A nozzle shortening design method based on the Mach number distribution of the initial expansion section wall

Abstract

The present invention provides a nozzle shortening design method based on the initial expansion section wall Mach number distribution, comprising the following steps: (1) giving the initial expansion section wall Mach number distribution; (2) calculating the nozzle core area flow field; (3) calculating the initial expansion section flow field; (4) giving the wave-breaking zone outlet boundary Mach number or flow angle distribution; (5) calculating the wave-breaking zone flow field; (6) the nozzle core area, initial expansion section, and wave-breaking zone obtained in steps 1 to 5 are spatially connected in sequence to form the flow field and profile of the entire nozzle. The present invention solves the problem that the characteristic lines at the initial expansion section wall surface are easy to intersect when designing the nozzle in the traditional method; solves the problem that the traditional method must be obtained by interpolation of solution points on both sides when generating the nozzle wall surface; achieves the effect of controlling the nozzle length by controlling the Mach number distribution; shortens the nozzle length under the same flow field quality, and due to the shortened length, the processing cost and processing difficulty are reduced; and the effective test area is increased.

Description

Technical Field

[0001] The invention relates to the technical field of wind tunnel nozzles, and in particular to a nozzle shortening design method based on the initial expansion section wall surface Mach number distribution. Background Art

[0002] Wind tunnel nozzles can provide high-quality supersonic flow for wind tunnel tests. The design of their wall profiles is directly related to the quality of the wind tunnel test flow field and is the key to the entire wind tunnel design. Affected by viscosity, the thickness of the supersonic nozzle wall boundary layer increases with the length. Under high Mach number conditions, it can even reach half of the exit radius, which has a great impact on the flow field quality, structural weight and processing accuracy; however, too short a length makes it difficult to ensure the uniformity of the exit flow field. To this end, it is necessary to develop corresponding design methods to balance the problem of shortening the wind tunnel nozzle and the uniformity of the exit flow field. From the aerodynamic principle, it can be seen that this goal can only be achieved by shortening the length of the core area of ​​the nozzle.

[0003] Traditional wind tunnel nozzle design methods include direct design of expansion section profile, using source flow area approximate calculation, arc plus straight line method approximation, etc. Among them, Busemann is a typical direct design of expansion section profile method. According to experience, this method uses a series of broken lines with gradually increasing slopes to replace the expansion section curve. At each turning point, the expansion wave is emitted, and the reflected waves on the nozzle axis or the curved wall intersect with each other to obtain the wave line. All broken lines are connected in sequence to form a continuous and smooth nozzle profile. However, this method has the following disadvantages:

[0004] 1) The nozzle expansion section profile of this method is designed based on experience;

[0005] 2) The expansion wave system at each level must be strictly organized. When the size of the broken line segment is large, the intersection of the expansion waves and their reflection on the wall will cause the expansion waves to not reflect at the turning point of the broken line segment, thereby causing the intersection of the characteristic lines at the wall, which in turn destroys the orderliness of the expansion wave system and causes irregularities in the nozzle's wave elimination exit flow field.

[0006] The Foelsch method is a typical method that uses the source flow area approximation calculation. The initial expansion section of this method approximates the sonic flow at the throat as the supersonic source flow at the turning point. The wave-breaking section profile is to transform the supersonic source flow into a parallel uniform flow that meets the design Mach number requirements, while eliminating all expansion waves on the wall. However, this method has the following disadvantages:

[0007] 1) The wall curve at the initial expansion section of the nozzle in this method is extremely dependent on experience;

[0008] 2) Due to the limitations of the source flow assumption, the designed nozzle is generally very long;

[0009] 3) The source flow area is directly connected to the uniform area, causing discontinuity in the axial velocity gradient and affecting the flow field quality.

[0010] The Sivells method obtains a nozzle profile with continuous curvature by setting the axial Mach number distribution, so that there is no concentrated compression wave inside the nozzle, which improves the flow field quality. However, this method has the following disadvantages:

[0011] 1) This method still uses the assumption of the source flow area, and the nozzle length is still very long;

[0012] 2) Setting the axial Mach number distribution dependent empirical formula;

[0013] 3) It is very easy for characteristic lines to intersect when calculating based on the axial Mach number distribution.

[0014] The arc plus straight line method is a relatively simple approximation method. This method replaces the turning point with an arc to reduce the flow separation caused by excessive curvature, and replaces the initial expansion section with a straight line. This method is simple in principle and easy to implement, but it cannot achieve good nozzle length control.

[0015] Zhao Yilong, Zhao Yuxin and others proposed a nozzle shortening design method based on the axis Mach number distribution in the article "Supersonic Profile Controllable Nozzle Design Method". This method abandons the assumption of the source flow area and directly controls the axis Mach number distribution through the sixth Bezier curve to shorten the nozzle core area. However, this method has the following disadvantages:

[0016] 1) The law of axis Mach number distribution depends on experience and there is no unified law;

[0017] 2) The calculated nozzle characteristic lines are very likely to intersect near the wall;

[0018] Xu Jinglei et al. proposed a nozzle shortening design method based on wall pressure distribution. This method controls the nozzle length by giving the pressure distribution of the entire expansion section of the nozzle. However, this method has the following disadvantages:

[0019] 1) The calculated nozzle cannot obtain uniform flow, so it is only applicable to aircraft nozzles;

[0020] 2) The total length of the nozzle cannot be controlled;

[0021] 3) As the Mach number increases, the characteristic line grid will become sparse, affecting the final calculation accuracy.

[0022] Comprehensive analysis shows that most of the current methods for nozzle shortening design rely on experience, and the operation is cumbersome, and the nozzle characteristic lines are prone to intersecting. Combining the principle of the nozzle design characteristic line method, the present invention provides a nozzle shortening design method based on the initial expansion section wall Mach number distribution, solves the problem of characteristic lines intersecting on the initial expansion section wall, and optimizes the final length of the nozzle according to the initial expansion section wall length. Summary of the invention

[0023] In view of the above-mentioned shortcomings of the prior art, the object of the present invention is to provide a nozzle shortening design method based on the initial expansion section wall Mach number distribution;

[0024] To achieve the above-mentioned purpose of the invention, the technical solution of the present invention is as follows:

[0025] A nozzle shortening design method based on the initial expansion section wall Mach number distribution comprises the following steps:

[0026] Step 1, giving the initial expansion section wall Mach number distribution;

[0027] Step 2, calculating the flow field in the core area of ​​the nozzle;

[0028] Step 3, calculating the flow field of the initial expansion section;

[0029] Step 4, giving the Mach number or flow angle distribution at the exit boundary of the wave-breaking zone;

[0030] Step 5, calculating the flow field in the wave-breaking area;

[0031] Step 6: The nozzle core area, initial expansion section, and wave-breaking area obtained in steps 1 to 5 are spatially connected in sequence to form the flow field and profile of the entire nozzle.

