Slender vortex filaments in the Boussinesq approximation

A model for the motion of slender vortex filaments is extended to include the effect of gravity. The model, initially introduced by Callegari and Ting [“Motion of a curved vortex filament with decaying vortical core and axial velocity,” SIAM J. Appl. Math. 35, 148–175 (1978)], is based on a matched asymptotic expansion in which the outer solution, given by the Biot–Savart law, is matched with the inner solution derived from the Navier–Stokes equations. Building on recent work by Harikrishnan et al. [“On the motion of hairpin filaments in the atmospheric boundary layer,” Phys. Fluids 35, 076603 (2023)], the Boussinesq approximation is applied such that the density variations only enter in the gravity term. However, unlike Harikrishnan et al. [“On the motion of hairpin filaments in the atmospheric boundary layer,” Phys. Fluids 35, 076603 (2023)], the density variation enters at a lower order in the asymptotic expansion and, thus, has a more significant impact on the self-induced velocity of the vortex filament. In this regime, which corresponds to the regime studied by Chang and Smith [“The motion of a buoyant vortex filament,” J. Fluid Mech. 857, R1 (2018)], the effect of gravity is given by an alteration of the core constant, which couples the motion of the filament to the motion within the vortical core, in addition to a change in the compatibility conditions (evolution equations), which determine the leading order azimuthal and tangential velocity fields in the vortex core. The results are used to explain certain properties of buoyant vortex rings, as well as qualitatively explore the impact of gravity on tornado-type atmospheric vortices

Slender vortex filaments in the Boussinesq Approximation

A model for the motion of slender vortex filaments is extended to include the effect of gravity. The model, initially introduced by Callegari and Ting (SIAM, J. of App. Math., (1978), vol. 35, pp. 148-175), is based on a matched asymptotic expansion in which the outer solution, given by the Biot-Savart law, is matched with the inner solution derived from the Navier-Stokes equations. Building on recent work by Harikrishnan et al (Phys. of Fluids, (2023), vol. 35) the Boussinesq approximation is applied such that the density variations only enter in the gravity term. However, unlike Harikrishnan et al. (2023) the density variation enters at a lower order in the asymptotic expansion, and thus has a more significant impact on the self-induced velocity of the vortex filament. In this regime, which corresponds to the regime studied by Chang and Smith (J. of Fl. Mech., (2018), vol. 857), the effect of gravity is given by an alteration of the core constant, which couples the motion of the filament to the motion within the vortical core, in addition to a change in the compatability conditions (evolution equations) which determine the leading order azimuthal and tangential velocity fields in the vortex core. The results are used to explain certain properties of bouyant vortex rings, as well as qualitatively explore the impact of gravity on tornado type atmospheric vorticies.Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in "Physics of Fluids 1 May 2024; 36 (5): 056604", and may be found at https://doi.org/10.1063/5.020502

On the motion of hairpin filaments in the atmospheric boundary layer

A recent work of Harikrishnan et al. [“Geometry and organization of coherent structures in stably stratified atmospheric boundary layers,” arXiv:2110.02253 (2021)] has revealed an abundance of hairpin-like vortex structures, oriented in a similar direction, in the turbulent patches of a stably stratified Ekman flow. In this study, hairpin-like structures are investigated by treating them as slender vortex filaments, i.e., a vortex filament whose diameter d is small when compared to its radius of curvature R. The corrected thin-tube model of Klein and Knio [“Asymptotic vorticity structure and numerical simulation of slender vortex filaments,” J. Fluid Mech. 284, 275 (1995)] is used to compute the motion of these filaments with the atmospheric boundary layer as a background flow. Our results suggest that the orientation of the hairpin filament in the spanwise direction is linked to its initial starting height under stable stratification, whereas no such dependency can be observed with the neutrally stratified background flow. An improved feature tracking scheme based on spatial overlap for tracking Q-criterion vortex structures on the direct numerical simulation data is also developed. It overcomes the limitation of using a constant threshold in time by dynamically adjusting the thresholds to accommodate the growth or deterioration of a feature. A comparison between the feature tracking and the filament simulation reveals qualitatively similar temporal developments. Finally, an extension of the asymptotic analysis of Callegari and Ting [“Motion of a curved vortex filament with decaying vortical core and axial velocity,” J. Appl. Math. 35, 148–175 (1978)] is carried out to include the effect of gravity. The results show that, in the regime considered here, a contribution from the gravity term occurs only when the tail of an infinitely long filament is tilted at an angle relative to the wall

On the motion of hairpin filaments in the atmospheric boundary layer

A recent work of Harikrishnan et al. [arXiv:2110.02253 (2021)] has revealed an abundance of hairpin-like vortex structures, oriented in a similar direction, in the turbulent patches of a stably stratified Ekman flow. The Ekman flow over a smooth wall is a simplified configuration of the Atmospheric Boundary Layer (ABL) where effects of both stratification and rotation are present. In this study, hairpin-like structures are investigated by treating them as slender vortex filaments, i.e., a vortex filament whose diameter $d$ is small when compared to its radius of curvature $R$. The corrected thin-tube model of Klein and Knio [J. Fluid Mech. (1995)] is used to compute the motion of these filaments with the ABL as a background flow. The influence of the mean background flow on the filaments is studied for two stably stratified cases and a neutrally stratified case. Our results suggest that the orientation of the hairpin filament in the spanwise direction is linked to its initial starting height under stable stratification whereas no such dependency can be observed with the neutrally stratified background flow. An improved feature tracking scheme based on spatial overlap for tracking $Q$-criterion vortex structures on the Direct Numerical Simulation (DNS) data is also developed. It overcomes the limitation of using a constant threshold in time by dynamically adjusting the thresholds to accommodate the growth or deterioration of a feature. A comparison between the feature tracking and the filament simulation reveals qualitatively similar temporal developments. Finally, an extension of the asymptotic analysis of Callegari and Ting [J. App. Math (1978)] is carried out to include the effect of gravity. The results show that, in the regime considered here, a contribution from the gravity term occurs only when the tail of an infinitely long filament is tilted at an angle relative to the wall

