An energy-stable Smoothed Particle Hydrodynamics discretization of the Navier-Stokes-Cahn-Hilliard model for incompressible two-phase flows

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Highlights

•The ﬁrst study of energy-stable ODE discretization and the energy-

stable fully discrete scheme by the SPH method for two-phase problems

•The scheme ensures the inheritance of momentum conservation and the

energy dissipation law from the PDE level to the ODE level, and then

to the fully discrete level.

•This novel method based on NSCH model enjoys the physical consis-

tency, and the detailed mathematical proof is also provided.

•The scheme helps increase the stability of the numerical method, which

allows much larger time step sizes than the traditional ISPH method.

•This energy-stable scheme not only alleviates tensile instability, but it

also captures the behavior of the interface and energy variation well.

arXiv:2210.11857v1 [physics.flu-dyn] 21 Oct 2022

An energy-stable Smoothed Particle Hydrodynamics

discretization of the Navier-Stokes-Cahn-Hilliard

model for incompressible two-phase ﬂows

Xiaoyu Fenga, Zhonghua Qiaob, Shuyu Suna,∗, Xiuping Wangc

aComputational Transport Phenomena Laboratory (CTPL), Division of Physical

Sciences and Engineering (PSE), King Abdullah University of Science and Technology

(KAUST), Thuwal, 23955-6900, Kingdom of Saudi Arabia

bDepartment of Applied Mathematics and Research Institute for Smart Energy, The

Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, Hong Kong,

cComputational Transport Phenomena Laboratory (CTPL), Division of Computer,

Electrical and Mathematical Sciences and Engineering (CEMSE), King Abdullah

University of Science and Technology (KAUST), Thuwal, 23955-6900, Kingdom of Saudi

Arabia

Abstract

Varieties of energy-stable numerical methods have been developed for incom-

pressible two-phase ﬂows based on the Navier-Stokes–Cahn–Hilliard (NSCH)

model in the Eulerian framework, while few investigations have been made

in the Lagrangian framework. Smoothed particle hydrodynamics (SPH) is a

popular mesh-free Lagrangian method for solving complex ﬂuid ﬂows. In this

paper, we present a pioneering study on the energy-stable SPH discretization

of the NSCH model for incompressible two-phase ﬂows. We prove that this

SPH method inherits mass and momentum conservation and the energy dis-

sipation properties at the fully discrete level. With the projection procedure

to decouple the momentum and continuity equations, the numerical scheme

meets the divergence-free condition. Some numerical experiments are carried

out to show the performance of the proposed energy-stable SPH method for

solving the two-phase NSCH model. The inheritance of mass and momentum

?https://ctpl.kaust.edu.sa

∗Corresponding author

Email addresses: xiaoyu.feng@kaust.edu.sa (Xiaoyu Feng),

zhonghua.qiao@polyu.edu.hk (Zhonghua Qiao), shuyu.sun@kaust.edu.sa (Shuyu

Sun), xiuping.wang@kaust.edu.sa (Xiuping Wang)

Preprint submitted to Submitted to XXX Journal October 24, 2022

conservation and the energy dissipation properties are veriﬁed numerically.

Keywords: Smoothed Particle Hydrodynamics, Energy stability,

Two-phase Flow, Navier-Stokes-Cahn-Hilliard model

1. Introduction

Initially proposed [1,2] for astrophysical problems, the smoothed particle

hydrodynamics (SPH) approach has become one of the most popular mesh-

free particle methods and it is now widely used in various applications, such as

multi-phase ﬂow [3], ﬂow in porous media [4], fracture and crack propagation

[5], lava ﬂow [6], magneto-hydrodynamics [7], the evolution of planets and

stars [8], and even movie special eﬀects [9]. Unlike the nodes in other mesh-

free methods [10], the SPH particles can be regarded not only mathematically

as interpolation points but also physically as material components, just like

the “embedded atoms" in molecular dynamics (MD) simulation [11].

SPH possesses many well-known appealing features. First of all, com-

pared to well-studied grid-based Eulerian numerical methods, SPH naturally

takes advantage of the Lagrangian framework, and it treats the convection

term eﬀectively and stably. Free surfaces, material interfaces, and moving

boundaries can all be traced naturally through this method without tracking

or reconstructing. This mesh-free feature facilitates applications like multi-

phase ﬂow and high-energy events like explosions, high-speed collisions, and

penetrations. Local mass conservation is automatically retained in SPH since

the mass is carried by each particle as a unique property of the particle. By

using the unique “color” property of each particle, it is much easier to describe

complex multi-component or multi-material phenomena, such as the hydro-

carbon components in oil and gas reservoirs. In addition, the SPH method

can handle large deformations and complicated geometry without causing

mesh distortion. With the progress of neighbor searching schemes and par-

allel computing strategies [12], SPH enjoys even greater computational per-

formance and potential. Some review literature for the SPH method can be

referred to [13,14].

In this paper, we consider incompressible two-phase ﬂows based on the

Navier-Stokes–Cahn–Hilliard (NSCH) model. Even though possessing a lot

of the appealing features mentioned above, SPH still faces a number of open

problems and challenges. First of all, the energy-stability and physical con-

sistency of SPH for the NS equation have not yet been rigorously studied.

