Standing Wave System Research Articles

Measurement of shock structure and shock–vortex interaction in underexpanded jets using Rayleigh scattering

The density field of underexpanded supersonic free jets issuing from a choked circular nozzle was measured using a Rayleigh scattering-based technique. This reliable and nonintrusive technique is particularly suitable for high-speed flows and is fundamentally superior to the intrusive probes and particle-based techniques such as laser Doppler velocimetry. A continuous wave laser and photon counting electronics were employed for time and phase-averaged density measurements. The use of dust-free air for the entrained flow allowed measurements in the shear layer region. The free jets were produced in the plenum to ambient pressure ratio range of 1.88–5.75, which corresponded to a fully expanded Mach number range of 0.99⩽Mj⩽1.8. A comparative study of schlieren photographs and time-averaged density data provided insight into the shock-cell structures. The radial profiles obtained at various axial stations covering a downstream distance of 10 jet diameters show the development of the jet shear layer and the decay of the shock–cells. The supersonic free jets produced screech sound. A phase-averaged photon counting technique, using the screech tone as the trigger source, was used to measure the unsteady density variation. The phase-averaged density data show the evolution of the large-scale turbulent vortices that are found to be modulated periodically along the flow direction. A comparison with previously obtained data showing near-field pressure fluctuation and convective speed of the organized vortices reveals many interesting dynamics. All quantities show regular spatial modulation. The locations of local maxima in density fluctuations are found to coincide with the high convective speed and the antinode points in the near-field pressure fluctuation. Interestingly, the periodicity of modulation is found to be somewhat different from the shock spacing. Instead it shows that the standing wave system, known to exist in the near-field pressure fluctuation, extends into the jet shear layer. The standing wave is formed between the downstream moving Kelvin–Helmholtz instability waves and the upstream propagating part of sound waves. A detailed field measurement of the unsteady density fluctuation was conducted for the Mj=1.19 and 1.42 jets for which the near-field pressure fluctuation data were obtained previously. The phase-matched, combined plots of the density fluctuation present inside the jet flow, and the pressure fluctuation present just outside the jet boundary provide a charming insight into the shock–vortex interaction leading to the sound wave generation.

Separation of regular waves by a transfer function method

This paper presents a transfer function method (TFM) which can separate a regular wave field into incident and reflected waves based on the linear wave theory. The TFM uses specific transfer functions and corresponding convolution integrals to separate time series data measured in a combined partial standing wave system into incident and reflected waves. After this separation, estimation of the reflection coefficient becomes very easy. All manipulations have been performed in time domain. Furthermore, this method does not involve the calculation of wave heights and/or phase differences. The present method is demonstrated through numerical sample and physical model experiments carried out in a wave flume. Compared with other methods, the TFM gives much better estimates of the incident wave heights for physical model experiments in this study.

Bed Forms Generated on Sandy Bottom by Oblique Standing Waves

This paper presents the results of a study of bed forms generated on a sandy bottom by an oblique standing wave resulting from obliquely incident long-crested waves on a fully reflecting vertical wall. Under antinodes of the oblique standing wave, the sand ripples have crests aligned perpendicular to the wall (type A); under nodes, the ripple crests are aligned with the wall and ride on the top of larger scale sandbars (type C), whereas “island” ripples of a honeycombed structure (type B) appear between nodes and antinodes of the oblique standing wave system. The type A and C ripples could be compared to those generated by long-crested progressive waves, whereas the sandbars at nodes, with ripples riding on them, were compared to those generated by two-dimensional standing waves. Results for the bed-form geometry of type A ripples, type C ripples, and sandbars, as well as type B “bottom islands,” are presented as a function of water particle semiexcursion, mobility number, and Shields parameter.

Modelling of a microwave discharge created by a standing surface wave

A standing surface wave discharge can be obtained by reflection of a TM00 surface wave from a short circuit. A standing wave system provides modulation of plasma characteristics. Wave propagation is studied assuming a plasma in which the electron density is modulated whereas the power for plasma maintenance and the electrical field inside the plasma do not depend on the density, as is the case for traveling wave discharges. Then, the authors determine the conditions for the generation of a standing surface wave pattern. Finally, a comparison of experimental results with the model shows that plasma characteristics can be suitably predicted.

