Method for solving zero-order aerodynamic parameters after oblique detonation wave based on gradient descent algorithm

Abstract

The invention discloses a method for solving zero-order aerodynamic parameters after oblique detonation waves based on a gradient descent algorithm, and belongs to the technical field of aerodynamics. According to the method, an iterative objective function is constructed on the basis of a control equation of a single-step chemical reaction, and a derivative of the objective function is solved. And solving to obtain a post-wave mass fraction and other aerodynamic parameters through a Newton iteration method. Compared with two different examples, the method provided by the invention is accurate and effective, and has a remarkable advantage in calculation speed. Furthermore, the method can also be used as an analysis means to research the mutual influence relationship among all parameters in detonation. In addition to forward solving, the method can also be developed into a reverse design means, and a corresponding wall angle is solved under the condition that a wave angle is given. Therefore, the method can be used as a quick and effective calculation and analysis tool under the condition that detonation gas composition is not concerned and detonation post-wave aerodynamic parameters are more concerned, and has a wide application prospect.

Description

Technical Field

[0001] The present invention belongs to the technical field of aerodynamic calculation, and particularly relates to a method for solving zero-order aerodynamic parameters after an oblique detonation wave based on a gradient descent algorithm. Background Technique

[0002] Detonation is an extreme combustion phenomenon in which an induced shock wave is tightly coupled with a chemical reaction. Therefore, the detonation process couples the interaction of hydrodynamic characteristics, shock waves, and thermochemical processes. To describe the chemical reaction process in detonation, there are two categories and three types of commonly used chemical reaction models. The first category is the simplified chemical reaction model, including the single-step chemical reaction model and the two-step chemical reaction model. The second category is the elementary / detailed chemical reaction model. Although the specific reaction processes of the three chemical reaction models are different, they all release heat based on a finite reaction rate (Arrhenius formula) and can capture the real detonation wave structure. In the calculation of detonation waves, generally, which model to use can be comprehensively considered according to the physical characteristics of the object under study, the research purpose, and the calculation ability.

[0003] There are usually three methods for solving the physicochemical processes with strong coupling between shock waves and chemical reactions. The first method is to assume that the specific heat ratio behind the wave is known. At the same time, the heat release in the chemical reaction of the shock wave is also known (Hu Z M, Zhou K, Peng J, et al. Shock relations in gases of heterogeneous thermodynamic properties[J]. Science China Technological Sciences, 2017, 60: 1050-1057). This method is simple and has a clear analytical solution. However, this method of pre-assuming the gas properties and heat release behind the wave is not accurate enough and is greatly affected by subjective selection. The second method is to solve based on chemical equilibrium, that is, to assume an infinite chemical reaction rate (Olivier H. A theoretical model for the shock stand-off distance in frozen and equilibrium flows[J]. Journal of Fluid Mechanics, 2000, 413: 345-353). In the calculation process, specific chemical reactions are not considered, but only the gas components at chemical equilibrium are concerned. A set of chemical equilibrium composition (CEC) programs developed by the National Aeronautics and Space Administration (NASA) is based on chemical equilibrium. Zhang Zijian (Zhang Z, Chihyung W E N, Zhang W, et al. A theoretical method for solving shock relations coupled with chemical equilibrium and its applications[J]. Chinese Journal of Aeronautics, 2022, 35(6): 47-62) has further developed a method for solving the oblique shock relations of chemical equilibrium on this basis. The third method is to solve the real chemical reaction kinetic process (Shi L, Shen H, Zhang P, et al. Assessment of vibrational non-equilibrium effect on detonation cell size[J]. Combustion Science and Technology, 2017, 189(5): 841-853). The advantage of this method is that it conforms to the real physicochemical process, and the disadvantage is that the calculation is too complex when using an overly complex chemical reaction model).

