Source function equation inverse function calculation method and supersonic nozzle profile calculation method

Abstract

The application discloses a source flow equation inverse function calculation method and an ultrasonic speed nozzle profile calculation method. E B E B E C E B E C The application solves the problems of huge calculation amount and poor flow field quality by combining the 3 / 4 / 5th order curve algorithm, so that unknown parameters can be obtained by calculation, and excellent flow field quality can be obtained by the 5th order curve algorithm.​​​​​​​​​

Description

Technical Field

[0001] This invention relates to the field of wind tunnel nozzle design, and more specifically, to a method for calculating the inverse function of the source-flow equation and a method for calculating the supersonic nozzle profile. Background Technology

[0002] The core algorithm for the design of the nozzle inner profile of a 2–8 Ma supersonic wind tunnel is the Sivells active flow method, or a method with special gas property corrections. The calculation steps are roughly as follows: first, design conditions are given; a series of empirical coefficients are proposed based on the design conditions; the inner profile of the nozzle is calculated by substituting the empirical coefficients into the Sivells algorithm; then, boundary layer correction is performed on the inner profile; CFD verification is performed using the corrected profile; when the verified nozzle flow field is unqualified, the empirical coefficients need to be re-given and the inner profile of the nozzle needs to be recalculated; this process is repeated continuously until the optimal combination of empirical coefficients is found to optimize the flow field at the nozzle exit.

[0003] Regarding Sivells' source-flow method, it can be roughly divided into three types: cubic, quartic, and quintic curve methods, each with its own advantages and disadvantages.

[0004] The calculation of a quintic curve requires six empirical coefficients to be manually specified: expansion angle η, throat curvature R, and Mach number M at point E. E Mach number M at point B B Location of point E (x) E and the position of point C x C These six empirical coefficients must be precise and reasonable to ensure the accuracy and rationality of the calculation results, which requires a lot of trial and error, and is also a huge burden on the computer.

[0005] When calculating a quartic curve, four empirical coefficients need to be manually specified: the expansion angle η, the throat curvature R, and the Mach number M at point E. E Mach number M at point B B Other coefficients can be obtained through calculation. These four empirical coefficients must be accurate and reasonable to ensure the accuracy and rationality of the calculation results; therefore, a certain number of trials are required.

[0006] The cubic curve, calculated using Sivells' source-flow method, has an internal surface shape that is close to that of the quartic curve, but with significantly less computation. The calculation requires two empirical coefficients: the expansion angle η and the throat curvature R; other coefficients can be calculated. However, the cubic curve calculation requires differentiating the inverse function of the source-flow equation, which, being a non-simple function, does not possess an analytic inverse function. Therefore, the cubic curve calculation method is difficult to apply in practice due to its computational complexity.

[0007] In summary, without experimental databases and sufficient experience, designing nozzles with high flow field quality requires a significant investment of human and material resources.

[0008] Therefore, it is necessary to develop a method for calculating the inverse function of the source-flow equation and a method for calculating the supersonic nozzle profile.

[0009] The information disclosed in the background section of this invention is intended only to enhance the understanding of the general background of this invention, and should not be construed as an admission or in any way implying that such information constitutes prior art known to those skilled in the art. Summary of the Invention

[0010] This invention proposes a method for calculating the inverse function of the source-flow equation and its supersonic nozzle profile. Based on the Sivells algorithm, it optimizes the Sivells algorithm into a composite algorithm that can achieve the same excellent flow field quality with minimal computational effort, without requiring a database. By effectively combining the 3rd, 4th, and 5th order curve algorithms, unknown empirical parameters can be obtained through calculation, thus eliminating the huge computational burden and achieving excellent flow field quality through the 5th order curve algorithm.

[0011] In a first aspect, embodiments of this disclosure provide a method for calculating the inverse function of a source-flow equation, including:

[0012] Determine the source-flow equation based on the velocity coefficient W and the Mach number M as the independent variable;

[0013] Calculate the first and second derivatives based on the source-flow equation;

[0014] Determine the first and second derivatives based on the positive and negative functions;

[0015] Calculate the first and second derivatives of the inverse function based on the source-flow equation.

[0016] Preferably, the source-flow equation in the form r = f(W) is:

[0017]

[0018] The source-flow equation in the form r = g(M) is:

[0019]

[0020] Where r1 is the source flow radius at M=1, and γ is the airflow adiabatic coefficient.

