The leakage flow rate through a sequence of labyrinth seal cavities, and the associated pressure and the circumferential velocity distributions are calculated for seals used in turbo machinery. Computational Fluid Dynamics is used to justify the use of bulk cavity variables, and to analyze the details of the flow in a single cavity under steady state, axisymmetric conditions. Periodic, analytic solutions of the continuity and circumferential momentum equations are obtained for the case of time dependent flow generated by a non-axisymmetric rotation of the shaft. The dynamic stiffness and damping coefficients necessary for the lateral stability analysis of the rotor are then calculated. The results compare reasonably well to experimentally obtained values.
Labyrinth seals are commonly found in ………. The correct prediction and control of this leakage is crucial for the efficient and economic operation of turbo machinery.
When the discharge flow coefficient is constant for all cavities, an analytical prediction of the leakage is possible. This prediction is compared to experimental results and to numerical results obtained by the use of more complex forms ………..
Analytical Prediction Techniques for Axisymmetric Flow in Gas Labyrinth Seals 
Labyrinth seals are commonly found in turbines and compressors. Their objective is to control gas leakage from high pressure regions to low pressure regions. The correct prediction and control of this leakage is crucial for the efficient and economic operation of turbomachinery. In this paper we present approaches for obtaining the above prediction in a simple analytical and explicit method. Both constant and pressure dependent flow coefficients are incorporated in the present study which extends to the higher inlet/outlet pressure differences. The results obtained with our methods compare favorably with the ones obtained by both numerical and experimental techniques. In many cases there is hardly a distinction between our results and the numerical prediction.
Calculation of Rotordynamic Coefficients for Straight-through Labyrinth Seals [ 2002]
The basic equations are derived for compressible flow in a straight-through gas labyrinth seal. The flow is assumed to be completely turbulent in the circumferential direction where the shear stresses in the boundary layers attached to the rotor and stator surfaces are determined by the Blasius correlation in smooth pipes. Zero-th-order and linearized first-order equations are developed for the perturbation flow generated by a small motion of the rotor about a centered position. The zero-th-order pressure distribution is found by satisfying the leakage equation while the zeroth-order circumferential velocity distribution is determined by satisfying the circumferential momentum equation. In this analysis we assume a radially linear circumferential velocity distribution in the core region between the two boundary layers. Several leakage models are discussed and compared. Periodic solution to the first-order equations is obtained describing the time dependent non-axisymmetric gas flow. Integration of the resultant pressure and shear stresses along and around the seal defines the reaction force developed by the seal and the corresponding rotordynamic stiffness and damping coefficients necessary for the lateral stability analysis of the rotor. The results of this analysis are then compared with existing experimental data and previous theoretical results.
Non-Newtonian and Viscoelastic Fluid Flows
Results obtained in a previous paper by Ballal and Rivlin on the forces associated with the slow flow of an incompressible non-Newtonian fluid, contained in the annular region between two infinite eccentric rotating cylinders, are applied to the calculation of the forces associated with the planetary motion of the inner cylinder about the axis of the outer cylinder. Either, neither, or both of the cylinders may rotate about their axes with constant angular velocities. As a limiting case the motion of an infinite cylinder is considered, in a half-space of incompressible non-Newtonian fluid bounded by a rigid plane, when the cylinder moves with constant velocity perpendicular to its length and parallel to the plane.
Results obtained in previous papers on the flow of an incompressible non -Newtonian fluid, satisfying the second-order Rivlin-Ericksen constitutive equation, in the annular region between infinite circular cylinders with parallel axe s, which results from the longitudinal steady motion of the inner cylinder, or t he existence of a longitudinal pressure gradient, or the steady rotation of the cylinders about their axes, are used to calculate the effect on the forces exert ed by the fluid on the cylinders when certain of these flows are superposed. As an asymptotic case, the forces exerted by the fluid on an infinite cylinder immersed in a half-space of the fluid bounded by a plane rigid wall are obtained when the cylinder moves parallel to the wall with a constant velocity inclined at an arbitrary angle to its length.
It has been observed by Mena that when a polymer solution flows through a tube of circular cross-section under a constant pressure gradient, the rate of discharge is increased if the tube is subjected to a longitudinal sinusoidal vibration. It is shown in the present paper how this effect can be predicted on t he basis of two different assumptions regarding the constitutive equation for th e fluid.
