CN112163312B - Method for carrying out numerical simulation on compressible flow problem through high-order WENO format reduction - Google Patents
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Abstract
The invention discloses a method for carrying out numerical simulation on compressible flow problems through high-order WENO format reduction, which comprises the following steps: s1, when calculating the numerical flux of a certain position, judging whether order reduction is needed according to a criterion for the given WENO format with highest precision; s2, if the order needs to be reduced, judging whether the order needs to be reduced or not according to the same criterion in the step S1 for the second-time high-precision WENO format after the order is reduced; s3, executing step S2 in a loop until WENO format with reduced order or lowest order precision is not needed; and then calculating the numerical flux of the position by adopting a WENO format with corresponding precision so as to further calculate and obtain a numerical simulation value of the position on the compressible flow problem containing the shock wave. The present invention can be used for the conservative difference format, and the amount of calculation is small because the format used is determined in advance.
Description
Technical Field
The invention relates to the technical field of numerical simulation of compressible flow, in particular to a method for carrying out numerical simulation on a compressible flow problem through order reduction in a high-order WENO format.
Background
In numerical simulation of compressible flow, discontinuity and other continuous small-scale structures may exist at the same time, and a Weighted-accuracy non-oscillatory (Weighted-essence-free) format can capture discontinuity and achieve high-order accuracy in a smooth area, and is commonly used for simulating the problems. Higher order formats generally allow for finer flow structures in the continuum of the flow field and more sharply capture the shock.
However, for the problem of strong non-linear discontinuity, non-physical oscillation is more likely to occur by adopting the high-order WENO format, and even the calculation may fail. One of the reasons is that the dissipation of the high order format is lower and thus oscillation may not be effectively suppressed. The second is that the wider templates they use may contain more than one discontinuity. Then all sub-templates may not be smooth and therefore cannot get an essentially oscillation-free solution. Third, Runge phenomenon in high-order interpolation or reconstruction. Numerical flux on an off-center wide template can lead to large numerical errors, which in turn affect stability. For the euler equation, the situation becomes worse due to the interaction of different feature directions. Severe numerical oscillations may occur in the solutions of the very high order formats and the calculations are more likely to diverge. For very high-order WENO formats, existing methods use recursive reduction to solve this problem, which requires reconstructed values of density and pressure at cell boundaries. However, reconstruction of higher-order WENO differential formats is typically done on the flux. Therefore, this method cannot be directly applied to the conservative differential format, and other methods need to be constructed.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: in view of the existing problems, a method for numerically simulating the compressible flow problem through the reduction of the order of the high-order WENO format is provided.
The technical scheme adopted by the invention is as follows:
a method for numerically simulating a compressible flow problem by reducing order in a higher-order WENO format, comprising the steps of:
s1, when calculating the numerical flux of a certain position, judging whether order reduction is needed according to a criterion for the given WENO format with highest precision;
s2, if the order needs to be reduced, judging whether the order needs to be reduced or not according to the same criterion in the step S1 for the second-time high-precision WENO format after the order is reduced;
s3, executing step S2 in a loop until WENO format with reduced order or lowest order precision is not needed; and then calculating the numerical flux of the position by adopting a WENO format with corresponding precision so as to further calculate and obtain a numerical simulation value of the position on the compressible flow problem containing the shock wave.
Further, the position of the flux of the numerical value is recorded as x j+1/2 WENO gridThe template used in formula is S ═ x j-m+1 ,x j-m+2 ,…,x j+m The criteria in step S1 and step S2 are that the following two requirements are satisfied simultaneously:
(1) when the number of the template points is more than or equal to 4, the smooth factor of the pressure intensity needs to meet the following requirements:
β (2m) ≤4β (2m-2) ；
smoothing factor beta (2m) All 2m points on the template S are needed for calculation;
smoothing factor beta (2m-2) All 2m-2 points of the template S with two end points removed are used for calculation;
(2) the pressure p and the velocity u need to satisfy:
i∈(j-m+1,…,j+m-1),l∈(j-m+1,…,j+m-2)；
wherein gamma is the gas specific heat ratio; h and delta t are respectively the space grid scale and the time step length of the current direction; d is the spatial dimension of the problem sought.
Further, the smoothing factor β (2m) Or smoothing factor beta (2m-2) The calculation formula of (a) is as follows:
wherein, beta (n) Represents the smoothing factor beta (2m) Or smoothing factor beta (2m-2) ；
x c Is the midpoint of the template, i.e. x j+1/2 ；
σ, δ is defined as follows:
and h is the space grid scale of the current direction.
