a two-stage fourth-order gas-kinetic scheme for ... · pdf filea two-stage fourth-order...

Commun. Comput. Phys.doi: 10.4208/cicp.OA-2017-0023

Vol. xx, No. x, pp. 1-27xxx 201x

A Two-Stage Fourth-Order Gas-Kinetic Scheme for

Compressible Multicomponent Flows

Liang Pan1,∗, Junxia Cheng1, Shuanghu Wang1 and Kun Xu2

1 Institute of Applied Physics and Computational Mathematics, Beijing, China.2 Department of Mathematics, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong.

Received 24 January 2017; Accepted (in revised version) 22 March 2017

Abstract. With the use of temporal derivative of flux function, a two-stage temporaldiscretization has been recently proposed in the design of fourth-order schemes basedon the generalized Riemann problem (GRP) [21] and gas-kinetic scheme (GKS) [28].In this paper, the fourth-order gas-kinetic scheme will be extended to solve the com-pressible multicomponent flow equations, where the two-stage temporal discretizationand fifth-order WENO reconstruction will be used in the construction of the scheme.Based on the simplified two-species BGK model [41], the coupled Euler equations forindividual species will be solved. Each component has its individual gas distributionfunction and the equilibrium states for each component are coupled by the physicalrequirements of total momentum and energy conservation in particle collisions. Thesecond-order flux function is used to achieve the fourth-order temporal accuracy, andthe robustness is as good as the second-order schemes. At the same time, the sourceterms, such as the gravitational force and the chemical reaction, will be explicitly in-cluded in the two-stage temporal discretization through their temporal derivatives.Many numerical tests from the shock-bubble interaction to ZND detonative waves arepresented to validate the current approach.

AMS subject classifications: (or PACS) To be provided by authors

Key words: Multicomponent flows, gas kinetic scheme, two-stage temporal discretization.

1 Introduction

The development of numerical methods for compressible multicomponent flows is im-portant in computational fluid dynamics. Over the past decades, significant progresseshave been made for the computations of multicomponent flows which are associated

∗Corresponding author. Email addresses: [email protected] (L. Pan), cheng [email protected] (J. X.Cheng), wang [email protected] (S. H. Wang), [email protected] (K. Xu)

http://www.global-sci.com/ 1 c©201x Global-Science Press

2 L. Pan et al. / Commun. Comput. Phys., xx (201x), pp. 1-27

with discontinuities and shock waves. One of the popular approaches is to solve an ex-tended system in which additional equations are introduced to the original Euler equa-tions in order to track different components. The additional equations can be the equa-tions for the volume fraction, mass fraction, and ratio of specific heats of the mixture[1, 6, 35]. In order to eliminate spurious oscillations and other computational inaccura-cies in the conservative methods, some non-conservative approaches which capture thecontact discontinuities by making use of additional non-conservative governing equa-tions were proposed [1, 2, 17, 27, 36]. Another approach is the sharp interface method.Each fluid is solved separately on each side of the interface by the method designed fora single-component flow. Interfaces between different fluids are captured by the level-set method [30] or front tracking method [38, 39]. Boundary conditions at interface aregiven by ghost-fluid method or application of exact Riemann solver at the interface [6,39].Although interface can be resolved sharply, it is difficult to apply these methods to theinterfaces associated with complex geometry.

The gas-kinetic scheme has been developed systematically for the compressible flowcomputations [42,43]. An evolution process from kinetic scale to hydrodynamic scale hasbeen constructed for the flux evaluation. The kinetic effect through particle free transportcontributes to the artificial dissipation for the capturing of shock waves, and the hydro-dynamic effect plays a dominant role for the capturing of resolved viscous and heat con-ducting solutions. In this sense, the gas-kinetic scheme is hybrid method of upwind andcentral difference, but with a smooth transition between these two limits. Due to the cou-pling of inviscid and viscous terms in the kinetic formulation, there is no fundamentalbarrier for the finite volume gas-kinetic scheme to capture Navier-Stokes solutions withstructured or unstructured meshes. With the discretization in particle velocity space, aunified gas-kinetic scheme has been developed for the transport process in the entireflow regimes from rarefied to continuum ones [11, 26, 44]. Recently, with the incorpo-ration of higher-order initial data reconstruction, a third-order gas-kinetic scheme hasbeen proposed in [22, 25, 29]. The flux evaluation is based on the time evolution solu-tion of flow variables from initial piece-wise discontinuous polynomials around a cellinterface. However, based on the time accurate evolution solution from a general initialcondition for the flux function, the gas-kinetic scheme becomes complicated for its fur-ther improvement of the order of the scheme, such as the construction of a fourth-orderflux function [24]. However, instead of developing one step time integration method, thetwo-stage Lax-Wendroff time stepping method in [21] provides an alternative frameworkto develop a fourth-order gas-kinetic scheme with a second-order flux function only [28].In comparison with the formal one-stage time-stepping third-order gas-kinetic solver,the fourth-order method not only reduces the complexity of the flux function, but alsoimproves the accuracy of the scheme, even though the third- and fourth-order schemeshave similar computation cost. Most importantly, the robustness of the fourth-order gas-kinetic scheme is as good as the second-order one.

