Vertical Symmetry Plane Research Articles

Kinematics of the deformation zone in plugless rolling of oil pipes

This article describes the process of plugless rolling of oil pipes. Correlations between caliber parameters and swaging were obtained. The mathematical model of the process of plugless rolling of pipes was made. Main parameters of the deformation zone were determined. A set of equations to determine the surface of the contact area of the round billet and the tool for two-roll and three-roll calibers was obtained. The velocity field of metal billets was presented. From the assumption that the axial component of the velocity vector changes according to the quadratic law and the metal entrance velocity in the deformation zone is connected with the exit velocity from the deformation zone by the reduction ratio, the dependence for the determination of this component of the velocity vector was obtained. The linear function of the radial component of the velocity vector was justified. From the relation of axial and radial components of the velocity vector by the tangent of the angle between the considered section and the vertical symmetry plane of the roll a dependence for the determination of the radial component of the velocity vector was defined. The third component of the velocity vector was determined subject to the incompressibility condition, which was defined for three-roll and two-roll calibers. Based on the determined components of the velocity vector, the components of the strain velocity tensor were defined for three-roll and two-roll calibers. The components of the strain velocity tensor were calculated for the case of rolling tubes in a three-roll reduction mill. The components of the strain velocity tensor were numerically estimated, and the components of the strain velocity tensor that may be neglected in engineering calculations were determined.

Maximum heat transfer reduction in a horizontal porous annular cavity induced by the attachment of symmetric radial baffles to the hot inner cylinder

A numerical model is developed to study the natural convection in a horizontal porous annulus with a heated inner cylinder and a colder outer cylinder. Sequential pairs of radial baffles are attached to the inner cylinder and arranged symmetrically relative to the vertical plane of symmetry. A numerical procedure based on the finite volume method is adopted to solve the system of coupled 2D Darcy–Brinkman–Boussinesq and energy equations. Numerical results show that the insertion of medium-sized baffles modifies the unicellular flow to a co-rotating multicellular flow and induces maximal heat transfer reduction when the baffles are located at appropriate angles measured from the bottom of the inner cylinder. The optimal conditions for maximal heat transfer reduction are determined as a function of the number, size and position of the pairs of baffles for the betterment of thermal insulation of a horizontal pipe. The best improvement for thermal insulation efficiency is obtained with a successive insertion of three pairs of baffles, wherein the maximal reduction of heat transfer can reach 12% as compared to the bare case without baffles. Subsequent insertion of four pairs of baffles generates Rayleigh–Bénard thermal instabilities in the upper part of the porous annulus, which is unfavorable for purposes of thermal insulation.

Effects of an upstream tetrahedron on the circular cylinder–flat plate juncture flow

A technique of installing a tetrahedron at the upstream corner of the circular cylinder–flat plate juncture is developed to control the characteristic horseshoe vortices appearing in the natural juncture flow. The Reynolds numbers based on the cylinder diameter are within the range of 500–2900. The flow patterns and time-averaged velocity fields in the vertical symmetry plane and a horizontal plane near the flat plate of the natural and tetrahedron-controlled juncture flows are examined by using the laser-assisted particle flow visualization method and particle image velocimetry in a towing water tank. The flow approaching the circular cylinder–flat plate juncture can induce a characteristic horseshoe vortical flow consisting of a single vortex, dual vortex, or triple vortex. These horseshoe vortices appearing in the natural case may be changed to a characteristic mode of vortical flow, reverse flow, or forward flow when a tetrahedron is installed at the upstream corner of the juncture. The appearance of the vortical flow, reverse flow, or forward flow mode depends on the geometric parameters of normalized axial length, expansion angle, and tilt angle as well as the flow parameter of the Reynolds number. The vortical flow mode appears at small axial length of tetrahedron. The forward flow mode appears at the large axial length of tetrahedron. When the forward flow mode appears, the boundary-layer upstream of the circular cylinder does not separate. Therefore, the horseshoe vortices induced in the natural juncture flow disappear. The data bank consists of the design parameters of axial length, tilt angle, and expansion angle of the tetrahedron, which is provided as a figure.

