LATTICE BOLTZMANN SIMULATION OF FLUID FLOW IN A LID DRIVEN CAVITY

Transcription

1 LATTICE BOLTZMANN SIMULATION OF FLUID FLOW IN A LID DRIVEN CAVITY M. Y. Gokhale, Ignatius Fernandes Maharashtra Institute of Technology, Pune 4 38, India University of Pune, India mukundyg@yahoo.co.in, ignatius4u@gmail.com Abstract -Numerical simulation of laminar fluid flow in a lid driven cavity is performed using lattice Boltzmann method. A square cavity with the top lid moving from left to right with a uniform velocity of. is considered. The influence of Reynolds number on the velocity profiles and the streamline plots is studied by considering a range of Reynolds numbers from to. Half-way bounce back boundary conditions are employed in the numerical simulation. The results obtained show a fine agreement with proven theories for fluid flow in a lid driven cavity. Keywords: lid driven cavity, Newtonian fluids, lattice Boltzmann method I. INTRODUCTION Fluid dynamics has evolved around solving Navier- Stokes (NS) equations, eventually, becoming the most important tool in understanding the physics of many complex fluid flow problems. However, in recent years, the lattice Boltzmann method (LBM) method has gained much attention for its ability to simulate fluid flows, and for its potential advantages over conventional numerical solution of the NS equations. The use of LBM is justified by standard benchmark problems which have been simulated by LBM and the results are shown to agree quite well with the corresponding NS solutions. Fluid flow in a lid driven cavity is very common in engineering and geophysical applications with huge research being carried out in this area. A number of researchers have contributes in this area. Mohamad [] gave an overview on application of Lattice Boltzmann method (LBM) for simulations of wide range of isothermal and non-isothermal fluid flows, also presenting the advantages of LBM over other conventional methods. Cox et al [5] developed a least squares finite element approach for the governing equations of generalized Newtonian and viscoelastic flows using a Carreau model. Garg et al [6] carried out simulation of laminar flow between finite rotating disks with a shroud. They used velocity and stream function formulation. Rogers et al [7] presented an experimental study of flow visualization in a rotating cavity with two steel discs and a peripheral polycarbonate shroud. Fox et al [8] presented an experimental investigation to simulate flow associated with two tubes arranged in the geometry of a cross. Alves et al [9] presented an experimental study of flow of a Newtonian flow and Boger fluids through sudden square-square contractions, giving a comparative results with finite volume method. Chai et al [] presented a multiple-relaxation-time lattice Boltzmann model for generalized Newtonian fluid. Mendu et al [] applied lattice Boltzmann method for simulation of two dimensional fluid flow in a square cavity driven by a periodically oscillating lid. They carried out the simulation for values, 4 and of Reynolds number. This paper simulates flow in a square lid driven cavity with respect to the Reynolds number and the aspect ratio of the problem. II. GOVERNING EQUATIONS The governing equations for fluid flow are the continuity equation and the Navier Stokes equations. For D steady, isothermal and incompressible laminar flow, these equations in Cartesian coordinates are as follows []: The continuity equation is The momentum equations are u v x y u u u p u u u v t x y x x y () (a) ISSN (Prin: , Volume-, Issue-, 4

2 v v v p v v u v t x y x x y (b) where is the relaxation time and is related to the kinematic viscosity by where u and v are the velocities in the x and y directions, respectively, is the viscosity, is the density and p is the pressure. f eq ( x, dx (.5) 3dt (5) i, in equation (5), is the corresponding equilibrium distribution function for DQ9 given by eq fi ( x, wi ( x, ( e. (, ) ( e. u(, ( u(,. u(, ) i u x t 4 i x x x (6) cs cs cs where u( is the velocity and wi is the weight coefficient with values Fig.. Geometric description of the problem III. NUMERICAL METHOD AND PROBLEM DESCRIPTION Lattice Boltzmann method In recent times, LBM has emerged as a new and effective approach of computational fluid dynamics and it has achieved considerable success in simulating fluid flows and heat transfer. The lattice Boltzmann method can be used to model hydrodynamic or mass transport phenomena by describing the particle distribution function ( giving the probability that a fluid f i particle with velocity ei enters the lattice site x at a time t. The subscript i represents the number of lattice links and i corresponds to the particle at rest residing at the center. The evolution of the particle distribution function on the lattice resulting from the collision processes and the particle propagation is governed by the discrete Boltzmann equation [] fi( x eidt, t d fi( i ( i,,.., 8 (3) where dt is the time step and i is the collision operator which accounts for the change in the distribution function due to the collisions. The Bhatnagar-Gross-Krook (BGK) model [3] is used for the collision operator eq i( fi( f ( i,,.., 8 i (4) 4/9 i w i /9 i,,3,4 (7) /36 i 5,6,7,8 Local particle density ( and local particle momentum u are given by [] 8 8 ( fi( and u( ei fi( i i. (8) The discrete particle velocities are defined as follows: c(,) i e i c(cos[( i ) / ],sin[( i ) / ] i,,3,4 (9) c (cos[( i 5) / / 4],sin[( i 5) / / 4]) i 5,6,7,8 Reynolds number, Re UL / v, where U and L is the characteristic velocity and characteristic length in macro-scale, respectively. Boundary Conditions The standard bounce back boundary conditions proposed by Zou et al [4] were applied on all stationary walls of the cavity. Here the distribution functions at the wall lattice node are assigned to be the distribution function of its opposite direction. The particle distribution functions on the left wall of the cavity are f f 3, 5 f7 f, f8 f6 (a) The particle distribution functions on the right side wall of the cavity are f3 f, 6 f8 3.3 Numerical Implementation f, f7 f5 (b) ISSN (Prin: , Volume-, Issue-, 4

3 v/u International Journal on Mechanical Engineering and Robotics (IJMER) Fluid flows through a lid driven cavity with the top lid moving from left to right with a uniform velocity ofu u. is considered. The velocities at all other nodes are assumed to be zero. The height of the cavity is H and the length of the horizontal sides is L. A uniform fluid density. is imposed initially. The solution procedure of the above LBM at each time step comprise streaming and collision steps, application of boundary conditions, calculation of particle distribution function followed by calculation of macroscopic variables. The numeric simulation is carried out for values, 5,, 5, and of Reynolds number. A MATLAB code is developed for a 9 9 grid. IV. RESULTS AND DISCUSSION Investigation is carried out to study the influence of Reynolds number on the velocity profiles and the streamline plots. A range of values from to is considered for Re. Fig presents the u-velocity profiles along y-axis along the geometric center of the cavity for various values of Re. As observed, the u-velocity starts at zero (on the boundary), decreasing to attain minimum negative value and then again increases to become zero at the center of the cavity. The u-velocity then increases to attain the maximum positive value. As the figure indicates, the basic trend of the u-velocity profiles remains the same for other values of Re, accept that the minimum negative value decreases with an increase in values of Re. Fig 3 presents the v-velocity profiles along x-axis at the geometric center of the cavity for various values of Re. As observed, the v-velocity starts from zero, increasing to reach the highest positive value and then decreases to become zero at the center of the cavity. The v-velocity then decreases to reach the minimum negative value, and then increasing to reach zero at the other side of the cavity. Similar trend is observed for other values of Re, The maximum v-velocity increases and the minimum v-velocity decreases with an increase in the values of Re. This is due to the rheological behavior of the fluid, since an increase in Re, increases the flow of the fluid. The maximum u-velocity is observed at the upper side of the cavity, as indicated by Fig, which is due to the effect of moving lid. The moving lid results in a stronger flow in the regions close to the lid. Fig - suggest that the velocities at the center of the cavity is almost zero indicating weak flow in the region. Fig.3 presents the streamline plots for various values of Re. For Re, the streamlines are observed towards right of the cavity, as seen from Fig 3a. The primary vortex is observed towards the center right of the cavity. Two weak eddies are formed at the left corners, as a result of weak secondary circulation. The primary circulation is concentrated towards the right vertical side of the cavity, with streamlines parallel to the side, indicating a weak circulation in the region. This observation can naturally be related to the value of Re being smaller. With an increase in the values of Re, the strength of the primary circulation is observed to increase, and covers bigger part of the cavity, as indicated by Fig 3a-f. The size of the two eddies increases and a third eddy is observed to form at the right bottom of the cavity, indicating secondary circulation in the regions. The location of the primary vortex shifts from right to the geometric center of the cavity. The size of the primary vortex is also seen to increase with Re, covering a bigger area at the center of the cavity. For higher values of Re, the primary circulation covers almost the entire cavity, with the streamlines becoming symmetric by the center of the cavity Re= Re=5 Re= Re=5 Re= Re= u/u (a) Re= Re=5 Re= Re=5 Re= Re= (b) Fig. Velocity profiles for various values of Re at the geometric center of the cavity. a) u-velocity profiles along y-axis. b) v-velocity profiles along x-axis. 3 ISSN (Prin: , Volume-, Issue-, 4