Further, the smoothing factor β (2m) Or smoothing factor beta (2m-2) The calculation formula of (a) is as follows:
β (n) ＝(b n ) 2 +|a n ·c n |
wherein, beta (n) Represents the smoothing factor beta (2m) Or smoothing factor beta (2m-2) ；
x c Is the midpoint of the template, i.e. x j+1/2 。
Further, a n ,b n ,c n Can be written as follows:
wherein k is the subscript of the leftmost end point of the template for the smoothing factor, and z is a n ,b n Or c n (ii) a Coefficient alpha n,l The values of (A) are as follows:
further, n > 3, and n is an integer.
In summary, due to the adoption of the technical scheme, the invention has the beneficial effects that:
1. the invention can adopt a very high-order WENO format in a continuous area, thereby obtaining a more precise flow field structure;
2. the WENO format with a relatively low order is started near the strong shock wave, so that the shock wave stability is good, the possibility of non-physical oscillation near the shock wave is low, and the calculation failure caused by the negative pressure is difficult to occur in the calculation;
3. the present invention can be used for the conservative difference format, and the amount of calculation is small because the format used is determined in advance.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the embodiments will be briefly described below, it should be understood that the following drawings only illustrate some embodiments of the present invention and therefore should not be considered as limiting the scope, and for those skilled in the art, other related drawings can be obtained according to the drawings without inventive efforts.
FIG. 1 is a block flow diagram of a method for numerically simulating a compressible flow problem via high-order WENO format reduction in accordance with the present invention.
Fig. 2 is a comparison graph of calculation results of several formats when the number of grid points is 200 and the calculation termination time T is 1.8 in the first calculation example of the present invention.
Fig. 3 is a comparison graph of several format calculation results when the number of grid points N is 400 and the calculation termination time T is 0.038 in calculation example two of the present invention.
Fig. 4a is a density cloud chart of the calculation result in the format of WENO5-M when the calculation area is [0,1] × [0,1], the grid point number is 400, and the calculation termination time T is 0.8 in the third calculation example of the present invention.
Fig. 4b is a density cloud chart of the calculation result in PORWENO9-S format when the calculation area is [0,1] × [0,1], the grid point number is 400, and the calculation termination time T is 0.8 in the third calculation example of the present invention.
Fig. 4c is a density cloud chart of the calculation result in PORWENO17-S format when the calculation area is [0,1] × [0,1], the grid point number is 400, and the calculation termination time T is 0.8 in the third calculation example of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the detailed description and specific examples, while indicating the preferred embodiment of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention. The components of embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.
Description of the drawings: for the solution of the conservation-oriented differential method of the euler equation, the general numerical solution steps are as follows:
and 4, starting to cycle the following time advancing process until the required termination time of the problem:
(1) for the interior points, numerical simulation calculation is carried out by adopting a space format;
(2) adopting proper numerical value boundary processing according to the boundary condition;
(3) and after the numerical calculation space of the inner points and the boundaries is dispersed, the method adopts a proper time format for advancing.
The invention mainly relates to a method for carrying out numerical simulation calculation by adopting a spatial format in the (1) th substep in the step 4, thereby the invention discloses a method for carrying out numerical simulation on a compressible flow problem by reducing the order through a high-order WENO format, which comprises the following steps:
s1, when calculating the numerical flux of a certain position, judging whether order reduction is needed according to a criterion for the given WENO format with highest precision;
s2, if the order needs to be reduced, judging whether the order needs to be reduced or not according to the same criterion in the step S1 for the second-time high-precision WENO format after the order is reduced;
s3, executing step S2 in a loop until WENO format with reduced order or lowest order precision is not needed; and then calculating the numerical flux of the position by adopting a WENO format with corresponding precision so as to further calculate and obtain a numerical simulation value of the position on the compressible flow problem containing the shock wave.
The main process of the present invention is illustrated by taking a one-dimensional euler equation as an example. The governing equation of the one-dimensional euler equation is:
U t +F(U) x ＝0；
U＝(ρ,ρu,E) T ，F＝(ρu,ρu 2 +p,u(E+p)) T ；
wherein rho, u, p and E are density, speed, pressure and total energy respectively; the state equation is:
wherein γ is a gas specific heat ratio. The semi-discrete conservation type difference form of the one-dimensional euler equation is:
wherein,
for numerical flux, it can be obtained from various reconstruction formats (e.g., the WENO format according to the present invention).
