Study of water waves with submerged obstacles using a vortex method with Helmholtz decomposition

The present study develops a 2‐D numerical scheme that combines the vortex method and the boundary integral method by a Helmholtz decomposition to investigate the interaction of water waves with submerged obstacles. Viscous effects and generation of vorticity on the free surface are neglected. The second kind of Fredholm integral equations that govern the strengths of vortex sheets along boundaries are solved iteratively. Vorticity is convected and diffused in the fluid via a Lagrangian vortex (blob) method with varying cores, using the particle strength exchange method for diffusion, with particle redistribution. A grid‐convergence study of the numerical method is reported. The inviscid part of the method and the simulation of the free‐surface motion are tested using two calculations: solitary wave propagation in a uniform channel and a moving line vortex in the fluid. Finally, the full model is verified by simulating periodic waves travelling over a submerged rectangular obstacle using nonuniform vortex blobs with a mapping of the redistribution lattice. Overall, the numerical model predicts the vortices' evolution and the free‐surface motion reasonably well. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical study of an inviscid incompressible flow through a channel of finite length

A two‐dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo‐symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions. Copyright © 2008 John Wiley & Sons, Ltd.     

Simulation on motion of particles in vortex merging process

In a two-phase flow, the vortex merging influences both the flow evolution and the particle motion. With the blobs-splitting-and-merging scheme, the vortex merging is calculated by a corrected core spreading vortex method （CCSVM）. The particle motion in the vortex merging process is calculated according to the particle kinetic model. The results indicate that the particle traces are spiral lines with the same rotation direction as the spinning vortex. The center of the particle group is in agreement with that of the merged vortex. The merging time is determined by the circulation and the initial ratio of the vortex radius and the vortex center distance. Under a certain initial condition, a stretched particle trail is generated, which is determined by the viscosity, the relative position between the particles and the vortex, and the asymmetrical circulation of the two merging vortices.     

Vortex motion of a disperse mixture

A study is made of the motion of fine spherical particles in a given steady vortex flow of an incompressible fluid. The results are given of an investigation into the Lyapunov stability of a particle trajectory coincident with the vortex axis, and of trajectories from which the distance to the vortex axis is determined by the condition of equality of the radial components of the force of the phase interaction and of the centrifugal force which acts on the particle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–169, November–December, 1984.     

Particle motion in a Taylor vortex

Here we present a study on the behavior of individual particles in the Taylor vortex. Two particle-fluid systems were tested: a cube with the edge length of 2 mm and the density of 0.13 g/cm³ (‘light particle’) in a working fluid of mineral oil (density of 0.86 g/cm³ and viscosity of 0.066 Pa.s); and a sphere with the diameter of 1.6 mm and the density of 2.2 g/cm³ (‘heavy particle’) in 90% glycerin/water (density of 1.23 g/cm³ and viscosity of 0.128 Pa.s). The Taylor–Couette device used for this study was featured with a short column (aspect ratio ≤ 6) and a wide gap (radius ratio ≤ 0.67). The interaction between the floating particle and Taylor vortices was investigated using a high speed camera and a particle image velocimetry (PIV) system. Moreover, computational fluid dynamics simulation was performed to calculate the liquid flow pattern and analyze the particle motion. Our results show that the particle behavior in the Taylor–Couette device is strongly dependent on the particle density and Reynolds number. With the increasing Reynolds number, four types of particle trajectories were sequentially identified from the light particle, including a circular trajectory on the surface of the inner cylinder, random shifting between the circular trajectory and oval orbit, a stable oval orbit in the annulus, and a circle along the vortex center. On the other hand, the heavy particle moves along a circular orbit and an oval orbit at low and high Reynolds numbers, respectively. Several unreported particle behaviors were also observed, such as the self-rotation of the particle when it moves along the above trajectories, the shifting axis of the oval orbit, etc. In addition, the PIV measurements show that the trapped particle can only influence the flow pattern locally around the particle. The study can help understand the particle behavior in a Taylor vortex better and therefore benefit applications of particle-laden Taylor vortex devices.     

Hydrodynamic characteristics of a vortex source performing translational motion in a multilayer heavy fluid

The problem of translational motion of a vortex source in a three-layer fluid bounded by a bottom from below is considered. The fluid in each layer is perfect, incompressible, heavy, and homogeneous. Based on the previously developed method, formulas for disturbed complex velocities of the fluid in each layer and the wave drag and lift force of the vortex source are obtained. The vortex motion is considered near the interface of two semi-infinite fluid media and in a two-layer fluid with different conditions at the boundary. In all cases, the hydrodynamic characteristics of the vortex source are given as functions of the Froude number. In a number of problems, these characteristics have discontinuities at the transition through the critical Froude numbers. The character of these discontinuities is studied analytically. Omsk Department of Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 140–146, September–October, 2000.     

