Research On Local Grid Refinement In Multiple-Relaxation-Time Lattice Boltzmann Method

The LBM is a new method for calculating fluid mechanics efficiently.It has unique advantages and It has been widely applied to fluid mechanics problems.LBM takes fluid as particles that can move along the regular lattice and collide with each other only at the lattice point.A series of velocity distribution functions are obtained by solving the collision-streaming equation.The macroscopic parameters such as pressure and velocity can be obtained by solving the moment equation of the distribution function.In the fluid,transfer of momentum,energy and mass are not synchronized,MRT model has multiple relaxation parameters,and the particle relaxation process is decomposed into several independent relaxation processes.By adjusting the parameters of the collision matrix,the transfer can be distinguished,improved computational stability and accuracy.In addition,the linear transformation matrix is constructed to makes the amount obtained after the collision and streaming has physical meaning.After the collision step is completed,the distribution function is transformed into the velocity space,and then the flow move from one grid point to next grid in discrete velocity direction.In the flow field,the physical quantity changes violently at the lobes,edges,etc.we use local grid refinement method to obtain accurate data such as forces and moments acting on the particles.Coarse grid can be used in the distance far away from the boundary.The calculation area is divided into coarse,buffer and refinement area.The distribution function transfer through the buffer boundary.Due to different lattice density,the data at interpolation points cannot be obtained by direct transfer.To minimize error,the cubic spline interpolation method is used to calculate.Different regions need to be interpolated in Lagrange interpolation method.The physical quantities such as density,velocity and stress should be kept continuous.Chapman-Enskog analysis is used to derive the transformation relation of the distribution function on the interface.The accuracy and numerical stability are improved by selecting appropriate relaxation parameters.The treatment of boundary condition plays an important role in LBM;it is directly related to the accuracy and efficiency of calculation.When the flow has periodicity in a particular direction,the periodic region is taken out as calculation region.For the complex surface boundary,the interpolation format is used to modify it.For the general straight boundary,the bounce back is used to modify the distribution function after the collision on the boundary.The lid-driven cavity is a classical example,and the simulation with Reynolds number of 1000 and 400 are carried out.In the left and right corner,the physical quantity changes violently.We need implement grid refinement in the left corner and right corner.The flow is in a static state initially,and after a long period of time,the flow reaches steady state.The velocity,pressure,vorticity and stress obtained from different grid structures are compared and analyzed.The results show that the data in refinement area is more accurate.The velocity curve can be better consistent with the benchmark,reduced the oscillation amplitude of the press and stress,and It's able to capture changes of stress,improved the quality of the simulation.For the Couette flow containing particles,using the UCG,UGF,LGR,and MGM to calculate.The structure of the program is collision first then streaming.Two examples are compared.In the fixed particle case,compared the drag and lift,the results are similar,can be used as the benchmark to compare with the moving particle case.In the moving particle case,the buffer and cage region move along with the particle.Compare drag force,lift force and torque.The results show that the fluctuation of force and torque in cage area is obviously smaller than other grid structure,and the calculation precision is improved,and the calculation time is less than the UGF.The contours of the stress are continuous on the interface;the correctness of the application of the local grid refinement method is confirmed.