Discrete Vortex Model Research Articles

Filamentary magnetohydrodynamic plasmas

A filamentary construct of magnetohydrodynamical plasma dynamics based on the Elsässer variables is developed. This approach is modeled after discrete vortex models of hydrodynamical turbulence, which cannot be expected in general to produce results identical to those based on a Fourier decomposition of the fields. In a highly intermittent plasma, the induction force is small compared to the convective motion, and when this force is neglected, the plasma vortex system is described by a Hamiltonian. A statistical treatment of a collection of discrete current-vorticity concentrations is given. Canonical and microcanonical statistical calculations show that both the vorticity and the current spectra are peaked at long wavelengths, and the expected states revert to known hydrodynamical states as the magnetic field vanishes. These results differ from previous Fourier-based statistical theories, but it is found that when the filament calculation is expanded to include the inductive force, the results approach the Fourier equilibria in the low-temperature limit, and the previous Hamiltonian plasma vortex results in the high-temperature limit. Numerical simulations of a large number of filaments are carried out and support the theory. A three-dimensional vortex model is presented as well, which is also Hamiltonian when the inductive force is neglected. A statistical calculation in the canonical ensemble and numerical simulations show that a nonzero large-scale magnetic field is statistically favored, and that the preferred shape of this field is a long, thin tube of flux. Possible applications to a variety of physical phenomena are suggested.

The role of coherent structures in bubble transport by turbulent shear flows

Using Auton's force law for the unsteady motion of a spherical bubble in inhomogeneous unsteady flow, two key dimensionless groups are deduced which determine whether isolated vortices or shear-layer vortices can trap bubbles. These groups represent the ratio of inertial to buoyancy forces as a relaxation parameter [tcy ] = ΔU2/2gx and a trapping parameter [Gcy ] = ΔU/VT where ΔU is the velocity difference across the vortex or the shear layer, x is streamwise distance measured from the effective origin of the mixing layer and VT is the terminal slip speed of the bubble or particle. It is shown here that whilst buoyancy and drag forces can lead to bubbles moving in closed orbits in the vortex flows (either free or forced), only inertial forces result in convergent trajectories. Bubbles converge on the downflow side of the vortex at a location that depends on the inertial and lift forces. It is important to note that the latter have been omitted from many earlier studies.A discrete-vortex model is used to simulate the large-scale unsteady flows within horizontal and vertical mixing layers between streams with velocity difference ΔU. Trajectories of non-interacting small bubbles are computed using the general force law. In the horizontal mixing layer it is found that Γ needs to have a value of about 3 to trap about 50% of the bubbles if Π is about 0.5 and greater if Π is less. The pairing of vortices actually enhances their trapping of bubbles. In the vertical mixing layer bubbles are trapped mainly within the growing vortices but bubbles are concentrated on the downflow side of the vortices as Γ and Π increase. In a companion paper we show that lateral dispersion of bubbles can be approximately described by an advective diffusion equation with the diffusivity about equal to the eddy viscosity, i.e. rather less than the diffusivity of heat or other passive scalars.

Distribution of Suspended Sediment over Wave‐Generated Ripples

Oscillatory flow over wave ripples is described by use of a discrete vortex model using the “cloud in cell” concept. The distribution of suspended sediment over the ripple bed is simulated by a Lagrangian model, following the track of individual sediment particles. A boundary‐layer model is used to determine the strength and the motion of the vortices very close to the ripple surface. The development of the boundary layer is described by the integrated momentum equation, assuming the boundary layer to be turbulent with a logarithmic velocity profile. In the boundary layer, turbulent diffusion is used to describe the distribution of suspended sediment. Outside the boundary layer along the ripple surface, advection with the velocity field resolved by the discrete vortex model determines the suspended‐sediment distribution. The simulated sediment concentrations are compared to measurements from the literature. The agreement between the time‐averaged simulated concentration profiles and the measurements from an oscillating water tunnel is satisfactory.

Discrete vortex representation of magnetohydrodynamics.

We present an alternative approach to statistical analysis of an intermittent ideal magnetohydrodynamics fluid in two dimensions, based on the hydrodynamic discrete vortex model applied to the Elsasser variables. The model contains negative temperature states which predict the formation of magnetic islands, but also includes a natural limit under which the equilibrium states revert to the familiar twin-vortex states predicted by hydrodynamic turbulence theories. Numerical dynamical calculations yield equilibrium spectra in agreement with the theoretical predictions.

