Numerical simulation of droplet deformation using spectral element method
arxiv: Physics::Fluid Dynamics
Spectral element methods are developed for solving two-phase flow problems of relevance to the power generation industry. In particular, an algorithm is presented for predicting the deformation of a Newtonian droplet accelerating from rest in a gas flow field. The dimensionless governing equations are written in terms of four dimensionless groups: Reynolds number, Weber number, and the viscosity and density ratios of the two fluids. The arbitrary Lagrangian-Eulerian (ALE) formulation is used to account for the movement of the mesh. Spectral element approximations are used to ensure a high degree of spatial accuracy. The computational domain is decomposed into two regions, one of which remains fixed in time while the other, located in the vicinity of the droplet, is allowed to deform within the ALE frame work. Transfinite mapping techniques are used to map the physical elements onto the computational element. Edges of elements on the free surface are described using an isoparametric mapping and blended into the elemental mapping. Surface tension is treated implicitly and naturally within the weak formulation of the problem. In addition to the movement of the mesh, the computation associated with each time step comprises an explicit treatment of the convection term and an implicit treatment of the linear terms. The generalized Stokes problem generated in the latter is solved using a nested preconditioned conjugate gradient method. The initial code development and the accuracy of the spectral element approximation is validated for the problem of flow over a solid sphere confined in a cylinder or in a uniform ambience. The problem of gas flow over a Newtonian droplet with different fluid properties is then investigated and the effects of changes in the values of the dimensionless groups on the deformation of the droplet is analysed. Finally, the Cross law is introduced to model the viscosity of non-Newtonian fluids and the problem of gas flow over a blood drop is simulated.
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