Steep Wave Front Research Articles

A dissipative finite element model for free surface flow

A dissipative finite element model applicable to one-dimensional free surface flow is presented. The classical Galerkin formulation seems to be inappropriate to simulate open channel flow when discontinuities are introduced in the flow domain. A weighted residual formulation based on discontinuous weighting functions is presented, which provides better results than the standard Galerkin approximation. Thus the problem of the undesirable noise in the vicinity of the discontinuities is overcome and the finite element method becomes capable of simulating steep wave fronts. This dissipative model is applied to continuous flow in a section of the Axios river in Greece and it is also applied to discontinuous flow in a rough rectangular horizontal channel. The computational results are compared with those obtained by finite difference schemes and with experimental data.

A dissipative finite element model for discontinuous unsteady flow in open channel

The purpose of this paper is to present a dissipative finite-element formulation applicable to discontinuous one-dimensional open-channel flow. The traditional Galerkin approximation seems to be inappropriate for simulation of open-channel flow when discontinuities are introduced in the flow domain; the solutions show undesirable oscillations in the vicinity of the discontinuity. The method presented herein is a weighted residual formulation based on discontinuous weighting functions which are not identical to the shape functions; it provides smoother and more accurate results than the standard Galerkin approximation, offering to the finite-element method the ability of simulating steep wave fronts. The dissipative model is applied to flow with surges in a rectangular horizontal channel. Computational results are compared with experimental data and with results obtained by finite-difference schemes.

The Use of Partial Discharge and Impulse Voltage Testing in the Evaluation of Interturn Insulation Failure of Large Motors

Several 13.2 kV, 21,000 HP synchronous motors failed as a result of interturn insulation breakdown. Investigation of motor failures indicated that the cause was related to high magnitude, steep fronted surge waves (References 1 and 2). Partial discharge (corona) related failures were discounted without any actual measurements. To confirm the validity of the previous investigation, partial discharge testing was conducted on four new 21,000 HP motors and on one failed 21,000 HP motor. Additionally, surge impulse testing was performed on the failed motor to assess interturn insulation endurance. This paper presents the results of above tests. It is concluded that failure of motors could not be caused by corona, Confirmation of previous investigation results indicated that partial discharge testing can be used as a diagnostic tool in the evaluation of interturn insulation of large motor.

PAB (Parabolic and Backwater) an unconditionally stable flood routing scheme particularly suited for real time forecasting and control

The paper describes a flood-routing scheme based upon the successive application of locally linearized parabolic equations in integral form and computation of the steady state backwater profiles (PAB). The method, which is unconditionally stable (a consequence of its integral form) is shown to be particularly suited to real-time forecasting and control. The method is generally applicable to flood routing problems in mild slope channels and rivers (apart from special cases where the inertial terms play an important role) and can also be used for routing steep front waves such as those produced by dam breaks.

Characteristics of pressure-wave propagation in a compliant tube with a fully collapsed segment

In order to model the fluid dynamics of Korotkoff sound generation when the artery under the cuff is fully collapsed during most of the heart cycle, the characteristics of pressure-wave propagation in a long silicone-rubber tube were studied experimentally. The central portion of this tube was designed to collapse to zero cross-sectional area as a result of high negative transmural pressure, thus simulating a collapsed artery. Propagation of a single half-sinusoidal pressure wave in and around this segment was studied in detail by pressure, velocity and tube-longitudinal-shape measurements.A very steep wave front (shock wave) capable of producing a short tapping sound was formed by an overtaking phenomenon in the fully collapsed tube segment and it propagated into the inflated tube distal to the collapsed segment. An empirical equation relating the flow rate penetrating into the collapsed segment, the incident-wave pressure and the external pressure Pc over the collapsed segment was obtained. This equation predicts that the pressure-wave propagation in a fully collapsed segment depends only on the flow rate into the collapsed segment.The initial internal pressure of the tube distal to the collapsed segment Pd is one independent variable in the high-cuff-pressure condition. The amplitude of the steep wave front and the shape of the pressure wave in the inflated tube distal to the collapsed segment are governed by Pc–Pd and the flow rate penetrating the collapsed segment. For the same flow rate, if Pc–Pd is lower than a critical value, the amplitude of the pressure in the distal tube decreases with increasing Pd because of positive pressure-wave reflection at the exit of the collapsed segment. On the other had, if Pc–Pd is higher than that value, no wave reflection occurs and the amplitude of the pressure wave is independent of Pd. In the latter case a severe constriction exists near the distal end of the collapsed segment, and flow occurs as two thin high-speed jets.

