This article addresses the fascinating 150 years history of the classical Helmholtz paper that laid the foundation of the vortex dynamics. Among general theorems on vortex motion, this memoir contains the special section on circular vortex filaments and axisymmetric vortex rings, in particular. The objective of this article is both to clarify some purely mathematical questions connected with the Dyson model of coaxial vortex rings in inviscid incompressible fluid and to provide a historical overview of achievements in experimental, analytical, and numerical studies of vortex rings interactions. The model is illustrated by several examples both of regular and chaotic motion of several vortex rings in an unbounded fluid.

Coaxial axisymmetric vortex rings: 150 years after Helmholtz

Modelling solute transport in shallow water with the lattice Boltzmann method

In the present paper, we describe the fascinating 150-year history of vortex dynamics, which started from the classical work by H. Helmholtz (1858). This work contains not only the general theorems on vortex motion, but also a special section devoted to axisymmetric circular vortex filaments and, in particular, to vortex rings. The aim of our investigation is both to clarify some purely mathematical questions connected with the Dyson model for coaxial vortex rings with a small circular cross-section in an ideal incompressible fluid and to provide a historical overview of the achievements in experimental, analytical, and numerical studies of the interaction of vortex rings. The model is illustrated by several examples of the motion of vortex rings. Finally, we present a complete classification of the interactions of two coaxial vortex rings in an ideal fluid.

Vortex rings: history and state of the art

The Lattice Boltzmann Method (LBM) was investigated to solve triangular cavity flow and free-surface problems in hydraulic dynamics. Some cases of triangular cavity flow and backward step flow were simulated to show the efficiency and stability of this method. Two-dimensional partial dam breaking problem and the propagation and diffraction of dam-break wave around rectangular and circular cylinder were numerically studied successfully. Excellent agreement was obtained between numerical predictions and available results.

Lattice Boltzmann simulations of triagular cavity flow and free-surface problems*

The three-dimensional interfacial waves due to a fundamental singularity steadily moving in a system of two semi-infinite immiscible fluids of different densities are investigated analytically. The two fluids are assumed to be incompressible and homogenous. There are three systems to be considered: one with two inviscid fluids, one with an upper viscous and a lower inviscid fluid, and one with an upper inviscid and a lower viscous fluid. The Laplace equation is taken as the governing equation for inviscid flows while the steady Oseen equations are taken for viscous flows. The kinematic and dynamic conditions on the interface are linearized for small-amplitude waves. The singularity immersed above or beneath the interface is modeled as a simple source in the inviscid fluid while as an Oseenlet in the viscous fluid. Based on the integral solutions for the interfacial waves, the asymptotic representations of wave profiles in the far field are explicitly derived by means of Lighthill’s two-stage scheme. An analytical solution is presented for the density ratio at which the maximum wave amplitude occurs. The effects of density ratio, immersion depth, and viscosity on wave patterns are analytically expressed. It is found that the wavelength of interfacial waves is elongated in comparison with that of free-surface waves in a single fluid.

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Interfacial waves due to a singularity in a system of two semi-infinite fluids

The closely coupled approach combined with the finite volume method (FVM) solver and the finite element method (FEM) solver is used to investigate the fluid-structure interaction (FSI) of a three-dimensional cantilevered hydrofoil in the water tunnel. The FVM solver and the coupled approach are verified and validated by comparing the numerical predictions with the experimental measurements, and good agreement is obtained concerning both the lift on the foil and the tip displacement. In the noncavitating flow, the result indicates that the growth of the initial incidence angle and the Reynolds number improves the deformation of the foil, and the lift on the foil is increased by the twist deformation. The normalized twist angle and displacement along the span of the hydrofoil for different incidence angles and Reynolds numbers are almost uniform. For the cavitation flow, it is shown that the small amplitude vibration of the foil has limited influence on the developing process of the partial cavity, and the quasi two-dimensional cavity shedding does not change the deformation mode of the hydrofoil. However, the frequency spectrum of the lift on the foil contains the frequency which is associated with the first bend frequency of the hydrofoil.

Fluid-structure interaction simulation of three-dimensional flexible hydrofoil in water tunnel

A finite-volume high-resolution numerical model for coupling the shallow water flows and pollutant diffusions was presented based on using a hybrid TVD scheme in space discretization and a Runge-Kutta method in time discretization. Numerical simulations for modelling dam-break, enlarging open channel flow and pollutant dispersion were implemented and compared with experimental data or other published computations. The validation of this method shows that it can not only deal with the problem involving discontinuities and unsteady flows, but also solve the general shallow water flows and pollutant diffusions.

High-resolution numerical model for shallow water flows and pollutant diffusions

The derivations are carried out for the velocity potentials of singularities moving with an arbitrary path either in the upper fluid or in the lower fluid with or without a horizontal bottom when two fluids are present. In such a case, the pressure distribution is no longer equal to a constant or zero at the free interface. Taking the influence of an upper fluid upon the lower fluid into consideration, a series of fundamental solutions in closed forms are presented in this paper.

Analytical solutions of singularities moving with an arbitrary path when two fluids are present

Inviscid coaxial interactions of two vortex rings, including head-on collisions and leapfrogging motions, are considered using a contour dynamics technique. Interactions of vortex rings with solid bodies are also investigated by combining the contour dynamics technique with a boundary integral equation method. Numerical results show that a clean, successful passage motion is possible for two vortex rings with not too thick cores. In both cases of head-on collisions and leapfrogging motions, very large core deformations are observed when two vortex rings get close to each other. A head-tail structure is formed in the later stage of a head-on collision of two fat vortices. Numerical results also show that a vortex ring will stretch and slow down when it moves toward a solid boundary, will shrink and speed up when it moves away from a solid boundary, and will either translate steadily or approach an oscillating asymptotic state when it is far away from any boundaries.

Coaxial interactions of two vortex rings or of a ring with a body

In this paper, a nonlinear, unsteady 3-D free surface problem of the oblique water exit of an axisymmetric body with a large water exit-angle was investigated by means of the perturbation method in which the complementary angle α of the water exit angle was chosen as a small parameter. The original 3-D problem was solved by expanding it into a power series of α and reduced to a number of 2-D problems. The integral expressions for the first three order solutions were given in terms of the complete elliptic functions of the first and second kinds. The zeroth-order solution didn't turn out to be a linear problem as usual but a nonlinear one corresponding to the vertical water exit for the same body. Computational results were presented for the free surface shapes and the forces exerted up to the second order during the oblique water exit of a series of ellipsoids with various ratios of length to diameter at different Froude numbers.

He You-sheng

Papers

Fluid-structure interaction simulation of three-dimensional flexible hydrofoil in water tunnel

High-resolution numerical model for shallow water flows and pollutant diffusions

Analytical solutions of singularities moving with an arbitrary path when two fluids are present

Coaxial interactions of two vortex rings or of a ring with a body

Perturbation solution to the nonlinear problem of oblique water exit of an axisymmetric body with a large exit-angle