A High-Precision Processing Method for Virtual Stratified Boundaries in the Lattice Boltzmann Method for Tree Grids

Abstract

The present invention proposes a high-precision processing method for virtual hierarchical boundaries in the tree-grid lattice Boltzmann method. On the premise of ensuring ultra-high computational efficiency, a high-precision interpolation format is adopted, the actual physical situation of the virtual boundary is considered, and the information transfer between fine and coarse grids is also involved, meeting the higher-precision calculation requirements and being applicable to all virtual boundary processing requirements for tree-grid LBM-based methods.

Description

Technical Field

[0001] The present invention relates to the technical field of computational fluid dynamics, and particularly to a high-precision processing method for virtual hierarchical boundaries in the tree-grid lattice Boltzmann method. Background Art

[0002] In engineering applications such as numerical simulation of airfoil flow around and internal flow in a cavity, it is a relatively common mesh generation method to use an adaptive tree grid, a special grid with a combination of coarse and fine grids. As Figure 1 shown, it shows the tree grids respectively used for numerical simulation of internal flow driven by a lid-driven cavity and airfoil flow around. It can be seen from the figure that the entire computational domain is divided into multiple regions with different grid sizes.

[0003] The lattice Boltzmann method (LBM) is a mesoscopic computational fluid dynamics algorithm that solves the distribution function based on the collision and migration theory of fluid molecules at grid points, thereby obtaining macroscopic physical quantities at grid points, such as velocity, density, pressure, etc. Figure 2 The left shows the collision and migration mode (evolution mode) of the lattice Boltzmann method. It can be seen that fluid molecules migrate in 8 directions along the grid lines, but it does not satisfy where c is the lattice velocity, that is, within one time step, it can only migrate to nearby grid points, and the migration distance is one grid step. If a uniform grid is considered, the grid perfectly fits the LBM evolution theory. However, if a special grid with a combination of coarse and fine grids like an adaptive tree grid is used, problems will occur because it is impossible to satisfy in the entire computational domain, and only hierarchical processing can be carried out. All grids with the same grid spacing are defined as the same level. Within the same level, the LBM evolution theory is fully satisfied. For the virtual sub-boundary, separate processing is required. As Figure 2 shown on the right, the black line is the virtual hierarchical boundary that separates the coarse and fine grids. Triangles represent the coincidence points of the coarse and fine grid surfaces, and circles represent hanging points. After understanding the data structure of the adaptive tree grid, the processing format of the virtual boundary needs to be considered next.

[0004] The applicant has been engaged in research on high-efficiency and high-precision algorithms for a long time. In the research of the lattice Boltzmann method (LBM) algorithm, the use of low-efficiency uniform rectangular grids has been abandoned, and an LBM optimization algorithm based on the tree-grid structure has been developed, which greatly improves the calculation efficiency while ensuring the calculation accuracy. The use of tree grids will inevitably encounter the problem of processing virtual boundaries (coarse-fine grid coincidence boundaries). In previous research work, the processing format of virtual boundaries has been developed. [1-2], but only relatively simple spatial and temporal interpolation is adopted from a pure mathematical perspective. The interpolation information sources used are few, and the actual physical conditions on the overlapping boundaries are not considered. In addition, only the information transfer from the fine grid to the coarse grid is considered. To meet the increasingly high requirements of practical application needs such as the numerical simulation of airfoil flow around and the internal flow of the engine cavity, and to improve the accuracy and efficiency of numerical simulation for specific engineering application backgrounds, during the numerical simulation process, it is necessary to develop High precision High precision and high efficiency computational methods. Through long-term research, it is found that there is still room for upgrading and optimizing the original virtual boundary treatment format. Summary of the Invention

[0005] To solve the problems existing in the prior art, meet the increasingly high requirements of practical application needs such as the numerical simulation of airfoil flow around and the internal flow of the engine cavity, and develop high-precision and high-efficiency computational methods, this application proposes a high-precision processing method for virtual hierarchical boundaries in the tree-grid lattice Boltzmann method. On the premise of ensuring ultra-high computational efficiency, a high-precision interpolation format is adopted, the actual physical conditions of the virtual boundary are considered, and at the same time, the information transfer between the coarse and fine grids is also involved, meeting the higher-precision computational requirements and being applicable to all virtual boundary processing requirements based on the tree-grid LBM. Furthermore, based on this high-precision processing method, corresponding numerical simulation methods for the flow around a square column, the internal flow driven by the lid in a square cavity, the flow around an airfoil, and the internal flow of the engine cavity are proposed, improving the accuracy and efficiency of the numerical simulation of the flow around the airfoil and the internal flow of the engine cavity.

