University of Twente

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1 University of Twente Internship Report Parameterstudy on a computational method for the prediction of noise production caused by sheet cavitation on a ship propeller Author: R.M. van Dijk Supervisors: Prof.Dr.Ir. H.W.M. Hoeijmakers Dr.Ir. H.C.J. van Wijngaarden January 6, 2014

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3 Contents 1 Introduction 5 2 Bubble dynamics Bubble size distribution Spectral analysis Gilmore Parameter study Bubble Dynamics Implemented model Results Conclusions Recommendations A Alternative Models 23 A.1 Rayleigh-Plesset A.2 Keller-Miksis A.3 Flynn

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5 Nomenclature B c h H k n p i p v p G0 p r R t u η c ν ρ ρ L σ φ Roman symbols Bulk constant Speed of sound Enthalpy difference Enthalpy difference at bubble wall Polytropic index Bulk constant Internal pressure Vapour pressure Initial gas pressure pressure at infinity Radial coordinate Bubble radius Time Radial velocity Greek symbols Sheet thickness at breakoff point Kinematic viscosity Density of medium Density of liquid Surface tension Velocity potential 3

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7 1 Introduction Cavitation is caused by the pressure of a fluid decreasing, locally, to a value below the vapour pressure of said liquid, i.e., the pressure at which the liquid starts to evaporate, in a process similar to the boiling of water. In flows around a propeller this happens because the pressure decreases along the wing shaped profile of the propeller, attaining a minimum on a fixed location, after which the pressure increases again. An attached sheet cavity forms in the area along the profile where the pressure is below the vapour pressure. Segments separating from this sheet are subjected to an increasing pressure field. Because of this, the cavity will diminish and disappear again. However, the time required for the vapour to condense is too long to cope with the pressure increase, resulting in the bubble being squashed by the surrounding water. This causes the pressure of the gas and vapour inside the cavity to increase to a pressure above the ambient pressure, due to the momentum of the water moving inwards. When the inwards motion stops, this pressure difference causes the bubble to expand outwards again, creating a cycle of collapses and recoveries. Once the pressure returns to the ambient pressure at the end of the profile, this cycle will dissipate slowly by, among others, viscous forces. At the moment of collapse the bubbles create powerful pressure pulses. These pressure pulses are strong enough to be harmful to the propeller and its surroundings. Within the scope of an internship at Marin a model for predicting the noise of these pressure pulses was studied. The model designed by Matusiak [4] uses data from a numerical simulation of a propeller. This model assumes that a decrease in size of an attached sheet cavity is caused by a segment of the cavity breaking off. These segments form bubbles which oscillate. The pressure caused by these bubbles is recorded and graphed on a decibel scale. To describe the bubble dynamics the model uses the Gilmore equation [2]. Matusiak s model, as implemented in MatLab by van Wijngaarden [7], was used in this study. Measurement computation computation with limit 200 SPL, db re 1 µ Pa Frequency band [Hz] Figure 1.1: Model compared with measurements The model Matusiak implemented limited the time in which the bubbles oscillated. If the equations 5

8 describing the motion of the bubbles are integrated numerically, the bubbles are found to oscillate for far greater periods, which would mean that Matusiak s model will underestimate the severity of the bubble oscillations. However, time integrating these equations without this limitation currently overestimates the strength of the pressure peaks of cavitation. The noise spectrum of Matusiak s model, with and without this oscillation limit, is compared with measurements. The results are given in figure 1.1. In it it is clear that the model without the limited number of oscillations overestimates the frequency range at which the oscillation are strongest while the model with the limit imposed underestimates the amount of noise generated. Purpose of the study was to identify the parameters which affect the power spectrum of the bubbles. These parameters can then be used to refine Matusiak s model such that it compares better with the measurements. To this end, first a short introduction of bubble acoustics is given in section 2. In this section the basic equations and assumptions behind the Gilmore equation used for modelling bubble dynamics are discussed. Then, in section 3, the sensitivity of the bubble dynamics model to several parameters is presented, viz. the density, bulk modulus, vapour and gas pressure, initial radius, polytropic constant and surface tension. The values of these parameters are chosen such that they bound the physically acceptable range of said parameters. After that parameters specific to the implemented model are analysed to determine their influence. Section 4 gives the conclusions and recommendations of the study. In Appendix A some of the alternative bubble dynamics models are described, including the assumptions for their derivation and the differences with the Gilmore equation. 6

