A method for modeling and optimizing the aerodynamic characteristics of a jet nozzle

Abstract

This application provides a method for modeling and optimizing the aerodynamic characteristics of a jet nozzle, belonging to the field of aero-engine technology. The method includes: determining the functional relationship between area ratio and Mach number based on isentropic flow theory, serving as the basic model for the nozzle's aerodynamic characteristics; dividing the flow state inside the nozzle based on different pressure ratios to obtain multiple operating conditions and state parameters under each condition; constructing a nozzle model based on the basic aerodynamic characteristic model and the state parameters under each operating condition, with input parameters including total pressure, external back pressure, throat area, and exit area, and output parameters including Mach number, airflow velocity, temperature, and flow rate under each flow condition; performing multi-condition simulation verification on the nozzle model; checking the accuracy of the nozzle model; and applying it to an actual nozzle control system for operating condition judgment. This application's solution improves the performance prediction and control accuracy of the nozzle system in high-altitude simulation tests and practical applications.

Description

Technical Field

[0001] This application relates to the field of aero-engine technology, and in particular to a method for modeling and optimizing the aerodynamic characteristics of a jet nozzle. Background Technology

[0002] Nozzle mechanism modeling and simulation techniques have a certain application foundation in aerospace, jet engine and other fields. Currently, nozzle modeling methods mainly rely on numerical calculations of fluid dynamics (such as computational fluid dynamics (CFD) technology), empirical formulas and simplified models. CFD technology can simulate the flow characteristics of airflow inside the nozzle in detail, including changes in parameters such as pressure, temperature and velocity, but this method usually has a large computational load and is difficult to adapt quickly to different operating conditions. Simplified models and empirical formulas can evaluate nozzle performance in a shorter time, but often ignore the complexity of the internal flow of the nozzle and have lower accuracy. In addition, some multiphysics coupled simulation methods attempt to combine aerodynamics and thermodynamics with factors such as the structural stress of the nozzle to improve accuracy, but this method is complex to model, has a large computational load, and is difficult to meet the performance requirements in real-time simulation. First, many existing models are based on specific operating conditions and cannot flexibly adapt to different operating environments, especially under variable operating conditions such as high pressure ratio, supersonic and subsonic speeds, where accuracy and stability cannot be guaranteed. Secondly, while existing numerical calculation methods can simulate flow states, they suffer from significant numerical instability and insufficient accuracy under complex flow phenomena such as shock waves and expansion waves, affecting the reliability and accuracy of simulation results. Thirdly, adjustments to existing models rely on static analysis or historical data, failing to provide real-time optimization under complex operating conditions, thus limiting the real-time control and optimization capabilities of nozzles. Finally, although existing technologies are continuously improving in computational accuracy, the problems of computational errors and numerical instability remain unresolved when switching between various operating conditions. Summary of the Invention

[0003] In view of this, embodiments of this application provide a method for modeling and optimizing the aerodynamic characteristics of a jet nozzle, which at least partially solves the problems of adaptability in jet nozzle modeling and inaccurate performance evaluation and simulation under complex working conditions in the prior art.

[0004] This application provides a method for modeling and optimizing the aerodynamic characteristics of a jet nozzle, the method comprising:

[0005] Based on the isentropic flow theory, the functional relationship between the area ratio and the Mach number is determined as the basic model of the nozzle aerodynamic characteristics.

[0006] Based on different pressure ratios and combined with gas dynamics theory, the flow state inside the nozzle is divided to obtain multiple working conditions and determine the state parameters under each working condition.

[0007] Based on the basic model of nozzle aerodynamic characteristics and state parameters under various operating conditions, a nozzle model is constructed. The input parameters of the nozzle model include total pressure, external back pressure, throat area and outlet area. The output parameters of the nozzle model include Mach number, airflow velocity, temperature and flow rate under various flow conditions.

[0008] Multi-condition simulation verification of the nozzle model;

[0009] Verify the accuracy of the nozzle model and apply it to the actual nozzle control system for condition judgment.

[0010] According to a specific implementation of this application, the functional relationship between the area ratio and the Mach number is expressed as follows:

[0011] ,

[0012] ,

[0013] Where A is the cross-sectional area, A t Let be the nozzle throat area, and q be the flow rate function. Here, is the velocity coefficient, Ma is the Mach number, and k is the specific heat ratio of the gas.

[0014] According to a specific implementation of an embodiment of this application, the plurality of operating conditions include a critical state, a subcritical state, and a supercritical state. The supercritical state includes a fully expanded state with no shock wave, a slightly expanded state with an expansion wave outside the tube, an over-expanded state with an oblique shock wave, and an over-expanded state with a normal shock wave.

[0015] According to a specific implementation of an embodiment of this application, the expression for the state parameter of the critical state includes:

[0016] Nozzle exit Mach number: ,

[0017] The ratio of outlet static pressure to inlet total pressure: ,

[0018] ,

[0019] Mass flow rate through the nozzle exit: ,

[0020] Among them, Ma e A is the nozzle exit Mach number. e p is the nozzle exit area, p0 is the total inlet pressure, p e For the outlet static pressure, p b The external back pressure at the outlet, π3 is the third characteristic pressure ratio, p b3 Let q be the outlet pressure under critical conditions. m,maxWhere T0 is the maximum mass flow rate, T0 is the total temperature, and K is a dimensionless constant. R is the gas constant.

[0021] According to a specific implementation of this application, the expression for the state parameters of the subcritical state includes:

[0022] The exit Mach number is calculated using a one-dimensional isentropic flow formula to determine the corresponding temperature.

[0023] ,

[0024] in, Let p be the gas velocity and p be the gas pressure. Let T be the gas density and T be the gas temperature. (a), (b), (c), (d), and (e) are the formula numbers, respectively.

[0025] Based on the total inlet pressure p0 and the external outlet back pressure p b Calculate the velocity v of an ideal gas ei :

[0026] ,

[0027] Among them, v ei For the ideal gas velocity, c P For the specific heat capacity at constant pressure, T t Temperature of the gas in the throat;

[0028] Based on the loss and the ideal gas velocity v ei To obtain the actual outlet airflow velocity v of the nozzle:

[0029] ,

[0030] Where v is the actual outlet airflow velocity. This is the loss coefficient;

[0031] The actual total outlet pressure p of the nozzle is obtained based on the pressure loss. ea :

[0032] ,

[0033] Where, p ea T represents the actual total pressure at the nozzle exit. e This refers to the nozzle exit temperature.