3D face recognition based on pose and expression invariant alignment

In this paper we present a novel pose and expression invariant approach for 3D face registration based on intrinsic coordinate system characterized by nose tip, horizontal nose plane and vertical symmetry plane of the face. It is observed that distance of nose tip from 3D scanner is reduced after pose correction which is presented as a quantifying heuristic for proposed registration scheme. In addition, motivated by the fact that a single classifier cannot be generally efficient against all face regions, a two tier ensemble classifier based 3D face recognition approach is presented which employs Principal Component Analysis (PCA) for feature extraction and Mahalanobis Cosine (MahCos) matching score for classification of facial regions with weighted Borda Count (WBC) based combination and a re-ranking stage. The performance of proposed approach is corroborated by extensive experiments performed on two databases: GavabDB and FRGC v2.0, confirming effectiveness of fusion strategies to improve performance.

ABOUT THE EVALUATION OF THE LONGITUDINAL FORCES LEVEL EFFECTING THE TRACK DISPLACEMENT AT TRANSIENT MODES OF TRAIN MOVEMENT

Purpose. Study the transient modes effect of movement on the track displacement for the freight train safety control is supposed in this paper. For this it is necessary to investigate the longitudinal dynamics of a train on the track displacement. Simultaneously to assess the longitudinal forces level of a track and rolling stock interaction. Methodology. The level of the longitudinal forces, effecting the track displacement, was evaluated using mathematical modeling of longitudinal vibrations of the trains at transient modes of motion caused by braking. It was considered that each train vehicle consists of a body (solid) and the wheel sets, connected with the body by friction bearings (inelastic link). It was believed that during the movement of each train vehicle the vertical plane of its symmetry coincident with the vertical plane of symmetry of the assembled rails and sleepers. At simulation it was also supposed that in the process of translational motion of the vehicle body wheels make pure rolling along the rail without slipping on it. Findings. In the results of calculations the values of the longitudinal forces at different types of braking were obtained (it is regenerative braking and pneumatic one) under quasi-static and shock transients. For this various initial state of clearances in the inter-car connections up to beginning of transient was considered. The level of dynamic additives to longitudinal forces of interaction between wheel and rail that are substantially depending on vehicle accelerations was assessed. Originality. The transient regimes effect of trains movement caused by braking on the level of the longitudinal forces of track and rolling stock interaction was investigated. The longitudinal load of freight trains with regenerative and pneumatic braking was researched. The effect of the initial state of the train and different modes of braking on a dynamic additive to the longitudinal forces of the interaction between the track and rolling stock, which may effect the displacement of assembled rails and sleepers, was estimated. Practical value. The obtained results can be used to select rational modes of braking of freight trains, especially on lengthy down grade, from the positions prevent possible track displacement.

MODULATION EFFECT OF VIBRATION FREQUENCY OF AN UNSPRUNG HEAVY MACHINE UNDER THE VARIABLE ROAD ADHESION CONDITIONS

Mobile heavy machines as unsprung vehicles exhibit low dissipation ability, hence the ride even at low speeds may give rise to intensive vibration. Particularly dangerous situations occur when the road wheels break away from the road surface due to the ’galloping’ effect, being the result of excited vertical and angular vibration of the machine frame in the vertical plane of symmetry. That implies a major restriction on the ride velocity, which negatively impacts on the machine performance. Vibrations thus produced are mostly in the low-frequency range and hence energy dissipation in tyres will reduce the vibration intensity in a minor degree only. The motion of tired wheels will always involve some slipping. While investigating the feasibility of increasing the efficiency of the vibration reduction systems, one ought to take into account the variable adhesion of road wheels due to different dynamic loading acting on the vehicle axles during the ride. Observations of unsprung machines during the ride suggest the occurrence of self-excited vibration. Mobile machines constitute dynamic systems, which can be governed by nonlinear, sometimes non-stationary differential equations of motion. Their stability also depends on intensity of external vibrations. This study investigates the motion of unsprung mobile machines, taking into account the dynamic processes in the driving system under the conditions of the variable adhesion of road wheels. The model of interaction between a tired wheel and the terrain takes into account the relationship between the road wheel adhesion factor and the slipping action. Mathcad supported by Matlab-Simulink and in the frequency domain – simulations in the time domain. The purpose of the simulation procedure was to find the causes of the vibration modulation frequency and determine the conditions triggering the occurrence of self-excited vibrations. Simulations are supported by the analysis of motion stability.