[0032] As a preferred embodiment, the given initial expansion section wall Mach number distribution in step 1 comprises the following steps:

[0033] The given initial expansion section wall Mach number distribution described in step 1 comprises the following steps:

[0034] 1.1) Establish a coordinate system with the origin as the coordinate, the horizontal axis x is the lateral distance coordinate of the nozzle, and the vertical axis Ma is the Mach number;

[0035] 1.2) Given 5 control points C i [x i ,Ma i ], where i = 1, 2, 3, 4, 5, and the initial expansion section wall Mach number distribution control coefficient Ma is given d ,L,yp1,yp2, f 1 、f2 ， x0,Ma0,where Ma d is the design Mach number, L is the length of the wall of the initial expansion section, yp1 is the slope at the control point C1, yp2 is the slope at the control point C5[x5,Ma5], f 1 、f 2 is the proportional coefficient of points C2 and C4 along the line segments at the starting and ending points;

[0036] 1.3) The default coordinates of the initial point C1[x1,Ma1] are C1[0,1.0], and the final point C5[L,Ma d ];

[0037] 1.4) Through the slope yp1 and the proportional coefficient f 1 Get the coordinates of C2;

[0038] 1.5) C3[x3,Ma3] is the middle point of segment C1C5, and its horizontal coordinate is f 1 L, ordinate is 1.0+yp1 f 1 L and Ma d The smaller value among

[0039] 1.6) Through the slope yp2 and the proportional coefficient f 2 Get the coordinates of C4;

[0040] 1.7) According to 1.1), 1.2), 1.3), 1.4), 1.5), and 1.6), the control points C1[0,1.0], C2[x2,Ma2], C3[x3,Ma3], C4[x4,Ma4], and C5[L,Ma d ], the distribution of the wall Mach number of the initial expansion section with the axial coordinate x is solved by the B-Spline curve.

[0041] As a preferred embodiment, the calculation of the nozzle core area flow field in step 2 includes the following steps:

[0042] 2.1) Given the number of throat boundary grid points n and the inlet parameter distribution, the throat boundary grid points are given equidistantly through the number of throat boundary grid points n, and the inner point in the characteristic line method is applied to solve the coordinates and flow field parameters of the intersection point of the right characteristic line emitted from the inlet starting point and the left characteristic line emitted from the adjacent throat boundary grid point, and the characteristic line method is applied in sequence to solve the coordinates and flow field parameters of the point to be solved where the right characteristic line emitted from the n-1th throat boundary grid point intersects the left characteristic line emitted from the throat boundary grid end point, wherein the inlet parameters refer to Mach number, pressure, density, velocity, velocity direction, and temperature, and the solution point obtained in this step is the inner point of the nozzle core area;

[0043] 2.2) Apply the axis point in the characteristic line method to solve the coordinates and flow field parameters of the point to be solved where the right characteristic line from the end point of the nozzle core area in 2.1) intersects the streamline from the end point of the throat boundary grid. The solution point obtained in this step is the axis point of the nozzle core area;

[0044] 2.3) Use the points in the nozzle core area and the axis points of the nozzle core area calculated in 2.1) and 2.2) as the previous solution points to replace the throat boundary grid points;

[0045] 2.4) Repeat the processes of 2.1) and 2.2) n-1 times, calculate the subsequent points in the nozzle core area and the axis points of the nozzle core area, and sequentially connect the line segment where the throat boundary grid point is located, the line segment where the axis point of the nozzle core area is located, and the line connecting the inlet starting point and the n-1th nozzle core area axis point to form the nozzle core area.

[0046] As a preferred embodiment, the calculation of the flow field of the initial expansion section in step 3 comprises the following steps:

[0047] 3.1) Given the number of wall points m in the initial expansion section;

[0048] 3.2) Based on the initial expansion section wall Mach number distribution obtained in step 1 as the constraint condition, and based on the right characteristic line where the line connecting the inlet starting point and the n-1th nozzle core area axis point obtained in step 2 is located as the boundary condition, the free boundary point in the characteristic line method is applied to solve the coordinates and flow field parameters of the point to be solved where the left characteristic line emitted from the point in the nozzle core area intersects with the streamline emitted from the initial point of the throat boundary grid. The solution point obtained in this step is the initial expansion section wall point;

[0049] 3.3) Compare the Mach number of the initial expansion section wall point calculated in 3.2) with the Mach number at the same x-coordinate on the initial expansion section wall Mach number distribution obtained in step 1. If the two are not within the specified error range, iterate forward in the influence domain of the point in the nozzle core area in 3.2) until the difference between the Mach number of the initial expansion section wall point and the initial expansion section wall Mach number distribution obtained in step 1 is within the error range;

[0050] 3.4) Apply the inner point in the characteristic line method, iteratively solve the intersection of the left characteristic line from the inner point of the nozzle core area and the right characteristic line from the wall point of the initial expansion section to obtain the coordinates and flow field parameters of the point to be solved, and then use this point as the previous solution point to replace the wall point of the initial expansion section to solve in sequence. The solution point obtained in this step is the inner point of the initial expansion section;

[0051] 3.5) Apply the axis point in the characteristic line method to iteratively solve the coordinates and flow field parameters of the point to be solved where the left characteristic line emitted from the end point of the nozzle core area axis in step 2.2) intersects with the right characteristic line emitted from the point in the initial expansion section calculated in 3.4). The solution point obtained in this step is the axis point of the initial expansion section;

[0052] 3.6) Take the right characteristic line where the initial expansion section wall point, the initial expansion section inner point, and the initial expansion section axis point calculated in 3.1) to 3.5) are located as the new boundary condition, and repeat 3.2) to 3.5) until the Mach number at the axis point on the characteristic line where the mth initial expansion section wall point is located is greater than the design Mach number M d The solution points obtained in this step are the subsequent points inside the initial expansion segment, the wall points of the initial expansion segment, and the axis points of the initial expansion segment;

[0053] 3.7) Select the middle point on the line connecting the m-1th initial expansion section wall point and the adjacent initial expansion section inner point, and repeat 3.2) to 3.5) with this point as the previous solution point until the Mach number at the m-1th axis point is the same as the design Mach number M d After comparison, within the error range, the new m-th initial expansion section wall point obtained in this step, the initial expansion section inner point on the characteristic line, and the initial expansion section axis are the initial expansion section outlet boundary;

[0054] 3.8) The curve segment where the initial expansion segment wall point obtained in 3.1) to 3.7) is located, the initial expansion segment outlet boundary, the line segment where the initial expansion segment axis point is located, and the line connecting the inlet starting point obtained in step 2 and the n-1th nozzle core area axis point constitute the initial expansion segment.