[0034] Based on the actual total outlet pressure p of the nozzle ea Calculate the corresponding Mach number:

[0035] ;

[0036] Calculate the nozzle exit flow rate. ,

[0037] Where, q m For export flow, λ e This is the nozzle exit velocity coefficient.

[0038] According to a specific implementation of this application, the expression for the state parameters of the fully expanded state without shock waves includes:

[0039] Outlet static pressure and characteristic pressure ratio: ,

[0040] ,

[0041] Where π1 is the first characteristic pressure ratio, p b1 The outlet pressure under fully expanded, shock-free conditions;

[0042] Ideal gas velocity v ei : ;

[0043] Actual nozzle exit airflow velocity v:

[0044] ,

[0045] Actual total pressure at nozzle exit: ;

[0046] Nozzle exit flow rate: ;

[0047] Nozzle exit temperature: .

[0048] According to a specific implementation of this application, the actual nozzle outlet airflow velocity v, Mach number, and nozzle outlet flow rate in the state parameters of the under-expanded state with an expansion wave are calculated using the same method as the actual nozzle outlet airflow velocity v, Mach number, and nozzle outlet flow rate in the state parameters of the fully expanded state without a shock wave.

[0049] The actual total outlet pressure of the nozzle in the state parameters of the under-expansion state with an expansion wave outside the nozzle is:

[0050] .

[0051] According to a specific implementation of this application, the expression for the state parameters of the over-expansion state with oblique shock waves includes:

[0052] Based on the static pressure flow function, the flow function equation, and the aerodynamic function, the post-shock Mach number Ma0 is obtained. The static pressure flow function is: ,

[0053] The flow function equation is: ,

[0054] The aerodynamic function is: ,

[0055] Where y is the static pressure flow function and Ma0 is the Mach number after the shock wave;

[0056] Ideal gas velocity v ei :

[0057] ;

[0058] Actual nozzle exit airflow velocity v: ;

[0059] Actual total pressure at nozzle exit: ;

[0060] Mach number: ;

[0061] Second characteristic pressure ratio: ;

[0062] Where π2 is the second characteristic pressure ratio.

[0063] According to a specific implementation of this application, the expression for the state parameters of the over-expansion state with a normal shock wave includes:

[0064] Mach number before shock wave:

[0065] ,

[0066] Where Ma1 is the Mach number before the shock wave;

[0067] Mach number after shock wave:

[0068] ;

[0069] Nozzle exit temperature:

[0070] ;

[0071] Ideal gas velocity v ei :

[0072] ;

[0073] Actual nozzle exit airflow velocity v: ;

[0074] Actual total pressure at nozzle exit: ;

[0075] Nozzle exit flow rate: .

[0076] According to a specific implementation of an embodiment of this application, the nozzle model includes:

[0077] The isentropic area ratio calculation module is used to solve for the corresponding ideal Mach number and subsonic Mach number based on the isentropic area ratio formula.

[0078] The operating condition judgment and outlet status calculation module is used to determine the current pressure ratio p0 / p b The relationship with π3, π2, and π1 determines the nozzle state, calculates the ideal gas velocity, actual outlet gas velocity, and total outlet pressure, and outputs the current operating condition type.

[0079] The static pressure judgment module is used to determine whether an expansion wave or shock wave exists based on the comparison between the outlet static pressure and the outlet external back pressure.

[0080] The outlet flow calculation module is used to calculate the outlet flow in real time based on the current aerodynamic conditions, using the flow calculation formula.

[0081] The outlet temperature calculation module is used to calculate the actual outlet temperature based on the outlet Mach number and to reflect the impact of the shock wave or expansion process on the thermodynamic state.

[0082] The single-condition speed comparison module is used to calculate the speed using the fixed-condition formula under the design state, and compare it with the output of the condition judgment and exit state calculation module to reflect the sudden change and response characteristics of the speed under shock wave or expansion wave.

[0083] The Mach number back-calculation module is used to back-calculate the actual Mach number based on the ratio of the total outlet pressure to the external back pressure, and to verify the response effect of the nozzle model under the action of normal shock wave or oblique shock wave.

[0084] Beneficial effects:

[0085] The jet nozzle aerodynamic characteristic modeling and optimization method in this application realizes the identification and dynamic simulation of subcritical, critical, and various supercritical flow states, effectively overcoming the problems of insufficient accuracy and poor real-time performance of traditional models in high pressure ratio and shock / expansion wave flow modeling. The simulation module can adapt to different environmental back pressure and inlet total pressure changes, outputting key parameters such as Mach number, velocity, temperature, and static pressure in real time, and accurately reflecting complex physical phenomena such as shock wave position changes and flow state transitions, providing reliable state discrimination basis and feedback support for nozzle control algorithms. Experiments have shown that this modeling method has high numerical stability and engineering adaptability, and can significantly improve the performance prediction and control accuracy of nozzle systems in high-altitude simulation tests and practical applications. Attached Figure Description

[0086] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0087] Figure 1 This is a schematic diagram of Laval nozzle parameters according to an embodiment of the present invention;

[0088] Figure 2 This describes the variation of the nozzle area ratio with Mach number according to an embodiment of the present invention.

[0089] Figure 3.1 This describes the flow state within a nozzle according to an embodiment of the present invention.

[0090] Figure 3.2 This is a schematic diagram illustrating the calculation of shock wave location according to an embodiment of the present invention;

[0091] Figure 4 This is a Simulink module for an integral nozzle according to an embodiment of the present invention;

[0092] Figure 5(a) shows the simulation results of the nozzle test outlet total pressure and operating condition indicators changing with the inlet total pressure;

[0093] Figure 5(b) shows the simulation results of the nozzle test exit Mach number and exit velocity changes;

[0094] Figure 5(c) shows the simulation effect of the nozzle outlet temperature change during the test;

[0095] Figure 5(d) shows the simulation effect of the nozzle test outlet flow rate change;

[0096] Figure 5(e) shows the simulation results of the change between the static pressure at the nozzle outlet and the external back pressure. Detailed Implementation

[0097] The embodiments of this application will now be described in detail with reference to the accompanying drawings.

