Multiple solutions of nonlinear coupled constitutive relation model and its rectification in non-equilibrium flow computation

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arXiv:2210.12716v2 [physics.comp-ph] 22 Nov 2022

Multiple solutions of nonlinear coupled constitutive relation model

and its rectiﬁcation in non-equilibrium ﬂow computation

Junzhe Caoa, Sha Liuab∗, Chengwen Zhongab , Congshan Zhuoab , Kun Xucd

aSchool of Aeronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

bInstitute of Extreme Mechanics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

cDepartment of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China

dShenzhen Research Institute, Hong Kong University of Science and Technology, Shenzhen, China

Abstract

In this study, the multiple solutions of Nonlinear Coupled Constitutive Relation (NCCR) model

are ﬁrstly observed and a way for identifying the physical solution is proposed. The NCCR model

proposed by Myong is constructed from the generalized hydrodynamic equations of Eu, and aims

to describe rareﬁed ﬂows. The NCCR model is a complicated nonlinear system. Many assumptions

have been used in the schemes for solving the NCCR equations. The corresponding numerical

methods may be associated with unphysical solution and instability. At the same time, it is hard

to analyze the physical accuracy and stability of NCCR model due to the uncertainties in the

numerical discretization. In this study, a new numerical method for solving NCCR equations is

proposed and used to analyze the properties of NCCR equations. More speciﬁcally, the nonlinear

equations are converted into the solutions of an objective function of a single variable. Under this

formulation, the multiple solutions of the NCCR system are identiﬁed and the criteria for picking

up the physical solution are proposed. Therefore, a numerical scheme for solving NCCR equations

is constructed. A series of ﬂow problems in the near continuum and low transition regimes with a

large variation of Mach numbers are conducted to validate the numerical performance of proposed

method and the physical accuracy of NCCR model.

Keywords: Supersonic/hypersonic rareﬁed ﬂow; Nonlinear coupled constitutive relation;

Numerical method for NCCR

∗Corresponding author

Email addresses: caojunzhe@mail.nwpu.edu.cn (Junzhe Caoa), shaliu.@nwpu.edu.cn (Sha Liua b ),

zhongcw@nwpu.edu.cn (Chengwen Zhonga b ), zhuocs@nwpu.edu.cn (Congshan Zhuoa b ), makxu@ust.hk (Kun Xucd)

Preprint submitted to Elsevier November 23, 2022

1. Introduction

With the development of hypersonic vehicle, spacecraft and micro-electromechanical system,

the non-equilibrium ﬂow receives more attention and developing corresponding numerical methods

is a challenging topic. In recent years, both the stochastic particle method [1, 2, 3] and Discrete

Velocity Method (DVM) [4, 5, 6, 7, 8, 9] are successfully developed. Meanwhile, a uniﬁed gas-kinetic

wave-particle method is proposed [10] on the framework of Uniﬁed Gas-Kinetic Scheme (UGKS).

It is a multiscale multi-eﬃciency preserving method, and is rapidly developed [11, 12, 13]. All of

above methods can accurately simulate ﬂows from the continuum ﬂow regime to the free molecular

ﬂow regime. However, eﬃciency problem exists in them. For example, the time step in the Direct

Simulation Monte Carlo (DSMC) method is seriously restricted in the near-continuum ﬂow regime.

Generally, the number of numerical particles should be prohibitively large when using stochastic

particle methods to simulate near-continuum ﬂows. On the other hand, the DVM meets the curse of

dimensionality, making the discrete velocity space expensive. The root cause of eﬃciency problem is

that when the non-equilibrium feature becomes stronger, there are rather more degrees of freedom to

be concerned about. It is worth noting that in macroscopic methods, less degrees of freedom are used

to describe the non-equilibrium feature, which results in that the Knudsen number (Kn, the ratio of

the molecular mean free path to the characteristic physical length scale) should not be too large when

using these methods, or the accuracy problem exists. From another point of view, in macroscopic

methods, the accuracy is sacriﬁced for eﬃciency. As a consequence, macroscopic methods are

suitable for simulating near-continuum ﬂows. Macroscopic methods, such as the Chapman-Enskog

method [14] and the method of moments [15], are well developed[16, 17, 18, 19] and extended to

polyatomic gases and multi-component mixture gases[20, 21].

As a kind of macroscopic methods, the Nonlinear Coupled Constitutive Relation (NCCR) model

is proposed by Myong [22, 23, 24], based on the generalized hydrodynamic equations of Eu [25]. In

recent years, the property of this model is studied from many points of view and is implemented

to many ﬂow mechanism studies and engineering applications. For example, the topology of the

NCCR model is studied by Myong [26] and the linear stability is proved by Jiang [27]. The model

is utilized to study complicated ﬂows like shock-vortex interaction [28, 29], dusty and granular

ﬂows [30]. The model is applied in the framework of discontinuous Galerkin by Xiao [31] and in

the framework of extended gas-kinetic scheme by Liu [32], where this model is further improved

according to one of recent conclusions in the non-equilibrium thermodynamics that the relaxation

time ratio between the real nonlinear one and the linear (near equilibrium) one is bounded[33, 34].

Furthermore, the NCCR model is extended to the multi-species gas ﬂow [35], thermodynamic non-

equilibrium ﬂow [36] and hypersonic reaction ﬂow [37]. The ability of NCCR model to describe the

hypersonic near-continuum ﬂow is systematically tested by Myong [38]. Diﬀerent from other kinds

of macroscopic methods, there are no high order spatial derivatives in the NCCR equations, which

reduces the diﬃculty in designing numerical methods and maintaining stability. However, this also

results in that the stress and heat ﬂux in NCCR equations are not expressed explicitly, and they

can only be obtained by solving this complicated nonlinear system.

