Spectral simulations of the reconnection process
of two vortices
Mémoire
Guillaume Beardsell
Maîtrise en génie mécanique
Maître ès sciences (M.Sc.)
Québec, Canada
Résumé
Ce mémoire investigue le phénomène de reconnexion visqueuse de deux tourbillons initiale-ment placés de façon orthogonale ou antiparallèle. Les équations de Navier-Stokes pour un fluide incompressible sont résolues directement (DNS) à l’aide d’un algorithme pseudospectral utilisant des expansions périodiques dans les trois directions cartésiennes. La condition de circulation nulle inhérente à cette méthode numérique est contournée en résolvant les équa-tions dans un repère tournant approprié. Une méthode simple utilisant des lignes de vorticité est proposée afin de calculer le pourcentage de reconnexion instantané η de deux tourbillons. Cette méthode est également employée pour séparer le champ de vorticité en ses composantes reconnectées et non-reconnectées, ce qui facilite l’identification visuelle des différentes étapes du processus de reconnexion. Finalement, l’échelle de temps de la reconnexion tourbillonnaire Trec est calculée pour différents nombres de Reynolds (500 ≤ Re ≤ 10000). Il est trouvé que
l’ordre de grandeur de Trec varie de façon continue de Trec ∼ Re−1 à Trec∼ Re−1/2 à mesure
Abstract
This work focuses on the viscous reconnection phenomenon of two vortex tubes that are ini-tially antiparallel or orthogonal to each other. The incompressible Navier-stokes equations are solved directly (DNS) using a Fourier pseudospectral algorithm with triply periodic boundary conditions. The associated zero-circulation constraint is circumvented by solving the govern-ing equations in a proper rotatgovern-ing frame of reference. A simple method usgovern-ing vortex lines is proposed to compute the instantaneous reconnection level η of two vortices. The proposed method is also used to split the vorticity field into its reconnected and non-reconnected parts, which allows for a clear and intuitive visual identification of the different reconnection phases. Finally, the Reynolds number dependence of the reconnection timescale Trec is investigated
for 500 ≤ Re ≤ 10000. The scaling is found to vary continuously as Re is increased from Trec ∼ Re−1 to Trec ∼ Re−1/2.
Table des matières
Résumé iii
Abstract v
Table des matières vii
Liste des tableaux ix
Liste des figures xi
Remerciements xv
Avant-propos xvii
1 Introduction 1
1.1 Vortex reconnection phenomenon . . . 1
1.2 Objectives of the master’s thesis . . . 4
1.2.1 First objective . . . 4
1.2.2 Second objective . . . 4
1.2.3 Third objective . . . 5
1.3 Thesis structure. . . 5
2 Preliminary concepts in spectral methods 7 2.1 Introduction to spectral methods . . . 7
2.1.1 Weighted residual methods . . . 7
2.1.2 Different types of spectral methods . . . 8
2.2 Fourier representation of a function . . . 9
2.2.1 Definition . . . 9
2.2.2 Precision . . . 10
3 Methodology 13 3.1 Problem definition . . . 13
3.1.1 Initial vortex configurations . . . 13
3.1.2 Vorticity distribution. . . 14
3.1.3 Incompressible Navier-Stokes equations . . . 15
3.2 Spectral solver . . . 16
3.2.1 Fourier expansion. . . 16
3.2.2 Solving the governing equations in the Fourier space . . . 16
3.2.4 Evaluation of the non-linear term . . . 17
3.2.5 Time integration . . . 19
3.2.6 Verification of the solver’s implementation . . . 20
3.3 Circumventing the zero-circulation constraint . . . 21
3.3.1 Zero-circulation constraint. . . 21
3.3.2 Solving the Navier-Stokes equations in a rotating reference frame . . . . 23
3.3.3 Solving the Navier-Stokes equations in a rotating reference frame using Fourier expansions . . . 25
3.3.4 Changing frames of reference . . . 26
3.3.5 Verification of the implementation of the rotating reference frame . . . . 27
3.4 Instantaneous reconnection level . . . 30
3.4.1 Definition . . . 30
3.4.2 Method using vortex lines . . . 31
3.4.3 Trajectory algorithm . . . 31
3.4.4 Verification of the method . . . 33
3.4.5 Validation of the method . . . 34
4 Paper 35 Investigation of the viscous reconnection phenomenon of two vortex tubes through spectral simulations 36 4.1 Introduction. . . 36
4.2 Initial configurations . . . 38
4.3 Computational method. . . 39
4.4 Instantaneous reconnection level . . . 43
4.5 Visualization of the evolving topology . . . 46
4.6 Comparative analysis of the reconnection process . . . 48
4.7 Comparison with alternative η estimators for the antiparallel configuration . . . 55
4.8 Reynolds number dependence of the reconnection timescale . . . 60
4.9 Conclusion . . . 62 4.10 Acknowledgments . . . 62 4.11 References . . . 63 63 5 Conclusion 65 Bibliographie 70
A Numerical integration scheme 75
B Conference paper 79
Liste des tableaux
3.1 Instantaneous reconnection level η obtained using different number of subsec-tions Ns for the reconnection of two antiparallel vortices at ReΓ = 9, 000 and
t∗ = 11.0. For each case, we indicate the number of reconnected Nrec and
non-reconnected Nnon−rec vortex lines as well as the total number of vortex lines
Nf = Nrec+ Nnon−rec. The last column shows the percentage of the circulation on
the (z = 0, x < π) half-plane that the algorithm has successfully identified as being reconnected or non-reconnected. . . 33
3.2 Same conditions as in Table 3.1 but we vary the Courant number C instead of the number of subsections Ns. . . 34
4.1 Instantaneous reconnection level η for the reconnection of two antiparallel vortices at ReΓ= 9, 000 and t∗ = 11.4 (Nq= 1024). The calculations are carried out using
Eq. (4.17) with different number of subsections Ns . For each case, we indicate the
number of reconnected Nrec and non-reconnected Nnon−rec vortex sub-tubes and
the total number of vortex lines integrated Nf = Nrec+ Nnon−rec. . . 46
A.1 Generic Butcher tableau. . . 75
A.2 Butcher tableau of the low-storage, explicit third-order Runge-Kutta method used in this work. . . 75
Liste des figures
1.1 Simple schematics of the reconnection process of two vortices. The gray area re-presents the viscous cancellation zone. The arrows specify the local orientation of
the vorticity vector. The time t progresses from top to bottom. . . 1
1.2 Probability density function P (α) of the angle α formed by two random vectors in three dimensions. This distribution was obtained by sweeping the four-dimensional domain of Eq. 1.1 with ∆θi= ∆φi= 0.01 and 0 ≤ θi, φi≤ 2π. . . 3
2.1 Plot of v(x) = esin(x). . . . 11
2.2 Amplitude of the Fourier coefficients ˆv(k) of the spectral representation of v(x) = esin(x) for N = 40. . . . 11
2.3 Maximum error on the domain ( x ∈]0, 2π[ ) for the Fourier representation of v(x) = esin(x) and different spatial resolutions (2 ≤ N ≤ 40). . . . 11
3.1 Orthogonal vortex configuration. . . 14
3.2 Perturbed antiparallel vortex configuration. . . 14
3.3 Compact Gaussian distribution. . . 15
3.4 Total enstrophy Etot in the periodic box during the reconnection of two orthogonal vortices mimicking the high-fidelity simulations of Boratav et al. (1992). The blue and red squares were extracted from Fig. 24 in their paper and are for evolutions at ReΓ= 696 and ReΓ= 2088respectively. The solid and dashed black lines were obtained at the same Reynolds numbers using the present spectral solver at the same resolution. . . 21
3.5 Desired initial configuration: Single straight vortex placed in a triple-periodic box. 21 3.6 Modified initial configuration: Single straight vortex embedded in constant back-ground vorticity in a triple-periodic box. . . 21
3.7 Initial vorticity profile of a single straight vortex in a cubic box of length Lq= 2π. The profile follows a compact Gaussian distribution (see Eq. (3.4)). The desired profile is shown in solid blue, while the modified profile associated with the periodic velocity constraint (zero net circulation) is shown in dashed red. . . 23
3.8 Initial reduced circulation (ruθ) of a single straight vortex in a cubic box of length Lq = 2π. The vorticity profile follows a compact Gaussian distribution (see Eq. (3.4)). The reduced circulation of the desired profile is shown in solid blue, while the reduced circulation of the modified profile is shown in dashed red which illustrates a centrifugally unstable flow field acording to the criterion (3.37) . . . . 23
3.10 Enstrophy spectra at t∗ = 0 for the unperturbed (blue) and perturbed (red) single
straight vortex configurations. The periodic domain is cubic with Nq = 256 and