9 2 Bubble dynamics 2.1 Bubble size distribution Bubble dynamics models are used to describe the motion of a single oscillating bubble. However, in the flow around the propeller many bubbles of varying sizes are formed. To simulate the different bubbles, a probability density function is used to create classes of bubbles, with a radius representative of the class. In the present study a beta-distribution is used to create dimensionless radii (defined by x = 2R/η c, with η c the maximum sheet thickness). The beta distribution is given by f(x) = m(1 x) m 1 (2.1) in which m is the fractal order of the distribution and x the dimensionless radius. The dimensionless radii are determined such that the number of bubbles per class is identical. The magnitude of the radii is determined by the thickness of the attached sheet cavity at the moment of break-off as R = xη c /2. These radii are determined for several radial stations on the propeller at several times during the rotation. The bubble distribution is also used to determine the average bubble volume, which is obtained by rewriting the distribution in terms of the bubble volume (using v = 4π/3x 3 ) and integrating from x = 0 to x = 1. This average bubble volume is used to determine the total number of bubbles formed when the sheet s size decreases. 2.2 Spectral analysis In the parameters study performed for this internship the effect of several parameters on the 1/3 octave band spectra is determined. These spectra are calculated by integrating the bubble dynamics model numerically, creating time series for the bubble motion and pressure. The time series of the pressure are transformed into a power spectrum using Fourier transformation. They are then averaged over the 1/3 octave bands to define the spectra. The graphs of these spectra are compared in the parameter study as a measure of the influence of the parameter on noise production. 2.3 Gilmore The first bubble dynamic models were only valid for incompressible flow. To describe cavitation for larger and faster oscillations, as it occurs in the flow around propellers, a model was required that incorporated the compressibility of the liquid. To this end, Gilmore developed a model which can be solved analytically for constant internal pressure and approximately for a linearly increasing pressure, if some assumptions are made. To this end he used the continuity and momentum equations, u = 1 Dρ ρ Dt (2.2) D u Dt = p ρ + ν ρ 2 u (2.3) where u is the velocity vector, t the time, ρ is the fluid density, φ is the velocity potential and ν is the viscosity. The total derivative used in the continuity equation is defined as D/Dt = / t + u / u. In 7

