Ideal Incompressible Fluid Research Articles

Новые интегрируемые случаи в задаче о движении тяжелого твердого тела в идеальной несжимаемой жидкости

Новые интегрируемые случаи в задаче о движении тяжелого твердого тела в идеальной несжимаемой жидкости

Modeling of particle dynamics in a vertical three – layer – mixer of bulk matters

The mathematical model, describing scattering dynamics of grain mixture particles at the top of the three-layer vertical mixer, equipped with cantilever shafts and impeller is given in the paper. The proposed model allows considering the aerodynamics of complex vortex flows, created by rotating cantilever shafts, and grain particle interaction with the screw.The aerodynamics model is based on the equations for ideal incompressible fluid, which are solved by the discrete vortex method. Grain particle movement is simulated within the particle dynamics model. For that, the equations of particle motion on the rotating helical surface were obtained. A simple criterion for the cantilever impeller operation efficiency in the mixing process, based on the introduced scattering coefficient was proposed.The proposed model allows conducting a computational experiment to determine the optimal impeller design parameters and the optimal mixer operating modes.The calculations confirmed the efficiency of installing cantilever shafts with impeller in the vertical mixer. The results of numerical calculations, performed on this model were used in the design and manufacture of vertical feed mill small-sized installation VFMSSI-04.

Complex singularity of a stokes wave

Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth can be described by a conformal map of the fluid domain into the complex lower half-plane. Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity. The increase of the scaled wave height $H/\lambda$ from the linear limit $H/\lambda=0$ to the critical value $H_{max}/\lambda$ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave. Here $H$ is the wave height and $\lambda$ is the wavelength. We simulated fully nonlinear Euler equations, reformulated in terms of conformal variables, to find Stokes waves for different wave heights. Analyzing spectra of these solutions we found in conformal variables, at each Stokes wave height, the distance $v_c$ from the lowest singularity in the upper half-plane to the real line which corresponds to the fluid free surface. We also identified that this singularity is the square-root branch point. The limiting Stokes wave emerges as the singularity reaches the fluid surface. From the analysis of data for $v_c\to 0$ we suggest a new power law scaling $v_c\propto (H_{max}-H)^{3/2}$ as well as new estimate $H_{max}/\lambda \simeq 0.1410633$.

Numerical modeling of cavitation characteristics and sensitivity curves for reversible hydraulic machinery

Abstract Details on the existence, extent and effects of cavitation can be helpful during the design stages of hydraulic machinery to minimize the effects of cavitation and optimize the design. Fluid flow in reversible hydraulic machinery elements is a complex three-dimensional problem. In this paper, an explicit numerical model based on finite element method (FEM) - dual reciprocity method (DRM) coupling is developed for the 2D simulation of the flow velocity and pressure distributions on the runner blade of a Francis-type reversible radial-axial hydraulic machine. The model is based on the assumptions of an ideal incompressible fluid and relative rotational motion. Instead of directly simulating the 3D flow, the proposed model is based on the analysis of specific 2D flows: fluid axial-symmetric motion and the stream function in an associated plane. This model is applied to reversible radial–axial hydraulic machinery that operates as a pump. The blade exhibits the basic profile NP205. The skeleton of this profile is defined by a quadratic equation, and its thickness function exhibits a NACA profile with a maximum relative thickness of five percent. The numerical results for different discharge values can be used to determine the cavitation characteristics and sensitivity curves for reversible hydraulic machinery.

The motion of a cluster of spherical particles in an ideal fluid

The motion of a cluster of an arbitrary number of spherical particles, attached to one another, in an ideal incompressible fluid is considered. Using the previously developed self-consistent field method, an expression is obtained for the virtual mass of the cluster in the form of an explicit function of the coordinates of all the particles. It is shown that, for special cases, the solution obtained is identical with the corresponding results known in the literature. For a statistically uniform particle distribution, using an averaging procedure over all the different possible configurations in the fluid inside a spherical volume, a simple analytic relation is obtained for the average value of the virtual mass of the spherical cluster as a function of its radius in the first approximation with respect to the volume fraction of the particles.

Strong expansion–contraction of a cavity in a fluid under acoustic action

The simplest model of the strong expansion–contraction of a spherical evacuated cavity in an unbounded ideal incompressible fluid under acoustic action is constructed. It is assumed that the cavity is formed at the instant of maximum dilatation of the fluid as the result of the breakdown in its continuity by elementary particles, and the amplitude of the induced acoustic oscillations exceed the mean pressure in the fluid by an order of magnitude or more. The trajectory of the motion of the interface is divided into three stages, in each of which it is assumed that there is a certain constant value of the pressure. This enables an approximate analytical solution of the problem to be obtained. It is shown that there is an agreement with the numerical solution of the initial problem that is sufficient for qualitative estimates. The solution obtained enables a parametric analysis of the expansion–contraction processes of a cavity to be carried out and enables the volume of calculations in the numerical modelling to be reduced when taking account of additional factors.

