Shock Waves (2019) 29:1083–1102https://doi.org/10.1007/s00193-018-0881-6

ORIG INAL ART ICLE

Limitation principle for computational fluid dynamics

C. Liu1 · G. Zhou4 ·W. Shyy2 · K. Xu1,2,3

Received: 26 May 2018 / Revised: 28 November 2018 / Accepted: 3 December 2018 / Published online: 3 January 2019© Springer-Verlag GmbH Germany, part of Springer Nature 2019

AbstractTheoretical gas dynamics uses the physical Knudsen number Knp, which is defined as the ratio of the particle mean freepath λ to the characteristic length scale L , to categorize the flow into different regimes. The Boltzmann equation is thefundamental equation for dilute gases, while the Navier–Stokes (NS) equations are used for the description of continuum flowat Knp ≤ 10−3. For computational fluid dynamics (CFD), the numerical resolution is limited by the discrete cell size and timestep. Therefore, we can define a cell Knudsen number Knc as the ratio of the particle mean free path λ to the cell size �x .In CFD, the numerical solution and the corresponding numerical flow regime are fully controlled by a numerical Knudsennumber Knn, which is a function of the physical Knudsen number Knp and the cell Knudsen number Knc. The limitationprinciple relates to the connections between Knn, Knp, and Knc. In this paper, based on the relationship between the modelingequation, cell resolution, and the physical structure thickness, we propose the division of numerical flow regimes. Accordingto the limitation principle, the range of validity of the NS equations is extended to max(Knp,Knc) ≤ 10−3. During a meshrefinement process, in some cases the NS equations alone may not be able to capture the flow physics once the large gradientsand high-frequency modes are resolved by numerical mesh size and time step. In order to obtain a physical solution in thecorresponding numerical scale efficiently, a multiscale method is preferred to identify the flow physics in the correspondingcell Knudsen number Knc, such as capturing hydrodynamic wave propagation in the coarse mesh resolution case and thekinetic particle transport in the fine mesh case. The unified gas-kinetic scheme (UGKS) is such a multiscale method forproviding continuum, near-continuum, and non-equilibrium solutions with a variation of cell Knudsen number. Numericalexamples with different physical Knudsen numbers are calculated under different cell Knudsen numbers. These results showthe mesh size effect on the numerical representation of a physical solution. In comparison with the NS and direct Boltzmannsolvers, the multiscale UGKS is able to capture flow physics in different regimes seamlessly with a variation of numericalresolution.

Keywords Cell Knudsen number · Grid refinement · Multiscale modeling · Non-equilibrium flow

Communicated by C.-H. Chang.

B K. Xumakxu@ust.hk

C. Liucliuaa@connect.ust.hk

G. Zhouzgz@pku.edu.cn

W. Shyyweishyy@ust.hk

1 Department of Mathematics, Hong Kong University ofScience and Technology, Kowloon, Hong Kong, China

2 Department of Mechanical and Aerospace Engineering, HongKong University of Science and Technology, Kowloon,Hong Kong, China

1 Introduction

For theoretical gas dynamics, the flow regime is definedaccording to the physical Knudsen number Knp, which isdefined as the ratio of particle mean free path λ to the char-acteristic scale of flow structure L , i.e., Knp = λ/L . Thephysical flow regimes are classified into rarefied (Knp ≥ 1),transitional (0.1 < Knp ≤ 1), near-continuum (0.001 <

Knp ≤ 0.1), and continuum (Knp ≤ 0.001). For compu-tational fluid dynamics (CFD), not only the flow physics,

3 HKUST Shenzhen Research Institute, Shenzhen 518057,China

4 College of Engineering, Peking University, Beijing 100871,China

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but also the limited numerical resolution needs to be prop-erly handled in the design of numerical methods [1]. Withthe numerical scale of cell size and time step included, thenumerical flow regime in the CFD solutionwill not be simplydetermined by Knp. The cell Knudsen number Knc needs tobe introduced, which is defined as the ratio of the mean freepath to cell size, i.e., Knc = λ/�x . The numerical Knudsennumber Knn, the ratio of the mean free path to the numeri-cal characteristic length scale Ln, i.e., Knn = λ/Ln, will bea function of the physical Knudsen number and cell Knud-sen number. The numerical solution and the correspondingregime will be directly determined by Knn. As a result, thehydrodynamic equations, such as the Navier–Stokes equa-tions and the Euler equations, are not only applicable in thephysical continuum regime, but also provide approximatesolutions for highly non-equilibriumflowfields once the non-equilibrium flow structure cannot be resolved by the meshsize and time step. In other words, the numerical resolutionaffects the numerical flow regime in the CFD solution.

The hydrodynamic equations have been widely used inthe study of turbulent flow and shear instability. However,with a continuous mesh refinement, such that the boundarylayer mesh in the hypersonic flow may go to the order ofparticle mean free path, we may need to obtain a more phys-ically consistent solution under such a refined mesh. In ourprevious study, for a flow field with high-frequency modesor large gradients, the local Knudsen number can becomelarge and non-equilibrium flow dynamics may emerge [2].For suchflowstructures, the physical flowfield shows a largerdissipation than the numerical NS flow field. During a meshrefinement process, in the direct numerical simulation (DNS)of turbulence flow and in the study of flow instability, high-frequency modes will appear in the numerical flow field. Aless dissipative hydrodynamic solver may result in a morechaotic flow field, which deviates from the real flow physics.Besides the modeling physics, the numerical algorithm willalso affect the numerical solution. The current numerical NSsolutions are mostly obtained through an operator-splittingapproach, i.e., theRiemann solver for the inviscid flux and thecentral difference for the viscous terms. The physical consis-tency in the operator-splitting approach for the NS equationsis questionable in the mesh refinement process as the advec-tion and dissipation merge in the small size or kinetic meanfree path scale [3]. The gas-kinetic scheme (GKS) constructsthe numerical flux from the kinetic equation and calculatesthe inviscid flux and viscous terms in a coupled way [4].Compared to the splitting approach, GKS is more physicallyreliable, especially under mesh refinement. However, insteadof sticking to solving the hydrodynamic equations, a multi-scale numerical method is required to capture flow physicsin the corresponding mesh resolution.

