Direct modeling for computational fluid dynamicsmakxu/PAPER/DMCFD.pdf · 2015. 7. 24. · Direct modeling for computational fluid dynamics 305 mesh size scale will be different.

Acta Mech Sin (2015) 31(3):303–318DOI 10.1007/s10409-015-0453-2

RESEARCH PAPER

Direct modeling for computational fluid dynamics

Kun Xu1

Received: 5 January 2015 / Revised: 5 February 2015 / Accepted: 23 March 2015 / Published online: 29 May 2015© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag BerlinHeidelberg 2015

Abstract All fluid dynamic equations are valid under theirmodeling scales, such as the particle mean free path andmean collision time scale of the Boltzmann equation and thehydrodynamic scale of the Navier–Stokes (NS) equations.The current computational fluid dynamics (CFD) focuseson the numerical solution of partial differential equations(PDEs), and its aim is to get the accurate solution of thesegoverning equations. Under such a CFD practice, it is hardto develop a unified scheme that covers flow physics fromkinetic to hydrodynamic scales continuously because thereis no such governing equation which could make a smoothtransition from the Boltzmann to theNSmodeling. The studyof fluid dynamics needs to go beyond the traditional numer-ical partial differential equations. The emerging engineeringapplications, such as air-vehicle design for near-space flightand flow and heat transfer in micro-devices, do require fur-ther expansion of the concept of gas dynamics to a largerdomain of physical reality, rather than the traditional dis-tinguishable governing equations. At the current stage, thenon-equilibrium flow physics has not yet been well exploredor clearly understood due to the lack of appropriate tools.Unfortunately, under the current numerical PDE approach, itis hard to develop such ameaningful tool due to the absence ofvalid PDEs. In order to constructmultiscale andmultiphysicssimulation methods similar to the modeling process of con-structing the Boltzmann or the NS governing equations, thedevelopment of a numerical algorithm should be based on thefirst principle of physical modeling. In this paper, instead offollowing the traditional numerical PDE path, we introduce

B Kun [email protected]

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

direct modeling as a principle for CFD algorithm develop-ment. Since all computations are conducted in a discretizedspacewith limited cell resolution, the flowphysics to bemod-eled has to be done in the mesh size and time step scales.Here, the CFD is more or less a direct construction of dis-crete numerical evolution equations, where themesh size andtime step will play dynamic roles in the modeling process.With the variation of the ratio between mesh size and localparticle mean free path, the scheme will capture flow physicsfrom the kinetic particle transport and collision to the hydro-dynamic wave propagation. Based on the direct modeling, acontinuous dynamics of flow motion will be captured in theunified gas-kinetic scheme. This scheme can be faithfullyused to study the unexplored non-equilibrium flow physicsin the transition regime.

Keywords Direct modeling · Unified gas kinetic scheme ·Boltzmann equation · Kinetic collision model ·Non-equilibrium flows · Navier–Stokes equations

1 Modeling for computational fluid dynamics

1.1 Limitation of current CFD methodology

All fluid dynamic equations are constructed by modelingflow physics with the implementation of physical laws, suchas mass, momentum, and energy conservation, in differentscales. With a variation of resolution to identify physicalreality, different governing equations have been obtained.The two most successful ones are the Boltzmann equationand the Navier–Stokes (NS) equations. These equations aremathematical representations of the flow physics in the cor-respondingmodeling scales. The current computational fluid

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304 K. Xu

dynamics (CFD) methodology is mostly based on the directdiscretization of these equations in a discretized space, i.e.,the so-called numerical partial differential equations (PDEs).In the numerical discretization, there is no longer a directaccount of the physicalmodeling scale. Evenwith the appear-ance of newscales, such as themesh size and time step, exceptthe truncation error, the dynamics from mesh size scale hasnever been seriously considered in CFD practice.

The current target of CFD is to recover the solution ofthe original PDEs as the mesh size and time step approachzero. Under such a CFDmethodology, the best result is hope-fully to get the exact solution of the governing equations. Butthe flow physics described by the numerical solution is stilllimited by the modeling physics of the original governingequations. In reality, due to the limited cell resolution, wecan never get the exact solution of the original governingequations due to the truncation error. Theoretically, we neverknow what the exact underlying governing equation of theCFD algorithm is, especially in the regions with unresolved“discontinuities”, and the uniqueness of the numerical solu-tion becomes a luxury requirement. For example, there aremany CFD algorithms for the same PDEs, such as the gigan-tic number of approximate Riemann solvers for the Eulerequations. If the design principle of the CFDmethod is basedon the limiting solutionwith vanishingmesh size, themissionof CFD can be never achieved. In addition to the assumptionof the flow physics, such as the smoothness of flow vari-ables, all PDEs are derived based on additional mathematicsfor simplification, such as shrinking a control volume to zerothickness in order to properly use the definition of deriva-tives. During this process, the peculiarity of the applicationof physical laws in different geometric configurations is lost.In other words, the geometric information is absent in thefluid dynamic equations. Unfortunately, a numerical schemedoes need a mesh, and the lost geometric information has tobe added back in the numerical discretization of the PDE,such as the implementation of geometrical conservation law.

For the Euler equations, due to the limited numerical cellresolution, it becomes impossible to capture the zero shockthickness of the equations. Theoretically, the best resolutiona scheme can have is themesh size.However, the shock thick-ness with numerical mesh size scale can be only recoveredfrom theEuler equationswith an additional amount of numer-ical dissipation. But, due to the absence of the dissipationin the Euler modeling, the lost physics, i.e., the dissipativemechanism in the mesh size scale, has to be created arti-ficially through the numerical procedure, such as all kindsof implicit and explicit dissipation in the Euler solvers. Withthe possible inconsistency of this kind of artificial dissipationfrom a physically required onewith a “physical” shock struc-ture in the mesh size scale, it will not be surprising to observeany unfavorable numerical behavior, such as the shock insta-bility in the Godunov method at strong shock cases, which is

Fig. 1 Schematic of hypersonic flow over a blunted body with regionsthat typically exhibit non-continuum effects [1,2]

a “black cloud” hanging over the clear CFD sky. Even target-ing on the inviscid Euler equations, all numerical schemeshave to add a non-equilibrium dissipative effect, where newgoverning equations have to be solved implicitly. The incom-patibility of physical modeling scale of PDEs and the meshsize scale has never been seriously studied in the current CFDmethodology.

