STOCHASTIC MODELING OF CAVITATION EROSION BASED ON THE CONCEPT OF RESONANT ENERGY ABSORPTION

Transcription

1 Cav03-GS Fifth International Symposium on Cavitation (cav2003) Osaka, Japan, November 1-4, 2003 STOCHASTIC MODELING OF CAVITATION EROSION BASED ON THE CONCEPT OF RESONANT ENERGY ABSORPTION Leszek Wilczyński Ship Design and Research Centre Gdańsk Ship Model Basin, POLAND ABSTRACT The paper dwells on the modeling of cavitation erosion regarded as the process of energy transfer from cavitating liquid towards the solid body structure. Both the characteristic features of the pressure impact generated during the cavitation collapse as well as the solid body response are considered. The crucial point of the proposed approach is that the erosion is the manifestation of the resonance between the pressure signal exciting the flowed body structure and the specific vibration properties of the excited solid body crystal structure. The presented erosion model enables qualitative explanation of the nonlinear erosion character, the occurrence and diversification of erosion incubation period and refers to the concept of cavitation intensity threshold level with regard to erosion. INTRODUCTION Cavitation induced erosion serves as the example of a process of energy transfer occurring in a systems driven far from equilibrium and reflecting the tendency of energy focusing from macroscopic scale to multi-degrees of freedom micro scale. Generally speaking, the process of cavitation erosion can be classified as irreversible and occurring in the dissipative physical system. Those qualitative features of the cavitation erosion phenomenon imply the certain approach towards its modeling. Besides, it has to be kept in mind that among the parameters influencing the course of the actual erosion process, those varying randomly in space and time are dominating. The problem of cavitation induced erosion focuses the interest of researchers since more than hundred years. The investigation of cavitation erosion originated from the problems connected with the operation of screw propellers of high-speed naval vessels. The first reported case was one of the first destroyers - HMS Daring launched in Thornycroft shipyard [1]. The energetic approach towards the cavitation erosion modeling can be found in the literature devoted to the problem. A combined empirical and analytical approach towards the quantitative prediction of cavitation erosion in the flow around a hydrofoil has been proposed in [7]. The wide range of analytical and numerical methods based on the consideration of energy transfer between the cavitating liquid and solid body structure and applied to the modeling of erosion phenomena can be found in [2, 3, 4, 5, 8, 9] The following paper presents an attempt to model the cavitation induced erosion as the consequence of irreversible, resonant absorption by solid body material of the energy released during the collapse of cavitation structures in the flow. Within the following paragraphs the qualitative features of the model are clarified, as well as its physical assumptions and hypotheses discussed with the special emphasis put on the conclusions arising from them. The considered physical system consists of cavitating liquid and solid body structure exposed to the impacts generated by the collapsing cavitation structures. The liquid is subjected to varying (or periodically alternating) flow conditions, at which negative and positive pressures are supposed to occur in the vicinity of the solid body boundary. In general, the pressure variation is the consequence of dynamic pressure change due to the character of flow, as well as the shape of a flowed body or acoustic (vibratory) excitations in the liquid volume. Incidentally this diversification is not trivial because in both cases the course of the erosion for a particular material can be different. According to the way of cavitation generation the course of erosion is differently related with the features of solid boundary surface and its internal structure. For example, it has been observed that polished samples of a particular material erode quicker than rough ones when cavitation is induced by acoustic excitation. On the other hand the erosion intensity changes oppositely when cavitation develops in the liquid flowing over the mentioned above kinds of material samples. As the result of the flow conditions different forms of cavitation develop and disappear close to the submerged solid body surface. When the process is enough systematic it is obvious that the solid body structure has to accumulate monotonously increasing amount of energy. and becomes driven from the equilibrium state. The energy of the moving or oscillating liquid is transformed into the potential energy of cavitation bubbles, then the collapsing bubbles release the energy, reaching the solid body structure. The energy released in the liquid by collapsing cavitation structures is dissipated. This 1