In CFD, we have the physical scale as well as the numer-ical scale. The physical scale is the scale of flow structure,

and the numerical scale is the scale of cell size and time step.Multiscale modeling, or more precisely multiple physical–numerical scale modeling, requires the recovery of correctflow physics on multiple physical scales and variable numer-ical scales. This means that a multiscale scheme shouldrecover the kinetic Boltzmann equation when the mesh sizeis on the order of the mean free path and recover the hydro-dynamic equations when the mesh size is much larger thanthe mean free path. Numerical multiscale modeling is morecomplicated than theoretical fluid dynamics. For theoreti-cal fluid dynamics, we do not have the numerical scale ofcell size and time step, and all discussion is based on thescale of physical flow structure in a continuous space andtime. For example, the Boltzmann equation is constructedon a small physical scale, such as the mean free path andparticle collision time, and it can recover large-scale flowphysics through the time accumulating solution to the largescale. Theoretically, the Boltzmann equation can also givelarge-scale hydrodynamic equations by asymptotic analysis[5]. Unfortunately, the theoretical analysis cannot be simplygeneralized to a discrete space and time in CFD. A numeri-cal scheme constructed on the kinetic scale of particle meanfree path, such as the direct Boltzmann solver and the directsimulationMonte Carlo (DSMC)method, may not be able togive accurate physical solution with a large mesh size scale,such as with a cell size being much larger than the parti-cle mean free path [6]. In order to recover large-scale flowphysics, the direct Boltzmann solver and DSMC still requirea mesh size on the order of the mean free path and obtain thelarge-scale solution through time accumulation. To captureflow physics with a changeable numerical scale, a multiscalenumerical scheme, such as the unified gas-kinetic scheme(UGKS), needs to be constructed. The UGKS is developedaccording to the discretized numerical scale and captures thesolutions in all flow regimes through the variation of the cellKnudsen number [4].

The paper is organized as follows. In Sect. 2, we studythe limitation principle of the flow regime. For the first time,we propose the division of the numerical flow field, based onwhich we study the numerical applicability of NS solvers.We also study the change of the numerical regime in a meshrefinement process. Themultiscale numerical schemeUGKSis introduced in Sect. 3. In Sect. 4, numerical tests are cal-culated to validate our argument and measure the deviationbetween NS solutions and multiscale UGKS solutions. Sec-tion 5 is the conclusion.

2 The limitation principle in flow simulation

This work follows our previous work on studying the validregime of fluid dynamic equations and numerical schemes.The previous work shows that the Navier–Stokes equations

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Limitation principle for computational fluid dynamics 1085

are valid for low-frequency continuum flow [2], and a directkinetic solver is only valid on the mean free path numericalscale [6]. In this work, we will study the numerically validregime for a NS solver and show the importance of the multi-scale property in a numerical scheme to capture the physicalsolution in the mesh refinement process. The numerical flowregime in CFD is characterized by the numerical Knudsennumber Knn, which depends on the physical Knudsen num-ber Knp, and the cell Knudsen number Knc. In this section,we study the numerical limitation principle through a sim-ple flow field and propose a general relationship between thethree Knudsen numbers.

Consider a simple physical flow field, whose normalizeddensity distribution follows the smooth step function

ρ(x) =⎧⎨

⎩

0 x ≤ − 1,erf(

√2π arctanh(x) + 1)/2 − 1 < x ≤ 1,

1 x > 1,

(1)

where erf(x) is the error function

erf(x) = 2√π

∫ x

0exp(− t2)dt,

and arctanh(x) is the inverse hyperbolic tangent functionarctanh(x) = 1

2 ln1+x1−x . We use the van Leer limiter to recon-

struct this physical flow field under three different numericalcell sizes �x = 0.05, 0.5, and 5. The physical Knudsennumber at x = 0 is

Knp = λ

ρ

∂ρ

∂x

∣∣∣∣x=0

= 2√2λ, (2)

where the physical characteristic length is defined by thelength scale of the flow structure ρ/|∇ρ| and the numer-ical characteristic length is defined as the length scale ofthe reconstructed flow structure ρn/|∇ρn|, and therefore, thenumerical Knudsen number is

Knn = λ

Ln= λ

ρn

∂ρn

∂x. (3)

The values of the three Knudsen numbers under three numer-ical resolutions are shown in Table 1. And the physical flowstructure and the numerical reconstructed flow structure are

Table 1 Values of three Knudsen numbers

Knudsen number �x = 0.05 �x = 0.5 �x = 5

Knp 2√2λ 2

√2λ 2

√2λ

Knn 2.8λ 1.8λ 0.2λ

Knc 20λ 2.0λ 0.2λ

shown in Fig. 1. It can be observed from the results that therelationship between the three Knudsen numbers follows asoft-minimum function [7],

Knn = softmin(Knp,Knc), (4)

which means that in the well-resolved region with Knp �Knc, the numerical Knudsen number converges to the phys-ical Knudsen number; when the flow field is under resolvedwith Knp � Knc, the numerical Knudsen number is con-trolled by the cell Knudsen number. For a partially resolvedregion, the numerical Knudsen number is close to andmaybeless than the physical and cell Knudsen number. However,the exact relationship between the three Knudsen numbersdepends on the physical flow field structure, numerical res-olution, and reconstruction technique. Based on the aboveobservation, we propose the following expression

Knn = 10ln(exp(lg(Knp))+exp(lg(Knc))). (5)

According to the Knudsen number relationship in the aboveequation, we propose the division of numerical flow regimesas shown in Fig. 2. The conventional division of the physicalflow regimes is given on the x-axis, which stands for Knp.The y-axis stands forKnc.Apoint P(x, y)on theflow regimeplane stands for the numerical solution of a flow field withKnp = x under a numerical resolution with Knc = y. Everypoint P on the numerical flow regime plane corresponds toa numerical Knudsen number

Knn,P = 10ln(exp(lg(x))+exp(lg(y))), (6)

based on which the numerical flow regime is divided. Thenumerical flow regime division shows that the numerical NSsolver is applicable on the numerical continuum regime withKnp ≤ 10−3 or Knc ≤ 10−3. The valid regime of numer-ical methods is given in Table 2. It can be observed fromthe numerical flow regime diagram Fig. 2 that in a meshrefinement process, the numerical flow regime will even-tually converge to the corresponding physical flow regime.Therefore, in order to capture the physically consistent flowfield and to obtain converged numerical solutions, a multi-scale numerical scheme is required.

