CN116415518A - WENO-based high-resolution high-precision shock wave intermittent capturing method - Google Patents

WENO-based high-resolution high-precision shock wave intermittent capturing method Download PDF

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CN116415518A
CN116415518A CN202310216211.8A CN202310216211A CN116415518A CN 116415518 A CN116415518 A CN 116415518A CN 202310216211 A CN202310216211 A CN 202310216211A CN 116415518 A CN116415518 A CN 116415518A
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许香照
宁建国
苏璇
任会兰
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Abstract

The invention discloses a WENO-based high-resolution high-precision shock wave intermittent capturing method, and belongs to the field of explosion mechanics. The implementation method of the invention comprises the following steps: introducing a local smoothness measurement factor with compact structure into the construction of the WENO format nonlinear weight; in the WENO format construction process, the smoothness information of all the sub templates is coupled into the whole calculation template through the global smoothness measurement factor, so that the quantitative representation of the smoothness of the whole template is realized; the WENO format can adaptively regulate and control the nonlinear weight of the template according to the flow field change by constructing an adaptive regulation and control coefficient, and nonlinear convex combination is carried out on the flux of the sub-template of the WENO format by adopting the nonlinear weight. The invention has the advantages of high precision, high resolution, low dissipation and high stability, and can improve the precision of tracking and identifying the shock wave discontinuities in the explosion impact, thereby realizing the numerical prediction of complex explosion impact problems comprising interaction of high-density ratio and high-pressure ratio strong shock waves and Jiang Xishu waves.

Description

WENO-based high-resolution high-precision shock wave intermittent capturing method
Technical Field
The invention relates to a WENO-based high-resolution high-precision shock wave intermittent capturing method, and belongs to the field of explosion mechanics.
Background
The explosion impact problem is widely applied in the fields of national defense and military industry, transportation industry, mine industry, civil construction engineering, machining and other production safety fields, and is highly valued in various fields. An important feature of the blast shock problem is the generation of high mach number blast products, jets, or extremely strong detonation waves, thus facing the complex problem of solving the high density ratio, high pressure ratio, strong shock waves interacting with Jiang Xishu waves. How to solve the problems of propagation of explosion shock waves and mutual collision with high efficiency and high resolution becomes a key point of research. The shock wave discontinuities which accurately track the propagation of high-intensity shock waves and the mutual collision process are the hot spot problems focused by current domestic and foreign scholars, and the shock wave discontinuity capturing method with high resolution and high precision is an effective way for realizing the accurate numerical simulation of the explosion impact process, and has practical production safety requirements, important scientific research and academic significance and engineering application value.
The high-resolution shock wave capturing format has important significance for numerical simulation of the flow field containing the impact discontinuity, not only can the scale of the grid be reduced, but also the complex wave system structure in the flow field can be well resolved. The weighted substantially non-oscillating (WENO) algorithm has gained popularity in recent years, being capable of both sharply capturing shock discontinuities and substantially not producing numerical oscillations, and being an ideal solution for flow fields containing shock discontinuities and complex features. However, along with the modern development of the application field and the enrichment of application scenes, higher requirements are put forward on the resolution of the logarithmic calculation format, and more sharp capturing of shock waves and clear resolution of complex microstructure after shock wave collision are required, so that more accurate numerical simulation of the explosion impact problem is realized. The traditional WENO format has large dissipation at shock wave discontinuous parts and precision reduction at high-order extreme points of smooth areas, so that shock wave discontinuities are smoothed, and complex vortex and fine bubble structures after shock wave collision cannot be accurately identified. In order to solve the problem, the invention provides a WENO-based high-resolution high-precision shock wave discontinuous capture method which can track shock wave discontinuous with high resolution so as to solve the complex problem of interaction between strong shock waves with high density ratio and high pressure ratio and Jiang Xishu waves generated by explosion shock wave collision.
Disclosure of Invention
Aiming at the problems that the existing numerical calculation method for capturing shock wave interruption has low calculation efficiency and large dissipation of shock wave interruption, the precision of the shock wave interruption is reduced at a high-order extreme point of a smooth area, and complex flow field structures such as vortexes generated by interaction of strong shock waves with high density ratio and high pressure ratio and Jiang Xishu waves are difficult to simulate in a high resolution mode, the invention mainly aims to provide a WENO format-based high-resolution high-precision shock wave interruption capturing method which comprises the following steps: the method has the advantages of realizing accurate solving of the shock wave interruption problem, improving the prediction precision of shock wave propagation, evolution and collision processes, improving the precision of the numerical calculation result of the complex flow field of the explosion impact, and further solving the engineering technical problems related to the explosion impact field. The invention has the advantages of high precision, high resolution, low dissipation and high stability.
The aim of the invention is achieved by the following technical scheme.
According to the WENO-based high-resolution high-precision shock wave intermittent capturing method disclosed by the invention, a local smoothness measurement factor with a compact structure is introduced into the structure of the nonlinear weight of the WENO format to quantitatively characterize the smoothness degree of the WENO format sub-template, so that the WENO format is easy to expand to a high order, and the capturing efficiency of the WENO format on the shock wave intermittent is improved on the premise of ensuring the capturing precision to be unchanged; in the construction process of the WENO format, the smoothness information of all the sub templates is coupled into the whole calculation template through a global smoothness measurement factor, the global smoothness measurement factor fully utilizes flow field information, the resolution of the WENO format is improved, and the solving precision of the WENO format at a high-order extreme point is further improved, so that the WENO format is more suitable for the accurate numerical simulation of complex flow field problems such as interaction of strong shock waves containing high density ratio and high pressure ratio and Jiang Xishu waves in explosion impact; in the under-smooth template regulation and control item of the WENO format nonlinear weight, the WENO format can adaptively regulate and control the nonlinear weight of the template according to the flow field change by constructing an adaptive regulation and control coefficient, nonlinear convex combination is carried out on the WENO format sub-template flux by adopting the nonlinear weight, the WENO format dissipation is reduced, the WENO format resolution is improved, false numerical oscillation is restrained, the numerical calculation is ensured to be carried out stably, and therefore sharp tracking capture of shock waves and accurate identification of curls and fine complex structures near contact discontinuities after shock wave collision are realized. The WENO-based high-resolution high-precision shock wave intermittent capturing method disclosed by the invention has the excellent characteristics of high precision, high resolution, low dissipation and high stability, and can improve the precision of tracking and identifying shock wave discontinuities in explosion impact, so that the numerical simulation of complex explosion impact problems including interaction of high-density ratio and high-pressure ratio strong shock waves and Jiang Xishu waves is realized, and the problems of propagation, evolution and collision of shock waves in the explosion impact process in the national defense and military field and the civil production safety field are accurately predicted.
