Propagation of three-dimensional acoustic perturbations in channels of variable cross sectional area at frequencies near the cutoff frequency

A study is made of the propagation of three-dimensional acoustic perturbations in a two-dimensional gas flow in axisymmetric channels of variable cross sectional area at frequencies near the cutoff frequency. The cross sectional area of the channel is assumed to vary slowly along its length. The obtained results are of interest, for example, from the point of view of estimating the level of noise carried through the gas ducts of a jet engine. The singularities of a solution found earlier in [1] (essentially the analog of the WKB approximation) are investigated. These singularities are due to the existence of turning points in the theory of the WKB approximation. The formalism of this theory [2] is used to develop a method for calculating the reflection coefficient of acoustic perturbations; in this approximation, reflection is found to be appreciable only at the channel sections where the frequency is near the cutoff frequency. Calculated examples are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 149–159, November–December, 1980.I am grateful to A. N. Kraiko, who pointed out the connection between the singularities investigated in the present paper and turning points in the theory of the WKB approximation and suggested that the formalism of this theory should be used to calculate the reflection coefficient. Discussions that I had with Kraiko during the work were very helpful.     

Propagation of disturbances in three-dimensional supersonic boundary layers

The propagation of disturbances in three-dimensional boundary layers under the conditions of a global and a local strong inviscid-viscous interaction is analyzed. A system of subcharacteristics is found based on the condition for the pressure-related subcharacteristic, and an algebraic relation that gives the propagation velocity of disturbances is obtained. The velocity of propagation of disturbances is calculated for two- and three-dimensional flows. The studied problem is of great importance for accurately formulating problems for three-dimensional unsteady boundary-layer equations and for constructing adequate computational models. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 116–127, May–June, 1999.     

Natural frequencies of circular arcs with varying cross section

This paper presents a transter matrix method for conveniently predicting the natural frequencies of circular arcs with varying cross section. The equations of in-plane vibration of an arc are written as a coupled set of first-order differential equations by using the transfer matrix of the arc. Once the transfer matrix has been determined, the natural frequencies and the mode shapes of the vibration are calculated in terms of the elements of the matrix for a given set of boundary conditions at the ends of the arc. The numerical results obtained by the method are compared with the experimental results for free-clamped arcs with tapered breadth, which are in good agreement with each other.     

Acoustic waves in a cylindrical duct with periodically varying cross section

The null field approach is used to study the propagation of acoustic waves in a rotationally symmetric hard-walled duct with a periodically varying cross section. For a radius that varies sinusoidally along the axial distance, numerical computations give the axial wave number and the passbands and stopbands of the modes of the duct. In particular, small passbands are seen to exist even for very large variations in radius, and probably all the way to the point where the duct is cut off. We also present some plots which show the pressure pattern inside the duct.     

Aerodynamic coefficients of non-conical bodies with a star-shaped transverse cross section

It is well known that, in a supersonic flow, the wave resistance of a body of non-round transverse cross section can be less than the resistance of an equivalent body of revolution with the same length and volume. Starting from 1959, when an exact solution was obtained to the problem of supersonic flow around conical bodies with a pyramidal system of flat discontinuities [1], a number of publications have appeared [2–5] developing this direction. Article [3] pointed out the possibility of achieving a flow with reflected shock waves, normal to the faces of a pyramidal body, by selection of the form of the leading edge. In [6, 7], using the Newton resistance law, bodies were constructed with a transverse cross section of a star-shaped form, having a wave resistance several times less than for an equivalent body of revolution. Just such forms, with certain limitations, have the least wave resistance and retain optimality with respect to the total resistance, taking approximate account of friction forces. Still two more exact solutions were then found, corresponding to flow around star-shaped bodies with regular and Mach interaction between shock waves [8, 9]. At a seminar of the Institute of Mechanics of Moscow State University, G. G. Chernyi advanced the postulation of the existence of certain classes of three-dimensional bodies not having the property of similitude and retaining optimality with respect to determined characteristics, for example, the resistance, the aerodynamic quality, or the torque, and stated partial problems of finding various forms of optimal bodies. Classes of bodies, optimal with respect to the resistance, were obtained within the framework of the Newton theory; the bodies consisted of helical surfaces, as well as of sections of planes and conical surfaces, formed by straight lines connecting the leading edges with a round contour. As a result of calculations using the Newton theory and experimental investigations it was established that bodies with a wedge-shaped nose part, with determined geometric parameters, have greater values of the lifting and of the aerodynamic quality than round cones [10]. The possibility of lowering the resistance and increasing the aerodynamic quality of aircraft by giving them shapes of the transverse cross section in the form of a star [11–14] leads to new investigations of three-dimensional bodies which retain optimality with respect to their aerodynamic characteristics, and are used in conjunction with bodies of revolution. This latter factor is of decisive importance with the use of such configurations as the nose part of the aircraft, or of a multi-step diffusor. The present article gives the results of an experimental investigation of flow around two classes of such bodies: multi-wedge and helical.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 127–132, November–December, 1974.     

Propagation of elastic waves in circular rods homogeneous over the cross section

Propagation of elastic waves in circular rods and in rods with a cylindrical cavity is numerically studied. The influence of the rod size on the wave propagation velocity is analyzed. A phenomenon of repeated rebound in short homogeneous rods is described.     

