Slender Vortex Research Articles

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.

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.

Helicity dynamics in viscous vortex links

The dynamics of two slender Hopf-linked vortex rings at vortex Reynolds numbers ( $Re \equiv \varGamma /\nu, \mathrm {circulation/viscosity}$ ) $2000$ , $3000$ and $4000$ is studied using direct numerical simulations of the incompressible Navier–Stokes equations. Under self-induction, the initially perpendicularly placed vortex rings approach each other and reconnect to form two separate vortex rings. The leading ring is closely cuddled and further undergoes secondary reconnection to form two even smaller rings. At high $Re$ , the leading ring and the subsequent smaller rings are unstable and break up into turbulent clouds consisting of numerous even smaller-scale structures. Although the global helicity $H$ remains constant before reconnection, it increases and then rapidly decays during reconnection – both the growth and decay rates increase with $Re$ . In the two higher $Re$ (i.e. 3000 and 4000) cases, $H$ further rises after the first reconnection and reaches a quasi-plateau with the asymptotic value continuously increasing with $Re$ – suggesting that $H$ for viscous flows is not conserved at very high $Re$ . Further flow analysis demonstrates that significant numbers of positive and negative helical structures are simultaneously generated before and during reconnection, and their different decay rates is the main reason for the complex evolution of $H$ . By examining the topological aspects of the helicity dynamics, we find that, different from $H$ , the sum of link and writhe ( $L_k+W_r$ ) continuously drop during reconnection. Our results also clearly demonstrate that the twist, which increases with $Re$ , plays a significant role in the helicity dynamics, particularly at high $Re$ .

Rotation suppresses giant-scale solar convection.

The observational absence of giant convection cells near the Sun's outer surface is a long-standing conundrum for solar modelers. We herein propose an explanation. Rotation strongly influences the internal dynamics, leading to suppressed convective velocities, enhanced thermal-transport efficiency, and (most significantly) relatively smaller dominant length scales. We specifically predict a characteristic convection length scale of roughly 30-Mm throughout much of the convection zone, implying weak flow amplitudes at 100- to 200-Mm giant cells scales, representative of the total envelope depth. Our reasoning is such that Coriolis forces primarily balance pressure gradients (geostrophy). Background vortex stretching balances baroclinic torques. Both together balance nonlinear advection. Turbulent fluxes convey the excess part of the solar luminosity that radiative diffusion cannot. We show that these four relations determine estimates for the dominant length scales and dynamical amplitudes strictly in terms of known physical quantities. We predict that the dynamical Rossby number for convection is less than unity below the near-surface shear layer, indicating rotational constraint.

Area waves on a slender vortex revisited

This paper considers the classic problem of the dynamics of axisymmetric waves on a rectilinear vortex, in the absence of viscosity. The waves alter the axial pressure distribution and thus generate axial flows which depend on the radial distribution of vorticity. To simplify this problem, models have been introduced which average over the cross-section and eliminate the radial dependence. One approach, pioneered by Lundgren and Ashurst (1989 J. Fluid Mech. 200 283–307), averages the momentum equation. Another averaging method, due to Leonard (1994 Phys. Fluids 6 765–77), focuses on the vorticity equation. The present paper takes a fresh look at the derivation of these two distinct models, which we refer to as the momentum wave model and vorticity wave model respectively, using the tools of differential geometry to develop a hybrid Eulerian–Lagrangian approach. We compare these models with area waves in the asymptotic limit of a slender vortex, with radial structure retained. Numerical calculations are presented to show the differences between waves in the full slender vortex system and those in the momentum and vorticity wave models. We also discuss modification of the vorticity wave model to allow an external irrotational flow, and simulations are presented where a vortex is subjected to uniform axial stretching. Our approach can also be developed to model more complicated configurations, such as occur during vortex collisions.

