CN114091376B - High-precision reconstruction correction shock wave capturing method based on subunit weighting format - Google Patents
High-precision reconstruction correction shock wave capturing method based on subunit weighting format Download PDFInfo
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Abstract
The invention discloses a high-precision reconstruction correction shock wave capturing method based on a subunit weighting format, which belongs to the technical field of shock wave capturing and comprises the following steps: s1, calculating the distribution of solution points and flux points of the high-order CPR and the flux derivative; s2, detecting the units with shock wave by shock wave detection method; s3, dividing the unit into subunits based on the flux points, and constructing a format with shock wave capturing capability on the subunits; s4, determining CPR, HNNW and calculation units of second-order format based on the value of the shock wave detection factor in the shock wave detection method; s5, the Riemann flux at the cell interface is calculated by interpolation polynomials corresponding to the calculation formats of the adjacent cells. The invention not only saves the calculation amount, but also can remarkably improve the capturing capability of the format on the shock wave, and can ensure that the final algorithm still has higher resolution near the shock wave while reducing the requirement of the algorithm on the detection accuracy.
Description
Technical Field
The invention relates to the technical field of shock wave capturing, in particular to a high-precision reconstruction correction shock wave capturing method based on a subunit weighting format.
Background
Computational Fluid Dynamics (CFD) is one of the important means for developing the research of Fluid mechanics mechanism, and plays an increasingly important role in the design and performance evaluation of aerospace aircrafts. With continuous refinement of design in engineering application, people have higher and higher precision requirements on CFD calculation results, and particularly when the precision flow field structure is simulated, a second-order precision algorithm is difficult to meet the requirements. Higher order precision algorithms require less computation than second order precision algorithms to achieve the same error level. The high-precision algorithm has the advantages of simulating a fine flow field structure, such as the advantages in various aspects of turbulence large vortex simulation, separation vortex simulation, direct simulation, pneumatic acoustic calculation and the like. Because of the method of Correction process via Reconstruction (CPR), when selecting a special Correction function, the method can be equivalent to a Discrete Galileo (DG) method, a Spectral Difference (SD) method or a Spectral Volume (SV) method, and the calculation amount is relatively small, which has received great attention in recent years.
In recent years, CPR methods have been developed in terms of free flow conservation, nonlinear stability, large vortex simulation, shock wave capture, and the like. However, the CPR method still has problems in shock wave capture, and further development is required. One method that has gained much attention is to mix the higher order format with the robust weighted class/lower order shock wave format, which can take full advantage of each format.
The difficulty of this shock wave capture method of CPR method is represented by: solution points (coordinate points stored by physical quantity when an equation is discrete) of the CPR method are distributed in a non-equidistant mode (distances between adjacent solution points at different positions are unequal), when the solution points are mixed with a traditional high-order finite difference format with better shock wave capturing capability and developed based on the equidistant solution points, the solution points of the two types of formats are not matched, and data conversion is required to be carried out repeatedly during format switching; secondly, when the CPR method is mixed with a second-order shock wave capturing format, the resolution ratio is greatly reduced due to the fact that the detection is not accurate enough near the shock wave, and how to guarantee the shock wave capturing and the high-resolution characteristic is difficult.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides a high-precision reconstruction correction shock wave capturing method based on a subunit weighting format, solves the problems in the background art, realizes the numerical simulation of the high-order CPR method on the flow of shock waves in computational fluid mechanics, not only saves the calculated amount, but also can obviously improve the capturing capability of the format on the shock waves, and can ensure that the final algorithm still has higher resolution near the shock waves while reducing the requirement of the algorithm on the detection accuracy.
