Waves in thin-walled elastic tubes containing an incompressible inviscid fluid

The effect of the non-linearity of the governing equations on the propagation of waves in fluid filled elastic tubes is investigated. Results are obtained by the method of characteristics for a particular form of pressure pulse applied at the end of a semi-infinite initially uniform tube. An expression is obtained for the distance along the tube at which shock formation is predicted. Two different hyperelastic materials whose elastic properties model those of biological tissue are considered for the tube walls. Numerical results are presented in graphical form.     

Hamilton principle for an inviscid compressible fluid model in Eulerian coordinates

It is shown that the well-known variational principles for the ideal compressible fluid model in Eulerian coordinates have the following deficiencies:

1. They are not related to the corresponding variational principles in Lagrangian coordinates;
2. The variation procedure in these variational problems does not lead to the equations of motion themselves in the Euler form; rather it leads to relations which correspond to definite classes of solutions of the Euler equations. Here allowance for the equations of the constraints imposed by the adiabaticity and continuity conditions limits the region of application of these variational principles to only potential flows;
3. More general results, involving flows other than potential, are achieved by artificial selection of certain additional constraint conditions imposed on the quantities being varied, and in this case additional clarification is required to ascertain whether any inviscid compressible fluid flow is the extremum of the corresponding variational problem.

A new formulation of the Hamilton principle for the inviscid compressible fluid in Eulerian coordinates is suggested which is free from these deficiencies.     

Statistical fluctuations of the charged surface of an inviscid fluid

A fluctuation mechanism of origin of the deformation of a charged liquid surface, which explains the experimental facts, is proposed. The Hamilton equations that describe the dynamics of an inviscid fluid with electrically charged free surface are formulated. Solutions of these equations, which describe the evolution of the free surface, are found. The spectra of the surface disturbances are investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 117–124, May–June, 1992.     

Steady axisymmetric flow with vorticity of an inviscid fluid in multistage turbomachines

The direct problem of steady axisymmetric flow of a gas with vorticity through a multistage turbomachine is formulated precisely and a generalized solution is constructed by a variational-difference method. The turbomachine is represented schematically by an annular channel in which there are fixed (₁) and rotating (₂) three-dimensional cascades and channels free of them (₀).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–15, October–December, 1981.     

Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface

The surface waves generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface are analytically investigated for two- and three-dimensional cases. The fluid is assumed to be inviscid, incompressible and homogenous. The inertial surface represents the effect of a thin uniform distribution of non-interacting floating matter. Four types of unsteady concentrated disturbances and two kinds of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an instantaneous/oscillating impulse on the surface, an initial impulse on the surface of the fluid, and an initial displacement of the surface. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the surface elevation are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motion for large time with a fixed distance- to-time ratio are derived by using the method of stationary phase. The effect of the presence of an inertial surface on the wave motion is analyzed. It is found that the wavelengths of the transient dispersive waves increase while those of the steady-state progressive waves decrease. All the wave amplitudes decrease in comparison with those of conventional free-surface waves. The explicit expressions for the freesurface gravity waves can readily be recovered by the present results as the inertial surface disappears.     

Weakly non-linear waves in a tapered elastic tube filled with an inviscid fluid

In the present work, treating the artery as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the longwave approximation, we have studied the propagation of weakly non-linear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equation admits a solitary wave-type solution with variable wave speed. It is observed that, the wave speed decreases with distance for positive tapering while it increases for negative tapering. It is further observed that, the progressive wave profile for expanding tubes (a>0) becomes more steepened whereas for narrowing tubes (a<0) it becomes more flattened.     

Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid

In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P₀ and an axial stretch λ_z and, in the course of blood flow, a finite time-dependent displacement is added to this initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given.     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.     

