Time-dependent Vorticity Research Articles

Objective molecular dynamics for atomistic simulation of macroscopic fluid motion

The method of Objective Molecular Dynamics (OMD) provides exact solutions of the equations of molecular dynamics for atoms filling all of space in non equilibrium situations. The method can be used to simulate families of incompressible, compressible and unsteady flows, also with time-dependent vorticity. In this paper we develop OMD as an efficient computational tool and introduce some of its applications. We apply the method to the evolution of compressible heat-conducting monoatomic gas under general incompressible flow, and to flow in a nanochannel having realistic atomistic solid boundaries. The macroscopic flows of OMD are exact solutions of the Navier-Stokes equation with Newtonian and Fourier models, and we make comparisons with these solutions. We also report the simulation of homoenergetic dilatational flow exhibiting condensation using OMD techniques.

Singular magnetic equilibria in the solar x-ray corona

Interlaced magnetic field line topologies are expected in the active x-ray solar coronal arches. In the presence of uniform fluid pressure the equilibrium involves magnetic stresses in equilibrium among themselves, i.e. the divergence of the magnetic stress tensor Mij is zero, ∂Mij/∂xj = 0. In the presence of an ideal infinitely conducting fluid the requirement reduces to the familiar force-free field equilibrium equation, ∇ × B = αB. This paper reviews and comments on the formal analytical demonstration that the equilibrium of almost all interlaced field line topologies involves surfaces of tangential discontinuity (TD). Consider, then, the static equilibrium of a randomly interlaced field extending in the z-direction through an infinitely conducting fluid throughout the space 0 < z < L between the end plates z = 0 and z = L. With the field anchored at both ends it is obvious that a stable equilibrium exists for all interlacing topologies. The equilibrium equation can be reduced to the form of the familiar 2D time-dependent vorticity equation. The solutions to the vorticity equation represent a topological set of measure zero compared with the set of all interlacing field line topologies. Lacking the special topology of the vorticity solutions, it follows that the equilibria of almost all interlaced topologies are described by the so-called weak solutions, containing TDs (current sheets). The spaces between the TDs are filled with a continuous field satisfying the vorticity equation. A brief discussion of the mathematical difficulties in constructing illustrative examples of weak solutions is assisted by recent effort at numerical simulation. In the real world resistive dissipation at incipient TDs appears to be the principal heat source for the solar x-ray corona.

Coherent motion of vortices driven by both dc and ac current in a Josephson junction ratchet array

We have studied the vortex dynamics in a ratchet array of Josephson junctions in the presence of magnetic field of 1/5 flux quantum per plaquette. The ratchet potential consists of both alternate critical currents for all the vertical junctions and alternate shunt capacitances for all the horizontal junctions. The vortices driven by an ac current in some parameters are found to show the directional motion as well as the asymmetric current–voltage characteristics. We use the time-dependent vorticity and the time-dependent vorticity–vorticity correlation function to analyze the motion of vortices on a few fractional Shapiro steps. We have found that vortices on a fractional Shapiro n / 5 -step move coherently through n plaquettes during a single ac cycle. The asymmetric features of the ratchet array gradually disappear as finite temperature increases.

The DRBEM solution of incompressible MHD flow equations

SUMMARYThis paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier‐Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function‐vorticity‐magnetic induction‐current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson‐type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time‐dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward‐facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Vortex Sound with Special Reference to Vortex Rings: Theory, Computer Simulations, and Experiments

This is a review paper on vortex sound with special reference to vortex rings and sound emission detected in experimental tests. Any unsteady vortex motion excites acoustic waves. From the fundamental conservation equations of mass, momentum and energy of fluid flows, one can derive a wave equation of aerodynamic sound. The wave equation of acoustic pressure can be reduced to a compact form, called the equation of vortex sound. This equation predicts sound generation by unsteady vortex motions. On the other hand, based on the matched asymptotic expansion (using the multipole expansions), one can derive a formula of wave pressure excited by time-dependent vorticity field localized in space. The theoretical predictions are compared with experimental observations and direct computer simulations. The systems considered are, head-on collision or oblique collision of two vortex rings, vortex-cylinder interaction, vortex-edge interaction, and others. Scaling laws of the pressure of emitted sound are predicted by the theory and compared with experimental observations. Comparison between theories and observations shows excellence of the theoretical predictions. Direct numerical simulations are reviewed briefly for sound generation by collisions of two vortex rings and that by a cylinder immersed in a uniform stream ( aeolian tones). Vortex sound in superfluid is also reviewed about experimental observations and computational studies on the basis of the Gross-Pitaevski equation.

