The focus of research works on cavitation has changed since the 1960s; the behaviour of a single bubble is no more the area of interest for most scientists. Its place was taken by the cavitating flow considered as a whole. Many numerical models of cavitating flows came into being within the space of the last fifty years. They can be divided into two groups: multifluid and homogeneous (i.e., single-fluid) models. The group of homogenous models contains two subgroups: models based on transport equation and pressure based models. Several works tried to order particular approaches and presented short reviews of selected studies. However, these classifications are too rough to be treated as sufficiently accurate. The aim of this paper is to present the development paths of numerical investigations of cavitating flows with the use of homogeneous approach in order of publication year and with relatively detailed description. Each of the presented model is accompanied by examples of the application area. This review focuses not only on the list of the most significant existing models to predict sheet and cloud cavitation, but also on presenting their advantages and disadvantages. Moreover, it shows the reasons which inspired present authors to look for new ways of more accurate numerical predictions and dimensions of cavitation. The article includes also the division of source terms of presented models based on the transport equation with the use of standardized symbols. Boldface lower-case letters refer to vectors while boldface capital letters and Greek lowercase letters refer to matrices.
This article presents a numerical method for determining tortuosity in porous beds consisting of randomly packed spherical particles. The calculation of tortuosity is carried out in two steps. In the first step, the spacial arrangement of particles in the porous bed is determined by using the discrete element method (DEM). Specifically, a commercially available discrete element package (PFC 3D ) was used to simulate the spacial structure of the porous bed. In the second step, a numerical algorithm was developed to construct the microscopic (pore scale) flow paths within the simulated spacial structure of the porous bed to calculate the lowest geometric tortuosity (LGT), which was defined as the ratio of the shortest flow path to the total bed depth. The numerical algorithm treats a porous bed as a series of four-particle tetrahedron units. When air enters a tetrahedron unit through one face (the base triangle), it is assumed to leave from another face triangle whose centroid is the highest of the four face triangles associated with the tetrahedron, and this face triangle will then be used as the base triangle for the next tetrahedron. This process is repeated to establish a series of tetrahedrons from the bottom to the top surface of the porous bed. The shortest flow path is then constructed geometrically by connecting the centroids of base triangles of consecutive tetrahedrons. The tortuosity values calculated by the proposed numerical method compared favourably with the values obtained from a CT image published in the literature for a bed of grain (peas). The proposed model predicted a tortuosity of 1.15, while the tortuosity estimated from the CT image was 1.14.
Tortuosity is one of the most elusive parameters of porous media. The fundamental question is whether it may be computed from the geometry only or the transport equations must be solved first. Elimination of the transport equations would significantly decrease the computation time and memory consumption and thus allow investigating larger samples. We compare the geometric to hydraulic tortuosity of a sphere-packed porous media. We applied the Discrete Element Method to generate a set of virtual beds based on experimental data taking into account the real porosity and particle distribution, the Lattice Boltzmann Method to compute the hydraulic tortuosity and geometrical approach, i.e. so-called Path Tracking Method, to calculate the geometrical tortuosity. Our study shows that the calculation time can be reduced from hours (if the LBM is used) to seconds (if the PTM is applied) without losing the accuracy of the final results. The relative error between average values of the tortuosity obtained for both used methods is less than 3%. We show that the applied geometrical method may serve as an attractive alternative to hydraulic tortuosity, particularly in large granular systems.
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