Second, the treatment of incompressibility in SPH remains tricky and chal-

lenging; in fact, most incompressibility treatments in the literature are not

energy stable. Third, to the best of our knowledge, there is not an energy-

stable SPH method for the NSCH system proposed in the literature.

Two common perspectives for the stability analysis of the SPH method

include: (1) tensile instability caused by disordered distributions of particles;

(2) the maximum time step that keeps the simulation stable. For the former,

a novel technique called “particle shifting” has been employed to alleviate

particle gathering [15]. For the second, to date, several attempts have been

made to study the energy variation of the SPH methods, which reveal that the

entropy-increasing (energy dissipation) methods possess better stability [7].

In many approaches, the energy dissipation was explicitly added by an artiﬁ-

cial viscosity term, but the artiﬁcial viscosity term can introduce additional

numerical errors (commonly known as numerical diﬀusion). For the treat-

ment of the incompressibility of the ﬂuid, two common approaches are used:

(1) approximately simulating incompressible ﬂows with small compressibility,

known as Weakly Compressible SPH (WCSPH); (2) simulating incompress-

ible ﬂows by enforcing the incompressibility, known as Incompressible SPH

(ISPH)[16]. However, energy stable treatment of incompressibility has not

fully been addressed in the literature. Recently, Sun and Zhu [17] proposed

an energy-stable SPH method for incompressible single-phase ﬂow that does

not use artiﬁcial viscosity terms, and it serves as the basis and inspiration

for this paper.

The interface, as a critical element of the two-phase ﬂuid system, can

be modeled either as a sharp interface [18,19,20] or as a diﬀuse interface

[21]. The NSCH model belongs to the diﬀuse interface approach, and it

obeys thermodynamically consistent energy dissipation laws. The same en-

ergy law is also desired to be retained in the discretized equations. Some tech-

niques, like convex-concave splitting [22,23] and the stabilizing approach, are

used to construct energy-stable schemes. Several advances in energy-stable

schemes include the scalar auxiliary variable method (SAV), the invariant

energy quadratization method (IEQ) [24], the exponential time diﬀerencing

method (ETD) [25], and the linear energy-factorization method (EF) [26,27].

The diﬀuse interface model based on the EoS is also proposed for modeling

complex two-phase and multi-component mixtures [28,29,30]. However, the

above studies are all based on a mesh and the Euler framework. Most of the

SPH two-phase methods are based on the sharp interface model [31]. Re-

searchers rarely looked into how to combine the SPH method and the diﬀuse

interface model together [32] and how to design an energy-stable scheme for

the system.

The rest of this paper is organized as follows. In Section 2, we review

the NSCH system and give a brief proof of energy law at the PDE level. In

Section 3, some basics of the SPH method are given and an energy-stable

ODE model is proposed based on the SPH framework. In Section 4, an

energy-stable fully discrete scheme is well developed. We provide the detailed

proof and derivation of the ODE system and fully discrete scheme. In Section

5, three numerical examples are presented to validate the scheme. Finally,

some concluding remarks are given in Section 6.

2. PDE model and its energy law

We now study a mixture of two immiscible, incompressible ﬂuids in a

conﬁned domain Ω⊂Rd(d= 2,3) and the NSCH model is introduced. To

identify the regions occupied by the two ﬂuids, we introduce a phase function

φ, such that

φ(x, t) = (1ﬂuid 1

−1ﬂuid 2

According to the idea of the diﬀuse interface and gradient ﬂow theory, one of

the most popular thermodynamic theories for inhomogeneous ﬂuids, the total

mixing energy has two contributions: Fb(φ)from the homogeneous part of the

ﬂuid and F∇(φ)from the inhomogeneity of the ﬂuid. The thermodynamic

behavior of the entire two-phase system is governed by the mixing energy

functional F(φ, ∇φ),

F(φ, ∇φ) = ZΩ

f(φ, ∇φ)dx=Fb(φ) + F∇(φ) = ZΩ

fb(φ)dx+ZΩ

f∇(φ)dx

=ZΩfb(φ) + λ

2k∇φk2dx,

(2.1)

where frepresents the energy density, and λdenotes the characteristic

strength of the phase mixing energy with respect to φ.λhas a relation

with the surface tension coeﬃcient σat the equilibrium state: λ=3σ

2√2ε,

where εis the capillary width of the interface thickness. The energy density

function from homogeneity, fb(φ), only depends on φlocally and fb(φ) =

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摘要：

?HighlightsTherststudyofenergy-stableODEdiscretizationandtheenergy-stablefullydiscreteschemebytheSPHmethodfortwo-phaseproblemsTheschemeensurestheinheritanceofmomentumconservationandtheenergydissipationlawfromthePDEleveltotheODElevel,andthentothefullydiscretelevel.ThisnovelmethodbasedonNSCHmodele...

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An energy-stable Smoothed Particle Hydrodynamics discretization of the Navier-Stokes-Cahn-Hilliard model for incompressible two-phase flows

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