Equi-vorticity Lines Research Articles

Numerical Study of the Kinematic Motions of a Solitary Wave Hitting an Inclined Plate

In this study, we numerically investigated the vortex patterns and free-surface deformations of a solitary wave interacting with a bottom-mounted plate at various inclinations in shallow water. Assuming two-dimensional, incompressible, and non-breaking conditions, we considered a viscous fluid based on the streamfunction-vorticity algorithm used in Navier–Stokes equations. We solved the governing equations by finite analysis, and used an averaged two-time step method to consider nonlinear conditions on the free surface. To fit the irregular boundaries and reveal the fine details of the vortex phenomenon, we applied a transient boundary-fitted grid along with a local-grid-refinement technique and examined the vortex mechanics around the submerged barrier. We then compared the numerical solutions obtained for the flow patterns with existing experimental observations to ensure the efficiency and accuracy of the numerical model. In addition, we systematically simulated other cases using this model to explore kinematic phenomena such as streamlines, equi-vorticity lines, streak lines, timelines, and path lines.

Numerical simulations and experimental visualizations of the vortex characteristics for a solitary wave interacting with a bottom-mounted vertical plate

This study is aimed to investigate numerically and experimentally the interaction of a solitary wave with a bottom-mounted vertical thin plate with focuses on the wave induced fluid kinematics and vortex flow patterns. A streamfunction–vorticity free-surface (SVFS) based fully nonlinear viscous wave model solved in a transient boundary-fitted coordinate system with locally overlaid grids is applied to study this wave-plate interaction problem. Qualitative comparisons between numerically generated flow patterns around the plate and the experimental observations, including images from both the particle-tracking and the laser-dye visualization methods, are presented. The shear-layer velocity profiles are compared quantitatively with the experimental measurements. The detailed flow characteristics and formed vortices including the effect of the shape of the plate top on the vortex formation and development are numerically investigated. For a thin vertical plate, the flow characteristics that are affected by the dimensionless parameters of incident-wave height and plate height are also investigated. The numerical visualizations illustrated by the virtual transport of particle-tracing, streamlines, and equi-vorticity lines are useful to understand the kinematic behaviors of the induced vortical motions. Furthermore, the pressure gradient, shear stress, and maximum wave force are examined to detail the hydrodynamic impacts on the submerged plate.

Cooled Vortex Street Characteristics

Common and different characteristics among the cooled vortex street, cooled Karman vortex street, and isothermal wavy wake in the cooled laminar wake above the isothermal or cooled circular cylinder exposing in the upward mainstream of mercury, air, or water at the Reynolds number Re = 44 are discussed. Employing the previous and new distributions, the cooled vortex street characteristics are elucidated, and the conclusions are obtained as follows. 1) In the cooled wake, i.e., the isothermal wavy wake, cooled Karman vortex street, and cooled vortex street, in any fluid, the negative buoyancy increases the amplitude of the meandering motion in the zero-streamlines, and decreases the distance between the neighboring streamlines near the zero-streamline. As a result, with increasing the cooling rate in any fluid, both the vortex spiral size and the velocity near the zero-streamline become large. 2) In any fluid, the isothermal wavy wake, cooled Karman vortex street, and cooled vortex street are qualitatively similar. However, the cooled vortex street is quantitatively different from both the isothermal wavy wake and cooled Karman vortex street, and has the extremely strong meandering zero-streamline, extremely large size vortex spiral, and velocity larger than the mainstream velocity. 3) The quantitative difference in the above is explained in the meandering streamlines, vortex spiral size, equivorticity lines, and isothermal lines of three kinds of wakes, i.e., the isothermal wavy wake, cooled Karman vortex street, and cooled vortex street, by introducing the enlargement ratio En. 4) In any fluid, the unstable transitional region exists always between the cooled Karman vortex street and the cooled vortex street. With increasing the cooling rate, when the wake goes over the transitional region, the cooled vortex street arises. 5) The cooled vortex street formation is extremely unique enough to give a separate name, i.e., the cooled vortex street.

Cooled Vortex Street with Large Vortex Spirals Generated by Cooling a Circular Cylinder at Low Reynolds Number: Effects of Fluid Kinds on Wake Patterns

Cooled wakes of mercury and water are numerically simulated, and computed results are shown and discussed. As a result, effects of fluid kinds on wake patterns, i.e., the streaklines, streamlines, equi-vorticity lines, and isothermal lines, are elucidated as follows. 1) The results are validated in mercury and air by comparing the mean Nusselt number in the isothermal wake with the previous result, and are supposed to be valid in water because the resolution in the isothermal lines is sufficient. 2) Effects of fluid kinds on the cooled wake and the cooled vortex street are extremely strong, because the dominant action is the temperature diffusion in mercury, and the temperature advection in water. 3) The temperature distribution is extremely different between mercury and water. In mercury, the temperature varies in the whole wake, and the isothermal lines draw almost concentric circles. In water, a number of the temperature islands are generated, and transport heat to the downstream. 4) The cooled vortex street is generated in both mercury and water. The shape of the vortex spiral is vague and is not circular in water, but rotates as a vortex with smooth and clear streaklines in mercury. 5) The motion of two separated shear layers of vorticity is extremely different between mercury and water. 6) The critical Richardson numbers for onsets of the cooled vortex street and symmetric wake in water are much larger than those in mercury.

