Detonation wave lateral expansion effect solving method based on generalized ZND theory

Abstract

The invention discloses a detonation wave lateral expansion effect solving method based on a generalized ZND theory. The method is suitable for calculating propagation parameters such as the curvature radius, the shock wave structure and the velocity loss of detonation waves under the lateral expansion effect. Solving flow field parameters after detonation waves through a generalized ZND theory, calculating an interface angle and a shock wave angle of lateral expansion of the detonation waves in combination with an oblique shock wave relational expression and a Prandtl-Mayer relational expression, fitting and calculating the curvature of the detonation waves through the geometrical relationship between the interface angle and the height of the detonation waves, updating the curvature radius of a generalized ZND model, and calculating the curvature of the detonation waves. And carrying out iterative calculation on the detonation wave propagation process until the solution is converged. The calculation method provided by the invention is accurate and reliable, has universality, and is closer to a steady-state solution of the detonation wave influenced by lateral expansion by considering the curvature influence, so that theoretical and technical supports are provided for the detonation mechanism and detonation propulsion related fields.

Description

Technical Field

[0001] The present invention relates to the field related to 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 Art

[0002] A detonation wave is a shock wave characterized by self-sustaining propagation, resulting from the coupling of a shock discontinuity and a violent chemical reaction. Zeldovich, von Neumann, Doering, and others (ZND) improved the CJ model in the mid-20th century and proposed the ZND model. The classic ZND model considers the release of chemical heat energy during detonation wave propagation. As the wavefront gas passes through a frozen leading shock wave and an induction zone where chemical reactions are slow, the chemical reaction rate increases dramatically, reaching a peak energy pulse before decaying. Finally, the reaction products approach equilibrium, and the entire region from the leading shock wave to the chemical reaction equilibrium constitutes the complete detonation structure. The classic ZND model does not account for area expansion, momentum, or heat loss during the chemical reaction process. As an idealized planar detonation model, it cannot investigate the effects of detonation wave curvature on CJ velocity and flow structure.

[0003] A detonation wave is followed by an expansion process. Under the influence of a pressure gradient, the product propagates laterally through the expansion wave, ultimately forming a curvature-laden detonation wave front. Extensive research has been conducted both domestically and internationally on detonation wave flow field modeling. However, the measurement technology for the detonation wave curvature radius is immature and lacks a link to the detonation wave flow field structure. This has led to relatively little theoretical research on the influence of detonation wave flow field structure on detonation wave propagation, hindering its widespread engineering application. Solving the lateral expansion effect of detonation waves in the generalized ZND theory reveals the universality of the influence of curvature on detonation wave propagation and discovers the inhibitory effect of the lateral expansion effect on the propagation velocity of detonation waves. This results in a calculated generalized CJ velocity that is closer to the true detonation wave velocity, enabling more accurate analysis of detonation wave propagation performance. Furthermore, research has been conducted on the influence of the geometric characteristics of the detonation wave flow field on detonation wave propagation. This analysis of the influence of the detonation wave flow field structure has profound engineering application value for the overall design and optimization of detonation engine combustion chambers. Summary of the Invention

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

[0005] The technical solution for achieving the objectives of the present invention is as follows: In a first aspect, the present invention provides a method for solving the lateral expansion effect of a detonation wave based on the generalized ZND theory. The method sets a finite number of iteration steps to obtain a converged solution for the generalized CJ velocity and curvature radius, comprising 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 at 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 flow deflection angle and Prandtl-Meyer expansion angle based on the gas state parameters behind the wave and the gas state parameters of the upper and lower layers;

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

[0011] Step 6: Based on the interface angle and the initially set detonation wave height, calculate the curvature radius and enter the next loop for solution.

[0012] In a second aspect, the present invention provides a computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the method described in the first aspect when executing the program.

[0013] In a third aspect, 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] In a fourth aspect, the present invention provides a computer program product, comprising a computer program, which implements the steps of the method described in the first aspect when executed by a processor.

