Inviscid Incompressible Fluid Research Articles

COAL DUST EMISSION PROBLEM

Purpose. The article aims to develop 2D numerical models for the prediction of atmospheric pollution during transportation of coal in the railway car, as well as the ways to protect the environment and the areas near to the mainline from the dust emission due to the air injection installation. Methodology. To solve this problem there were developed numerical models based on the use of the equations of motion of an inviscid incompressible fluid and mass transfer. For the numerical integration of the transport equation of the pollutant the implicit alternating-triangular difference scheme was used. For numerical integration of the 2D equation for the velocity potential the method of total approximation was used. The developed numerical models are the basis of established software package. On the basis of the constructed numerical models it was carried out a computational experiment to assess the level of air pollution when transporting bulk cargo by rail when the railway car has the air injection. Findings. 2D numerical models that belong to the class «diagnostic models» were developed. These models take into account the main physical factors affecting the process of dispersion of dust pollution in the atmosphere during transportation of bulk cargo. The developed numerical models make it possible to calculate the dust loss process, taking into account the use of the air injection of the car. They require a small cost of the computer time during practical realization at the low and medium power machines. There were submitted computational calculations to determine pollutant concentrations and the formation of the zone of pollution near the train with bulk cargo in «microscale» scale taking into account the air curtains. Originality. 2D numerical models taking into account the relevant factors influencing the process of dispersion of pollutants in the atmosphere, and the formation of the zone of pollution during transportation of bulk cargo by rail were created. A way to protect the atmosphere from pollution during the emission of bulk cargoes from the rail car, which is based on the principle of the air injection, was developed. Practical value. The efficient numerical models which can be used in the development of environmental protection measures in the operation of railway transport were considered. The proposed model allows calculating 2D dynamics of wind flow, taking into account the installed air injection, and mass transfer process of pollutants in the atmosphere.

Symmetries and conservation laws of the Euler equations in Lagrangian coordinates

We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries of the action functional and generate corresponding conservation laws in Lagrangian coordinates. We clarify and demonstrate relationships between symmetries and the classical balance laws of energy, linear momentum, center of mass, angular momentum, and the statement of vorticity advection. Using a newly obtained scaling symmetry, we obtain a new conservation law for the Euler equations in Lagrangian coordinates in n-dimensional space. The resulting integral balance relates the total kinetic energy to a new integral quantity defined in Lagrangian coordinates. This relationship implies an inequality which describes the radial deformation of the fluid, and shows the non-existence of time-periodic solutions with nonzero, finite energy.

THE ASSESSMENT MODELS OF AIR POLLUTION DURING TRANSPORTATION OF BULK CARGO

Purpose. The scientific work is concentrated on development of 3D, 2D numerical models for the prediction of atmospheric pollution during transport of bulk cargo in the railway car. Methodology. To solve this problem numerical models were developed, based on the use of the motion equations of inviscid incompressible fluid and mass transfer, to determine the field of wind velocity near the cars and dispersion of dust in the atmosphere. For the numerical integration of the pollutant transport equation implicit alternating-triangular difference scheme was used. When constructing a difference scheme splitting of the transport equation is carried out that allows us to construct an efficient algorithm for solving a differential problem. Unknown value of the pollutant concentration at every step of splitting is determined by the explicit scheme – the method of point-to-point computation, which provides a simple numerical implementation of splitting equations. For numerical integration of the 3D equation for the velocity potential method of Richardson is applied. For numerical integration of the 2D equation for the velocity potential the method of total approximation is applied. The developed numerical models are the basis of established software package. On the basis of the constructed numerical models a computational experiment to assess the level of air pollution when demolition of coal dust from the gondola car was carried out. Findings. 3D, 2D numerical models that belong to the class «diagnostic models» were developed. These models take into account the main physical factors affecting the process of dust pollution dispersion in the atmosphere during transportation of bulk cargo, but require small costs of the computer time in the practice at the low and medium power machines. These models are used for serial calculations of various situations of scenarios related to issues of environmental protection and pollution intensity diagnostics for different weather conditions. Computational calculations to determine pollutant concentrations and formation of pollution zone near the cars with bulk cargo in «microscale» scale were submitted. Originality. 3D, 2D numerical models were created. They allow taking into account the relevant factors, influencing the process of pollutants dispersion in the atmosphere, and formation of the pollution zone during transport of bulk cargo by rail. Practical value. Efficient numerical models «diagnostic models» for rapid calculation of the atmosphere pollution level during transportation of bulk cargo by rail were considered. Models can be used in the development of environmental protection measures at the operation of rail transport. Proposed model allows calculating 3D, 2D hydrodynamics of wind flow and mass transfer process of pollutants in the atmosphere.

