Lattice Boltzmann Simulation For Stirred Flow And Drag Reduction Of Cylinder

With the development of technology, the high-speed operation could be carried out by the computer. With the birth of computer, the superiorities of the numerical methods have been exhibited. A lot of softwares with a variety of numerical methods make the original complex calculations simplified. The computational fluid dynamics, as the organic union of the computer technology, classic fluid mechanics and numerical computation methods,becomes the third effective method to investigate the hydrokinetics. Computational fluid dynamics is widely applied in the field of science and engineering. In the past twenty years, Lattice Boltzmann method came forth and progressed rapidly as a numerical computation method. It has broad prospects for future.In 1988, McNamara and Zanetti proposed an improvement of lattice gas automata (LAG) method, Lattice Boltzmann method came into being. Lattice Boltzmann method not only inherits the advantages of lattice gas automata, but also overcomes the shortcomes of the lattice gas automata.Lattice Boltzmann method is an effective numerical method for simulation of the fluid. A series of the complex flow problems which can not be solved by macroscopic equations can be resolved by this method. Lattice Boltzmann method is better than other numerical method. 1. The complex interactions between the different fluid interfaces could be easy handled using Lattice Boltzmann method. 2. In a single time step, each grid point of the fluid is localized. The processes of the evolution of the fluid are not complex. Highly parallel computing could be achieved using computer machine. 3. The pressure could be calculated directly with the correlative data. Based on the advantages above, many scholars investigate the Lattice Boltzmann method deeply.The evolution equation of the Lattice Boltzmann method is as follows where the expressions of the f_α, f_α^eq and e_αare different with different models. However, f_α, f_α^eq and e_αin any model satisfy the following equations and the equilibrium distribution function satisfies the Maxwell-Boltzmann distribution Use the Taylor expansion,The Navier-Stokes equation can be recovered by Lattice Boltzmann method if Chapman-Enskog expansion is used.Usually, the time scales and the space scales are selected as t₀, t₁ =εt₀, t₂ =ε²t₀and Then Use the Taylor expansion again at point ( x , t), the equations corresponding to the different time scales could be obtained. Combining with the equilibrium distribution function and the equations of the two time scales, the Navier-Stokes equations can be recovered.There are only simple particle dynamics in Lattice Boltzmann model, namely the collision and streaming. The evolution equations can be rewritten two equations. Collision: StreamingThe treatment for boundary is very important for a numerical method. It is also an important part of the Lattice Boltzmann method. Among these methods to treat the boundary, the bounce-back scheme is the most basic. Half-Way bounce-back scheme is a second-order accuracy method. It is a modified method based on bounce-back scheme. For the complex boundary, we need consider more factors.The fictitious equilibrium distribution function can be expressed as follows: here u≡u ( x ,t)is the velocity of the point x , u^* is a fictitious velocity. Then the linear interpolation is used to calculate the distribution function at point x + e_αδt.Combing with the equilibrium distribution function at point x in the D2Q9 model, the fictitious equilibrium distribution function f_α( x + e_αδt ,t) can be calculated.The parameter q is defined as , The unknown quantities u^* in the above formula can be calculated using the formula as follows. This method to treat the boundary is a second-order accuracy method.Another second-order accuracy method adopts quadratic interpolation to calculate the distribution functionThis method is often used in MRT model. There is none relationship between the treatment for the boundary conditions and the collision of the fluid. This method is also can be used in SRT model.In this paper, the D2Q9 model of lattice Boltzmann method is used to simulate the incompressible viscous fluid. Some numerical results are obtained. Numerical simulation of flow stirred in closed cavity.The Stirred reactor has complicated process. The complex geometries of the stirrer can be handled by lattice Boltzmann method.Through the simulations of the stirrer in different shapes of the closed cavity, some conclusions are obtained.1. It is more easily for us to grasp the stirred process in the traditional circular cavity.2. The shape of the cavity will has impact on the results of the stirring.3. If the traditional circular cavity is used, The energy loss is smaller.4. The results will be more satisfying if the traditional circular cavity is used as mixed containers.5. The force coefficients are very different from the general movement of flow around a cylinder. The periodic of the drag coefficient is changed. The flow around a main cylinder with affiliated cylindersFlow around a cylinder is a classic problem to study the hydrodynamics. The numerical simulation of flow around a cylinder with affiliated cylinders is investigated in the fourth chapter of this paper. There are also some conclusions:1. The drag coefficient of the cylinder will decrease if the affiliated cylinder is placed near the wake region of the main cylinder.2. Compared with a way of placing an affiliated cylinder, drag coefficient will reduce more if two affiliated cylinders are placed in the wake region of the main cylinder.3. The vortex can be better suppressed if two affiliated cylinders are placed near the wake of the main cylinder.4. It is difficulty to suppress the vortex behind the main cylinder if the affiliated cylinders are placed away from the main cylinder. But it also could reduce the drag coefficient.5. When x_α> 2, the vortex behind the main cylinder comes into being. But the structure of the vortex can be changed after the vortex past the affiliated cylinders.