3 Multiscale method: unified gas-kineticscheme

In this section, following the direct modeling methodology,we present amultiscale UGKS that couples the particle trans-port and collision based on the local flow physics on the timestep scale [4,8]. The UGKS is constructed starting from thekinetic equation

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Fig. 1 Diagram for physicalflow field (solid line) andreconstructed numerical flowfield (dashed line) for awell-resolved smooth flow (a), apartially resolved flow (b), andan under-resolved flow withlarge gradient (c)

–0.10 –0.05 0.05 0.10

0.2

0.4

0.5

0.6

0.7

0.8Well Resolved Flow

Physical flow structure

Numerical reconstructed flow

a

–1.0 –0.5 0.5 1.0

0.2

0.4

0.6

0.8

1.0

Partially Resolved Flow

b–10 –5 5 10

0.2

0.4

0.6

0.8

1.0

Under Resolved Flow

c

Flow Regimes of Physical and Numerical Flow Field

Numerical Rarefied Regime

(Dotted line )

Numerical Transitional Regime

(Dashed line ----)

Numerical Continuum Regime

(Solid line )

Mesh R

efinement P

ath

Partially-Resolved

Partially-Resolved

Partially-Resolved

Under-Resolved

Under-Resolved

Under-Resolved

Fully-Resolved

Fully-Resolved

FullyR

esolved

10-2 10-1 100 101 102 103 104 105Kn-1

P

x

x >

x

>>

Kn-1 c

Physical Rarefied Regime Physical Transitional Regime Physical Continuum Regime

RegimeBoundaries

RarefiedRegime

TransitionalRegime

ContunuumRegime

Fig. 2 Division of flow regimes for physical and numerical flow fields.Horizontal axis is the reciprocal physical Knudsen number and verticalaxis is the reciprocal cell Knudsen number. Both axes use a logarithmicscale. A point P(x, y) on this plane stands for the numerical solution of

a flow field with Knp = x under a numerical resolution with Knc = y.The mesh refinement path shows the numerical regime changing duringa mesh refinement process

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Limitation principle for computational fluid dynamics 1087

Table 2 Valid regime of numerical methods

Numerical method Valid flow regime

Navier–Stokes solver Knp ≤ 10−3 or Knc ≤ 10−3

Direct Boltzmann solver Knc ≥ 1

UGKS All flow regime

∂ f

∂t+ v · ∇x f = Q, (7)

where f (x, t, v) is the velocity distribution function and Qis the collision term. The phase space is discretized as

X × V =∑

i, j

�i j =∑

i, j

�xi × �v j ,

where �xi is the i th numerical cell in physical space, �vi

is the j th numerical cell in velocity space, and �i j is theintersection of �xi and �vi , i.e., �i j = �xi × �v j . Theevolution of the cell-averaged distribution function

fi j = 1

|�i j |∫

�i j

f (x, t, v)dvdx (8)

is coupled with the evolution of cell-averaged macroscopicconservative variables

Wi = 1

|�i |∫

�i

⎛

⎝ρ

ρUρE

⎞

⎠ dx. (9)

Note the above fi j andWi are cell-averaged variables with avariable cell size. The governing equation for fi j is differentfrom the Boltzmann equation, which is valid in the kineticscale only, such as the scale less than the particle mean freepath.

The evolution equation for the cell-averaged velocity dis-tribution function is

f n+1i j = f ni j − 1

|�i |∫ tn+1

tn

∮

∂�i

v · n f∂�i (t, v j )dsdt

+βnQn + (�t − βn)Qn+1, (10)

and the evolution equation for the conservative variable is

Wn+1i = Wn

i − 1

|�i |∫ tn+1

tn

∮

∂�i

ψv · n f∂�i (t, v)dsdt, (11)

where ψ = (1, u, v, w, 12v

2)T is the conservative moments.The above two equations are based on the direct modelingof conservation laws in a physical scale of cell size and time

step. In the following formulation, we choose Qn to be theBoltzmann collision termand Qn+1 to be theShakhovmodel.The explicit Boltzmann collision term is calculated by thefast spectral method [9], and the evolution equation for thedistribution function then becomes

f n+1i j =

(

1 + �t − βn

τ n+1

)−1

×[

f ni j − 1

|�i |∫ tn+1

tn

∮

∂�i

v · n f∂�i (t, v j )dsdt

+βnQ( f n, f n) + �t − βn

τ n+1 f +(n+1)i j

]

, (12)

where the post-collision distribution function f + is definedas

f + = g

(

1 + (1 − Pr)c · q(

c2m

kBT− 5

)m

5pkBT

)

, (13)

with the Prandtl number Pr, the peculiar velocity c, and theheat flux q. The detailed formulation of βn can be found in[8]. The local Maxwellian distribution g(x, t, v) is obtainedfrom the local conservative flow variables. The numericalfluxes for the distribution function and conservative variablesare calculated from the integral solution f∂�i (t, v) of thekinetic equation (7) with the Shakhov collision term,

f∂�i (t, v) = 1

τ

∫ tn+t

tnf +(x′, t ′, v)e−(t−t ′)/τdt ′

+ e−t/τ f0(x∂�i − vt, v), (14)

where x′ = x∂�i − u(t − t ′) is the particle trajectory and f0is the distribution function at time tn . Based on the integralsolution (14), the UGKS couples two effects in the particletransport for the numerical flux evaluation. The particle freetransport and collision are connected based on the ratio oflocal time step to the particle collision time, which is a func-tion of local cell Knudsen number Knc. The UGKS recoversthe direct kinetic solver in the numerical rarefied regime andrecovers the hydrodynamic solvers in the numerical contin-uum regime. With the change of numerical scale, the UGKScan always capture the numerically resolved flow physics.Compared to the direct Boltzmann solver, the UGKS is moreefficient, since the numerical scale of UGKS is not restrictedto be on the mean free path level. Compared to the hydrody-namic solver, the UGKS is applicable in all numerical andphysical flow regimes. UGKS can provide reliable solutionson different numerical scales and is capable of providinga physically consistent cell-converged solution in a meshrefinement process.

