CN113591345B - Explosion reaction flow high-precision prediction method based on generalized Riemann solver - Google Patents
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Abstract
The invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, and belongs to the technical field of explosion mechanics. The invention is realized by combining a two-step four-step Lax-Wendroff method and a conservation interface method. The two-step four-order Lax-Wendroff method combines the simplicity characteristic of the Runge-Kutta algorithm and the space-time coupling property of a generalized Riemann solver, can realize high-precision construction of numerical flux, and can reduce the time pushing step of the four-order precision algorithm by half. The conservation interface approach was developed based on a combined algorithm of the level set approach and the virtual fluid approach. The level set method can rapidly and accurately process the topology change of the interface and accurately capture the evolution process of the multi-medium interface; the virtual fluid method can skillfully convert the multi-medium problem into a single-medium problem, and effectively inhibit the non-physical oscillation of the multi-medium interface problem; the conservation interface method can improve the problem of non-conservation of the multi-medium interface and improve the reliability of the prediction result.
Description
Technical Field
The invention relates to a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, and belongs to the technical field of explosion mechanics.
Background
Explosion is accompanied by severe chemical reaction, and relates to hydrodynamic phenomena of extreme conditions such as high speed, high temperature, high pressure and the like, and relates to a process of mutual coupling of various media such as gas, liquid, solid and the like. The research of explosion phenomenon plays an important role in designing and damage evaluation of weapon and ammunition in the national defense industry field, disaster prevention and reduction of explosion accidents in the industry field, and the like. The experiment is an effective method for researching explosion phenomenon, but has the defects of high risk, high cost, high equipment requirement and the like. Therefore, with the rapid development of computer science, particularly after the massive parallel processing system is widely applied, the numerical prediction method with the advantages of safety, economy, easy operation and the like has been developed as an important means for researching explosion phenomenon.
For fluid problems involving multiple media coupling to each other, the location of the multi-media interface can be determined by a kinematic interface tracking technique. The Front tracking method can accurately capture the position of the interface, but the calculation process is very complicated. The VOF method can ensure conservation in the process of processing the topology change of the interface, but is difficult to realize high-order precision. The level set method not only can accurately describe the topological change process of the interface, but also can rapidly realize high-order precision, so that the level set method is widely applied to various fields such as fluid calculation, image processing, computer vision and the like.
The level set approach is often combined with the virtual fluid approach to transform the multi-media problem into a single-media problem through the construction of the virtual fluid. The level set/virtual fluid method can effectively suppress non-physical oscillation of the predicted result, but has the significant defect of lack of conservation. The conservation interface method improves the traditional level set/virtual fluid method, the level set method is used for tracking the multi-medium interface, and the problem of non-conservation can be effectively solved and the reliability of the prediction result can be improved through the modified finite volume method discrete control equation.
Solving the numerical flux is a key step of the finite volume method, and the Generalized Riemann Problem (GRP) method is used as a second-order expansion format of the Godunov method, and has strong space-time coupling. The basic idea is to use a piecewise linear function to represent the solution of the computational element boundary, and to construct the numerical flux by solving the generalized Riemann problem. The two-step four-step Lax-Wendroff method based on the generalized Riemann solver fuses the simplicity characteristic of the Runge-Kutta algorithm and the space-time coupling property of the generalized Riemann solver, can realize four-step precision by advancing two steps in the time direction, can effectively improve the calculation efficiency and the compactness, and is mainly applied to the problem of single-medium fluid movement.
Disclosure of Invention
Aiming at the problems of lower calculation efficiency, conservation loss and the like in the explosion reaction flow prediction method in the prior art, the invention discloses a generalized Riemann solver-based explosion reaction flow high-precision prediction method, which aims at solving the technical problems that: the high-precision numerical flux construction method based on the generalized Riemann solver is realized, and the calculation efficiency of the high-precision prediction method is improved; meanwhile, the problem of non conservation is effectively solved, the detonation and the detonation process are accurately predicted, and the technical problem of related engineering in the field of explosion mechanics is further solved. Related engineering technical problems in the field of explosion mechanics include weapon ammunition design and damage assessment and disaster prevention and reduction of explosion accidents.
The invention aims at realizing the following technical scheme:
the invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, which is realized by combining a two-step four-step Lax-Wendroff method and a conservation interface method. And solving the topology change of the detonation interface and the deflagration interface by using a level set method, and rapidly and accurately tracking the interface evolution process. The virtual fluid method is utilized to construct virtual fluid in the detonation interface and a computing unit near the detonation interface, so that the multi-medium problem is converted into a single-medium problem. And constructing a conservation type discrete format of the explosion reaction flow by utilizing a conservation interface method, and constructing a conservation amount exchange term along the interface by solving the Riemann problem containing detonation waves and deflagration waves. And combining a two-step four-step Lax-Wendroff method and a conservation interface method to construct a high-precision numerical flux solving method based on a generalized Riemann solver. The method can improve the calculation efficiency of the high-precision prediction method of the explosion reaction flow; meanwhile, the problem of non-conservation of a multi-medium interface is effectively solved, and accurate prediction of detonation and deflagration processes of compressible reaction fluid is realized.
The invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, which comprises the following steps:
step 1: establishing a rectangular coordinate system, defining the space step length of the x direction and the y direction, and setting (M+1) ×N+1 calculation units in a calculation area, namely units (i, j), wherein i=0, 1, …, M, j=0, 1, … and N.
Step 2: the level set function of all the calculation units in the calculation area at the initial time and the fluid physical quantities of the unburned gas and the burned gas are defined according to the initial conditions, including density, movement speed in the x direction, movement speed in the y direction and pressure.