The purpose of the invention is realized by the following scheme:
a high-precision reconstruction correction shock wave capturing method based on a subunit weighting format comprises the following steps:
s1, distributing solution points on grid cells as Gauss-Legendre integral points, wherein flux points are one more than solution points according to the number of flux points in each dimension, the distance between adjacent flux points is Gauss integral weight, and the flux points at two ends are distributed on a cell interface; the flux derivative of the high-order CPR method is obtained by derivation based on Lagrange interpolation polynomial of the flux point, and the flux derivative of the solution point is corrected through Radau polynomial by the difference between the Riemann flux and the unit discontinuous flux;
s2, detecting the units with shock wave by shock wave detection method;
s3, dividing the unit into sub-units based on the flux points, and constructing a format with shock wave capturing capacity on the sub-units: the first format is a high-order hybrid non-equidistant nonlinear weighting format HNNW, and flux derivatives are calculated based on a differential operator of a solution point flux point hybrid; the flux at the flux point is Riemann flux, and the left and right values of the physical quantity at the flux point are obtained through non-equidistant nonlinear weighted interpolation based on the solution point; the second format is a second order format, calculating flux derivatives based on two flux points adjacent to the solution point; the left and right values of the physical quantity at the flux point are obtained through restricted first-order polynomial distribution;
s4, determining CPR, HNNW and calculation units of second-order format based on the value of the shock wave detection factor in the shock wave detection method;
s5, the Riemann flux at the cell interface is calculated by interpolation polynomials corresponding to the calculation formats of the adjacent cells.
Further, in step S1, the mesh cells include quadrilateral mesh cells of a two-dimensional structural mesh.
Further, in step S2, the shock wave detection method is specifically a shock wave detection method based on the highest modal attenuation.
Further, in step S3, the calculating the flux derivative based on the difference operator of the solution point flux point hybrid type includes the sub-steps of: two flux points adjacent to the solution point and the solution point outside the flux point are used to form the template point for calculating the flux derivative.
Further, in step S4, the calculation unit includes a problem unit, and the problem unit adopts a high-order non-equidistant non-linear weighting format HNNW and/or a second-order format calculation.
Further, comprising the sub-steps of: and adopting a second-order format for the problem units in the small area containing the shock waves, and adopting a high-order HNNW format for the rest problem units.
The beneficial effects of the invention include:
because the solution point of the HNNW and the second-order format is coincident with the solution point of the CPR, the shock wave capturing format of the CPR method based on the sub-unit HNNW and the second-order format limit does not need data conversion, thereby saving the calculation amount.
The HNNW format with strong shock wave capturing capability and the second-order format with strong shock wave capturing capability are adopted, so that the shock wave capturing capability of the formats can be obviously improved, and meanwhile, the HNNW still has high-order precision in a smooth area, so that the requirement of the algorithm on the detection accuracy is greatly reduced, and finally, the algorithm still has high resolution near the shock wave.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a diagram of the distribution of solution points and flux points on a quadrilateral grid cell in an embodiment of the present invention;
fig. 2 is a template point for the fifth-order HNNW5 to calculate the flux derivative in an embodiment of the present invention;
FIG. 3 is a block diagram of the template points for second order format calculation of flux derivatives in an embodiment of the invention.
Detailed Description
All features disclosed in all embodiments in this specification, or all methods or process steps implicitly disclosed, may be combined and/or expanded, or substituted, in any way, except for mutually exclusive features and/or steps.
The technical problems, technical concepts, working principles and working processes to be solved by the present invention are further described in detail and fully with reference to the accompanying drawings 1 to 3.
The design concept of the invention is as follows: the scheme provided by the invention aims to provide a shock wave capturing technical scheme based on a subunit non-equidistant weighting format of a high-order CPR method, and can realize numerical simulation of the high-order CPR method on flow including shock waves in computational fluid mechanics. In order to achieve the purpose, the scheme provided by the invention firstly constructs a Hybrid non-equidistant non-linear Weighted (HNNW) format and a second-order format of high-order solution point flux points based on solution points which are distributed in a non-equidistant way in a calculation space, and then divides a calculation unit into a CPR unit, an HNNW unit and a second-order format unit by a shock detection method, so that a problem unit in a small region containing shock waves is calculated by adopting the second-order format, and the rest problem units are calculated by adopting the HNNW format with high resolution characteristic and shock wave capturing capability, thereby realizing effective shock wave capturing and high resolution.