[0098] The following specific examples illustrate the implementation of this application. Those skilled in the art can easily understand other advantages and effects of this application from the content disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. This application can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of this application. It should be noted that, in the absence of conflict, the following embodiments and features in the embodiments can be combined with each other. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0099] It should be noted that various aspects of embodiments within the scope of the appended claims are described below. It will be apparent that the aspects described herein can be embodied in a wide variety of forms, and any particular structure and / or function described herein is merely illustrative. Based on this application, those skilled in the art will understand that one aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects set forth herein can be used to implement the device and / or practice the method. Additionally, this device and / or method can be implemented using structures and / or functionalities other than one or more of the aspects set forth herein.

[0100] It should also be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of this application. The illustrations only show the components related to this application and are not drawn according to the number, shape and size of the components in actual implementation. In actual implementation, the form, quantity and proportion of each component can be arbitrarily changed, and the layout of the components may also be more complex.

[0101] Furthermore, specific details are provided in the following description to facilitate a thorough understanding of the examples. However, those skilled in the art will understand that the described aspects can be practiced without these specific details.

[0102] This application provides a method for modeling and optimizing the aerodynamic characteristics of a jet nozzle, as described below. Figure 1 See Figure 5(e) for a detailed description.

[0103] In one embodiment, a method for modeling and optimizing the aerodynamic characteristics of a jet nozzle is provided, the method comprising:

[0104] Based on the isentropic flow theory, the functional relationship between the area ratio and the Mach number is determined as the basic model of the nozzle aerodynamic characteristics.

[0105] Based on different pressure ratios and combined with gas dynamics theory, the flow state inside the nozzle is divided to obtain multiple working conditions and determine the state parameters under each working condition.

[0106] Based on the basic model of nozzle aerodynamic characteristics and state parameters under various operating conditions, a nozzle model is constructed. The input parameters of the nozzle model include total pressure, external back pressure, throat area and outlet area. The output parameters of the nozzle model include Mach number, airflow velocity, temperature and flow rate under various flow conditions.

[0107] Multi-condition simulation verification of the nozzle model;

[0108] Verify the accuracy of the nozzle model and apply it to the actual nozzle control system for condition judgment.

[0109] In practice, firstly, based on the nozzle structural parameters and boundary conditions (including total pressure, total temperature, external back pressure, and area ratio), an isentropic area ratio formula and a flow state discrimination model are established. Secondly, based on the pressure ratio, seven typical flow states of the nozzle under different operating conditions are classified, and the corresponding aerodynamic parameter calculation methods are derived for each. A nozzle simulation module is built in the Simulink environment, including characteristic pressure ratio judgment and static pressure analysis functions, to realize real-time response to complex flow states and simulation of operating condition switching. By setting the inlet total pressure change process, simulation tests are carried out to verify the model's response accuracy under key states such as shock wave occurrence, Mach number transition, and sudden change in outlet pressure. Finally, the simulation results are compared with theoretical analysis to confirm the model's usability and engineering application value, and to provide a dynamic operating condition judgment basis for the nozzle system control strategy design.

[0110] In one embodiment, the functional relationship between the area ratio and the Mach number is expressed as follows:

[0111] ,

[0112] ,

[0113] Where A is the cross-sectional area, A t Let be the nozzle throat area, and q be the flow rate function. Here, is the velocity coefficient, Ma is the Mach number, and k is the specific heat ratio of the gas.

[0114] Specifically, to establish a basic model of the nozzle's aerodynamic characteristics, it is necessary to analyze the flow state and characteristics within the nozzle. Commonly known conditions include: the total pressure p0 and total temperature T0 of the inlet airflow, and the external back pressure p at the nozzle outlet. b And outlet temperature T, area ratio A e / A t (Here and in the following text, the subscripts e and t represent the parameters at the nozzle exit and throat, respectively); the distribution of parameters is as follows: Figure 1As shown. By writing the continuity equation for the throat and any cross-section, we can obtain the formula for the isentropic area ratio, i.e.:

[0115] (1)

[0116] (2)

[0117] Where A is the cross-sectional area, A t Let be the nozzle throat area, q be the flow rate function, λ be the velocity coefficient, Ma be the nozzle exit Mach number, and k be the gas specific heat ratio.

[0118] This formula can only be used when there is no shock wave between the throat and section A. Of course, section A can be located in the supersonic or subsonic range of the nozzle. From the above formula, it can be seen that for a given gas area ratio, it only depends on Ma, and its variation follows the pattern shown below. Figure 2 As shown in the figure. It can be seen from the figure that to generate a certain Ma at the nozzle exit cross-section... e The supersonic airflow at (nozzle exit Mach number) corresponds to a nozzle area ratio of A. e / A t It is unique. Furthermore, each area ratio corresponds to two Mach numbers: one for subsonic airflow and one for supersonic airflow. According to the isentropic area ratio formula, a certain pipe area ratio is required to establish a supersonic airflow at a specific Mach number at the nozzle exit. However, even with the required pipe area ratio, whether supersonic flow can be achieved depends, according to gas dynamics theory, on the total pressure p0 at the nozzle inlet and the external back pressure (experimental chamber pressure) p at the outlet. b The following analysis will determine the flow patterns that may occur within a Laval nozzle with a given area ratio.

[0119] Furthermore, the multiple operating conditions include critical state, subcritical state and supercritical state. The supercritical state includes fully expanded state with no shock wave, under-expanded state with expansion wave outside the tube, over-expanded state with oblique shock wave and over-expanded state with normal shock wave.

[0120] In one embodiment, the expression for the state parameter of the critical state includes:

[0121] Nozzle exit Mach number: ,

[0122] The ratio of outlet static pressure to inlet total pressure: ,

[0123] ,

[0124] Mass flow rate through the nozzle exit: ,

[0125] Among them, Ma e A is the nozzle exit Mach number. e p is the nozzle exit area, p0 is the total inlet pressure, p e For the outlet static pressure, p b The external back pressure at the outlet, π3 is the third characteristic pressure ratio, p b3 Let q be the outlet pressure under critical conditions. m,max Where T0 is the maximum mass flow rate, T0 is the total temperature, and K is a dimensionless constant. R is the gas constant.