[0055] As a preferred embodiment, the step 4 provides the Mach number or flow angle distribution at the outlet boundary of the wave-breaking zone, including the following steps:

[0056] 4.1) According to the design Mach number M d The Mach angle is , and the Mach line is emitted at the m-1th initial expansion section axis point calculated in step 3;

[0057] 4.2) Directly give the exit Mach number Ma on the Mach line described in 4.1) e Or the flow angle θ, k wave-breaking zone outlet boundary points are given directly on the wave-breaking zone outlet boundary, where k=m-1.

[0058] As a preferred embodiment, the calculation of the flow field in the wave-breaking zone in step 5 includes the following steps:

[0059] 5.1) Use the characteristic line method to solve the coordinates and flow field parameters of the exit boundary point of the wave breaking zone where the streamlines emitted from the points on the exit boundary of the initial expansion section obtained in step 3 intersect with the Mach lines obtained in step 4;

[0060] 5.2) Apply the inner point in the characteristic line method to solve the coordinates and flow field parameters of the point to be solved where the streamline emitted from the point on the outlet boundary of the initial expansion section obtained in step 3 intersects with the right-going characteristic line emitted in the opposite direction from the outlet boundary point of the wave-breaking zone obtained in step 5.1). The solution point obtained in this step is the inner point of the wave-breaking zone;

[0061] 5.3) The point in the wave-breaking zone calculated in 5.2) is used as the previous solution point to replace the boundary point of the wave-breaking zone outlet, and 5.2) is repeated until the right-moving characteristic line sent from the subsequent point in the wave-breaking zone in the reverse direction intersects with the streamline sent from the end point of the initial expansion section wall at the point to be solved, and the wall point in the characteristic line method is used to solve the coordinates and flow field parameters of the point to be solved. The solution point obtained in this step is the wall point of the wave-breaking zone;

[0062] 5.4) The wall points and inner points of the wave-breaking zone calculated in 5.2) and 5.3) are used as the previous solution points to replace the initial expansion section exit boundary points;

[0063] 5.5) Repeat 5.1) to 5.4) until the streamlines emitted from the wall points of the wave-breaking zone intersect with the Mach lines obtained in step 4 at the kth wave-breaking zone exit boundary point. The solution points obtained in this step are the subsequent points in the wave-breaking zone and the wall points of the wave-breaking zone;

[0064] 5.6) The curve segment where the wall points of the wave-breaking zone obtained in 5.1) to 5.5) are located, the outlet boundary of the wave-breaking zone obtained in step 4 and the outlet boundary of the initial expansion section obtained in step 3 constitute the wave-breaking zone.

[0065] The beneficial effects of the present invention are:

[0066] 1) It solves the problem that the characteristic lines at the wall of the initial expansion section are easy to intersect in the traditional method of nozzle design;

[0067] 2) It solves the problem that the traditional method must interpolate the solution points on both sides when generating the nozzle wall;

[0068] 3) Controlling the nozzle length by controlling the Mach number distribution;

[0069] 4) Under the same flow field quality, the nozzle designed by the present invention is shorter than the nozzle designed by the traditional method;

[0070] 5) Due to the shortened length, the processing cost and difficulty are reduced;

[0071] 6) Due to the shortening of the length, the thickness of the boundary layer at the nozzle exit is reduced and the effective test area is increased. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 It is a nozzle structure diagram of the present invention;

[0073] Figure 2 This is a flow chart of the nozzle shortening design method based on the initial expansion section wall Mach number distribution;

[0074] Figure 3 This is the principle diagram of the Mach number distribution on the wall of the initial expansion section;

[0075] Figure 4 This is the grid principle diagram of the nozzle core area;

[0076] Figure 5 It is the principle diagram of the characteristic lines of points in the core area of ​​the nozzle;

[0077] Figure 6 It is the principle diagram of the characteristic line of the axis point;

[0078] Figure 7 This is the grid schematic diagram of the initial expansion section;

[0079] Figure 8 This is the principle diagram of the characteristic line of the wall point of the first initial expansion section;

[0080] Fig. 9 It is the principle diagram of the characteristic line of the wall point of the subsequent initial expansion section;

[0081] Fig.10 This is the grid principle diagram of the wave-breaking area;

[0082] Fig.11 This is the schematic diagram of the wave-breaking zone outlet boundary;

[0083] Fig.12 It is the principle diagram of characteristic lines of points in the wave-breaking area;

[0084] Fig.13 It is the principle diagram of characteristic lines of wall points in the wave-breaking area;

[0085] Among them, 1-nozzle core area, 2-initial expansion section, 3-initial expansion section wall, 4-initial expansion section outlet boundary, 5-wave breaking zone, 6-wave breaking zone wall, 7-axis, 8-wave breaking zone outlet boundary, 9-initial expansion section wall Mach number distribution control line, 10-initial expansion section Mach number distribution interpolation curve, C1-initial expansion section wall Mach number distribution first control point, C2-initial expansion section wall Mach number distribution second control point, C3-initial expansion section wall Mach number distribution third control point, C4-initial expansion section wall Mach number distribution fourth control point, C5-initial expansion section wall Mach number distribution fifth control point, C + is the left characteristic line, C - is the right characteristic line and C0 is the streamline. DETAILED DESCRIPTION

[0086] The following describes the embodiments of the present invention through specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. Example 1

[0087] like Figure 1 As shown, Figure 1 It is a nozzle structure diagram of the present invention; wherein,

[0088] 1 represents the core area of ​​the nozzle, 2 represents the initial expansion section, 3 represents the wall of the initial expansion section, 4 represents the exit boundary of the initial expansion section, 5 represents the wave-breaking zone, 6 represents the wall of the wave-breaking zone, 7 represents the axis, and 8 represents the exit boundary of the wave-breaking zone; R represents the radius, and x represents the horizontal coordinate; the dotted line where R is located represents the coordinate axis of the radius, and the dotted line where x is located represents the coordinate axis of the horizontal coordinate; point A…A n represents the nozzle throat inlet section point, n is the number of inlet section points, determined by the design conditions; point B m represents the end point of the initial expansion section wall, where m is the number of initial expansion section wall points, and m is determined by the design conditions; point D n-1 …D n+m-1’ Indicates the axis point of the initial expansion section; point E k Represents the end point of the exit boundary of the wave-breaking zone, where k=m-1.