Separation scaling for viscous vortex reconnection

Abstract

On singularity formation via viscous vortex reconnection

Recognizing the fact that the finite-time singularity of the Navier–Stokes equations is widely accepted as a key issue in fundamental fluid mechanics, and motivated by the recent model of Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019a, pp. 930–967; J. Fluid Mech., vol. 870, 2019b, R1) on this issue, we have performed direct numerical simulation (DNS) for two colliding slender vortex rings of radius . The separation between the two tipping points and the scale of the core cross-section are chosen as ; the vortex Reynolds number ( ) ranges from 1000 to 4000. In contrast to the claim that the core remains compact and circular, there is notable core flattening and stripping, which further increases with – akin to our previous finding in the standard anti-parallel vortex reconnection. Furthermore, the induced motion of bridges arrests the curvature growth and vortex stretching at the tipping points; consequently, the maximum vorticity grows with substantially slower than the exponential scaling predicted by the model – implying that, for this configuration, even physical singularity is unlikely. Our simulations not only shed light on the longstanding question of finite-time singularities, but also further delineate the detailed mechanisms of reconnection. In particular, we show for the first time that the separation distance before reconnection follows 1/2 scaling exactly – a significant DNS result.

A Compact Eulerian Representation of Axisymmetric Inviscid Vortex Sheet Dynamics

AbstractA classical problem in fluid mechanics is the motion of an axisymmetric vortex sheet evolving under the action of surface tension, surrounded by an inviscid fluid. Lagrangian descriptions of these dynamics are well‐known, involving complex nonlocal expressions for the radial and longitudinal velocities in terms of elliptic integrals. Here we use these prior results to arrive at a remarkably compact and exact Eulerian evolution equation for the sheet radius r(z, t) in an explicit flux form associated with the conservation of enclosed volume. The flux appears as an integral involving the pairwise mutual induction formula for vortex loop pairs first derived by Helmholtz and Maxwell. We show how the well‐known linear stability results for cylindrical vortex sheets in the presence of surface tension and streaming flows [A. M. Sterling and C. A. Sleicher, J. Fluid Mech. 68, 477 (1975)] can be obtained directly from this formulation. Furthermore, the inviscid limit of the empirical model of Eggers and Dupont [J. Fluid Mech. 262 205 (1994); SIAM J. Appl. Math. 60, 1997 (2000)], which has served as the basis for understanding singularity formation in droplet pinchoff, is derived within the present formalism as the leading‐order term in an asymptotic analysis for long slender axisymmetric vortex sheets and should provide the starting point for a rigorous analysis of singularity formation. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

Vortex Shedding from Airfoils in Reverse Flow

The vortex shedding characteristics of three airfoils held at static angles of attack through 360 deg are presented with a focus on reverse flow (). Wind tunnel testing was performed on one airfoil with a sharp trailing edge (NACA 0012) and two airfoils featuring a blunt trailing edge (ellipse and DBLN-526). Time-resolved particle image velocimetry and smoke flow visualization were used to identify three reverse flow wake regimes: slender body vortex shedding, turbulent, and deep stall vortex shedding. The slender body regime is present for low angles of attack and low Reynolds numbers. In the turbulent regime, separation occurs in reverse flow at the sharp aerodynamic leading edge of a NACA 0012, whereas flow separation occurs further down the chord of airfoils with a blunt geometric trailing edge. The deep stall vortex shedding frequency was measured using unsteady force balance measurements. The Strouhal number (based on the projected diameter of the airfoils) was found to be for , which is well below the value of for a corresponding cylinder. The results of the work presented here provide fundamental insight for rotor applications where flow separation and vortex shedding due to reverse flow can lead to unsteady loading, vibrations, and fatigue.