Vortex induced vibration of circular cylinder

The flow field around a vortex induced vibrating cylinder is analyzed by the numerical simulation using the discrete vortex model. When the vortex shedding locks in the vibration, the separated shear layers sway in a body like a pendulum. The locked-in is not the necessary condition of the vortex induced vibration, but the vibration develops when the vortex does not shed synchronously with the vibration. At the unsynchronous state, the rolling up of the separated shear layers are prevented and promoted alternatively by the cylinder movement at every few cycles of the vibration. The energy which encourages the movement of the cylinder is supplied when the rolling up of the shear layer is promoted by the movement. The effect of the splitter plate in the wake was also investigated and the result of the simulation compared with the flow patterns experimentally obtained from the smoke wire method. The simulated flow patterns agree well with the one which were experimentally obtained.

Large Eddy Structures Revisited in Elliptic Vortex Model.

A new discrete vortex model is developed and applied to reveal the essential feature of the vortex pairing, a key process to turbulent mixing layer evolution. The model accounts for the finite size of each eddy as a circulation monopole plus a vortex quadrupole and describes a self-rotating elliptic vortex deformable in a stretching flow field in the presence of other vortices. It is demonstrated that the coalescence of pairing vortices takes place, through interaction between a pair of quadrupoles, when the magnitude of the stretching rate in the eddies exceeds that of the self-rotating angular velocity. The streakline pattern outside the eddies, calculated for a temporally developing mixing layer, is in fairly good agreement with existing numerical calcuation results, and is thus expected to provide a powerful analytical tool for simulating turbulent mixing layers.

Numerical simulation of jet‐forced flow in a circular reservoir using discrete and random vortex methods

AbstractThis paper describes a Biot–Savart discrete vortex model for simulating the flow patterns which occur when a single high‐velocity inflow jet is used to stir the fluid within a circular container. The first stage of the model consists of conformally mapping the circular perimeter of the container onto a rectangle by means of a Schwarz–Christoffel transformation. A potential flow solution is then obtained for the flow inside the rectangle and this is transformed to give the potential flow inside the circle. In the second stage of the simulation, discrete vortices are added at the inlet of the physical system in order to model the inflow shear layers. Velocity components resulting from the discrete vortices and their images in the walls of the cylinder are superimposed on the uniform potential flow solution. The positions of the vortices are updated using a Lagrangian tracking procedure. Viscous effects are incorporated through the use of random walks. From the results it is shown that the discrete vortex method does predict qualitatively the important features of jet‐forced reservoir flow.

The Flow of the Rotational and Translational Oscillations of a Flat Plate.

Alternating vortices are generated by an oscillating plate when a plate is abruptly started to oscillate from rest. It is an interesting problem to examine the flow around the plate. We analyze the flow around a flat plate acting in a combination of rotational and translational oscillations in a stationary fluid, using a discrete vortex model with the conformal mapping method. Numerical calculations are performed for various angles in rotational oscillation, amplitudes in translational oscillation, and frequencies and phase angles in both oscillations, Vortex patterns and unsteady forces are obtained. As a result, vortex clusters stay near the plate. The thrust on the plate is generated by these clusters.

A physical simulation model for turbulent mixing layer combustion

A previously proposed discrete vortex model is applied to reveal the underlying physics of diffusion combustion in vortical flow fields generated by pairing vortices. The model accounts for the finite size of each localized vorticity region as a circulation monopole plus a vortex quadrupole, and describes a self-rotating elliptic vortex deformable in a stretching flow due to the presence of other vortices and flame. The coalescence of pairing vortices can be recognized as a resonance phenomenon which takes place, through interaction between a pair of quadrupoles, when the magnitude of the stretching rate in the vorticity regions exceeds that of the self-rotating angular velocity. The Shvab-Zeldovich formulation is used for combustion modeling. Results of simulation performed for a temporally evolving mixing layer are in fairly good agreement with existing numerical calculation results. The effect of thermal expansion is incorporated in the form of volume source at the flame front, so that changes from constant density case attribute to the expanding flow from the flame. The burning rate is reduced since the flow opposes the diffusive fuel and oxidizer fluxes toward the flame. The distance of separation between flame segments is enlarged, resulting in an rapid increase in the flame length and mixing layer thickness. This has another important significance in the light of the present model because the effect of vortex quadrupole decays faster than that of circulation monopole away from each vortex core and the preservation of localized vorticity regions through the repeated processes of vortex pairing and coalescence is inhibited. In addition to the commonly mentioned increased viscosity due to temperature rises, this mechanism is proposed to provide a physical explanation why turbulent free shear layer with combustion assumes a simpler structure in a shorter time.

