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CN-121980847-A - Serial calculating method for dispersion characteristic of finite difference format

CN121980847ACN 121980847 ACN121980847 ACN 121980847ACN-121980847-A

Abstract

A serial calculation method for the dispersion characteristics of finite difference format includes such steps as determining the minimum value and maximum value of angular label bias in all template lattice points and the weight coefficient corresponding to each template lattice point based on the finite difference format of discrete flow term in computational fluid mechanics, sequentially calculating the maximum absolute value M of angular label bias of template lattice point, extended weight array and dispersion equivalent weight array, constructing the calculation equation of dispersion characteristic function coefficient, calculating the dispersion characteristic function coefficient, and obtaining dispersion characteristic function based on the dispersion characteristic function coefficient. The method calculates the dispersion characteristic function by constructing the dispersion equivalent weight array, and solves the problem that the format dispersion characteristic in the full-wave number range is difficult to carry out systematic theoretical analysis and inaccurate calculation of the dispersion characteristic in the existing method.

Inventors

  • KANG JIAN

Assignees

  • 中国航天空气动力技术研究院

Dates

Publication Date
20260505
Application Date
20251223

Claims (10)

  1. 1. A serial calculation method of chromatic dispersion characteristics in finite difference format, characterized by comprising: Determining minimum value of corner mark bias in all template grid points in finite difference format based on finite difference format of discrete flow terms in computational fluid dynamics Corner mark offset maximum value in all template grid points And a weight coefficient corresponding to each template grid point; Offset minima based on corner markers Sum-angle sign offset maximum value Calculating the maximum absolute value M of the template grid point corner mark bias; Offset minima based on corner markers Maximum value of angle sign offset And maximum absolute value M, calculate the extension weight array Wherein The k expansion weight in the expansion weight array; based on the extended weight array, a dispersion equivalent weight array is calculated Wherein The k-th dispersion equivalent weight in the dispersion equivalent weight array; constructing a dispersion characteristic function coefficient calculation equation based on the dispersion equivalent weight array, calculating a first dispersion characteristic function coefficient based on the dispersion characteristic function coefficient calculation equation Coefficient of second dispersion characteristic function The M-th dispersion characteristic function coefficient ; Based on first dispersion characteristic function coefficient Coefficient of second dispersion characteristic function The M-th dispersion characteristic function coefficient Obtaining a dispersion characteristic function 。
  2. 2. The method for serially calculating the dispersion characteristics of the finite difference format according to claim 1, wherein the finite difference format of discrete terms of flow in the computational fluid dynamics is a linear finite difference format used on a uniform Cartesian grid, and specifically comprises the following steps: Wherein, the For the derivative value of the jth grid point, i.e. the finite difference format of the 1 st derivative of the jth grid point, For the distance between two adjacent grid points, The function value representing the j + k th grid point, And the weight coefficient corresponding to the kth template grid point.
  3. 3. The serial calculation method of finite difference format dispersion characteristics according to claim 1, wherein the calculation formula of the maximum absolute value M of the template grid point corner mark offset is: Wherein, max is a maximum value function, And Respectively, minimum value of angle sign bias Sum-angle sign offset maximum value Is the absolute value of (c).
  4. 4. A serial calculation method of dispersion characteristics in finite difference format according to claim 1, wherein the calculation of the expansion weight array Specifically, each extended weight in the extended weight array is calculated so as to obtain the extended weight, and the calculation method of each extended weight in the extended weight array is that when In the time-course of which the first and second contact surfaces, When (1) Or (b) In the time-course of which the first and second contact surfaces, I is the i-th expansion weight in the expansion weight array Corresponding subscripts; And the weight coefficient corresponding to the ith template grid point.
  5. 5. The method for serially calculating the dispersion characteristics of the finite difference format according to claim 1, wherein the calculating the dispersion equivalent weight array comprises the specific steps of calculating each dispersion equivalent weight in the dispersion equivalent weight array to obtain the dispersion equivalent weight array, and the calculating method of each equivalent weight in the dispersion equivalent weight array comprises the following steps: wherein i is the i-th dispersion equivalent weight in the dispersion equivalent weight array Corresponding subscripts; And The i-th expansion weight and the i-th expansion weight in the expansion weight array are respectively.
  6. 6. The method for serial calculation of finite difference format dispersion characteristics according to claim 1, wherein the dispersion characteristic function coefficient calculation equation is: wherein M is the maximum absolute value of the template grid point corner mark offset, Is the kth dispersion equivalent weight in the dispersion equivalent weight array.
  7. 7. The method for serial calculation of finite difference format dispersion characteristics according to claim 6, wherein the first dispersion characteristic function coefficient is calculated based on a dispersion characteristic function coefficient calculation equation Coefficient of second dispersion characteristic function The M-th dispersion characteristic function coefficient The specific process comprises the steps of carrying out descending order value on k in a dispersion characteristic function coefficient calculation equation, sequentially enabling k= M, k =M-1, the number of the dispersion characteristic function coefficients to be equal to or higher than the number of the dispersion characteristic function coefficients to be equal to or lower than the number of the dispersion characteristic function coefficient calculation equations, obtaining M dispersion characteristic function coefficient calculation equations, combining all dispersion characteristic function coefficient calculation equations, and solving a first dispersion characteristic function coefficient Coefficient of second dispersion characteristic function The M-th dispersion characteristic function coefficient 。
  8. 8. A serial calculation method of a finite difference format dispersion characteristic according to claim 1, wherein said dispersion characteristic function The calculation formula is that Wherein M is the maximum absolute value of the template grid point corner mark offset, Is the k+1 dispersion characteristic function coefficient, Is wave number.
  9. 9. A computer program product, characterized in that the computer program product comprises a computer program which, when executed by a processor, implements the steps of a serial calculation method of the dispersion characteristics of a finite difference format according to any one of claims 1 to 8.
  10. 10. A processor, wherein the processor is configured to run a program, and wherein the program, when run, performs a method for serially calculating a dispersion characteristic in a finite difference format as claimed in any one of claims 1 to 8.