The effect is calculated of a superposed sinusoidal vibration on the rat e of discharge of an incompressible, isotropic, non-Newtonian fluid, which is in plane Poiseuille flow, under a constant pressure gradient, between two parallel plates. The calculations are made both when the vibration is longitudinal and when it is transverse. Calculations are also made in the case of Poiseuille flow through a straight pipe of circular cross-section, when a sinusoidal vibration , either longitudinal or rotational, about the axis of the pipe is superposed.
An incompressible, isotropic, non-Newtonian fluid undergoes plane Poiseuille flow between two parallel plates, or through a pipe of circular cross-section, as a result of a uniform time-independent pressure gradient. The effect is studied of superposed vibrations of the boundaries, which are not necessarily purely sinusoidal, on the mean rate of discharge of the fluid. The calculations a re carried out in detail for a particular constitutive equation of the Rivlin-Ericksen type, with the assumption that the fluid is slightly non-Newtonian. It i s seen that in addition to a change in the mean rate of discharge which arises a s an interaction of the vibration with the pressure gradient, there may also occur a change in the mean rate of discharge which arises from the interaction of t he harmonic components of the vibration and may be independent of the pressure gradient.
The effect is discussed of a superposed, longitudinal, sinusoidal vibration on the flow, under a constant pressure gradient, of a slightly non-Newtonian fluid in a straight pipe of non-circular cross-section. It is found that a flow in transverse planes is produced. This is the superposition of steady and sinusoidal flows. The stream-function corresponding to the steady flow is obtained in the case when the pipe has a rectangular cross-section. The discussion is based on a Rivlin-Ericksen constitutive equation in which only "second-order" non -Newtonian terms are present.
An incompressible viscoelastic fluid is contained between parallel rigid plates, which at some instant of time are subjected to a translational velocity which is then held constant. The dependence on time of the resulting flow field in the fluid is calculated. Similar calculations are carried out when the fluid is contained in an infinitely long circular cylinder, which is set to longitudinal motion, or in rotation.
It is seen that for a certain broad class of viscoelastic fluids the transfer of momentum, or angular momentum, from the boundary to the interior of the fluid takes place by a mechanism, which is essentially diffusive in character. For another broad class of fluids, of which the Maxwellian fluid is a special ca se, the transfer of momentum results from the reflection back and forth of a velocity shock wave. These reflections take place at the boundaries in the case of run-up between parallel plates and at the boundary and axis in the case of run- up, or spin-up, in a circular cylinder.
The transfer of momentum from the solid walls of a liquid filled cavity to the fluid contained within is examined in a simplified geometry. The liquid filler is taken to be an incompressible viscoelastic fluid. Special emphasis is given to the case of a Maxwellian fluid. The following problems are considered :
i) The fluid is contained between parallel rigid plates, which at some time are subjected to a translational velocity (parallel to the plates) which is then held constant.
ii) The rigid plates of problem i are subjected to an instantaneous increase of angular velocity about an axis perpendicular to the plates.
It is seen that for a Maxwellian fluid the transfer of momentum takes pl ace as a result of reflections back and forth of a velocity discontinuity wave. The secondary flow generated behind the wave front in problem ii is examined in detail and it is found that it has a direction opposite to that generated in a Newtonian fluid.
The transfer mechanism of momentum from the boundaries into the interior of a viscoelastic fluid is examined in the case when the boundary consists of two parallel plates.
Special emphasis is given to the case of a Maxwellian fluid. It is seen that an increase in the azimuthal angular velocity at the boundaries progresses into the interior of the liquid in the form of a velocity discontinuity wave. Simultaneously, an axial circulatory flow pumps liquid from the low rotation region near the middle plane into the high rotation region near the boundaries. De tailed calculations are given for the initial stage of evolution.
Pressure measurements in flows of highly viscous and elastic fluids are of practical importance in many manufacturing processes. Problems may arise during such pressure measurements, since high fluid viscosity and elasticity result in excessive dynamic response time of the pressure measuring systems as well as in some distortion. This is true for systems that consist of manometers as well as pressure transducers. In this work we develop an analytical model for the pressure pulse transmission in columns of viscoelastic fluids leading to pressure transducers. Basic equations are derived and analytical solutions are illustrated for a square wave pulse. Predictions of the model can be utilized to interpret correctly pressure transducer readings in fluid systems exhibiting viscoelastic behavior.
Pressure measurements in flows of highly viscous and elastic fluids are of practical importance in polymer processing and rheology systems. Special problems arise during such pressure measurements. High fluid viscosity results in excessive dynamic response time (rise time) of the pressure measuring systems. This is true for systems that consist of manometers as well as pressure transducers attached to the base of a small hole at the wall. We model the dynamic response and examine related disturbing effects. These systematic errors in pressure measurements include hole-pressure effects, instabilities in cavity flow, and the time lag of the disturbance wave. We consider static and dynamic flow systems of polymer solution ( PIB/C14/PB Boger fluid) to study these problems and show that instantaneous pressure measurements in these systems can effectively be performed.
