Mathematical modeling method for cold flow rocket

Abstract

The invention belongs to the field of cold flow rocket control systems, and relates to a mathematical modeling method for a cold flow rocket, which comprises the following steps: establishing a physical model of the cold flow rocket; calculating the pressure intensity and volume of high-pressure gas and the volume of residual water; calculating resistance loss, and calculating nozzle outlet flow velocity according to a Bernoulli equation; the thrust is calculated according to a Tsiolkovsky rocket equation; calculating the total mass of the rocket, and calculating the overall motion acceleration, speed and height of the rocket according to the Newton second law; judging whether the height of the rocket is zero or not, and if not, returning to continue circulation until the rocket lands; if yes, landing is considered, and circulation is ended. According to the method, the state parameters and the performance parameters of the cold flow rocket in the whole flight process can be calculated, mapping and simulation of the whole launching process of the cold flow rocket are achieved in a digital space, and the method has the advantages of being efficient, real-time, accurate in whole-process simulation and accurate.

Description

Technical Field

[0001] The invention belongs to the field of cold flow rocket control systems, and in particular relates to a mathematical modeling method for cold flow rockets. Background Art

[0002] In recent years, the technology of reusable rockets has been developing continuously, and its core technology is the rocket control system. As an environmentally friendly and low-cost rocket configuration, the research and development of the control system of cold flow rockets helps to improve the flight stability, accuracy and rocket recovery ability of rockets, and provides technical and theoretical support for the development of reusable rocket technology.

[0003] The first step in the development of a cold-flow rocket control system is to establish an accurate mathematical model of the control object, that is, to calculate the changes in the state parameters and performance parameters of the rocket during flight according to the composition and physical mechanism of the cold-flow rocket, and obtain its overall flight parameters, which can be used to verify the feasibility of the control system. At the same time, establishing an accurate mathematical model of a cold-flow rocket can help designers choose appropriate design parameters during the rocket design process. However, the current research and invention of cold-flow rockets only stays at the structural design stage, and does not involve the research on mathematical modeling and simulation of cold-flow rockets. Therefore, in the field of cold-flow rockets, which is full of exploratory and educational significance, the introduction of digital modeling and simulation methods will undoubtedly bring about a profound change. Summary of the invention

[0004] The purpose of the present invention is to provide a mathematical modeling method for a cold flow rocket, which can calculate the state parameters and performance parameters of the cold flow rocket during the entire flight process, and realize the mapping and simulation of the entire launch process of the cold flow rocket in the digital space, with the characteristics of high efficiency, real-time, full-process simulation and accuracy.

[0005] The present invention is achieved through the following technical solutions:

[0006] The invention discloses a mathematical modeling method for a cold flow rocket, comprising:

[0007] S1. Establishing a physical model of a cold flow rocket; the physical model of the cold flow rocket includes a high-pressure water bottle, a pipeline, a valve, and a nozzle; wherein the outlet of the high-pressure water bottle is connected to the nozzle through a pipeline, and the valve is installed in the pipeline;

[0008] S2, set the unit time step; initialize the overall parameters of the rocket and the propulsion system parameters;

[0009] S3, based on the unit time step and propulsion system parameters of S2, calculate the high-pressure gas pressure in the high-pressure water bottle at the current moment;

[0010] S4, considering the resistance loss, obtaining the height difference from the liquid surface to the outlet of the high-pressure water bottle, and calculating the outlet flow rate of the nozzle according to the Bernoulli equation based on the high-pressure gas pressure in the high-pressure water bottle obtained in S3;

[0011] S5, calculating the cold flow rocket thrust based on the nozzle outlet flow velocity obtained in S4;

[0012] S6, judging whether the water is sprayed completely; if so, the propulsion system stops working and proceeds to the next step; if not, proceeds to the next step;

[0013] S7, calculating the total mass, acceleration, rocket speed and rocket altitude of the rocket at the current moment;

[0014] S8, judging whether the cold flow rocket can take off. If the thrust of the cold flow rocket at the initial moment is less than the gravity, it cannot take off and the cycle ends; if the thrust of the cold flow rocket at the initial moment is greater than the gravity, proceed to the next step;

[0015] S9, judging whether the rocket height is zero, if it is zero, it is regarded as landing and the cycle ends; if it is not zero, proceeding to S10;

[0016] S10, continuously repeat the process of steps S2-S7 and S9 until the rocket lands.