In the current CFD, due to the separation between thenumerical discretization and the physical modeling, in mostcases the PDEs are blindly numerically treated. For exam-ple, in the direct numerical discretization of the Fourier’s law,where the heat flux is proportional to temperature gradient,the mesh size used can be from 10−10 m upto 1 m, 1 km, oreven 1 light year! How could we believe that such a law isstill valid for such a mesh size scale? The gigantic number ofexperiments clearly demonstrate the invalidity of Fourier’slaw for microscale heat transport [3]. Instead of separatingnumerical discretization from physical modeling, the CFDshould aim to model the flow physics in the mesh size scaledirectly and figure out the physical law at such a scale. Fig-ure 1 presents the experiment of a hypersonic flow passingthrough a blunt body. If the flow physics is to be describedusing a mesh size scale, at different locations different flowphysics will be identified, such as the equilibrium flow in theupstream region, highly non-equilibrium flow in the shockregion, the rarefied flow in the trailing edge, and turbulentflow in the wakes. In terms of mesh size resolution, differentflow physics appear locally in different regions.

One example which cannot be properly treated in the cur-rent CFD methodology is the flow simulation of a flightvehicle passing through the whole atmosphere environment.With a reasonable number of mesh points around a vehi-cle, such as 109 mesh points in total, at different altitudesthe number of particles and their dynamic description in the

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Direct modeling for computational fluid dynamics 305

mesh size scale will be different. At altitudes below 40 km,inside each control volume of mesh size scale, there are agigantic number of particles, the flow can be described by theNS equations for the accumulating wave effect. At altitudesabove 80 km, the mesh size may come to be compatible withthe particle mean free path, the Boltzmann equation or thedirect simulation Monte Carlo (DSMC) method can be usedfor capturing the particle movement and collision. However,in the transition regime between 40 and 80 km, the number ofparticles inside each numerical cell varies significantly andthe corresponding physics has both particle and wave effect.There is basically no such a valid governing equation at thismesh size scale. Theoretically, wemay refine themesh size tothe particlemean free path and apply the Boltzmann equationeverywhere. But it just becomes a brute force approach andit is unrealistic under current computational resources. It istrue that the Boltzmann equation is valid in all flow regimesfrom free molecular to the continuum NS solution. But thisstatement is based on the assumption of fully resolving theBoltzmann physics in the mean free path scale. In the contin-uum flow regime, we do not have the luxury of constructinga mesh in the mean free path scale. The aim of CFD is todescribe the flow physics accurately in a most efficient way.

1.2 CFD modeling

Following the current CFDmethodology, it becomes difficultto develop a multiple scale method if there is no such validgoverning equation for all scales. A few distinct governingequations with specific modeling scales may not be adequateto present a complete picture of gas dynamics. The reason forthe existence of a few distinguishable governing equations,such as the NS and the Boltzmann, is that it is relatively easyto do the modeling at these scales. The Boltzmann equationseparates the particle transport and collision, which is a validmodeling in the mean free path scale for dilute gas. TheNS equations describe the accumulating collisional effectof a gigantic number of particles with wave propagation inthe hydrodynamic scale. The NS equations are valid in thesituation that there is a linear relationship between stress andstrain. In the scale between the above two limits, the non-equilibrium flow behavior appears, which encounters greatdifficulty in its modeling. Actually, how to describe the non-equilibriumflowandwhat kind of flowvariables to be trackedhere are not so clear.

Even without a valid governing equation for all scales, westill need to study the flow dynamics in different regimes.Fortunately, the CFD provides us an alternative way todesign numerical algorithms, i.e., the so-called constructionof discrete governing equations through direct modeling, seeFig. 2. The principle of direct modeling is not to solve anyspecific equation, but to construct a flow evolution model inthe mesh size scale. With the variation of the ratio between

the mesh size and the local particle mean free path, the mod-eling should be able to capture different flow physics fromthe kinetic scale particle collision and transport to the hydro-dynamic scale wave propagation.

The unified gas-kinetic scheme (UGKS) has been devel-oped under such a direct modeling principle, where acontinuous description of flow physics from kinetic to hydro-dynamic scale is recovered in its numerical algorithm [3–11].The success of the UGKS is due to the automatic adoption ofa crossing scale modeling of flow physics in the numericalflux construction. The specific solution used locally for theupdate of flow variables depends on the ratio between thelocal time step and the local particle collision time. With thevariation of this ratio, theUGKSprovides a smooth transitionof different scale flow physics. In the following section, thebasic idea underlying the UGKS method will be introduced,which is followed by the analysis and applications in variousflow regimes.

2 Unified gas kinetic scheme

2.1 General methodology

Physically, different flow regimes are defined through theKnudsen number, which is defined as the ratio of the particlemean free path to the characteristic length scale, such as thecontinuum (Kn � 10−3), transition (10−3 < Kn < 10),and free molecular ones (Kn � 10). Numerically, the com-putation takes place in a discretized space. With the currentcomputer power, the mesh size used in an engineering appli-cation is limited, such as 103 × 103 × 103 grid points in thephysical space.With such amesh distribution around a flyingvehicle, the flowdynamics to be identified inside each controlvolume depends on the local mesh resolution and the particlemean free path. With a large variation of the mesh size overthe particle mean free path, a unified scheme aims to capturethe corresponding flow physics in different flow regime.

The construction of the unified scheme is similar tothe modeling process in the derivation of theoretical fluiddynamic equations, but without shrinking the control volumeto zero. Different mesh size scale in terms of local particlemean free path will notify different transport phenomena.The accuracy requirement in a physical modeling dependson the information needed for a practical engineering appli-cation, and the availability of computational resources. If thecell size comes to a scale of particle mean free path every-where, the Boltzmann modeling physics, such as transportand collision, can be directly used to construct the scheme[12,13]. If the cell size and time step are much larger thanthe particle mean free path and particle collision time, thecorresponding physics due to the accumulation of particlestransport and collision needs to be modeled.

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306 K. Xu

Fig. 2 Direct CFD modeling in mesh size scale

The main ingredients in the unified scheme are the mod-eling of flow transport across a cell interface and inner cellcollision. Themodeling solution covers the evolution processfrom the initial free transport to thefinal hydrodynamicswavepropagation. More specifically, the time evolution solutionmodeling in the algorithm includes the non-equilibrium par-ticle transport and the equilibriumNS solutionwithin a singleformulation. The weights of the contribution from the kineticand hydrodynamic parts in the flux transport depend on theratio of time step to the local particle collision time. There-fore, the numerical governing equation underlying the unifiedscheme depends on the space and time resolution. The Boltz-mann equation is recovered in the kinetic scale. The evolutionsolution with the accumulation of particle collisions capturesthe flow physics in the transition regime.