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4 Numerical experiments

In this section, five numerical experiments will be used todemonstrate the multiscale solutions under different meshresolutions. The examples include the density sine wavepropagation, 1-D shock–interface interaction, 2-D shock–interface interaction, shock–bubble interaction, and force-driven Poiseuille flow. For the density wave propagation andthe shock–interface interaction problems, the NS solutionsare comparedwith themultiscale results under differentmeshresolutions. It can be observed that when the cell Knudsennumber is small, the NS solutions agree with the multiscaleresults, which means that the NS equations give reasonablesolutions on large space and time scales. Relatively large dif-ferences between the NS and multiscale solutions appear asthe cell Knudsen number increases in the mesh refinementprocess, and the NS equations do not provide an accuratephysical solution in such a process. The shock–bubble inter-action problem shows that for a fixed cell resolution, as thelocal Knudsen number increases, the single-scale NS solu-tions deviate from the multiscale solutions. Even for smallKnudsen numbers, there are still deviations between the NSand multiscale solutions in the high-order moment, suchas the heat flux inside a shock layer. The study of force-driven Poiseuille flow concludes that conventional heat fluxmodeling, i.e., Fourier’s law, is incomplete for a flow underthe influence of a large external force field. The heat fluxinduced by the external force qF

x can be on the same orderas the Fourier heat flux induced by the temperature gradi-ent in the cases of Knp ≥ 10−2 for the Knudsen numberand Fx ≥ 10−2 for the normalized external force. With thedecreasing Knudsen number and external force, the force-induced heat flux qF

x decays linearly with respect to theKnudsen number and external force qF

x ∝ Knp · Fx . Even in

the continuum regime, the Navier–Stokes–Fourier equationsare not adequate to provide a valid description of physicalphenomena. These examples show the multiple scale natureof gas dynamics and the necessity of a multiscale method.

4.1 Density wave propagation

In the next three test cases,we compare theNS andmultiscalesolutions under different cell Knudsen numbers. First, westudy the propagation of a density wave in argon gas. Theinitial condition is set as

(ρ,U , p) = (5 sin(30x) + sin(x) + 10, 1.0, 0.5).

The computational domain is [0, 2π ] with a periodic bound-ary condition. The variable hard sphere (VHS) model is usedwith the viscosity temperature dependency index ω = 0.81.The Knudsen number is set to be Knp = 5 × 10−2. Thevelocity space is truncated from − 5 to 5 with 200 veloc-ity grids to minimize the velocity integration error. A coarsemesh with 150 cells and a fine mesh with 1000 cells in thephysical computational domain are adopted. The UGKS isused to obtain the physically consistent cell-averaged solu-tions. The NS solutions are obtained by the GKS [4]. Forany quantity Q, the deviation between the UGKS and NSsolutions is calculated as

Dev = ‖QNS − QUGKS‖L2

‖QUGKS‖L2. (15)

The initial density distribution is shown in Fig. 3, and theinitial condition is numerically resolved on bothmeshes. Thedensity distribution and the cell Knudsen number at t = 4πare shown in Figs. 4 and 5, and the deviations are listed inTable 3.

0 1 2 3 4 5 6x

4

6

8

10

12

14

16

Den

sity

Initial density distribution ( =2 /150)

a

0 1 2 3 4 5 6x

4

6

8

10

12

14

16

Den

sity

Initial density distribution ( x=2 /1000)

b

Fig. 3 The initial density distribution of the density wave propagation on coarse mesh (a) and on fine mesh (b)

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Limitation principle for computational fluid dynamics 1089

0 1 2 3 4 5 6x

9

9.5

10

10.5

11

Den

sity

Density distribution at t=4 ( x=2 /150)

NS (GKS)Multiscale (UGKS)

Dev=7.1514 10-8

a

0 1 2 3 4 5 6x

0.02

0.021

0.022

0.023

0.024

0.025

0.026

0.027

0.028

0/x

Local Knudsen number at t=4

NS (GKS)Kinetic (UGKS)

b

Fig. 4 The density distribution at t = 4π on coarse mesh (a), and the cell Knudsen number at t = 4π on coarse mesh (b)

0 1 2 3 4 5 6x

9

9.5

10

10.5

11

Den

sity

Density distribution at t=4 ( x=2 /1000)

NS (GKS)Kinetic (UGKS)

1.4 1.5 1.6 1.7

10.92

10.94

10.96

10.98

11

11.02

11.04

Dev=1.2902 10-6

a

0 1 2 3 4 5 6x

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0/x

Local Knudsen number at t=4

NS (GKS)Multiscale (UGKS)

b

Fig. 5 The density distribution at t = 4π on fine mesh (a), and the cell Knudsen number at t = 4π on fine mesh (b)

Table 3 Parameters underdifferent meshes, deviationbetween NS and multiscalesolution for the density wavepropagation

Density wave propagation Knc τ/�t Dev

Fine mesh ∼ 2 × 10−1 ∼ 3 × 10−2 1.29 × 10−6

Coarse mesh ∼ 2 × 10−2 ∼ 3 × 10−3 7.15 × 10−8

On a relatively coarse mesh, the NS solution agrees wellwith the UGKS solution. On such a numerical cell sizescale, the cell Knudsen number is relatively small Knc ∼1 × 10−2, and the cell-averaged velocity distribution isclose to the local equilibrium. Under the limitation prin-ciple, the numerical solution is the one corresponding tothe cell Knudsen number. When the grid points in physi-cal space increase to 1000, the local non-equilibrium effectin the high-frequency mode appears. Under such a resolu-tion, the local flow physics is resolved by the mesh size,

and the numerical Knudsen number for the solution is thesame as the physical Knudsen number. The NS solutiondeviates from the physical one. The results show that thephysical solution is more dissipative than the NS solutiondue to the inclusion of particle transport mechanisms in sucha scale. So, even with a mesh-converged solution, the NSsolution still does not capture the physical solution in thecorresponding resolution. The UGKS can accurately recoverthe attenuation of high-frequency waves in the experiments[10].

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1090 C. Liu et al.

4.2 Shock–interface interaction

In the shock–interface interaction case, non-equilibriumeffect beyond the NS modeling will emerge when the shockwave impinges upon the interface with large gradients in theflow variables. The NS solutions will be compared with themultiscale solutions under different cell resolutions. On therelative coarse mesh case, the cell Knudsen number is small,and the NS solver will have the same results as the multi-scale ones, because under the limitation principle this is thebest result a multiscale method can have under such a meshresolution. When the cell size decreases and cell Knudsennumber increases, the NS solutions deviate from the multi-scale ones, which is due to the limitation of NS modeling incapturing the non-equilibrium phenomena.