4 International Journal on Mechanical Engineering and Robotics (IJMER) (a) (b) (c) (d) (e) Fig.3 Streamline plots for various values of Re. a) Re= b) Re=5 c) Re= d) Re=5 e) Re= f) Re=. (f) 4 ISSN (Prin: , Volume-, Issue-, 4

5 This indicated that, at higher values of Re, though secondary circulation also increases, the flow is stronger and covers entire cavity. V. CONCLUSION This paper investigated the influence of Reynolds number on the flow properties of Newtonian fluid in a lid driven cavity. The velocity profiles and streamline plots are presented for different values of Re ranging from to. Reynolds number is observed to govern the flow properties of the fluid having great impact on the velocity profiles and the streamline patterns. The velocity profiles show a significant variation with Re, with the maximum and minimum values of the velocity profiles changing with Reynolds number. The location of the primary vortex is observed to shift from right to the center of the cavity. The primary circulation covers the entire cavity for higher values of Re, with the size of the primary vortex increasing to cover larger area at the center. REFERENCES [] A. Mohamad, Lattice Boltzmann method, Springer-Verlag London Limited. [] Shiyi Chen, Gary D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 3 (998) [3] P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision process in gasses, Phys. Rev. 94 (954) [4] Qisu Zou, Xiaoyi He, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. Fluids 9 (6), June 997. [5] Cox C. L., Chen T. F., Lee H. C., Tung K. L., Least-squares finite element method for generalized Newtonian and viscoelastic flows, Applied Numerical Mathematics 6 () 4 4. [6] Sousa P.C, Coelho P.M., Oliveira M.S.N., Alves M.A., Three dimensional flow of Newtonian and Boger fluids in square-square contractions, J. Non-Newtonian Fluid Mech. 6 (9) 39. [7] Chai Zhenhua, Shi Baochang, Gou Zhaoli, Rong Fumei, Multiple-relaxation time lattice Boltzmann model for generalized Newtonian fluid flows, J. Non-Newtonian Fluid Mech. 66 () [8] Garg Vijay K, Szeri A. Z., Fluid flow and heat transfer between finite rotating disks, Int. J. of heat and fluid flow, Vol. 4, No., June 993. [9] Rogers R. H. et al, Flow and heat transfer in a rotating cavity with a radial inflow of fluid, Int. J. of Heat and Fluid flow, Vol. 6, N. 4, December 985. [] Fox T. A., Toy N., Fluid flow at the center of a cross composed of tubes, Int. J. Heat and Fluid Flow, Vol. 9, No., March 998. [] Mendu Siva Subrahmanyam, Das P.K., Fluid flow in a cavity driven by an oscillating lid-a simulation by lattice Boltzmann method, European Journal of Mechanics B/Fluids 39 (3) [] U. Ghia, K. N. Ghia, C. T. Shin, High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method, Journal Of Computational Physics 48, (98). 5 ISSN (Prin: , Volume-, Issue-, 4