Surface waves at the interface between an inviscid fluid and a dipolar gradient solid

This paper is about the dispersion analysis of surface waves propagating at the interface between an inviscid fluid and a higher gradient homogeneous elastic solid modelled as a dipolar gradient continuum. In order to compare the results, a second gradient model is also evaluated. The analysis is carried out by finding the roots of the secular equation, and by carefully studying their physical meaning. As it is well known, higher gradient continua are dispersive, i.e. phase and group velocities are frequency dependent. As a consequence, the existence of surface waves will indeed depend on frequency. In order to investigate the behaviour of surface waves in this specific fluid–solid configuration, a complete dispersion analysis is performed, with a particular focus on the frequency range in which the phase velocity of shear waves is lower than the speed of waves of the fluid. Surface waves of the type Leaky Rayleigh and Scholte–Stoneley are observed in this frequency range. This work extends the knowledge on surface waves in the case of higher gradient solids and applications of these results can be found in the field of non-destructive damage evaluation in micro structured materials, composites, metamaterials and biological tissues.     

求解无粘可压Euler方程组的虚拟流方法

首先将三阶Godunov型半离散中心迎风格式推广到四阶,之后再将该新的四阶半离散中心迎风格式与Level Set方法以及虚拟流方法结合起来,成功地处理了非反应激波问题和多介质流中的爆轰间断问题。由于Level Set函数能隐式地追踪到界面的位置,而虚拟流的构造能隐式地捕捉到界面的边界条件,故而本文的方法可以很自然地推广到多维情况。     

Nonlinear statics and dynamics of a simply supported nonuniform tube conveying an incompressible inviscid fluid

Nonlinear static and dynamic behaviour of a simply supported fluid-conveying tube, which has a constant inner diameter and a variable thickness is analysed analytically and numerically. Nonlinear static bending is considered in two loading cases: (i) a tube subjected to supercritical axial compressive forces acting at its edges or (ii) a tube loaded by concentrated bending moments, which provide a symmetrical (with respect to the mid-span) shape of a tube. The nonlinear governing equations of motions are derived by using Hamilton's principle. The elementary plug flow theory of an incompressible inviscid fluid is adopted for modelling a fluid–structure interaction. The flow velocity is taken as the sum of a principal constant ‘mean’ velocity component and a fairly small pulsating component. Firstly, eigenfrequencies and eigenmodes of a deformed tube are found from linearised equations of motions. Then resonant nonlinear oscillations of a tube about its deformed static equilibrium position in a plane of static bending are considered. A multiple scales method is used and a weak resonant excitation by the flow pulsation is considered in a single-mode regime and in a bi-modal regime (in the case of an internal parametric resonance) and the stability of each of them is examined. The brief parametric study of these regimes of motions is carried out.     

Investigation of trajectories of inviscid fluid particles in two-dimensional rotating boxes

The particle trajectories of inviscid fluid flow within two-dimensional rotating (elliptic, triangular, and square) boxes are numerically investigated. The source panel method is employed to represent the instantaneous potential interior flow field, and the Runge–Kutta method is used to track the fluid particles. The analytic solutions for the fluid trajectories for the elliptic box are used to verify the numerical accuracy of the method. The numerical error can be reduced to the level of the round-off error if the panels are properly configured and an appropriate number of panels is used. The stagnation of the particles at the corners of the triangular box is successfully predicted with this method. The corner of the square box is found to be a singularity. A logarithmic complex potential is proposed to account for the singularity, using which the stagnation of the particles at the corner in the square box is also captured. The natural frequency of the particles in the rotating elliptic box is constant throughout the flow domain, and the fluid trajectories are epitrochoidal curves. In the triangular box and the square box, the natural frequency strongly depends on the particle position, and the particle trajectories are similar to epitrochoidal curves. In general, the trajectory patterns depend only on the box rotating frequency and the natural frequency of the fluid particle motion.      