[0089] A nozzle shortening design method based on the initial expansion section wall Mach number distribution comprises the following steps:

[0090] Step 1, giving the initial expansion section wall surface 3 Mach number distribution;

[0091] Step 2, calculating the flow field in the nozzle core area 1;

[0092] Step 3, calculating the flow field of the initial expansion section 2;

[0093] Step 4, giving the Mach number or flow angle distribution at the exit boundary of the wave-breaking zone;

[0094] Step 5, calculating the flow field in the wave-breaking area 5;

[0095] Step 6: The nozzle core area 1, the initial expansion section 2, and the wave-breaking area 5 obtained in steps 1 to 5 are spatially connected in sequence to form the flow field and profile of the entire nozzle.

[0096] like Figure 2 As shown, Figure 2 This is a flow chart of the nozzle shortening design method based on the initial expansion section wall Mach number distribution;

[0097] The flowchart is from left to right and from top to bottom: given inlet parameters and design Mach number; step 1 given initial expansion section wall 3 Mach number distribution; step 2 calculate nozzle core area 1 flow field; step 3 calculate initial expansion section 2 flow field; whether the last axis 7 point of initial expansion section 2 meets the outlet Mach number Mach e , if satisfied, proceed to step 4, otherwise return to step 3 for iterative correction calculation; step 4 sets the Mach number / flow angle of the outlet boundary 8 of the wave-breaking zone; step 5 calculates the flow field of the wave-breaking zone 5; step 6 connects the nozzle core area 1, the initial expansion section 2, and the flow field of the wave-breaking zone 5 in sequence; outputs the flow field and profile of the nozzle.

[0098] Figure 3 is the principle diagram of the Mach number distribution on the wall of the initial expansion section;

[0099] 9 represents the initial expansion section wall Mach number distribution control line, 10 represents the initial expansion section Mach number distribution interpolation curve; Ma represents the Mach number, Ma d is the design Mach number, L is the length of the initial expansion section, x is the horizontal coordinate; the dotted line where Ma is located represents the coordinate axis of the Mach number, and the dotted line where x is located represents the coordinate axis of the horizontal coordinate; C1 represents the first control point of the initial expansion section wall Mach number distribution, C2 represents the second control point of the initial expansion section wall Mach number distribution, C3 represents the third control point of the initial expansion section wall Mach number distribution, C4 represents the fourth control point of the initial expansion section wall Mach number distribution, and C5 represents the fifth control point of the initial expansion section wall Mach number distribution.

[0100] The given initial expansion section wall surface Mach number distribution described in step 1 comprises the following steps:

[0101] 1.1) Establish a coordinate system with the origin as the coordinate, the horizontal axis x is the lateral distance coordinate of the nozzle, and the vertical axis Ma is the Mach number;

[0102] 1.2) Optimally give 5 control points Ci [x i ,Ma i ](where i = 1,2,3,4,5) and the initial expansion section wall surface Mach number distribution control coefficient Ma is given d ,L,yp1,yp2, f 1 、f 2 ， x0,Ma0,where Ma d is the design Mach number, L is the length of the initial expansion wall 3, yp1 is the slope at the control point C1, yp2 is the slope at the control point C5[x5,Ma5], f 1 、f 2 is the proportional coefficient of the line segment between points C2 and C4 along the starting point and the end point, x0 is the horizontal coordinate of the vertex A on the nozzle throat, which defaults to 0.0, Ma0 is the Mach number of the vertex on the nozzle throat, which defaults to 1.0, C1[x1,Ma1] is the first control point of the initial expansion section wall 3 Mach number distribution, C2[x2,Ma2] is the second control point of the initial expansion section wall 3 Mach number distribution, C3[x3,Ma3] is the third control point of the initial expansion section wall 3 Mach number distribution, C4[x4,Ma4] is the fourth control point of the initial expansion section wall 3 Mach number distribution, C5[x5,Ma5] is the fifth control point of the initial expansion section wall 3 Mach number distribution;

[0103] 1.3) The default coordinates of the initial point C1[x1,Ma1] are C1[0,1.0], and the final point C5[L,Ma d ]

[0104] (1)

[0105] 1.4) Through the slope yp1 and the proportional coefficient f 1 Get the coordinates of C2

[0106]

[0107] Where temp is the weighting coefficient;

[0108] 1.5) C3[x3,Ma3] is the middle point of segment C1C5, and its horizontal coordinate is f 1 L, ordinate is 1.0+yp1 f 1 L and Ma d The smaller value among

[0109]

[0110] 1.6) Through the slope yp2 and the proportional coefficient f 2 Get the coordinates of C4

[0111]

[0112] 1.7) According to 1.1), 1.2), 1.3), 1.4), 1.5), and 1.6), the control points C1[0,1.0], C2[x2,Ma2], C3[x3,Ma3], C4[x4,Ma4], and C5[L,Ma d ], the distribution of the initial expansion section wall Mach number 3 with the axial coordinate x is solved by the B-Spline curve.

[0113] 3. If Figure 4 As shown, Figure 4 It is the grid principle diagram of nozzle core area 1;

[0114] 1 represents the core area of ​​the nozzle, 7 represents the axis; R represents the radius, and x represents the horizontal coordinate; the dotted line where R is located represents the coordinate axis of the radius, and the dotted line where x is located represents the coordinate axis of the horizontal coordinate; point AA1…A n represents the nozzle throat boundary grid, n is the number of throat boundary grid points, which is determined by the design conditions; points D1…D n-1 Indicates the axis point of the nozzle core area; point O n1 …O 11 O n2 …O n(2n-3) represents the point in the core area of ​​the nozzle; C + Indicates the left characteristic line, C - represents the right characteristic line and C0 represents the streamline.