Energy balance analysis of nonlinear combustion instability

To be of practical value, analytical models for nonlinear rocket motor combustion instability must adequately represent: 1) steep fronted waves, 2) limit cycle operation, and 3) triggering phenomena. Inclusion of all these effects in an approximate analysis based on perturbation expansion procedures requires retention of terms to at least the third order in the perturbation parameter representing the system amplitude. In this paper, the acoustic energy balance method is extended into the nonlinear regime and combined with a simplified geometrical representation of the wave structure to greatly simplify this procedure. A practical approximate model for axial pressure fluctuations in a tubular rocket motor results. Effects of nonlinear combustion and nonisentropic energy losses in the steep wave fronts are represented. Since the relative amplitudes of the Fourier components that comprise the traveling shock waves may be assumed to remain effectively fixed near amplitude limiting conditions, great mathematical simplifications accrue. The behavior of the system is governed by a simple polynomial expression which gives the rate of change of the composite system amplitude. The coefficients are integral expressions taken over the chamber volume and its bounding surfaces. The familiar exponential growth rate model appears as the first-order (linear) limiting case. The model demonstrates all of the nonlinear characteristics observed in motor data and shows promise as an easily used diagnostic tool. It has been successfully employed to simulate actual experimental results from pulser tests in inert chambers and from sonic end-vent burner tests.

Multiple-wavelength phase-shifting interferometry

This paper describes a method to enhance the capability of two-wavelength phase-shifting interferometry. By introducing the phase data of a third wavelength, one can measure the phase of a very steep wave front. Experiments have been performed using a linear detector array to measure surface height of an off-axis parabola. For the wave front being measured the optical path difference between adjacent detector pixels was as large as 3.3 waves. After temporal averaging of five sets of data, the repeatability of the measurement is better than 25-A rms (λ = 6328 A).

Damping mechanisms and shock-like transitions in the human arterial tree

The unidirectional nature of blood flow in mammalian arteries encourages the modelling of propagating pressure and flow pulses in the arterial tree by means of a one-dimensional mathematical approximation. It has been shown that this approximation yields realistic results in the proximal as well as in the distal regions of a simulated arterial conduit, providing that the damping induced by the viscoelasticity of the vessel walls as well as by the viscosity of the blood are properly taken into account. Often, models were formulated on the basis of an elastic formulation of the arterial wall properties and of an approximation for the damping due to blood viscosity which is derived from parabolic velocity profiles (“Poiseuille model”). Yet, such models are known to produce shock-like transitions in the propagating pulses which are not observed in man under physiological conditions. The viscoelastic damping characteristics of the vessel walls are such that they reduce the tendency of shock formation in the model. This can be shown with the aid of a wave front expansion, from which criteria for the steepening of wave fronts are derived. The application of the results to the human arterial system shows that shock-like waves are not to be expected under normal conditions. However, in case of a pathologically increased pressure rise at the root of the aorta, shock-like transitions may still develop in the periphery. Such circumstances can occur, e. g., in cases of severe aortic valve insufficiency (without aortic stenosis). Furthermore, the damping characteristics associated with blood viscosity are shown to impede mathematical shock formation in the strict sense a priori. Steepening of a wave front according to the criteria mentioned above still may occur, however, with increasing steepening this process is gradually counterbalanced by the dissipative mechanism induced by viscous friction. Finally, the relative influence of the damping due to wall viscoelasticity and blood viscosity is discussed as a function of the relevant parameters describing the geometry and pulsatility of the typical blood flow conditions.

Gemesis of Korotkoff Sound at High Cuff Pressure

The genesis of Korotkoff sound at high cuff pressure has been made clear by the model expreimental study using the thin walled silicon rubber tube being modified to have zero internal cross sectional area as an artery ehen compressed by the relatively higher external pressure than the internal one. The sound is generated by the steep pressure wave front formed in the course of pressure wave propagation through the completely collapsed tube segment under the cuff. The amplitude of the pressure wave penetrating into the collapsed tube segment and that of the steep wave front reaching to the distal end of the segment, measured from the artery also weakens correspondingly and, at some Peb, the sound becomes unaudible. The pressure wave form depends strongly on the transmural pressure at the distal end of the collapsed segment.