[0021] Preferably, the first derivative of the source-flow equation r = f(W) is:

[0022]

[0023] The second derivative of the source-flow equation r = f(W) is:

[0024]

[0025] Preferably, the first derivative of the source-flow equation r = g(M) is:

[0026]

[0027] The second derivative of the source-current equation r = g(M) is:

[0028]

[0029] Preferably, the first and second derivatives of the positive and negative functions are as follows:

[0030]

[0031]

[0032]

[0033]

[0034] Preferably, the first and second derivatives are based on the inverse function of the source-flow equation r = f(W):

[0035]

[0036]

[0037]

[0038] Preferably, the first and second derivatives are based on the inverse function of the source-flow equation r = g(M):

[0039]

[0040]

[0041] Secondly, this disclosure also provides a method for calculating the profile of a supersonic nozzle, including:

[0042] Determine the boundary conditions and the initial values ​​of the throat curvature and the expansion segment angle;

[0043] Based on the initial values ​​of the throat curvature and the expansion segment angle, a cubic curve calculation is performed using the inverse function calculation method of the source-flow equation to obtain the Mach number M at point E. E Mach number M at point B B ;

[0044] Based on the initial values ​​of the laryngeal curvature and the dilation segment angle, ME M B Perform four curve calculations to obtain the position x of point E. E Location of point C (x) C ;

[0045] Based on the initial values ​​of the laryngeal curvature and the dilation segment angle, M E M B x E x C Perform five curve calculations to obtain the five curve profiles and then perform boundary layer correction.

[0046] CFD verification is performed on the corrected profile. If the flow field quality does not meet the standards, the expansion section angle and / or throat curvature are updated, and the above steps are repeated until the flow field quality meets the standards and the calculation is completed.

[0047] Preferably, the fourth-order curve calculation is performed based on the source-flow equation inverse function calculation method.

[0048] Preferably, the fifth-order curve calculation is performed based on the source-flow equation inverse function calculation method.

[0049] Preferably, the boundary conditions include the nozzle exit Mach number, nozzle size, total pressure before and after the nozzle, and static pressure requirements.

[0050] Its beneficial effects are as follows:

[0051] To improve flow field quality while reducing computational load, this invention combines 3rd, 4th, and 5th order curve algorithms into a composite algorithm to calculate unknown empirical parameters. This not only eliminates the huge computational load but also allows for excellent flow field quality through the 5th order curve algorithm.

[0052] To solve the system of cubic / quadratic curve equations simultaneously, the first and second derivatives of the inverse functions of the source and flow functions must be determined. Using the mathematical relationship between the first and second derivatives of positive and inverse functions, the inverse function of the source and flow functions can be skipped. The analytical solution relationship between the first and second derivatives of the inverse functions of the source and flow functions can be directly obtained, and the cubic curve equation system can be effectively solved based on this relationship.

[0053] The method of the present invention has other features and advantages that will be apparent from or will be set forth in detail in the accompanying drawings and following detailed description, which together serve to explain the particular principles of the invention. Attached Figure Description

[0054] The above and other objects, features and advantages of the present invention will become more apparent from the more detailed description of exemplary embodiments of the invention in conjunction with the accompanying drawings, wherein the same reference numerals generally represent the same parts.

[0055] Figure 1 A flowchart illustrating the steps of a method for calculating the inverse function of the source-flow equation according to an embodiment of the present invention is shown.

[0056] Figure 2 A flowchart illustrating the steps of a supersonic nozzle profile calculation method according to an embodiment of the present invention is shown.

[0057] Figure 3 A feature line diagram of the Sivells nozzle algorithm according to an embodiment of the present invention is shown. Detailed Implementation

[0058] Preferred embodiments of the invention will now be described in more detail. While preferred embodiments of the invention are described below, it should be understood that the invention can be implemented in various forms and should not be limited to the embodiments set forth herein.

[0059] To facilitate understanding of the solutions and effects of the embodiments of the present invention, two specific application examples are given below. Those skilled in the art should understand that these examples are merely for the purpose of understanding the present invention, and any specific details therein are not intended to limit the present invention in any way.

[0060] Example 1

[0061] Figure 1 A flowchart illustrating the steps of a method for calculating the inverse function of the source-flow equation according to an embodiment of the present invention is shown.

[0062] like Figure 1 As shown, the method for calculating the inverse function of the source-flow equation includes: Step 101, determining the source-flow equation based on the velocity coefficient W and the Mach number M as the independent variable; Step 102, calculating the first-order and second-order derivatives based on the source-flow equation; Step 103, determining the first-order and second-order derivatives based on the positive and negative functions; Step 104, calculating the first-order and second-order derivatives based on the inverse function of the source-flow equation.

[0063] In one example, the source-flow equation in the form r = f(W) is:

[0064]

[0065] The source-current equation in the form r = g(M) is:

[0066]

[0067] Where r1 is the source flow radius at M=1, and γ is the airflow adiabatic coefficient.

