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 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. Main Features: core of vorticity and axial velocity stagnation point reverse flow or “recirculation bubble” 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 • 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 • Combustion burner • Introduction • Swirling flow • Vortex breakdown • Applications • Classification • Problematic Numerical method 2D vortex breakdown 3D vortex breakdown Active open-loop control Summary and perspectives Tornado • 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 • Billant et al. (1998) • Faler & Leibovich (1977)

6. Pipe 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)… Problematic • 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. 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 Overview

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 • 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. 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. 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 Direct Numerical Simulation (DNS) • Introduction • Numerical method • Flow configuration • DNS • RPM • Arc-length continuation 2D vortex breakdown 3D vortex breakdown Active open-loop control Summary and perspectives

15. Steady solutions with b.c.can be found by the iterative procedure:un+1=F(un), whereF(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 (!) Recursive Projection Method (RPM)

16. Newton iterations 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 Qonto subspaceP and its orthogonal complement Q, respectively: • u= PN(p,q) + QF Reconstruct solution: un+1= p+q=PN(p,q)+QF Initial stateun • n  n +1 DNS un+1 =F(un) • Picard • iterations F(un) Subspace Q =I-P Subspace P of few slow & unstable eigenmodes Convergence? • no • yes Steady stateus Shroff et al. (1993)

17. 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 Arc-length continuation • 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 • Transcritical bifurcation (inviscid) • Viscous effects • Resolution test J. Kostas

19. 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 Axisymmetric vortex breakdown: review Beran & Culick (1992) • Open flow • ?

20. 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 onW W1=S12- the “critical swirl” . Solution q1 determined up to a multiplicative constant q1= Aq’1 Transcritical bifurcation (inviscid) open flow Vyazmina et al. (2009)

21. Wang & Rusak (1997) showed in a pipe:regular expansion is invalid near W1=S12 Vyazmina et al. (2009): non-homogeneous expansionfor open flow Viscous effects: asymptotics of an 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. A2M1+ADW’M2+n’ W1M3=0, 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 • Obtain solution q1= Aq’1

23. Viscous effects: numerical simulations Re=1000

24. Resolution N1: NR =127; Nx =257 Other resolutions: N2=2N1; N3=3N1; N4=4N1 Point C: comparisonN1 andN4 Importance of the resolution for high Re • ? • 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. 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. 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 • Mathematical formulation • Spiral vortex breakdown Lim & Cui (2005)

27. 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) 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

28. 2D axisymmetric state is stable to axisymmetric perturbations 3D perturbations? 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 • 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 )

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

30. 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 • Theoretical expectations • Numerical results

31. 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) An imposed pressure gradient: review for a pipe • 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)

32. Carrying out the similar non-homogeneous asymptotic analysis with two competitive small parameters: n andbusing dominant balance (n=e2n’,b =e2b ’) 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. Pressure gradient: Theoretical expectations • Schematic bifurcation surface

33. 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

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

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

36. 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 Sand b is more complex than a double fold Summary

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

38. Perspectives: Supercritical Hopf bifurcation • Introduction • Numerical method 2D vortex breakdown 3D vortex breakdown Active open-loop control Summary and perspectives • Summary • Perspectives

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

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