Method for solving lateral expansion effect of detonation wave based on generalized znd theory

Abstract

The application discloses a kind of solving methods of detonation wave lateral expansion effect based on generalized ZND theory, the method is suitable for calculating the curvature radius of detonation wave under the action of lateral expansion, shock structure and velocity loss etc.Propagation parameter;The flow field parameter after detonation wave is solved by generalized ZND theory, the interface angle and shock angle of detonation wave lateral expansion are calculated by combining oblique shock relationship and Prandtl-Meyer relationship, the curvature of detonation wave is calculated by the geometric relationship fitting of interface angle and detonation wave height, the curvature radius of generalized ZND model is updated, the propagation process of detonation wave is iteratively calculated until convergence is solved.The calculation method provided by the application is accurate and reliable, has universality, by considering curvature influence, more close to the steady-state solution of detonation wave affected by lateral expansion, so as to provide theoretical and technical support for detonation mechanism and detonation propulsion related field.

Description

Technical Field

[0001] This invention relates to the fields of detonation mechanism and detonation propulsion, and in particular to a method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory. Background Technology

[0002] Detonation waves are shock waves that couple with a violent chemical reaction process and possess self-sustaining propagation characteristics. Zeldovich, Von Neumann, Doering (ZND) and others improved the CJ model in the mid-20th century, proposing the ZND model. The classical ZND model considers the release of chemical thermal energy during detonation wave propagation. The wavefront gas passes through a frozen leader shock wave, then through a slowly reacting induction zone, where the chemical reaction rate increases dramatically, reaching an energy pulse peak before decaying. Finally, the reaction products tend towards equilibrium. The entire interval from the leader shock wave to the chemical reaction equilibrium state constitutes the complete detonation structure. However, the classical ZND model does not consider area expansion, momentum loss, and heat loss during the chemical reaction process; it is an idealized planar detonation model and cannot study the influence of detonation wave curvature on CJ velocity and flow structure.

[0003] Following the detonation wave, an expansion process occurs. Under the influence of the pressure gradient, the products propagate laterally through the expansion wave, ultimately forming a curved detonation wave front. Currently, extensive research is being conducted both domestically and internationally on detonation wave flow field modeling. However, the technology for measuring the radius of curvature of the detonation wave is not yet mature and lacks a connection with the structure of the detonation wave flow field. This results in relatively limited theoretical research on the influence of the flow field structure on the detonation wave propagation process, hindering its engineering application. This study solves for the lateral expansion effect of the detonation wave according to the generalized ZND theory, revealing the universal law governing the influence of curvature on the detonation wave propagation process. It also discovers the suppressive effect of the lateral expansion effect on the propagation velocity of the detonation wave, making the calculated generalized CJ velocity closer to the actual detonation wave velocity, enabling more accurate analysis of detonation wave propagation performance. Simultaneously, research is conducted on the influence of the geometric characteristics of the detonation wave flow field on its propagation, analyzing the impact of the flow field structure. This has profound engineering application value for the overall design and optimization of the detonation engine combustion chamber. Summary of the Invention

[0004] The purpose of this invention is to provide a method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory. This method can accurately calculate the curvature and velocity of detonation waves under stable propagation conditions, thereby providing theoretical and technical support for the fields of detonation mechanism and detonation propulsion.

[0005] The technical solution to achieve the objective of this invention is as follows: Firstly, this invention provides a method for solving the lateral expansion effect of detonation waves based on generalized ZND theory. This method sets a finite number of iteration steps to obtain convergent solutions for the generalized CJ velocity and radius of curvature, including the following steps:

[0006] Step 1: Input the initial state parameters according to the design requirements;

[0007] Step 2: Based on the initial state parameters, calculate the propagation process under different detonation wave velocities to find the generalized CJ velocity;

[0008] Step 3: Calculate the detonation wave propagation process based on the CJ velocity;

[0009] Step 4: Calculate the oblique shock wave deflection angle and Prandtl-Mayer expansion angle based on the back gas state parameters and the upper and lower gas state parameters;

[0010] Step 5: Establish the detonation wave flow field and calculate the interface angle;

[0011] Step 6: Calculate the radius of curvature based on the interface angle and the initially set detonation wave height, and proceed to the next iteration to solve.

[0012] In a second aspect, the present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of the method described in the first aspect.

[0013] Thirdly, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method described in the first aspect.

[0014] Fourthly, the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the method described in the first aspect.

