scholarly journals LAGRANGIAN FORMALISM IN PROBLEMS OF SMALL OSCILLATIONS OF VORTEX FLOWS AND ITS CONNECTION WITH THE VARIATIONAL PRINCIPLE FOR IDEAL INCOMPRESSIBLE HYDRODYNAMICS OF VORTEX LINES

2019 ◽  
Vol 47 (1) ◽  
pp. 74-77
Author(s):  
V.F. Kopiev ◽  
S.A. Chernyshev
Keyword(s):  
Incompressible Fluid ◽  
Quadrupole Moment ◽  
Displacement Field ◽  
Vortex Flow ◽  
Lagrangian Formalism ◽  
Lagrangian Mechanics ◽  
Vortex Flows ◽  
Lagrange Equations ◽  
Small Perturbations ◽  
Lagrangian Variables

The paper discusses the description of vortex flows of an ideal incompressible fluid based on the formalism of Lagrangian mechanics. Using the displacement field of liquid particles as a generalized coordinate, we write out the Lagrangian describing the dynamics of small perturbations (Kopiev, Chernyshev, 2018). The corresponding Lagrange equations are the equation for the displacement field (Drazim, Reid, 1981): This equation is equivalent to the Helmholtz equation for vorticity perturbations. The displacement field is defined as the difference in the positions of liquid particles on trajectories in disturbed and undisturbed flows. Although this definition is given in terms of Lagrangian variables associated with liquid particles, the displacement field itself is an Euler variable, expressed through velocity and vorticity perturbations. An example of using Lagrangian to solve the problem of conservation of the quadrupole moment of a vortex flow is considered. Using the Noether theorem, conditions on a stationary flow are obtained, under which the quadrupole moment of small perturbations of this flow is an integral of motion (Kopiev, Chernyshev, 2018). It is shown that these conditions are satisfied for the jet flows uniform along the longitudinal coordinate. The result obtained is important in aeroacoustics due to the fact that the quadrupole moment of the vortex flow represents the main term of the decomposition of a compact acoustic source in Machnumber (Lighthill, 1952; Crow, 1970; Kopiev, Chernyshev, 1995). The generalization of these results to the nonlinear case is considered. The Lagrangian is obtained for an arbitrary nonlinear displacement field: nowhere Gis Green’s function of the Laplace equation. The corresponding Lagrange equations coincide with the differential equations describing the nonlinear dynamics of the displacement field (Drazin, Reid, 1981). Expansion of the Lagrangian in small perturbations to quadratic terms gives the Lagrangian of the linear system. The question of the relationship of the proposed approach to the description of the dynamics of an incompressible fluid and known approaches based on the formalism of Lagrangian mechanics with the coordinates of liquid particles as generalized coordinates (Chapman, 1978; Goncharov, Pavlov, 2008; Kuznetsov, Ruban, 1998) is considered. It is shown that the transformation of the Lagrangian obtained in (Kuznetsov, Ruban, 1998) to the Lagrangian can be carried out by transforming Lagrangian variables (coordinates of liquid particles) to Eulerian variables (displacement field). This study was supported by the Russian Science Foundation, project No. 17-11-01271.

One-dimensional model of fourth order for rods with loading on lateral boundary: The case of rectangular cross section

2016 ◽  
Vol 22 (12) ◽  
pp. 2269-2287 ◽  
Author(s):  
Erick Pruchnicki
Keyword(s):  
Cross Section ◽  
Fourth Order ◽  
Displacement Field ◽  
Rectangular Cross Section ◽  
Transverse Dimension ◽  
Lateral Boundary ◽  
Dimensional Model ◽  
Equilibrium Equations ◽  
Lagrange Equations ◽  
One Dimensional

We propose deducing from three-dimensional elasticity a one dimensional model of a beam when the lateral boundary is not free of traction. Thus the simplification induced by the order of magnitude of transverse shearing and transverse normal stress must be removed. For the sake of simplicity we consider a beam with rectangular cross section. The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod and we truncate the potential energy at the fourth order. By considering exact equilibrium equations, the highest-order displacement field can be removed and the Euler–Lagrange equations are simplified.

scholarly journals Dynamics of laminar flows with coaxial oppositely-rotating layers

Vestnik MGSU ◽  
2019 ◽  
pp. 332-346
Author(s):  
Andrey L. Zuikov
Keyword(s):  
Reynolds Number ◽  
Active Zone ◽  
Vortex Flow ◽  
Stokes Equations ◽  
Laminar Flows ◽  
Radial Velocities ◽  
Navier Stokes ◽  
Recirculation Region ◽  
Vortex Flows ◽  
Navier Stokes Equations