An incompressible Newtonian fluid is contained in the annular region between two infinite cylinders, one or both of which rotate with constant angular velocities about their respective axes. The first-order inertial correction to t he forces exerted by the fluid on the cylinders is obtained in explicit algebraic form. The results are applied to the related problem in which the inner cylinder executes a planetary motion about the axis of the outer cylinder. They are also applied to the problem of the transverse sedimentation of a long cylinder i n a half-space of fluid bounded by a rigid wall. Certain anomalies, which arise in this case, are noted.
The development of embedded functions to represent the mean velocity and total enthalpy distributions in the wall layer of a supersonic turbulent boundary layer is considered. The asymptotic scaling laws (in the limit of large Reynolds number) for high speed compressible flows are obtained to facilitate eventual implementation of the embedded functions in a general prediction method. A s elf-consistent asymptotic structure is derived, as well as a compressible law of the wall in which the velocity and total enthalpy are logarithmic within the overlap zone, but in the Howath-Dorodnitsyn variable. Simple outer region turbulence models are proposed (some of which are modifications of existing incompressible models) to reflect the effects of compressibility. As a test of the methodology and the new turbulence models, a set of self-similar outer region profiles is obtained for constant pressure flow; these are then coupled with embedded functions in the wall layer. The composite profiles thus obtained are compared directly with experimental data, and good agreement is obtained for flows with Mac h numbers up to 10.
With increasing mainstream Mach number, viscous dissipation becomes a progressively important influence in high-speed compressible turbulent boundary layers. An asymptotic analysis is carried out for high Reynolds numbers and Mach numbers of order 1, and it is shown that viscous dissipation gives rise to important terms in the solution of the total enthalpy equation. For simplicity, the case of supersonic flow over an adiabatic wall is considered. An expression for the adiabatic wall temperature is derived. It is shown that the asymptotic analysis constrains the types of turbulence models that can be used to represent the effects of viscous dissipation. A simple algebraic turbulence model is propos ed and comparisons with measured total enthalpy profile data show good agreement .
An asymptotic analysis of the compressible turbulent boundary-layer equations is carried out for large Reynolds numbers and mainstream Mach numbers of order one. A self-consistent two-layer asymptotic structure is described wherein the time-mean velocity and total enthalpy are logarithmic within the overlap zone but in terms of the Howarth-Dorodnitsyn variable; the proposed structure lead s to a compressible law of the wall for high-speed turbulent flows with surface heat transfer. Simple outer region algebraic turbulence models are formulated t o reflect the effects of compressibility. To test the proposed asymptotic structure and turbulence models, a set of self-similar outer region profiles for velocity and total enthalpy is obtained for constant pressure flow; these are combined with wall-layer profiles to form a set of composite profiles valid across the entire boundary layer. A direct comparison with experimental data shows good agreement over a wide range of conditions for flows with and without surface heat transfer.
An asymptotic analysis of the equations describing supersonic turbulent flow over an adiabatic wall is carried out for high Reynolds numbers, Re, and mainstream Mach numbers, Me = O (1). A general expression for the adiabatic-wall temperature is derived. The asymptotic theory constrains the types of turbulence models that are suitable to represent the effects of viscous dissipation. A simple algebraic turbulence model is proposed and comparisons with measured total enthalpy profile data show good agreement, capturing the overshoot observed in total enthalpy near the boundary-layer edge.
In this report we consider a cylindrical canister undergoing spinning motion at a constant rate about its longitudinal axis and simultaneously a rotation at a constant coning rate about an axis which forms a fixed angle with its longitudinal axis. The canister is completely filled with a highly viscous liquid which is fully spun-up and consequently moves together with the canister as a rigid body. The equations determining the motion of the liquid relative to t he solid casing, due to perturbations to the coning and spinning rates, are derived. These equations are solved and subsequently the shearing stresses and the resultant moments are evaluated. The effect of the flat end walls is ignored in the present analysis. In addition we derive the equations governing the motion of the entire canister so that we may be able to discuss the stability of the canister motion. A brief stability analysis for the zero order approximation indicates that the system is stable in this approximation.