[0017] Further, in S2, the unit time step dt is set;

[0018] The overall parameters of the rocket include the rocket empty weight m e , total mass of the rocket m r , rocket height H r , speed v r , acceleration a r ;

[0019] The propulsion system parameters include the volume of high-pressure gas V0, the volume of water in the high-pressure water bottle V1, the pressure of high-pressure gas p0, the diameter of each cross section, the local resistance loss coefficient, and the rocket thrust F.

[0020] Further, the diameters of each cross section include the high-pressure water bottle outlet diameter d1, the pipeline diameter d2, and the nozzle diameter d3;

[0021] The local resistance loss coefficient includes the local resistance loss coefficient ζ1 at the outlet of the high-pressure water bottle, the local resistance loss coefficient ζ2 at the valve and the local resistance loss coefficient ζ3 at the nozzle.

[0022] Further, S3 is specifically: according to the volume flow rate at the previous moment Multiply the unit time step dt to get the volume of the sprayed water Then the volume of gas in the high-pressure water bottle is increased and The value of is equal; ΔV0 is taken as the cumulative increase of gas volume, expressed as:

[0023]

[0024] t represents the current moment, t-1 represents the previous moment, and the high-pressure gas pressure P0 at the current moment is calculated according to the ideal gas state equation; the ideal gas state equation is:

[0025]

[0026] Among them, the superscript t0 represents the physical quantity at the initial moment, V0 t0 Represents the volume of high pressure gas at the initial moment, P0 t0 Indicates the pressure of high-pressure gas at the initial moment.

[0027] Further, S4 is specifically as follows: obtaining the valve local resistance coefficient ζ2 according to the valve opening, obtaining the local resistance loss coefficient ζ1 of the high-pressure water bottle outlet and the local resistance loss coefficient ζ3 at the nozzle according to the cross-sectional shape;

[0028] According to the high-pressure water bottle outlet velocity v1 at time t-1 t-1 Calculate the Reynolds number and the drag coefficient along the way, and calculate the local resistance loss h f1 and the resistance loss along the way h f2 The sum of the two is the resistance loss h f ;

[0029] The nozzle outlet flow rate at time t is calculated according to the Bernoulli equation.

[0030] Further, the Bernoulli equation is:

[0031]

[0032] Among them, p0 is the pressure of high-pressure gas in the high-pressure water bottle, ρ is the density of water, g is the acceleration of gravity, H0 is the height difference from the liquid surface to the outlet of the high-pressure water bottle, H1 is the height difference from the outlet of the high-pressure water bottle to the outlet of the nozzle, v0 is the speed of the liquid in the high-pressure water bottle relative to the rocket, and the default value is 0; p atm is the atmospheric pressure, v3 is the nozzle outlet velocity, h f is the resistance loss during the flow process.

[0033] Furthermore, the calculation expression of resistance loss is:

[0034]

[0035] Among them, λ is the resistance coefficient along the way, ζ1 is the local resistance loss coefficient at the outlet of the high-pressure water bottle, ζ2 is the local resistance loss coefficient of the valve, ζ3 is the local resistance loss coefficient at the nozzle, v3 is the nozzle outlet flow velocity, v1 is the high-pressure water bottle outlet velocity; d2 is the pipeline diameter, and L is the total length of the pipeline.

[0036] Furthermore, the drag coefficient along the way is related to the Reynolds number R e The size is related to the resistance coefficient along the way:

[0037]

[0038] The expression for the Reynolds number is:

[0039]

[0040] Among them, υ is the kinematic viscosity of water, v1 is the velocity of the liquid at the outlet of the high-pressure water bottle; L is the total length of the pipeline.

