Numerical Simulation of Lid-Driven Cavity Flow by the …€¦ · · 2012-10-16Numerical...
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1 st Regional Conference on Mechanical Engineering-RCME2011 March 2011, Islamic Azad University – East Tehran Branch, Iran 1 RCME2011-02-0042 Numerical Simulation of Lid-Driven Cavity Flow by the Lattice Kinetic Scheme Mohammad Hossein Borghei 1 , Morteza Heydari 2 , S. Mostafa Hosseinalipour 3 , Mehdi Navidbakhsh 4 1 M.Sc. Student, Iran University of Science and Technology; [email protected] 2 PhD Student, Iran University of Science and Technology; [email protected] 3 Associate Professor, Iran University of Science and Technology; [email protected] 4 Associate Professor, Iran University of Science and Technology; [email protected] Abstract A new numerical method for incompressible Newtonian fluid flows based on the lattice Boltzmann method (LBM) is proposed. In this method, the relaxation time in the BGK collision term is kept at unity taking account of numerical stability. Compared with the lattice Boltzmann method, the lattice kinetic scheme (LKS) can save the computer memory since there is no need to store the density distributions. The implementation of the boundary condition is direct and just the same as the convectional Navier-Stokes solvers. In this study, the lattice kinetic scheme is applied to two dimensional lid- driven cavity flow as a benchmark problem in fluid dynamics. Obtained results have been compared very well with single relaxation time (SRT) results and also available data in the literature. The simulations indicate that the method can be used for different practical fluid flows very well. Keywords: lattice kinetic, lattice Boltzmann method, lid-driven cavity, single relaxation time. Introduction In recent years, the lattice Boltzmann method has been developed into an alternative promising tool for fluid mechanics. It has been widely used in many kinds of complex flows such as turbulent flows, multiphase flows, and microflows [1]. However, there are still some items in need of further study. One is the collision model. The Bhatnagar-Gross-Krook (BGK) model with a single relaxation time is usually used for the collision term. The shortcomings of the BGK model are pointed out in the works of d’Humieres [2] and Lallemand and Luo [3]. The other is the boundary condition. The bounce-back scheme in the LBM was originally taken from the LGA method. Although this heuristic scheme is very simple to implement, it is found to be the first order in the numerical accuracy at the boundaries [4,5]. In order to improve the numerical accuracy, other boundary treatments have been proposed. It appears, however, that the extension of these treatments to the complex boundary surface is difficult. Chen et al. [6] proposed a boundary condition using a second-order extrapolation scheme of the distributions in the flow to obtain the unknown particle distribution functions on the boundaries. When the flow problems with complex geometries, especially in the three dimensions are encountered, the determination of the unknown particle directions is troublesome. All the implementations are not so direct since on the boundaries the macroscopic variables, not the density distributions, are given. Related to these two difficulties, a lattice kinetic scheme for the incompressible viscous flows was developed by Inamuro [7]. This scheme is based on the idea that if the dimensionless relaxation time in the LBM with the BGK model is set to unity, the macroscopic variables such as velocity components and density instead of the density distribution functions become the dependent variables in the computation so that the numerical stability can be obtained for relatively high Reynolds number flows; nevertheless one is able to determine the fluid viscosity using a constant parameter appearing in the additional term of the equilibrium distribution function. As compared to the standard LBM, this scheme can save computer memory because there is no need to store the density distribution functions. The implementation of the boundary condition is very easy since on the boundaries only the macroscopic variables rather than the density distributions are needed as for the conventional Navier- Stokes solvers. This feature is very useful when the flow problems with complex geometry are concerned. Peng et al. [8] developed this idea for simulation of fluid flows on arbitrary meshes using technique of Taylor series expansion and least-square-based lattice Boltzmann method. Recently, Yoshino et al. [9] have employed this method to simulate incompressible non- Newtonian fluid flows. In this paper, a lattice kinetic scheme for incompressible viscous flows is used to simulate two dimensional lid-driven cavity flows in laminar region. To verify the accuracy of the scheme, obtained results have been compared with available standard results. Theory Before introducing lattice kinetic scheme, we will give a brief description about the original lattice Boltzmann method. A. Lattice Boltzmann method The evolution equation for the density distribution t x f , in the two dimensions with the particle velocity e can be written as )] , x ( ) , x ( [ ) , x ( ) , e x ( t f t f t f t t t f eq ð- ð- ð= ðD ð+ ðD ð+ (1) where t is the single relaxation time, eq f is the corresponding equilibrium density distribution function, Click to buy NOW! P D F - X C h a n g e V i e w e r w w w . d o c u - t ra c k . c o m Click to buy NOW! P D F - X C h a n g e V i e w e r w w w . d o c u - t ra c k . c o m
Transcript of Numerical Simulation of Lid-Driven Cavity Flow by the …€¦ · · 2012-10-16Numerical...