A viscous vortex particle method for deforming bodies with application to biolocomotion

Bio‐inspired mechanics of locomotion generally consist of the interaction of flexible structures with the surrounding fluid to generate propulsive forces. In this work, we extend, for the first time, the viscous vortex particle method (VVPM) to continuously deforming two‐dimensional bodies. The VVPM is a high‐fidelity Navier–Stokes computational method that captures the fluid motion through evolution of vorticity‐bearing computational particles. The kinematics of the deforming body surface are accounted for via a surface integral in the Biot–Savart velocity. The spurious slip velocity in each time step is removed by computing an equivalent vortex sheet and allowing it to flux to adjacent particles; hence, no‐slip boundary conditions are enforced. Particles of both uniform and variable size are utilized, and their relative merits are considered. The placement of this method in the larger class of immersed boundary methods is explored. Validation of the method is carried out on the problem of a periodically deforming circular cylinder immersed in a stagnant fluid, for which an analytical solution exists when the deformations are small. We show that the computed vorticity and velocity of this motion are both in excellent agreement with the analytical solution. Finally, we explore the fluid dynamics of a simple fish‐like shape undergoing undulatory motion when immersed in a uniform free stream, to demonstrate the application of the method to investigations of biomorphic locomotion. Copyright © 2008 John Wiley & Sons, Ltd.     

Wake dynamics of external flow past a curved circular cylinder with the free-stream aligned to the plane of curvature

The fundamental mechanism of vortex shedding past a curved cylinder has been investigated at a Reynolds number of 100 using three-dimensional spectral/hp computations. Two different configurations are presented herein: in both cases the main component of the geometry is a circular cylinder whose centreline is a quarter of a ring and the inflow direction is parallel to the plane of curvature. In the first set of simulations the cylinder is forced to transversely oscillate at a fixed amplitude, while the oscillation frequency has been varied around the Strouhal value. Both geometries exhibit in-phase vortex shedding, with the vortex cores bent according to the body's curvature, although the wake topology is markedly different. In particular, the configuration that was found to suppress the vortex shedding in absence of forced motion exhibits now a primary instability in the near wake. A second set of simulations has been performed imposing an oscillatory roll to the curved cylinder, which is forced to rotate transversely around the axis of its bottom section. This case shows entirely different wake features from the previous one: the vortex shedding appears to be out-of-phase along the body's span, with straight cores that tend to twist after being shed and manifest a secondary spanwise instability. Further, the damping effect stemming from the transverse planar motion of the part of the cylinder parallel to the flow is no longer present, leading to a positive energy transfer from the fluid to the structure.     

Umströmung beschleunigter und verzögerter Partikeln

Zusammenfassung Die Bewegung von Partikeln ist in technischen Anlagen fast ausschließlich instationär. Insbesondere dann, wenn die Partikeln in Schwärmen auftreten, muß man aufgrund von Zusammenstößen zwischen den Partikeln oder von Partikeln mit einer festen Wand erwarten, daß im periodischen Wechsel Bewegungsabschnitte mit Beschleunigung und Verzögerung auftreten. Diese Bewegungsvorgänge sind für die Umströmung und folglich auch für damit verbundene Wärme- und Stoffübergangsprozesse sowie chemische Reaktionen von großer Bedeutung.Aus diesem Grunde wurden in Teil 1 dieses Berichtes die instationären Bewegungsvorgänge bei Beschleunigung näher untersucht. Besonderes Augenmerk sollte auf die Entstehung und die weitere Entwicklung der Wirbel gerichtet werden, die hinter den Partikeln auftreten.Die mathematische Beschreibung des Strömungsfeldes erfolgt für den Fall der linearen Partikelbewegung unter Vernachlässigung der Schwerkraft. Die numerische Lösung der Differentialgleichungen erfolgte mit Hilfe der Methode der finiten Elemente.Die berechneten Stromlinienfelder vermitteln einen umfassenden Einblick in die Umströmung der Partikeln. Sie lassen bei der Beschleuigung aus dem Ruhezustand die hinausgezögerte Entstehung und dann folgende Entwicklung der Wirbel erkennen. Sowohl die Länge der Wirbel als auch der Ablösepunkt der Strömung von der Partikeloberfläche wurden in Abhängigkeit von der Zeit dargestellt. Ferner wurde der Widerstandsbeiwert bestimmt. Die entsprechenden Ergebnisse für die verzögerte Bewegung von Partikeln werden in Teil 2 dieses Berichtes diskutiert.