Comment on "Calculation of Asymmetric Vortex Separation on Cones and Tangent Ogives Based on a Discrete Vortex Model"

The boundary value problem for vortex separation at zero sideslip on cones and tangent ogives is set up by means of a discrete vortex model. The nonlinear algebraic equations for the boundary value problem admit multiple, physically feasible solutions, including the symmetric and asymmetric vortex solutions. Multiple solutions are proposed as an alternative explanation of the existence of asymmetric vortex separation at zero sideslip.

Decaying, two-dimensional, Navier-Stokes turbulence at very long times

Two-dimensional Navier-Stokes turbulent decay has been followed numerically for very long times. The code is spectral, satisfies periodic boundary conditions, and does not make use of hyperviscosity or any small-scale smoothing. The resolution is (512) 2, the initial Reynolds number based on the box dimension is about 14000, and the run continues for over 190 initial eddy turnover times. Isolated vortices form, and the turbulence has become highly intermittent in the manner seen by McWilliams and Brachet et al., for times greater than about 30. However, it is not the case that merger of like-signed vortices stops; it only slows down. By t = 210, the final merger is complete, and the vorticity distribution is dominated by one large vortex of either sign. The negative (positive) vortices each occupy about two percent of the box area, and together account for over 98 percent of the total enstrophy. Their alignment suggests the formation of an Ewald lattice with a basic cell containing two point vortices. The ratio of enstrophy to energy continues to decrease monotonically, and the picture is consistent with a “selective decay” process, as described some time ago. A not-entirely-understood phenomenon is the concentration of the vorticity into two cores, suggestive of a negative-temperature state of the discrete line vortex model.

Investigations of wake flows of a flat plate in steady, oscillatory and combined flows

Numerical study on near wake flows of a flat plate in three kinds of oncoming flows is made by using the discrete vortex model and improved vorticity creation method. For steady oncoming flow, both gross and detailed features of the wake flow are calculated and discussed. Then, in harmonic oscillatory oncoming flow two different wake flow patterns withK c=2,4 and 10 are obtained respectively. Our results present a new wake flow pattern for lowKc numbers (Kc<5) describing vortex shedding, pairing and moving in a period of the oscillatory flow starting from rest. The calculated drag and inertia force coefficients are closer to experimental data from the U-tube than the previous results of vortex simulation. For in-line combined oncoming flow the vortex lock-in and dynamic characteristics are simulated. The results are shown to be in good agreement with experiments.

Viscous flow simulation using the discrete vortex model—the diffusion velocity method

Abstract A new numerical model is presented for viscous flow simulation using the discrete vortex model. The viscous diffusion is produced by the vortices' movement induced by the diffusion velocity introduced in this paper. It is shown that our proposed method is valid, even at Re values below the lower limit of applicability of the random vortex method and that our solutions are much smoother. In addition, our method is clearly more dependent on Re than the conventional discrete vortex method, and since our method can treat the development, separation and reattachment of the boundary layer, it dispenses with the boundary layer theory.

A discrete vortex analysis of flow around a vibrating cylinder with a splitter plate

The mechanism of flow-induced vibration of a circular cylinder with a splitter plate was investigated by means of numerical simulation using the discrete vortex model. Since the splitter plate prevents interaction of the upper and lower separated shear layers, the interaction between the shear layers is not important, but the interaction between the flapping of the shear layers and the cylinder vibration is important. Consequently the lift on the cylinder fluctuates synchronously with the vibration frequency at any velocity. The phase lag between the lift on the cylinder and the displacement of the cylinder itself increases from 0 to π as the velocity increases. Therefore the aerodynamic damping is always negative and consequently the self-excited vibration can be induced at any velocity. It should be noted that in this case the aerodynamic damping estimated by the quasi-steady assumption is positive. The negative damping is caused caused by the vortex shedding and its reattachment to the splitter plate, which are synchronized with the vibration of the cylinder. The development and synchronization processes of the vibration of a cylinder without a splitter plate are also discussed.

Flow Around and Forces on a Pipeline Near a Scoured Bed in Steady Current

Abstract Flow around a pipeline placed initially on a flat, erodible bed was investigated at five characteristic stages of the progressive scour process in currents. The velocities, both in the streamwise direction and in the direction normal to the bed, were measured. A two-color flow visualization technique was used to detect the time-dependent flow structures. The forces on the cylinder were recorded simultaneously with the flow visualization. In the theoretical part of the study, a discrete vortex model (namely, the cloud-in-cell model) is employed to predict the flow. The early results appear to reveal the gross features of the flow shown by the experiments.

