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Nonlinear terms: second order Adams-Bashforth extrapolation
The nonlinear terms have to be evaluated at instant tn+1. Two main approaches can be considered. A first method would consist in writing the term NL (m) n+1 using variables G (p).
n+1 relative to all mode p at time tn+1 (implicit nonlinear term). This would result in a fully coupled nonlinear system which could be solved using for instance Newton-Raphson iterations. The alternative used in the present code, Helix, is an explicit formulation for the nonlinear terms. It consists in writing the nonlinear term at time tn+1 using an extrapolation from its values at times tn and tn−1. This has the advantage that at each time step the system to be solved becomes linear. The extrapolation is obtained via second order Adams-Bashforth scheme: NL (m) n+1 = 2NL(m) n − NL(m) n−1 + O(t2).
Fourier decomposition along ‘
The Fourier series in equation (2.36) and (2.37) are truncated to a finite number of complex mode |m| < M. Since modes with negative m can be direclty obtained as complex conjugates, we consider only the positive m. The standard 2/3 dealiasing requires thecomputation of 3M/2 modes. This is done by discretising ‘ at N = 3M physical angles given by ‘j = j (j = 0, …,N − 1), ‘N = ‘0 = 0, (3.10) where = 2/N. The first M first complex modes are effectively used, the higher modes m = M, …, 3M/2 − 1 being dismissed at each time step.
Irregular meshes
We create a radial grid mesh where points are not necessarily evenly spaced. We aim at placing most of the grid points within the region where the vortices are supposed to be localised. In the present context such a region is either an annulus or a disk 0 R1 r R2, bound by the lower and upper radii R1 and R2. This ensures that the dynamics and the core structure of the vortices are accurately captured in that region. In order to reduce the total number of grid points, we connect the above refined region to regions with a coarser grid, corresponding to the regions where the flow is likely to be potential. A typical arrangement of the regions is depicted in figure 3.1. In the refined region (region 2), we use N2 regularly spaced grid points, spaced by a distance = (R2 − R1) /N2 set at ri = R1+(i − N1) for i = N1, …,N1+N2. In the coarse mesh regions (regions 1 and 3), we use N1 and N3 grid points. The cells next to the refined regions are imposed to be of size (see red ticks in figure 3.1) while for the others we set fixed contraction/expansion rates i: one chooses the grid spacings hi ri+1 − ri such that:
1 = hi/hi−1 for i < N1 − 1 (3.11).
3 = hi/hi−1 for i N1 + N2 + 1. (3.12).
We require the i parameters to be close to 1 (we choose in practice 0.9 < i < 1.1) in order to control the discretisation errors (see section 3.2.3 below). In practice, radii R1,R2,Rext and the number N2 of grid points in region 2 are first chosen, which fixes . For region 3, the expansion rate 3 is then determined as the closest value below 1.1 which ensures that the outer boundary at r = Rext is a grid point. Similarly for region 1, when present, the contraction rate 1 is the closest value above 0.9 ensuring that the centre of the domain r = 0 is a grid point. The number of grid points N1,N3 of regions 1 and 3 are then an output of this procedure. Since in region 3, the following sequence holds r (N1 + Np) = R2.
The ODE system governing the helical vortex filaments
If we determine the motion of the particular fluid particle located at time t at position r0 i (t), we are able to recover the position and motion of the whole filament Hi because of the helical symmetry.
Multipolar decomposition of the fields in the ? plane
In order to characterize a vortex in the ? plane, the vorticity component !B is not sufficient, for instance, to evaluate the vortex circulation: except at point A, !B is not the vorticity component perpendicular to the plane ?. Consequently, it is preferable to separate the vorticity field into a component orthogonal to the plane ?, namely !BA, and two in-plane polar components ! and ! defined by:
!BA (M1) ! (M1) · eBA, (5.16).
! (M1) ! (M1) · e, (5.17).
! (M1) ! (M1) · e .
Characterization of the vortex core structure
Examples of vortex core structures in the ? plane are depicted for large and small pitch L in figures 5.4-b and 5.5-b respectively. For a vortex of large pitch (figure 5.4-b), the core structure is close to axisymmetric as for two-dimensional vortex. For small pitch, strong interactions between successive turns occur, causing the core structure to deviate from axisymmetry because of the self induced strain field (see figure 5.5-b). We introduce here the two most relevant quantities used to characterise the vortex core shape: the core size which can be quantified from the axisymmetric component, and the ellipticity from the quadripolar component.
