CN107133418B - Earth fluid material advection transportation simulation method based on alternative TVD differential algorithm - Google Patents

Earth fluid material advection transportation simulation method based on alternative TVD differential algorithm Download PDF

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CN107133418B
CN107133418B CN201710383303.XA CN201710383303A CN107133418B CN 107133418 B CN107133418 B CN 107133418B CN 201710383303 A CN201710383303 A CN 201710383303A CN 107133418 B CN107133418 B CN 107133418B
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刘哲
林磊
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Abstract

The invention discloses an earth fluid substance advection transportation simulation method based on an alternative TVD differential algorithm, which comprises the steps of constructing an advection transportation equation of earth fluid substances, and further carrying out numerical value dispersion on the advection transportation equation to obtain a dispersion equation of earth fluid substance transportation; resolving a discrete equation for earth fluid matter transportation into steps equal to the dimension of the discrete equation; acquiring flow field data of the earth fluid substances, inputting the flow field data into a discrete equation for conveying the earth fluid substances to perform numerical iterative computation, and predicting the concentration of the earth fluid substances in each preset iterative time period; in the numerical iterative computation process, different types of total variation reduction limiting functions are alternately used on each dimension according to the number of steps equal to the dimension of the dimension in different preset iteration time periods to compute the concentration of the fluid substances in the next time period, and finally, the change of the concentration of the fluid substances in the advection transportation process of the fluid substances in the earth is simulated and predicted.