[0004] Existing methods are difficult to balance accuracy and efficiency, especially unable to provide a fast analysis tool for engineering applications. Therefore, there is an urgent need for a method to solve the parameters after the detonation wave that can not only reflect the dynamic characteristics of chemical reactions but also have high computing power. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for solving the zero-order aerodynamic parameters after an oblique detonation wave based on the gradient descent algorithm to achieve the solution of the aerodynamic parameters after the detonation wave. Considering the advantages of single-step / two-step chemical reactions compared with elementary / detailed chemical reactions and combining with the third solution method, a solution method based on single-step chemical reactions can be constructed. By constructing the objective function and its derivative and combining with the gradient descent algorithm to achieve efficient iteration, the problems of complex calculation and slow convergence of traditional methods are solved, and at the same time, forward solution and reverse design are supported. This solution method can take into account relatively real physicochemical processes without overly complex computational complexity, and can provide an effective computational means for the study of detonation waves.

[0006] To achieve the above-mentioned invention purpose, the present invention provides the following technical solutions.

[0007] A method for solving the zero-order aerodynamic parameters after an oblique detonation wave based on the gradient descent algorithm, comprising the following steps:

[0008] 1) According to the control equations of detonation, a theoretical solution method is obtained through mathematical derivation, and the theoretical calculation results are compared with the simulation calculation results to verify the correctness of the theoretical solution method; the control equations of detonation are as follows:

[0009] (1)

[0010] (2)

[0011] (3)

[0012] (4)

[0013] Among them, equations (1), (2), (3), and (4) are respectively the mass conservation, momentum conservation, energy conservation, and chemical reaction equations of detonation; the four equations form the control equation set of detonation; in the formula, subscripts 1 and 2 respectively represent before and after the detonation wave, represents the density before the wave, represents the density after the wave; represents the normal velocity before the wave, represents the normal velocity after the wave; represents the pressure before the wave, represents the pressure after the wave; represents the enthalpy value before the wave, represents the enthalpy value after the wave; is the energy release of a chemical reaction; represents the pre-wave temperature, represents the post-wave temperature; the chemical reaction process before the wave is usually 1, represents the chemical reaction process after the wave; refers to the pre-exponential factor, generally a constant; is the activation energy; is the pre-wave gas constant, is the post-wave gas constant; among them, the pressure can be calculated according to the ideal gas state equation:

[0014] (5)

[0015] The calculation formula for the enthalpy value is as follows:

[0016] (6)

[0017] Among them, is the specific heat ratio of the pre-wave gas, is the specific heat ratio of the post-wave gas; the energy release relationship with the chemical reaction process Z:

[0018] (7)

[0019] Among them, represents all the energy released when the chemical reaction is completely completed; the calculation formula for the normal velocity is as follows:

[0020] (8)

[0021] Among them, is the angle of the wedge surface, is the angle of the detonation wave;

[0022] 2) According to the control equation of detonation, select the objective function and take its derivative; specifically including:

[0023] 2.1) Dividing equation (4) by equation (1) can obtain:

[0024] (9)

[0025] Rewrite equation (9) from an equation into the form of a function:

[0026] (10)

[0027] Observing equation (10), it can be known that the objective function f is related to multiple variables. The present invention selects the chemical reaction process Taking it as the independent variable and the remaining variables as the dependent variables; thus, solving the problem of aerodynamic parameters after the detonation wave can be transformed into solving the solution of the function f with respect to the independent variable;

[0028] 2.2) Solve the total derivative of the function f with respect to the independent variable :

[0029] Since there is more than one variable in the function f, according to the chain rule of differentiation, solve the total derivative of the function f with respect to the independent variable ;

[0030] (11)

[0031] Among them, the partial derivative of the function f with respect to the independent variable is:

[0032] (12)

[0033] The partial derivative of the function f with respect to the variable is:

[0034] (13)

[0035] The partial derivative of the variable with respect to is:

[0036] (14)

[0037] The partial derivative of the variable with respect to is:

[0038] (15)

[0039] The partial derivative of the variable with respect to is:

[0040] (16)

