Eulerian methods multiphase flows

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

Since a multiphase flow usually takes place in a confined volume, the desire to have a mathematical description based on a fixed domain renders the Eulerian method an ideal one to describe the flow field. The Eulerian approach requires that the transport quantities of all phases be continuous throughout the computational domain. As mentioned before, in reality, each phase is time-dependent and may be discretely distributed. Hence, averaging theorems need to be applied to construct a continuum for each phase so that the existing Eulerian description of a single-phase flow may be extended to a multiphase flow. 

For multiphase systems a rough distinction can be made between systems with separated flows and those with dispersed flows. This classification is not only important from a physical point of view but also from a computational perspective since for each class different computational approaches are required. For multiphase systems involving multiphase flow both Eulerian, mixed Eulerian-Lagrangian, and two-material free surface methods can be used. An excellent review on models and numerical methods for multiphase flow has been presented by Stewart and Wendroff (1984). A similar review with emphasis on dilute gas-particle flows has been presented by Crowe (1982). 

Discussions in Chapter 2 may be referred to for explanations of the various symbols. It is straightforward to apply such conservation equations to single-phase flows. In the case of multiphase flows also, in principle, it is possible to use these equations with appropriate boundary conditions at the interface between different phases. In such cases, however, density, viscosity and all the other relevant properties will have to change abruptly at the location of the interface. These methods, which describe and track the time-dependent behavior of the interface itself, are called front tracking methods. Numerical solution of such a set of equations is extremely difficult and enormously computation intensive. The main difficulty arises from the interaction between the moving interface and the Eulerian grid employed to solve the flow field (more discussion about numerical solutions is given in Chapters 6 and 7). 

Ferry, J. Balachandar, S. 2001 A fast Eulerian method for two-phase flow. International Journal of Multiphase Flow 27, 1199-1226. 

Subramaniam S Lagrangian—Eulerian methods for multiphase flows. Prog Energy Combust Sci 39 215-245, 2013. 

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