Lq = 2π.. . . 28
3.11 Enstrophy spectra for the perturbed single vortex configuration. The initial spec-trum (t∗ = 0) is shown in black. The blue and red curves denote the spectra at
t∗ = 100 for the simulation carried out in the proper rotating frame of reference (red) and for the simulation carried out in the inertial frame of reference (blue). The periodic domain is cubic with Nq = 256 and Lq = 2π. The simulation is
performed at ReΓ= 20, 000. . . 28
3.12 Evolution of a single vortex embedded in fine-scale perturbations at ReΓ = 20000
and using no correction for the background vorticity. The periodic domain is cubic with Nq = 256and Lq= 2π.. . . 29
3.13 Same conditions as in Fig. 3.12 but the simulation is carried out in the proper rotating reference frame Ω. . . 29
3.14 Schematics of a reconnection event where some circulation remains non-reconnected. 30
3.15 Collocation cube surrounding the position of interest x = (x, y, z). x0= (x0, y0, z0)
satisfies x0 < x, y0< y and z0 < z, while x1 = (x0+ ∆x, y0+ ∆y, z0+ ∆z). . . 32
4.1 Orthogonal vortex configuration. . . 38
4.2 Perturbed antiparallel vortex configuration. . . 38
4.3 Evolution of the instantaneous reconnection level η for the orthogonal vortex confi-guration, with η obtained using Eq. (4.17). Both simulations are performed at ReΓ = 2000and with the same local spatial resolution. The domain size is Lq = 2π
for the solid line (blue online) and Lq = 4π for the dashed line (red online). . . 42
4.4 Evolution of the maximum vorticity magnitude in the domain kω∗k
max for the
orthogonal vortex configuration. Same conditions as in Fig. 4.3 . . . 42
4.5 Enstrophy spectra at t∗ = t
1, with t1 being chosen such that it corresponds to
the highest value of the left-hand side of Eq. (4.16). The solid lines (red online) show the spectra of the low-resolution simulations, with Nq = 256for the low-ReΓ
cases (circles) and Nq = 512 for the high-ReΓ cases (crosses). The dashed lines
(blue online) show the spectra of the high-resolution simulations (Nq = 512for the
low-ReΓ cases and Nq = 1024for the high-ReΓ cases) . . . 43
4.6 Identification of the seeding and receiving surfaces, denoted as S1 and S2 respec-tively, for the antiparallel configuration. The dark gray structures (red and blue online) contain the non-reconnected parts of Vortex A and B, while the light gray structures (yellow online) contain the reconnected parts. The arrows show the di-rection of the local vorticity vector. The black grid illustrates the partitioning of S1 into Ns = 200subsections. Selected vortex lines are plotted in black. . . 44
4.7 Identification of the seeding and receiving surfaces for the orthogonal configuration. Same as in Fig. 4.6, but here Ns= 400. . . 45
4.8 Vorticity isosurfaces showing the evolution of the topology of the perturbed an-tiparallel vortex system at ReΓ = 2000 (a,b,c) and ReΓ = 9000 (d,e,f). Yellow
isosurfaces contain the reconnected parts ωrec of the vortices while the blue and
red isosurfaces contain the non-reconnected parts ωnon−rec. Black isosurfaces (e,f)
contain the looping structures ωloop, i.e., vorticity that is neither reconnected nor
non-reconnected. Isosurface values are kω∗k = [0.1 0.25 0.5 1.0] . . . . 49
4.9 Vorticity isosurfaces showing the evolution of the topology of the orthogonal vortex system at ReΓ= 2000(a,b,c) and ReΓ= 10, 000(d,e,f). Yellow isosurfaces contain
the reconnected parts ωrec of the vortices while the blue and red isosurfaces contain
the non-reconnected parts ωnon−rec. Isosurface values are kω∗k = [0.1 0.25 0.5 1.0] . 50
4.10 Three-dimensional energy spectra for the evolution of the antiparallel vortex confi-guration at different times corresponding to those shown in Fig. 4.8. The solid black, solid red and dashed black lines corresponds to Fig. 4.8 a,d, Fig. 4.8 b,e and Fig. 4.8 c,f respectively. . . 51
4.11 Same as in Fig. 4.10, but for the orthogonal vortex configuration (Fig. 4.9). . . 51
4.12 Evolution of the instantaneous reconnection level η. Dark gray (blue online) and light gray (red online) lines denote the antiparallel and orthogonal configurations respectively. Solid lines show the evolution of our low-Re cases (ReΓ = 2000)
and dashed lines our high-Re cases (ReΓ = 9000 for the antiparallel case and
ReΓ= 10, 000 for the orthogonal case). . . 52
4.13 Closeup of the antiparallel configuration depicted in Fig. 4.8e (ReΓ = 9000, t∗ =
11.0, η = 43.9%). The trajectories of two selected vortex lines are shown in white. Isosurface values are kω∗k = [0.5 1 3 5] . . . . 53
4.14 Isocontours of different quantities on the dividing plane (x = π) of the configuration depicted in Fig.4.8e (reconnection of two antiparallel vortices at ReΓ = 9000 and
t∗ = 11.0). Dashed lines show the projection on the x = π plane of the vortex lines
depicted in Fig. 4.13. Crosses indicate their intersection with the x = π plane. . . . 53
4.15 Same as in Fig. 4.14, but for the dividing plane of Fig.4.8b (reconnection of two antiparallel vortices at ReΓ= 2000 and t∗ = 11.6). . . 54
4.16 Simple model of the vortex topology during the reconnection process for the anti-parallel configuration. Non-reconnected parts of the vortices are shown in dark gray (blue and red online). Reconnected parts are shown in light gray (yellow online) (the thin yellow lines depict freshly reconnected filaments while the thick yellow lines represent the bridges). The arrows specify the local orientation of the vorticity vector. . . 55
4.17 Evolution of different estimators of the instantaneous reconnection level η for the perturbed antiparallel configuration at different Reynolds numbers. The solid line is obtained using Eq. (4.17), while the black and the light-gray (red online) dashed lines are computed using relations (4.26) and (4.29) respectively. . . 56
4.18 Two types of reconnection events that can happen during the evolution of the antiparallel configuration. Non-reconnected parts of the vortices are shown in dark gray (blue and red online), while reconnected parts are shown in light gray (yellow online). Looping structures are shown in black. . . 57
4.19 Improved model of the vortex topology during the reconnection process for the antiparallel configuration. Non-reconnected parts of the vortices are shown in dark gray (blue and red online), while reconnected parts are shown in light gray (yel-low online) (the thin yel(yel-low lines depict freshly reconnected filaments while the thick yellow lines represent the bridges). A vortex ring structure resulting from non-centered reconnection events is shown in black. The arrows specify the local orientation of the vorticity vector. . . 58
4.20 Closeup view of the configuration shown in Fig. 4.8f in order to better show the interaction between the remnants (ReΓ = 9000, t∗ = 13.2, η = 87.6%). Yellow
isosurfaces contain the reconnected parts of the vortices ωrec, while blue and red
isosurfaces contain the non-reconnected parts ωnon−rec. Black isosurfaces contain
the looping structures ωloop, i.e., vorticity that is neither reconnected nor
non-reconnected. kω∗
k = [0.15 0.4 0.8 1.5]. . . 59 4.21 Reynolds number dependence of the reconnection timescale Trec for the
orthogo-nal (red stars) and antiparallel (blue crosses) vortex configurations. The magenta circles show Trec for a modified antiparallel configuration having a stronger initial
perturbation ( = 0.3). . . 60
5.1 Schematics of a vortex line (curved dashed line). The positions x1 and x2 are
located on the vortex line and separated by a distance ∆s. The vorticity vectors at these two positions (ω1 and ω2) form an angle ∆θ. . . 67
Remerciements
Tout d’abord, j’aimerais remercier mon directeur de recherche, le professeur Guy Dumas. Au fil de mes quelques années à le côtoyer, j’ai pu apprécier sa profonde connaissance de la mécanique des fluides, sa passion pour la recherche ainsi que sa grande sagesse. J’aimerais aussi remercier mon codirecteur, le professeur Louis Dufresne, dont la précieuse aide a été essentielle à la réussite de ce projet. À travers nos interactions, ces deux mentors m’ont entre autres appris à apprécier la mécanique des fluides non seulement pour la panoplie d’applications qu’elle engendre, mais également pour sa beauté intrinsèque. Leur passion fut contagieuse et je les en remercie. Je garde un excellent souvenir des rencontres animées et instructives que nous avons eues tous les trois.