Implementation and validation of a slender vortex filament code: its application to the study of a four-vortex wake model

A computational code EZ-vortex is developed for the motion of slender vortex filaments of closed or open shape. The integro-differential equations governing the motion of the vortex centrelines are either the Callegari and Ting equations, which are the leading order solution of a matched asymptotic analysis, or equivalent forms of these equations. They include large axial velocity and nonsimilar profiles in the vortical cores. The fluid may be viscous or inviscid. This code is validated both against known solutions of these equations and results from linear stability analyses. The linear and non-linear stages of a perturbed two-vortex wake and of a four-vortex wake model are then computed

Shrinkage characterization and compensation for 3DPC

3D concrete printing is an additive manufacturing process in which elongated beads are assembled in layers to form 3D parts. In cementitious materials, water evaporation as well as the setting of the material results in a volumetric shrinkage of the printed structures. This imposed shrinkage strain, commonly referred to as eigenstrain, is the source of residual progressive stress evolution within the printed parts. Consequently, the accumulation of stress during fabrication and drying may induce cracks, buckling, and surface defects in the final product. Evaluating and modeling the effect of shrinkage on the printed beads is therefore essential to optimize the machine path and process parameters in order to mitigate such issues and guarantee the integrity of the printed parts. In particular, this paper focuses on the development of a compensation strategy such that the final geometry correctly approximates the target geometry. The proposed approach relies on the experimental characterization of the displacement field and hence the total strain by using Digital Image Correlation performed on in-situ imaging of the process. The history of the eigenstrain strain (i.e., shrinkage) has been measured on flat rectangular thin-walled walls, and then used in a mechanical model as an imposed strain. Resulting geometrical distortions have been validated against experiments. On this basis, a simple compensation strategy is proposed consisting in correcting the initial machine path by the opposite of the computed distortions when the structure is subjected to shrinkage. Several examples on various part geometries are presented and discussed. A fast one-dimensional mechanical model named QuadWire proposed recently is being used for numerical simulations, as the final objective is to create large database to train neural network algorithms in order to apply this compensation strategy in real-time during the printing process

Mouvement et dynamique des filaments et des anneaux tourbillons de faible épaisseur

A vortex filament is a particular rotationnel flow in which the vorticity is located onlyaround a three dimensional curve called central line of the filament. Solving the flow meansfinding the evolution of this vorticity region, called vortex filament, that is to say finding themotion of the central line. This dissertation proceed as Callegari and Ting to obtain thisequation of motion in adding some remarks and comments and in giving higher order terms ofthe expansion. We point out the importance of singular Biot and Savart integral expansionsperformed with matched asymptotic expansions. Then, these results are used to justify cut-offmethods and to generalise them to the case with viscosity and axial velocity. We completeWidnall and Sullivan’s linear study of a perturbed circular vortex ring in giving the oscillationperiod of the different modes and in comparing these results with a numerical simulation ofthe equation of motion . Finally, we study the oscillations of a straight vortex filament and thestability of parallel counter-rotating vortex filaments.Un filament tourbillon est un cas particulier d’écoulement rotationnel, pour lequel lavorticité se trouve uniquement autour d’une courbe tridimensionnel, dite fibre centrale dufilament. Calculer l’écoulement, c’est déterminer l’évolution de cette zone de vorticité, ditefilament tourbillon, c’est à dire trouver le mouvement de la fibre centrale. Cette thèse reprendla démarche de Callegari et Ting pour obtenir l’équation d’évolution de la fibre centrale d’unanneau tourbillon en la complétant avec divers remarques et commentaires et en donnant desordres supérieurs du développement. Nous mettons en valeur le développement des intégralessingulières de Biot et Savart à l’aide de la méthode des développements asymptotiquesraccordés. Puis, les résultats obtenus sont utilisés pour justifier des méthodes de coupure etpour les généraliser au cas visqueux et avec vitesse axiale. Nous complétons les résultats deWidnall et Sullivan sur l’étude linéaire d’un anneau circulaire perturbé en donnant la périoded’oscillation des différents modes et en comparant ces résultats avec une simulationnumérique de l’équation d’évolution. Enfin, nous étudions les oscillations d’un filament droitet la stabilité de filaments parallèles contrarotatifs

Cookbook asymptotics for spiral and scroll waves in excitable media

Algebraic formulas predicting the frequencies and shapes of waves in a reaction-diffusion model of excitable media are presented in the form of four recipes. The formulas themselves are based on a detailed asymptotic analysis (published elsewhere) of the model equations at leading order and first order in the asymptotic parameter. The importance of the first order contribution is stressed throughout, beginning with a discussion of the Fife limit, Fife scaling, and Fife regime. Recipes are given for spiral waves and detailed comparisons are presented between the asymptotic predictions and the solutions of the full reaction-diffusion equations. Recipes for twisted scroll waves with straight filaments are given and again comparisons are shown. The connection between the asymptotic results and filament dynamics is discussed, and one of the previously unknown coefficients in the theory of filament dynamics is evaluated in terms of its asymptotic expansion. (C) 2002 American Institute of Physics