Skip to main content

Full text of "Applications of the Lattice Boltzmann Method to Complex and Turbulent Flows"

See other formats


NASA/CR-2002-211659 
ICASE Report No. 2002-19 




Applications of the Lattice Boltzmann Method to 
Complex and Turbulent Flows 

Li-Shi Luo 

ICASE, Hampton, Virginia 

Dewei Qi 

Western Michigan University, Kalamazoo, Michigan 

Lian-Ping Wang 

University of Delaware, Newark, Delaware 




July 2002 



The NASA STI Program Office ... in Profile 



Since its founding, NASA has been dedicated 
to the advancement of aeronautics and space 
science. The NASA Scientific and Technical 
Information (STI) Program Office plays a key 
part in helping NASA maintain this 
important role. 

The NASA STI Program Office is operated by 
Langley Research Center, the lead center for 
NASA's scientific and technical information. 
The NASA STI Program Office provides 
access to the NASA STI Database, the 
largest collection of aeronautical and space 
science STI in the world. The Program Office 
is also NASA's institutional mechanism for 
disseminating the results of its research and 
development activities. These results are 
pubhshed by NASA in the NASA STI Report 
Series, which includes the following report 
types: 

• TECHNICAL PUBLICATION. Reports of 
completed research or a major significant 
phase of research that present the results 
of NASA programs and include extensive 
data or theoretical analysis. Includes 
compilations of significant scientific and 
technical data and information deemed 

to be of continuing reference value. NASA's 
counterpart of peer-reviewed formal 
professional papers, but having less 
stringent limitations on manuscript 
length and extent of graphic 
presentations. 

• TECHNICAL MEMORANDUM. 
Scientific and technical findings that are 
preliminary or of specialized interest, 
e.g., quick release reports, working 
papers, and bibliographies that contain 
minimal annotation. Does not contain 
extensive analysis. 

• CONTRACTOR REPORT. Scientific and 
technical findings by NASA-sponsored 
contractors and grantees. 



• CONFERENCE PUBLICATIONS. 
Collected papers from scientific and 
technical conferences, symposia, 
seminars, or other meetings sponsored or 
cosponsored by NASA. 

• SPECIAL PUBLICATION. Scientific, 
technical, or historical information from 
NASA programs, projects, and missions, 
often concerned with subjects having 
substantial public interest. 

• TECHNICAL TRANSLATION. Enghsh- 
language translations of foreign scientific 
and technical material pertinent to 
NASA's mission. 

Specialized services that complement the 
STI Program Office's diverse offerings include 
creating custom thesauri, building customized 
data bases, organizing and publishing 
research results . . . even providing videos. 

For more information about the NASA STI 
Program Office, see the following: 

• Access the NASA STI Program Home 
Page at http://www.sti.nasa.gov 

• Email your question via the Internet to 
help@sti.nasa.gov 

• Fax your question to the NASA STI 
Help Desk at (301) 621-0134 

• Telephone the NASA STI Help Desk at 
(301) 621-0390 

• Write to: 

NASA STI Help Desk 

NASA Center for AeroSpace Information 

7121 Standard Drive 

Hanover, MD 21076-1320 



NASA/CR-2002-211659 
ICASE Report No. 2002-19 




Applications of the Lattice Boltzmann Method to 
Complex and Turbulent Flows 

Li-Shi Luo 

ICASE, Hampton, Virginia 

Dewei Qi 

Western Michigan University, Kalamazoo, Michigan 

Lian-Ping Wang 

University of Delaware, Newark, Delaware 

ICASE 

NASA Langley Research Center 
Hampton, Virginia 

Operated by Universities Space Research Association 




Prepared for Langley Research Center 
under Contract NAS 1-97046 



July 2002 



Available from the following: 

NASA Center for AeroSpace Information (CASI) National Technical hiformation Service (NTIS) 

7121 Standard Drive 5285 Port Royal Road 

Hanover, MD 21076-1320 Springfield, VA 22161-2171 

(301) 621-0390 (703) 487-4650 



APPLICATIONS OF THE LATTICE BOLTZMANN METHOD TO COMPLEX AND 

TURBULENT FLOWS 

LI-SHI LUO*, DEWEI Qlt, AND LIAN-PING WANG* 

Abstract. We briefly review the method of the lattice Boltzmann equation (LBE). We show the three- 
dimensional LBE simulation results for a non-spherical particle in Couette flow and 16 particles in sedimen- 
tation in fluid. We compare the LBE simulation of the three-dimensional homogeneous isotropic turbulence 
flow in a periodic cubic box of the size 128^ with the pseudo-spectral simulation, and flnd that the two 
results agree well with each other but the LBE method is more dissipative than the pseudo-spectral method 
in small scales, as expected. 