Gravity waves over porous bottoms

Water wave interactions with porous seabeds of granular material were investigated theoretically and experimentally. An assessment on the nature of the resistances in a general, unsteady, porous flow model showed that Darcy's law (linear resistance only) has a very limited application under a natural wave environment; in most cases, inertial and nonlinear resistances should be considered. The case of a linear wave system over a rigid porous bed of finite depth was solved to facilitate comparisons with the experiment. The solutions on wave damping and on wave kinematics (change of wave period or wave length) are quite different from those of available solutions. The maximum wave damping was found to occur when the dissipative resistance (velocity related) is equal to the nondissipative resistance (acceleration related). In the experiment, twelve different porous gravel beds with three water depths (a total of 36 cases) were tested under a standing wave system. The virtual mass coefficient, C a, determined from the experiment data was 0.46 rather than zero as assumed in some publications. The predicted values of this study agree much better with the experiment data than those of the available solutions. The tests were also extended to two extreme cases: beds with thickness of a single grain size and a bed of fine sand. In both cases, the wave kinematics are only slightly affected. However, the damping characteristics are significantly altered. In the former case, the boundary layer effect apparently could no longer be ignored, whereas in the latter case the movement of sand particles caused the rate of damping to increase by several orders of magnitude.

Hydrodynamic coefficients of a monolithic circular offshore structure

AbstractExpressions for both the rectilinear and rotational inertial and damping coefficients for a circular monolithic tower of uniform radius are derived. The analysis matches the fluid velocity, derived from potential theory, with the structural velocity in sway. That is, the motions of the tower are assumed to be in a vertical plane. The analysis is then applied to a tower composed of (lumped‐mass) elements, where the expressions for the added‐mass and damping coefficients are shown to be functions of wave number. The added‐mass is shown to be a product of two wave systems: a travelling wave system, which is responsible for the radiation damping, and a standing wave system, called the evanescent system, which is attached to the structure.The added‐mass of the evanescent system is negative for small wave numbers, while that of the travelling waves is positive. The negative sign simply means that the inertial force of the evanescent waves is 180° out of phase with that of the travelling system. Furthermore, it is shown that the contributions of the two wave systems to the total added‐mass of the structure counteract each other, resulting in a total added‐mass which varies gradually with the wave number.Finally, the analysis is applied to an experiment, and results of the analysis and the experiment are found to agree rather well.

UNDISTORTED FROUDE MODEL FOR SURF ZONE SEDIMENT TRANSPORT

Small scale movable bed wave tank experiments were carried out according to undistorted Froude model laws with the sediment fall time, H/wT, as the governing parameter for scaling the model sediment. Four questions addressed in this study included: (a) the ability to reproduce larger scale model results for both erosional and accretive conditions, (b) the effects of more realistic concave upward initial beach profiles instead of the more usual planar initial slopes, (c) the criterion for onshore-offshore sediment transport, and (d) the capability of the model to simulate post-storm recovery. Based on a comparison with large scale results of Saville (1957), it was found that the model provided good agreement for erosive conditions. For accretive conditions, the results were less conclusive although the general patterns of profile change were similar. The final beach profiles resulting from concave upward initial profiles were found to be substantially different from those for an initially planar profile. It appears that the initially planar profile unrealistically affects the breaker type and results in a more pronounced longshore bar and offshore slopes that are steeper than found in nature. Tests conducted to evaluate the criterion separating onshore-offshore transport suggested a higher value of the fall time parameter, H/wT, than was originally proposed by Dean (1973); this is interpreted to be due to scale effects in most of the model data used in the original development. Tests to simulate post-storm recovery were affected by the presence of "reflection bars" associated with a partial standing wave system. The reflection bars appear to strongly affect the sediment transport limiting the post-storm profile recovery. The most effective recovery was induced by continually changing wave conditions to maintain the wave breakpoint slightly landward of the bar crest.

Erratum: Dynamic analysis of a suspended pump in a vertical well connected to the ocean

In the design of a salt-water intake system which is connected to the ocean, a numerical model is developed to simulate the hydraulic and structural dynamic responses to the ocean's wave action. On the basis of available data, the design wave height is determined from the condition for breaking. For such a complex system, wherein several natural periods are contributing to the total response, a range of wave periods is considered. The standing wave system resulting from the reflection of a progressive wave train from a vertical wall (clapotis) is modelled and the hydrodynamic response of the system is computed. To limit water level fluctuations in the well, a strong damping of the water column is provided by constricting the flow through an orifice plate installed at the bottom of the pump well. The hydrodynamic force on the suspended pumps due to nearby submerged turbulent jet flow is described. A damped single-degree-of-freedom oscillator is employed to represent the structural dynamics of the pumps. Ex...

Dynamic analysis of a suspended pump in a vertical well connected to the ocean

In the design of a salt-water intake system which is connected to the ocean, a numerical model is developed to simulate the hydraulic and structural dynamic responses to the ocean's wave action. On the basis of available data, the design wave height is determined from the condition for breaking. For such a complex system, wherein several natural periods are contributing to the total response, a range of wave periods is considered. The standing wave system resulting from the reflection of a progressive wave train from a vertical wall (clapotis) is modelled and the hydrodynamic response of the system is computed. To limit water level fluctuations in the well, a strong damping of the water column is provided by constricting the flow through an orifice plate installed at the bottom of the pump well. The hydrodynamic force on the suspended pumps due to nearby submerged turbulent jet flow is described. A damped single-degree-of-freedom oscillator is employed to represent the structural dynamics of the pumps. Extensive experiments are required to supplement this work in order to describe the complex system fully. Keywords: dynamic analysis, salt-water intake, waves, hydrodynamic force, suspended pump, submerged jet, oscillator.