Temporal evolution of a single helical vortex
In the following, we study the temporal evolution of a single generic helical vortex with enforced helical symmetry in an unbounded incompressible medium. The numerical domain is a disk of nondimensional radius Rext = 3, meshed by Nr×N grid points where Nr = 512 and N = 384 typically. All the computations are run with the initial condition procedure described in section 6.2. The chosen values for the pitch are L = 0.25, 0.5, 0.75, 1, 2, 3 and the Reynolds number is fixed at Re = 5000.
The vortex evolution is mainly characterised by four stages of evolution. During the first stage the vortex goes through a rapid relaxation process where it adapts its structure to the strain field it is subject to (section 6.4). The second stage is a slow diffusion process where the vortex grows in size and rotates quasi steadily at angular velocity ( ), i.e. a quasi-equilibrium state (section 6.5). This generalises the inviscid equilibrium flow induced by one helical vortex (Kuibin and Okulov, 1998). At low pitch, a third stage occurs for large time: successive coils merge and the vortex progressively loses its helical structure and becomes a cylindrical vorticity layer. For all values of the pitch, the system finally evolves towards a columnar axisymmetric Gaussian vortex.
Relaxation towards quasi-equilibria for a single vortex
This section illustrates the first evolution stage, where the system rapidly relaxes towards a quasi-equilibrium state. This temporal evolution is only presented for a single helical vortex of pitch L = 0.25, of core size a0 = 0.06, at Re = 5000. Results are similar for a single vortex at other pitch values and Reynolds numbers, as well as for a polygonal array of helical vortices.
The whole adaptation process of the helical vortex is the so-called relaxation process. For an array of vortices, the initial fields (6.24)–(6.25) are not an equilibrium of the Euler equations due to the presence of an external strain. For a single vortex, this also holds since a self-induced strain is present, originating from local curvature as well as induction due to remote vorticity. The vortex thus adapts its structure to this self-induced strain field: snapshots in figure 6.3 show how the initial axisymmetric vorticity distribution evolves towards an elliptic one within the core, while the very weak peripheral vorticity region displays a more complex evolution which is associated to the damping of inertial waves by viscosity. This relaxation process is thus very similar to what is observed in two dimensions for co-rotative vortices (Le Dizès and Verga, 2002) and counter rotatives vortices (Sipp et al., 2000).
Core size of a helical vortex
The core size a of the helical vortex is computed using the technique based on the Gaussian fit of the axisymmetric part of the helical vorticity (see section 5.4.1). When L > 1, the two-dimensional diffusion law is a fair approximation for the core size evolution, as can be seen in figure 6.6. When L < 1, the core size increases less than its two-dimensional counterpart. The strong increase of a observed for L = 0.25 around = 130 corresponds to a core size as large as a 0.32, where the notion of core size becomes questionable.
Self-similar solutions
On figure 6.7-a, the axisymmetric part of the helical vorticity ! (0) BA (see 5.3) is plotted for a set of equally spaced times. A self-similar behaviour is identified: while the Gaussian’s amplitude decreases in time, the radial spreading increases accordingly. When rescaled as the profiles collapse (figure 6.7-b) onto the Gaussian curve ˜! (0) BA () = exp(−2). This self-similarity was already observed on numerically computed rotor wakes by Ali and Abid (2014). The profiles of u (0) H , the axisymmetric part of quantity uH, are presented in figure 6.8-a. These profiles are also spreading in time (figure 6.8-b). However, as evidenced in figure 6.8-a, the amplitude of the velocity deficit remains approximately constant in time: based on equation (6.20) this implies that the maximum of vorticity varies as 1/. When plotted as a function of with the same variable and normalised by its absolute maximum value, the curves collapse on a single one close to ˜u (0) H () = −e−2 (figure 6.8-b).
Helix radius rA and angular velocity
Let us characterize the global dynamics by tracking point A (rA( ), A( )) in the 0- plane where !B is maximum and by providing the angular velocity of the vortex ( ).
The temporal evolution of the helix radius rA is plotted in figure 6.9-a for different values of L at Re = 5000. For all values of L considered, the helix radius first increases. The explanation of this radial drift is not clear at the moment and is believed to be related to the conservation of the angular momentum with a core size increasing in time. However, for larger times, rA reaches a maximum and then decreases (see figure 6.9-c). For small L, the maximum is reached at a critical time where the vortex successive coils are about to merge, as depicted in figure 6.10. For larger L, this mechanism is no more active, but since rA should asymptotically be zero, a maximum is still reached. As seen in figure 6.11, the same argument about the successive coils about to come into contact does not hold to explain this maximum of rA.
The angular velocity of the vortex is plotted as a function of time in figures 6.9-b and d. It is compared to the values obtained with the corresponding temporal simulation using vortex filaments along with the cut-off theory introduced in chapter 4.
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
