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The research group at City University London has developed its own computational fluid dynamics code which has been applied to the simulation of cavitation in nozzle flows. The model accounts for the detailed dynamics of collapsing bubbles and has been recently extended to include thermal and compressibility effects as well as the dynamics of non-spherical bubbles formed in the core of vortical structures. Combination of experiments and CFD calculations in a number of nozzle configurations aiming to characterise the effect of cavitation on engine performance and the durability of the fuel system with respect to erosion damage, have also been performed. Typical of the complexity of the injector modelling studies so far undertaken within the research group at City University London is the result of the fluid flow computation shown in Figure 1.

At TU Delft prediction methods were initially directed towards potential flow boundary element methods. With the development of multiphase Reynolds Averaged Navier Stokes (RANS) method, the computational direction has now been shifted towards the development and validation of these codes for cavitation dynamics problems. Experiments on fundamental cavitation dynamics have been performed in the Delft University and MARIN high speed cavitation tunnels, where instrumentation and experience is shared with the Fluid Dynamics Laboratory of Prof. Westerweel, also at the Technical University of Delft.

Existing commercial or open source computational fluid dynamics (CFD) codes are available to the group through the Fluid Dynamics Laboratory at Delft. The group, contrary to the approach of City University London, has chosen not to develop its own multiphase CFD code, but instead has liaised with groups that specialise in multiphase CFD developments. To this end, a close cooperation exists with the development group of Prof. Hoeijmakers at the Twente University in The Netherlands, where a compressible multiphase Euler code has been developed in a previous joint project with the Technical University of Delft’s Chair of Propulsion and Resistance.

About Cavitation Modelling

RANS/LES cavitation models may treat vapour either as discrete vapour bubble whose trajectory is calculated on a Lagrangian frame of reference or as a continuum media. The liquid phase flow is described in the Eulerian frame of reference by the Navier-Stokes equations, taking into account the effect of the liquid phase volume fraction and the momentum and energy exchange source terms between the two-phases. Additionally, different turbulence models (many RANS models as well as LES model) can be used to simulate the effects of turbulence.

Cavitation is usually assumed to be initiated by cavitation nuclei which subsequently grow into bubbles. These nuclei are artificially created (randomly generated) to simulate those pre-existing within the bulk of the flow. The size of the nuclei is sampled from a probability density function (PDF). Once the pressure of the liquid falls below its vapour pressure, the volume under tension is identified and the most probable locations for bubble nuclei formation are calculated randomly by sampling from a distribution function which considers the current bubble volume fraction, the volume and the non-dimensional tension of each Eulerian cell. The bubble velocity is calculated according to Newton’s second law of motion. The forces arising from the bubbles’ interaction with the continuous phase are assessed and the new velocity for each bubble is calculated.

Following the assumptions of classical bubble dynamics, the response of a bubble to its variable surrounding pressure field is calculated by integrating the classical Rayleigh-Plesset equation. In this approach the effects of local turbulence, relative velocity and surrounding bubbles can be also considered. Regarding the behaviour of the contaminant gas inside each bubble, a polytropic coefficient can be included which, depending on the bubble wall velocity, switches between adiabatic and isothermal behaviour within the bubble. The effect of heat transfer and thermal bubble dynamics can be also included. Due to their motion within the liquid, bubbles deform as a consequence of the shear forces exerted upon them. The origin of these shear forces is local turbulence and the relative velocity of the bubble to that of the liquid flowing around the bubble. According to the magnitude of the local turbulent kinetic energy dissipation, break-up can occur and in this case the resulting size of the daughter bubbles is generated by a PDF.

For direct numerical simulation of cavitation bubbles the most suitable approach is given by sharp-interface models. The Level Set and VOF techniques represent a reasonably easy way to define and track a sharp interface. Adaptive local grid refinement is available and will be used for enhanced resolution at the interface.