10 this equation it has already been assumed that the flow is irrotational, allowing the velocity to be written as u = φ. This assumption is valid for all spherically symmetric bubble models. Mass conservation, Eq. 2.2, is used to write the last term on the right hand side of the momentum equation, Eq 2.3 as a product of the viscosity ν and the compressibility, Dρ/Dt. Since both the viscosity term and the compressibility term are small, their product can be neglected. As a result of this, for the Gilmore equations, the viscosity will only appear in the boundary condition. Eq. 2.3 is integrated by making two additional assumptions: Firstly, that at infinity the pressure p is constant and that the velocity potential and velocity vanish, and secondly, that the density is a function of pressure only. Eq. 2.3 then becomes φ t u2 = with u the local velocity and with the integration bounds p, the pressure at the bubble wall and p, the pressure far from the bubble. Due to these assumptions no integration constant is necessary in Eq. 2.4 and the expression on the right hand side can be expressed as h(p), the enthalpy difference between the liquid at p and p. The Kirkwood-Bethe assumption, i.e., the assumption that the quantity r(h + u 2 /2) is propagated outward with a variable velocity c + u, with c the local speed of sound, is rewritten in terms of the total derivative to obtain the following: p p dp ρ (2.4) D u2 [r(h + Dt 2 )] + c u2 [r(h + )] = 0 (2.5) r 2 Using the momentum and continuity equations, rewritten for spherically symmetric flows; to remove either all derivatives of t or r yields, Du Dt = h r u r + 2u r = 1 Dh c 2 Dt u r + 2u r (1 u/4c + u2 /4c 2 ) = 1 1 u/c c h r + 1 c h + u/c (1 r 1 u/c ) r Dh Dt (1 u c )ch(1 + u c ) rcdu Dt 3 2 cu2 (1 u 3c ) = 0 The first of these equations gives the requirements for the initial pressure field. The second equation, applied at the bubble wall, becomes the Gilmore equation: (1 Ṙ(t) c(r) )R(t) R(t) (1 Ṙ(t) 3c Ṙ(t) )R(t)Ṙ(t) = (1 + )H(R) + (1 Ṙ c(r) (2.6) (2.7) c(r) ) R (2.8) c(r)ḣ(r) In which R is the radius of the bubble wall. The terms H, Ḣ and c depend on the assumptions made for the pressure. In case of isentropic compression and a linearly increasing pressure, they become: H = n [ p + B ( P + B ] n 1 ρ L P + B ) n 1 n 1 c = c 0 ( pv + p G0 (R 0 /R) 3k 2σ/R + B p (t) + B ) n 1 2n D Ḣ = p V + Dt + B H D [ ρ ( P + B p V + Dt + B ) n 1 Ṙ pv + Dt + B n + ρ L R P + B ] 1 [ n 2σ (2.9) ] 3k R 3kp R G0 with n and B the constants of Tait s law for compressibility, p V the vapour pressure, p G the gas pressure and k the polytropic constant. The term P is used for the internal bubble pressure. The subscript 0 is used to denote the initial value and D is used to describe the linearly increasing pressure on the liquid side of the bubble wall, D = (p 0 p v )/t t, with t t the travel time from the initial location of the bubble to the trailing edge and p 0 the pressure at the trailing edge. This is the equation used by Matusiak [4] and the model by van Wijngaarden [7] evaluated in this report. Some alternative models are discussed in appendix A. R 0 8

11 3 Parameter study 3.1 Bubble Dynamics In this section the parameters which influence the Gilmore equation are evaluated. The effect of each parameter on the 1/3 octave band spectrum is plotted for several values. The density was varied for ±10% of the reference density. The polytropic index k was varied as 1 and 1.4, which determines the behaviour of the bubble as either isothermal or adiabatic, respectively The constants B and n were varied for ±10% from the values found in the study of Matusiak [4], and 7, respectively. The bubble radius was varied over several orders, from R 0 = O(1e 4)m to R 0 = O(1e 2)m. The initial bubble volume was taken constant. This caused the power output of each class of bubbles to be nearly in the same decibel range, which would not be the case if only one of each type of bubbles were to be evaluated. The vapour pressure was varied from its approximate value at T = 0 C to its approximate value at T = 20 C, which are 600 P a and 2300 P a respectively. The initial gas pressure was assumed to be either equal to the vapour pressure or equal to the atmospheric pressure. The surface tension was either assumed to be that of water (σ = N/m) or zero. Parameters are evaluated separately (or in pairs where required) to see the effect of each individual parameter. The default values for the parameters are given in Table 3.1 Parameter Value Density ρ [kg/m 3 ] Polytropic index k 1.4 [ ] Bulk constant B [P a] Bulk constant n 7 [ ] Initial bubble radius R 0 1 [mm] Vapour pressure p v 857 [P a] Initial gas pressure p g0 857 [P a] Surface tension σ [N/m] Table 3.1: Default values used in calculations The parameters have been ordered by their influence into strong and weak parameters by the magnitude of the effect they have on the power spectrum. The results will be discussed in the following sections. 9

