Vortex Ring Model Research Articles

Shock wave interaction with a naturally generated vortex ring

The vortex ring is usually analytically modeled in computational studies involving vortex ring and shock wave interaction. However, incompressibility assumptions of the isolated vortex ring model mostly limits its applicability to the lower Mach number regime. To overcome this limitation, a compressible vortex ring is generated using a pulse jet exiting a circular nozzle, and its interaction with a shock wave is analyzed. The shock–vortex interaction and resulting sound wave generation is simulated by solving the compressible axisymmetric Navier–Stokes equations. A characteristics-based filter is used to isolate acoustic and hydrodynamic disturbances in the flow field. A reconfigured computational domain is used to negate the aftereffects of the nozzle on the flow field. The strong interaction of compressible vortex rings at higher Mach numbers with different shock wave Mach numbers in the flow field is studied including resultant acoustic wave generation.

Comparison of Vortex Cut and Vortex Ring Models for Toroidal Bubble Dynamics in Underwater Explosions

The jet impact from a collapsing bubble is an important mechanism of structural damage in underwater explosions and cavitation erosion. The Boundary Integral Method (BIM) is widely used to simulate nonspherical bubble dynamic behaviors due to its high accuracy and efficiency. However, conventional BIM cannot simulate toroidal bubble dynamics, as the flow field transforms from single-connected into double-connected. To overcome this problem, vortex cut and vortex ring models can be used to handle the discontinuous potential on the toroidal bubble surface. In this work, we compare these two models applied to toroidal bubble dynamics in a free field and near a rigid wall in terms of bubble profile, bubble gas pressure, and dynamic pressure induced by the bubble, etc. Our results show that the two models produce comparable outcomes with a sufficient number of nodes in each. In the axisymmetric case, the vortex cut model is more efficient than the vortex ring model. Moreover, we found that both models improve in self-consistency as the number of bubble surface elements (N) increases, with N=300 representing an optimal value. Our findings provide insights into the numerical study of toroidal bubble dynamics, which can enhance the selection and application of numerical models in research and engineering applications.

Dynamic inflow model for a floating horizontal axis wind turbine in surge motion

Abstract. Floating offshore wind turbines may experience large surge motions, which can cause blade–vortex interaction if they are similar to or faster than the local wind speed. Previous research hypothesized that this blade–vortex interaction phenomenon represented a turbulent wake state or even a vortex ring state, rendering the actuator disc momentum theory and the blade element momentum theory invalid. This hypothesis is challenged, and we show that the actuator disc momentum theory is valid and accurate in predicting the induction at the actuator in surge, even for large and fast motions. To accomplish this, we develop a dynamic inflow model that simulates the vorticity–velocity system and the effect of motion. The model's predictions are compared to other authors' results, a semi-free-wake vortex ring model, other dynamic inflow models, and CFD simulations of an actuator disc in surge. The results show that surge motion and rotor–wake interaction do not result in a turbulent wake or vortex ring state and that the application of actuator disc momentum theory and blade element momentum theory is valid and accurate when applied correctly in an inertial reference frame. In all cases, the results show excellent agreement with the higher-fidelity simulations. The proposed dynamic inflow model includes a modified Glauert correction for highly loaded streamtubes and is accurate and simple enough to be easily implemented in most blade element momentum models.

A Modified Free Wake Vortex Ring Method for Horizontal-Axis Wind Turbines

A modified free-wake vortex ring model is proposed to compute the dynamics of a floating horizontal-axis wind turbine, which is divided into two parts. The near wake model uses a blade bound vortex model and trailed vortex model, which is developed based on vortex filament method with straight lifting lines assumption. By contrast, the far wake model is based on the vortex ring method. The proposed model is a good compromise between accuracy and computational cost, for example when compared with more complex vortex methods. The present model is used to assess the influence of floating platform motions on the performance of a horizontal-axis wind turbine rotor. The results are validated on the 5 MW NREL rotor and compared with other aerodynamic models for the same rotor subjected to different platform motions. The results show that the proposed method is reliable. In addition, the proposed method is less time consuming and has similar accuracy when comparing with more advanced vortex based methods.