[0126] Specifically, for the critical state, at an appropriate pressure ratio As the airflow accelerates within the contraction zone, it reaches a Mach number of Ma in the throat. t =1, then decelerate within the expansion phase until the exit Mach number Ma. e <1, and the outlet static pressure p e = p b This flow state is called the critical state of the Laval nozzle. The change in static pressure of the airflow along the nozzle axis is as follows: Figure 3.1 The curve b in the figure shows the critical state. The characteristic of the critical state is Ma. t = 1, Ma e <1, p e = p b (Fully expanded), with no shock waves inside the nozzle, and neglecting friction, the entire flow inside the nozzle can be considered isentropic. Let the exit pressure under the critical state be p. b3 That is, the pressure ratio under critical conditions for It is evident that when At this point, the flow in the nozzle is in a critical state. The relevant parameters under the critical state are calculated as follows:

[0127] Nozzle exit Mach number e From the isentropic area ratio formula (1), we can obtain that:

[0128] outlet static pressure p e Total pressure of imports The ratio: (3)

[0129] because ,so It's about the area ratio. The function;

[0130] Mass flow rate through the nozzle exit (mass flow rate of fluid through the nozzle or any control surface):

[0131] (4)

[0132] Where K is a dimensionless constant. R is the gas constant, therefore the flow is isentropic, and the mass flow rate reaches its maximum value. .

[0133] In one embodiment, the expression for the state parameters of the subcritical state includes:

[0134] The exit Mach number is calculated using a one-dimensional isentropic flow formula to determine the corresponding temperature.

[0135] ,

[0136] in, Let p be the gas velocity and p be the gas pressure. Let T be the gas density and T be the gas temperature. (a), (b), (c), (d), and (e) are the formula numbers, respectively.

[0137] Based on the total inlet pressure p0 and the external outlet back pressure p b Calculate the velocity v of an ideal gas ei :

[0138] ,

[0139] Among them, v ei For the ideal gas velocity, c P For the specific heat capacity at constant pressure, T t Temperature of the gas in the throat;

[0140] Based on the loss and the ideal gas velocity v ei To obtain the actual outlet airflow velocity v of the nozzle:

[0141] ,

[0142] Where v is the actual outlet airflow velocity. This is the loss coefficient;

[0143] The actual total outlet pressure p of the nozzle is obtained based on the pressure loss. ea :

[0144] ,

[0145] Where, p ea T represents the actual total pressure at the nozzle exit. e This refers to the nozzle exit temperature.

[0146] Based on the actual total outlet pressure p of the nozzle ea Calculate the corresponding Mach number:

[0147] ;

[0148] Calculate the nozzle exit flow rate. ,

[0149] Where, q m For export flow, λ e This is the nozzle exit velocity coefficient.

[0150] In practical implementation, the subcritical state refers to the state where the flow within the nozzle is entirely subsonic. For example, when... At that time, there is no flow within the entire nozzle, the static pressure equals the total pressure and remains constant along the tail nozzle, such as Figure 3.1 The straight line parallel to the s direction is shown in the figure. Figure 3.1 Line 1 in the diagram represents a limiting case of the subcritical state.

[0151] when At that time, the airflow accelerates within the nozzle's contraction section, and remains at Mach 6 at the throat. t <1, then decelerate within the expansion section, until exit Ma e <1, p e =p b ,like Figure 3.1 Curve a shown represents a subcritical flow state. Therefore, the characteristic of the subcritical state is Ma. t <1, Ma e <1, p e =p b The airflow expands completely within the nozzle, resulting in subsonic flow throughout the nozzle. The relevant parameters for the subcritical state are calculated as follows:

[0152] The exit Mach number is calculated using a one-dimensional isentropic flow formula to determine the corresponding temperature.

[0153] (5)

[0154] In the formula Let p be the gas velocity and p be the gas pressure. Let T be the gas density and T be the gas temperature.

[0155] Then from p0 and p b Calculate the velocity of an ideal gas :

[0156] (6)

[0157] In the formula For the specific heat capacity at constant pressure, T t This refers to the temperature of the gas in the throat.

[0158] Considering the actual flow losses (such as pressure losses), the actual nozzle exit gas velocity is:

[0159] (7)

[0160] Where φ is the loss coefficient (generally taken as 0.95~1), the actual total pressure at the nozzle exit is then calculated based on the pressure loss:

[0161] (8)

[0162] In the formula p ea T represents the actual total pressure at the nozzle exit. e This refers to the nozzle exit temperature.

[0163] The corresponding Mach number can be calculated using the pressure ratio formula:

[0164] (9)

[0165] Nozzle exit flow rate:

[0166] (10)

[0167] in This is the nozzle exit velocity coefficient.

[0168] In one embodiment, the expression for the state parameters of the fully expanded, shock-free state includes:

[0169] Outlet static pressure and characteristic pressure ratio: ,

[0170] ,

[0171] Where π1 is the first characteristic pressure ratio, p b1 The outlet pressure under fully expanded, shock-free conditions;

[0172] Ideal gas velocity v ei : ;

[0173] Actual nozzle exit airflow velocity v:

[0174] ,

[0175] Actual total pressure at nozzle exit: ;

[0176] Nozzle exit flow rate: ;

[0177] Nozzle exit temperature: .

[0178] In practical implementation, for the supercritical state, when At this point, the flow inside the nozzle is called the supercritical state. The airflow accelerates in the converging section of the nozzle, reaching Ma at the throat. t =1, then the flow within the expansion section is based on Depending on the size, there may be several situations as follows.

[0179] (1) No shock wave in the fully expanded state:

[0180] The airflow continues to accelerate within the expansion section, reaching Ma at the exit. e >1, and at the same time, the airflow reaches full expansion at the nozzle exit, p e =p b There are no shock waves within the entire expansion section, nor are there any shock waves or expansion waves outside the outlet. The static pressure changes along the nozzle as follows: Figure 3.1 The curve f is shown in the figure. Let p be the outlet pressure under this condition. b1 That is, the pressure ratio under this state is This situation is the design state; the pressure ratio under this state is recorded. = .