Wave Response during Hydrostatic and Geostrophic Adjustment. Part II: Potential Vorticity Conservation and Energy Partitioning

Abstract This second part of a two-part study of the hydrostatic and geostrophic adjustment examines the potential vorticity and energetics of the acoustic waves, buoyancy waves, Lamb waves, and steady state that are generated following the prescribed injection of heat into an isothermal atmosphere at rest. The potential vorticity is only nonzero for the steady class and depends only on the spatial and time-integrated properties of the injection. The waves contain zero net potential vorticity, but undergo a time-dependent vorticity exchange involving latent and relative vorticities. The energy associated with a given injection may be partitioned distinctly among the various wave classes. The characteristics of this partitioning depend on the spatiotemporal detail of the injection, as well as whether the imbalance is generated by injection of heat, mass, or momentum. Spatially, waves of a scale similar to that of the injection are preferentially excited. Temporally, an extended duration injection preferentially filters high-frequency waves. An instantaneous injection, that is, the temporal Green’s function, contains the largest proportions of the high-frequency waves. The proportions of kinetic, available elastic, and available potential energies that are carried by the various waves are functions of the homogeneous system. For example, deep buoyancy waves of small horizontal scale primarily contain equal portions of available potential and vertical kinetic energy. The steady state contains more available potential energy than kinetic energy at small horizontal scale, and vice versa. These qualities of the wave energetics illustrate the mechanisms that characterize the physics of each wave class. The evolution and spectral partitioning of the energetics following localized warmings identical to those in Part I are presented in order to illustrate some of these basic properties of the energetics. For example, a heating lasting longer than a few minutes does not excite acoustic waves. However, Lamb waves of wide horizontal scale can be excited by a heating of several hours. The first buoyancy waves to be filtered by an extended duration heating are those of the deepest and narrowest structure that have a frequency approaching the buoyancy frequency. The energetics of the steady state depends only on the spatial and time-integrated properties of the warming. However, the energetics and transient evolution toward a given steady state depend on the temporal properties of the warming and may differ widely.

Fluid–body interaction in the presence of uniform vorticity and density gradient

We consider the case of a rigid or deformable body moving in a linear shear flow of an inviscid, incompressible and inhomogeneous fluid with a uniform density gradient. The body is impulsively introduced into the nonuniform ambient flow field possessing time-dependent vorticity. In order to make the analysis amenable, it is further assumed that the size of the body is small with respect to the inhomogeneity of the ambient flow. By integrating the Euler equation it is shown that analytic expressions can be obtained for the hydrodynamical reactions (forces and moments) acting on the body which determine its trajectories. In addition to the traditional purely inertial quadratic terms, three new types of interaction modes are thus revealed: Vorticity-inertia, density gradient-inertia, and vorticity-density gradient. The first two modes have been considered separately just recently but the third one is presented here for the first time. It is finally demonstrated that all three modes of interaction can be obtained from a relatively straightforward unified approach.

Dynamics of a Viscous Vortex Ring

The evolution of a viscous vortex ring through the use of the earlier obtained solution of the Stokes equations in the form of time-dependent vorticity distribution is studied. In the long-time limit this distribution transforms into the classical self-similar Phillips’ distribution and for t → 0 it reduces to a delta-function. Also, this distribution satisfies the condition of the total impulse conservation and for early times its leading-order approximation inside the viscous vortex core is described by the Oseen-Lamb vortices. The integral transforms method is used to derive the corresponding stream function and the translation velocity of the ring. The obtained stream function behaves similarly to the vorticity distribution: at large times it transforms into Phillips’ result and for t → 0 it reduces to a circular line vortex. The predicted velocity agrees with the asymptotic drift velocity at ∞ → t , the Saffman result for the rings with small crosssections, and with the available experimental data.