When trying to solve these complicated nonlinear equations, because of their complexity, the

convergency property of classical iteration methods is hard to be analyzed, such as the ﬁxed point

iteration method and Newton’s method. And actually they do not even converge. In Ref. [23, 24],

a decomposed solving method is proposed by Myong, which is the most popular method for solving

the NCCR equations currently. In Myong’s method, the three-dimensional problem is simpliﬁed

approximately into three one-dimensional non-interfering problems in x, y, z directions. Though

accurate solution on each direction can be obtained, coupled terms between diﬀerent directions

are not taken into consideration, which results in deviation. Aiming at Myong’s method, Jiang

comments that the most unsatisﬁed feature is the computational instability induced directly by an

unphysical negative-density phenomenon, particularly in some expansion regions. In Ref. [39], the

ﬁxed point iteration method is modiﬁed, which brings about better stability. But the convergency

property of this method is still hard to analyze and indeed its result is not accurate in tests. If the

result of Modiﬁed Fixed Point Iteration (MFPI) method is taken as the precondition of Newton’s

method, the result is accurate and convergency property is better. This is the coupling solving

method proposed by Jiang. However, the convergency property of this method is still hard to

be analyzed. From another perspective, because these solving methods are approximate, it is a

hard task to test the performance of NCCR model for describing non-equilibrium ﬂows, whose

development is hence restricted.

In this study, aiming at this problem, a new method is proposed to convert the problem of

solving the multi-variable complicated nonlinear system into solving an objective function of a

single variable. Under this foamulation, it is much more easy to analyze this complicated nonlinear

system. The multiple solutions of NCCR equations are observed and the criteria for picking up the

physical solution are proposed. An iterative solving method is then proposed without assumptions.

A series of numerical test cases in the near continuum and low transition regimes with a large

variation of Mach numbers are conducted. The numerical performance of proposed method and the

physical accuracy of NCCR model are validated. In the test cases, results of experiments, DSMC

method, UGKS method and Discrete UGKS (DUGKS) method are taken as references. Other

macroscopic methods are also utilized for comparison, including the classical methods for solving

Navier-Stokes(NS) equations (whether the bulk viscosity is considered or not and whether the slip

boundary condition is implemented or not). The MFPI method for solving NCCR equations is

implemented in these test cases as well, because this method has good stability and it is desirable

to test its accuracy.

The remainder of this paper is organized as follows: the nondimensionalization system of this

paper and the corresponding governing equations are introduced in Sec.2, along with a short analysis

about the NCCR model. Sec.3 is the construction of the proposed solving method and multiple

solutions are exhibited. The numerical test cases are conducted in Sec.4. The conclusions are in

Sec.5.

2. Nondimensionalization, governing equations and NCCR model

This study is done under the framework of Stanford university unstructured (SU2) open source

solver [40, 41]. The stress and heat ﬂux in the viscous ﬂux are calculated by NCCR equations, and

the inviscid ﬂux is computed by the Kinetic Inviscid Flux (KIF)[42] or AUSM+ -up scheme[43].

The nondimensionalization system should be introduced ﬁrstly,

ϕ=ϕpre

ϕref

,(1)

where ϕdenotes macroscopic values to be used in the nondimensionalized equations, ϕpre denotes

values before the nondimensionalization and ϕref denotes reference values, as Tab.2.

The governing equation used in this paper is

∂tΨ+∇ · Fc− ∇ · Fv=0,(2)

where Ψdenotes the conserved value vector, Fcdenotes the inviscid ﬂux vector and Fvdenotes

Table I: Reference values in the nondimensionalization

Value name Reference value

Length lref

Pressure pref

Density ρref

Temperature Tref

Velocity Uref =ppref /ρref

Speciﬁc energy eref =U2

ref

Stress Πref =pref

Heat ﬂux Qref =ρref eref Uref

Gas constant Rref =eref /Tref

Heat capacity (constant pressure) Cp,ref =Rref

Dynamic viscosity µref =ρref Uref lref

Heat conductivity kref =Cp,ref µref

the viscous ﬂux vector. The equation can be expanded into







ρU

ρE





t

+∇ · 





ρU

ρUU +pI

(ρE +p)U





− ∇ · 





Π+ ∆I

(Π+ ∆I)·U+Q







=0,(3)

where Iis unit tenser and ∆ is the excess normal stress, which is related to the bulk viscosity. It is

perceived to have direct correlation with the rotational relaxation time of diatomic gas and be able

to describe rotational relaxation eﬀect to some extent [44]. In NS equations, the linear constitutive

relation is as follows:

Πij = 2µ∂U<i

∂xj>

Qi=µCp

∂T

∂xi

∆ = µb

∂Ui

∂xi

(4)

where Pr denotes the Prandtl number, Cp=Rγ/(γ−1), µbis the bulk viscosity, and symbol <·>

denotes the second-order symmetric and trace-free tensor:

A<ij>=1

2(Aij +Aji)−1

3δijAkk,(5)

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arXiv:2210.12716v2[physics.comp-ph]22Nov2022Multiplesolutionsofnonlinearcoupledconstitutiverelationmodelanditsrectiﬁcationinnon-equilibriumﬂowcomputationJunzheCaoa,ShaLiuab∗,ChengwenZhongab,CongshanZhuoab,KunXucdaSchoolofAeronautics,NorthwesternPolytechnicalUniversity,Xi’an,Shaanxi710072,ChinabInsti...

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Multiple solutions of nonlinear coupled constitutive relation model and its rectification in non-equilibrium flow computation

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