The national defense and military field comprises high-speed/ultra-high-speed warhead structure optimization, aerial explosion, near-ground explosion, underwater explosion, ultra-high-speed collision and energy-gathering jet flow; the civil production safety field comprises transportation industry, mine industry, civil construction engineering and mechanical processing.
The invention discloses a WENO-based high-resolution high-precision shock wave intermittent capturing method, which comprises the following steps:
in a Cartesian coordinate system, a numerical simulation model for a compressible flow field is established for the problem of explosion impact to be simulated, and the numerical simulation model is initialized and set. The initialization setting comprises determining a calculation domain, performing grid division, setting initial flow field parameters and initializing grid physical quantities, wherein the initial flow field parameter settings comprise initial conditions and initial boundary condition settings.
Step two: from the CFL stability conditions, a time step Δt is determined.
The CFL stability condition is as shown in formula (1):
Figure BDA0004114996960000021
wherein CFL is a stability condition coefficient, |u c | max Is the maximum value of the speed of the full physical field under the current time step:
Figure BDA0004114996960000022
where c is the local sound velocity.
Step three: and updating the grid physical quantity in the calculation area according to the compressible flow field model equation in the calculation area divided by the explosion impact problem to be simulated. The compressible flow field model equations include a continuous equation, a momentum equation, and an energy equation. The physical quantities include: fluid density ρ, velocity u, v, pressure p. According to the calculated dimension of the explosion impact problem to be simulated, a one-dimensional Euler equation and a two-dimensional Euler equation are respectively adopted as compressible flow field model equations to update and calculate the grid physical quantity in the domain.
For the one-dimensional compressible flow problem, a one-dimensional Euler equation shown in formula (3) is adopted as a compressible flow field model equation:
Figure BDA0004114996960000031
where x is a spatial variable and t is a temporal variable. The fluid density ρ, the velocity u, the pressure p, the total energy per unit length e and the gas coefficient γ constitute a conservation vector variable u and a flux vector f:
Figure BDA0004114996960000032
for the two-dimensional compressible flow problem, a two-dimensional Euler equation as shown in formula (5) is adopted as a compressible flow field model equation:
Figure BDA0004114996960000033
the conservation vector variables u, the x-direction flux vector f and the y-direction flux vector g are respectively as follows:
Figure BDA0004114996960000034
wherein u and v are x-direction speed and y-direction speed respectively, and the physical meaning represented by other variables is the same as that of the formula (4).
Step four: and (3) updating the boundary grid physical quantity by adopting boundary conditions required by the explosion impact problem to be simulated according to the updated physical quantity in the step (III), thereby obtaining the virtual grid node physical quantity required by the construction of the WENO format. The boundary conditions include: an inflow boundary condition, an outflow boundary condition, a fixed wall reflection boundary condition.
The inflow boundary condition is uniform inflow, namely the physical quantity at the inflow is constant, and the inflow boundary condition is determined according to the problem of explosion impact to be simulated. The physical quantity includes density ρ, velocity u, v, pressure p.
The outflow boundary condition is a reflection-free boundary condition, as shown in formula (7):
Figure BDA0004114996960000035
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0004114996960000036
is the reciprocal of the direction along the boundary normal.
The fixed wall reflection boundary condition is as shown in formula (8):
Figure BDA0004114996960000037
wherein v is τ For tangential velocity, v n =0 is the normal velocity.
Step five: and (3) performing space dispersion on the compressible flow field model equation in the step three by using a WENO format to obtain a semi-discrete finite difference form of the compressible flow field model equation.
The compressible flow field model control equation comprises a one-dimensional Euler equation and a two-dimensional Euler equation, which can be decoupled and expressed by a unified scalar form equation. The scalar form equation is:
Figure BDA0004114996960000041
where u is the constancy and f is the corresponding flux vector.
The semi-discrete finite difference form is:
Figure BDA0004114996960000042
wherein N is the total grid number of the calculated area, deltax is the grid spacing, u j (t) is u (x) j Approximation of t), uniform grid node coordinates are: x is x j ＝jΔx,j＝0,...,N。
Figure BDA0004114996960000043
The numerical flux is obtained by reconstructing a Lax-Friedrichs vector flux splitting method and a WENO format. The Lax-Friedrichs vector flux splitting method is shown in the step six; the WENO format is shown in the step seven.
Step six: and dividing the flux of each grid node into positive and negative fluxes by using a Lax-Friedrichs vector flux dividing method, and inhibiting non-physical oscillation at a strong break. The positive flux is f + (u) the negative flux is f - (u)。
The Lax-Friedrichs vector flux splitting method is shown in a formula (11), and the fluxes of all grid nodes are split into positive fluxes and negative fluxes, so that non-physical oscillation at strong discontinuities is inhibited.
Figure BDA0004114996960000044
Where α is the maximum value of u in the solution domain.
