Sharp interface approaches and deep learning techniques for multiphase flows

Abstract

We present a review on numerical methods for simulating multiphase and free surface flows. We focus in particular on numerical methods that seek to preserve the discontinuous nature of the solutions across the interface between phases. We provide a discussion on the Ghost-Fluid and Voronoi Interface methods, on the treatment of surface tension forces that avoid stringent time step restrictions, on adaptive grid refinement techniques for improved efficiency and on parallel computing approaches. We present the results of some simulations obtained with these treatments in two and three spatial dimensions. We also provide a discussion of Machine Learning and Deep Learning techniques in the context of multiphase flows and propose several future potential research thrusts for using deep learning to enhance the study and simulation of multiphase flows.

Introduction

Multiphase flows are ubiquitous in the physical and life sciences and their simulations are used in a wide array of applications. Examples where simulations are key include the study of cavitation in the naval industry, the understanding of solidification processes of most metals, the analysis of boiling flows in nuclear plants, the discovery of novel surfactant and liquid delivery into the lung, the prediction of porous media flows for oil and gas operations, and even in the art, where simulations are used to produce believable flows in computer graphics.

The simulation of multiphase flows presents several challenges. First, one must keep track of the interface between phases; this interface evolves in time and can experience changes in topology. Second, the large variations in some key physical quantities, say the fluid's density, velocity or the pressure, can only be represented as jump conditions at the mesoscopic and macroscopic scales so that numerical methods must enforce such a representation. Third, the multiscale nature of multiphase flows, where fine features develop locally in some fluid regions, and the limitation of computational resources demand that numerical methods on adaptive grids and parallel architecture be considered.

Early work on the treatment of multiphase flows can be traced back to the work of the immersed boundary method of Peskin [103], [102]. Although multiphase flows per se were not considered, the use of a discrete δ-function to solve blood flows in an immersed elastic interface paved the way to extending to arbitrary domains the discretization of partial differential equations in rectangular domains. Unverdi et al. used this formulation in tandem with the front-tracking method to introduce a multiphase incompressible flow solver. In this case, the interface between phases was allowed to undergo large deformations including changes in topology. In [129] and [19], the authors used the level-set method to capture the interface motion instead of the front tracking approach, easing the treatment of topological changes. However, even if the front tracking and the level-set methods are sharp interface representations as opposed to phase-field models for example, which represent the interface between phase as a numerical mushy zone, the use of a δ-function formulation smears the physical quantities across the interface. In this work, we focus on ‘sharp’ interface methods, i.e. schemes that numerically preserve the discontinuity in discontinuous quantities.

In what follows, we consider a computational domain $\mathrm{\Omega }={\mathrm{\Omega }}^{-}\cup {\mathrm{\Omega }}^{+}$, where ${\mathrm{\Omega }}^{-}$ and ${\mathrm{\Omega }}^{+}$ are separated by an interface Γ. The fluid's motion is modeled by the incompressible Navier–Stokes equations:

$$\rho (\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u})=-\mathrm{\nabla }p+\mu \mathrm{\Delta }\mathbf{u}+\mathbf{f}\text{in }{\mathrm{\Omega }}^{-}\cup {\mathrm{\Omega }}^{+}\text{,}$$

$$\mathrm{\nabla }\cdot \mathbf{u}=0\text{in }{\mathrm{\Omega }}^{-}\cup {\mathrm{\Omega }}^{+}\text{,}$$

where t is the time, $\mathbf{u}=(u,v,w)$ is the velocity field, p is the pressure and f includes the external forces such as gravity. We consider here the case of a fluid with uniform viscosity μ and uniform density ρ in each subdomain ${\mathrm{\Omega }}^{-}$ and ${\mathrm{\Omega }}^{+}$. Appropriate boundary conditions are imposed on Γ and on the boundary of the computational domain, ∂Ω. These will be discussed in details in the remaining of this manuscript.

The framework we use to solve those equations is the standard projection approach introduced by Chorin in 1967 [20], which consists of the following three steps. Compute an intermediate velocity field ${\mathbf{u}}^{⁎}$ using the momentum equation (1) and ignoring the pressure gradient term, e.g. in the case where the temporal derivative is discretized with a forward Euler step:

$$\rho (\frac{{\mathbf{u}}^{⁎}-{\mathbf{u}}^n}{\mathrm{\Delta }t}+{\mathbf{u}}^n\cdot \mathrm{\nabla }{\mathbf{u}}^n)=\mu \mathrm{\Delta }{\mathbf{u}}^{⁎}+\mathbf{f}.$$

The second step is based on the Helmoltz–Hodge decomposition that decomposes a twice continuously differentiable bounded vector field ${\mathbf{u}}^{⁎}$ into a divergence-free component ${\mathbf{u}}^{n+1}$ and a curl-free component ∇Φ:

$${\mathbf{u}}^{⁎}={\mathbf{u}}^{n+1}+\mathrm{\nabla }\mathrm{\Phi },\text{ or }{\mathbf{u}}^{n+1}={\mathbf{u}}^{⁎}-\mathrm{\nabla }\mathrm{\Phi },$$

where Φ is referred to as the Hodge variable. Imposing the incompressibility constraint on ${\mathbf{u}}^{n+1}$ gives an equation for Φ:

$$\mathrm{\nabla }\cdot \mathrm{\nabla }\mathrm{\Phi }=\mathrm{\nabla }\cdot {\mathbf{u}}^{⁎}.$$

The boundary conditions and how to impose them will be described throughout this manuscript.

In section 2, we review the level-set and the particle-level-set methods; in section 3, we discuss the level-set method on Quad-/Oc-tree Cartesian grids in parallel as well as how to solve free surface flows; in section 4, we discuss numerical methods related to solving multiphase flows; section 5 is focused on free surface flows and section 6 discusses Machine and Deep Learning opportunity for multiphase flows. A short conclusion is given in section 7.