Curved surface boundary algorithm for multiphase flow numerical simulation

Abstract

The invention discloses a curved surface boundary algorithm for multiphase flow numerical simulation. The curved surface boundary algorithm comprises the following steps: 1, carrying out mesh refinement between a physical solid wall boundary point and a nearest fluid point; 2, executing a collision step and a whole-course rebound operation under the fine grid; and 3, converting the particle distribution function rebounded under the fine grid into a particle distribution function under the coarse grid, wherein the particle distribution function is an unknown particle distribution function to be solved returning to the flow field from the boundary of the curved surface. Compared with a curved surface boundary algorithm based on interpolation, the algorithm can well keep mass conservation, and the precision and stability in multiphase flow simulation are improved. And the method has relatively important application value in practical engineering problems related to multi-phase flow curved surface boundary calculation.

Description

Technical Field

[0001] The present invention relates to the fields of statistical physics, fluid mechanics and computer numerical simulation, and specifically relates to a curved boundary algorithm for multiphase flow numerical simulation. Background Art

[0002] Multiphase flows widely exist in nature and industrial and agricultural applications, such as flowing rivers, falling raindrops, inkjet printing, spray cooling systems, fuel cell systems, and condensation heat transfer systems. Studying the basic theories and laws of multiphase flows is of great significance for promoting the development of industry and agriculture. When solving complex multiphase flow problems, boundary conditions play an indispensable and important role, and complex curved boundaries are ubiquitous, and the treatment of curved boundaries is somewhat challenging. When using the lattice Boltzmann method (LBM) for multiphase flow simulation, in order to more accurately capture the boundaries of complex geometries and improve the simulation accuracy, researchers have successively developed some curved boundary algorithms. Filippova and Hanel [1] proposed a boundary algorithm with second-order accuracy that uses linear interpolation by constructing a solid-wall virtual equilibrium distribution function. However, when the fluid points near the boundary are very close to the solid wall, its numerical stability is poor. Mei et al. [2] proposed an improved curved boundary algorithm for the problems existing in the FH algorithm. When the actual boundary is too close to the solid-wall lattice points, the velocity interpolation boundary lattice points are set as the nearest fluid nodes, thereby increasing the stability range of the algorithm. Bouzidi et al. [3] proposed a boundary algorithm that combines linear interpolation or quadratic interpolation with a bounce-back scheme. This algorithm does not require constructing virtual lattice points for extrapolation, but calculates the unknown function during the simulation in segments during the bounce-back process. To avoid discontinuities in boundary treatment, Yu et al. [4] proposed a unified version of the curved boundary algorithm, using two-step sequential interpolation to avoid discontinuities in boundary treatment. However, they introduce viscosity-related slip at the boundary. Ginzburg and d’Humières [5]A multi-reflection boundary algorithm is proposed, which can be regarded as an enhanced interpolation bounce-back scheme. They mainly determine the interpolation coefficients by performing a second-order Chapman-Enskog expansion on the interpolation distribution function. At present, these surface algorithms have made important contributions to improving the numerical simulation accuracy and stability of multiphase flows. However, research shows that there is a mass non-conservation problem when these interpolation-based surface algorithms are used for multiphase flow simulation, which may seriously affect the stability of the simulation and the accuracy of the simulation results, thus limiting their application in practical engineering. Therefore, in order to accurately simulate multiphase flow problems involving curved boundaries using LBM, there is an urgent need to develop a new surface boundary algorithm with high computational accuracy and good mass conservation. Summary of the Invention

[0003] Aiming at the problems existing in the existing algorithms, the present invention proposes a new surface boundary algorithm for numerical simulation of multiphase flows. First, the grid is refined between the physical solid wall boundary points and the nearest fluid points, and then the collision step and the full bounce-back operation are performed under the fine grid. Finally, the particle distribution function bounced back under the fine grid is converted into the particle distribution function under the coarse grid, which is the unknown particle distribution function returning to the flow field from the curved boundary to be sought. This algorithm can better maintain mass conservation and improve the accuracy and stability in multiphase flow simulation.

[0004] The technical solution adopted by the present invention is as follows: A surface boundary algorithm for numerical simulation of multiphase flows includes the following steps:

[0005] ① Refine the grid between the physical solid wall boundary points and the nearest fluid points, and establish the relationship between various physical quantities between the coarse and fine grids:

[0006]

[0007] Among them, x A is the grid point shared by the coarse and fine grids, Q is the refinement factor (its value is equal to the distance from the shared point x A to the physical solid wall), f i represents the particle distribution function, and ρ and u represent the fluid density and velocity respectively. The superscripts f and c of each physical quantity represent the fine grid and the coarse grid respectively.