J’aimerais également remercier mes collègues actuels et passés du Laboratoire de Mécanique des Fluides Numérique (LMFN) : Matthieu Boudreau, Carl-Anthony Beaubien, Sébastien Bourget, Philippe Côté, Étienne Gauthier, Olivier Gauvin-Tremblay, Rémi Gosselin, Marine Heschung, Steve Julien, Thomas Kinsey, Simon Lapointe, Julie Lefrançois, Mathieu Olivier, Christian Perron, Marc-André Plourde Campagna et Jean-Christophe Veilleux. Merci pour les nombreux et fructueux échanges ainsi que pour la bonne humeur qui règne en permanence au laboratoire.
Je souhaite aussi souligner les bourses octroyées par le Fonds de Recherche du Québec - Nature et technologies (FRQNT) et le Conseil de Recherches en Sciences Naturelles et en Génie (CRSNG), qui ont été d’une importance capitale dans l’accomplissement de ce projet en me permettant de m’y consacrer pleinement.
Sur une note plus personnelle, j’aimerais remercier Catherine pour m’avoir appuyé tout au long de cette aventure et avoir toujours su me faire rire, même lorsque c’était nuageux. Enfin, un immense merci à mes parents, les meilleurs au monde, pour votre présence, votre aide et vos inestimables conseils.
Avant-propos
Chapter4presents the manuscript entitled Investigation of the viscous reconnection phenome-non of two vortex tubes through spectral simulations that was initially submitted on August 18, 2015 to the journal Physics of Fluids. In this thesis, we present the revised version of the manuscript which was submitted on January 29, 2016. As the first author, I developed and implemented the method that is proposed in this paper. I also performed the numerical simulations and carried out their analysis. I wrote the first draft which was then revised by my codirector Professor Louis Dufresne and my director Professor Guy Dumas, respectively the first and second coauthors.
Chapitre 1
Introduction
The viscous reconnection of two vortex tubes is a fundamental process occurring in a variety of phenomena ranging from large-scale interactions of aircraft wake vortices (Spalart(1998)) to fine-scale mixing in turbulence (Hussain (1983,1986)). Despite being the focus of various studies in the past, this phenomenon still gather attention as its characterization and quan-tification have remained vague (Kida and Takaoka (1994); Hussain and Duraisamy (2011)). The goal of this thesis is to contribute to the existing knowledge on this topic by carrying out numerical simulations of simple vortex configurations undergoing a reconnection process.
1.1
Vortex reconnection phenomenon
Vortex reconnection happens when two vortex filaments like those shown in Fig. 1.1 locally become so close that viscous diffusion cancels the opposite-signed vorticity of both filaments (the gray area in this figure). This results in the reconnection of the remaining parts of the filaments on both sides of the viscous cancellation zone (Wu et al.(2007)).
𝑡
Figure 1.1 – Simple schematics of the reconnection process of two vortices. The gray area re-presents the viscous cancellation zone. The arrows specify the local orientation of the vorticity vector. The time t progresses from top to bottom.
Attention to the vortex reconnection phenomenon sparked from the late-stage evolution of the Crow instability (seeCrow(1970)) of two antiparallel vortex tubes, which is characterized by their collision and subsequent reconnection into vortex rings. In this work, vortex reconnection refers to a change in the topology of the vorticity lines, in accordance with the vorticity reconnection concept of Kida and Takaoka (1994). Therefore, vortex reconnection requires viscosity by the Kelvin-Helmholtz theorems.
One of the questions that we want to investigate in this work is the impact of the Reynolds number on the reconnection process. At high Reynolds numbers, carrying out DNS becomes very costly. The choice of the initial vortex configurations is therefore aimed at minimizing the required computational resources for a given vortex Reynolds number ReΓ ≡ Γ/ν. For
example, since the reconnection of aircraft wake vortices is of great practical interest, it would be interesting to study the evolution of two antiparallel vortices embedded in fine-scale tur-bulence. However, simulating this whole vortex system using DNS would be very costly, as one has to wait for the instability to grow and use a rather large computational domain. As a consequence, we look for simpler vortex configurations leading to reconnection in a shorter time frame and taking place in a more limited physical space.
Most of the experimental studies of the reconnection phenomenon have focused on highly or-dered vortical motions (Kida and Takaoka(1994)), such as colliding vortex rings and counter-rotating vortex tubes shed from a lifting surface. Similarly, the vast majority of the numerical studies investigating viscous vortex reconnection employ one of three simple configurations : vortex rings (Ashurst and Meiron(1987);Kida et al.(1989,1991);Aref and Zawadzki(1991)), orthogonal vortices (Melander and Zabusky (1988); Zabusky and Melander (1989); Zabusky et al.(1991);Boratav et al. (1992)) and antiparallel vortices with a perturbation (Pumir and Kerr (1987); Melander and Hussain (1988); Kerr and Hussain (1989); Shelley et al. (1993);
Hussain and Duraisamy(2011);van Rees et al.(2012);Kerr(2013)). In this thesis, we choose to concentrate our efforts on the orthogonal and the perturbed antiparallel configurations fol-lowing the approaches ofBoratav et al.(1992) andHussain and Duraisamy(2011) respectively. The choice of the antiparallel configuration can be easily justified by the practical importance of better characterizing the wake vortex dynamics behing a lifting surface, in which reconnection often plays a crucial role (Spalart (1998)). For its part, the orthogonal configuration might seem to be more of academic interest. However, as noted byBoratav et al.(1992), interactions between orthogonal vortices are more likely than interactions between antiparallel vortices in a random (e.g. turbulent) three-dimensional field. To prove this point, we consider two straight vortices randomly placed in relation to each other. The angle between two vectors in three dimensions is given by
α = cos−1(cos(θ1) cos(θ2) + sin(θ1) sin(θ2) + sin(φ1) sin(φ2)), (1.1)
where θ1, θ2 are the azimuthal angles taken from the spherical coordinates representation
of the two vectors, while φ1, φ2 are the polar angles. Fig. 1.2 shows the probability density
function P (α) for the angle α between two random vortices. This distribution was obtained by sweeping the four-dimensional domain of Eq. (1.1) with ∆θ = ∆φ = 0.01 and 0 ≤ θi,φi≤ 2π.
As it can be seen, vortex encounters are ten times more likely to happen around α = 90° than around α = 180° in a random field. Therefore, the evolution of the orthogonal vortex configuration is very relevant from a physical point of view as it is likely to be a generic phenomenon in turbulent flows.
0 50 100 150 0 0.002 0.004 0.006 0.008 0.01
Figure 1.2 – Probability density function P (α) of the angle α formed by two random vectors in three dimensions. This distribution was obtained by sweeping the four-dimensional domain of Eq.1.1 with ∆θi = ∆φi = 0.01 and 0 ≤ θi,φi ≤ 2π.
1.2
Objectives of the master’s thesis
The global objective of this master’s thesis is to contribute to the existing knowledge of the viscous reconnection process of two vortex tubes through direct numerical simulations (DNS) of the incompressible Navier-Stokes equations. The specific objectives of this thesis are threefold :
. to develop a simple and effective way to accurately evaluate the degree of reconnection of two vortices ;
. to find a way to use the triply-periodic spectral method for non-zero circulation flows ; . to compare the reconnection processes of the orthogonal and the perturbed antiparallel
vortex configurations.
The first two objectives are more methodological in nature while the third focuses on the physics involved.