Description

Serial calculating method for dispersion characteristic of finite difference format Technical Field The invention relates to a serial calculation method of dispersion characteristics of a finite difference format, belonging to the field of computational aerodynamics. Background In computational fluid dynamics, in particular branching disciplines in which compressible flows need to be considered, such as computational aerodynamics, shock waves are a flow phenomenon that requires special handling. Because of discontinuities in flow parameters (e.g., pressure, density, velocity, temperature, etc.) before and after the shock wave, calculation of aerodynamic model equations using a linear format can cause non-physical oscillations in the numerical solution near the shock wave. Such non-physical oscillations may also diffuse from the vicinity of the shock wave to the surrounding flow field and even diverge the calculations so that numerical simulations cannot be performed. In order to make the results of numerical simulations as close as possible to the physical phenomenon that actually occurs, it is necessary to consider the problem of how to effectively deal with the flow parameter discontinuities generated by the shock waves. The shock wave capturing format based on the finite difference method is one of the methods for processing the discontinuity of the flow parameters before and after the shock wave is widely used at present. Among the high-precision high-resolution shock wave capturing formats, a method is widely used, wherein a calculation result of a linear finite difference format is used as a basis, a plurality of linear formats are used for calculation firstly, then weight coefficients corresponding to the linear formats are calculated according to flow field data, and finally weighted average of the calculation results of the linear formats is used as a final calculation result. When the finite difference format on the uniform grid described above calculates the convection term in the Euler equation or the Navier-Stokes equation, two errors occur, one error representing the phase error of the numerical solution, which is called dispersion, and the other error representing the amplitude error of the numerical solution, which is called dissipation. In order for the numerical solution to reflect the flow phenomenon as truly as possible, it is often desirable to optimize the linear format so that the dispersion and dissipation errors it introduces are as small as possible. Accordingly, a corresponding study of dispersion characteristics is required. There are three methods currently used to calculate the dispersion characteristics of the linear format. The first is to write the dispersion function directly by definition, starting from theory. Although the method can give an accurate expression, the expression form is not suitable for unified theoretical analysis, and the expression is difficult to design a high-precision high-resolution shock wave capture format. The second method is a half-numerical value half-theory method, a model equation is firstly solved by using a format to be analyzed, then the chromatic dispersion characteristics of the format are determined according to a numerical solution, the method can analyze various formats (including linear formats and nonlinear formats), but can only analyze a limited number of formats and can only give approximate numerical values, and when strict mathematical demonstration is required for the chromatic dispersion characteristics of an infinite number of formats, the method cannot be used. The third is a theoretical method, in which the theoretical expression obtained by the first method is subjected to taylor expansion in the vicinity of the wave number of 0, and then finite terms are intercepted for analysis. This method is based on theory, which gives a more accurate theoretical result for the case of low wavenumbers, but because it truncates the higher-order terms of wavenumbers, it may give an incorrect inference of the chromatic dispersion characteristics of the format if the wavenumbers are so large that the effect of the higher-order terms becomes non-negligible. Therefore, the conventional method is difficult to perform unified theoretical analysis on the format dispersion characteristics in the full-wave number range, the dispersion characteristic calculation is inaccurate, and the subsequent design of a high-precision shock wave capturing format is difficult to support. Disclosure of Invention The invention solves the technical problem of overcoming the defects of the prior art and providing a serial calculation method of the chromatic dispersion characteristics of a finite difference format. The method calculates the dispersion characteristic function by constructing the dispersion equivalent weight array, and solves the problem that the format dispersion characteristic in the full-wave number range is difficult to carry out systematic theo