[0015] Compared with the prior art, the present invention has the following significant advantages:

[0016] (1) Compared with the traditional ZND theory, the analysis of detonation structure is no longer restricted by the assumption that there is no momentum loss in the detonation zone. The influence of curvature is analyzed by adding an area expansion term to the continuity equation. At the same time, the analysis of the stable detonation velocity must also take the detonation structure into consideration. When determining the stable detonation state by analyzing the detonation structure, a specific detonation velocity must be selected to maintain a continuous solution across the sonic 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 package reaction, the calculated chemical reaction heat under adiabatic conditions is closer to the actual chemical reaction result; at the same time, according to the size of the chemical reaction process, the gas specific heat ratio changes according to the gas composition and temperature, thereby reducing the error caused by the calculation of the fixed specific heat ratio.

[0018] (3) The present invention adopts the numerical solution of oblique shock wave and Prandtl-Meyer expansion wave in the fixed coordinate system of detonation wave, which avoids the rigidity problem in the process of analyzing nonlinear equations, optimizes the solution process and improves the calculation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

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

[0022] Figure 4 A comparison chart of the velocity loss results between the solution method of the present invention and Xiao et al.

[0023] Figure 5 This is a comparison chart of the velocity loss results between the solution method of the present invention and that 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 layer gases, p2e—pressure after the detonation wave, p2i—pressure after the oblique shock wave, and p3e—final pressure of the Prandtl-Meyer expansion. DETAILED DESCRIPTION

[0025] The present invention proposes a method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory. The method is suitable for calculating propagation parameters such as the curvature radius, shock wave structure, and velocity loss of detonation waves under the action of lateral expansion. The present invention can adopt different chemical reaction mechanisms and use methods such as the generalized ZND theory, oblique shock wave relationship, Prandtl-Meyer relationship, and curvature fitting to model and calculate the detonation wave flow field under weak constraints. The generalized ZND theory is used to solve the flow field parameters after the detonation wave, and the oblique shock wave relationship and Prandtl-Meyer relationship are combined to calculate the interface angle and shock wave angle of the lateral expansion of the detonation wave. The detonation wave curvature is calculated by fitting the geometric relationship between the interface angle and the detonation wave height, and the curvature radius of the generalized ZND model is updated. The detonation wave propagation process is iteratively calculated until the solution converges. The calculation method proposed by the present invention is accurate, reliable, and universal. By considering the influence of curvature, it is closer to the steady-state solution of the detonation wave affected by lateral expansion, thereby providing theoretical and technical support for the fields related to detonation mechanism and detonation propulsion.

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

[0027] Combine Figure 1 The present 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] The initial state parameters include: initial temperature of upper and lower gas layers and premixed combustible gas, initial pressure (wavefront flow field remains consistent), equivalence ratio and detonation wave height.

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

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

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

[0033] In order to study the effect of lateral expansion of detonation waves under generalized ZND, the area expansion source term is added to the reaction zone governing equation. The reaction zone governing equation including the area expansion source term is as follows:

[0034]

[0035] Where p is the pressure in the reaction zone, M is the incoming flow Mach number, ρ is the density in the reaction zone, and w is the airflow velocity in the reaction zone. is the thermal property, η is the acoustic parameter, Y k is the mole fraction of the kth group component, W k is the molar mass of the kth group component, is the mass net yield of the kth group of components. It indicates the speed of heat release during chemical reaction, which is related to the number of molecules of the product and the chemical reaction rate. wα is the source term of the area expansion of the pressure gradient. The relationship between α, curvature radius and detonation velocity is given by the following formula

[0036]

[0037] where R s is the radius of curvature, A is the expansion area of ​​the detonation wave, U is the propagation velocity of the detonation wave, when j=1 it is a cylindrical wave, and when j=2 it is a spherical wave.