1.2.1 First objective
First, we intend to develop, implement and validate a simple method using vortex lines to accurately estimate the instantaneous degree of reconnection of two vortices, usable both for the orthogonal and the perturbed antiparallel configurations. For the reconnection of two anti-parallel vortices and under certain assumptions, some authors use a simple surface-integrated quantity to approximate the degree of reconnection (Melander and Hussain,1988; Kerr and Hussain,1989;Shelley et al.,1993;van Rees et al.,2011;Hussain and Duraisamy,2011;Kerr,
2013), which is sometimes termed circulation transfer (van Rees et al.,2011;Hussain and Du-raisamy,2011). We will use this approximation to verify the implementation of our method.
1.2.2 Second objective
Second, we aim to find a way to use the triply-periodic spectral method for non-zero circulation flows. Fourier series can only represent velocity fields for which the circulation on the periodic box is null. When attempting to represent configurations with a non-zero circulation, there is apparition of a small background vorticity. It is possible to decrease the intensity of this spurious vorticity by increasing the size of the periodic box. However, it was observed by
Pradeep and Hussain (2004) that even for large computational domains, a single vortex is erroneously rendered centrifugally unstable by the (inviscid) Rayleigh criterion. We therefore look for a way to circumvent the zero-circulation constraint.
1.2.3 Third objective
Third, we want to compare the reconnection processes of the orthogonal and the perturbed antiparallel vortex configurations. It has already been noted that these two initial configura-tions follow very similar reconnection processes (Boratav et al.,1992;Zabusky and Melander,
1989). However, we intend to explore these similarities in greater depth by directly comparing the two evolutions, using the tools outlined in the first objective. Also, the orthogonal configu-ration having a finite (non-zero) circulation, results stemming from the second objective will be critical to accurately simulate the physics at play.
1.3
Thesis structure
After having introduced the context of this study in this chapter, we outline some important aspects of spectral methods and Fourier series in Chapter 2. In Chapter 3, we detail the me-thodology used in this work. We present the initial configurations and the governing equations (Sec. 3.1), the spectral solver that we employ (Sec. 3.2) as well as the method that we use to circumvent the zero-circulation constraint (Sec. 3.3).
The aim of the three first chapters is to complete the information given in the paper stemming from this work, which is presented in Chapter 4. By its nature, Chapter 4 is therefore self-supporting and readers already familiar with this subject might consider reading the paper first. Finally, the main results and future research avenues are discussed in Chapter 5. Note that while the paper is presented with its own bibliography, the complete set of references can be found at the end of the thesis.
Chapitre 2
Preliminary concepts in spectral
methods
2.1
Introduction to spectral methods
The information presented in this section provides a brief background on the concepts and terminology of spectral methods. It is mostly based on the following books : Spectral Me-thods : Fundamentals in Single Domains (Canuto et al.(2007a)), Spectral Methods : Evolution to Complex Geometries and Applications to Fluid Dynamics (Canuto et al.(2007b)) and Nu-merical Analysis of Spectral Methods : Theory and Application (Gottlieb and Orszag(1977)). The interested reader is referred to these textbooks for a comprehensive review of the different spectral methods and their applications.
2.1.1 Weighted residual methods
Weighted residual methods (WRM) aim to find an approximate solution to a system of ordi-nary differential equations (ODE) or partial differential equations (PDE) using a finite sum of trial functions (Finlayson and Scriven(1966)). For example, suppose that we want to solve
D(u(x)) = p(x) x0 ≤ x ≤ x1 (2.1)
for u(x), where D() is a linear operator. We first approximate the solution by u(x)≈ PNu(x) =
N −1
X
k=0
akφk(x) , (2.2)
where φk(x) are the trial functions—chosen in advance for a given problem—and the ak are
In order to find these coefficients, we define the residual
R(x) = D(PNu(x))− p(x) 6= 0 . (2.3)
We force the residual to be null in an average sense, i.e., Z x1
x=x0
ψjR(x)dx = 0 j = 0, 1, . . . , N − 1 , (2.4)
where ψj are the test functions, cleverly chosen in order to simplify the system of equations
arising from (2.4). The choices of both φk and ψj define a particular WRM.
Spectral methods are an important subclass of WRM in which the trial functions φk are
non-zero over the whole domain [x0, x1]. The use of spectral methods to solve PDE systems
was first investigated by Steven Orszag in a series of papers starting in 1969. Amongst other things, this pioneer explored periodic domains using Fourier series, finite and infinite domains using polynomial expansions (Chebyshev, Legendre, etc.), as well as the resolution of strongly nonlinear problems using pseudospectral methods.
2.1.2 Different types of spectral methods
Collocation
In a collocation method, the test functions ψj are Dirac delta functions centered at the
col-location points, the latter depending on the particular choice of trial functions φk. Therefore,
this method forces the residual (2.4) to be null at the collocation points. Galerkin
In a Galerkin method, the test functions ψj are non-zero over the whole domain. Also, the
trial functions φk are chosen to individually satisfy the boundary conditions. In this work,
we employ a Galerkin method as we use Fourier series in the three Cartesian directions on a triply periodic domain.
Tau
Tau methods also use test functions ψj that are non-zero over the entire domain. In contrast
to Galerkin methods, Tau methods do not require the trial functions φkto individually satisfy
the boundary conditions. As a consequence, the boundary conditions must be explicitly in-corporated into the system of equations to be solved. An example of a Tau method would be to use Chebyshev polynomials as trial functions on a domain with Dirichlet and/or Neumann boundary conditions.
2.2
Fourier representation of a function
2.2.1 Definition
Let u(x) be a function that we want to approximate using a complex Fourier series. If u(x) is continuous, periodic, and of bounded variation on 0 ≤ x < 2π , then its complex Fourier series representation converges uniformly (see Canuto et al.(2007a)) and therefore
u(x) = Su(x)≡ +∞ X k→−∞ ˜ u(k)eikx. (2.5)
Using the orthogonality relation Z 2π x=0 eimxe−inxdx = 2π if m = n 0 otherwise (2.6)
along with Eq. (2.5), one can find the different Fourier coefficients ˜ u(k) = 1 2π Z 2π x=0 u(x) e−ikxdx k∈ ] − ∞, + ∞[ . (2.7)
However, since Eq. (2.5) implies to store an infinite array of coefficients ˜u(k), we therefore consider the discrete (troncated) Fourier expansion
u(x)≈ PNu(x)≡ N/2−1 X k=−N/2 ˆ u(k)eikx. (2.8)
We now look for a simple way to compute the N coefficients ˆu(k) from Eq.(2.8). Using the discrete set of points
xj ≡
2πj
N , j = 0, 1, . . . , N − 1 , (2.9)
which will be referred to as the collocation points, it is possible to derive the following relation 1 N N −1 X j=0 eipxjdx = 1 if p = Nm, m = 0, ± 1, ± 2, . . . 0 otherwise , (2.10)
which is the discrete version of the orthogonality relation (2.6). Multiplying Eq. (2.8) with Eq. (2.10), one finds
ˆ u(k) = 1 N N −1 X j=0 u(xj) e−ikxj k∈ [−N/2, N/2[ , (2.11)
which is the discrete equivalent of (2.7). Note that PNu(xj) has been replaced with u(xj) in
(2.11) as it can be shown that (seeCanuto et al.(2007a))
PNu(xj) = u(xj) j = 0, 1, . . . , N − 1 . (2.12)
2.2.2 Precision
Consider another function v(x), periodic on 0 ≤ x < 2π and having continuous derivatives v(p)(x) for p = 0,1, . . . ,n − 1. Then, integration by parts of Eq. (2.7) yields
ˆ v(k) = 1 2π(ik)n Z 2π x=0 v(n)(x) e−ikxdx k∈ ] − ∞, + ∞[ (2.13) provided that v(n)(x) is integrable so that the right-hand side of (2.13) has a finite value.
