Investigations of the speed of sound in a cell variable influence in modeling of the flow in a plane channel and flow of the circle cylinder stream of viscous liquid calculating by the method of lattice Boltzmann equations

Main Article Content

A. A. Ostapenko
O. N. Bulanchuck
G. G. Bulanchuck


In this work the technique of modeling of two-dimensional flows of viscous liquid by method of lattice Boltzmann equations when sound speed in a cell isn't a fixed value is investigated. Such approach helps to avoid method’s instability and to control the size of a settlement cell, a step on time and Makh number. Ways of setting of some types of boundary conditions are investigated and the new way of setting of a condition of constant pressure is offered. To test proposed algorithm we simulates flow of a viscous liquid in open channel and fluid flow past a circular cylinder. The numerical solutions of the flow in a channel were compared with the corresponding results obtained by the finite element method in the Comsol Multiphysics package. The form of the Karman vortex path after the circular cylinder is shown and it is discussed the dependence between the flow pattern and Reynolds number. To analyze the accuracy of unsteady vortex flows we propose to calculate Strukhal's number in a new way based on the horizontal vortex distance. In the paper there is also investigated the dependence between the numerical result’s accuracy and local Makh number. It is shown that results can be obtained with the relative error less than 10% and the local Makh number should be less than 0.3.

Article Details

Materials Physics


Succi S. (1991). The Lattice Boltzmann Equation: A New Tool for Computational

Fluid-Dynamics. Physica D: Nonlinear Phenomena, 47, 219-230.

Strang G. (1973). An Analysis of The Finite Element Method. Englewood-Cliffs:

Prentice Hall.

Eymard R., Gallouet T., Herbim R. (2000). Finite volume methods. Techniques of

Scientific Computing, 7, 713-1020.

Belocerckovskyy S.M., Skobelev B.U. (1993). Discrete vortex method and

turbulence. Novosibirsk: ITPM. (in Rus)

Ogami Y. (1991). Viscous flow simulation using the discrete vortex model - the

diffusion velocity method. Computers & Fluids, 19(3), 433-441.

Monaghan J.J. (1988). An introduction to SPH. Computer Physics

Communications, 48, 88-96.

Kupershtoh A.L. (2012). 3D modeling with the lattice Boltzmann Method on GPU.

Vuchislitelnue metodu i programmirovanie (Numerical methods and programming), 13, 130-

(in Rus)

Grazyna K. (2006). The numerical solution of the transient heat conduction

problem using the lattice Boltzmann method. Scientific Research of the Institute of

Mathematic and Computer Science, 11, 23-30.

Leclaire S., Pellerin N., Reggio M., Trepanier J.-Y. (2013). Enhanced equilibrium

distribution functions for simulating immiscible multiphase flows with variable density ratios

in a class of lattice Boltzmann models. International Journal of Multiphase Flow, 57, 159-168.

Favier J., Revell A. , Pinelli A. (2014). A lattice Boltzmann-immersed boundary

method to simulate the fluid interaction with moving and slender flexible objects. Journal of

Computational Physics, 261, 145-161.

Anderl D., Bogner S., Rauh C., Rude U., Delgado A. (2014). Free surface lattice

Boltzmann with enhanced bubble model. Computers and Mathematics with Applications,

(2), 331-339.

Coupanec E. (2010). Boundary conditions for the lattoce Boltzmann method. Mass

conserving boundary conditions for moving walls. Trondheim: Norwegian University of

Science and Technology. Department of Energy and Process Engineering.

Mussa M. (2008). Numerical Simulation of Lid-Driven Cavity Flow Using the

Lattice Boltzmann Method. Applied Mathematics, 13, 236-240.

Hong X., Di W., Yuhe S. (2013). Research of Micro-Rectangular-Channel Flow

Based on Lattice Boltzmann Method. Research Journal of Applied Science, Engineering and

Technology, 6(14), 2520-2525.

Wolf-Gladrow D. (2005). Lattice-Gas Cellular Automata and Lattice Boltzmann

Models - An Introduction. Bremerhaven: Alfred Wegener Institute for Polar and Marine.

Sucop M. (2006). Lattice Boltzmann Modeling. An Introduction for Geoscientists

and Engineers. Miami: Springer.

Rettinger C. (2013). Fluid Flow Simulation using the Lattice Boltzmann Method

with multiple relaxation times. Erlander: Friedrich-Alexander Universuty ErlanderNuremberg.