Soundproofing Performance Evaluation of Panels Made of Fibers of Acrylonitrile Butadiene Styrene Copolymer (ABS)

Abstract Industrial development and intensive mechanization of all production processes, multiplication and diversification of transport, development of traffic flows in urban and extra-urban areas, all lead to the increased noise in terms of intensity, frequency, duration of exposure to noise. In these circumstances, a global interest goes to the monitoring and analysis of the road traffic noise maps and to the noise and vibration sources. Thus, several European and national directives on the road traffic noise and beyond have been introduced, including guides and noise assessment methods. The paper aims to present the acoustic behavior of composite panels made from chips resulted in the milling process of acrylonitrile butadiene styrene (ABS), in terms of sound proofing performance in connection with the factors which affect the sound insulation of noise level. The panels with the sizes of 300x300x15 up to 20 mm were produced by hot pressing the ABS chips, at a temperature of 100 - 110 °C for 10 minutes. The link between chips was achieved by melting point of acrylonitrile. The experimental tests investigated: the influence of the distance between source and the noise barrier formed by the composite material made from ABS; the influence of the noise level emitted by the source; the influence of the distance between the sound barrier and the receptors placed collinear with the source; the influence of the position of receptors against the lateral ends of the barrier; the influence of the barrier angle relative to vertical plane of symmetry. The research highlighted the acoustic performance of composites made from ABS waste, innovative products, which can be considered an ecological, technological and economic solution in obtaining the core of acoustic barrier structures. Increased thickness of ABS fiberboard or combined with other absorbent material can lead to the increase of sound absorption coefficient and to obtain superior acoustic performances.

A study of flow fields in step-down street canyons

A step-down street canyon is a street canyon in which the upwind building height ( $$H_{u}$$ ) is greater than the downwind building height ( $$H_{d}$$ ) ( $$H_{u}>H_{d}$$ ). Here, the effect of downwind building height and canyon width on the flow structure in isolated step-down canyons is investigated through wind-tunnel measurements. The measurements were acquired along the vertical symmetry plane of the model buildings using two-dimensional particle image velocimetry for normal approach flow. For the present study, $$H_{u}$$ was kept constant at $$120$$ mm, and $$H_{d}$$ was increased in increments of $$\approx $$ 0.08 $$H_{u}$$ , to span the range: $$0.08\le H_{d}/H_{u}\le 1$$ . The configuration $$H_{d}/H_{u} \approx 1$$ corresponds to a deep canyon. The footprints of the buildings were square, with the widths ( $$W$$ ) and lengths ( $$L$$ ) being, $$W (= L) \approx 32$$ mm. Four different street-canyon widths ( $$S$$ ) were considered, with $$S/W \approx 2.5, 2, 1.5, 1$$ . This resulted in a total of 48 test cases, with 12 cases for every street-canyon width. The flow topology in the near-wake of an isolated tall building ( $$H_{d}=0$$ ) is characterised by a bow-shaped structure comprising the vortex core, saddle point, and ground originating shear layer. For $$S/W \approx 2.5, 2$$ , and $$1.5$$ , increasing the downwind building height from $$H_{d}/H_{u} \approx 0.08$$ to $$1$$ resulted in the in-canyon flow structure transitioning from wake dominated to deep canyon wake interference regimes. Similar increase of the downwind building height for $$S/W \approx 1$$ resulted in the flow structure transitioning from wake dominated to deep canyon skimming flow regime. The results indicate that in step-down canyons formed by tall and slender buildings, momentum transport into and out of the canyon around the building sidewalls plays a crucial role in the determining the overall flow patterns in the canyon.

