The solutions for three cases of retarded flow of a rigid-viscous material are given. These cases are: (a) Sliding parallel plates; (b) Couette flow between concentric cylinders; and (c) axial flow through a circular pipe. For Couette flow two types of loading-unloading paths are found. The results of this analysis explain some of the discrepancies between experimental results and theoretical predictions for a Bingham material.

Results are presented on an analytical study of the flow field existing upstream of a blade row, where the axial flow is subsonic and the relative flow is supersonic. The flow model used as a basis for the calculations assumes isentropic flow, and considers the case where the suction surface is a circular arc in the entrance region. The results clearly show the unique dependence of the flow through a blade row upon the geometry of the entrance region. Using the results, the complete flow field in the entrance region and upstream of the blade row can be constructed easily.

The equations of small motion of a straight cantilever beam attached to the rim of a rotating disk are determined assuming the Bernoulli-Euler theory of bending and the Saint Venant theory of torsion are valid, the mass and elastic axes coinciding and retaining all inertia terms. Influence of the secondary inertia terms on the fundamental torsional and lateral frequencies is then examined at two angular settings for a uniform beam having a length to disk-radius ratio in the range usually encountered in gas-turbine buckets and axial-flow compressor blades.

It is suggested that flow patterns without axial symmetry may occur when an axial-compressor stage operates in a steady, uniform stream, provided that the blade airfoil characteristics exhibit certain peculiarities. For example, blades acting close to the stall may show nonlinearity, hysteresis, and/or time lags, and these may cause asymmetric flow patterns. Conditions necessary for existence of such phenomena are analyzed by consideration of an annulus of infinitely many blades and very large radius. It is shown that rotational-flow effects due to vortex shedding by the blades must be considered. The induced velocities resulting from asymmetrical loadings are calculated. The final step of the calculation is to show that these induced effects may actually support the circulation distribution that produces them. This step is only partly taken in the present paper. It is shown that time lag in the development of lift of a blade may produce such “eigen-solutions.” Detailed study of the results of non-linearity in the blade characteristics is left for another investigation.

The Lorenz theory has been extended to the viscous, laminar, incompressible flow through axial-flow turbo-machines with infinitesimal blade spacing. Expressions are derived for the velocity components, pressure, power input and output for arbitrary blade surfaces. A numerical example is presented and the flow variables and blade surfaces are plotted.

This paper presents a summary of the basic principles of the supersonic compressor and an analytical study of the types of this compressor most likely to prove successful. This study assumes rather idealized conditions, ignoring such variables as radial effects and flow irregularities. Hence the results are not intended as design criteria. They should, however, give a fair guide to what may be expected of the various cases considered; their outstanding limitations, their range of operation, and their performance expectations. Conclusions are that good pressure ratios with high efficiencies may be attained; that the compressor having a normal shock in the stator is superior to that having a normal shock in the rotor; and that the largest restriction on this normal-shock-in-stator compressor is the Mach number limitation in the rotor.

In this paper a theoretical method is presented to determine the flow of a perfect (frictionless, incompressible) fluid across a cascade built up of arbitrary airfoils. The presentation refers to compressor blading but is immediately applicable to turbine blading. The lift-coefficient curve as a function of angle of attack (or deflection-angle curve) of an arbitrary cascade can be determined in about 10 man-hours. The complete investigation of all possible cascades built up of a given airfoil (any solidity and any stagger) requires about 60 man-hours. The procedure is graphical and standardized to the extent that no mathematical knowledge is required of the computer. Agreement between theory and tests is excellent (see Figs. 19 and 20). An extension of the method is indicated whereby the detailed pressure and velocity distribution of the flow can be obtained.

When a stream of air is partially stopped by an inserted temperature probe, the temperature increase due to the conversion of kinetic energy affects the reading of the probe. The fraction of the total kinetic temperature rise which is registered by the probe, i.e., the so-called “recovery factor” of the probe, is a function of a number of variables. Tests dealing with the effect of probe shape and air velocity on this recovery factor, and with the influence of radiation on the accuracy of the measurements, are reported in this paper. Bare-wire probes gave recovery factors of approximately 0.65 in transverse flow and, in axial flow, approached 0.87 as the air velocity increased (in good agreement with theoretical predictions for flow over flat plates). With a spherical enlargement at the thermocouple junction, recovery approached 0.75. Recovery of twisted-wire couples varied from 0.72 to 0.83. A reduced-scale model of the Franz probe was found unsatisfactory after extensive study. Two simpler probes were developed, having high recovery (above 0.98 as velocity approaches sonic) and satisfactory insensitivity to yaw and radiation errors.