Description

Earth fluid material advection transportation simulation method based on alternative TVD differential algorithm
Technical Field
The invention belongs to the field of fluid dynamics numerical simulation and calculation, and particularly relates to an earth fluid material advection transport simulation method based on an alternative TVD differential algorithm.
Background
The numerical mode is an important tool for researching and forecasting movement of rivers, oceans and the like and transportation of substances in the movement in the nature, and with the development of computer technology and social economy, the numerical mode plays an increasingly important role in scientific research, business forecasting, environmental management and planning of lakes, reservoirs, oceans and the like.
Advection is the main process of carrying materials in the earth's fluids for spatial transport, and can be expressed as equation (1). In the numerical mode, the equation (1) is usually discretized by a difference method to obtain an advection difference equation in the form of the equation (2), and then the equation (2) is subjected to numerical iteration to obtain the spatial distribution of the substance concentration c and the change with time.
Figure BDA0001305687130000011
Wherein c is the concentration of a certain transported substance, and u, v and w are flow rates in x, y and z directions respectively.
Figure BDA0001305687130000012
Wherein the content of the first and second substances,
Figure BDA0001305687130000013
the concentration of the substance at the half-layer grid is shown, delta t is a discrete time step, delta x, delta y and delta z are grid intervals in the x direction, the y direction and the z direction respectively, the index superscript n represents the number of time steps, and the subscripts i, j and k represent grid indexes in the x direction, the y direction and the z direction respectively.
In the difference equation (2), how to calculate the concentration of the substance at the half-layer grid
Figure BDA0001305687130000014
Is the main difference between different differential formats. For example, the windward format uses the upstream grid concentration, and the mid-range format uses the average concentration of the upstream and downstream grids. Although the algorithms are simple and convenient to calculate, the algorithms have the problems of low precision or strong numerical dispersion, so that the simulation result is high in distortion and the stability of the model is influenced.
Total Variation subtraction (TVD) differential formats are a class of high-precision frequency-dispersion-free advection differential algorithms that are widely used in ocean and atmospheric numerical models and related research. The TVD algorithm is mainly characterized in that a high-precision item of back diffusion is added on the basis of an original low-precision windward format, the precision of a advection format is improved, and meanwhile, a limiting function is introduced to control the strength of the added high-precision item, so that the total variation of a numerical value format is kept not increased in the whole forward integration process, and the numerical value frequency dispersion of the high-precision format is effectively restrained.
The algorithm for the TVD advection format is as follows (taking the x direction as an example):
as shown in equation (3), in calculating
Figure BDA0001305687130000021
When in use, a high-precision correction term is added on the basis of the windward format,and adding a limiting function psi (r) before the correction termi+1/2) Thereby obtaining:
Figure BDA0001305687130000022
wherein the content of the first and second substances,
Figure BDA0001305687130000023
ensuring total variation of calculation result after adding limiting function
Figure BDA0001305687130000024
Remain unchanged, i.e.:
TV(cn+1)≤TV(cn) (4)
the constraint function is made to satisfy the following condition:
Figure BDA0001305687130000025
wherein r represents a concentration gradient ratio, i.e.
Figure BDA0001305687130000026
Or
Figure BDA0001305687130000027
Although there are multiple TVD limiting functions, a single limiting function cannot control the strength of the high-precision terms added to the TVD format, which results in the pseudo-effects of value diffusion (e.g., Minmod, Van Leer, MUSCL, HSIMT) and back diffusion (e.g., Superbee) in the above TVD value format. In addition, the TVD format can ensure that the total variation in one-dimensional case is not increased, while in multi-dimensional case it is less desirable, and there is still a numerical dispersion. Therefore, there are unsatisfactory simulation results when the existing TVD advection differential format is used to simulate the material transport in the earth fluid for a long time.
Disclosure of Invention
In order to solve the defects of the prior art, the invention provides a simulation method for mass advection transportation of earth fluid based on an alternative TVD differential algorithm, and provides a differential method-TVDal (TVD with Alternating transmitters) which effectively weakens the pseudo-effect of numerical diffusion and anti-diffusion and solves the problem of a multidimensional frequency dispersion advection format, and the method is applied to simulation and prediction of mass transportation in earth fluid.
The invention discloses an earth fluid material advection transportation simulation method based on an alternative TVD differential algorithm, which comprises the following steps:
constructing an advection transport equation of the earth fluid substance, and further performing numerical value dispersion on the advection transport equation to obtain a dispersion equation of the earth fluid substance transport; the dimensionality of the discrete equation of earth fluid matter transport is at least two-dimensional;
resolving a discrete equation for earth fluid matter transportation into steps equal to the dimension of the discrete equation;
acquiring flow field data of the earth fluid substances, inputting the flow field data into a discrete equation for conveying the earth fluid substances to perform numerical iterative computation, and predicting the concentration of the earth fluid substances in each preset iterative time period;
in the numerical iterative computation process, different types of total variation reduction limiting functions are alternately used on each dimension according to the number of steps equal to the dimension of the dimension in different preset iteration time periods to compute the concentration of the fluid substances in the next time period, and finally, the change of the concentration of the fluid substances in the advection transportation process of the fluid substances in the earth is simulated and predicted.
Further, based on the construction method of the variation reduction format, the advection transport equation of the earth fluid substances is subjected to numerical value dispersion.
Further, the total variation mitigation clipping limit function includes a limit function of numerical diffusion properties and a limit function of numerical anti-diffusion properties.
Further, the earth fluid species concentration is a function of time and space.
Further, the process of solving the advection transport equation of the earth fluid substance is as follows:
and finally accumulating the derivative functions of the concentration of the earth fluid substances to the time to obtain the advection transport equation of the earth fluid substances after the derivative functions of the concentration of the earth fluid substances to the direction parameters of the characterization space are multiplied by the flow velocity of the direction parameters of the corresponding characterization space.
Wherein the limiting function of the numerical diffusion property is a Minmod function.
The limiting function of the numerical diffusion property is a Van Leer function.
The limiting function of the numerical diffusion property is the muslc function.
The limiting function of the numerical diffusion property is the HSIMT function.
The limiting function of the numerical back-diffusion property comprises a Superbee function.
In practical calculation, there are several commonly used classic TVD limiting functions, mainly:
Superbee：
ψ(r)Superbee＝max[0,min(2r,1),min(r,2)](6)
Van Leer(or Harmonic)：
ψ(r)VanLeer＝(r+|r|)/(r+1) (7)
MUSCL：
Figure BDA0001305687130000031
Minmod：
ψ(r)Minmod＝max[0,min(r,1)](9)
HSIMT：
ψ(r)HSIMT＝max[0,min(2r,2,β)](10)
wherein the content of the first and second substances,
Figure BDA0001305687130000041
compared with the prior art, the invention has the beneficial effects that:
(1) the method does not change the original formula and increase extra calculation amount, but alternately uses algorithms with different properties at different time steps, so that the numerical diffusion and anti-diffusion pseudo effects of the traditional TVD advection difference algorithm are greatly weakened, and the error of a simulation result is smaller.
