Vortex Sheet Research Articles

Estimating the non-dimensional energy of vortex rings by modelling their roll-up

The non-dimensional energy of starting vortex rings typically converges to values around 0.33 when they are created by a piston-cylinder or a bluff body translating at a constant speed. To explore the limits of the universality of this value and to analyse the variations that occur outside of those limits, we present an alternative approach to the slug-flow model to predict the non-dimensional energy of a vortex ring. Our approach is based on the self-similar vortex sheet roll-up described by Pullin (J. Fluid Mech., vol. 88, 1978, pp. 401–430). We derive the vorticity distribution for the vortex core resulting from a spiralling shear layer roll-up and compute the associated non-dimensional energy. To demonstrate the validity of our model for vortex rings generated through circular nozzles and in the wake of disks, we consider different velocity profiles of the vortex generator that follow a power law with a variable time exponent $m$ . Higher values of $m$ indicate a more uniform vorticity distribution. For a constant velocity ( $m=0$ ), our model yields a non-dimensional energy of ${{E}^{{*}}}=0.33$ . For a constant acceleration ( $m=1$ ), we find ${{E}^{{*}}}=0.19$ . The limiting value $m \rightarrow \infty$ corresponds to a uniform vorticity distribution and leads to ${{E}^{{*}}}=0.16$ , which is close to values found in the literature for Hill's spherical vortex. The radial diffusion of the vorticity within the vortex core results in the decrease of the non-dimensional energy. For a constant velocity, we obtain realistic vorticity distributions by radially diffusing the vorticity distribution of the Pullin spiral and predict a decrease of the non-dimensional energy from 0.33 to 0.28, in accordance with experimental results. Our proposed model offers a practical alternative to the existing slug flow model to predict the minimum non-dimensional energy of a vortex ring. The model is applicable to piston-generated and wake vortex rings and only requires the kinematics of the vortex generator as input.

Vorticity generation and conservation on generalised interfaces in three-dimensional flows

This article presents a three-dimensional theory of vorticity creation on generalised interfaces, including both non-slip and free-slip boundaries, which generalises a previous two-dimensional formulation (Terrington et al., J. Fluid Mech., vol. 890, 2020, p. A5). Under this description, vorticity may be created on a boundary by the inviscid relative acceleration between fluid elements on each side of the boundary, driven by either tangential pressure gradients or body forces. Viscosity acts to transfer circulation between the vortex sheet representing the slip velocity on the interface, and the fluid interior, but is not responsible for the creation of vorticity on the interface. This formulation also describes a principle of vorticity conservation for interfacial and free-surface flows: in many flow configurations, the net generation of vorticity on the interface is zero, and the total circulation remains constant throughout flow evolution.

Including acoustic modes in the vortex-sheet eigenbasis of a jet.

Vortex-sheet (V-S) models of jets are widely used to describe the dynamics of modes, such as the Kelvin-Helmholtz instability and guided acoustic waves. However, the absence of the free-stream acoustic modes in the V-S eigenspectrum is seldom pointed out in the literature. This family of modes is important if, for example, one is interested into the problems of sound emission or flow-acoustic interactions. In this work, it is shown how a distantly confined jet may be used as a surrogate problem for the free jet, in which the free-stream acoustic waves appear as a set of discrete modes. Comparing the modes observed in the free jet with those of the distantly confined jet, we show that other than the free-stream acoustic modes, the eigenvectors and eigenvalues converge exponentially with the wall distance. The proposed surrogate problem, thus, efficiently reproduces the dynamics of the original problem while accounting for the dynamics of the free-stream acoustic eigenmodes.