Step seven: spatial dispersion was performed using the WENO format. Introducing a local smoothness measurement factor with a compact structure into the construction of the WENO format nonlinear weight, quantitatively characterizing the smoothness of a neutron template in a calculation domain grid through the local smoothness measurement factor, so that the WENO format is easy to expand to a high order, and the capturing efficiency of the WENO format on shock wave interruption is improved on the premise of ensuring that the capturing is unchanged; in the WENO format construction process, a global smoothness measurement factor is constructed through linear combination of local smoothness measurement factors, all sub-template information is coupled into the whole calculation template through the global smoothness measurement factor, so that quantitative representation of the overall template smoothness is realized, the global smoothness measurement factor fully utilizes flow field information, the WENO format resolution is improved, the precision of the WENO format at a high-order extreme point is further improved to achieve optimal-order convergence, and the WENO format is more suitable for accurate numerical simulation of a complex flow field containing interaction of high-density-ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves in explosion impact problems; in the under-smooth template regulation item of the WENO format nonlinear weight, the WENO format can carry out self-adaptive regulation on the template weight according to flow field change by constructing the self-adaptive regulation coefficient. Nonlinear weights of all sub templates are obtained through cooperative regulation and control of local smoothness measurement factors, global smoothness measurement factors and self-adaptive regulation and control coefficients, nonlinear convex combination is carried out on WENO format sub template flux by adopting the nonlinear weights, WENO format dissipation is reduced, WENO format resolution is improved, false numerical oscillation is restrained, numerical calculation is ensured to be carried out stably, sharp tracking capture of shock waves and accurate identification of curls and fine complex structures near contact discontinuities after shock wave collision are achieved, and numerical simulation of complex explosion impact problems including interaction of high-density ratio and high-pressure ratio strong shock waves and Jiang Xishu waves is achieved.
Preferably, in the seventh step, the 5-order WENO format is adopted, so that efficiency is considered, and meanwhile, high-resolution high-precision shock wave intermittent capturing precision can be ensured.
When the WENO format in the step seven adopts a 5-order WENO format, the implementation method in the step seven is as follows:
step 7.1: given an overall template t= { x of 5 points j-2 ,x j-1 ,x j ,x j+1 ,x j+2 Dividing into 3 sub-templates T 0 ＝{x j-2 ,x j-1 ,x j }，T 1 ＝{x j-1 ,x j ,x j+1 }，T 2 ＝{x j ,x j+1 ,x j+2 }。
Step 7.2: construction sub-template T k Third order reconstruction flux of (2)
Figure BDA0004114996960000051
Figure BDA0004114996960000052
Wherein f j J=0, …, N is the flux on the solution domain mesh node. c kn Is Lagrange interpolation coefficient, can be uniquely obtained through Taylor series expansion, and the third-order reconstruction flux after the result is introduced into the formula (12)
Figure BDA0004114996960000053
The specific form is as follows:
Figure BDA0004114996960000054
step 7.3: the method is characterized in that a local smoothness measurement factor with a compact structure is introduced into the structure of the nonlinear weight of the WENO format, the smoothness of a neutron template in a calculation domain grid is quantitatively represented through the local smoothness measurement factor, so that the WENO format is easy to expand to a high order, and the capturing efficiency of the WENO format on shock wave interruption is improved on the premise of ensuring that the capturing is unchanged.
The local smoothing factor IS k As shown in equation (14):
Figure BDA0004114996960000055
Figure BDA0004114996960000061
Figure BDA0004114996960000062
step 7.4: in the construction process of the WENO format, a global smoothness measurement factor is constructed through linear combination of local smoothness measurement factors, all sub-template information is coupled into the whole calculation template through the global smoothness measurement factor, so that quantitative representation of the overall template smoothness is realized, the global smoothness measurement factor fully utilizes flow field information, the resolution of the WENO format is improved, the precision of the WENO format at a high-order extreme point is further improved to achieve optimal-order convergence, and the WENO format is more suitable for accurate numerical simulation of a complex flow field containing interaction of high-density-ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves in the explosion impact problem;
the global smoothness metric factor η is shown in formula (17):
Figure BDA0004114996960000063
step 7.5: in the under-smooth template regulation item of the WENO format nonlinear weight, the WENO format can carry out self-adaptive regulation on the template weight according to flow field change by constructing the self-adaptive regulation coefficient. The WENO format dissipation is reduced, the WENO format resolution is improved, false numerical oscillation is restrained, and stable numerical calculation is ensured.
The construction process of the self-adaptive regulation and control coefficient lambda is as follows:
selecting a minimum value of deviation between the reference submodule weight and the ideal weight:
Figure BDA0004114996960000064
reference weight
Figure BDA0004114996960000065
Normalized reference weight->
Figure BDA0004114996960000066
The construction is as follows:
Figure BDA0004114996960000067
wherein the method comprises the steps of
Figure BDA0004114996960000068
Constructing an adaptive regulation and control coefficient lambda through the ideal weight of the submodule and the normalized reference weight:
Figure BDA0004114996960000071
the final expression for obtaining the adaptive regulation coefficient λ is:
Figure BDA0004114996960000072
step 7.6: the nonlinear weights of all the sub templates are obtained through the cooperative regulation and control of the local smoothness measurement factor in the step 7.3, the global smoothness measurement factor in the step 7.4 and the self-adaptive regulation and control coefficient in the step 7.5, the nonlinear weights are adopted to carry out nonlinear convex combination on WENO format sub template flux, the WENO format dissipation is reduced, the WENO format resolution is improved, false numerical oscillation is restrained, and numerical calculation is ensured to be carried out stably, so that sharp tracking capture of shock waves and accurate identification of curls and fine complex structures near contact discontinuities after shock collision are realized, and numerical simulation of complex explosion impact problems comprising interaction of high-density-ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves is realized.
The nonlinear weight
Figure BDA0004114996960000073
As shown in equation (22):
Figure BDA0004114996960000074
wherein C is k Ideal weights for sub-templates, (C) 0 ,C 1 ,C 2 )＝(1/10,6/10,3/10)。
Step 7.7: reconstructing flux from the three sub-templates obtained in step 7.2
Figure BDA0004114996960000075
And the nonlinear weight corresponding to each sub-template obtained in the step 7.6 can obtain the five-order WENO reconstruction flux +.>
Figure BDA0004114996960000076
The five-order WENO reconstruction flux
Figure BDA0004114996960000077
As shown in formula (23):
Figure BDA0004114996960000078
step 7.8: moving the whole template leftwards by one unit, and repeating the steps 7.1 to 7.7 to obtain the numerical flux
Figure BDA0004114996960000079
The new format constructed by the present invention is named NWENO-Z format.
