Cavitation Compliance Research Articles

Modeling and scaling of cavitation compliance in Venturi

Understanding cavitation compliance in Venturi is important for research on system oscillations in fluid machinery, which describes the responsiveness of the cavity size to alternating inflow parameters. To consider both Venturi geometries and inflow conditions in evaluating the cavitation compliance, the cavitation region is modeled as a large vapor bubble, based on the experimental characteristics of the attached cavity characterizing the cavity size. Steady-state numerical solutions of cavity sizes show the influencing trends by geometrical and inflow parameters consistent with experiments. According to the analytical solution approximating the numerical solution, effective cavitation number σe and effective length ratio L̃* are defined to unify these influencing parameters. Based on the causes of cavitation pressure drops at the throat indicated by σe, cavitation compliance can be categorized into two regimes. Regime I features a smaller σe, and the dynamic pressure dominates in the pressure drops, theoretically resulting in L̃*∝σe−9. Regime II features a larger σe, and the bend-flow inertia and dynamic pressure are comparable, resulting in L̃*∝σe−2 in contrast. The power-law predictions of cavitation compliance are validated by experiments and literature.

The Similarity Law of Cavitation Number on Cavitation Instability in Liquid Rocket Inducer

Abstract The influence of inducer rotation speed on the propagation characteristics of rotating cavitation and on the unsteady characteristics of cavitation in rotating cavitation and cavitation surge were examined using a three-bladed inducer, which is named THK inducer, to develop the rocket engine turbopump that can operate stably over a wide operating range. The novel point of the paper is focusing on the cavitation itself, which is the cause of cavitation instabilities. The paper conducted two types of experiments using an inducer and single hydrofoils, and the dimensional and nondimensional unsteady characteristic, which is the frequency of unsteady cavitation and the Strouhal number, were evaluated. Three types of rotating cavitation and cavitation surge were observed in the inducer. For a given cavitation number, the Strouhal number of rotating cavitation and the frequency of unsteady cavitation in cavitation surge are independent of the inducer rotation speed, respectively. The characteristic of rotating cavitation corresponds to that of cavitations arising in hydrofoils, but its characteristic of cavitation surge does not correspond. The dynamic characteristic of cavitation compliance in cavitation surge is similar to that of several rocket engine turbopumps. Therefore, it was proved that rotating cavitation is a cavity oscillation and cavitation surge is a system oscillation. Additionally, in a flow field with the same flow coefficient and different impeller rotation speed, the unsteady characteristics of cavitation instabilities are equal if the cavitation number is equal, which is named the “similarity law of cavitation number on cavitation instability.”

Establishment of Prediction Model for Cavitation Surge Frequency and Onset in an Inducer Considering Dynamic Characteristics of Cavitation Compliance and Mass Flow Gain Factor

Abstract An accurate prediction method of cavitation surge is desired to design a reliable turbopump for rocket engines that allow complex operational sequences such as controlled reentry and landing. Therefore, the paper aims to develop a novel model that enables accurate predictions of both frequency and onset of cavitation surge by considering dynamic characteristics of cavitation compliance K and mass flow gain factor M. The paper conducts both experimental and numerical studies in an inducer known to cause cavitation surge. First, characteristics of cavitation surge including frequency and occurrence conditions are experimentally surveyed. Secondary, K and M for the studied inducer are numerically obtained by steady RANS simulations. Then, the one-dimensional model is initially established based on the premises acquired by performed experiments and applied to cavitation surge predictions using quasi-statically evaluated K and M. As a result of comparisons with experiments, the cavitation surge frequency is adequately evaluated by the established model, whereas the cavitation surge onset fails to be estimated only using quasi-static parameters. The new model is thus proposed by including dynamic characteristics of K and M, and it is mathematically clarified that phase properties of K and M may play a key role to trigger cavitation surge. By presuming adequate values of dynamic characteristics based on past literatures, consequently, the developed model succeeds in accurate predictions of the cavitation surge onset in the experiment. This evidences that dynamic characteristics of K and M are essential to predict cavitation surge quantitatively.