Transport of Heavy Particles in a Three-Dimensional Mixing Layer

Particle transport in a three-dimensional, temporally evolving mixing layer has been calculated using large eddy simulation of the incompressible Navier-Stokes equations. The initial fluid velocity field was obtained from a separate simulation of fully developed turbulent channel flow. The momentum thickness Reynolds number ranged from 710 in the initial field to 4460 at the end of the calculation. Following a short development period, the layer evolves nearly self-similarly. Fluid velocity statistics are in good agreement with both the direct numerical simulation results of Rogers and Moser (1994) and experimental measurements of Bell and Mehta (1990). Particles were treated in a Lagrangian manner by solving the equation of motion for an ensemble of 20,000 particles. The particles have the same material properties as in the experiments of Hishida et al. (1992), i.e., glass beads with diameters of 42, 72, and 135 μm. Particle motion is governed by drag and gravity, particle-particle collisions are neglected, and the coupling is from fluid to particles only. In general, the mean and fluctuating particle velocities are in reasonable agreement with the experimental measurements of Hishida et al. (1992). Consistent with previous studies, the Stokes number (St) corresponding to maximum dispersion increases as the flow evolves when defined using a fixed fluid timescale. Definition of the Stokes number using the time-dependent vorticity thickness, however, shows a maximum in dispersion throughout the simulation for St ≈ 1.

Sound Generation in Oblique Collision of Two Vortex Rings

A theory of vortex sound formulated in the form of multipole expansions is applied to the oblique collision of two vortex rings at right angles. Using the theoretical formula for the far-field acoustic pressure excited by a time-dependent localized vorticity distribution, the coefficients of the multipole modes of the wave pressure are estimated by the numerical data of computer simulation. Time evolution of the vorticity field is obtained by solving a viscous incompressible vorticity equation with a vorticity-potential method developed for a three-dimensional vorticity field in a bounded domain. Numerical simulations are carried out for two kinds of vortex rings with the same core parameters but different ring radii, and the details of the vortex motion during the oblique collision are studied numerically. Computed main-mode amplitudes of the wave pressure excited by the vortex motion are found to be consistent with those of the experimentally observed acoustic wave not only qualitatively but also quanti...

Numerical convergence of the random vortex method for complex flows

Vortex methods rely principally on a discretization of the continuous two-dimensional time dependent vorticity field into a large number of vortex blobs, whose position and strength determine the underlying velocity field. In this paper, the convergence of the random vortex method (RVM) for a complex flow is studied in function of three discretization parameters. Two of these parameters are related to the spatial discretization of the vorticity, i.e. G(sheet or blob strength) and h (sheet length or core radius of a blob) and the third one to the discretization of time, ?t i.e. . Two principal events are observed. First, the computation works but the convergence is not attained. Second, the computation fails. The first behaviour is attributed to a lack of accuracy while the second is attributed to a lack of numerical stability. Once the stability conditions are satisfied, decreasing the value of the parameters always leads to convergence.

Oblique collision of two vortex rings and its acoustic emission

A theory of vortex sound is formulated in the form of multipole expansions and an explicit formula is presented for the wave pressure excited by a time-dependent vorticity field localized in space. This is applied to the case of the oblique collision of two vortex rings at right angles, in which higher-order terms are important to represent asymmetric emission. The vortex motion and the generated waves are also studied experimentally and numerically. The initial setup of the two vortices is arranged so that they come into contact by their own motions and perform a reconnection of the vortex lines

THE TRANSIENT MOTION OF A SPHERICAL FLUID DROPLET

Abstract A numerical method is developed for investigation of the unsteady motion of a spherical fluid droplet under the influence of gravity. This study extends previous work valid for creeping flow to moderate Reynolds number. The unsteady flow fields inside and outside of the fluid sphere are described by the two-dimensional, axisymmetric Navier-Stokes equations in the form of vorticity and stream function, along with the equation of motion of the droplet. The governing equations are approximated by a central difference and a second-order upwind difference, and are solved iteratively using the Gauss-Siedel and secant methods. Numerical results of the time-dependent vorticity, stream function and drop velocity are presented for a water droplet moving through air and for an air bubble rising in water. The steady state drop velocity and the drag coefficient at various Reynolds numbers are examined, and they are shown to agree very well with previous results.

