CN111159958A - Method for acquiring guaranteed physical characteristics of states on two sides of multi-medium coupling problem interface - Google Patents

Method for acquiring guaranteed physical characteristics of states on two sides of multi-medium coupling problem interface Download PDF

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CN111159958A
CN111159958A CN201911256999.5A CN201911256999A CN111159958A CN 111159958 A CN111159958 A CN 111159958A CN 201911256999 A CN201911256999 A CN 201911256999A CN 111159958 A CN111159958 A CN 111159958A
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CN111159958B (en
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刘铁钢
于长胜
冯成亮
曾志强
张孝涛
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Beihang University
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Abstract

The invention discloses a method for acquiring the physical property of states on two sides of an interface, which is suitable for the problem of symmetrical multi-medium coupling, provides an interface speed and pressure intensity matching equation, and an equation-based method for acquiring the state values and state derivative values on two sides of a vibration-free interface. The method can keep the characteristics of speed and pressure balance at the interface, accords with the physical characteristics of a real multi-medium coupling problem, and has important practical application value.

Description

Method for acquiring guaranteed physical characteristics of states on two sides of multi-medium coupling problem interface
Technical Field
The invention belongs to the technical field of computational fluid mechanics, and particularly relates to a method for acquiring guaranteed physical characteristics of states on two sides of an interface, which is suitable for a symmetric multi-medium coupling problem.
Background
In numerical simulation of multi-medium coupling problems such as underwater explosion simulation and the like, how to keep the real physical characteristics of speed and pressure balance at an interface is a very important technical difficulty, which is expressed in that if the relation between the pressure and the speed balance at the interface is not satisfied in the design process of a numerical format, a non-physical phenomenon of speed or pressure dislocation occurs in a numerical simulation result at the interface.
In the multimedia coupling problem, simulating the motion process at the interface by solving the multimedia Riemann problem at the interface is an important method, such as a modified virtual medium method (MGFM). Therefore, obtaining the flow field states on the two sides of the multi-medium interface through numerical solution is indispensable in the design of a numerical method and is also a research difficulty in the multi-medium coupling problem. Generally, the current assignment method for flow field state values on two sides of a multi-medium interface mainly utilizes a difference value approximation method for the flow field state values at grid points or directly assigns values by using grid point values near the interface. The methods have good effect in processing hyperbolic conservation law equations without source terms. However, when the method is applied to the problem of high-dimensional water gas explosion, due to the influence of a source item in a control equation, the continuous physical characteristics of the speed and the pressure at the interface cannot be maintained in calculation, so that the speed and the pressure at the interface cannot be matched in a numerical simulation result, and errors of the speed and the pressure at the interface are accumulated continuously in long-time numerical simulation.
In view of the above problems, the present invention aims to provide an equation describing states of two sides of an interface, and based on the equation, a difference value is approximated by selecting an appropriate physical quantity to obtain a flow field state value and a state derivative value of the two sides of the interface. The technology can keep the characteristics of speed and pressure balance at the interface, accords with the physical characteristics of a real multi-medium coupling problem, and has important practical application value.