Key words, lattice Boltzmann method, turbulent flow, 3D homogeneous isotropic turbulence, spectral 
method, non-spherical particulate suspensions 

Subject classification. Fluid Mechanics 

1. Introduction. More than a decade ago, the lattice-gas automata (LGA) [5, 24, 6] and the lattice 
Boltzmann equation (LBE) [17, 12, 2, 22] were proposed as alternatives for computational fluid dynamics 
(CFD). Since their inception, the lattice-gas and lattice Boltzmann methods have attracted much interest 
in the physics community. However, it was only very recently that the LGA and LBE methods started 
to gain the attention from CFD community. The lattice-gas and lattice Boltzmann methods have been 
particularly successful in simulations of fluid flow applications involving complicated boundaries or/and 
complex fluids, such as turbulent external flow over complicated structures, the Rayleigh- Taylor instability 
between two fluids, multi-component fluids through porous media, viscoelastic fluids, free boundaries in flow 
systems, particulate suspensions in fluid, chemical reactive flows and combustions, magnetohydrodynamics, 
crystallization, and other complex systems (see recent reviews [3, 16] and references therein). 

Historically, models of the lattice Boltzmann equation evolved from the lattice-gas automata [5, 24, 6]. 
Recently, it has been shown that the LBE is a special discretized form of the continuous Boltzmann equation 
[8, 9]. For the sake of simplicity without loss of generality, we shall demonstrate an a priori derivation of 
the lattice Boltzmann equation from the continuous Boltzmann equation with the single relaxation time 
(Bhatnagar-Gross-Krook) approximation [1]. The Boltzmann BGK equation can be written in the form of 
an ordinary differential equation: 



^^f^\f=\f'''^ ^"^-(4^-p 



29 



(1.1) 



where Dt = dt + $-'V, f = f{x, ^, t) is the single particle distribution function, A is the relaxation time, 
and /(") is the Boltzmann distribution function in i?-dimensions, in which p, u and 9 = kBT/m are the 
macroscopic density of mass, the velocity, and the normalized temperature, respectively, T, ks and m are 
temperature, the Boltzmann constant, and particle mass. The macroscopic variables are the moments of the 



*ICASE, Mail Stop 132C, NASA Langley Research Center, 3 West Reid Street, Building 1152, Hampton, Virginia 23681- 
2199 (email address: luo@icase.edu). This research was supported by the National Aeronautics and Space Administration under 
NASA Contract No. NASl-97046 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, 
Virginia 23681-2199. 

tDepartment of Paper and Printing Science and Engineering, Western Michigan University, Kalamazoo, Michigan 49008. 

■t-Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716. 



distribution function / with respect to the molecular velocity ^: 

p0 = \j{i- uf fdi = \j{i- uf /(°) di . 

Equation (1.1) can be formally integrated over a time interval 5t: 



(1.2a) 
(1.2b) 
(1.2c) 



fix + ^5t, t t + 5t)= e-'^l^ fix, t t) + ^e-^'/^ j ' e*'/^ /(«) (a; + ^i', ^, i + t') dt' . (1. 



3) 



Assuming that 5t is small enough and /(") is smooth enough locally, and neglecting the terms of the order 
0((5|) or smaller in the Taylor expansion of the right hand side of (1.3), we obtain 



fix + ^St, ^, t + St)- fix, ^, t) = --[fix, ^, t) - /(°) ix, ^, t)] , 



(1.4) 



where t = X/5t is the dimensionless relaxation time. The equilibrium /(") can be expanded as a Taylor 



series in u up to u 



f 



(eq) 



(27r0)^/2 ''''P I 20 



6> 26>2 26 



(1.5) 



To obtain the Navier-Stokes equations, the hydrodynamic moments ip, pu, and p9) and their fluxes 
must be preserved in finite discretized momentum space {^a|a = 1, 2, . . . , &}, i.e.. 



a a 

a a 

P^ = l E(^« - «)' f- = l E(^« - «)' /a'"^ ' 



(1.6a) 
(1.6b) 

(1.6c) 



where /„ = faix, t) = Wa fix, $,a, t) [8, 9]. It turns out that these moments can be evaluated exactly in 
discretized momentum space by using Gaussian-type quadrature [8, 9, 23]. 

