Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic <Emphasis Type="Italic">S</Emphasis>-Model

The nonisothermal steady rarefied gas flow driven by a given pressure gradient (Poiseuille flow) or a temperature gradient (thermal creep) in a long channel (pipe) of an arbitrary cross section is studied on the basis of the linearized kinetic S-model. The solution is constructed using a high-order accurate conservative method. The numerical computations are performed for a circular pipe and for a cross section in the form of a regular polygon inscribed in a circle. The basic characteristic of interest is the gas flow rate through the channel. The solutions are compared with previously known results. The flow rates computed for various cross sections are also compared with the corresponding results for a circular pipe.     

长波在弯曲管道中的传播——Ⅰ变截面弯管中长波传播的基本分析

本文研究长波在三维变截面弯管中的传播问题.通过建立正交曲线坐标系,以波数k和管道横截面的特征半径a的乘积ka作为小参数,对波动方程进行无量纲处理,用正则摄动法,把三维的Helmholtz方程化为二维的Laplace(或Poisson)方程和一维的Webster方程.并分析了管道的几何参数(横截面面积、管道中心线的曲率和挠度)对复速度势渐近展开的各阶项的影响.文中指出,横截面面积的变化首先影响浙近解的零阶项.在横截面的形状具有某种对称性时,管道中心线的曲率首先影响渐近解的二阶项,而挠度首先影响渐近解的三阶项.最后,给出了长波在弯曲圆管中传播的实例.     

Numerical Simulation of Detonation Excitation at Shock Wave Interaction with Dust Cloud in Channel

The study of detonation ability of reactive particle gas mixtures is necessary to prevent industrial explosions in industries where dispersed powders are used. The present paper focuses on numerical simulation of the shock wave interaction with semiinfinite aluminum dust cloud, which is situated inside a plane channel. The cloud fills entirely or partly the channel cross‐section and has initially a rectangular shape. The scenarios of detonation initiation in the cloud are determined depending on the incident shock wave amplitude values. The processes of transformation and spreading of finite width clouds under weak incident shock wave action (when the particles do not ignite) are investigated. The types of an oblique shock wave reflection from the plane of symmetry in the cloud are analyzed. The processes of particle ignition and detonation structure formation at strong incident shock wave action are investigated. Nonstationary periodic fuctuations take place in the detonation flow due to transversal wave effect. Nevertheless the detonation structure established propagates in quasistationary regime. If the incident shock wave is attenuated with a rarefaction wave then the detonation formation fails at clouds of insufficient width.     

On a Control Problem for a Linear System with Measurements of a Part of Phase Coordinates

We consider a three-dimensional unsteady flow with a rotating detonation wave arising in an annular gap of an axially symmetric engine between two parallel planes perpendicular to its symmetry axis. The corresponding problem is formulated and studied. It is assumed that there is a reservoir with quiescent homogeneous propane–air combustible mixture with given stagnation parameters; the mixture flows from the reservoir into the annular gap through its external cylindrical surface toward the symmetry axis, and the parameters of the mixture are determined by the pressure in the reservoir and the static pressure in the gap. The detonation products flow out from the gap into a space bounded on one side by an impermeable wall that is an extension of a side of the gap. Through a hole on the other side of the gap and through a conical output section with a half-opening angle of 45°, the gas flows out from the engine into the external space. We formulate a model of detonation initiation by energy supply in which the direction of rotation of the detonation wave is defined by the position of the energy-release zone of the initiator with respect to the solid wall situated in a plane passing through the symmetry axis. After a while, this solid wall disappears (burns out). We obtain and analyze unsteady shock-wave structures that arise during the formation of a steady rotating detonation. The analysis is carried out within single-stage combustion kinetics by the numerical method based on the Godunov scheme with the use of an original software system developed for multiparameter calculations and visualization of flows. The calculations were carried out on the Lomonosov supercomputer at Moscow State University.     

Stability loss in an infinite plate with a circular inclusion under uniaxial tension

Loss of stability under uniaxial tension in an infinite plate with a circular inclusion made of another material is analyzed. The influence exerted by the elastic modulus of the inclusion on the critical load is examined. The minimum eigenvalue corresponding to the first critical load is found by applying the variational principle. The computations are performed in Maple and are compared with results obtained with the finite element method in ANSYS 13.1. The computations show that the instability modes are different when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. As the Young’s modulus of the inclusion approaches that of the plate, the critical load increases substantially. When these moduli coincide, stability loss is not possible.     

