Two-dimensional Inviscid Fluid Research Articles

Resonant excitation of single Kelvin–Helmholtz high-order waves in a magnetized electron fluid vortex

Thanks to the isomorphism between the drift-Poisson and Euler equations, inviscid two-dimensional fluid experiments can be performed in magnetized, single-component plasmas in Penning–Malmberg traps. Within this analogy, a trapped electron plasma column is equivalent to a two-dimensional vortex. Here, we focus our attention on the generation of V-states, i.e. $l$ -fold symmetric rotating vorticity patches where the deformation with respect to the circular cross-section has reached the nonlinear regime. We detail a linear theoretical analysis and devise an experimental routine to generate V-states through the precise excitation of single Kelvin–Helmholtz perturbations in a magnetized electron plasma. This technique makes use of suitable multipolar rotating electric fields, which are shown to be able to select the desired wavemode. In particular, with rotating fields, a hardware limitation in the highest accessible mode is removed and nonlinear Kelvin–Helmholtz waves of generic order $l$ can be attained, which pave the way for further investigations on the evolution and stability properties of V-states. Systematic experimental results for the selective mode growth in the linear and nonlinear regimes up to saturation and collapse are discussed.

Transition to instability of the leapfrogging vortex quartet

The point-vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter α defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value α=ϕ−2, where ϕ is the golden ratio. This value had been hypothesized based on careful numerics by Tophøj and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.

Large-amplitude membrane flutter in inviscid flow

We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. We apply small initial deflections and track their exponential decay or growth and subsequent large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric, except when the deformations are unrealistically large. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically at intermediate values of mass density. As mass density increases, the motions are increasingly aperiodic, and the amplitudes increase and spatial and temporal frequencies decrease. As mass density decreases from the periodic regime, the amplitudes decrease and spatial and temporal frequencies increase until the motions become difficult to resolve numerically. With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow.

On the dynamics of a free surface of an ideal fluid in a bounded domain in the presence of surface tension

We derive a set of equations in conformal variables that describe a potential flow of an ideal two-dimensional inviscid fluid with free surface in a bounded domain. This formulation is free of numerical instabilities present in the equations for the surface elevation and potential derived in Dyachenko et al. (Plasma Phys. Rep. vol. 22 (10), 1996, pp. 829–840) with some restrictions on analyticity relieved, which allows to treat a finite volume of fluid enclosed by a free-moving boundary. We illustrate with a comparison of numerical simulations of the Dirichlet ellipse, an exact solution for a zero surface tension fluid. We demonstrate how the oscillations of the free surface of a unit disc droplet may lead to breaking of one droplet into two when surface tension is present.

On the global stability of a beta-plane equation

We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as beta-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a "double null form" that annihilates interactions between spatially coherent waves and a lemma for Fourier integral operators which allows us to control a strong weighted norm.

Evolution of a Vortex in a Strain Flow.

Experiments and vortex-in-cell simulations are used to study an initially axisymmetric, spatially distributed vortex subject to an externally imposed strain flow. The experiments use a magnetized pure electron plasma to model an inviscid two-dimensional fluid. The results are compared to a theory assuming an elliptical region of constant vorticity. For relatively flat vorticity profiles, the dynamics and stability threshold are in close quantitative agreement with the theory. Physics beyond the constant-vorticity model, such as vortex stripping, is investigated by studying the behavior of nonflat vorticity profiles.

The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

Flapping propulsion using a fin ray

AbstractWe calculate optimal driving motions for a fin ray in a two-dimensional inviscid fluid, which is a model for caudal fin locomotion. The driving is sinusoidal in time, and consists of heaving, pitching and a less-studied motion called ‘shifting’. The optimal phases of shifting relative to heaving and pitching for maximum thrust power and efficiency are calculated. The optimal phases undergo jumps at resonant combinations of fin ray bending and shear moduli, and are nearly constant in regions between resonances. In two examples, pitching- and heaving-based motions converge with the addition of optimal shifting. Shifting provides an order-one increase in output power and efficiency.

Hydroelastic waves on fluid sheets

AbstractNonlinear travelling waves on a two-dimensional inviscid fluid sheet are investigated when the sheet is bounded above and below by two thin elastic plates. Symmetric and antisymmetric solution branches are identified, together with a pair of bifurcation branches. It is shown that far along the branches the solutions approach limiting configurations that correspond to static solutions valid in the absence of fluid forcing.

