Non-dimensional Quantities Research Articles

Nonlinear frequency shifts in quasispherical-cap shells: Pitch glide in Chinese gongs

The large-amplitude vibrational behavior of a shallow spherical-cap shell is investigated theoretically using simple arguments. The results of this approximate analysis are expressed in the form of appealingly simple nondimensional quantities. It is shown that the frequency of the fundamental mode of such a shell falls by as much as 50% as the vibration amplitude is increased to approximately the shell depth. For even larger amplitudes the frequency rises, and it exceeds the small-amplitude frequency when the amplitude is more than about twice the shell depth. The influence of shell thickness is considered and increasing thickness is shown to decrease the frequency shift. This analysis is shown to account for the pitch glide behavior of certain Chinese opera gongs.

Trapezoidal side-channel spillways

Steady flows in trapezoidal, prismatic side-channel spillways are analysed using a hydraulic approach. Distinction between channels of small and moderate bottom slopes is made. All results are presented in typical nondimensional quantities, by which an immediate application is made possible. Key words: open channel flow, spillway, gradually varied flow, discharge supply, hydraulics.

Dynamic Response of Voltage-Driven Induction Machines

Abstract When operated at rated frequency, typical induction machines exhibit simple first or second order dynamic response depending upon machine size and load inertia and damping. At reduced frequency much more complex behavior can occur including the possiblity of small signal instability. This paper presents a general analysis of small signal dynamic behavior and introduces non-dimensional quantities which allow easy estimation of the region of poor dynamic response for specific machines. Graphical results which illustrate the regions of instability and poor damping are presented. The influence of machine parameters is described and the trends to be expected for different size and types of machines are discussed. In particular, 1t 1s shown that all machines pass through a minimum damping region as the frequency 1s reduced and that larger, more efficient machines will often exhibit small signal instability. An important result of the analysis is that all of the modifications made 1n producing “high eff...

On the Contact Heat Transfer with Melting : 1st Report: Experimental Study

The phase change process of a melting solid pressed on a hot heat transfer surface was investigated experimentally, as a basic research of the heat storage process in a latent heat TES (thermal energy storage) capsule. Along with measurements for various conditions, the physical parameters deciding the phenomena were analyzed, normalizing the basic equations and boundary conditions. It was shown as the result that the nondimensional heat flux expressed as a function of the Stefan number and the nondimensional contact pressure. Based on the experimental results, the relation among the bove nondimensional quantities was shown by a simple formula.

An analytical-experimental estimation of elastic linear zone for photoelastic mode I SIF extraction

The traction continuity condition is used to delineate the elastic linear zone (ELZ) for the extraction of the Mode I stress intensity factor (SIF) from photoelastic data. The procedure requires an estimation, with the aid of experimental data, of some non-dimensional quantities which are the ratios of unknown parameters of the LEFM equations and the corresponding ones for an infinite body problem. Furthermore, the crack tip nonlinear zone is assumed to admit of plastic deformation with small-scale yielding. The estimation of the ELZ thus obtained seems to predict well the experimentally observed one.

Deficient pore pressures in an eroding soil mass

The pore pressure response of a semi-infinite, fully saturated soil mass of finite thickness eroded at a constant rate is examined. Using Terzaghi's classical consolidation theory, an expression is derived for the deficient pore pressure as a function of depth and time. The resulting equation is mildly nonlinear with a moving top boundary. Closed form analytical solutions are obtained for various bottom boundary conditions and are presented in nondimensional quantities. The results indicate that the development of deficient pore pressures is dependent on (i) the rate of soil removal; (ii) the swelling potential of the soil mass; (iii) the ratio of the thickness of soil layer(s) removed to its original thickness; and (iv) the nature of the bottom boundary. Substantial negative pore pressures are likely to persist in the residual mass, even close to the eroded surface, although free water is made available at the top. The plots presented in this paper will be helpful for ascertaining the erosion–swelling characteristics of a soil mass that has been or is being eroded. Examples are presented to illustrate the development and dissipation of deficient pore pressures due to swelling. Keywords: constant rate, deficient pore pressures, depth factor, dredging, erosion–swelling ratio, isochrones, Laplace transform, moving boundary, swelling.