In the level set method, the level set function is a sign distance function, denoted by phi. Let the regions phi > 0 and phi < 0 represent the distribution regions of the two fluids of unburned gas and burned gas, respectively, and the position phi=0 represents the detonation or deflagration interface. At the initial time, the value of the level set function φ is defined using the following formula:
wherein Γ is 0 A distribution curve representing the initial time interface, d (x, y) representing the calculation unit center (x, y) to the initial interface Γ 0 Is omega 1 、Ω 2 The areas where unburned gas and burned gas are distributed are shown, respectively.
According to the initial conditions, the density of unburned gas and burnt gas on both sides of the initial interface, the movement speed in the x direction, the movement speed in the y direction and the pressure are defined.
Step 3: a control equation set of compressible reactive fluid is established and boundary conditions are set.
Step 3.1: a system of control equations is established for the compressible reactive fluid.
The CJ model is applied to build a system of control equations for compressible fluids, i.e., regardless of chemical reactivity, detonation and deflagration interfaces are considered as jump discontinuities, chemical reactants are immediately converted to products along the interfaces, and heat is released. The unburned gas and burned gas satisfy the euler equation:
where ρ, u, v, p, E represent the density, the movement speed in x-direction, the movement speed in y-direction, the pressure, and the total energy per unit mass of gas, respectively. The sum of internal and kinetic energy is always expressed, i.eWhere e represents the internal energy per unit mass of gas. The equation of state of the unburned gas and the burnt gas satisfies:
p＝(γ-1)ρ(e-q), (2)
where γ represents the specific heat ratio and q represents the heat released by the chemical reaction.
Step 3.2: a control equation set boundary condition for the compressible reactive fluid is set.
In the construction of high-order precision numerical fluxes, physical quantities to surrounding computing units are required. Therefore, in order to update the constancy around the boundary of the computation area, the computation area needs to be extended to three layers.
Left-right boundary:
for the incoming flow boundary condition, the conservation amount of the continuation unit is set as:
where i=1, 2,3, j=0, 1,2, …, N, u= (ρ, ρu, ρv, ρe) T 。U -1,j 、U -2,j 、U -3,j Representing the first, second and third layer constancy of the left boundary of the calculation region extending to the left, U M+1,j 、U M+2,j 、U M+3,j Representing the first, second and third layer constancy of the right boundary of the calculated area extending to the right,the specific numerical value of (3) is required to be set according to the actual working condition.
For the outflow boundary condition, the conservation amount of the continuation unit is set as:
where i=1, 2,3, j=0, 1,2, …, N. That is, the conservation amount of the layer closest to the left boundary or the right boundary in the calculation region is directly assigned to the conservation amount of the continuation unit.
For the fixed wall boundary condition, the conservation amount of the extension unit is set as follows:
where i=1, 2,3, j=0, 1,2, …, N. I.e. calculating the mass, y-direction momentum of the first, second and third layers closest to the left or right boundary in the region and always assigning corresponding physical quantities to the first, second and third layers extending outwards respectively; and assigning opposite numbers of x-direction momentums of the first, second and third layers closest to the left boundary or the right boundary in the calculated region to corresponding physical quantities of the first, second and third layers extending outward, respectively.
Upper and lower boundaries:
for the incoming flow boundary condition, the conservation amount of the continuation unit is set as:
where i=0, 1,2, …, M, j=1, 2,3.U (U) i,N+1 、U i,N+2 、U i,N+3 Representing the first, second and third layer constancy of the upper boundary extension of the calculation region, U i,-1 、U i,-2 、U i,-3 Representing the first, second and third layer constancy of the downward continuation of the lower boundary of the computation area,the specific numerical value of (3) is required to be set according to the actual working condition.
For the outflow boundary condition, the conservation amount of the continuation unit is set as:
where i=0, 1,2, …, M, j=1, 2,3. That is, the conservation amount of the layer closest to the upper boundary or the lower boundary in the calculation region is directly assigned to the conservation amount of the continuation unit.
For a fixed wall boundary, the conservation amount of the continuation unit is set as follows:
where i=0, 1,2, …, M, j=1, 2,3. I.e. calculating the mass, x-direction momentum of the first, second and third layers closest to the upper or lower boundary in the region and always assigning corresponding physical quantities to the first, second and third layers extending outwards respectively; and assigning opposite numbers of y-direction momentums of the first, second and third layers closest to the upper or lower boundary in the calculated region to corresponding physical quantities of the first, second and third layers, respectively, extending outward.
Step 4: a time step is calculated to update the unburned and burned gas constancy.
Calculating a time step that satisfies the CFL condition:
wherein CFL represents CFL coefficient, and the value range is (0, 1); Δx and Δy represent the spatial step sizes in the x-direction and y-direction, respectively; u (u) i,j And v i,j Representing the velocity of movement of the fluid in the x-direction and the y-direction, respectively, within the calculation unit (i, j); c i,j Representing the speed of sound of the fluid in the computing unit (i, j).
Step 5: and updating the level set function according to the advection equation, and re-initializing to solve the evolution process of the detonation or deflagration interface.
Updating the level set function according to the advection equation shown in the following formula:
φ t +μφ x +νφ y ＝0, (4)
where μ and ν represent the velocity of motion of the level set function along the x-direction and y-direction, respectively, and are defined by the following system of equations:
wherein D represents the velocity of movement of the detonation or deflagration interface, N x And N y Representing the components of the unit normal direction vector N along the x-direction and the y-direction, respectively.
Performing time dispersion on equation (4) by using a third-order TVD (transient voltage direct-current) range-Kutta method, and performing partial derivative phi on level set function in x direction and y direction x And phi y It is required to obtain the modified Godunov method. The spatial derivative phi can be found by x And phi y ：
Wherein the method comprises the steps ofS (φ) represents a sign function of the level set function φ.
Wherein the method comprises the steps of
To improve the calculation efficiency, the level set function needs to be updated by equation (4) only in one to four layers of calculation units near the interface.