Step eight: and (3) carrying the reconstructed numerical flux in the NWENO-Z format back to the semi-discrete finite difference form of the model equation in the step four, and performing time propulsion by adopting a three-order TVD (transient voltage detector) range-Kutta method to realize the propagation, evolution and collision process numerical calculation of the explosion shock wave. Accurately capturing shock wave interruption to obtain grid physical quantity of the (n+1) th time step.
The three-order TVD finger-Kutta method is shown in a formula (24):
Figure BDA0004114996960000081
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0004114996960000082
step nine: according to the set termination time of the to-be-simulated explosion impact problem, time pushing is carried out according to the third-order TVD range-Kutta method in the step seven before the termination time is reached, each time step needs to be iterated three times, each iteration adopts the NWENO-Z format in the step seven to carry out space dispersion until the termination time is reached, sharp tracking identification of shock waves and capturing of curling and fine complex structures after the shock waves collide are realized, and accurate prediction of propagation, evolution and collision processes of the explosion impact waves is realized.
The beneficial effects are that:
1. the WENO-based high-resolution high-precision shock wave intermittent capturing method disclosed by the invention can be used for accurately solving the shock wave intermittent problem, improving the prediction precision of shock wave propagation, evolution and collision processes, improving the precision of the numerical calculation result of the complex flow field of explosion impact, and further solving the engineering technical problems related to the explosion impact in the national defense and military field and the civil production safety field.
2. According to the WENO-based high-resolution high-precision shock wave intermittent capturing method disclosed by the invention, a local smoothness measurement factor with a compact structure is introduced into the structure of the nonlinear weight of the WENO format, and the smoothness of a neutron template in a calculation domain grid is quantitatively represented through the local smoothness measurement factor, so that the WENO format is easy to expand towards a high order, and the capturing efficiency of the WENO format on the shock wave intermittent is improved on the premise of ensuring that the capturing is unchanged.
3. The invention discloses a WENO-based high-resolution high-precision shock wave intermittent capturing method, which is characterized in that in the WENO format construction process, a global smoothness measurement factor is constructed through linear combination of local smoothness measurement factors, all sub-template information is coupled into a whole calculation template through the global smoothness measurement factor, so that quantitative characterization of the whole template smoothness is realized, the global smoothness measurement factor fully utilizes flow field information, the resolution of the WENO format is improved, the precision of the WENO format at a high-order extreme point is further improved, so that optimal-order convergence is achieved, and the WENO format is more suitable for accurate numerical simulation of a complex flow field containing interaction of high-density-ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves in the explosion impact problem.
4. According to the WENO-based high-resolution high-precision shock wave intermittent capturing method disclosed by the invention, in an under-smooth template regulation item of the WENO format nonlinear weight, the WENO format can carry out self-adaptive regulation on the template weight according to flow field changes by constructing the self-adaptive regulation coefficient, so that the dissipation of the WENO format is reduced, the resolution of the WENO format is improved, false numerical oscillation is suppressed, and stable numerical calculation is ensured.
5. The invention discloses a WENO-based high-resolution high-precision shock wave intermittent capturing method, which is characterized in that nonlinear weights of all sub-templates are obtained through cooperative regulation and control of local smoothness measurement factors, global smoothness measurement factors and self-adaptive regulation and control coefficients, nonlinear convex combination is carried out on WENO format sub-template flux by adopting the nonlinear weights, WENO format dissipation is reduced, WENO format resolution is improved, false numerical oscillation is restrained, stable numerical calculation is ensured, sharp tracking capturing of shock waves and accurate identification of curls and fine complex structures near contact discontinuities after shock wave collision are realized, and numerical simulation of complex explosion impact problems comprising interaction of high-density ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves is realized.
Drawings
Fig. 1 is a flow chart corresponding to a high-resolution high-precision shock wave intermittent capturing method based on WENO.
Fig. 2 is a schematic diagram of a mining strip-shaped explosive package axial detonation wave collision problem model in example 1.
Fig. 3 is a density distribution graph of the axial detonation wave collision problem of the mining bar shaped charge of example 1 at time t=0.038 s for each method.
FIG. 4 is a simplified model of the mine tunnel explosion problem of example 2.
Fig. 5 is a cloud graph of density distribution of the simplified model of the mine cavity explosion problem of example 2 at time t=0.2 s for each method.
Fig. 6 is a partial magnified density cloud of dual mach reflection structures of the simplified model of the mine cavity explosion problem of example 2 at time t=0.2 s for each method.
Detailed Description
The present invention will be described in detail with reference to the accompanying drawings and examples. At the same time, the technical problems and advantages solved by the technical proposal of the invention are also described, and it is pointed out that the described embodiment is only intended to facilitate the understanding of the invention and does not have any limiting effect
Example 1:
in the embodiment 1 of the invention, the axial detonation wave collision of the mining strip-shaped explosive package is taken as an example, and the numerical simulation is carried out on the collision of two density shock waves. Solving the problem that the field contains sharp shock waves and contact discontinuities is a typical detonation wave problem. Compared with the traditional method, the method can more sharply catch shock wave interruption and simulate mining axial detonation collision with higher resolution; and the pressure of each point on the full calculation area can be output, so that a basis is provided for the research on the axial detonation wave collision energy collection characteristics of the mining strip-shaped explosive package. Other safety protection fields can use the pressure as a damage judgment basis to carry out safety evaluation analysis on protection engineering. The invention provides a WENO-based high-resolution high-precision shock wave intermittent capturing method, which is described in detail below by taking the axial detonation wave collision numerical simulation of a mining strip-shaped explosive package as an example, and comprises the following steps:
step one, in a Cartesian coordinate system, aiming at a numerical simulation model of the axial detonation wave collision problem of the mining strip-shaped explosive package, a schematic diagram of the axial detonation wave collision problem model of the mining strip-shaped explosive package in the embodiment 1 is shown in fig. 2. And initializing and setting the numerical simulation model. And setting a calculation region as x epsilon [0,1], establishing an Euler rectangular coordinate system, and dividing the x direction of the calculation region into 400 grids. The calculation end time is set to t=0.038. Initializing grid physical quantity according to the problem of axial detonation wave collision of the mining strip-shaped explosive package, wherein the initial conditions are as follows:
Figure BDA0004114996960000101
step two: the CFL coefficient is set to 0.5, and the time step Deltat is determined according to the CFL stability condition.