Numerical investigation of toroidal bubble dynamics in a compressible fluid based on boundary integral method

The external liquid compressibility cannot be ignored because the speed of the bubble jet emerging at the end of bubble collapse reaches hundreds of meters per second. Additionally, when the bubble jet penetrates the surface of a bubble, a toroidal bubble forms and the singly connected flow domain changes to a doubly connected topology. As the Biot–Savart law is based on the assumption of incompressibility, the vortex ring model is very difficult to extend to compressible fluids. This paper describes the use of the boundary integral method to establish a numerical model of a toroidal bubble, considering the external liquid compressibility and the internal gas wave effect. A cut is introduced into the fluid domain so that it can be considered as singly connected, with the discontinuity of velocity across this cut equal to the circulation of the flow. Furthermore, the initial bubble condition is calculated by the volume acceleration model. The numerical model is validated through comparisons with experimental data from underwater explosions. The numerical results are found to correlate well with the experimental results. Then, the influence of buoyancy parameters and the internal gas wave effect on toroidal bubble dynamics in a gravitational field is investigated.

Bubble-sphere interaction beneath a free surface

The bubble-sphere interaction beneath a free surface is investigated experimentally and numerically, which is associated with underwater explosion and cavitation. We use the electric discharge technique to generate an oscillating bubble that is set above a movable sphere and below a free surface. The transient fluid-structure interaction (FSI) is recorded by high-speed imaging. We identify three distinctive bubble jet impact patterns, i.e., (i) jet impacts on the sphere top with water layer between them; (ii) the bubble partly envelops the sphere at first and then the jet impacts on the sphere top without water layer; (iii) the jet impacts on an annular ring, splitting the bubble into a toroidal bubble and a new singly-connected bubble. Such transient FSI events are modeled by the boundary integral method and the foregoing bubble splitting is modeled by an improved vortex ring model. The bubble, free surface, and sphere dynamics in experiments are found to be captured extremely well by numerical simulations. The bubble dynamic behaviors are associated with three dimensionless parameters and a systematic study of these parameters are provided. Lastly, the two types of cavitation phenomena on sphere surface are discussed and the detailed physical mechanisms are analyzed.

The dynamic wake of an actuator disc undergoing transient load: A numerical and experimental study

Abstract The currently most used theory for rotor aerodynamics — Blade Element Momentum is based on the assumption of stationary wake conditions. However, an unsteady rotor loading results in an unsteady wake flow field. This work aims to study the impact of an unsteady actuator disc on the wake flow field using a free wake vortex ring model. The numerical results are compared to a wind tunnel measurement, where the wake flow of an actuator disc model undergoing transient load was obtained. The numerical results complement the experimental work while providing information such as the vorticity field and contributions from different vortex elements. The velocity at different locations is compared between the experimental and numerical results. The observed velocity peaks in the experimental results are also observed in the numerical results. A steeper ramp time results in a steeper velocity transient slope, and in turn in a larger amplitude of peak values. It is revealed that the rolling-up processes is the main cause for the velocity difference at various locations and in the three cases by decomposing velocity induced by different vortex element.

Numerical study on motion of the air-gun bubble based on boundary integral method

To simulate the motion of the non-spherical bubble generated by air gun, boundary integral method is used in conjunction with the basic theory of the air gun. The time length of the bubble aeration τ (i.e., the close time of the air gun), the utilization efficiency of the gas inside the air gun chamber η, the heat transfer coefficient α and other key parameters of the air gun are employed to control the motion of the bubble. In the simulation of a single air gun bubble with buoyancy, the vortex ring model is used to simulate the motion of the toroidal bubble after jet impacts, and it is extended to the multiple vortex rings in the simulation of the interaction between two clustered air gun bubbles. Some new findings are summarized as follows: (i) Under the effect of the buoyancy, the shape of the bubble becomes non-spherical at later phase of bubble collapse, but the deformation seems to have little effect on the far-field pressure. (ii) For the clustered air gun bubbles, the primary-to-bubble pulse ratio will reach a maximum when the standoff distance d is about 1.6 times maximum radius of a single bubble. (iii) Primary-to-bubble pulse ratio is relative large at a small firing depth of the air gun H (the vertical distance from the bubble center to the free surface).