[0006] The technical solution of the present invention is as follows:

[0007] A high-precision processing method for virtual hierarchical boundaries in the tree-grid lattice Boltzmann method, characterized in that: for the hanging points h and overlapping points o in the virtual hierarchical boundaries in the tree-grid lattice Boltzmann numerical simulation, the coarse grid points hc1, hc2, hc3, hc4, fine grid points hf1, hf2, hf3, hf4 for constructing the hanging point h, and the coarse grid points oc1, oc2, oc3, oc4 and fine grid points of1, of2, of3, of4 for constructing the overlapping point o are respectively determined;

[0008] For the overlapping point o, the macroscopic variable function considering half a time step 0.5Δt for the corresponding fine grid is:

[0009]

[0010] And the distribution function of the overlapping point o in the virtual hierarchical boundary after half a time step is

[0011]

[0012] where is the equilibrium distribution function, is the non - equilibrium distribution function, and the non - equilibrium distribution function of the coincidence point o in the virtual stratified boundary after half a time step is

[0013]

[0014] For the coincidence point o, the macroscopic variable function corresponding to the coarse grid considering a time step Δt is:

[0015]

[0016] The distribution function of the coincidence point o in the virtual stratified boundary after one time step is

[0017]

[0018] where the non - equilibrium distribution function of the coincidence point o in the virtual stratified boundary after one time step is

[0019]

[0020] For the hanging point h, the macroscopic variable function corresponding to the fine grid considering half a time step 0.5Δt is:

[0021]

[0022] And the distribution function of the hanging point h in the virtual stratified boundary after half a time step is

[0023]

[0024] where the non - equilibrium distribution function of the hanging point h in the virtual stratified boundary after half a time step is

[0025]

[0026] For the hanging point h, the macroscopic variable function corresponding to the coarse grid considering a time step Δt is:

[0027]

[0028] The distribution function of the hanging point h in the virtual stratified boundary after one time step is

[0029]

[0030] where the non - equilibrium distribution function of the hanging point h in the virtual stratified boundary after one time step is

[0031]

[0032] Beneficial effects

[0033] The present invention can solve the problem of virtual boundary treatment encountered in numerical simulation based on the tree - grid LBM algorithm. Compared with the old format developed in previous studies, the new format further improves the calculation accuracy on the premise of ensuring high calculation efficiency. In practical applications such as airfoil design, engine design, flow around a square cylinder, and internal flow driven by a lid in a square cavity, better application effects can be achieved.

[0034] The additional aspects and advantages of the present invention will be partly given in the following description, partly will become obvious from the following description, or will be understood through the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] The above - mentioned and / or additional aspects and advantages of the present invention will become obvious and easy to understand from the description of the embodiments in conjunction with the following drawings, wherein:

[0036] Figure 1 The tree - grid of internal flow in a square cavity and flow around an airfoil is shown;

[0037] Figure 2 The LBM evolution mode and the virtual hierarchical boundary in the tree - grid structure are described;

[0038] Figure 3 The comparison of the information sources of interpolation points between the new and old formats is described; (a) old format (b) new format;

[0039] Figure 4 The instantaneous streamline diagrams at different moments within a complete cycle of flow around a square cylinder (Re = 150) are shown; (a) t0 = 13.0217s, (b) t1 = 13.0397s, (c) t2 = 13.0576s, (d) t3 = 13.0756s, (e) t4 = 13.0936s (f) t5 = 13.11148s

[0040] Figure 5 The streamline diagram of the internal flow driven by a lid in a square cavity (Re = 5000, steady flow) is shown. DETAILED DESCRIPTION OF THE INVENTION

[0041] The present invention aims at the problem of virtual boundary treatment encountered in the numerical simulation of complex shapes based on the tree - grid Lattice Boltzmann method (LBM) algorithm, and proposes a new high - precision processing method for the virtual hierarchical boundary in the tree - grid lattice Boltzmann method. In practical engineering calculations, it can not only handle external flow fields such as flow around an airfoil, but also be applicable to the numerical simulation of internal flow fields in cavities with curved physical boundaries.