Figure BDA0004114996960000102
Step three: in the calculation area x epsilon [0,1] divided by the axial detonation wave collision problem of the mining strip-shaped explosive package in the embodiment 1, the grid physical quantity in the calculation area is updated according to the compressible flow field model equation. The compressible flow field model equations include a continuous equation, a momentum equation, and an energy equation. The physical quantities include: fluid density ρ, velocity u, pressure p. According to the calculation dimension of the axial detonation wave collision problem of the mining bar-shaped explosive package in the implementation 1, a one-dimensional Euler equation is adopted as a compressible flow field model equation to update and calculate the grid physical quantity in the domain:
step four: according to the updated physical quantity in the third step, aiming at the problem of axial detonation wave collision of the mining strip-shaped explosive package in the embodiment 1, the boundary grid physical quantity is updated by adopting a fixed wall reflection boundary condition on the left side and the right side.
Step five: and (3) performing space dispersion on the compressible flow field model equation in the step three by using a WENO format to obtain a semi-discrete finite difference form of the compressible flow field model equation. The compressible flow field model control equation is a one-dimensional Euler equation in the problem of axial detonation wave collision of the 1-mining bar-shaped explosive package, and numerical flux is obtained by reconstructing a Lax-Friedrichs vector flux splitting method and a WENO format
Figure BDA0004114996960000103
The Lax-Friedrichs vector flux splitting method is shown in the step six; the WENO format is shown in the step seven.
Step six: and dividing the flux of each grid node into positive and negative fluxes by using a Lax-Friedrichs vector flux dividing method, and inhibiting non-physical oscillation at a strong break. The positive flux is f + (u) the negative flux is f - (u)。
Step seven: spatial dispersion was performed using the WENO format. Introducing a local smoothness measurement factor with a compact structure into the construction of the WENO format nonlinear weight, quantitatively characterizing the smoothness of a neutron template in a calculation domain grid through the local smoothness measurement factor, so that the WENO format is easy to expand to a high order, and the capturing efficiency of the WENO format on shock wave interruption is improved on the premise of ensuring that the capturing is unchanged; in the WENO format construction process, a global smoothness measurement factor is constructed through linear combination of local smoothness measurement factors, all sub-template information is coupled into the whole calculation template through the global smoothness measurement factor, so that quantitative representation of the overall template smoothness is realized, the global smoothness measurement factor fully utilizes flow field information, the WENO format resolution is improved, the precision of the WENO format at a high-order extreme point is further improved to achieve optimal-order convergence, and the WENO format is more suitable for accurate numerical simulation of a complex flow field containing interaction of high-density-ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves in explosion impact problems; in the under-smooth template regulation item of the WENO format nonlinear weight, the WENO format can carry out self-adaptive regulation on the template weight according to flow field change by constructing the self-adaptive regulation coefficient. Nonlinear weights of all the sub templates are obtained through cooperative regulation and control of the local smoothness measurement factors, the global smoothness measurement factors and the self-adaptive regulation and control coefficients, nonlinear convex combination is carried out on WENO format sub template flux by adopting the nonlinear weights, WENO format dissipation is reduced, WENO format resolution is improved, false numerical oscillation is restrained, stable numerical calculation is ensured, and therefore numerical simulation of the mining strip-shaped explosive package axial detonation wave collision problem is achieved.
Step eight: and (3) carrying the reconstructed numerical flux in the NWENO-Z format back to the semi-discrete finite difference form of the model equation in the step four, and performing time propulsion by adopting a three-order TVD (transient voltage detector) range-Kutta method to realize the propagation, evolution and collision process numerical calculation of the explosion shock wave. Accurately capturing shock wave interruption to obtain grid physical quantity of the (n+1) th time step.
Step nine: according to the method, the time is advanced according to the three-order TVD range-Kutta method in the step seven before the ending time is reached, three iterations are needed for each time step, each iteration adopts the NWENO-Z format in the step seven to carry out space dispersion until the ending time is reached, and the tracking identification of detonation wave collision is realized.
And (3) analysis of calculation results:
FIG. 3 is a density distribution graph of the axial detonation wave collision problem for the mining strip shaped charge of example 1. According to the numerical calculation result of the embodiment 1, the WENO-based high-resolution high-precision shock wave intermittent capturing method disclosed by the invention can be used for well simulating the axial detonation wave collision problem of the mining strip-shaped explosive package. In particular, fig. 3 can observe a "valley" structure around x=0.75, with the result of the new method NWENO-Z of the invention closest to "Reference"; the "peak" structure near x=0.78 is most acute with NWENO-Z capture. The invention can effectively realize the accurate numerical simulation calculation of the problem of the extreme shock wave interruption in the impact process, and provides reliable basis for evaluation in the fields of mining safety engineering protection and the like.
Example 2:
in the embodiment 2 of the invention, the problem of explosion of a mine tunnel is taken as an example, the problem of reflection of detonation waves on an inclined plane is simulated, and the identification capability of the invention on a complex double Mach structure is mainly examined. In order to simplify the model, in this embodiment, oblique shock waves are propelled in a regular rectangular mine tunnel, and a dual mach structure evolved by collision reflection with the wall surface of the mine tunnel is simulated, and a simplified model diagram of the mine tunnel explosion problem in this embodiment 2 is shown in fig. 4. At the initial moment, the intersection point of the 10 Mach oblique shock wave with the inclination angle of 60 degrees and the lower boundary is at the position of x=1/6, and the oblique shock wave advances and evolves into a double Mach complex flow field structure along with time. Compared with the traditional method, the method can better distinguish the complex double Mach structure after the explosion of the ore-drawing tunnel and can more sharply capture the evolved vortex structure, and can provide more reliable and accurate analysis for the safety evaluation of the ore-drawing tunnel protection engineering. The present invention provides a high resolution and high precision shock wave intermittent capturing method based on the WENO, wherein the steps one, two, three, four, nine are different from the embodiment 1, and the other steps are the same as the embodiment 1.