The results obtained in this report are in agreement with the experiment al work of Miller . We show that the despin moment is independent of and depends nonlinearly on and . The viscosity dependence of the despin moment is also found to be similar to the one described in . We also show that there exists an overturning moment due to the asymmetric axial flow which under some conditions may tend to increase the coning angle . We consider the work presented in this report as a necessary ground work for a more complete and precise stability analysis of the system. For a complete analysis we should evaluate the forces and resultant moments due to the pressure forces of the liquid and incorporate them in the stability discussion.
Consider a cylindrical canister spinning at a constant rate about its longitudinal axis while simultaneously rotating at a constant coning rate about another axis, which forms a fixed liquid. When the motion of the canister is perturbed, the liquid moves relative to the walls. This relative motion of the liquid generates internal pressure and shear forces which influence the stability of the canister motion. In this paper we summarize earlier results about the internal shear moments acting on the system and we calculate the internal pressure moments. Our model neglects the effects of the flat walls of the cylinder and produces results which are in agreement with the experimental work of Miller . We show that the despin moment is independent of and depends nonlinearly on and The viscosity dependence of the despin moment is also similar to t he one described in . The lateral moments generated by the pressure forces and by the shear stresses depend linearly on
Wave Phenomena in Inviscid Fluid Flow
This paper describes the behavior of large amplitude, long gravity waves as they move over a horizontal bed into a region where the flow is steady but sheared in a vertical direction. A new class of exact solutions to the nonlinear hydraulic flow equations is derived. These solutions describe progressing wave s and are sufficiently general to allow both the shape of the free surface at any instant and the shear profile of the undisturbed flow to be specified. The waves are examples of neutrally stable disturbances in the sense that they neither grow nor decay in amplitude, although, like simple waves on an un sheared flow, they can break.
The presence of an ambient sheared flow can greatly affect the behavior of gravity waves in the atmosphere and the oceans. Most of the analyses which describe these effects are, however, valid only when the governing equations can be formally linearized, or when the flow is steady relative to the wave. In this paper we describe the behavior of a new class of long gravity waves as they propagate over a horizontal bed into a region where the flow is steady but sheared in a vertical direction.
This study discusses the evolution of long gravity waves on shear flows. Although the paper is concerned mainly with finite amplitude neutrally stable flows, which contain a critical level, a new representation is given for the unstable mode solutions of the linearized equations. From these solutions it appear s that focusing instabilities, usually associated with nonlinear viscous effects, can occur even in linear inviscid theory.
For finite amplitude disturbances the analysis is restricted to polygonal shear profiles and only the neutrally stable solutions are considered. The theory is presented in detail for a simple two-layer profile which can support a critical mode. At small Froude numbers the critical mode is essentially an internal wave. This limiting solution also describes critical flows between parallel rigid boundaries when there is no body force.
The finite amplitude solutions are generalizations of the classical s imp le wave solutions for un sheared flows. As in the classical case, those waves ca n break but it is found that the conditions under which they break can be marked ly different for shear flows. Calculations for the particle trajectories are al so presented. These trajectories differ from the usual Kelvin's cat's eye pattern in that they are, in general, no longer closed.
Finally, it is observed that there are many other barotropic flows for which the governing equations can be reduced to a form equivalent to the shallow water equations discussed here. A list of such flows is given.
The use of controlled sound, generated by unidirectional secondary sources, in canceling primary noise of low frequency in long ducts is investigated. It is shown that for broadband primary noise, say four octaves, the simple arrangement of two secondary speakers used in many experimental studies is not efficient. This inefficiency manifests itself by a very low available output for part of the frequency band (0(16 times that of a single speaker) and by an unacceptably slow decay of the cross modes which may confuse the sensing device. The plane waves and the cross modes generated by rectangular or circular loudspeakers, which are mounted on a duct wall, are investigated. It is seen that an increase o f the size of the loudspeaker reduces its efficiency and at the same time slows down the decay of the cross modes. Consequently, the inefficiency of a particular unidirectional arrangement cannot be corrected by the use of larger speakers. The plane wave generated by a secondary source consisting of four speakers is calculated. It is shown that such an arrangement generates unidirectional sound of amplitude at least 1(3 times that of a single speaker over a frequency range of octaves.
The use of controlled sound, generated by unidirectional secondary sources, in canceling primary noise of low frequency in long ducts is investigated. It is shown that for broadband primary noise, say four octaves, the simple arrangement of two secondary speakers used in many experimental studies is not efficient. This inefficiency manifests itself by a very low available output for part of the frequency band ( 0.16 times that of a single speaker) and by an unacceptably slow decay of the cross modes which may confuse the sensing device.