The BGK-based numerical methods for the multicomponent flow have also been pro-posed in recent years. By incorporating a conservative γ-model [1] into the gas-kinetic

L. Pan et al. / Commun. Comput. Phys., xx (201x), pp. 1-27 3

scheme, a second-order γ-model gas-kinetic scheme for the compressible multicompo-nent flow was proposed in [14, 15], in which the material interface with different ratiosof specific heat is considered as a contact discontinuity. Based on the BGK equation foreach component with its own equilibrium state, a second-order gas-kinetic scheme formulticomponent flow was presented in [23, 41]. In this approach, the equilibrium statesof different components are coupled in space and time due to the intensive particle col-lisions, and they share the common macroscopic flow velocities and temperatures. Inorder to describe the collision effects between different species in the gas-mixtures moreaccurately, many BGK-type collision models were proposed, including multiple-BGK op-erator model and single-BGK operator model. Based on the multiple operator model, akinetic scheme was proposed for inhomogeneous flow [18], in which self-collisions andcross-collisions were taken into account explicitly by implementing different collision op-erators. In the single operator model, only one global collision operator is used for eachcomponent to take account of both self-collision and cross-collision effects, and the typ-ical example is AAP model [3]. Based on this model, a unified gas-kinetic scheme formulticomponent flow is constructed [40].

In this paper, a fourth-order gas-kinetic scheme for the compressible multicompo-nent flows is presented based on the two-stage temporal discretization method [21] andthe fifth-order WENO reconstruction. The two-species Euler equations will be solvedthrough the flux function for each individual component, and the flux function is evalu-ated from the two-species BGK model [41], in which these two species share the commonvelocity and temperature due to the intensive momentum and energy exchange throughparticle collision in the Euler limit. The time dependent numerical fluxes can be obtainedby taking moments of the gas distribution function for each species at a cell interface.With the numerical flux function and its temporal derivative, the two-stage temporal dis-cretization can be implemented to design a fourth-order scheme for the multi-componentflow. Compared with the traditional high-order scheme, the complexity of the currentscheme is reduced greatly, and the robustness is as good as the second-order schemes.The time derivative of the source term in the gravitational force and the chemical reac-tive flows will be included as well in the design of the fourth-order scheme. Numericaltests from the shock-bubble interaction to the ZND detonative waves are presented tovalidate the current approach.

This paper is organized as follows. The two-stage fourth-order temporal discretiza-tion method is reviewed in Section 2. In Section 3, two-component BGK model and thecorresponding gas-kinetic scheme are presented. Numerical examples are presented tovalidate the scheme in Section 4. The last section is the conclusion.

2 Two-stage fourth-order temporal discretization

The design of high order accurate methods has been one of main themes for compressiblefluid dynamics. Many numerical methods have been developed in the past decades, such


as essentially non-oscillatory (ENO) [12], weighted essentially non-oscillatory (WENO)[16], discontinuous Galerkin (DG) [5, 32] methods, etc. Most of these methods are basedon the first order Riemann flux function and use the Runge-Kutta approach [10] to achievehigh order accuracy in time. In order to achieve a fourth-order temporal accuracy, fourRunge-Kutta steps are needed. Here, based on the time-dependent high-order flux func-tion, a two-stage temporal discretization can be developed for a fourth-order scheme forthe Euler and Navier-Stokes equations [21, 28]. The following is a brief review of thetwo-stage fourth-order method.

Consider the following time-dependent equation

∂w

∂t=L(w), (2.1)

with the initial condition at tn, i.e.,

w(t= tn)=wn, (2.2)

where L is an operator for spatial derivative of flux. The time derivatives are obtainedusing the Cauchy-Kovalevskaya method,

∂wn

∂t=L(wn),

∂

∂tL(wn)=

∂

∂wL(wn)L(wn).

Introducing an intermediate state at t∗= tn+∆t/2,

w∗=wn+1

2∆tL(wn)+

1

8∆t2 ∂

∂tL(wn), (2.3)

the corresponding temporal derivatives are obtained as well for the intermediate stagestate,

∂w∗

∂t=L(w∗),

∂

∂tL(w∗)=

∂

∂wL(w∗)·L(w∗).

Then, the state w can be updated with the following formula,

wn+1=wn+∆tL(wn)+1

6∆t2

( ∂

∂tL(wn)+2

∂

∂tL(w∗)

). (2.4)

It can be proved that the above time stepping method with Eq. (2.3) and Eq. (2.4) providesa fourth-order temporal accurate solution for w(t) at t= tn+∆t. More details about thetwo-stage discretization can be found in [21]. Thus, a fourth-order temporal accuracy canbe achieved from the two-stage discretization through Eq. (2.3) and Eq. (2.4).

Consider the following conservation laws

∂w

∂t+

∂f(w)

∂x=0.


The semi-discrete form of a finite volume scheme can be written as

∂wi

∂t=Li(w)=−

1

∆x(fi+1/2−fi−1/2),

where wi are the cell averaged conservative variables, fi+1/2 are the fluxes at the cellinterface x = xi+1/2, and ∆x is the cell size. With the temporal derivatives of the flux,which can be given by the Lax-Wendroff type flow solvers, the two-stage fourth-orderschemes have been developed [21, 28]. Similarly, for the conservation laws with sourceterms

∂w

∂t+

∂f(w)

∂x=S(w),

the corresponding operator can be denoted as

∂wi

∂t=Li(w)=−

1

∆x(fi+1/2−fi−1/2)+S(wi). (2.5)

The temporal derivatives of source St(wi) can be expressed as the combination of (wi)t,which is given by Eq. (2.5). Thus, the two-stage temporal discretization can be directlyextended for the conservation laws with source terms.