Local well‐posedness and zero‐αlimit for the Euler‐αequations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐αequations of inviscid incompressible fluid flows in. We also establish the convergence rate of the solutions of the Euler‐αequations to the corresponding solutions of the Euler equations as the regularization parameterαapproaches 0 in. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical analysis of the dynamics of distributed vortex configurations

A numerical algorithm is proposed for analyzing the dynamics of distributed plane vortex configurations in an inviscid incompressible fluid. At every time step, the algorithm involves the computation of unsteady vortex flows, an analysis of the configuration structure with the help of heuristic criteria, the visualization of the distribution of marked particles and vorticity, the construction of streamlines of fluid particles, and the computation of the field of local Lyapunov exponents. The inviscid incompressible fluid dynamic equations are solved by applying a meshless vortex method. The algorithm is used to investigate the interaction of two and three identical distributed vortices with various initial positions in the flow region with and without the Coriolis force.

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.

Drift due to two obstacles in different arrangements

We study the drift induced by the passage of two cylinders through an unbounded extent of inviscid incompressible fluid under the assumption that the flow is two-dimensional and steady in the moving frame of reference. The goal is to assess how the resulting total particle drift depends on the parameters of the geometric configuration, namely, the distance between the cylinders and their angle with respect to the direction of translation. This problem is studied by numerically computing, for different cylinder configurations, the trajectories of particles starting at various initial locations. The velocity field used in these computations is expressed in closed form using methods of the complex function theory and the accuracy of calculations is carefully verified. We identify cylinder configurations which result in increased and decreased drift with respect to the reference case when the two cylinders are separated by an infinite distance. Particle trajectories shed additional light on the hydrodynamic interactions between the cylinders in configurations resulting in different drift values. This ensemble of results provides insights about the accuracy of models used to study biogenic transport.

On the computation of measure-valued solutions

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm ofmeasure-valued solutionsfor these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

A refined analytical model for earthquake-induced sloshing in half–full deformable horizontal cylindrical liquid containers

The coupled response of elastic deformable liquid containers of horizontal-cylindrical shape under external seismic excitation is examined, through an analytical methodology, assuming inviscid-incompressible fluid and irrotational-flow conditions. In particular, the case of a half–full horizontal-cylindrical deformable container is examined, considering an analytical series-type solution for the velocity potential function that describes the liquid motion under external excitation. This mathematical analysis extends the solution methodology presented in previous publications of the senior author, taking into account full coupling between sloshing and wall deformation in a rigorous manner, where wall deformation is considered through a sinusoidal assumed-shape function. In the mathematical formulation, the velocity potential is decomposed into three parts: (a) a first part, which represents liquid motion that follows the external excitation, (b) a “convective part”, representing liquid motion associated with free surface elevation (sloshing), and (c) a third part caused by the wall deformation. Using an elegant mathematical manipulation, the coupled transient overall response of the liquid-container system is obtained in an efficient manner. Numerical results are presented in terms of the principal natural frequencies of the coupled system, as well as the system response under strong seismic input, and emphasize on the effects of container aspect ratio on the dynamic behavior of the system. The mathematical formulation for the case of long cylinders results in a simplified model, identical to the simplified “physical model” presented in a previous publication.

Wavelet characterization of 2D turbulence and intermittency in magnetized electron plasmas

A study of the free relaxation of turbulence in a two-dimensional (2D) flow is presented, with a focus on the role of the initial vorticity conditions. Exploiting a well-known analogy with 2D inviscid incompressible fluids, the system investigated here is a magnetized pure electron plasma. The dynamics of this system are simulated by means of a 2D particle-in-cell code, starting from different spiral density (vorticity) distributions.A wavelet multiresolution analysis is adopted, which allows the coherent and incoherent parts of the flow to be separated. Comparison of the turbulent evolution in the different cases is based on the investigation of the time evolution of statistical properties, including the probability distribution functions and structure functions of the vorticity increments. It is also based on an analysis of the enstrophy evolution and its spectrum for the two components.In particular, while the statistical features assess the degree of flow intermittency, spectral analysis allows us not only to estimate the time required to reach a state of fully developed turbulence, but also estimate its dependence on the thickness of the initial spiral density distribution, accurately tracking the dynamics of both the coherent structures and the turbulent background. The results are compared with those relevant to annular initial vorticity distributions (Chen et al 2015 J. Plasma Phys. 81 495810511).

Mechanical balance laws for fully nonlinear and weakly dispersive water waves

The Serre–Green–Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is known to describe accurately the wave motion at the surface of an incompressible inviscid fluid in the case when the fluid flow is irrotational and two-dimensional. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre–Green–Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre–Green–Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling. One consequence of the present analysis is that the energy loss appearing in the shallow-water theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre–Green–Naghdi equations using a finite-element discretization coupled with an adaptive Runge–Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre–Green–Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling.

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.

Minimal Geodesics Along Volume-Preserving Maps, Through Semidiscrete Optimal Transport

We introduce a numerical method for extracting minimal geodesics along the group of volume-preserving maps, equipped with the $L^2$ metric, which as observed by Arnold [Ann. Inst. Fourier (Grenoble), 16 (1966), pp. 319--361] solve the Euler equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier [Comm. Pure Appl. Math., 44 (1991), pp. 375--417], numerically implemented through semidiscrete optimal transport. It is robust enough to extract nonclassical, multivalued solutions of Euler's equations, for which the flow dimension---defined as the quantization dimension of Brenier's generalized flow---is higher than the domain dimension, a striking and unavoidable consequence of thismodel [A. I. Shnirelman, Geom. Funct. Anal., 4 (1994), pp. 586--620]. Our convergence results encompass this generalizedmodel, and our numerical experiments illustrate it for the first time in two space dimensions.