We can derive the nine-velocity athermal LBE model on a square lattice in two-dimensions 

faiXi + eJt, t + 5t)- faiXi, t) = --[Uxi, t) - fi^'Hxi, t)] , (1.7) 

T 

where the equihbrium fa , the discrete velocity set {ca}, and the weight coeflicients {wa} are given by 

' 4/9, a = 0, 

(1.8b) 



1+ ' "^ ' + 
(0, 0), a = 



(±1, 0)c, (0, ±l)c, a = l, 2, 3,4, 
(±1, ±l)V2c, a = 5, 6, 7, 8, 



4/9, 
1/9, 
1/36, a = 5, 6, 7 



a = 1, 2, 3, 4, 



and c = Sx/St- Equation (1.7) involves only local calculations and uniform communications to the nearest 
neighbors. Therefore it is easy to implement and natural to massively parallel computers. 



The (incompressible) Navier-Stokes equation derived from the above LBE model is: 

pdtu + pu-Vu = -VP + pvV'^u, (1.9) 

with the isothermal ideal gas equation of state, the viscosity, and the sound speed given by 

P = cIp, v=(^-^cl5t, c, = -^c. (1.10) 

It should be noted that the factor —1/2 in the above formula for v accounts for the numerical viscosity due 
to the second order derivatives of /„. This correction in v formally makes the LGA and LBE methods second 
order accurate. Similarly, we can derive the six-velocity and seven-velocity models on a triangular lattice in 
two-dimensions, and the twenty-seven- velocity models on a cubic lattice in three-dimensions [9] . 

There have been some significant progress made recently to improve the lattice Boltzmann method: (i) 
the generalized lattice Boltzmann equation with multiple relaxation times which overcomes some shortcom- 
ings of the lattice BGK equation [14]; (ii) use of grid refinement [4] and body-fitted mesh [10, 7] with inter- 
polation/extrapolation techniques; (iii) adaptation of unstructured grid by using the finite element method 
or the characteristic Galerkin method; (iv) application of implicit scheme for steady state calculation and 
multi-grid technique to accelerate convergence (see a recent review [16] for further references). 

In what follows we shall demonstrate the applications of the LBE method to simulate the fiow of non- 
spherical particulate suspensions in fiuid and homogeneous isotropic turbulence in a periodic box. 

2. LBE Simulation of Flows of Non- Spherical Particulate Suspensions. The fiow of particulate 
suspensions in fiuid is difiicult to quantify experimentally and to simulate numerically in some cases. Yet 
the fiow of particulate suspensions is important to industrial applications such as fiuidized beds. There have 
been some successful simulations of the fiow of spherical suspensions by using conventional CFD methods, 
such as the finite element method. However, the simulation of the fiow of non-spherical suspensions still 
remains as a challenge to the conventional CFD methods. Recently the LBE method has been successfully 
applied to simulate the fiow of non-spherical suspensions in three-dimensions [19, 20]. The success of the 
LBE method to this problem relies on the fact that the LBE method can easily handle the particle-fiuid 
interfaces [15], and accurately evaluate the force on the particle due to the fiuid fiow [18]. 

We first simulate a single non-spherical particle in the Couette fiow. The equilibrium states in a non- 
spherical particulate suspension in a 3D Couette fiow are simulated for a particle Reynolds number up to 
320. Particle geometries include prolate and oblate spheroids, cylinders and discs. We show that the inertial 
effect at any finite Reynolds number qualitatively changes the rotational motion of the suspension, contrary 
to Jeffery's theory at zero Reynolds number [13]. At a non-zero Reynolds number, a non-spherical particle 
reaches an equilibrium state in which its longest and shortest axes are aligned perpendicular and parallel 
to the vorticity vector of the fiow, respectively. This equilibrium state is unique, dynamically stable, fully 
determined by the inertial effect, the maximum energy dissipation state. Systems of either fifty cylinders or 
fifty discs in Couette fiow are also simulated. Multi-particle interactions significantly change the equilibrium 
orientation of solid particle. The effect is stronger for cylinders than for discs. The details of this work will 
be reported elsewhere [21]. 