[0042] Taking the flow around a square column and the lid-driven flow in a square cavity as examples respectively, the high-precision processing method of the new virtual hierarchical boundary is described below.

[0043] Example 1: Numerical simulation of flow around a square column:

[0044] When performing the numerical simulation of the flow around a square column, after performing CFD operations such as establishing the three-dimensional model of the square column and dividing the tree grid, the virtual boundary is processed through the following process:

[0045] As Figure 3 (a) shows, corresponding to the source of the interpolation point information required by the previous virtual hierarchical boundary processing format; Figure 3 (b) shows, corresponding to the source of the interpolation point information required by the new optimized virtual hierarchical boundary processing format. Compared with the old format, first, the new format uses more points to construct the virtual hierarchical points, and the interpolation format is more accurate. Second, in the old format, the velocity and density of the hierarchical virtual boundary points are solved through the interpolation format, and then the distribution function of the virtual hierarchical points is extrapolated on the fine grid side through the extrapolation method. The disadvantage of this is that it fails to consider factors that are more in line with physical reality such as the equilibrium distribution function and the non-equilibrium distribution function. And only the information of the fine grid is used during extrapolation, ignoring the information of the coarse grid, while the new optimized processing format proposed by the present invention solves both of these problems.

[0046] Figure 3 Describes the virtual hierarchical boundary (thick line) in the tree grid LBM numerical simulation. As can be observed from Figure 3 (b), points h and o are the hanging point and the coincident point in the virtual hierarchical boundary respectively. Points hc1, hc2, hc3, hc4 are the coarse grid points used to construct the hanging point h. Points hf1, hf2, hf3, hf4 are the fine grid points used to construct the hanging point h. At the same time, points oc1, oc2, oc3, oc4 are the coarse grid points used to construct the coincident point o. Points of1, of2, of3, of4 are the fine grid points used to construct the coincident point o.

[0047] The new format is specifically introduced below taking the coincident point as an example. Since the construction of the virtual hierarchical boundary needs to consider the interpolation of both time and space, first consider half a time step (corresponding to the fine grid) 0.5Δt, and the macroscopic variable function of this point can be constructed according to formula (1)

[0048]

[0049] Regarding the time interpolation of half a time step in formula (1), taking the randomly selected coarse grid point oc2 as an example, the construction process is specifically described, as shown in formula (2)

[0050]

[0051] Therefore, the distribution function of the coincidence point o in the virtual hierarchical boundary after half a time step is obtained from formula (3)

[0052]

[0053] wherein, the equilibrium distribution function can be calculated according to formula (4), the macroscopic physical quantities used therein are constructed by formula (1), and the non-equilibrium part is calculated by formula (5)

[0054]

[0055] Therefore, after one time step with respect to point o, formulas (1), (3), and (5) become formulas (6), (7), and (8)

[0056]

[0057] Construct the required coarse and fine grid points according to the same principle, and construct the macroscopic physical quantities and distribution functions of the hanging point h. The detailed process will not be elaborated here, and only the relevant formulas are given

[0058]

[0059]

[0060]

[0061]

[0062] Table 1. Comparison of the calculation results of the flow around a square cylinder using the new format (Re = 150)

[0063]

[0064] Table 1 presents the comparison of the calculation results of the flow around a square cylinder with the old and new virtual boundary formats at Re = 150. Since the flow has evolved into a periodic flow, the average drag coefficient Cd-mean and another parameter, the Strouhal number St, which measures the periodic flow, are listed in the table. It can be seen from the table that the new format improves the calculation accuracy on the premise of ensuring the high computational efficiency of the tree-grid LBM algorithm. This is because, in the construction process of the new format, not only a more accurate mathematical interpolation format is adopted, but also the actual physical situation is considered. In addition, the new format also satisfies the information transfer between the coarse and fine grids. Since the new format is more complex than the old format, the calculation time required for each time step is slightly more. However, because the calculation results after each time step are more accurate, the iterative speed is locally accelerated. Therefore, generally, the calculation time required for the same example on the same machine is slightly more than that required by the old format. But it has little impact on the high computational efficiency of the tree-grid LBM algorithm. Because in previous studies, it was found that compared with the uniform rectangular grid, the use of the tree grid can improve the computational efficiency by 5-20 times through different grid refinement strategies.