[0041] Further, S5 is specifically: according to the Tsiolkovsky rocket equation, the cold flow rocket thrust F is calculated as follows:

[0042] F=A3ρv3 2

[0043] Among them, A3 is the nozzle outlet area, ρ is the density of water, and v3 is the nozzle outlet flow rate.

[0044] Further, S6 is specifically as follows: if the water in the high-pressure water bottle is sprayed out, that is, V1 = 0, the propulsion system stops working, the high-pressure gas pressure P0 is equal to the external atmospheric pressure, the nozzle outlet flow rate v3 = 0, and the propulsion system thrust F = 0;

[0045] S7 is specifically:

[0046] The total mass of the cold stream rocket continues to decrease:

[0047]

[0048] Where q3 is the volume flow rate of the ejected water, expressed as q3=A3v3. Subtracting the mass of the ejected water in each cycle can calculate the total mass of the rocket at the current moment.

[0049] According to Newton's second law, for cold flow rockets, we have:

[0050]

[0051] Among them, m r is the mass of the rocket, a r is the rocket acceleration, v r is the rocket speed, g is the acceleration due to gravity, H ris the rocket height;

[0052] The above differential equations are all solved by Euler’s method;

[0053] Take the unit time step as dt, use t to represent the current moment, t-1 to represent the previous moment, and calculate the flight parameters of the cold flow rocket at the current moment.

[0054] Compared with the prior art, the present invention has the following beneficial technical effects:

[0055] The invention discloses a mathematical modeling method for a cold flow rocket, which completes pipeline calculation according to the Bernoulli equation, calculates the flight parameters of the cold flow rocket by Newton's second law, and realizes dynamic simulation of the complete flight process of the cold flow rocket by iterative calculation. The invention completes the mathematical modeling technology of the cold flow rocket and fills the gap in the current technology in this field.

[0056] Taking into account factors such as the drag loss along the way, the local drag loss coefficient, the effect of valve opening on valve drag loss, the pressure change of high-pressure gas in the dynamic process, and the volume of water in the high-pressure water bottle, and solving the differential equations through the Euler method, an accurate dynamic real-time mathematical model of the cold flow rocket was established;

[0057] Through mathematical modeling, starting from the design parameters of the cold flow rocket, the operation process of the cold flow rocket, flight speed, mass, altitude changes, etc. are obtained, providing data support for the design, manufacture and control of the cold flow rocket. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 A structural diagram of a cold flow rocket provided by the present invention;

[0059] Among them, 1. fairing; 2. guidance chamber; 3. rocket skin; 4. high-pressure water bottle; 5. pipeline; 6. valve; 7. nozzle;

[0060] A is the cross section of the water surface in the high-pressure water bottle; B is the cross section of the high-pressure water bottle outlet, C is the valve cross section, and D is the nozzle outlet cross section;

[0061] Figure 2 Provide the overall idea of ​​mathematical modeling for cold stream rockets;

[0062] Figure 3 It is a flow chart of the mathematical modeling code implementation of cold stream rocket;

[0063] Figure 4 Figure 1 is a simulation result of a cold flow rocket of an embodiment; Figure a is a curve showing the change of rocket acceleration over time; Figure b is a curve showing the change of rocket height over time; Figure c is a curve showing the change of rocket speed over time; and Figure d is a curve showing the change of rocket gravity and thrust over time. DETAILED DESCRIPTION

[0064] In order to make the purpose, technical solution and advantages of the present invention more clear, the following is further described in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention, that is, the embodiments described are only part of the embodiments of the present invention, not all embodiments.

[0065] The detailed description of the embodiments of the present invention provided in the following drawings is not intended to limit the scope of the invention claimed for protection, but merely represents a selected embodiment of the present invention. Based on the drawings and embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work are within the scope of protection of the present invention.

[0066] It should be noted that the terms "comprises", "includes" or any other variations are intended to cover non-exclusive inclusion, so that a process, element, method, article or equipment that includes a series of elements includes not only those elements, but also includes other elements not explicitly listed, or also includes elements inherent to the process, element, method, article or equipment.