3 Two-component BGK model and gas-kinetic scheme

In this paper, the two-component flows are considered. Similar with the single compo-nent flow, the BGK-type model can be constructed as simplification of Boltzmann equa-tions for the multicomponent flows. Existing BGK-type gas mixture models can be classi-fied into two categories, i.e. multiple-BGK operator model [18] and single-BGK operatormodel [3]. However, in this paper we are targeting on the multicomponent Euler equa-tions, a simplified single BGK operator model will be used.

3.1 Two-species Euler equations and two-species BGK model

In this paper, we mainly focus on the two-species Euler equations [41] for the multicom-ponent flows, which can be expressed as

ρ1

ρ2

ρUρVρE

t

+

ρ1Uρ2U

ρU2+pρUV

U(ρE+p)

x

+

ρ1Vρ2VρUV

ρV2+pV(ρE+p)

y

=0, (3.1)


where ρ1 and ρ2 are the densities for each component, ρ = ρ1+ρ2 is the total density,p= p1+p2 is the total pressure, ρE = ρ1E1+ρ2E2 is total energy, and (U,V) is averagedflow velocity. The corresponding two-component BGK model [41] can be written as

∂ fi

∂t+u·∇ fi =

gi− fi

τ, (3.2)

where fi, i=1,2 is the gas distribution function of each component, and τ is the collisionterm, the compatibility for this two-component BGK model is given as follows

∑i

∫(gi− fi)φidudξi =0, (3.3)

where φ1 = (1,0,u,v, 12(u

2+v2+ξ21)) and φ2 = (0,1,u,v, 1

2(u2+v2+ξ2

2)). The equilibriumstates gi in this multicomponent model share the common velocity and temperature dueto the intensive momentum and energy exchange in the Euler limit, and can be writtenas

gi =ρi

(λ0

π

) Ki+2

2e−λ0((u−u0)

2+ξ2i ),

where the average common velocity and temperature satisfy the following conservativerequirement

ρ1U1+ρ2U2=(ρ1+ρ2)U0,

ρ1V1+ρ2V2=(ρ1+ρ2)V0, (3.4)

ρ1E1+ρ2E2=(ρ1+ρ2)

2(U2

0+V20 )+

(K1+2)ρ1+(K2+2)ρ2

λ0.

In the framework of finite volume scheme, to update the flow variables, the numericalflux is constructed based on the integral solution of gas distribution function from theBGK equation at a cell interface

fi(xj+1/2,t,u,v,ξ)=1

τ

∫ t

0gi(x′,y′,t′,u,v,ξ)e−(t−t′)/τdt′+e−t/τ fi0(−ut,−vt,u,v,ξ), (3.5)

where xj+1/2=(0,0) is the location of the cell interface, xj+1/2= x′+u(t−t′) and yj+1/2=y′+v(t−t′) are the trajectory of particles, fi0 is the initial gas distribution function, andgi is the corresponding equilibrium state. According to Eq. (3.5), the time dependent gasdistribution function fi(xj+1/2,t,u,v,ξ) at the cell interface xj+1/2 can be expressed as

fi(xj+1/2,t,u,v,ξ)=(1−e−t/τ)gi+((t+τ)e−t/τ−τ)(aiu+biv)gi

+(t−τ+τe−t/τ)Aigi

+e−t/τgir [1−(τ+t)(airu+birv)−τAir)]H(u)

+e−t/τgil [1−(τ+t)(ail u+bilv)−τAil)](1−H(u)). (3.6)


Based on the spatial reconstruction of macroscopic flow variables, which will be pre-sented in the following section, the equilibrium states gil and gir for each component canbe determined. The conservative variables Wi0 and the equilibrium state gi0 at the cellinterface can be given according to the compatibility condition Eq. (3.3) as follows

∫ψigi0dudvdξi =Wi0=

∫

u>0ψigildudvdξi+

∫

u<0ψigirdudvdξi ,

where ψi =(1,u,v, 12(u

2+v2+ξ2i )). The coefficients related to the spatial derivatives and

time derivative aik,bik,Aik,k= l,r and ai,bi,Ai in gas distribution function Eq. (3.6) can bedetermined according to the spatial derivatives and compatibility condition. More detailsof the multi-component gas-kinetic scheme can be found in [41].

3.2 Two-stage temporal discretization for multicomponent flow

With the gas distribution function at the cell interface, the time dependent numericalfluxes for each component obtained, and the semi-discrete form of the finite volumescheme for Eq. (3.1) can be written as

∂Wnjk

∂t=L(Wn

jk)=−1

∆x(Fj+1/2,k(W

n,t)−Fj−1/2,k(Wn,t))

−1

∆y(Gj,k+1/2(W

n,t)−Gj,k−1/2(Wn,t)), (3.7)

where W =(ρ1,ρ2,ρU,ρV,ρE), Fj+1/2,k and Gj,k+1/2 are the total numerical fluxes in the xand y directions respectively. For example, Fi+1/2,j can be written as

Fj+1/2,k(Wn,t)=F1

j+1/2,k(Wn,t)+F2

j+1/2,k(Wn,t), (3.8)

where

F1j+1/2,k(W

n,t)=

Fρ1

0Fρ1U1

Fρ1V1

Fρ1E1

=

∫uφ1 f1dudξ1,

and

F2j+1/2,k(W

n,t)=

0Fρ2

Fρ2U2

Fρ2V2

Fρ2E2

=

∫uφ2 f2dudξ2.


