Helix radius rA and angular velocity

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Table of contents

1 Introduction 
1.1 Context
1.1.1 Renewable energy: wind turbines
1.1.2 Helicopter wakes and VRS
1.1.3 The ANR project: HELIX
1.2 Flows behind rotors
1.2.1 Wind turbine wakes: the near- and far-wake
1.2.2 Experiments and numerical computations
1.3 Helically symmetric vortices and their instabilities
1.4 Goal and personal contributions
1.5 Outline
2 Navier-Stokes equations for helical flows 
2.1 Helical symmetry
2.1.1 Expression of differential operators for helical fields
2.1.2 Incompressibility for helical fields
2.2 Governing equations for helically symmetric flows
2.3 Spectral formulation of the governing equations
2.3.1 Modes m 6= 0
2.3.2 Modes m = 0
2.4 Boundary conditions at r = 0
2.4.1 Boundary conditions at r = 0 for modes m 6= 0
2.4.2 Boundary conditions at r = 0 for mode m = 0
2.5 Boundary condition at r = Rext
2.5.1 Modes m 6= 0
2.5.2 Modes m = 0
3 DNS code with enforced helical symmetry : HELIX 
3.1 Temporal scheme for the dynamical equations
3.1.1 Temporal derivative: 2nd order backward Euler scheme
3.1.2 Nonlinear terms: second order Adams-Bashforth extrapolation .
3.1.3 Viscous terms: implicit scheme
3.1.4 General form
3.2 Spatial discretisation
3.2.1 Fourier decomposition along ‘
3.2.2 Irregular meshes
3.2.3 Finite differences along the radial direction r
3.3 Discrete system of equations for m 6= 0
3.3.1 Modified Poisson equation for (m)
3.3.2 Modified Helmholtz equations for ! (m) B , u (m) B.
3.4 Discrete system of equations for m = 0
3.4.1 Computation of the streamfunction (0)
3.4.2 Modified Helmholtz equations for u (0) ‘ , u (0) B
4 Vortex filaments: cut-off theory 
4.1 The ODE system governing the helical vortex filaments
4.1.1 Computation of the velocity using the Biot-Savart law
4.1.2 Motion of the intersecting point
4.2 Numerical integration for the helical vortex system
4.3 Numerical validation
5 Characterization of helical vortices 
5.1 Vortex position and angular velocity
5.1.1 Helix radius rA
5.1.2 Angular velocity
5.2 Framework description
5.2.1 Definition of ? and its associated basis
5.2.2 Relationships between planes 0 and ?
5.3 Multipolar decomposition of the fields in the ? plane
5.4 Characterization of the vortex core structure
5.4.1 Vortex core size a
5.4.2 Ellipticity
6 Quasi-equilibrium solutions for helical vortices 
6.1 Invariant quantities
6.1.1 Global Invariant quantities
6.1.2 Local conservation laws for the inviscid case
6.1.3 Local conservation laws in the viscous case
6.2 Initial conditions for a generic time evolution
6.3 Temporal evolution of a single helical vortex
6.4 Relaxation towards quasi-equilibria for a single vortex
6.5 Quasi equilibrium stage
6.5.1 Relationship between R, uH and !B
6.5.2 Core size of a helical vortex
6.5.3 Self-similar solutions
6.5.4 Helix radius rA and angular velocity
6.5.5 Ellipticity μ and major axis angle e
6.6 Streamline topology
6.6.1 Streamline topology in the laboratory frame
6.6.2 Streamline topology in the rotating frame
6.7 Particle transport by a helical vortex
6.7.1 Equations for the particle motion in the rotating frame
6.7.2 Particle initialisation and simulation
6.7.3 Results for the passive case: St = 0
6.7.4 Preliminary results for the inertial cases St 6= 0
6.8 Late evolution: coil merging and axisymmetrisation
7 Modal decomposition of the core structure: comparisons with asymptotic theory 
7.1 Extraction of the multipolar profile from the DNS data
7.1.1 Choice of plane ?
7.1.2 Expression of the theoretical velocity field in the rotating frame
7.1.3 Getting the parameters of the monopolar contribution
7.2 Paper: Internal structure of vortex rings and helical vortices
8 Linear Stability analysis in the helical framework 
8.1 Basic state solutions in the rotating frame: frozen quasi-equilibrium solutions
8.2 Perturbation equations in the helical and rotating framework
8.3 The Arnoldi method
8.3.1 Initial condition: random noise
8.3.2 Time stepper approach
8.3.3 The Arnoldi algorithm
8.3.4 Time-stepping and orthogonalisation
8.3.5 Recovery of the temporal frequency ! of the modes when t is chosen too big
8.4 Validation of the Arnoldi implementation in fixed frame
8.4.1 Linear modes of the Batchelor or q-vortex
8.4.2 Linear modes of the Carton-McWilliams shielded vortex
8.5 Validation of the Arnoldi algorithm for rotating basic state
9 Linear helical stability: results 
9.1 A single helical vortex
9.2 Two helical vortices
9.2.1 Influence of L on the dominant mode
9.2.2 Point vortex analogy
9.2.3 Vortex ring array analogy
9.2.4 Influence of the core size a
9.2.5 Influence of Reynolds number
9.3 Two helical vortices with a central hub vortex
9.4 Three helical vortices with a central hub vortex
10 Nonlinear evolution in the helical framework 
10.1 Leapfrogging and merging of vortices
10.2 General mechanism of two helical vortices
10.3 Results for two helical vortices
10.3.1 Influence of the reduced pitch L
10.3.2 Influence of the Reynolds number Re
10.3.3 Influence of the core size a
10.3.4 Cut-off theory
10.4 Merging of two helical vortices
11 Linear stability analysis with respect to general perturbations 
11.1 Stability analysis
11.1.1 Frozen helically symmetric base flow
11.1.2 Equations for the perturbations
11.1.3 Numerical code HELIKZ
11.2 HELIKZ results
11.2.1 Long wavelength and mutual induction instability
11.2.2 Elliptical instability
Appendices 
Appendix A Rate of strain tensor for helically symmetric flows. 
Appendix B Boundary conditions at the axis. 
B.1 Symmetry of the Fourier coefficients
B.2 Regularity constraints on scalar fields
B.3 Regularity constraints on vector field components ( ± 1)
Appendix C HELIX code: Discretisation of first and second derivatives at second order accuracy on irregular meshes 
Appendix D Velocity of a set of helical filaments for the cut-off theory 
D.1 Induced velocity by a helical vortex j on vortex i 6= j
D.2 Self induced velocity of a helical vortex
Appendix E Vortex characterisation 
E.1 Vortex characterisation: two-dimensional interpolations
E.1.1 One dimensional interpolation: Chebyshev polynomials
E.2 Vortex characterisation: nonlinear least square method
Appendix F Linear stability in the rotating frame 
Appendix G Paper: Instabilities in helical vortex systems: linear analysis and nonlinear dynamics