[0067] like Figure 1 As shown, the cold flow rocket includes: fairing 1, guidance chamber 2, rocket skin 3, high-pressure water bottle 4, pipeline 5, valve 6, nozzle 7. Among them, A is the water surface section in the high-pressure water bottle 4, which is also the section where the high-pressure gas is located, B is the outlet section of the high-pressure water bottle 4, C is the section of the valve 6, and D is the outlet section of the nozzle 7. The outlet of the high-pressure water bottle 4 is connected to the nozzle 7 through the pipeline 5, and the valve 6 is installed in the pipeline 5.

[0068] The operating principle of the cold flow rocket is: the high-pressure gas in the bottle sprays water out, and according to Newton's third law, the sprayed water reacts on the cold flow rocket, thereby pushing the cold flow rocket up. In order to meet the needs of subsequent control and enable the cold flow rocket to maintain a steady state level and take off, land and recover, this modeling method can calculate the thrust data, flow data of the propulsion system, and the overall height and weight data of the cold flow rocket in real time.

[0069] In order to improve the accuracy of mathematical modeling, the following points need to be considered:

[0070] (1) Water is an ideal fluid, which is incompressible and non-viscous during flow;

[0071] (2) Local resistance loss will occur at the outlet of the water bottle, the valve, and the nozzle, and the corresponding local resistance coefficient needs to be calculated;

[0072] (3) Consider the resistance loss along the entire pipeline 5;

[0073] (4) The flow process conforms to the Bernoulli equation and flow continuity equation;

[0074] (5) There is no external interference during the flight of the cold flow rocket.

[0075] like Figure 2 As shown, the overall idea of ​​mathematical modeling of the cold flow rocket is as follows:

[0076] (1) Calculate the resistance loss and calculate the nozzle outlet flow rate according to the Bernoulli equation;

[0077] (2) Calculate thrust according to Tsiolkovsky rocket equation;

[0078] (3) Calculate the total mass of the rocket and calculate the acceleration, velocity, and altitude of the rocket based on Newton's second law of motion;

[0079] (4) Update the calculation of high-pressure gas pressure, volume and residual water volume.

[0080] (5) After takeoff, determine whether to land. If not, return to the first step and continue the loop calculation; if so, the loop ends.

[0081] The Bernoulli equation is a classic equation in fluid mechanics pipeline calculation. To improve modeling accuracy, the present invention calculates the local resistance loss and the resistance loss along the way during the water flow in the cold flow rocket, and calculates the rocket nozzle outlet flow rate by the Bernoulli equation. In the cold flow rocket, according to the Bernoulli equation:

[0082]

[0083] Among them, p0 is the pressure of the high-pressure gas above the water in the high-pressure water bottle 4, ρ is the density of water, g is the acceleration of gravity, H0 is the height difference from the liquid surface to the outlet of the high-pressure water bottle 4, that is, the distance between the cross sections AB, H1 is the height difference from the outlet of the high-pressure water bottle 4 to the nozzle outlet, that is, the distance between the cross sections BD, v0 is the speed of the water in the high-pressure water bottle 4 relative to the rocket (the default is 0), p atm is the atmospheric pressure, v3 is the nozzle outlet velocity, h f is the resistance loss during the flow process.

[0084] In the Bernoulli equation, the resistance loss h during the flow process f Including local resistance loss h f1 and the resistance loss along the way h f2 .

[0085] The local resistance loss:

[0086]

[0087] Among them, i = 1, 2, 3; ζ1, v1 are the local resistance loss coefficient and the outlet velocity of the high-pressure water bottle 4 respectively; ζ2, v2 are the local resistance loss coefficient and the velocity in the pipeline of the valve respectively, here, it is considered that v1 = v2; ζ3, v3 are the local resistance loss coefficient and the outlet velocity of the nozzle respectively. The local resistance loss coefficient is obtained by looking up the table.