In order for the level set function to maintain the properties of the distance function, it is necessary to reinitialize by the following equation:
step 6: the geometrical parameters of the detonation or deflagration interface are defined according to the values of the level set function.
Let Γ denote the detonation or deflagration interface,the proportions of the left, right, lower and upper boundaries of the calculation unit (i, j) cut by the interface Γ are represented, respectively, and specific values can be obtained by operator calculation of the level set function. For fluids with phi > 0, A is defined by the following equation:
wherein the method comprises the steps of
For fluids with phi < 0, the interfacial cut ratio A passes through equation A - ＝1-A + The result is obtained, wherein superscript-sum+ respectively represents the interface cutting proportion geometric parameters corresponding to the fluid with phi < 0 and phi > 0.
The volume fraction α of each fluid can also be calculated by the operator of the level set function, α being defined for a fluid with a value of φ > 0 by the following equation:
wherein the method comprises the steps of
For fluids with phi < 0, the volume fraction alpha passes through the equation alpha - ＝1-α + The result is obtained, wherein the superscript-and+ represent phi < 0 and phi, respectivelyFluid corresponding to a volume fraction geometry of > 0.
Step 7: the virtual fluid is constructed according to a continuation equation within the computing unit near the interface.
Constructing a physical quantity state of the virtual fluid according to the extension equation:
V τ ±N·▽V＝0, (12)
wherein V represents physical quantity to be extended, including density, movement speed in x direction, movement speed in y direction and pressure; n represents a unit normal direction vector along the interface, represented by the equationObtaining; τ represents the manual time step, defined as: />In equation (12), the +sign is used to calculate the physical quantity of fluid with phi > 0 in the virtual cell; the number is used to calculate the physical quantity of the fluid with phi < 0 in the virtual unit.
Performing time dispersion on equation (12) by using a third-order TVD (transient voltage direct-current) range-Kutta method, and calculating a spatial derivative by using a windward formatAnd +.>
Step 8: in a computing unit near the detonation or deflagration interface, a Riemann problem containing the detonation wave or deflagration wave is constructed and solved, and the physical quantity states of the two sides of the detonation wave or deflagration wave are obtained through computation, wherein the physical quantity states comprise density, movement speed, pressure and specific internal energy.
In a computing unit near the interface, initial conditions of the Riemann problem are constructed along the method direction according to the states of the real fluid and the virtual fluid. According to the mass, momentum and energy conservation equations, the detonation wave or the wave front and wave back fluid of the detonation wave satisfy the following relation:
where subscripts 0 and 1 represent the corresponding physical quantities of the unburnt gas and the post-combusted gas, respectively, before the detonation wave or deflagration wave, and w represents the velocity of movement of the fluid relative to the detonation wave or deflagration wave. The fluid states of the shock wave front and the shock wave back need to meet the shock wave relation; the fluid states of the sparse wave front and the sparse wave back need to satisfy the Riemann invariant as a constant.
When CJ detonation occurs, rimand solution consists of non-reactive waves (shock waves or sparse waves), contact discontinuities, sparse waves, and CJ detonation waves; when strong detonation occurs, the Riemann solution consists of non-reactive waves, contact discontinuities, and strong detonation waves. When deflagration occurs, riemann solution consists of non-reactive waves, contact discontinuities, deflagration waves and non-reactive waves. According to shock wave, sparse wave, contact discontinuity and detonation wave or deflagration wave relation, solving the physical quantity states of two sides of each wave system, including density, movement speed, pressure and specific internal energy.
The physical quantity states of the two sides of each wave system in the Riemann solution of the detonation wave or the deflagration wave are solved through an iteration method, and a dichotomy is preferably selected, so that the dichotomy solving process is complex, but has strong robustness.
Step 9: based on the detonation wave and the Riemann solution of the detonation wave, a conservation amount exchange term of the unburned gas and the burned gas along the interface is constructed.
For detonation conditions, the detonation wave velocity is calculated from the Riemanne solution of the detonation wave:
wherein ρ is r And u r Respectively representing the density and the speed of the unburnt gas before detonation wave;and->Representing the density and velocity, respectively, of the burnt gas after detonation.
For unburned gas, the interface term is calculated by:
wherein ρ is 1 、u 1 、v 1 、p 1 、E 1 Respectively representing the density of the unburnt gas before detonation wave, the movement speed in the x direction, the movement speed in the y direction, the pressure and the total energy of the gas per unit mass;the length of the interface cut in the calculation unit (i, j) is represented by the following equation:
the interface term of the combusted gas is directly valued as the opposite number of interface terms of the unburnt gas, thereby ensuring that mass, momentum and energy conservation are met along the detonation interface.
The situation of detonation is slightly different from detonation, and since the Riemann solution containing the deflagration wave consists of four waves, the side edge of the deflagration wave is provided with one non-reactive wave, so that the deflagration problem can be solved under the condition that the wave speed of the deflagration wave is known.
For unburned gas, the interface term can be calculated by:
wherein ρ is 0 、u 0 、p 0 、E 0 Representing the density, fluid movement speed, pressure and total energy of the gas per unit mass of the unburnt gas before the deflagration wave in Riemann solution respectively; u (u) 1 、v 1 、u r The x-direction movement speed, the y-direction movement speed and the method direction movement speed of the unburned gas in the initial conditions of the Riemann problem are respectively represented; d represents the deflagration wave velocity.
The interface terms of the combusted gas are directly valued as the opposite number of interface terms of the unburned gas, thereby ensuring that mass, momentum and energy conservation are met along the deflagration interface.
Step 10: and constructing a conservation type discrete format of the control equation set according to the corrected finite volume method.