[0008] ② Perform LBM collision at the grid point x A in the fine grid:

[0009]

[0010] Among them, f i eq represents the equilibrium particle distribution function.

[0011] ③Perform the full bounce operation without interpolation in the fine grid, and obtain the particle distribution function in the fine grid after collision using Equation (3). Through flow + collision bounce, it is obtained as follows:

[0012] The particle distribution function after collision:

[0013] f i f (x c , t + δt f ) = f i +,f (x A , t) (4),

[0014]

[0015] where x c is the physical solid wall grid point in the fine grid. Then, perform the flow step operation to obtain the unknown particle distribution function in the fine grid:

[0016]

[0017] ④Finally, convert the obtained particle distribution function in the fine grid back to the particle distribution function in the coarse grid:

[0018]

[0019] The present invention discloses a curved surface boundary algorithm for multiphase flow numerical simulation. This algorithm can better maintain mass conservation and improve the accuracy and stability in multiphase flow simulation. It has relatively important application value in practical engineering problems involving multiphase flow curved surface boundary calculation. To verify the model of the present invention, numerical simulation of the phenomenon of a droplet on a stationary curved surface boundary is carried out here. Figure 1 、 2 shows the comparison of the mass loss calculated using the solution of the present invention with the result obtained by linear interpolation under hydrophilic and hydrophobic conditions. As can be seen from Figure 1 , as the evolution process proceeds, the mass of the droplet calculated using the linear interpolation algorithm [1] (IBC) has become 0.92% of the initial mass at 15,000 time steps, and by 50,000 time steps, the mass has decreased to 0.01% of the initial mass, indicating that the flow field has all become gas phase. However, the result calculated using the solution of the present invention has well maintained mass conservation. Figure 2 shows that as the evolution process proceeds, the mass of the droplet calculated using the linear interpolation algorithm has become 132% of the initial mass at 5,000 time steps, and by 50,000 time steps, the mass has increased to 695% of the initial mass, indicating that the flow field has all become liquid phase. However, the result calculated using the solution of the present invention can still well maintain mass conservation. ThroughFigure 1 , 2 It is illustrated that the solution of the present invention improves the simulation accuracy, thus being able to better maintain mass conservation. Figure 3 , 4 shows the dynamic evolution process of droplet simulation using the solution of the present invention. From Figure 3 and Figure 4 it can be seen that the results obtained by the solution of the present invention are conserved whether on a hydrophilic floor or a hydrophobic floor, while the linear interpolation scheme represented by the white dotted line will cause a drastic change in mass whether on a hydrophilic floor or a hydrophobic floor. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 is the mass comparison between the solution of the present invention and the linear interpolation scheme on a hydrophilic floor in the embodiment of the present invention;

[0021] Figure 2 is the mass comparison between the solution of the present invention and the linear interpolation scheme on a hydrophobic floor in the embodiment of the present invention;

[0022] Figure 3 is the mass dynamic evolution of the solution of the present invention and the linear interpolation scheme on a hydrophilic floor in the embodiment of the present invention;

[0023] Figure 4 is the mass dynamic evolution of the solution of the present invention and the linear interpolation scheme on a hydrophobic floor in the embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0024] The following further elaborates on the content of the present invention in conjunction with the drawings and embodiments, but it is not a limitation to the present invention.

[0025] Embodiment:

[0026] A droplet is located at the boundary of a stationary surface, including the following steps:

[0027] A flow field with a droplet located at the boundary of a stationary surface is established, and relevant parameters are set. The multiphase flow simulation uses the multiphase model based on chemical potential proposed by Wen et al. [6] The size of the simulated flow field is 200×250. Periodic boundary conditions are adopted for the upper, lower, left, and right of the flow field. The solution of the present invention and the linear interpolation scheme are respectively used to simulate between the droplet and the circular solid. In the simulation, the relaxation time is τ = 0.8 and the temperature is T = 0.6.

[0028] ① Mesh refinement is performed between the physical solid wall boundary points and the nearest fluid points, and the relationships between various physical quantities between the coarse and fine meshes are established:

[0029]

[0030] where x AThe shared grid points are for the coarse and fine grids, Q is the refinement factor (its value is equal to the distance from the shared point x A to the physical solid wall), f i represents the particle distribution function, and ρ and u represent the fluid density and velocity respectively. The superscripts f and c of each physical quantity represent the fine grid and the coarse grid respectively.

[0031] ② Perform the LBM collision at the grid point x A in the fine grid:

[0032]

[0033] where f i eq represents the equilibrium particle distribution function.