Using the Riemann-Lebesgue lemma (see Gottlieb and Orszag (1977)), it can be deduced from Eq. (2.13) that
ˆ
v(k) 1/kn k→ ∞ (2.14)
as well as
max|PNv(x)− v(x)| ∼ O(1/Nn−1) N → ∞. (2.15)
Following Eq. (2.15), if v(x) is periodic on 0 ≤ x < 2π, infinitely differentiable and that its derivatives are continuous, then the decay rate of the Fourier coefficients ˆv(k) must be faster than algebraic for k sufficiently large. This is what is called “spectral convergence”, which corresponds in fact to an exponential convergence rate. For example, let
v(x) = esin(x), (2.16)
which is shown in Fig. 2.1. Since (2.16) is periodic on 0 ≤ x < 2π, infinitely differentiable and that its derivatives are continuous, we expect the coefficients ˆv(k) to decay faster than algebraically. This is indeed what we observe in Fig.2.2, which uses a log-log scale. Similarly, as shown on Fig.2.3, the maximum error on the domain is seen to decay faster than algebraically as the resolution (number of modes) is increased. On Fig.2.2, it can be seen that ˆv(k) ∼ 10−15
for k& 15. This is due to the machine precision of the computer number format that we use. Since we employ a double-precision representation, we have ≈ 10−16. This error completely
contaminates the computed values of the higher wavenumbers (k & 15). As seen on Fig.2.3, this in turn imposes a minimum limit of O ∼ (10−14) on the maximum error on the domain.
As we have just seen, one of the most remarkable characteristics of spectral approximation is the exponential convergence rate of the spectral coefficients. This means that if someone obtains a fair approximation using a certain discretization, increasing the resolution a little more will yield a very precise approximation. For well-suited problems, spectral methods are therefore unrivaled when a very high precision is sought, which is typically what one aims for when carrying out DNS.
0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3
Figure 2.1 – Plot of v(x) = esin(x).
100 101
10−15 10−10 10−5 100
Figure 2.2 – Amplitude of the Fourier coefficients ˆv(k) of the spectral represen-tation of v(x) = esin(x) for N = 40.
100 101
10−15 10−10 10−5 100
Figure 2.3 – Maximum error on the do-main ( x ∈]0,2π[ ) for the Fourier represen-tation of v(x) = esin(x) and different
Chapitre 3
Methodology
3.1
Problem definition
3.1.1 Initial vortex configurations
Orthogonal vortex configuration
The orthogonal vortex configuration shown in Fig. 3.1 is made of two orthogonally-offset, straight vortex tubes initially separated by a distance b0.
Antiparallel vortex configuration with a perturbation
The other initial configuration, shown in Fig. 3.2, consists of two anti-parallel vortices with an initial perturbation designed to force the collision of the two vortices and their subsequent reconnection. Following the approach of Hussain and Duraisamy(2011), we separate the un-perturbed cores by a distance b0 = 1.62 and we set a sinusoidal perturbation on the vortex
centroids
r = [ xc+ p cos(α) sin(z), yc+ p sin(α) sin(z), z ] (3.1)
where xc and yc correspond to the positions of the unperturbed vortex centroids, p = 0.2 and
α = 60◦. For both the orthogonal and the antiparallel configurations, nondimensional time t∗ is given by
t∗ ≡ t b02/Γ
(3.2) while nondimensional vorticity ω∗ is defined as
ω∗ ≡ ω
kωkmax,0
Figure 3.1 – Orthogonal vortex configu-ration.
Figure 3.2 – Perturbed antiparallel vor-tex configuration.
where kωkmax,0 is the maximum vorticity magnitude in the domain at the initial time t ∗ = 0.
Note that throughout this work, k · k denotes the Euclidian norm.
3.1.2 Vorticity distribution
The vorticity profile of each vortex, shown in Fig.3.3, is chosen to exhibit a compact Gaussian distribution (seeMelander and Hussain (1988)), i.e.,
ω(r) = α 1− exp−R κ/r exp [R/(r − R)] if r ≤ R 0 if r > R , (3.4)
where R is the radius of the vortex, κ is a shape parameter and α is a scaling factor. In this work, we choose κ = 2.56 (seeMelander et al.(1987)) and R = 0.5. The vortex circulation Γ is given by : Γ = Z 2π θ=0 Z R r=0 α R2 1− exp−R κ/r exp [R/(r − R)] r dr dθ u 0.863905 α R2. (3.5) 14
Therefore, the parameter α can be expressed as a function of R and the desired vortex circu-lation Γ :
α u 1.15753 Γ
R2 (3.6)
When simulating the interaction between two or more vortices, the compactness of Eq. (3.4) prevents the vortices from overlapping initially. If we were using a non-compact distribution, the vortices would have no choice but to be slightly overlapped, which is counter-intuitive as we expect the vortices to experience significant deformation before their respective vorticity diffuse into each other. Also, all the derivatives of Eq. (3.4) exist and are continuous, therefore allowing for an exponential convergence of the spectral representation of Eq. (3.4) (see Sec. 2.2.2).
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Figure 3.3 – Compact Gaussian distribution.
3.1.3 Incompressible Navier-Stokes equations
Fluid dynamics can be described with differential equations expressing the principles of mass, energy and momentum conservation. In this work, we solve the incompressible Navier-Stokes equations for a Newtonian fluid in Cartesian coordinates
∂u
∂t +∇ · (uu) = − 1
ρ∇p + ν ∇
2u, (3.7)
along with the incompressibility constraint
∇ · u = 0 , (3.8)
where u is the velocity vector, ρ is the fluid density, p is the static pressure and ν is the kinematic viscosity. We assume that the properties of the fluid are constant, i.e., ν(x,t) = ν and ρ(x,t) = ρ. Providing appropriate boundary and initial conditions, Eqs. (3.7) and (3.8) form a complete set of equations for p and the three components of u.
For further information on the derivation of these equations, the interested reader is referred to textbooks such as Fundamentals of Fluid Mechanics by Munson et al. (2012) and Incom-pressible Flow by Panton(2013).
3.2
Spectral solver
3.2.1 Fourier expansion
We pose a Fourier expansion for the velocity field u(x,t)≈ PNu(x,t)≡
X
k
ˆ
u(k,t)eik·x, (3.9)
where the wavenumber vector k is given by k={ kx, ky, kz}, kq= mq 2π Lq − Nq 2 ≤ mq< Nq 2 and the position vector x by
x={ x, y, z} xq = mq
2π Nq
0≤ mq< Nq (3.10)
with q being an index standing for one of the three Cartesian coordinates and L representing (Lx, Ly, Lz), the size of the periodic domain.
3.2.2 Solving the governing equations in the Fourier space
The Fourier transform of Eq. (3.7) yields d dt+ νkkk 2 ˆ u = −ikˆp + ˆf, (3.11) where ˆ f =− \∇ · (uu). (3.12)
3.2.3 Elimination of the pressure term
The pressure term of Eq. (3.11) can be eliminated by taking the divergence of Eq. (3.7) (see
Orszag and Patterson(1972); Canuto et al.(2007b)) :
∇ ·∂u∂t − ∇ · f = −∇ · ∇p + ν ∇ · ∇2u. (3.13)
Due to the incompressibility constraint (3.8), the time derivative vanishes ∇ ·∂u
∂t = ∂k~ek · ∂tui~ei = ∂t ∂| {z }iui
=0
= 0 , (3.14)
as well as the viscous term
ν∇ · ∇2u= ν ∂
k~ek · ∂j∂jui~ei = ν ∂j∂j ∂iui
| {z }
=0
= 0 . (3.15)
Therefore, Eq. (3.13) simplifies to
∇ · f = ∇2p . (3.16)
Taking the Fourier transform of Eq. (3.16), i.e.,
ik· ˆf =−kkk2p ,ˆ (3.17)
one finds an explicit relation for ˆp : ˆ
p =−i(k· ˆf)
kkk2 . (3.18)
Using Eq. (3.18), Eq. (3.11) can be rewritten without the pressure : d dt + νkkk 2 ˆ u = −k(k· ˆf) kkk2 + ˆf. (3.19)
From Eq. (3.19), it can be infered that the role of the pressure is to project the non-linear term onto the basis of functions satisfying the incompressibility constraint (Eq. (3.8)).
3.2.4 Evaluation of the non-linear term
Pseudospectral approximation
In the spectral domain, the non-linear term ˆf amounts to a convolution sum ˆ
f(k) = X
m+n=k
im ˆu(m) ˆu(n) , (3.20)
where m, n and k are wavenumber vectors. For each wavenumber k, the sum in Eq. (3.20) requires O(N3) operations. Since there are N3 wavenumbers, the total operation count for
the non-linear term amounts to O(N6), which is prohibitively expensive. To circumvent this
transform (FFT), the total operation count drops to O(N3log
2N ). However, the evaluation
of the non-linear term in the physical space may introduce aliasing errors, hence the term pseudospectral approximation.