Buckling analysis of a ring under the action of internal pressure in a cylindrical shell

Buckling analysis of a thin cylindrical shell stiffened by rings with T-shaped cross section under the action of uniform internal pressure in the shell is performed. An annular plate stiffened over the outer edge by a circular beam is used as the ring model. The classical ring model, which is a beam with a T-shaped cross section, is inappropriate in this problem, since in the case of the loss of stability, buckling deformations are localized on the ring surface. The beam model does not allow one to find the critical pressure that corresponds to such a loss of stability. In the first approximation, the problem of the loss of stability of the annular plate connected with the shell is reduced to solving the boundary value problem for finding eigenvalues of the annular plate bending equation. Approximate formulas for determining critical pressure are obtained under the assumption that the plate width is much smaller than its inner radius. The results found using the Rayleigh method and the shooting method differ slightly from each other. It has been demonstrated that the critical pressure for rings with rectangular cross section is higher than that for rings with a T-shaped cross section.     

Properties of Detonation Waves

The phenomenon of unphysical wave propagation speeds sometimes occurs in numerical computations of detonation waves on coarse grids. The strong detonation wave splits into two parts, a weak detonation which travels with the speed of one cell per time step and an ordinary shock wave. We analyse a simplified set of equations and look for travelling wave solutions. It is shown that the solution depends on the dimensionless number Kr = μK/Qρ₁. Here μ is the viscosity, K is the rate of reaction, Q is the heat release available in the process and ρ₁ is the density at the unburnt state. It is shown that the density peak of the travelling wave depends on Kr and also, that if Kr is sufficiently large there is no travelling wave solution. The erroneous behaviour above is explained as an effect of the artificial viscosity necessarily inherent in the numerical methods when coarse grids are used. To prevent this unphysical behaviour we suggest the use of an ‘artificial rate of reaction’ such that the actual value of Kr used in the numerical method retains its correct physical value.     

Electric Field-induced dynamical evolution of spiral wave in the regular networks of Hodgkin-Huxley neurons

An additional gradient force is often used to simulate the polarization effect induced by the external field in the reaction-diffusion systems. The polarization effect of weak electric field on the regular networks of Hodgkin-Huxley neurons is measured by imposing an additive term V_E on physiological membrane potential at the cellular level, and the dynamical evolution of spiral wave subjected to the external electric field is investigated. A statistical variable is defined to study the dynamical evolution of spiral wave due to polarization effect. In the numerical simulation, 40000 neurons placed in the 200 × 200 square array with nearest neighbor connection type. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical value, otherwise, spiral wave keeps alive completely. On the other hand, breakup of spiral wave occurs as the intensity of electric field exceeds the critical value in the presence of weak channel noise, otherwise, spiral wave keeps robustness to the external field completely. The critical value can be detected from the abrupt changes in the curve for factors of synchronization vs. control parameter, a smaller factor of synchronization is detected when the spiral wave keeps alive.     

The flow of a combustible mixture with a Chapman–Jouguet detonation wave around a cone

The flow around a circular cone under conditions of Chapman–Jouguet (CJ) self-sustaining detonation is investigated in the classical formulation of an infinitely thin detonation wave (DW) in an inviscid and non-heat conducting combustible mixture. Such conditions for flow around a cone are remarkable in several respects. In 1959, Chernyi and Kvashnina had already shown that, for supersonic flows of a combustible mixture around a cone, CJ detonation, as in the case of a wedge, is not only possible for a strictly fixed cone angle (the “CJ” angle) but also for angles smaller than this (including a zero angle, that is, when there is no cone). In the case of flow around a wedge with an angle that is less than the corresponding CJ angle, a centred rarefaction wave that turns the supersonic flow in the required direction borders on the CJ DW. In the case of a cone, a conical rarefaction flow also borders on the CJ DW. However, if, in the plane case, a uniform supersonic flow adjoins the centred rarefaction wave along its boundary C⁺-characteristic then the conical rarefaction flow is bounded by a conical shock wave (SW), bordered up to the surface of the cone by a conical compression flow. Only for a zero cone angle does the SW degenerate into the C⁺-characteristic and the conical compression flow degenerates into a uniform supersonic flow. The configuration obtained in the general case became the first example of a self-similar solution with two SW of “one family” (the first of them is the DW) diverging from a single point. The results of calculations carried out in this paper with the construction of streamlines and the characteristics of the two families give a fairly complete representation of the above mentioned features of the flows considered.     

Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels

Unsteady rarefied gas flows in narrow channels accompanied by shock wave formation and propagation were studied by solving the Boltzmann kinetic equation. The formation of a shock wave from an initial discontinuity of gas parameters, its propagation, damping, and reflection from the channel end face were analyzed. The Boltzmann equation was solved using finite differences. The collision integral was calculated on a fixed velocity grid by a conservative projection method. A detector of shock wave position was developed to keep track of the wave front. Parallel computations were implemented on a cluster of computers with the use of the MPI technology. Plots of shock wave damping and detailed flow fields are presented.     

Magnetogasdynamic effects on the growth of transverse acoustic waves in a reacting gas

The growth of small disturbances is studied in an electrically conducting and reacting gas permeated by a magnetic field. The model extends to the magnetogasdynamic case some aspects of the theory of spinning detonation waves. It is shown that the stratification introduced by the applied magnetic field constrains the orientation of the propagation and that whilst the rate of wave growth is significantly impeded by the magnetic field it is not totally inhibited.     