[0015] Compared with the prior art, the significant advantages of the present invention are as follows:

[0016] (1) Compared with the traditional ZND theory, when analyzing the detonation structure, it is no longer restricted by the assumption that there is no momentum loss in the detonation zone. The curvature effect is analyzed by adding an area expansion term to the continuity equation. At the same time, the detonation structure must also be considered when analyzing the steady detonation velocity. When determining the steady detonation state by analyzing the detonation structure, a specific detonation velocity must be selected to solve for a continuous solution across the sound speed singularity, so that the generalized CJ velocity solution is closer to the actual detonation velocity.

[0017] (2) The present invention adopts a detailed chemical reaction mechanism. Compared with the single-step Arrhenius formula of the total reaction, the calculated heat of chemical reaction under adiabatic conditions is closer to the actual chemical reaction result. At the same time, according to the magnitude of the chemical reaction process, the specific heat ratio of the gas changes according to the gas composition and temperature, thereby reducing the error caused by the calculation of the constant specific heat ratio.

[0018] (3) The present invention adopts the numerical solution of oblique shock wave and Prandtl-Mayer expansion wave in fixed coordinate system of detonation wave, avoids rigidity problem in nonlinear equation analysis, optimizes the solution process and improves computational efficiency. Attached Figure Description

[0019] Figure 1 This is a flowchart illustrating a method for solving the lateral expansion effect of detonation waves based on generalized ZND theory, according to an embodiment of the present invention.

[0020] Figure 2 This is a diagram of the detonation wave flow field structure involved in this invention.

[0021] Figure 3 This is a cloud map of the detonation wave flow field involved in this invention.

[0022] Figure 4 This is a comparison chart of the solution method of this invention and the velocity deficit results of Xiao et al.

[0023] Figure 5 This is a comparison chart of the solution method of this invention and the velocity deficit results of Liu et al.

[0024] right Figure 2 The detonation wave flow field structure diagram is marked as follows: U—detonation wave velocity, p1—pressure of premixed combustible gas and upper and lower gas layers, p2e—pressure after detonation wave, p2i—pressure after oblique shock wave, p3e—final pressure of Prandtl-Mayer expansion. Detailed Implementation

[0025] This invention proposes a method for solving the lateral expansion effect of detonation waves based on generalized ZND theory. This method is applicable to calculating propagation parameters such as the radius of curvature, shock structure, and velocity deficit of detonation waves under lateral expansion. This invention can employ different chemical reaction mechanisms and utilizes generalized ZND theory, oblique shock wave relations, Prandtl-Mayer relations, and curvature fitting methods to model and calculate the flow field of detonation waves under weak constraints. The flow field parameters after the detonation wave are solved using generalized ZND theory. The interface angle and shock wave angle of the lateral expansion of the detonation wave are calculated using the oblique shock wave relations and Prandtl-Mayer relations. The curvature of the detonation wave is calculated by fitting the geometric relationship between the interface angle and the detonation wave height, updating the radius of curvature of the generalized ZND model. The detonation wave propagation process is iteratively calculated until the solution converges. The calculation method proposed in this invention is accurate, reliable, and universal. By considering the influence of curvature, it more closely approximates the steady-state solution of the detonation wave under the influence of lateral expansion, thus providing theoretical and technical support for the fields of detonation mechanism and detonation propagation.

[0026] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. The present invention discloses preferred embodiments, but not all embodiments. Other embodiments obtained without inventive step based on the present invention are all within the protection scope of the present invention.

[0027] Combination Figure 1 This invention discloses a method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory, which is further explained here:

[0028] Step 1: Input the initial state parameters according to the design requirements.

[0029] Initial state parameters include: initial temperature of the upper and lower gas layers and the premixed combustible gas, initial pressure (keeping the wavefront flow field consistent), equivalence ratio and detonation wave height.

[0030] At the same time, different chemical reaction mechanism files are invoked according to fuel and oxidant requirements.

[0031] Step 2: Based on the initial state parameters, calculate the propagation process under different detonation wave velocities to find the generalized CJ velocity.

[0032] In the fixed coordinate system of the detonation wave, the flow in the chemical reaction zone is subsonic (η>0). For CJ detonation, the flow at the end of the reaction zone approaches the freezing speed of sound (η=0). The flow in the chemical reaction zone of the detonation wave is the process of gas flow increasing from subsonic to sonic speed behind the Von Neumann surface.