Direct numerical simulation of a spatially developing compressible plane mixing layer: flow structures and mean flow properties

AbstractThe spatially developing compressible plane mixing layer with a convective Mach number of 0.7 is investigated by direct numerical simulation. A pair of equal and opposite oblique instability waves is introduced to perturb the mixing layer at the inlet. The full evolution process of instability, including formation of $ \mrm{\Lambda} $-vortices and hairpin vortices, breakdown of large structures and establishment of self-similar turbulence, is presented clearly in the simulation. In the transition process, the flow fields are populated sequentially by $ \mrm{\Lambda} $-vortices, hairpin vortices and ‘flower’ structures. This is the first direct evidence showing the dominance of these structures in the spatially developing mixing layer. Hairpin vortices are found to play an important role in the breakdown of the flow. The legs of hairpin vortices first evolve into sheaths with intense vorticity then break up into small slender vortices. The later flower structures are produced by the instability of the heads of the hairpin vortices. They prevail for a long distance in the mixing layer until the flow starts to settle down into its self-similar state. The preponderance of slender inclined streamwise vortices is observed in the transversal middle zone of the transition region after the breakup of the hairpin legs. This predominance of streamwise vortices also persists in the self-similar turbulent region, though the vortices there are found to be relatively very weak. The evolution of both the mean streamwise velocity profile and the Reynolds stresses is found to have close connection to the behaviour of the large vortex structures. High growth rates of the momentum and vorticity thicknesses are observed in the transition region of the flow. The growth rates in the self-similar turbulence region decay to a value that agrees well with previous experimental and numerical studies. Shocklets occur in the simulation, and their formation mechanisms are elaborated and categorized. This is the first three-dimensional simulation that captures shocklets at this low convective Mach number.

Motion and structure of atmospheric mesoscale baroclinic vortices: dry air and weak environmental shear

AbstractA strongly tilted, nearly axisymmetric vortex in dry air with asymmetric diabatic heating is analysed here by matched asymptotic expansions. The vortex is in gradient wind balance, with vortex Rossby numbers of order unity, and embedded in a quasi-geostrophic (QG) background wind with weak vertical shear. With wind speeds of $60{{\ndash}}120~\mathrm{km} ~{\mathrm{h} }^{\ensuremath{-} 1} $, such vortices correspond to tropical storms or nascent hurricanes according to the Saffir–Simpson scale. For asymmetric heating, nonlinear coupling of the evolution equations for the vortex tilt, its core structure, and its influence on the QG background is found. The theory compares well with the established linear theory of precessing quasi-modes of atmospheric vortices, and it corroborates the relationship between vortex tilt and asymmetric potential temperature and vertical velocity patterns as found by Jones (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 821–851) and Frank & Ritchie (Mon. Weath. Rev., vol. 127, 1999, pp. 2044–2061) in simulations of adiabatic tropical cyclones. A relation between the present theory and the local induction approximation for three-dimensional slender vortex filaments is established.

Alteration of helical vortex core without change in flow topology

The abrupt expansion of the slender vortex core with changes in flow topology is commonly known as vortex breakdown. We present new experimental observations of an alteration of the helical vortex core in wall bounded turbulent flow with abrupt growth in core size, but without change in flow topology. The helical symmetry as such is preserved, although the characteristic parameters of helical symmetry of the vortex core transfer from a smooth linear variation to a different trend under the influence of a non-uniform pressure gradient, causing an increase in helical pitch without changing its sign.

Experimental and numerical investigation of turbulent convection in a rotating cylinder