Figure 1 shows a 3D LBE simulation of sixteen cylindrical particles falling under the infiuence of gravity. 
The left figure illustrates the time evolution of the entire system of sixteen particles, while the right figure 
demonstrates the formation of inverted T configurations in the sedimentation, which has been observed 
experimentally. To the best of our knowledge, this phenomenon was first reproduced numerically by the 
LBE direct numerical simulation [20]. 



^\x 



^. 









gs* 



.•p^ 



t^ 






*> 




mf: 



>L. 






^ 



If 



Fig. 1. 3D LBE simulation of particles sedimentation in fluid. Particle size is D = 12 and L = 24. System size is 
Nx X Ny X Nz = 140 X 150 x 35. The averaged single-particle Re « 16.9. (left) Evolution of 16 particles (from left to right and 
top to bottom), (right) Formation of inverted T configurations which are also observed in experiment. 

Table 3.1 
Parameters in lattice Boltzmann and pseudo-spectral simulations: L is the length of box side; N^ is the system size; v is 
the viscosity; u' is the RMS fluctuation of the initial velocity field; dt is the time step size; T is total integration time, K^\ is 
the Taylor microscale Reynolds number; and M is the Mach number. 



Method 


L 


7V3 


V 


w' 


dt 


T 


ReA 


M 


Spectral 


27r 


1283 


0.01189 


0.993311 


0.002 


2 


35.0 





LBE 


128 


1283 


0.009869 


0.040471 


1 


1000 


35.0 


0.0687 



3. LBE Simulation of 3D Homogeneous Isotropic Turbulence. Homogeneous isotropic turbu- 
lence in a three-dimensional periodic cubic box remains as a stand problem in the field of direct numerical 
simulation of turbulence. Due to the simplicity of the boundary conditions, the pseudo-spectral method can 
be easily used to simulate the fiow. Because of its accuracy, the pseudo-spectral result is often used as a 
benchmark standard. Here the LBE simulation of the fiow is compared with the pseudo-spectral simulation. 



The parameters of the simulation are given in Table 3.1. The initial condition is a random velocity field 





: (a) 




Of 


-1:^ 




Energy Spectrum 


■1 


I' 


/ 








\ 


1 




< 

■3 


r 










m 


■4 


r 










Um 




: 


o 


Lattice Boltzniaiin 




I %s cKT 


-S 












ffli mA ily^ 












10 


20 30 To 



Mean kinetic energy K and dissipation rate e 




K Speclral 



e Spectral 

K Lattice Boltzmaim 

O e Lattice Boltzmaim 



O.S 



Fig. 2. LBE vs. Pseudo-spectral DNS of 3D homogeneous isotropic turbulence. System size is 128^. Re;^ = 35. (a) The 
energy spectrum E{k) as a function of time, (b) The decay of the mean kinetic energy K and dissipation rate e. The results 
from the LBE simulation are scaled according to the dimensions used in the spectral simulation. 



with a Gaussian distribution and a compact energy spectrum 



k ^ 
E{k) oc -— exp 
fco 



1' 



The boundary conditions are periodic in three dimensions. The Taylor microscale Reynolds number is defined 
as 



Re> 



2K{t = 0) A 
3 v 



,X 



where K{t = 0) = {uq/2)v = (3'«|^g/2)y is the volume averaged kinetic energy (of the initial zero-mean 
Gaussian velocity field uq with RMS component Wrms), and A is the transverse Taylor microscopic scale: 

A 



^A5 



vui, 



s/e, 



where e is the dissipation rate. 

Figure 2 shows the energy spectrum E{k) as function of time, and the time evolution of the mean kinetic 
energy K and dissipation rate e. The lattice Boltzmann results (symbols) are compared with the pseudo- 
spectral results (lines). The LBE results agree well with the pseudo-spectral results. Obviously the LBE 
method is more dissipative, especially at high wave numbers k > |fcmax, where fcmax = ^N, and TV is the 
number of mesh nodes in each direction. This is because the LBE method is only second order accurate in 
space and time and thus more dissipative than the pseudo-spectral method. 