ESTABLISHING MOBILE ACOUSTIC COMMUNICATION SYSTEM (MACS) SAFETY ON USS NAUTILUS

ABSTRACTUSS Nautilus (SSN‐571) was decommissioned on 3 March 1980. In her later years in service she served as a test bed for many programs which investigated multiple aspects of future sonar designs. The Mobile Acoustic Communication System (MACS) was one of the last major experiments to be performed in Nautilus. This paper describes a unique data collection and presentation system with large numbers of strain gauges that was used to establish structural safety for the MACS antenna and Nautilus operations. The system utilized real‐time processing of digital data with a minicomputer and an instantaneous display of certain nondimensional quantities on two television monitors. This data system performed excellently and allowed Nautilus safely to conduct deep dive, full speed, and emergency blow operations.

Heat Transfer in Surface-Cooled Objects Subject to Microwave Heating

Several investigators in microwave bioeffects research have exposed biological preparations to intense microwave fields, while at the same time cooling the sample with flowing water. We examine the heat transfer characteristics of this situation, to estimate the maximum temperature increase and thermal time constants that might be encountered in such an experiment. The sample is modeled as a uniform sphere, cylinder, or slab subject to uniform heating, which is located in an unbounded coolant flow. The heat transfer is determined by the Biot and Reynolds numbers (which reflect the geometry, fluid flow, and material thermal properties of the system) the temperature rise is governed by the heat conduction equation coupled with external convection. The results are expressed in terms of nondimensional quantities, from which the thermal response of a heated object of arbitrary size can be determined. At low coolant flow rates, the maximum temperature rise can be biologically significant, even for relatively small objects (of millimeter radius) exposed to moderate levels of microwave energy (with a SAR of ca. 100 mW/g). The results are valid also where the coolant is a gas or a liquid different from water, the only restriction being on the Reynolds number of the flow.

Wetting Layer and Void Fraction Nonunif ormity in a Liquid-Metal MHD Generator

THE question of the effect of a liquid layer on the walls of an MHD channel in the case of uniform void fraction distribution in the flow core was first considered several years ago. * More recently an analytic solution for high Hartmann numbers was obtained,2 which led to the conclusion that this layer does not have a significant effect on the efficiency of large generators. This paper postulates an analytic model which makes it possible to estimate the effect of a void fraction nonuniformity, in the presence of the wetting layer on the walls, on the generator performance. Contents Statement of the Problem The flow of a conducting fluid in the channel of an MHD generator with nonconducting walls z= ±a and conducting walls y= ±d/2 is considered. It is assumed that d>a and the channel length L>a. The fluid is taken to be a homogeneous liquid-gas mixture. The assumption of * 'homogeneity makes it possible to treat the mixture as a single-phase fluid, the properties of which are a function of void fraction a. It is also assumed that a. can vary in the magnetic field direction and a (z) is a known function. Let us introduce the nondimensional quantities (designated by an asterisk) using the following expressions V=VQU* (1)

Alternative methods for wake quality assessment

Results of research carried out at B.S.R.A. and elsewhere have indicated that the main causes of unsteady cavitation and the resulting after-end vibration are the non-uniformity of flow into the propeller and the heavy propeller-loading. The need to have criteria for the assessment of flow quality from a given velocity-distribution is evident, and in recent years efforts have been made to devise such criteria as well as to improve the flow quality. As part of the Propeller-Excited Vibration project recently carried out by B.S.R.A., research has been conducted into methods of assessing wake quality; the Authors summarise the results of this research. The methods of assessment may be divided into two groups: heuristic and analytic. Heuristic methods generally use only the axial velocity component and assess the flow quality by means of the variation of this component alone. On the other hand, analytic methods require all three components of the velocity, and their variations, and assess the flow quality by means of non-dimensional quantities related to flow kinematics and flow geometry. It is shown that, with present wake-survey techniques, analytic methods cannot be applied adequately because of insufficient data although their potential for representing flow quality is high. Criteria based on heuristic methods are therefore proposed, and the development by B.S.R.A. of such criteria, for assessing the quality of both the wake and the wake/propeller combination, is described and discussed. The basis of the develoment is the experience gained from ships in service with known vibration characteristics. Compliance with these criteria is expected to reduce the possibility of high levels of pressure due to cavitation, though it cannot be guaranteed that vibrations will not occur; however, if the wake criteria are not met, alterations to the afterbody form should be considered.