-200 -100 0 100 200x

1

1.5

2

2.5

3

3.5

y

Initial density distribution

Fig. 6 Initial condition of the shock–contact interaction

Table 4 Parameters under different meshes, deviation between NS andmultiscale solution for the shock–interface interaction

Shock–interface interaction Knc τ/�t Dev

Fine mesh ∼ 5 ∼ 50 3.24 × 10−4

Coarse mesh ∼ 0.5 ∼ 5 1.47 × 10−4

For the 1-D shock–interface interaction problem, theworking gas is argon modeled by VHS. As shown in Fig. 6,initially a normal shock wave with M = 3.0 is situated withits density barycenter at x = 0. The upstream initial condi-tion is set as

(ρ, u, T ) ={

(1.0, 2.74, 1.0) x ≥ − 210,(2.0, 2.74, 0.5) x < − 210.

The x-axis is normalized by the shock upstream mean freepath, which means the Knudsen number is one. The NS solu-tions are compared with the multiscale solutions under a finemesh �x = 0.2 and a coarse mesh �x = 2. For UGKS, thevelocity domain is [− 8, 8] covered by 100 velocity points.The solutions at t = 80 are shown in Fig. 7 when the inter-face moves to the origin and interacts with the shock wave.As shown in Table 4, a larger deviation is observed under afine mesh since the NS equations are not able to recover thephysical structure of the shock wave, which is supposedlyresolved in such a fine mesh case. Under a coarse mesh, theshock wave structure is not resolved by the numerical resolu-tion, and the physical shock wave is replaced by a numericaldiscontinuity, such as the shock capturing scheme. There-fore, both the NS and the multiscale method give the sameresult under the limitation principle.

05005-x

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Den

sity

Density distribution ( x=0.2 )

NS solution (GKS)Kinetic solution (UGKS)

Dev=3.24 10-4

a

05005-x

1.5

2

2.5

3

3.5

4

4.5

Den

sity

Density distribution ( x=2 )

NS solution (GKS)Multiscale solution (UGKS)

Dev=1.47 10-4

b

Fig. 7 Density of the shock–contact interaction at t = 80. The solutions of NS and multiscale method are compared under a fine mesh (a) and acoarse mesh (b)

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Limitation principle for computational fluid dynamics 1091

X

Y

-0.2 -0.1 0 0.1 0.2

-0.4

-0.2

0

0.2

0.4 54321

a

-0.2 -0.1 0 0.1 0.2x

2

3

4

5

6

7

8

9

10

11

Den

sity

Density distribution along y=0

NS SolutionMultiscale solution

-0.12 -0.11 -0.1 -0.09 -0.08 -0.07

3

4

5

6

7

8

9

10

11

Dev=1.6 10-3

Knc 1

b

-0.5 0 0.5 1 1.5x

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Den

sity

Density distribution along y=0

NS solution (GKS)Multscale solution (UGKS)

-0.15 -0.1 -0.05 0

3

3.5

4

4.5

5

5.5

6

Dev=3.15 10-5

Knc 0.1

c

Density

Fig. 8 Left figure shows the initial condition of RM instability (a). Right figures show the density distribution along y = 0 at t = 0.026. The NSsolutions and multiscale solutions are compared under a fine mesh (b) and a coarse mesh (c)

4.3 Richtmyer–Meshkov instability

The Richtmyer–Meshkov (RM) instability is caused by ashock wave passing through a contact interface. Vorticeswill be generated during the passage of the shock waveand will trigger interface instability. In the following, the2-D RM instability is numerically studied on different timescales. The interaction between shock and interface is stud-ied on the time scale t ∼ 10τ , and the development ofthe instability is studied on the time scale t ∼ 105τ . TheNS solutions are calculated by GKS, while the UGKS pro-vides the multiscale solutions. At the starting time, the initialcondition is shown in Fig. 8a. The computational domain is[− 0.2, 0.2] × [− 0.5, 0.5], the top and bottom are imposedwith periodic boundary conditions, and the left and rightare imposed with inflow/outflow boundary conditions. The

working gas is argon modeled by VHS, and the Knudsennumber is Knn = 5 × 10−3. Initially, a shock wave withM = 3.0 is located at x = − 0.15, and a contact interface islocated at x = 0 as

(ρ, u, T ) ={

(1.0, 0, 1.0) x < sin(2π y + 1.5π),

(5.0, 0, 0.2) x > sin(2π y + 1.5π).

The shock wave is pre-calculated, and the fully devel-oped shock structure is used as the initial condition. ForUGKS, the velocity domain is [− 5, 5] × [− 5, 5], and48 × 32 velocity grids are used. The NS and multiscalesolutions are calculated under a fine mesh 240 × 600 withKnc ∼ 1 and a coarse mesh 40 × 100 with Knc ∼0.1. The density distribution at t = 0.026 ≈ 18τ alongy = 0 is shown in Fig. 8b, c. Under the coarse mesh,

123

1092 C. Liu et al.

Table 5 Parameters under different meshes, deviation between NS andmultiscale solution for the Richtmyer–Meshkov instability

Richtmyer–Meshkov instability Knc τ/�t Dev

Fine mesh ∼ 1 ∼ 50 1.6 × 10−3

Coarse mesh ∼ 0.1 ∼ 5 3.15 × 10−5

the NS solutions agree well with the multiscale solutionswith Dev = 3.15 × 10−5, while a large deviation canbe observed under a fine mesh with Dev = 1.6 × 10−3.The data are presented in Table 5. The UGKS solutionis limited by the limitation principle in the coarse meshcase.

Next, we calculate the development of the interface insta-bility for a longer time scale. The computational domain is[− 0.5, 1.0] × [− 0.5, 0.5] covered by 150× 100 cells in thephysical domain. TheKnudsen number is Knp = 1.0×10−4,and the Mach number of the shock wave is M = 1.3. Thecontact discontinuity located at x = 0 is

(ρ, u, T ) ={

(1.0,− 0.255, 1.0) x < sin(2π y + 1.5π),

(2.0,− 0.255, 0.5) x > sin(2π y + 1.5π).