Cooled Vortex Street Generated by Cooling a Circular Cylinder in Low Reynolds Number Wake (3rd Report, Effect of Fluid Kinds on Wake Patterns)

Cooled wakes of mercury and water are numerically simulated, and computed results are shown and discussed. As a result, effects of fluid kinds on wake patterns, i.e. the streaklines, streamlines, equi-vorticity lines, and isothermal lines, are elucidated as follows : (1) Effects of the fluid kinds on the cooled wake and the cooled vortex street are extremely strong, because the dominant action is the temperature diffusion in mercury, and the temperature advection in water. (2) The temperature distribution is extremely different between mercury and water. In mercury, the temperature varies in the whole wake, and the isotheral lines draw concentric circles. In water, a number of the temperature islands are generated, and advect heat to the downstream. (3) The cooled vortex street occurs in mercury and water. The shape of the vortex spiral of water is vague and is not circular. The vortex spiral of mercury rotates as a vortex with smooth and clear streaklines. (4) The motions of two separated shear layers of vorticity are extremely different between mercury and water. (5) The Richardson numbers of onsets of the cooled vortex street and symmetric wake in water are extremely larger than those in mercury.

COMPUTER-AIDED FLOW VISUALIZATION AND VORTICITY BALANCE IN THE LAMINAR WAKE OF A CIRCULAR CYLINDER

The two-dimensional Navier-Stokes equations were used for the solution of laminar vortex shedding behind a circular cylinder at a Reynolds number of 106 using the finite element technique. Streamlines, equivorticity lines and filament lines have been used for the wake visualization, while the pressure distribution in the wake is presented in the form of isobars. Useful conclusions have been drawn from the superposition of different flow visualization patterns, a technique used extensively herein. The strengths of the laminar vortices were calculated at various locations from the vorticity distribution, confirming the hypothesis that maximum strength occurs at the end of the formation region. A vorticity balance in the cylinder wake was conducted, comparing the circulation influx into the wake over a period and the strength of the vortices. The time-dependent streamwise fluid velocities were calculated from the present simulation at various points in the wake, and are presented on the same diagram with velocities measured experimentally at similar conditions. The validity of alternative definitions for the vortex formation length was examined and the velocity and displacement of the newly formed vortex as function of time was calculated.

The oblique rise of a viscous vortex ring toward a deformable free surface

A vortex ring in a viscous, incompressible fluid rises obliquely at an angle of 45° toward a deformable free surface. The mathematical description of this flow situation is a time-dependent nonlinear free-surface problem that has been solved numerically for a three-dimensional laminar flow with the aid of the Navier-Stokes equations. Boundary-fitted coordinates were used that include adaptive gridding and grid refinement. For two different Froude numbers at Reynolds number 100, results are presented on the encounter of the vortex ring with the free surface, the deformation of the free surface, the decay of the primary vortices, the generation of surface vorticity and secondary vortices, and reconnection with the free surface. These results are presented in the form of streamlines and equivorticity lines in the plane of symmetry, contour lines of surface elevation, and streamlines and equivorticity lines on the free surface. The two cases reveal a novel ring reconnection at the free surface in the form of a cylindrical sheet.

Numerical Investigation of Response and Wake Characteristics of a Vortex-Excited Cylinder in a Uniform Stream

The numerical solution of vortex-excited oscillations of a circular cylinder is described herein, at Reynolds numbers below 130, where the wake behind the cylinder is fully laminar. The mathematical model of the problem consists of the Navier-Stokes equations, in the formulation where the stream function and the vorticity are the field variables. The finite element technique was favoured for the numerical solution of the Navier-Stokes equations. The parameters of the cylinder-elastic support arrangement considered in the numerical computation were identical to those of an experimental investigation at similar conditions, for the sake of comparison of the results. From the numerical results, the cylinder oscillation frequency to be slightly lower than the cylinder natural frequency in air below the lock-in region, and slightly higher and above the lock-in region. Outside the lock-in region the cylinder displacement and the hydrodynamic forces are modulated, although the numerical model is two-dimensional. For all Reynolds numbers considered, the time history of the cylinder displacement and of the hydrodynamic forces acting on the cylinder were calculated. The time-dependent velocities of the fluid in the stream- and cross-flow directions were calculated at some nodal points downstream of the cylinder from the stream function values. The effect of cylinder oscillation on the vortex spacing was examined by plotting the equivorticity lines. The results of this investigation were found to be in good agreement with the experimental evidence, except at reduced velocities above the lock-in region.