2 serves as the necessary, but not sufficient condition of erosion appearance. The energy flux meets on the way of its propagation the solid body, thus the further course of the process depends on the body structure features. The way the solid body responds to the cavitation induced pressure impacts determines the sufficient condition for erosion development. The erosion itself has a nonlinear character. The nonlinearity stands in that the superposition of the combined action of cavitation events inducing erosion does not coincide with the result of the superposition of their separate effects. When erosion starts the solid body structure, considered as the system that dissipates energy, acts like an amplifier, i.e. moves away from the reference state and evolves rapidly towards new regime characterizing with different energy dissipation rate. Besides the non-linearity, the most conspicuous feature of the phenomenon of cavitation induced erosion is its stochastic character. The macroscopically manifesting erosion of the particular solid body specimen is usually preceded by the incubation period. It means that during a certain period, the action of the imploding cavitation structures brings no visible by naked eye influence on the solid body boundary. Thus, within the model, the incubation period should be regarded as the time, during which the solid body internal structure is modified under the action of pressure impacts on the material surface. The energy flux is then accumulated by the solid body and once more gains the form of potential energy (similarly as in the case of the generation of the bubbles or other cavitation forms). When the level of this potential energy locally exceeds the bonding energy of the crystal lattice then the probability of the local separation of the solid body spalls increases. CAVITATION EROSION MODEL Let consider the collapse of the individual cavitation bubble occurring in the vicinity of the solid body boundary as the single cavitation erosion event. Macroscopically manifesting erosion is the effect of the occurrence of sufficiently big number of erosion events. From the energetic point of view the structure of the solid boundary, exposed to the action of erosion events is a band type structure. Therefore, the energy emitted by the collapsing cavitation forms (being the superposition of erosion events) cannot be absorbed in a continuous way. The band structure of the vibration energy spectrum of the solid body crystal lattice is responsible for that the pressure impacts are absorbed selectively In other words, only selected frequencies are visible for the solid body. When the range of crystal lattice vibration energy bands, which fit to the frequencies encountered in the energy flux resulting from cavitation collapses increases, then the absorption becomes more effective and it can be stated that there occurs a resonance between them. This situation cannot last too long because the energy is constantly supplied to the system and cannot be accumulated endlessly. When the energy bands become saturated the system looks for another effective ways of energy accumulation or dissipation. One of them may be the separation of small portions of the solid body. The occurrence of resonance conditions is the crucial hypothesis of the proposed model explaining how the energy of cavitation collapse convert into work necessary to break the solid body bonding strength. The energy of a cavitation collapse is absorbed locally with different rates and with different characteristic frequencies. This occurs due to the non-homogeneity of the crystal lattice (e.g. lattice imperfections). In general the cavitation collapse energy absorption should be considered as local because of: granular character of the material, material crystal lattice imperfections, material surface irregularities, short duration of the cavitation collapse event, limitation of the solid body area affected by the pressure impact. The energy transfer from collapsing cavitation structures towards the crystal lattice results in the progressive saturation of vibration energy levels. The crucial features of this transfer process are impact magnitude and frequency. The process itself is a random one. The following quantities carry the features of random variables: the moment of the pressure impact appearance, the rate and quantities of the pressure impacts, the frequency of energy amounts released during the erosion event, the location of the impact with respect to the solid body material surface, the material susceptibility to erosion (internal lattice imperfections, state of the surface, surface and internal material micro-cracks). The course of the erosion is determined by the mutual relation between the rate of energy flux from cavitating liquid towards the material structure, the range of material crystal structure vibration energy bands and the rate of dissipation of intra-crystalline vibration energy. Within the model the erosion serves as the manifestation of the necessary arising of the energy dissipation channels for the physical system consisting of cavitating liquid and solid body. The channels of energy dissipation associated with erosion become available to the considered physical system when it is driven far from equilibrium. The modeled system being additionally a dissipative one evolves in the non-linear fashion. The choice of the energy dissipation channels associated with material separation brings to the system the entropy jump. When the channels of energy dissipation, which do not impose the discontinuous entropy increase are enough effective and additionally allow the system to follow the steady state behavior then the material does not erode. The steady state can be then defined by the simple equation: dm = 0 (1) dt where m denotes the mass of the flowed body. The erosion begins when the steady state of energy dissipation within the given crystal lattice cannot be longer maintained on the way of the continuous entropy increase. The situation is illustrated in the Fig. 1. The particular stages of the erosion process are associated with the characteristic features of the entropy time course. 2