[0068] In one example, the first derivative based on the source-flow equation r = f(W) is:

[0069]

[0070] The second derivative of the source-flow equation r = f(W) is:

[0071]

[0072] In one example, the first derivative based on the source-flow equation r = g(M) is:

[0073]

[0074] The second derivative of the source-current equation r = g(M) is:

[0075]

[0076] In one example, the first and second derivatives of the positive and inverse functions are as follows:

[0077]

[0078]

[0079]

[0080]

[0081] In one example, the first and second derivatives of the inverse function of the source-flow equation r = f(W) are shown:

[0082]

[0083]

[0084] In one example, the first and second derivatives of the inverse function of the source-flow equation r = g(M) are:

[0085]

[0086]

[0087] Specifically, given the source-flow equations based on the velocity coefficient W and the Mach number M as independent variables, they are presented as formulas (1) and (2) in the form of r = f(W) and r = g(M). The inverse functional forms of the source-flow equations can be written as W = F(r) and M = G(r), and in fact, there are no analytical expressions for W = F(r) and M = G(r).

[0088] Calculate the first-order derivatives based on the source-flow equations r = f(W) and r = g(M), i.e. and and the second derivative, i.e. and

[0089] Given the first and second derivative relationships based on the positive and negative functions, respectively, as formulas (7)-(10), substituting the first and second derivatives of the source-flow equations r=f(W) and r=g(M) into the relationships based on the first and second derivatives of the positive and negative functions, we can obtain the first derivatives of the inverse functions W=F(r) and M=G(r) of the source-flow equations. and and second derivative and The analytical solution is given by formulas (11)-(14) based on the first and second derivatives of the inverse functions W=F(r) and M=G(r) of the source-flow equation.

[0090] Example 2

[0091] Figure 2 A block diagram of a supersonic nozzle profile calculation device according to an embodiment of the present invention is shown.

[0092] like Figure 2 As shown, the supersonic nozzle profile calculation device includes:

[0093] Step 201: Determine the boundary conditions and the initial values ​​of the throat curvature and the expansion segment angle;

[0094] Step 202: Based on the initial values ​​of throat curvature and expansion segment angle, perform cubic curve calculation using the inverse function calculation method of the source-flow equation to obtain the Mach number M at point E. E Mach number M at point B B ;

[0095] Step 203, based on the initial values ​​of laryngeal curvature and dilation segment angle, M E M B Perform four curve calculations to obtain the position x of point E. E Location of point C (x) C ;

[0096] Step 204, based on the initial values ​​of laryngeal curvature and dilation segment angle, M E M B x E x C Perform five curve calculations to obtain the five curve profiles and then perform boundary layer correction.

[0097] Step 205: Perform CFD verification on the corrected profile. If the flow field quality does not meet the standard, update the expansion section angle and / or throat curvature, and repeat the above steps until the flow field quality meets the standard and the calculation is completed.

[0098] In one example, a fourth-order curve is calculated based on the inverse function of the source-flow equation.

[0099] In one example, a fifth-order curve is calculated based on the inverse function of the source-flow equation.

[0100] In one example, the boundary conditions include the nozzle exit Mach number, nozzle size, total pressure before and after nozzle, and static pressure requirements.

[0101] Specifically, the boundary conditions are determined, including the nozzle exit Mach number, nozzle size, total pressure before and after the nozzle, and static pressure requirements, and the initial values ​​of throat curvature and expansion section angle are determined.

[0102] Figure 3 A feature line diagram of the Sivells nozzle algorithm according to an embodiment of the present invention is shown.

[0103] like Figure 3 As shown, the Sivells nozzle algorithm consists of TI, EG, BA, IE, and BC lines. Finally, the TG and AD lines, representing the nozzle's internal profile, are obtained using the characteristic line equation. Based on the initial conditions of throat curvature R and the initial value η of the expansion segment angle, a total of two empirical parameters, these are substituted into the Sivells active flow cubic curve method for calculation.

[0104] Substituting Sivells' source-flow multiple curve method, the system of equations mainly consists of the following 12 equations:

[0105]

[0106]

[0107]

[0108]

[0109] The boundary conditions for Sivells cubic curves with source flow are:

[0110] W′ I and W″ I It is known that

[0111]

[0112]

[0113]

[0114] After substituting the above boundary conditions, the velocity coefficient W at point E can be obtained from the system of equations. E And the Mach number M at point B B .

[0115] Based on the initial conditions of throat curvature R and initial value of expansion segment angle η, the velocity coefficient W at point E is calculated using the cubic curve method. E And the Mach number M at point B B A total of four empirical parameters were used, and the calculation was performed using the fourth-order curve method. Since W... E and M B Since the cubic curve has already been calculated, this value is reasonable.

[0116] The boundary conditions for a Sivells source-current quartic curve are:

[0117] W′ I 、W″ I W E M B It is known that

[0118]

[0119]

[0120]

[0121] After substituting the above boundary conditions, the system of equations can be solved to obtain the absolute coordinates x of point E. E and the absolute coordinates x of point C C .