The solutions at t = 18.6 ≈ 2.4 × 105τ are shown inFigs. 9 and 10, where the density, cell Knudsen number, vor-ticity magnitude, and streamline are plotted together with theNS solution (upper) and the UGKS solution (lower). On suchlarge space and time scales, the NS solutions agree well withthe multiscale solutions.

4.4 Shock–bubble interaction

In this section, we study the process of a shock wave inter-acting with a dense cold bubble to show the capability of

NS equations in describing the flow with large gradients.The initial condition is shown in Fig. 11. A shock wave issituated with its density barycenter at x = − 1.0, travel-ing in the positive x-direction into a flow field at rest with(ρ,U , p) = (1.0, 0, 0.5). Around (x, y) = (0.5, 0), a densecircular bubble is placed with constant pressure p = 0.5.The computational domain is [− 2, 3] × [− 1, 1], and theinflow/outflow boundary is imposed. The velocity space hasa domain [− 8, 8] × [− 8, 8] covered by 100 × 100 velocitygrid points.

First, we set the Mach number of the shock wave to beM = 1.3 and set the density of the bubble to be

ρ(x, y) = 1 + 0.4 exp(− 16(x2 + y2)). (16)

TwodifferentKnudsen numbers are considered relative to thebubble radius, namely Knp = 1.0 × 10−4 and Knp = 0.3.For Knp = 1.0 × 10−4 case, 250 × 100 cells are used inthe physical domain. The solutions of UGKS at t = 1.3are shown in Fig. 12 when the shock wave is about topass the bubble. This situation shows large density andtemperature gradients in the shock layer which can leadto strong non-equilibrium flow behavior. Figure 13 showsthe comparison between the NS and UGKS solutions, andthe agreements in the density and temperature profiles havebeen confirmed at a small cell Knudsen number. For high-order moments, such as the x-direction heat flux, a slightdeviation still can be observed inside the shock–bubbleinteraction region. For Knp = 0.3, the physical domain isdivided into 100 × 40 mesh points. Solutions at t = 1.3are shown in Fig. 14. For this case, the cell Knudsennumber is relatively large, and the non-equilibrium flowcan be resolved. The NS and the multiscale solutions indensity and temperature deviate from each other in thiscase.

Density Distribution

10.50-0.5

0

0.1

0.2

0.3

0.4

0.5

1.6

1.8

2

2.2

2.4

2.6

2.8

3

10.50-0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

Navier-Stokes Solution(GKS)

Multiscale Solution(UGKS)

a

Cell Knudsen Number10.50-0.5

0

0.1

0.2

0.3

0.4

0.5

10.50-0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

2

2.5

3

3.5

4

10-3

Navier-Stokes Solution(GKS)

Multiscale Solution(UGKS)

b

Fig. 9 The density (a) and cell Knudsen (b) at t = 18.6, with the NS solution (upper) and the multiscale solution (lower)

123

Limitation principle for computational fluid dynamics 1093

Vorticity Magnitude10.50-0.5

0

0.1

0.2

0.3

0.4

0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

10.50-0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

Multiscale Solution(UGKS)

Navier-Stokes Solution(GKS)

a

10.50-0.5

0.1

0.2

0.3

0.4

0.5

Streamline

10.50-0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

Navier-Stokes Solution(GKS)

Multiscale Solution(UGKS)

b

Fig. 10 The vorticity (a) and streamline (b) at t = 18.6, with the NS solution (upper) and the multiscale solution (lower)

Fig. 11 Initial condition for theshock–bubble interactionprocess

10.8

0.60.95-2 0.4

1

-1.5 0.2

1.05

-10-0.5

1.1

0 -0.2

Initial density distribution

1.15

0.5 -0.4

1.2

1 -0.61.5

1.25

2 -0.8

1.3

2.5 -13

1.351.4

1.45

Next, we increase the gradients of the flow variables byincreasing the Mach number of the shock wave to M = 2.0and a bubble density distribution

ρ(x, y) = 1 + 1.5 exp(− 16(x2 + y2)). (17)

The solutions at small Knudsen number Knp = 1.0 × 10−4

and at t = 1.0 are shown in Fig. 15. For this case, theReynolds number relative to the bubble radius is greaterthan 103. The NS density and temperature profiles agreewell with the multiscale solutions, which have a small incre-ment in the deviation in comparison with M = 1.3 andKnp = 1.0 × 10−4 case. For the heat flux, a large devia-tion can be clearly observed, especially for the x-directionalheat flux with 43% deviation. When the Knudsen number is

increased to Knp = 0.3, a relatively large deviation in allflow variables can be observed in Fig. 16.

For the shock–bubble interaction at a relatively large cellKnudsen number, the flowphysics is not inNS regime and theNS equations do not provide accurate solutions. At a smallcell Knudsen number, the low-order moments from the NS,such as density and temperature, agree with the multiscalesolution due to the cell-averaged effect under the limitationprinciple. However, even at a small cell Knudsen number,the high-order moments, such as the heat flux, will not bewell predicted by NS equations at the region with large flowgradient. Therefore, even in the near-continuum flow regime,the multiscale numerical method can provide more accuratephysical solutions under the same cell resolution as the NSsolver.

123

1094 C. Liu et al.

Fig. 12 Two-dimensional plotsof the fields of density,temperature, and heat fluxes att = 1.3

11-2

Density

0

2

02

3

-14

10.5-2

0 0

1

2-14

1.5

1-1-2

00

X-direction heat flux

0

10-3

2-14

1

1-2-2

00

Y-direction heat flux

0

10-4

2-14

2

Temperature

4.5 Force-driven Poiseuille flow

In the following, through force-driven Poiseuille flow, westudy the capability of the NS equations in the descriptionof non-equilibrium flow physics, even in the continuum flowregime. This example clearly indicates the multiscale natureof the flow physics and theNSmodeling is not fully completefor the physical solution.

Poiseuille flow is a flow confined between two parallelisothermal plates. A constant body force is applied in thex-direction along the channel. A periodic boundary conditionis taken in the flow direction. The simulated gas is a hardsphere monatomic gas. The initial condition is set as ρ0 =1.0, ux = 0, T = 1. In the following, the solutions at avariety of Knudsen numbers and an external force will beobtained. The Knudsen number is defined by the ratio of themean free path to the wall distance, Knp = λ/Ly . The resultsfrom DSMC and NS have been previously presented [11].