Disclosure of Invention
In order to solve the defects of the prior art, the invention provides a method for acquiring the physical preservation characteristics of the states on two sides of the interface, which is suitable for the symmetric multi-medium coupling problem, and mainly embodies the proposal of an interface speed and pressure matching equation and a method for acquiring the state values and state derivative values on two sides of the vibration-free interface based on the equation. The specific technical scheme of the invention is as follows:
a method for acquiring the physical preservation characteristics of the states at two sides of a multi-medium coupling problem interface is disclosed, wherein the state values of an m-dimensional symmetric multi-medium coupling problem at the nth time step are known as follows:
Figure BDA0002310549050000021
wherein m is the dimension of the problem to be solved, and m is 1,2, 3;
Figure BDA0002310549050000022
the method is characterized in that the method is a column vector formed by density, pressure and speed on an ith grid point at an nth time step, N is the total number of grids, an interface is positioned between a grid j and a grid j +1, linear distribution of the left side and the right side of the interface of the nth time step, namely state values at the two sides of the interface and space derivative values of the state are obtained, and the method is characterized by comprising the following specific steps of:
step S1: establishing state values and state derivative values at two sides of the interface to satisfy a pressure balance equation and a speed balance equation:
Figure BDA0002310549050000023
where ρ is1，P1，u1，c1Respectively the density, pressure, velocity and sound velocity distribution, rho, of the medium flow field on the left side of the interface2，P2，u2，c2Respectively the density, pressure, speed and sound velocity distribution of the medium flow field on the right side of the interface, wherein r is a space coordinate, and r iscdIs the spatial coordinate of the interface, rcd m-1u1Is the velocity integral, r, of the media flow field on the left side of the interfacecd m-1u2Is the integral quantity of the velocity of the media flow field on the right side of the interface,
Figure BDA0002310549050000024
representing the rate of change of velocity at the right side of the interface with the interface,
Figure BDA0002310549050000025
representing the rate of change of the velocity at the left side of the interface with the interface,
Figure BDA0002310549050000026
indicating the rate of change of pressure on the right side of the interface with the interface,
Figure BDA0002310549050000027
indicating the rate of change of pressure at the left side of the interface with the interface,
Figure BDA0002310549050000028
the spatial derivative of the pressure on the right side is indicated,
Figure BDA0002310549050000029
the spatial derivative of the left-hand pressure is represented,
Figure BDA00023105490500000210
the spatial derivative of the right-hand velocity integral is represented,
Figure BDA00023105490500000211
a spatial derivative representing a left-hand velocity integral;
step S2: discretizing the equation of the step S1 to obtain the speed u at the interfacemidAnd pressure PmidThe expression is:
Figure BDA0002310549050000031
wherein u ismid，PmidIs the velocity and pressure at the interface, ρj，Pj，uj，cjDivided into density, velocity, pressure and speed of sound, p, at grid point jj+1，Pj+1，uj+1，cj+1Density, velocity, respectively, at grid point j +1,Pressure and speed of sound, rjSpatial coordinates of grid point j, rj+1The spatial coordinates for grid point j + 1;
step S3: obtaining the space derivative value P of the left pressure intensity of the interfaceL' sum velocity space derivative value uL′：
Figure BDA0002310549050000032
Figure BDA0002310549050000033
Figure BDA0002310549050000034
Wherein the content of the first and second substances,
Figure BDA0002310549050000035
sign (x) is a sign function, min (x, y) is a minimum function, PL' is the value of the spatial derivative of the pressure on the left side of the interface, uL'is the value of the space derivative of the velocity at the left side of the interface, P'j，(rm-1u)′j，uj' is P, r at grid jm-1The values of the spatial derivatives of u,
Figure BDA0002310549050000036
the pressure spatial derivative approximations to the left and right of grid point j respectively,
Figure BDA0002310549050000037
approximate values of the spatial derivatives of the velocity integral quantity on the left side and the right side of the grid point j are respectively obtained;
step S4: obtaining the density value rho on the left side of the interface by using the isentropic relationLAnd the value of the density spatial derivative ρL：
Figure BDA0002310549050000041
Figure BDA0002310549050000042
Wherein, PB,1，γ1Is the equation of state parameter of the left-hand medium,
Figure BDA0002310549050000043
step S5: obtaining the space derivative value P of the right pressure intensity of the interfaceR' sum velocity space derivative value uR′：
Figure BDA0002310549050000044
Figure BDA0002310549050000045
Figure BDA0002310549050000046
Wherein, PR' is the value of the spatial derivative of the pressure on the right side of the interface, uR'is the space derivative value of the velocity on the right side of the interface, P'j+1，(rm-1u)′j+1，uj+1' is P, r at grid j +1m-1The spatial derivative values of u, u;
step S6: obtaining the density value rho on the right side of the interface by using the isentropic relationRAnd the value of the density spatial derivative ρR′：
Figure BDA0002310549050000047
Wherein, PB,2，γ2Are the parameters of the equation of state of the medium on the right,
Figure BDA0002310549050000051
step S7: the linear distribution of the left and right states of the interface is obtained as follows:
Figure BDA0002310549050000052
the invention has the beneficial effects that:
1. providing an equation describing the physical characteristics of the constant balance of the speed and the pressure at the interface, wherein the equation comprises a speed balance equation and a pressure balance equation and provides a theoretical basis for obtaining the real derivative information of the states at two sides of the interface;
2. the method is compatible with a speed balance equation and a pressure balance equation at the interface, and further obtains state derivative information at two sides of the interface by using the state values at two sides of the interface obtained by the method and utilizing a minimized error technology;
3. the method can be applied to processing the multi-medium coupling problem, has the advantages of keeping the speed and pressure balance at the multi-medium interface, reducing numerical value oscillation by selecting proper physical quantity to carry out differential solution and minimizing variation, and provides support for long-time numerical simulation.