In the cutting unit (i, j), the finite volume dispersion of the unburned gas and the burned gas, respectively, is corrected to obtain a discrete form of the control equation set:
wherein the geometrical parameter alpha of the interface i,j 、Obtaining according to the step 6; conservation amount of fluid along interface exchange term X (Γ i,j ) Obtaining according to the steps 7-9; F. g represents the numerical flux of the computational cell boundary along the x-direction and the y-direction, respectively.
Step 11: the numerical flux high-precision construction of detonation and deflagration problems is realized by combining a two-step four-step Lax-Wendroff method and a conservation interface method, and the numerical flux high-precision construction is substituted into a modified finite volume discrete format to update the conservation quantities of unburned gas and burnt gas. The two-step four-step Lax-Wendroff method can reduce the time pushing step of a four-step precision algorithm by half, so that the calculation efficiency is remarkably improved, but the method is mainly applied to the problem of single-medium fluid movement. It is therefore desirable to apply the level set method and virtual fluid method, accurately capture the multi-media interface locations, and translate the multi-media problem into a single-media problem. The conservation interface approach was developed based on a combined algorithm of the level set approach and the virtual fluid approach. The level set method can rapidly and accurately process the topology change of the interface and accurately capture the evolution process of the multi-medium interface; the virtual fluid method can skillfully convert the multi-medium problem into a single-medium problem, thereby effectively inhibiting the non-physical oscillation of the multi-medium interface problem; by solving the Riemann problem containing detonation waves or deflagration waves, the conservation interface method can improve the non-conservation defect of the explosion reaction flow problem and improve the reliability of the prediction results of the detonation and deflagration processes. Therefore, the two-step four-step Lax-Wendroff method is combined with the conservation interface method, so that the high-precision prediction of the detonation and the deflagration process is realized, and the calculation efficiency of the high-precision prediction of the detonation and the deflagration process is remarkably improved.
Defining an area where the unburned gas is located by a level set method by utilizing the step 5; using step 7, the multi-media detonation and deflagration problems are converted by virtual fluid methods into single media problems with unburned and burned gases.
Step 11.1: based on average constancy in the computing unitAnd calculating the conservation amount at the cell boundary +.>And->Obtaining the time derivative of the variable by HWENO reconstruction technique and generalized Riemann solver>And
step 11.2: calculating the average conservation quantity after half a time stepCalculating the conservation amount at the cell boundaryAnd->The HWENO reconstruction technique and the generalized Riemann solver are applied again to obtain the time derivative ++>And->
Step 11.3: and constructing the numerical flux of the boundary of the calculation unit according to the time derivative of the numerical flux, substituting the numerical flux into a finite volume discrete format, and updating the unburnt gas and the conservation quantity of the burnt gas.
Step 12: judging the calculation time t of the nth step according to the time step delta t n Relation to termination time T, if T n +Deltat is less than or equal to T, T is n+1 ＝t n +Δt, returning to step 3, and updating the conservation amount U at the next time n+1 The method comprises the steps of carrying out a first treatment on the surface of the Otherwise Δt=t-T n Returning to the step 3, obtaining the conservation amount at the termination time, and calculating the mass, the movement speed in the x direction, the movement speed in the y direction and the pressure at the termination time according to a state equation, and ending the cycle.
And (3) outputting a level set function, a mass, a movement speed in the x direction, a movement speed in the y direction and a pressure of all the calculation units at the termination moment, predicting the position of a detonation or deflagration interface and the areas where unburned gas and burnt gas are located, and calculating the mass, the movement speed in the x direction, the movement speed in the y direction and the pressure of fluid in the areas.
Further comprising step 13: predicting detonation and deflagration problems by utilizing the steps 1 to 12, so that the calculation efficiency of the high-precision prediction method is improved; meanwhile, the problem of non conservation is effectively solved, reliable prediction of detonation and the detonation process is realized, and the technical problem of related engineering in the field of explosion mechanics is further solved.
Related engineering technical problems in the field of explosion mechanics include weapon ammunition design and damage assessment and disaster prevention and reduction of explosion accidents.
The beneficial effects are that:
1. the invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, which is characterized in that a two-step four-step Lax-Wendroff method based on the generalized Riemann solver is used for solving the numerical flux of a boundary of a computing unit, and the two-step four-step Lax-Wendroff method is combined with the simplicity characteristic of a Runge-Kutta algorithm and the space-time coupling property of the generalized Riemann solver, so that the prediction method can improve the computing efficiency and stability of the predicted detonation and deflagration process.
2. The invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, which is characterized in that a conservation interface method is applied to obtain a conservation type calculation format, a modified finite volume method discrete control equation is adopted, and a Riemann problem containing detonation waves or deflagration waves is solved to solve a conservation constant exchange term, so that the conservation of quality, momentum and energy along an interface can be realized. Therefore, the prediction method can overcome the defect that the traditional level set/virtual fluid method lacks conservation, and improve the accuracy of predicting detonation and deflagration processes.
3. The invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, which is used for tracking a detonation or deflagration interface by using a level set method, and the level set method can realize high-order precision efficiently and rapidly and can be simply and effectively expanded to a high-dimensional space. Therefore, the prediction method can quickly and accurately capture the topology change process of detonation or deflagration interface.
4. The invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, which is used for constructing the states of unburned gas and burnt gas in a virtual unit by using a virtual fluid method and can skillfully convert a multi-medium problem into a single-medium problem. Therefore, the invention can effectively restrain the non-physical oscillation near the interface of the multi-media.
Drawings
FIG. 1 is a schematic diagram of a modified finite volume method;
FIG. 2 is a schematic illustration of a virtual fluidic method;
FIG. 3 is a schematic diagram of a Riemann solution structure containing detonation waves (left view: CJ detonation, right view: strong detonation);
FIG. 4 is a schematic diagram of a Riemann solution structure including a deflagration wave;
FIG. 5 is a plot of Riemann solution profile (density, velocity, and pressure profile in sequence from left to right) along a deflagration interface at an initial time;
fig. 6 is a flowchart of a method for predicting the high precision of the explosion reaction flow based on the generalized Riemann solver according to the present embodiment.