Lift calculations based on accepted wake models for animal flight are inconsistent and sensitive to vortex dynamics

There are three common methods for calculating the lift generated by a flying animal based on the measured airflow in the wake. However, these methods might not be accurate according to computational and robot-based studies of flapping wings. Here we test this hypothesis for the first time for a slowly flying Pacific parrotlet in still air using stereo particle image velocimetry recorded at 1000 Hz. The bird was trained to fly between two perches through a laser sheet wearing laser safety goggles. We found that the wingtip vortices generated during mid-downstroke advected down and broke up quickly, contradicting the frozen turbulence hypothesis typically assumed in animal flight experiments. The quasi-steady lift at mid-downstroke was estimated based on the velocity field by applying the widely used Kutta–Joukowski theorem, vortex ring model, and actuator disk model. The calculated lift was found to be sensitive to the applied model and its different parameters, including vortex span and distance between the bird and laser sheet—rendering these three accepted ways of calculating weight support inconsistent. The three models predict different aerodynamic force values mid-downstroke compared to independent direct measurements with an aerodynamic force platform that we had available for the same species flying over a similar distance. Whereas the lift predictions of the Kutta–Joukowski theorem and the vortex ring model stayed relatively constant despite vortex breakdown, their values were too low. In contrast, the actuator disk model predicted lift reasonably accurately before vortex breakdown, but predicted almost no lift during and after vortex breakdown. Some of these limitations might be better understood, and partially reconciled, if future animal flight studies report lift calculations based on all three quasi-steady lift models instead. This would also enable much needed meta studies of animal flight to derive bioinspired design principles for quasi-steady lift generation with flapping wings.

The motion of a 3D toroidal bubble and its interaction with a free surface near an inclined boundary

The numerical modelling of 3D toroidal bubble dynamics is a challenging problem due to the complex topological transition of the flow domain, and physical and numerical instabilities, associated with jet penetration through the bubble. In this paper, this phenomenon is modelled using the boundary integral method (BIM) coupled with a vortex ring model. We implement a new impact model consisting of the refined local mesh near the impact location immediately before and after impact, and a surgical cut at a high resolution forming a smooth hole for the transition from a singly connected to doubly connected form. This enables a smooth transition from a singly connected bubble to a toroidal bubble. The potential due to a vortex ring is reduced to the line integral along the vortex ring. A new mesh density control technique is described to update the bubble and free surfaces, which provides a high mesh quality of the surfaces with the mesh density in terms of the curvature distribution of the surface. The pressure distribution in the flow field is calculated by using the Bernoulli equation, where the partial derivative of the velocity potential in time is calculated using the BIM model to avoid numerical instabilities. Experiments are carried out for the interaction of a spark generated bubble with a free surface near a boundary, which is captured by using a high speed camera. Our numerical results agree well with the experimental images, for the bubble and free surface shapes for both before and after jet impact. New results are analyzed for the interaction of a toroidal bubble with a free surface near a vertical boundary and a sloping boundary, at both negative and positive angles to the vertical, without and with buoyancy, respectively. After jet impact, the bubble becomes a bubble ring, whose cross section is much thinner at the distal side from the boundary. It subsequently breaks into a crescent shaped bubble. The free surface displays singular features at its intersection with an inclined boundary.

Validation of engineering dynamic inflow models by experimental and numerical approaches

The state of the art engineering dynamic inflow models of Pitt-Peters, Øye and ECN have been used to correct Blade Element Momentum theory for unsteady load prediction of a wind turbine for two decades. However, their accuracy is unknown. This paper is to benchmark the performance of these engineering models by experimental and numerical methods. The experimental load and flow measurements of an unsteady actuator disc were performed in the Open Jet Facility at Delft University of Technology. The unsteady load was generated by a ramp-type variation of porosity of the disc. A Reynolds Averaged Navier-Stokes (RANS) model, a Free Wake Vortex Ring (FWVR) model and a Vortex Tube Model (VTM) simulate the same transient load changes. The velocity field obtained from the experimental and numerical methods are compared with the engineering dynamic inflow models. Velocity comparison aft the disc between the experimental and numerical methods shows the numerical models of RANS and FWVR model are capable to predict the velocity transient behaviour during transient disc loading. Velocity comparison at the disc between the engineering models and the numerical methods further shows that the engineering models predict much faster velocity decay, which implies the need for more advanced or better tuned dynamic inflow models.

Verifying the Blade Element Momentum Method in unsteady, radially varied, axisymmetric loading using a vortex ring model

AbstractAlthough the Blade Element Momentum method has been derived for the steady conditions, it is used for unsteady conditions by using corrections of engineering dynamic inflow models. Its applicability in these cases is not yet fully verified. In this paper, the validity of the assumptions of quasi‐steady state and annuli independence of the blade element momentum theory for unsteady, radially varied, axi‐symmetric load cases is investigated. Firstly, a free wake model that combines a vortex ring model with a semi‐infinite cylindrical vortex tube was developed and applied to an actuator disc in three load cases: (i) steady uniform and radially varied, (ii) two types of unsteady uniform load and (iii) unsteady radially varied load. Results from the three cases were compared with Momentum Theory and also with two widely used engineering dynamic inflow models—the Pitt‐Peters and the Øye for the unsteady load cases. For unsteady load, the free wake vortex ring model predicts different hysteresis loops of the velocity at the disc or local annuli, and different aerodynamic work from the engineering dynamic inflow models. Given that the free wake vortex ring model is more physically representative, the results indicate that the engineering dynamic inflow models should be improved for unsteady loaded rotor, especially for radially varied unsteady loads. © 2016 The Authors. Wind Energy Published by John Wiley & Sons, Ltd.

