A mathematical modeling method for cold-flow rockets

Abstract

The present invention belongs to the field of cold-flow rocket control systems and relates to a mathematical modeling method for cold-flow rockets, comprising: establishing a physical model of the cold-flow rocket; calculating the pressure, volume, and residual water volume of high-pressure gas; calculating drag loss and nozzle exit velocity based on the Bernoulli equation; calculating thrust based on the Tsiolkovsky rocket equation; calculating the total mass of the rocket and calculating the overall acceleration, velocity, and altitude of the rocket based on Newton's second law; determining whether the rocket altitude is zero; if not, returning to a loop and continuing until the rocket lands; if so, the rocket is deemed to have landed, terminating the loop. The present invention can calculate the state parameters and performance parameters of the cold-flow rocket throughout its flight, and achieves mapping and simulation of the entire cold-flow rocket launch process in digital space, with the characteristics of high efficiency, real-time, full-process simulation, and precision.

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] Reusable rocket technology has continued to develop in recent years, with rocket control systems being a core technology. Cold-stream rockets are environmentally friendly and low-cost rocket configurations. The development of control systems for these rockets helps improve flight stability, accuracy, and recovery capabilities, providing technical and theoretical support for the development of reusable rocket technology.

[0003] The first step in developing a cold-stream rocket control system is to establish a precise mathematical model of the control object. This involves calculating the changes in the rocket's state and performance parameters during flight based on the rocket's composition and physical mechanisms, and obtaining its overall flight parameters, which can be used to verify the control system's feasibility. Furthermore, establishing a precise mathematical model of the cold-stream rocket can help designers select appropriate design parameters during the rocket design process. However, current research and development of cold-stream rockets remains limited to the structural design stage and does not involve the study of mathematical modeling and simulation of cold-stream rockets. Therefore, the introduction of digital modeling and simulation methods will undoubtedly bring about a profound transformation in this exploratory and educational field. Summary of the Invention

[0004] The purpose of the present invention is to provide a mathematical modeling method for cold flow rockets, which can calculate the state parameters and performance parameters of the cold flow rockets during the entire flight process, and realize the mapping and simulation of the entire launch process of the cold flow rockets in 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 present invention discloses a mathematical modeling method for a cold flow rocket, comprising:

[0007] S1. Establish 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 of S2 and the propulsion system parameters, calculate the high-pressure gas pressure in the high-pressure water bottle at the current moment;

[0010] S4. Considering the resistance loss, obtain the height difference between the liquid surface and the outlet of the high-pressure water bottle. Based on the high-pressure gas pressure in the high-pressure water bottle obtained in S3, calculate the outlet flow rate of the nozzle according to the Bernoulli equation.

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

[0012] S6. Determine whether the water spraying is complete; 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, speed, and altitude of the rocket at the current moment;

[0014] S8. Determine whether the cold stream rocket can take off. If the initial thrust of the cold stream rocket is less than the gravity, the rocket cannot take off, and the cycle ends. If the initial thrust of the cold stream rocket is greater than the gravity, proceed to the next step.

[0015] S9, determine whether the rocket altitude is zero, if it is zero, it is considered to have landed and the loop ends; if it is not zero, proceed to S10;

[0016] S10. Repeat steps S2-S7 and S9 until the rocket lands.

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

[0018] The overall parameters of the rocket include the rocket's empty weight m e , total rocket mass 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 section, the local resistance loss coefficient, and the rocket thrust F.

[0020] Furthermore, the diameters of each cross section include the high-pressure water bottle outlet diameter d1, the pipe 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] Furthermore, S3 is specifically: according to the volume flow at the previous moment Multiply the unit time step dt to get the volume of the sprayed water The volume of gas in the high-pressure water bottle is increased and The value of is equal; take ΔV0 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 Indicates 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 query, and 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 query table;

[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 velocity at time t is calculated according to the Bernoulli equation.

[0030] Furthermore, the Bernoulli equation is:

[0031]

[0032] Among them, p0 is the high-pressure gas pressure 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 nozzle outlet, 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 flow velocity at the nozzle outlet, v1 is the velocity at the outlet of the high-pressure water bottle; 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] Where, 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] Furthermore, S5 is specifically as follows: According to the Tsiolkovsky rocket equation, the cold flow rocket thrust F is calculated as follows:

[0042] F=A3ρv3 2

[0043] Where A3 is the nozzle outlet area, ρ is the density of water, and v3 is the nozzle outlet flow velocity.

[0044] Furthermore, 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 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 altitude;

[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 present invention discloses a mathematical modeling method for a cold-flow rocket. The method uses the Bernoulli equation to calculate pipelines, Newton's second law to calculate cold-flow rocket flight parameters, and iterative calculation to achieve dynamic simulation of the complete flight process of the cold-flow rocket. The present invention completes the mathematical modeling technology of the cold-flow rocket, filling 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 during the dynamic process, and the volume of water in the high-pressure water bottle, an accurate real-time mathematical model of the cold flow rocket dynamics was established by solving the differential equations using the Euler method.