3 Fig. 1.The entropy of the liquid-solid body system in the resonant model of the cavitation induced erosion The system prior to erosion start remains in steady state but it is constantly shifted away from equilibrium. With the increase of the amount of accumulated energy (local density of crystal lattice vibration energy) the measure of the steady states of the system tends to zero. Consequently the probability of system remaining in steady state also evolves to zero and probability of system unsteady behavior increases. As the system is dissipative it evolves towards an attractor state. In case of erosion such a state can be defined as: dm dt = const. 0 In the language of dynamical systems theory the period of erosion incubation and the period of erosion with constant mass loss rate should be considered as the chains of steady states and the erosion rate acceleration, as well as its attenuation represent the unsteady behavior of the system. The qualitative features of evolution of the eroding system in the phase space ( m m& ) presented in the Fig. 2. (2), are The occurrence of incubation period is the consequence of the randomness of the characteristic factors influencing the erosion course, as well as of the initial relation between the crystal lattice energy absorption bands and characteristic frequencies of the energy flux released during collapse of cavitation forms. When the vibration energy bands characteristic for the initial form of crystal lattice do not fit to the frequencies of the impacts from the cavitation collapse, then the channels for dissipation remain invisible. At this stage of the process, prior to energy absorption, the structure of the material undergoes modification aiming at the creation of the channels sufficient for energy dissipation. The change of the bands range can be realized for example on the way of the local material hardening. The initial stage of the cavitation induced erosion process, i.e. incubation can be thus divided into two parts. The first one of them consists in the action of the impacts aiming at the local modification of the physical features of the solid body. The structure of the solid body changes in such a way that the range of vibration energy bands of the crystal lattice layers located closest to the surface starts to coincide with the frequencies encountered in the cavitation induced pressure impacts spectrum. The second stage of incubation begins when the crystal lattice becomes ready to accumulate energy from cavitation collapses. Depending on the mutual relation between the pressure impacts spectrum and characteristic physical properties of the solid body the stage of accumulation may start without earlier material treatment, i.e. when the dissipation channels already fit to the impact spectrum from the beginning of the process. The features of the solid body structure determine also the effectiveness of the process of the internal vibration energy dissipation. The balance between the energy fluxes: from cavitating liquid towards the crystal structure and from the surface crystal layers into the solid body interior indicates that there is no erosion. The crystal layers located the closest to the solid body surface serve as a kind of energetic buffer. When the rate, at which the energy bands become saturated (depending on the number and density of cavitation impacts) exceeds the rate of energy dissipation within the solid body internal structure then the balance is disturbed and the system becomes unstable. This can be understand as analogous to the overheated liquid. For such a state there always exists a finite perturbation that the system cannot sustain. The schematic view of the evolution of the way of cavitation impacts energy dissipation is presented in the series of Figs. 3 a-d Fig. 2.The evolution of the system facing cavitation induced erosion. The point moving along the curve represents the instantaneous state of the system. Because of the randomness of the physical factors determining the course of erosion the shaded area is more appropriate to represent the set of allowed states. 3

4 Fig 3a. A pre-incubation period. The vibration energy bands of crystal structure of a solid body remain invisible (do not fit to the frequencies encountered in the energy spectrum of cavitation collapse generated impacts). The energy flux released during the cavities collapse E c cannot be effectively absorbed by the solid body. Only a small portion of energy E t is transmitted into the body. The remaining portion E r is reflected The energy conservation implies that E c =E r +E t Fig. 3b. The change of material properties. Consequently the energy flux from cavitating liquid penetrates the most outer layers of solid body crystal structure. The rate of dissipation increases and the erosion incubation begins. Fig. 3c. The solid body crystal structure energy bands fit to the cavitation generated output. The rate of dissipation increases. The structure of solid body becomes visible for the pressure impacts. The saturation of the energy bands starts. Fig. 3d. Erosion - release of the energy stored within the vibration energy bands. The behavior of the system can be interpreted analogous