The steady-state form of large-amplitude internal solitary waves

A new numerical scheme for obtaining the steady-state form of an internal solitary wave of large amplitude is presented. A stratified inviscid two-dimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver, where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear time-dependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed.

Fluid-structure interaction of two bodies in an inviscid fluid

The interaction of two arbitrary bodies immersed in a two-dimensional inviscid fluid is investigated. Given the linear and angular velocities of the bodies, the solution of the potential flow problem with zero circulation around both bodies is reduced to the determination of a suitable Laurent series in a conformally mapped domain that satisfies the boundary conditions. The potential flow solution is then used to determine the force and moment acting on each body by using generalized Blasius formulas. The current formulation is applied to two examples. First, the case of two rigid circular cylinders interacting in an unbounded domain is investigated. The forces on two cylinders with prescribed motion (forced-forced) is determined and compared to previous results for validation purposes. We then study the response of a single “free” cylinder due to the prescribed motion of the other cylinder (forced-free). This forced-free situation is used to justify the hydrodynamic benefits of drafting in aquatic locomotion. In the case of two neutrally buoyant circular cylinders, the aft cylinder is capable of attaining a substantial propulsive force that is the same order of magnitude of its inertial forces. Additionally, the coupled interaction of two cylinders given an arbitrary initial condition (free-free) is studied to show the differences of perfect collisions with and without the presence of an inviscid fluid. For a certain range of collision parameters, the fluid acts to deflect the cylinder paths just enough before the collision to drastically affect the long time trajectories of the bodies. In the second example, the flapping of two plates is explored. It is seen that the interactions between each plate can cause a net force and torque at certain instants in time, but for idealized sinusoidal motions in irrotational potential flow, there is no net force and torque acting at the system center.

Power-series solution for the two-dimensional inviscid flow with a vortex and multiple cylinders

The problem of a point vortex and N fixed cylinders in a two-dimensional inviscid fluid is studied and an analytical-numerical solution in the form of an infinite power series for the velocity field is obtained using complex analysis. The velocity distribution for the case of two cylinders is compared with the existing results of the problem of a vortex in an annular region which is conformally mapped onto the exterior of two cylinders. Limiting cases of N cylinders and the vortex, being far away from each other are studied. In these cases, “the dipole approximation” or “the point-island approximation” is derived, and its region of validity is established by numerical tests. The velocity distribution for a geometry of four cylinders placed at the vertices of a square and a vortex is presented. The problem of vortex motion with N cylinders addressed in the paper attracted attention recently owing to its importance in many applications. However, existing solutions using Abelian function theory are sophisticated and the theory is not one of the standard techniques used by applied mathematicians and engineers. Moreover, in the N ≥ 3 cylinder problem, the infinite product involved in the presentation of the Schottky–Klein prime function must also be truncated. So, the approach used in the paper is simple and an alternative to existing methods. This is the main motivation for this study.

Statistical mechanics of two-dimensional turbulence

The statistical mechanical description of two-dimensional inviscid fluid turbulence is reconsidered. Using this description, we make predictions about turbulent flow in a rapidly rotating laboratory annulus. Measurements on the continuously forced, weakly dissipative flow reveal coherent vortices in a mean zonal flow. Statistical mechanics has two crucial requirements for equilibrium: statistical independence of macro-cells (subsystems) and additivity of invariants of macro-cells. We investigate these requirements in the context of the annulus experiment. The energy invariant, an extensive quantity, should thus be additive, i.e. the interaction energy between a macro-cell and the rest of the system (reservoir) should be small, and this is verified experimentally. Similarly, we use additivity to select the appropriate Casimir invariants from the infinite set available in vortex dynamics, and we do this in such a way that the exchange of micro-cells within a macro-cell does not alter an invariant of a macro-cell. A novel feature of the present study is our choice of macro-cells, which are continuous phase-space curves based on mean values of the streamfunction. Quantities such as energy and enstrophy can be defined on each curve, and these lead to a local canonical distribution that is also defined on each curve. The distribution obtained describes the anisotropic and inhomogeneous properties of a flow. Our approach leads to the prediction that on a mean streamfunction curve there should be a linear relation between the ensemble-averaged potential vorticity and the time-averaged streamfunction, and our laboratory data are in good accord with this prediction. Further, the approach predicts that although the probability distribution function for potential vorticity in the entire system is non-Gaussian, the distribution function of micro-cells should be Gaussian on the macro-cells, i.e. for curves defined by mean values of the streamfunction. This prediction is also supported by the data. While the statistical mechanics approach used was motivated by and applied to experiments on turbulence in a rotating annulus, the approach is quite general and is applicable to a large class of Hamiltonian systems, including drift-wave plasma models, Vlasov–Poisson dynamics, and kinetic theories of stellar dynamics.

