A numerical simulation method and device for the piston oil injection cooling process

Abstract

The present invention relates to the technical field of numerical simulation, and particularly to a numerical simulation method and device for the piston oil injection cooling process. The method includes: establishing a large density ratio gas-liquid two-phase flow SPH calculation model; establishing a heat flux boundary heat transfer calculation model; establishing a calculation model for the pressure-type outlet boundary, including the establishment of a buffer region, the arrangement of seed points, the introduction and update of virtual mass, a particle generation mechanism, and a particle deletion mechanism; the seed points are used to represent the templates of the generated particles; performing numerical simulation on the piston oil injection cooling process based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the calculation model for the pressure-type outlet boundary. Through the present application, it is possible to perform numerical simulation on the piston oil injection cooling process by adopting the weak compressibility assumption, reducing the calculation amount of the numerical simulation.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical simulation, and more particularly, to a numerical simulation method and device for the piston oil injection cooling process. Background Art

[0002] In the aerospace industry, the cylinder drives the piston to vibrate at high frequency and high speed. The mixture of oil and air flows inside the piston, and the heat dissipation characteristics of oil injection are of great concern. When using a closed boundary, the large reciprocating vibration of the piston causes a large range of changes in the particle volume, which in turn affects the weakly compressible assumption in the traditional SPH numerical simulation process.

[0003] The existing solution is to connect a box with a size much larger than the pipeline to the piston vibration pipeline, so that the volume change caused by the piston vibration will not cause a large range of changes in the particle volume within the overall computational domain, and it still complies with the weakly compressible assumption. However, this solution will lead to a sharp increase in the amount of calculation, and even make the simulation of the piston oil injection cooling process impossible to proceed. Summary of the Invention

[0004] In view of this, the purpose of this application is to provide a numerical simulation method and device for the piston oil injection cooling process, which can numerically simulate the piston oil injection cooling process using the weakly compressible assumption and reduce the amount of calculation in the numerical simulation.

[0005] In a first aspect, an embodiment of this application provides a numerical simulation method for the piston oil injection cooling process, and the method includes:

[0006] Establish a large density ratio gas-liquid two-phase flow SPH calculation model; the continuity equation between the same type of fluids in the large density ratio gas-liquid two-phase flow SPH calculation model adopts a simplified delta-term smoothed pressure field; an artificial viscosity term is introduced into the momentum equation in the SPH calculation model; the relationship between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model is explicitly established by using the weakly compressible assumption and introducing the Tait equation of state and the background pressure term;

[0007] Establish a heat flux boundary heat transfer calculation model;

[0008] Establish a calculation model for the pressure-type outlet boundary, and the calculation model for the pressure-type outlet boundary includes the establishment of a buffer region, the arrangement of seed points, the introduction and update of virtual mass, a particle generation mechanism, and a particle deletion mechanism; the seed points are used to represent the templates of the generated particles;

[0009] Based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the calculation model for the pressure-type outlet boundary, numerically simulate the piston oil injection cooling process.

[0010] In a possible implementation, the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model is established through the following steps:

[0011] Introduce the multiphase delta term to obtain the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model:

[0012] ;

[0013] ;

[0014] where is the dissipation term of the density of the i-th particle, is a constant, is the arithmetic mean of the smoothing length of the i-th particle and the smoothing length of the j-th particle searched in the computational domain, is the arithmetic mean of the sound speed of the i-th particle and the sound speed of the j-th particle searched in the computational domain, is the distance vector difference between the i-th particle and the j-th particle searched in the computational domain, is the Hamiltonian operator of the i-th particle, is the kernel function between the i-th particle and the j-th particle searched in the computational domain, is the volume of the j-th particle searched in the computational domain, is the density of the i-th particle, is the density of the j-th particle searched in the computational domain, is the physical relationship between the i-th particle and the j-th particle searched in the computational domain, is the physical property of the i-th particle, is the physical property of the j-th particle searched in the computational domain.

[0015] In a possible implementation, the momentum equation in the large density ratio gas-liquid two-phase flow SPH calculation model includes:

[0016] The discretization of the momentum equation in the large density ratio gas-liquid two-phase flow SPH calculation model for the pressure gradient term is expressed by the following formula:

[0017] ;

[0018] where is the pressure gradient term of the i-th particle, is the pressure of the i-th particle, is the -th particle searched in the computational domain.

[0019] The momentum equation in the large density ratio gas-liquid two-phase flow SPH calculation model is expressed for the velocity change by the following formula:

[0020] ;

[0021] wherein, is the change in the velocity of the i-th particle caused by the artificial viscosity term, is a constant, is the velocity vector difference between the velocity vector of the i-th particle and the velocity vector of the j-th particle searched within the computational domain.

[0022] In a possible implementation manner, the relationship between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model is established through the following steps:

[0023] Introduce the Tait equation of state EOS to obtain the relationship between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model:

[0024] ;

[0025] ; ;

[0026] wherein, is the pressure of the particle, is the reference density, is the reference sound speed, is the adiabatic coefficient of the particle, is the density of the particle, is the background pressure, , is the density change of the particle, is the maximum velocity in the piston fuel injection scenario, is the maximum pressure in the piston fuel injection scenario.

[0027] In a possible implementation manner, the heat flux boundary heat transfer calculation model includes:

[0028] Calculate the surface identification parameter of the particle in the piston fuel injection scenario ;

[0029] If the surface identification parameter of the particle in the piston fuel injection scenario is less than the preset value, then the i-th particle is a solid surface particle;

[0030] Based on the solid surface particles, calculate the temperature change caused by the heat flux input or output at the solid boundary.

[0031] In a possible implementation, the calculation model of the pressure-type outlet boundary includes:

[0032] The area pointed to by the positive normal direction of the pressure-type outlet boundary is the buffer area; the area pointed to by the opposite direction corresponding to the positive normal direction of the pressure-type outlet boundary is the internal area;

[0033] Extend a preset distance along the direction pointed to by the corresponding positive normal direction starting from the pressure-type outlet boundary , to obtain a particle addition and deletion layer; arrange seed points in the particle addition and deletion layer, and the interval distance of the seed points is positively correlated with the size of the particles.

