Предмет исследования: работа посвящена исследованию ламинарного течения несжимаемой жидкости, в котором движущиеся в цилиндрической трубе спутные коаксиальные слои вращаются в противоположных направлениях. В литературе течение получило название контрвихревого. Цели: совершенствование теоретической модели контрвихревого ламинарного течения. В турбу лентном диапазоне течение характеризуется интенсивным перемешиванием движущейся среды и гашением ее механической энергии. Оба свойства находят практическое применение: первое-в технологиях, включающих смешивание неоднородных и многофазных сред; второе-для гашения механической энергии потоков жидкостей или газов в высоконапорных гидротехнических водосбро сах и для подавления шума авиадвигателей, гребных винтов. Теоретическое исследование ламинарных контрвихревых течений позволяет выявить общие физические закономерности их гидродинамики. Материалы и методы: в основу теоретической модели ламинарного контрвихревого течения положен метод разложения дифференциальных уравнений Навье-Стокса в ряды Фурье-Бесселя. Результаты: получены уточненная теоретическая модель ламинарного контрвихревого течения, основанная на снятии указанного допущения, и уточненные формулы расчета радиально-продольных распределений азимутальных и аксиальных скоростей в исследуемом течении в виде рядов или произведений рядов Фурье-Бесселя. Распределения азимутальных и аксиальных скоростей представлены графически в виде их профилей. Выводы: повышена точность аналитического расчета полей скоростей в контрвихревых течениях при малых числах Рейнольдса.

The article is devoted to the theoretical study of laminar flows with the coaxial layers rotating in opposite directions moving along the pipe. These flows have a wide practical application potential in technologies of mixing multiphase and heterogeneous media in microbiology, chemistry, ecology, heat engineering, power engineering, civil engineering and engine and rocket science. Such flows have a complicated three-dimensional structure. The theoretical model of the test flow is based on the Navier – Stokes's equations and Fourier – Bessel's method of expansion of differential equations. The article presents the formulas and graphs showing the radial-axial distributions of tangential, axial and radial flow velocities, stream functions and viscous vortex components. The authors made the theoretical analysis of the kinematic structure of such flows.

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.

No abstract