Scramjet Inlet Control by Off-Body Energy Addition: A Virtual Cowl

The theory of energy addition to hypersonic airflow off the vehicle to increase air mass capture and reduce spillage in scramjet inlets at Mach numbers below the design value is explored. The heated region creates a virtual cowl and deflects flow streamlines into the inlet. Optimization studies are performed with a two-dimensional inviscid fluid code. The best location of the energy addition region is near the intersection of the nose shock of the vehicle with the continuation of the cowl line, and slightly below that line. In that case, the shock generated by the heating is close to the shock that is a reflection of the vehicle nose shock off the imaginary solid surface-extension of the cowl. Effects of the size and shape of the energy addition region on inlet performance are also studied.

Free-surface wave damping due to viscosity and surfactants

The steady wave pattern developed on the free surface in inviscid two-dimensional fluid domain is well known. In this work, we investigate the influence of a thin film of insoluble material on an air–liquid interface on the wave characteristics. We employ a quasi-potential approximation and assume that the main flow is potential, while the effect of viscosity and surfactants enters the problem via Bernoulli equation. Thus, the problem is formulated in terms of a single potential function. The formulation presented in this work can be extended to three-dimensional problems. The wave amplitude consistent with the approximations made, may be moderate. We use our formulation to study a pressure distribution moving with a constant velocity, on a two-dimensional deep water air–liquid interface, on which there exists a monolayer film of surface active agents. The problem is solved numerically by a boundary integral equation method and numerical solutions are presented.

Scaling Properties of an Inviscid Mean-Motion Fluid Model

An inviscid two-dimensional fluid model with nonlinear dispersion that arises simultaneously in coarse-grained descriptions of the dynamics of the Euler equation and in the description of non-Newtonian fluids of second grade is considered. The scaling of the equilibrium states of this model for conserved energy and enstrophy retains the corresponding scaling for the Euler equations on the large scales and at the same time greatly deemphasizes the importance of small scales. This is the first clear demonstration of the beneficial effect of nonlinear dispersion in the model, and should highlight its utility as a subgrid model in more realistic situations.

Complete phase diagram for coherent vortex formation in a two-dimensional inviscid fluid in an annulus

By calculating the two-dimensional solutions and the second-order bifurcation analysis of the mean field equations of a two-dimensional ideal fluid, we are able to complete the phase diagram of coherent vortex formation in an annulus, which we have partially calculated in a previous paper [Phys. Rev. E 50, 2022 (1994)]. The complete phase diagram demonstrates the advantage of the statistical mechanics approach to determine the final equilibrium directly from the initial condition.

Phase diagram for coherent vortex formation in the two-dimensional inviscid fluid in circular geometries.

We calculate the equilibrium states of a two-dimensional inviscid fluid in disk and annular geometries using the mean field equations that respect all conservation laws of the Euler equations. Axisymmetric vorticity distributions and their bifurcations to asymmetric solutions are calculated for a wide range of system parameters. Approximate zero-temperature compact vortices are also constructed and compared to symmetric states. From these results, the parameter ranges leading to the formation of a coherent vortex by the long time evolution from given initial conditions can be predicted without knowledge of the intervening dynamics. Applications to models of Jupiter’s great red spot and experiments on one-component plasmas in a magnetic field are discussed.

Collapse and winding of an asymmetric annulus of vorticity

The dynamics of an asymmetric annulus of vorticity in an incompressible, inviscid twodimensional fluid are experimentally studied using a pure electron plasma. A strict fluid analogy requires that the plasma system behave like an ideal fluid in a frictionless cylindrical container. For certain parameters the asymmetric annulus undergoes a complex evolution which is quite different from that of a symmetric annulus. During the first ‘active’, phase the symmetries grow until the annulus collapses, leaving a large vortex at the device centre. In the second, ‘passive’, phase the remainder of the annulus winds around this central vortex into an ever tighter spiral. Finally, slow shear instabilities destroy the structure of the highly evolved spiral.

Comment on “diffeomorphism groups and quantized vortex filaments”

In this paper it is shown that a new “quasiclassical” concept of intensive (|α| ⪡ 1) point vortices can be introduced within the framework of the quantum hydrodynamics of an incompressible inviscid twodimensional fluid, bypassing “surprising” result of Goldin et al.