Specifically, as a preferred embodiment 2 of the present invention, the step one, the step two, the step three, the step four, and the step nine are:
step one, under a Cartesian coordinate system, pushing oblique shock waves in a regular rectangular mine cavity, simulating a double Mach structure evolved by collision reflection with the wall surface of the mine cavity, and establishing a numerical simulation model, as shown in fig. 4. And initializing and setting the numerical simulation model. An Euler rectangular coordinate system is established and the calculated area is determined to be x epsilon [0,4], y epsilon [0,1] according to the embodiment 2. The calculation region x-direction is divided into 1200 grids, and the y-direction is divided into 400 grids, which are 480000 grids in total. The calculation end time is set to t=0.2. Grid physical quantity initialization is performed according to the preferred embodiment 2 of the present invention, and the flow field initial conditions are set as follows:
Figure BDA0004114996960000121
wherein U is L The physical field distribution in the calculation domain on the left side of the oblique shock wave; u (U) R The physical field distribution in the domain is calculated on the right side of the oblique shock wave; h (x, t) is a ramp position function at any instant.
U L ＝(ρ,u,v,p) T ＝(8.0,7.1447,-4.125,116.5) T (28)
U R ＝(ρ,u,v,p) T ＝(1.4,0,0,1) T (29)
Figure BDA0004114996960000123
Step two: the CFL coefficient is set to 0.3, and the time step Deltat is determined according to the CFL stability condition.
Figure BDA0004114996960000122
Step three: in the calculation region x epsilon [0,4] and y epsilon [0,1] divided in the preferred embodiment 2, the grid physical quantity in the calculation domain is updated according to the compressible flow field model equation. The compressible flow field model equations include a continuous equation, a momentum equation, and an energy equation. The physical quantities include: fluid density ρ, velocity u, v, pressure p. According to the calculation dimensions of the preferred embodiment 2, the physical quantity of the grid in the calculation domain is updated using a one-dimensional euler equation as the compressible flow field model equation.
Step four: according to the updated physical quantity in the third step, for the embodiment 2, inflow and outflow boundary conditions are respectively adopted on the left side and the right side; the upper boundary is the exact solution of shock wave propagation; the lower boundary is divided into two parts, wherein the 0< x <1/6 part adopts shock wave propagation accurate value, and the 1/6< x <4 part adopts solid wall boundary condition. And updating the physical quantity of the boundary grid through the boundary condition setting.
Step nine: according to the calculation end time t=0.2 set in the preferred embodiment 2, time pushing is performed according to the third-order TVD range-Kutta method in step seven before the end time is reached, each time step is iterated three times, each iteration adopts the NWENO-Z format in step seven to perform spatial dispersion until the end time is reached, sharp tracking capture of shock waves and precise identification of curls and fine complex structures near contact discontinuities after shock wave collision are achieved, and high-resolution numerical simulation of the reflection process of detonation waves on the inner wall of a mine hole in the mine hole explosion problem is completed.
And (3) analysis of calculation results:
fig. 5 is a density distribution cloud of the simplified model of the mine tunnel explosion problem of example 2. Fig. 6 is a partial magnified density cloud of dual mach reflection structures of the simplified model of the mine cavity explosion problem of example 2 at time t=0.2 s for each method. According to the numerical calculation result of the embodiment 2, the high-resolution high-precision shock wave intermittent capturing method based on the WENO can simulate the reflection process of detonation waves on the inner wall of a mine hole in the mine hole explosion problem well. Specifically, fig. 5 shows that the contour line of the result of the new method NWENO-Z is the smoothest, and the complex flow field containing the reflection collision of detonation waves on the inner wall surface of the mine cavity can be solved more stably; region I of FIG. 6 shows that the vortex structure captured by the NWENO-Z method is sharper than the conventional method, and the shock wave captured by the NWENO-Z method is the most clear and noiseless in region II. The example 2 proves that the invention can effectively realize the accurate numerical simulation calculation of the problem of the extreme shock wave interruption in the impact process and provide reliable basis for the safety engineering protection evaluation of mining engineering and the like.
The foregoing detailed description has set forth the objects, aspects and advantages of the invention in further detail, it should be understood that the foregoing description is only illustrative of the invention and is not intended to limit the scope of the invention, but is to be accorded the full scope of the invention as defined by the appended claims.

