Dynamics of laminar flows with coaxial oppositely-rotating layers

Abstract

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.

Structure of laminar flows with oppositely-rotating coaxial layers

“…In the article presents the results of theoretical studies of the structure of counter-vortex flows based on recent publications [10][11][12][13][14][15]. In these publications in the cylindrical coordinate system r -x ( Fig.…”

“…In these publications in the cylindrical coordinate system r -x ( Fig. 1), the steady (∂/∂twhere t is the current time) symmetric relative to the pipe axis (∂/∂) flow of incompressible fluid in the laminar range is described by the Navier -Stokes's equations [16]…”

“…Decisions of the system of equations (2) with the boundary conditions (3) are given in the above publications [10][11][12][13][14][15] which in chronological order improve and complement each other. But in general, the most comprehensive solutions for today are the radial-axial distributions of azimuthal and axial velocities given in monograph [10] in the form of Fourier -Bessel's series or the products of Fourier -Bessel's series…”

Flow kinematics with oppositely rotating coaxial layers

“…Their wide spread is associated with a number of important properties inherent to such flows compared with conventional axial flows. Swirled flows are used in processes that ensure atomization of liquid fuel [1,2], mixing [3] and dispersion of fluids, forming aerosols, flames [4] and other technological processes. On the basis of deep and extensive studies of swirling flows, the devices have been started to be created that make it possible to repeatedly intensify the processes of energy, mass and heat exchange in heat engineering, hydraulic machine engineering, aircraft building, mechanical engineering and in a number of other industries [5,6].…”

Stability of counter vortex flows in hydraulic engineering construction

“…The hydraulic characteristics of counter vortex flows were studied in [8]. An analytical study of the main flow field based on the Oseen model, is given in [9].…”