Cavity Detachment from a Wedge with Rounded Edges and the Surface Tension Effect

Cavity flow around a wedge with rounded edges was studied, taking into account the surface tension effect and the Brillouin–Villat criterion of cavity detachment. The liquid compressibility and viscosity were ignored. An analytical solution was obtained in parametric form by applying the integral hodograph method. This method gives the possibility of deriving analytical expressions for complex velocity and for potential, both defined in a parameter plane. An expression for the curvature of the cavity boundary was obtained analytically. By using the dynamic boundary condition on the cavity boundary, an integral equation in the velocity modulus was derived. The particular case of zero surface tension is a special case of the solution. The surface tension effect was computed over a wide range of the Weber number for various degrees of cavitation development. Numerical results are presented for the flow configuration, the drag force coefficient, and the position of cavity detachment. It was found that for each radius of the edges, there exists a critical Weber number, below which the iterative solution process fails to converge, so a steady flow solution cannot be computed. This critical Weber number increases as the radius of the edge decreases. As the edge radius tends to zero, the critical Weber number tends to infinity, or a steady cavity flow cannot be computed at any finite Weber number in the case of sharp wedge edges. This shows some limitations of the model based on the Brillouin–Villat criterion of cavity detachment.

Variants of the Hodograph Method for Solving a System of Two Quasilinear Equations

Строится решение задачи Коши для системы двух квазилинейных однородных уравнений в частных производных первого порядка при помощи метода годографа, позволяющего преобразовать решение квазилинейных уравнений в частных производных первого порядка к решению некоторого линейного дифференциального уравнения в частных производных второго порядка с~переменными коэффициентами. Показано, что различные варианты метода годографа - стандартного, на основе закона сохранения и обобщенного метода годографа, позволяющие строить решение задачи Коши в неявной форме, в конечном итоге, приводят к одному и тому же результату и отличаются лишь объемом технической работы. Доказательство осуществляется путем вычисления инвариантов Лапласа для канонической формы линейного дифференциального уравнения в частных производных второго порядка. В случае, когда уравнения допускают явную связь исходных переменных с инвариантами Римана и соответствующее линейное уравнение метода годографа позволяет указать явную форму функции Римана - Грина, описан способ построения явного решения на линиях уровня неявного решения. Задача Коши для системы двух квазилинейных уравнений в частных производных первого порядка сводится к задаче Коши для некоторой системы обыкновенных дифференциальных уравнений. В качестве примера приведено точное неявное решение для системы слабо-нелинейных уравнений. Все рассмотренные методы и способ построения явного решения можно применять для уравнений гиперболического и эллиптического типов. В случае гиперболических уравнений возможно построение автомодельных и разрывных решений (после добавления условий на разрывах), а также решений многозначных по пространственной координате (если такие решения допускаются постановкой задачи). Несмотря на то, что на заключительном этапе метода задачу Коши для обыкновенных дифференциальных уравнений приходится решать численно, никаких аппроксимаций уравнений в частных производных, типичных для конечно-разностного метода, метода конечных элементов, метода конечных объемов и т. п. не используется. Метод является точным в том смысле, что погрешность вычислений связана лишь с точностью интегрирования обыкновенных дифференциальных уравнений.

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

Extraction of density-layered fluid from a porous medium

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a simple finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Such nonlinear profiles cannot be considered with the eigenfunction approach. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear case there are certain values of flow rate for which a steady solution does not occur. References Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 9th ed. National Bureau of Standards, Washington, 1972. Bear, J. and Dagan, G. Some exact solutions of interface problems by means of the hodograph method. J. Geophys. Res. 69(8):1563–1572, 1964. doi:10.1029/JZ069i008p01563 Bear, J. Dynamics of fluids in porous media. Elsevier, New York, 1972. https://store.doverpublications.com/0486656756.html COMSOL Multiphysics. COMSOL Multiphysics Programming Reference Manual, version 5.3. https://doc.comsol.com/5.3/doc/com.comsol.help.comsol/COMSOL_ProgrammingReferenceManual.pdf Farrow, D. E. and Hocking, G. C. A numerical model for withdrawal from a two layer fluid. J. Fluid Mech. 549:141–157, 2006. doi:10.1017/S0022112005007561 Henderson, N., Flores, E., Sampaio, M., Freitas, L. and Platt, G. M. Supercritical fluid flow in porous media: modelling and simulation. Chem. Eng. Sci. 60:1797–1808, 2005. doi:10.1016/j.ces.2004.11.012 Lucas, S. K., Blake, J. R. and Kucera, A. A boundary-integral method applied to water coning in oil reservoirs. ANZIAM J. 32(3):261–283, 1991. doi:10.1017/S0334270000006858 Meyer, H. I. and Garder, A. O. Mechanics of two immiscible fluids in porous media. J. Appl. Phys., 25:1400–1406, 1954. doi:10.1063/1.1721576 Muskat, M. and Wycokoff, R. D. An approximate theory of water coning in oil production. Trans. AIME 114:144–163, 1935. doi:10.2118/935144-G GNU Octave. https://www.gnu.org/software/octave/doc/v4.2.1/ Yih, C. S. On steady stratified flows in porous media. Quart. J. Appl. Maths. 40(2):219–230, 1982. doi:10.1090/qam/666676 Yu, D., Jackson, K. and Harmon, T. C. Disperson and diffusion in porous media under supercritical conditions. Chem. Eng. Sci. 54:357–367, 1999. doi:10.1016/S0009-2509(98)00271-1 Zhang, H. and Hocking, G. C. Axisymmetric flow in an oil reservoir of finite depth caused by a point sink above an oil-water interface. J. Eng. Math. 32:365–376, 1997. doi:10.1023/A:1004227232732 Zhang, H., Hocking, G. C. and Seymour, B. Critical and supercritical withdrawal from a two-layer fluid through a line sink in a bounded aquifer. Adv. Water Res. 32:1703–1710, 2009. doi:10.1016/j.advwatres.2009.09.002 Zill, D. G. and Wright, W. S. Differential Equations with Boundary-value problems, 8th Edition. Brooks Cole, Boston USA, 2013.