Investigation of a laboratory mechanical system with fibre and pulley

Experimental measurements focused on the investigation of a fibre behaviour are performed on an assembled weigh-fibre-pulley-drive mechanical system. The fibre is driven with one drive and is led over a pulley. On its other end there is a prism-shaped steel weight, which moves in prismatic linkage on an inclined plane. The angle of inclination of the inclined plane could be changed. The position of the weight is asymmetric with respect to the vertical plane of drive-pulley symmetry. Drive exciting signals can be of a rectangular, a trapezoidal and a quasi-sinusoidal shape and there is a possibility of variation of a signal rate. Dynamic responses of the weight and the fibre (time histories of the weight position, of the drive position and of the force acting in the fibre) are measured. The same system is numerically investigated by means of a multibody model. The influence of the model parameters on the coincidence of results of experimental measurements and the simulations results are evaluated. The simulations aim is to create a phenomenological model of a fibre, which will be utilizable in fibre modelling in the case of more complicated mechanical or mechatronic systems.

Vertical vibrations of composite bridge/track structure/high-speed train systems. Part 2: Physical and mathematical modelling

A theory of one-dimensional physical and mathematical modelling of the composite (steel-concrete) bridge/track structure/highspeed train system is developed including viscoelastic suspensions of rail-vehicles with two two-axle bogies each, non-linear Hertz contact stiffness and one-sided contact between wheel sets and rails, the viscoelastic and inertia features of the bridge, the viscoelastic track structure on and beyond the bridge, approach slabs, and random vertical track irregularities. Compared to the state-of-the-art, the physical model developed in the study accurately reproduces dynamic processes in the considered system. Division of the system into the natural subsystems, a method of formulation of the equations of motion partly in implicit form and the finite element method are applied. Vibrations in the vertical plane of symmetry are described by more than nine matrix equations of motion with constant coefficients. Couplings and non-linearity are hidden in the generalized load vectors. The equations of motion are integrated using the implicit Newmark average acceleration method with linear extrapolation of the interactions between the subsystems.

Smoke visualization under heat transfer by free convection on horizontal cylinder surface

The method of visualizing a dynamic field in conditions of free convection by introducing a laminar smoke stream into the area which is vertically lower than the existing thermal and dynamic boundary layers has been proposed and examined in the paper. The suggested method is simple, obvious, cheap and fast enough, that allows advising it for applying in the study of free-convective heat transfer problems using air as a heat-transfer agent. As a result of the conducted research, a set of assumptions, giving the basis to calculations and methods of analyzing experiment results has been demonstrated. In particular, among them there can be singled out the following: the laminarity of a heat-transfer agent motion mode, the laminarity of a thermal tracing and boundaries of its sustainable occurrence over a hot surface, the growth of boundary layer thickness in the cylinder stern, symmetry of washing cylinders from different sides from the vertical plane of symmetry, and others. Juxtaposing the results with the data, represented in the literature, shows a high correspondence.

Modal Stability Control of a Wheeled Heavy Machine

Mobile heavy machines as unsprung vehicles exhibit low dissipation ability, hence the ride even at low speeds may give rise to intensive vibration, or even to the galloping effect due to enhanced vertical and angular vibration causing the road wheels to break away from the roadway. As these are mostly low-frequency vibrations, energy dissipation in wheel tires will reduce the vibration intensity in a minor degree only. As a result, the speed of a travel has to be reduced, which adversely affects the cost-efficiency of those machines. Observations of an unsprung machine during its forward ride lead us to formulate the hypothesis of reduced machines stability while it moves in the vertical plane of symmetry. The work [2] confirms the view that the vertical motion of an unsprung heavy machine while it moves forward is implemented under the condition of stability limit, mostly due to stiffness of the dynamic system in the horizontal direction. Mobile machines are dynamic systems, governed by nonlinear and often non-stationary differential equations of motion. Their stability is also dependent on the intensity of acting excitations.