[0034] ③ Perform the full-range bounce-back operation without interpolation in the fine grid. Using the particle distribution function after collision in the fine grid obtained by equation (3), it is obtained through flow + collision bounce-back:

[0035] f

[0036] f i f (x c , t + δt f ) = f i +,f (x A , t) (4),

[0037]

[0038] where x c is the physical solid wall grid point in the fine grid. Then, perform the flow step operation to obtain the unknown particle distribution function in the fine grid:

[0039]

[0040] ④ Finally, convert the obtained particle distribution function in the fine grid back to the particle distribution function in the coarse grid:

[0041]

[0042] In the simulation example of the droplet located at the static curved surface boundary, two simulations of the hydrophilic bottom plate and the hydrophobic bottom plate were carried out respectively, where θ = 60° is the hydrophilic bottom plate and θ = 120° is the hydrophobic bottom plate. Figure 1 、 2The quality comparison between the solution of the present invention and the linear interpolation solution is given. It can be seen from the figure that, whether in the hydrophilic bottom plate or the hydrophobic bottom plate, the mass calculated by the solution of the present invention is conserved, with almost no mass loss. For the linear interpolation solution on the hydrophilic bottom plate, at the 20,000th time step, the mass is almost zero, while on the hydrophobic bottom plate, at the 20,000th time step, the mass has reached about 200% of the initial value. In addition, we plot the dynamic evolution diagrams at the 5,000th and 10,000th time steps during the simulation process, as Figure 3 and 4 shown. It can still be seen from the figure that, regardless of whether it is the 5,000th or 10,000th time step, the mass of the droplet calculated by the solution of the present invention is conserved, while the linear interpolation solution will cause the mass of the droplet to increase or decrease, resulting in a huge error.

[0043] In summary, by adopting a curved boundary algorithm for multiphase flow numerical simulation proposed by the present invention, the unknown particle distribution function at the curved boundary can be accurately obtained, and the mass conservation of the flow field can be well maintained. Through the present invention, the problems of large mass loss affecting the stability of the simulation and the accuracy of the results in multiphase flow simulation involving curved boundaries can be effectively solved, and it has relatively important application value in practical engineering problems involving the calculation of multiphase flow curved boundaries.

[0044] References:

[0045] [1] FILIPPOVA O, HANEL D. Grid refinement for lattice - BGK models[J]. J Comput Phys, 1998, 147(1): 219 - 28.

[0046] [2] MEI R W, LUO L S, SHYY W. An accurate curved boundary treatment in the lattice Boltzmann method[J]. J Comput Phys, 1999, 155(2): 307 - 30.

[0047] [3] BOUZIDI M, FIRDAOUSS M, LALLEMAND P. Momentum transfer of a Boltzmann - lattice fluid with boundaries[J]. Phys Fluids, 2001, 13(11): 3452 - 9.

[0048] [4] YU D, MEI R, LUO L-S, et al. Viscous flow computations with the method of lattice Boltzmann equation[J]. Prog Aero Sci, 2003, 39(5): 329-67.

[0049] [5] Irina G, Dominique D. Multireflection boundary conditions for lattice Boltzmann models[J]. Physical Review E, 2003, 68(6):

[0050] [6] WEN B, ZHOU X, HE B, et al. Chemical-potential-based lattice Boltzmann method for nonideal fluids[J]. Phys Rev E, 2017, 95(6): 063305.

Claims

1. A surface boundary algorithm for multiphase flow numerical simulation, characterized in that it comprises the following steps: 1.1) Refine the grid between the physical solid wall boundary point and the nearest fluid point, and establish the relationship between the physical quantities between the coarse and fine grids: Among them, x A is the shared grid point of the coarse and fine grids, Q is the refinement factor (its value is equal to the shared point x A distance to the physical solid wall), f i represents the particle distribution function, ρ and u represent the fluid density and velocity respectively. The superscripts f and c of each physical quantity represent the fine grid and the coarse grid respectively. 1.2) At a grid point x in the fine grid A Perform LBM collision at: in, represents the equilibrium particle distribution function.

2. Obtain the particle distribution function of the surface boundary based on step 1. The specific steps are as follows: 2.1) Perform a full-bounce operation without interpolation in the fine grid, and use equation (3) to obtain the particle distribution function after collision in the fine grid, and obtain it through flow + collision rebound: in, x c is the physical solid wall grid point in the fine grid. Then, the flow step operation is performed to obtain the unknown particle distribution function in the fine grid: 2.2) Finally, the obtained particle distribution function in the fine grid is converted back to the particle distribution function in the coarse grid:

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