Detailed Description
For a better description of the objects and advantages of the present invention, the present invention will be further described with reference to the accompanying drawings and examples.
Example 1:
the application of the embodiment in the double deflagration wave fusion example.
As shown in fig. 6, the embodiment discloses a method for predicting the explosion reaction flow with high precision based on a generalized riman solver, which comprises the following specific implementation steps:
step 1: the calculation area is set to (0, 2 m) × (0, 2 m), and the mesh size is set toI.e. 400 x 400 grids are arranged within the calculation area. The calculation units are denoted (i, j), where i=0, 1, …,399, j=0, 1, …,399.
Step 2: the level set function at the initial time is taken as:
the densities of the unburned gas and the burned gas, the movement speed in the x direction, the movement speed in the y direction, and the pressure at the initial time are taken as follows:
| ρ | u | v | p | |
| unburned gas | 1.0kg/m 3 | 0m/s | 0m/s | 1.0×10 5 Pa |
| Burned gas | 0.142168kg/m 3 | 0m/s | 0m/s | 9.45695×10 4 Pa |
Table 1
Step 3: both the unburned gas and the burned gas satisfy the euler equation, i.e., equation set (1). In order to close the control equation set, the state equations of the unburned gas and the burnt gas, that is, equation (2), are given. And the parameters of the state equation are taken as follows:
| γ | q | |
| unburned gas | 1.4 | 2.0×10 6 J/kg |
| Burned gas | 1.4 | 0 |
Table 2
The left and right boundaries and the upper and lower boundaries are set as fixed wall boundary conditions:
where i=1, 2,3, j=0, 1,2, …,399.
Where i=0, 1,2, …,399, j=1, 2,3.
Step 4: the time step is calculated according to equation (3), where the CFL coefficient is taken to be 0.4.
Then, step 5 to step 10 are performed, wherein the level set function updating process in step 5, the Riemann solution solving process in step 7 and the conservation quantity exchange term solving process in step 9 all need to know the propagation speed of the deflagration wave, and the deflagration wave speed is calculated according to the following formula:
where V, p, ρ are the method direction movement velocity, pressure and density of the unburned gas, respectively.
In step 8, the Riemann problem containing the deflagration wave is solved by applying a dichotomy iteration, and the density, the speed and the pressure of the unburnt gas before the deflagration wave in the Riemann solution at the initial moment can be calculated, wherein the density, the speed and the pressure are as follows:
| ρ 0 | u 0 | p 0 |
| 1.157kg/m 3 | 55.594m/s | 1.227×10 5 Pa |
TABLE 3
Fig. 5 is a plot of the initial time along the deflagration interface, with density, velocity and pressure profiles in sequence from left to right, showing that the Riemann solution consists of four waves, in sequence from left to right: shock, contact discontinuity, deflagration wave, and shock.
And then, step 11, solving the boundary value flux of the calculation unit by using a two-step four-step Lax-Wendroff method, substituting the calculated boundary value flux into a finite volume discrete format, and updating the unburnt gas and the conservation quantity of the burnt gas. The numerical flux high-precision construction of detonation and deflagration problems is realized by combining a two-step four-step Lax-Wendroff method and a conservation interface method, and the numerical flux high-precision construction is substituted into a modified finite volume discrete format to update the conservation quantities of unburned gas and burnt gas. The two-step four-step Lax-Wendroff method can reduce the time pushing step of a four-step precision algorithm by half, so that the calculation efficiency is remarkably improved, but the method is mainly applied to the problem of single-medium fluid movement. It is therefore desirable to apply the level set method and virtual fluid method, accurately capture the multi-media interface locations, and translate the multi-media problem into a single-media problem. The conservation interface approach was developed based on a combined algorithm of the level set approach and the virtual fluid approach. The level set method can rapidly and accurately process the topology change of the interface and accurately capture the evolution process of the multi-medium interface; the virtual fluid method can skillfully convert the multi-medium problem into a single-medium problem, thereby effectively inhibiting the non-physical oscillation of the multi-medium interface problem; by solving the Riemann problem containing detonation waves or deflagration waves, the conservation interface method can improve the non-conservation defect of the explosion reaction flow problem and improve the reliability of the prediction results of the detonation and deflagration processes. Therefore, the two-step four-step Lax-Wendroff method is combined with the conservation interface method, so that the high-precision prediction of the detonation and the deflagration process is realized, and the calculation efficiency of the high-precision prediction of the detonation and the deflagration process is remarkably improved.
Then, step 12 to step 13 are performed, and finally, the level set function, the mass, the movement speed in the x direction, the movement speed in the y direction and the pressure in the calculation region at different moments are output.
And (3) analysis of calculation results:
the double deflagration waves initially propagate outwards respectively, and as time progresses, the two bubbles gradually start to merge. The invention discloses a high-precision prediction method for explosion reaction flow based on a generalized Riemann solver, which combines a two-step four-step Lax-Wendroff method and a conservation interface method, can accurately describe the topology change of a deflagration interface, predicts the fluid motion process of unburned gas and burnt gas with high precision, and can effectively improve the calculation efficiency.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
Claims (10)
1. A high-precision prediction method for explosion reaction flow based on a generalized Riemann solver is characterized by comprising the following steps of: comprises the following steps of the method,
step 1: establishing a rectangular coordinate system, defining the space step length in the x direction and the y direction, and setting (M+1) ×N+1 calculation units in a calculation area, namely units (i, j), wherein i=0, 1, …, M, j=0, 1, … and N;
step 2: defining a level set function of all calculation units in the initial moment calculation area and fluid physical quantities of unburned gas and burned gas according to initial conditions, wherein the level set function comprises density, x-direction movement speed, y-direction movement speed and pressure;
step 3: establishing a control equation set of the compressible reaction fluid and setting boundary conditions;
step 4: calculating a time step for updating the conservation quantity of unburned gas and burned gas;
step 5: updating the level set function according to the advection equation, and re-initializing to solve the evolution process of detonation or deflagration interface;
step 6: defining geometrical parameters of detonation or deflagration interfaces according to the numerical value of the level set function;
step 7: constructing a virtual fluid in a computing unit near the interface according to a continuation equation;
step 8: constructing and solving a Riemann problem of detonation wave or deflagration wave in a computing unit near the detonation or deflagration interface, and computing to obtain physical quantity states of the detonation wave or deflagration wave on two sides, including density, movement speed, pressure and specific internal energy;
step 9: constructing conservation quantity exchange items of unburned gas and burnt gas along an interface according to the detonation wave and the Riemann solution of the detonation wave;
step 10: constructing a conservation type discrete format of a control equation set according to the corrected finite volume method;
step 11: the numerical flux high-precision construction of detonation and deflagration problems is realized by combining a two-step four-step Lax-Wendroff method and a conservation interface method, and the numerical flux high-precision construction is substituted into a corrected limited volume discrete format to update the conservation amounts of unburned gas and burnt gas; the two-step four-step Lax-Wendroff method can reduce the time advancing step of a four-step precision algorithm by half, so that the calculation efficiency is remarkably improved, but the method is mainly applied to the problem of single-medium fluid movement; therefore, a level set method and a virtual fluid method need to be applied to accurately capture the multi-media interface position and convert the multi-media problem into a single-media problem; the conservation interface method is developed based on a combined algorithm of a level set method and a virtual fluid method; the level set method can rapidly and accurately process the topology change of the interface and accurately capture the evolution process of the multi-medium interface; the virtual fluid method can skillfully convert the multi-medium problem into a single-medium problem, thereby effectively inhibiting the non-physical oscillation of the multi-medium interface problem; by solving the Riemann problem containing detonation waves or deflagration waves, the conservation interface method can improve the non-conservation defect of the explosion reaction flow problem and improve the reliability of the prediction results of the detonation and deflagration processes; therefore, a two-step four-step Lax-Wendroff method is combined with a conservation interface method, so that high-precision prediction of detonation and deflagration processes is realized, and meanwhile, the calculation efficiency of the high-precision prediction of detonation and deflagration processes is remarkably improved;
step 12: judging the calculation time t of the nth step according to the time step delta t n Relation to termination time T, if T n +Deltat is less than or equal to T, T is n+1 ＝t n +Δt, returning to step 3, and updating the conservation amount U at the next time n+1 The method comprises the steps of carrying out a first treatment on the surface of the Otherwise Δt=t-T n Returning to the step 3, obtaining the conservation amount at the termination time, and calculating the mass, the movement speed in the x direction, the movement speed in the y direction and the pressure at the termination time according to a state equation, wherein the cycle is ended;
and (3) outputting a level set function, a mass, a movement speed in the x direction, a movement speed in the y direction and a pressure of all the calculation units at the termination moment, predicting the position of a detonation or deflagration interface and the areas where unburned gas and burnt gas are located, and calculating the mass, the movement speed in the x direction, the movement speed in the y direction and the pressure of fluid in the areas.
2. The high-precision prediction method for the explosion reaction flow based on the generalized Riemann solver as claimed in claim 1, which is characterized by comprising the following steps of: further comprising step 13: predicting detonation and deflagration problems by utilizing the steps 1 to 12, so that the calculation efficiency of the high-precision prediction method is improved; meanwhile, the problem of non conservation is effectively solved, reliable prediction of detonation and the detonation process is realized, and the technical problem of related engineering in the field of explosion mechanics is further solved;
related engineering technical problems in the field of explosion mechanics include weapon ammunition design and damage assessment and disaster prevention and reduction of explosion accidents.
3. A method for predicting the blast reaction flow with high precision based on a generalized riman solver as claimed in claim 1 or 2, characterized in that: the implementation method of the step 2 is that,
in the level set method, the level set function is a symbol distance function, denoted by phi; let the areas phi > 0 and phi < 0 represent the distribution areas of the two fluids of unburned gas and burned gas, respectively, and the position phi=0 represents the detonation or deflagration interface; at the initial time, the value of the level set function φ is defined using the following formula:
wherein Γ is 0 A distribution curve representing the initial time interface, d (x, y) representing the calculation unit center (x, y) to the initial interface Γ 0 Is omega 1 、Ω 2 Respectively represent areas where unburned gas and burned gas are distributed;
according to the initial conditions, the density of unburned gas and burnt gas on both sides of the initial interface, the movement speed in the x direction, the movement speed in the y direction and the pressure are defined.
