Parameterstudy on a computational method for the prediction of noise production caused by sheet cavitation on a ship propeller

1.1 Introduction Cavitation is the evaporation of liquid when its pressure decreases to below the vapour pressure. In ows around a propeller this behaviour can cause an attached sheet cavity to form on the surface of the propeller. Bubbles separating from this sheet can collapse violently, causing large pressure uctuations. These pressure uctuations can be harmful to the propeller and its surroundings, e.g., cause erosion and cause large amounts of noise. A noise prediction model designed by Matusiak uses the sheet cavitation size, as computed by a boundary element method, as input for determining the sizes and number of bubbles. This is used as input for a bubble dynamics model, the results of which are used to predict the noise these pressure pulses cause. The results of this model were evaluated for different initial conditions, and compared to measurements obtained from a ship propeller. 1.2 Bubble Dynamics model The bubble dynamics model used in Matusiak's model is the Gilmore equation, which is a second order ODE describing the relation of the bubble radius, the bubble wall velocity and acceleration for a bubble in a compressible uid, subjected to a linearly increasing pressure field. This model describes a single oscillating gas filled bubble. To determine the effect of multiple, differently sized bubbles, the solution of several bubbles are superimposed, with the results given a random time offset. The bubbles are distributed accros a range of sizes using a dimensionless distribution using a sheet thickness as computed by the boundary element method. The resulting pressure uctuations are transformed into a 1=3 octave band spectrum for comparison. This is done for both the single bubbles and the superimposed solutions to determine the in uence of several initial condition both on a single bubble and on the complete method. 1.3 Results The most important initial conditions for the bubble dynamics were found to be the initial bubble radius and the initial pressure of the gas in the bubble. The first of these determined both the noise levels of the spectrum and the frequency of its compo- nents, with larger bubbles having stronger pressure uctuations and lower frequencies. The initial gas affected the total amount of noise produced by the bubble, with higher initial gas pressure resulting in stronger oscillations, and to a lesser extend the frequency where most noise was produced, with higher pressure resulting in a maximum occuring at a lower frequency. Other initial con- ditions involved the surface tension, the polytropic index, the density and the compressibility coeffcients of Tait's equation for the liquid. These all had a more limited effect. The effect of several initial conditions and parameters exclusive to the superposition method on the bubble behaviour was also evaluated. The largest impact on the results was caused by the number of bubble oscillations for which the ODE-solver was run and the factor determining the sheet thickness used to compute the bubble size and volume. A larger number of oscillations resulted in higher frequency 1 noise produced by the bubbles, converging to a solution for approximately 35 oscillations and more. Increasing the maximum sheet thickness factor caused the noise production to occur at a lower frequncy, which also converged to a single solution for higher values of the factor. 1.4 Conclusions and recommendations From the observed results of the initial conditions and parameters, the following can be concluded: - A larger bubble radius shifts the maximum noise power to lower frequency ranges and increases the power. - The surface tension affects small bubbles (radius < 1 mm) and causes an increase in the amplitude of the oscillations. - The energy relation, either adiabatic or isothermal, mainly in uences 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. - 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 have similar solutions. - The fractal bubble size 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 - Converged results can be obtained using 35 or more oscillations. Recommendations for the improvement of the impelemtation are as follows: - Validate the bubble size distribution - Implement a method to determine the initial gas pressure from equilibrium. - Implement and compare different bubble dynamics models. f Limit the computation time by implementing a method to stop the bubble dynamics model when the bubble noise becomes insignificant.