2.2 Numerical evolution equations

The unified scheme is a direct modeling in a discretizedspace. The “governing” equation is the numerical algorithmitself. Since the modeling is on the mesh size and time stepscale, there is no reason to “get” the so-called PDEs.

The discretized space is divided into control volume,i.e., Ω i, j,k(x) = �x�y�z with the cell sizes (�x) =xi+1/2, j,k − xi−1/2, j,k,�y = yi, j+1/2,k − yi, j−1/2,k , and�z = zi,i,k+1/2 − zi, j,k−1/2 in a physical space. The tem-poral discretization is denoted by tn for the n-th time step.The particle velocity space is discretized by Cartesian meshpoints with velocity spacing �u, �v, and �w with a vol-ume Ω l,m,n(u), around the center of the (l,m, n)-velocityinterval at (ul , vm, wn). The fundamental flow variable in adiscretized space is the cell-averaged gas distribution func-tion in a control volume (i, j, k), at time step tn , and aroundparticle velocity (ul , vm, wn),

f (xi , y j , zk, tn, ul , vm, wn) = f ni, j,k,l,m,n

= 1Ωi, j,k(x)Ωl,m,n(u)ÚΩi, j,kÚΩl,m,n f (x, y, z, tn, u, v, w)dxdu,

where dx = dxdydz and du = dudvdw.

The time evolution of a gas distribution function in thecomputational space is due to the particle transport throughcell interface and the particle collisions inside each cell,which re-distributes particles in the velocity space. The directmodeling in such a space gives

f n+1i, j,k,l,n,m = f ni, j,k,l,m,n +1

Ωi, j,k Útn+1

tn

∑

r=1ur f̂r (t)�Srdt

+ 1Ωi, j,k Ú

tn+1

tn ÚΩi, j,k Q( f )dxdt, (1)

where f̂r is the time-dependent gas distribution function ata cell boundary, which is integrated along the surfaces ofthe control volume Ωi, j,k, ur is the particle velocity compo-nent normal to the cell interface,�Sr is the r -th cell interfacearea, and Q( f ) is the particle collision term, which redistrib-utes the particle in the velocity space due to collision. Theabove equation is the fundamental governing equation in adiscretized space. It is an evolution equation in the mesh sizescale. The dynamics underlying the above equation dependson the size of the control volume, where the modeling of theinterface flux and collision term inside each cell depends onthe scale of control volume.

Equation (1) can be considered an integral form of theBoltzmann equation, but it is beyond the validity regime ofthe Boltzmann equation on the kinetic scale if the inter-face flux is properly modeled instead of direct streamingof particles. On the other hand, if Eq. (1) is considereda direct modeling; the Boltzmann equation can be derivedfrom it with a specific modeling scale on the size of con-trol volume. The Boltzmann equation is a consequence ofthe physical modeling in the particle mean free path andcollision time scale. Under such a scale, the particle freetransport and collision can be separated in the Boltzmannequation. However, in the above equation, the mesh size andtime step can go much beyond the kinetic scale. Under amuch enlarged scale, such as tens of particles mean freepath, the time evolution of a gas distribution function at acell interface will not take free transport, and the collisioneffect inside each cell could be an accumulation of multipleparticle collisions. Therefore, in terms of physical modeling

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Direct modeling for computational fluid dynamics 307

the above equation is more general than the Boltzmann equa-tion, such as the continuity assumption is not needed forflow variables in Eq. (1). Instead of using the integral equa-tion, a direct discretization of the Boltzmann equation willuse the upwind or particle free transport for the flux eval-uation. But the interface flux in the above equation has togo beyond the kinetic scale transport if the NS solution inthe hydrodynamic limit needs to be recovered. Equation (1)is a representation of physical law in the mesh size scale,which could include different scale flow physics in com-parison with the Boltzmann modeling. The quality of thescheme depends on the modeling of interface flux and theinner cell collision term, which are closely related to themesh size.

If we take conservative moments ψα on Eq. (1), i.e.,

ψ =(ψ1, ψ2, ψ3, ψ4, ψ5

)�=

(1, u, v, w,

1

2(u2+v2+w2)

)�,

where du = dudvdw is the volume element in the phasespace. Due to the conservation of conservative variables dur-ing particle collisions, the update of conservative variablesbecomes

Wn+1i, j,k = Wni, j,k +1

Ωi, j,k Útn+1

tn

∑

r=1�Sr · Fr (t)dt, (2)

whereW is the volume averaged conservative mass, momen-tum, and energy densities inside each control volume, and Fis the flux for the corresponding macroscopic flow variablesacross the cell interface. These fluxes for macroscopic flowvariables can be obtained from a time-dependent gas dis-tribution function as well. Now the fundamental governingequations of the unified scheme are theEqs. (1) and (2). Thesetwo equations are the governing equations for the descriptionof flow motion in all flow regimes, where the flow physicssolely depends on the time evolution of the gas distributionfunction at a cell interface and the particle collision insideeach control volume. Equations (1) and (2) are the directmodeling equations; physically there is no any error intro-duced. The updates of the gas distribution function and theconservative flow variables depend on the modeling of inter-face flux and the inner cell particle collision term. In general,no conservative moments can be taken as well on Eq. (1),such as the rotational or vibrational energy, and the corre-sponding macroscopic equations will have additional sourceterms.

The central task of the unified scheme is to model a time-dependent gas distribution function at a cell interface, whichis to recover gas evolution process in the mesh size scale,with a variation of the ratio between the time step and theparticle collision time. In order to model the gas evolutionprocess for the flux construction and collision term, the gas-

kinetic Bhatnagar–Gross–Krook (BGK) model, the Shakhovmodel, the ellipsoid statistical BGK (ES-BGK) model, theRykov model for diatomic gases, and even the full Boltz-mann equation, can be used to do the modeling. Basically,a local time evolution solution of the gas distribution func-tion at a cell interface and the particle collision inside eachcell need to be supplied to the above numerical governingequations.