[0038] At the end of the reaction zone, the gas velocity reaches the speed of sound, and the acoustic parameters approach zero. According to classical CJ ​​theory, the sonic surface coincides with the chemical equilibrium surface, where all chemical reactions are complete, leading to singular solutions when the chemical reactions reach equilibrium. To eliminate singular solutions in the governing equations, we assume that the sonic surface is reached when chemical reactions and curvature have equal effects on the detonation wave parameters.

[0039] Considering the influence of curvature, the generalized CJ theory uses the denominator and numerator of equations (Equation 1), (Equation 2), and (Equation 3) to be zero at the same time as the regularity condition to calculate the expected solution of the detonation wave velocity.

[0040] When the curvature radius is known, the post-von Neumann airflow state is calculated by assuming the detonation wave velocity. When equations (1), (2), (3), and (4) are integrated, the regularity condition may not be met when the airflow velocity reaches the speed of sound.

[0041] The detonation wave velocity is adjusted and the above process is iterated until the CJ velocity that satisfies the regularity condition at the sonic singular point is obtained.

[0042] Step 3: Calculate the detonation wave propagation process based on the CJ velocity.

[0043] The generalized CJ velocity found in step 2 is used as the non-ideal detonation wave propagation velocity. By integrating the reaction zone control equations (Equation 1), (Equation 2), (Equation 3), and (Equation 4) containing the area expansion source term, the state parameters of the gas under different reaction processes are calculated.

[0044] Step 4: Calculate the oblique shock wave flow deflection angle and Prandtl-Meyer expansion angle based on the gas state parameters behind the wave and the gas state parameters of the upper and lower layers.

[0045] S41. The final state of the detonation wave propagation process calculated in step 3 is used as the initial value of the expansion process. By calculating different specific volumes, the expansion differential equation from the CJ surface to the slip line is:

[0046]

[0047] Where δ is the Prandtl-Meyer angle, M 3e is the Mach number of the expanding gas, ρ 3e Density of the expanding gas.

[0048] The discrete form of the trapezoidal numerical integration formula is used for Equation 6, and the integral is approximated by multiplying the average value of the function value between two adjacent points by the step size. The Prandtl-Meyer angle and the gas state parameters after the expansion wave are determined based on the final stable state of the detonation wave.

[0049] S42. In the fixed detonation wave coordinate system, the solution for the flow deflection angle δ is:

[0050]

[0051] Where β is the oblique shock wave angle, which can be determined by the arc sine of the ratio of the normal velocity of the incoming flow to the incoming flow velocity, w 2i is the normal velocity of the gas after the oblique shock wave, u1 is the incoming flow velocity, w 1i is the normal velocity of the incoming flow. The second half of the left side of (Eq. 7) is the angle between the oblique shock wave angle and the flow deflection angle. It is calculated by the inverse tangent 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.

[0052] During the oblique shock compression process, the normal Mach number of the incoming gas defines the theoretical range of the oblique shock flow deflection angle. To prevent the oblique shock wave from degenerating into a Mach wave, the normal Mach number must not be less than the local speed of sound at the minimum angle. Ignoring oblique shock shedding, the maximum angle represents a normal shock wave, and the normal velocity of the incoming gas is the propagation velocity of the detonation wave. The oblique shock flow deflection angle and post-oblique shock gas state parameters are determined based on the detonation wave reaching its final steady state.

[0053] Step 5: Establish the detonation wave flow field structure and calculate the interface angle.

[0054] The piecewise cubic Hermite interpolation method is used to construct a smooth polynomial function using the discrete data points solved by the numerical solution in step 4. The cubic polynomial constructed by the present invention is:

[0055] f(x)=a i (xx i )3 +b i (xx i ) 2 +c i (xx i )+d i #(Equation 8)

[0056] The coefficients ai, bi, ci, and di are determined by the difference between the function value and the derivative value, and x i The horizontal coordinate value of the starting point.