[0088] According to the flow continuity equation, A1v1=A3v3, where A1 and A3 are the outlet area of ​​the high-pressure water bottle 4 and the outlet area of ​​the nozzle respectively, and v1=v3A3 / A1 is obtained, thus establishing the relationship between v1 and v3.

[0089] Among the local resistance losses, ζ2 is related to the valve opening.

[0090] The expression of the resistance loss along the way is:

[0091]

[0092] Among them, λ is the resistance coefficient along the way, L is the total length of pipeline 5, and d2 is the diameter of pipeline 5.

[0093] The drag coefficient along the way is related to the Reynolds number Re, and the drag coefficient along the way is specifically:

[0094]

[0095] The expression for the Reynolds number is:

[0096]

[0097] Among them, υ is the kinematic viscosity of water, v1 is the outlet velocity of the high-pressure water bottle 4; and L is the total length of the pipeline 5.

[0098] The resistance loss, according to the principle of head superposition, can be obtained as:

[0099]

[0100] According to the above calculation, substituting it into the Bernoulli equation, it becomes:

[0101]

[0102] The nozzle outlet velocity v3 can be calculated according to the above formula.

[0103] According to the Tsiolkovsky rocket equation, the cold flow rocket thrust F is calculated as:

[0104]

[0105] Among them, Q is the mass flow rate per unit time, q3 is the volume flow rate per unit time, and A3 is the nozzle outlet area.

[0106] If the water in the high-pressure water bottle is sprayed out, that is, V1=0, the propulsion system stops working, the high-pressure gas pressure P0 is equal to the external atmospheric pressure, the nozzle outlet flow rate v3=0, and the propulsion system thrust F=0.

[0107] Based on the above calculations, the thrust provided by the cold flow rocket propulsion system under the current working conditions is obtained. The entire flight process of the cold flow rocket is a dynamic process, which requires updating the remaining water volume, high pressure gas volume and pressure in the high pressure water bottle 4, and calculating the rocket flight parameters including acceleration, speed, altitude and mass through Newton's second law. Take the unit time step as dt, use t to represent the current moment, and t-1 to represent the previous moment. Specifically, it is:

[0108] (1) The volume of water in the high-pressure water bottle 4 is continuously reduced:

[0109] dV1=-q3dt

[0110] Wherein, V1 is the volume of water in the high-pressure water bottle 4.

[0111] (2) The volume V0 of the high-pressure gas in the high-pressure water bottle 4 continues to increase.

[0112] dV0=-dV1=q3dt

[0113] The pressure of the high-pressure gas in the high-pressure water bottle 4 is constantly decreasing. Assuming that the high-pressure gas is adiabatically expanding, according to the ideal gas state equation:

[0114]

[0115] Among them, the superscript t0 represents the physical quantity at the initial moment, and ΔV0 is the cumulative increase in gas volume.

[0116] According to the volume flow rate at the previous moment Multiply the unit time step dt to get the volume of the sprayed water Then the volume of gas in high pressure water bottle 4 is increased and The value of is equal; ΔV0 is taken as the cumulative increase of gas volume, expressed as:

[0117]

[0118] (3) The total mass of the cold-flow rocket continues to decrease:

[0119]

[0120] (4) The acceleration, speed, and altitude of a cold-stream rocket are constantly changing. According to Newton's second law, for a cold-stream rocket:

[0121]

[0122] Among them, m r is the mass of the rocket, a r is the rocket acceleration, v r is the rocket speed, g is the acceleration due to gravity, H r is the rocket altitude.

[0123] The above differential equations are all solved by the Euler method to calculate the gas parameters in the high-pressure water bottle and the flight parameters of the cold flow rocket at the current moment; the calculated high-pressure gas pressure P0 and the volume V0 of water in the high-pressure water bottle are used as the parameters required for the Bernoulli equation calculation at the next moment, and the iterative cycle is continuously repeated until the rocket lands.

[0124] like Figure 3 As shown, the specific simulation process is as follows:

[0125] 1) Set the unit time step dt;

[0126] 2) Initialize the overall parameters of the rocket, including the rocket's empty weight m e , total mass of the rocket m r , rocket height H r , speed v r , acceleration a r ;

[0127] Initialize the propulsion system parameters, including the volume of high-pressure gas V0, the volume of water in the high-pressure water bottle 4 V1, the pressure of high-pressure gas p0, and the diameter d of each section i , local resistance loss coefficient ζ i , rocket thrust F;

[0128] 3) Calculate the gas volume V0 and pressure P0 above the high-pressure water bottle 4. Based on the volume flow rate at the last moment Multiply the unit time step dt to get the volume of the sprayed water The volume of the gas increases Calculate the cumulative increase of gas ΔV0, and calculate the high-pressure gas pressure at the current moment according to the ideal gas state equation

[0129] 4) Considering the resistance loss, the nozzle outlet flow rate is calculated according to the Bernoulli equation, specifically:

[0130] According to the valve opening, query the relevant data to obtain the valve local resistance coefficient ζ2, and according to the cross-sectional shape, query the table to obtain ζ1 and ζ3;

[0131] According to the outlet velocity v1 of high-pressure water bottle 4 at time t-1 t-1 Calculate the Reynolds number and the drag coefficient along the way, and calculate the local resistance loss h f1 and the resistance loss along the way h f2 The sum of the two is the resistance loss h f .

[0132] According to the Bernoulli equation, the nozzle outlet flow velocity v3 at time t is calculated t , volume flow and mass flow rate Q t .

[0133] 5) Calculate the rocket thrust F according to the Tsiolkovsky rocket equation;

[0134] 6) Determine whether the water has been sprayed. If so, the propulsion system stops working and proceeds to the next step. If not, proceed to the next step;

[0135] 7) Calculate the total mass of the rocket at the current moment m r t ;

[0136] 8) Calculate the acceleration a according to Newton's second law r ;

[0137] 9) Calculate the current speed and altitude according to the Euler method;

[0138] 10) Determine whether it can take off. If the initial rocket thrust F t0 Less than gravity m r to g, it cannot take off and the cycle ends. If the initial rocket thrust F t0 Greater than gravity m r to g, it takes off successfully and proceeds to the next step;

[0139] 11) After takeoff, determine whether the altitude is 0. If it is 0, it is considered as landing and the loop ends; if it is not 0, proceed to the next step;

[0140] 12) Repeat steps 3)-9) and 11) continuously, iterating the cycle until the rocket lands.

[0141] Steps 6) and 10) are code protection. In step 6), when the water is sprayed, the thrust should be 0, the water volume in the high-pressure water bottle is 0, and the high-pressure gas pressure is equal to the ambient pressure. Without this code protection, the high-pressure gas will not be equal to the ambient pressure immediately during the calculation, and the water volume in the high-pressure water bottle will become negative, which is not in line with reality. In step 10), when the initial thrust is less than gravity and the aircraft cannot take off throughout the entire process, the altitude should always be 0. Without this code protection, the acceleration will be negative from the beginning, and the altitude will become negative, which is also not in line with reality.

[0142] During the first cycle, the judgment of step 6) must be yes; it is only necessary to judge whether takeoff is possible in the first cycle, and step 10) is no longer required in subsequent cycles.

[0143] The simulation results are described in detail below through an embodiment.

[0144] Take the valve as fully open, initialize the water volume V1 in the high-pressure water bottle 4, the high-pressure gas volume V0 in the high-pressure water bottle 4, the initial pressure of the high-pressure gas is p0, the diameter of the pipeline 5 is d2, the outlet diameter of the high-pressure water bottle 4 is d1, and the nozzle diameter is d3. After simulation, the following results are obtained, and the iteration step dt is initialized. Rocket empty weight M e (minus the weight of water).

[0145] The following table shows the specific parameters of each physical quantity:

[0146]

[0147] The simulation results are as follows Figure 4 As shown, Figure a is the curve of height changing with time; Figure b is the curve of acceleration changing with time; Figure c is the curve of speed changing with time; Figure d is the curve of thrust / cold flow rocket total gravity changing with time. It can be seen from Figure a that the flight height first increases and then decreases during the whole process until landing. It can be seen from Figures b and c that the rocket continues to accelerate after takeoff, and the acceleration is greater than zero. As time increases, it can be seen from Figure d that the gravity of the rocket continues to decrease, and the thrust also continues to decrease. The difference between its thrust and gravity first decreases and then increases. Therefore, the acceleration in Figure b decreases first and then increases. The maximum flight height is 16.4m, and the total flight time is 5s. At 2.25s, all the water is sprayed and the thrust disappears.

[0148] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in the relevant field should understand that the specific implementation methods of the present invention can still be modified or replaced by equivalents. Any modification or equivalent replacement that does not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A mathematical modeling method for cold flow rockets, characterized in that: include: S1. Establishing a physical model of a cold-flow rocket; the physical model of the cold-flow rocket comprises a high-pressure water bottle (4), a pipeline (5), a valve (6), and a nozzle (7); wherein the outlet of the high-pressure water bottle (4) is connected to the nozzle (7) through the pipeline (5), and the valve (6) is installed in the pipeline (5); S2, set the unit time step; initialize the overall parameters of the rocket and the propulsion system parameters; S3, based on the unit time step of S2 and the propulsion system parameters, calculate the pressure of the high-pressure gas in the high-pressure water bottle (4) at the current moment; S4, taking into account the resistance loss, obtaining the height difference from the liquid surface to the outlet of the high-pressure water bottle (4), and calculating the outlet flow rate of the nozzle (7) according to the Bernoulli equation based on the high-pressure gas pressure in the high-pressure water bottle (4) obtained in S3; S5, calculating the cold flow rocket thrust based on the outlet flow velocity of the nozzle (7) obtained in S4; S6, judging whether the water is sprayed completely; if so, the propulsion system stops working and proceeds to the next step; if not, proceeds to the next step; S7, calculating the total mass, acceleration, rocket speed and rocket altitude of the rocket at the current moment; S8, judging whether the cold flow rocket can take off. If the thrust of the cold flow rocket at the initial moment is less than the gravity, it cannot take off and the cycle ends; if the thrust of the cold flow rocket at the initial moment is greater than the gravity, proceed to the next step; S9, judging whether the rocket height is zero, if it is zero, it is regarded as landing and the loop ends; if it is not zero, proceeding to S10; S10, continuously repeat the process of steps S2-S7 and S9 until the rocket lands.

2. A mathematical modeling method for cold flow rockets according to claim 1, characterized in that: In S2, the unit time step dt is set; The overall parameters of the rocket include the rocket empty weight m e , total mass of the rocket m r , rocket height H r , speed v r , acceleration a r ; The propulsion system parameters include the volume of high-pressure gas V0, the volume of water in the high-pressure water bottle V1, the pressure of high-pressure gas p0, the diameter of each cross section, the local resistance loss coefficient, and the rocket thrust F.

3. A mathematical modeling method for cold flow rockets according to claim 2, characterized in that: The diameters of each cross section include the outlet diameter d1 of the high-pressure water bottle (4), the diameter d2 of the pipeline (5) and the diameter d3 of the nozzle (7); The local resistance loss coefficient includes the local resistance loss coefficient ζ1 at the outlet of the high-pressure water bottle (4), the local resistance loss coefficient ζ2 at the valve (6) and the local resistance loss coefficient ζ3 at the nozzle (7).

4. A mathematical modeling method for cold flow rockets according to claim 1, characterized in that: S3 is specifically: according to the volume flow q3 at the previous moment t-1 Multiply the unit time step dt to get the volume of the sprayed water dV1 t-1 , then the volume of gas in the high-pressure water bottle increased by dV0 t , dV1 t-1 With dV0 t The value of is equal; ΔV0 is taken as the cumulative increase of gas volume, expressed as: t represents the current moment, t-1 represents the previous moment, and the high-pressure gas pressure P0 at the current moment is calculated according to the ideal gas state equation; the ideal gas state equation is: Among them, the superscript t0 represents the physical quantity at the initial moment, V0 t0 Represents the volume of high pressure gas at the initial moment, P0 t0 Indicates the pressure of high-pressure gas at the initial moment.

5. A mathematical modeling method for cold flow rockets according to claim 1, characterized in that: S4 specifically includes: obtaining the valve local resistance coefficient ζ2 according to the valve opening, obtaining the local resistance loss coefficient ζ1 at the outlet of the high-pressure water bottle (4) and the local resistance loss coefficient ζ3 at the nozzle (7) according to the cross-sectional shape; According to the outlet velocity v1 of the high-pressure water bottle (4) at time t-1 t-1 Calculate the Reynolds number and the drag coefficient along the way, and calculate the local resistance loss h f1 and the resistance loss along the way h f2 The sum of the two is the resistance loss h f ; The nozzle outlet flow rate at time t is calculated according to the Bernoulli equation.

6. A mathematical modeling method for cold flow rockets according to claim 5, characterized in that: The Bernoulli equation is: Wherein, p0 is the pressure of the high-pressure gas in the high-pressure water bottle (4), ρ is the density of water, g is the acceleration of gravity, H0 is the height difference from the liquid surface to the outlet of the high-pressure water bottle (4), H1 is the height difference from the outlet of the high-pressure water bottle (4) to the outlet of the nozzle, v0 is the speed of the liquid in the high-pressure water bottle (4) relative to the rocket, and the default value is 0; atm is the atmospheric pressure, v3 is the nozzle outlet velocity, h f is the resistance loss during the flow process.

7. A mathematical modeling method for cold flow rockets according to claim 5, characterized in that: The calculation expression of resistance loss is: Among them, λ is the resistance coefficient along the way, ζ1 is the local resistance loss coefficient at the outlet of the high-pressure water bottle (4), ζ2 is the local resistance loss coefficient of the valve, ζ3 is the local resistance loss coefficient at the nozzle, v3 is the nozzle outlet flow rate, v1 is the outlet velocity of the high-pressure water bottle (4); d2 is the pipeline diameter, and L is the total length of the pipeline.

8. A mathematical modeling method for cold flow rockets according to claim 5, characterized in that: Along-the-line drag coefficient and Reynolds number R e The size is related to the resistance coefficient along the way: The expression for the Reynolds number is: Wherein, υ is the kinematic viscosity of water, v1 is the velocity of the liquid at the outlet of the high-pressure water bottle (4), and L is the total length of the pipeline.

9. A mathematical modeling method for cold flow rockets according to claim 1, characterized in that: S5 is specifically: According to the Tsiolkovsky rocket equation, the cold flow rocket thrust F is calculated as follows: <h2 style=";text-align:left;direction:ltr">F=A3ρv3<h2 style=";text-align:left;direction:ltr"> 2 Among them, A3 is the nozzle outlet area, ρ is the density of water, and v3 is the nozzle outlet flow rate.

10. A mathematical modeling method for cold flow rockets according to claim 1, characterized in that: S6 is specifically as follows: if the water in the high-pressure water bottle is sprayed out, that is, V1 = 0, the propulsion system stops working, the high-pressure gas pressure P0 is equal to the external atmospheric pressure, the nozzle outlet flow rate v3 = 0, and the propulsion system thrust F = 0; S7 is specifically: The total mass of the cold stream rocket continues to decrease: Where q3 is the volume flow rate of the ejected water, expressed as q3=A3v3. Subtracting the mass of the ejected water in each cycle can calculate the total mass of the rocket at the current moment. According to Newton's second law, for cold flow rockets, we have: Among them, m r is the mass of the rocket, a r is the rocket acceleration, v r is the rocket speed, g is the acceleration due to gravity, H r is the rocket height; The above differential equations are all solved by Euler’s method; Take the unit time step as dt, use t to represent the current moment, t-1 to represent the previous moment, and calculate the flight parameters of the cold flow rocket at the current moment.

Images