In specific application, the technical scheme of the invention is explained in detail as follows:
(1) on a quadrilateral grid unit of a two-dimensional structure grid, the distribution solution points are Gauss-Legendre integral points, flux points are one more than solution points according to the number of the flux points in each dimension, the distance between adjacent flux points is Gauss integral weight, the flux points at two ends are distributed on a unit interface, the condition that the number of solution points in each direction is 5 is given in figure 1, the solution points are circular points in the figure, and the flux points are square points in the figure. The flux derivative of the high-order CPR method is obtained by derivation based on Lagrange interpolation polynomial of the flux point, and the flux derivative of the solution point is corrected through Radau polynomial according to the difference between the Riemann flux and the unit discontinuous flux.
(2) The shock detection technique selects the detection method based on the highest modal attenuation (MD). Units in which shock waves are likely to exist are detected by a shock wave detection technology, and are called problem units.
(3) The cell is split into sub-cells based on the flux points, as is the grey background cell in fig. 1. A format with shock wave capture capability is constructed on the subunit. The first format is a high-order non-equidistant non-linear weighting format (HNNW), which calculates the flux derivative based on a difference operator of a solution point flux point hybrid, that is, two flux points adjacent to the solution point and a solution point outside the flux point are used to form a template point for calculating the flux derivative. Using fifth order CPR as an example, fig. 2 shows the template points of the calculated flux derivative for the second solution point of a cell. The flux at the flux point is Riemann flux, and the left and right values of the physical quantity at the flux point are obtained through non-equidistant nonlinear weighted interpolation based on the solution point. The second format is a second order format, where the flux derivative is calculated based on two flux points adjacent to the solution point, as shown in fig. 3. The left and right values of the physical quantity at the flux point are obtained by a restricted first order polynomial distribution.
(4) The problem unit is computed using a high-order HNNW and/or a second-order format. In a specific embodiment, the calculation can be divided into three regions from small to large based on the value of the shock detection factor, and the calculation is performed by using CPR, HNNW and second-order formats respectively. The principle is that a second-order format is adopted for problem units in a small area containing shock waves, and a high-order HNNW format is adopted for other problem units.
(5) The Riemann flux at the cell interface is calculated by interpolation polynomials corresponding to the calculation formats of adjacent cells.
In the embodiment of the invention, the provided brand-new mixed high-order non-equidistant nonlinear weighting format of the subunits improves and guarantees the high resolution of the final format, and data conversion is not needed between the main unit and the subunits. And the problem unit is divided into two formats for calculation based on the shock wave detection factor, a second-order format is adopted near a shock wave region to ensure the robustness of the final format capture, and a high-resolution characteristic of the final format is ensured by the problem unit outside the shock wave by adopting a high-order HNNW.
Example 1: a high-precision reconstruction correction shock wave capturing method based on a subunit weighting format comprises the following steps:
s1, distributing solution points on grid cells as Gauss-Legendre integral points, wherein flux points are one more than solution points according to the number of flux points in each dimension, the distance between adjacent flux points is Gauss integral weight, and the flux points at two ends are distributed on a cell interface; the flux derivative of the high-order CPR method is obtained by derivation based on Lagrange interpolation polynomial of the flux point, and the flux derivative of the solution point is corrected through Radau polynomial by the difference between the Riemann flux and the unit discontinuous flux;
s2, detecting the units with shock wave by shock wave detection method;
s3, dividing the unit into sub-units based on the flux points, and constructing a format with shock wave capturing capacity on the sub-units: the first format is a high-order non-equidistant non-linear weighting format HNNW, and flux derivatives are calculated based on a differential operator of a solution point flux point mixed type; the flux at the flux point is Riemann flux, and the left and right values of the physical quantity at the flux point are obtained through non-equidistant nonlinear weighted interpolation based on the solution point; the second format is a second order format, calculating flux derivatives based on two flux points adjacent to the solution point; the left and right values of the physical quantity at the flux point are obtained through restricted first-order polynomial distribution;
s4, determining CPR, HNNW and calculation units of second-order format based on the value of the shock wave detection factor in the shock wave detection method;
s5, the Riemann flux at the cell interface is calculated by interpolation polynomials corresponding to the calculation formats of the adjacent cells.