4. A method for predicting the flow of explosion reaction with high precision based on a generalized Riemann solver as claimed in claim 3, which is characterized in that: the implementation method of the step 3 is that,
step 3.1: establishing a control equation set of a compressible reaction fluid;
establishing a control equation set of the compressible fluid by using the CJ model, namely taking no chemical reaction rate into consideration, taking detonation and a deflagration interface as jump discontinuities, immediately converting chemical reactants into products along the interface, and releasing heat; the unburned gas and burned gas satisfy the euler equation:
wherein ρ, u, v, p, E represent the density, the movement speed in x-direction, the movement speed in y-direction, the pressure, and the total energy per unit mass of gas, respectively; the sum of internal and kinetic energy is always expressed, i.eWherein e represents the internal energy per unit mass of gas; the equation of state of the unburned gas and the burnt gas satisfies:
p＝(γ-1)ρ(e-q), (2)
wherein γ represents the specific heat ratio, q represents the heat released by the chemical reaction;
step 3.2: setting boundary conditions of a control equation set of the compressible reaction fluid;
in the construction process of high-order precision numerical flux, physical quantities to surrounding computing units are required to be used; therefore, in order to update the conservation quantity near the boundary of the calculation region, the calculation region needs to be extended to three layers;
left-right boundary:
for the incoming flow boundary condition, the conservation amount of the continuation unit is set as:
where i=1, 2,3, j=0, 1,2, …, N, u= (ρ, ρu, ρv, ρe) T ；U -1,j 、U -2,j 、U -3,j Representing the first, second and third layer constancy of the left boundary of the calculation region extending to the left, U M+1,j 、U M+2,j 、U M+3,j Representing the first, second and third layer constancy of the right boundary of the calculated area extending to the right,the specific numerical value of (2) is required to be set according to the actual working condition;
for the outflow boundary condition, the conservation amount of the continuation unit is set as:
wherein i=1, 2,3, j=0, 1,2, …, N; namely, the conservation quantity of the layer closest to the left boundary or the right boundary in the calculation region is directly assigned to the conservation quantity of the extension unit;
for the fixed wall boundary condition, the conservation amount of the extension unit is set as follows:
wherein i=1, 2,3, j=0, 1,2, …, N; i.e. calculating the mass, y-direction momentum of the first, second and third layers closest to the left or right boundary in the region and always assigning corresponding physical quantities to the first, second and third layers extending outwards respectively; and assigning opposite numbers of x-direction momentums of the first, second and third layers closest to the left boundary or the right boundary in the calculation region to corresponding physical quantities of the first, second and third layers extending outward, respectively;
upper and lower boundaries:
for the incoming flow boundary condition, the conservation amount of the continuation unit is set as:
where i=0, 1,2, …, M, j=1, 2,3; u (U) i,N+1 、U i,N+2 、U i,N+3 Representing the first, second and third layer constancy of the upper boundary extension of the calculation region, U i,-1 、U i,-2 、U i,-3 Representing the first, second and third layer constancy of the downward continuation of the lower boundary of the computation area,the specific numerical value of (2) is required to be set according to the actual working condition;
for the outflow boundary condition, the conservation amount of the continuation unit is set as:
where i=0, 1,2, …, M, j=1, 2,3; namely, the conservation amount of the layer closest to the upper boundary or the lower boundary in the calculation region is directly assigned to the conservation amount of the extension unit;
for a fixed wall boundary, the conservation amount of the continuation unit is set as follows:
where i=0, 1,2, …, M, j=1, 2,3; i.e. calculating the mass, x-direction momentum of the first, second and third layers closest to the upper or lower boundary in the region and always assigning corresponding physical quantities to the first, second and third layers extending outwards respectively; and assigning opposite numbers of y-direction momentums of the first, second and third layers closest to the upper or lower boundary in the calculated region to corresponding physical quantities of the first, second and third layers, respectively, extending outward.
5. The high-precision prediction method for the explosion reaction flow based on the generalized Riemann solver as set forth in claim 4, which is characterized in that: the implementation method of the step 4 is that,
calculating a time step that satisfies the CFL condition:
wherein CFL represents CFL coefficient, and the value range is (0, 1); Δx and Δy represent the spatial step sizes in the x-direction and y-direction, respectively; u (u) i,j And v i,j Representing the velocity of movement of the fluid in the x-direction and the y-direction, respectively, within the calculation unit (i, j); c i,j Representing the speed of sound of the fluid in the computing unit (i, j).
6. The high-precision prediction method for the explosion reaction flow based on the generalized Riemann solver as set forth in claim 5, which is characterized in that: the implementation method of the step 5 is that,
updating the level set function according to the advection equation shown in the following formula:
φ t +μφ x +νφ y ＝0, (4)
where μ and ν represent the velocity of motion of the level set function along the x-direction and y-direction, respectively, and are defined by the following system of equations:
wherein D represents the velocity of movement of the detonation or deflagration interface, N x And N y Components of the unit normal direction vector N along the x-direction and the y-direction, respectively;
performing time dispersion on equation (4) by using a third-order TVD (transient voltage direct-current) range-Kutta method, and performing partial derivative phi on level set function in x direction and y direction x And phi y The method needs to be calculated according to a corrected Godunov method; the spatial derivative phi can be found by x And phi y ：
Wherein the method comprises the steps ofS (phi) represents a sign function of the level set function phi;
wherein the method comprises the steps of
In order to improve the calculation efficiency, only the level set function needs to be updated through an equation (4) in one to four layers of calculation units near the interface;
in order for the level set function to maintain the properties of the distance function, it is necessary to reinitialize by the following equation:
7. the high-precision prediction method for the explosion reaction flow based on the generalized Riemann solver as set forth in claim 6, which is characterized in that: the implementation method of the step 6 is that,
let Γ denote the detonation or deflagration interface,the proportions of the left boundary, the right boundary, the lower boundary and the upper boundary of the computing unit (i, j) cut by the interface Γ are respectively represented, and specific numerical values can be obtained through the operator operation computation of the level set function; for fluids with phi > 0, A is defined by the following equation:
wherein the method comprises the steps of
For fluids with phi < 0, the interfacial cut ratio A passes through equation A - ＝1-A + Obtaining, wherein superscript-sum+ respectively represents interface cutting proportion geometric parameters corresponding to fluids with phi < 0 and phi > 0;
the volume fraction α of each fluid can also be calculated by the operator of the level set function, α being defined for a fluid with a value of φ > 0 by the following equation:
wherein the method comprises the steps of
For fluids with phi < 0, the volume fraction alpha passes through the equation alpha - ＝1-α + The values are obtained, wherein the superscript-sum+ represents the geometric parameters of the volume fractions corresponding to the fluids phi < 0 and phi > 0, respectively.
