Isostatic Recovery and the Strain Rate Dependent Viscosity of the Earth's Mantle

This paper is concerned with the interpretation of isostatic recovery data in terms of the flow properties of the earth's mantle. A hydrodynamic analysis is first presented that allows straightforward calculation of the relaxation time for isostatic recovery within a mantle in which the viscosity varies continuously with depth. However, it transpires that no curve of this type (i.e., choice of a reference viscosity and a rate of change of viscosity with depth) can of itself adequately explain the available observational data from the Fennoscandian and Laurentide ice sheets and the pluvial Lake Bonneville. Proceeding onward it is then demonstrated that the strain rates within such flows are in fact greater than the critical strain rate envisaged by Weertman (1970) in his theoretical rheological model of the mantle. Below this critical value, diffusion creep is the dominant flow process, and the flow can be modeled by a Newtonian viscosity. But above this value, dislocation glide takes over, and the viscosity exhibits a decrease with increasing strain rate. This feature is then incorporated into the theoretical model, and the isostatic recovery data are interpreted in such a way as to provide experimental values of the strain rate dependent viscosity that can be compared with the values in Weertman's rheological model. It is demonstrated that the data become most self-consistent and exhibit the most satisfactory agreement with Weertman's model when the increase of mantle viscosity with depth is given roughly by exp (5 × 10^(−4)z), where z is the depth in kilometers. Thus in addition, the analysis would appear to provide some verification of Weertman's model of the mantle flow properties. It is further demonstrated that the much larger increase of viscosity with depth predicted by McConnell (1968) and others from previous analyses of isostatic recovery data is an artifice induced by the nature of such flows in which the strain rate decreases with depth; this led to an apparent increase of viscosity that is much larger than the actual variation

On the Unsteady, Dynamic Response of Phase Changes in Hydraulic Systems

This paper is concerned with the unsteady, dynamic behavior of hydraulic systems and, in particular, with the dynamic characteristics of internal flows involving phase-change and two-phase flows. This emphasis is motivated by the large number of different flows of this kind which exhibit "active" dynamic characteristics (see Section 3) and therefore have the potential to cause instability in the whole hydraulic system of which they are a part (see Section 4). We begin, first, with a discussion of the form and properties of dynamic transfer functions for hydraulic systems. Then, following the discussion of a number of examples we present an analysis leading to the transfer function for a simple phase-change and demonstrate its "active" dynamic character

Some Cavitation Experiments with Dilute Polymer Solutions

A previous paper (Brennen (1968b)) reported the observation and analysis of wave patterns on the surface of fully developed cavities behind a series of axisymmetric headforms in No.2 water tunnel at Ship Division, NPL. Comparison of theory and experiment appeared to confirm that these waves, which appeared a short distance after separation, grew in amplitude as they were convected downstream and then under certain conditions broke up into turbulence, were the amplified result of a select frequency instability in the separated or cavity surface boundary layer. The small vertical tunnel (figure 1) was employed to study and extend observation of the same phenomenon to smaller headforms and Reynolds numbers. An additional intention was to investigate the effect of small quantities of polymer additive on the behaviour of this instability. But, a more dramatic phenomenon was manifest with the addition of these drag-reducing chemicals, leaving the original objective unattainable

The Unsteady, Dynamic Characterization of Hydraulic Systems with Emphasis on Cavitation and Turbomachines

The ability to analyze and model the unsteady, dynamic response of hydraulic systems has arisen from the need for operational stability, flexibility and controlled transient behavior. Following a general discussion, this paper discusses experiments and analysis directed toward identification of the dynamic response of hydraulic pumps, both cavitating and non-cavitating. It is shown that rather modest amounts of cavitation cause the pump to become dynamically active and therefore capable of exciting instabilities and resonance with the hydraulic system