[0034] In a possible implementation, the particle deletion mechanism includes:

[0035] Calculate the position vector of the i-th particle and the position vector of the center point of the pressure-type outlet boundary , and obtain the vector difference ;

[0036] Through the rotation matrix R of the pressure-type outlet boundary and the vector difference , map the i-th particle onto the standard plane to obtain the position coordinates of the i-th particle on the standard plane ( , , );

[0037] According to the position coordinates on the standard plane ( , , ), determine the area where the i-th particle is located;

[0038] If the area where the i-th particle is located is outside the particle addition and deletion layer, delete the i-th particle.

[0039] In a possible implementation, the particle generation mechanism includes:

[0040] Assign a virtual mass M to each seed point s, and the initial value of the virtual mass M of the seed point s is 0;

[0041] Interpolate the surrounding flow field of the pressure-type outlet boundary through the density, volume, and velocity of all particles at the pressure-type outlet boundary to obtain the velocity and density at the interpolation point I;

[0042] According to the velocity and density at the interpolation point I and the virtual mass of the seed point s at the current time step , generate the virtual mass of the seed point s at the next time step ;

[0043] If , then generate the position information and flow field information of the new particle Q according to the position information of the seed point s; where is the floor function of is the floor function of

[0044] In a second aspect, the embodiments of the present application further provide a numerical simulation device for the piston fuel injection cooling process. The numerical simulation device for the piston fuel injection cooling process includes:

[0045] A building module, configured to build a large density ratio gas-liquid two-phase flow SPH calculation model; in the large density ratio gas-liquid two-phase flow SPH calculation model, the continuity equation between the same type of fluids adopts a simplified delta term smoothed pressure field; an artificial viscosity term is introduced into the momentum equation in the SPH calculation model; the connection between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model is explicitly established by adopting the weak compressibility assumption and introducing the Tait equation of state and the background pressure term;

[0046] The building module is further configured to build a heat flux boundary heat transfer calculation model;

[0047] The building module is further configured to build a calculation model for the pressure-type outlet boundary. The calculation model for the pressure-type outlet boundary includes the establishment of a buffer region, the arrangement of seed points, the introduction and update of virtual mass, a particle generation mechanism, and a particle deletion mechanism; the seed points are used to represent the templates of the generated particles;

[0048] A numerical simulation module, configured to perform numerical simulation on the piston fuel injection cooling process based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the calculation model for the pressure-type outlet boundary.

[0049] In a possible implementation manner, the building module is specifically configured to establish the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model through the following steps: introduce a multiphase delta term to obtain the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model: ; ; where is the dissipation term of the density of the i-th particle, is a constant, is the arithmetic mean of the smoothing length of the i-th particle and the smoothing length of the j-th particle searched in the calculation domain, is the arithmetic mean of the sound speed of the i-th particle and the sound speed of the j-th particle searched within the computational domain, is the difference vector of the distances between the i-th particle and the j-th particle searched within the computational domain, is the Hamiltonian operator of the i-th particle, is the kernel function between the i-th particle and the j-th particle searched within the computational domain, is the volume of the j-th particle searched within the computational domain, is the density of the i-th particle, is the density of the j-th particle searched within the computational domain, is the physical relationship between the i-th particle and the j-th particle searched within the computational domain, is the physical property of the i-th particle, is the physical property of the j-th particle searched within the computational domain.

[0050] In a third aspect, an embodiment of the present application further provides an electronic device, including: a processor, a storage medium, and a bus. The storage medium stores machine-readable instructions executable by the processor. When the electronic device runs, the processor communicates with the storage medium through the bus, and the processor executes the machine-readable instructions to perform the steps of the numerical simulation method for the piston fuel injection cooling process as described in any item of the first aspect.

[0051] In a fourth aspect, an embodiment of the present application further provides a computer-readable storage medium. A computer program is stored on the computer-readable storage medium. When the computer program is run by a processor, it performs the steps of the numerical simulation method for the piston fuel injection cooling process as described in any item of the first aspect.

[0052] An embodiment of the present application provides a numerical simulation method and device for a piston fuel injection cooling process. The method includes: establishing a large density ratio gas-liquid two-phase flow SPH calculation model; the continuity equation between the same type of fluids in the large density ratio gas-liquid two-phase flow SPH calculation model adopts a simplified delta-term smoothed pressure field, the momentum equation introduces an artificial viscosity term, and the relationship between the density term and the pressure term is explicitly established by adopting the weak compressibility assumption and introducing the Tait equation of state and the background pressure term; establishing a heat flux boundary heat transfer calculation model; establishing a calculation model for the pressure-type outlet boundary, including the establishment of a buffer region, the arrangement of seed points, the introduction and update of virtual mass, the particle generation mechanism, and the particle deletion mechanism; the seed points are used to represent the templates of the generated particles; performing a numerical simulation of the piston fuel injection cooling process based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the calculation model for the pressure-type outlet boundary. Through the present application, the numerical simulation of the piston fuel injection cooling process can be performed by adopting the weak compressibility assumption, reducing the computational amount of the numerical simulation. Brief Description of the Drawings

[0053] To more clearly illustrate the technical solutions of the embodiments of the present application, the drawings required for the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present application, and therefore should not be regarded as limiting the scope. For those of ordinary skill in the art, without creative efforts, other relevant drawings can also be obtained based on these drawings.