(2) According to the method, the alternate limiting function and the discrete equation are split and combined, and the numerical dispersion problem of material transport simulation under the condition of multidimensional calculation of the TVD advection format is effectively solved based on the obtained multidimensional frequency dispersion-free advection algorithm.
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The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the application and, together with the description, serve to explain the application and are not intended to limit the application.
FIG. 1 is a schematic flow diagram of the present invention;
FIG. 2(a) is a substance concentration distribution diagram at an initial time;
fig. 2(b) is a Superbee simulation result based on the conventional TVD algorithm;
fig. 2(c) is HSIMT simulation results based on the conventional TVD algorithm;
FIG. 2(d) is a Superbee simulation result of a conventional TVD algorithm combined with a discrete equation splitting algorithm;
FIG. 2(e) is the HSIMT simulation result of the conventional TVD algorithm combined with the discrete equation splitting algorithm;
FIG. 2(f) is a simulation of the method of the present invention.
Detailed Description
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
The dimension of the discrete equation for earth fluid matter transport is at least two dimensions, which is exemplified below by the dimension of the discrete equation for earth fluid matter transport being two dimensions:
as shown in fig. 1, the earth fluid material advection transportation simulation method based on the alternating TVD differential algorithm of the present invention includes:
step 1: and constructing an advection transport equation of the earth fluid substance, and further performing numerical value dispersion on the advection transport equation to obtain a dispersion equation of the earth fluid substance transport.
The advection transport equation of the earth fluid substance is constructed as formula (1):
Figure BDA0001305687130000051
carrying out numerical value dispersion on the advection transport equation to obtain a dispersion equation of the earth fluid substance transport, wherein the dispersion equation is a formula (2):
Figure BDA0001305687130000052
wherein the content of the first and second substances,
Figure BDA0001305687130000053
the concentration of the substance at the half-layer grid is shown, delta t is discrete time step, delta x and delta y are grid intervals in the x direction and the y direction respectively, the index superscript n represents the number of the time step, and the subscripts i and j represent grid indexes in the x direction and the y direction respectively.
Step 2: the discrete equation of earth fluid matter transport is decomposed into steps equal to its dimensions.
And step 3: acquiring flow field data of the earth fluid substances, inputting the flow field data into a discrete equation for conveying the earth fluid substances to perform numerical iterative computation, and predicting the concentration of the earth fluid substances in each preset iterative time period;
in the numerical iterative computation process, different types of total variation reduction limiting functions are alternately used on each dimension according to the number of steps equal to the dimension of the dimension in different preset iteration time periods to compute the concentration of the fluid substances in the next time period, and finally, the change of the concentration of the fluid substances in the advection transportation process of the fluid substances in the earth is simulated and predicted.
Specifically, information of the flow velocity u and the flow velocity v is input, a discrete equation is iterated, parity of iteration steps is judged, if the number of the iteration steps is odd, the following step a) is adopted for calculation, and if the number of the iteration steps is even, the step b) is adopted for calculation.
Step a):
the discrete equation (2) is decomposed into two steps of calculation of an equation (3-1) and an equation (3-2):
Figure BDA0001305687130000054
Figure BDA0001305687130000055
wherein the content of the first and second substances,
Figure BDA0001305687130000056
the superscripts com and dif indicate that the TVD limiting function of the numerical back-diffusion property (e.g., Superbee) and numerical diffusion property (e.g., Minmod) is used to calculate the concentration of the species at the half-layer lattice
Figure BDA0001305687130000057
The discrete process is divided into (3-1) and (3-2), and advection processes in the x direction and the y direction are calculated in sequence: firstly, when calculating the advection process in the x direction, firstly, using a TVD restriction function with numerical back diffusion property to calculate and obtain an intermediate variable c; substituting the intermediate variable c into the differential equation in the y direction to calculate the advection process in the y direction, and calculating the concentration c of a new time step by using a TVD limiting function with numerical diffusion propertyn+1。
Step b):
calculating in two steps (4-1) and (4-2) in the same way, but firstly calculating the advection process in the y direction, and calculating to obtain an intermediate variable c by using a TVD limiting function with numerical back diffusion property; and substituting the intermediate variable c into the differential equation in the x direction to calculate the advection process of the x equation, and calculating to obtain the material concentration of a new time step by using a TVD limiting function of numerical diffusion property.
Figure BDA0001305687130000061
Figure BDA0001305687130000062
Wherein the content of the first and second substances,
Figure BDA0001305687130000063
the superscripts com and dif indicate that the TVD limiting function of the numerical back-diffusion property (e.g., Superbee) and numerical diffusion property (e.g., Minmod) is used to calculate the concentration of the species at the half-layer lattice
Figure BDA0001305687130000064
And finally, judging whether the integration duration reaches a set value of the mode, if not, returning to the step 2 for continuation, and if so, terminating the calculation.
The present invention can be further explained by the following simulation results.
1. Simulation experiment: the method and other methods are applied to simulate the material advection process of a given certain concentration distribution.
2. Simulation results
In a horizontal two-dimensional sea area, there is a reciprocating flow in the 45 ° direction with a period of 12 hours, the distribution of the concentration of the substance at the initial moment is given as the distribution shown in fig. 2(a), and the simulation results after 100 periods obtained by using different advection difference algorithms are respectively shown in fig. 2(b) -fig. 2 (f). Wherein, fig. 2(b) and 2(c) are simulation results based on Superbee and HSIMT of the conventional TVD algorithm, and fig. 2(d) and 2(e) are simulation results of Superbee and HSIMT of the conventional TVD algorithm combined with the discrete equation splitting algorithm; FIG. 2(f) shows the simulation results of the method of the present invention.
As can be seen from fig. 2(a) -2 (f), the simulation result based on the conventional TVD algorithm has significant numerical dispersion, the numerical dispersion is improved after the algorithm is split by using a discrete equation, but the numerical anti-diffusion and diffusion pseudo-effect still exists.
In conclusion, the method provided by the invention effectively weakens the numerical diffusion and anti-diffusion pseudo effects and solves the problem of the material advection simulation method of the TVDal which is a difference algorithm of a multidimensional frequency dispersion advection format. The method effectively weakens the numerical diffusion and the back diffusion by alternately using the traditional TVD limiting functions with the opposite pseudo effects, reduces the error of the advection simulation result, and simultaneously, splits and calculates the multidimensional discrete equation to ensure that the multidimensional advection simulation is the generation of infinite dispersion, and the improvement can be seen through the simulation result of the material advection experiment.
Although the embodiments of the present invention have been described with reference to the accompanying drawings, it is not intended to limit the scope of the present invention, and it should be understood by those skilled in the art that various modifications and variations can be made without inventive efforts by those skilled in the art based on the technical solution of the present invention.