[0065] Figure 4 The instantaneous streamline diagrams at different times during a complete cycle of the flow around a square cylinder are shown, and the evolution process (Karman vortex street) of the vortex shedding in the wake region with attached vortices at Re = 150 can be observed.

[0066] Example 2: Numerical simulation of the internal flow driven by the lid of a square cavity:

[0067] When conducting the numerical simulation of the internal flow driven by the lid of a square cavity, after performing CFD operations such as establishing the three-dimensional model of the lid of the square cavity and dividing the tree grid, the virtual boundary is processed in the same manner as in Example 1. The comparison of the results is shown in Table 2, and the streamline diagram of the internal flow driven by the lid of the square cavity is as Figure 5 shown.

[0068] It can be seen that the processing method proposed in the present invention is applicable to the numerical simulation of the flow around a square cylinder and the numerical simulation of the internal flow driven by the lid of a square cavity.

[0069] Table 2. Comparison of the calculation results of the internal flow driven by the lid of a square cavity using the new format (Re = 5000 (steady flow), 9500 (unsteady periodic flow))

[0070]

[0071]

[0072] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. Numerical simulation of flow around a square column, the numerical simulation including establishing a three-dimensional model of the square column, dividing a tree grid, and virtual hierarchical boundary processing, characterized in that: In the numerical simulation process, the tree-grid lattice Boltzmann method is adopted, and the following method is used to process the virtual stratified boundary: For the hanging points h and the coincident points o in the virtual stratified boundary in the tree-grid lattice Boltzmann numerical simulation, the coarse grid points hc1, hc2, hc3, hc4, the fine grid points hf1, hf2, hf3, hf4 for constructing the hanging point h, and the coarse grid points oc1, oc2, oc3, oc4 and the fine grid points of1, of2, of3, of4 for constructing the coincident point o are determined respectively; For the coincident point o, the macroscopic variable function under a half time step of 0.5Δt for the corresponding fine grid is: The distribution function of the coincidence point o in the virtual hierarchical boundary after half a time step is where is the equilibrium distribution function, is the non-equilibrium distribution function, and the non-equilibrium distribution function of the coincidence point o in the virtual stratified boundary after half a time step is For the coincident point o, the macroscopic variable function under a time step of Δt for the corresponding coarse grid is: The distribution function of the coincident point o in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the coincident point o in the virtual stratified boundary after one time step is For the hanging point h, the macroscopic variable function under a half time step of 0.5Δt for the corresponding fine grid is: The distribution function of the hanging point h in the virtual hierarchical boundary after half a time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after a half time step is For the hanging point h, the macroscopic variable function under a time step of Δt for the corresponding coarse grid is: The distribution function of the hanging point h in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after one time step is 2. Numerical simulation of the internal flow driven by the top cover of a square cavity. The numerical simulation includes establishing a three-dimensional model of the top cover of the square cavity, dividing tree grids, and virtual hierarchical boundary processing, and is characterized in that: In the numerical simulation process, the tree-grid lattice Boltzmann method is adopted, and the following method is used to process the virtual stratified boundary therein: For the hanging points h and the coincident points o in the virtual stratified boundary in the tree-grid lattice Boltzmann numerical simulation, the coarse grid points hc1, hc2, hc3, hc4, the fine grid points hf1, hf2, hf3, hf4 for constructing the hanging point h, and the coarse grid points oc1, oc2, oc3, oc4 and the fine grid points of1, of2, of3, of4 for constructing the coincident point o are determined respectively; For the coincident point o, the macroscopic variable function under a half time step of 0.5Δt for the corresponding fine grid is: The distribution function of the coincidence point o in the virtual hierarchical boundary after half a time step is wherein is the equilibrium distribution function, is the non-equilibrium distribution function, and the non-equilibrium distribution function of the coincidence point o in the virtual stratified boundary after half a time step is For the coincident point o, the macroscopic variable function under a time step of Δt for the corresponding coarse grid is: The distribution function of the coincident point o in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the coincident point o in the virtual stratified boundary