The effects of an axial rotation on the turbulent convective flow because of an adverse temperature gradient in a water-filled upright cylindrical vessel are investigated. Both direct numerical simulations and experiments applying stereoscopic particle image velocimetry are performed. The focus is on the gathering of turbulence statistics that describe the effects of rotation on turbulent Rayleigh–Bénard convection. Rotation is an important addition, which is relevant in many geophysical and astrophysical flow phenomena.A constant Rayleigh number (dimensionless strength of the destabilizing temperature gradient) Ra = 109 and Prandtl number (describing the diffusive fluid properties) σ = 6.4 are applied. The rotation rate, given by the convective Rossby number Ro (ratio of buoyancy and Coriolis force), takes values in the range 0.045 ≤ Ro ≤ ∞, i.e. between rotation-dominated flow and zero rotation. Generally, rotation attenuates the intensity of the turbulence and promotes the formation of slender vertical tube-like vortices rather than the global circulation cell observed without rotation. Above Ro ≈ 3 there is hardly any effect of the rotation on the flow. The root-mean-square (r.m.s.) values of vertical velocity and vertical vorticity show an increase when Ro is lowered below Ro ≈ 3, which may be an indication of the activation of the Ekman pumping mechanism in the boundary layers at the bottom and top plates. The r.m.s. fluctuations of horizontal and vertical velocity, in both experiment and simulation, decrease with decreasing Ro and show an approximate power-law behaviour of the shape Ro0.2 in the range 0.1 ≲ Ro ≲ 2. In the same Ro range the temperature r.m.s. fluctuations show an opposite trend, with an approximate negative power-law exponent Ro−0.32. In this Rossby number range the r.m.s. vorticity has hardly any dependence on Ro, apart from an increase close to the plates for Ro approaching 0.1. Below Ro ≈ 0.1 there is strong damping of turbulence by rotation, as the r.m.s. velocities and vorticities as well as the turbulent heat transfer are strongly diminished. The active Ekman boundary layers near the bottom and top plates cause a bias towards cyclonic vorticity in the flow, as is shown with probability density functions of vorticity. Rotation induces a correlation between vertical vorticity and vertical velocity close to the top and bottom plates: near the top plate downward velocity is correlated with positive/cyclonic vorticity and vice versa (close to the bottom plate upward velocity is correlated with positive vorticity), pointing to the vortical plumes. In contrast with the well-mixed mean isothermal bulk of non-rotating convection, rotation causes a mean bulk temperature gradient. The viscous boundary layers scale as the theoretical Ekman and Stewartson layers with rotation, while the thermal boundary layer is unaffected by rotation. Rotation enhances differences in local anisotropy, quantified using the invariants of the anisotropy tensor: under rotation there is strong turbulence anisotropy in the centre, while near the plates a near-isotropic state is found.

Similarity solutions for viscous cavitating vortex cores

Theoretical analysis of the dynamic behavior of cavitating vortices requires knowledge of the velocity distribution in the vortex core. For that reason an existing similarity model for slender axisymmetric non-cavitating vortex cores has been extended with the viscous boundary conditions for a cavitating vortex. Cavitating similarity solutions exist for a vortex of which both core diameter and circulation grow with the square root of the axial coordinate. Results for this model are presented for varying cavitation numbers and Reynolds numbers. It is shown that viscosity may have a significant influence on the velocity distribution around a cavitating vortex.

On an analogy to the area–velocity relation of gasdynamics in slender vortices

Inviscid compressible flow in a slender longitudinal vortex with the axis parallel to the main flow direction is discussed. The Euler equations for steady, axially symmetric flow are simplified for the neighbourhood of the axis of the vortex. The resulting relations, expressing the angular velocity in terms of the axial mass flow, when recast and integrated, represent an analogy to the area–velocity and the area-Mach number relation of one-dimensional gasdynamics. By restricting the analysis to the flow in a Rankine vortex with constant stagnation enthalpy, the static pressure on the axis can be related to the free-stream pressure far away from the axis and the maximum value of the azimuthal velocity. The influence of a radial wake-like distribution of the axial velocity component on the axial pressure is discussed. Its dependence on variable external boundary conditions is given. A condition is formulated, which has to be satisfied for the formation of a free stagnation point on the axis.

A coaxial vortex ring model for vortex breakdown

A simple–yet plausible–model for B-type vortex breakdown flows is postulated; one that is based on the immersion of a pair of slender coaxial vortex rings in a swirling flow of an ideal fluid rotating around the axis of symmetry of the rings. It is shown that this model exhibits in the advection of passive fluid particles (kinematics) just about all of the characteristics that have been observed in what is now a substantial body of published research on the phenomenon of vortex breakdown. Moreover, it is demonstrated how the very nature of the fluid dynamics in axisymmetric breakdown flows can be predicted and controlled by the choice of the initial ring configurations and their vortex strengths. The dynamic intricacies produced by the two ring + swirl model are illustrated with several numerical experiments.