2.3 Physical modeling for interface flux and inner cellcollision

In order to construct the interface gas distribution functionand the inner cell collision in the mesh size and time stepscales, we need to understand the kinetic equation first, andmodel the flow physics in other scales. The Boltzmann equa-tion describes the time evolution of the density distributionof a dilute monatomic gas with binary elastic collisions. Forspace variable x ∈ R3, particle velocityu = (u, v, w)t ∈ R3,the Boltzmann equation reads:

∂ f

∂t+ u · ∇x f = Q( f, f ), (3)

where f := f (x, t, u) is the time-dependent particles dis-tribution function in the phase space. The collision operatorQ( f, f ) is a quadratic operator consisting of a gain term anda loss term,

Q( f, f ) = ÚR3ÚS2B(cos θ, |u − u∗|) f (u′∗) f (u′)dΩdu∗︸ ︷︷ ︸Q+

− ν(u) f (u)︸ ︷︷ ︸Q−

,

where

ν(u) = ÚR3ÚS2B(cos θ, |u − u∗|) f (u∗)dΩdu∗,is the collision frequency. Here, u and u∗ are the pre-collisionparticle velocities, while u′ and u′∗ are the correspondingpost-collision velocities. Conservation of momentum andenergy yield the follow relations

u′ = u + u∗2

+ |u − u∗|2

Ω = u + |r |Ω − ur2

,

u′∗ =u + u∗

2− |u − u∗|

2Ω = u∗ − |ur| − ur

2,

where ur = u − u∗ is the relative pre-collision velocity andΩ is a unit vector in S2 along the relative post-collisionvelocity u′ − u′∗. The collision kernel B(cos θ, |u − u∗|)is nonnegative and depends on the strength of the relativevelocity and deflection angle. For hard sphere molecules,

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308 K. Xu

the collision kernel B = |ur |σ = |ur |d2/4, where d isthe molecular diameter. For (η − 1)-th inverse power-law,the collision kernel is a power-law function of the relativevelocity

B = |ur |σ = cα(θ)|ur |α, α = η − 5η − 1 ,

and according to the Chapman–Enskog expansion [12], theviscosity coefficient follows

μ = 5m(RT/π)1/2(2mRT/κ)2/(η−1)

8A2(η)Γ(4 − 2

η−1) ,

A2(η) = Ú∞0 sin2 χW0dW0.Most kinetic model equations replace the Boltzmann col-

lision term inEq. (3)with a relaxation-type source term S( f ),

S( f ) = M̃( f ) − fτs

,

where M̃( f ) maps f to the corresponding modified equilib-rium state, where the ES-BGK [14] and Shakhov [15] are twopopular ones, which can be combined as well [16]. Here, wewill concentrate on the full Boltzmann and Shakhov modelequations to construct UGKS.

The Shakhov model can be written as,

ft + u · ∇x f = M̃( f ) − fτs

, (4)

where

M̃( f ) =M( f )[1 + (1 − Pr)c · q

(c2

RT− 5

)/(5pRT )

],

=M( f ) + τsg1( f ),

where M( f ) is the Maxwellian distribution function, c is thepeculiar velocity, and q is the heat flux. Although the kineticmodels are much simpler than the full Boltzmann equation,they share the similar asymptotic property [12] in the hydro-dynamic regime, which means both equations recover theEuler and NS equations when the Knudsen number is small.

In UGKS, the interface flux plays a dominant role to cap-ture the flow evolution in different scales from kinetic up tothe NS ones. For example, in the 1D case, depending on thescale of �x and �t , the solution at the interface f j+1/2,k isconstructed from an evolution solution of the kinetic model(Eq. (4)). Without loss of generality, the cell interface isassumed to be at x j+1/2 = 0 and tn is assumed to be 0,

f (0, t, uk, ξ) = 1τs Ú

t

0M̃(x ′, t ′, uk, ξ)e−(t−t

′)/τsdt ′

+ e−t/τs f0(−ukt, uk, ξ), (5)

where x ′ = −uk(t − t ′) is the particle trajectory andf0(−ukt, uk, ξ) is the gas distribution function at time t = 0.In order to determine fully the evolution solution, the ini-tial condition and the equilibrium states around the cellinterface have to be modeled. The basic ingredient in theabove equation is that the initial non-equilibrium state decaysexponentially due to particle collision, which presents a gasevolution process from the kinetic scale, such as particle freetransport, to the hydrodynamical scale evolution, with theemerging of equilibrium solution due to intensive particlecollisions. In the hydrodynamic limit, the NS solutions canbe recovered from the integration of the above equilibriumstate. Besides the above two limits, the above modeling alsopresents the physics in the whole transition regime, whichdepends on the ratio of t/τ . Equation (5) is the direct model-ing for the interface gas distribution function, which can beused to evaluate the interface fluxes for the update of Eqs. (1)and (2). Theoretically, we may integrate the flux based onthe solution equation (5) over a local time step �t local, thendivide it by �t local to get an average flux. This averaged fluxrepresents the dynamics of the local mesh size scale. Then,this local averaged flux can be used explicitly for the updateof flow variables with a uniform time step �t over the wholedomain for an unsteady flow computation. In this way, theconstraint on the local flow physics equation (5) due to thedirect adoption of a global uniform small time step, whichis determined by the Courant–Friedrichs–Lewy (CFL) con-dition on the smallest cell size, can be released.

Now we have two choices for the collision term mod-eling inside each control volume, which can be the fullBoltzmann collision term Q( f n, f n) and themodel equation(M̃( f n+1)− f n+1)/τ n+1s . Depending on the flow regime, theUGKS uses a time step �t which varies significantly rela-tive to the local particle collision time. Starting fromageneralinitial distribution function, for the account of particle colli-sion only inside each control volume, the evolution solutionsfrom the full Boltzmann collision term and the kinetic modelequation will become identical after a few collision times. Inotherwords, for a single binary collision, there are differencesbetween the solutions from the full Boltzmann collision termand themodel equation. However, whenmany collisions takeplace within a time step inside each control volume, thesedifferences diminish. Therefore, the real place where the fullBoltzmann collision term is useful is the region of highlynon-equilibrium flow and with a time step being at or lessthan the local particle collision time [17,18]. This is reason-able because when a few collision are taken into accountwithin a time step, the solution will not be sensitive to theindividual particle collision [19]. As a result, we can modelthe collision term in Eq. (1) from the combination of the fullBoltzmann collision term [17,18] and Shakhov model [15],

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Direct modeling for computational fluid dynamics 309

Q( f ) = AQ( f nj , f nj )k + BM̃( f n+1j )k − f n+1j,k

τ n+1s, (6)

where the coefficients A and B in the above modeling needsto satisfy the following constraints,

(1) A + B = 1 in order to have a consistent collision termtreatment.