12 Strong parameters Variables which have a larger impact are the radius of the bubbles, the polytropic constant, k, the surface tension and the vapour and the initial gas pressure,. The bubble size is the most important parameter found in this study, as it determines the frequency range in which most of the noise is produced. For this computation the results have been calculated with the assumption of a constant bubble volume; leading to many small bubbles and only few large ones. As can be seen in Fig. 3.1, the multitude of small bubbles create a similar power output as the intermediate sized bubbles, albeit at a different frequency. The largest bubble class still makes more noise than the others. 160 SPL, db re 1 µ Pa Frequency band [Hz] R 0 = 0.1mm R 0 = 1mm R 0 = 10mm Figure 3.1: 1/3 Octave Band Spectrum for different radii The polytropic index determines whether the system is adiabatic or isothermal. For the latter, the value of k is 1, for the former it is 1.4. The adiabatic system provides less high frequency noise, while the peak remains in the same bandwidth. 120 SPL, db re 1 µ Pa Frequency band [Hz] k = 1 k = 1.4 Figure 3.2: 1/3 Octave Band Spectrum of the polytropic index The surface tension was either assumed to be negligible or assumed to be that of water, σ = N/m. The surface tension has maximum effect on bubbles when they are small. The effect on the bubble dynamics was contrary to expectations. For a bubble of radius R 0 = 0.1mm, neglecting the surface tension increased the amplitude of the bubble vibrations by an order of magnitude, with the expectation being that it would lower it instead. The effect of the surface tension becomes small for bubbles of sizes R 0 1 mm. The effect of surface tension on the power spectrum of the radiated 10

13 100 SPL, db re 1 µ Pa without surface Frequency tensionband With [Hz] surface tension Figure 3.3: 1/3 Octave band spectrum with and without surface tension pressure is displayed in figure 3.3 for a bubble radius of R 0 = 0.1mm, its power is nearly 10 db lower over the entire frequency range if the surface tension is assumed to be zero. The power increase could be caused by the additional stiffness from the surface tension. For the vapour pressure the values for T = 0 C and T = 20 C were evaluated: 600 P a and 2300 P a, respectively. The initial gas pressure is assumed to be either equivalent to the vapour pressure or to the pressure far from the bubble, p atm = P a. For both assumptions the difference caused by the vapour pressure is mostly in the high frequency bands (above 1600 Hz), while the latter assumption causes the magnitude of the power spectrum to increase over the entire range and its peak to shift to a lower frequency (see Fig 3.4(a)). However, for p G0 = P atm, the bubble (with initial radius R 0 = 1mm) is seen in figure 3.4(b) to be growing just after t = 0, despite the initial condition for bubble wall velocity being Ṙ = 0. While the bubble is inherently unstable, the initial growth of the bubble in the increasing pressure is counter-intuitive and it is assumed that, for this value, the behaviour of the bubble is physically unrealistic. 11

14 150 SPL, db re 1 µ Pa bubble radius [mm] p v = 600; p G0 = 600 p v = 2300; p G0 = 2300 Frequency band [Hz] (a) 1/3 Octave band p v = 600; p G0 = p atm p v = 2300; p G0 = p atm time [ms] (b) initial motion Figure 3.4: Vapour and initial gas pressure 12