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In order to illustrate embodiments of the present invention or technical solutions in the prior art more clearly, the drawings which are needed in the embodiments will be briefly described below, so that the features and advantages of the present invention can be understood more clearly by referring to the drawings, which are schematic and should not be construed as limiting the present invention in any way, and for a person skilled in the art, other drawings can be obtained on the basis of these drawings without any inventive effort. Wherein:
FIG. 1 is a flow chart of a state value and state derivative value acquisition technique for both sides of an interface according to the present invention;
FIG. 2 is a schematic view of the distribution of the state of the region near the interface at the nth time;
FIG. 3 is an example of a one-dimensional spherical symmetric cavitation explosion using the method of the present invention;
fig. 4(a) is a calculated t-0.01807072 second velocity profile;
fig. 4(b) is a calculated t-0.01807072 second pressure profile.
Detailed Description
In order that the above objects, features and advantages of the present invention can be more clearly understood, a more particular description of the invention will be rendered by reference to the appended drawings. It should be noted that the embodiments of the present invention and features of the embodiments may be combined with each other without conflict.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, however, the present invention may be practiced in other ways than those specifically described herein, and therefore the scope of the present invention is not limited by the specific embodiments disclosed below.
FIG. 1 is a flow chart of a state value and state derivative value acquisition technique for both sides of an interface according to the present invention; fig. 2 is a schematic view of the state distribution of the region near the interface at the nth time.
For the m (m ═ 1,2,3) dimensional symmetry problem, the governing equation of a non-viscous compressible rigid gas in euler coordinates is:
Figure BDA0002310549050000061
Figure BDA0002310549050000062
where U is a constant, F is a flux, Ψ is a symmetric source term, ρ is a density, U is a velocity, P is a pressure, E is a total energy, E is an internal energy, m is a dimension of the problem, and E ═ E (ρ, P) ═ E- γ P (γ -1) ρ E- γ PBIs a rigid gas state equation, gamma is a specific heat ratio, PBIs a pressure parameter.
