Bubble size distribution prediction for large-scale ship flows: Model evaluation and numerical issues Alejandro M. Castro ⇑ , Pablo M. Carrica IIHR-Hydroscience and Engineering, C. Maxwell Stanley Hydraulics Laboratory, The University of Iowa, Iowa City, IA 52242-1585, United States article info Article history: Received 8 November 2012 Received in revised form 30 July 2013 Accepted 7 August 2013 Available online 14 August 2013 Keywords: Polydispersed flow Bubbly flow Multigroup discretization Ship flow Bubble breakup Bubble coalescence abstract Prediction of the bubble size distribution in the wake of a ship is important to analyze its acoustic signa- ture. To achieve CFD simulation of dynamic ships with moving control surfaces and rotating propellers in waves, a robust implementation is paramount. In this work a mass conserving multigroup discretization strategy of the Boltzmann transport equation for polydispersed bubbly flows is presented, as well as an analysis of available breakup and coalescence models. Modifications of the discrete equations for the fixed pivot method at the boundaries are introduced that guarantee exact bubble mass conservation. The role of the time stepping scheme in the conservation of mass and number of bubbles is discussed. Though the conservation properties of the discrete system of equations are satisfied provided they are solved exactly, in practice an iterative procedure must be used since the ODE’s are non-linear. Three iter- ative schemes are proposed and they are analyzed in terms of robustness and efficiency. Breakup, coales- cence and dissolution models are analyzed from the numerical point of view. Available models of breakup and coalescence are studied finding appropriate choices for ship applications. Other models are appropri- ate as well, but are more costly numerically. As appropriate for ship applications, an extension to the model of Prince and Blanch for salt water is proposed and analyzed. The final model is tested against experimental data and computations by other researchers, and convergence properties in bubble size dis- cretization is studied. It is found that for salt water the final steady state is dependent on the initial con- dition since there is a range of sizes for which coalescence and breakup are both negligible. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The acoustic signature of a ship strongly depends on its bubbly wake, which can be characterized by measuring acoustic attenua- tion and/or scattering. In addition, the bubbly wake shows at the free surface as a characteristic signature of ‘‘white water’’. The detection of the bubbly wake from underwater or from satellites can lead to the identification of surface ships. It is for this reason that the two-phase bubbly flow around surface ships has attracted increasing attention recently, though the first studies date from be- fore World War II (Borowski et al., 2008). This has motivated the development of computational tools for bubbly flows around ships. A model for monodispersed bubbly flows around ships is pre- sented by Carrica et al. (1998) and further extended to polydis- persed bubbly flows in Carrica et al. (1999). Additional improvements to the model with application to ship flows are pre- sented by Moraga et al. (2008) and Ma et al. (2011). The impor- tance of using polydispersed models for bubbly flows around ships has been pointed out by Carrica et al. (1999). Given the wide range of hydrodynamic conditions around a ship spanning from a quasi-potential flow away from the ship, a strong shear-driven tur- bulent flow in the ship’s boundary layer and the highly unsteady stern flow, several bubble size distribution shapes can be found at different points in the domain. This not only justifies the need for polydispersed models but also for numerical methods that al- low for any arbitrarily shaped bubble size distribution. Therefore, this work adopts the multigroup approach presented by Carrica et al. (1999), while the fixed pivot method by Kumar and Ram- krishna (1996) is used to obtain a discrete approximation of the intergroup transfer terms. The main disadvantage of this approach, however, is the large number of additional unknowns introduced with the consequent computational cost. Single phase CFD computations of ship flows have improved considerably over the past few years. Successful applications in- clude large amplitude motions (Carrica et al., 2008), self-propul- sion with discretized propellers (Lübke, 2005; Carrica et al., 2010; Castro et al., 2011) and maneuvering with movable con- trolled appendages (Pankajakshan et al., 2002; Carrica et al., 2012). On the other hand, the modeling of bubbly flows around ships has not kept the same pace of progress and more work is needed to improve the numerical methods and physical models. 0301-9322/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2013.08.001 ⇑ Corresponding author. Tel.: +1 319 335 5237; fax: +1 319 335 5238. E-mail address: alejandro-castro@uiowa.edu (A.M. Castro). International Journal of Multiphase Flow 57 (2013) 131–150 Contents lists available at ScienceDirect International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow