aiaa.paperone-step aeroacoustics simulation using lattice boltzmann method

8/11/2019 Aiaa.paperOne-Step Aeroacoustics Simulation Using Lattice Boltzmann Method

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AIAA JOURNALVol. 44, No. 1, January 2006

One-Step Aeroacoustics SimulationUsing Lattice Boltzmann Method

X. M. Li, R. C. K. Leung, and R. M. C. So

Hong Kong Polytechnic University, Hong Kong, Peoples Republic of China

ThelatticeBoltzmannmethod(LBM)is a numerical simplificationof theBoltzmannequationof thekinetictheory

of gases that describes fluid motions by tracking the evolution of the particle velocity distribution function based

on linear streaming with nonlinear collision. If the BhatnagarGrossKrook (BGK) collision model is invoked, the

velocity distribution function in this mesoscopic description of nonlinear fluid motions is essentially linear. This

intrinsic feature of LBM can be exploited for convenient parallel programming, which makes itself particularly

attractive for one-step aeroacoustics simulations. It is shown that the compressible NavierStokes equations and

the ideal gas equation of state can be correctly recovered by considering the translational and rotational degrees of

freedom of diatomic gases in the internal energy and using a multiscale ChapmanEnskog expansion. Assuming

two relaxation times in the BGK model allows the temperature dependence of the first coefficient of viscosity of

diatomicgases to be replicated. ThemodifiedLBM model is solved using a two-dimensional 9-discretized and a two-

dimensional 13-discretized velocity lattices. Three cases are selected to validate the one-step LBM aeroacoustics

simulation. They are the one-dimensional acoustic pulse propagation, the circular acoustic pulse propagation,

and the propagation of acoustic, vorticity, and entropy pulses in a uniform stream. The accuracy of the LBMis established by comparing with direct numerical simulation (DNS) results obtained by solving the governing

equations using a finite difference scheme. The tests show that the proposed LBM and the DNS give identical

results, thus suggesting that the LBM can be used to simulate aeroacoustics problems correctly.

Nomenclature

cp, cv = specific heats at constant pressureand constant volume

c0 = speed of soundD = spatial dimensionDT,DR = translational and rotational degrees of freedom

of particle motione = internal energy

Fex = external body forcef = particle velocity distribution functionfeq = MaxwellianBoltzmann equilibrium

distribution functionfneq = nonequilibrium particle distribution functionh = enthalpyJ = approximation forQkB = Boltzmann constantl = lengthM = Mach numberMn = molecular massn = particle number densityPr = Prandtl numberp = thermodynamic pressureQ = collision operatorR = gas constantRe = Reynolds numberrm = particle separationS0 = Sutherland constant

Received 7 February 2005; revision received 20 July 2005; acceptedfor publication 26 July 2005. Copyright c 2005 by R. C. K. Leung andR. M. C. So. Published by the American Institute of Aeronautics and Astro-nautics, Inc., with permission. Copiesof this papermay bemadefor personalor internal use, on condition that the copier pay the $10.00 per-copy fee tothe Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA01923; include the code 0001-1452/06 $10.00 in correspondence with theCCC.

Ph.D. Student, Department of Mechanical Engineering, Hung Hom,Kowloon.

AssistantProfessor,Departmentof Mechanical Engineering,Hung Hom,Kowloon. Senior Member AIAA.Chair Professorand Head, Department of Mechanical Engineering,Hung

Hom, Kowloon. Fellow AIAA.

T = temperatureTref = Sutherland law reference temperaturet = timeu = velocity vectoru, v = velocity components alongxandy directions = specific heat ratio,cp/cv = Kronecker delta = Knudsen number

= thermal conductivity = thermal diffusivity = second coefficient of viscosity = first coefficient of viscosity = index of repulsion = velocity vector of fluid particle = densitym = effective particle diameter() = collision cross section, 1, 2, eff = relaxation times = viscous stress tensor = dissipation rate = collision invariants = spatial angle = collision frequency

= modulusSubscripts

i = indices of discrete lattice, = indices0 = reference variables = mean flow quantities

Superscripts

= average values = peak values

I. Introduction

N OISE reduction is an important part of engineering design inthe transportation industries. Airplane, automobiles, and trains78


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LI, LEUNG, AND SO 79

all produce noise that disturbs passengers, operators, and the sur-rounding communities. Examples of current interest include air-frame noise, cavity acoustics, jet screech, sonic boom, cabin noise,and noise generated by blade/vortex interactions. In particular, theneed to meet more stringent community noise-level standards hasresulted in recent attention given to the relatively new field of time-domain computational aeroacoustics (CAA), which focuses on theaccurate prediction of aerodynamic sound generated by airframecomponents and propulsion systems, as well as on its propagation

and far-field characteristics. Both aspects of the problem, that is,sound generation and propagation, are extremely demanding froma time-domain computation standpoint due to the large number ofgrid points and small time steps that are typically required. There-fore, if realistic aeroacoustics simulations are to become more fea-sible, higher-order accurate and optimized numerical schemes haveto be sought to reduce the number of grid points required per wave-length while still ensuring tolerable levels of numerically induceddissipation and dispersion.

There are two major categories of CAA simulation methodol-ogy, namely, hybrid methods and one-step or direct simulations. 1,2

In hybrid methods, unsteady computational fluid dynamics simu-lations, such as direct numerical simulation (DNS) or large eddysimulation, are used to replicate the spacetime properties of thenoise-generating flow. The solutions are then treated as equivalent

noise sources with distributed strength and used as inputs for a sec-ond calculation of the noise propagation to the far field. The noisecalculation is usually achieved by exploiting Lighthills acousticanalogy,3 or its derivative (see Ref. 4), by solving the linearizedEuler equation (see Ref. 5), or by acoustic/viscous decompositiontechniques.6,7 Usually two different meshes are required due to thedifferent length scales of the flow and sound fields. Consequently,hybrid methods are not able to handle any interaction of the flowwith the noise it generates and are, therefore, only good for noise-generation prediction. One-step simulations attempt to resolve boththe unsteady flow and the sound in just one calculation. The com-putational mesh required is not only able to cover the source andobserver regions, but also capable of resolving the different lengthscales of the two regions. The calculated flow and sound fields arerequired to leavethe computationaldomain smoothly, andthere is nononphysical disturbances created by the domain boundaries. 8 Thisis commonly achieved by laying artificial absorbing buffer regionsaround the entire domain to eliminate the outgoing waves beforethey touch the boundaries. High computational cost is required tosatisfy all of these requirements for accurate one-step simulations.Even then, one-step simulations are still preferred because they canprovide thedetails of soundgeneratingmechanisms,as well as flowsound interactions in complex flows.

Recent reviews of computational aeroacoustics have been givenby Tam1 and Wells and Renaut,9 who discuss various numericalschemes that are currently used in CAA. These include, among oth-ers, the dispersion-relation-preserving scheme of Tam and Webb,10

the family of higher-order compact differencing schemes of Lele, 11

the method for minimization of group velocity errors due to

Holberg,12

and the essentially nonoscillatory scheme.13

The firstthree schemes are all centered nondissipative schemes, a propertythat is desirable for linear wave propagation. However, the inherentlack of numericaldissipation mayresult in spurious numericaloscil-lations and instability in practical applications involving general ge-ometries, approximate boundary conditions, or nonlinear features.In the dispersion-relation-preserving approach, for instance, artifi-cial selective damping has to be employed under these conditions. 1

Although quite robust, standard upwind and upwind-biased formu-lations may be undesirable for situations involving linearwave prop-agation due to their excessive dissipation. To overcome this diffi-culty, higher-order upwind or essentially nonoscillatory approacheswere proposed.14 These spatial semidiscretizations are typicallycombined with high-order explicit time-integration methods suchas the multistage RungeKutta procedure (see Ref. 15). In addition

to the spatial and temporal discretizations, another critical aspectin CAA simulations is the accurate treatment of the physical andcomputational boundary conditions. Recent reviews of radiation,

outflow, and wall boundary treatments are provided by Colonius 8

and Tam.16 In most of these formulations, a continuum model isused to derive the governing equations from the NavierStokes andenergy equations.

The Boltzmann equation (BE) describes the evolution of the par-ticle velocity distribution function based on the free streaming andcollisions of particles.17,18 The macroscopic quantities of the fluid,such as density, momentum, internal energy, and energy flux, aredefined via moments of the distribution function. These relations

constitute the kinetic equations. There are three major differencesbetween the NavierStokes equations and the BE. First, the BE isapplicable even if the medium could not be considered as contin-uous, such as in simulating rarefied gas flows or multiphase andmulticomponent flows. Second, the BE provides clear physical def-initions for the equation of state of the fluid, the viscous stress, andthe heat conduction from the molecular transport viewpoint. Forthe NavierStokes equations, these considerations could not be de-rived directly from the continuum model. In general, the perfect gasequation of state, the Stokes viscous hypothesis, and the Fourierheat conduction relation have to be introduced to solve the equa-tions. Third, there is a large timescale disparity between these twokinds of equations. In most fluids of practical interest, the velocitydistribution function has a timescale equal to that of the collision in-terval time of the particles, which is of the order of 108

109 sfor

most practical ranges of pressure and temperature.19 On the otherhand, the macroscopic quantities in the NavierStokes equations,such as density, velocity, pressure, and temperature, are affectedby the long timescale of the velocity distribution function, whichis of the order of about 104 s. Consequently, the BE has a muchsmaller timescale than the NavierStokes equations. However, theBE is relatively simple compared to the NavierStokes equations.Therefore, the numerical code could exploit the intrinsic features ofparallelism. This makes it especially useful to model problems withcomplicated boundary conditions and with multiphase interfaces.

The lattice Boltzmann method (LBM) is derived from the latticegas automata.18 As a simplified version of the BE, the LBM is dis-crete in phase (orvelocity) space.It wasproposedas an alternativetothe conventional computational fluid dynamics techniques20,21 morethan a decadeago. A variety of LBM models have been proposed fordifferent hydrodynamic systems, such as single component hydro-dynamics, multiphase and multicomponent flows, particle suspen-sions in fluid, reaction-diffusion systems, and flow through porousmedia.22 In kinetic theory of gases, the evolution of a fluid is de-scribed by thesolutions to the continuous BE. Development of LBMfor single-phase compressible flow, such as air, has received partic-ular attention because its solution recovers the macroscopic NavierStokes equations in the asymptotic limit of Knudsen number (seeRefs. 2326). When a fully discrete particle velocity model wasused, where space and time were discretized on a square lattice,internal energy of the particles was fixed. Therefore, the LBM wasonly used to simulate isothermal flows and low Mach number flowsin the incompressible limit.27,28

There have been attempts to incorporate the effects of tempera-

ture variations in the LBM to simulate compressible flows and wavepropagation. Sun29 introduced a potential energy term into the ex-pressionof particle energy in theLBMon hexagonal lattice forshockwave simulations. The potential energy term was not dependent ontemperature but arbitrarily prescribed to recover the correct localspecific heat ratio of all fluid particles. Thus modified, the LBMhas serious limitations. The numerical results compared rather fa-vorably with analytical solutions. This was accomplished by settingthe relaxation time to unity. However, under this assumption, thefirst coefficient of viscosity would vary linearly with temperatureand Sutherland law could not be recovered correctly. Palmer andRector30 attempted to incorporate the temperature effect by sepa-rately modeling the internal energy as a scalar field using a seconddistribution in addition to the isothermal LBM calculation. The en-ergy is then properly accounted for in the evaluation of the density

and the momentum of the fluid particles. Good agreement was ob-tained in several thermal convection test cases but the applicabilityof their model to aeroacoustics simulations is questionable because


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80 LI, LEUNG, AND SO

the equation of state is not explicitly recovered. Tsutahara et al.31

and Kang et al.32 attempted to include the particle rotational energyinto a modified LBM by solving an additional distribution functionthat is the product of velocity distribution and a rotational energyterm. This model assumed monoatomic gasparticles andgave rise toa constant specific heat ratio of 1.67. They calculated shock reflec-tion and Aeolian tone generated by a circular cylinder and obtainedqualitative agreement with DNS results. However, this was at theexpense of having to specify 21 discrete velocities on a hexagonal

lattice.This paper focuses on the development of a one-step aeroacous-tics simulation schemeusing the LBM for two-dimensional flows. Ifthe aeroacoustics calculations were to be correct, the ideal gas equa-tion of state and the temperature dependence of the first coefficientof viscosity of the gas have to be recovered properly. Therefore,it is important to show that the LBM could satisfy these require-ments. Furthermore, the credibility of the one-step LBM simulationof aeroacoustics has to be established, either by comparing the cal-culations with theoretical solutions or with DNS results.

In the next section is given a brief description of the set of un-steady compressible NavierStokes equations and their solution us-inga DNStechnique where thegoverning equations aresolved usinghigher-order schemes for spatial differencing and for time marching.This is followed by a discussion of the continuous Boltzmann equa-

tion and the kinetic energy equations for different particle velocityorders. It is shown that the set of unsteady compressible NavierStokes equations for real diatomic gas could be fully recovered withthe specific heat ratio given by= 1.4 after the ChapmanEnskogexpansion has been assumed for the expansion of the distributionfunction in the BE and the internal energy expression has beenproperly modified. In Sec. IV, the recovery of the correct temper-ature dependence of the first coefficient of viscosity as describedby the Sutherland law is achieved provided the relaxation time forthe collision model has been suitably adjusted to include the ef-fect of the weak repulsive potential. Two velocity lattice models,a two-dimensional 9-discrete-velocity lattice (D2Q9) and a two-dimensional 13-discrete-velocity lattice (D2Q13), are introducedtogether with a higher-order finite difference scheme in Sec. V. Val-idations of the LBM against some basic acoustic pulses where ana-lytical solutions are available are given in Sec. VI. In addition, theLBM solutions are also compared with one-step DNS solutions of anumber of aeroacoustics problems. All comparisons show that theproposed LBM could be used to carry out one-step aeroacousticssimulations with equal accuracy as the DNS.

II. One-Step DNS Aeroacoustics Simulation

A one-step aeroacoustics simulation means that the sound fieldgenerated from unsteady flows and its propagation are calculatedwithout having to carry out the calculations of aerodynamics andacoustics fields separately. Because the flow and acoustics fields arecalculated simultaneously in a single computation, further inves-tigations of the aerodynamic sound-generation mechanisms couldbe easily carried out. In aeroacoustics analysis, the fluid is usually

modeled as a continuum and its unsteady dynamics is essentiallydetermined by the fundamental principles of the conservation ofmass, momentum, and energy at each fluid point in the continuum.The acoustic fluctuations are considered as weak pressure or den-sity fluctuations that propagate to the stationary regions of the samecontinuum where the effects of flow unsteadiness have completelydecayed. The governing equations for the fluid flow are the un-steady compressible NavierStokes and energy equations; they aregiven as

t+ ( u )

x= 0 (1)

(u )

t+ ( u u )

x= p

x+

x(2)

( h)

t+ (hu )

x=

t+ u p

x

+ +

x

T

x

(3)

where uandx are the velocity component and position coordinatein the direction, pis pressure, is fluid density, and summationover repeated indices and is assumed. The viscous stress and the dissipation rateare given by

= 2

S+ 13

S

, = S

S= 1

2u

x + u

x

(4)

where Stokes hypothesis with zero bulk viscosity is invoked to de-duce the viscous stresses,is the first coefficient of viscosity of thefluid, and is the fluid thermal conductivity. The viscosity andthermal conductivity are regarded as functions of temperature.Equations (1 4) are closed once the thermodynamic coupling ofthe internal energy, the pressure and the density is specified by theideal gas equation of state, that is,

p= RT (5)

whereR = cp cvis the gas constant and cpand cv are the specificheats at constant pressure and constant volume, respectively.

The conventional approach for aeroacoustics simulation proceedsby truncating the governing equations, or their linearized forms, inthe spatial domain. The temporal evolutions are resolved by meansof finite difference or finite volume techniques. On the other hand,if DNS is used to simulate aeroacoustics problems instead of theconventional methods, Eqs. (15) are solved directly. This way, itis possible to deduce and analyze the flowfield in detail. Becausethe acoustics quantities could be as small as 104 106 of themean flow quantities, high-accuracy schemes are required if thesound field are to be resolved correctly with minimum numericalerrors. This imposes a rather heavy penalty on the DNS scheme. Inview of the credibility and accuracy of the DNS scheme, it is usedin the present study as a benchmark to assess the proposed LBMscheme. The DNS solutions are obtained by solving Eqs. (15) us-ing a sixth-order spatial scheme and a fourth-order RungeKutta

time-marching scheme. Because the boundary conditions are asso-ciated with the aeroacoustics problems to be calculated, they willbe discussed when the specific cases are analyzed. Details of theDNS scheme and its solutions of specific aeroacoustics problemsare given elsewhere33; therefore, they will not be repeated here.

III. One-Step LBM Aeroacoustics Simulation

To simulate aeroacoustics problems correctly using a one-stepLBM approach, it is necessary to demonstrate that Eqs. (15) can berecovered properly. This requires the ideal gas equation of state with= 1.4 andto be recovered correctly from the LBM formulation.Anattempt to accomplish this is carried outin thefollowing sections.

Continuous BE

Thebasis of theLBMis theconnectionbetweenthe BEdescribingthe kinetic theory of gases and the macroscopic equations of fluidflow. In kinetic theory, a simple dilute gas, such as air, is representedas a cloud of particles and is fully described by a continuous particledistribution function,34 f(x, , t), which is the probability of find-ing a gas particle at location x moving with microscopic velocityat time t. The tracking of f gives rise to a mesoscopic descrip-tion of the fluid. This description is an intermediate step betweenthe macroscopic continuum model and the microscopic description,which treats the fluid as an avalanche of discrete interacting gasmolecules. For a dilute gas in which only binary collisions betweenparticles occur, the evolution of the distribution function is governedby the continuous BE,

f

t +

f

+Fex

f

=Q(f, f)

=

d3

d()| 1|[f()f(1) f( )f(1)] (6)


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Theleft-handside of Eq. (6) describes the motions of streamingpar-ticles. The variable Fexindicates external body force due to gravityor of electromagnetic origin. Because the rate of particle collisionis not affected by the external body force, Fex = 0 is assumed in thepresent formulation as a first attempt to solve Eq. (6). The operatorQaccounts for the binary particle collision occurring withina differ-ential collision cross section (), which transforms the velocitiesfrom the (incoming) space{, 1} to the (outgoing) space{, 1}.For elastic collisions, the mass, momentum, and kinetic energy of

the particles are conserved. Consequently, Q must possess exactlyfive collision invariantss ( ),s = 0, 1, 2, 3, 4, in the sense that Q(f, f)s ( ) d

3= 0

The elementarycollisioninvariants are 0 = 1, (1, 2, 3) = and4 = ||2, which are proportional to mass, momentum, and kineticenergy of the fluid, respectively.

The nonlinear integrodifferential equation (6) completely de-scribes the spatiotemporal behavior of a dilute gas, but it is quitedifficult to solve dueto the complicated mathematical structureofQfor the closure problem.34 Nevertheless, some measurable macro-scopic averages of the flow can be defined from the velocity mo-ments of the distribution function, for example, the zero- and first-

order moments will give and u, or

=

f d (7)

u=

fd (8)

The definition of the fluid internal energy e needs further consid-eration of the molecular nature of the fluid. As will be indicated,realization of the diatomic nature of the fluid molecules is crucial toa successful recoveryof theequationof state for a perfect gas, whichis the key to a correct estimate of the first coefficient of viscosityand, hence, proper simulations of aeroacoustics problems.

A monoatomic gas model was commonly assumed in most pre-vious LBM treatments. In this model, each gas particle supportsonly translational motion with only a DTdegree of freedom. Thisassumption does notappear to be appropriate foraeroacoustics com-putation because the fluid medium of interest is mostly air, which ismainly composed of diatomic nitrogen and oxygen gases. Generally,a polyatomic gas particle can undergo rotational motion with an ad-ditionalDR degree of freedom. This indicates that both translationaland rotational kinetic energies of polyatomic gas particles should betaken into account in a proper definition of the macroscopic internalenergy. The total number of degrees of freedom is DT+ DR = 5 fordiatomic gas such as air.35 From the statistical mechanics point ofview, the kinetic energy should be equally distributed by all degreesof freedom of gas particle motions. Following the arguments de-scribed in Appendix A, the macroscopic internal energy of the fluid

can be defined by the second velocity moment as

e + 12

|u|2 = DT+ DRDT

1

2 f||2 d (9)

Similarly, the fluid energy flux can be defined by the third velocitymoment as

e + p + 12

|u|2

u= DT+ DRDT

1

2 f||2d (10)

Integration of Eq. (9) suggests an explicit internal energy definitione = (DT+ DR )RT/2 for diatomic gas. Actually, it can be shownthat, with the definitions of macroscopic fluid variables describedin Eqs. (710), the compressible NavierStokes equations and theperfect gas equation of state can be completely recovered from theBE and certain microscopic collision models through the ChapmanEnskog expansion (see Ref. 36).

Collision Model and the ChapmanEnskog Expansion

The collision operator Qcontains all of the details of the binaryparticle interactions, but it is very difficult to evaluate due to thecomplicated structure of the integral. Simpler expressions for Qhave been proposed. The idea behind that replacement is that thevast amount of details of the particle interactions is not likely toinfluence significantly the values of many experimentally measuredmacroscopic quantities.34,35 It is expected that the fine structure ofQ(f, f) can be replaced by a blurred image based on a simpler

operatorJ(f), which retains only the qualitative and average prop-erties of the true operator. Furthermore, the H theorem shows thatthe average effect of collisions is to modify fby an amount propor-tional to the departure from the local MaxwellianBoltzmann equi-librium distribution feq , which is expressed inDspatial dimensionsas

feq = [/(2RT)D/2] exp(| u|2/2RT) (11)

Therefore, the collision operator is approximated as J(f) =( f feq ). In case of a fixed collision interval, that is, = 1/, the well-known BhatnagarGrossKrook37 (BGK), orsingle-relaxation-time (SRT) model for monoatomic gas is recov-ered, and Eq. (6) is expressed as the BoltzmannBGK kinetic

model,

f

t+ xf= 1

(f feq ) (12)

whereis the time taken for a nonequilibrium fto approach feq .The relaxation time is much smaller than the free movement

time of the particle. It is the basic timescale in the BE. The dispar-ity in the Boltzmann and macroscopic timescales indicates that allmacroscopic quantities have converged to local equilibrium statesat a very fast rate. The two disparate timescales in fact facilitate thederivation of the macroscopic compressible NavierStokes equa-tions and their transport coefficients from the kinetic model ofEq. (12) by means of the ChapmanEnskog expansion (see Ref. 36).In essence, it is a standard multiscale expansion in which time andspatial dimensions are rescaled with the Knudsen number as asmall expansion parameter, so that

t1= t, t2= 2t, x1= x

t=

t1+ 2

t2,

x=

x1(13)

and the distribution function fis expanded as

f= feq + fneq = f(0) + f(1) + 2 f(2) + O(3) (14)

The Knudsen number is the ratio of the mean free path betweentwo successive particle collisions and the characteristic spatial scaleof the fluid system. When O(1)or larger, the gas in the systemunder consideration can no longer be considered as a fluid. WhenEq. (14) is inserted into Eq. (12), terms with the same order ofare collected, the resulting equation are multiplied with s ( ), andsubsequent integration is performed over d3 in the velocity space,the following conservation laws are obtained:

t+ (u )

x= 0 (15)

( u )

t+ ( u u )

x=

x

2

DT+ DRe

+ x

u

x+ u

x

+

x

u

x

(16)


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t

e + 1

2u2

+

x

e + 2e

DT+ DR+ 1

2u2

= x

e

x

+

x

u

u

x+ u

x

+ x

u

xu

(17)

It is clear that Eqs. (1517) are the macroscopic conservationequations needed for a flow governed by the compressible NavierStokes equations with the first coefficient of viscosity, the secondcoefficient of viscosity, and the thermal diffusivity defined as

= ( 1)e (18) = ( 1)2e (19) = ( 1)e (20)

In Eqs. (16) and (17), the terms associated with the second viscos-ity, that is,( u/x), are generally small compared to the otherterms in practical flows38; therefore, they could be neglected in thefollowing analysis. The relation between and in the presentformulation is

= cv / Pr. Comparing these expressions with the

macroscopic energy conservation Eq. (3) leads to cp = and con-sequently a Prandtl number of unity in the present formulation. Adifferent Prandtl number could be assigned by scaling the value(DT+ DR )/DTin Eqs. (9) and (10), but it was not attempted in thepresent paper.

The fluid properties are all dependent on , which is a functionofTbecause the relaxation phenomenon depends on T. Therefore,the temperature behavior of the fluid properties is, to a great extent,governed by the relation betweenandT. In the following section,the ideal gas equation is derived first, and this is followed by aderivation of based on a SRT model for J. The need for anothermodel besides the SRT is shown.

Equation of State and the Specific Heat Ratio

If the pressure and the ratio of specific heats are defined asp = 2e/(DT+ DR )and = (DT+ DR + 2)/(DT+ DR ), respec-tively, then the ideal gas equation of state follows:

p = ( 1)e = RTwhich is identical to Eq. (5). Since DT+ DR = 5, = 1.4 andthe ideal gas equation of state are recovered correctly using theChapmanEnskog expansion to facilitate the derivation of the un-steady compressible NavierStokes equations.

In most numerical simulationsof aerodynamics and aeroacousticsbased on macroscopic conservation laws, the value of is usuallyestimated from the Sutherland law. The law of viscosity based onSutherlands model of intermolecular force potential shows that thedependence of on Ttakes the following form:

= 516

12m

Mn kB T1 + S0/ T

TTref

32Tref/ T+ S0/ T

1 + S0/ T(21)

whereTrefis a reference temperature and S0 is the Sutherland con-stant, equal to 111, 107, and 139 K for air, nitrogen, and oxygen,respectively (see Ref. 39). The error associated with this approxi-mation is within 24% over a temperature range of 2101900 K. Ifthe LBM scheme were to be credible, it should recover correctly;otherwise, the Reynolds number effect of unsteady flows would beincorrectly captured.

According to Eq. (12), a SRT model is tacitly assumed for . Inother words, a rigid-sphere collision model is assumed, and the ki-netic model admits a relaxation time for particle velocity expressedas38

54

= 54

|| = 54 12 n2m ||

= 54

2

1

n2m

8kB TMn

(22)

where is the average mean free path, is the time interval ofparticle collision and || = (8kB T/Mn )is the magnitude of themean particle velocity. This gives a relation between and T andallows the determination of from Eq. (18). Combining Eqs. (5)and (18) gives = ( 1)e= p= RT. Simple manipula-tion then shows that

5

16

1

2m

Mn kB T (23)

Equation (23) clearly indicates thathas a differentTdependence

compared to Eq. (21), the Sutherland law. This means that real gaseffects oncannot be modeled properly from the microscopic SRTmodel. If were to reflect the correct temperature dependence asindicated in Eq. (21), another model forhas to be proposed.

IV. Recovery of the Correct FirstCoefficient of Viscosity

Thephenomenon of fluid viscosity could be attributed to momen-tum transfer between gas particles before and after collisions. Thedistributions of momenta of the particles depend on the momentumof each particle when they are far separated, as well as the inter-actions of intermolecular potentials when two particles are in closeencounter. The intermolecular potential represents the contributionsof intermolecular attraction and repulsion to the potential function.According to Ferziger and Kaper,39 from the kinetic theory point

of view, SRT is equivalent to the adoption of a rigid-sphere modelin which the short-range force potential behaves as if a Dirac-deltafunction with finite repulsion at the separation when two rigid par-ticles are in contact (particle separation rm = m ). The model yieldsan exaggerated potential change at rm m and predicts poorly thetemperature dependenceof the macroscopic fluidproperties. Suther-land (see Ferziger and Kaper39)then suggests to include a weak butrapidly decaying repulsive potential, (m/rm) (being the indexof repulsion), in the interaction and successfully provides a morerealistic description of the dependence ofon temperature, such asgiven by Eq. (21). The effects of this weak potential might be morepronounced in the relaxation of a diatomic gas due to its more com-plicated molecular structure. The question then is how to accountfor this weak potential effect. As a first attempt, it is proposed to

include the relaxation times associated with both the intermolecularpotential and the weak repulsive potential in the estimate of forthe BGK collision model invoked in Eq. (12).

It is assumed that in Eq. (12) can be replaced by an effectiverelaxation time effand that this could be determined from a com-bination of the relaxation times associated with the intermolecularpotential 1 and the weak repulsive potential 2. The relation be-tweeneff,1, and2 together with separate expressions for 1 and2 have to be determined. According to Ferziger and Kaper,

39 1could be estimated from the rigid-sphere model. Therefore, 1 isgiven by Eq. (22). It can be rewritten as

1 (5/4) 1

T (24)

The determination of2 is much more complicated because it de-pends on the particle velocity as well as on the physical nature of

the gas under consideration. It could be argued that because theweak potential is a long-range potential, 2 could be postulated tobe proportional to the average approach velocity of the particles,that is,

2 || T 12 (25)The dependence of2on T is, therefore, known; however, the exactrelation has yet to be determined. Its determination will becomeclear after a thorough comparison of the derived has been madewith the Sutherland law.

With the functional form known for 1 and 2, the next step isto determine effso that the LBM scheme would yield a correctin addition to being able to recover the ideal gas equation of state.How to relate effto 1and 2can be gleaned from an examinationof the derivation of as shown in Eq. (23) and Sutherland law. Itis obvious that an SRT based on the rigid-sphere model would givean incorrect representation for . The discrepancy is in the T de-pendence. If the correct dependence on Twere to be recovered, the


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Fig. 1 Variations of first coefficient of viscosity with temperature and

its comparison with Sutherland law.

right-hand side of Eq. (23) should be multiplied by a factor propor-tional to 1/(1 + S0/T). Because 1/2 is proportional to 1/ T, thissuggests representing effby eff= 1/(1 + 1/2). Consequently, thesimplified collision operator Jcan be expressed as

J(f) = (1/eff)(f feq ) = 1/[12/(1 + 2)](f feq )(26)

Theprocedure used to derive Eq.(23) canbe repeated using Eq.(26)for J(f). The result is

= RTeff= RT

1

1+ 1

2

5

16

1

2m

Mn kB T

1 + 1/2(27)

A comparison of Eq. (27) with Eq. (21) clearly shows that 1/2should be related to S0/ Tby the simple expression 1/2 = S0/ T.This is the relation for the determination of2. Thus derived, theLBM scheme will yield a that has the proper dependence on T.

A comparison of the Sutherland law with the derived viscosityrelations can now be made. Figure 1 shows the variations of fortwo common diatomic gases, nitrogen and oxygen, as calculatedusing Eqs. (23) and (27) and the Sutherland law. It is evident thatthe SRT model overpredicts the values of for a wide range oftemperature with errors ranging from 28% at high temperature to70% at low temperature. On the other hand, the two-relaxation-time(1and2) model yields results that are in excellent agreement withthose given by the Sutherland law. Theoretical analysis40 showsthat the acoustic properties of a fluid can be fully and accuratelyresolved by a multiple-relaxation-time (MRT) model in which [in Eq. (12)] for each velocity moment can be adjusted separately.However, adoption of an MRT model would require complicated

programming to adjust for each velocity moment, thus, giving riseto higher computational cost. As will be seen in later comparisonwith DNS and theoretical results, the present model will also yieldfully and accurately the acoustic properties of the fluid with muchless complication in programming for the same problem.

V. Numerical Scheme

Instead of tracking the evolutions of the primitive variables inthe flow solutions in conventional numerical flow simulations, themethod of LBM solves only the evolution of f as prescribed inEq. (12). This equation is first discretized in a velocity space usinga finite set of velocity vectors {i } in the context of the conservationlaws27 such that

fi

t = i

x fi

1

efffi f

eq

i (28)

where fi (x, t) = f(x, i , t) is the distribution function associatedwith theth discrete velocityi and f

eq

i is the corresponding equi-

librium distribution function in the discrete velocity space. Thecon-tinuous local Maxwellian feq may be rewritten up to the third orderof the velocity after a Taylor expansion in u and can be expressedin the discrete velocity space as

feq = Ai

1 + u

+ ( u)2

22 u

2

2 ( u)u

2

22

+( u)3

63 +Ou

4

2 (29)

whereu = (u, v) and= RT. The weighting factors Ai are depen-dent on the lattice model selected to represent the discrete velocityspace. They are evaluated from the constraints of local macroscopicvariables [Eqs. (710)] in the lattice with Ndiscrete velocity sets,as

=N

i = 1f

eq

i , e +1

2|u|2 = DT+ DR

DT

Ni = 1

1

2 f

eq

i ||2

u=N

i = 1i f

eq

i

e + p + 1

2|u|2

u= DT+ DR

DT

Ni = 1

1

2 f

eq

i |i |2i (30)

For thetwo-dimensional diatomic gasflow considered in thepresentstudy, twodiscretevelocitysets areattempted forthe lattice, namely,a D2Q9 model and a D2Q13 model. For the D2Q9 model (Fig. 2a),

0= 0i = c(cos[(i 1)/4], sin[(i 1)/4]), i = 1, 3, 5, 7

i =

2c(cos[(i 1)/4], sin[(i 1)/4]), i = 2, 4, 6, 8A0= 1 + 2 2 3

A1= A3= A5= A7= 2 + A2= A4= A6= A8= ( /2)2 14

For the D2Q13 model (Fig. 2b),

0= 0i= c(cos[(i 1)/4], sin[(i 1)/4]), i = 1, 3, 5, 7

i =

2c(cos[(i 1)/4], sin[(i 1)/4]), i = 2, 4, 6, 8i = 2c(cos[(i 1)/2], sin[(i 1)/2]), i = 9, 10, 11, 12

a) b)

Fig. 2 Lattice velocity models: a) D2Q9 and b) D2Q13.


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A0= 1 (5/2)+ [3/2 + 2]2

A1= A3= A5= A7= (2/3)+ (1 )2

A2= A4= A6= A8= [3/4 + (1/2)]2

A9= A10= A11= A12= (1/24)+ (1/8)2

It is evident from later comparisons that the D2Q13 model yieldsmore accurate results; hence, it is preferred.

The collision process represented by J on the right-hand sideof Eq. (28) is evaluated locally at every time step, whereas thestreaming process represented by the convective derivatives of fiare evaluated by a sixth-order finite difference scheme.11 The time-dependent term on the left-hand side of Eq. (28) is calculated bytime marching using a second-order RungeKutta scheme. No nu-merical filter is used for the following cases in the present study.These treatments have minimal effects on the numerical viscosityfor small quantity disturbances, and so a one-step numerical simu-lation of the BE could lead to realistic prediction of aeroacousticsproblems where the results are essentially identical to those derivedfrom similar DNS solutions and theoretical results.

VI. Results and Discussion

Many practical aeroacoustics simulations aim to predict soundradiation created by unsteady flows and their interactions with solidboundaries. Correct simulation of wave propagation is an impor-tant measure of the success of a numerical model for aeroacousticssimulation. With all viscous terms neglected, Eqs. (1) and (2) re-duce to the Euler equation that supports three modes of waves,namely, acoustic, vorticity, and entropy waves. The acoustic wavesare isotropic, nondispersive, nondissipative, and propagate with thespeed of sound c = (RT). The vorticity and entropy waves arenondispersive, nondissipative, and propagate in the same directionof the mean flow with the same velocity of the flow. Propagationsof the three types of waves are selected for the validation of thepresent LBM scheme. The accuracy of the LBM aeroacoustics sim-ulation isassessedby comparingthe LBMcalculationswith theDNSresults obtained by using convectional finite difference method33

to solve the two-dimensional fully compressible unsteady NavierStokes equations. A measure of the difference between the LBMand DNS results of a macroscopic variable b is expressed in termsof the L pintegral norm, that is,

L p(b) =

1

M

Mb = 1

|bLBM,j bDNS,j |p1/p

(31)

for any integer pand its maximum

L(b) = maxj

|bLBM,j bDNS,j | (32)

Three example test cases are carried out to validate the proposed

LBM. These are 1) the propagation of a plane pressure pulse instationary fluid in a tube, 2) the propagation of a circular pulse instationary fluid, and 3) simulations of an acoustic, an entropy, anda vortex pulse convected with subsonic uniform mean flow velocityu0. In the first two cases, the nondimensional parameters for thelength, time, density, velocity, pressure, and temperature are speci-fied as L 0, L 0/c0,0,c0, c

20, and T0, and the Reynolds number is

defined by Re = 0L 0c0/. For the LBM, the pressure is impliedin the kinetic equation and can be deduced from the state equation,p = ( 1)e. Thenormalized internal energyand sound speed aregiven by e = T/[ ( 1)]and c = T. In the third case, the nondi-mensional parameters for time, density, velocity, pressure, and tem-perature areL 0/u0, 0, u0, u

20,andT0, respectively, andM= u0/c0

is the Mach number. The LBM and DNS solutions are compared inall three cases. In case 3, analytical solutions are also available10;these, too, will be shown for comparisons with the LBM and DNSsolutions. All physical quantities in the following discussion aredimensionless, except where specified.

Case 1: Propagation of a Plane Pressure Pulse

This one-dimensional problem aims to validate the accuracy androbustness of the proposed LBM and, at the same time, to assess theefficiency of the proposed lattice models. The distribution functionfi (x, t) = f(x, i , t) is developing with the collision function andthe streaming function of Eq. (28). The initial fluid state is definedas a small plane pressure fluctuation in the center of a tube, suchthat

=

(33a)

u= 0 (33b)v= 0 (33c)

p= p + exp(ln2 x2/0.082) (33d)where mean field density and pressure are given by = 1 andp = 1/, the pulse amplitude is set to 4 106, 16 106,and 100 106, respectively, and Re = 5000 is specified in thiscase. The computational domain size of the tube is 5 x 5 by0 y 2.Auniformgridofsize0.02 0.02 is adopted. Slip bound-aryconditions areapplied on theupperand lowertube surfaces in theDNS calculation. The stability criterion31 of the collision term re-quires that the time step shouldbe t< /2; therefore, t= 0.0001is chosen for the present LBM computations. Two buffer zones arespecified in the DNS calculation to simulate a true nonreflectinginlet and outlet boundary condition.33 For all LBM calculations,the gradient of the distribution function on all boundaries are set tozero. When all disturbances are far away from the numerical bound-aries, these conditions can ensure that there is essentially no errorcontribution coming from the boundary treatment.

In case 1, the initial conditions Eq. (33) essentially combine oneacoustic wave and one entropy wave. These two waves overlap thedensity fluctuations with each other only for the initial state. Aftertheacoustic wave leavesthe center area, thedensity fluctuations cre-ated by the entropy wave would appear in the center. Figure 3 showsthe fluctuations along the centerline of the tube at t = 1.0 and 3.0for the case = 100 106. The LBM and DNS simulations show aslight difference in the density at the center when the acoustic pulse

propagates toward computational boundaries. (Further calculationscarried out after the manuscript had been accepted showed that theslight difference at the center was due to an error in setting the initialvalue of f at that point.)However,the density distribution in thecen-tral region is essentially the same. The two positive density fluctua-tion peaks areleaving the center witha propagation speed c = 1, andthe amplitude of this density fluctuation is = 4 105 (t= 3.0).At the same corresponding positions, there are two pressure fluctu-ation peaks with a value ofp = 4 105. Actually these two wavesare the exact acoustic waves because the transmission speed is thephysical sound speed andthe amplitudes follow theacoustic relationp = c2. These results show that the proposed LBM can replicate

the correct acoustic waves and the calculated macroscopic quantitiesare developing correctly, just as the DNS solution indicates.

This is evident from a comparison of the calculated

Lp(p)

(pressure). Figure 4 shows the time-dependent difference in the be-havior ofL 1(p),L 2(p), andL(p) ( = 4 106). Bothlattice D2Q9 and D2Q13 solutions are reported. The differences be-tweentheLBMandDNSsolutionsusingtheD2Q13latticearemuchsmaller than those using the D2Q9 lattice. For example, consider theL 2(p) value, thedifferenceobtained forthe D2Q13lattice is about1011, whereas the corresponding value for the D2Q9 lattice is closeto 109. This shows that the D2Q13 lattice could effect an improve-ment in Lp(p) of two orders of magnitude when only four morediscrete velocities are specified. In view of this, only the D2Q13lattice model results are presented in the following discussion.

The pulseamplitude effect on the differenceL 2(p) is comparedin Fig. 5. The criterion of a Taylor expansion on a Maxwellian dis-tribution requires that the flow speed u to be much smaller than theparticle speed, and the error of this expansion would occur in theterm ofO(u4/2). When the pulse amplitude is smaller, the dis-turbanceu is also smaller. This would lead to a smaller error termO(u4/2). Therefore,L 2(p) would be smaller for = 4 106


8/12


a) b)

Fig. 3 Density, pressure, and velocityu fluctuations along the x axis at a)t = 1.0 and b)t = 3.0 for =100 106:, LBM (D2Q13) and , DNS.