[0033] To investigate the effect of lateral expansion of detonation waves under generalized ZND, a source term for area expansion was added to the governing equations of the reaction zone. The governing equations of the reaction zone containing the area expansion source term are as follows:

[0034] (1)

[0035] (2)

[0036] (3)

[0037] (4)

[0038] in For the pressure in the reaction zone, , Density of the reaction zone The airflow velocity in the reaction zone, It is hot in nature. For acoustic parameters, For the first Mole fraction of components For the first Molar mass of the components For the first Net yield of the component by mass. Thermal properties ( (This indicates that the rate of heat release during a chemical reaction is related to the number of molecules of the products and the rate of the chemical reaction.) This is the source term for the area expansion of the pressure gradient. The relationship between the radius of curvature and the detonation velocity is given by the following formula.

[0039] (5)

[0040] in Let be the radius of curvature. Let U be the detonation wave expansion area, U be the detonation wave propagation speed, j = 1 is a cylindrical wave, and j = 2 is a spherical wave.

[0041] At the end of the reaction zone, the airflow velocity reaches the speed of sound, and the acoustic parameters tend to zero. According to classical CJ ​​theory, the sound velocity surface coincides with the chemical equilibrium surface at which all chemical reactions have fully occurred, which will lead to singular solutions when the chemical reactions reach equilibrium. To eliminate the phenomenon of singular solutions in the governing equations, we assume that the sound velocity surface is reached when the chemical reactions and curvature have the same effect on the detonation wave parameters.

[0042] Considering the influence of curvature, the generalized CJ theory uses the fact that the denominator and numerator of equations (1), (2), and (3) are both 0 as a regularization condition to calculate the expected solution of the detonation wave velocity.

[0043] Given the radius of curvature, the von Neumann airflow state is calculated by assuming the detonation wave velocity. Equations (1), (2), (3), and (4) are integrated. When the airflow velocity reaches the speed of sound, the regularity condition may not be met.

[0044] Adjust the detonation wave velocity and iterate the above process until the CJ velocity that satisfies the regularity condition at the sound speed singularity is obtained.

[0045] Step 3: Calculate the detonation wave propagation process based on the CJ velocity.

[0046] Using the generalized CJ velocity found in step two as the propagation velocity of the non-ideal detonation wave, the state parameters of the gas under different reaction processes are calculated by integrating the control equations of the reaction zone containing the area expansion source term (Equations (1), (2), (3), and (4)).

[0047] Step 4: Calculate the oblique shock wave deflection angle and Prandtl-Mayer expansion angle based on the back gas state parameters and the upper and lower gas state parameters.

[0048] S41. Using the final state of the detonation wave propagation process calculated in step three as the initial value for the expansion process, and calculating different specific volumes, the expansion differential equation from the CJ surface to the slip line is:

[0049] (6)

[0050] in For Prunt-Meyer Point, The Mach number of the expanding gas. Density of expanding gas.

[0051] The discrete form of equation (6) is adopted using the trapezoidal numerical integration formula, and the integral is approximated by multiplying the average of the function values ​​of two adjacent points by the step size. The Prandtl-Mayer angle and the gas state parameters after the expansion wave are determined based on the final steady state of the detonation wave.

[0052] S42. Flow deflection angle in a fixed coordinate system of the detonation wave. The solution method is as follows:

[0053] (7)

[0054] in The angle of the oblique shock wave can be determined by the arcsine of the ratio of the incoming flow normal velocity to the incoming flow velocity. The normal velocity of the gas after the oblique shock wave is... For the incoming flow velocity, The normal velocity of the incoming flow is calculated by taking the arctangent of the ratio of the normal velocity of the gas after the oblique shock wave to the tangential velocity of the gas after the oblique shock wave, which is the angle between the oblique shock wave angle and the flow deflection angle in the latter half of the left side of equation (7).

[0055] In the compression process of oblique shock waves, the normal Mach number of the incoming gas defines the theoretical range of the oblique shock wave flow deflection angle. To prevent the oblique shock wave from degenerating into a Mach wave, the normal Mach number cannot be less than the local speed of sound when the wave angle is at its minimum. Ignoring the issue of oblique shock wave separation, the wave angle is at its maximum when it is a normal shock wave, and the normal velocity of the incoming gas is the propagation speed of the detonation wave. The oblique shock wave flow deflection angle and the gas state parameters after the oblique shock wave are determined based on the final steady-state of the detonation wave.

[0056] Step 5: Establish the detonation wave flow field structure and calculate the interface angle.