4. Conclusions and Discussion. The above simulations were performed on a Beowulf cluster of 
Pentium CPUs. For the simulation of the particulate suspension, the code consists two part: the lattice 
Boltzmann method for the fiuid and molecular dynamics (MD) for the solid particles [19]. Even though 
the MD part of the code is not yet parallelized, the speed of the code still scales well with the number of 
CPUs up to 32 CPUs when the system size is 64^ and with fifty particles. Presently we can easily simulate 
a system of a few hundred particles on our Beowulf system. 



As for the simulation of the 3D homogeneous isotropic turbulence, the LBE code without optimization 
has the same speed as the spectral code with a Beowulf cluster of eight CPUs (about Is per time step). 
However, we do expect the LBE code will scale linearly with the number of CPUs, but not the spectral code. 

Our current research includes particulate suspension in fluid with high volume fraction of particles, vis- 
coelastic and non-Newtonian fluids, and forced or free-decay homogeneous isotropic turbulence in a periodic 
cube by using the lattice Boltzmann method on massively parallel computers. 

REFERENCES 

[1] P. L. Bhatnagar, E. p. Gross, and M. Krook, A model for collision processes in gases. I. Small 
amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), pp. 511- 
525. 
[2] H. Chen, S. Chen, and W. H. Matthaeus, Recovery of the Navier-Stokes equations using a lattice- 
gas Boltzmann method, Phys. Rev. A, 45 (1992), pp. R5339-5342. 
[3] S. Chen and G. D. Doolen, Lattice Boltzmann method for fluid flows , Annu. Rev. Fluid Mech., 30 

(1998), pp. 329-364. 
[4] O. FiLiPPOVA AND D. Hanel, Grid refinement for lattice-BGK models, J. Comput. Phys., 147 (1998), 

pp. 219-228. 
[5] U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice-gas automata for the Navier-Stokes equation, 

Phys. Rev. Lett., 56 (1986), pp. 1505-1508. 
[6] U. Frisch, D. d'Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J. -P. Rivet, 
Lattice gas hydrodynamics in two and three dimensions. Complex Systems, 1 (1987), pp. 649-707. 
[7] X. He and G. Doolen, Lattice Boltzmann method on curvilinear coordinates system: Flow around a 

circular cylinder, J. Computat. Phys., 134 (1997), pp. 306-315. 
[8] X. He and L.-S. Luo, A priori derivation of the lattice Boltzmann equation, Phys. Rev. E, 55 (1997), 
pp. R6333-R6336. 

[9] , Theory of the lattice Boltzmann equation: From the Boltzmann equation to the lattice Boltzmann 

equation, Phys. Rev. E, 56 (1997), pp. 6811-6817. 
[10] X. He, L.-S. Luo, and M. Dembo, Some progress in lattice Boltzmann method. Part L Nonuniform 
mesh grids, J. Computat. Phys., 129 (1996), pp. 357-363. 

[11] , Some Progress in the lattice Boltzmann method. Reynolds number enhancement in simulations, 

Physica A, 239 (1997), pp. 276-285. 
[12] F. J. HiGUERA, S. Succi, AND R. Benzi, Lattice gas-dynamics with enhanced collisions, Euro- 

phys. Lett., 9 (1989), pp. 345-349. 
[13] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. R. London Ser. 

A, 102 (1922), pp. 161-179. 
[14] P. Lallemand and L.-S. Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, 
isotropy, Galilean invariance, and stability Phys. Rev. E, 61 (2000), pp. 6546-6562. 

[15] , Lattice Boltzmann method for moving boundary problem, submitted to J. Computat. Phys. (2001). 

[16] L.-S. Luo, The lattice-gas and lattice Boltzmann methods: Past, present, and future, in Proceedings 
of International Conference on Applied Computational Fluid Dynamics, Beijing, China, October 
17-20, 2000, edited by J.-H. Wu and Z.-J. Zhu (Beijing, 2000), pp. 52-83. 
[17] G. R. McNamara and G. Zanetti, Use of the Boltzmann equation to simulate lattice-gas automata. 