[0054] Figure 1 Shows a flowchart of a numerical simulation method for a piston oil injection cooling process provided by an embodiment of the present application;

[0055] Figure 2 Shows a schematic structural diagram of a pressure-type outlet boundary provided by an embodiment of the present application;

[0056] Figure 3 Shows a schematic structural diagram of a numerical simulation device for a piston oil injection cooling process provided by an embodiment of the present application;

[0057] Figure 4 Shows a schematic structural diagram of an electronic device provided by an embodiment of the present application. Detailed Description of the Embodiments

[0058] To make the objectives, technical solutions, and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. It should be understood that the drawings in the present application only serve the purpose of illustration and description, and are not used to limit the protection scope of the present application. Additionally, it should be understood that the schematic drawings are not drawn to actual scale. The flowcharts used in the present application show the operations implemented according to some embodiments of the present application. It should be understood that the operations in the flowchart may not be implemented in sequence, and steps without logical context relationships may be reversed or implemented simultaneously. Moreover, those skilled in the art can add one or more other operations to the flowchart or remove one or more operations from the flowchart under the guidance of the content of the present application.

[0059] In addition, the described embodiments are only a part of the embodiments of the present application, rather than all of the embodiments. The components of the embodiments of the present application described and shown in the drawings here can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present application provided in the drawings is not intended to limit the scope of the present application to be protected, but only represents the selected embodiments of the present application. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without creative efforts belong to the protection scope of the present application.

[0060] In order to enable those skilled in the art to use the content of this application, in combination with a specific application scenario, namely the "numerical simulation technology field", the following embodiments are provided. For those skilled in the art, without departing from the spirit and scope of this application, the general principles defined here can be applied to other embodiments and application scenarios. Although this application is mainly described around the "numerical simulation technology field", it should be understood that this is only an exemplary embodiment.

[0061] It should be noted that the term "including" will be used in the embodiments of this application to indicate the existence of the features stated thereafter, but does not exclude the addition of other features.

[0062] The following will provide a detailed description of a numerical simulation method for the piston fuel injection cooling process provided by the embodiments of this application.

[0063] Refer to Figure 1 As shown, it is a schematic flowchart of a numerical simulation method for the piston fuel injection cooling process provided by the embodiments of this application. The following will provide a detailed description of each step:

[0064] S101. Establish a large density ratio gas-liquid two-phase flow SPH calculation model.

[0065] In the embodiment of this application, for the continuity equation between the same type of fluids in the large density ratio gas-liquid two-phase flow SPH calculation model, a simplified delta-term smoothed pressure field is adopted; for the momentum equation in the SPH calculation model, an artificial viscosity term is introduced; the relationship between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model is explicitly established by adopting the weak compressibility assumption and introducing the Tait equation of state and the background pressure term. The continuity equation between the same type of fluids, the relationship between the density term and the pressure term, the discretization of the pressure gradient term in the momentum equation, and the velocity change in the momentum equation in the large density ratio gas-liquid two-phase flow SPH calculation model are established for the particles in the internal region.

[0066] Specifically, the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model is established through the following steps:

[0067] Introduce a multiphase delta term to obtain the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model:

[0068] ;

[0069] ;

[0070] Among them, is the dissipation term of the density of the i-th particle, is a constant (usually taken as 0.1), is the arithmetic mean of the smoothing length of the $i$-th particle and the smoothing length of the $j$-th particle searched within the computational domain, is the arithmetic mean of the sound speed of the $i$-th particle and the sound speed of the $j$-th particle searched within the computational domain, is the distance vector difference between the $i$-th particle and the $j$-th particle searched within the computational domain, is the Hamiltonian operator of the $i$-th particle, is the kernel function between the $i$-th particle and the $j$-th particle searched within the computational domain, is the volume of the $j$-th particle searched within the computational domain, is the density of the $i$-th particle, is the density of the $j$-th particle searched within the computational domain, is the physical property relationship between the $i$-th particle and the $j$-th particle searched within the computational domain, is the physical property attribute of the $i$-th particle, is the physical property attribute of the $j$-th particle searched within the computational domain.

[0071] Specifically, the momentum equation in the SPH calculation model for gas-liquid two-phase flow with a large density ratio includes:

[0072] Step 1. The discretization of the pressure gradient term in the momentum equation of the SPH calculation model for gas-liquid two-phase flow with a large density ratio is expressed by the following formula:

[0073] ;

[0074] where, is the pressure gradient term of the $i$-th particle, is the pressure of the $i$-th particle, is the pressure of the $j$-th particle searched within the computational domain.

[0075] Step 2. The representation of the velocity change in the momentum equation of the SPH calculation model for gas-liquid two-phase flow with a large density ratio is as follows:

[0076] ;

[0077] where, is the change in the velocity of the $i$-th particle caused by the artificial viscosity term, is a constant (usually taken as 0.05), is the velocity vector difference between the velocity vector of the $i$-th particle and the velocity vector of the $j$-th particle searched within the computational domain.

[0078] Specifically, the relationship between the density term and the pressure term in the SPH calculation model for gas-liquid two-phase flow with a large density ratio is established through the following steps:

[0079] Introduce the Tait equation of state EOS to obtain the relationship between the density term and the pressure term in the SPH calculation model of gas-liquid two-phase flow with a large density ratio:

[0080] ;

[0081] ; ;

[0082] where, is the pressure of the particle, is the reference density, is the reference sound speed, is the adiabatic coefficient of the particle, is the density of the particle, is the background pressure, (this condition is used to satisfy the compressibility assumption if applicable), is the density change of the particle, is the maximum velocity in the piston fuel injection scenario, is the maximum pressure in the piston fuel injection scenario.

[0083] In the embodiment of the present application, generally, for liquid particles, the value of γ is taken as 7, and for gas particles, the value of γ is taken as 1.4.

[0084] S102. Establish a heat flux boundary heat transfer calculation model.

[0085] Specifically, the heat flux boundary heat transfer calculation model includes:

[0086] Step 1. Calculate the surface identification parameter of the particle in the piston fuel injection scenario ; if the surface identification parameter of the particle in the piston fuel injection scenario is less than a preset value (such as 2.25), then the i-th particle is a solid surface particle;

[0087] Specifically, calculate the surface identification parameter of the particle in the piston fuel injection scenario through the following formula :

[0088] ;

[0089] where, is the surface particle identification parameter of the i-th particle, is the distance vector difference between the i-th particle and the j-th particle searched in the computational domain, is the Hamiltonian operator of the i-th particle, is the kernel function between the i-th particle and the j-th particle searched in the computational domain, is the volume of the j-th particle searched in the computational domain.