15 Weak Parameters Plots for the effect of the variations of the density and the bulk modulus constants B and n are given in Fig As can be seen from this figure, the influence of these parameters is limited. The density was varied from -10% to +10 of the reference density of water, kg/m 3, which gives a range for the density greater than occurs naturally. The results are given in Fig. 3.5(a) The constants of Tait s equation of state, B and n were varied over ±10% of the values found in Matusiak s study [4]. These variations are of the order of the maximum possible range found in reality. In Figure 3.5(b) the combinations for the values of B and n are given. It is clear that the effect is minor SPL, db re 1 µ Pa SPL, db re 1 µ Pa Frequency band [Hz] Frequency band [Hz] 90% 100% 110% B = 2.7e8, n = 6.3 B = 3e8, n = 7 B = 3.3e8, n = 7.7 (a) Density (b) B and n Figure 3.5: 1/3 Octave Band Spectrum of the weak parameters ρ L,B and n 3.2 Implemented model In the previous part, the behaviour of individual bubbles or bubble classes was treated, focussing on the parameters involved in the Gilmore equation. In this section the effect of parameters used for the implemented version are discussed. The fluid properties are chosen such that they describe the conditions under which the measurements of the ship s propeller were taken. This propeller is the same as the one modelled in the numerical simulation used in the model. In the implemented version, data from numerical simulations was used to determine the volume of bubbles and a bubble distribution was proposed [4] from which the (dimensionless) radii of the bubbles were determined. This was done such that the number of bubbles per bubble class was the same for all classes [7]. This leads to several additional parameters: The number of bubble classes was varied from 1 to 6 classes The fractal distribution parameter of the bubble classes is varied from 1 to 9 The void fraction of the attached sheet cavity was varied from 10 to 100% The maximum number of oscillations the bubble model is allowed to evaluate is varied from 3.5 to oscillations The factor F s used to determine the sheet thickness η c is varied from 0.6 to 3.0 The default values used are given in Table 3.2. The number of bubble classes determines the number of different radii which are used in the analysis. This parameter has a limited influence when more than two classes are used, as can be seen in Figure 13

16 Parameter Value Number of bubble classes N c 3 [ ] fractal distribution m 9 [ ] void fraction β 0.8 [ ] maximum number of oscillations τ c 3.5 [ ] sheet thickness factor F s 1.8 [ ] Table 3.2: default values in the implemented model 3.6. This is interesting because in the study of the bubble dynamics model a strong dependency of the bubble dynamics on the bubble radius was found. However, the mean radius found using the current bubble distribution is independent of the number of bubble classes. Coupled with the fact that the bubble volume separating from the attached sheet cavity is constant, the influence of the number of bubble classes is limited. As the number of classes goes up so does the difference between the minimum and maximum radius; with each class containing fewer bubbles. SPL, db re 1 µ Pa Frequency band [Hz] N c = 1 N c = 2 N c = 3 N c = 4 N c = 5 N c = 6 Figure 3.6: 1/3 Octave band spectrum for multiple bubble classes The radii of bubbles in the implemented model is determined using a fractal probability function, Eq. 2.1 This distribution is used to determine the radii such that each bubble class contains the same number of bubbles. Matusiak used a fractal distribution constant of 9. The resulting power spectrum for m = 1 and m = 9 is given for both 1 bubble class and 3 bubble classes in Figure 3.7. With the bubble radius having a strong influence on the bubble dynamics one would expect the fractal order to have a major effect on the implemented model because it varies the bubble sizes over a range of the same order as done in the earlier analysis (with the largest radius at m = 1 and the smallest at m = 9). The effect however, is much smaller. This probably has to do with the bubble distribution, in which the number of bubbles per bubble class is assumed to be equal. The variation found in figure 3.7 is more comparable to the ratio of the maxima of the radii found in each class, reinforcing the notion that the limited effect is caused by the bubble distribution. As shown in Fig. 3.8, varying the void fraction primarily causes a frequency independent scaling. This is logical, as the number of bubbles spawned from the attached sheet cavity is, among others, dependant on the void fraction of the sheet cavity. The increase is similar for all bubble classes and their respective pressure influences and thus, the void fraction is independent of frequency. The number of oscillations of the cavitating bubbles was limited in the original implemented model and the MatLab variant, probably to limit the computation time which was quite expensive when the program was first developed. The limit uses a typical collapse time due to a pressure jump, τ = R 0 (ρ L /p 0 ), which for linear increasing pressure becomes τ c = (3τ/2) 2/3 (t t ) 1/3. This limit might be appropriate for large bubbles which oscillate slowly and thus move away from the blade in a few collapses, or for small bubbles in free oscillation, which would dampen out due to dissipative effects. However, in the bubble dynamic model, the bubbles were seen to oscillate for much longer. The number 14