The first test has a Knudsen number Knp = 0.1 and accel-eration Fx = 1.26 × 10−1. The physical domain is dividedinto 51 cells in the y-direction, and velocity domain [− 6, 6]3is divided by32 velocity points in x, z-direction and 64 veloc-ity points in y-direction. For this set of parameters, the timestep is about 100 times smaller than the local collision time;therefore, the Boltzmann collision term takes effect in (10).It can be observed in Figs. 17, 18, and 19 that the UGKSsolutions agree well with DSMC results in the prediction ofdensity, velocity, temperature, stress, and heat flux. The NS

solution is calculated byGKSwith a first-order slip boundarycondition

uslip = αλ∂u

∂ y, Tslip = β

2γ

γ + 1

λ

Pr

∂T

∂ y,

withα = 1.11 andβ = 1.13. In theflow resolved calculation,the non-equilibrium effect is not well captured by the NSequations at all, especially the stress and heat flux.

Next, we reduce theKnudsen number and external force toFx = Knp = 5×10−2 and Fx = Knp = 2×10−2. For smallKnudsen number and small accelerations, an asymptoticsolution from the Bhatnagar–Gross–Krook (BGK) equationis derived by Aoki et al. [12]. The UGKS solutions are com-pared with the second-order asymptotic solutions and NSsolutions.As shown inFig. 20, the asymptotic solution agreeswellwith theDSMCsolution, especially forKnp = 2×10−2,while the NS solutions always predict zero heat flux in theflow direction.

Next, we study the Poiseuille flow under a relatively largeexternal force. The calculation is performedwith three sets ofparameters: Knp = 0.1, Fx = 1, Knp = 5×10−2, Fx = 0.5,and Knp = 2 × 10−2, Fx = 0.2. The x-directional fluxresults are shown in Fig. 21. It is shown that the heat fluxinduced by the external force is proportional to the first orderof the Knudsen number, which is on the same order as theFourier heat flux induced by the temperature gradient. There-fore, with the external force field, the Fourier heat flux in the

123

Limitation principle for computational fluid dynamics 1095

Cell Knudsen number

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.5

2

2.5

310-3

a

Heat flux

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.510-4

b

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Den

sity

Density distribution along x=0

NS (GKS)Kinetic (UGKS)

Dev=1.14 10-8

c

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

0.7

0.8

0.9

1

1.1

1.2

1.3

Tem

pera

ture

Temperature distribution along x=0

NS (GKS)Multiscale (UGKS)

Dev=8.92 10-9

d

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

X-d

irect

ion

heat

flux

10-4 qx distribution along x=0

NS (GKS)Kinetic (UGKS)

Dev=0.0534

e

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-1

0

1

2

3

4

5

6

Y-d

irect

ion

heat

flux

10-5 qy distribution along y=-0.4

NS (GKS)Kinetic (UGKS)

Dev=0.0026

f

Fig. 13 Results of the shock–bubble interaction with M = 1.3 and Knp = 1.0 × 10−4 at t = 1.3. Top two figures show the cell Knudsennumber (a) and total heat flux (b). c–f Comparison between NS solution profiles (solid line) and UGKS solution profiles (dash line)

123

1096 C. Liu et al.

Cell Knudsen number

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

a

Heat flux

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

b

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Den

sity

Density distribution along y=0

NS (GKS)Kinetic (UGKS)

Dev=8.59 10-5

c

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

1

1.05

1.1

1.15

1.2

1.25

1.3

Tem

pera

ture

Temperature distribution along y=0

NS (GKS)Multiscale (UGKS)

Dev=4.16 10-5

d

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

X-d

irect

ion

heat

flux

qx distribution along y=0

NS (GKS)Kinetic (UGKS)

Dev=0.0239

e

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-2

0

2

4

6

8

10

12

Y-d

irect

ion

heat

flux

10-3 qy distribution along y=-0.4

NS (GKS)Kinetic (UGKS)

Dev=0.0327

f

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Fig. 14 Results of the shock–bubble interaction with M = 1.3 and Knp = 0.3 at t = 1.3. Top two figures show the cell Knudsen number (a) andtotal heat flux (b). c–f Comparison between NS solution profiles (solid line) and UGKS solution profiles (dash line)

123

Limitation principle for computational fluid dynamics 1097

Cell Knudsen number

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

a

X-direction heat flux

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

510-3

b

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

0

1

2

3

4

5

6

7

Den

sity

Density distribution along y=0

NS (GKS)Multscale (UGKS)

Dev=3.45 10-7

c

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Tem

pera

ture

Temperature distribution along y=0

NS (GKS)Multscale (UGKS)

Dev=1.81 10-7

d

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-0.5

0

0.5

1

1.5

2

2.5

3

X-d

irect

ion

heat

flux

10-3 qx distribution along y=0

NS (GKS)Multscale (UGKS)

Dev=0.43

e

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-1

0

1

2

3

4

5

6

Y-d

irect

ion

heat

flux

10-4 qy distribution along y=-0.4

NS (GKS)Multscale (UGKS)

Dev=0.0068

f

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Fig. 15 Results of the shock–bubble interaction with M = 2 and Knp = 1.0× 10−4 at t = 1.0. Top two figures show the cell Knudsen number (a)and total heat flux (b). c–f Comparison between NS solution profiles (solid line) and UGKS solution profiles (dash line)

123

1098 C. Liu et al.

Cell Knudsen number

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

a

Heat flux

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

b

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Den

sity

Density distribution along y=0NS (GKS)Multiscale (UGKS)

Dev=2.17 10-4

c

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Tem

pera

ture

Temperature distribution along y=0NS (GKS)Multiscale (UGKS)

Dev=5.79 10-4

d

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X-d

irect

ion

heat

flux

qx distribution along y=0

NS (GKS)Multiscale (UGKS)

Dev=0.1247

e

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3x

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Y-d

irect

ion

heat

flux

qy distribution along y=-0.5

NS (GKS)Multiscale (UGKS)

Dev=0.0642

f

Fig. 16 Results of the shock–bubble interaction with M = 2 and Knp = 0.3 at t = 1.0. Top two figures show the cell Knudsen number (a) andtotal heat flux (b). c–f Comparison between NS solution profiles (solid line) and UGKS solution profiles (dash line)