[0090] Step 2: Calculate the temperature change caused by the heat flux input or output at the solid boundary based on the solid surface particles.

[0091] Specifically, the temperature change caused by the heat flux input or output at the solid boundary is calculated based on the solid surface particles through the following formula:

[0092] ;

[0093] ;

[0094] ;

[0095] Wherein, is the heat flux of the th solid surface particle, is the heat flux input or output at the solid boundary, is the temperature change caused by the heat flux input or output at the solid boundary, is the density of the th solid surface particle, is the specific heat capacity, is the th solid surface particle area, is the th solid surface particle mass, is the th solid surface particle Hamiltonian operator, is the th solid surface particle and the kernel function between the jth particle searched in the computational domain, is the volume of the jth particle searched in the computational domain, is the th solid surface particle volume.

[0096] In the embodiments of the present application, after determining the surface area, through the definition formula of heat flux , the formula for the temperature change caused by the heat flux input or output at the solid boundary can be obtained. Wherein, is the heat conduction coefficient (the energy per unit area flowing into or out of the solid domain through the surface), is the temperature gradient of the particle, is the normal vector of the particle, is the heat flux input or output at the boundary.

[0097] S103: Establish a calculation model for the pressure-type outlet boundary.

[0098] In the embodiments of the present application, the calculation model of the pressure-type outlet boundary includes the establishment of a buffer region, the arrangement of seed points, the introduction and update of virtual mass, the particle generation mechanism, the particle deletion mechanism, the continuity equation between like fluids, the relationship between the density term and the pressure term, the discretization of the momentum equation for the pressure gradient term, and the velocity change of the momentum equation. Among them, the continuity equation between like fluids, the relationship between the density term and the pressure term, the discretization of the momentum equation for the pressure gradient term, and the velocity change of the momentum equation in the calculation model of the pressure-type outlet boundary are established for the particles in the buffer region.

[0099] Here, in the calculation model of the pressure-type outlet boundary, the region pointed to by the positive normal direction of the pressure-type outlet boundary is the buffer region; the region pointed to by the opposite direction corresponding to the positive normal direction of the pressure-type outlet boundary is the internal region; starting from the pressure-type outlet boundary and extending a preset distance along the direction pointed to by the corresponding positive normal direction , a particle addition and deletion layer is obtained; seed points are arranged in the particle addition and deletion layer, and the interval distance of the seed points is positively correlated with the size of the particles. Among them, is related to the kernel function W and is usually taken as 2 or 3. is the smoothing length. Refer to Figure 2 shown, which is a schematic structural diagram of the pressure-type outlet boundary provided by the embodiment of the present application.

[0100] Specifically, the continuity equation between like fluids, the relationship between the density term and the pressure term, and the formula corresponding to the velocity change of the momentum equation in the calculation model of the pressure-type outlet boundary are the same as those of the continuity equation between like fluids, the relationship between the density term and the pressure term, and the velocity change of the momentum equation in the SPH calculation model of the gas-liquid two-phase flow with a large density ratio;

[0101] Specifically, the discretization of the pressure gradient term in the calculation model of the pressure-type outlet boundary is represented by the following formula:

[0102] ;

[0103] Among them, is the set pressure value at the pressure-type outlet boundary.

[0104] Specifically, the calculation model of the pressure-type outlet boundary further includes a velocity correction formula to eliminate the influence of the tangential velocity:

[0105] ;

[0106] Among them, is the velocity vector calculated by the momentum equation for the velocity change in the calculation model of the pressure-type outlet boundary at step n, is the positive normal direction of the outlet boundary. is the corrected velocity vector.

[0107] Specifically, the particle deletion mechanism includes:

[0108] Step 1: Calculate the position vector of the i-th particle and the position vector of the center point of the pressure-type outlet boundary to obtain the vector difference .

[0109] Specifically, the rotation matrix R is determined through the following steps:

[0110] Set the rotation matrix for the pressure-type outlet boundary to rotate once along the x-axis as , the rotation matrix for rotating once along the y-axis as , and the rotation matrix for rotating once along the z-axis as ;

[0111] ; ;

[0112] ;

[0113] ;

[0114] where is the rotation matrix of the pressure-type outlet boundary, is the rotation angle of the pressure-type outlet boundary.

[0115] Here, the above formula indicates that the rotation matrix is the matrix multiplication of a series of single-degree-of-freedom rotation matrices. Since it involves specific operation steps and has no unified form, the subscripts, numbers, and corresponding angles of the right-side rotation matrix are not indicated.

[0116] Step 2: Map the i-th particle to the standard plane through the rotation matrix R of the pressure-type outlet boundary and the vector difference to obtain the position coordinates of the i-th particle on the standard plane ( , , ).

[0117] Specifically, through the rotation matrix R of the pressure-type outlet boundary and the vector difference , map the i-th particle to the standard plane to obtain the position coordinates of the i-th particle on the standard plane ( , , ), including:

[0118] Substitute the rotation matrix R of the pressure-type outlet boundary and the vector difference into the following formula to obtain the position coordinates of the i-th particle on the standard plane ( , , );

[0119] ;

[0120] Among them, is the position coordinate vector of the i-th particle mapped to the standard plane, is the unit vector of the x-axis of the local coordinate system on the standard plane, is the axis of the local coordinate system on the standard plane is the unit vector of the z-axis of the local coordinate system on the standard plane.

[0121] Step 3: Determine the region where the i-th particle is located according to the position coordinates ( , , ) on the standard plane.