The decay of a pair of point vortices in a viscous fluid

The decay of two counter-rotating point vortices of equal strength in a viscous incompressible fluid is studied by solving the Navier–Stokes equations with a finite-difference technique. Solutions for Reynolds numbers 10, 50, and 100 are compared with a model presented by Cantwell and Rott. Equivorticity lines and the decay rate of translational velocity and circulation of the half-plane are presented, together with the diverging paths of the vortex pair.

An Analysis of Transient Flow in a Clean Room by a Modified GSMAC Finite Element Method

A method of numerical simulation has been developed to delineate guidelines for the ventilation design of a clean room. This paper presents a modified finite element analysis for solving unsteady incompressible Navier-Stockes equations. This scheme is based on the algorithm of the HSMAC (highly simplified marker and cell) method and the general idea of orthogonal decomposition. Unsteady flow simulations in terms of stream lines, equipressure lines and equivorticity lines are obtained on a color graphic display

Numerical solution for laminar two-dimensional flow about a fixed and transversely oscillating cylinder in a uniform stream

A finite element solution is presented for laminar 2-dimensional flow past a fixed and transversely oscillating cylinder. The solution of vortex street development behind the cylinder is obtained when the cylinder remains fixed. Then the cylinder which is assumed elastically mounted is allowed to oscillate in the cross-flow direction under the action of the fluctuating lift force. The grid system is translated with the cylinder at each time step and the field variables are interpolated to new grid locations. The computer results predict the lock-in phenomenon which occurs when oscillation frequency is near the natural vortex-shedding frequency. Particular attention was given to flow configuration, by means of streamlines, filament lines, and equi-vorticity lines. The effect of cylinder oscillation on lift and drag forces and separation angles is extracted from the numerical results. Computer results at Reynolds numbers 106 and 115 are in good agreement with experimental measurements conducted by the author.

An analysis of transient flow in clean room by an improved GSMAC finite element method.

A clean room has recently become indispensable equipment for various kinds of industries. In order to design a clean room, it is important to consider improvement of the working environment and energy conservation. The air flow of a clean room is very complex. It consists of recirculation flows, and the direction of mean air flow is changed locally. A method of numerical simulation have been developed to delineate a guideline for ventilation design of a clean room. This paper presents an improved finite element analysis for solving unsteady incompressible Navier-Stokes equations. This scheme is based on the algorithm of the HSMAC (Highly simplified marker and cell) method in SOLA and the general idea of orthogonal decomposition. It has actually been applied to the case of the lid-riven square cavity flow at Re=10 000, and of air flow in a conventional clean room. Unsteady flow simulations in terms of stream lines, equipressure lines, and equivorticity lines are obtained on a color graphic display.

円柱伴流内の渦列の渦度分布

The vorticity distributions in the vortex street behind a circular cylinder were measured by using the hot-wire technique in a low-speed wind tunnel. At intermediate distances from the cylinder, the vorticity distribution of a vortex in the vortex street was similar to that of an isolated viscous vortex which had a Gaussian distribution. The equivorticity lines in a cross section of a vortex were nearly concentric circles. Farther downstream, the vorticities wcich were concentrated around a vortex center diffused into the surroundings. Hovever, the vorticity distribution around the vortex center was still similar to that of the isolated viscous vortex. The circulation of a vortex in the vortex street was decreased as the distance from the cylinder increased. Farther downstream, when two consecutive vortices in the primary vortex street merged, the resultant circulation corresponded approximately to twice the basic vortex.

Aerodynamic interactions between the steady cascade loading and inflow gust vorticity. Vorticity effect in turbine rotor cascade.

The aim of the present paper is to show that the aerodynamic interactions between the gust vorticity and steady loading of a rotor cascade play a significantly important role, particularly in a turbine cascade by regarding the viscous wake rows shed by upstream nozzle cascade as inviscid sinusoidal gust. Through some notable deformations of equi-vorticity lines within rotor passage shearing of equi-vorticity lines over the rotor surface and its downstream, unsteady secondary flow by pair vortics etc are shown to be well explained. Further significant influences of the vorticity on the surface pressure distributions, the tangential and axial fluid forces and moment are clarified by numerical examples in the circumferential velocity and inflow gust phase.

