Lattice Boltzmann Equation for Laminar Boundary Flow

A simple method based on the lattice Boltzmann equatio n is presented for the evaluati on of the velocity profile of fluid flows near walls or in the vicinity of the interface between two fluids. The met hod is applied to fluid flow near a wall, to channel flow, and to the transition zone between two fluids flowing parallel to each other in opposite direct ions. The results show good agreement with micrody namical lattice gas simulati ons and with classical fluid dynamics. 1. The lattice boundary layer p roblem Since the pioneer ing work by Hardy, Pomeau, and de P azzis in 1973 [1,2]' Wolfr am in 1983 [3J, and mostly since the recent int rodu ction of the hexagonal lattice gas by Frisch, Hass lacher, and Pomeau [4], lat t ice gas methods have evolved bo th in efficiency and complexity (an extensive introduction to the subject can be found in [5]). T he theor eti cal and computational developme nt of the field has been so exte nsive in the last coup le of years that it has given rise to applications in var ious areas of physics [6J . Lat ti ce gases share common operational features with cellular automata and so ar e mos t easi ly implemented on parallel machines, in particular for fluid dynamical pro blems at large Reynolds nu mb ers [7J which require high computat ional performances. On the ot her hand, there exists a variety of operationally simple problems of valuable physica l interest that can be solved wit h modest computational means for which small computers provid e sufficient power . For the class of pr oblems considered here, t he lat t ice gas flow description can be reduced to a "one-dimens ional" formula t ion; therefore such problems can be solved wit h low power computational techniques. @ 1989 Complex Systems Publications, Inc. 318 Lattice Boltzmann Equation for Laminar Boundary Flow The formation and growth of boundary layers is of cru cial importance in fluid dynamical flows, in particular as t heir occurrence triggers the development of turbulence at high Reynolds numbers . For viscous flow (at low Rey nolds number) , boundary layer pr oblems can be solved within the limi ts of reasonable approximations . Such problems so appear as an interesting test for the validity of the lat tice gas method and their solutions are a prerequisite to the understanding of mor e complex flows and of the three-dimensionalization in the transitio n to tur bulence. The pur pose of the present work is to show that laminar boundary flow can be treated efficient ly, that is, simply and economically, by the lattice gas method . The basic idea is the following: consider that a lattice gas, initially in homogeneous un idirectional motion, is suddenly put in contact with a wall. All lattice gas nodes in any layer parallel to the flow direct ion have the same particle distribution and, the system being translationally invariant, it suffices to perform one-dimensional computat ion to evaluate the velocity profil e. T he wall effects on the flow velocity are propagat ed by the particl es at the microscopic "t hermal velocity," whereas t he flow profil e modifi cations pr opagate via particle interactions, i.e. , at much lower speed. Interactions with the wall will first be felt on the first layer of the gas (i.e., the layer adjacent to the wall); at the next time st ep, they will be felt on the first and second layers , and progressively the successive lattice layers will be interactively involved. More pr ecisely, we consider a gas (density d) flowing parallel to a wall wit h free flow velocity Uo. Momentum is first exchanged between the wall and the first layer: the flow is slowed down in that layer due to velocity reversal of the particles colliding with the wall. T he first layer will come to a state of local equilibrium acquiring velocity U parallel to the wall with U < Uo. All layers beyond the first one remain at the free flow velocity Uo. During the second t ime st ep , the transport of the particles from the first layer will affect the second layer of the gas: particles in the second layer are now slowed down due to t he lower veloci ty in the first layer. The new equilibrium populations in the first and second layers ar e computed. The third ste p in the process will affect the third layer of the gas; however, the fresh values of the second layer computed from the second time step will exert an influ ence on the first layer so that equilibrium values for the first two layers must be up dated before that of the third be evaluated. At each time step of the process , the new equilibrium values of each underlying layer are updated from the fresh values obtained for the upper and lower adjacent layers a t the previous step and the next upper layer is included in the computation. At any given time, all the layers that have reached equilibrium and have a velocity value U ~ O.99Uo are considered to belong to the bo undary layer. Because of the discreteness of the lat ti ce, the exact value of the boundary layer thickness 8 must be evaluated (in general) by interpolation. The boundary layer thickness growth is much slower than the increase in the number of layers: the boundary layer thickness grows as the square root of the number of time st eps [8], whereas the number of layers is equal to the number of ti me steps. All the layers that have not Paul Lavallee, Jean Pierre Boon , and Alain Noullez Double collision =>jp,\ ---...'* Trip le collis ion I ~-< H\