[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 water surface cross-section in the high-pressure water bottle; B is the outlet cross-section of the high-pressure water bottle; 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 flowchart of the mathematical modeling code implementation of the 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 solutions and advantages of the present invention more clear, the following is a further detailed description 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 intended to limit the present invention. That is, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments.

[0065] The detailed description of the embodiment of the present invention provided in the following figures is not intended to limit the scope of the claimed invention, but merely represents a selected embodiment of the present invention. All other embodiments derived by those skilled in the art based on the figures and embodiments of the present invention without inventive effort shall fall 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 apparatus 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 apparatus.

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

[0068] The cold-stream rocket operates on the principle that high-pressure gas within a bottle ejects water. According to Newton's third law, the ejected water reacts on the cold-stream rocket, propelling it upward. To meet subsequent control requirements and enable the cold-stream rocket to maintain a steady level and perform takeoff, landing, and recovery, this modeling method can calculate the propulsion system's thrust and flow data, as well as the rocket's overall height and weight, in real time.

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

[0070] (1) Water is an ideal fluid, 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 the 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 using Newton's second law;

[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 used in fluid mechanics pipeline calculations. To improve modeling accuracy, this paper calculates the local resistance loss and along-the-line resistance loss experienced by water flowing in a cold-flow rocket, and uses the Bernoulli equation to calculate the flow velocity at the rocket nozzle outlet. 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] Where i = 1, 2, 3; ζ1 and v1 are the local resistance loss coefficient and velocity at the outlet of high-pressure water bottle 4, respectively; ζ2 and v2 are the local resistance loss coefficient of the valve and the velocity in the pipe, respectively. Here, v1 is assumed to be v2; ζ3 and v3 are the local resistance loss coefficient at the nozzle and the velocity at the nozzle outlet, 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] Wherein, λ is the resistance coefficient along the way, L is the total length of the pipeline 5, and d2 is the diameter of the 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] Wherein, 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 can be obtained according to the principle of head superposition:

[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 operating 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, velocity, altitude and mass using Newton's second law. The unit time step is dt, with t representing the current time and t-1 representing the previous time. Specifically, it is expressed as:

[0108] (1) The volume of water in the high-pressure water bottle 4 continues to decrease:

[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 expanding adiabatically, according to the ideal gas state equation:

[0114]

[0115] Wherein, 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; take ΔV0 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, velocity, and altitude of a cold-stream rocket are constantly changing. According to Newton's second law, for a cold-stream rocket, we have:

[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 of water V0 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 carried out until the rocket lands.

[0124] like Figure 3 As shown in the figure, 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 rocket mass m r , rocket height H r , speed v r , acceleration a r ;

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

[0128] 3) Calculate the volume V0 and pressure P0 of the gas above the high-pressure water bottle 4. Based on the volume flow rate at the previous moment Multiply the unit time step dt to get the volume of the sprayed water The volume of gas increases Calculate the cumulative increase in gas ΔV0 and calculate the current high-pressure gas pressure based on 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] Obtain the valve local resistance coefficient ζ2 by querying relevant data based on the valve opening, and obtain ζ1 and ζ3 by looking up the table based on the cross-sectional shape;

[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] Calculate the nozzle outlet velocity v3 at time t according to the Bernoulli equation 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 spraying is finished. 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 using the Euler method;

[0138] 10) Determine whether takeoff is possible. If the initial rocket thrust F t0 Less than gravity m r to g, then it cannot take off and the cycle ends. If the initial rocket thrust F t0 Greater than gravity m r to g, then take off successfully and proceed 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) repeatedly until the rocket lands.

[0141] Steps 6) and 10) are code protection. In step 6), when the water spray is finished, 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 pressure will not immediately equal the ambient pressure during the calculation, and the water volume in the high-pressure water bottle will become negative, which is not realistic. In step 10), when the initial thrust is less than gravity, and takeoff is impossible, the altitude should remain 0. Without this code protection, the acceleration will be negative from the beginning, and the altitude will become negative, which is also not realistic.

[0142] During the first cycle, the judgment in 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 high-pressure water bottle 4, the high-pressure gas volume V0 in high-pressure water bottle 4, the initial pressure of high-pressure gas is p0, the diameter of pipe 5 is d2, the outlet diameter of high-pressure water bottle 4 is d1, and the nozzle diameter is d3. After simulation, the following results are obtained and the iteration step size dt is initialized. Rocket empty weight M e (excluding water weight).

[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 shows the altitude versus time; Figure b shows the acceleration versus time; Figure c shows the velocity versus time; and Figure d shows the thrust / total gravity of the cold-flow rocket versus time. Figure a shows that the flight altitude first increases and then decreases until landing. Figures b and c show that the rocket accelerates continuously after takeoff, with acceleration greater than zero. As time passes, Figure d shows that the rocket's gravity decreases, and so does the thrust. The difference between thrust and gravity first decreases and then increases, resulting in the acceleration decreasing and then increasing in Figure b. The maximum flight altitude was 16.4 meters, and the total flight time was 5 seconds. At 2.25 seconds, all the water was sprayed, and the thrust disappeared.