For the semi-discretized finite volume scheme Eq. (3.7), the operator

L(Wnjk)=−

1

∆x(Fj+1/2,k(W

n,t)−Fj−1/2,k(Wn,t))

−1

∆y(Gj,k+1/2(W

n,t)−Gj,k−1/2(Wn,t)),

falls into the two-stage temporal discretization, such as Eqs. (2.3) and (2.4) directly. Toobtain the time derivatives of the flux function at tn and t∗ = tn+∆t/2 with the correctphysics, the flux function should be approximated as a linear function of time withina time interval. Let’s first introduce the following notation, where the fluxes in the xdirection is chosen as example,

Fj+1/2,k(Wn,δ)=

∫ tn+δ

tn

Fj+1/2,k(Wn,t)dt,

where the total fluxes Fj+1/2,k(Wn,t) are given in Eq. (3.8). In the time interval [tn,tn+∆t],

the flux is expanded as the following linear form

Fj+1/2,k(Wn,t)=Fn

j+1/2,k+∂tFnj+1/2,k(t−tn). (3.9)

The coefficients Fnj+1/2,k and ∂tF

nj+1/2,k can be determined by solving the following linear

system

Fj+1/2,k(Wn,tn)∆t+

1

2∂tFj+1/2,k(W

n,tn)∆t2=Fj+1/2,k(Wn,∆t),

1

2Fi+1/2(W

n,tn)∆t+1

8∂tFj+1/2,k(W

n,tn)∆t2=Fj+1/2,k(Wn,∆t/2).

(3.10)

Similarly, Fj+1/2,k(W∗,t∗) and ∂tFj+1/2,k(W

∗,t∗) for the intermediate state can be con-structed. For the two-dimensional computation, the corresponding fluxes in the y-directioncan be obtained as well.

3.3 Spatial reconstruction

With the semi-discrete form of the finite volume scheme Eq. (3.7), the high-order schemefor the multicomponent flow will be given by the two-stage fourth-order temporal dis-cretization and the fifth-order WENO spatial reconstruction. The spatial reconstructionfor the gas-kinetic scheme contains two parts, i.e. initial data reconstruction and recon-struction for equilibrium.

In this paper, the fifth-order WENO method [16] is used for the initial data reconstruc-tion. Assume that W are the macroscopic flow variables that need to be reconstructed. Wj

are the cell averaged values, and Wrj ,W l

j are the two values obtained by the reconstruction

at two ends of the j-th cell. The fifth-order WENO reconstruction is given as follows

Wrj =

2

∑k=0

wkwrk, W l

j =2

∑k=0

wkwlk,


where all quantities involved are taken as

wr0=

1

3Wj+

5

6Wj+1−

1

6Wj+2, wl

0=11

6Wj−

7

6Wj+1+

1

3Wj+2,

wr1=−

1

6Wj−1+

5

6Wj+

1

3Wj+1, wl

1=1

3Wj−1+

5

6Wj−

1

6Wj+1,

wr2=

1

3Wj−2−

7

6Wj−1+

11

6Wj, wl

2=−1

6Wj−2+

5

6Wj−1+

1

3Wj,

and wm,wm, m=0,1,2 are the nonlinear weights and can be written as follows

wJSm =

αJSk

∑2p=0αJS

p

, αJSm =

dm

(ǫ+βm)2, wJS

m =αJS

s

∑2p=0 αJS

p

, αJSm =

dm

(ǫ+βm)2,

where

d0= d2=3

10, d1= d1=

3

5, d2= d0 =

1

10, ǫ=10−6,

and βm is the smooth indicator, and more details for its construction can be found in [16].After the initial date reconstruction, the reconstruction of equilibrium part is pre-

sented. For the cell interface xj+1/2, the reconstructed variables at both sides of the cellinterface are denoted as Wl ,Wr. According to the compatibility condition, which will begiven later, the macroscopic variables at the cell interface is obtained and denoted as W0.The conservative variables around the cell interface can be expanded as

W(x)=W0+S1(x−x∗)+1

2S2(x−x∗)

2+1

6S3(x−x∗)

3+1

24S4(x−x∗)

4,

where x∗=0 is the local coordinate of cell interface. With the following conditions,

∫

Ij+m

W(x)=Wj+m, m=−1,··· ,2,

the derivatives are given by

Wx =S1=[−

1

12(Wj+2−Wj−1)+

5

4(Wj+1−Wj)

]/∆x.

With the initial data reconstruction and the reconstruction for the equilibrium, the gas-distribution function can be fully determined. The numerical flux and its temporal deriva-tives can be evaluated by the procedure presented above.

3.4 Numerical procedure

With spatial reconstruction and the two-stage temporal discretization, the procedure ofthe fourth-order scheme for the multicomponent flows is given as follows:


1. With the fifth-order WENO reconstruction, (ρni ,(ρiUi)

n,(ρiVi)n,(ρiEi)

n), i= 1,2 arereconstructed for each species.

2. Construct the gas distribution function fi(xj+1/2,t,u,v,ξ), i=1,2 based on the spatialreconstruction for each species, and calculate the time dependent flux Fj+1/2,k(W

n,t)according to Eq. (3.8).

3. Modify the cell averaged variables Wnjk according to Eq. (3.4), update W∗

jk at t∗ =

tn+∆t/2 by

W∗jk =Wn

jk−1

∆x

[Fj+1/2,k(W

n,∆t/2)−Fj−1/2,k(Wn,∆t/2)

]

−1

∆y

[G j,k+1/2(W

n,∆t/2)−G j,k−1/2(Wn,∆t/2)

],

and compute the fluxes and their derivatives by Eq. (3.10) for future use,

Fj+1/2,k(Wn,tn), Gj,k+1/2,(W

n,tn), ∂tFj+1/2,k(Wn,tn), ∂tGj,k+1/2(W

n,tn).