Example 2: on the basis of embodiment 1, in step S1, the mesh cells include quadrangular mesh cells of a two-dimensional structural mesh.
Example 3: in step S2, based on embodiment 1, the shock wave detection method is specifically a shock wave detection method based on the highest modal attenuation.
Example 4: on the basis of embodiment 1, in step S3, the calculating the flux derivative based on the difference operator of the solution point flux point mixture type includes the sub-steps of: two flux points adjacent to the solution point and the solution point outside the flux point are used to form the template point for calculating the flux derivative.
Example 5: on the basis of embodiment 1, in step S4, the calculation unit includes a problem unit, and the problem unit adopts a high-order non-equidistant non-linear weighting format HNNW and/or a second-order format calculation.
Example 6: on the basis of the embodiment 5, the method comprises the following substeps: and adopting a second-order format for the problem units in the small area containing the shock waves, and adopting a high-order HNNW format for the rest problem units.
The parts not involved in the present invention are the same as or can be implemented using the prior art.
The above-described embodiment is only one embodiment of the present invention, and it will be apparent to those skilled in the art that various modifications and variations can be easily made based on the application and principle of the present invention disclosed in the present application, and the present invention is not limited to the method described in the above-described embodiment of the present invention, so that the above-described embodiment is only preferred, and not restrictive.
Other embodiments than the above examples may be devised by those skilled in the art based on the foregoing disclosure, or by adapting and using knowledge or techniques of the relevant art, and features of various embodiments may be interchanged or substituted and such modifications and variations that may be made by those skilled in the art without departing from the spirit and scope of the present invention are intended to be within the scope of the following claims.
Claims (4)
1. A high-precision reconstruction correction shock wave capturing method based on a subunit weighting format is characterized by comprising the following steps:
s1, distributing solution points on grid cells as Gauss-Legendre integral points, wherein flux points are one more than solution points according to the number of flux points in each dimension, the distance between adjacent flux points is Gauss integral weight, and the flux points at two ends are distributed on a cell interface; the flux derivative of the high-order CPR method is obtained by derivation based on Lagrange interpolation polynomial of the flux point, and the flux derivative of the solution point is corrected through Radau polynomial by the difference between the Riemann flux and the unit discontinuous flux;
s2, detecting the units with shock wave by shock wave detection method;
s3, dividing the unit into sub-units based on the flux points, and constructing a format with shock wave capturing capacity on the sub-units: the first format is a high-order hybrid non-equidistant nonlinear weighting format HNNW, and flux derivatives are calculated based on a differential operator of a solution point flux point hybrid; the flux at the flux point is Riemann flux, and the left and right values of the physical quantity at the flux point are obtained through non-equidistant nonlinear weighted interpolation based on the solution point; the second format is a second order format, calculating flux derivatives based on two flux points adjacent to the solution point; the left and right values of the physical quantity at the flux point are obtained through restricted first-order polynomial distribution;
s4, determining CPR, HNNW and calculation units of second-order format based on the value of the shock wave detection factor in the shock wave detection method; in step S4, the calculating unit includes a problem unit, and the problem unit adopts a high-order non-equidistant non-linear weighting format HNNW and/or a second-order format calculation; and comprises the substeps of: adopting a second-order format for the problem units in the region with small shock waves, and adopting a high-order HNNW format for the rest problem units;
s5, the Riemann flux at the cell interface is calculated by interpolation polynomials corresponding to the calculation formats of the adjacent cells.
2. The method for high-precision reconstruction-modified shock wave capturing based on sub-unit weighting format according to claim 1, wherein in step S1, the grid cells comprise quadrilateral grid cells of a two-dimensional structural grid.
3. The method for high-precision reconstruction-modified shock wave capturing based on subunit weighting format according to claim 1, wherein in step S2, the shock wave detection method is a shock wave detection method based on highest modal attenuation.
4. The method for high-precision reconstruction-modified shock wave capturing based on subunit weighting format according to claim 1, wherein in step S3, the calculating the flux derivative based on the difference operator of solution point flux point hybrid type includes the sub-steps of: two flux points adjacent to the solution point and the solution point outside the flux point are used to form the template point for calculating the flux derivative.
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