[0057] When the pressure ratio on both sides of the slip line is the same, the interpolation value that matches the angle of the oblique shock wave pole curve and the angle of the Prandtl-Meyer expansion wave pole curve is the interface angle of the slip line.

[0058] Step 6: Based on the interface angle and the initially set detonation wave height, calculate the curvature radius and enter the next loop for solution.

[0059] Combine Figure 2 、 Figure 3 By setting the components and states 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. The upstream and downstream sliding line functions and the detonation wave head function can be solved when the detonation wave height is known. The sliding line function and the detonation wave head function intersect at the three wave points. Therefore, the lateral circle tangent to the three linear functions can be solved through the above functional relationship. The radius of the lateral circle is solved by the radius formula (Equation 9) to be the curvature radius of the detonation wave.

[0060]

[0061] Where S is the area of ​​the triangular region formed by the intersection of the slip line function and the detonation wave head function, b and c are the lengths of the slip line function in the triangular region, and a is the detonation wave height.

[0062] Update the initial curvature radius and iterate the above process until the maximum step size is reached or a converged solution is obtained.

[0063] like Figure 4As shown, this is a comparison chart of the effect of curvature on the propagation velocity of detonation waves in the solution method of the present 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 embodiment is the addition of inert gas Ar to the H2 / O2 mixture, the equivalence ratio is 2H2 / O2 / 3Ar, the initial pressure is 20 kPa, and the initial temperature is 300 K. The red line is the calculation result of Xiao et al., and the blue line is the calculation result of the present invention. It has been verified that the calculation results of Xiao et al. are in good agreement.

[0064] The shock wave angle and interface angle at different equivalence ratios in the present method are compared with the experimental results of Zhou Zhulin et al. (Zhou Zhulin, Liu Weidong, Liu Shijie, et al., Self-sustaining mechanism of detonation waves based on the influence of lateral expansion. Journal of Aerospace Power, 2013, 28(09):1976-1974). In this example, the initial temperature is 300K, the initial pressure is 101kPa, and the H2 / Air equivalence ratio is 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 relatively small and within a reasonable range, verifying the rationality of the numerical solution for the interface angle of oblique shock waves and Prandtl-Meyer expansion waves.

[0065] Table 1 Verification of calculation results of shock wave angle and interface angle at different equivalence ratios

[0066]

[0067] like Figure 5As shown in the figure, the effect of different detonation wave heights on the detonation wave propagation velocity in the solution method of the present invention is compared with the results of Liu et al. The embodiment is an initial temperature of 300K, an initial pressure of 0.5MPa, and a stoichiometric C10H20 / Air. The detonation wave heights are 0.03m, 0.035m, and 0.04m respectively. When the detonation wave height is 0.03m, the detonation wave curvature is greater than the critical curvature, which is a non-physical solution and is consistent with the detonation wave extinction 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.). The velocity losses calculated in other cases are 5.31% and 4.45% respectively. The red line is the calculation result of Liu et al., and the blue line is the calculation result of the present invention, which is in good agreement with the calculation result of Liu et al.

[0068] The embodiments in the drawings described above are only some embodiments, and the present invention is not limited to the above 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 detonation waves based on the generalized ZND theory, characterized in that: Setting a finite number of iteration steps to obtain a converged solution for the generalized CJ velocity and curvature radius includes the following steps: Step 1: Input the initial state parameters according to the design requirements; Step 2: Based on the initial state parameters, calculate the propagation process at 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: Calculate the oblique shock wave flow deflection angle and Prandtl-Meyer expansion angle based on the gas state parameters behind the wave and the gas state parameters of the upper and lower layers; Step 5: Establish the detonation wave flow field and calculate the interface angle; Step 6: Based on the interface angle and the initially set detonation wave height, calculate the curvature radius and enter the next loop for solution.

