Wave flow of a liquid film together with a flow of gas

The wave flow of a thin layer of viscous liquid in conjunction with a flow of gas was considered in a linear formulation earlier [1, 2]. In this paper the problem of the wave flow of a liquid film together with a gas flow is solved in a nonlinear setting. On this basis relationships are derived for calculating the parameters of the film and the hydrodynamic quantities.Ivanovo. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 12–18, January–February, 1972.     

2D shallow water flow model for the hydraulic jump

A flow model is presented for predicting a hydraulic jump in a straight open channel. The model is based on the general 2D shallow water equations in strong conservation form, without artificial viscosity, which is usually incorporated into the flow equations to capture a hydraulic jump. The equations are discretised using the finite volume method. The results are compared with experimental data and available numerical results, and have shown that the present model can provide good results. The model is simple and easy to implement. To demonstrate the potential application of the model, several hydraulic jumps occurring in different situations are simulated, and the predictions are in good agreement with standard solution for open channel hydraulics. Copyright © 1999 John Wiley & Sons, Ltd.     

Wave flow of a liquid film in a horizontal separator

The calculation of the motion of separated moisture in a linear horizontal separator is made on the basis of the analysis of the development of the waves in a flow of a thin layer of liquid along a vertical surface without allowance for the transverse flow of mass [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–176, March–April, 1985.     

The hydraulic jump in circular jet impingement and in other thin liquid films

The circular hydraulic jump exhibits behavior quite different from that commonly observed in planar jumps. Here we examine experimentally some of the causes and consequences of those differences. We suggest that surface tension plays a dominant role in establishing the shape of the circular jump for impinging jets. The importance of surface tension is a direct result of the thinness of the liquid films normally encountered in circular jump configurations. A sequence of instabilities appears in the jump's structure as the subcritical liquid film becomes thicker and surface tension effects decrease. These conclusions are corroborated by experiments on thin planar films which result in unusual jump structures, like those seen in circular jumps. In addition, we show that the standard momentum balance for the circular jump is effective only at relatively low supercritical Froude numbers or at low ratios of downstream to upstream depth. Typical values of those parameters for circular jumps are often quite large relative to the usual values for planar open-channel flows.List of Symbols d jet diameter - D fictitious downstream drag force - Fr _d jet Froude number, u _f / gd - Fr _h supercritical film Froude number, u _f d ²/8r _j gh ³ - Fr _s subcritical film Froude number, u _f d ²/8r _s gs ³ - g gravitational body force - h local thickness of liquid sheet - p hydrostatic pressure - r radius measured from jet stagnation point - r _j radius at which hydraulic jump begins - r _s radius at which subcritical depth equals s - R radius of curvature of jump interface - Re _d Reynolds number of the jet, u _f d/v. - s liquid sheet thickness after hydraulic jump - u(r,y) radial velocity distribution in liquid film - u _f velocity of impinging jet - _h depth average velocity for sheet of thickness h, u _f d ²/8rh - y distance normal to the wall - We Weber number of jump, s pg/ Greek letters v liquid kinematic viscosity - liquid density - surface tension     

Statistical characteristics of the pressure fluctuation in a hydraulic jump

The author examines experimental data on the correlation in an essentially inhomogeneous, statistically stationary pressure fluctuation field at the bottom of a turbulent flow in the region of a hydraulic jump. Certain data relating to other statistical characteristics of the field in question (in particular, the one-dimensional distribution laws) may be found in [1].The author thanks O. F. Vasil'ev for supervising this research and also V. A. L'vov, N. S. Poleshchuk, and V. V. Zykov for making and adjusting the apparatus for analyzing the oscillograms, and Z. V. Danilov for compiling the programs and processing the experimental data.     

Construction of the underground contour of a hydraulic structure with constant flow velocity sections

The underground contour of an embedded rectangular dam, whose corners are rounded in accordance with curves of constant flow velocity and whose water-permeable base is underlain by a confining layer with a curvilinear roof characterized by a constant flow velocity, is constructed. The corresponding boundary value problem is solved by means of the semi-inverse use of the velocity hodograph method. The results of the numerical calculations are given and the effect of the main determining parameters of the model on the shape and dimensions of the underground contour of the dam and the curvilinear confining layer is analyzed. The limiting cases in which the water-permeable base of the dam has an unbounded thickness, namely, a streamlined apron with a horizontal insert and streamlined sheet piling (tooth), are investigated in detail.     