[0115] The calculation of the flow field in the nozzle core region 1 described in step 2 includes the following steps:

[0116] 2.1) Given the number of throat boundary grid points n and the inlet parameter distribution, the throat boundary grid points AA1…A are given equidistantly through the number of throat boundary grid points n. n , use the interior point in the characteristic line method to solve the intersection point O of the right characteristic line from the inlet starting point A and the left characteristic line from the adjacent throat boundary grid point A1 n1 The coordinates and flow field parameters of the throat are solved by repeatedly applying the characteristic line method to the n-1 throat boundary grid point A. n-1 The right characteristic line and the throat boundary grid end point A n The point O where the left characteristic line intersects 11The coordinates and flow field parameters of the inlet parameters refer to the Mach number, pressure, density, speed, speed direction, temperature, and the solution point O obtained in this step n1 …O 11 It is the inner point of the nozzle core area 1;

[0117] like Figure 5 It is the principle diagram of the characteristic line of the inner point of the nozzle core area 1;

[0118] Points A and A1 represent the previous solution points, and point O n1 represents the point to be solved; line segment AA1 represents the line connecting the previous solution points A and A1; arc segment C - , C + , C0 represent the right characteristic line, left characteristic line and streamline respectively; dotted line AO n1 、A1O n1 , C0 represent the calculated approximate lines of the right characteristic line, the left characteristic line and the streamline respectively.

[0119] 2.2) Apply the axis point in the characteristic line method to solve the end point O in the nozzle core area 1 in 2.1) 11 The right characteristic line and the throat boundary grid end point A n The coordinates and flow field parameters of the point D1 to be solved where the emitted streamlines intersect. The solution point D1 obtained in this step is point 7 on the axis of the nozzle core area 1;

[0120] like Figure 6 It is the principle diagram of the 7-point characteristic line of the axis; among them,

[0121] 7 represents the axis; point A n , O 11 represents the previous solution point, point D1 represents the point to be solved; arc segment C - , C + Represent the right characteristic line and the left characteristic line respectively; the dotted line O 11 D1, A n O 11 , C0 represent the calculated approximate lines of the right characteristic line, the left characteristic line and the streamline respectively.

[0122] 2.3) The inner point O of the nozzle core area 1 calculated in 2.1) and 2.2) is n1 …O 11 Point D1 of the nozzle core area 1 axis is used as the previous solution point to replace the throat boundary grid point AA1...A n ;

[0123] 2.4) Repeat the process of 2.1) and 2.2) n-1 times to calculate the subsequent point O in the core area 1 of the nozzle n2 …O n(2n-3) Point D at 7 of the nozzle core area 1 axis n-1, where the throat boundary grid points AA1…A n Line segment AA n 、Nozzle core area 1 axis 7 points D1...D (n-1) Line segment D1D (n-1) , the inlet starting point A and the n-1th nozzle core area 1 axis 7 point D n-1 AD n-1 It constitutes the nozzle core area 1.

[0124] like Figure 7 is the grid principle diagram of the initial expansion section 2; where:

[0125] 2 represents the initial expansion section, 3 represents the wall of the initial expansion section, 4 represents the exit boundary of the initial expansion section, and 7 represents the axis; x represents the abscissa; the dotted line where x is located represents the coordinate axis of the abscissa; point A represents the initial point of the nozzle throat inlet section; point D n-1 D n …D n+m-1 represents the axis point of the initial expansion section, where n is the number of inlet section points, m is the number of initial expansion section wall points, n and m are determined by the design conditions; point B1B2…B m Indicates the wall point of the initial expansion section; point O (n+1)1 O (n+1)2 …O (m+2)(2n-3) Indicates the point in the initial expansion section; point B m 'O (m+2)1 'O (m+2) '…O (m+2)(2n-3) 'D n+m-1 ' is the intermediate result in the characteristic line method solution; C + Indicates left characteristic line, C - represents the right characteristic line and C0 represents the streamline.

[0126] The calculation of the flow field of the initial expansion section 2 described in step 3 includes the following steps:

[0127] 3.1) Given the number m of three points on the wall of the initial expansion section;

[0128] 3.2) Based on the initial expansion section wall 3 Mach number distribution obtained in step 1 as a constraint condition, and based on the inlet starting point A obtained in step 2 and the n-1 nozzle core area 1 axis 7 point D n-1 AD n-1 The right characteristic line where the nozzle is located is used as the boundary condition, and the free boundary point in the characteristic line method is used to solve the point O in the nozzle core area 1. n1 The coordinates and flow field parameters of the point B1 to be solved where the left characteristic line issued intersects with the streamline issued by the initial point A of the throat boundary grid. The solution point B1 obtained in this step is the 3 points on the wall of the initial expansion section;

[0129] like Figure 8 It is the principle diagram of the three-point characteristic lines on the wall of the first initial expansion section;

[0130] Point A represents the initial point of the throat entrance section, point O n1 represents the previous solution point, point B1 represents the point to be solved; arc segment C - , C + , C0 represent the right characteristic line, left characteristic line and streamline respectively; dotted line AO n1 , O n1 B1 represents the calculated approximate lines of the right characteristic line and the left characteristic line respectively; Ma(x) on the streamline C0 represents the Mach number distribution about the horizontal coordinate x.

[0131] like Fig. 9 It is the principle diagram of the three-point characteristic lines of the wall surface of the subsequent initial expansion section; among them,

[0132] Point B1, O (n+1)1 represents the previous solution point, point B2 represents the point to be solved; arc segment C - , C + , C0 represent the right characteristic line, left characteristic line and streamline respectively; dotted line B1O (n+1)1 , O (n+1)1 B2 represents the calculated approximate lines of the right characteristic line and the left characteristic line respectively.