[0122] Specifically, determine the region where the i-th particle is located according to the position coordinates ( , , ), including:

[0123] Assume that the position where z = 0 is the position of the pressure-type outlet boundary, and let S be the projection range of the shape of the pressure-type outlet boundary on the plane where z = 0. If , then the region where the i-th particle is located is the buffer region; among them, is the shape function determined by the shape of the pressure-type outlet boundary, used to measure the distance from the projection point of the i-th particle on the plane where z = 0 to the outer contour of the pressure-type outlet boundary; if , then the region where the i-th particle is located is outside the particle addition / removal layer; if the i-th particle is not in the buffer region and not outside the particle addition / removal layer, then the region where the i-th particle is located is the internal region.

[0124] Step 4: If the region where the i-th particle is located is outside the particle addition / removal layer, delete the i-th particle.

[0125] Specifically, the particle generation mechanism includes:

[0126] Step 1: Assign a virtual mass M to each seed point s, and the initial value of the virtual mass M of the seed point s is 0.

[0127] Step 2: Interpolate the flow field around the pressure-type outlet boundary through the density, volume, and velocity of all particles at the pressure-type outlet boundary to obtain the velocity and density 。

[0128] Specifically, the surrounding flow field of the pressure-type outlet boundary is interpolated through the density, volume, and velocity of all particles at the pressure-type outlet boundary to obtain the velocity at the interpolation point I and density , including:

[0129] Substitute the density, volume, and velocity of all particles at the pressure-type outlet boundary into the following formula to obtain the velocity at the interpolation point I and density :

[0130] ;

[0131] where , , is the velocity of the j-th particle, is the density of the j-th particle, is the kernel function between the interpolation point I and the j-th particle, is the volume of the j-th particle.

[0132] Step 3. Generate the virtual mass of the seed point s at the next time step according to the velocity and density at the interpolation point I, and the virtual mass of the seed point s at the current time step.

[0133] Specifically, generate the virtual mass of the seed point s at the next time step according to the velocity and density at the interpolation point I, and the virtual mass of the seed point s at the current time step, including:

[0134] Substitute the velocity and density at the interpolation point I, and the virtual mass of the seed point s at the current time step into the following formula to obtain the virtual mass of the seed point s at the next time step;

[0135] ;

[0136] where is the current time step, is the area of the interpolation point, is positively correlated with , is the particle size, dim is the spatial dimension, is the time step.

[0137] Step 4. If , then generate the position information and flow field information of the new particle Q according to the position information of the seed point s; where is the floor function of is the floor function of

[0138] Specifically, generating the position information and flow field information of the new particle Q according to the position information of the seed point s includes:

[0139] Substitute the position information of the seed point s into the following formula to obtain the position information and flow field information of the new particle Q;

[0140] ;

[0141] where is the position information of the new particle Q, is the position information of the seed point s, is the size of the particle, is the current time step, is the velocity of the new particle Q, is the velocity of the interpolation point I, is the density of the interpolation point I, is the temperature of the interpolation point I, is the temperature of the new particle Q, is the density of the new particle Q, is the pressure of the new particle Q, is the equation of state, and the flow field information of the new particle Q includes , , , .

[0142] S104. Conduct a numerical simulation on the piston oil injection cooling process based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the pressure-type outlet boundary calculation model.

[0143] In the embodiments of the present application, obtain the pressure-type outlet boundary position set by the numerical simulation user for the fluid calculation domain in the piston fuel injection scenario, the solid geometry model of the piston (supporting formats such as stl, stp, and obj), and the motion form (including velocity setting and displacement setting); establish a buffer region within the fluid calculation domain of the solid geometry model based on the establishment process of the buffer region in the pressure-type outlet boundary; discretize the fluid calculation domain into particles; at the initial moment, search for the particles in the internal region from the discretized particles; and extrapolate the target particles among the particles in the internal region along the direction indicated by the positive normal of the pressure-type outlet boundary to obtain the particles in the buffer region; generate the particles to be numerically simulated at the pressure-type outlet boundary; search for the neighboring particles of each particle and establish a linked list; call the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the calculation model of the pressure-type outlet boundary, and calculate the time derivative term of the control equation of each particle based on the established linked list; perform leapfrog time stepping on the particles, and delete the particles according to the particle deletion mechanism; generate particles according to the particle generation mechanism; if the current time is less than the end time of the numerical simulation, then jump to generating the particles to be numerically simulated at the pressure-type outlet boundary and continue to execute.

[0144] Here, for the list-type input, first determine the interval range to which the motion time t belongs And , and then use Lagrangian interpolation to obtain the input value of the motion form:

[0145] .

[0146] Among them, Is the velocity at step n, Is the time at step n, Is the time at step n + 1, Is the velocity at step n + 1, Is the time range , The interpolated velocity within.

[0147] Here, the mathematical form of the leapfrog time stepping is as follows:

[0148] ;

[0149] .

[0150] Among them, the superscript represents the time step, the fraction represents the physical quantity at the intermediate time step, and the variable with subscript 0 represents the variable defined at the intermediate time step.

[0151] Based on the same inventive concept, an embodiment of the present application further provides a numerical simulation device for the piston oil injection cooling process corresponding to the numerical simulation method of the piston oil injection cooling process. Since the principle of solving problems by the device in the embodiment of the present application is similar to that of the above-mentioned numerical simulation method of the piston oil injection cooling process in the embodiment of the present application, the implementation of the device can refer to the implementation of the method, and the repeated parts will not be described again.

[0152] Referring to Figure 3 As shown in the figure, it is a schematic diagram of a numerical simulation device for the piston oil injection cooling process provided by an embodiment of the present application. The numerical simulation device for the piston oil injection cooling process includes:

[0153] A building module 301, configured to build a large density ratio gas-liquid two-phase flow SPH calculation model; in the continuity equation between the same type of fluids in the large density ratio gas-liquid two-phase flow SPH calculation model, a simplified delta term smoothing pressure field is adopted; an artificial viscosity term is introduced into the momentum equation in the SPH calculation model; the relationship between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model is explicitly established by adopting the weak compressibility assumption and introducing the Tait equation of state and the background pressure term;

[0154] The building module 301 is further configured to build a heat flux boundary heat transfer calculation model;

[0155] The building module 301 is further configured to build a calculation model for the pressure type outlet boundary. The calculation model for the pressure type outlet boundary includes the establishment of a buffer region, the arrangement of seed points, the introduction and update of virtual mass, a particle generation mechanism, and a particle deletion mechanism; the seed points are used to represent the templates of the generated particles;

[0156] A numerical simulation module 302, configured to perform numerical simulation on the piston oil injection cooling process based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the calculation model for the pressure type outlet boundary.