[0181] Outlet static pressure and characteristic pressure ratio:

[0182] (11)

[0183] It can be seen that when At this point, the flow inside the nozzle is in a supercritical state, and the airflow reaches full expansion at the nozzle exit. Its characteristic is Ma t =1, Ma e >1, p e =p b outlet static pressure = Ideal total export pressure = The ideal outlet airflow velocity is;

[0184] (12)

[0185] Considering the losses in actual flow (such as pressure loss), the actual outlet airflow velocity of the nozzle... , Given the nozzle velocity loss coefficient (typically taken as 0.95~1), the actual total nozzle exit pressure is:

[0186] (13)

[0187] Due to Ma t =1, the flow rate reaches its maximum value, the flow rate through the nozzle exit:

[0188] (14)

[0189] Temperature calculation:

[0190] (15)

[0191] (2) Expansion wave exists outside the tube in the under-expansion state:

[0192] when At that time, the airflow accelerates to Mach 1 at the outlet during the expansion phase. e >1, the airflow is not fully expanded inside the nozzle, i.e., p e >p b Therefore, the supersonic airflow generates an expansion beam at the nozzle exit. Within this pressure ratio range, changes in back pressure do not affect the flow inside the nozzle because external disturbances propagate at the speed of sound, while the flow at the nozzle exit is supersonic. Its flow characteristics are Ma t =1, Ma e >1. This is usually referred to as under-expanding flow state. For example... Figure 3.1 The curve g is shown in the figure. The calculation methods for the outlet gas velocity, Mach number, and flow rate through the nozzle are the same as those in (1) the fully expanded state without shock waves, and the outlet pressure p e >p b p e =p b1 This corresponds to the flow state with an expansion wave at the nozzle in the supercritical state. The total outlet pressure is calculated using the same formula (8).

[0193] (3) Oblique shock waves exist in the over-expansion state:

[0194] remember Figure 3.1 The outlet pressure corresponding to state d in the middle curve is p b2 That is, the pressure ratio under this state is ,when At this pressure ratio range, the airflow accelerates to Mach 1 at the outlet during the expansion phase. e >1, the airflow will generate an oblique shock wave at the outlet, such as Figure 3.1 As shown by curve e in the figure. The pressure after the oblique shock wave is equal to the external back pressure, and the shock wave intensity is determined by the pressure ratio. Decide.

[0195] In this state, the throat Mach number is critical, and the Mach number drops sharply after the shock wave inside the nozzle. To calculate the exit Mach number, the flow balance equation must first be established for the nozzle throat and exit cross-section:

[0196] (16)

[0197] Where p t Let be the pressure at the throat, and y be the static pressure flow function. Also, because... ,so

[0198] (17)

[0199] From the flow function equation:

[0200] (18)

[0201] And aerodynamic functions:

[0202] (19)

[0203] By combining the solutions, we can obtain the subsonic Ma0 solution, where Ma0 is the Mach number after the shock wave; then, we can use the subsonic solution to determine the velocity of the ideal gas.

[0204] (20)

[0205] After obtaining the ideal velocity, calculate the gas velocity and total outlet pressure after loss according to formulas (7) and (8), and finally calculate the actual Mach number according to formula (9).

[0206] As the pressure ratio increases, the shock wave intensifies, and the shock wave angle gradually increases. When the shock wave angle reaches 90°, i.e., when the oblique shock wave becomes a normal shock wave, the ratio of the pressure behind the shock wave to the total pressure is denoted as... ,like Figure 3.1 The curve d in the figure is shown. This type of flow is usually called the overexpansion state. This state corresponds to the supercritical flow state with shock waves at the nozzle.

[0207] pressure ratio It can be determined based on the shock wave relation, i.e.

[0208] (twenty one)

[0209] We can obtain:

[0210] .

[0211] (4) In the over-expansion state, there is a normal shock wave:

[0212] when At this pressure ratio range, a shock wave will be generated within the nozzle expansion section. This shock wave can be considered as a result of the pressure ratio increasing. The continuous increase in pressure causes the normal shock wave to move continuously into the pipe. Before the shock wave in the expansion section, it accelerates to supersonic speed, resulting in a decrease in pressure. After passing through the normal shock wave, the pressure increases. The subsonic airflow behind the shock wave decelerates and increases in pressure in the expansion section until it reaches Ma at the outlet. e <1, p e =p b The pressure ratio at this point changes along the axis as follows: Figure 3.1 Curve c in the figure is shown. This situation corresponds to the flow state with shock waves inside the tube in the supercritical state, and its flow characteristics are as follows: Ma at the throat. t =1, .

[0213] In one-dimensional flow, given the nozzle area ratio, incoming total pressure, and back pressure, the shock wave location within the nozzle can be calculated using the following method. Let A... s Represents the cross-sectional area where the shock wave is located, such as Figure 3.2 As shown, based on the condition that the airflow pressure at the outlet section equals the back pressure, the continuity equation is applied to the critical section and the outlet section, i.e. ,in ,so ,Depend on The nozzle exit value can be obtained by referring to the aerodynamic function table. and Ma e Then, using the continuity equation again, the total pressure recovery factor σ through the shock wave can be calculated: The Mach number Ma before the shock wave can be found from the normal shock wave table. s Since the flow between the throat and the shock front is adiabatic isentropic, the continuity equation yields... In the formula, A s The area of ​​the cross section where the shock wave is located.

[0214] Flow parameter calculation: Here, the shock wave is close to the nozzle, which should be distinguished from the oblique shock wave in the over-expansion state of case (3). In case (3), the gas can still be accelerated in the expansion section after experiencing the shock wave. However, in the nozzle shock wave, the gas enters the experimental chamber directly after the Mach number drops sharply at the outlet, without further acceleration. Therefore, there will be discontinuities in this case compared to the oblique shock wave in the over-expansion state of case (3) and the no-shock state in case (1). At this time, the Mach number before the shock wave is calculated first:

[0215] (twenty three)

[0216] Then, based on shock wave theory, the Mach number after the shock wave is obtained:

[0217] (twenty four)

[0218] After obtaining the Mach number after the shock wave, the temperature can be calculated using formula (15), the gas velocity can be calculated using formula (20), the total outlet pressure can be calculated using formula (8), and the outlet flow rate can be calculated using formula (10).

[0219] In one embodiment, the actual nozzle exit velocity v, Mach number, and nozzle exit flow rate in the state parameters of the under-expanded state with an expansion wave are calculated using the same method as the actual nozzle exit velocity v, Mach number, and nozzle exit flow rate in the state parameters of the fully expanded state without a shock wave.

[0220] The actual total outlet pressure of the nozzle in the state parameters of the under-expansion state with an expansion wave outside the nozzle is:

[0221] .