(2) The scheme is stable in the whole flow regime.(3) In the rarefied flow regime, the scheme gives the Boltz-

mann solution.(4) In the continuum regime, the scheme can efficiently

recover the NS solutions.

One of the choices can be A = 1 if�t � τ ; otherwise A = 0.The unified scheme does not require that the time step to beless than the particle collision time. Therefore, the unifiedscheme can use a scale-dependent collision term. The fullBoltzmann collision term plays a role only in a small subsetof the collision process. Even with the choice of Shakhovcollision model only (A = 0, B = 1), the unified schemecan still present reasonable and accurate results in the wholeflow regime. Based on the above constraints, we may alsouse the following choice

A = β, B = (1 − β), (7)

with

β = e− �tτr min⎛

⎝1,1

τr supu∈ϒ∣∣∣ Q( f, f )f−M

∣∣∣

⎞

⎠ , (8)

where Υ := {u ∈ R3|( f − M( f ))(u) �= 0}. The abovechoice presents a smooth transition from the Boltzmann col-lision term to the kinetic model equation. The transitionparameter β is chosen based on the following two reasons[29]:

(1) Based on the numerical experiments, we find when theratio �t/τr becomes large, the Shakhov model performssimilarly as theBoltzmann collision term. It is reasonableto use the Shakhov model to approximate the Boltzmannoperator when the time step is large. Thus, β contains anexponential term e−�t/τr .

(2) Both the Boltzmann collision term and Shakhov modelare stiff operators. An implicit part must be includedto stabilize the scheme, especially when the distribu-tion function is highly non-equilibrium. Therefore, β

contains the term supu∈Υ∣∣∣ Q( f, f )f−M

∣∣∣ which indicates thedeviation of the local distribution from the correspond-ing equilibrium. For highly non-equilibrium f , the term

supu∈Υ∣∣∣ Q( f, f )f−M

∣∣∣ is large such that more weight is put onthe implicit part to stabilize the scheme.

Practically, many other simplified models for the determi-nation of A and B can be used as well in the numericalcomputations. Even with A = 0 and B = 1, all simulationsare still acceptable [5,6]. The choice of (A = 1, B = 0)is not applicable due to the following reasons. Firstly, thecalculation of the full Boltzmann collision term is too timeconsuming and it is hard to make it implicit. Fortunately, theexplicit form can be faithfully used in the rarefied regimewith the time step being less than the particle collision time.Secondly, it is not needed physically to use the full Boltz-mann collision term when the time step is larger than theparticle collision time.

After determining coefficients A and B, the collision termin Eq. (6) can be supplied to Eq. (1) for the evaluation of theinner cell collision term.With the modeling of both interfacegas distribution function equation (5) and inner cell colli-sion equation (6), the numerical procedure for UGKS is thefollowing,

(1) Update the conservative flow variables through Eq. (2)and evaluate the equilibrium inside each cell at the nexttime step;

(2) Update the gas distribution function in Eq. (1), where thecollision term can be treated implicitly.

3 Analysis of unified scheme

The traditional continuum and rarefied flow simulations arebased on solving different governing equations, such as theNS and direct Boltzmann solver. The UGKS provides asmooth transition for gas dynamics from the kinetic to thehydrodynamic scales. The solutions obtained from UGKSdepend on the ratio of the time step (or local time step) to thelocal particle collision time. In the following, we are goingto analyze properties of the UGKS.

3.1 Dynamic coupling between different scales

The UGKS is a multiscale method to simulate both rarefiedand continuumflowwith the update of bothmacroscopic flowvariables equation (2) and the microscopic gas distributionfunction equation (1). Instead of solving different governingequations, the UGKS captures the flow physics through thedirect modeling of a scale dependent evolution solution forthe flux evaluation. In the continuum flow regime, intensiveparticle collisionwith�t τ will drive the gas system closeto the equilibrium state. Therefore, the part based on the inte-gration of the equilibrium state in Eq. (5) will automatically

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310 K. Xu

play a dominant role. It can be shown that the integrationof the equilibrium state presents precisely an NS gas distri-bution function when �t τ . Because there is one-to-onecorrespondence between macroscopic flow variables and theequilibrium state, the integration of the equilibrium state canbe also fairly considered as a macroscopic component ofthe scheme to capture the flow physics in hydrodynamicscale. In the free molecule limit with inadequate particlecollisions, the integral solution at the cell interface will auto-matically present a purely upwind scheme, where the particlefree transport f0 in Eq. (5) will give the main contribu-tion when �t � τ . This is precisely the modeling usedin the derivation of the Boltzmann equation. Therefore, theUGKS captures the flow physics in the rarefied regime aswell.

In UGKS simulation, the ratio between the time step �tto the local particle collision time τ can cover a wide rangeof values. Here, the time step is determined by the maximumparticle velocity, such as �x/max(|u|), which is equivalentto

�t = CFL �x|U | + c ,

where CFL is the CFL number, c is the sound speed, and�x is the local mesh size. On the other hand, the particlecollision time is defined by

τ = μp

= ρ|U |�xpRe

,

where Re = ρ�x |U |/μ is the cell’s Reynolds number. Withthe approximation,

|U | + c c,

which is true for low speed flow and is approximately correcteven for hypersonic flow because the flow velocity is on thesame order as sound speed; we have

�t

τ= Re

M,

where M is the Mach number. So, in the region close toequilibrium, even for a modest cell Reynolds number, suchas 10, the local time step for the UGKS can be much largerthan the local particle collision time. The above equation canbe written in the following form

�t = τKn

,

where Kn = M/Re is the local cell Knudsen number.So, in a computation which covers both continuum and

rarefied regimes, the ratio of time step over local parti-cle collision time can be changed significantly. For steadystate calculation, the use of local time step will enhancethe efficiency of the scheme further for the steady statecalculations.

3.2 Asymptotic preserving property

The numerical scheme, which is capable of capturing thecharacteristic behavior in different scales with a fixed dis-cretization in both space and time, is called an asymptoticpreserving (AP) scheme [20,21]. Specifically, for a gas sys-tem, it requires the scheme to recover the NS limit in a fixedtime step and mesh size as the Knudsen number goes tozero. A standard explicit scheme for kinetic equation alwaysrequires the space and time discretizations to resolve thesmallest scale in the system, such as the particle mean freepath and collision time. It causes the scheme to be extremelyexpensive when the system is close to the continuum limit.In recent years, many studies contributed to the develop-ment of AP schemes. It has been shown that the delicatetime and space discretizations should be adopted in order toachieve AP property [22]. From a physical point of view,the continuum limit is achieved through intensive particlecollisions. The local velocity distribution function evolvesrapidly to an equilibrium state. Based on this fact, it is clearthat anyplausible approximation to the collisionprocessmustproject the nonequilibrium data to the local equilibrium onein the continuum limit. Previous results show that an effec-tive condition for recovering the correct continuum limit isthat the numerical scheme projects the distribution functionto the local equilibrium, which has a discrete analogue of theasymptotic expansion for the continuous equations. In thesestudies, implicit time discretization is implemented to meetrequirement of the numerical stability and AP property.