Effects of the Centrifugal Pump Outlet Blade Angle on Its Internal Flow Field Characteristics under Cavitation Condition

The outlet blade angle is a key geometrical parameter that governs how the impeller directly influences centrifugal pump performance. Therefore, a reasonable angle selection is crucial. To investigate the effects of the outlet blade angle on centrifugal pump internal flow field characteristics under cavitation conditions, this study employs a combination of a modified SST k-ω turbulent model with a Zwart-Gerber-Belamri cavitation model to perform transient flow simulations. Outlet blade angles of 15°, 20°, 25°, 30°, and 35° were tested. The results indicated that as the outlet blade angle increased, the relative liquid flow angle, vapor volume, and corresponding fraction first increased, then decreased, and finally increased again. Meanwhile, the distribution scope of each stall vortex on the suction surface became smaller, then larger, and smaller again, whereas the scopes of the pressure surfaces grew constantly as the outlet blade angle increased. Pressure fluctuations at all monitored points in the volute became weaker over time, and variations in the pressure fluctuations alternated with the outlet blade angle. The main frequency amplitude increased and the frequency doubling decreased as the outlet blade angle increased. Although the energy corresponding to the main frequency was unstable, it consistently held the dominant position. The duration that cavitation compliance was less than 0 first decreased and then increased as the outlet blade angle increased. For the impeller with an outlet blade angle of 25°, stall vortices accounted for the smallest regions, and the duration of negative cavitation compliance was minimized. In this case, the overall performance of the centrifugal pump was optimal.

Dynamic characteristics and stability evaluation of cavitation-induced flow instability in a conical diffuser

Abstract It is known that cavitation surge, which is a self-excited vibration caused by cavitation volume fluctuation, may occurs in a draft tube of a water turbine when it is operated under overload. In this study, a cavitation surge was generated by a swirling flow into a Venturi tube, and its dynamic characteristics and stability were evaluated using a one-dimensional model and a transfer matrix. In addition, the variation of the cavitation volume was calculated using the binarization process with a high speed camera and compared with CFD. As a result, it is possible to evaluate the period of the cavitation volume fluctuation by the binarization process, but it is difficult to evaluate its volume quantitatively. The magnitudes of the cavitation compliance and the mass flow gain factor decrease with the increase of the cavitation coefficient. The magnitudes of the mass flow gain factor also tend to decrease with decreasing swirl number. The phase of the mass flow gain factor tends to delay as the cavitation coefficient increases. It is found that the phase of the mass flow gain factor and the cavitation compliance are also important parameters for the stability.

Prediction of Cavitation Surge Onset Point by One-Dimensional Stability Analysis

Cavitation instabilities such as rotating cavitation and cavitation surge often occur in high speed turbopumps. It has been shown by a stability analysis that the cause of cavitation instabilities is explained by the positive mass flow gain factor, representing the increase of cavity volume against the decrease (increase) of flow rate (incidence angle). The cavitation compliance, representing the increase of cavity volume in response to the decrease of inlet pressure, determines the frequency of instabilities. However, one-dimensional stability analysis cannot be directly used for the prediction of the onset point of cavitation surge when quasi-steady assumption is applied. In the present study, unsteady characteristics, i.e. the phase lag in the response of cavity volume against the flow rate/inlet pressure fluctuations, are taken into account in a stability analysis of cavitation surge in the form of lag element. The onset criterion considering the time lag is newly proposed for one-dimensional stability analysis, and the criterion is validated by comparisons with a two-dimensional stability analysis based on a singularity method applied to a free streamline theory.

Low-frequency oscillations in hydroturbines caused by cavitation together with the phase transitions effects

Among the unsteady phenomena taking place in hydroturbines one can separate oscillations caused by a cavitation bubble behind the runner. Usually a concept of cavitation compliance is used to model the cavity pulsations (Chen et al.). There was demonstrated the destabilizing effect of the diffuser and swirl. Kuibin et al. had found that the effect of swirl on stability is much smaller than that described by Chen et al. In the present work, the same analysis is repeated, but taking into account influence of the evaporation and condensation on dynamics of the cavity appearing in the draft tube. During analysis a simplified spherical cavity form was considered.

Dynamic Model-Based Identification of Cavitation Compliance and Mass Flow Gain Factor in Rocket Engine Turbopump Inducers

Abstract Cavitation dynamics continue to pose a significant risk in the development and operation of launch vehicle (LV) propulsion systems. In addition to generating unsteady loads that can directly damage turbopump hardware, cavitation dynamics often couple with LV fluid feed systems, producing system wide POGO instability that can cause catastrophic failures. Despite its importance, the current understanding of cavitation dynamics, and especially pump transfer matrices, is limited. Given the relatively sparse amount of inducer transfer matrix data available, there is a critical need for more in-depth characterization of the cavitation dynamics in turbopump inducers to avoid POGO instability. This paper defines and validates a new reduced-order approach to infer key parameters such as cavitation compliance, K, and mass flow gain factor, M, from simple, single sensor unsteady pressure measurements during inducer inlet pressure ramps. The utility of this approach is demonstrated for a range of inducer geometries reported in the literature. The results are in agreement with experimental data and the paper provides a new capability supporting the assessment of LV POGO instability.