Aliasing errors
It can be shown (Canuto et al.(2007a)) that using a pseudospectral approximation leads to ˆ f∗(k) = X m+n=k im ˆu(m) ˆu(n)+ X m+n=k±N im ˆu(m) ˆu(n) = ˆf(k) + X m+n=k±N im ˆu(m) ˆu(n) . (3.21) The last term in Eq. (3.21) is the aliasing error. The next section is concerned with minimizing this error using the phase-shift technique.
Phase-shift technique
The phase-shift technique, which was developed byPatterson and Orszag(1971), can be used to completely remove the aliasing error. For simplicity, assume that we want to compute the following one-dimensional non-linear term :
ˆ
s(k) =− \u(x)v(x) = X
m+n=k
ˆ
u(m)ˆv(n). (3.22)
First, we compute the transform on a grid shifted by the factor ∆ in the physical space :
u∆(x j) = N/2−1 X k=−N/2 ˆ u(k)eik(xj+∆), v∆ j = N/2−1 X k=−N/2 ˆ v(k)eik(xj+∆), j = 0,1, . . . ,N − 1 . (3.23) We then compute s∆(xj) = u∆(xj)v∆(xj) j = 0,1, . . . ,N− 1 (3.24) and ˆ s∆(k) = 1 N N −1 X j=0 s∆(xj)e−ik(xj+∆). (3.25)
Substituting Eqs. (3.23) and (3.24) in Eq. (3.25), one obtains
ˆ s∆(k) = X m+n=k ˆ u(m)ˆv(n) + e±iN ∆ X m+n=k±N ˆ u(m)ˆv(n) . (3.26) 18
Choosing ∆ = π/N, the fully dealiased coefficients ˆs(k) can be computed using ˆ s(k) = 1 2 ˆ s∆(k) + ˆs∗(k), (3.27) where ˆ s∗(k) = 1 N N −1 X j=0 u(xj)v(xj)e−ikxj = X m+n=k ˆ u(m)ˆv(n) + X m+n=k±N ˆ u(m)ˆv(n) (3.28) is the aliased evaluation of the non-linear term. Therefore, the aliasing contributions to the non-linear term can be eliminated completely at the cost of two evaluations of the convolution sum.
Random phase-shift technique
In three dimensions, eight evaluations of the convolution sum are required in order to fully remove the aliasing error (Canuto et al.(2007b)), which is quite expensive knowing that FFTs account for most of the computational cost of pseudospectral methods. However, Rogallo
(1977) observed how the phase-shift technique can be incorporated at practically no cost in a time integrating scheme such as the four-stage, third-order explicit Runge-Kutta scheme used in this work. This method — called the random phase-shift technique — does not completely remove the aliasing errors but greatly reduces them. That is the method implemented in the present spectral solver. Its effectiveness has been monitored and asserted throughout this study by looking at the high-wavenumber amplitude of kinetic and enstrophy spectra.
3.2.5 Time integration
Integrating Factor Technique
The viscous term is treated exactly with the integrating factor technique (Rogallo (1977)). This technique can be applied to any system of equations of the form
∂ y(x,t)
∂t + P (x) y(x,t) = Q(x) . (3.29)
We first pose
For Eq. (3.19), we have P (k) = νkkk2, (3.31) hence M (k,t) = eR0t(ν kkk 2)dt = e(ν kkk2)t. (3.32)
Multiplying (3.19) by (3.32), one obtains d dt e(ν kkk2)t uˆ = e(ν kkk2)t fˆ− k(k· ˆf) kkk2 ! . (3.33)
The viscous term is now implicitly treated and thus exactly integrated. Explicit third-order Runge-Kutta method for the non-linear term
The RHS of (3.33) is integrated using an explicit, third-order, four-stage, low-storage Runge-Kutta method. The details of this numerical scheme are given inAppendix A.
3.2.6 Verification of the solver’s implementation
The implementation of the spectral solver was verified by reproducing simulations from Bo-ratav et al. (1992), who investigated the reconnection process of two orthogonal vortices. In order to achieve high Reynolds numbers using a limited spatial resolution, they solved the hyperviscous Navier-Stokes equations
∂u ∂t +∇ · (uu) = − 1 ρ∇p + ν ∇ 2u− ν 4∇4u, (3.34)
with ν4 chosen such that ν/ν4 = 500. The interested reader is referred to the subsection
Hyperviscosityof Chap.5for a brief introduction on the subject. The periodic box is a cube of length Lq= 2πand the spatial resolution is uniform with Nq= 64. For two different Reynolds
numbers, Fig.3.4shows the evolution of the total enstrophy in the computational domain Etot≡ 1 2 Z V– ω 2d V– . (3.35)
The squares in the figure correspond to data extracted from Fig. 24 in their paper, while the lines were obtained using our spectral solver. It can be seen that the agreement is excellent. This, among several other things, gives us great confidence that the implementation of our spectral solver is bug-free and adequate.
0 2 4 6 4 6 8 10 12 14
Figure 3.4 – Total enstrophy Etot in the periodic box during the reconnection of two
ortho-gonal vortices mimicking the high-fidelity simulations of Boratav et al.(1992). The blue and red squares were extracted from Fig. 24 in their paper and are for evolutions at ReΓ = 696
and ReΓ = 2088 respectively. The solid and dashed black lines were obtained at the same
Reynolds numbers using the present spectral solver at the same resolution.
3.3
Circumventing the zero-circulation constraint
3.3.1 Zero-circulation constraint
Assume that we want to study the evolution of a single straight vortex in a triple-periodic box, as shown in Fig. 3.5.
Figure 3.5 – Desired initial configura-tion : Single straight vortex placed in a triple-periodic box.
Figure 3.6 – Modified initial configura-tion : Single straight vortex embedded in constant background vorticity in a triple-periodic box.
Because this configuration is periodic for vorticity but not for velocity, the present Fourier representation of this velocity field corresponds in fact to the velocity field associated with another vorticity distribution. In other words, the velocity field associated with a homoge-neous, finite-circulation vorticity field is not homogeneous. The modified vorticity distribution consists of the original vorticity field embedded in a constant background vorticity, as depic-ted in Fig. 3.6. This is due to the fact that a periodic velocity field (assumed by our Fourier
expansions) must be circulation-free when evaluated on its periodic boundaries. Therefore, the surface integral of vorticity must equal zero in the x-y plane of Fig.3.6. The amplitude of the background vorticity is thus such that the total circulation on the periodic box is zero, allowing periodicity for velocity :
ωbg,z =−
Γz
LxLy
. (3.36)
Figure 3.7 shows the desired and the modified initial vorticity profiles for a single straight vortex in a cubic domain of length Lq = 2π. The desired profile is shown in solid blue, while the
modified profile is shown in dashed red. It can be seen that both profiles are very similar in this example, which is quite representative and which will be used in Sec.3.3.5to demonstrate the potential impact of this spurious background vorticity. The two curves have a vertical offset of (∆ω/ω(0)) = 5.5 × 10−3, which corresponds to the background vorticity obtained from
Eq.(3.36).Pradeep and Hussain(2004) found that the zero-circulation constraint imposed by the triply-periodic expansions for velocity could strongly influence the dynamics of even the most simple vortex configurations. Indeed, they showed that a straight isolated vortex may be rendered unstable following the (inviscid) Rayleigh criterion
∂ (ruθ)2
∂r < 0 , (3.37)
where uθ is the circumferential velocity. This criterion stipulates that in the absence of
vis-cosity, an axisymmetric flow satisfying Eq. (3.37) is subject to a centrifugal instability for an infinitesimal perturbation (Drazin and Reid(2004)). We plot the left-hand side of Eq. (3.37) in Fig.3.8for both the desired and the modified configurations. It can be seen that according to Eq. (3.37), most of the computational domain is rendered centrifugally unstable in the case of the modified initial configuration (dashed red line). As we will see in Sec.3.3.5, a certain level of perturbation is necessary to trigger the centrifugal instability in a real fluid as viscous effects tend to stabilize the flow (Drazin and Reid(2004)). Note that previous work byDufresne and Winckelmans(2005) had shown that the presence of this background vorticity does not signi-ficantly affect the reconnection dynamics of two orthogonal vortices. However, it is possible to completely circumvent the zero-circulation constraint with negligible computational cost, as detailed in the next section.