Bifurcation of an elliptic vortex ring

Time-dependent vorticity fields of elliptic vortex rings of aspect ratios 2, 3 and 4 were measured by means of hot-wire anemometry. The time evolution of their vorticity fields was analyzed and the processes of vortex ring formation, advection, interaction and decay, and the mechanism of vortex bifurcation are studied. The following crosslinking model is proposed: A thick vortical region composed of many equivalent vortex filaments with distributed cores is initially formed at the orifice and they behave as inviscid filaments. The elliptic ring deforms and the end parts of its major axis get closer. Then, the vortex filaments interact at the touching point and the ring partially bifurcates. Almost simultaneously, turbulent spot appears at this point, and propagates around the ring cross section, thus preventing further bifurcation. And it becomes a turbulent blob. This model is also supported by numerical simulation by a high-order vortex method and the Navier-Stokes solution.

Cross-linking of two vortex rings

The mechanism for cross-linking two vortex rings has been investigated experimentally. Phase-locked velocity measurements using x-type hot wires were carried out point by point over the entire flow field, and time-dependent vorticity fields were educed. It is found that a new pair of vortex tubes is created, which connects the interacting vortex rings and grows gradually, while the vorticity intensity of the main rings decreases. This is considered to be a type of bridging phenomenon of the vortex tubes, as proposed by Kida and Takaoka [Phys. Fluids 30, 2911 (1987)] in their numerical simulation.

渦輪の干渉におけるブリッジ現象

Bridging phenomenon in crosslinking two vortex rings has been studied. Phase locked measurements of velocity using X-type hot wire were carried out over the whole flow field trasversing point by point, and time dependent vorticity fields were educed. Creation of new pair of vortex tubes was found in the interacting rtex rings.

The propagation of vorticity in a viscoelastic fluid

A layer of constant vorticity exists in an infinite space of incompressible isotropic viscoelastic fluid for which the shear stress for rectilinear shearing flows depends linearly on the history of the velocity gradient. At some instant of time the forces maintaining this flow are removed. The subsequent time-dependent vorticity field is calculated explicitly in the particular cases when the fluid is Newtonian and when it is Maxwellian. The limiting case in which the vorticity layer becomes a vortex sheet is also calculated.

Wake Calculations around 2-Dimensional Elliptic Cylinders by Time-Dependent Vorticity Transport Equation

A new method to predict near wake flow is proposed where the time dependent vorticity transport equation and the boundary layer equation are used as governing equations. The flow field is supposed to be that made by a sudden move of a body to which a vorticity layer is attached. The attached vorticity layer is equivalent to the steady boundary layer. The flow field is solved by the time marching method. The two-equations model is invoked for the turbulent closure. The wakes of three 2-dimensional elliptic cylinders are calculated which are significantly separated and recirculating. The wake flow of a flat plate is calculated, in advance, to make sure the propriety of the present method. As far as the near wake flows are concerned, satisfactory results are obtained within a reasonable computing time. Recirculating and fluctuating flow fields are nicely simulated. The viscous drag is also calculated by making use of the results of wake calculations.

A general finite-difference formulation with application to Navier-Stokes equations

A general method for deriving a finite-difference equation from a partial differential equation with the use of local model solutions is presented. The method, which is based on the evaluation of local functions, is illustrated by applying to a time-dependent vorticity equation. The main advantage of the present approach is that there is no apparent stability limitation on grid size and time step. Time dependent Navier-Stokes equations are solved with the present formulation for a Couette flow with constant suction and a square cavity flow problems. Steady-state, convergent solutions are obtained with grid sizes and time steps much larger than those determined from the conventional stability criteria for the forward time central difference method. The local model solutions used in the present paper were designed to produce steady-state solution; care must be taken to interpret the transient solution obtained from the present model with large time step.

A numerical investigation of western boundary currents in subtropical and subpolar gyres with a barotropic model

Dynamics of western boundary currents in the subtropical and subpolar gyres are studied as a source-sink flow of barotropic fluid by means of numerical integration of the time-dependent non-linear vorticity equation. The bottom topography consists of a continental shelf of uniform slope (120 km wide) parallel to the straight western coast and a flat bottom of uniform depth.