Coupled instability of compressed thin-walled rods with a closed transverse cross section

Dnepropetrovsk Civil Engineering Institute, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 29, No. 2, pp. 62–68, February, 1993.     

Rigorous theory for transient capillary imbibition in channels of arbitrary cross section

This article addresses a classical fluid mechanics problem where the effect of capillary action on a column of viscous liquid is analyzed by quantifying its time-dependent penetrated length in a narrow channel. Despite several past studies, a rigorous mathematical formulation of this inherently unsteady process is still unavailable, because these existing works resort to a crucial assumption only valid for mildly transient systems. The approximate theories use an integral approach where the penetration is described by equating total force acting on the domain to rate of change of total momentum. However, while doing so, the viscous resistance under temporally varying condition is assumed to be same as the resistance created by a quasi-steady velocity profile. Thus, leading order error appears due to such approximation which can only be true when the variation in time is not strong enough causing negligible transient deviation in the hydrodynamic quantities. The present paper proposes a new way to solve this problem by considering the unsteady field itself as an unknown variable. Accordingly, the analysis applies an eigenfunction expansion of the flow with unknown time-dependent amplitudes which along with the unsteady intrusion length are calculated from a system of ordinary differential equations. A comparative exploration identifies the situation for which the integral approach and the rigorous technique based on eigenfunction expansion deviate from each other. It also reveals that the two methods differ substantially in short-time dynamics at the initial stage. Then, an asymptotic perturbation shows how the two sets of results should coincide in their long-time behavior. In this way, the findings will provide a comprehensive understanding of the physics behind the transport phenomenon.     

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–10, January–February, 1973.     

Direct numerical modeling of three-dimensional turbulent flows in pipes of circular cross section

Turbulent viscous incompressible flows in pipes of circular cross section are subjected to a detailed numerical investigation on the Reynolds number interval from 2000 to 10 000. The evolution of the motion and the distribution of the average characteristics in the limiting regimes are studied. A detailed comparison shows good general agreement between the results of the calculations and the known experimental data. Possible causes of the differences are discussed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 14–26, November–December, 1994.     

On bounds for the torsional stiffness of shafts of varying circular cross section

We state upper and lower bound formulas for the torsional stiffness of shafts of varying circular cross section, in accordance with the classical Michell formulation of this problem, through use of the principles of minimum potential and complementary energy. The general results are used to obtain explicit first-approximation bounds which, for the limiting case of the cylindrical shaft, reproduce the known elementary exact results. It is conjectured that the first-approximation lower bound is significantly closer to the exact result than the first-approximation upper bound.A report on work supported by the Office of Naval Research, Washington, D.C.     

Asymptotic drag formulas for rapidly rotating radial channels of rectangular cross section

Taking, as an example, the solution of the drag problem of a slitlike channel, the change in flow characteristics with increase in intensity of rotation is shown. The form of the asymptotic drag formulas is determined for rapidly rotating channels with sides in a finite ratio, at Rossby numbers of the order of unity. Good agreement with experimental data is found. The final form of the asymptotic solution, valid at small Rossby numbers, is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 42–49, November–December, 1978.This considerably restricts the range of permissible values of E for channels elongated in the direction of the axis of rotation.     

Characterization of convective heat transfer in channels of small cross section using digital interferometry

Forced convection in channels of small cross sectional dimensions has been recommended as an effective heat removal method for electronic components and packages. Many of the experimental results reported in the literature on the heat transfer performance of small-cross section channels are of contradicting nature, even though some generally agreeing results are also found. One of the probable reasons for the deviations, as suggested by investigators, is the intrusive nature of measurement techniques, which interferes with the flow field. Hence a non intrusive measurement technique is preferable for temperature measurement in small channels. The present work is aimed at developing an interferometric method for convective heat transfer measurement in a liquid medium flowing through channels of small cross sectional dimensions, with hydraulic diameters ranging from 12 to 3 mm and characterizing the nature of fluid flow and heat transfer in these channels. Mach–Zehnder interferometric arrangement is used to obtain the temperature distributions in water flowing through the channels, which are further analyzed digitally to obtain the local heat transfer coefficients and Nusselt numbers. The results are compared and contrasted with classical results for channel flow and heat transfer, and attempt has been made to interpret the variations and deviations observed. The experimental study has been performed under different fluid velocities in the laminar flow regime, and under various wall heat fluxes corresponding to the heat dissipation range expected in microelectronic devices. Parametric variations for the heat transfer performance have been obtained and correlated using the experimental results.     

Transient stresses in axisymmetric bodies of varying area

The purpose of this paper is to report on an experimental and theoretical investigation of one-dimensional stress-wave propagation in axisymmetric bodies of varying area. An experimental investigation of elastic waves produced by the axial collision of strikers with truncated 7075 aluminum cones and conical shells with a half angle of 5 deg was performed using an air gun. Strain distributions along the 5-deg conical specimens were obtained by strain gages mounted on the specimen. The method of characteristics was utilized in the theoretical analysis of the propagation of elastic stress waves. Procedures of numerical integration along the characteristic directions are established and carried out for all the experimentally tested cases on a digital computer. Good agreement between the data and the results of calculations based on the analysis was obtained.