Asymptotic preserving for the NS solution in the contin-uum limit is a preferred property for all kinetic schemes.Before designing such an AP method, we have to real-ize that the continuum and non-continuum flow behaviordepends closely on our numerical cell resolution. Specifi-cally, it depends on the ratio of numerical cell size to theparticle mean free path. It is basically meaningless to talkabout AP method without sticking to the discretized spaceresolution. The Boltzmann equation itself is a dynamicalmodel in particle mean free path and mean collision timescale. The direct upwind discretization of the transport partof the kinetic equation can not go beyond such a model-ing scale, where �x needs to be on the same order as theparticle mean free path. If �x is on the scale of hundredsof particle mean free path, this free transport discretizationis problematic. Certainly, the flow physics from the kineticto the hydrodynamic scale can be still captured by solvingthe kinetic equation through a brute force approach. Then,

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Direct modeling for computational fluid dynamics 311

the efficiency will become a problem because in many casesthe brute force approach with the resolution up to mean freepath everywhere is unnecessary and too expensive. In mostapplications, we may not need to get such detailed informa-tion of a flow system. Practically, the cell size with respectto the mean free path varies significantly in different regionsaround a flying vehicle at different altitudes. As a result, withthe variation of the ratio between the cell size and particlemean free path, the corresponding physical behavior needsto be captured. The free transport for the interface flux hasto be avoided for a valid AP method. Unfortunately, it seemsthat most current kinetic AP solvers use the free transportmechanism without doubt. More analysis about these kindof AP schemes can be found in Ref. [23].

The distinguishable feature of the UGKS is that a time-dependent solution of the kinetic model equation with theinclusion of collision effect is used for the flux evaluationat a cell interface. This solution itself covers different flowregimes from the initial free molecular transport to the NSformulation. The real solution used depends on the ratio oftime step to the particlemean collision time. In the continuumflow limit, due to the massive particle collision, the contribu-tion from the free transport part f0 disappears. The UGKSwill pick up the NS solution automatically. For the updateof the conservative flow variables, the unified scheme recov-ers the GKS for the NS solution in the hydrodynamic limit[24,25].

In the regionwith comparable values of time step and localparticle collision time such as in the transitional flow regime,both the kinetic scale free transport and the hydrodynamicNS evolution will contribute to its dynamical evolution. TheUGKS provides a continuous transition with the variationof the ratio �t/τ . The UGKS has been validated throughextensive benchmark tests [11].

4 Extension of unified framework to othertransport process

Themultiscalemodeling in the unified scheme can be appliedto many other transport equations, such as radiation andneutron transports, plasma simulation, and the electron andphonon transport process in semiconductors. In radiationtransport, the governing equation is a linear kinetic equa-tion. Similar to the Boltzmann equation, the dynamics of theradiation equation is driven by a balance between photonfree transport and interaction with material medium. Sincethe collision frequencies vary by several orders of magni-tude through the optical thin or thick material, equations ofthis type will exhibit multiscale phenomena, such as thosein the rarefied and continuum flow regimes. For the neutronand radiation transport, many AP schemes have been pro-posed. The UGKS provides a framework for the construction

of schemes covering multiple scale transport mechanism.Recently, Mieussens applied the methodology of the UGKSto the radiative transfer equation [26]. While such a prob-lem exhibits purely diffusive behavior in the optically thick(or small Knudsen) regime, it is proven that the UGKS isstill asymptotic preserving in this regime, and captures thefree transport regime as well. Moreover, he modified thescheme to include a time implicit discretization of the limitdiffusion equation, and to correctly capture the solution incase of boundary layers. Contrary to many AP schemes, theUGKS-type AP method for radiative transfer is based ona standard finite volume approach, and it does not use anydecomposition of the solution or staggered grids. It providesa general framework for the solution of transport equation.Along the same line, recently the UGKS has been extendedto solve gray radiative transfer equations [27]. The recentapplications of the UGKS for phonon heat transfer and mul-tiple frequency radiative transport are very successful aswell.

The UGKS presents a modeling for the transition fromfree molecular transport to the NS solutions. This NS solu-tion is the same as the direct numerical simulation (DNS)approach for the turbulence, where all scales in the hydrody-namic regime are well resolved. As the mesh size becomeseven larger, the flow structures with eddy will appear insideeach numerical cell, which cannot be fully resolved by themesh size resolution. As shown in Fig. 3, there are bothlaminar and turbulent flows in the order of mesh size scale.Therefore, instead of averaging on the NS equations, such asthe most approaches for turbulent modeling [28], we wouldlike to propose a continuous extension of the UGKS from theNS up to the unresolved turbulent modeling,

Fig. 3 The representation of turbulent flownumericallywithmesh sizescale [28]

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312 K. Xu

f (x, t,u, ξ) = 1τt Ú

t

0e−(t−t ′)/τt Ppdf(x′, t ′,u′, ξ)dt ′

+ et/τt[1

τ Út0e−(t−t′)/τ g(x′, t ′,u′, ξ)dt ′

+ e−t/τ f0(x − ut,u, t, ξ)].

(resolved to unresolved turbulent modeling)

(1 − e−t/τt )Ppdftur + e−t/τt f nslaminar,

where τt is the turbulent relaxation time. Ppdf is the “equi-librium” state approached by the turbulent flow, such as theprobability density function (PDF) of the large eddy simula-tion (LES). The above equation presents a transition processfrom the laminar to turbulent description. The real solutionused for the flux evaluation depends on the flow structureand cell size resolution. The turbulent relaxation time needsto be constructed through the modeling of the time scale forthe energy dissipation.

5 A few applications for non-equilibrium flows

TheUGKShas been used in a gigantic amount of engineeringapplications [11]. Here we only present two cases to demon-strate the usefulness of the UGKS.