Boundary controllability of incompressible Euler fluids with Boussinesq heat effects

This paper deals with the boundary controllability of inviscid incompressible fluids for which thermal effects are important. They will be modeled through the so-called Boussinesq approximation. In the zero heat diffusion case, by adapting and extending some ideas from J.-M. Coron and O. Glass, we establish the simultaneous global exact controllability of the velocity field and the temperature for 2D and 3D flows. When the heat diffusion coefficient is positive, we present some additional results concerning exact controllability for the velocity field and local null controllability of the temperature.

Axisymmetric shape of gas bubbles in a uniform flow: Numerical study of bifurcation by analytic continuation

This paper investigates axisymmetric shapes of an incompressible inviscid fluid bubble in a uniform flow. Approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method. Bifurcation and turning points are located by applying a special type of Hermite–Padé approximation.

Time-fractional effect on pressure waves propagating through a fluid filled circular long elastic tube

The pressure waves propagating through an incompressible inviscid fluid that moves in a circular cylindrical long elastic tube are considered. The reductive perturbation method is used to derive the KdV equation from the hydrodynamic equations of the system. The Euler–Lagrange variational technique described by Agrawal has been applied to formulate the time-fractional KdV equation. The derived time-fractional KdV equation is solved by employing the variational-iteration method represented by He. The effects of the tube and fluid parameters and the time fractional order on the propagation of pressure waves are investigated.

Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models

We discuss general incompressible inviscid models, including the Euler equations, the surface quasi-geostrophic equation, incompressible porous medium equation, and Boussinesq equations. All these models have classical unique solutions, at least for short time. We show that they have real analytic Lagrangian paths. More precisely, we show that as long as a solution of any of these equations is in a class of regularity that assures Hölder continuous gradients of velocity, the corresponding Lagrangian paths are real analytic functions of time. The method of proof is conceptually straightforward and general, and we address the combinatorial issues head-on.

The effects of surface tension on the initial development of a free surface adjacent to an accelerated plate

When a vertical rigid plate is uniformly accelerated horizontally from rest into an initially stationary layer of inviscid incompressible fluid, the free surface will undergo a deformation in the locality of the contact point. This deformation of the free surface will, in the early stages, cause a jet to rise up the plate. An understanding of the local structure of the free surface in the early stages of motion is vital in many situations, and has been developed in detail by King & Needham (J. Fluid Mech., vol. 268, 1994, pp. 89–101). In this work we consider the effects of introducing weak surface tension, characterized by the inverse Weber number $\mathscr{W}$, into the problem considered by King & Needham. Our approach is based upon matched asymptotic expansions as $\mathscr{W}\rightarrow 0$. It is found that four asymptotic regions are needed to describe the problem. The three largest regions have analytical solutions, whilst a numerical method based on finite differences is used to solve the time-dependent harmonic boundary value problem in the last region. Our results identify the local structure of the jet near the vicinity of the contact point, and we highlight a number of key features, including the height of this jet as well as its thickness and strength. We also present some preliminary experimental results that capture the spatial structure near the contact point, and we then show promising comparisons with the theoretical results obtained within this paper.

Numerical resolution of Euler equations through semi-discrete optimal transport

Geodesics along the group of volume preserving diffeomorphisms are solutions to Euler equations of inviscid incompressible fluids, as observed by Arnold. On the other hand, the projection onto volume preserving maps amounts to an optimal transport problem, as follows from the generalized polar decomposition of Brenier. We present, in the first section, the framework of semi-discrete optimal transport, initially developed for the study of generalized solutions to optimal transport and now regarded as an efficient approach to computational optimal transport. In a second and largely independent section, we present numerical approaches for Euler equations seen as a boundary value problem: knowing the initial and final positions of some fluid particles, reconstruct intermediate fluid states. Depending on the data, we either recover a classical solution to Euler equations, or a generalized flow for which the fluid particles motion is non-deterministic. See Minimal Geodesics along Volume Preserving Maps, through Semi-Discrete Optimal Transport, Merigot and Mirebeau 2015, for the proofs of the results presented in these proceedings.

An efficient finite element approach to examine the free vibration characteristics of liquid tankages in space launch vehicles

This paper deals with the development of an efficient finite element method to study the dynamic characteristics of structures containing fluid. Structural and fluid domains are discretized with 8-node plate/shell element and 20 node brick element respectively. The inviscid incompressible fluid is assumed under irrotational motion. Proper care is taken to eliminate the pressure at fluid–structure interfaces. Mass due to fluid–structure interaction is merged with structural mass and expressed the added mass in terms of inverted fluid stiffness matrix with pre- and post- multiplication of area coefficient matrix. An appropriate node numbering scheme is adopted for idealizing the structural domain and subsequently the fluid domain to gain the computational efficiency and reduction in the storage required for the added mass matrix. The adequacy of the present approach is demonstrated by considering the free vibrations of clamped cantilever plate and clamped-free open-ended cylindrical tanks and partially filled liquid tankages having common bulk head.