[0148] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in the 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 by 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 includes 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 high-pressure gas pressure in the high-pressure water bottle (4) at the current moment; S4, taking into account the resistance loss, obtain the height difference between the liquid surface and the outlet of the high-pressure water bottle (4), and calculate the outlet flow rate of the nozzle (7) based on the high-pressure gas pressure in the high-pressure water bottle (4) obtained in S3 according to the Bernoulli equation; S5, calculating the cold flow rocket thrust based on the outlet flow velocity of the nozzle (7) obtained in S4; S6. Determine whether the water spraying is complete; 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, speed, and altitude of the rocket at the current moment; S8. Determine whether the cold stream rocket can take off. If the initial thrust of the cold stream rocket is less than the gravity, the rocket cannot take off, and the cycle ends. If the initial thrust of the cold stream rocket is greater than the gravity, proceed to the next step. S9, determine whether the rocket altitude is zero, if it is zero, it is considered to have landed and the loop ends; if it is not zero, proceed to S10; S10, repeating steps S2-S7 and S9 until the rocket lands; S5 is specifically: Calculate the cold flow rocket thrust according to the Tsiolkovsky rocket equation F , the calculation formula is: in, is the nozzle outlet area, is the density of water, is the nozzle outlet flow rate; S6 is specifically: if the water in the high-pressure water bottle is sprayed out, , the propulsion system stops working and the high-pressure gas pressure Equal to the external atmospheric pressure, the nozzle outlet flow rate , propulsion system thrust ; S7 specifically: The total mass of the cold stream rocket continues to decrease: in, is the volume flow rate of the sprayed water, expressed as , by subtracting the mass of the ejected water in each cycle, the total mass of the rocket at the current moment can be calculated; According to Newton's second law, for cold flow rockets, we have: in, is the mass of the rocket, is the rocket acceleration, is the rocket speed, is the acceleration due to gravity, is the rocket altitude; The above differential equations are all solved by Euler’s method; Take the unit time step as ,use Indicates the current moment, Indicates the previous moment, and calculates the flight parameters of the cold flow rocket at the current moment.

2. A mathematical modeling method for cold flow rockets according to claim 1, characterized in that: In S2, set the unit time step ; The overall parameters of the rocket include the rocket's empty weight , total rocket mass , rocket height ,speed , acceleration ; Propulsion system parameters include high pressure gas volume 、The volume of water in the high-pressure water bottle , high pressure gas pressure , the diameter of each section, the local drag loss coefficient, the rocket thrust F .

3. A mathematical modeling method for cold flow rockets according to claim 2, characterized in that: The diameter of each section includes the outlet diameter of the high-pressure water bottle (4) 、Pipeline (5) diameter and nozzle (7) diameter ; The local resistance loss coefficient includes the local resistance loss coefficient at the outlet of the high-pressure water bottle (4) , local resistance loss coefficient of valve (6) and the local resistance loss coefficient at the nozzle (7) .

4. The mathematical modeling method for cold flow rocket according to claim 1, characterized in that: S3 is specifically: according to the volume flow at the previous moment Multiply by the unit time step 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; take is the cumulative increase in gas volume, expressed as: Indicates the current moment, Indicates the previous moment, and calculates the high-pressure gas pressure at the current moment according to the ideal gas state equation ; The ideal gas state equation is: Among them, the superscript represents the physical quantity at the initial moment, represents the volume of high-pressure gas at the initial moment, Indicates the pressure of high-pressure gas at the initial moment.

5. The mathematical modeling method for cold flow rocket according to claim 1, characterized in that: S4 is specifically: obtain the valve local resistance coefficient according to the valve opening query , according to the cross-sectional shape, the local resistance loss coefficient of the high-pressure water bottle (4) outlet is obtained by looking up the table and the local resistance loss coefficient at the nozzle (7) ; According to the outlet velocity of the high-pressure water bottle (4) at time t-1 Calculate the Reynolds number and the drag coefficient along the way, and calculate the local resistance loss respectively and along-the-line resistance loss The sum of the two is the resistance loss ; The nozzle outlet flow velocity 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: ； in, is the high pressure gas pressure in the high pressure water bottle (4), is the density of water, is the acceleration due to gravity, is the height difference from the liquid surface to the outlet of the high-pressure water bottle (4), is the height difference between the outlet of the high-pressure water bottle (4) and the outlet of the nozzle, is the velocity of the liquid in the high-pressure water bottle (4) relative to the rocket, and the default value is 0; is the atmospheric pressure, is the nozzle outlet flow rate, is the resistance loss during the flow process.

7. The mathematical modeling method for cold-flow rockets according to claim 5, characterized in that: The calculation expression of resistance loss is: ； in, is the resistance coefficient along the path, is the local resistance loss coefficient at the outlet of the high-pressure water bottle (4), is the local resistance loss coefficient of the valve, is the local resistance loss coefficient at the nozzle, is the nozzle outlet flow rate, is the outlet velocity of the high-pressure water bottle (4); is the pipe diameter, is the total length of the pipeline.

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

Images