More details of the reconstruction can be found in [28].

4. Reconstruct intermediate values (ρ∗i ,(ρiUi)∗,(ρiVi)

∗,(ρiEi)∗), i=1,2, which are pro-

vided by W∗jk according to Eq. (3.4).

5. Construct the gas distribution function fi(xj+1/2,t,u,v,ξ), i=1,2 in the time interval[t∗,t∗+∆t] for each species, and calculate

∂tFj+1/2,k(W∗,t∗), ∂tGj,k+1/2(W

∗,t∗),

where the derivatives are determined by Eq. (3.8) and Eq. (3.10).

6. Update Wn+1ij by

Wn+1jk =Wn

jk−∆t

∆x[F n

j+1/2,k−Fnj−1/2,k]−

∆t

∆y[G n

j,k+1/2−Gnj,k−1/2],

where F ni+1/2,j and G n

j,k+1/2 are the numerical fluxes and expressed as

Fnj+1/2,k =Fj+1/2,k(W

n,tn)+∆t

6

[∂tFj+1/2,k(W

n,tn)+2∂tFj+1/2,k(W∗,t∗)

],

Gnj,k+1/2=Gj,k+1/2(W

n,tn)+∆t

6

[∂tGj,k+1/2(W

n,tn)+2∂tGj,k+1/2(W∗,t∗)

].

For each flux, the Gaussian quadratures are used in the tangential direction.


4 Numerical tests

For the inviscid flow, the collision time τ takes

τ=ǫ∆t+C∣∣∣

pl−pr

pl+pr

∣∣∣∆t,

where ε=0.05, C=1, pl and pr denote the pressure on the left and right sides of the cellinterface. The reason for including artificial dissipation through the additional term in theparticle collision time is to enlarge the kinetic scale physics in the discontinuous regionfor the construction of a numerical shock structure through the particle free transport. Forthe single-component flows, a simple stability analysis was given in [28]. The scheme isbasically stable with CFL number around 0.5. In this paper, the CFL number takes a fixedvalue of 0.4 for the multi-component flows.

4.1 Accuracy test

In this case, the advection of density perturbation is used to validate the accuracy of thescheme. The initial condition is set as follows

ρi(x)=0.5+0.1sin(πx), γi=1.4, i=1,2,

U(x)=1, p(x)=1, x∈ [0,2].

The periodic boundary condition is adopted in this case. Due to the identical specificheat ratio, the analytic solution for the total density ρ=ρ1+ρ2 is

ρ(x,t)=1+0.2sin(π(x−t)).

This case is also used to validate the recovery of the current scheme to the single-component flow. In the computation, a uniform mesh with N points and a fixed CFLnumber CFL=0.4 are used for different meshes. As analyzed in [28], with the fifth-orderspatial reconstruction, the leading truncation error in the inviscid case from the fourth-order GKS is O(∆x5+∆t4). With the fixed CFL number, we have ∆t=c∆x and the leadingterm becomes O(∆x5+∆t4)∼O(∆x5+c4∆x4). The L1 and L2 errors and orders at t= 2are presented in Table 1. With the mesh refinement, the expected order of accuracy isobtained by the current scheme.

4.2 One-dimensional Riemann problem

The one-dimensional Sod problem [17, 20] is presented to validate the current scheme.The domain [0,0.5] is occupied by component 1 and [0.5,1] is occupied by component 2.The initial condition given as follows

{(ρ,U,p,γ)=(1,0,1,5/3), x≤0.5

(ρ,U,p,γ)=(0.125,0,0.1,1.4), x>0.5.


Table 1: Accuracy test for the advection of density perturbation.

mesh L1 error convergence order L2 error convergence order

10 1.50979E-002 1.20377E-002

20 4.29672E-004 5.1349 3.38105E-004 5.1539

40 1.31369E-005 5.0315 1.02931E-005 5.0377

80 4.07813E-007 5.0095 3.19290E-007 5.0106

160 1.27248E-008 5.0021 9.96301E-009 5.0021

320 3.99352E-010 4.9938 3.12669E-010 4.9938

The computational domain is [0,1], a uniform mesh with 200 points is used in the com-putation and the non-reflective boundary condition is used at both ends of the compu-tational domain. The numerical solutions and exact solutions for the distributions of thetotal density ρ, density for each species ρ1, ρ2, total pressure p, average velocity U andaverage specific heat ratio γ=(ρ1γ1+ρ2γ2)/(ρ1+ρ2) are given in Fig. 1 respectively. Thenumerical results agree well with the exact solutions.

4.3 Shock-bubble interaction

In this case, the well-known shock-bubble interaction problem are considered, whichinvolves the collision of a shock wave in air with a circular gas bubble [13,31,33,34]. Theschematics of the initial condition is given in Fig. 2. Initially, there is a planar leftward-moving Mach 1.22 shock wave in air located at x=275mm traveling towards a stationarygas bubble with center (xc,yc) = (225,44.5)mm and of radius r = 25mm lying in front ofit. The computational domain for the shock tube has a size (x,y)∈ [0,445]×[0,89]mm2 .In the computation, 1530×400 uniform mesh points are used. The solid wall boundarycondition on the top and bottom is used, and the non-reflecting boundary condition onthe left and right is adopted. Two kinds of initial conditions are considered:

1. The first case is the interaction of shock and helium bubble, and the initial conditionis given as follows:

(ρ,u,v,p,γ)=(1.225kg/m3 ,0,0,1.01325×105Pa,1.4) Pre-shock air,

(ρ,u,v,p,γ)=(1.686kg/m3 ,−113.5m/s,0,1.59×105 Pa,1.4) Post-shock air,

(ρ,u,v,p,γ)=(0.167kg/m3 ,0,0,1.01325×105Pa,1.667) Bubble helium.