17 SPL, db re 1 µ Pa SPL, db re 1 µ Pa Frequency band [Hz] m = 1 m = 3 m = 5 m = 7 m = 9 (a) N c = Frequency band [Hz] m = 1 m = 3 m = 5 m = 7 m = 9 (b) N c = 1 Figure 3.7: 1/3 Octave band spectrum for multiple fractal distributions SPL, db re 1 µ Pa Frequency band [Hz] β = 0.1 β = 0.2 β = 0.3 β = 0.4 β = 0.5 β = 0.6 β = 0.7 β = 0.8 β = 0.9 β = 1 Figure 3.8: 1/3 Octave band spectrum for different void fractions of oscillations were therefore varied. The results are displayed in Figure 3.9 for several oscillation limits. For the larger limits the parameters are seen to converge. Big drawback of the extended oscillation limits is that the time required for the simulation is approximately linearly dependent on the number of time steps. The dimensionless radius, as defined in section 2, is used to determine the number of bubbles forming from the separating segment and is used for scaling the bubble radius. The maximum sheet thickness η c in this relation is computed from the data of a numerical simulation. It is determined using the volume change of the attached sheet cavity and a factor, F s. In Matusiak s study [4] this factor has a value of 1.8. The results of varying this factor is displayed in figure This factor, together with the fractal distribution are responsible for determining the bubble radii used in the simulation. For higher values of F s the spectrum is shifted to a lower frequency range, while the power output is not influenced strongly. 15

18 PS db, re 1 µpa (1/3 octave band) /3 Octave Band Spectrum osc. lim. = 3.5 osc. lim. = 11.1 osc. lim. = 35 osc.lim. = osc. lim. = Frequency band [Hz] Figure 3.9: 1/3 Octave band spectrum for several oscillation limits 180 SPL, db re 1 µ Pa Frequency band [Hz] F s = 0.6 F s = 1.2 F s = 1.8 F s = 2.4 F s = 3 Figure 3.10: 1/3 Octave band spectrum for several factors 3.3 Results With the effects of the parameters known they can be chosen such that the simulation compares best with the measurements. The best values are given in table 3.3, with the number of oscillations limited by the time of the blade passage. These values are compared with the measurements in figure The comparison has improved with respect to the original computations but the results from the simulation still appear to be shifted to a higher frequency when compared to the measurements. 16

19 Parameter Value Void fraction β 0.4 [-] fractal distribution m 1 [-] Sheet factor F s 3 [-] Number of classes 3 [-] Number of oscillations 110 [-] Table 3.3: Best parameters Measurement Computation 200 SPL, db re 1 µ Pa Frequency band [Hz] Figure 3.11: 1/3 Octave band spectrum measurement and best parameters 17

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21 4 Conclusions Aim of this study was to find the parameters which affected the power spectrum created by the model and use them to improve the results from the model of Matusiak [4] and van Wijngaarden [7]. The parameters are used with the optimal values as obtained in the parameter study to obtain the best comparison with the measurements. The power spectrum found from this simulation is not an exact match but provides an estimation of the power spectrum emitted by the vessel. If these values, when used for a simulation based on the results obtained from other vessels, provide results which compare similarly to measurements of these vessels this method is useful for the prediction of the noise production of propellers. Bubble parameters In section 2 the bubble dynamics model, the background to the bubble size distribution and the octave band spectrum were explained. In this section the parameters relevant to the bubble dynamics were introduced. These parameters were analysed in section 3. The parameters which showed a strong influence are listed below. A larger bubble radius shifts the maximum power to lower frequency ranges and increases the power. The surface tension affects small bubbles (radius < 1 mm) and caused an increase in the amplitude of the oscillations. The energy relation, either adiabatic or isothermal, mainly influences the behaviour at higher frequencies, reducing high frequency power output for adiabatic behaviour. The vapour pressure causes an increase in the power output at high frequencies. With the initial gas pressure assumed to be atmospheric the bubble dynamics are unrealistic. The parameters which have little to no influence are the density of the liquid and the constants from Tait s Law, B and n. Although varied over a range greater than which occurs in nature, the spectra were not affected significantly. Model parameters The effects of these parameters on the power output of the bubble dynamics model was limited by the implementation, which used the values found from the conditions in which the ship whose propeller was used in the numerical simulation operated. The implementation used the size distribution and maximum sheet thickness to determine the size of the bubbles. This method led to a spectrum largely determined by the formation of the bubbles as prescribed by the numerical simulation of the propeller. The parameters which were used to alter the spectra were: Increasing the void fraction was found to increase the magnitude of the spectrum but did not significantly affect the frequency distribution For more than two bubble classes the results were found to converge to a solution. 19