Claims (8)

1. A WENO-based high-resolution high-precision shock wave intermittent capturing method is characterized by comprising the following steps of: comprises the following steps of the method,
firstly, establishing a numerical simulation model for a compressible flow field aiming at the problem of explosion impact to be simulated under a Cartesian coordinate system, and initializing and setting the numerical simulation model; the initialization setting comprises determining a calculation domain, performing grid division, setting initial flow field parameters and initializing grid physical quantities, wherein the initial flow field parameter settings comprise initial conditions and initial boundary condition settings;
step two: determining a time step delta t according to the CFL stability condition;
step three: in a calculation area divided by the explosion impact problem to be simulated, updating the grid physical quantity in the calculation area according to a compressible flow field model equation; the compressible flow field model equation comprises a continuous equation, a momentum equation and an energy equation; the physical quantities include: fluid density ρ, velocity u, v, pressure p; according to the calculated dimension of the explosion impact problem to be simulated, respectively adopting a one-dimensional Euler equation and a two-dimensional Euler equation as compressible flow field model equations to update and calculate the grid physical quantity in the domain;
step four: according to the updated physical quantity in the step three, updating the boundary grid physical quantity by adopting boundary conditions required by the explosion impact problem to be simulated, thereby obtaining virtual grid node physical quantity required by the construction of WENO format; the boundary conditions include: inflow boundary conditions, outflow boundary conditions, fixed wall reflection boundary conditions;
step five: performing space dispersion on the compressible flow field model equation in the third step by using a WENO format to obtain a semi-discrete finite difference form of the compressible flow field model equation;
step six: dividing each grid node flux into positive and negative fluxes by using a Lax-Friedrichs vector flux dividing method, and inhibiting non-physical oscillation at a strong break; the positive flux is f + (u) the negative flux is f - (u)；
Step seven: performing space dispersion by adopting a WENO format; introducing a local smoothness measurement factor with a compact structure into the construction of the WENO format nonlinear weight, quantitatively characterizing the smoothness of a neutron template in a calculation domain grid through the local smoothness measurement factor, so that the WENO format is easy to expand to a high order, and the capturing efficiency of the WENO format on shock wave interruption is improved on the premise of ensuring that the capturing is unchanged; in the WENO format construction process, a global smoothness measurement factor is constructed through linear combination of local smoothness measurement factors, all sub-template information is coupled into the whole calculation template through the global smoothness measurement factor, so that quantitative representation of the overall template smoothness is realized, the global smoothness measurement factor fully utilizes flow field information, the WENO format resolution is improved, the precision of the WENO format at a high-order extreme point is further improved to achieve optimal-order convergence, and the WENO format is more suitable for accurate numerical simulation of a complex flow field containing interaction of high-density-ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves in explosion impact problems; in the under-smooth template regulation item of the WENO format nonlinear weight, the WENO format can carry out self-adaptive regulation on the template weight according to flow field change by constructing a self-adaptive regulation coefficient; nonlinear weights of all sub templates are obtained through cooperative regulation and control of local smoothness measurement factors, global smoothness measurement factors and self-adaptive regulation and control coefficients, nonlinear convex combination is carried out on WENO format sub template flux by adopting the nonlinear weights, WENO format dissipation is reduced, WENO format resolution is improved, false numerical oscillation is restrained, numerical calculation is ensured to be carried out stably, sharp tracking capture of shock waves and accurate identification of curls and fine complex structures near contact discontinuities after shock wave collision are realized, and numerical simulation of complex explosion impact problems including interaction of high-density ratio and high-pressure ratio strong shock waves and Jiang Xishu waves is further realized; the constructed new format is named NWENO-Z format;
step eight: carrying the reconstructed numerical flux of the NWENO-Z format back to the semi-discrete finite difference form of the model equation in the fourth step, and performing time propulsion by adopting a three-order TVD (transient voltage detector) range-Kutta method to realize the propagation, evolution and collision process numerical calculation of the explosion shock wave; accurately capturing shock wave interruption to obtain grid physical quantity of the (n+1) th time step;
step nine: according to the set termination time of the to-be-simulated explosion impact problem, time pushing is carried out according to the third-order TVD range-Kutta method in the step seven before the termination time is reached, each time step needs to be iterated three times, each iteration adopts the NWENO-Z format in the step seven to carry out space dispersion until the termination time is reached, sharp tracking identification of shock waves and capturing of curling and fine complex structures after the shock waves collide are realized, and accurate prediction of propagation, evolution and collision processes of the explosion impact waves is realized.
2. The high-resolution high-precision shock wave intermittent capturing method based on WENO in claim 1, wherein the method is characterized by comprising the following steps of: the CFL stability condition is as shown in formula (1):
Figure FDA0004114996950000021
wherein CFL is the stability condition coefficient, |u c ∣ max Is the maximum value of the speed of the full physical field under the current time step:
Figure FDA0004114996950000022
where c is the local sound velocity.
3. The high-resolution high-precision shock wave intermittent capturing method based on WENO as claimed in claim 2, wherein the method is characterized by comprising the following steps of: the implementation method of the third step is that,
for the one-dimensional compressible flow problem, a one-dimensional Euler equation shown in formula (3) is adopted as a compressible flow field model equation:
Figure FDA0004114996950000023
wherein x is a spatial variable and t is a temporal variable; the fluid density ρ, the velocity u, the pressure p, the total energy per unit length e and the gas coefficient γ constitute a conservation vector variable u and a flux vector f:
Figure FDA0004114996950000024
for the two-dimensional compressible flow problem, a two-dimensional Euler equation as shown in formula (5) is adopted as a compressible flow field model equation:
Figure FDA0004114996950000025
the conservation vector variables u, the x-direction flux vector f and the y-direction flux vector g are respectively as follows:
Figure FDA0004114996950000031
wherein u and v are x-direction speed and y-direction speed respectively, and the physical meaning represented by other variables is the same as that of the formula (4).
4. The high-resolution high-precision shock wave intermittent capturing method based on WENO as claimed in claim 3, wherein the method is characterized by comprising the following steps of: in the fourth step, the first step is performed,
the inflow boundary condition is uniform inflow, namely the physical quantity at the inflow position is constant, and the inflow boundary condition is determined according to the problem of explosion impact to be simulated; the physical quantity comprises density rho, speeds u and v and pressure p;
the outflow boundary condition is a reflection-free boundary condition, as shown in formula (7):
Figure FDA0004114996950000032
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA0004114996950000033
is the reciprocal in the direction of the boundary normal;
the fixed wall reflection boundary condition is as shown in formula (8):
Figure FDA0004114996950000034
wherein v is τ For tangential velocity, v n =0 is the normal velocity.
5. The WENO-based high-resolution and high-precision shock wave intermittent capturing method as claimed in claim 4, wherein the method is characterized by comprising the following steps of: in the fifth step, the first step is to carry out the process,
the compressible flow field model control equation comprises a one-dimensional Euler equation and a two-dimensional Euler equation, which can be decoupled and expressed by a unified scalar form equation; the scalar form equation is:
Figure FDA0004114996950000035
where u is the constancy and f is the corresponding flux vector;
the semi-discrete finite difference form is:
Figure FDA0004114996950000036
wherein N is the total grid number of the calculated area, deltax is the grid spacing, u j (t) is u (x) j Approximation of t), uniform grid node coordinates are: x is x j ＝jΔx,j＝0,…,N；
Figure FDA0004114996950000037
The numerical flux is obtained by reconstructing a Lax-Friedrichs vector flux splitting method and a WENO format; the Lax-Friedrichs vector flux splitting method is shown in the step six; the WENO format is shown in the step seven.