Observations of Cavitating Flow

This paper will present a review of some of the recent advances in our understanding of the dynamics and acoustics of cavitating flows. We focus first on the individual events which evolve from a single travelling nucleus and describe observations of the intricate micro-fluid-mechanics which affect both the bubble shape and the subsequent emission of noise. These phenomena have important consequences in terms of their implications for the scaling of cavitation damage and noise. We also present calculations of the interaction between the individual traveling bubbles and the irrotational flow outside of the boundary layer of the headform. Comparisons of predicted and experimentally observed bubble shapes show qualitative agreement but further work is necessary to understand the details of the interactions between the viscous boundary layer and the bubble. To model the processes of cavitation inception, noise and damage it is necessary to generate a model of the cavitation event rate which can then be coupled with the consequences of the individual events. In the second part of this paper we describe recent efforts to connect the observed event rates to the measured distributions of cavitation nuclei in the oncoming stream. Such studies necessarily raise questions regarding the nuclei distributions in water tunnels and in the ocean and it would seem that we still know little of the nuclei population dynamics in either context. This is illustrated by a few observations of the population dynamics in a particular facility. The third subject addressed in this paper is the question of the noise produced by an individual travelling cavitation event. It is shown that the distortions in the shape of cavitation bubbles leads to acoustic impulses which are about an order of magnitude smaller than those predicted by the spherical bubble dynamics of the Rayleigh-Plesset equation. However, at the higher cavitation numbers, the upper bound on the experimental impulses scales with speed and size much as one would expect from the spherical bubble analysis. Initially, as the cavitation number is decreased, the impulse increases as expected. But, beyond a certain critical cavitation number, the noise again decreases in contrast to the expected increase. This phenomenon is probably caused by two effects, namely the interaction between events at the higher event densities and the reduction in the impulse due to a change in the dominant type of cavitation event. From the single event we then move to the larger scale structures and the interactions which occur when the density of the events becomes large and individual bubbles begin to interact. One of the more important interaction phenomena which occur results from the behaviour of a cloud of cavitation bubbles. Most previous theoretical studies of the dynamics of cavitating clouds have been linear or weakly non-linear analyses which have identified the natural frequencies and modes of cloud oscillation but have not, as yet, shown how a cloud would behave during the massively non-linear response in a cavitating flow. We present non-linear calculations which show the development of an inwardly propagating shock wave during the collapse phase of the motion. These observations confirm the earlier speculation of Mørch and his co-workers

Cloud cavitation with particular attention to pumps

In many cavitating liquid flows, when the number and concentration of the bubbles exceeds some critical level, the flow becomes unsteady and large clouds of cavitating bubbles are periodically formed and then collapse when convected into regions of higher pressure. This phenomenon is known as cloud cavitation and when it occurs it is almost always associated with a substantial increase in the cavitation noise and damage. These increases represent serious problems in devices as disparate as marine propellers, cavitating pumps and artificial heart valves. This lecture will present a brief review of the analyses of cloud cavitation in simplified geometries that allow us to anticipate the behavior of clouds of cavitation bubbles and the parameters that influence that behaviour. These simpler geometries allow some anticipation of the role of cloud cavitation in more complicated flows such as those in cavitating pumps

Cloud cavitation : observations, calculations and shock waves

A recent significant advance in our understanding of cavitating flows is the importance of the interactions between bubbles in determining the coherent motions, dynamic and acoustic, of the bubbles in a cavitating flow. This lecture will review recent experimental and computational findings which confirm that, under certain conditions, the collapse of clouds of cavitating bubbles involves the formation of bubbly shock waves and that the focussing of these shock waves is responsible for enhanced noise and potential damage in cloud cavitation. The recent experiments of Reisman et al. (1998) complements the work begun by Mørch and Kedrinskii and their co-workers and demonstrates that the very large impulsive pressures generated in bubbly cloud cavitation are caused by shock waves generated by the collapse mechanics of the bubbly cavitatting mixture. Two particular types of shocks were observed: large ubiquitous global pressure pulses caused by the separation and collapse of indiviual clouds from the downstream end of the cavitation and much more localized local pressure pulses which occur much more randomly within the bubbly cloud. One of the first efforts to model cloud cavitation was due to vanWijngaarden (1964) who linked basic continuity and momentum equations for the mixture with a Rayleigh-Plesset equation for the bubble size in order to study the behavior of a bubbly fluid layer next to a solid wall. In the 1980s there followed a series of papers on the linearized dynamics of clouds of bubbles (for example, d’Agostino et al. 1983, 1988, 1989). But highly non-linear processes such as the formation of shock waves require computational efforts which are capable of resolving these phenomena in both time and space. A valuable first effort to do this was put forward by Kubota et al. (1992) but by limiting the collapse of individual bubbles they prevented the formation of the large pressure pulses associated with bubble collapse. Wang et al. (1994, 1995) and Reisman et al. (1998) present accurate calculations of a simple spherical cloud subject to a low pressure episode and show that, for a large enough initial void fraction, the collapse occurs as a result of the formation of a shock wave on the surface of the cloud and the strengthening of this shock by geometric focussing as the shock propagates inward. This review will discuss other efforts to investigate these phenomena computationally. Wang and Brennen (1997, 1998) have extended the one-dimensional methodology used for the spherical cloud to investigate the steady flow of a bubbly, cavitating mixture through a onvergent/divergent nozzle. Under certain parametric conditions, the results are seen to model the dynamics of flashing within the nozzle. Moreover, it is clear from these steady flow studies that there are certain conditions in which no steady state solution exists and it is speculated that the flow under those conditions may be inherently unstable. Of course, it has frequently been experimentally observed that cavitating nozzle flows can become unstable and oscillate violently. Finally, we will also describe recent efforts (Colonius et al. 1998) to extend the code to two and three space dimensions. A simple example of such a calculation is the collision of a plane pressure pulse with a cylindrical or spherical cloud of bubbles. When the pressure pulse is negative, the growth and subsequent collapse of the cloud is particularly interesting and is seen to involve the formation and propagation of a shock waves within the cloud. Moreover, the non-linear scatterring of the pressure waves into the far field provides valuable information. The long term objective is to develop computational techniques and experience which would allow practical calculation of much more complex bubbly flows such as occur on hydrofoils, on propellers and in pumps where there is a real need for CFD methodologies which allow calculation of the noise and damage potential of these flows

