Fluid Dynamics of Cavitation and Cavitating Turbopumps

This chapter is a general introduction to cavitation. Various features of cavitating flows are analyzed on the basis of the Rayleigh-Plesset equation. They concern not only the simple configuration of a single spherical bubble but also complex cavitating flows as those observed in cavitating turbopumps. Scaling rules, erosive potential, thermodynamic effect, supercavitation, traveling bubble cavitation, cavitation modeling are some of the topics addressed here. They are examined through this simple, basic equation which proves to be a quite useful tool for a first approach of real cavitation problems.

This paper summarizes a set of lectures given on the hydrodynamics and cavitation of pumps presented at CISM in July, 2005. The lectures are based on my book entitled “Hydrodynamics of Pumps” (Brennen 1994) published jointly by Concepts ETI and Oxford University Press and available on the internet at http://caltechbook.library.caltech.edu/22/01/pumps.htm. The author is very grateful to Concepts ETI for permission to utilize large fractions of that book in this summary of the lectures. Readers who wish to explore the subject matter in more detail are encouraged to consult the original book.

This article describes about various types of cavitation instabilities including rotating cavitation and alternate blade cavitation. In section 1, general characteristics of cavitating flow through turbopump inducers are reviewed. In section 2, rotating cavitation in 3- and 4-bladed inducers are discussed focusing on the effect of the number of blades. Section 3 is intended to show the relationship between the operating condition and various types of cavity oscillations in an inducer.

Three types of stability analysis of cavitating flows through inducers are presented. The first is a one-dimensional analysis in which the characteristics of cavitation are represented by two factors that should be evaluated by other appropriate methods. This method has been used for the analysis of rotating cavitation and cavitation surge, as well as more general instabilities, surge and rotating stall to show the similarity and difference of the characteristics and the mechanisms of those instabilities. The second is a stability analysis of two-dimensional cavitating flow using a closed blade surface cavity model. This model has been used to clarify various types of cavitation instabilities. The third one was developed to understand a cavitation instability associated with the degradation of pressure performance due to cavitation.

Avoidance of cavitation instabilities is essential for the design of reliable inducers. Two possible methods are examined in this article, i.e., the leading edge sweep, and the casing enlargement at the inlet. Although they do have certain effects for the avoidance, it is becoming clear that in certain cases these methods are not sufficient to completely suppress cavitation instabilities. So, the suppression of cavitation instabilities is still an important problem to be solved.

In addition to blade surface cavitation, tip leakage and backflow cavitations occur in real inducers. Although most of cavitation instabilities can be explained by considering only blade surface cavitation, these types of cavitation might play an important role especially for the suppression of instabilities. Those cavitations are three-dimensional in its nature and this makes it difficult to analyze them theoretically even without cavitation. Under this situation not much is known about these cavitation types. In this document, research efforts into unsteady characteristics of those cavitation types are described, although they are by no means complete as yet.

The linearized dynamics of the flow in cavitating axial helical inducers and centrifugal turbopomp impellers is investigated with the purpose of illustrating the impact of the dynamic response of cavitation on the rotordynamic forces exerted by the fluid on the rotors of whirling turbopumps. The flow in the impellers is modeled as a fully-guided, incompressible and inviscid liquid. Cavitation is included through the boundary conditions on the suction sides of the blades, where it is assumed to occur uniformly in a small layer of given thickness and complex acoustic admittance, whose value depends on the void fraction of the vapor phase and the phase-shift damping coefficient used to account for the energy dissipation. Constant boundary conditions for the total pressure are imposed at the inlet and outlet sections of the impeller blade channels. The unsteady governing equations are written in rotating “body fitted” orthogonal coordinates, linearized for small-amplitude whirl perturbations of the mean steady flow, and solved by modal decomposition. In helical turbopump inducers the whirl excitation and the boundary conditions generate internal flow resonances in the blade channels, leading to a complex dependence of the lateral rotordynamic fluid forces on the whirl speed, the dynamic properties of the cavitation region and the flow coefficient of the machine. Multiple subsynchronous and supersynchronous resonances are predicted. At higher levels of cavitation the amplitudes of these resonances decrease and their frequencies approach the rotational speed (synchronous conditions). On the other hand, application of the same approach indicates that no such resonances occur in whirling and cavitating centrifugal impellers and that the rotordynamic fluid forces are almost insensitive to cavitation, consistently with the available experimental evidence. Comparison with the scant data from the literature indicates that the present theory correctly captures the observed features and parametric trends of rotordynamic forces on whirling and cavitating turbopump impellers. Hence there are reasons to believe that it can usefully contribute to shed some light on the main physical phenomena involved and provide practical indications on their dependence on the relevant flow conditions and parameters.

A hyperbolic two-phase flow model involving five partial differential equations is built for liquid-gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves into the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast only at specific locations, typically at evaporation fronts. Doing so, metastable states are involved for locations far from these fronts. Specific numerical hyperbolic and relaxation solver are built to solve the non-conservative system. Computational tests are done in 1D and 2D and are compared to experimental observations.