8. The high-precision prediction method for the explosion reaction flow based on the generalized Riemann solver as set forth in claim 7, which is characterized in that: the implementation method of the step 7 is that,
constructing a physical quantity state of the virtual fluid according to the extension equation:
wherein V represents physical quantity to be extended, including density, movement speed in x direction, movement speed in y direction and pressure; n represents a unit normal direction vector along the interface, represented by the equationObtaining; τ represents the manual time step, defined as:in equation (12), the +sign is used to calculate the physical quantity of fluid with phi > 0 in the virtual cell; -number for calculating the physical quantity of the fluid with phi < 0 in the virtual unit;
performing time dispersion on equation (12) by using a third-order TVD (transient voltage direct-current) range-Kutta method, and calculating a spatial derivative by using a windward formatAnd +.>
The implementation method of the step 8 is that,
constructing initial conditions of the Riemann problem along the method direction in a computing unit near the interface according to the states of the real fluid and the virtual fluid; according to the mass, momentum and energy conservation equations, the detonation wave or the wave front and wave back fluid of the detonation wave satisfy the following relation:
wherein subscripts 0 and 1 represent the corresponding physical quantities of the detonation wave or the pre-unburned gas and the post-combusted gas, respectively, and w represents the movement speed of the fluid relative to the detonation wave or the deflagration wave; the fluid states of the shock wave front and the shock wave back need to meet the shock wave relation; the fluid states of the sparse wave front and the sparse wave back need to meet the Riemann invariant as a constant;
when CJ detonation occurs, riemann solution consists of non-reactive waves, contact discontinuities, sparse waves and CJ detonation waves; when strong detonation occurs, riemann solution consists of non-reactive waves, contact discontinuities and strong detonation waves; when deflagration occurs, riemann solution consists of non-reactive waves, contact discontinuities, deflagration waves and non-reactive waves; according to shock waves, sparse waves, contact discontinuities and detonation waves or deflagration wave relation formulas, solving physical quantity states of two sides of each wave system, wherein the physical quantity states comprise density, movement speed, pressure and specific energy;
the implementation method of the step 9 is that,
for detonation conditions, the detonation wave velocity is calculated from the Riemanne solution of the detonation wave:
wherein ρ is r And u r Respectively representing the density and the speed of the unburnt gas before detonation wave;and->Respectively representing the density and the speed of the burned gas after detonation wave;
for unburned gas, the interface term is calculated by:
wherein ρ is 1 、u 1 、v 1 、p 1 、E 1 Respectively representing the density of the unburnt gas before detonation wave, the movement speed in the x direction, the movement speed in the y direction, the pressure and the total energy of the gas per unit mass;the length of the interface cut in the calculation unit (i, j) is represented by the following equation:
the interface item of the burnt gas is directly valued as the opposite number of the interface item of the unburnt gas, thereby ensuring that mass, momentum and energy conservation are satisfied along the detonation interface;
the situation of detonation is slightly different from detonation, and because the Riemann solution containing the deflagration wave consists of four waves, the side edge of the deflagration wave is provided with one more nonreactive wave, so that the deflagration problem can be solved under the condition that the wave speed of the deflagration wave is known;
for unburned gas, the interface term is calculated by:
wherein ρ is 0 、u 0 、p 0 、E 0 Representing the density, fluid movement speed, pressure and total energy of the gas per unit mass of the unburnt gas before the deflagration wave in Riemann solution respectively; u (u) 1 、v 1 、u r The x-direction movement speed, the y-direction movement speed and the method direction movement speed of the unburned gas in the initial conditions of the Riemann problem are respectively represented; d represents the deflagration wave velocity;
the interface term of the burnt gas is directly valued as the opposite number of the interface term of the unburnt gas, thereby ensuring that mass, momentum and energy conservation are satisfied along the deflagration interface;
the step 10 is implemented by the method that,
in the cutting unit (i, j), the finite volume dispersion of the unburned gas and the burned gas, respectively, is corrected to obtain a discrete form of the control equation set:
wherein the geometrical parameter alpha of the interface i,j 、Obtaining according to the step 6; conservation amount of fluid along interface exchange term X (Γ i,j ) Obtaining according to the steps 7-9; F. g represents the numerical flux of the computational cell boundary along the x-direction and the y-direction, respectively.
9. The high-precision prediction method for the explosion reaction flow based on the generalized Riemann solver as claimed in claim 8, which is characterized in that: the implementation method of the step 11 is that,
defining an area where the unburned gas is located by a level set method by utilizing the step 5; converting the multi-media detonation and deflagration problems into single-media problems with respect to unburned and burned gases by virtual fluid methods using step 7;
step 11.1: based on average constancy in the computing unitAnd calculating the conservation amount at the cell boundary +.>Andobtaining the time derivative of the variable by HWENO reconstruction technique and generalized Riemann solver>And->
Step 11.2: calculating the average conservation quantity after half a time stepAnd calculating the conservation amount at the cell boundary +.>And->The HWENO reconstruction technique and the generalized Riemann solver are applied again to obtain the time derivative ++>And
step 11.3: constructing the numerical flux of the boundary of the calculation unit according to the time derivative of the numerical flux, substituting the numerical flux into a finite volume discrete format, and updating the unburned gas and the conservation quantity of the burned gas;
10. the high-precision prediction method for the explosion reaction flow based on the generalized Riemann solver as claimed in claim 9, which is characterized in that: the physical quantity states of the two sides of each wave system in the Riemann solution of the detonation wave or the deflagration wave are solved through an iteration method, and a dichotomy is selected, so that the dichotomy solving process is complex, but has strong robustness.
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