Evaluation of a Dynamic Transfer Matrix for a Hydraulic Turbine

Abstract It is well known that hydraulic machines experience various types of flow instabilities causing a negative influence on the system under off-design operations. The transfer matrix method correlating the flow properties in upstream and downstream of hydraulic machines is widely adopted as a first step to investigate dynamical characteristics of flow. Transfer matrix elements are the key to understand hydraulic system stability. This study focuses on measurements of transfer matrix elements for a hydraulic turbine. The oscillations of the flowrate are produced by two flow exciters located in upstream and downstream of the turbine, and evaluated from the fluctuations of the pressure difference across two streamwise locations. It is shown that the transfer matrices are successfully evaluated at part load and full load operations in the presence and absence of cavitation. In particular, cavitation compliance and mass flow gain factor, which determine the dynamical response of cavitation to the change of pressure and flowrate, are calculated from the measured transfer matrix elements. The absolute value of both cavitation compliance and mass flow gain factor is found to increase with respect to the decrease of the cavitation number. The phase of the mass flow gain factor is delayed as the excitation frequency increases. This suggests that hydraulic systems may be stabilized when the oscillation frequency increases. As a result of stability analyses, it is demonstrated that the mass flow gain factor plays a crucial role, especially in the full load cavitation surge.

On low-frequency unsteady phenomena caused by cavitation in hydroturbines

When hydroturbines operate at part load or over load, intense pressure and vibration pulsations may occur. One of the instability mechanisms is due to the presence of a cavitation bubble (or cloud) behind the runner. Chen et al. [1] the phenomenon is modeled using the concept of cavitation compliance. They demonstrated the destabilizing effect of the diffuser and swirl. In the present work, the influence of two factors on the coefficients of the characteristic equation, whose solution yields natural frequencies and oscillation increments, is revealed. They are the vorticity distribution and the size of the cavitation cavity behind the runner. It is shown that the effect of swirl on stability is much smaller than that described in [1] and rapidly decreases with the growth of cavitation bubble size.

Numerical investigation of the cavitation dynamic parameters in a Francis turbine draft tube with columnar vortex rope

The cavitation developed in the Francis turbine draft tube is known to be a source of the hydraulic instability, especially under the conditions with columnar cavitation vortex rope occurring in the draft tube. The hydraulic system will suffer from the risk of the self-excitation and the high pressure pulsations because of the dynamic features of the complex cavitation vortex rope. The links between the cavitation dynamic characteristics and the hydraulic system stability are important for an accurate system instability prediction. For this purpose, the cavitating flows in a Francis turbine operating under the overload conditions are simulated by using the Zwart-Gerber-Belamri (ZGB) cavitation model. The cavitation dynamic parameters in different cavitation development stages and the evolutions under various overload conditions are analyzed and compared in this paper. It is shown that the cavitation dynamic parameters, including the cavitation compliance, the mass flow gain factor and the wave speed, can well represent the cavitation characteristics with the vortex rope evolution. The predicted cavitation dynamic parameters are in a reasonable agreement with other available results, with an obvious dependency on the loads. The results obtained in this study are relevant to the system stability analysis.

Analysis of the Francis turbine upper-part-load pulsation Part II – Mechanism of self-excitation