Prediction of air pollution from motor transport on city streets and districts

Purpose. Development of applied numerical model for prediction of atmospheric pollution rate on streets and districts of a city taking into account chemical transformations of pollutants. Methodology. To solve hydrodynamic task of determining velocity field of wind flow in street the method of discrete vortices was used, in the city district – the method of separation flows vortex of ideal incompressible fluid, for solution equation of pollutant transfer - alternating triangular implicit difference scheme. Findings.An efficient numerical model using the type of «street canyons»for prediction of air quality on city streets and districts with emissions from motor transport considering chemical transformations of pollutants was designed in the work. Originality.The numerical model, which allows taking into account impact of buildings on pollutants dispersion and requiring a small consumption of computer time during practical realization was created. The advantage of the model is the possibility of rapid calculation of emissions dispersion in the street with considering the chemical reactions of pollutants. Practical value. The developed numerical model can be used in practice during the planning of new highways in new urban areas or in the renovation of old ones, for a series of calculations that require search of different variants for arrangement of buildings, highways, under certain weather conditions.

Nonstationary cylindrical vortex in an ideal fluid

Equations of rotationally symmetric motion of an ideal incompressible fluid are considered. A class of solutions to these equations, described by a hyperbolic equation of the fourth order with one space variable, for which an initial boundary-value problem is formulated, is distinguished. The new class of exact solutions of the Euler equations was used to describe the a nonstationary cylindrical vortex in an ideal fluid.

Hydrodynamic theory of magnetic fields due to electron currents in a straight wire

In previous articles in this journal, the author presented hydrodynamic models of the electron as a sink and the positron as a source for the continuous flow of a primordial incompressible ideal fluid from sources to sinks in ordinary three-dimensional space generating the electrostatic field. The return flow from a sink to a source occurs through a wormhole tube (called an ether string) in four-dimensional space-time. It was assumed that the finite core of a primitive electron and the core of a primitive positron would be trapped at the boundary surface of the cylindrical core of a linear vortex in the fluid. The interaction between the flow component into the electron sink and the vortex field would wind the ether string linking the electron and positron cores into a helix (called an ether spring) located at the surface of the vortex core. Loosely bound electrons in metal atoms can migrate through the lattice of a metal crystal while still remaining linked by an ether spring to the positron in the proton in the original atomic nucleus. When a piece of metal is broken into two or more parts, some of the free electrons in each part may remain linked to protons in another part. These electrons are referred to as externally linked conduction electrons. When such electrons are induced to move along a straight wire, a magnetic field is generated around the wire. The magnetic field is due to regions of vorticity in the primordial fluid around the wire. These regions result from a criss-crossing of fixed and moving ether springs. The fixed ether springs link conduction electrons in metal objects outside the wire to positive ions in the wire. The axes of ether springs linked to the moving conduction electrons in the wire are straight and inclined forward through a small Lorentz angle so that they cross over the fixed straight ether springs linked to positive ions in the wire. It can be shown that the overlapping ether springs produce regions with a vorticity with vorticity vectors tangent to circles centered on the straight wire axis. Alternating currents will cause these regions of vorticity to move outward away from the wire corresponding to radio waves.

Double layer air current during the high speed train flow-around

Purpose. Investigation of the jet stream mechanism during the high-speed passenger train flow around. On the basis of theoretical studies to determine the distribution of air flow velocity field, the pressure on the upper layer of the double layer air current, the pressure at the boundary of both layers of double-layer air current. Methodology. To solve the problem on the top layer of air current the Zhukovsky method and the method of theory of ideal incompressible fluid jets was applied. Then to solve the boundary layer equations the pressure at the boundary of both layers was determined, the Navier - Stokes equation, which takes into account the influence of the top layer on the bottom layer was set up. Airflow in the upper region is a potential, stationary; the air in the vicinity is considered as the compressible one. Findings. The distribution of the velocity field for both layers of air current was obtained. The pressure on the upper layer of the double layer air flow was defined. Originality. The problem of a double layer air current formed by the high-speed train movement consisting of the boundary layer above the car surfaces and the air current with no viscosity was studied for the first time. Practical value. The problem under study is used to provide the safety of high-speed train traffic.