[0222] In one embodiment, the expression for the state parameters of the over-expansion state with oblique shock waves includes:

[0223] Based on the static pressure flow function, the flow function equation, and the aerodynamic function, the post-shock Mach number Ma0 is obtained. The static pressure flow function is: ,

[0224] The flow function equation is: ,

[0225] The aerodynamic function is: ,

[0226] Where y is the static pressure flow function and Ma0 is the Mach number after the shock wave;

[0227] Ideal gas velocity v ei :

[0228] ;

[0229] Actual nozzle exit airflow velocity v: ;

[0230] Actual total pressure at nozzle exit: ;

[0231] Mach number: ;

[0232] Second characteristic pressure ratio: ;

[0233] Where π2 is the second characteristic pressure ratio.

[0234] In one embodiment, the expression for the state parameters of the over-expansion state with a normal shock wave includes:

[0235] Mach number before shock wave:

[0236] ,

[0237] Where Ma1 is the Mach number before the shock wave;

[0238] Mach number after shock wave:

[0239] ;

[0240] Nozzle exit temperature:

[0241] ;

[0242] Ideal gas velocity v ei :

[0243] ;

[0244] Actual nozzle exit airflow velocity v: ;

[0245] Actual total pressure at nozzle exit: ;

[0246] Nozzle exit flow rate: .

[0247] In one embodiment, the nozzle model includes:

[0248] The isentropic area ratio calculation module is used to solve for the corresponding ideal Mach number and subsonic Mach number based on the isentropic area ratio formula.

[0249] The operating condition judgment and outlet status calculation module is used to determine the current pressure ratio p0 / p b The relationship with π3, π2, and π1 determines the nozzle state, calculates the ideal gas velocity, actual outlet gas velocity, and total outlet pressure, and outputs the current operating condition type.

[0250] The static pressure judgment module is used to determine whether an expansion wave or shock wave exists based on the comparison between the outlet static pressure and the outlet external back pressure.

[0251] The outlet flow calculation module is used to calculate the outlet flow in real time based on the current aerodynamic conditions, using the flow calculation formula.

[0252] The outlet temperature calculation module is used to calculate the actual outlet temperature based on the outlet Mach number and to reflect the impact of the shock wave or expansion process on the thermodynamic state.

[0253] The single-condition speed comparison module is used to calculate the speed using the fixed-condition formula under the design state, and compare it with the output of the condition judgment and exit state calculation module to reflect the sudden change and response characteristics of the speed under shock wave or expansion wave.

[0254] The Mach number back-calculation module is used to back-calculate the actual Mach number based on the ratio of the total outlet pressure to the external back pressure, and to verify the response effect of the nozzle model under the action of normal shock wave or oblique shock wave.

[0255] In practical implementation, based on the aforementioned theoretical analysis, it is known that the Laval nozzle will experience various typical flow states under different pressure ratios, ranging from subcritical, critical to supercritical (including expansion waves and shock waves), and each state can be precisely defined by characteristic pressure ratios (π3, π2, π1). The theory states that, given a Mach number Ma, the nozzle area ratio A... e / A tIt is the only certainty; however, whether the corresponding Mach number can actually be reached depends on the inlet total pressure p0 and the outlet back pressure p. b The ratio of area ratio to pressure ratio is used for determination. Therefore, the flow state of the nozzle is jointly determined by the area ratio and the pressure ratio. From the above analysis, it can be seen that the Laval nozzle has three characteristic pressure ratios. , and The flow state inside the Laval nozzle is divided into 4 regions and 7 working states by characteristic pressure ratio parameters.

[0256] Based on this theory, this embodiment builds a Laval nozzle modeling module covering the above-mentioned multiple operating conditions in the Matlab / Simulink environment. Its structure and function strictly correspond to the theoretical model, and its framework... Figure 4 As shown. The entire modeling process includes the following key functional modules:

[0257] The isentropic area ratio calculation module is used to solve for the corresponding ideal Mach number and subsonic Mach number based on the isentropic area ratio formula, which serves as the basis for judging the critical conditions of the working condition.

[0258] The operating condition judgment and outlet state calculation module combines the flow state classification logic in theory to determine the current pressure ratio p0 / p b The relationship with π3, π2, and π1 determines the nozzle state, calculates the ideal gas velocity, actual outlet gas velocity, and total outlet pressure, and outputs the current operating condition type.

[0259] The static pressure judgment module is used to determine whether there is an expansion wave or shock wave based on the comparison between the outlet static pressure and the outlet external back pressure, so as to provide a basis for subsequent velocity correction and state switching.

[0260] The outlet flow calculation module is used to calculate the outlet flow in real time based on the current aerodynamic conditions, using the flow calculation formula.

[0261] The outlet temperature calculation module is used to calculate the actual outlet temperature based on the outlet Mach number and to reflect the impact of the shock wave or expansion process on the thermodynamic state.

[0262] The single-condition speed comparison module is used to calculate the speed using the fixed-condition formula under the design state, and compare it with the output of the condition judgment and exit state calculation module to reflect the sudden change and response characteristics of the speed under shock wave or expansion wave.

[0263] The Mach number back-calculation module is used to back-calculate the actual Mach number based on the ratio of the total outlet pressure to the external back pressure, and to verify the response effect of the nozzle model under the action of normal shock wave or oblique shock wave.

[0264] Through the synergistic effect of the above modules, a complete modeling process was achieved, starting from the area ratio-Mach number theoretical formula, to working condition identification, flow characteristic calculation, and actual parameter output. This model can dynamically respond to changes in inlet total pressure, accurately identify the shock wave movement position and different flow states, and output key aerodynamic parameters in real time, verifying the feasibility and engineering applicability of the theoretical analysis in the simulation environment.

[0265] In practical implementation, multi-condition simulation verification mainly includes: setting the back pressure to an altitude of 15km and increasing the inlet total pressure p0 from 0 Pa to 120,000 Pa, ensuring the model remains stationary at the three characteristic pressure points π3, π2, and π1 to verify its dynamic response capability under different conditions. Key observations include: the trend of outlet total pressure change; changes in Mach number, velocity, and temperature within the nozzle; and the rationality of the simulated shock wave occurrence and position migration.