after one time step is For the hanging point h, the macroscopic variable function under a half time step of 0.5Δt for the corresponding fine grid is: The distribution function of the hanging point h in the virtual hierarchical boundary after half a time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after a half time step is For the hanging point h, the macroscopic variable function under a time step of Δt for the corresponding coarse grid is: Distribution function of the hanging point h in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after one time step is 3. Numerical simulation of airfoil flow around, the numerical simulation includes establishing an airfoil model, dividing tree grids, and virtual hierarchical boundary treatment, characterized in that: In the numerical simulation process, the tree-grid lattice Boltzmann method is adopted, and the following method is used to process the virtual stratified boundary therein: For the hanging points h and the coincident points o in the virtual stratified boundary in the tree-grid lattice Boltzmann numerical simulation, the coarse grid points hc1, hc2, hc3, hc4, the fine grid points hf1, hf2, hf3, hf4 for constructing the hanging point h, and the coarse grid points oc1, oc2, oc3, oc4 and the fine grid points of1, of2, of3, of4 for constructing the coincident point o are determined respectively; For the coincidence point o, the macroscopic variable function corresponding to the fine grid considering a half time step of 0.5Δt is as follows: The distribution function of the coincidence point o in the virtual hierarchical boundary after half a time step is where is the equilibrium distribution function, is the non-equilibrium distribution function, and the non-equilibrium distribution function of the coincidence point o in the virtual stratified boundary after half a time step is For the coincidence point o, the macroscopic variable function corresponding to the coarse grid considering a time step of Δt is as follows: The distribution function of the coincidence point o in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the coincidence point o in the virtual stratified boundary after one time step is For the hanging point h, the macroscopic variable function corresponding to the fine grid considering a half time step of 0.5Δt is as follows: The distribution function of the hanging point h in the virtual hierarchical boundary after half a time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after a half time step is For the hanging point h, the macroscopic variable function corresponding to the coarse grid considering a time step of Δt is as follows: Distribution function of the hanging point h in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after one time step is 4. A numerical simulation method for the internal flow in an engine cavity, the numerical simulation including establishing an engine cavity model, dividing a tree grid, and performing virtual hierarchical boundary processing, characterized in that: During the numerical simulation process, the tree-grid lattice Boltzmann method is adopted, and the following method is used to process the virtual stratified boundary therein: For the hanging point h and the coincidence point o in the virtual stratified boundary in the tree-grid lattice Boltzmann numerical simulation, the coarse grid points hc1, hc2, hc3, hc4 for constructing the hanging point h, the fine grid points hf1, hf2, hf3, hf4, as well as the coarse grid points oc1, oc2, oc3, oc4 and the fine grid points of1, of2, of3, of4 for constructing the coincidence point o are determined respectively; For the coincidence point o, the macroscopic variable function corresponding to the fine grid considering a half time step of 0.5Δt is as follows: The distribution function of the coincidence point o in the virtual hierarchical boundary after half a time step is where is the equilibrium distribution function, is the non-equilibrium distribution function, and the non-equilibrium distribution function of the coincidence point o in the virtual stratified boundary after half a time step is For the coincidence point o, the macroscopic variable function corresponding to the coarse grid considering a time step of Δt is as follows: The distribution function of the coincident point o in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the coincidence point o in the virtual stratified boundary after one time step is For the hanging point h, the macroscopic variable function corresponding to the fine grid considering a half time step of 0.5Δt is as follows: The distribution function of the hanging point h in the virtual hierarchical boundary after half a time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after a half time step is For the hanging point h, the macroscopic variable function corresponding to the coarse grid considering a time step of Δt is as follows: Distribution function of the hanging point h in the virtual hierarchical boundary after one time step is Among them, the non-equilibrium distribution function of the hanging point h in the virtual stratified boundary after one time step is

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