When a Francis turbine operates at partial load or very high load, the swirling flow in the draft tube may cause objectionable oscillations of pressure and power. The cavitating core of the vortex plays an important role in these pulsations. The present paper deals with a class of self-excited oscillations of the entire water column in the power plant; self-excitation means that at least one eigenvalue of the hydraulic system becomes unstable. A one-dimensional (1D) model in frequency domain explains how the normal damping is eliminated. Oscillation power is provided in regions whose flow gain in streamwise direction has a component in phase with pressure. The model contains a module for the dynamic transmission behavior of the cavitating vortex; it represents the response of the cavity size to variations of the local pressure and swirl. The sensitivity to pressure changes (the ‘cavitation compliance’) controls the natural frequencies but cannot cause instability whereas the response to swirl changes (‘mass flow gain’) may supply net oscillation energy and thus cause instability. Both influences act all along the cavitating part of the vortex; it is crucial that the variation of runner exit swirl can propagate along the vortex only with the axial velocity of the fluid. The oscillation energy balance depends on the wavelength of swirl variation, i.e. the combination of axial velocity and oscillation frequency. All instabilities of this class are ‘breathing’ pulsations, synchronous within one cross section. In the simplest case with the lowest natural frequency the pressure variation is roughly synchronous in the whole draft tube; for this mode (full-load surge) a lumped-parameter model may be adequate. By contrast, the upper-part-load pulsation occurs in a more complex eigenmode; a distributed-parameter model version is required to represent the essential features. The draft tube pressure oscillation has two quarter waves and a pressure node within the cavitation zone. The pressure at both ends of the draft tube cavitation zone has roughly opposite phase. Difficulties to transpose the stability between reduced-scale model and prototype are explained using the 1D model, as well as some influence of the runner hub shape and of the upstream conduit. Damping at the runner explains why the pulsation is limited to low-head turbines.

Investigation of cavitation instabilities in a centrifugal pump based on one-element theory

In order to investigate the cavitation instabilities in a centrifugal pump, the unsteady cavitating flow in a low-specific speed centrifugal pump was simulated by using a modified SST k-ω turbulence model combined with Kubota cavitation model. The intrinsic and system instabilities of cavitating flow were comprehensively analyzed and interpreted. For the cavity patterns formation process in a centrifugal pump, the separated flow is dominant instead of re-entrant jet flow. The transient incidence angle, cavitation number, cavitation compliance and mass flow gain factor of the middle streamline were addressed in the centrifugal pump at design point based on classical one-element theory in order to characterize the system instability. The results show that the bubbles are static stability at cavitation inception stage, which can reduce pressure fluctuations by varied cavity volume and hard to develop to rotating cavitation. It performs static instability at the stage of development because of the positive-feedback between cavity volume and flow rates, the decreasing of bubble volume would lead to the increasing of flow rate, which can induce undesirable cavitation instabilities.

One-Dimensional Analysis Method for Cavitation Instabilities of a Rotating Machinery

Rotating cavitation is an important problem, which makes it difficult to design reliable rotating machines. In this study, a simple analysis method that tried to evaluate the cavitation instabilities of a rotating machinery by using one-dimensional (1D) system analysis software was attempted. In this method, cavitation compliance and mass flow gain factor are distributed in each flow path of the inducer. Analysis results show that cavitation instabilities, including rotating phenomena, exist. With the evolved analysis model, effects of various parameters on the eigenvalues of the system were investigated. Analysis results agreed with inducer test results qualitatively. Furthermore, by the analysis considered whirl motion of the rotor, effects of it on cavitation instabilities were investigated.

Transposition of Francis turbine cavitation compliance at partial load to different operating conditions

Francis turbines operating in part load conditions experience a swirling flow at the runner outlet leading to the development of a precessing cavitation vortex rope in the draft tube. This cavitation vortex rope changes drastically the velocity of pressure waves traveling in the draft tube and may lead to resonance conditions in the hydraulic circuit. The wave speed being strongly related to the cavitation compliance, this research work presents a simple model to explain how it is affected by variations of operating conditions and proposes a method to transpose its values. Even though the focus of this paper is on transpositions within the same turbine scale, the methodology is also expected to be tested for the model to prototype transposition in the future. Comparisons between measurements and calculations are in good agreement.

Experimental investigation of the mass flow gain factor in a draft tube with cavitation vortex rope

At off-design operating operations, cavitating flow is often observed in hydraulic machines. The presence of a cavitation vortex rope may induce draft tube surge and electrical power swings at part load and full load operations. The stability analysis of these operating conditions requires a numerical pipe model taking into account the complexity of the two-phase flow. Among the hydroacoustic parameters describing the cavitating draft tube flow in the numerical model, the mass flow gain factor, representing the mass excitation source expressed as the rate of change of the cavitation volume as a function of the discharge, remains difficult to model. This paper presents a quasi-static method to estimate the mass flow gain factor in the draft tube for a given cavitation vortex rope volume in the case of a reduced scale physical model of a ν = 0.27 Francis turbine. The methodology is based on an experimental identification of the natural frequency of the test rig hydraulic system for different Thoma numbers. With the identification of the natural frequency, it is possible to model the wave speed, the cavitation compliance and the volume of the cavitation vortex rope. By applying this new methodology for different discharge values, it becomes possible to identify the mass flow gain factor and improve the accuracy of the system stability analysis.