A method for acquiring the physical preservation characteristics of the states at two sides of a multi-medium coupling problem interface is disclosed, wherein the state values of an m-dimensional symmetric multi-medium coupling problem at the nth time step are known as follows:
Figure BDA0002310549050000063
wherein m is the dimension of the problem to be solved, and m is 1,2, 3;
Figure BDA0002310549050000064
the column vector of density, pressure and velocity at the ith grid point at the nth time step, N is the total number of grids. The interface is located between grid j and grid j +1, and as shown in fig. 2, linear distributions of the left and right sides of the nth time step interface, that is, state values at both sides of the interface and spatial derivative values of the state, are obtained, which is characterized by comprising the following specific steps:
a method for acquiring the physical preservation characteristics of the states at two sides of a multi-medium coupling problem interface is disclosed, wherein the state values of an m-dimensional symmetric multi-medium coupling problem at the nth time step are known as follows:
Figure BDA0002310549050000071
wherein m is the dimension of the problem to be solved, and m is 1,2, 3;
Figure BDA0002310549050000072
the method is characterized in that the method is a column vector formed by density, pressure and speed on an ith grid point at an nth time step, N is the total number of grids, an interface is positioned between a grid j and a grid j +1, linear distribution of the left side and the right side of the interface of the nth time step, namely state values at the two sides of the interface and space derivative values of the state are obtained, and the method is characterized by comprising the following specific steps of:
step S1: establishing state values and state derivative values at two sides of the interface to satisfy a pressure balance equation and a speed balance equation:
Figure BDA0002310549050000073
where ρ is1，P1，u1，c1Respectively the density, pressure, velocity and sound velocity distribution, rho, of the medium flow field on the left side of the interface2，P2，u2，c2Respectively the density, pressure, speed and sound velocity distribution of the medium flow field on the right side of the interface, wherein r is a space coordinate, and r iscdIs the spatial coordinate of the interface, rcd m-1u1Is the velocity integral, r, of the media flow field on the left side of the interfacecd m-1u2Is a boundaryThe integral of the velocity of the right-hand media flow field,
Figure BDA0002310549050000074
representing the rate of change of velocity at the right side of the interface with the interface,
Figure BDA0002310549050000075
representing the rate of change of the velocity at the left side of the interface with the interface,
Figure BDA0002310549050000076
indicating the rate of change of pressure on the right side of the interface with the interface,
Figure BDA0002310549050000077
indicating the rate of change of pressure at the left side of the interface with the interface,
Figure BDA0002310549050000078
the spatial derivative of the pressure on the right side is indicated,
Figure BDA0002310549050000079
the spatial derivative of the left-hand pressure is represented,
Figure BDA00023105490500000710
the spatial derivative of the right-hand velocity integral is represented,
Figure BDA00023105490500000711
a spatial derivative representing a left-hand velocity integral;
step S2: discretizing the equation of the step S1 to obtain the speed u at the interfacemidAnd pressure PmidThe expression is:
Figure BDA0002310549050000081
wherein u ismid，PmidIs the velocity and pressure at the interface, ρj，Pj，uj，cjDivided into density, velocity, pressure and speed of sound, p, at grid point jj+1，Pj+1，uj+1，cj+1Density, velocity, pressure and speed of sound, r, at grid point j +1, respectivelyjSpatial coordinates of grid point j, rj+1The spatial coordinates for grid point j + 1;
step S3: obtaining the space derivative value P of the left pressure intensity of the interfaceL' sum velocity space derivative value uL′：
Figure BDA0002310549050000082
Figure BDA0002310549050000083
Figure BDA0002310549050000084
Wherein the content of the first and second substances,
Figure BDA0002310549050000085
sign (x) is a sign function, min (x, y) is a minimum function, PL' is the value of the spatial derivative of the pressure on the left side of the interface, uL'is the value of the space derivative of the velocity at the left side of the interface, P'j，(rm-1u)′j，uj' is P, r at grid jm-1The values of the spatial derivatives of u,
Figure BDA0002310549050000086
the pressure spatial derivative approximations to the left and right of grid point j respectively,
Figure BDA0002310549050000087
approximate values of the spatial derivatives of the velocity integral quantity on the left side and the right side of the grid point j are respectively obtained;
step S4: obtaining the density value rho on the left side of the interface by using the isentropic relationLAnd the value of the density spatial derivative ρL：
Figure BDA0002310549050000091
Wherein, PB,1，γ1Is the equation of state parameter of the left-hand medium,
Figure BDA0002310549050000092
step S5: obtaining the space derivative value P of the right pressure intensity of the interfaceR' sum velocity space derivative value uR′：
Figure BDA0002310549050000093
Figure BDA0002310549050000094
Figure BDA0002310549050000095
Wherein, PR' is the value of the spatial derivative of the pressure on the right side of the interface, uR'is the space derivative value of the velocity on the right side of the interface, P'j+1，(rm-1u)′j+1，uj+1' is P, r at grid j +1m-1The spatial derivative values of u, u;
step S6: obtaining the density value rho on the right side of the interface by using the isentropic relationRAnd the value of the density spatial derivative ρR′：
Figure BDA0002310549050000096
Wherein, PB,2，γ2Are the parameters of the equation of state of the medium on the right,
Figure BDA0002310549050000101
step S7: the linear distribution of the left and right states of the interface is obtained as follows:
Figure BDA0002310549050000102
for the convenience of understanding the above technical aspects of the present invention, the following detailed description will be given of the above technical aspects of the present invention by way of specific examples.