Martinez D.O., Matthaeus W.H., Chen S., Montgomery D.C. (1994). Comparison

of spectral method and lattice Boltzmann simulations of two-dimentional hydrodynamics.

Physics of Fluids, 6(3), 1285-1298.

Mohamad A.A. (2011). Lattice Boltzmann Method. Fundamentals and Engineering

Applications with Computer Codes. London: Springer.

Agrawal T., Lin C. (2013). Implementation of an incompressible lattice Boltzmann

model on GPU to simulate Poiseuille flow. Proceeding of the Forties National Conference on

Fluid Mechanics and Fluid Power, 40, 991-995.

Horwitz J.A. (2013). Lattice Boltzmann Simulations of Multiphase Flows.

Illinois: University of Illinois at Urnana-Champaign.

Wang L. (2008). Direct simulation of viscous flow in a wavy pipe using the lattice

Boltzmann approach. Engineering Systems Modelling and Simulation, 1(1), 20-29.

Turinov A.I., Avramenko A.A., Basok B.I., Davudenko B.V. (2011). Microflows

modeling with the lattice Boltzmann method. Promushlennaya teplotehnika (Industrial heat

engineering), 33(2), 11-18. (in Rus)

Zaharov A.M., Senin D.S., Grachev E.A. (2014). Lattice Boltzmann modeling with

multirelaxationals times. Vuchislitelnue metodu i programmirovanie (Numerical methods and

programming), 15, 644-657. (in Rus)

Bikulov D.A., Senin D.S., Demin D.S., Dmitriev A.V., Grachev N.E. (2012).

Lattice Boltzmann Method on GPU claster. Vuchislitelnue metodu i programmirovanie

(Numerical methods and programming), 13, 13-19. (in Rus)

Samoilov D.A., Gubkin A.S. (2014). Numerical posibilities of the lattice

Boltzmann method. Vestnik Tiumenskogo Gosudarstvenogo Universiteta. FizikoMatematicheskie

nauki (Bulletin of Toumen state university. Physical and mathematical

science), 7, 83-91. (in Rus)

Liapin I.I. (2003). Introduction to the kinetic equations theory. Ekaterinburg:

UGTU. (in Rus)

Aslan E. (2014). Investigation of the Lattice Boltzmann SRT and MRT Stability for

Lid Driven Cavity Flow. International Journal of Materials, Mechanics and Manufacturing,

(4), 317-324.

Bulanchuk G., Bulanchuk O., Ostapenko A. (2015). Stability investigation of the

two-dimensional nine-vectors model of the lattice Boltzmann method for fluid flows in a

square cavity. Vestnik Kharkovskogo Nacionalnogo Universiteta im. V.N.Karazina. Seria:

Matematicheskoe modelirovanie. Informacionnue technologii. Avtomatizirovannue sistemu

upravlenia (Bulletin of Kharkov national University named after V.N.Karazin. Series:

Mathematical modeling. Information technology. Automated control system), 28, 113-125.

Franse M. (1999). Large Scale Lattice-Boltzmann Simulations. The Netherlands,

Enschede: PrintPartners Ipskamp.

He X. (1997). Lattice Boltzmann Model for the Incompressible Navier – Stokes

Equation. Journal of statistical physics, 88, 927–944.

Succi S. (2001). The Lattice Boltzmann Equation for Fluid Dinamics and Beyond.

Oxford: Univercity Press.

Latt J., Chopard B., Malaspinas O., Deville M., Michler A. (2008). Straight

velocity boundaries in the lattice Boltzmann method. Physical Review, 77, 1-17.

Belocerckovskyy O.M., Belocerckovskyy S.O., Gushin V.A. (1984). Numerical

modeling of the unsteady periodic flow of the viscous fluid past a cilinder. Vuchislitelnaya

matematica i matematicheskaya fizika (Computational mathematics and mathematical

physics), 24(8), 1207-1216.

Schlichting H. (1974). Boundary layer theory. Moskow: Nauka. (in Rus)

Cheng P. (1973). Tear flows. Moskow: Nauka. (in Rus)

Fomin G.M. (1974). About the vortexes circulations ans Carman track velocity.

Uchenue zapiski CAGI (Scientific notes of TSAGA), 2(4), 99-102. (in Rus)