Claims (10)

1. An earth fluid material advection transportation simulation method based on an alternative TVD differential algorithm is characterized by comprising the following steps:
constructing an advection transport equation of the earth fluid substance, and further performing numerical value dispersion on the advection transport equation to obtain a dispersion equation of the earth fluid substance transport;
the advection transport equation of the earth fluid substance is constructed as formula (1):
Figure DEST_PATH_IMAGE002
（1）
wherein the content of the first and second substances,cin order to achieve a certain concentration of the transported substance,u、vare respectively asx、yThe flow velocity in the direction of the flow,
carrying out numerical value dispersion on the advection transport equation to obtain a dispersion equation of the earth fluid substance transport, wherein the dispersion equation is a formula (2):
Figure DEST_PATH_IMAGE004
（2）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE006
，
Figure DEST_PATH_IMAGE008
，
Figure DEST_PATH_IMAGE010
is the concentration of the substance at the half-layer grid,
Figure DEST_PATH_IMAGE012
in the form of discrete time steps of the time,
Figure DEST_PATH_IMAGE014
、
Figure DEST_PATH_IMAGE016
are respectively asx、yGrid spacing of directions, numbering superscriptnIndicating the number of time steps, subscriptsi、jRespectively representx、yGrid number of directions;
resolving a discrete equation for earth fluid matter transportation into steps equal to the dimension of the discrete equation;
acquiring flow field data of the earth fluid substances, inputting the flow field data into a discrete equation for conveying the earth fluid substances to perform numerical iterative computation, and predicting the concentration of the earth fluid substances in each preset iterative time period;
in the numerical iterative computation process, different types of total variation reduction limiting functions are alternately used on each dimension according to the number of steps equal to the dimension of each dimension in different preset iteration time periods to compute the concentration of the fluid substances in the next time period, and finally, the variation of the concentration of the fluid substances in the advection transportation process of the fluid substances in the earth is simulated and predicted.
2. The alternating TVD differential algorithm based earth fluid matter advection transport simulation method of claim 1, wherein the earth fluid matter advection transport equation is numerically discretized based on a construction method of a variation subtraction format.
3. The alternating TVD differential algorithm based earth fluid material advection transport simulation method of claim 1, wherein the total variation mitigation restriction function comprises a restriction function of numerical diffusion properties and a restriction function of numerical anti-diffusion properties.
4. The alternating TVD differential algorithm based earth fluid material advection transport simulation method of claim 1, wherein earth fluid material concentration is a function with respect to time and space.
5. The earth fluid material advection transport simulation method based on the alternating TVD differential algorithm of claim 4, wherein the process of solving the advection transport equation of the earth fluid material is:
and finally accumulating the derivative functions of the concentration of the earth fluid substances to the time to obtain the advection transport equation of the earth fluid substances after the derivative functions of the concentration of the earth fluid substances to the direction parameters of the characterization space are multiplied by the flow velocity of the direction parameters of the corresponding characterization space.
6. The alternating TVD differential algorithm based earth fluid material advection transport simulation method of claim 3, wherein the limiting function of numerical diffusion properties is a Minmod function.
7. The alternating TVD differential algorithm based earth fluid material advection transport simulation method according to claim 3, wherein the limiting function of the numerical diffusion property is a Van Leer function.
8. The alternating TVD differential algorithm based earth fluid material advection transport simulation method according to claim 3, wherein the limiting function of numerical diffusion property is a MUSCL function.
9. The alternating TVD differential algorithm based earth fluid material advection transport simulation method of claim 3, wherein said limiting function of numerical diffusion properties is HSIMT function.
10. The alternating TVD differential algorithm based earth fluid material advection transport simulation method according to claim 3, wherein the limiting function of the numerical back-diffusion property comprises a Superbee function.
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