Cavitation Influence in 1D Part-load Vortex Models

Residual swirl in the draft tube of Francis turbines may cause annoying low- frequency pulsation of pressure and power output, in particular during part-load operation. A 1D analytical model for these dynamic phenomena would enable simulation by some conventional method for computing hydraulic transients. The proper structure of such a model has implications for the prediction of prototype behaviour based on laboratory tests.The source of excitation as well as the dynamic transmission behaviour of the draft tube flow may both be described either by lumped or distributed parameters. The distributed version contains more information and, due to limited possibilities of identification, some data must be estimated. The distributed cavitation compliance is an example for this dilemma. In recent publications, the customary assumption of a constant wave speed has produced dubious results. The paper presents a more realistic model for distributed compressibility. The measured influence of the Thoma number is applied with the local cavitation factor. This concept is less sensitive to modelling errors and explains both the Thoma and Froude number influence.The possible effect of the normally unknown non-condensable gas content in the vortex cavity is shortly commented. Its measurement in future tests is recommended. It is also recommended to check the available analytical vortex models for possible dispersion effects.

Bondgraphs model on cavitating pump system

Cavitation phenomena in a pump unit as well as internal flows can be analysed by CFD codes with meaningful precision. However, it is difficult to analyse the dynamic characteristics of cavitating pump systems such as cavitation compliance and mass flow gain factor. The characteristic parameters on pump cavitation are studied through CFD calculations on cavity volume and the results are related the system bondgraphs in a lumped parameter system. The cavitation compliance can be represented by 1-port C element and mass flow gain factor can be represented not only by 1-port R element as flow resistance but also by 1-port I element as fluid inertia.

Influence of Pipe Length on Cavitation Surge Frequency in a Cascade

Cavitation surge is a system instability that occurs in pumping machineries in which the unsteady characteristics depend on system components such as the pipe length. On the other hand, under typical boundary conditions of numerical simulations of internal flow, it is supposed that a uniform flow extends infinitely upstream and downstream from the boundaries. Therefore, a numerical simulation of the influence of the pipe length of fluid machinery cannot be realized just by changing the position of a boundary of the computational region. In the present study, a boundary condition that takes into account the pipe length was used to numerically simulate unsteady cavitation in a three-blade cyclic cascade. The resulting cavitation surge frequency became lower when the upstream pipe length becomes longer, which is consistent with previous findings. Also, the resulting cavitation surge frequency showed good agreement with empirical frequencies of the actual cavitation surge of pumps, although the empirical frequency is a rough prediction because it covers various system components. Additionally, the theoretical frequencies were predicted by estimating quasi-steady and local cavitation compliances from the present results. The theoretical frequency takes into account the upstream pipe length. As a result, local cavitation compliance was relatively effective for prediction of the theoretical cavitation surge frequency.

Experimental investigation of the local wave speed in a draft tube with cavitation vortex rope

Hydraulic machines operating in a wider range are subjected to cavitation developments inducing undesirable pressure pulsations which could lead to potential instability of the power plant. The occurrence of pulsating cavitation volumes in the runner and the draft tube is considered as a mass source of the system and is depending on the cavitation compliance. This dynamic parameter represents the cavitation volume variation with the respect to a variation of pressure and defines implicitly the local wave speed in the draft tube. This parameter is also decisive for an accurate prediction of system eigen frequencies. Therefore, the local wave speed in the draft tube is intrinsically linked to the eigen frequencies of the hydraulic system. Thus, if the natural frequency of a hydraulic system can be determined experimentally, it also becomes possible to estimate a local wave speed in the draft tube with a numerical model.In the present study, the reduced scale model of a Francis turbine (v=0.29) was investigated at off-design conditions. In order to measure the first eigenmode of the hydraulic test rig, an additional discharge was injected at the inlet of the hydraulic turbine at a variable frequency and amplitude to excite the system. Thus, with different pressure sensors installed on the test rig, the first eigenmode was determined. Then, a hydro-acoustic test rig model was developed with the In-house EPFL SIMSEN software and the local wave speed in the draft tube was adjusted to obtain the same first eigen frequency as that measured experimentally. Finally, this method was applied for different Thoma and Froude numbers at part load conditions.