Example 1
Taking the one-dimensional spherical symmetrically compressible water cavity explosion process as an example, as shown in fig. 3, wherein the inner sphere is ultrahigh pressure gas, and the outer sphere is normal pressure compressible liquid water;
the total calculation area is a one-dimensional spherical symmetrical area with the length of 12 meters, x is set as a space coordinate, namely x belongs to (0,12), the grid is 9999 points which are uniformly distributed, and the radius r of the initial bubble is0＝0.401m；
Initial value of gas velocity ugas,00.0, initial pressure value pgas,08290.91, density is initialized to ρgas,01.27, the equation of state is the complete gas equation of state:
Figure BDA0002310549050000103
where ρ is density, p is pressure, e is internal energy, γgas1.4 is the ideal gas specific heat ratio;
the initial value of the water velocity of the compressible liquid is ugas,00.0, initial pressure value pgas,01.0, initial density value is ρgas,01.0, the state equation is the Tait state equation: p ═ Nwater-1)ρe-Nwater*BwaterWherein the state constant Nwater＝7.0，Bwater＝3000.0。
To illustrate the effect of the interface state acquisition technique of the present invention, the results are shown in fig. 4 for a profile of pressure and velocity at 0.01803772 seconds, where the dotted rectangle line represents the calculation results of the MGFM method to which the present technique is not applied, and the dotted triangle line represents the calculation results of the MGFM/ASC method to which the present technique is applied.
In the present invention, unless otherwise expressly stated or limited, the terms "mounted," "connected," "secured," and the like are to be construed broadly and can, for example, be fixedly connected, detachably connected, or integrally formed; can be mechanically or electrically connected; either directly or indirectly through intervening media, either internally or in any other relationship. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
In the present invention, unless otherwise expressly stated or limited, "above" or "below" a first feature means that the first and second features are in direct contact, or that the first and second features are not in direct contact but are in contact with each other via another feature therebetween. Also, the first feature being "on," "above" and "over" the second feature includes the first feature being directly on and obliquely above the second feature, or merely indicating that the first feature is at a higher level than the second feature. A first feature being "under", beneath and "under" a second feature includes the first feature being directly under and obliquely under the second feature, or simply means that the first feature is at a lesser elevation than the second feature.