CHARACTERISTICS OF THE JET IMPACT DURING THE INTERACTION BETWEEN A BUBBLE AND A WALL

The dynamics of a toroidal bubble splitting near a rigid wall in an inviscid incompressible fluid is studied in this paper. The boundary integral method is adopted to simulate the bubble motion. After the jet impact, the vortex ring model is used to handle the discontinued potential of the toroidal bubble. When the toroidal bubble is splitting, topology changes are made tear the bubble apart. Then, the vortex ring model is extended to multiple vortex rings to simulate the interaction between two toroidal bubbles. A typical case is discussed in this study. Besides, the velocity fields and pressure contours surrounding the bubble are used to illustrate the numerical results. An annular high pressure region is generated at the splitting location, and the maximum pressure may be much higher than the jet impact. More splits may happen after the first split.

Experimental and numerical study on the dynamic pressure caused by the bubble jet

The high speed liquid jet is an important mechanism of damage to hydraulic machinery by cavitation bubbles, as well as damage to vessels by underwater explosion bubble. In this study, the bubble motion near a wall and the pressure impulse are investigated through experimental and numerical methods. In the experiment, the bubble is generated by the electric discharge, and a high speed camera is used to capture the bubble motion. Numerical studies are conducted using the boundary element method, and the vortex ring model is adopted to deal with the discontinued potential of the toroidal bubble. Calculated results show excellent agreement with experimental observations. Meanwhile, the dynamic pressure caused by the bubble in the flow domain is calculated by an auxiliary function, which improves the accuracy of the results. A highly localized pressure region will be generated on the wall by the bubble jet. The optimal stand-off parameter (the ratio of the distance the bubble center at inception from the wall to the maximum bubble radius) for a most damaging jet formation is around 0.9.

Optimal reconstruction of inviscid vortices

We address the question of constructing simple inviscid vortex models that optimally approximate realistic flows as solutions of an inverse problem. Assuming the model to be incompressible, inviscid and stationary in the frame of reference moving with the vortex, the ‘structure’ of the vortex is uniquely characterized by the functional relation between the streamfunction and vorticity. It is demonstrated how the inverse problem of reconstructing this functional relation from data can be framed as an optimization problem which can be efficiently solved using variational techniques. In contrast to earlier studies, the vorticity function defining the streamfunction–vorticity relation is reconstructed in the continuous setting subject to a minimum number of assumptions. To focus attention, we consider flows in three-dimensional axisymmetric geometry with vortex rings. To validate our approach, a test case involving Hill's vortex is presented in which a very good reconstruction is obtained. In the second example, we construct an optimal inviscid vortex model for a realistic flow in which a more accurate vorticity function is obtained than produced through an empirical fit. When compared with available theoretical vortex-ring models, our approach has the advantage of offering a good representation of both the vortex structure and its integral characteristics.

Modelling of confined vortex rings

This paper is focused on the investigation of vortex rings evolving in a tube. A new theoretical model for a confined axisymmetric vortex ring is developed. The predictions of this model are shown to be in agreement with available experimental data and numerical simulations. The model combines the viscous vortex ring model, developed by Kaplanski & Rudi (Phys. Fluids, vol. 17, 2005, 087101), with Brasseur’s (PhD thesis, Stanford University) approach to deriving a wall-induced streamfunction correction. Using the power-law assumption for the time variation of the viscous length of the vortex ring, the time variations of the main integral characteristics, circulation, kinetic energy and translational velocity are obtained. Direct numerical simulation (DNS) is used to test the range of applicability of the model and to investigate new physical features of confined vortex rings recently reported in the experimental study by Stewart et al. (Exp. Fluids, vol. 53, 2012, pp. 163–171). The model is shown to lead to a very good approximation of the spatial distribution of the Stokes streamfunction, obtained by DNS. The vortex signature and the time evolution of the energy of the vortex are also accurately predicted by the model. A procedure for fitting the model with realistic vortex rings, obtained by DNS, is suggested. This opens the way to using the model for practical engineering applications.

