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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. 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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. 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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. 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