Mixed Eulerian-Lagrangian method

Udaykumar, H. S., Shyy, W., and Rao, M. M., ELAFINT A mixed Eulerian-Lagrangian method for fluid flows with complex and moving boundaries. Int. J. Num. Methods Fluids 22,691 (1996). 

The single-fluid approach has been undertaken utilizing Eulerian methods such as VOF, LS, and the mixed Eulerian-Lagrangian methods [11, 12], For most interface capturing schemes, which use single-fluid formulation, an additional equation is solved to obtain the interface evolution and topology. This equation governs the advection of a variable that can be attributed to the interface. The equation of interface motion is... 

The single-fluid approach has been undertaken utilizing Eulerian methods such as the VOF, level set, and mixed Eulerian-Lagrangian methods [10, 11]. 

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). 

The Eulerian equations of motion are more useful for numerical solution of highly distorted fluid flow than are Lagrangian equations of motion. Multicomponent Eulerian calculations require equations of state for mixed cells and methods for moving mass and its associated state values into and out of mixed cells. These complications are avoided by Lagrangian calculations. Harlow s particle-in-cell (PIC) method uses particles for the mass movement. The first reactive Eulerian hydrodynamic code EIC (Explosive-in-cell) used the PIC method, and it is described in reference 2. The discrete nature of the mass movement introduced pressure and temperature variations from cycle to cycle of the calculation that were unacceptable for many reactive fluid dynamic problems. A one-component continuous mass transport Eulerian code developed in 1966 proved useful for solving many one-component problems of interest in reactive fluid dynamics. The need for a multicomponent Eulerian code resulted in a second 2DE code, described in reference 4. Elastic-plastic flow and real viscosity were added in 1976. The technique was extended to three dimensions in the 1970 s and the resulting 3DE code is described in Appendix D. 

The intra-cell processes are common to all PDF codes, and are treated the same in both Eulerian and Lagrangian PDF codes.8 On the other hand, inter-cell processes are treated differently in Eulerian PDF codes due to the discrete representation of space in terms of x . In PDF codes, fractional time stepping is employed to account for each process separately. Methods for treating chemical reactions and mixing are described in Section 6.9. Thus we will focus here on the treatment of inter-cell processes in Eulerian PDF codes. 

Murthy et al [63] were the first who used the sliding grid method for the simulation of unsteady flows in mixing vessels. The model formulation used with moving meshes is of the arbitrary-Lagrangian-Eulerian (ALE) t rpe [70]. In this particular approach the flow domain is divided into two cylindrical, non-overlapping sub-domains, each gridded as a separate block. The outer... 

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