The plane wave generated by a secondary source consisting of four speakers is calculated. It is shown that such an arrangement generates unidirectional sound of amplitude at least 1.3 times that of a single speaker over a frequency range of octaves
The propagation of sound waves i n a circular cylindrical duct filled with an inhomogeneous liquid is investigate d. Both the density of the liquid and its sound speed are taken to vary with the axial distance from a point source mounted on the wall of the duct. Two different types of inhomogeneity are studied. First the reflection and transmission of acoustic waves at the interfaces of three layers of constant but distinct acoustic properties are calculated. The second type of inhomogeneity consists of a finite layer of variable density and variable sound speed sandwiched between two semi-infinite layers of constant but distinct acoustic properties. The reflection and transmission characteristics of the layer depend on the impedance of the interface. An initial value problem for the determination of this frequency-dependent impedance is formulated and its solution is obtained numerically for a specific example.
In this work, we find the appropriate form of a scalar valued function L , of an arbitrary number of certain linear differential geometric objects (up to the second order tensors) and their first derivatives so that the functional formed by integration of the function L over an arbitrary sub domain d of the domain D under consideration, is invariant under infinitesimal Lorentz transformations.
This work was performed while I was a graduate student at Lehigh University's Center for the Application of Mathematics. The help and guidance of Prof. Dom Edelen was very important to me. This started my long time association and friendship with Dom.
In this paper heat conducting micropolar fluids are introduced as an extension of the theory of micropolar fluids. Constitutive equations appropriate t o describe the thermal and mechanical response of micropolar fluids are constructed. The heat conduction equation is derived and the field equations are obtained. The solution to the problem of Poiseuille flow through a channel with flat walls is given. This is my first paper, done while I was a graduate student at Notre Dame University.
Consider a trace-free, real symmetric matrix A. For the case, such a matrix represents an infinitesimal strain tensor corresponding to an isochoric de formation or a stress deviatoric tensor. In this note, we derive an orthogonal transformation which transforms A to a matrix B whose diagonal elements are zero . For the special case of matrices this is equivalent to determining the orientations of the pure shear axes. To my knowledge such an algorithm has not been previously published, despite the fact that its existence has always been assumed.
This paper continues the investigation of large amplitude waves in bounded media started by Cekirge & Varley (19 73). It describes the early stages of the deformation produced in an elastic slab contained between two parallel plane interfaces when the normal traction at one of them changes discontinuously. During the subsequent deformation, energy is radiated across these interfaces to adjacent elastic materials. Typically, the disturbance in the slab could be caused by the arrival of a constant strength shock wave traveling through an adjacent material or when the slab, which forms the front part of a composite material, impacts some other elastic material.
It is assumed that the dynamic response of the slab can be approximated by that of one of the model materials introduced in the first part of this study . It is shown that this is possible for a whole host of materials. These include polycrystalline solids, metals when subjected to high pressure, water, explosive products, gases, and yarns as well as elastic-plastic, rigid-plastic and rig id-elastic materials. The results reported are obtained by showing that for the se model materials a simple, but exact, representation can be found that describes the interaction of a centered wave with any wave traveling in the opposite direction. The arbitrary functions occurring in this representation are then found for the special case when this opposite traveling wave is the wave reflected from an interface with some other elastic material during the arrival of the centered wave. The limiting cases of a perfectly free interface, a perfectly rigid interface, and an interface with Hookean material are analyzed in great detail.
Although the terminology used in this paper is that associated with nonlinear elastodynamics, the results are directly applicable to any system whose response is described by the nonlinear wave equation. For example, the slab could represent a layer of nonlinear dielectric embedded in some other nonlinear dielectric and the disturbance could be generated by the arrival of an electromagnetic shock. Alternatively, the slab could represent sea water which is bounded by air from above and by rock from below while the disturbance is produced by a sudden motion of the water/rock interface.
The early phases of propagation of a large amplitude electromagnetic disturbance in a nonlinear dielectric slab embedded between two linear media are investigated by the method of characteristics. The disturbance is triggered by the arrival of an electromagnetic shock wave at one of the interfaces. Reflection and transmission of an arbitrary signal when it arrives at one of the slab boundaries is characterized in terms on nonlinear reflection and transmission coefficients for the interface. No restrictions are placed on the form of the constitutive laws of the material comprising the slab.
By introducing, for the nonlinear dielectric, a class of model equations, an exact solution to the characteristic equations, which describes the interaction of a centered wave with an arbitrary oncoming signal, is obtained. Solutions for the electromagnetic fields are derived for the special case in which the incident disturbance interacts with the reflected signal from the slab interface. A particular case of the disturbance propagating across a nonmagnetic slab is also examined.
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