[0041] The partial derivative of the variable with respect to is:

[0042] (17)

[0043] The partial derivative of the variable with respect to is:

[0044] (18)

[0045] The partial derivative of the function f with respect to the variable The partial derivative of is:

[0046] (19)

[0047] So far, substituting the above calculation formulas (12)-(19) of each derivative into equation (11) can complete the solution of the total derivative of the function f with respect to the independent variable ;

[0048] 3) After solving the objective function and its derivative, according to the idea of the Newton iteration method, construct the iterative solution format of function (10);

[0049] (20)

[0050] where, is the value of the independent variable of the function at the k-th time, represents the value of the independent variable at the (k + 1)-th time, represents the damping factor, a value set to prevent the iteration from being too oscillatory, generally taking a value between [0, 1]; is the objective function, is the derivative of the objective function;

[0051] 4) After having the iterative solution format of function f, the aerodynamic parameters after the detonation wave can be solved for the given calculation conditions; the calculation conditions should include the following parameters: the incoming flow pressure , the incoming flow Mach number , the incoming flow temperature , the incoming flow velocity , the chemical reaction progress , the specific heat ratio of the incoming flow , the wall wedge angle , the total heat release of the gas , the activation energy of the chemical reaction , the pre-exponential factor of the chemical reaction and other parameters;

[0052] 5) After knowing the calculation conditions, arbitrarily give an initial chemical reaction progress , and based on this initial value, solve the next chemical reaction progress value according to equation (16); among them, the objective function is calculated according to equation (6), and the derivative is calculated according to equation (7); the initial chemical reaction progress generally takes a value between [0, 1];

[0053] 6) Calculate the absolute value of the difference between the updated chemical reaction progress and the chemical reaction progress of the previous step;

[0054]

[0055] Error is used to measure the difference between two iterations of the independent variable, and is generally used as a basis for judging whether it has converged;

[0056] 7) Given the convergence condition C, set the convergence condition according to the accuracy required by the specific problem. Generally, the convergence condition is set to a smaller value;

[0057] 8) Combined with the convergence condition of the iterative difference 7) in 6), determine whether the calculation has reached convergence; when the error is less than the convergence condition C, it is determined that the calculation has reached convergence and the iteration is stopped; otherwise, it has not reached convergence and needs to continue iteration;

[0058] 9) If the convergence condition is not met, the updated independent variable Bring it into step 5), and repeat 5)-8) until convergence;

[0059] 10) If convergence has been reached, the loop ends and the independent variable at this time is Substitute the control equations (1)-(4) to calculate the aerodynamic parameters after the detonation wave.

[0060] Compared with the prior art, the present invention has the following advantages:

[0061] 1. Aiming at the problem of solving shock wave coupled chemical reaction, the present invention proposes a calculation method based on Newton iteration method and gradient descent algorithm. By calculating the objective function and its derivative, an iterative relationship is constructed to realize efficient iterative solution of aerodynamic parameters after detonation wave. It effectively solves the problem of solving shock wave coupled chemical reaction, i.e. detonation wave.

[0062] 2. The present invention has a fast computing speed, breaking the dilemma of complex traditional calculations and slow convergence, and can quickly output parameters to meet the rapid analysis needs in the fields of aerospace, energy and power, and greatly shorten the research and development cycle of related equipment and systems. At the same time, the present invention combines single-step chemical reactions with real physical and chemical processes to avoid errors caused by subjective assumptions or neglect of details in traditional methods, providing reliable data for the design optimization of aircraft engines, missile engines, etc., and ensuring optimal product performance.

[0063] 3. Compared with computer simulation calculations, the present invention overcomes the shortcomings of long solution time and failure to obtain a unique theoretical solution, realizes the integration of theory and practice, and not only helps academic research, but also provides optimization solutions for engineering applications. In addition, the present invention supports forward solution and reverse design, and can be widely used in aerospace, energy and power, national defense and military, industrial combustion and other fields, providing a comprehensive solution to the detonation wave problem, and has a good guiding significance for the solution and analysis of detonation waves. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 Schematic diagram of an embodiment of the present invention.