Aerodynamic sources of acoustic resonance in a duct with baffles

Experimental and numerical investigations of the generation of resonant sound by flow in a duct containing two sets of baffles and the “feedback” of the sound on the vortex shedding process are reported. The experiments are conducted in a wind tunnel and the numerical simulations are used to predict the sources of resonant sound in the flow. The resonant sound field, which is principally longitudinal, is calculated by the finite element method and a discrete-vortex model is used to predict the observed separated flow. Analysis of the passage of a single point vortex past a baffle indicates that the amount of acoustic energy generated is a function of the phase of the acoustic cycle at which the vortex passes the baffle. A more elaborate model simulates the growth of vortex clouds through the clustering of elemental vortices shed from an upstream baffle, tracks the passage of these vortex clouds past a downstream baffle, predicts the generation of acoustic energy using Howe's theory of aerodynamic sound, and accounts for the feedback of sound on the vortex shedding. Comparison is made between the predicted time-dependent structures and the observed flow structures using smoke visualization. The vortex cloud model predicts the flow conditions under which net acoustic energy is generated by the flow and therefore when resonance can be sustained; the results are consistent with the occurrence of peaks in the observed resonant sound pressure levels.

Discrete vortex simulation for flow around a circular cylinder with a splitter plate

A discrete vortex model was developed to investigate unsteady flow in a wake of a circular cylinder with a splitter plate. In the model the splitter plate was expressed by a series of sources such as to cancel out the flow across it. Various flow features which had been observed experimentally by many researchers were successfully reproduced in the simulation. The vortices suddenly rolled up between the cylinder and the plate when the distance between them was beyond the critical value. The critical distance was hardly affected by the size of the plate when the plate was along the flow, on the other hand it increased proportionally to the size of the plate when the plate was normal to the flow. The critical distance was about a half of the experimental results at Reynold number 10 4–2×10 4, because the length of the vortex formation region in the simulation was shorter than in the experiments. It was also clarified that the various physical quantities such as the base pressure coefficient, the fluctuating lift coefficient, the Strouhal number and the position of the separation point, etc., were systematically varied according to the change of the flow pattern.

A numerical analysis of the unsteady flow past bluff bodies

The paper describes in detail a relatively sophisticated numerical approach, using the Boundary Element Method in conjunction with the Discrete Vortex Model, to represent the complex unsteady flow field around a bluff body with separating shear layers. Important steps in the numerical analysis of this challenging problem are discussed and a performance evaluation algorithm established. Of considerable importance is the effect of computational parameters such as number of elements representing the geometry, time-step size, location of the nascent vortices, etc., on the accuracy of results and the associated cost.

Visualization analysis of the starting flow around a pitching airfoil. 3rd Report. Numerical simulation using a discrete vortex approximation.

A discrete vortex model is introduced in the calculation of the starting flows past a two-dimensional elliptic airfoil oscillating in pitch at large incidences. The potential flow past an obstacle is represented by a fixed number of bound vortices distributed along the surface and the viscous wake by an increasing number of free vortices, which are shed from the leading and trailing edges with a regular time interval and convected downstream under the action of the free stream velocity and the vortex-induced velocity. Every time stage of the calculated wake is visualized by means of point vortex plots and streamline tracings. In parallel, the instantaneous hydrodynamic loads exerted on the oscillating airfoil are estimated by using the Blasius formulae for the unsteady regime. The tested experimental parameters are the reduced pitching frequency ranging between 0.1 and 1.0 and the amplitude of oscillation between 7°and 15°. The mean incidence is fixed at 30°and the initial incidence at the minimals. Some comparison is made with the purely experimental streamlines presented in the first report.

A discrete vortex analysis of flow around a vibrating cylinder with a splitter plate

The mechanism of flow-induced vibration of a circular cylinder with a splitter plate was investigated by means of numerical simulation using the discrete vortex model. Since the splitter plate prevents interaction of the upper and lower separated shear layers, the interaction between the shear layers is not important, but the interaction between the flapping of the shear layers and the cylinder vibration is important. Consequently the lift on the cylinder fluctuates synchronously with the vibration frequency at any velocity. The phase lag between the lift on the cylinder and the displacement of the cylinder itself increases from 0 to π as the velocity increases. Therefore the aerodynamic damping is always negative and consequently the self-excited vibration can be induced at any velocity. It should be noted that in this case the aerodynamic damping estimated by the quasi-steady assumption is positive. The negative damping is caused caused by the vortex shedding and its reattachment to the splitter plate, which are synchronized with the vibration of the cylinder.The development and synchronization processes of the vibration of a cylinder without a splitter plate are also discussed.