[0266] Specifically, this includes: conducting simulation tests on the Simulink mechanism module for the nozzle, setting the back pressure to an environment at an altitude of 15 km, and setting the inlet total pressure p0 to increase from 0 Pa to 120000 Pa, where p0 is maintained at three characteristic pressures corresponding to the characteristic pressure ratio for a period of time, testing whether the module can reflect the airflow velocity and temperature under different operating conditions. Figure 5(a) shows the outlet total pressure p. ea As can be seen from Figure 5(a) regarding the variation of the inlet total pressure p0, the operating conditions corresponding to the three characteristic pressure ratios of Flag are correct. The outlet total pressure is not equal to the inlet total pressure between characteristic pressure ratios π3 and π1 (4~14s). Under the condition of π2 (8~10s), the outlet total pressure p0... ea Slightly higher than the total inlet pressure p0, there is a sudden change near π1 (14s), which simulates the sudden change in the deceleration pressure of the shock gas at the nozzle exit. From the above analysis, it can be seen that the nozzle module's response to the total outlet pressure is basically correct.

[0267] As can be seen from the Mach number and velocity changes in Figure 5(b), during the rise of the total inlet pressure p0, the flow is entirely subsonic before the characteristic pressure ratio π2 (8s). After π2 (10s), it begins to flow supersonicly. However, as the shock wave gets closer to the outlet, the airflow velocity begins to slow down until there is a sudden change near π1 (14s). This simulates the situation where the oblique shock wave at the nozzle outlet causes the gas to decelerate. In addition, since the temperature changes after the gas flow accelerates inside the nozzle, the Mach number and velocity cannot be completely correlated, but the overall trend is the same, which is consistent with the situation in the ideal analysis. From the above analysis, it can be seen that the nozzle module's response to the Mach number and gas velocity under typical flow conditions is basically correct. In addition, there are also changes in temperature (Figure 5(c)), flow rate (Figure 5(d)), and outlet static pressure (Figure 5(e)).

[0268] In one embodiment, to verify the accuracy of the model and assist in the design of the control algorithm, the simulation results are compared with the theoretical expected trends to confirm that the model can accurately simulate: shock wave position movement; flow state switching under different pressure ratios; and the sensitive response of various physical quantities (such as temperature, flow rate, and static pressure) to the characteristic pressure ratio. Finally, the simulation module is used in the actual nozzle control system to provide data support and operating condition judgment basis for the control algorithm.

[0269] Specifically, this includes: to verify the accuracy and engineering applicability of the established nozzle model, comparing and analyzing the simulation results with the theoretical expected trends, focusing on a systematic evaluation from three aspects: shock wave position movement, flow state switching, and the response of key physical quantities to the characteristic pressure ratio.

[0270] During the simulation, the total pressure p0 at the nozzle inlet was gradually adjusted while maintaining the external back pressure p. b The simulation was conducted under constant pressure ratios, observing the changes in total pressure, Mach number, and velocity at the nozzle exit. The simulation results show that when the pressure ratio is close to the characteristic pressure ratio π2-π1, the model accurately captures the generation and movement of the shock wave, manifested as abrupt changes in airflow velocity and Mach number. Especially when the pressure ratio gradually approaches π1, the simulation results show a sudden change in the airflow at the nozzle exit from supersonic to subsonic speeds, consistent with the theoretical expectation that the shock wave will move to the nozzle exit and generate a normal shock wave. Furthermore, within different pressure ratio ranges, the model can correctly identify the seven flow states of the nozzle (subcritical, critical, fully expanded, under-expanded, over-expanded with oblique shock waves, normal shock waves, etc.) and automatically switch states based on the trends in Mach number and static pressure. This demonstrates the model's strong adaptability and accuracy in flow state identification. The simulation outputs of physical quantities such as temperature, flow rate, and static pressure all show a sensitive response to changes in the characteristic pressure ratio. The temperature changes significantly before and after the shock wave, the mass flow rate tends to saturate after reaching the critical state, and the deviation between the static pressure and the external back pressure clearly reflects whether the current state of full expansion has been reached.

[0271] The aforementioned trends are highly consistent with theoretical calculations, further verifying the numerical reliability and physical consistency of the model. Finally, based on the verified simulation module, it was embedded into the Simulink control framework of the actual nozzle control system as the system's dynamic state monitoring unit. The Mach number, exit pressure, and characteristic pressure ratio flags provided by the simulation module can serve as input or feedback variables for the control algorithm, providing real-time and reliable data support for the nozzle system's adjustment strategy. This approach not only enhances the control system's ability to identify complex flow states but also strengthens the system's steady-state performance and dynamic robustness under varying operating conditions and high pressure ratios, providing a solid foundation for subsequent intelligent nozzle control.

[0272] The embodiments provided by this invention aim to solve the key problem that general models cannot simultaneously handle complex flow calculations and computational speed. By establishing a model in the Matlab / Simulink environment that flexibly adapts to different pressure ratios and flow states, the performance of the nozzle can be accurately described under various typical flow conditions. Simultaneously, the solution of this invention optimizes the calculation process for complex flow states, employing more efficient algorithms and numerical methods, significantly improving computational efficiency and simulation accuracy, ensuring high-precision simulations under high pressure ratios and supersonic conditions. Furthermore, the solution of this invention adds temperature-pressure coupling and static pressure calculation modules, further improving the simulation accuracy of the thermodynamic processes inside the nozzle, enabling the nozzle model to more accurately reflect the influence of temperature and pressure changes on the flow. Through these technical improvements, this invention not only improves the accuracy and adaptability of nozzle modeling but also addresses the shortcomings of existing technologies in real-time response and numerical stability, providing more reliable support for nozzle design and control.