The schlieren-type images of density distributions at t=32,52,82,245 and 427µs forthis case are presented in Fig. 3.

2. The second case is the interaction of shock and R22 bubble, and the initial condition


x

tota

l den

sity

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1

200 cellsexact sloution

x

pres

sure

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1


x

dens

ity 1

0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1


x

velo

city

0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1


x

dens

ity 2

0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4


x

γ

0.2 0.4 0.6 0.81.3

1.4

1.5

1.6

1.7

1.8


Figure 1: One-dimensional Riemann solution with 200 mesh points and at the output time t= 0.2. The totaldensity, density for each species, average velocity, total pressure, and average specific heat ratio distributions.

Figure 2: Shock bubble interaction: the schematics of initial condition.


Figure 3: Shock bubble interaction (air-helium interaction): the schlieren-type images of density of the experi-mental results [13] (left) and numerical results (right) at t=32,52,82,245 and 427µs.


Figure 4: Shock bubble interaction (air-R22 interaction): the schlieren-type images of density of the experimentalresults [13] (left) and numerical results (right) at t=115,187 and 247µs.

is given as follows

(ρ,u,v,p,γ)=(1.225kg/m3 ,0,0,1.01325×105 Pa,1.4) Pre-shock air,

(ρ,u,v,p,γ)=(1.686kg/m3 ,−113.5m/s,0,1.59×105 Pa,1.4) Post-shock air,

(ρ,u,v,p,γ)=(3.863kg/m3 ,0,0,1.01325×105 Pa,1.25) Bubble R22.

The schlieren-type images of density distributions at t = 115,187,247,342,417 and1020µs for the air-R22 interaction are given in Fig. 4 and Fig. 5.

The time is measured relative to the time where the incident shock first hits the upstreambubble wall. The numerical results reproduce the large-scale structure of the experimentsdescribed in [13]. To get the flow features, the diagnosis plot of the space-time locationsof the incident shock, the refracted shock, the transmitted shock, the upstream edge ofbubble, the downstream edge of bubble, and the air jet head up to t=250µs, are given inFig. 6, and the results agree well with those appeared in the literatures [31, 33, 34].


Figure 5: Shock bubble interaction (air-R22 interaction): the schlieren-type images of density of the experimentalresults [13] (left) and numerical results (right) at t=342,417 and 1020µs.

4.4 Multi-material triple-point problem

The multi-material triple-point problem was presented in [9,19], which was widely usedto validate to the performance of Lagrangian and ALE methods with large mesh de-formation. A rectangular [0,7]×[0,3] domain is split among three materials, which areshown in Fig. 7. The domain D1∪D2 is occupied by component 1 and D3 is occupied bycomponent 2. The initial condition is given as follows

γ=1.5, ρ=1, p=1, U=0, V=0, (x,y)∈D1,

γ=1.5, ρ=0.125, p=0.1, U=0, V=0, (x,y)∈D2,

γ=1.4, ρ=1, p=0.1, U=0, V=0, (x,y)∈D3.

The non-reflective boundary conditions are used at all boundaries. Left from x = 1, ahigh-pressure material is located in D1, which generates a shock wave propagating tothe right. The rest of the domain is split by the y = 1.5 into two parts i.e. D2∪D3. As


XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

* * * * * * *

++++++++++++++++++

++++++++++

+++

x(mm)

time(

µs)

0 20 40 600

50

100

150

200

250

V_diV_RV_SV_TV_jV_ui

X*+

* * * *

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

X

++++++++++++++++++++++++++

x(mm)

time(

µs)

0 20 40 600

50

100

150

200

250

V_diV_TV_SV_RV_ui

*X

+

Figure 6: Shock bubble interaction: diagram for the air-helium (left) and air-R22 (right) interaction cases witha schematic showing the points used to construct the diagram: Vs for incident shock; VR for refracted shock:VT for transmitted shock; Vui for upstream edge of bubble, Vdi for downstream edge of bubble, and Vj for airjet head.

x

y

1 2 3 4 5 6

1

2

3

D1

D2

D3

Figure 7: Multi-material triple-point problem: schematic for the initial condition.

there is the same pressure across the initial horizontal line, at the beginning no wavesare generated there. However, due to different fluid properties different shock speedsare generated by the pressure gradient over the vertical line. Therefore, a vortex appears,and swirls all three materials around the triple point. In the computation, the uniformmesh with ∆x = ∆y = 1/50,1/100 and 1/200 are used, and the density distributions att = 3.5 and 5 are presented in Fig. 8. With the mesh refinement, the Kelvin-Helmholtzinstability is clearly observed.


Figure 8: Multi-material triple-point problem: the distribution of total density with the mesh size ∆x=∆y=1/50(top), 1/100 (middle), and 1/200 (bottom) at t=3.5 (left) and 5 (right).