[0133] 3.3) Compare the Mach number of point B1 on the wall of the initial expansion section calculated in step 3.2) with the Mach number at the same x coordinate on the Mach number distribution of the wall of the initial expansion section obtained in step 1. If the two are not within the specified error range, then the point O in the nozzle core region 1 in step 3.2) is n1 Iterate forward in the influence domain until the difference between the Mach number of point B1 on the wall of the initial expansion section and the Mach number distribution of the wall of the initial expansion section obtained in step 1 is within the error range;

[0134] 3.4) Apply the interior point in the characteristic line method to iteratively solve the interior point O of the nozzle core area 1 n2 The left characteristic line issued intersects with the right characteristic line issued by point B1 on the wall of the initial expansion section to obtain the point O to be solved. (n+1)1 The coordinates and flow field parameters of this point O (n+1)1 As the previous solution point, replace the initial expansion section wall 3 point B1 and solve it in sequence. The solution point O obtained in this step (n+1)1 O (n+1)2 …O (n+1)(2n-3) That is, the point inside the initial expansion section 2; the principle of the point inside the initial expansion section 2 is the same as Figure 5 The principle is the same as that in the principle diagram of the point characteristic line in the core area 1 of the nozzle shown, and will not be repeated here;

[0135] 3.5) Apply the axis point in the characteristic line method to iteratively solve the end point D of the axis 7 of the nozzle core area 1 in step 2.2) (n-1) The left characteristic line emitted is consistent with the point O in the initial expansion section 2 calculated in 3.4). (n+1)(2n-3) The point D to be solved where the right characteristic line intersects n The coordinates and flow field parameters of the solution point D obtained in this step n That is, the initial expansion section 2 axis 7 points; the initial expansion section 2 axis 7 points principle is the same as Figure 6 The principle is the same as that in the 7-point characteristic line principle diagram of the axis shown, and will not be repeated here;

[0136] 3.6) The initial expansion section wall point B1 and the initial expansion section 2 inner point O calculated in 3.1) to 3.5) are (n+1)(2n-3) , Initial expansion section 2 axis 7 point D n The right characteristic line is used as the new boundary condition, and 3.2) to 3.5) are repeated until the calculation reaches point B on the wall of the mth initial expansion section. m Axis 7 point D on the characteristic line n+m-1 The Mach number on the d The solution point obtained in this step is point O in the subsequent initial expansion segment 2. (n+2)1 O (n+2)2 …O (m+2)(2n-3) , 3 points on the wall of the initial expansion section B1B2…B m , Initial expansion section 2 axis 7 point D n D n+1 …D n+m-1 ;

[0137] 3.7) At point B on the wall of the m-1th initial expansion segment m-1 and the adjacent point O in the initial expansion section 2 (m+1)1 Select the middle point O on the line connecting (m+1)1 ', and with B m-1 O (m+1)1 '…D n+m-2 Repeat 3.2) to 3.5) for the previous solution point until the m-1th axis 7 point D n+m-1 The Mach number on the surface and the design Mach number M d After comparison, the calculated B m 'O (m+2)1 'O (m+2)2 '…O (m+2)(2n-3) 'D n+m-1 'Replace B m O (m+2)1 O (m+2)2 …O (m+2)(2n-3) D n+m-1 The new m-th initial expansion section wall 3 points B obtained in this stepm and point O in the initial expansion section 2 on the characteristic line (m+2)1 O (m+2)2 …O (m+2)(2n-3) , Initial expansion section 2 axis 7 point D n+m-1 That is the initial expansion section outlet boundary 4;

[0138] 3.8) The three points B1B2…B on the wall of the initial expansion section obtained in 3.1) to 3.7) m The curve segment AB1B m , initial expansion section outlet boundary 4, initial expansion section 2 axis 7 point D n D n+1 …D n+m-1 Line segment D n D n+m-1 , the inlet starting point A obtained in step 2 and the axis 7 point D of the n-1th nozzle core area n-1 AD n-1 The initial expansion section 2 is formed.

[0139] In step 4, the Mach number or flow angle distribution of the outlet boundary of the wave-breaking zone is given, and the following steps are included:

[0140] 4.1) According to the design Mach number M d The Mach angle of point D at axis 7 of the m-1th initial expansion segment 2 calculated in step 3 n+m-1 Mach lines are emitted at

[0141] 4.2) Directly give the exit Mach number Ma on the Mach line described in 4.1) e Or flow angle θ, directly give k points E1E2…E at the outlet boundary 8 of the wave-breaking zone. (k-1) E k , where k=m-1;

[0142] like Fig.11 It is the schematic diagram of the exit boundary 8 of the wave-breaking zone 5; among them,

[0143] 8 represents the exit boundary of the wave-breaking zone; point Q 1(2n-1) , E1 represents the previous solution point, point E2 represents the point to be solved; line segment E1E2 represents the Mach line; arc segment C - , C0 represent the right characteristic line, left characteristic line and streamline respectively; the dotted line Q 1(2n-1) E1, Q 1(2n-1) E2 represents the calculated approximate lines of the right characteristic line and streamline respectively.

[0144] like Fig.10 It is the principle diagram of the 5-grid in the nozzle wave-breaking area; among them,

[0145] 4 represents the exit boundary of the initial expansion section, 5 represents the wave-breaking zone, 6 represents the wall of the wave-breaking zone, 7 represents the axis, and 8 represents the exit boundary of the wave-breaking zone; x represents the horizontal coordinate; the dotted line where x is located represents the coordinate axis of the horizontal coordinate; point B m represents the end point of the initial expansion section wall, where m is the number of initial expansion section wall points, and m is determined by the design conditions; point D n+m-1 represents the end point of the initial expansion section axis, n is the number of inlet section points, which is determined by the design conditions; point E1E2…E (k-1) E k represents the exit boundary point of the wave-breaking zone, where k=m-1; point Q 12 Q 13 …Q (k-2)2 Represents the point in the wave elimination zone; point Q 11 Q 21 …Q (k-1)1 Represents the wall point of the wave elimination zone; C + Indicates the left characteristic line, C - represents the right characteristic line and C0 represents the streamline.

[0146] The calculation of the flow field in the wave-breaking zone 5 in step 5 includes the following steps:

[0147] 5.1) Use the characteristic line method to solve the point O on the initial expansion section outlet boundary 4 obtained in step 3 (m+2)(2n-3) The coordinates and flow field parameters of the 8th point E1 at the exit boundary of the wave-breaking zone where the emitted streamline intersects with the Mach line obtained in step 4;

[0148] 5.2) Apply the interior point method in the characteristic line method to solve the point O on the initial expansion section outlet boundary 4 obtained in step 3 (m+2)2 The point Q to be solved where the streamline intersects the right-hand characteristic line emitted in the opposite direction from point E1 at the exit boundary of the wave-breaking zone obtained in 5.1) 1(2n-1) The coordinates and flow field parameters of the solution point Q obtained in this step 1(2n-1) That is the point inside the wave-breaking zone 5;

[0149] like Fig.12 It is the principle diagram of the characteristic line of the point in the wave-breaking area 5;

[0150] Point O (m+2)2 , E1 represents the previous solution point, point Q 1(2n-1) Represents the point to be solved; line segment O (m+2)2 E1 represents the previous solution point O (m+2)2 , E1; arc segment C - , C + , C0 represent the right characteristic line, left characteristic line and streamline respectively; the dotted line Q 1(2n-1) E1, O (m+2)2 Q 1(2n-1) , C +They represent the calculated approximate lines of the right characteristic line, streamline and left characteristic line respectively.

