water Article Modeling of the Free-Surface Vortex-Driven Bubble Entrainment into Water Ryan Anugrah Putra 1,2,* and Dirk Lucas 1 1 Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Institute of Fluid Dynamics, Bautzner Landstr. 400, 01328 Dresden, Germany; [email protected] 2 Department of Mechanical & Industrial Engineering, Faculty of Engineering, Universitas Gadjah Mada, Jalan Grafika No. 2, Yogyakarta 55281, Indonesia * Correspondence: [email protected] or [email protected] Received: 20 December 2019; Accepted: 3 March 2020; Published: 5 March 2020 Abstract: The recently developed GENTOP (Generalized Two Phase Flow) concept, which is based on the multifield Euler-Euler approach, was applied to model a free-surface vortex—a flow situation that is relevant for hydraulic intake. A new bubble entrainment model has been developed and implemented in the concept. In general, satisfactory agreement with the experimental data can be achieved. However, the gas entrainment can be significantly affected by several parameters or models used in the CFD (Computational Fluid Dynamics) simulation. The scale of curvature correction Cscale in the turbulence model, the coefficient in the entrainment model Cent, and the assigned bubble size to be entrained have a significant influence on the gas entrainment rate. The gas entrainment increases with higher Cscale values, which can be attributed to the stronger rotation captured by the simulation. A smaller bubble size gives higher gas entrainment, while a larger bubble size leads to a smaller entrainment. The results also show that the gas entrainment can be controlled by adjusting the entrainment coefficient Cent. Based on the modeling framework presented in this paper, further improvement of the physical modeling of the entrainment process should be done. Keywords: multiphase flow; bubble entrainment; free-surface vortex; rotating flow; GENTOP 1. Introduction A free-surface vortex (see Figure1) may exist in a wide range of scales; it can be as small as a “bathtub vortex” [1–3] or can be as big as an ocean whirlpool [4]. The topic of a free-surface vortex is often found in the discussion of hydropower plants, nuclear reactors and other applications using pumps. The supply of water for irrigation, domestic, industry, and power generation is usually taken from rivers or reservoirs through an intake that is located near the surface [5]. Insufficient submergence (a short distance between the water surface and an intake) may lead to the formation of a free-surface vortex that can induce gas entrainment into the intake [5]. A free-surface vortex and its associated gas entrainment may lead to several operational and safety problems [5–9]. They may cause mechanical damage and loss of performance in fluid machinery such as turbines and pumps [6,7]. A swirl in a sump leads to rotational flow in a pipe, which may reduce the performance of the pump [8,9]. If such flow is unsteady, it may also cause fluctuating loads on pump bearings [8]. The gas entrainment induced by a free-surface vortex will reduce the delivery of a pump (1% air reduces the efficiency of a centrifugal pump by 5–15%) [6]. This reduction may cause a severe problem such as the overtopping of a dam [3], which may lead to a safety hazard and cause loss of life [7]. Water 2020, 12, 709; doi:10.3390/w12030709 www.mdpi.com/journal/water Water 2020, 12, 709 2 of 22 Water 2020, 12, x FOR PEER REVIEW 2 of 23 Figure 1. Free-surface vortex observed in Akkats hydropower station, Sweden. The image is taken Figure 1. Free-surface vortex observed in Akkats hydropower station, Sweden. The image is taken from [10]. from [10]. 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In/ wet-wellthe BWRs [36, the,37 ].gas entrainment due to a free surface vortexComputational may occur at the fluid suction dynamics inlet from (CFD) a condensation may help to chamber/wet design a safer-well process, [36,37]. minimizing the aforementionedComputational risks fluid associated dynamics with (CFD) gas entrainmentmay help to duedesign to aa free-surfacesafer process vortex., minimi Generally,zing the aforementionedthe previous CFD risks works associated available with in thegas literature entrainment can bedue divided to a free into-surface two parts: vortex. single-phase Generally, andthe previoustwo-phase CFD computations. works available In a in single-phase the literature simulation, can be divided the deformation into two parts: of thesingle free-phase surface and is two not- phaseconsidered computations. and the free In surface a single is defined-phase assimulation, a free slip boundarythe deformation [38,39]. Whenof the the free mesh surface resolution is not is consideresufficientd and and the the appropriate free surface turbulence is defined as model a free is slip used, boundary the velocity [38,39 fields]. When can the be calculatedmesh resolution using issingle-phase sufficient and simulation, the appropriate as reported turbulence by [17,40 model]. However, is used, to the the velocity best of ourfields knowledge, can be calculated the estimation using singleof the- gasphase entrainment simulation rate, as has reported never been by performed[17,40]. However, by a single-phase to the best CFD. of Inour addition, knowledge, the direct the estimation of the gas entrainment rate has never been performed by a single-phase CFD. In addition, the direct observation of a free-surface vortex from the single-phase simulation is not possible. A post-processing method is required to judge the occurrence and the location of the vortex, e.g., Q criterion [38]: Water 2020, 12, 709 3 of 22 observation of a free-surface vortex from the single-phase simulation is not possible. A post-processing method is required to judge the occurrence and the location of the vortex, e.g., Q criterion [38]: 1 Q = W2 S2 > 0, (1) 2 k − k where Q is the second invariant of velocity gradient tensor, W is the vorticity tensor, and S is the strain rate tensor. The above equation states that a free surface vortex exists when the strength of rotation is bigger than the local strain rate [38]. In the case of two-phase simulation, usually, the volume of fluid (VOF) model is employed [20,40,41]. The computation is performed in a fixed grid solving only one momentum equation, which is shared by both fluids [42]. Generally, a very fine mesh is required to resolve the interface, e.g., [40].