A node-moving algorithm with application to Burgers equation and the Moltz problem

Burgers' equation models nonlinear wave behaviour; a typical solution exhibits a steep wavefront which is difficult to represent or resolve using traditional finite difference or finite element methods. By using a finite element mesh having dynamic nodes good results for Burgers' equation are obtained using very few elements. The method developed yields a one to one mapping from some initial nodal distribution (usually uniform) to a different nodal distribution reflecting properties of the solution. The method is applied to the Moltz problem, which is a two-dimensional potential problem where there is a boundary singularity. The initial uniform mesh changes to one in which the nodes concentrate around the singularity, resulting in a more accurate solution.

Numerical techniques for solving nonlinear instability problems in solid rocket motors

The results of an investigation to select a satisfactory numerical method for calculating the propagation of steep-fronted, shock-like waveforms in a solid rocket motor combustion chamber are presented. A number of different numerical schemes were evaluated by comparing the results obtained for three related but simpler problems: the shock-tube problem, the linear wave equation, and nonlinear wave propagation in a closed tube. The most promising method, a combination of the Lax-Wendroff, Hybrid, and Artificial Compression techniques, was incorporated into an existing nonlinear combustion instability program. The capability of the modified program to treat steep-fronted wave instabilities in low-smoke tactical motors was verified by solving a number of motor test cases with disturbance amplitudes as high as 80% of the mean pressure.

Transient Effects of Nonlinear Wave Propagation in Magnetized Plasmas

The evolution of a steep wavefront propagating perpendicular to an externally applied homogeneous magnetic field is considered. The propagation is accompanied by the excitation of low-frequency waves. When the nonlinear interaction is taken into account we observe the evolution of a modulational instability trailing the wavefront.

On the Transient Effects of Nonlinear Wave Propagation in Magnetized Plasmas

Approximate analytical solutions are obtained for the set of coupled nonlinear partial differential equations which govern the evolution of a steep wavefront propagating perpendicular to an externally applied homogeneous magnetic field. Several of the features noted in a numerical solution of these equations by Pécseli and Thomsen are predicted by these approximate solutions.

Distribution of switching surges in the line-end coils of cable-connected motors

When specifying the winding insulation of electrical machines it is necessary to know the electrical stresses to which the winding will be subjected. In the case of interturn insulation the most severe stresses are caused by steep-fronted voltage waves produced by circuit-breaker closure. To some extent such steepfronted waves are reduced in severity by the time they reach the machine terminals by distortion and attenuation in the cable connecting the machine to the circuit breaker. In the studies described the interturn voltages in the line-end coils of cable-connected machines are evaluated. The equivalent circuits for the machine winding, the cable, and the energising source are obtained and compounded into an equivalent circuit for the complete system. The interturn voltages in the line-end coils of the machine are obtained from solutions of this equivalent circuit. Studies are made with a number of different cable lengths and types, and for a number of energisation source representations. Single-pole and 3-pole simultaneous closure of the circuit breaker are considered. The use of series inductors at the circuit breaker, or shunt capacitors at the motor terminals, for reducing the severity of the interturn voltages is also investigated. The results obtained indicate that the type of cable insulation and the length of cable are both significant in influencing the magnitude of interturn voltages. Energisation source impedance has a large effect. Comparison of seriesinductance and shunt-capacitance methods of surge protection reveals that both methods are effective, but that both produce other unwanted effects.

The propagation of solitary waves and bores over a porous bed

Amplitude attenuation for the solitary wave propagating over a porous bed of finite or infinite depth is derived assuming irrotational motion in the fluid above. Velocity components in the porous bed are derived for small-amplitude solitary waves and bores. Attenuation rates are compared with known results for frictional dissipation and the conclusion drawn that a porous bed contributes very significantly to wave damping. The motion in the porous bed due to a pressure distribution simulating the passage of a bore is analyzed and its influence on the dissipation processes in a bore discussed. Fluidization of the porous bed under bores and steep fronted waves is considered.

Physical and pathophysiological effects of blast

Blast is a general term used to convey the effect produced when an explosive is detonated in any medium. Blast-produced injury implies those detrimental changes occurring in an organism whilst it is being subjected to the pressure field produced by an explosion, whether such changes are produced directly or indirectly by the explosive phenomena. This article examines briefly the physical and pathophysiological effects of blast resulting from detonations in air and in water. Air blast is further subdivided into blast from conventional and from nuclear explosions. Detonation produces a high speed chemical decomposition of a solid or liquid explosive into gas. Almost instantaneously the space previously occupied by the explosive is filled with gas and there is release of large amounts of thermal energy. The hot gaseous products develop a very high pressure which is transmitted to the surrounding medium and propagated in all directions as a shock wave, travelling at about the speed of sound. Typically this is a steep-fronted wave rising in a few microseconds and decaying over a period of milliseconds, depending upon the nature of the explosive and the medium which surrounds it. These also determine the character of the shock pulse and the subsequent phenomena.