The Influence of Leading-Edge Tubercles on the Wake Flow Dynamics of a Marine Rudder

The impact of two tubercle leading-edge (TLE) modifications on the turbulent wake of a representative marine rudder at Reynolds number 2.26×106 was analysed numerically using Detached-Eddy Simulations. TLE have been shown to alter the flow profile over aero/hydrofoils through the generation of streamwise counter-rotating vortex pairs behind the tubercles, which can enhance the lifting performance. This paper studies the formation of these vortex pairs and their impact on the wake structures behind the rudder to find out if vortex interaction can reduce the tip vortex. The tubercles enhanced lift for angles of attack (AOA) 10o and above, but at the cost of a large drag penalty that reduced the rudders' lift-to-drag ratio. The formation of the distinctive streamwise counter-rotating vortex pairs behind the tubercles was shown. Due to the inherent spanwise flow component of finite-span lifting surfaces the vortices were generated at unequal strength and only positive vortices were maintained in the wake. The vortices facilitated flow compartmentalisation over the rudder suction side which broke up the trailing-edge vortex sheet and confined the spanwise flow separation over the rudder surface as AOA increased. The tubercles confined flow separation closer to the rudder tip which caused a tip-offloading effect that minimised the initial tip vortex strength. Large elements of streamwise counter-rotating vorticity formed around the localised stall cells of the TLE rudders that interacted with the tip vortex downstream, introducing elliptical instabilities further weakening the tip vortex and changing its trajectory.

Turbulent entrainment in viscoelastic fluids

Direct numerical simulations (DNS) of turbulent fronts spreading into an irrotational flow region are used to analyse the turbulent entrainment mechanism for viscoelastic fluids. The simulations use the FENE-P fluid model and are initiated from DNS of isotropic turbulence with Weissenberg and turbulence Reynolds numbers varying in the ranges $1.30 \le Wi \le 3.46$ and $206 \le Re_{\lambda }^{0} \le 404$, respectively. The enstrophy dynamics near the turbulent/non-turbulent interface (TNTI) layer, that separates regions of turbulent and irrotational flow, includes a new mechanism – the viscoelastic production – caused by the interaction between the vorticity field and the polymer stresses. This term can be a sink or a source of enstrophy in the turbulent core region of the flow, depending on the Weissenberg number, and contributes to the initial growth of the enstrophy in the viscous superlayer, together with the viscous diffusion, which is the only mechanism present for Newtonian fluids. For low and moderate Weissenberg numbers the scaling of the TNTI layer is similar to the scaling of TNTI layers for Newtonian fluids, but this is no longer the case at high Weissenberg numbers where the enstrophy tends to be concentrated into thin vortex sheets instead of vortex tubes. Finally, it is shown that the substantial decrease of the entrainment rates observed in turbulent flows of viscoelastic fluids, compared with Newtonian fluids, is caused by a reduction of the surface area and fractal dimension of the irrotational boundary, originated by the depletion of ‘active’ scales of motion in the fluid solvent caused by the viscoelasticity.

Dissipative Euler Flows for Vortex Sheet Initial Data without Distinguished Sign

AbstractWe construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a “turbulence” zone which grows linearly in time around the vortex sheet. As a by‐product, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Nonlinear evolution of two vortex sheets moving separately in uniform shear flows with opposite direction

<abstract><p>It has been considered that two close vortex sheets become unstable and evolve simultaneously when sufficiently strong uniform shears exist. However, Moore (Mathematika, 1976) suggested in his linear analysis that a vortex sheet evolves just as if the other vortex sheet were absent under certain conditions. In the current study, we investigate how the two vortex sheets evolve in the nonlinear region when they satisfy Moore's condition. We also consider density stratification, which is not included in Moore's analysis. Moore's estimate is only valid within linear theory; however, a motion suggested by Moore appears even in the nonlinear regime when Moore's condition is satisfied. We found that there is a case that a vortex sheet hardly deforms, even though the other sheet becomes unstable and largely deforms. We also show that there is a case that Moore's analysis is not effective even the condition is satisfied when a density instability exists in the system.</p></abstract>

Quantized vortices in pionic superfluid

The interplay between classical vorticity being the main undisputed source of polarization in heavy-ion collisions (HIC) and quantized vortices is considered. The vortex tubes emerging in the rotating pionic (super) fluid polarize the baryons in their cores and explain the emerging global polarization. The appearance of vortices in the region separating participants and spectators in non-central HIC is similar to that for sliding layers of liquid helium. From the other side, it is also the region where the classical vorticity was earlier found to be large forming the vortex sheets. The formation of tubes manifests a threshold at certain critical vorticity implying the vanishing polarization at lower energies. For central HIC the compact jet-like flows may lead to formation of vortex rings related to local polarization. The P-odd momentum correlations for their experimental investigation are suggested. The role of shear and viscosity in the emergence of polarization is discussed.