Investigation of the Flow Structure in Step-Up Street Canyons—Mean Flow and Turbulence Statistics

A step-up street canyon is a characteristic urban element composed of two buildings in which the height of the upwind building ( $$H_\mathrm{u}$$ ) is less than the height of the downwind building ( $$H_\mathrm{d}$$ ). Here, the effect of canyon geometry on the flow structure in isolated step-up street canyons is investigated through isothermal wind-tunnel measurements. The measurements were acquired along the vertical symmetry plane of model buildings using two-dimensional particle image velocimetry (PIV) for normal approach flow. The building-height ratios considered were: $$H_\mathrm{d}/ H_\mathrm{u} \approx 3$$ , and $$H_\mathrm{d}/ H_\mathrm{u} \approx 1.67$$ . For each building-height ratio, the along-wind lengths (L) of the upwind and downwind buildings, and the street-canyon width (S) were kept constant, with $$L \approx S$$ . The cross-wind widths (W) of the upwind and downwind buildings were varied uniformly from $$W/S \approx 1$$ through $$W/S \approx 4$$ , in increments of $$W/S \approx 1$$ . The objective of the work was to characterize the changes in the flow structure in step-up canyons as a function of W/S, for fixed L, S, and $$H_\mathrm{d}/H_\mathrm{u}$$ values. The results indicate that the in-canyon flow structure does not vary significantly for $$H_\mathrm{d}/H_\mathrm{u} \approx 3$$ for the W/S values considered. Qualitatively, for $$H_\mathrm{d}/H_\mathrm{u} \approx 3$$ , the upwind building behaves as an obstacle in the upwind cavity of the downwind building. In contrast, the flow patterns observed for the $$H_\mathrm{d}/H_\mathrm{u} \approx 1.67$$ configurations are unique and counter-intuitive, and depend strongly on building width (W/S). For $$W/S \approx 1$$ and $$W/S \approx 2$$ , the effect of lateral flow into the canyon is so prominent that even the mean flow patterns are highly ambiguous. For $$W/S \approx 3$$ and 4, the flow along the vertical symmetry plane is more shielded from the lateral flow, and hence a stable counter-rotating vortex pair is observed in the canyon. In addition to these qualitative features, a quantitative analysis of the mean flow field and turbulence stress field is presented.

Field Measurements of Wind Turbine Wakes with Lidars

AbstractField measurements of the wake flow produced from a 2-MW Enercon E-70 wind turbine were performed using three scanning Doppler wind lidars. A GPS-based technique was used to determine the position of the wind turbine and the wind lidar locations, as well as the direction of the laser beams. The lidars used in this study are characterized by a high spatial resolution of 18 m, which allows the detailed characterization of the wind turbine wake. Two-dimensional measurements of wind speed were carried out by scanning a single lidar over the vertical symmetry plane of the wake. The mean axial velocity field was then retrieved by averaging 2D scans performed consecutively. To investigate wake turbulence, single lidar measurements were performed by staring the laser beam at fixed directions and using the maximum sampling frequency. From these tests, peaks in the velocity variance are detected within the wake in correspondence of the turbine top tip height; this enhanced turbulence could represent a source of dangerous fatigue loads for downstream turbines. The spectral density of the measured velocity fluctuations shows a clear inertial-range scaling behavior. Then, simultaneous measurements with two lidars were performed in order to characterize both the axial and the vertical velocity components. For this setup, the two velocity components were retrieved only for measurement points for which the two laser beams crossed nearly at a right angle. Statistics were computed over the sample set for both velocity components, and they showed strong flow fluctuations in the near-wake region at turbine top tip height, with a turbulence intensity of about 30%.