Fig. 4 Time history of the differenceLp(p).

than for = 16 106 and100 106. This resultis clearly demon-strated in Fig. 5, which shows that the performance of the LBM isbetter with smaller fluctuations than those large fluctuations.

Case 2: Propagation of a Circular Pulse

If an initial circular pulse is imparted to a uniform fluid, thefluctuations thus created would propagate equally in all directions.This means that, at any time, the pulse would remain circular inshape. However, the lattice velocity model restricts the particles to

Fig. 5 Time history of the differenceL2(p)with D2Q13.

move in certain discrete directions, such as 0, /4, /2, and(Fig. 2). To test the ability of the LBM with a D2Q13 lattice modelto replicate the symmetry property of the circular pulse, it is usedto simulate a circular initial pressure pulse in a uniform fluid. Thedistribution is defined as

= (34a)u= 0 (34b)


9/12


p-p

a)

p-p

b)

Fig. 6 Pressure and velocity fluctuations at a) t = 2.5 and b) t = 5.0:

, positive levels and - - - -, negative levels.

v= 0 (34c)p= p + exp[ln2 (x2 + y2)/0.22] (34d)

where = 1,p = 1/, = 16 106, andR e = 5000. The com-putational domain is 20 20, and the grid size is 0.05 0.05. BothLBM and DNS are used to simulate this problem.

The contours of the pressure and u fluctuations are shown inFig. 6. The upper-half of the computation domain is the LBM so-

lution and the lower-half the DNS solution at the same time. Forpressure fluctuations, six contours are equally distributed between0.4 106 and 0.4 106; for velocity fluctuations, six contoursare equally distributed between 1 106 and 1 106. It is clearthat the contours of the LBM solution have no discernible differencewith those of the DNS result. This is despite that the LBM solu-tion is derived from a velocity lattice model D2Q13, where particlevelocities are specified for discrete directions only. This shows thatthe LBM simulation is just as valid as the DNS result.

Case 3: Simulations of Acoustic, Entropy, and Vortex Pulses

The acoustic, entropy, and vorticity pulses are basic fluctuationsin aeroacoustics problems.In this case, thepulses aredeveloping in auniform mean flow. Basically, only the acoustic pulse is propagating

with the sound speed; the entropy pulse and the vortex pulse wouldmove with the mean flow. The initial conditions are defined as

= + 1exp{ln2 [(x+ 1)2 + y2]/0.22}

+ 2exp{ln2 [(x 1)2 + y2]/0.42} (35a)

u= u + 3yexp{ln2 [(x 1)2 + y2]/0.42} (35b)

v= v + 3(x 1) exp{ln2 [(x 1)2 + y2]/0.42} (35c)

p= p + (1/M2)1exp{ ln2 [(x+ 1)2 + y2]/0.22} (35d)where M= 0.2 and 1 = 0.0001, 2 = 0.001, and 3 = 0.001. Themean field has density, speed, andpressuregiven by = 1, u = 1and v

=0, and p

=1/(M2), andR e

=1000. These pulse mod-

els follow the definitions of Tam (see Ref. 10). The acoustic pulse isinitialized at the pointx= 1,y = 0. The entropypulse and the vor-ticitypulseare initialized atx= 1,y = 0. The computational domain

p-p

a)

p-p

b)

Fig. 7 Pressure and velocity fluctuations at a) t= 1.0 and b) t= 1.5:, positive levels and - - - -, negative levels.

is 10 x 10 by 10 y 10, and the grid size is 0.05 0.05.For this problem witha relatively large mean flow, the mean flow ef-fect could become critical because the symmetry lattice coefficientsarebased on theassumption that theflow speed is much smaller thanthe particle speed. An improvement to the proposed LBM model isgiven in Appendix B to address this problem.

The pressure and the u velocity fluctuation contours are shown

in Fig. 7 att= 1.0 and 1.5. Both LBM and DNS results are showntogether. The initial acoustic pulse causes disturbances. Because themean flow speed is defined as 1.0, the center of the pulse has movedtox= 0,y = 0 at t= 1.0, and tox= 0.5,y = 0 at t= 1.5. Theacous-tic pressure fluctuation is propagating with c = 1/M= 5 and exhibitcircles of radii equal to5 and 7.5 at the samemoment. The uvelocityfluctuation is symmetric about the xaxis. For the vorticity pulse,the center of the vortex would move tox= 2,y = 0 att= 1.0 and tox= 2.5,y = 0 at t= 1.5.In Fig. 7,the upper-half ofthe domainis theLBM solution; the lower-half the domain is the DNS solution. Forpressure fluctuations, six contours are equally distributed between5 105 and 5 105; for velocity fluctuations, six contours areequally distributed between 5 105 and 5 105. The distribu-tion of the u velocity fluctuation from this pulse would give thesame absolute fluctuations that propagate along the negativexaxis.

The pressure anduvelocity fluctuations are in Fig. 8. The LBM andDNS simulations are essentially identical, and they agree well withthe analytical inviscid solutions.10

The same case with Re = 100 is also calculated to investigatethe effect of viscosity on the LBM simulation. The pressure andvelocity fluctuations at t= 1.0 are compared in Fig. 9, where thedistributions along6 x 0 are shown. The asterisks representthe LBM solution, whereas the DNS result is given by the dottedline. The solid line shows the analytical inviscid solution. Again,LBM and DNS give essentially the same solution and are closeto the inviscid result. There is a discernible viscous effect on theacoustic pulse, which is essentially a disturbance generated fromthe viscous effect on the entropy pulse. TheLp differences forpressure andu velocity fluctuations are given in Table 1.

There are two macroscopic velocity scales in case 3, namely,the mean flow velocity and the fluctuation propagation velocity. Toassess the correctness of the LBM in resolving small-fluctuationpropagation in a mean flow, it is worthwhile to study the spreading


10/12


Fig. 8 Pressure and velocity fluctuation distributions along x axis att =1.0andRe= 1000: ,analytical inviscidsolution;, LBM solution;and, DNS solution.

Fig. 9 Pressure anduvelocity fluctuation distributions alongxaxis att = 1.0 andRe = 100: , analytical inviscid solution;, LBM solution;and, DNS solution.

of the acoustic pulse. Figure 10 shows the decay of acoustic pulsepeak in LBM and DNS solutions at Re = 100 and 1000. The ana-lytical result is also shown. For an acoustic pulse spreading in twodimensions, the local intensity of the waveIshould be proportionalto 1/rdue to conservation of total energy e = I 2r, where ris theradial distance. In the absence of viscosity, the intensity bears a re-lationship with instantaneous pressure peak amplitude A as I

A2.

