Multi-phase fluid flow simulation by using peridynamic differential operator

Abstract

The problems of multi-phase fluid flows are often encountered in engineering. In this study, a non-local numerical model of multi-phase fluid flows in the Lagrangian description is developed. Based on the peridynamic theory, a peridynamic differential operator is proposed which can convert any arbitrary order of differentials into their integral form without calculating the peridynamic parameters. Therefore, the Navier-Stokes equations including the surface tension forces are reformulated into their integral form. Subsequently, an updated Lagrangian algorithm for solving the multi-phase fluid flow problems is proposed. Besides, the particle shifting technology and moving least square algorithm are also adopted to avoid the possible tension instability. Finally, several benchmark multi-phase fluid flow problems such as two-phase hydrostatic problem, two-phase Poiseuille flow, and 2D square droplet deformation are solved to validate the proposed non-local model. It can be concluded from the current study that the peridynamic differential operator can be applied as an alternative method for multi-phase fluid flow simulation.

The extended peridynamic model for elastoplastic and/or fracture problems

“…Overall, a good agreement is found between our computed responses and the results obtained in literature. 74 Note also that Figure 9 depicts consistent numerical results for two different approximation domain sizes in (18).…”

“…where Δx is the spacing in a uniform grid and r denoting the size of approximation domain. For small deformation problems considered in this article, it is found that the size of approximation domain r in (18) does not influence the results significantly. An example will be provided later in Section 7.1.1.…”

“…[11][12][13][14][15][16] Recently, Reference 17 proposed a peridynamic differential operator to construct nonlocal solutions of differential equations, thus extending the application of peridynamics to more areas. [18][19][20][21] The original bond-based peridynamic formulation by Silling, 4 however, can only describe a fixed Poisson's ratio. Moreover, the force density grows linearly with bond stretch until a threshold value, where the connection between two material points vanishes.…”

“…It is thus commonly adopted for modeling fracture processes 11–16 . Recently, Reference 17 proposed a peridynamic differential operator to construct nonlocal solutions of differential equations, thus extending the application of peridynamics to more areas 18–21 . The original bond‐based peridynamic formulation by Silling, 4 however, can only describe a fixed Poisson's ratio.…”