By coupling the Donnell–Mushtari shell equations to an analytical inviscid fluid solution, the linear dynamics of a rotating cylindrical shell with a corotating axial fluid flow is studied. Previously discovered mathematical singularities in the flow solution are explained here by the physical phenomenon of blocking. From a reference frame moving with the traveling waves in the shell wall, the flow is identical to the flow in a rigid varicose tube. When the ratio of rotation rate to flow velocity approaches a critical value, the phenomenon of blocking creates a stagnation region between the humps of the wall. Since the linear model cannot account for this phenomenon, the solution blows up.

It is shown that, in certain regions of parameter space, travelling wave solutions in rotating shells containing co-rotating inviscid fluid become indeterminate. This may render the determination of the flutter speed impossible, or the solution nonphysical.

The stability of circular cylindrical shells with supported ends in compressible, inviscid axial flow is investigated. Nonlinearities due to finite-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory; the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied at the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear potential flow theory is applied to describe the fluid-structure interaction. Both annular and unbounded external flow are considered by using two different sets of boundary conditions for the flow beyond the shell length: (i) a flexible wall of infinite extent in the longitudinal direction, and (ii) rigid extensions of the shell (baffles). The system is discretized by the Galerkin method and is investigated by using a model involving seven degrees-of-freedom, allowing for traveling-wave response of the shell and shell axisymmetric contraction. Results for both annular and unbounded external flow show that the system loses stability by divergence through strongly subcritical bifurcations. Jumps to bifurcated states can occur well before the onset of instability predicted by linear theory, showing that a linear study of shell stability is not sufficient for engineering applications.

In this paper the dynamical characteristics are discussed of a cluster of cylinders, the extremities of which are interconnected by beam-like connectors, modeled here by sets of linear springs as formulated in Part 1 of this study; resistance to both in-plane and out-of-plane deformation of the connectors is taken into account. The cylinders are subjected to axial flow, and hydrodynamic coupling in cylinder motions is taken into account. The system eigenfrequencies and modal shapes are calculated for different conditions of structural and hydrodynamic coupling. The effect of increasing flow on the dynamics of the system is also studied, up to and beyond the threshold of fluid-elastic instabilities.

This paper presents an analytical model for the dynamics of a cluster of flexible cylinders, the extremities of which are structurally interconnected, e.g., through “end plates” or other kinds of structural connectors. These connectors are modeled by sets of translational and rotational springs, such that resistance to both in-plane and out-of-plane deformation of the connectors is fully taken into account. The dynamics of the system in vacuum is examined here, and especially the effect of varying the different spring stiffnesses on the system eigenfrequencies. This provides a reference base and the necessary analytical tools for the study of the system in axial flow , which is presented as Part 2 of this work.

This paper studies theoretically the dynamical behavior of a flexible slender cylinder in pulsating axial flow. The dynamics of the system in steady, unperturbed flow are examined first. For various sets of boundary conditions the eigenfrequencies of the system at any given flow velocity are determined, and the critical flow velocities are established, beyond which buckling (divergence) would occur. The behavior of the system in pulsating flow is examined next, establishing the existence of parametric resonances. The effects of the mean flow velocity, boundary conditions, dissipative forces, and virtual (hydrodynamic) mass on the extent of the parametric instability zones are then discussed.

This paper examines experimentally the dynamical behavior of a flexible slender cylinder in axial flow, perturbed harmonically in time. Parametric resonance oscillations were found to exist over certain ranges of frequencies and amplitudes of flow-velocity perturbations. The most prominent of the resonances, in these experiments, were associated with the second-mode principal primary resonance, and were studied extensively. Agreement with theory was found to be quite good.

This paper describes experiments in which flexible cylinders placed centrally within a narrow cylindrical flow channel were subjected to axial flow. The dynamical behavior of such cylinders, either cantilevered or supported at both ends, is described, both in liquid and in simulated two-phase flows; the effect of several parameters, such as the annular confinement, was investigated. It was observed that, with increasing flow, the cylinder is subject sequentially to instabilities of increasing mode number, confinement severely destabilizing the system. The experimental observations are compared to a theoretical model for the dynamics of such systems. In the case of liquid flow, agreement is qualitatively good and quantitatively fair—keeping in mind the experimental difficulties. In the case of two-phase flows, theory and experiment do not agree.

This paper develops a general theory for the dynamics of slender, nonuniform axisymmetric beams subjected to either internal or external flow, or to both simultaneously. The effect of the boundary layer of the external flow is taken into account in the formulation. Typical solutions of the equations of motion are presented for cantilevered conical beams in external flow and for beams with a conical internal flow passage. Such systems lose stability at sufficiently high flow velocity, internal or external, either by flutter or by buckling. The effect of several parameters is investigated. For internal flow, the internal and external shape, whether uniform or conical, and the density of the surrounding fluid have sometimes unexpected effects on stability; e.g., tubular beams lose stability at lower internal flow when immersed in water than when in air. For external flow the effects of conicity, free end shape and boundary-layer thickness are investigated; the latter has a strong stabilizing influence, such that simple theory neglecting this effect results in serious error.