---

Fluid flow around accelerated and decelerated particlesPart 1

Fluid flow around particles or the motion of particles in a fluid is in general of unsteady state. Therefore the relative fluid motion around a particle with spherical shape has been studied applying theoretical-numerical methods.Unsteady state motion of the particles is due to collision when moving in swarms or when they collide with a wall. Prior to collision particle motion is decelerated and after collision they will be accelerated because of elastic properties of particle and wall materials. In many cases particle motion occurs in combination with heat transfer.It is well known, that, for steady state conditions, behind a particle vortex formation is observed. Vortex formation is expected to occur for unsteady state conditions in a different way. When the particle is accelerated, vortex formation is expected to be delayed. With decelerated motion vortex formation is expected to take place at a much smaller relative velocity and the vortex should envelope the particle for certain values of deceleration.The theoretical investigation is based on several assumptions. The particle is assumed to move with prescribed velocity on a linear path prior to constant acceleration or deceleration. Finite element methods are applied for solution of the differential equations describing the unsteady state motion.Streamlines give a comprehensive picture of the studied fluid motion. They reveal the expected delay of vortex formation for the case of acceleration and premature formation for the case of deceleration, when motion occurs at a Reynolds-number beyond a certain value. In this case, the vortex will move around the particle and finally completely envelope it. Below this value of the Reynolds-number the vortex, established under unsteady state motion, prior to onset of deceleration, will decade.From the determined fields of streamlines the length of the vortices and the angle of detachment of the fluid from particle surface have been determined for several values of acceleration and deceleration. From the calculated velocity fields the local shear stress and the local and mean values of the resistance factor have been derived.The paper is presented in two parts. The first part includes the physical fundamentals of unsteady state motion of particles, the numerical methods applied for the solution of the problems, and a discussion of results obtained for accelerated particles. The second part includes the discussion of results obtained for decelerated particles.

     

Motion analysis of a spinning satellite system with off-centre tanks partially filled with liquid

The sloshing problem for, a spherical tank partially filled with liquid is analysed in this paper. The study is based on the goveming, equations of fluid dynamics and the Euler's equations of systems with the influences of tank off-centering, fluid vortices and the Coriolis' acceleration on the motion states of the systems taken into consideration. In the study, we adopt the concept of uniform vortex motion of fluid generalized by Pfeiffer and apply the boundary element method (BEM) to the calculation of the natural frequence and the velocity field of the liquid sloshing. The motion characteristics of the flow-solid spinning system is then analysed. The project is supported by the National Natural Science Foundation of China and Ministry of Astronautics.     

Implementation and validation of a slender vortex filament code: Its application to the study of a four‐vortex wake model

A computational code EZ‐vortex is developed for the motion of slender vortex filaments of closed or open shape. The integro‐differential equations governing the motion of the vortex centre lines are either the Callegari and Ting equations, which are the leading order solution of a matched asymptotic analysis, or equivalent forms of these equations. They include large axial velocity and nonsimilar profiles in the vortical cores. The fluid may be viscous or inviscid. This code is validated both against known solutions of these equations and results from linear stability analyses. The linear and non‐linear stages of a perturbed two‐vortex wake and of a four‐vortex wake model are then computed. Copyright © 2004 John Wiley & Sons, Ltd.     

The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

Suspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a time-dependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudo-spectral method in space. The coupled system is integrated in time with a fourth-order Runge–Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner et al. (2015, Journal of Fluid & Structures 52, 166–180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterized by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.     

Direct numerical simulation of a particle-laden low Reynolds number turbulent round jet

Direct numerical simulation method is used for the investigating of particle-laden turbulent flows in a spatially evolution of low Reynolds number axisymmetric jet, and the Eulerian–Lagrangian point-particle approach is employed in the simulation. The simulation uses an explicit coupling scheme between particles and the fluid, which considers two-way coupling between the particle and the fluid. The DNS results are compared well with experimental data with equal Reynolds number (R_e = 1700). Our objects are: (i) to investigate the correlation between the particle number density and the fluctuating of fluid streamwise velocity; (ii) to examine whether the three-dimensional vortex structures in the particle-laden jet are the same as that in the free-air jet and how the particles modulate the thee-dimensional vortex structures and turbulence properties with different Stokes number particles; (iii) to discover the particle circumferential dispersion with different Stokes number particles. Our findings: (i) all the particles, regardless of their particle size, tend to preferentially accumulate in the region with large-than-mean fluid streamwise velocity; (ii) the small Stokes number particles take an important part in the modulation of three-dimensional vortex structures, but for the intermediate and larger sized particles, this modulation effect seems not so apparent; (iii) the particle circumferential dispersion is more effective for the smaller and intermediate sized particles, especially for the intermediate sized particles.     