SPAN LENGTH INFLUENCE ON DYNAMIC RESPONSE OF SELECTED BRIDGE UNDER HIGH-SPEED TRAIN

The paper presents a methodology of finite element (FE) modelling and simulation of the bridge – track – moving high-speed train system using CAE systems. Two composite (reinforce-concrete – steel) bridges were considered. The span length was equal to 15 and 21 meters, respectively. Bridges selected for the study belong to the proposed series of bridges with the span length of 15 to 27 meters stepped by 3 meters. Full symmetry of the bridges was assumed. RC platform was homogenized since the rebars were distributed quasi-uniformly in the specified platform sections. The FE model of a bridge superstructure consisted of 4-node shell elements (main beams) and 8-node 48 DOF solid elements (reinforced concrete platform). RAIL_TRACK and RAIL_TRAIN LS-DYNA’s modules were applied for simulating the moving train – track interaction. Ballasted track with the rectilinear rail-line axis was taken into consideration. German ICE-3 train running at velocity of 200–300 km/h was selected as a representative for the study. All mass components of the train FE model were treated as rigid bodies. Symmetric vibrations of the train units were assumed with respect to the main longitudinal vertical plane of symmetry of the system. Nodal displacement and longitudinal normal stress in shell elements were registered during the FE analysis. The results were depicted in the form of time histories for selected velocities. In addition, extreme values of vertical deflections and normal stress were compiled and presented a function of train velocity. It allowed to assess the dynamic response of the bridge depending on its span length. Contours of resultant displacement for the RC platform was also presented.

An Alternative Three-phase Transmission Lines Model in Phase Domain

The objective of this paper is to show an alternative model of a non-transposed three-phase transmission line with a vertical symmetry plane in phase domain. Due the line physical characteristics, it can be represented by a system consisting of a single?phase and a two-phase line. In this system, the equations describing the behavior of the values in single-phase line terminals are known, while the equations for two-phase line to be obtained. Using a transformation matrix written explicitly according to three-phase line parameters, it is possible to obtain the currents and voltages in phase domain of two-phase line. Then, modal values of three-phase line are converted into phase domain and thus obtain the analytical model for this line. To verify the performance of this model, it was used to simulate the energization of a 440 kV three-phase line and the results were compared to results obtained using a classical model.

Thermal stratification in an internal combustion engine due to wall heat transfer measured by laser-induced fluorescence

Laser-induced fluorescence of toluene was used to image spatial fluctuations of gas temperature in an optically accessible engine. These thermal inhomogeneities develop due to wall heat-transfer and convection during the compression stroke. They are known to be important for slowing heat release when operating the engine in compression auto-ignition (CAI) mode. The engine had a pent-roof four-valve head typical for automotive spark-ignited engines and a flat-top piston with a window. Measurements were performed in the central vertical symmetry plane of the cylinder with the engine motored and fed with nitrogen.Toluene was seeded homogeneously into the intake gas. Fluorescence was excited at 248nm and detected spectrally integrated. Toluene fluorescence decreases strongly with increasing temperature. Estimating the absolute in-cylinder temperature from isentropic compression along the measured pressure trace, we found the magnitude of this decrease in the engine to be consistent with literature data from heated flow-cell measurements. This calibration allowed for determination of the spatial fluctuations in temperature against the multi-cycle average temperature. Precision error, mostly from laser mode fluctuations, was between 1.6 and 4.0K depending on crank angle.The results show that the temperature–fluctuation field transitions during the compression stroke from small, evenly distributed inhomogeneities to much greater fluctuations, mostly localized near walls, but convecting into the cylinder center very late in the stroke. There are subtle differences in the spatial structure of the near-wall fluctuation field between cylinder head and piston top. Compiled from the entire imaged area, at top dead-center the standard deviation of the fluctuations was 1.9% of the temperature differential between gas phase and wall, consistent with corresponding literature data.

Verification and Validation of a thermal stratification experiment CFD simulation

Abstract Thermal stratification and striping are observed in many piping systems including those of nuclear power plants. Periodic occurrences of these thermal transients lead to fatigue and may induce undesirable failures and deformations to the piping. The Thermal Hydraulic Laboratory of the Centro de Desenvolvimento da Tecnologia Nuclear/Comissao Nacional de Energia Nuclear (CDTN/CNEN) conducts an experimental and numerical project simulating the thermal stratified flows in piping systems of pressurized water reactors (PWR) to obtain some understanding on these phenomena. Experiments were carried out in a test section simulating the steam generator injection nozzle of a PWR. A numerical simulation of one experiment was performed with the commercial finite volume Computational Fluid Dynamic code CFX 13.0. A vertical symmetry plane along the pipe was adopted to reduce the geometry in one half, reducing mesh element size and minimizing processing time. The RANS two equations RNG k–ɛ turbulence model with scalable wall function and the full buoyancy model were used in the simulation. In order to properly evaluate the numerical model it was performed a Verification and Validation (V&V) process. Numerical uncertainties due to mesh refinement and time step were evaluated. This validation process showed the great importance of a proper quantitative evaluation of numerical results. In past studies qualitative evaluations were considered enough and numerical results like the one presented here could be considered satisfactory for the prediction of thermal stratified flows. However, with the present V&V study it was possible to identify objectively the strengths and weaknesses of the model.

Determination and Stability Analysis of Equilibrium Configurations of Objects Suspended From Multiple Aerial Robots

This work addresses the problem for determining the position and orientation of objects suspended with n cables from n aerial robots. This is actually the direct kinematics problem of the 3D cable system. First, the problem is formulated based on the static equilibrium condition. Then, an analytic algorithm based on resultant elimination is proposed to determine all possible equilibrium configurations of the planar 4-bar linkage. As the nonlinear system can be reduced to a polynomial equation in one unknown with a degree 8, this algorithm is more efficient than numerical search algorithms. Considering that the motion of a 3D cable system in its vertical planes of symmetry can be regarded as the motion of an equivalent planar 4-bar linkage, the proposed algorithm is used to solve the direct kinematics problem of objects suspended from multiple aerial robots. Case studies with three to six robots are conducted for demonstration. Then, approaches for stability analysis based on Hessian matrix are developed, and the stability of obtained equilibrium configurations is analyzed. Finally, experiments are conducted for validation.

On the existence and evolution of a spanwise vortex in laminar shallow water dipoles

The present work investigates the existence and evolution of a spanwise vortex at the front of shallow dipolar vortices. The vortex dipoles are experimentally generated using a double flap apparatus. Particle image velocimetry measurements are performed in a horizontal plane and in the vertical symmetry plane of the flow. The dynamics of such vortical structures is investigated through a parametric study in which both the Reynolds number Re=U0D0/ν∈[90,470] and the aspect ratio α=h/D0∈[0.075,0.7], associated with the shallowness of the flow, are varied, where U0 is the initial velocity of the vortex dipole, D0 is the initial diameter, h is the water depth, and ν is the kinematic viscosity of the fluid. The present experiments confirm the numerical results obtained in a companion paper by Duran-Matute et al. [Phys. Fluids 22, 116606 (2010)], namely that the flow remains quasi parallel with negligible vertical motions below a critical value of the parameter α2Re. By contrast, for large values of α2Re and α≲0.6, a three-dimensional regime is observed in the shape of an intense spanwise vortex generated at the front of the dipole. The present study reveals that the early-time motion and dynamics of the spanwise vortex do not scale on the unique parameter α2Re but is strongly influenced by both the aspect ratio and the Reynolds number. A mechanism for the generation of the spanwise vortex is proposed. For α≳0.6, a third regime is observed, where the spanwise vortex is replaced by a vorticity tongue.