[0122] Based on the initial conditions of throat curvature R and initial value of expansion segment angle η, the velocity coefficient W at point E is calculated using the cubic curve method. E And the Mach number M at point B B The absolute coordinates x of point E obtained from the 4th-order curve calculation E and the absolute coordinates x of point C C A total of 6 empirical parameters were used, and the calculation was performed using the fifth-order curve method. Since x... E and x C Since the value has already been calculated for the quartic curve, this value is reasonable.

[0123] The boundary conditions for a Sivells source-flow quintic curve are:

[0124] W′ I 、W″ I W E M B x E x C It is known that

[0125]

[0126]

[0127]

[0128] After substituting the above boundary conditions, the equations can be used to solve for the nozzle profile based on the 5th-order curve.

[0129] CFD verification is performed on the profile. If the flow field quality does not meet the standards, the expansion section angle η and / or throat curvature R are updated. The above steps are repeated until the flow field quality meets the standards and the calculation is completed.

[0130] This method utilizes the relationship between the derivatives of positive and inverse functions in differential equations to derive the first and second derivative expressions of the source and current functions and their inverses. This makes the system of equations representing the IE and BC lines in the cubic curve solvable, and allows the determination of M using the cubic curve. E and M B This facilitates subsequent calculations.

[0131] By combining the 3rd, 4th, and 5th order curve algorithms into a composite algorithm, unknown empirical parameters can be obtained through calculation, eliminating the need for extensive manual trial and error. This reduces the 6-cycle iterative calculation process to 2 cycles, decreasing the computational load by 3-4 orders of magnitude. Simultaneously, the 5th order curve algorithm yields excellent flow field quality, such as a Mach number standard deviation σ ≤ 0.005 and an axial Mach number gradient dMa. C The ratio of / dX≤0.025 and the area ratio of the uniform rhomboid region κ≥0.8 are both better than the national standard.

[0132] Those skilled in the art should understand that the above description of the embodiments of the present invention is only intended to illustrate the beneficial effects of the embodiments of the present invention, and is not intended to limit the embodiments of the present invention to any of the examples given.

[0133] The various embodiments of the present invention have been described above. These descriptions are exemplary and not exhaustive, nor are they limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments.

Claims

1. A method for calculating the profile of a supersonic nozzle, characterized in that, include: Determine the boundary conditions and the initial values ​​of the throat curvature and the expansion segment angle; Based on the initial values ​​of the throat curvature and the expansion segment angle, a cubic curve calculation is performed using the inverse function calculation method of the source-flow equation to obtain the Mach number at point E. Mach number at point B ; Based on the initial values ​​of the laryngeal curvature and the dilation segment angle, , Perform four curve calculations to obtain the position of point E. Point C ; Based on the initial values ​​of the laryngeal curvature and the dilation segment angle, , , , Perform five curve calculations to obtain the five curve profiles and then perform boundary layer correction. CFD verification is performed on the corrected profile. If the flow field quality does not meet the standards, the expansion section angle and / or throat curvature are updated, and the above steps are repeated until the flow field quality meets the standards and the calculation is completed. The methods for calculating the inverse function of the source-flow equation include: Determined based on velocity coefficient And Mach number The source-flow equation is the independent variable; Calculate the first and second derivatives based on the source-flow equation; Determine the first and second derivatives based on the positive and negative functions; Calculate the first and second derivatives of the inverse function based on the source-flow equation.

2. The supersonic nozzle profile calculation method according to claim 1, wherein, The fourth-order curve is calculated based on the inverse function calculation method of the source-flow equation.

3. The supersonic nozzle profile calculation method according to claim 1, wherein, The fifth-order curve is calculated based on the inverse function calculation method of the source-flow equation.

4. The supersonic nozzle profile calculation method according to claim 1, wherein, by The source-flow equations of the form are as follows: （1） by The source-flow equations of the form are as follows: （2） in, for The source radius at that point, is the airflow adiabatic coefficient.

5. The supersonic nozzle profile calculation method according to claim 4, wherein, Based on source-flow equations The first derivative is: （3） Based on source-flow equations The second derivative is: （4）。 6. The method for calculating the profile of a supersonic nozzle according to claim 5, wherein, Based on source-flow equations The first derivative is: （5） Based on source-flow equations The second derivative is: （6）。 7. The supersonic nozzle profile calculation method according to claim 6, wherein, Based on the first and second derivatives of the positive and negative functions: （7） （8） （9） （10）。 8. The method for calculating the profile of a supersonic nozzle according to claim 7, wherein, Based on the source-flow equation The first and second derivatives of the inverse function: （11） （12）。 9. The method for calculating the profile of a supersonic nozzle according to claim 7, wherein, Based on the source-flow equation The first and second derivatives of the inverse function: （13） （14）。

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