Therefore, the analytical result should be a straight line in Fig. 10with slope equal to 1

2. For Re = 1000, the amplitudes are very

close to the inviscid solution, indicating the acoustic propagation is

Table 1 Lp difference and the effect of Reynolds numbera

Trial L1 L2 L

L p =

1

N

6


11/12


energy has been modified to include both the translational and ro-tational degrees of freedom of the particles. The collision model ismodified to take into account the relaxation times associated withthe intermolecular potential and the weak repulsive potential. Withthese modifications, the ideal gas equation of state with = 1.4 isrecovered exactly and the first coefficient of viscosity of the gas hasthe correct temperature dependence. Consequently, the set of un-steady compressible NavierStokes equations is fully recovered foraeroacoustics simulations. Two lattice (D2Q9 and D2Q13) models

were used to carry out calculations for diatomic gases. A sixth-orderfinite difference schemeis used to evaluatethe streamingterm, andasecond-order time-marching technique is used to calculate the time-dependent term in the BE, whereas the collision term is evaluatedlocally. Three test cases were used to validate the LBM. They arethe one-dimensional acoustic pulse propagation, the circular acous-tic pulse propagation, and the propagation of acoustic, vorticity, andentropypulsesinauniformmeanflow.Theaccuracyofthemethodisestablished by comparing the calculations with analytical solutionsand with DNS results obtained with a sixth-order spatial schemeand a fourth-order RungeKutta time-marching scheme. All com-parisons show that the LBM aeroacoustics simulation possesses thesame accuracyas DNSsolutions forCAA andis a simpler numericalmethod.

It is especially convenient to use LBM to simulate aeroacoustics

problems numerically. First, the inherent linearity of the govern-ing Boltzmann equation and the numerical efficiency of the LBMallow the intrinsic features of parallelism23 to be exploited. Sec-ond, the LBM offers new opportunities for kinetic-type nonreflect-ing boundary conditions. These two difficulties are common inmost CAA but could be overcome with further developments ofLBM. Finally, the results of test cases 13 reflect that the presenttwo-relaxation-time LBM scheme correctly captures the physicalsound speed for 0.2 M 1.0. This contrasts with the extremelynarrow Mach number range (M 0) for conventional SRT LBMscheme. For the sake of completeness, it is important to investigatewhether the present LBM scheme can capture the high sound speedin the limit ofM< 0.2. This issue will be discussed in a companionpaper.

Appendix A: Definition of Internal Energy

The internal energye is defined by the following relation:

e + 12

|u|2 = DT+ DRDT

1

2 f||2 d (A1)

In general, the particle distribution function f can be decom-posed into an equilibrium part and a nonequilibrium part, that is,f= feq + fneq . The nonequilibrium fneq is required to satisfiedthe nullity requirement for moments of different velocity orders,that is,

fneq d= fneq d

= fneq 2 d

=0 (A2)

Therefore, with use of Eq. (11) for feq , Eq. (A1) can be expressedas

e + 12

|u|2 = DT+ DRDT

1

2 feq||2 d

= DT+ DRDT

1

2

(2RT)D/2exp

| u|

2

2RT

||2 d (A3)

The total number of degrees of freedom for diatomic gas motionsis always DT+ DR = 5 (where DT= 3 and DR = 2). Therefore,the right-hand side of Eq. (A3) is a function of temperature alone.The soletemperature effect on e is realized inthe redistributionof theparticle momentum due to particle collision and should be indepen-dent of the mean flow velocity carrying the particles.31 Therefore,e should be the same irrespective of whether u = 0. Integration of

Eq. (A3) in three-dimensional simulations (D = 3) leads to

e = DT+ DR3

0

4r21

2

1

(2RT)32

exp

|r|

2

2RT

|r|2 dr

= DT+ DR3

3

2RT

= DT+ DR

2 RT (A4)

wherer= u.In a two-dimensional simulation, only two planar translational

motions are allowed, thus giving DT= 2. If similar arguments forthe temperaturedependence in the three-dimensional caseis applied,then, withr= uin two dimensions,

e = DT+ DR2

0

2r1

2

(2RT)exp

|r|

2

2RT

|r|2 dr

= DT+ DR2

RT (A5)

Evidently, the definition of e in the present LBM holds for bothtwo- and three-dimensional flows. Note that the integration resultsof Eqs. (A3) and (A5) perfectly match the classical equipartitiontheorem of the kinetic theory of gases, which states that, for a poly-atomic gas, each degree of freedom equally contributesR T/2tothe

total amount of internal energy.

Appendix B: LBM with Mean Flow

The Taylor expansion of the local Maxwellian, Eq. (11), in thediscrete velocity space is given by Eq. (29). The assumption ofsymmetry coefficients of the lattice requires that the velocity u tobe much smaller than the particle speed. When the effective flowspeed is large, the truncation error in this expansion would increase.For the present problem, the macroscopic flow velocity is about thesame as the mean flow velocityu. This means that the fluctuationvelocityu u is much smaller than the particle speed. When thelocal Maxwellian is rewritten with this interpretation, Eq. (11) canbe expressed as

feq

=

(2RT)D/2 exp|( u) (u u)|

2

2RT

(B1)

where u uis the relativeflow velocity based on the mean flow andit is much smaller than the particle speed. When the lattice velocityspeed is defined based on the total part of u, the coefficient ofthe lattice still satisfies the symmetry assumption. Therefore, theTaylor expansion of feq [Eq. (29)] can be expressed as

feq = Ai

1 + ( u) (u u)

+ [( u) (u u)]2

22 (u u)

2

2

[( u) (u u)](u u)2

22 + [( u) (u u)]

3

63

(B2)

This expansion can be used with the earlier symmetry lattice to dis-cretize the velocity u u. The discretized velocities in the lattice aredefined asT= (0, 0), (1, 0), (0, 1), . . . ; therefore, the real particlespeed becomes= u + T. The original Boltzmann equation in thediscretized space can then be written as

fi

t= (T i+ u) fi

1

fi feqi

(B3)

It follows that the kinetic equations will become

=

fdT (B4)

(u u) = T f dT (B5)

e + 12

|(u u)|2 = DT+ DRDT

1

2 f|T|2 dT (B6)


12/12


e + p + 1

2|(u u)|2

(u u) = DT+ DR

DT

1

2 f|T|2TdT

(B7)

Whenu = (0, 0), Eqs. (B4B7) reduce to Eqs. (710). If there is aneffective macroscopic mean flow, only the relative velocityu uisconsidered in theTaylor expansion andthe kineticequations. Again,the relative velocity is much smaller than the particle speed, and sothe symmetry coefficients can be used but the evolution equation is

given by Eq. (B3). This method is proven useful in case 3.

Acknowledgment

Support given by the Research Grants Council of the Govern-ment of theHong Kong Special AdministrativeRegion under GrantsPolyU5174/02E, PolyU5303/03E, and PolyU1/02C is gratefully ac-knowledged.

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D. GaitondeAssociate Editor

aiaa.paperone-step aeroacoustics simulation using lattice boltzmann method

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