Eulerian-Lagrangian methods

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 IB technique [22] is a combined Eulerian-Lagrangian method, in the sense that the flow is solved using a Eulerian approach on a fixed Cartesian mesh, and the interface is represented by a discrete set of points and advected in a Lagrangian... 

The Eulerian-Eulerian method is more popular because memory storage requirements and computer power demand depend on the number of computational cells considered instead of the number of particles. Hence, the Eulerian-Eulerian approach can be applied to cases for low and high superficial gas velocities. The disadvantage of using the Eulerian-Eulerian method is that the bubble-bubble and bubble-liquid interactions cannot be considered as straightforward as the Eulerian-Lagrangian method, and models for these interactions are typically applied. 

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

In the Eulerian-Lagrangian methods, Eulerian parts of the equations are generally solved with methods described previously for the quasi-single-fluid models. The bubble trajectory equations are solved by direct integration [33,74] of (9.51) and (9.52). The solution procedure employed by Johansen and Boysan to solve the Eulerian-Lagrangian model [33] can be summarized as follows ... 

Xie J, Zhong WQ, Jin BS, Shao YJ, Huang YJ Eulerian—Lagrangian method for three-dimensional simulation of fluidized bed coal gasification, Adv Powder Technol 24 382-392, 2013. 

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