Обтекание колеблющегося крыла потоком идеальной несжимаемой жидкости

Обтекание колеблющегося крыла потоком идеальной несжимаемой жидкости

Effects of inertia and stratification in incompressible ideal fluids: pressure imbalances by rigid confinement

AbstractConsequences of density stratification are studied for an ideal (Euler) incompressible fluid, confined to move under gravity between rigid lids but otherwise free to move along horizontal directions. Initial conditions that generate horizontal pressure imbalances in a laterally unbounded domain are examined. The aim is to show analytically the existence of classes of initial data for which total horizontal momentum evolves in time, even though only vertical forces act on the fluid in this set-up. A simple class of such initial conditions, leading to momentum evolution, is identified by systematic asymptotic expansions of the governing inhomogeneous Euler equations in the small-density-variation limit. These results for Euler equations are compared and confirmed with long-wave asymptotic models, which can handle arbitrary density variations and provide closed-form mathematical expressions for limiting cases. In particular, the role of wave dispersion arising from the fluid inertia is captured by the long-wave models, even for short-time dynamics emanating from initial conditions outside the models’ asymptotic range of validity. These results are compared with direct numerical simulations for variable-density Euler fluids, which further validate the numerical algorithms and the analysis.

Poromechanics approach for modeling closed-cell porous materials with soft matrices

The main purpose of this paper is the formulation of a modeling framework for closed-cell porous materials by means of the nowadays classical approach of poromechanics. The cavities being saturated with a single fluid phase, Biot’s theory drastically simplifies since no fluid diffusion is present in this case. Nevertheless, the fluid contribution to the macroscopic behavior is still present and the fluid mass conservation controls the pore pressure that contributes to the total stress at the macro-scale. A constitutive law for the saturating fluid is introduced that encompasses both ideal gas and incompressible fluids as particular cases. The approach is built around the unified framework of continuum thermodynamics which is crucial in setting the convenient form for the state laws. As the finite strain range is of concern, the modeling strongly depends on the Eulerian porosity law adopted in the spatial configuration. Two model examples are discussed with parametric studies to show the effectiveness of the proposed approach.

Interaction of Alfven waves in the linearized system of magnetohydrodynamics for an incompressible ideal fluid

Interaction of Alfven waves in the vicinity of multiple characteristic roots is considered. The asymptotic behavior for solutions of the Cauchy problem for the linearized equations of magnetohydrodynamics with highly oscillating initial data is obtained.

Initial stage of the inclined impact of a smooth body on a thin fluid layer

The two-dimensional problem of the oblique impact of a free rigid body with a smooth flat bottom on a thin layer of an ideal incompressible fluid is considered. The initial stage of the impact when the body motion is accompanied by the formation of jets on the boundary of the body-fluid contact zone is investigated. The problem is solved jointly, i.e., the fluid flow initiated by the body motion and the motion of the body itself are determined simultaneously. A priori the “body-fluid” contact zone is unknown and determination of its time evolution represents a significant difficulty and the method of asymptotic matched expansions is used to overcome this difficulty. A system of integro-differential equations is obtained and the motion of the body under the action of hydrodynamic loads is investigated numerically on the basis of this system. It is shown that the hydrodynamic force exerted on the body during the impact is maximum precisely in the initial stage; therefore, the motion of the body varies fairly significantly in time considered.

On the topology of steady-state solutions of hydrodynamic and vortex consequences of the Vlasov equation and the Hamilton-Jacobi method

INTRODUCTIONThe Vlasov–Poisson and Vlasov–Maxwell equations were successfully applied to describe plasma in[1–3]. These equations were reduced to MHDtypeequations by using a hydrodyn amic substitution in [4–8].For the equations of an ideal incompressible fluid,Arnold proved a theorem on the structure of steadystate solutions based on the presence of two commuting vector fields [9]. Kozlov generalized this construction to a contractible fluid [10]. In this paper, we investigate the possibility of such constructions for the Vlasov–Poisson and Vlasov–Maxwell equations.1. THE VLASOV–POISSON EQUATIONSThese equations describe the motion of electrons,electrostatic plasma, or selfgravitating objects, suchas galaxies or galactic clusters. Consider the followingslightly more general case of the Vlasov–Poisson system for gravitational electrostatics [11]:(1)Here, the α denote types of particles,

The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.

Vortex rings: history and state of the art

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.

Real beads on virtual strings: Charged particles on magnetic field lines

We discuss a similarity between the drift of a charged particle inside a slowly moving solenoid and the motion of a fluid element in an ideal incompressible fluid. This similarity can serve as a useful instructional example to illustrate the concepts of magnetic field lines and magnetic confinement.

Bodies without secondary flows in their 3D boundary layers

There is no secondary flow in a boundary layer where streamlines coincide with the surface geodesic curves and earlier this coincidence was mathematically proven for constant pressure surfaces in ideal incompressible fluid. Two 3D inverse problems on determination of such surfaces are solved here: Design of a surface between the given body bow and an initially undetermined body cylindrical central part; Fitting such a surface with an initially undetermined bow to the given body cylindrical part.