Drawdown Analysis of Slopes by Numerical Method

A numerical procedure is used to analyze slopes subjected to transient free surface seepage. The procedure combines a finite element scheme with a method of slope-stability analysis. The finite element scheme allows computations of time dependent locations of free surface, and the stability analysis method permits computation of factor of safety. The final results are plotted in terms of charts expressed through a number of nondimensional quantities. It is found that the value of the lowest factor of safety by the proposed procedure is approximately equal to that from the conventional assumption of sudden drawdown. The procedure can be found to be more economical if the designer can adopt drawdown ratios less than unity. The numerical procedures can provide an improved understanding of the behavior of slopes during transient drawdown conditions.

On normal impact of an infinite elastic-plastic thin-walled plate by a finite elastic rod

The problem of the transverse collision of a finite circular cylinder (flat-ended metal projectile) of length L and radius R 0 , moving with the initial velocity V 0 , with an immobile thin infinite plate, is considered. It is assumed that the plate is in a pure shear state. The one-dimensional waves both in the plate (cylindrical elastic-plastic waves) and in cylinder (plane longitudinal elastic waves) are investigated. Graphs of the peak dimensionless shear stress at the point of impact versus four non-dimensional quantities representing the material and geometrical properties of the plate and the projectile, are presented. These graphs allow the projectile speed just sufficient to initiate ejection of the slug to be estimated.

Parametric Influences on the Response of Structural Shells

The effects on structural response produced by variation of geometric, loading, and material property parameters have been investigated for the specific case of a fixed-ended cylinder subjected to a “frontal cosine” distribution of impulsive loading (although many qualitative results can be expected to apply to other cases). Responses were determined using the REPSIL code, which employs a finite difference method to solve the partial differential equations for large deflection, anelastic shell motions. Graphical representations of results in terms of six nondimensional quantities make it possible to obtain an overview of the influence of various system parameters on structural response and to draw certain conclusions regarding their relative importance. An empirical model based on REPSIL code data has been developed which permits rapid estimation of cylinder responses over a useful range of system parameters.

An analysis of ignition in the laminar boundary layer with unsteady fuel injection

The present work is a study of the problem of ignition in the boundary layer on a flat plate, where a cold gaseous fuel is injected uniformly from the porous surface into the boundary layer, diffuses and reacts with the oncoming hot oxidizer diluted with an inert gas. The effects of various physical parameters on the ignition time are examined. A one-step unopposed global reaction following the Arrhenius rate law is assumed. The unsteady non-similar two-dimensional boundary layer equations are solved for the case where the fuel injection velocity is proportional to t α ( t is the time and α is the arbitrary nonfractional number) by the series expansion method. It is found that the nondimensional ignition time at a certain location (nondimensional) is characterized by the four nondimensional quantities, i.e., the nondimensional wall temperature, injection rate, heat of combustion, and activation energy. The ignition time is shown to decrease with the increase in the wall temperature, the injection rate and the heat of combustion, and the decrease in the activation energy. The main stream temperature has remarkable influences on the ignition time as well. It is demonstrated that the trends of the effects of these parameters on the ignition time are analogous to those on the ignition length in the case of the steady injection rate.