4.5 Rayleigh-Taylor instability

Rayleigh-Taylor instability happens on an interface between fluids with different densi-ties when an acceleration is directed from the heavy fluid to the light one. The instabilityhas a fingering nature, i.e. bubbles of light fluid rising into the ambient heavy fluid andspikes of heavy fluid falling into the light fluid. This case is given to validate the perfor-mance of the current scheme with sources. The upper domain with y>0.5 is occupied bycomponent 1 and the lower domain with y<0.5 is occupied by component 2. The initialcondition of this problem is given as follows

{(ρ,U,V,p,γ)=(2,0,−0.025ccos(8πx),2y+1,1.4), y≤0.5,

(ρ,U,V,p,γ)=(1,0,−0.025ccos(8πx),y+3/2,5/3), y>0.5,

where c =√

γpρ is the sound speed. The initial condition was originally given for the

single-component flow [37]. The computational domain is [0,0.25]×[0,1]. The reflective


y

x

0.1 0.2

0.2

0.4

0.6

0.8

y

x

0.1 0.2

0.2

0.4

0.6

0.8

y

x

0.1 0.2

0.2

0.4

0.6

0.8

y

x

0.1 0.2

0.2

0.4

0.6

0.8

Figure 9: Rayleigh-Taylor instability: the distribution of total density with the mesh ∆x = ∆y = 1/640 att=1.75,2,2.25 and 2.5.

y

x

0.1 0.2

0.2

0.4

0.6

0.8

y

x

0.1 0.2

0.2

0.4

0.6

0.8

y

x

0.1 0.2

0.2

0.4

0.6

0.8

y

x

0.1 0.2

0.2

0.4

0.6

0.8

Figure 10: Rayleigh-Taylor instability: the distribution of total density with the mesh ∆x = ∆y = 1/1280 att=1.75,2,2.25 and 2.5.

boundary conditions are imposed for the left and right boundaries. At the top bound-ary, the flow values are set as (ρ,U,V,p,γ) = (1,0,0,2.5,1.4). At the bottom boundary,they are (ρ,U,V,p,γ)=(2,0,0,1,5/3). The source terms on the right side of the governingequations are S(w) = (0,0,ρ,ρV), which can be directly incorporated into the two-stagediscretization. In the computation, the uniform mesh with ∆x=∆y=1/640 and 1/1280are used. The distribution of total density with the mesh are presented at t=1.75,2,2.25and 2.5 in Fig. 9. The development of the instability from the initial condition, over thewell known mushroom shape, to the final complicated form containing secondary insta-


t

grou

th-r

ate

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

∆x=∆y=1/1280∆x=∆y=1/640

Figure 11: Rayleigh-Taylor instability: growth rate is defined as the difference of the y coordinates of the lowestfragment and the highest fragment.

bilities are observed. The growth of the instability from the beginning of the simulationto t=2.5 are presented in Fig. 11.

4.6 ZND detonation

As a direct application of the current approach, the ZND model for the detonation isstudied in this section. The study of detonation waves has been undertaken theoreti-cally and computationally for over a century, and the ZND model, which is proposed byZel’dovich, von Neumann, and Doering, has come to be a standard model. Based on thetwo-species Euler equations, the one-dimensional ZND model for the reaction flows canbe expressed as

ρ1

ρ2

ρUρE

t

+

ρ1Uρ2U

ρU2+pU(ρE+p)

x

=

−K(T)ρ1

K(T)ρ1

0K(T)Q0ρ1

,

where ρ1 and ρ2 are the densities for reactant and product, Q0 is the heat release, andK(T) is the chemical reaction rate, which is a function of temperature. In this model, thereactant is converted to the product by a one-step irreversible reactive rule governed byArrhenius kinetics. The factor K(T), which depends on the temperature, is given by

K(T)=K0Tαe−E+T,

where K0 is a positive constant and E+ is the activation energy. For simplicity, we assumethat α=0 and the gas constant R is normalized to unity. More details of the ZND solutionfor the reacting compressing Euler equations can be found in [7].


In the ZND model, it consists of a nonreactive shock followed by a reaction zone, andboth the shock and the reaction zone travel at a constant speed D. Given the specific heatratio γ and the heat release Q0, the Chapman-Jouguet value DCJ can be determined. Theoverdrive factor f , which is relates the shock speed D of a given detonation wave to theChapman-Jouguet value DCJ , is defined as

f =( D

DCJ

)2.

Here f is one of the factors that determine the stability of the detonative front. Anotherimportant parameter in the ZND solution is the half-reaction length L1/2, which is de-fined as the distance for half-completion of the reactant starting from the shock front.Usually, the reaction prefactor K0 is selected so that the half-reaction length is unity. Fromthe Arrhenius formula, the half-reaction length is defined as

L1/2=∫ 1

1/2

D−U

ZK0Tαe−E+TdZ,

where D is the speed of the shock, and U is the postshock flow speed, and Z is the massfraction, which is defined as

Z=ρ1

ρ1+ρ2.

In this section, one-dimensional ZND detonations are considered. The denotationterms S(W) = (−K(T)ρ1,K(T)ρ1,0,K(T)Q0ρ1)

T is simply considered as source terms intwo-species Euler equations. Thus, the operator L(Wn) in the semi-discretized finitevolume scheme Eq. (3.7) can be written as

L(Wn)=−1

∆x(Fj+1/2(W

n,t)−Fj−1/2(Wn,t))+S(Wn).

The two-stage temporal discretization can be applied in the ZND detonation.