5.1 Shock structure simulation

The shock structure is one of the most important test case forthe non-equilibrium flow. In this calculation, a nonuniformmesh in physical domain is used, such as a fine mesh in theupstream and a relative coarse mesh in the downstream [29].In addition, a local time step is used in order to get the station-ary solution more efficiently. The UGKS with the inclusionof the full Boltzmann collision term will be tested. The para-meters to determine the switching between full Boltzmannand Shakhovmodels in the current UGKSdepends on the rel-ative values of the local time step and particle local collisiontime.

The shock wave of argon gas with Lennard–Jones poten-tial at M = 5 is calculated by UGKS and is compared with amolecular dynamics simulation of Ref. [30]. Figures 4 and 5show the shock wave structure and the distribution functionsinside the shock layer.

5.2 Lid-driven cavity flow

In the cavity flow case [31], the gaseous medium consistsof monatomic molecules of argon with mass, m = 6.63 ×10−26 kg. The variable hard sphere (VHS) model is used,with a reference particle diameter of d = 4.17 × 10−10 m.In the current study, the wall temperature is kept at the same

x /l-6 -4 -2 0 2 4 6

0

0.2

0.4

0.6

0.8

1

MD-data(temperature)

MD-data(density)

UGKS(density)

UGKS(temperature)

UGKS(velocity)

V ’T ’ n’

A=Δt,B=0A= Δβ

βt,

B=(1− )Δt A=0,B=Δt

Fig. 4 Normalized number density, temperature and velocity distrib-utions from UGKS (symbols) and molecular dynamics (MD) solutions(lines) [30]

reference temperature of Tw = T0 = 273 K, and the up wallvelocity is kept fixed at Uw = 50 m/s. Maxwell’s diffusionmodel with full accommodation is used for the boundarycondition. In the following test cases, a nonuniform mesh isused in order to capture the boundary layer effect. The gridpoint follows, in the x-direction

x = (10−15s+6s2)s3−0.5, s = (0, 1, . . . , N )/2N . (9)

A similar formula is used in the y-direction.The first few tests are in the rarefied and transitional

regime, where the UGKS solutions are compared withDSMC ones. Figures 6, 7, 8 show the results from UGKSand DSMC solutions of Ref. [31] at Knudsen numbers 10, 1,and 0.075. The computational domain for Kn = 10 andKn = 1 cases is composed of 50 × 50 nonuniform meshin the physical space and 72 × 72 × 24 points in the veloc-ity space. Because of the decreasing Knudsen number, themesh size over the particle mean free path can be muchenlarged. The computational domain for Kn = 0.075 caseis composed of 23 × 23 nonuniform mesh in physical spaceand 32 × 32 × 12 points in the velocity space. Because ofthe use of non-uniform of mesh and the local time step,Fig. 8 also includes the switching interface between theuse of the full Boltzmann collision term and the Shakhovmodel.

In order to validate the AP property of the current schemein the continuum limit, the case at Knudsen number 1.42 ×10−4 or Re = 1000 is tested. The computational domainfor Re = 1000 is composed of 61 × 61 nonuniform meshin physical space and 32 × 32 points in the velocity space.In both cases, the freedom of molecules is restricted in a 2D

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Direct modeling for computational fluid dynamics 313

Vx

PD

F

-1000 0 1000 2000 3000

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

UGKS-Boltzmann

MD

UGKS-Boltzmann

MD

UGKS-Boltzmann

MD

UGKS-Boltzmann

MD

Vx

PD

F

-2000 -1000 0 1000 2000 3000

0

0.05

0.1

0.15

0.2

Vx

PD

F

-2000 0 2000 4000

0

0.05

0.1

0.15

Vx

PD

F

-1000 0 1000 2000 3000

0

0.05

0.1

0.15

Fig. 5 The distribution function ÚÚ f dvdwn at locations of density n′ = 0.151, 0.350, 0.511, and 0.759. UGKS solutions (lines) and MD solution(symbols) [30]

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1276275.6275.2274.8274.4274273.6273.2272.8272.4272271.6271.2

Kn=10.0 argona

Y (X )

U/U

w(V

/Uw)

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

0.3

0.4

0.5

U -Y(DSMC)U -Y(UGKS)

V -X(UGKS)V -Y(DSMC)

V-X

U-Y

b

Fig. 6 Cavity flow at Kn = 10. a Temperature contours, black lines DSMC, white lines and background UGKS. b U-velocity along the centralvertical line and V-velocity along the central horizontal line, circles DSMC, line UGKS

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314 K. Xu

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1275.6275.2274.8274.4274273.6273.2272.8272.4272271.6

Kn=1.0 argona

0 0.2 0.4 0.6 0.8 1-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

V-X

U-Y

b

U/U

w(V

/Uw)

U -Y(DSMC)

U -Y(UGKS)

V -X(UGKS)

V -Y(DSMC)

Y (X )

Fig. 7 Cavity flow at Kn = 1. a Temperature contours, black lines DSMC, white lines and background UGKS. b U-velocity along the centralvertical line and V-velocity along the central horizontal line, circles DSMC, line UGKS

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

275274.75274.5274.25274273.75273.5273.25273272.75272.5

A=Δt,B=0

A=0,B=Δt

a

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

b

c

0 0.2 0.4 0.6 0.8 1-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

U-Y(DSMC)

U-Y(UGKS)

V-X(UGKS)

V-Y(DSMC)

V-X

U-Y

d

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

U/U

w(V

/Uw)

Y (X )

Fig. 8 Cavity flow at Kn = 0.075. a Temperature contours with domain interface for different collision models, black lines DSMC, white linesand background UGKS. b Computational mesh in physical space. c Heat flux, dash lines DSMC, solid lines UGKS. d U-velocity along the centralvertical line and V-velocity along the central horizontal line, circles DSMC, line UGKS

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Direct modeling for computational fluid dynamics 315

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

176.565.554.543.532.521.510.5

Y

U/U

w

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X

V/U

w

0 0.2 0.4 0.6 0.8 1-0.6

-0.4

-0.2

0

0.2

0.4

X

P

0 0.2 0.4 0.6 0.8 1

0

0.02

0.04

0.06

0.08

0.1 UGKSNS solution(Ghia) UGKS

NS solution(Ghia)

UGKSNS solution(Ghia)

UGKSNS solution(Ghia)

Y

P

0 0.2 0.4 0.6 0.8 10

0.05

0.1

Fig. 9 Cavity flow at Kn = 1.42 × 10−4 and Re = 1000. (left) Velocity stream lines with temperature background; (right) U-velocity along thecentral vertical line, V-velocity along the central horizontal line, pressure along the central vertical line, and pressure along the central horizontalline, circles NS solution, line UGKS

space in order to get the flow condition close to the 2D incom-pressible flow limit. Also, the non-slip boundary condition isimposed in the calculation. Figure 9 shows the UGKS resultsand reference NS solutions [32]. This clearly demonstratesthat the UGKS converges to the NS solutions accurately inthe hydrodynamic limit.