[0157] In a possible implementation manner, the building module 301 is specifically configured to establish the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model through the following steps: introducing a multiphase delta term to obtain the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model: ; ; where is the dissipation term of the density of the i-th particle, is a constant, is the arithmetic mean of the smoothing length of the i-th particle and the smoothing length of the j-th particle searched in the calculation domain, is the arithmetic mean of the sound speed of the i-th particle and the sound speed of the j-th particle searched in the calculation domain, is the distance vector difference between the i-th particle and the j-th particle searched within the computational domain, is the Hamiltonian operator of the i-th particle, is the kernel function between the i-th particle and the j-th particle searched within the computational domain, is the volume of the j-th particle searched within the computational domain, is the density of the i-th particle, is the density of the j-th particle searched within the computational domain, is the physical property relationship between the i-th particle and the j-th particle searched within the computational domain, is the physical property attribute of the i-th particle, is the physical property attribute of the j-th particle searched within the computational domain.

[0158] In a possible implementation manner, the momentum equation in the large density ratio gas-liquid two-phase flow SPH calculation model includes:

[0159] The discretization of the pressure gradient term in the momentum equation in the large density ratio gas-liquid two-phase flow SPH calculation model is represented by the following formula:

[0160] ;

[0161] Among them, is the pressure gradient term of the i-th particle, is the pressure of the i-th particle, is the pressure of the j-th particle searched within the computational domain.

[0162] The representation of the velocity change in the momentum equation in the large density ratio gas-liquid two-phase flow SPH calculation model by the following formula:

[0163] ;

[0164] Among them, is the change in the velocity of the i-th particle caused by the artificial viscosity term, is a constant, is the velocity vector difference between the velocity vector of the i-th particle and the velocity vector of the j-th particle searched within the computational domain.

[0165] In a possible implementation manner, the establishing module 301 is specifically configured to establish the connection between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model through the following steps: Introduce the Tait equation of state EOS to obtain the connection between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model:

[0166] ; ;

[0167] ;

[0168] Among them, is the pressure of the particle, is the reference density, is the reference sound speed, is the adiabatic coefficient of the particle, is the density of the particle, is the background pressure, , is the density change of the particle, is the maximum speed in the piston fuel injection scenario, is the maximum pressure in the piston fuel injection scenario.

[0169] In a possible implementation manner, the establishment module 301 is specifically configured to establish a heat flux boundary heat transfer calculation model through the following steps: Calculate the surface recognition parameter of the particle in the piston fuel injection scenario ; If the surface recognition parameter of the particle in the piston fuel injection scenario is less than the preset value, then the i-th particle is a solid surface particle; Calculate the temperature change caused by the heat flux input or output at the solid boundary based on the solid surface particle.

[0170] In a possible implementation manner, the establishment module 301 is specifically configured to establish the calculation model of the pressure-type outlet boundary through the following steps, including: The area pointed to by the positive normal of the pressure-type outlet boundary is the buffer area; The area pointed to by the opposite direction corresponding to the positive normal of the pressure-type outlet boundary is the internal area; Extend a preset distance from the pressure-type outlet boundary along the direction pointed to by the corresponding positive normal

[0171] to obtain a particle addition and deletion layer; Arrange seed points in the particle addition and deletion layer, and the interval distance of the seed points is positively correlated with the size of the particle. to obtain the vector difference ; Through the rotation matrix R of the pressure-type outlet boundary and the vector difference ; Map the i-th particle to the standard plane through the rotation matrix R of the pressure-type outlet boundary and the vector difference , and obtain the position coordinates of the i-th particle on the standard plane ( , , ); According to the position coordinates ( , , ), determine the region where the i-th particle is located; if the region where the i-th particle is located is outside the particle addition / removal layer, then delete the i-th particle.

[0172] In a possible implementation, the establishing module 301 is specifically configured to establish the particle generation mechanism through the following steps, including: assigning a virtual mass M to each seed point s, where the initial value of the virtual mass M of the seed point s is 0; interpolating the surrounding flow field of the pressure-type outlet boundary through the density, volume, and velocity of all particles at the pressure-type outlet boundary to obtain the velocity and density at the interpolation point I; according to the velocity and density at the interpolation point I, and the virtual mass of the seed point s at the current time step, generate the virtual mass of the seed point s at the next time step; if , then generate the position information and flow field information of the new particle Q according to the position information of the seed point s; where is the floor of , and is the floor of .

[0173] The present application provides a numerical simulation device for the piston fuel injection cooling process. The device includes: an establishing module 301 for establishing a large density ratio gas-liquid two-phase flow SPH calculation model; the continuity equation between like fluids in the large density ratio gas-liquid two-phase flow SPH calculation model adopts a simplified delta-term smoothed pressure field; an artificial viscosity term is introduced into the momentum equation in the SPH calculation model; the connection between the density term and the pressure term in the large density ratio gas-liquid two-phase flow SPH calculation model is explicitly established by adopting the weak compressibility assumption and introducing the Tait equation of state and the background pressure term; the establishing module 301 is further configured to establish a heat flux boundary heat transfer calculation model; the establishing module 301 is further configured to establish a calculation model for the pressure-type outlet boundary, and the calculation model for the pressure-type outlet boundary includes the establishment of a buffer region, the arrangement of seed points, the introduction and update of virtual mass, a particle generation mechanism, and a particle deletion mechanism; the seed points are used to represent the templates of the generated particles; a numerical simulation module 302 for numerically simulating the piston fuel injection cooling process based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the calculation model for the pressure-type outlet boundary. Through the present application, the piston fuel injection cooling process can be numerically simulated by adopting the weak compressibility assumption, reducing the computational amount of the numerical simulation.