22 The fractal distribution was found to affect the spectrum similar to the bubble radius but to a lesser degree, i.e., for a lower fractal order the mean radius of the bubble distribution was larger. Increasing sheet thickness factor shifts the spectrum to a lower frequency range For higher number of oscillations the results converge. Reasonable results can be obtained using 35 or more oscillations. 4.1 Recommendations In the process of investigating the sensitivity of the bubble dynamics model to the parameters discussed, there were several issues, whose assessment were outside the scope of this work. These will be addressed in this section. First of these was the bubble distribution used. The current one was used by Matusiak because the results from the simulation best matched the measurements. In the current model by van Wijngaarden, a bubble distribution with an equal number of bubbles per bubble class was used to compute the field pressure. The influence of bubble sizes on the field pressure is large, therefore the bubble distribution used should be validated. Second was the initial gas pressure. Together with the vapour pressure this affected the bubble oscillations, however for large values of the gas pressure (for atmospheric pressure for certain) this assumption caused the bubble to grow in the decreasing pressure field. This is not a realistic occurrence as this growth would already have affected the attached cavity that the bubbles are assumed to separate from. For a more accurate representation, it would seem logical that the gas pressure must be derived from a (quasi) equilibrium state. Proposed is to implement a method, which determines the initial gas pressure at separation from the sheet cavity from equilibrium conditions, into the model and compare the results with the present ones. Third is the bubble dynamics model used. The approximation as derived by Gilmore is useful for single oscillating bubbles in simple pressure fields, in the current derivation for constant or linear pressure fields only. The pressure field near the propeller surface is approximated as a linear field. Other models are available with both a less limited choice in pressure fields and with equations for the bubble temperature. Proposed is to use the current method with different bubble dynamics models and compare the results. Lastly is the computational effort required. The current model requires a method which stops the simulation when the oscillations have damped out to levels which do not significantly contribute to the noise spectrum of the total model. This is not a necessity, but would be useful for reduction of the time required to compute the simulation, especially for the larger bubbles, whose oscillations dampen relatively quick. Summarizes, the recommendations are: Validate the bubble distribution Implement a method to determine the initial gas pressure from equilibrium Implement and compare different bubble dynamics models Limit the computational time by implementing a method to stop the bubble dynamics model when bubble noise becomes insignificant. 20

23 Bibliography [1] Hugh G Flynn. Cavitation dynamics. i. a mathematical formulation. The Journal of the Acoustical Society of America, 57:1379, [2] Forrest R Gilmore. The growth or collapse of a spherical bubble in a viscous compressible liquid [3] Joseph B Keller and Michael Miksis. Bubble oscillations of large amplitude. The Journal of the Acoustical Society of America, 68:628, [4] Jerzy Matusiak. Pressure and noise induced by a cavitating marine screw propeller. PhD thesis, Technical Research Centre of Finland, [5] MS Plesset. The dynamics of cavitation bubbles. J. appl. Mech, 16(3): , [6] Lord Rayleigh. Viii. on the pressure developed in a liquid during the collapse of a spherical cavity. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34(200):94 98, [7] E v. Wijngaarden. Implementation of Matusiak s model for cavitation-induced radiated noise. Technical Report RD, Marin,