In the present invention, the terms "first", "second", "third", and "fourth" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance. The term "plurality" means two or more unless expressly limited otherwise.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (1)

1. A method for acquiring the physical preservation characteristics of the states at two sides of a multi-medium coupling problem interface is disclosed, wherein the state values of an m-dimensional symmetric multi-medium coupling problem at the nth time step are known as follows:
Figure FDA0002310549040000011
wherein m is the dimension of the problem to be solved, and m is 1,2, 3;
Figure FDA0002310549040000012
the method is characterized in that the method is a column vector formed by density, pressure and speed on an ith grid point at an nth time step, N is the total number of grids, an interface is positioned between a grid j and a grid j +1, linear distribution of the left side and the right side of the interface of the nth time step, namely state values at the two sides of the interface and space derivative values of the state are obtained, and the method is characterized by comprising the following specific steps of:
step S1: establishing state values and state derivative values at two sides of the interface to satisfy a pressure balance equation and a speed balance equation:
Figure FDA0002310549040000013
where ρ is1，P1，u1，c1Respectively the density, pressure, velocity and sound velocity distribution, rho, of the medium flow field on the left side of the interface2，P2，u2，c2Respectively the density, pressure, speed and sound velocity distribution of the medium flow field on the right side of the interface, wherein r is a space coordinate, and r iscdIs the spatial coordinate of the interface, rcd m-1u1Is the velocity integral, r, of the media flow field on the left side of the interfacecd m-1u2Is the integral quantity of the velocity of the media flow field on the right side of the interface,
Figure FDA0002310549040000014
representing the rate of change of velocity at the right side of the interface with the interface,
Figure FDA0002310549040000015
representing the rate of change of the velocity at the left side of the interface with the interface,
Figure FDA0002310549040000016
indicating the rate of change of pressure on the right side of the interface with the interface,
Figure FDA0002310549040000017
indicating the rate of change of pressure at the left side of the interface with the interface,
Figure FDA0002310549040000018
the spatial derivative of the pressure on the right side is indicated,
Figure FDA0002310549040000019
the spatial derivative of the left-hand pressure is represented,
Figure FDA00023105490400000110
the spatial derivative of the right-hand velocity integral is represented,
Figure FDA00023105490400000111
a spatial derivative representing a left-hand velocity integral;
step S2: discretizing the equation of the step S1 to obtain the speed u at the interfacemidAnd pressure PmidThe expression is:
Figure FDA00023105490400000112
wherein u ismid，PmidIs the velocity and pressure at the interface, ρj，Pj，uj，cjDivided into density, velocity, pressure and speed of sound, p, at grid point jj+1，Pj+1，uj+1，cj+1Density, velocity, pressure and speed of sound, r, at grid point j +1, respectivelyjSpatial coordinates of grid point j, rj+1The spatial coordinates for grid point j + 1;
step S3: obtaining the space derivative value P of the left pressure intensity of the interfaceL' sum velocity space derivative value uL′：
Figure FDA0002310549040000021
Figure FDA0002310549040000022
Wherein，
Figure FDA0002310549040000023
sign (x) is a sign function, min (x, y) is a minimum function, PL' is the value of the spatial derivative of the pressure on the left side of the interface, uL'is the value of the space derivative of the velocity at the left side of the interface, P'j，(rm-1u)′j，uj' is P, r at grid jm-1The values of the spatial derivatives of u,
Figure FDA0002310549040000024
the pressure spatial derivative approximations to the left and right of grid point j respectively,
Figure FDA0002310549040000025
approximate values of the spatial derivatives of the velocity integral quantity on the left side and the right side of the grid point j are respectively obtained;
step S4: obtaining the density value rho on the left side of the interface by using the isentropic relationLAnd the value of the density spatial derivative ρL′：
Figure FDA0002310549040000026
Wherein, PB,1，γ1Is the equation of state parameter of the left-hand medium,
Figure FDA0002310549040000027
step S5: obtaining the space derivative value P of the right pressure intensity of the interfaceR' sum velocity space derivative value uR′：
Figure FDA0002310549040000031
Figure FDA0002310549040000032
Wherein, PRIs the right side pressure of the interfaceStrong value of the spatial derivative uR'is the space derivative value of the velocity on the right side of the interface, P'j+1，(rm- 1u)′j+1，uj+1' is P, r at grid j +1m-1The spatial derivative values of u, u;
step S6: obtaining the density value rho on the right side of the interface by using the isentropic relationRAnd the value of the density spatial derivative ρR′：
Figure FDA0002310549040000033
Wherein, PB,2，γ2Are the parameters of the equation of state of the medium on the right,
Figure FDA0002310549040000034
step S7: the linear distribution of the left and right states of the interface is obtained as follows:
Figure FDA0002310549040000035
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CN114254573A (en) * 2021-12-17 2022-03-29 北京航空航天大学 High-order processing method for multi-medium interface with symmetric source items
CN114254573B (en) * 2021-12-17 2024-05-24 北京航空航天大学 Multi-medium interface high-order processing method with symmetry source item

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