Non-destructive testing of multilayer medium by the method of velocity of elastic waves hodograph

The method of velocity of elastic waves hodograph, aimed at non-destructive testing of structurally heterogeneous composite materials and products based on them, as well as multilayer products and constructions, is considered. The theoretical basis for determining the propagation velocity of elastic waves in a multilayer medium by the hodograph method is given. Based on the studies, recommendations are given for determining the propagation velocity of elastic waves in each individual layer of a multilayer medium, which allows non-destructive testing of the physicomechanical characteristics of each layer of a multilayer medium. It is shown that in addition to simple multiple reflections in a homogeneous medium, in a multilayer medium with parallel interfaces consisting of two or more layers, complex types of multiple reflected waves and mixed waves (reflected-refracted and refracted-reflected) can arise. The main task of applying the low-frequency ultrasonic method is to determine the acoustic parameters of the propagation of elastic waves (velocities, amplitudes, spectra). The main methods for determining the elastic wave velocities are considered, based on the hodograph equation of the indicated reflected waves in a multilayer medium.

Time Optimal Trajectories for a Mobile Robot Under Nonsliding and Radius-of-Turn Constraints

The time-optimal path problem for a point mass mobile robot is considered. Given initial and target states, we seek the time optimal path subject to the following constraints: (1) A limitation on its maximal linear acceleration; (2) a speed-dependent nonsliding condition; and (3) a minimal radius of turn. The paper formulates and analyzes the time optimal path problem using standard optimal control formulation with extensive use of the classical Hodograph method. Based on the analysis, the time optimal path consists of five path primitives. Numerical solutions are obtained to support and illustrate the analysis.

Impulsive Motion Inside a Cylindrical Cavity

Experimental studies of supercavitating models moving at speeds in the range from 400 m/s to 1000 m/s revealed a regime of bouncing motion, in which the rear part of an axisymmetric body periodically bounces against the free boundaries of the supercavity. The impulsive force generated by the impacts is the main concern in this paper. The analysis is performed in the approximation of two-dimensional potential flow of an ideal and incompressible liquid with negligible surface tension effects. The primary interest of the study is to determine the added mass taking into account the shape of the cavity. The theoretical study is based on the integral hodograph method, which makes it possible to obtain analytic expressions for the flow potential and for the complex velocity in an auxiliary parameter plane and obtain a parametric solution to the problem. The problem is reduced to a system of two integro-differential equations in two unknowns: the velocity magnitude on the cavity boundary and the slope of the velocity angle to the body. The equations are solved numerically using the method of successive approximations. The obtained results show that the added mass of an arc impacting a cylindrical cavity depends heavily on the arc angle. As the angle tends to zero or the radius of the cavity tends to infinity, the obtained solution predicts the added mass corresponding to a plate impacting a flat free surface.

DAGESTAN

For the Dagestan seismic network consisting of nine analog and seven digital seismic stations in 2013, the values of the maximum and reliable earthquake detection range are given depending on the energy class КР. On their basis, the contours of the energy representation of earthquakes with Кmin=5, 6, 7 are constructed, and earthquakes with КР³8 are recorded without loss in the entire area of responsibility of the Dagestan branch of GS RAS within the coordinates: j=41.0–44.0°N, l=45.0–49.0°Е. The coordinates of hypocenters were determined by the hodograph method using local hodographs. The classification of earthquakes, as before, was carried out in T.G. Rautian energy classes. 684 earthquakes with КР=4.2–12.7 and total energy of SЕ=2.13×1012 were recorded in total. The notable events were the Georgian-Dagestan and Kichi-Hamri earthquakes, the last of which was described in a separate article of this yearbook. On the basis of the map of the epicenters of all earthquakes, seismicity was described in detail in seven regions.

Stability of Periodic Motions and Synthesis of Relay Sampled Data Control Systems

This article is devoted to research and design of relay systems with control of data sampling. It is shown that the time sample has a significant effect on the parameters of periodic oscillations. We propose an exact method for analyzing periodic modes in digital self-oscillatory control systems with a two-position relay element and a linear piecewise-linear part is proposed. The proposed approach extends the phase hodograph method to the class of systems operating in discrete time. Two approaches have been developed to assess the stability of periodic motions in such systems. In the first approach, a discrete representation of a plant is considered and areas of stability are defined for each possible limit cycle. The sampling of the control system causes a delay in the switching of the relay in a batch mode in comparison with the continuous case. The second approach assumes the replacement of a discrete system by an equivalent continuous system with a time delay. Further, the asymptotic orbital stability of self-oscillations in a relay control system (RCS) with a delay is estimated. We consider the linearization of relay systems with digital control of the input signal. It is also shown that when linearizing a relay element in a digital RCS using a useful signal, the relay transfer ratio will belong to a certain range of values. Synthesis of corrective devices for relay control systems with regard to digital implementation has been reviewed. At the stage of optimization of parameters of the relay control system, the sample is taken into account. The model example demonstrates an advantage in the synthesis of digital technologies. It is shown that when optimizing the controller parameters with regard to time discretization, it was possible to provide the desired frequency of self-oscillations, which ensures the best accuracy of the tracking mode.