[0065] Figure 2 Flowchart of an embodiment of the present invention.

[0066] Figure 3 Iterative solution process diagram provided by an embodiment of the present invention.

[0067] Figure 4 Simulation result extraction diagram provided by an embodiment of the present invention. Detailed implementation manners

[0068] The following will describe in detail the specific implementation manners of the present invention in conjunction with the attached drawings. For purposes of explanation rather than limitation, specific details are set forth in the following description to help understand the present invention comprehensively.

[0069] The following embodiments will describe in detail the calculation method for solving the parameters after the detonation wave according to the present invention in conjunction with the attached drawings:

[0070] Refer to Figure 1 , this figure is a schematic diagram of the problem of solving the aerodynamic parameters after the detonation wave provided by the present invention. Among them, the red solid line represents the detonation wave, and each variable with subscript 1 represents the oncoming flow conditions of the air flow, and each variable with subscript 2 represents the aerodynamic parameters of the air flow after the wave. Therefore, the problem can be described as solving parameters such as under the conditions of known oncoming flow pressure , oncoming flow velocity , density , oncoming flow temperature , chemical reaction progress , wall wedge angle and other parameters;

[0071] Refer to Figure 2 , this figure is a flowchart of the theoretical solution method for the aerodynamic parameters after the detonation wave provided by the present invention;

[0072] Step 1. Given the inflow parameters and chemical reaction parameters, the following calculation conditions are given according to the inflow parameters and chemical reaction parameters in the literature (Wenshuo Zhang, Zijian Zhang, Zonglin Jiang, Xin Han, Yunfeng Liu, Chun Wang; Numerical investigation of free oblique detonation wave induced by non-intrusive energy deposition. AIP Advances 1 December 2021; 11 (12): 125119. https: / / doi.org / 10.1063 / 5.0073035):

[0073]

[0074] In the formula, represents the inflow pressure, represents the inflow temperature, represents the density, represents the inflow velocity, represents the total heat release of the gas, represents the activation energy of the chemical reaction, represents the wall wedge angle.

[0075] Step 2. Arbitrarily give an initial chemical reaction process .

[0076] Step 3. Combine the calculation conditions given in Step 1 and the initial independent variables given in Step 2 to solve the governing equations (1)-(4). Obtain the aerodynamic parameters after the initial detonation wave:

[0077]

[0078] Step 4. Solve the objective function (6) and its derivative (7) at this time based on Step 3. The objective function , and the derivative .

[0079] Step 5. According to the objective function (6) and its derivative (7), combined with the iterative update formula (16), solve the independent variables for the next step. Among them, the damping factor is given as 0.5. Calculate the independent variables for the next step as .

[0080] Step 6. Calculate the updated independent variable and the chemical reaction process The difference between them. Through calculation, it can be known that error = 0.3.

[0081] Step 7: Given the convergence condition. To ensure the accuracy of the calculation, in this example, the given convergence condition C is 1×10 -5 . Only when the difference between independent variables is less than this convergence condition will the iteration terminate.

[0082] Step 8: Compare that the difference between independent variables calculated in Step 6 is less than the convergence condition given in Step 7, 0.3 > 1×10 -5 . Therefore, the convergence condition is not reached.

[0083] Step 9: Continue the iterative solution, using as the initial independent variable for the next iteration, and repeat Steps 2 to 8 until the convergence condition is reached. After calculation, the iteration reaches the convergence condition at the 17th step. The complete iterative flowchart is as shown in Figure 3 .

