Lattice Boltzmann Simulation Of Flow With Moving Boundary

Lattice Boltzmann Method (LBM) is a new mesoscopic numerical method, which is first proposed by McNamara in 1988. LBM is originated from Lattice Gas Automata (LGA). The applications of real number and Boltzmann equation are the sign of LBM origination. LBM is suitable for parallel computing for that particle distribution function is only concerned with its neighboring node information. LBM is easy to program for that the evolution equation of particle distribution function is linear. Because the boundary treatment is very simple, LBM can be used in complex problems.Two-dimensional nine-velocity (D2Q9) model is used in this paper. The evolution equation is written as fαe q is the equilibrium distribution function and its expression is as follows Whereρis macroscopical density,ωαis weighting coefficient, c s is lattice sound speed, and their values in D2Q9 model are as followsThe macroscopical pressure, density and speed are determined by the following formulasThe evolution equation of LBM is consists of two computational processes collision: streaming:The collision is also called local collision because it occurs only between neighboring nodes. The streaming process is the distribution functions of post collision transfer to the adjacent nodes in accordance with the particle microscopical speed direction. The fluid particle distribution function near the boundary is transferred from its neighboring solid node with virtual distribution function. The computational scheme proposed by Lallemand is used in this paper [100]. By applying Chapman-Enskog expansion technique, the macroscopic governing equation can be obtained from the LBM microscopic evolution equation. The differential format of time and distribution function is expressed asExpand the first term of LBM evolution equation in the left side, incorporate some rational assumes, and abide mass and momentum conservation, then macroscopic governing equation is derived.In numerical simulations, initial condition and boundary condition are the important factors that influence computational stability and convergence. Boundary treatment is especially important in LBM simulation. Some simple boundary treatment schemes include bounce back scheme, periodic boundary, and full developed boundary. If the macroscopic physical condition is known, the microscopic variables can be computed by equilibrium distribution function. The distribution function on the boundary can be divided into equilibrium part and non-equilibrium part. The evolution format of non-equilibrium is similar with that of equilibrium part. The virtual equilibrium distribution function contains the effect of moving boundary.The first numerical simulation is about flow past two side-by-side arranged rotational circular cylinders. This simulation is based on multi-body flow. The rotation of circular cylinder can suppress the vortex shedding in the wake. The suppression of vortex shedding is concerned with rotational speed ratioαand spacing ratio T / D. Critical rotational speed ratio is the focus of this problem. The rotational speed ratios in present study are 0.5, 1.0, 1.5 and 2.0, and the spacing ratio is over the range 1.2~3.0. Based on the analysis of the results, several conclusions are obtained1. For small rotational speed ratio, the vortex pattern of small spacing ratio generally is single bluff body pattern, and the vortex pattern of large spacing ratio generally is anti-phase symmetric pattern.2. For large rotational speed ratio, the vortex pattern of small spacing ratio mainly is single bluff body pattern or vortex shedding is suppressed completely, and the vortex pattern of large spacing ratio mainly is anti-phase symmetric pattern.; 3. The flow rate between the two circular cylinders is concerned with rotational speed ratio and spacing ratio. It increases as the rotational speed ratio decreases and the spacing ratio increases.4. The fluid pattern is depended on the flow rate between the cylinders. For more flow rate, fluid is easy to form vortex and shed from cylinder. The vortex pattern is bias or symmetric pattern. For less flow rate, the vortex pattern generally is single bluff body pattern.The second numerical simulation is about flow past circular cylinder near a moving wall. This simulation is based on the background of moving car. The Reynolds number Re and spacing ratio G / D are two main important parameters. The determination of critical spacing ratio is the focus of present study. In present simulation, the Reynolds number is 600, and the spacing ratio is over the range of 0 .1≤G /D≤1.0. Under different spacing ratio, the velocity in the spacing, the vortex shedding pattern, lift and drag coefficients, Strouhal number will exhibit different characteristics. Based on detailed analysis, some useful conclusions are obtained. The method of determining critical spacing ratio is concluded1. The velocity in the cross section can determine critical spacing ratio. As two local maximum values are equal, the spacing ratio reaches critical condition.2. Vortex shedding pattern also can determine critical spacing ratio. Under critical spacing ratio, vortex shedding from lower side of cylinder is suppressed completely. While the vortex from lower side begins to shed, the spacing ratio reaches critical condition.3. Drag coefficient can determine critical spacing ratio. As spacing ratio increases and the drag coefficient has two local maximum values in a period, the spacing ratio reaches critical condition.4. As spacing ratio reaches critical condition, the Strouhal number becomes mild.The third numerical simulation is about flow past periodic deforming circular cylinder. This case is based on fishes'self-adaptive behavior. The object of present study is to check if the deformation can suppress vortex shedding and reduce drag. The ratio of deforming magnitude relative to radius is 0.05, 0.075, 0.1 and 0.15. The ratio of deforming frequency relative to natural vortex shedding frequency is over the range 0.8~2.0. Several conclusions are concluded1. Small deforming frequency and deforming magnitude have little effect on drag.2. For small deforming frequency, drag decreases as deforming magnitude increases. For large deforming frequency, drag increases as deforming magnitude increases, and drag increases as deforming frequency increases.3. For small deforming frequency and magnitude, vortex shedding mainly is 2S pattern. For large deforming magnitude, vortex shedding mainly is P+S pattern or 2P pattern.4. As ar =0.15 and fr =1.2, the drag is less than that of an isolated cylinder. This confirms that the deformation has the function of drag reduction.