Coupling of a free wake vortex ring near-wake model with the Jensen and Larsen far-wake deficit models

This paper presents a simple physical model to improve the currently used far-wake deficit models in the wind industry. The main improvement is deemed on the determination of the wake deficit in the near-wake. A Vortex Ring Model (VRM) is used to calculate the induced velocities in the near-wake, which are then coupled to the Jensen far-wake model and the Larsen far-wake model based on the concept of Eddy Viscosity (EV). The inviscid near-wake VRM is based on the shedding of discrete tip vortex rings released from a uniformly loaded actuator disc. The model is validated against wind tunnel measurements from experiments with a two- bladed turbine and a circular metal mesh with a uniform porosity to represent an actuator disc. The VRM shows a good agreement with the experimental data with respect to the wake deficit evolution. The VRM is coupled with two well-known engineering type far-wake models: the Jensen and Larsen wake deficit models. The results of the coupling of the VRM and the more elaborated Larsen far-wake model are compared against a 3D Large Eddy Simulation (LES) CFD model. This comparison shows the effect of different near-wake models on the development of centreline velocities in the far-wake. The centreline velocity deficit predicted by the VRM-Larsen model more closely matches LES calculations in comparison with the reference Larsen model.

Study on splitting of a toroidal bubble near a rigid boundary

The splitting of a toroidal bubble near a rigid boundary is commonly observed in experiments, which is a quite complex phenomenon in bubble dynamics and still not yet well understood. In present study, the bubble splitting phenomenon is studied using the boundary integral method. The vortex ring model is extended to multiple vortex rings to simulate the interaction between two toroidal bubbles after splitting. Buoyancy and non-buoyancy cases are investigated numerically in this study. Numerical results with buoyancy effects show favorable agreement with the experimental observations, which validates the present model. Generally, the first split of the toroidal bubble occurs when an annular “sideways jet” collides with the other side of the bubble. After the toroidal bubble splitting, some new phenomena are found as follows: (i) An annular high pressure region is generated at the splitting location, and the maximum pressure is associated with the velocity differences between the two sides therein just before splitting. (ii) The total volume varies continuously, while the two sub-bubbles vary differently in volume after splitting. (iii) The sideways jet continues propagating on a sub-bubble surface, which would cause more splits or partial breakup of the splash film into droplets. This may be an important reason for the formation of bubble cloud and the rough bubble surface in the rebounding process.

Numerical analysis of the bubble jet impact on a rigid wall

The main characteristic of the bubble dynamics near a rigid wall is the development of a high speed liquid jet, generating highly localized pressure on the wall. In present study, the bubble dynamic behaviors and the pressure impulses are investigated through experimental and numerical methods. In the experiment, the dynamics of a spark-generated bubble near a steel plate are captured by a high-speed camera with up to 650,000 frames per second. Numerical studies are conducted using a boundary integral method with incompressible assumption, and the vortex ring model is introduced to handle the discontinued potential of the toroidal bubble. Meanwhile, the pressure on the rigid wall is calculated by an auxiliary function. Calculated results with two different stand-off parameters show excellent agreement with experimental observations. A double-peaked or multiple-peaked structure occurs in the pressure profile during the collapse and rebounding phase. Generally, the pressure at the wall center reaches the first peak soon after the jet impact, and the second peak is caused by the rapid migration of the bubble toward the wall, and the subsequent peaks may be caused by the splashing effect and the rebounding of the toroidal bubble. At last, both agreements and differences are found in the comparison between the present model and a hybrid incompressible–compressible method in Hsiao et al. (2014). The differences show that the compressibility of the flow is another influence factor of the jet impact. However, the main features of the jet impact could be simulated using the present model.

Bubble jet impact on a rigid wall of different stand-off parameters

One of the key features of the dynamics of a bubble near a rigid wall is the development of a high liquid jet, generating highly localized pressure on the wall. In present study, the boundary integral method is employed to simulate this phenomenon, and the vortex ring model is introduced to handle the discontinued potential of the toroidal bubble. Meanwhile, the pressure induced in the whole process is calculated by an auxiliary function. The effect of the stand-off parameter on the bubble dynamics and the pressure on the wall is investigated, and a double-peaked structure occurs in the pressure profile after the jet impact in some cases, which is associated with the jet impact and the high internal pressure inside the bubble.