Unsteady laminar convection in uniformly heated vertical pipes

In this paper an exact solution is presented for the problem of unsteady laminar convective flow under a pressure gradient along a vertical pipe. We have obtained the solution of the problem on the basis of the assumption that the velocity and buoyancy profiles far from the pipe entrance do not change with the height, and the entry lengths have been ignored. The wall of the pipe is heated or cooled uniformly. We have discussed both the cases, when buoyancy forces act together with the pressure gradient or in opposite direction.In the case when the upflow is heated (or a downflow is cooled) the velocity and thermal boundary layers are formed for sufficiently large Rayleigh numbers. In the second case which has been discussed in detail (when the upflow is cooled or the downflow is heated) we have found the critical value of the Rayleigh number R = R, beyond which the velocity profile and the temperature profile become unsteady and turbulent in all the cases. In the case of the elliptical cylinder R, increases up to 1730 as the ellipticity is increased while in the case of the co-axial pipes this Rayleigh number increases as the gap c between the cylinders is decreased (if c = a/b = 1·2 then R, = 60762, but decreases to 1 when c = 4). Besides this, the time required to reach steady state increases as the Rayleigh number increases in both circular and elliptical pipes; it also increases when the eccentricity is decreased. The cases discussed by Morton (1960 and Dalip Singh (1965 are particular cases of the results derived below.In this investigation we have dealt with the following ducts: (i) circular tubes, (ii) elliptical tubes and (iii) co-axial tubes. The general solutions for both velocity and temperature fields have been found for the case when the pressure gradient is an arbitrary function of time, with an arbitrary heat source also present. Particular cases when both the parameters are absohte constants have been discussed in detail.We have made use of finite transforms very frequently; especially for the case of an elliptical tube, a new transform involving Mathieu functions developed by Gupta (1964 has been used. A few new infinite series have been summed with the help of this transform.Various non-dimensional quantities (for both the cylinders) such as the Nusselt number, volume flux and rate of heat transfer have been found when the pressure gradient and source of heat generation are absolute constants.

Model and dimensional analysis of prestressed concrete hollow beam

The paper describes the model and dimensional analysis of a prestressed concrete beam. Non-dimensional quantities are formed as invariant of similarity to represent quantities having the same stresses and strains in the original and the model.

Upward Liquid Flow in a Small Tube into which Air Streams : 3rd Report, Pipe Friction I

The pressure gradient of a bubble and slug flow after the confluence of air and liquid in a vertical small tube, and the condition of existence of a fish scale type slug flow, are investigated. For the decision of nondimensional empirical equations of pressure gradient, the following nondimensional quantities are taken into consideration, that is, supplied air -liquid ratio, liquid Reynolds number, Froude number and, Weber number. The maximum pressure gradient of the fish scale type slug flow is affected by the length and diameter of bubbles and the surface tension. It is very difficult to analyze the effect of surface tension on a two-phase flow with surface active agent solution. It is because the dynamic surface tension is very different from the static surface tension.

Entrance Loss for Turbulent Flow without Swirl between Parallel Discs

This paper is concerned with the investigation on the entrance loss which occurs in a radial turbulent flow between parallel discs. Because the theoretical determination of the entrance loss is almost impossible, the author has constituted an empirical equation by using nondimensional quantities - the space between the discs, the radius of rounded corner at entry boundary etc.- which have large influences on the entrance loss. This equation will be usefull for the solution of practical problems.

On the motion generated by a plate vibrating in a stratified fluid

We initiate a study of acoustic streaming in a stratified fluid. A set of equations describing nondimensional field quantities is first derived. By introducing successive approximations, we next derive equations for the primary and the secondary motion. These equations are briefly discussed.

EXPECTED DISCHARGE OP IRREGULAR WAVE OVERTOPPING

An experiment was carried out on the overtopping of mechanically generated irregular waves over vertical walls. The experimental discharge was almost in agreement with the expected discharge which had been calculated with the wave height histogram and the data of regular wave overtopping based on the principle of linear summation. The expected values of overtopping discharge were calculated for various laboratory data, which had been represented in a unified form of non-dimensional quantities. The calculation has yielded two diagrams of expected overtopping discharge, one for the sea wall of vertical wall type and the othei for the sea wall covered with artificial concrete blocks.