = p+q=PN(p,q)+QF
n  n +1
Initial state un
DNS
un+1 =F(un)
Newton
iterations
Picard
iterations
F(un)
Subspace
Q =I-P
Subspace P
of few slow &
unstable
eigenmodes
Convergence? no
yes
Steady state us
Shroff et al. (1993)

17. Arc-length continuation
Continuation of a branch of steady solution with respect to the parameter l:
F(u,l)=0, where in our case
We assume that the solution curve u(l) is a multi-valued
function of l
At l= lc
Pseudo – arc length condition
Full system
Newton
iterations
RPM procedure:
Picard iteration in Q
Newton in other

18. 2D (axisymmetric) vortex breakdown
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

19. 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 effect
Beran & Culick (1992)
Open flow ?

20. Transcritical bifurcation (inviscid) open flow
Base flow : Grabowski inlet profile q0(r)=(ux0(r),ur0(r),uq0(r))
Small disturbance analysis q(x,r)=q0(r) +eq1(x,r)+…, q1(x,r)=(ux1(x,r),ur0(x,r),uq0(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
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)

21. Viscous effects: asymptotics of an open flow
Wang & Rusak (1997) showed in a pipe: regular expansion is invalid near W1=S12
Vyazmina et al. (2009): non-homogeneous expansion for open flow
Introduction
Numerical method
Axisymmetric vortex breakdown
Transcritical bifurcation (inviscid)
Viscous effect
Resolution test
Three-dimensional vortex breakdown
Active open-loop control
Summary and perspectives
W=W1+eDW’, n=e2n’,
with DW’=O(1), n’=O(1)
q(x,r)=q0(r)+ e q1(x,r)+ e 2 q2(x,r) + …
q1= Aq’1
Linearization of
Navier-Stokes
e : L ur1=0
e 2: L ur2=s(q1,q0),
Fredholm alternative
Amplitude equation:
A2M1+ADW’M2+n’ W1M3=0,
with

22. Viscous effect: asymptotics of an open flow
Introduction
Numerical method
Axisymmetric vortex breakdown
Transcritical bifurcation (inviscid)
Viscous effect
Resolution test
3D vortex breakdown
Active open-loop control
Summary and perspectives
A2M1+ADW’M2+n’ W1M3=0,
Obtain solution q1= Aq’1

23. Viscous effects: numerical simulations Re=1000

24. 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 %

25. Importance of the resolution for high Re: point A
N1 error 3.8 %
N2 error 0.69 %
N3 error %
Point B:
N1 error 2.47 %
N2 error 0.36 %
N3 error 0.07 %
Point C:
N1 error 8.1 %
N2 error 0.9 %
N3 error 0.17 %

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. Three-dimensional vortex breakdown
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
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)

29. 3D vortex breakdown: mathematical formulation
Introduction
Numerical method
2D vortex breakdown
Three-dimensional vortex breakdown
Mathematical formulation
Spiral vortex breakdown
Active open-loop control
Summary and perspectives
2D axisymmetric state is stable to axisymmetric perturbations
3D perturbations?
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, w - complex frequency;
the growth rate s=Re(-i w )
the frequency n=Re(-i w )

30. 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

31. Effect of the external pressure gradient
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)

33. External axial pressure gradient
Introduction
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Theoretical expectations
Numerical results
Summary and perspectives
Idea:
An application of the external axial pressure gradient works as the tube convergence
help to achieve a viscous columnar solution at at S >Scn 2

34. Pressure gradient: Theoretical expectations
Carrying out the similar non-homogeneous asymptotic analysis with two competitive small parameters: n and b using dominant balance (n=e2n’, b =e2 b ’) we obtain the amplitude equation in the form
A2M1-ADW’M2+n’ W1M3-b ’ M4=0,
M4 did not calculated, since there is not analytical solution for the adjoint problem.
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

36. Pressure gradient: numerical results Re=1000
Does the steady solution exist down to b =0?
No, in the case Re=1000 N3 N2 N2 N1 N3 N1 N3 N3 N3 N2 N3
Favorable pressure gradient
delays vortex breakdown

37. Summary and perspectives
Introduction
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
Summary
Perspectives
Summary and perspectives

38. 2D: Bifurcation due the viscosity: numerical and theoretical analysis.
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 b can delay or even prevent vortex breakdown;
Bifurcation with respect to S and b is more complex than a double fold

39. Investigation of the stability of the solution
Perspectives
Introduction
Numerical method
2D vortex breakdown
3D vortex breakdown
Active open-loop control
Summary and perspectives
Summary
Perspectives
Computations at higher Reynolds numbers
to find vortex breakdown-free state at S >Scn 2
Asymptotic analysis with two competitive parameters n and b, 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
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

41. Hopf bifurcation and period doublings  perspectives
 Chaotic dynamics ?

42. Merci pour votre attention!