4.6.1 Stable ZND detonation

In this case, the stable ZND detonation is considered [4]. The initial condition for theunburnt preshock state and the parameters are given as follows

(ρ,U,p)unburnt=(1,0,1),

γunburnt=γburnt=1.2, Q0=50, E+=50,

K0=145.68913, f =1.8.

The postshock burnt state can be obtained using the Chapman-Jouguet condition [7].According to the parameters given above, the length of the half-reaction zone L1/2 is


x

dens

ity

920 925 930 935

2

4

6

8

10

10 cells for L1/2 20 cells for L1/2exact solution

x

pres

sure

920 925 930 935

20

40

60

80

10 cells for L1/2 20 cells for L1/2exact solution

Figure 12: ZND detonation: numerical solutions (symbols) and exact solutions (solid lines) of the density ρ andpressure p for the stable ZND detonation at t=100.

time

fron

t pre

ssur

e

0 20 40 60 80 100

60

70

80

90

100

10 cells for L1/220 cells for L1/2

Figure 13: ZND detonation: the pressure history at the shock front for the stable ZND detonation with 10 and20 cells for L1/2.

normalized to unity. In the computation, the computational domain is x∈ [0,1000], andthe shock is initially positioned at x = 200. The uniform meshes with 10 and 20 pointsper L1/2 are used. Numerical and exact solutions of the density ρ and pressure p forthe stable ZND detonation are presented at t = 100 in Fig. 12. The numerical solutionsagree well with exact solutions. The pressure history at the shock front is an importantphysical quality for the ZND model, and it should exhibit small fluctuations and decayas time evolves for the stable ZND detonation. The pressure history at the shock front forthis case with 10 and 20 cells for L1/2 are presented in Fig. 13, and the stable ZND profilesare obtained as time evolves.

4.6.2 Unstable ZND detonation

In this case, the unstable ZND detonation is considered, which is first studied in [8]. Theinitial condition for the unburnt preshock state and the parameters are given as follows:


x

dens

ity

850 900 950 1000 1050 1100

2

4

6

8

10

x

dens

ity

850 900 950 1000 1050 1100

2

4

6

8

10

x

pres

sure

850 900 950 1000 1050 1100

20

40

60

80

x

pres

sure

850 900 950 1000 1050 1100

20

40

60

80

Figure 14: ZND detonation: numerical solutions of total density ρ and pressure p for the unstable ZNDdetonation at t=90 and 100.

(ρ,U,p)unburnt=(1,0,1),

γunburnt=γburnt=1.2, Q0=50, E+=50,

K0=230.75, f =1.6.

The postshock burnt state is obtained by the Chapman-Jouguet condition [7] as well.According to the parameters given above, the length of the half-reaction zone L1/2 isnormalized to unity. In the computation, the computational domain is x∈ [0,1200], andthe shock is initially positioned at x= 200. The uniform meshes with 20 points per L1/2

are used. The density ρ and pressure p for the unstable ZND detonation at t = 90 and100 are presented in Fig. 14. For the unstable detonative wave, the pressure at the shockfront presents a regular periodic pulsating detonation with a maximum shock pressure.The pressure history at the shock front for the unstable ZND detonation with 20 cells perL1/2 is presented in Fig. 15, and the local maximum and minimum pressure agree with the


x

fron

t pre

ssur

e

0 20 40 60 80

50

60

70

80

90

100

110

120reference Lian20 cells for L1/2

time

fron

t pre

ssur

e

0 20 40 60 80

50

60

70

80

90

100

110

120

reference Bourlioux20 cells for L1/2

Figure 15: ZND detonation: the pressure history at the shock front for the unstable ZND detonation with 20cells for L1/2.

reference data given by Lian and Xu [23]. Due to the ”start-up” numerical incompatibility[23], different initial data will present different pressure history profile. The pressurehistory with another initial ZND detonation wave profile is also presented in Fig. 15, andthe pressure history agrees with the reference data given by Bourlioux [4].

5 Conclusion

In this paper, a fourth-order gas-kinetic scheme is proposed for the compressible multi-component flows based on the two-stage temporal discretization and fifth-order WENOreconstruction. The equations to be solved are the multi-component Euler equations.The time accurate flux function for individual species is obtained from a simplified multi-component gas-kinetic BGK model, where the equilibrium states of the two-species sharethe common local macroscopic velocity and temperature. For each species, the time-dependent gas distribution function can be obtained from the integral solution of indi-vidual BGK model, which provides the flux and time-derivative of the flux function for


the development of two-stage fourth-order scheme. This is the first time a higher-ordergas-kinetic scheme has been developed for the multicomponent flow computation. Atthe same time, the source term effects, such as external gravitational force and chemi-cal reaction, have been explicitly included in construction of the higher-order scheme.Many numerical examples, such as the shock-bubble interaction and the detonative ZNDwaves, have been used to validate the current approach. The numerical tests clearly showthe advantages of the higher-order schemes, where more detailed flow structures can beobserved with a relative coarse mesh. The current two-stage fourth-order scheme is ef-ficient in comparison with other four-stage Runge-Kutta time stepping methods, andthe fourth-order scheme has the same robustness as the second-order one. The schemedeveloped in the current paper will be very useful for the study of flows with highlycomplicated structures under severe flow conditions, such as high speed reactive flows.

Acknowledgments

The work of L. Pan is supported by China Postdoctoral Science Foundation(2016M600065). The work of J.X. Cheng and S.H. Wang is supported by NSAF(U1630247). The work of K. Xu is supported by HKUST (IRS16SC42, PROVOST13SC01,SBI14SC11) and NSFC (91330203, 91530319).

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