Track formation in a liquid hydrogen ultrasonic bubble chamber

Track sensitivity to minimum ionising particles has been demonstrated in liquid hydrogen using only an intense ultrasonic field. Carefully designed transducer systems are shown to be capable of producing pressure amplitudes >;2.8 atm in a standing wave system in liquid hydrogen. The growth of bubbles to visible size (0.1 mm) in less than 0.2 ms, and their collapse in less than 15 ms, indicates that rapid cycling rates of 50–100 pulses per second may be feasible with this technique.

On the mass transport induced by oscillatory flow in a turbulent boundary layer

Oscillatory flow in a turbulent boundary layer is modelled by using a coefficient of eddy viscosity whose value depends upon distance from a fixed boundary. A general oscillatory flow is prescribed beyond the layer, and the model is used to calculate the mass transport velocity induced by this within the layer. The result is investigated numerically for a representative distribution of eddy viscosity and the conclusions interpreted in terms of the mass transport induced by progressive and standing waves. For progressive waves, the limiting value of the mass transport velocity at the outer edge of the layer is the same as for laminar flow. For standing waves, the limiting value is reduced relative to its laminar value but, within the lowermost 25% of the layer, there is a drift which is reversed relative to the limiting value. This is considerably stronger than its counterpart in the laminar case and, in view of the greater thickness of the turbulent layer, it may make a dominant contribution to the net movement of loose bed material by a standing wave system.

The Acoustic Force on a Liquid Droplet in a Stationary Sound Field

A stationary sound field exerts on a spherical inclusion a net acoustic force that tends to propel the sphere toward the nodes or antinodes of the standing wave system. With sufficient acoustic pressure amplitudes, it is possible to trap certain liquid droplets near the antinodes. Measurements are presented of the acoustic pressure amplitudes required to trap small, individual liquid droplets of various liquids near a pressure antinode of a stationary sound field in a cylindrical container filled with water. The liquids investigated were toluene, hexane, benzene, carbon tetracholoride, chlorobenzene, and paraldehyde. The driving frequency was near 41.7 kHz, the droplet size on the order of 500 μ, and the wavelength in the a direction about 8 cm. It was found that pressure amplitudes on the order of five bars were required to trap most droplets. The experimental results agree well with theory if the compressibility of the liquid droplets is taken into consideration. [Work supported by the Office of Naval Research.]

Non-linear effects in a focused underwater standing wave acoustic system

Intermodulation and sub-harmonic production in a roughly focused standing wave system are studied in relation to cavitation. It is found that when two slightly different frequencies are applied, the difference frequency is radiated at a level which increases rapidly as cavitation commences. Sub-harmonic generation appears to be dependent on the formation of bubbles as cavitation sets in, and does not occur with acoustic pressures in the primary wave insufficient to produce cavitation. If the water is degassed, sub-harmonic generation does not occur even at acoustic pressures well above those which are sufficient in ordinary tap water.

Electron densities from satellite antenna impedance

Impedance of satellite antennas has been measured throughout the orbits of several satellites by means of a standing wave system, which is a lumped-circuit equivalent to the slotted-line impedance-measurement technique. Impedance data from a recent satellite, ranging in altitude from 240 to 1,000 km (kilometers), have been used to calculate the electron density of the ionosphere. On the sunny side of the earth, this density was found to be a function of the latitude and altitude; on the dark side, it was found to be nearly a constant.

New Method for the Calibration of a Plane Hydrophone

The high-frequency calibration of a plane disk hydrophone of large diameter by the usual plane-wave reciprocity method is a difficult task, requiring a large anechoic tank and precise mechanical alignment gear. A new and absolute calibration technique utilizing the radiation pressure of a standing wave system is described. The calibration curve of a barium titanate hydrophone of diameter 1.5 in. and of resonant frequency 1.0 mc is presented over the frequency range of 300 to 5000 kc.