A vortex sheet based analytical model of the curled wake behind yawed wind turbines

Motivated by the need for compact descriptions of the evolution of non-classical wakes behind yawed wind turbines, we develop an analytical model to predict the shape of curled wakes. Interest in such modelling arises due to the potential of wake steering as a strategy for mitigating power reduction and unsteady loading of downstream turbines in wind farms. We first estimate the distribution of the shed vorticity at the wake edge due to both yaw offset and rotating blades. By considering the wake edge as an ideally thin vortex sheet, we describe its evolution in time moving with the flow. Vortex sheet equations are solved using a power series expansion method, and an approximate solution for the wake shape is obtained. The vortex sheet time evolution is then mapped into a spatial evolution by using a convection velocity. Apart from the wake shape, the lateral deflection of the wake including ground effects is modelled. Our results show that there exists a universal solution for the shape of curled wakes if suitable dimensionless variables are employed. For the case of turbulent boundary layer inflow, the decay of vortex sheet circulation due to turbulent diffusion is included. Finally, we modify the Gaussian wake model by incorporating the predicted shape and deflection of the curled wake, so that we can calculate the wake profiles behind yawed turbines. Model predictions are validated against large-eddy simulations and laboratory experiments for turbines with various operating conditions.

Active tail flexion in concert with passive hydrodynamic forces improves swimming speed and efficiency

Fish typically swim by periodic bending of their bodies. Bending seems to follow a universal rule; it occurs at about one-third from the posterior end of the fish body with a maximum bending angle of about $30^{\circ }$ . However, the hydrodynamic mechanisms that shaped this convergent design and its potential benefit to fish in terms of swimming speed and efficiency are not well understood. It is also unclear to what extent this bending is active or follows passively from the interaction of a flexible posterior with the fluid environment. Here, we use a self-propelled two-link model, with fluid–structure interactions described in the context of the vortex sheet method, to analyse the effects of both active and passive body bending on the swimming performance. We find that passive bending is more efficient but could reduce swimming speed compared with rigid flapping, but the addition of active bending could enhance both speed and efficiency. Importantly, we find that the phase difference between the posterior and anterior sections of the body is an important kinematic factor that influences performance, and that active antiphase flexion, consistent with the passive flexion phase, can simultaneously enhance speed and efficiency in a region of the design space that overlaps with biological observations. Our results are consistent with the hypothesis that fish that actively bend their bodies in a fashion that exploits passive hydrodynamics can at once improve speed and efficiency.

The study on the wake of a CLT propeller under different advance coefficients

In this paper, the impact of advance coefficients on the characteristics of the CLT propeller wake is investigated and the large eddy simulation turbulence model is employed. A wide range of advance coefficients (0.1∼0.9) are calculated by solving the unsteady Navier-Stokes equations. The verification is performed and the results from the calculations are compared with experimental data. Under smaller advance coefficients, the wake is separated into three flow regimes: free stream, slipstream, and vortices between two regions. The boundaries among the flow regimes are indistinct under bigger advance coefficients. The perturbations induced by the rotating blades get smaller as the advance coefficient increases. Then, the vortical structures represented by the Q isosurface show that the endplate tip vortices merge with the secondary tip vortices, and the merged tip vortices will re-merge downstream. The inner trailing sheet vortices remain further than the outer ones under all advance coefficients. Finally, the trajectories (including the radius, the distance between two consecutive vortices, the pitch) of the vortex-system are analyzed in detail and fitted with different functions. It is recommended that proper considerations should be used for the establishment of the reliable wake model due to the various characteristics of the trajectory.