[0273] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for modeling and optimizing the aerodynamic characteristics of a jet nozzle, characterized in that, The method includes: Based on the isentropic flow theory, the functional relationship between the area ratio and the Mach number is determined as the basic model of the nozzle aerodynamic characteristics. Based on different pressure ratios and combined with gas dynamics theory, the flow state inside the nozzle is divided to obtain multiple working conditions and determine the state parameters under each working condition. The multiple working conditions include critical state, subcritical state and supercritical state. The supercritical state includes fully expanded state without shock wave, under-expanded state with expansion wave outside the tube, over-expanded state with oblique shock wave and over-expanded state with normal shock wave. Based on the basic model of nozzle aerodynamic characteristics and state parameters under various operating conditions, a nozzle model is constructed. The input parameters of the nozzle model include total pressure, external back pressure, throat area and outlet area. The output parameters of the nozzle model include Mach number, airflow velocity, temperature and flow rate under various flow conditions. Multi-condition simulation verification of the nozzle model; Verify the accuracy of the nozzle model and apply it to the actual nozzle control system for condition judgment; The nozzle model includes: The isentropic area ratio calculation module is used to solve for the corresponding ideal Mach number and subsonic Mach number based on the isentropic area ratio formula. The operating condition judgment and outlet status calculation module is used to determine the current pressure ratio p0 / p b The relationship between π3, π2, and π1 is used to determine the nozzle state, calculate the ideal gas velocity, actual outlet gas velocity, and total outlet pressure, and output the current operating condition type; where p0 is the total inlet pressure, p b For the external back pressure at the outlet, π3 is the third characteristic pressure ratio, π2 is the second characteristic pressure ratio, and π1 is the first characteristic pressure ratio; The static pressure judgment module is used to determine whether an expansion wave or shock wave exists based on the comparison between the outlet static pressure and the outlet external back pressure. The outlet flow calculation module is used to calculate the outlet flow in real time based on the current aerodynamic conditions, using the flow calculation formula. The outlet temperature calculation module is used to calculate the actual outlet temperature based on the outlet Mach number and to reflect the impact of the shock wave or expansion process on the thermodynamic state. The single-condition speed comparison module is used to calculate the speed using the fixed-condition formula under the design state, and compare it with the output of the condition judgment and exit state calculation module to reflect the sudden change and response characteristics of the speed under shock wave or expansion wave. The Mach number back-calculation module is used to back-calculate the actual Mach number based on the ratio of the total outlet pressure to the external back pressure, and to verify the response effect of the nozzle model under the action of normal shock wave or oblique shock wave.

2. The method for modeling and optimizing the aerodynamic characteristics of a jet nozzle according to claim 1, characterized in that, The expression for the functional relationship between the area ratio and the Mach number is: ， ， Where A is the cross-sectional area, A t Let be the nozzle throat area, and q be the flow rate function. Here, is the velocity coefficient, Ma is the Mach number, and k is the specific heat ratio of the gas.

3. The method for modeling and optimizing the aerodynamic characteristics of a jet nozzle according to claim 2, characterized in that, The expressions for the state parameters of the critical state include: Nozzle exit Mach number: , The ratio of outlet static pressure to inlet total pressure: , ， Mass flow rate through the nozzle exit: , Among them, Ma e A is the nozzle exit Mach number. e p is the nozzle exit area. e For the outlet static pressure, p b3 Let q be the outlet pressure under critical conditions. m,max Where T0 is the maximum mass flow rate, T0 is the total temperature, and K is a dimensionless constant. R is the gas constant.

4. The method for modeling and optimizing the aerodynamic characteristics of a jet nozzle according to claim 3, characterized in that, The expressions for the state parameters of the subcritical state include: The exit Mach number is calculated using a one-dimensional isentropic flow formula to determine the corresponding temperature. ， in, Let p be the gas velocity and p be the gas pressure. Let T be the gas density and T be the gas temperature. (a), (b), (c), (d), and (e) are the formula numbers, respectively. Based on the total inlet pressure p0 and the external outlet back pressure p b Calculate the velocity v of an ideal gas ei : ， Among them, v ei For the ideal gas velocity, c P For the specific heat capacity at constant pressure, T t Temperature of the gas in the throat; Based on the loss and the ideal gas velocity v ei To obtain the actual outlet airflow velocity v of the nozzle: ， Where v is the actual outlet airflow velocity. This is the loss coefficient; The actual total outlet pressure p of the nozzle is obtained based on the pressure loss. ea : ， Where, p ea T represents the actual total pressure at the nozzle exit. e This refers to the nozzle exit temperature. Based on the actual total outlet pressure p of the nozzle ea Calculate the corresponding Mach number: ； Calculate the nozzle exit flow rate. , Where, q m For export flow, λ e This is the nozzle exit velocity coefficient.

5. The method for modeling and optimizing the aerodynamic characteristics of a jet nozzle according to claim 4, characterized in that, The expressions for the state parameters of the fully expanded, shock-free state include: Outlet static pressure and characteristic pressure ratio: , ， Where, p b1 The outlet pressure under fully expanded, shock-free conditions; Ideal gas velocity v ei : ; Actual nozzle exit airflow velocity v: ， Actual total pressure at nozzle exit: ; Nozzle exit flow rate: ; Nozzle exit temperature: .

6. The method for modeling and optimizing the aerodynamic characteristics of a jet nozzle according to claim 5, characterized in that, The actual nozzle exit velocity v, Mach number, and nozzle exit flow rate in the under-expansion state with an expansion wave outside the nozzle are calculated using the same method as the actual nozzle exit velocity v, Mach number, and nozzle exit flow rate in the fully expanded state without a shock wave. The actual total outlet pressure of the nozzle in the state parameters of the under-expansion state with an expansion wave outside the nozzle is: 。 7. The method for modeling and optimizing the aerodynamic characteristics of a jet nozzle according to claim 5, characterized in that, The expressions for the state parameters of the over-expansion state with oblique shock waves include: Based on the static pressure flow function, the flow function equation, and the aerodynamic function, the post-shock Mach number Ma0 is obtained. The static pressure flow function is: , The flow function equation is: , The aerodynamic function is: , Where y is the static pressure flow function and Ma0 is the Mach number after the shock wave; Ideal gas velocity v ei : ； Actual nozzle exit airflow velocity v: ; Actual total pressure at nozzle exit: ; Mach number: ; Second characteristic pressure ratio: ; Where, p b2 The outlet pressure is under oblique shock wave conditions in an over-expansion state.

8. The method for modeling and optimizing the aerodynamic characteristics of a jet nozzle according to claim 7, characterized in that, The expressions for the state parameters of the over-expansion state with a normal shock wave include: Mach number before shock wave: ， Where Ma1 is the Mach number before the shock wave; Mach number after shock wave: ； Nozzle exit temperature: ； Ideal gas velocity v ei : ； Actual nozzle exit airflow velocity v: ; Actual total pressure at nozzle exit: ; Nozzle exit flow rate: .

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