Numerical Simulation Of Flow Around Prisms Using The Lattice Boltzmann Method

In this paper, Lattice Boltzmann Method (LBM) and its application in the simulation of flows past prisms are studied. As a numerical method of computational fluid mechanics, it has been developed very fast, and been applied in simulating many kinds of fluid flows.1. Lattice Boltzmann methodFirstly, Lattice Gas Automa method is introduced in this paper, and then the developing process from Lattice Gas Automa method to Lattice Boltzmann method is reviewed. After that, the basic principles of the LBM are expressed. Taking D2Q9 model as an example, we carried out a detailed derivation of the evolution of LBM, derived dynamical equation. The curve boundary treatment with second order accuracy is briefly introduced. We also give a computational method of force evaluation in the Lattice Boltzmann method for curve boundary.For two-dimensional nine-velocity (D2Q9) incompressible lattice Boltzmann model, the Boltzmann equation can be discretized in space x and time t intoIn the above equation, eα(α=0, 1,…8) is the particle velocity in the a direction, fα(x,t) is the density distribution function along the a direction, fα(eq)(x,t) is its corresponding equilibrium distribution function, x is the spatial position vector, and t is the time,τis the dimensionless relaxation time andδt is time step.The equilibrium distribution function of D2Q9 model is:in the above equation, fo ; else Equation (1) can be computed by the following two steps: Collision step: Streaming step:Where ~fαdenote the post-collision state of the distribution function. One can notice that the collision is completely local and the distribution function of a lattice is only affected by neighboring ones in the streaming step.In spite of numerous improvements in the LBM over the last several years, one important issue that has not been systematically studied is the accurate determination of the fluid dynamic force acting on curved boundaries.Consider an arbitrary curved wall which separates the solid region from the fluid region, let xw, xf and xb be the intersections of the boundary with various lattice links, the boundary node in the fluid region, and that in the solid region, respectively. Then,Δcan be defined asIt is well understood that the bounce-back boundary condition satisfies the no-slip boundary condition with second-order accuracy at the location ofΔ=1/2. But this is only true with simple boundaries of straight line parallel to the lattice grid. For a curved boundary, simply placing the boundary halfway between two nodes will alter the geometry on the grid level and degrade the accuracy of the numerical results.On the boundary it is important for us to define distribution function at xb in order to get the value at xf. Chapman-Enskog expansion for post- collision distribution function on the right-hand side of Eq. (3b) is carried out on the boundary. It has been proved that a second-order accurate no-slip boundary condition can be achieved by this method.Using the momentum-exchange method, the total force on the boundary can be computed with:2. Simulation of Flow around prisms with the lattice Boltzmann methodSecondly, several flows around prisms are simulated using the Lattice Boltzmann method. For the simulation of real flow, we use D2Q9 model to investigate four cases of flow past prisms. For case 1, one single prism is located at the center of the channel , we show the streamline contours, vortices contours,and present the Karman vortex, then compute the lift coefficient, drag coefficient, Strouhal numbers etc. For the case 2, we simulate the linear shear flow over a single prism; compare the evolution of flow with different velocity gradient. For case 3: two prisms arranged side by side in the center of the channel, the flow features at different spacing ratios are investigated. For case 4: we compute the flow around two prisms at different distance in tandem arrangement are simulated, and show the variations of flow patterns. Numerical results are compared with the results of experiment as well as that of other numerical method. Our results show that lattice Boltzmann method is accuary and has the potential ability to simulate the fluid flows.