The perturbation on initial binding energy for a Majda-CJ combustion model

We investigate the perturbed Riemann problem for a scalar Chapman–Jouguet combustion model – the perturbation on initial binding energy. Under the entropy conditions, we obtain the unique solutions in a neighbourhood of the origin (t?>?0) on the (x,?t) plane. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. That is, the perturbation may transform a Chapman–Jouguet detonation into a strong detonation or a weak deflagration following a shock wave; a strong detonation into a weak deflagration following a shock wave; a Chapman–Jouguet deflagration into a weak deflagration.     

Interaction of oblique deflagration and shock for two-dimensional steady adiabatic combustion system

The interaction of an oblique deflagration wave and an oblique shock wave for two-dimensional steady adiabatic combustion system is analyzed. Using the shock wave polar and combustion wave polar, we exhibit the construction of the solutions. It is found that the deflagration remains if the shock is weak. However, the shock transforms the deflagration into a detonation(DDT) if it is strong or stops the deflagration if it is proper.     

Acoustic eigenoscillations near thin-walled obstacles in an annular cylindrical channel

Under analytical and numerical studies are the acoustic eigenoscillations near thinwalled obstacles in homogeneous circular cylindrical channels. The acoustic eigenoscillations are described by the Neumann problem for the Laplace operator. Using the representation theory of the symmetry groups in the solution space, it is shown that, for a large class of thin-walled obstacles in the circular channels, there always exists a pure point spectrum immersed into the continuous spectrum of the self-adjoint extension of the Laplace operator corresponding to the homogeneous Neumann problem. The dependencies are obtained of the eigenfrequencies on the geometric parameters of the thin-walled obstacles in a homogeneous cylindrical circular channel. The form of the eigenfunctions is investigated. The influence is discussed of the geometric characteristic of the domain on the frequencies, number, and form of eigenoscillations.     

Peristaltic mechanism of a Maxwell fluid in an asymmetric channel

The peristaltic flow of a Maxwell fluid in an asymmetric channel is studied. Asymmetry in the flow is induced by taking peristaltic wave train of different amplitudes and phase. The viscoelasticity of the fluid is induced in the momentum equation. An analytic solution is obtained through a series of the wave number. The leading velocity term denotes the Newtonian result. The first and second order terms are the viscoelastic contribution to the flow. Expressions for stream function and longitudinal pressure gradient are obtained analytically. Numerical computations have been performed for the pressure rise per wavelength and discussed.     

Second order wave loads on large monolithic offshore structures

This paper considers the wave loads on large monolithic offshore structures. The second order wave force formulae developed by Rahman and Heaps, applicable to large circular cylinders in waves, are extended to evaluate the overturning moments on large circular cylinders. The theory is then applied to square section caissons in waves, to predict the wave loads on these structures. These calculations are performed using the exact form of the second order velocity potential, φ₂, with arbitrary wave number, k₂, and the approximate form of φ₂, with twice the value of the wave number of the first order velocity potential. The second order analytical predictions are compared with available experimental data for various ranges of wave parameters for both circular and square caissons in large amplitude waves.     

Wave structure interaction problems in three-layer fluid

Wave structure interaction problems in a three-layer fluid having an elastic plate covered free surface are studied in a three-dimensional fluid domain in both the cases of finite and infinite water depths. Wave characteristics are analyzed from the dispersion relation of the associated wave motion, and approximate results are derived in both the cases of deep water and shallow water waves. Further, the expansion formulae and the associated orthogonal mode-coupling relations are derived for the velocity potentials for the wave structure interaction problems in channels of finite and infinite depths. The utility of the expansion formulae is demonstrated by (1) deriving the source potentials associated with the wave structure interaction problems in a three-layer fluid medium of finite and infinite water depths and (2) analyzing the wave scattering by a partially frozen crack in a floating ice sheet in the three-layer fluid medium in a three-dimensional channel of finite water depth. Various results derived can be used to deal with acoustic wave interaction with flexible structures and other wave structure interaction problems of similar nature arising in different branches of physics and engineering.     

Approximate Analytical and Numerical Solutions of the Stationary Ostrovsky Equation

Approximate stationary solutions of the Ostrovsky equation describing long weakly nonlinear waves in a rotating liquid are constructed. These solutions may be regarded as a periodic sequence of arcs of parabolas containing Korteweg-de Vries solitons at the junctures. Results of numerical computations of the dynamics of the approximate solutions obtained from the nonstationary Ostrovsky equation are presented. It is found that, in the presence of negative dispersion, the shape of a stationary wave is well predicted by the approximate theory, whereas the calculated wave velocity differs slightly from the theoretical value. The stationary solutions in media with positive dispersion are evidently unstable (at least for sufficiently strong rotation), and numerical computations demonstrate a complicated picture of nonstationary destruction.