Introduction. The work relates to the scientific foundations of hydraulic and energy construction and is devoted to the study of laminar flows with coaxial oppositely-rotating layers. In the literature, such flows are called counter-vortex. In the turbulent range, counter-vortex flows are characterized by intensive mixing of the medium, which is widely used in the technologies of mixing non-natural and multi-phase media in thermal and atomic energy, in systems of mass- and heat transfer, in chemistry and microbiology, ecology, engine and rocket production. The aim of the theoretical study is to study the physical laws of the hydrodynamics of counter-vortex flows. Research methods. The theoretical Navier-Stokes equations and continuity equation are the basis of the theoretical model of the laminar counter-vortex flow. Results. Assuming the radial velocities are much less than the azimuthal and axial velocities and taking the Oseen approximation, the solution of the Navier - Stokes equations is obtained as Fourier - Bessel series or products of Fourier - Bessel series. In particular, the following were obtained: formulas for calculating the radial-longitudinal distributions of the normalized azimuthal, axial and radial velocities in the flow under study, the velocities are presented graphically in the form of radial profiles; equations for the calculation of current lines and viscous vortex fields, which are also presented in the form of graphs, were obtained. The two-layer and four-layer counter-vortex flows are considered. The analysis of the obtained theoretical results is performed. Conclusions. On the axis at the beginning of the active zone, the formation of a return flow with significant negative velocities is characteristic. This leads to the formation of a recirculation region, the mass exchange between which and the external flow is absent. Cascades of concentric vortexes of such high intensity that are not found in streams of a different nature are generated in the active zone. Calculation formulas include exp (-λ2x/Re) exponent multiplied by Reynolds number in degree b = 0 or b = -1, therefore increasing Reynolds number when b = 0 leads to proportional transfer of arbitrary characteristic counter-vortex flow down the pipe; and at b = -1, the bias of characteristic is accompanied by a proportional decrease in its scale.

scholarly journals Numerical modeling of plasma oscillations with consideration of electron thermal motion

2018 ◽  
pp. 194-206
Author(s):  
А.А. Фролов ◽  
Е.В. Чижонков
Keyword(s):  
Traveling Wave ◽  
Numerical Experiments ◽  
Plasma Temperature ◽  
Thermal Motion ◽  
Initial Region ◽  
Small Perturbations ◽  
Plasma Oscillations ◽  
Lagrangian Variables ◽  
Increasing Temperature ◽  
Good Agreement

Исследовано влияние теплового движения электронов на плоские нерелятивистские нелинейные плазменные колебания. Численно и аналитически показано, что при учете теплового движения колебания трансформируются в бегущую волну. При этом амплитуда волны растет с ростом температуры, что способствует выносу энергии из первоначальной области локализации колебаний. Для численного моделирования построена схема метода конечных разностей на основе эйлеровых переменных. При использовании лагранжевых переменных для приближения малых возмущений получены распределения максимумов электронной плотности в зависимости от температуры плазмы. Аналитические результаты находятся в хорошем соответствии с численными экспериментами. The effect of electron thermal motion on plane nonrelativistic nonlinear plasma oscillations is studied. It is shown numerically and analytically that when the thermal motion is taken into account, the oscillations are transformed to a traveling wave. At the same time, the wave amplitude grows with increasing temperature, which promotes the removal of energy from the initial region of oscillation localization. A finite-difference scheme is proposed for the numerical simulation on the basis of Eulerian variables. When using the Lagrangian variables to approximate small perturbations, the distributions of electron density maxima are obtained depending on the plasma temperature. The obtained analytical results are in good agreement with numerical experiments.

Constrained Lagrangian Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Lagrange Equation ◽  
Constrained Systems ◽  
Lagrangian Mechanics ◽  
Lagrange Equations ◽  
Minimal Set ◽  
Holonomic Constraints ◽  
Constrained Optimisation ◽  
Constraint Equations ◽  
Linearly Independent ◽  
Set Up

This chapter builds on the previous two chapters to tackle constrained systems, using Lagrangian mechanics and constrained variations. The first section deals with holonomic constraint equations using Lagrange multipliers; these can be used to reduce the number of coordinates until a linearly independent minimal set is obtained that describes a constraint surface within configuration space, so that Lagrange equations can be set up and solved. Motion is understood to be confined to a constraint submanifold. The variational formulation of non-holonomic constraints is then discussed to derive the vakonomic formulation. These erroneous equations are then compared to the central Lagrange equation, and the precise nature of the variations used in each formulation is investigated. The vakonomic equations are then presented in their Suslov form (Suslov–vakonomic form) in an attempt to reconcile the two approaches. In addition, the structure of biological membranes is framed as a constrained optimisation problem.

scholarly journals Noncommutative Effects on the Fluid Dynamics and Modifications of the Friedmann Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Kai Ma
Keyword(s):  
Fluid Dynamics ◽  
Poisson Bracket ◽  
Mass Density ◽  
Equations Of Motion ◽  
Friedmann Equation ◽  
Noncommutative Space ◽  
Lagrangian Formalism ◽  
Noncommutative Algebra ◽  
Symmetry Properties ◽  
Lagrangian Variables