[0151] 5.3) The point Q in the wave-breaking zone 5 calculated in 5.2) is 1(2n-1) As the previous solution point, replace the exit boundary point 8 of the wave-breaking zone, E1, and repeat 5.2) until the next point Q in the wave-breaking zone 5 12 The right characteristic line issued in the reverse direction is at the end point B of the initial expansion section wall 3 m The streamlines emitted intersect at the point to be solved Q 11 , and apply the wall point in the characteristic line method to solve the point Q 11 The coordinates and flow field parameters of the solution point Q obtained in this step 11 These are the 6 points on the wall of the wave-breaking area;

[0152] like Fig.13 It is the principle diagram of the 6-point characteristic lines on the wall of the wave-breaking area; among them,

[0153] Click Q 11 , Q 22 Represents the previous solution point, point Q 21 Represents the point to be solved; line segment Q 11 Q 22 Represents the previous solution point Q 11 , Q 22 The line connecting the arc segment C - , C + , C0 represent the right characteristic line, left characteristic line and streamline respectively; the dotted line Q 21 Q 22 , C + Represent the calculated approximate lines of the right characteristic line and the left characteristic line respectively.

[0154] 5.4) The 6 points Q on the wall of the wave-breaking zone calculated in 5.2) and 5.3) are 11 and point Q in wave elimination zone 5 12 …Q 1(2n-1) As the previous solution point, replace the initial expansion section exit boundary 4 point B m O (m+2)1 O (m+2) …O (m+2)(2n-2) ;

[0155] 5.5) Repeat 5.1) to 5.4) until the 6th point Q on the wall of the wave-breaking area (k-1)1 The streamlines emitted intersect with the Mach lines obtained in step 4 at point E at the exit boundary of the kth wave-breaking zone. k The solution point obtained in this step is the point Q in the subsequent wave-breaking zone 5. 12 Q 13 …Q 22 Q 23 …Q (k-2)1 Q (k-2)2, 6 point Q on the wall of the wave elimination zone 11 Q 21 …Q (k-1)1 ;

[0156] 5.6) The 6 points Q on the wall of the wave-breaking zone obtained in 5.1) to 5.5) 11 Q 21 …Q (k-1)1 Curve segment B m Q (k-1) 1E k , the wave-breaking zone outlet boundary 8 obtained in step 4 and the initial expansion section outlet boundary 4 obtained in step 3 constitute the wave-breaking zone 5.

[0157] The above embodiments are merely illustrative of the principles and effects of the present invention, and are not intended to limit the present invention. Anyone familiar with the art may modify or alter the above embodiments without departing from the spirit and scope of the present invention. Therefore, all equivalent modifications or alterations made by a person of ordinary skill in the art without departing from the spirit and technical ideas disclosed by the present invention shall still be covered by the claims of the present invention.

Claims

1. A nozzle shortening design method based on the initial expansion section wall Mach number distribution, characterized in that: The steps include: Step 1, giving the initial expansion section wall Mach number distribution; Step 2, calculating the flow field in the core area of ​​the nozzle, includes the following steps: 2.1) Given the number of throat boundary grid points n and the inlet parameter distribution, the throat boundary grid points are given equidistantly through the number of throat boundary grid points n, and the inner point in the characteristic line method is applied to solve the coordinates and flow field parameters of the intersection point of the right characteristic line emitted from the inlet starting point and the left characteristic line emitted from the adjacent throat boundary grid point, and the characteristic line method is applied in sequence to solve the coordinates and flow field parameters of the point to be solved where the right characteristic line emitted from the n-1th throat boundary grid point intersects the left characteristic line emitted from the throat boundary grid end point, wherein the inlet parameters refer to Mach number, pressure, density, velocity, velocity direction, and temperature, and the solution point obtained in this step is the inner point of the nozzle core area; 2.2) Apply the axis point in the characteristic line method to solve the coordinates and flow field parameters of the point to be solved where the right characteristic line from the end point of the nozzle core area in 2.1) intersects the streamline from the end point of the throat boundary grid. The solution point obtained in this step is the axis point of the nozzle core area; 2.3) Use the points in the nozzle core area and the axis points of the nozzle core area calculated in 2.1) and 2.2) as the previous solution points to replace the throat boundary grid points; 2.4) Repeat the process of 2.1) and 2.2) n-1 times, calculate the subsequent points in the nozzle core area and the axis points of the nozzle core area, and sequentially connect the line segment where the throat boundary grid point is located, the line segment where the axis point of the nozzle core area is located, and the line connecting the inlet starting point and the n-1th nozzle core area axis point to form the nozzle core area; Step 3, calculating the flow field of the initial expansion section, includes the following steps: 3.1) Given the number of wall points m in the initial expansion section; 3.2) Based on the initial expansion section wall Mach number distribution obtained in step 1 as the constraint condition, and based on the right characteristic line where the line connecting the inlet starting point and the n-1th nozzle core area axis point obtained in step 2 is located as the boundary condition, the free boundary point in the characteristic line method is applied to solve the coordinates and flow field parameters of the point to be solved where the left characteristic line emitted from the point in the nozzle core area intersects with the streamline emitted from the initial point of the throat boundary grid. The solution point obtained in this step is the initial expansion section wall point; 3.3) Compare the Mach number of the initial expansion section wall point calculated in 3.2) with the Mach number at the same x-coordinate on the initial expansion section wall Mach number distribution obtained in step 1. If the two are not within the specified error range, iterate forward in the influence domain of the point in the nozzle core area in 3.2) until the difference between the Mach number of the initial expansion section wall point and the initial expansion section wall Mach number distribution obtained in step 1 is within the error range; 3.4) Apply the inner point in the characteristic line method, iteratively solve the intersection of the left characteristic line from the inner point of the nozzle core area and the right characteristic line from the wall point of the initial expansion section to obtain the coordinates and flow field parameters of the point to be solved, and then use this point as the previous solution point to replace the wall point of the initial expansion section to solve in sequence. The solution point obtained in this step is the inner point of the initial expansion section; 3.5) Apply the axis point in the characteristic line method to iteratively solve the coordinates and flow field parameters of the point to be solved where the left characteristic line emitted from the end point of the nozzle core area axis in step 2.2) intersects with the right characteristic line emitted from the point in the initial expansion section calculated in 3.4). The solution point obtained in this step is the axis point of the initial expansion section; 3.6) Take the right characteristic line where the initial expansion section wall point, the initial expansion section inner point, and the initial expansion section axis point calculated in 3.1) to 3.5) are located as the new boundary condition, and repeat 3.2) to 3.5) until the Mach number at the axis point on the characteristic line where the mth initial expansion section wall point is located is greater than the design Mach number M d The solution points obtained in this step are the subsequent points inside the initial expansion segment, the wall points of the initial expansion segment, and the axis points of the initial expansion segment; 3.7) Select the middle point on the line connecting the m-1th initial expansion section wall point and the adjacent initial expansion section inner point, and repeat 3.2) to 3.5) with this point as the previous solution point until the Mach number at the m-1th axis point is the same as the design Mach number M d After comparison, within the error range, the new m-th initial expansion section wall point obtained in this step, the initial expansion section inner point on the characteristic line, and the initial expansion section axis are the initial expansion section outlet boundary; 3.8) The curve segment where the initial expansion segment wall point obtained in 3.1) to 3.7) is located, the initial expansion segment outlet boundary, the line segment where the initial expansion segment axis point is located, and the line connecting the inlet starting point obtained in step 2 and the n-1th nozzle core area axis point constitute the initial expansion segment; Step 4, giving the Mach number or flow angle distribution at the exit boundary of the wave-breaking zone; Step 5, calculating the flow field in the wave-breaking area; Step 6: The nozzle core area, initial expansion section, and wave-breaking area obtained in steps 1 to 5 are spatially connected in sequence to form the flow field and profile of the entire nozzle.