A multifrequency instability of cavitating inducer systems

Recent testing of high speed cavitating turbopump inducers has revealed the existance of more complex instabilities than the previously-recognized cavitating surge and rotating cavitation. This paper explores one such instability which is uncovered by considering the effect of a downstream asymmetry such as a volute on a rotating disturbance similar to (but not identical to) that which occurs in rotating cavitation. The analysis uncovers a new instability which may be of particular concern because it occurs at cavitation numbers well above those at which conventional surge and rotating cavitation occur. This means that it will not necessarily be avoided by the conventional strategy of maintaining a cavitation number well above the performance degradation level. The analysis considers a general surge component at an arbitrary frequency, ω, present in a pump rotating at frequency, Ω, and shows that the existence of a discharge asymmetry gives rise not only to beat components at frequencies, Ω − ω and Ω + ω (as well as higher harmonics) but also to circumferentially-varying components at all these frequencies. In addition, these interactions between the frequencies and the basic and complementary modes lead to “coupling impedances” that effect the dynamics of each of the basic frequencies. We evaluate these coupling impedances and show not only that they can be negative (and thus promote instability) but also are most negative for surge frequencies just a little below Ω. This implies potential for an instability involving the coupling of a basic mode with a frequency around 0.9Ω and a low frequency complementary mode about 0.1Ω. We also examine how such an instability would be manifest in unsteady pressure measurements at the inlet to and discharge from a cavitating pump and establish a “footprint” for the recognition of such an instability

A Review of Cavitation Uses and Problems in Medicine; Invited Lecture

There are an increasing number of biological and bioengineering contexts in which cavitation is either utilized to create some desired effect or occurs as a byproduct of some other process. In this review an attempt will be made to describe a cross-section of these cavitation phenomena. In the byproduct category we describe some of the cavitation generated by head injuries and in artifical heart valves. In the utilization category we review the cavitation produced during lithotripsy and phacoemulsification. As an additional example we describe the nucleation suppression phenomena encountered in supersaturated oxygen solution injection. Virtually all of these cavitation and nucleation phenomena are critically dependent on the existence of nucleation sites. In most conventional engineering contexts, the prediction and control of nucleation sites is very uncertain even when dealing with a simple liquid like water. In complex biological fluids, there is a much greater dearth of information. Moreover, all these biological contexts seem to involve transient, unsteady cavitation. Consequently they involve the difficult issue of the statistical coincidence of nucleation sites and transient low pressures. The unsteady, transient nature of the phenomena means that one must be aware of the role of system dynamics in vivo and in vitro. For example, the artificial heart valve problem clearly demonstrates the importance of structural flexibility in determining cavitation occurrence and cavitation damage. Other system issues are very important in the design of in vitro systems for the study of cavitation consequences. Another common feature of these phenomena is that often the cavitation occurs in the form of a cloud of bubbles and thus involves bubble interactions and bubble cloud phenomena. In this review we summarize these issues and some of the other characteristics of biological cavitation phenomena

Some Current Advances in Cavitation Research

Several recent experimental and analytical investigations of cavitating flows have revealed new phenomena which clearly affect how we should view cavitation growth and collapse and the strategies used to ameliorate its adverse effects. On the scale of individual bubbles it is now clear that the dynamics and acoustics of single bubbles are severely affected by the distortion of the bubble by the flow. This distortion depends on the typical dimension and velocity of the flow (as well as the Reynolds number) and therefore the distortion effects are very important in the process of scaling results up from the model to the prototype. The first part of the lecture will discuss the implications of these new observations for the classic problem of scale-up. Another recent revelation is the importance of the interactions between bubbles in determining the coherent motions, dynamic and acoustic, of a cloud of cavitation bubbles. The second part of the lecture focusses on these cloud cavitation effects. It is shown that the collapse of a cloud of cavitating bubbles involves the formation of a bubbly shock wave and it is suggested that the focussing of these shock waves is responsible for the enhanced noise and damage in cloud cavitation. The paper describes experiments and calculations conducted to investigate these phenomena in greater detail as part of an attempt to find ways of ameliorating the most destructive effects associated with cloud cavitation