2. The method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory according to claim 1 is characterized in that: In step 1, the initial state parameters that need to be input include: initial temperature, initial pressure, equivalence ratio and detonation wave height of the upper and lower gas layers and premixed combustible gas.

3. The method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory according to claim 1 is characterized in that: In steps 2 and 3, the detonation propagation process is solved by the differential governing equations of the area expansion source term, which are: Where p is the pressure in the reaction zone, M is the incoming flow Mach number, ρ is the density in the reaction zone, and w is the airflow velocity in the reaction zone. is the heat, η is the acoustic parameter, Y k is the mole fraction of the kth group component, W k is the molar mass of the kth group component, is the mass net output rate of the kth group of components; It indicates the speed of heat release during the chemical reaction, which is related to the number of molecules of the product and the chemical reaction rate; wα is the source term of the area expansion of the pressure gradient, and the relationship between α, curvature radius, and detonation velocity is given by the following formula where R s is the radius of curvature, A is the expansion area of ​​the detonation wave, U is the propagation velocity of the detonation wave, when j=1 it is a cylindrical wave, and when j=2 it is a spherical wave.

4. The method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory according to claim 3 is characterized in that: In step 2, the generalized CJ velocity is determined based on: Considering the influence of curvature, the generalized CJ theory uses the denominator and numerator of equations (Equation 1), (Equation 2), and (Equation 3) to be zero as a regularity condition to calculate the expected solution of the detonation wave velocity. When the curvature radius is known, the detonation wave velocity is assumed and the post-Von Neumann airflow state is calculated by the shooting method. When the airflow velocity reaches the speed of sound, the solutions that do not meet the regularity condition are eliminated until a generalized CJ velocity solution that meets the regularity condition at the sonic singularity is obtained.

5. The method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory according to claim 1 is characterized in that: In step 4, the method for calculating the oblique shock wave flow deflection angle and the Prandtl-Meyer expansion angle includes the following steps: S41. The final state of the detonation wave propagation process calculated in step 3 is used as the initial value of the expansion process. By calculating different specific volumes, the expansion differential equation from the CJ surface to the slip line is: Where δ is the Prandtl-Meyer angle, M 3e is the Mach number of the expanding gas, ρ 3e Density of the expanding gas; Integrate Equation 6 and determine the Prandtl-Meyer angle and the gas state parameters after the expansion wave according to the detonation wave reaching the final stable state; S42. The theoretical range of the oblique shock wave flow deflection angle is defined by the normal Mach number of the incoming gas flow. When the wave angle is minimum, the normal Mach number cannot be less than the local sound speed. When the wave angle is maximum, it is a normal shock wave, and the normal velocity of the incoming gas flow 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 detonation wave reaching the final stable state.

6. The method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory according to claim 1 is characterized in that: In step 5, the method for calculating the interface angle is as follows: The piecewise cubic Hermite interpolation method is used to construct a smooth polynomial function from the discrete data points. When the pressure ratio on both sides of the slip line is the same, the interpolation value that matches the angle of the oblique shock wave pole curve and the angle of the Prandtl-Meyer expansion wave pole curve is used as the interface angle of the slip line.

7. The method for solving the lateral expansion effect of detonation waves based on the generalized ZND theory according to claim 1 is characterized in that: In step six, the curvature radius is calculated as follows: Calculate the interface angles upstream and downstream of the detonation wave in step 5. Knowing the height of the detonation wave, the upstream and downstream sliding line functions and the detonation wave head function can be solved to solve the curvature radius R of the detonation wave. s : Where S is the area of ​​the triangle region formed by the intersection of the slip line function and the detonation wave head function, b and c are the lengths of the slip line function in the triangle region, and a is the detonation wave height; Update the initial curvature radius and iterate the above process until the maximum step size is reached or a converged solution is obtained.

8. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the method according to any one of claims 1 to 7 are implemented.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps of the method according to any one of claims 1 to 7 are implemented.

10. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 7 are implemented.

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