1st Regional Conference on Mechanical Engineering-RCME2011March 2011, Islamic Azad University – East Tehran Branch, Iran
1
RCME2011-02-0042
Numerical Simulation of Lid-Driven Cavity Flow by the Lattice Kinetic Scheme
Mohammad Hossein Borghei1, Morteza Heydari2, S. Mostafa Hosseinalipour3, Mehdi Navidbakhsh4
1 M.Sc. Student, Iran University of Science and Technology; [email protected] PhD Student, Iran University of Science and Technology; [email protected]
3 Associate Professor, Iran University of Science and Technology; [email protected] Associate Professor, Iran University of Science and Technology; [email protected]
AbstractA new numerical method for incompressible Newtonianfluid flows based on the lattice Boltzmann method(LBM) is proposed. In this method, the relaxation timein the BGK collision term is kept at unity taking accountof numerical stability. Compared with the latticeBoltzmann method, the lattice kinetic scheme (LKS)can save the computer memory since there is no need tostore the density distributions. The implementation ofthe boundary condition is direct and just the same as theconvectional Navier-Stokes solvers. In this study, thelattice kinetic scheme is applied to two dimensional lid-driven cavity flow as a benchmark problem in fluiddynamics. Obtained results have been compared verywell with single relaxation time (SRT) results and alsoavailable data in the literature. The simulations indicatethat the method can be used for different practical fluidflows very well.Keywords: lattice kinetic, lattice Boltzmann method,lid-driven cavity, single relaxation time.
IntroductionIn recent years, the lattice Boltzmann method has beendeveloped into an alternative promising tool for fluidmechanics. It has been widely used in many kinds ofcomplex flows such as turbulent flows, multiphaseflows, and microflows [1]. However, there are still someitems in need of further study. One is the collisionmodel. The Bhatnagar-Gross-Krook (BGK) model witha single relaxation time is usually used for the collisionterm. The shortcomings of the BGK model are pointedout in the works of d’Humieres [2] and Lallemand andLuo [3]. The other is the boundary condition. Thebounce-back scheme in the LBM was originally takenfrom the LGA method. Although this heuristic schemeis very simple to implement, it is found to be the firstorder in the numerical accuracy at the boundaries [4,5].In order to improve the numerical accuracy, otherboundary treatments have been proposed. It appears,however, that the extension of these treatments to thecomplex boundary surface is difficult. Chen et al. [6]proposed a boundary condition using a second-orderextrapolation scheme of the distributions in the flow toobtain the unknown particle distribution functions onthe boundaries. When the flow problems with complexgeometries, especially in the three dimensions areencountered, the determination of the unknown particledirections is troublesome. All the implementations arenot so direct since on the boundaries the macroscopicvariables, not the density distributions, are given.
Related to these two difficulties, a lattice kineticscheme for the incompressible viscous flows wasdeveloped by Inamuro [7]. This scheme is based on theidea that if the dimensionless relaxation time in theLBM with the BGK model is set to unity, themacroscopic variables such as velocity components anddensity instead of the density distribution functionsbecome the dependent variables in the computation sothat the numerical stability can be obtained for relativelyhigh Reynolds number flows; nevertheless one is able todetermine the fluid viscosity using a constant parameterappearing in the additional term of the equilibriumdistribution function. As compared to the standardLBM, this scheme can save computer memory becausethere is no need to store the density distributionfunctions. The implementation of the boundarycondition is very easy since on the boundaries only themacroscopic variables rather than the densitydistributions are needed as for the conventional Navier-Stokes solvers. This feature is very useful when the flowproblems with complex geometry are concerned.
Peng et al. [8] developed this idea for simulation offluid flows on arbitrary meshes using technique ofTaylor series expansion and least-square-based latticeBoltzmann method. Recently, Yoshino et al. [9] haveemployed this method to simulate incompressible non-Newtonian fluid flows.
In this paper, a lattice kinetic scheme forincompressible viscous flows is used to simulate twodimensional lid-driven cavity flows in laminar region.To verify the accuracy of the scheme, obtained resultshave been compared with available standard results.
TheoryBefore introducing lattice kinetic scheme, we will give abrief description about the original lattice Boltzmannmethod.
A. Lattice Boltzmann methodThe evolution equation for the density distribution
txf , in the two dimensions with the particle velocity
e can be written as
)],x(),x([
),x(),ex(tftftftttf
eq (1)
where t is the single relaxation time, eqf is thecorresponding equilibrium density distribution function,
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