Numerical research on the instabilities of CLT propeller wake

Optimising contracted and load tip (CLT) propellers has proven difficult due to the lack of research on flow fields. In this study, the hydrodynamics and wake of the P1727 propeller, a CLT propeller, are investigated using large eddy simulation for three advance coefficients. Fluctuations in the thrust and torque are minimal, with a frequency of up to 1000 Hz, as indicated by the simulations, which are verified based on experimental data and study on spatial and temporal convergences. Instantaneous, phase-averaged, and azimuthal-averaged flow fields are used to analyse propeller wake, including the tip vortices of endplate, secondary tip vortices, merged and trailing sheet vortices and the hub vortex. Based on previous works, we present a novel mutual inductance mechanism for the instabilities in CLT propeller wake. The merged vortices formed by merging between the tip vortices of endplate and the secondary tip vortices are strengthened, stretched by the outer trailing vortices that are caused by the crack of trailing sheet vortices, then the reinforced merged vortices trigger the leapfrogging phenomenon and induce the outer instabilities. While the inner trailing vortices, or root vortices, remain coherent, they undergo contraction and approach the hub vortex. Thereafter, they intertwine with the spiraling hub vortex, acting as the dominant mutual inductance mode of the inner CLT propeller wake.

Complex singularity analysis for vortex layer flows

We study the evolution of a 2D vortex layer at high Reynolds number. Vortex layer flows are characterized by intense vorticity concentrated around a curve. In addition to their intrinsic interest, vortex layers are relevant configurations because they are regularizations of vortex sheets. In this paper, we consider vortex layers whose thickness is proportional to the square-root of the viscosity. We investigate the typical roll-up process, showing that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterizes the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics. We display the inner core's self-similar structure, with the scale factor depending on the Reynolds number. We reveal the presence of complex singularities in the solutions of Navier–Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff–Rott singularity seems to suggest that vortex layers, in the limit $Re\rightarrow \infty$ , behave differently from vortex sheets.

A computational model of the swimming dynamics of a fish-like body in two dimensions

The undulatory swimming of a fish-like body is investigated by using an inviscid vortex shedding model in two dimensions. The body and separated vortices from the edge of the body are described by vortex sheets. We extend the model to include the forward and lateral motions as well as the net torque on the body and develop a new numerical method for a self-propelled deforming body. The numerical computation of the model successfully demonstrates the self-propulsion of the body and the formation of pairs of anti-rotating vortices shed from the body. At the starting stage, the body accelerates rapidly and turns the angle of incidence to some degree. The forward velocity reaches an asymptotic mean with oscillations, and the lateral velocity oscillates uniformly over time. The results of the model are in agreement with previous full numerical simulations. Furthermore, we examine the effects of the recoil motions of lateral translation and rotation of a body. The wake pattern and intensity significantly differ from the constraint of the recoil motions. It is found that free swimming requires less input power and affords more efficient energetic performance than the motions without recoil and with lateral translation. A body without recoil requires a large amount of power and is energetically inefficient. This shows that both recoil motions enhance the swimming efficiency.

Vortical structures and turbulence characteristics in an isothermal channel flow at supercritical pressure

Due to effect of the variations in the thermo-physical properties, the role of vortical structures in wall-bounded turbulence characteristics at supercritical pressure becomes particularly complex. In this work, we investigate the dynamics of vortical structures in an isothermal turbulent channel flow at supercritical pressure by direct numerical simulation. We find that the streaks near the cold (respectively, hot) wall are strengthened (respectively, weakened), which are associated with the enhanced (respectively, decreased) frequency of bursting events induced by the higher (respectively, lower) mean values of density and viscosity near the cold (respectively, hot) wall. Additionally, the typical hairpin vortex is modulated due to the effect of the baroclinic field and the bursting events near the wall, and accompanied by the generation of the sheet-like, tube-like vortical structures and the wavepackets. Consequently, both the volume fractions of vortex sheets and vortex tubes increase in comparison with the incompressible flow, especially near the cold wall. Further, we confirm that, for the variable property cases, the mechanism underlying the increased contribution of vortex sheets to the enstrophy production in the viscous sublayer is the stronger vortex stretching than the incompressible counterpart. The turbulence anisotropy is found to be increased due to the increase in the mean shear and the overlap between the large energy-containing eddies and the dissipative small scales. For the downward flow with mixed convection, buoyancy mainly weakens the anisotropy in the viscous sublayer near the cold wall due to the larger mean density.