to the laser action, especially when the inversion of the saturation of energy bands occurs. The rate of entropy production encounters discontinuity. The most crucial features of the resonance model of cavitation induced erosion can be recapitulated as follows: the energy released during the collapse of cavitation structures gains the form of pressure impacts and meets on the way of its propagation the solid body crystal structure, the process of the energy transfer from liquid to solid body is determined by amplitude and frequency of pressure field variations, from energetic point of view the crystal lattice of a solid body characterizes with band type structure with unique vibration energy bands ready to absorb the cavitation collapse energy propagating through the liquid volume, the first stage of cavitation erosion consists in the exploration by the system consisting of cavitating liquid and solid body the possible energy dissipation channels, when the crystal lattice vibration energy bands do not coincide with the spectrum of the pressure impact resulting from cavitation collapse then the material properties of the solid body are modified (e.g. on the way of local surface hardening) in order to gain the possibility of energy absorption and dissipation, the erosion incubation period begins, the system becomes driven from equilibrium but due to the balanced process of energy dissipation is remains stable, the equilibrium between the energy flux from the collapsing cavities and the ability of the solid body crystal structure to accumulate and dissipate the energy is maintained, the incubation stage of erosion develops, absorbing the increasing amount of energy the solid body structure becomes driven far from equilibrium what reduces the measure of available steady states to zero, brings instability and non-linearity to the process of energy dissipation, the system approaches to the end of incubation period, the system faces the resonance between the collapse impacts and crystal lattice vibration, the probability of following by the system the evolution with continuous rate of entropy production reduces to zero, the separation of solid body portions starts and creates the way for attaining the new steady state (which is the characteristic feature of dissipative systems), in the first period of erosion the excess of accumulated energy is released, especially when the inversion of the saturation of energy bands has occurred during the incubation period, the system behaves like an amplifier and the erosion acceleration occurs, the system evolves towards a new steady state (attractor) and after the erosion acceleration period the comparatively small attenuation of erosion rate may occur, finally the new steady state is attained, the new balance between the energy flux from cavitating liquid towards the solid body and the energy consumed to break the crystal lattice bonding strength as well as its part being accumulated and dissipated within the solid body interior is created, the constant mass loss period of erosion begins, 4

5 because the pressure impacts generated during the cavities collapse and acting on the solid body surface characterize with the random features and the local properties of the crystal structure are to a certain degree random too thus the way the steady erosion state is attained is the course of stochastic process, the parameters of the probability density functions describing the quantities determining the erosion phenomenon and their evolution define the response of the solid body structure and influence the way of attaining the erosion steady state. QUALITATIVE SUBSTANTIATION OF THE MODEL The following part of the paper contains the considerations regarding the qualitative conclusions that can be raised from the assumptions of the resonant energy absorption model of cavitation erosion. The basic aim of the model is to explain the following features of the erosion course: the occurrence and instability of the erosion incubation period, the existence of the erosion threshold level of cavitation intensity, the diversity of the response of the material to various forms of cavitation as well as different ways of cavitation generation, Let consider the possible relations between the impact energy spectrum and the range of crystal lattice energy bands. In general two qualitatively different situations can occur. The first one of them consists in that the range of material structure vibration energy bands coincide with the frequencies encountered in the cavitation collapse pressure spectrum. In this case the saturation of vibration bands of the solid body material can start immediately as soon as the collapse impacts reach the solid body surface. The further course of the process depends on the material (crystal lattice) properties. When the vibration energy can be transmitted and dissipated towards the solid body interior then erosion will not start until the balance between the energy fluxes is possible to maintain. In this case the idea of cavitation intensity threshold level can find the substantiation. The cavitation intensity threshold is connected with the balance between the crystal structure energy bands saturation and release. When the balance cannot be maintained then it can be stated that the cavitation erosion threshold level has been achieved. The excess of the vibration energy, which cannot be propagated into the solid body interior is