Eulerian prediction of the fluid/particle correlated motion in turbulent two-phase flows

An approximate equation governing the turbulent fluid velocity encountered along discrete particle path is used to derive the fluid/particle turbulent moments required for dispersed two-phase flows modelling. Then, closure model predictions are compared with results obtained from large-eddy simulation of particle fluctuating motion in forced isotropic fluid turbulence.     

Spherical Vortex Structures with a Core and a Shell

In this paper we construct and investigate the vortex structure consisting of a spherical vortex (vortex core) inside a spherical vortex layer (shell). A partial case of this structure is a spherical vortex with uniformly helical motion of the fluid within the core and the shell. The strengths of the helical flows in the core and the shell are generally different. The case of identical strengths is analyzed in detail. The streamline pattern is presented. The vortex velocity limit at which the vortex does not collapse is found. This proves to be less by a factor 1.7 than the analogous quantity for a vortex without a shell and 4 times lower than the maximum velocity of the Hill vortex.     

Interaction between shed vorticity, free surface waves and forced roll motion of a two-dimensional floating body

A vortex tracking method is used to study forced harmonic roll motion of a two-dimensional floating body with sharp corners. The effect of free surface waves is incorporated. Errors due to numerical and physical problems are discussed. The presented test cases show that free surface waves have an effect on the vortex generation, whereas the free shear layers do not have a noticeable influence on the free surface waves. The results show that roll damping due to eddymaking and wave generation cannot be separated.     

A comparison between a collocation and weak implementation of the rigid‐body motion constraint on a particle surface

In this work, we implemented and compared two different methods to impose the rigid‐body motion constraint on a solid particle moving inside a fluid. We consider a fictitious domain method to easily manage the particle motion. As the solid as well as the fluid inertia are neglected, the particle can be discretized through its boundary only. The rigid‐body motion is imposed via Lagrange multipliers on the boundary. In the first method, such constraints are imposed in discrete points on the boundary (collocation), whereas in the second the constraint is imposed in a weak way on elements dividing the particle surface. Two test problems, that is, a spherical and an ellipsoidal particle in a sheared Newtonian fluid, are chosen to compare the methods. In both cases, the analysis is carried out in 2D as well as in 3D. The results show that for the collocation method an optimal number of collocation points exist leading to the smallest error. However, small variations in the optimal value can generate large deviations. In the weak implementation, the error is only mildly affected by the number of elements used to discretize the particle boundary and by the Lagrange multiplier's interpolation space. A further analysis is carried out to study the effect of an approximated integration of weak constraints. A comparison between the two methods showed that the same accuracy can be achieved by using less constraints if the weak discretization is used. Finally, the rigid‐body motion imposed via weak constraints leads to better conditioned linear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Two-dimensional approach to the motion of a red blood cell in a plane couette flow of plasma

In relation to microrheology of blood, a theoretical approach to the motion of a red blood cell in a plane Couette flow between two parallel plates is made with emphasis on effects of wall. The red blood cell is assumed to be an elliptic cylindrical particle with a thin, inextensible membrane moving like a tank-tread along its perimeter and to contain a Newtonian fluid inside. Fluid motions are analysed numerically both inside and outside the particle on the basis of the Stokes equations, using the finite element method.A quasi-static equilibrium condition leads to the solution for the motion of the particle. It is shown that two types of motion exist (a stationary orientation motion and a flipping motion), depending on the viscosity ratio of inner to outer fluid, the axis ratio of the elliptic cylinder and the ratio of particle size to channel width. The results are applied to capillary blood flow.     

圆柱周围湍流和含沙多相流的SPH数值模拟

湍流和多相流是流体力学中最具挑战性的两个主题,湍流多相流的实验和数值模拟更是一项艰巨的挑战。此外,对颗粒干沉积方面的多相流、多尺度、多物理耦合特征的风沙流的综合实地观测仍然很少。因此,本文综合考虑湍流、多相流与多物理耦合等方面,采用以圆柱为干扰物产生对流涡流的强制干扰技术,以塔克拉玛干沙漠地带中和田至若羌铁路的过沙桥桥墩为研究背景。为摆脱有限元软件中由网格大变形或失真引起的各种问题,采用SPH方法的宏观界面追踪和微观单点追踪相结合的方式,初步揭示了以单相对流涡流为风场背景的含沙多相流环境下的圆柱周围复杂的流场变化以及对颗粒干沉积运动的影响。采用数值模拟与现场实验相结合的方式,着重对计算域边界壁面和圆柱壁面对空气单相流中对流涡流的成形运动及其特征分析、两相流中对流涡流在圆柱周围的夹沙运动模拟及其特性分析、两相流中对流涡流的夹沙率以及边界壁湍流对沙粒干沉积效率展开分析研究。