Studies of perturbed three vortex dynamics

It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated with completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the dynamics of the integrable planar three vortex system, it is shown that the admissible class of perturbed systems is broad enough to include three vortices in a half plane, three coaxial slender vortex rings in three space, and “restricted” four vortex dynamics in a plane. Included are two basic categories of results for admissible perturbations: (i) general theorems for the persistence of invariant tori and periodic orbits using Kolmogorov-Arnold-Moser- and Poincaré-Birkhoff-type arguments and (ii) more specific and quantitative conclusions of a classical perturbation theory nature guaranteeing the existence of periodic orbits of the perturbed system close to cycles of the unperturbed system, which occur in abundance near centers. In addition, several numerical simulations are provided to illustrate the validity of the theorems as well as indicating their limitations as manifested by transitions to chaotic dynamics.

Trapped slender vortex filaments in statistical equilibrium

AbstractSystems of nearly parallel, slender vortex filaments in which angular momentum is conserved are an important simplification of the Navier‐Stokes equations where turbulence can be studied in statistical equilibrium. We study the canonical Gibbs distribution based on the Klein‐Majda‐Damodaran (KMD) (Klein et al. [1995]) model and find a divergence in the mean square vortex position from that of the point vortex model of Onsager [1949] at high temperature. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Improved Harmonic Large-Angle Fuselage Aerodynamics Model

Introduction F LIGHT vehicle simulation, guidance and control law design, and flight performance analysis methods commonly use aerodynamic models based on tabulated force and moment coefficients. Inputs for these models are generally derived from wind-tunnel test data and/or computational fluid dynamics (CFD) results. However, wind-tunnel data are often only available for a limited range of pitch and yaw angles, and CFD computations for a large number of arbitrary attitudes can be expensive. Rotorcraft simulation models require aerodynamic inputs at extreme attitudes and commonly use trigonometric high-angle models to provide smoothly varying data in these regimes.1,2 Other applications for extreme high-angle aerodynamic models are ground-handling simulations, where the wind may come from any azimuth, flight performance analysis at high angles,2 and trajectory analysis of tumbling bodies. This Engineering Note presents a trigonometric, quasi-steady, fuselage aerodynamics model that is valid at all aerodynamic angles with a range of ±180 deg yaw and ±90 deg pitch. This model is based on the C81 high-angle equation (HAE) model found in some rotorcraft simulations1 but with improvements to the usability of the original model. The HAE model uses aerodynamic coefficient inputs at key values of pitch and yaw angles that may come from wind-tunnel tests, CFD analysis, or other sources. A set of sinusoidal functions is used to estimate the values for all six wind-reference fuselage aerodynamic forces and moments at any combination of pitch angle and yaw angle. The HAE model is desirable in that the equations are continuous and easily computed, give the proper trends at all attitudes, include some cross-coupling effects, and require relatively few inputs. The quasi-steady nature of the HAE model makes it unsuitable for situations where time-dependent flow characteristics are significant. This model is not suitable where high accuracy is required and is not expected to capture sudden changes in flow characteristics at critical angles such as the onset of stall or slender forebody vortex shedding. To address this issue, a blending scheme is often used to transition between wind tunnel data (or other high-quality data) at low angles and the HAE model at high angles.1 This Note addresses three limitations with the Ref. 1 model. First, the original model assumes lateral–directional coefficients are con-

A critical pattern of crossflow around a slender

Topological structure of a slender crossflow was discussed with topological analysis. It is pointed that the development of slender vortices leads to the change of topological structure about cross flow, and a critical flow pattern will appear. There is a high-order singular point in this critical flow pattern. And the index of the high-order singular is −3/2. The topological structure of this singular point is instable, so bifurcation will occur and the topological structure of flowfield will be changed by little disturbance.