0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1
Figure 3.7 – Initial vorticity profile of a single straight vortex in a cubic box of length Lq = 2π. The profile follows a compact Gaussian distribution (see Eq. (3.4)). The desired
profile is shown in solid blue, while the modified profile associated with the periodic velocity constraint (zero net circulation) is shown in dashed red.
0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 3.8 – Initial reduced circulation (ruθ) of a single straight vortex in a cubic box of length
Lq = 2π. The vorticity profile follows a compact Gaussian distribution (see Eq. (3.4)). The
reduced circulation of the desired profile is shown in solid blue, while the reduced circulation of the modified profile is shown in dashed red which illustrates a centrifugally unstable flow field acording to the criterion (3.37)
3.3.2 Solving the Navier-Stokes equations in a rotating reference frame
As noted by Otheguy et al. (2006), the modified initial configuration depicted in Fig. 3.6
corresponds to the desired configuration (Fig. 3.5) seen from a rotating reference frame with an angular velocity
Ωz= ωbg,z =−
Γz
LxLy
Therefore, solving the Navier-Stokes equations in a rotating frame with angular velocity Ω for the modified initial configuration (Fig.3.6) is equivalent to solving the Navier-Stokes equations in the inertial reference frame for the desired initial configuration (Fig.3.5).
If the initial vorticity field contains vorticity in the three Cartesians directions, the background vorticity is given by ωbg =− Γx LyLz ˆ ex− Γy LxLz ˆ ey− Γz LxLy ˆ ez. (3.39)
Using Eq. (3.39), Eq. (3.38) can be generalized to any vorticity field Ω=−ωbg 2 = Γx 2 LyLz ˆ ex+ Γy 2 LxLz ˆ ey+ Γz 2 LxLy ˆ ez. (3.40)
We want to solve the Navier-Stokes equations in a rotating reference frame, i.e., ∂ uR ∂t +∇ · (uRuR) =− 1 ρ∇p + ν∇ 2u R− 2 Ω × uR− Ω × (Ω × r) , (3.41)
where r is the three-dimensional position relative to the center of rotation xΩ. The subscript
( )R refers to a quantity computed in the rotating reference frame, while ( )I means that the quantity is computed in the inertial reference frame.
As detailed inTritton(1988), the centrifugal term (Ω×(Ω×r)) of Eq. (3.41) can be expressed as the gradient of a potential :
Ω× (Ω × r) = −∇ 1 2kΩk
2r02
, (3.42)
where r0 is the distance to the rotation axis, as depicted in Fig.3.9. r0 is related to r by the
geometrical relation r0= r− ΩΩ· r kΩk2 . (3.43)
Figure 3.9 – Geometrical relation between r, r0 and Ω.
Using Eq. (3.42) , the centrifugal term of Eq. (3.41) can be included in a modified pressure pm= p− 1 2kΩk 2 r02. (3.44)
Using Eq. (3.44), Eq. (3.41) simplifies to ∂ uR ∂t + 1 ρ∇pm− ν∇ 2u R=−∇ · (uRuR)− 2 Ω × uR. (3.45)
Thus, the only term that needs to be taken into account explicitly is the Coriolis term (2 Ω × uR).
3.3.3 Solving the Navier-Stokes equations in a rotating reference frame
using Fourier expansions
First, we express the velocity vector, the pressure and the right-hand side of Eq. 3.45 using three-dimensional Fourier expansions :
uR= X k ˆ uRei k·x, (3.46) pm = X k ˆ pmei k·x, (3.47) − ∇ · (uRuR)− 2 Ω × uR ≡ fR = X k ˆ fRei k·x. (3.48)
Substituting (3.46), (3.47) and (3.48) into (3.45) and using the orthogonality relation (2.10), one obtains ∂ ∂t + νkkk 2 ˆ uR = ˆfR− 1 ρik ˆpm. (3.49)
As detailed in Sec. 3.2.3, an explicit relation for the modified pressure can be found by taking the divergence of Eq. (3.45) and transforming it in the Fourier space :
ˆ
pm =−i
k· ˆfR
Note that because the divergence of the Coriolis term is non-null, i.e., ∇ · (Ω × uR) = (∇ × Ω)
| {z }
=0
·uR− Ω · (∇ × uR) = −Ω · ωR6= 0 , (3.51)
this term does affect the modified pressure through ˆfR in Eq. (3.50).
It is now possible to rewrite Eq. (3.49) without using the pressure coefficients : ∂ ∂t+ νkkk 2 ˆ uR = ˆfR− k k· ˆfR kkk2 . (3.52)
In addition to Otheguy et al. (2006), Roy et al. (2008) also used a rotating reference frame to circumvent the zero-circulation constraint imposed on a non-homogeneous velocity field. However, a number of recent studies (Joly and Reinaud (2007);Labbé et al.(2007);Colonius and Taira (2008); Duraisamy and Lele (2008); Pradeep and Hussain (2010); Chatelain and Koumoutsakos (2010); Gautier et al. (2013); Hussain and Stout (2013)) still set aside the triply periodic spectral algorithm for configurations with a non-zero circulation, citing the spurious instabilities highlighted by Pradeep and Hussain (2004). We believe that proper communication of the rotating reference frame technique would restore the use of the triply periodic spectral algorithm in the vortex dynamics community. This is precisely one of the contributions of the present thesis.
3.3.4 Changing frames of reference
For post-processing purposes, it can be interesting to switch back from the rotating reference frame Ω to the inertial frame of reference. The relations to compute the different physical quantities in the inertial reference frame are given by
ω(x) = ωR(xR)− 2Ω , (3.53) u(x) = uR(xR) + Ω× r , (3.54) and p(x) = pm(xR) + 1 2kΩk 2 r02, (3.55)
with the position vector x given by
x= xΩ+ RΩ(xR− xΩ) , (3.56)
where RΩ is the rotation matrix formed from the three basic rotation matrices Ri(θi) (Rao
(2005)) :
RΩ= Rx(Ωxt) Ry(Ωyt) Rz(Ωzt) . (3.57)
To find u or p, it is therefore necessary to define the center of rotation xΩ. Remember that
since the centrifugal term was included in a modified pressure (see Sec. 3.3.2), there was no need to define the center of rotation xΩin order to solve the equations of motion (3.52). As a
consequence, there is no prescribed location for xΩ. This might seem strange, however recall
that the domain is periodic in all three directions and therefore the very notion of an absolute center of rotation is somewhat contradictory. Note that there is no need to define the center of rotation to compute ω(x), which is the main quantity that we will use to carry out the different physical analyses presented in Chap. 4.
3.3.5 Verification of the implementation of the rotating reference frame
The aim of this subsection is to show that the implementation of the rotating reference frame is successful and adequate to circumvent the zero-circulation constraint.
Evolution of a slightly perturbed single straight vortex
In order to trigger the artificial centrifugal instability described in Sec. 3.3.1, Pradeep and Hussain (2004) embedded a single vortex in fine-scale “turbulence”. In this work, we aim to reproduce these results in order to verify that the rotating reference frame effectively suppresses the artificial instability. The initial configuration is made of a single vortex having the vorticity profile given by Eq. (3.4) and placed in a cubic periodic box of length Lq = 2πwith an uniform
resolution Nq= 256. For this configuration, nondimensional time t∗ is given by
t∗ ≡ t
R2/Γ. (3.58)
We also set random fine-scale perturbations centered around kkk = 20, as shown on Fig.3.10. Note that the kinetic energy of the perturbation Ek,pert is very small compared to the total
kinetic energy Ek,tot :
Ek,pert
Ek,tot ≈ 10
−3. (3.59)
We performed two simulations at ReΓ = 20,000; one in the inertial frame of reference (see
Fig. 3.12) and the other in the proper rotating frame of reference Ω (see Fig. 3.13). While the artificial centrifugal instability is clearly seen to grow in the first, the perturbations are quickly dampened in the second. This is also reflected in the evolution of their respective
enstrophy spectra, shown on Fig. 3.11for t∗ = 100. The amplitude of the fine-scale vorticity
structures is much greater for the simulation carried out in the inertial reference frame (in blue). Although this is not quite a formal verification, these results give us good confidence that the implementation of the rotating reference frame is successful and adequate to circumvent the zero-circulation constraint.