123

Limitation principle for computational fluid dynamics 1099

0 0.2 0.4 0.6 0.8 1Y

0.995

1

1.005

1.01

1.015

1.02Density Distribution

DSMC SolutionUGKS SolutionNS Solution

a

0 0.2 0.4 0.6 0.8 1Y

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

U

Velocity Distribution

DSMC SolutionUGKS SolutionNS Solution

b

0 0.2 0.4 0.6 0.8 1Y

1.02

1.025

1.03

1.035

1.04

1.045

1.05

T

Temperature Distribution

DSMC SolutionUGKS SolutionNS Solution

c

0 0.2 0.4 0.6 0.8 1Y

1.038

1.039

1.04

1.041

1.042

1.043

1.044

1.045

1.046

1.047

1.048

p

Pressure Distribution

DSMC SolutionUGKS SolutionNS Solution

d

Fig. 17 The a density, b x-velocity, c temperature, d pressure distribution for the force-driven Poiseuille flow at steady state with Knudsen numberKnp = 0.1 and acceleration Fx = 0.126. The solid lines are the UGKS solutions, the dashed lines are the NS solutions and symbols stand forDSMC solutions

NS equations is not complete in the description of energytransport, even in the continuum flow regime [13]. This testcase indicates the importance of a multiscale method for thecapturing of intrinsically multiple scale flow dynamics. Thelimitation principle presented in this paper is for the multi-scale method. For the NS solver, the quality of the solution isfar away from the limiting solution constrained by the limi-tation principle.

5 Conclusion

For computational fluid dynamics, due to the limited andchangeable numerical resolution, the analysis of the validflow regime of numerical methods should go beyond theconventional asymptotic analysis. In this paper, we pro-pose a numerical Knudsen number Knn, which is controlledby the physical Knudsen number Knp and cell Knudsen

number Knc through a soft-minimum-type function Knn =softmin(Knp,Knc). According to the numerical Knudsennumber, a division of the numerical flow regime is proposedfor the first time. It can be observed from the numericalflow regime division that the numerical NS solver is appli-cable in the numerical continuum regime with Knp ≤ 10−3

or Knc ≤ 10−3. However, in the process of mesh refine-ment, for a multiscale method the numerical flow regimewill eventually converge to the physical flow regime. Dueto the dependence of flow regime on the mesh resolution,a single-scale scheme for the modeling equations, such asthe hydrodynamic solver and direct Boltzmann solver, mayfail to capture the physical flow field in the correspondingmesh size scale. The multiscale property is essential for anefficient and accurate calculation. As a multiscale numericalscheme, the UGKS provides a physically consistent solutionaccording to the numerical resolution.

123

1100 C. Liu et al.

0 0.2 0.4 0.6 0.8 1Y

1.035

1.04

1.045

1.05

1.055

1.06

1.065

p xx

Stress distribution pxx

DSMC SolutionUGKS SolutionNS Solution

a

0 0.2 0.4 0.6 0.8 1Y

1.037

1.0372

1.0374

1.0376

1.0378

1.038

1.0382

1.0384

1.0386

1.0388

p yy

Stress Distribution pyy

DSMC SolutionUGKS SolutionNS Solution

b

0 0.2 0.4 0.6 0.8 1Y

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

p xy

Stress Distribution pxy

DSMC SolutionUGKS SolutionNS Solution

c

0 0.2 0.4 0.6 0.8 1Y

1.038

1.0385

1.039

1.0395

1.04

1.0405

1.041

1.0415

1.042

1.0425

p zz

Stress Distribution pzz

DSMC SolutionUGKS SolutionNS Solution

d

Fig. 18 The stress distribution for the force-driven Poiseuille flow at steady state: a pxx , b pyy , c pxy , d pzz . The solid lines are the UGKS solutions,the dashed lines are the NS solutions and symbols stand for DSMC solutions

0 0.2 0.4 0.6 0.8 1Y

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Qx

X-directional heat flux

DSMC SolutionUGKS SolutionNS Solution

a

0 0.2 0.4 0.6 0.8 1y

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Qy

Y-directional heat flux

DSMC SolutionUGKS SolutionNS Solution

b

Fig. 19 The heat flux distribution for the force-driven Poiseuille flow at steady state: a x-directional heat flux Qx , b y-directional heat flux Qy .The solid lines are the UGKS solutions, the dashed lines are the NS solutions and symbols stand for DSMC solutions

123

Limitation principle for computational fluid dynamics 1101

0 0.2 0.4 0.6 0.8 1Y

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Qx

10-3 X-directional heat flux (Kn=0.05)

Asymptotic Solution O(Kn2)UGKS SolutionNS Solution O(Kn)

a

0 0.2 0.4 0.6 0.8 1Y

-2

-1

0

1

2

3

4

5

6

7

8

Qx

10-4 X-directional heat flux (Kn=0.02)

Asymptotic Solution O(Kn2)UGKS SolutionNS Solution O(Kn)

b

Fig. 20 x-directional heat flux with a Knudsen number Knp = 0.05 and acceleration Fx = 0.05, b Knudsen number Knp = 0.02 and accelerationFx = 0.02. The UGKS solutions (solid lines) are compared with the asymptotic solutions (symbols) and NS solutions (dashed lines)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Y

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Qx

X-directional heat flux (Kn=0.1, Fx=1)

UGKS SolutionNS Solution

a

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Y

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Qx

X-directional heat flux (Kn=0.05, Fx=0.5)

UGKS SolutionNS Solution

b

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Y

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Qx

X-directional flux (Kn=0.02, Fx=0.2)

UGKS SolutionNS Solution

c

Fig. 21 x-directional heat flux with relative large external force.a Knudsen number Knp = 0.1 and acceleration Fx = 1.0, b Knud-sen number Knp = 0.05 and acceleration Fx = 0.5, c Knudsen number

Knp = 0.02 and acceleration Fx = 0.2. The solid lines representUGKSsolutions and dashed lines stand for the NS solutions

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1102 C. Liu et al.

Acknowledgements The current research was supported by the HongKong Research Grant Council (16206617, 16207715) and the NationalScience Foundation of China (11772281, 91530319).

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