We propose a new approach in Lagrangian formalism for studying the fluid dynamics on noncommutative space. Starting with the Poisson bracket for single particle, a map from canonical Lagrangian variables to Eulerian variables is constructed for taking into account the noncommutative effects. The advantage of this approach is that the kinematic and potential energies in the Lagrangian formalism continuously change in the infinite limit to the ones in Eulerian formalism and hence make sure that both the kinematical and potential energies are taken into account correctly. Furthermore, in our approach, the equations of motion of the mass density and current density are naturally expressed into conservative form. Based on this approach, the noncommutative Poisson bracket is introduced, and the noncommutative algebra among Eulerian variables and the noncommutative corrections on the equations of motion are obtained. We find that the noncommutative corrections generally depend on the derivatives of potential under consideration. Furthermore, we find that the noncommutative algebra does modify the usual Friedmann equation, and the noncommutative corrections measure the symmetry properties of the density function ρ(z→) under rotation around the direction θ→. This characterization results in vanishing corrections for spherically symmetric mass density distribution and potential.

Aplicación del formalismo de Lagrangianos Controlados en la regulación de velocidad del motor de inducción trifásico

2021 ◽  
Author(s):  
◽  
Yaima González Acosta
Keyword(s):  
Mechanical Systems ◽  
Electrical Machines ◽  
Lagrangian Formalism ◽  
Control Law ◽  
Lagrange Equations ◽  
Two Phase ◽  
Main Research ◽  
Three Phase ◽  
First Time ◽  
Selection Of

In this work the Controlled Lagrangian Formalism applied to electrical machines is explored for the first time. It begins with an analysis of the purely mechanical systems, once understood, the study is carried out on a two-phase induction motor, this implying a greater degree of complexity because there is no reference that has done it before. Finally, this study is expanded to the three-phase motor, this being the main research object of the project. The main guide used was the Bloch article cite bloch2000controlled on the analysis of mechanical systems. Regarding the procedure, the first thing that is done is the selection of the generalized coordinates, the Lagrangian is proposed and the model is obtained from it through the Euler-Lagrange equations, followed by that the symmetries are identified (which in the case of MI is especially interesting because these symmetries are obvious from the choice of coordinates) and the configuration space is divided into vertical and horizontal directions, the horizontal directions are redefined and the Controlled Lagrangian is proposed. Finally, generalized forces are sought, using Noether's Theorem as support and thus establishing the control law. The development to obtain the Controlled Lagrangian and the control law is done in detail, explaining each step of the procedure and using specific algebraic methods of this formalism that are strongly based on the geometric structure of the variety of configuration. The results obtained are an approach in the direction of Controlled Lagrangians applied electrical machines.

Transition of Taylor–Görtler vortex flow in spherical Couette flow

1983 ◽  
Vol 132 ◽  
pp. 209-230 ◽  
Author(s):  
Koichi Nakabayashi
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Vortex Flow ◽  
Basic Flow ◽  
Taylor Number ◽  
Vortex Flows ◽  
Spherical Couette Flow ◽  
Görtler Vortices ◽  
Inner Sphere ◽  
Spherical Couette

The critical Taylor number, phenomena accompanying the transition to turbulence, and the cellular structure of Taylor–Görtler vortex in the flow between two concentric spheres, of which the inner one is rotating and the outer is stationary, are investigated using three kinds of flow-visualization technique. The critical Taylor number generally increases with the ratio β of clearance to inner-sphere radius. For β [les ] 0.08, the critical Taylor number in spherical Couette flow is smaller than in circular Couette flow, but vice versa for β > 0.08. A pair of toroidal Taylor–Görtler vortices occurs first around the equator at the critical Reynolds number Rec (or critical Taylor number Tc). More Taylor–Görtler vortices are added with increasing Reynolds number Re. After reaching the maximum number of vortex cells, as Re is increased, the number of vortex cells decreases along with the various transition phenomena of Taylor–Görtler vortex flow, and the vortex finally disappears for very large Re, where the turbulent basic flow is developed. The instability mode of Taylor–Görtler vortex flow depends on both β and Re. The vortex flows encountered as Re is increased are toroidal, spiral, wavy, oscillating (quasiperiodic), chaotic and turbulent Taylor–Görtler vortex flows. Fourteen different flow regimes can be observed through the transition from the laminar basic flow to the turbulent basic flow. The number of toroidal and/or spiral cells and the location of toroidal and spiral cells are discussed as a means to clarify the spatial organization of the vortex. Toroidal cells are stationary. However, spiral cells move in relation to the rotating inner sphere, but in the reverse direction of its rotation and at about half its speed. The spiral vortices number about six, and the spiral angle is 2–10°.

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