Phys. Rev. Lett., 61 (1988), pp. 2332-2335. 
[18] R. Mei, D. Yu, W. Shyy, and L.-S. Luo, Force evaluation in the lattice Boltzmann method involving 

curved geometry, Phys. Rev. E, 65 (2002), 041203. 
[19] D. Qi, Lattice- Boltzmann simulations of particles in non-zero-Reynolds-number flows , J. Fluid Mech., 

385 (1999), pp. 41-62. 
[20] , Simulations of fluidization of cylindrical multiparticles in a three-dimensional space, Int. J. Mul- 
tiphase Flow, 27 (2001), pp. 107-118. 
[21] D. Qi AND L.-S. Luo, Inertial effect of non-spherical suspension in 3D Couette flow, submitted to 

Phys. Rev. Lett. (2001). 
[22] Y. QiAN, D. d'Humieres, and P. Lallemand, Lattice BGK models for Navier-Stokes equation, 

Europhys. Lett., 17 (1992), pp. 479-484. 
[23] X. Shan and X. He, Discretization of the velocity space in the solution of the Boltzmann equation, 

Phys. Rev. Lett., 80 (1998), pp. 65-68. 
[24] S. Wolfram, Cellular automaton fluids 1: Basic theory, J. Stat. Phys., 45 (1986), pp. 471-526. 



REPORT DOCUMENTATION PAGE 



Form Approved 
0MB No. 0704-0188 



Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 
gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 
collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson 
Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503. 



1. AGENCY USE ONLYfieave blank) 



2. REPORT DATE 

July 2002 



3. REPORT TYPE AND DATES COVERED 

Contractor Report 



4. TITLE AND SUBTITLE 

APPLICATIONS OF THE LATTICE BOLTZMANN METHOD TO 
COMPLEX AND TURBULENT FLOWS 



6. AUTHOR(S) 

Li-Shi Luo, Dewei Qi, and Lian-Ping Wang 



5. FUNDING NUMBERS 

C NASl-97046 
WU 505-90-52-01 



7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

ICASE 

Mail Stop 132C 

NASA Langley Research Center 

Hampton, VA 23681-2199 



8. PERFORMING ORGANIZATION 
REPORT NUMBER 

ICASE Report No. 2002-19 



9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 

National Aeronautics and Space Administration 
Langley Research Center 
Hampton, VA 23681-2199 



10. SPONSORING/MONITORING 
AGENCY REPORT NUMBER 

NASA/CR-2002-211659 
ICASE Report No. 2002-19 



11. SUPPLEMENTARY NOTES 

Langley Technical Monitor: Dennis M. Bushnell 

Final Report 

To appear in the Lecture Notes in Computational Science and Engineering, Vol. 21, 2002. 



12a. DISTRIBUTION/AVAILABILITY STATEMENT 

Unclassified-Unlimited 

Subject Category 34 

Distribution: Nonstandard 

Availability: NASA-CASI (301) 621-0390 



12b. DISTRIBUTION CODE 



13. ABSTRACT (Maximum 200 words) 

We briefly review the method of the lattice Boltzmann equation (LBE). We show the three-dimensional LBE 
simulation results for a non-spherical particle in Couette flow and 16 particles in sedimentation in fluid. We compare 
the LBE simulation of the three-dimensional homogeneous isotropic turbulence flow in a periodic cubic box of the 
size 128^ with the pseudo-spectral simulation, and find that the two results agree well with each other but the LBE 
method is more dissipative than the pseudo-spectral method in small scales, as expected. 



14. SUBJECT TERMS 

lattice Boltzmann method, turbulent flow, 3D homogeneous isotropic turbulence, 
spectral method, non-spherical particulate suspensions 



15. NUMBER OF PAGES 

12 



16. PRICE CODE 

A03 



17. SECURITY CLASSIFICATION 
OF REPORT 

Unclassified 



18. SECURITY CLASSIFICATION 
OF THIS PAGE 

Unclassified 



19. 



SECURITY CLASSIFICATION 
OF ABSTRACT 



20. LIMITATION 
OF ABSTRACT 



NSN 7540-01-280-5500 



Standard Form 298(Rev. 2-89) 

Prescribed by ANSI Std. Z39-18 
298-102