converted into the work of breaking the crystal lattice bonding strength. The character of the response (violent or slow) depends on what range of energy bands becomes saturated from the beginning of the process. When the higher bands become saturated earlier than lower then the response (erosion) will be more violent. The absorption of the energy leads to inversion of vibration energy levels occupancy. The release of the accumulated energy can thus be more influencing the solid body structure. The system stability is limited (as in the case of overheated liquid) and it can behave like an amplifier while small impact propagating to the crystal lattice leads to release of considerable amount of accumulated, damaging energy. On the other hand, when the lower energy bands become saturated earlier then the probability of enough vibration energy concentration exceeding the available flux into the body interior and crystal structure bonding strength is lower and consequently the erosion is slower. From practical application point of view the most crucial problem is to recognize adequately the sufficient and necessary conditions of the violent course of the erosion. The second possibility is that the bands do not fit to the spectrum of the cavitation collapse outcome. Then the preincubation stage of erosion process, i.e. modification of solid body material properties precedes the adequate erosion incubation period. When the propagation of the cavitation collapse impacts becomes possible then the erosion incubation begins. Once more the concept of cavitation intensity threshold finds the substantiation. The assumption stating that the cavitation induced erosion consists in occurrence of resonance between the energy flux from cavitation collapse impacts and local crystal lattice vibration energy bands enables the explanation of the incubation period as well as the instability of the eroding system behavior during incubation. The occurrence of cavitation induced erosion in the smallest scale, specific for this phenomenon requires the coincidence of two events, i.e. the pressure impact propagating from the flow towards the solid body and the appropriate response of the solid body to this impact. Therefore the macroscopic manifestation of erosion is the result of superposition of two randomly varying physical fields: the first one describing the pressure impact and the second one describing the solid body ability to absorb the energy output from collapsing cavitation structures. The possible relations between those two fields are presented in the Fig 4 a-e. The energy-frequency spectrum of the pressure impacts generated by collapsing cavitation structures is represented by the curve and the vibration energy levels of the material structure is depicted with two bands Fig 4a The dominating frequencies in the collapse output spectrum, i.e. those carrying the maximum energy amount are less than the lowest vibration energy bands specific for the material. The saturation of energy bands is a relatively slow process which can be balanced with the energy dissipation inside the solid body. The material is erosion resistant or in the other words the cavitation intensity threshold is not achieved. Cavitation brings no effect to the material. 5

6 Fig. 4d The energy is absorbed on the way of the saturation of higher energy bands. The incubation period is short and cavitation intensity threshold level is achieved quickly. The inversion of the saturation of vibration energy bands causes that the system is driven far from equilibrium and the erosion being the attractor state can be achieved in the way of small (finite) disturbance. The release of the stored energy leading to material separation is most probable. Fig. 4b The frequencies corresponding to maximum energy flux from the cavitating liquid towards the solid fit to the lowest available vibration energy bands. The material absorbs the cavitation collapse energy and depending on its physical properties the further dissipation of energy can takes either the form of material separation of internal dissipation. The erosion process is relatively slow with long incubation period. The energy concentration sufficient to break the crystal structure bonding strength is a rare event. Fig 4c The maximum of the energy flux is located between the low and high energy bands specific for the solid body structure. The saturation of both lower and higher bands occurs with the comparable probability. The erosion incubation period is shorter than in case 4b. The action of cavitation can be more violent and threshold cavitation intensity is achieved comparatively earlier. Material separation depends on the properties of the process of internal energy dissipation. Fig.4e The frequencies of impact energy spectrum do not fit to the vibration energy bands of the solid body. The action of the cavitation generated impacts leads to the modification of local material properties (e.g. hardening) resulting in the appearance of higher vibration energy bands. The interaction of the cavitation induces impacts may result, depending on the material properties in occurrence of the situation presented in Figs. 4b, 4c, or 4d. The local properties of the solid body surface are randomly diversified. Due to the way of the material crystallization and particular properties of its surface the physical field representing the material should be regarded as the stochastic one. Therefore the