Tones in the Acoustic Far Field of Jets in the Upstream Direction

The presence of tones in the acoustic far field of isothermal round jets for angles relative to the flow direction is investigated for Mach numbers between and 2 based on data from both experiments and large-eddy simulations. For all Mach numbers, as observed in the jet near-nozzle region in previous studies, peaks are found in the far-field pressure spectra regardless of the nozzle-exit boundary-layer properties. They emerge near the cutoff frequencies of the free-stream upstream-propagating guided jet modes predicted by a vortex sheet model, sharply for but more weakly for lower Mach numbers. The variations of their characteristics with the radiation angle are shown. As the angle increases, the peaks are more prominent, and the most apparent ones are related to lower azimuthal modes. Thus, peaks associated with the axisymmetric mode strongly predominate in the upstream direction.

Two-dimensional incompressible micropolar fluid models with singular initial data

This paper deals with the interaction between microstructures and the appearance or persistence of singular configurations in the Cauchy problem for the two-dimensional model of incompressible micropolar fluids. We analyze the case of null angular viscosity and singular initial data, including the possibility of vortex sheets or measures as initial data in Morrey spaces. Through integral techniques we establish the existence of weak solutions local or global in time. In addition, the uniqueness and stability of these solutions is analyzed.

Helical Contour Dynamics

The equations of motion for an incompressible flow with helical symmetry (invariance under combined axial translation and rotation) can be expressed as nonlinear evolution laws for two scalars: vorticity and along-helix velocity. A metric term related to the pitch of the helix enters these equations, which reduce to two-dimensional and axisymmetric dynamics in appropriate limits. We take the vorticity and along-helix velocity component to be piecewise constant. In addition to this vortex patch, a vortex sheet develops when the along-helix velocity is nonzero. We obtain a contour dynamics formulation of the full nonlinear equations of motion, in which the motion of the boundary is computed in a Lagrangian fashion and the velocity field can be expressed as contour integrals, reducing the dimensionality of the computation. We investigate the stability properties of a circular vortex patch along the axis of the helix in the presence of a vortex sheet and along-helix velocity. A linear stability calculation shows that the system is stable when the initial vortex sheet is zero, but can be stable or unstable in the presence of a vortex sheet. Using contour dynamics, we examine the nonlinear evolution of the system, and show that nonlinear effects become important in unstable cases.

The Centroid-Deformation Decomposition for Buoyant Vortex Patch Motion

The motion of a two-dimensional buoyant vortex patch, i.e. a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid, deformation of its boundary and the strength of a vortex sheet which is essential to enforce pressure continuity across the boundary. The equations for the centroid are derived by a linear momentum analysis and that for the sheet strength by applying Euler's equations on the boundary, while the boundary deformation is studied in the centroid-fixed frame. A complicated coupled set of equations is obtained which, to the best of our knowledge, has not been derived before. The evolution of the sheet strength is obtained as an integral equation. The equations are also examined in the limit of a patch of vanishing size or a buoyant point vortex.

Acoustic and hydrodynamic power of wave scattering by an infinite cascade of plates in mean flow

We study the balance of outgoing and incoming power for acoustic wave scattering by a cascade of flat blades in uniform subsonic mean flow. In nonzero mean flow the Kutta condition at the trailing edge of the cascade plates leads to the production of vorticity in the form of unsteady vortex sheets attached at the trailing edges, which results in a hydrodynamic power flux and the exchange of energy between the unsteady vorticity and the acoustic field. Although the scattering problem itself has been subject to much previous research, a comprehensive study of the composition of the outgoing energy flux for cascade scattering has received little previous attention. As an application of our detailed study we show how the time-averaged energy balance can be used to explain certain symmetries of the field in the angle of incidence with respect to the cascade face when there is no mean flow and the effect of zero acoustic reflection at certain angles of incidence with mean flow. We also show that, depending on the angle of incidence of the acoustic wave and the velocity of the mean flow, the vortex sheets can either absorb or generate acoustic energy. This can, for certain parameter values, result in a net increase in outgoing sound power over the incident sound power, as we demonstrate on a number of numerical examples that are computed using the semi-analytical solution described in recent work by Maierhofer and Peake (J. Sound Vib., vol. 481 (2020), 115418).