100 101 102
10−30 10−20 10−10 100
Figure 3.10 – Enstrophy spectra at t∗ = 0 for the unperturbed (blue) and perturbed (red) single straight vortex configurations. The periodic domain is cubic with Nq= 256and Lq = 2π.
10−5
100 101 102
10−10 100
Figure 3.11 – Enstrophy spectra for the perturbed single vortex configuration. The initial spectrum (t∗ = 0) is shown in black. The blue and red curves denote the spectra at t∗ = 100
for the simulation carried out in the proper rotating frame of reference (red) and for the simulation carried out in the inertial frame of reference (blue). The periodic domain is cubic with Nq= 256and Lq = 2π. The simulation is performed at ReΓ= 20,000.
(a) t∗= 0
(b)t∗= 51
(c) t∗= 100
(d)t∗= 193
Figure 3.12 – Evolution of a single vor-tex embedded in fine-scale perturbations at ReΓ = 20000 and using no correction
for the background vorticity. The perio-dic domain is cubic with Nq = 256 and
Lq = 2π.
(a)t∗= 0
(b)t∗= 51
(c) t∗= 100
(d)t∗= 193
Figure 3.13 – Same conditions as in Fig. 3.12but the simulation is carried out in the proper rotating reference frame Ω.
3.4
Instantaneous reconnection level
3.4.1 Definition
As we saw in Section1.1, vortex reconnection involves a change in the topology of the recon-necting vortex filaments. Fig. 3.14 schematizes the reconnection process of two antiparallel vortices using two filaments per vortex, each filament having the same circulation. We define the instantaneous reconnection level η of two vortices as the percentage of the circulation of a vortex that has been transferred to the other, i.e.,
η ≡ Γrec
Γrec+ Γnon−rec ≡
Γrec
Γtot
, (3.60)
where Γtot is the total vortex circulation, while Γrec and Γnon−rec are its reconnected and
non-reconnected parts respectively.
According to this figure, it is not clear what is the precise value of η when the filaments are experiencing viscous vorticity cancellation, schematized by the gray area. Using two filaments per vortex, one can only deduce that 0% < η < 50%. However, this does not mean that η does not have a precise value. Remember that the vorticity field is solenoidal, i.e.,
∇ · ω = ∇ · (∇ × u) = 0 . (3.61)
Therefore, all the circulation entering the gray area in Fig. 3.14 must be leaving the same area at any time, just like the inflow rate must balance the outflow rate in a container full of incompressible fluid. This implies that if we used an infinity of vortex filaments instead of four, then all of them would either be reconnected or non-reconnected.
𝑡
Figure 3.14 – Schematics of a reconnection event where some circulation remains non-reconnected.
3.4.2 Method using vortex lines
In order not to repeat ourselves, we refer to Sec.4.4for the description of the original method we propose to compute η using vortex lines. It is therefore suggested to read, at least, this section in the paper presented in Chap. 4 before going through Secs. 3.4.3 and 3.4.4 below which provide more details on the implementation of this diagnostic as well as its validation.
3.4.3 Trajectory algorithm
At a given position, the magnitude and orientation of the vorticity vector is computed with a trilinear interpolation using the eight adjacent collocation points. We first find the fraction xf = (xf,yf,zf), defined as
xf = (x− x0)/(x1− x0)
yf = (y− y0)/(y1− y0)
zf = (z− z0)/(z1− z0) ,
(3.62) where x = (x,y,z) is the current position. x0 = (x0,y0,z0) and x1 = (x1,y1,z1) are the
collocation points shown on Fig. 3.15, with x0 chosen such that x0 < x, y0 < y and z0 < z
and x1= (x0+ ∆x, y0+ ∆y, z0+ ∆z).
The approximated vorticity is then given by
ω(x) u a ω(x0,y0,z0) + b ω(x1,y0,z0) + c ω(x0,y1,z0) + d ω(x1,y1,z0)
+e ω(x0,y0,z1) + f ω(x1,y0,z1) + g ω(x0,y1,z1) + h ω(x1,y1,z1) ,
(3.63) where the weights a, b, c, d, e, f, g and h are given by
a = (1− xf)(1− yf)(1− zf) b = xf(1− yf)(1− zf) c = (1− xf) yf(1− zf) d = xf yf(1− zf) e = (1− xf)(1− yf) zf f = xf(1− yf) zf g = (1− xf) yf zf h = xf yf zf. (3.64)
𝒙
1𝒙
0𝒙
𝑥
𝑦
𝑧
∆𝑥
∆𝑦
∆𝑧
Figure 3.15 – Collocation cube surrounding the position of interest x = (x,y,z). x0 =
(x0,y0,z0) satisfies x0 < x, y0 < y and z0 < z, while x1= (x0+ ∆x, y0+ ∆y, z0+ ∆z).
To integrate the trajectories of the vortex lines, we use a fourth-order Runge-Kutta scheme : xn+1= xn+ (hn/6)(k1+ 2k2+ 2k3+ k4) (3.65) with k1 = ω(xn) k2 = ω(xn+ (hn/2)k1) k3 = ω(xn+ (hn/2)k2) k4 = ω(xn+ hnk3). (3.66)
The step size hn is continuously adjusted using a CFL-type condition
hn≡
C∆x max(ω(xn))
. (3.67)
In this work, we choose the Courant number C = 0.5, as detailed in the next section. A vortex line is discarded as soon as the local vorticity magnitude gets too small (kω∗
k ≤ 10−5).
Table 3.1 – Instantaneous reconnection level η obtained using different number of subsections Ns for the reconnection of two antiparallel vortices at ReΓ = 9,000 and t∗ = 11.0. For each
case, we indicate the number of reconnected Nrec and non-reconnected Nnon−rec vortex lines
as well as the total number of vortex lines Nf = Nrec + Nnon−rec. The last column shows
the percentage of the circulation on the (z = 0,x < π) half-plane that the algorithm has successfully identified as being reconnected or non-reconnected.
Ns C Nrec Nnon−rec Nf η (%) Γrec+ Γnon−rec Γz,0 (%) 4× 102 0.5 9 14 23 60.715 99.998 1× 104 0.5 199 400 599 67.964 99.982 1× 106 0.5 19,489 40,876 60,365 66.059 99.950 1× 108 0.5 1,942,099 4,089,793 6,031,89 66.044 99.946
3.4.4 Verification of the method
For the reconnection of two antiparallel vortices at ReΓ = 9,000and t∗= 11.0, we approximate
the instantaneous reconnection level η using different number of subsections Ns(see Table3.1)
and different Courant numbers C (see Table 3.2).
As it can be seen in Table 3.1, the proposed method is convergent even for low numbers of subsections. For the rest of this work, we choose Ns= 106 (of the order of 50,000 atual vortex
lines followed through the given vortex tube) so as to obtain an extremely reliable approxima-tion of η. Table 3.1 also shows that the number of filaments tracked Nf is significantly lower
than the number of subsections Ns
Nf
Ns ≈ 6% . (3.68)
This is because most of the computational domain is irrotationnal therefore there is no vorticity line to track. The last column of Table3.1shows the percentage of the circulation on the (z = 0, x < π) half-plane that the algorithm has successfully identified as being reconnected or non-reconnected. For all cases, this number is essentially 100%, meaning that all the vortex lines associated with a non-negligible fraction of the vortex circulation were successfully tracked. From Table3.2, one can see that the different Courant numbers C tested did not significantly impact the values of η computed (∆η < 0.1%). We therefore choose C = 0.5 for the rest of this work. This is very conservative but since the computation of the vortex lines is very cheap with respect to the simulation time, this is deemed reasonable.
Table 3.2 – Same conditions as in Table3.1 but we vary the Courant number C instead of the number of subsections Ns.
Ns C Nrec Nnon−rec Nf η (%) Γrec+ Γnon−rec Γz,0 (%) 1× 106 0.1 19,008 40,106 59,114 66.121 99.928 1× 106 0.5 19,489 40,876 60,365 66.059 99.950 1× 106 1.0 19,876 41,325 61,201 66.085 99.887
3.4.5 Validation of the method
In Sec. 4.7, we test the validity of the simple model depicted in Fig. 4.16 by comparing the results obtained using either the proposed method or that of Eq. (4.26). As it can be seen on Figs.4.17a and 4.17b, the agreement is very good for the low-Re cases, which validates both the simple model of Fig.4.16 and the method proposed in this section.