[0084] Step 10: After convergence, solve the remaining parameters behind the detonation wave according to the converged independent variables. According to the calculation results, the converged parameters behind the detonation wave are:

[0085]

[0086] The following uses the theoretical calculation method of the parameters behind the detonation wave proposed by the present invention and compares it with the computer simulation results. The effectiveness of this method is illustrated by comparing the errors of the results. The oncoming flow parameters calculated by computer simulation are consistent with the parameters in the specific implementation manner of the present invention. The calculated results are shown in Figure 4 . Extract the results of the simulation calculation and the theoretical calculation and compare them in Table 1. It is not difficult to see from Table 1 that the errors between the aerodynamic parameters behind the detonation wave solved by the theoretical calculation method proposed by the present invention and the results obtained by computer simulation are small. Therefore, the calculation method proposed in this paper is feasible.

[0087] Table 1

[0088]

[0089] In addition, in terms of the calculation time consumption, Table 2 compares the number of solution steps and the calculation time consumption of the computer simulation calculation and the solution method proposed by the present invention. It is not difficult to find by comparison that the number of solution steps of the method proposed by the present invention is only 0.0497% of that of the computer simulation, and the calculation time consumption is only 0.0259% of that of the computer simulation.

[0090] Table 2

[0091]

[0092] Through the above comparative analysis, it can be seen that the theoretical calculation method of post-detonation wave parameters proposed by the present invention has the advantages of high accuracy and short calculation time compared with the computer simulation calculation method, indicating the effectiveness and superiority of the theoretical calculation method of post-detonation wave parameters provided by the present invention, and having good guiding significance for related research such as the solution and analysis of detonation waves.

[0093] In summary, the present invention combines mathematical algorithms with practical technical problems in the physical field. By constructing a solution method based on single-step chemical reactions and combining the gradient descent algorithm to achieve efficient iteration, the calculation speed is improved, and the aerodynamic parameters after the detonation wave can be quickly obtained, meeting the requirements for a rapid analysis tool in engineering applications. For example, in the real-time monitoring and adjustment of the combustion process of aeroengines, key parameters can be quickly provided to assist in decision-making. The present invention considers a relatively real physicochemical process and can more accurately reflect the actual state after the detonation wave and obtain more accurate aerodynamic parameters compared with methods that assume known post-wave parameters or only solve based on chemical equilibrium. The present invention makes up for the deficiencies of computer simulation calculations, obtains a unique theoretical solution, provides a reliable theoretical basis and calculation means for the research and analysis of detonation waves, helps to deeply understand the physical mechanism of detonation waves, and promotes the development of related theories in aerodynamics. This method can be applied to fields such as aeroengine design and aircraft research in aerospace, internal combustion engine and gas turbine R & D in energy and power, and chemical production and industrial furnace combustion optimization in industrial combustion.

[0094] The above embodiments are only preferred embodiments of the present invention and should not be considered as limiting the scope of implementation of the present invention. All equivalent changes and improvements made in accordance with the scope of the application of the present invention shall still fall within the scope covered by the patent of the present invention.