The Frequency Dependence of Acoustic Cavitation

In the past it has been necessary to infer the frequency dependence of acoustic cavitation from the work of different investigators at different isolated frequencies, the experimental techniques and the condition of the liquid varying markedly from investigator to investigator. The results of such inference are open to considerable doubt. The present investigation has determined the dependence upon frequency of the acoustic pressure required for the onset of cavitation under conditions such that the only variable was the acoustic frequency. The method utilized a standing wave system set up between a quartz crystal projector and a calibrated barium titanate hydrophone used as a reflector. The liquid sample, protected by a polyethylene box with thin windows, was inserted in the standing wave system. The output of the quartz projector was raised until cavitation began, as evidenced by visible bubbles, non-linearity in the hydrophone response, or cavitation noise picked up by an auxiliary hydrophone. The variation with frequency of the acoustic pressure required for the onset of cavitation was determined in water and several other liquids through the range of 300 kc/sec to 5 mc/sec.

Ultrasonic Luminescence of Liquids

Luminescence in various liquids under ultrasonic irradiation has been reported by many observers, but there is little or no agreement among their results. The present preliminary experiments were conducted in ten liquids, using a photomultiplier tube and galvanometer to measure the intensity of luminescence. A vertical ultrasonic beam of frequency 1 mc/sec and intensity up to 10 w/cm2 was employed successively as a progressive wave train and as a standing wave system. Luminescence was observed in the following liquids, listed in order of decreasing intensity: aqueous solution of KCl, aqueous solution of MgSO4, aqueous solution of H2O2, distilled water, fresh tap water, aqueous solution of bromine, aqueous solution of CCl4, glycerin. No luminescence was observed in nitrobenzene or in transformer oil. Almost all of the luminescence originated in the fountain at the free surface of the liquid in the case of progressive waves and just below the reflector in the case of standing waves. The intensity of luminescence increased with increasing ultrasonic intensity.

A New Method for the Calibration of a Plane Hydrophone

The calibration of a plane-disk hydrophone of 112-inch diameter over a wide frequency range by the usual plane-wave reciprocity method would be a very difficult task, requiring a large anechoic tank and precise mechanical alignment gear. A new and absolute method has been developed utilizing the radiation pressure of a standing wave system. The air-backed barium titanate hydrophone was suspended in oil above a quartz crystal projector by means of a sensitive Joly balance spring. The hydrophone thus acted simultaneously as a reflector to set up standing waves, as an electromechanical transducer, and as a device to measure radiation pressure. The weight, buoyant force, spring force, and radiation pressure combined to trap the hydrophone in any one of a number of discrete equilibrium positions, while the radiation pressure tended to align the hydrophone face parallel to the face of the quartz projector. The acoustic pressure was calculated from the spring extension, and the electrical output of the hydrophone was measured with a Ballantine voltmeter. In this manner the sensitivity of the hydrophone was determined over the frequency range 300 kc/sec to 5 Mc/sec. For a hydrophone of 1 Mc/sec resonant frequency, the calibration showed peaks at 1 Mc/sec, 3 Mc/sec, and 5 Mc/sec, and the sensitivity varied from a high of −116 db to a low of −156 db ref 1 volt/(dyne/cm2).

Surface Vibration Patterns of Piezoelectric Radiators

An experimental study of the surface vibration patterns of piezoelectric ultrasonic radiators, radiating into liquids, is being made. A quartz X-cut crystal having one surface polished to optical quality is used as one mirror of a Michelson interferometer, with the unpolished surface radiating into a liquid of acoustic impedance ρ1c1, different from that of the quartz, ρc. Examination of the fringes allows the entire surface of the quartz to be inspected directly. Other experimenters have used somewhat similar methods in air. A sonic standing wave system can be used to enhance the result. When the reflection coefficient between quartz and fluid approaches unity (radiation into air) only some five percent of the surface vibrates with appreciable amplitude; this area increases when the reflection coefficient is 0.9 (methyl alcohol). There is a corresponding decrease in amplitude of vibration, as predicted by theory. These measurements are at sonic frequencies of 0.5 and 1.0 mc. A stroboscope of variable phase is contemplated; this might allow direct measurement of the amplitudes of vibration over the whole surface of the crystal at all stages of the vibration.

Apparatus for Cavitation Studies

Cavitation in the body of a liquid, throughout a region well removed from all boundary surfaces, has been produced by means of a converging sound wave. The cavitation phenomenon can thus be studied as a function of the state and properties of the liquid, without regard to the possible effects of minute irregularities or adsorbed gases on the surface of the sound generator. The method employed is to bring one or more sound beams of moderate intensity to a common focus within the liquid by means of a parabolic mirror. At the sixty-kilocycle operating frequency of the magnetostriction transducers which serve as the projectors, attenuation in the liquid medium is small; nevertheless the use of a controlled resonant standing-wave system permits the experiments to be conducted in a tank of reasonable laboratory size not equipped with any special energy absorbing devices. Moreover, the necessary driving power is reduced. The apparatus provides for measurement of the cavitation threshold of various liquids as a function of temperature, hydrostatic pressure, and pulse length. This paper will describe the equipment and present some preliminary results.