[0174] Such as Figure 4As shown in the figure, an electronic device 400 provided by an embodiment of the present application includes: a processor 401, a memory 402, and a bus. The memory 402 stores machine-readable instructions executable by the processor 401. When the electronic device runs, the processor 401 communicates with the memory 402 through the bus. The processor 401 executes the machine-readable instructions to perform the steps of the numerical simulation method for the piston fuel injection cooling process as described above.

[0175] Specifically, the above-mentioned memory 402 and processor 401 can be general-purpose memory and processor, and no specific limitation is made here. When the processor 401 runs the computer program stored in the memory 402, it can execute the numerical simulation method for the piston fuel injection cooling process as described above.

[0176] Corresponding to the numerical simulation method for the piston fuel injection cooling process, an embodiment of the present application also provides a computer-readable storage medium. A computer program is stored on the computer-readable storage medium. When the computer program is run by a processor, it executes the steps of the numerical simulation method for the piston fuel injection cooling process as described above.

[0177] Those skilled in the art can clearly understand that for the convenience and simplicity of description, the specific working processes of the systems and devices described above can refer to the corresponding processes in the method embodiments, and will not be repeated in this application. In the several embodiments provided in this application, it should be understood that the disclosed systems, devices, and methods can be implemented in other ways. The device embodiments described above are only illustrative. For example, the division of the modules is only a logical function division, and there may be other division methods in actual implementation. For another example, multiple modules or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed mutual coupling or direct coupling or communication connection can be through some communication interfaces. The indirect coupling or communication connection of the devices or modules can be in an electrical, mechanical, or other form.

[0178] The modules described as separate components may or may not be physically separated. The components displayed as modules may or may not be physical units, that is, they can be located in one place or distributed to multiple network units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0179] In addition, in each embodiment of the present application, the functional units can be integrated into one processing unit, or each unit can exist physically alone, or two or more units can be integrated into one unit.

[0180] When the above-mentioned function is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a non-volatile computer-readable storage medium executable by a processor. Based on such an understanding, the technical solution of the present application, in essence, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium and includes several instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the information processing method described in various embodiments of the present application. The aforementioned storage medium includes: various media such as USB flash drives, mobile hard disks, ROM, RAM, magnetic disks, or optical discs that can store program codes.

[0181] The above is only the specific implementation manner of the present application, but the protection scope of the present application is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present application can easily think of changes or substitutions, which should all be covered by the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A numerical simulation method for piston oil spray cooling process, characterized in that: The method comprises: A SPH calculation model for gas-liquid two-phase flow with a large density ratio is established; the continuity equation between similar fluids in the SPH calculation model for gas-liquid two-phase flow with a large density ratio adopts a simplified delta term smoothing pressure field; the momentum equation in the SPH calculation model introduces an artificial viscosity term; the connection between the density term and the pressure term in the SPH calculation model for gas-liquid two-phase flow with a large density ratio is explicitly established by adopting a weak compressibility assumption and introducing the Tait state equation and background pressure term; Establish a heat flux boundary heat transfer calculation model; Establishing a calculation model of a pressure-type outlet boundary, the calculation model of the pressure-type outlet boundary includes the establishment of a buffer area, the arrangement of seed points, the introduction and update of virtual mass, the generation of particle mechanisms, and the deletion of particle mechanisms; the seed points are used to characterize the template of the generated particles; Numerical simulation of the piston oil injection cooling process is performed based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the pressure type outlet boundary calculation model; The momentum equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow includes: The momentum equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow is expressed by the following formula for the discretization of the pressure gradient term: ； in, is the pressure gradient term of the ith particle, is the pressure of the ith particle, is the first The pressure of a particle; is the density of the ith particle, is the density of the jth particle searched in the computational domain, is the Hamiltonian operator of the ith particle, is the kernel function between the ith particle and the jth particle searched in the computational domain, is the volume of the jth particle searched in the computational domain, is the physical property relationship between the i-th particle and the j-th particle searched in the computational domain; The momentum equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow is expressed by the following formula for the velocity change: ； in, is the change in velocity of the ith particle caused by the artificial viscosity term, is a constant, is the velocity vector difference between the velocity vector of the i-th particle and the velocity vector of the j-th particle searched in the computational domain, is the arithmetic mean of the smooth length of the i-th particle and the smooth length of the j-th particle searched in the computational domain, is the arithmetic mean of the sound velocity of the ith particle and the sound velocity of the jth particle searched in the computational domain, is the distance vector difference between the i-th particle and the j-th particle searched in the computational domain.

2. The numerical simulation method for piston oil spray cooling process according to claim 1, characterized in that: The continuity equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow is established by the following steps: By introducing the multiphase delta term, the continuity equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow is obtained: ； ； in, is the dissipation term of the density of the ith particle, is a constant, is the arithmetic mean of the smooth length of the i-th particle and the smooth length of the j-th particle searched in the computational domain, is the arithmetic mean of the sound velocity of the ith particle and the sound velocity of the jth particle searched in the computational domain, is the distance vector difference between the i-th particle and the j-th particle searched in the computational domain, is the Hamiltonian operator of the ith particle, is the kernel function between the ith particle and the jth particle searched in the computational domain, is the volume of the jth particle searched in the computational domain, is the density of the ith particle, is the density of the jth particle searched in the computational domain, is the physical property relationship between the ith particle and the jth particle searched in the computational domain, is the physical property of the ith particle, is the physical property of the jth particle searched in the computational domain.