Based on the above simulations, we can confidently usethe UGKS in the whole flow regime. In the near continuumregime, it will be interesting to use the UGKS to test thevalidity of the NS solution. Before the development of theUGKS, an accurate gas-kinetic scheme (GKS) for the NSsolutionswas constructed and validated thoroughly [25]. Thecomparison between the solutions from the UGKS and GKSis basically a comparison of the governing equations of theUGKS and the NS ones. In the following, we test the cavitycase at Re = 5, 20, and 40,which are shown in Fig. 10.At theabove Reynolds numbers, the velocity profiles between theUGKS and GKS are basically the same. But the heat flux cankeep differences between the UGKS and GKS. As shown inthese figures, the heat flux from the UGKS is not necessarilyperpendicular to the temperature contour level, which is thebasic assumption of Fourier’s law.We believe that the UGKSprovides a reliable physical solution in comparison with NS.The UGKS will become an indispensable tool in the studyof non-equilibrium flow in the near continuum flow regime.

6 Conclusion

This paper reviews the direct modeling as a general frame-work for the CFD algorithm development and the con-

struction of the unified gas kinetic scheme. The principleunderlying the direction modeling is that the mesh sizeand time step will actively participate in (describe) the gasevolution, and the aim of CFD is basically to capture the cor-responding gas dynamics in the mesh size scale. The unifiedscheme is constructed under such a principle and provides amultiple scale gas evolution modeling for flow simulations.The flow dynamics in the UGKS provides a continuous spec-trum of flow motion from the rarefied to the continuum one.The cell size used in the UGKS can range from the par-ticle mean free path to the hydrodynamic dissipative layerthickness in different locations for the capturing of both theBoltzmann and NS solutions when needed. This CFD prin-ciple is distinguishable from the existing CFD methodology,where a direct discretization of the partial differential equa-tions is usually adopted. In the traditional CFD approach,the cell size and time step are basically separated from theflow dynamics. The numerical mesh size seems to introduceerror only. The dependence of the flow dynamics on themesh resolution can be easily understood once we realizethat all existing fluid dynamics equations are derived basedon the physical modeling in their specific scales. Here weonly change the modeling scale to the mesh size and timestep. Certainly, we can use the Boltzmann equation or evenmolecular dynamics all the time to resolve the flow physicsin the smallest scale everywhere, but this is not necessaryand is not practical at all in the low transition and contin-uum flow regime. The aim of science is to figure out themost efficient and consistent way to describe the nature. TheCFD should present the flow dynamics in different scales assimple as possible, but not simpler. For a gas flow without

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316 K. Xu

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 275.5275.25275274.75274.5274.25274273.75273.5273.25273272.75272.5272.25272

Re=5.0 Kn=2.85×10-2 Re=5.0 Kn=2.85×10-2

UGKSa

X

Y

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

275.5275.25275274.75274.5274.25274273.75273.5273.25273272.75272.5272.25272

GKSb

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1274.8274.671274.543274.414274.286274.157274.029273.9273.771273.643273.514273.386273.257273.129273

Re=20.0 Kn=7.12×10-3 Re=20.0 Kn=7.12×10-3UGKSc

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1274.8274.671274.543274.414274.286274.157274.029273.9273.771273.643273.514273.386273.257273.129273

GKSd

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1274.75274.628274.505274.383274.26274.138274.015273.893273.77273.648273.526273.403273.281273.158273.036

Re=40.0 Kn=3.56×10-3 Re=40.0 Kn=3.56×10-3UGKSe

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1274.75274.628274.505274.383274.26274.138274.015273.893273.771273.648273.526273.403273.281273.158273.036

GKSf

Fig. 10 Cavity simulation using the UGKS and GKS at Re = 5, 20, and 40 with the plots of temperature contour and heat flux. Left columnUGKS; right column GKS

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Direct modeling for computational fluid dynamics 317

discernible scale separation, it will become hard if not impos-sible to construct a multiscale method from a few governingequations with distinguishable modeling scales. The multi-ple scale turbulent problem will not be solved if targeting onthe “averaging” of the NS equations only (static approach)without developing a dynamic multiscale modeling princi-ple.

The key in the unified scheme is the modeling of a time-and scale-dependent evolution solution for the interface fluxand inner cell collision in the update of both macroscopicflow variables and microscopic gas distribution function.This time evolution solution covers different flow regimes,from kinetic to hydrodynamic. The solution used locally forthe numerical flow evolution depends on the ratio of the timestep to the local particle collision time. The current studyclearly indicates that the UGKS is a valuable and indispens-able tool for flow study, especially for the flow simulationwith the co-existenceof both continuumand rarefied regimes.The direct modeling principle can be naturally used to sim-ulate many multiscale transport dynamics, such as radiation,neutron, and phonon transport. More detailed constructionof the kinetic schemes and their applications can be found ina recent monograph [11].

The traditionalCFDprinciple, i.e., the so-called numericalpartial differential equations with emphasis on the truncationerror analysis and modified equations, has to be re-examinedcarefully. The final destiny of the direct modeling is toprovide a continuous spectrum governing equations with avariation of scales. Besides the traditional partial differentialequations for flow description, the discrete dynamic schemeprovides another kindof governing equationswhichmaygivea more faithful description of gas dynamics.

Acknowledgments Theworkwas supported byHongKongResearchGrant Council (Grants 621011,620813 and 16211014) and HKUST(IRS15SC29 and SBI14SC11).

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Direct modeling for computational fluid dynamicsAbstract1 Modeling for computational fluid dynamics1.1 Limitation of current CFD methodology1.2 CFD modeling

2 Unified gas kinetic scheme2.1 General methodology2.2 Numerical evolution equations2.3 Physical modeling for interface flux and inner cell collision

3 Analysis of unified scheme3.1 Dynamic coupling between different scales 3.2 Asymptotic preserving property

4 Extension of unified framework to other transport process5 A few applications for non-equilibrium flows5.1 Shock structure simulation5.2 Lid-driven cavity flow

6 ConclusionAcknowledgmentsReferences