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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 in a swirling flow* * Bifurcations d’un écoulement tournant Directeurs de thèse: Jean-Marc Chomaz et Peter Schmid 13 juillet 2010

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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.

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**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.

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**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

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**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)

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**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…

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**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;

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**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

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**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

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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)

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**Grabowski profile (matches experiments of Mager (1972))**

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

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**Flow configuration: open lateral boundary**

Traction-free Boersma et al. (1998) Ruith et al. (2003)

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**Flow configuration: open outlet boundary**

Convective outlet conditions (steady state) Ruith et al. (2003)

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**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

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**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 (!)

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**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)

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**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

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**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

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**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 ?

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**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)

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**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

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**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

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**Viscous effects: numerical simulations Re=1000**

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**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 %

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**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 %

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**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

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**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)

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**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)

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**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 )

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**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

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**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

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**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)

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**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

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**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

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**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

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**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

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**Summary and perspectives**

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

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**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

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**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

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**Perspectives: Supercritical Hopf bifurcation**

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

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**Hopf bifurcation and period doublings perspectives**

Chaotic dynamics ?

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**Merci pour votre attention!**

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