Magnetic Field Generation by Ablation Waves or Shocks Propagating in Inhomogeneous Plasma

A simple model is given for magnetic field generation by steep steady-profile wavefronts such as ablation waves or shocks propagating across initially field-free nonuniform plasmas or fluids. The model enables one to obtain approximate values for the field penetration through the ablation layer into the interior of a laser target or pellet. These fields could also be produced and measured in laboratory ionizing shock tubes. (AIP)

One dimensional elasto-plastic wave interaction and boundary reflections

Elasto-plastic wave propagation in a one dimensional rod is treated by utilizing finite element analysis. The time dependence is handled by means of a variable time step numerical integration method. Elasto-plastic reflections from a fixed boundary and interaction of plastic waves are discussed. Theoretical expressions are presented for determining the stress response for a material exhibiting bilinear hardening behavior during elasto-plastic wave interaction and boundary reflection. These expressions are employed to evaluate predictions made by the numerical procedure. In the computations, steep wave fronts are approximated by ramps. The steepness of these ramps may be increased by refining the finite element grid. Excellent agreement is obtained between numerical and theoretical elastic and inelastic stress response levels.

Theoretical investigation of nonlinear three-dimensional instabilities in liquid rockets with real nozzles

An approximate analytical technique has been developed for the solution of nonlinear three-dimensional, transverse and axial combustion instability problems that are frequently observed in liquid-propellant rocket motors. This theory is an extension and generalization of previous analyses, which could analyze either transverse or axial instabilities in liquid combustors with quasi-steady nozzles, to the practical situations of three-dimensional instabilities in combustors with conventional DeLaval nozzles. Unlike the quasi-steady nozzle, the presence of a conventional nozzle imposes restrictions upon the behavior of both the amplitudes and phases of the oscillations at the nozzle entrance plane. The applicability of the solution technique, which is based on the Galerkin method, is demonstrated by analyzing the nonlinear stability of a cylindrical liquid-rocket combustor with uniform injection of propellants at one end and a conventional nozzle at the other end. Crocco's pressure sensitive time-lag model is used to describe the unsteady combustion process. Calculated results for transverse and axial instabilities indicate that the developed model can predict the transient behavior and nonlinear wave shapes that have been observed during unstable motor operation as well as limit-cycle amplitudes and frequencies typical of unstable motor operation. These results establish the relationship that exists between the resulting instability (i.e., waveform, final amplitude, and final frequency), the combustion parameters (i.e., interaction index, n, and time-lag, \\ ̄ gt), and the chamber Mach number and length-to-diameter ratio. Results indicate that the limit-cycle amplitude increases with increasing sensitivity of the combustion process to pressure oscillations. For transverse instabilities, calculated pressure waveforms exhibit sharp peaks and shallow minima, and the frequency of oscillation is always within a few percent of the frequency of one of the chamber's acoustic modes. For axial instabilities, the theory predicts the presence of a steep-fronted wave moving back and forth along the combustor. In both cases calculations of pressure and velocity perturbations at the nozzle entrance plane show that the approximation to the nozzle boundary condition is very good. The studies presented in this paper represent significant progress toward the development of a unified nonlinear theory for the solution of general three-dimensional, transverse and axial combustion instability problems.

Application of the implicit method to surges in open channels

A technique is proposed by which continuous numerical computation of flows with surges in open channels can be handled with an implicit formulation. The one‐dimensional gradually varied flow equations of continuity and momentum are written in finite difference form for the point (I + ½, J + θ) in the x, t plane, where I is the distance counter, J is the time counter, and θ is a weighting parameter that is varied from 0.5 to 1.0. A computer program was written to solve the problem in which the discharge hydrograph is given at the upstream end of a channel and a stage‐discharge relationship is specified at the downstream end. This program was applied to different flow situations in laboratory channels and natural rivers. It was found that θ = 0.5 produced results with a steep wave front but with considerable numerical instability. The use of θ = 1.0 gave results with considerable diffusion of the wave front. The use of θ = 0.60 produced results that had a steep wave front and only a minor degree of numerical instability. This value of θ resulted in good agreement between calculated and measured hydrographs at the downstream end of the channel in most applications. However, the optimum value of θ depends on channel friction and the steepness of the wave front.