[0057] Using the piecewise cubic Hermite interpolation method, a smooth polynomial function is constructed from the discrete data points obtained through the numerical solution in step four. The cubic polynomial constructed in this invention is as follows:

[0058] (8)

[0059] The coefficients ai, bi, ci, and di are determined by the difference estimation of the function value and the derivative value. This is the x-coordinate of the starting point.

[0060] When the pressure ratios on both sides of the slip line are the same, the interpolation value that matches the angle of the oblique shock wave polar curve and the angle of the Prandtl-Mayer expansion wave polar curve is the interface angle of the slip line.

[0061] Step 6: Calculate the radius of curvature based on the interface angle and the initially set detonation wave height, and proceed to the next iteration to solve.

[0062] Combination Figure 2 , Figure 3 By setting the composition and state of the upper and lower gas layers, an idealized flow model of the double-sided expansion of the detonation wave is established, and the interface angles of the upstream and downstream of the detonation wave are calculated. Given the height of the detonation wave, the upstream and downstream slip line functions and the detonation wave front function can be solved. The slip line function and the detonation wave front function intersect at three wave points. Therefore, the peripheral circle tangent to the three linear functions can be solved by the above functional relationship. The radius of the peripheral circle is the radius of curvature of the detonation wave, which can be solved by the radius formula (9).

[0063] (9)

[0064] Where S is the area of ​​the triangular region formed by the intersection of the slip line function and the detonation wave front function, b and c are the lengths of the slip line function in the triangular region, and a is the height of the detonation wave.

[0065] Update the initial radius of curvature and iterate the above process until the maximum step size is reached or a convergent solution is obtained.

[0066] like Figure 4 The figure shows a comparison between the effect of curvature on the detonation wave propagation velocity in the solution method of this invention and the results of Xiao et al. (Qiang Xiao, Matei I. Radulescu, Dynamics of hydrogen–oxygen–argon cellular detonations with a constant mean lateral strain rate. Combustion and Flame, 2020, 215: 437-457.). The example in this invention involves adding inert gas Ar to an H2 / O2 mixture with an equivalence ratio of 2H2 / O2 / 3Ar, an initial pressure of 20 kPa, and an initial temperature of 300 K. The red line represents the calculation results of Xiao et al., and the blue line represents the calculation results of this invention. Verification shows that the results agree well with those of Xiao et al.

[0067] The shock wave angle and interface angle under different equivalence ratios in the solution method of this invention are compared with the experimental results of Zhou Zhulin et al. (Zhou Zhulin, Liu Weidong, Liu Shijie, et al., Self-sustaining mechanism of detonation wave based on lateral expansion influence. Journal of Aerospace Power, 2013, 28(09):1976-1974.). The embodiment is that the initial temperature is 300K and the initial pressure is 101kPa, and the H2 / Air equivalence ratios are 0.80 and 1.00, respectively. As can be seen from Table 1, the error between the experimental results in this paper and the literature is small and within a reasonable range, which verifies the rationality of the numerical solution method for the interface angle of oblique shock waves and Prandtl-Mayer expansion waves.

[0068] Table 1. Verification of Calculation Results of Shock Angle and Interface Angle under Different Equivalence Ratios

[0069]

[0070] like Figure 5The figure shows a comparison between the results of Liu et al. and the results of the solution method of this invention regarding the influence of different detonation wave heights on the detonation wave propagation velocity. The example used was a stoichiometric C10H2O / Air solution with an initial temperature of 300K and an initial pressure of 0.5MPa. The detonation wave heights were 0.03m, 0.035m, and 0.04m. When the detonation wave height was 0.03m, the detonation wave curvature was greater than the critical curvature, indicating a nonphysical solution, which is consistent with the detonation wave quenching results of Liu et al. (Liu Qiuyue, Wang Fang, et al., Layered detonation propagation subject to bilateral expansion: Role of the air layer height. Physics of Fluids, 2025, 37(4): 046125.). In the other cases, the calculated velocity losses were 5.31% and 4.45%, respectively. The red line represents the calculation results of Liu et al., and the blue line represents the calculation results of this invention, showing good agreement with the calculation results of Liu et al.

[0071] The embodiments shown in the accompanying drawings are only some examples, and the present invention is not limited to these embodiments. Those skilled in the art can make improvements within the scope of protection of the present invention.