3. The numerical simulation method for piston oil spray cooling process according to claim 1, characterized in that: The connection between the density term and the pressure term in the SPH calculation model of the large density ratio gas-liquid two-phase flow is established through the following steps: The Tait equation of state EOS is introduced to obtain the relationship between the density term and the pressure term in the SPH calculation model of the large density ratio gas-liquid two-phase flow: ； ； ； in, is the pressure of the particle, is the reference density, is the reference sound speed, is the adiabatic coefficient of the particle, is the density of the particle, is the background pressure, , is the density change of the particle, is the maximum speed in the piston spraying scenario, is the maximum pressure in the piston oil injection scenario.

4. The numerical simulation method for piston oil spray cooling process according to claim 1, characterized in that: The heat flux boundary heat transfer calculation model includes: Calculate the surface identification parameters of particles in the piston spray scene ; If the surface identification parameters of particles in the piston spray scene If it is less than the preset value, the i-th particle is a solid surface particle; The temperature change caused by the heat flux input or output from the solid boundary is calculated based on the solid surface particles.

5. The numerical simulation method for piston oil spray cooling process according to claim 1, characterized in that: The calculation model of the pressure-type outlet boundary includes: The area pointed to by the positive normal of the pressure-type outlet boundary is the buffer area; the area pointed to by the opposite direction corresponding to the positive normal of the pressure-type outlet boundary is the internal area; Extending a preset distance from the pressure outlet boundary in the direction pointed by the corresponding positive normal , a particle adding and deleting layer is obtained; seed points are arranged in the particle adding and deleting layer, and the spacing distance of the seed points is positively correlated with the size of the particles.

6. The numerical simulation method for piston oil spray cooling process according to claim 5, characterized in that: The particle deletion mechanism includes: Calculate the position vector of the i-th particle The position vector of the center point of the pressure profile outlet boundary The difference between ; The rotation matrix R of the pressure outlet boundary, the vector difference , map the i-th particle to the standard plane, and obtain the position coordinates of the i-th particle on the standard plane ( , , ); According to the position coordinates on the standard plane ( , , ), determine the area where the i-th particle is located; If the region where the i-th particle is located is outside the particle adding and deleting layer, the i-th particle is deleted.

7. The numerical simulation method for piston oil spray cooling process according to claim 5, characterized in that: The particle generation mechanism includes: Assign a virtual mass M to each seed point s, where the initial value of the virtual mass M of the seed point s is 0; The velocity at the interpolation point I is obtained by interpolating the flow field around the pressure-type outlet boundary through the density, volume and velocity of all particles at the pressure-type outlet boundary. and density ; According to the velocity at the interpolation point I and density , the virtual mass of the seed point s at the current time step , generates the virtual mass of the seed point s at the next time step ; like , then the position information and flow field information of the new particle Q are generated according to the position information of the seed point s; wherein, for Round down, for Round down.

8. A numerical simulation device for piston oil spray cooling process, characterized in that: The device comprises: Establishing a module for establishing a SPH calculation model for gas-liquid two-phase flow with a large density ratio; the continuity equation between similar fluids in the SPH calculation model for gas-liquid two-phase flow with a large density ratio adopts a simplified delta term smoothing pressure field; the momentum equation in the SPH calculation model introduces an artificial viscosity term; the connection between the density term and the pressure term in the SPH calculation model for gas-liquid two-phase flow with a large density ratio is explicitly established by adopting a weak compressibility assumption and introducing the Tait state equation and background pressure term; The establishment module is also used to establish a heat flux boundary heat transfer calculation model; The establishment module is further used to establish a calculation model of the pressure-type outlet boundary, wherein the calculation model of the pressure-type outlet boundary includes the establishment of a buffer area, the arrangement of seed points, the introduction and update of virtual mass, the generation of particle mechanisms, and the deletion of particle mechanisms; the seed points are used to characterize the template of the generated particles; A numerical simulation module, used for numerically simulating the piston oil injection cooling process based on the large density ratio gas-liquid two-phase flow SPH calculation model, the heat flux boundary heat transfer calculation model, and the pressure type outlet boundary calculation model; The momentum equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow includes: The momentum equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow is expressed by the following formula for the discretization of the pressure gradient term: ； in, is the pressure gradient term of the ith particle, is the pressure of the ith particle, is the first The pressure of a particle; is the density of the ith particle, is the density of the jth particle searched in the computational domain, is the Hamiltonian operator of the ith particle, is the kernel function between the ith particle and the jth particle searched in the computational domain, is the volume of the jth particle searched in the computational domain, is the physical property relationship between the i-th particle and the j-th particle searched in the computational domain; The momentum equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow is expressed by the following formula for the velocity change: ； in, is the change in velocity of the ith particle caused by the artificial viscosity term, is a constant, is the velocity vector difference between the velocity vector of the i-th particle and the velocity vector of the j-th particle searched in the computational domain, is the arithmetic mean of the smooth length of the i-th particle and the smooth length of the j-th particle searched in the computational domain, is the arithmetic mean of the sound velocity of the ith particle and the sound velocity of the jth particle searched in the computational domain, is the distance vector difference between the i-th particle and the j-th particle searched in the computational domain.

9. The numerical simulation device for piston oil spray cooling process according to claim 8, characterized in that: The establishment module is specifically used to establish the continuity equation in the large density ratio gas-liquid two-phase flow SPH calculation model through the following steps: By introducing the multiphase delta term, the continuity equation in the SPH calculation model of the large density ratio gas-liquid two-phase flow is obtained: ； ； in, is the dissipation term of the density of the ith particle, is a constant, is the arithmetic mean of the smooth length of the i-th particle and the smooth length of the j-th particle searched in the computational domain, is the arithmetic mean of the sound velocity of the ith particle and the sound velocity of the jth particle searched in the computational domain, is the distance vector difference between the i-th particle and the j-th particle searched in the computational domain, is the Hamiltonian operator of the ith particle, is the kernel function between the ith particle and the jth particle searched in the computational domain, is the volume of the jth particle searched in the computational domain, is the density of the ith particle, is the density of the jth particle searched in the computational domain, is the physical property relationship between the ith particle and the jth particle searched in the computational domain, is the physical property of the ith particle, is the physical property of the jth particle searched in the computational domain.

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