Simulating air entrainment and vortex dynamics in a hydraulic jump

The air entrainment characteristics of three separate Froude number hydraulic jumps are investigated numerically using an unsteady RANS, realizable k–ε turbulence model, with a Volume of Fluid treatment for the free surface. Mean velocity profiles, average void fraction, and Sauter mean diameter compare favorably with experimental data reported in literature. In all simulations, time-averaged void fraction profiles show good agreement with experimental values in the turbulent shear layer and an accurate representation of interfacial aeration at the free surface. Sauter mean diameter is well represented in the shear layer, and free surface entrainment results indicate bubble size remains relatively unchanged throughout the depth of the jump. Several different grid resolutions are tested in the simulations. Significant improvements in void fraction and bubble size comparison are seen when the diameter to grid size ratio of the largest bubbles in the shear layer surpasses eight. A three-dimensional simulation is carried out for one Froude number jump, showing an improvement in the prediction of entrained air and bubble size compared with two-dimensional results at a substantial increase in computation time. An analysis of three-dimensional vorticity shows a complex interaction between spanwise and streamwise vortical structures and entrained air bubbles. The jump is similar to a turbulent mixing layer, constrained by the free surface, with vortex pairing and subsequent fluctuations in free surface elevation. Downstream fluctuations of the toe are associated with a roll up of the primary spanwise vortex, fluctuations of the free surface, and counter-rotating streamwise vortex pairs. The action of these flow structures is likely responsible for the improvement in three-dimensional results.     

Equalizing plane incompressible fluid flow through a rotating circular lattice of thin curved profiles with radial direction at the inlet

The problem of equalizing the two dimensional flow of an incompressible fluid through a plane lattice of radial plates has been solved approximately by Mitrokhin [1]. In the following we present an approximate solution of the same problem for a cascade of thin curved blades which have a radial direction at the entrance. The bases of the proposed solution are the same assumptions as used in [1]. The formulas of Mitrokhin for the cascade of radial plates are a particular case which derives from the formulas obtained for the flow equalization radius and energy losses associated with separated flow.     

Linear stability in the flow of a liquid film concurrent with a gas flow

The two-phase flow of liquid films are often encountered in practice, but the number of theoretical papers devoted to this problem is limited. The problem of the linear stability of a viscous liquid film subjected to a gas flow has been formulated in [1] and, in somewhat different form, in [2]. The linear stability of plane-parallel motion in films has been studied analytically in [1–8] for some limiting cases. The range of validity of the analytic approaches remains an open question. Therefore, an exact numerical analysis of flow stability over a fairly broad range is required. In the present paper a separate solution of the problem for the gas and the liquid is shown to be possible. The Orr-Sommerfeld equation has been integrated numerically, and the results are compared to the results of analytic calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–146, January–February, 1976.The author is grateful to É. É. Markovich for directing the work and to V. Ya. Shkadov for his interest in the work and many useful comments.     

Vorticity jump across a shock in nonuniform flow

A new relation is derived for the vorticity just downstream of a shock wave when the upstream flow is nonuniform. Aside from the vorticity contribution from a curved shock, there is an amplified upstream vorticity contribution and terms associated with upstream Mach number and stagnation enthalpy gradients, along the shock, that may be present even if the upstream flow is irrotational.     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

Mechanical action of a traveling hydraulic jump on a vertical cylinder

The technique and the results of an experimental investigation of the longitudinal force component and the vertical coordinate of the point of its application when bore-type waves impinge on a vertical cylinder are presented. The bore was generated by removing a vertical gate providing a difference of free surface levels in a channel with a smooth horizontal bottom. It was found that the presence of a free surface in the incident flow has a considerable effect on the force and its moment about the channel bottom.     

含圆形嵌体弹性平面中径向裂纹问题的超奇异积分方程方法

根据含圆形嵌体平面问题在极坐标下的弹性力学基本解,使用Betti互换定理,在有限部积分意义下将问题归结为两个以裂纹岸位移间断为基本未知量、对于Ⅰ型和Ⅱ型问题相互独立的超奇异积分方程,对含圆形嵌体弹性平面中的径向裂纹问题进行了研究.根据有限部积分原理,建立了问题的数值算法.计算结果表明,嵌体半径、裂纹位置及材料剪切弹性模量等都对裂纹应力强度因子具有较为明显的影响.     

Experimental investigation flow structure and hydraulic resistance of a cylindrical duct with injection a fan slot jet

The paper describes results of an experimental study of pressure and velocity fields arising during normal injection of a radial slot jet into ducted flow. The experiments were carried out for slots of two different widths and for injection parameters varying in a broad range. The pressure profile along the duct length plotted in generalized coordinates was found to be quite a universal distribution. Experimental correlations for the minimum rarefaction in the separation region behind the injected jet were obtained, and comparison was made with the results of simplest numerical analysis. Results of measurements of local hydraulic losses are presented for the duct section where the normal injection of the slot jet was organized. The experimental data are shown to be underestimated compared with the results predicted by the theory of perfect mixing for a ducted flow with mass supply. The possible reasons of hydraulic losses coefficient behavior are discussed.