Claims

1. A method for solving the lateral expansion effect of a detonation wave based on generalized ZND theory, characterized in that, To obtain convergent solutions for the generalized CJ velocity and radius of curvature by setting a finite number of iteration steps, the following steps are included: Step 1: Input the initial state parameters according to the design requirements; Step 2: Based on the initial state parameters, calculate the propagation process under different detonation wave velocities to find the generalized CJ velocity; Step 3: Calculate the detonation wave propagation process based on the CJ velocity; Step 4: Based on the state parameters of the back gas and the upper and lower gas layers, calculate the oblique shock wave deflection angle and Prandtl-Mayer expansion angle, specifically: S41. Using the final state of the detonation wave propagation process calculated in step three as the initial value for the expansion process, and calculating different specific volumes, the expansion differential equation from the CJ surface to the slip line is: （6） wherein is the Prandtl-Meyer angle, is the expansion gas Mach number, is the expansion gas density; Integrating equation (6), the Prandtl-Mayer angle and gas state parameters after the expansion wave are determined based on the detonation wave reaching its final stable state. S42. The theoretical range of the deflection angle of the oblique shock wave is defined by the normal Mach number of the incoming gas; when the wave angle is at its minimum, the normal Mach number cannot be less than the local speed of sound; when the wave angle is at its maximum, it is a normal shock wave; the normal velocity of the incoming gas is the propagation velocity of the detonation wave; the deflection angle of the oblique shock wave and the gas state parameters after the oblique shock wave are determined based on the detonation wave reaching its final steady state. Step 5: Establish the detonation wave flow field and calculate the interface angle; the method for calculating the interface angle is as follows: Using the piecewise cubic Hermite interpolation method, a smooth polynomial function is constructed from the discrete data points obtained. Under the condition that the pressure ratios on both sides of the slip line are the same, the interpolation value that matches the angle of the oblique shock wave polar curve and the angle of the Prandtl-Mayer expansion wave polar curve is the interface angle of the slip line. Step 6: Calculate the radius of curvature based on the interface angle and the initially set detonation wave height, and proceed to the next iteration to solve.

2. The method of claim 1, wherein the generalized ZND theory-based solution of the lateral expansion effect of a detonation wave is characterized by, In step one, the initial state parameters that need to be input include: the initial temperature, initial pressure, equivalence ratio, and detonation wave height of the upper and lower gas layers and the premixed combustible gas.

3. The method of claim 1, wherein the generalized ZND theory based solution of the lateral expansion effect of a detonation wave is characterized by, In steps two and three, the detonation propagation process is solved using the differential governing equations of the area expansion source term. These differential governing equations are: （1） （2） （3） （4） Where p is the pressure in the reaction zone, , Density of the reaction zone The airflow velocity in the reaction zone, For popularity, For acoustic parameters, For the first Mole fraction of components For the first Molar mass of the components For the first Net yield of components; heat This indicates that the rate of heat release during a chemical reaction is related to the number of molecules of the products and the rate of the chemical reaction. This is the source term for the area expansion of the pressure gradient. The relationship between the radius of curvature and the detonation velocity is given by the following formula. （5） wherein is the radius of curvature, is the detonation wave expansion area, is the detonation wave propagation velocity, = 1 is a cylindrical wave, = 2 is a spherical wave.

4. The method of claim 3, wherein, In step two, the determination criterion for the generalized CJ velocity is as follows: Considering the influence of curvature, the generalized CJ theory uses the simultaneous zeroing of the denominator and numerator in equations (1), (2), and (3) as a regularization condition to calculate the expected solution of the detonation wave velocity. Given the radius of curvature, the detonation wave velocity is assumed, and the Von Neumann airflow state is calculated by the target shooting method. When the airflow velocity reaches the speed of sound, solutions that do not meet the regularization condition are eliminated until the generalized CJ velocity solution that satisfies the regularization condition at the sound speed singularity is obtained.

5. The method of claim 1, wherein, In step six, the method for calculating the radius of curvature is as follows: The interface angle of the detonation wave upstream and downstream in step five is calculated, and the height of the detonation wave is known, so the slip line function upstream and downstream and the wave head function of the detonation wave can be solved, and the curvature radius of the detonation wave is solved : （7） Where S is the area of ​​the triangular region formed by the intersection of the slip line function and the detonation wave front function, b and c are the lengths of the slip line function in the triangular region, and a is the height of the detonation wave; Update the initial radius of curvature and iterate the above process until the maximum step size is reached or a convergent solution is obtained.

6. A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps of the method described in any one of claims 1-5.

7. A computer readable storage medium having stored thereon a computer program, characterized in that, The program, when executed by the processor, implements the steps of the method of any one of claims 1-5.

8. A computer program product comprising a computer program, characterized in that, The computer program, when executed by the processor, implements the steps of the method of any one of claims 1-5.

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