A computational method for simulating transient motions of an incompressible inviscid fluid with a free surface

A new numerical method has been developed for the analysis of unsteady free surface flow problems. The problem under consideration is formulated mathematically as a two-dimensional non-linear initial boundary value problem with unknown quantities of a velocity potential and a free surface profile. The basic equations are discretized spacewise with a boundary element method and timewise with a truncated forward-time Taylor series. The key feature of the present paper lies in the method used to compute the time derivatives of the unknown quantities in the Taylor series. The use of the Taylor series expansion has enabled us to employ a variable time-stepping method. The size of time increment is determined at each time step so that the remainders of the truncated Taylor series should be equal to a given small error limit. Such a variable time-stepping technique has made a great contribution to numerically stable computations. A wave-making problem in a two-dimensional rectangular water tank has been analysed. The computational accuracy has been verified by comparing the present numerical results with available experimental data. Good agreement is obtained.     

Motion of inviscid fluid tongues and bubbles in a Hele-Shaw cell occupied by a viscoelastic fluid

Problems of steady-state motion of an inviscid fluid tongue or bubble in a Hele-Shaw cell occupied by a viscoelastic fluid are considered. The problems generalize the well-known Saffman-Taylor problems [1, 2] in the approximation in which the pressure jump on the interface between the two fluids depends on the normal velocity of the interface [4]. In the zero approximation the results obtained by means of an iteration technique are in good agreement with the analytic solutions obtained by Saffman and Taylor [1, 2].__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 117–123. Original Russian Text Copyright © 2005 by Entov, Kolganov, and Kolganova.     

Entropy jump across an inviscid shock wave

The Shock jump conditions for the Euler equations in their primitive form are derived by using generalized functions. The shock profiles for specific volume, speed, and pressure and shown to be the same, however, density has a different shock profile. Careful study of the equations that govern the entropy shows that the inviscid entropy profile has a local maximum within the shock layer. We demonstrate that because of this phenomenon, the entropy propagation equation cannot be used as a conservation law.This research was supported in part under NASA Contract No. NAS1-19480, while the second author was in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681-0001, U.S.A.     

On energy focusing with multiple sources in an inviscid field

A concept has been proposed for focusing underwater energy. An array of independently detonated spherical TNT charges has been suggested to achieve this aim. The goal of the present research effort is to model this system and identify optimal array geometry and detonation timing. A reduced-order model must be developed due to the highly iterative nature of most optimization algorithms. The fast computational speed required by the optimization problem renders the use of the sophisticated numerical solvers impractical. An analytical, physics-based model must be developed to satisfy the computational time and accuracy demands of the multiple charge problem. This model must capture shock propagation speed, pressure–time histories throughout the fluid domain, and shock wave interaction phenomena. Such a model is presented herein.     

Bifurcation and chaos of a flag in an inviscid flow

A two-dimensional model is developed to study the flutter instability of a flag immersed in an inviscid flow. Two dimensionless parameters governing the system are the structure-to-fluid mass ratio M^⁎ and the dimensionless incoming flow velocity U^⁎. A transition from a static steady state to a chaotic state is investigated at a fixed M^⁎=1 with increasing U^⁎. Five single-frequency periodic flapping states are identified along the route, including four symmetrical oscillation states and one asymmetrical oscillation state. For the symmetrical states, the oscillation frequency increases with the increase of U^⁎, and the drag force on the flag changes linearly with the Strouhal number. Chaotic states are observed when U^⁎ is relatively large. Three chaotic windows are observed along the route. In addition, the system transitions from one periodic state to another through either period-doubling bifurcations or quasi-periodic bifurcations, and it transitions from a periodic state to a chaotic state through quasi-periodic bifurcations.     

The distortion of a non-aligned magnetic field by the flow of an inviscid conducting fluid past a circular cylinder

The distortion of a magnetic field by the flow of a conducting fluid past a cylinder of the same permeability is found for small and infinite values of the magnetic Reynolds number. For small values good agreement is obtained with the results of Seebass and Tamada when the flow is aligned with the field at large distances from the body.For infinite magnetic Reynolds number, all the lines of force are dragged into the cylinder and upstream and downstream wake regions are present on the axis of the flow.     

Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.