examples of the mutual relations between the collapse impacts and material characteristics presented in Figs 4a-e should be considered as the statistically dominating and the appropriate pdfs have to ascribed to the parameters of both fields. The process of generation pressure impacts during the collapse of cavitation structures in the flow is the continuous and mass- type. Because the generation cavitation collapse impacts in the vicinity of a flowed body carries distinct features of stochastic process, the input faced by the solid surfaces is diversified. Depending on the parameters of the impact distribution function and its time evolution the solid body prepares its response. The deterministic core of the process (that means the reproducible and controlled erosion events) fulfills the laws given by physical constrains: the energy has to be conserved and the dissipation (or the entropy increase faced by the system) has to be as slow as possible. The state of the solid body structure prior to the erosion development is quasi-stable and far from equilibrium. From the point of view of system evolution theory it serves as the negative nodal point. That means that approaching to the final stage of erosion incubation the solid body becomes extremely sensitive to small perturbations and its behavior after critical perturbation does not follow the steady state evolution scheme. Because the system is dissipative it evolves towards the certain 6

7 attractor. In case of cavitation induced erosion such an attractor can be defined as is the state of constant rate of mass loss. The way the attractor is attained is diversified. Physical factors determining the occurrence of the cavitation induced erosion of a particular flowed body can be gathered into two groups: those associated with the flow and those specific for the material. The above classification brings certain simplifications into the description of the erosion phenomenon. On the other hand such an approach enables construction of the stochastic process reflecting the physical mechanism of cavitation erosion according to the following scheme: the randomized input signal (the series of pressure impacts) undergoes randomized modulation (response of the solid body structure to the impacts) and with certain probability leads to the erosion. The time-variable pressure distribution defined on the specified area of the solid body surface can be regarded as the input signal and the local solid body material response to the pressure impact as the signal modulation. In the most cases of cavitation induced erosion occurring in the flows encountered in the practice both the input signal and its modulation remain random. The randomness of the input signal consists in the chaotic diversification of the magnitude and the frequency of the impacts resulting from the collapse of cavitation bubbles. For the purpose of erosion process modeling it can be assumed that at the particular point of the considered region of a solid body surface the instantaneously recorded pressure is described by the probability density function. The series of such recordings form a discrete stochastic process, which specification is one of the aims of erosion modeling. The exact form of the instantaneous signal distribution serves as the model parameter. The character of the instantaneous impact pdf can be either deduced from the general physical constraints specific for the collapse process or determined empirically. The simplest assumptions concerning the process of generation of pressure impacts can be formulated as follows: the moments of appearance of the individual erosion events (single impacts) form the process with independent increments, i.e. the waiting time dividing the successive events gains the exponential pdf, the location of the individual erosion event with respect to the solid body surface is the random variable described by uniform distribution, the magnitude of the individual impact is characterized by the Poisson pdf, the frequency of the individual impact satisfies the normal distribution. The above assumptions serve only as the initial approximation of the process. They express the mutual independence of the particular erosion events observed at the certain point of the solid body surface in the course of the process. This can be accepted only when the individual point of the surface is considered. The first two assumptions are the most robust. They do not reflect for example the occurrence of the collective, simultaneous collapse of the large number of bubbles. Such an approach does not take into account the variation of the global flow parameters, which influences the characteristic parameters of the stochastic pressure field. Therefore in order to model the phenomenon more adequately the parameters of the stochastic pressure field have to be related with the variation of the flow. This can be carried out on the way of correlation of parameters of probability density functions describing the instantaneous distribution of the impacts on the solid body surface and the moments of their appearance with the prediction of cavitation forms in the flow around a propeller/rudder. The action of the pressure impacts should be thus considered as the random field which varies in time coherently with the changes in cavitation forms appearance. Determination of this variation is preceded with the calculation of cavitation characteristics of the flow. The