Recording the flux of the value
Is in the position x j+1/2 The template used in the WENO format is S ═ { x ═ x j-m+1 ,x j-m+2 ,…,x j+m The criteria in step S1 and step S2 are that the following two requirements are satisfied simultaneously:
(1) when the number of the template points is more than or equal to 4, the smooth factor of the pressure intensity needs to meet the following requirements:
β (2m) ≤4β (2m-2) ；
smoothing factor beta (2m) All 2m points on the template S are needed for calculation;
smoothing factor beta (2m-2) All 2m-2 points of the template S with two end points removed are used for calculation;
this requirement (1) is based on the smoothness of the pressure on the template, and can effectively suppress the occurrence of non-physical oscillations.
(2) The pressure p and the velocity u need to satisfy:
i∈(j-m+ 1,…,j+m-1),l∈(j-m+ 1,…,j+m-2)；
wherein gamma is the gas specific heat ratio; h and delta t are respectively the space grid scale and the time step length of the current direction; d is the spatial dimension of the problem sought. This requirement (2) is based on a control equation of pressure, and can effectively prevent the occurrence of negative pressure in the calculation process.
One group of WENO formats that may be used includes: ninth-order WENO9-S, seventh-order WENO7-S, and fifth-order WENO5-M formats. The main process of the present invention is as follows: in calculating the flux of each value
When the requirement is met, firstly, taking m to be 5, judging whether the two requirements are met simultaneously according to a criterion, and if the requirement is met, adopting a WENO9-S format; and if not, changing M to 4, judging whether the two requirements are met at the same time according to the criterion, if so, adopting WENO7-S, otherwise, adopting a WENO5-M format.
The smoothing factor beta (2m) Or smoothing factor beta (2m-2) The following two calculation formulas can be adopted:
the first method comprises the following steps:
wherein, beta (n) Represents the smoothing factor beta (2m) Or smoothing factor beta (2m-2) ；
x c Is the midpoint of the template, i.e. x j+1/2 ；
σ, δ is defined as follows:
and h is the space grid scale of the current direction.
And the second method comprises the following steps:
β (n) ＝(b n ) 2 +|a n ·c n |
wherein, beta (n) Represents the smoothing factor beta (2m) Or smoothing factor beta (2m-2) ；
x c Is the midpoint of the template, i.e. x j+1/2 。
For the second, further, a n ,b n ,c n Can be written as follows:
wherein k is the subscript of the leftmost end point of the template for the smoothing factor, and z is a n ,b n Or c n (ii) a Coefficient alpha n,l The values of (A) are as follows:
in fact, the value of n in the table may be n > 3, and n is an integer, and the corresponding parameter value may be deduced, and the table should not be used to limit the present invention.
The features and properties of the present invention are described in further detail below by specific examples. The time format in the calculation adopts a 3-order TVD Runge Kutta method, WENO format combinations used in the space format are WENO (2r-1) -S, WENO (2r-3) -S, … and WENO7-S, WENO5-M, and for convenience of description, the format of the invention constructed by reducing the order through the high-order WENO format is marked as PORWENO (2r-1) -S format with the highest order being (2 r-1). For comparison, the results of WENO5-M are also given. A local Lax-Friedrichs splitting method and a characteristic projection technology are adopted in the calculation, and in addition, the CFL number is uniformly 0.25.
The first calculation example: shock wave and entropy wave interaction problem
The control equation of the problem is a one-dimensional Euler equation, and a shock wave with a Mach number of 3 and an entropy wave are mutually interfered to induce high-frequency vibration. A comparison graph of calculation results of several formats when the grid point number is N200 and the calculation termination time T is 1.8 is shown in fig. 2, and since the problem has no analytic solution, the calculation result of the classic five-order WENO format when N8000 is used as a reference accurate solution, which is represented by a solid line "Exact" in fig. 2; w5 represents the calculation result of WENO5-M, and W (2r-1) represents the result of PORWENO (2r-1) -S, (2r-1) ≧ 9. As can be seen from FIG. 2, the PORWENO (2r-1) -S format constructed in the present invention achieves better results in the high frequency oscillation region.