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25 A Alternative Models A.1 Rayleigh-Plesset Bubbles and cavitation have been studied for a long time and several models of varying complexity have been proposed. One of the first models for bubble dynamics was derived by Rayleigh in 1917 [6] for a single spherical bubble in an infinite domain, R R RṘ = p(r) p ρ L (A.1) in which R is the bubble radius, p(r) the pressure at the bubble wall and p is the pressure far from the bubble. ρ L is used to denote the liquid density and Newton s notation is used for derivatives, i.e., ẋ = dx/dt. Rayleigh s equation assumes an incompressible fluid and neglects surface tension and viscosity. Equation A.1 was later adapted by Plesset [5] to describe the growth and collapse of bubbles, taking p as a function of time. The function can easily be rewritten to include effects of viscosity and surface tension, thus forming the equation known as the generalized Rayleigh-Plesset equation, R R RṘ = p i p 2σ R 4ν R Ṙ ρ L (A.2) in which σ is the surface tension, nu the viscosity and p i the internal pressure of the bubble. A.2 Keller-Miksis The Keller-Miksis model was developed after finding that earlier methods, e.g. Rayleigh-Plesset and Gilmore, failed to accurately describe the dynamics of the bubble for acoustic fields near the eigenfrequency of the bubble, leading to large oscillations. This model accounted for the acoustic radiation [3], unlike its predecessors. In its derivation the speed of sound is assumed constant. The density is assumed constant in space after using mass conservation to include compressibility. The Keller-Miksis equation is: ( ) ( ) 4ν R R(Ṙ ρ c) = 12Ṙ3 + Ṙ (R) c 3 2Ṙ2 + 4νṘ L ρ L R + 2σ ρ L R (R) +RṘ (R) + 2 ( 1 + Ṙ c ) g ( t + R ) (A.3) c In which is the Laplace-operator, 2 and its time derivative. In the derivation of this equation the wave equation is integrated, resulting in an arbitrary function g. in further derivation this function is required to conform to 2g = c φ(0, t), with φ the incident field. For further details, see Keller-Miksis 1980 [3] 23

26 A.3 Flynn Flynn formulated a model for use on the computer for large amplitude oscillations to obtain reliable estimates for the order of magnitude of quantities relevant in bubbles growing to a maximum and then collapsing violently. For his derivation he made several assumptions, listed below: Spherical bubble. Diffusion excluded, but not evaporation and condensation. No effects of translation velocity. Uniform pressure distribution inside the bubble. Bubble particle velocity is linear function of r. Density of liquid is assumed constant where compressibility would otherwise affect heat conduction. The viscosity relations are assumed to be those of an incompressible fluid. in reality p affects the bubble after a finite time, yet the model assumes this time to be infinitesimal. Unlike other methods described in this report, Flynn s method accounts for heat conduction and other thermal effects in both the bubble and the surrounding medium. Flynn obtained a differential equation via a derivation following Herring [1]: R du dt U 2 = 1 ρ e q [ P (t) p (t) + R(1 u) dp ] dt (A.4) In which all terms have been made dimensionless by dividing them by a reference value. In the equation, P (t) is the pressure at the boundary wall, which needs further description. This equation is modified using factors found by Herring and Trilling. The result resembles the Gilmore equation when the latter is derived for a constant far field presssure. With P (t) = p c (t) 4νṘ/R 2σ/R and p (t) = 1 f(t) Eq A.4 becomes: [ R(1 Ṙ) 1 + 4ν ] R + 3 ρ eq R 2 (1 Ṙ = 1 ρ eq 3 )Ṙ2 [ {(1 + Ṙ)[p 2σ c(t) + f(t) 1] (1 + Ṙ2 R + 4νṘ R ] ) + R(1 Ṙ)dp c dt } (A.5) In this equation ρ eq is the density of the liquid in equilibrium state and p c is the pressure inside the cavity, for which Flynn wrote p c (t) = p g (t) + p v (T ), with T a function of time only. After this, Flynn derived equations for the thermodynamics within the bubble, which is too extensive for this report to include. 24