Process of formation of a vortex street in the wake behind a flat plate

The process of the formation of a vortex street in the wake behind a flat plate set parallel to a uniform flow was investigated in a low speed wind tunnel. The vorticity distributions in the wake were calculated from the measured velocities using Taylor's hypothesis.Just behind the plate, the equi-vorticity lines were nearly parallel to the free stream. At locations somewhat downstream, sinusoidal contour lines appeared near the wake center. Further downstream, some closed contour lines appeared in the figures mapped. The arrangement of the closed lines suggests the existence of a vortex street. The maximum value for vorticity in a wave length of the fundamental velocity fluctuation decreased in the downstream direction; the concentration of vorticity, however, increased in a region the further downstream it was. Meanwhile, the value for circulation obtained by the surface integral of vorticity within the closed contours of a vortex increased until the vortex street was established.

Calculations of edgetone flow with forced longitudinal oscillations

Numerical calculations were carried out to study the effect of forced, symmetric, longitudinal flow oscillations on the inherent, strongly antisymmetrical oscillations of a previously studied edgetone flow at a Reynolds number of 450. The flow consists of a two-dimensional jet issuing from a nozzle and impinging on a body with a wedge-shaped leading edge. The flow is assumed to be incompressible, laminar and two-dimensional, and a finite-difference vorticity/stream-function formulation of the Navier-Stokes equations is employed. Three cases were considered with various combinations of forcing frequency and amplitude. It was found that for the two cases with large forcing amplitudes, the naturally dominant flow frequencies lock-in to the forcing frequency and its harmonics. In the third case the forcing amplitude was smaller and lock-in was not observed but the forced oscillations still had a significant impact on the flow. Mode competition between symmetric and antisymmetric modes is discussed for the three cases along with the manner in which the jet vortical structure is altered as a function of time and space. Results for all three cases are presented in the form of computer drawn equivorticity lines and plots of frequency spectra for the jet oscillations and for the pressure on the wedge.

GSMAC-A new finite element method for unsteady incompressible viscous flow problems. 2nd report Flow in a square cavity at high Reynolds numbers.

The unsteady flowfields in a square cavity at high Reynolds numbers are investigated numerically by solving the unsteady Navier-Stokes equation through the GSMAC finite element method which includes two important concepts: orthogonal decomposition and cycle-to-cycle self adjustment. Unsteady flow simulations in terms of streamlines, isobars, equi-vorticity lines and equienergy lines are obtained at Reynolds numbers of 5000 and 104. The conclusions are as follows; (1) The influence of the spiral motion of the main vortex to the shapes and sizes of secondary vortex are explained. (2) It is observed from the results that the flows in a square cavity at a Reynolds number of 104 show turbulent-like behavior. (3) The GSMAC finite element method requires only the same amount of CPU time and storage as a standard finite difference method and shows the great capability of solving unsteady flow problems.

Numerical study on viscous shear flow past a circular cylinder using integro-differential formulation

Numerical solutions of the unsteady Navier-Stokes equations are presented for the two dimensional flow about a cylinder immersed in a simple shear flow by using the integrodifferential formulation. Selected numerical results for Reynolds numbers 40 and 80 shear parametersK=0.1 and 0.2 are presented and compared with finite-difference results. Plots of streamlines and equi-vorticity lines and time histories of lift, drag, stagnation point and separation angles are given. It was found that two separation bubbles exist periodically behind the lowvelocity side of the cylinder; a main bubble and a secondary one, while there is only one bubble behind the high-velocity side. Numerical results showed that the lift force acts in the direction towards the low-velocity side of the free stream and its magnitude varies periodically with the nondimensional time. In addition, the magnitude of the lift force is approximately proportional to the shear parameterK.

A numerical study on the blood flow in an artery with a rough surface. (1st report Steady flow)

The acting shear stress and pressure on the vascular wall are considered as being a source of vascular lession, and are investigated numerically from the hydrodynamic point of view. As a model of the blood flow in an artery with a rough surface, the rigid channel with roughness elements in the wall is considered, and blood is assumed to be Newtonian fluid. Using this model, the Navier-Stokes's momentum equation and the equation of continuity are solved by the finite difference method. The calculation is performed for the Reynolds numbers of 50, 100, 200, 400, 800, 1000 and 2000. And the shear stress at the wall, the stream lines and the equi-vorticity lines are presented.

The flow of a viscous incompressible fluid past a normal flat plate at low and intermediate Reynolds numbers: the wake

The Navier-Stokes equations are solved numerically for the steady separated flow past a normal flat plate for Reynolds numbers in the range 0.1 ≤ R ≤ 20. Eddy dimensions together with the position of the vortex centre are presented and compared with the few other estimates and predictions available. Streamlines and equivorticity lines are also given. The main result of interest is the extremely good comparison with experimental results over this range of Reynolds numbers. The method of solution is based on an artificial time-dependent procedure using a distorted time. Results are given only for the steady-state flow.