6. The WENO-based high-resolution and high-precision shock wave intermittent capturing method as claimed in claim 5, wherein the method is characterized by comprising the following steps of: in the sixth step, the Lax-Friedrichs vector flux splitting method is shown in a formula (11), each grid node flux is split into positive flux and negative flux, and non-physical oscillation at a strong break is restrained;
Figure FDA0004114996950000041
where α is the maximum value of u in the solution domain.
7. The high-resolution high-precision shock wave intermittent capturing method based on WENO as claimed in claim 6, wherein the method is characterized by comprising the following steps of: when the WENO format in the step seven adopts a 5-order WENO format, the implementation method in the step seven is as follows:
step 7.1: given an overall template t= { x of 5 points j-2 ,x j-1 ,x j ,x j+1 ,x j+2 Dividing into 3 sub-templates T 0 ＝{x j-2 ,x j-1 ,x j }，T 1 ＝{x j-1 ,x j ,x j+1 }，T 2 ＝{x j ,x j+1 ,x j+2 }；
Step 7.2: construction sub-template T k Third order reconstruction flux of (2)
Figure FDA0004114996950000042
Figure FDA0004114996950000043
Wherein f j J=0, …, N is the flux on the solution domain mesh node; c kn Is Lagrange interpolation coefficient, can be uniquely obtained through Taylor series expansion, and the third-order reconstruction flux after the result is introduced into the formula (12)
Figure FDA0004114996950000044
The specific form is as follows:
Figure FDA0004114996950000045
step 7.3: introducing a local smoothness measurement factor with a compact structure into the construction of the WENO format nonlinear weight, quantitatively characterizing the smoothness of a neutron template in a calculation domain grid through the local smoothness measurement factor, so that the WENO format is easy to expand to a high order, and the capturing efficiency of the WENO format on shock wave interruption is improved on the premise of ensuring that the capturing is unchanged;
the local smoothing factor IS k As shown in equation (14):
Figure FDA0004114996950000046
Figure FDA0004114996950000047
Figure FDA0004114996950000051
step 7.4: in the construction process of the WENO format, a global smoothness measurement factor is constructed through linear combination of local smoothness measurement factors, all sub-template information is coupled into the whole calculation template through the global smoothness measurement factor, so that quantitative representation of the overall template smoothness is realized, the global smoothness measurement factor fully utilizes flow field information, the resolution of the WENO format is improved, the precision of the WENO format at a high-order extreme point is further improved to achieve optimal-order convergence, and the WENO format is more suitable for accurate numerical simulation of a complex flow field containing interaction of high-density-ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves in the explosion impact problem;
the global smoothness metric factor η is shown in formula (17):
Figure FDA0004114996950000052
step 7.5: in the under-smooth template regulation item of the WENO format nonlinear weight, the WENO format can carry out self-adaptive regulation on the template weight according to flow field change by constructing a self-adaptive regulation coefficient; the WENO format dissipation is reduced, the WENO format resolution is improved, false numerical oscillation is restrained, and stable numerical calculation is ensured;
the construction process of the self-adaptive regulation and control coefficient lambda is as follows:
selecting a minimum value of deviation between the reference submodule weight and the ideal weight:
Figure FDA0004114996950000053
reference weight
Figure FDA0004114996950000054
Normalized reference weight->
Figure FDA0004114996950000055
The construction is as follows:
Figure FDA0004114996950000056
constructing an adaptive regulation and control coefficient lambda through the ideal weight of the submodule and the normalized reference weight:
Figure FDA0004114996950000057
the final expression for obtaining the adaptive regulation coefficient λ is:
Figure FDA0004114996950000061
step 7.6: the nonlinear weights of all the sub templates are obtained through the cooperative regulation and control of the local smoothness measurement factor in the step 7.3, the global smoothness measurement factor in the step 7.4 and the self-adaptive regulation and control coefficient in the step 7.5, nonlinear convex combination is carried out on WENO format sub template flux by adopting the nonlinear weights, the WENO format dissipation is reduced, the WENO format resolution is improved, false numerical oscillation is restrained, and numerical calculation is ensured to be carried out stably, so that sharp tracking capture of shock waves and accurate identification of curls and fine complex structures near contact discontinuities after shock collision are realized, and numerical simulation of complex explosion impact problems comprising interaction of high-density ratio and high-pressure-ratio strong shock waves and Jiang Xishu waves is realized;
the nonlinear weight omega k NZ As shown in equation (22):
Figure FDA0004114996950000062
wherein C is k Ideal weights for sub-templates, (C) 0 ,C 1 ,C 2 )＝(1/10,6/10,3/10)；
Step 7.7: reconstructing flux from the three sub-templates obtained in step 7.2
Figure FDA0004114996950000063
And the nonlinear weight corresponding to each sub-template obtained in the step 7.6 can obtain the five-order WENO reconstruction flux +.>
Figure FDA0004114996950000064
The five-order WENO reconstruction flux
Figure FDA0004114996950000065
As shown in formula (23):
Figure FDA0004114996950000066
step 7.8: moving the whole template leftwards by one unit, and repeating the steps 7.1 to 7.7 to obtain the numerical flux
Figure FDA0004114996950000067
The new format of the construct is named NWENO-Z format.
8. The high-resolution high-precision shock wave intermittent capturing method based on WENO as claimed in claim 7, wherein the method is characterized by comprising the following steps of: in the step eight, the step of,
the three-order TVD finger-Kutta method is shown in a formula (24):
Figure FDA0004114996950000071
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure FDA0004114996950000072
CN202310216211.8A 2023-03-08 2023-03-08 WENO-based high-resolution high-precision shock wave intermittent capturing method Pending CN116415518A (en)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116611369A (en) * 2023-07-17 2023-08-18 中国空气动力研究与发展中心计算空气动力研究所 Interpolation method and device based on smoothness magnitude and candidate template point number
CN116611369B (en) * 2023-07-17 2023-09-29 中国空气动力研究与发展中心计算空气动力研究所 Interpolation method and device based on smoothness magnitude and candidate template point number

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