Lattice Boltzmann Method For Simulating The Fluid Flows Around Cylinders

This thesis covers two main parts: the theoretical research ofLattice Boltzmann Method, and application of LB method inengineering, the simulation of flow fields around square cylinder indifferent situatioin.1. The model of Lattice Boltzmann MethodThe method of lattice Boltzmann equation (LBE) solves themicroscopic kinetic equation for particle distribution function f ( x , ξ , t),where ξ is the particle velocity, in phase space ( x , ξ ) and time t,from which the macroscopic quantities (flow mass density ρ andvelocity u) are obtained through moment integration of f ( x , ξ , t).Its major notion is to use the simple and regular microscopic particlemovement to instead the complex macroscopic phenomena. Forexample we introduce the foundation of Lattice Boltzmann model indetail for two-dimensional nine-velocity (D2Q9) incompressible latticeBoltzmann model.For two-dimensional nine-velocity (D2Q9) incompressible latticeBoltzmann model, the Boltzmann equation can be discretized in space xand time t intof α ( x + eαδ t ,t+δt)?fα(x,t)=?τ1[fα(x,t)?fα( eq)(x,t)] (1)In the above equation, eα (α=0,1,…8) is the particle velocity in thea direction, fα(x,t) is the density distribution function along the adirection, fα(eq)(x,t) is its corresponding equilibrium state, x is the spatialposition vector, and t is the time, τ is the dimensionless relaxation timeand δ t is time step. The equilibrium distribution function of D2Q9model is:( ) 1 3( ) 9 ( )2 322 2fα eq= ρωα ??? + eα ? u + eα ? u ?u ???, (2)in the above equation, when α =0, ω α=4/9;else when α = 1,2,3,4,ω α=1/9;else α = 5,6,7,8, ω α=1/36.Equation (1) can be computed by the following two steps,Collision step: ~f α (x ,t)= fα(x,t)?τ1 fα[(x,t)?fα( eq )(x,t)] (3a)Streaming step: f α ( x + eαδ t ,t+δt)=~fα(x,t) (3b)Where ~fα denote the post-collision state of the distributionfunction. One can notice that the collision is completely local and thedistribution function of a lattice is only affected by neighboring ones inthe streaming step.Now we give a computational method of force evaluation in theLattice Boltzmann method involving curved geometry:Consider an arbitrary curved wall which separates the solid regionfrom the fluid region. Let xw, xf and xb be the intersections of theboundary with various lattice links, the boundary node in the fluidregion, and that in the solid region, respectively. Then, ? can bedefined asf wf b? = xx ??xx (4)It is well understood that the bounce-back boundary conditionsatisfies the no-slip boundary condition in second-order accuracy atthe location of ? =1/2.On the boundary it is important for us to define distributionfunction at xb in order to get the value at xf. To construct ~f α (xb,t)based upon some know information in the surrounding,Chapman-Enskog expansion for post-collision distribution function onthe right-hand side of Eq. (3b) is carried out on the boundary. It has beenproved that a second-order accurate no-slip boundary condition can beachieved by this method.Using the momentum-exchange method, the total force on theboundary can be computed with:0[ ( b , ) ( b t , )][1 ( b t)]all xbF α f α t f α α t wααδ δ≠= ∑ ∑e x + x + e ? x +e (5)2. Simulation of Flow past square cylinder with LB MethodFor the simulation of actual flow, we use D2Q9 investigate fourcases of flow past square cylinders in this paper. For case 1, one singlesquare cylinder is located at the center of the channel , we describe thestreamline contour, vortices contours,simulate the Karman vortex,then compute the lift coefficient, drag coefficient, Strouhal numbersetc. For the case 2, simulate the flow past a cylinder of rectangularcross-section;compute the change of Strouhal numbers varying withthe side ratio. For case 3: two square cylinders arranged side by side inthe center of the channel, the flow features at different spacing ratiosare studied. For case 4: we compute the linear shear flow over a squarecylinder, compare the evolution of flow with different velocitygradient. The results of the simulation including the streamlines,vorticity contours, lift and drag coefficients etc. are agreed with thoseof available literatures, and show that LB method and itsmomentum-exchange method can achieve accurate results and obtainthe reasonable flow in detail.