Claims

1. A method for solving the zero-order aerodynamic parameters after oblique detonation wave based on gradient descent algorithm, characterized in that The following steps are involved: 1) According to the control equation of knock, a theoretical solution method is obtained through mathematical deduction, and the theoretical calculation results are compared with the simulation calculation results to verify the correctness of the theoretical solution method; The control equation of the knock is as follows: (1) (2) (3) (4) Among them, equations (1)(2)(3) (4) are respectively the mass conservation, momentum conservation, energy conservation and chemical reaction equations of detonation; the four equations constitute the control equation group of detonation; where subscripts 1 and 2 represent the detonation wave front and wave back, respectively. represents the wavefront density, represents the post-wave density; represents the wavefront normal velocity, represents the normal velocity behind the wave; represents the wavefront pressure, Indicates the post-wave pressure; represents the wavefront enthalpy, represents the post-wave enthalpy; It is the energy released by chemical reactions; represents the wavefront temperature, represents the post-wave temperature; The chemical reaction progress of the wavefront is usually 1, Indicates the progress of chemical reactions behind the wave; is the pre-factor, which is usually a constant; is the activation energy; is the wavefront gas constant, is the post-wave gas constant; where the pressure can be calculated from the gas state equation: (5) The calculation formula of enthalpy is as follows: (6) in, is the specific heat ratio of the gas at the wavefront, is the ratio of the specific heats of the gas behind the wave; energy released Relationship with chemical reaction process Z: (7) in, Represents the total energy released when the chemical reaction is completely completed; the normal velocity is calculated as follows: (8) in, is the angle of the wedge surface, is the angle of the blast wave; 2) According to the control equation of knock, select the objective function and derive it; specifically including: 2.1) Dividing equation (4) by equation (1) yields: (9) Rewrite equation (9) from an equation into a function form: (10) Observe formula (10) to get the objective function f related to multiple variables, select the chemical reaction process As the independent variable, the other variables are dependent variables; so far, the problem of solving the aerodynamic parameters after the detonation wave is transformed into solving the function f for the independent variable untie; 2.2) Solve the function f with respect to the independent variable The total derivative of : Since there is more than one variable in the function f, according to the chain rule, we can solve the function f for the independent variable The total derivative of (11) Therefore, the function f has The partial derivative of is: (12) Function f on variables The partial derivative of is: (13) variable right The partial derivative of is: (14) variable right The total derivative of is: (15) variable right The total derivative of is: (16) variable right The total derivative of is: (17) variable right The partial derivative of is: (18) Function f on variables The partial derivative of is: (19) At this point, the calculation formulas (12)-(19) of the above derivatives are substituted into equation (11) to complete the function f with respect to the independent variable Solve the total derivative of ; 3) After solving the objective function and its derivatives, the iterative solution format of function (10) is constructed according to the idea of ​​Newton's iteration method; (20) in, is the value of the function's independent variable at the kth time, represents the value of the k+1th independent variable, represents the damping factor; is the objective function, is the derivative of the objective function; 4) After the iterative solution format of the function f is obtained, the aerodynamic parameters after the detonation wave are solved according to the given calculation conditions; 5) After the calculation conditions are known, any given initial chemical reaction process Based on this initial value, the next step of chemical reaction process is solved iteratively according to formula (16). Value; where the objective function is calculated according to formula (6) and the derivative is calculated according to formula (7); 6) Calculate the updated chemical reaction progress Chemical reaction progress with the previous step The absolute value of the difference between Error is used to measure the difference between two iterations of the independent variable as a basis for judging whether it has converged; 7) Given the convergence condition C, set the convergence condition according to the accuracy required by the specific problem, and set the convergence condition to a smaller value; 8) Combine the iterative difference error in step 6) and the convergence condition C in step 7) to determine whether the calculation has reached convergence; when the error is less than the convergence condition C, the calculation is judged to have reached convergence and the iteration is stopped; otherwise, the calculation has not reached convergence and it needs to continue iteration; 9) If the convergence condition is not met, the updated independent variable Bring it into step 5), and repeat 5)-8) until convergence; 10) If convergence has been reached, the loop ends and the independent variable at this time is Substitute the control equations (1)-(4) to calculate the aerodynamic parameters after the detonation wave.

2. A method for solving zero-order aerodynamic parameters after oblique detonation wave based on gradient descent algorithm as claimed in claim 1, characterized in that In step 3), the damping factor The value range of is between [0,1] to ensure the stability of the iterative process.

3. A method for solving zero-order aerodynamic parameters after oblique detonation wave based on gradient descent algorithm as claimed in claim 1, characterized in that In step 4), the calculation conditions include the following parameters: incoming flow pressure , the incoming flow Mach number , inflow temperature , flow velocity , chemical reaction process , heat ratio of incoming flow , wall wedge angle , the gas completely releases heat , the activation energy of a chemical reaction , the pre-exponential factor of a chemical reaction .

4. A method for solving zero-order aerodynamic parameters after oblique detonation wave based on gradient descent algorithm as claimed in claim 1, characterized in that In step 5), the initial chemical reaction progresses Takes a value between [0,1].

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