results of this calculation serve as the input for the erosion model. This type of correlation is regarded as the global one and does not impose the changes in the pdfs describing the magnitude and frequency of the individual pressure impact. The second type of correlations between the parameters of stochastic pressure field acting on the solid body surface arises from the interaction between the individual bubbles collapses. In this case the magnitude and frequency of the impacts as well as their number changes in relation to the case when all the bubbles collapse independently. The tendency of group collapse of the cavitation bubbles can be taken into account by appropriate setting of parameters of pdfs describing the individual event. The frequency pdf can gain the form different from gaussian, mostly due to the necessary corrections of the flatness and the skewness coefficients. CONCLUDING REMARKS The presented above approach enables recognition of two regimes of damping of the cavitation released energy. The first one can be described as thermodynamic and remains insensitive to small perturbations. The second one, occurring above a certain threshold level of cavitation intensity leads to qualitatively different evolution of the physical system consisting of the solid body material and cavitating liquid. As the consequence of the permanent action of the pressure pulses generated during cavitation collapses the solid body material remains locally at the state, which is driven far from the equilibrium. The characteristic feature of such a state is a nonlinear response to small perturbation. The solid body structure driven far from equilibrium acts as an amplifier. When the system remains close to equilibrium its evolution does not impose the necessity of consideration of its internal structure. The far from equilibrium states characterize with high local energy density and the only way for the system to evolve is to diversify its structure. The particular solid body structure can absorb the specified amount of energy. The initial stage of the process, preceding the material separation consists in the absorption and accumulation of the cavitation collapse energy and on the other hand on its internal dissipation. When the energy accumulation is impossible then the system needs the appropriate channels for energy propagation. The local modification of material properties occurs leading to generation of such channels. Because the cavitation collapse energy is 7

8 absorbed selectively initially the available energy bands become saturated. The possible situations are as follows: 1. There are no available dissipation channels the emission and absorption frequencies do not fit to each others, there is no resonance and no erosion, the solid body doest not notice the effects of cavitation, the state can be represented by the point in the phase space, but this point does not serve as the attractor state and the system gains the tendency to abandon it as becomes more distant from the equilibrium state, 2. The lower vibration energy bands are available, what results in relatively long erosion incubation period, 3. The higher energy bands are available, the erosion incubation period is short, the erosion is a violent process. The diversification of the material structure and physical properties suppresses the erosion intensity. The necessary condition of the collapse energy transfer leading to the erosion is the occurrence of resonant energy absorption by high energy bands. The crucial parameters determining the course of the process are as follows: The magnitude of the energy emitted during the cavitation collapse The energy dissipation rate inside the solid body (phonons propagation) the feature of the given crystal structure The frequency spectrum characterizing the energy flux from liquid to solid body. The adequate modeling of the erosion should take into account the randomization of above features. [8] Pereira F. et al., Cavitation erosion statistical analysis of transient cavities, International Symposium on Cavitation CAV 95 Deauville 1995 [9] Pereira F., Avellan F., Dupont Ph., Prediction of cavitation erosion: an energy approach, Journal of Fluids Engineering, Dec 1998 ACKNOWLEDGMENTS The paper has been elaborated within the frames of the EU 5th Framework Program project EROCAV Erosion of Ship Propellers and Rudders the Influence of Cavitation on Material Damages, Project no. GRD REFERENCES [1] Barnaby Kenneth C., Basic Naval Architecture, Hutchinsons Scientific and Technical Publications, London, [2] Berchiche N., Franc J.P., Michel J.M. A cavitation erosion model for ductile materials PASADENA 2001 [3] Fortes-Patella R. Reboud J-L.,A new approach to evaluate the cavitation erosion power, Journal of Fluids Engineering, June 1998 [4] Fortes-Patella R., et. al., Cavitation erosion mechanism: numerical simulations of the interaction between pressure waves and solid boundaries, 4th Int. Symp. on Cavitation PASADENA 2001 [5] Fortes-Patella R., Reboud J-L., Energetical approach and impact efficiency in cavitation erosion, 3rd Int. Symp. on Cavitation GRENOBLE 1998 [6] Franc J-P, Michel J-M., et al., From pressure pulses measurements to mass losss prediction: the analysis of a method, 2nd Int. Symp. on Cavitation, TOKYO 1994 [7] Kato H., et al., Possibility of quantitative prediction of cavitation erosion without model test, International Symposium on Cavitation CAV 95 Deauville