2. The nozzle shortening design method based on the initial expansion section wall Mach number distribution according to claim 1 is characterized in that: The given initial expansion section wall Mach number distribution described in step 1 comprises the following steps: 1.1) Establish a coordinate system with the origin as the coordinate, the horizontal axis x is the lateral distance coordinate of the nozzle, and the vertical axis Ma is the Mach number; 1.2) Given 5 control points C i [x i ,Ma i ], where i = 1, 2, 3, 4, 5, and the initial expansion section wall Mach number distribution control coefficient Ma is given d ,L,yp1,yp2, f 1 、f 2 ， x0,Ma0,where Ma d is the design Mach number, L is the length of the wall of the initial expansion section, yp1 is the slope at the control point C1, yp2 is the slope at the control point C5[x5,Ma5], f 1 、f 2 is the proportional coefficient of points C2 and C4 along the line segments at the starting and ending points; 1.3) The default coordinates of the initial point C1[x1,Ma1] are C1[0,1.0], and the final point C5[L,Ma d ]; 1.4) Through the slope yp1 and the proportional coefficient f 1 Get the coordinates of C2; 1.5) C3[x3,Ma3] is the middle point of segment C1C5, and its horizontal coordinate is f 1 L, ordinate is 1.0+yp1 f 1 L and Ma d The smaller value among 1.6) Through the slope yp2 and the proportional coefficient f 2 Get the coordinates of C4; 1.7) According to 1.1), 1.2), 1.3), 1.4), 1.5), and 1.6), the control points C1[0,1.0], C2[x2,Ma2], C3[x3,Ma3], C4[x4,Ma4], and C5[L,Ma d ], the distribution of the wall Mach number of the initial expansion section with the axial coordinate x is solved by the B-Spline curve.

3. The nozzle shortening design method based on the initial expansion section wall Mach number distribution according to claim 2 is characterized in that: In step 4, the Mach number or flow angle distribution of the outlet boundary of the wave breaking zone is given, which includes the following steps: 4.1) According to the design Mach number M d The Mach angle is , and the Mach line is emitted at the m-1th initial expansion section axis point calculated in step 3; 4.2) Directly give the exit Mach number Ma on the Mach line described in 4.1) e Or the flow angle θ, k wave-breaking zone outlet boundary points are given directly on the wave-breaking zone outlet boundary, where k=m-1.

4. The nozzle shortening design method based on the initial expansion section wall Mach number distribution according to claim 3 is characterized in that: The calculation of the flow field in the wave-breaking zone in step 5 includes the following steps: 5.1) Use the characteristic line method to solve the coordinates and flow field parameters of the exit boundary point of the wave breaking zone where the streamlines emitted from the points on the exit boundary of the initial expansion section obtained in step 3 intersect with the Mach lines obtained in step 4; 5.2) Apply the inner point in the characteristic line method to solve the coordinates and flow field parameters of the point to be solved where the streamline emitted from the point on the outlet boundary of the initial expansion section obtained in step 3 intersects with the right-going characteristic line emitted in the opposite direction from the outlet boundary point of the wave-breaking zone obtained in step 5.1). The solution point obtained in this step is the inner point of the wave-breaking zone; 5.3) The point in the wave-breaking zone calculated in 5.2) is used as the previous solution point to replace the boundary point of the wave-breaking zone outlet, and 5.2) is repeated until the right-moving characteristic line sent from the subsequent point in the wave-breaking zone in the reverse direction intersects with the streamline sent from the end point of the initial expansion section wall at the point to be solved, and the wall point in the characteristic line method is used to solve the coordinates and flow field parameters of the point to be solved. The solution point obtained in this step is the wall point of the wave-breaking zone; 5.4) The wall points and inner points of the wave-breaking zone calculated in 5.2) and 5.3) are used as the previous solution points to replace the initial expansion section exit boundary points; 5.5) Repeat 5.1) to 5.4) until the streamlines emitted from the wall points of the wave-breaking zone intersect with the Mach lines obtained in step 4 at the kth wave-breaking zone exit boundary point. The solution points obtained in this step are the subsequent points in the wave-breaking zone and the wall points of the wave-breaking zone; 5.6) The curve segment where the wall points of the wave-breaking zone obtained in 5.1) to 5.5) are located, the outlet boundary of the wave-breaking zone obtained in step 4 and the outlet boundary of the initial expansion section obtained in step 3 constitute the wave-breaking zone.

Images