Example two: double detonation wave problem
The control equation of the problem is a one-dimensional Euler equation, and the initial value condition is as follows:
this problem has a high requirement for format stability, and the direct use of the very high-order WENO format can cause calculation failures. Both of these formats yield robust and reliable results after the inventive reduced order method is employed. Fig. 2 shows a comparison of several calculation results in the format where the number of grid points N is 400 and the calculation termination time T is 0.038, and both ends adopt the reflection boundary condition. Since there is no analytical solution to this problem, the calculation result of the classic five-order WENO format with N8000 is used as the reference Exact solution, which is represented by the solid line "Exact" in fig. 3; w5 represents the calculation result of WENO5-M, and W (2r-1) represents the result of PORWENO (2r-1) -S, (2r-1) ≧ 9. Although the numerical results for the different formats are very close, their difference can be seen near x-0.78. From the enlarged view herein, it can be seen that PORWENO (2r-1) -S format of the present invention configuration achieves a higher peak value than WENO5-M format.
Example three: two-dimensional Riemann problem
The governing equation of the problem is a two-dimensional Euler equation with initial value conditions
Fig. 4a, 4b, and 4c show density cloud charts of calculation results of WENO5-M format, PORWENO9-S, and PORWENO17-S when the calculation area is [0,1] × [0,1], the grid point number is 400, and the calculation termination time T is 0.8, respectively. PORWENO9-S and PORWENO17-S achieve finer flow field configurations than the WENO5-M format.
As can be seen from the above, the present invention has the following beneficial effects:
1. the invention can adopt a very high-order WENO format in a continuous area, thereby obtaining a more precise flow field structure;
2. the WENO format with a relatively low order is started near the strong shock wave, so that the shock wave stability is good, the possibility of non-physical oscillation near the shock wave is low, and the calculation failure caused by the negative pressure is difficult to occur in the calculation;
3. the present invention can be used for the conservative difference format, and the amount of calculation is small because the format used is determined in advance.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.
Claims (5)
1. A method for numerically simulating a compressible flow problem by reducing order in a higher-order WENO format, comprising the steps of:
s1, when calculating the numerical flux of a certain position, judging whether order reduction is needed according to a criterion for the given WENO format with highest precision;
s2, if the order needs to be reduced, judging whether the order needs to be reduced or not according to the same criterion in the step S1 for the second-time high-precision WENO format after the order is reduced;
s3, executing step S2 in a loop until WENO format with reduced order or lowest order precision is not needed; then, calculating the numerical flux of the position by adopting a WENO format with corresponding precision so as to further calculate and obtain a numerical simulation value of the position to the compressible flow problem containing shock waves;
noting the position of the numerical flux as x j+1/2 The template used in the WENO format is S ═ { x ═ x j-m+1 ，x j-m+2 ，…，x j+m The criteria in step S1 and step S2 are that the following two requirements are satisfied simultaneously:
(1) when the number of the template points is more than or equal to 4, the smooth factor of the pressure intensity needs to meet the following requirements:
β (2m) ≤4β (2m-2) ；
smoothing factor beta (2m) All 2m points on the template S are needed for calculation;
smoothing factor beta (2m-2) All 2m-2 points of the template S with two end points removed are used for calculation;
(2) the pressure p and the velocity u need to satisfy:
wherein gamma is the gas specific heat ratio; h and delta t are respectively the space grid scale and the time step length of the current direction; d is the spatial dimension of the problem sought.
2. The method of claim 1, wherein the smoothing factor β is a measure of a compressibility flow problem that is numerically modeled by a higher order WENO format reduction (2m) Or smoothing factor beta (2m-2) The calculation formula of (a) is as follows:
wherein, beta (n) Represents the smoothing factor beta (2m) Or smoothing factor beta (2m-2) ；
x c Is the midpoint of the template, i.e. x j+1/2 ；
σ, δ is defined as follows:
and h is the space grid scale of the current direction.
3. The method of claim 1, wherein the smoothing factor β is a measure of a compressibility flow problem that is numerically modeled by a higher order WENO format reduction (2m) Or smoothing factor beta (2m-2) The calculation formula of (a) is as follows:
β (n) ＝(b n ) 2 +|a n ·c n |
wherein beta is (n) Represents the smoothing factor beta (2m) Or smoothing factor beta (2m-2) ；
x c Is the midpoint of the template, i.e. x j+1/2 。
4. The method of claim 3, wherein a is a numerical simulation of compressible flow problems via higher-order WENO format reduction n ，b n ，c n Can be written as follows:
wherein k is the subscript of the leftmost end point of the template for the smoothing factor, and z is a n ，b n Or c n (ii) a Coefficient alpha n，l The values of (A) are as follows:
5. the method of claim 4, wherein n > 3 and n is an integer, and wherein n is an integer.
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