Empirical Equation of the Mach Number as a Function of the Stagnation Pressure Ratio for a Quasi-One-Dimensional Compressible Flow

In the present work for a quasi-one-dimensional isentropic compressible flow model, an empirical equation of the Mach number is constructed as a function of the stagnation pressure ratio for an analytical equation that algebraic procedures cannot invert. The Excel 2019 Solver tool was applied to calibrate the coefficients and exponents of the empirical equation during its construction for the Mach number range from 1 to 10 and 1 to 5. A specific heat ratio from 1.1 to 1.67 and the generalized reduced gradient iterative method were used to minimize the sum of squared error, which was set as the objective function. The results show that for Mach 1 to 10, an error of less than 0.063% is obtained, and for Mach 1 to 5, an error of less than 0.00988% is obtained. It is concluded that the empirical equation obtained is a mathematical model that reproduces the trajectories of the inverted curves of the analytical equation studied.


INTRODUCTION
The numerical characteristics for compressible flow in convergent-divergent nozzles vary mainly in the flow direction.They can be considered given by its approximation as quasi-one-dimensional isentropic flow.Isentropic flow assumes a steady state, a perfect gas, and a non-viscous fluid [1][2][3].
It should be noted that for ideal flows, quasi-onedimensional models have applications in wind tunnels, nozzles, and diffusers, among others [2][3][4].Reference is made to some wind tunnel tests, such as the experi-mental study on the influence of the length of the spike on the aerodynamic characteristics of blunt bodies for various angles of attack and Mach and Reynolds numbers [15].Study of experimental tests on the calibration of a wind tunnel, where a calibration model is used that represents a generic configuration of a winged missile [16.17] presents comparative results for the standard AGARD-B model in the transonic speed range tested in six wind tunnels.That model is a simple configuration of a high-speed deltawinged air-plane.As well as in [18], they also report the experimental evaluation of eight SZ-1500 series airfoils for low Reynolds numbers tested in a wind tunnel.
A shock wave is an intense pressure wave that travels faster than the speed of sound.The normal shock wave occurs in the diverging section for a quasi-onedimensional flow and under the nozzle's inlet and outlet pressure conditions.In the region where the shock occurs, the process is irreversible, and the entropy increases; in that region of the shock, the process is not considered an isentropic process, and whose shock thickness is extremely thin on the order of 10 -5 cm according to Anderson [2].
The analysis for a steady flow without heat transfer, for a control volume of a shock wave, considers the principles of conservation of mass, energy, momentum, and state.Furthermore, by intercepting the Fanno and Rayleigh curves, the limits of the thermodynamic parameters are defined before and after the shock [1,19,20].The first analyses of these relations are due to Rankine (1870) and Hugoniot (1887), known as Rankine-Hugoniot relations.

Normal shock wave in a nozzle
A simple scheme of a convergent-divergent nozzle is shown in Fig. 1a; in the position of local area ratio A x /A * and stagnation pressure ratio P oy /P ox , a normal shock wave is located, where A x is the cross-sectional area in the shock position and A * the throat area.The geometric parameter of the convergent is the half angle β, and of the divergent is the half angle α.At the inlet of the nozzle is the stagnation pressure P 0 (total pressure), whose pressure has the same magnitude in the combustion chamber, in the convergent, and the divergent until the shock position at x, where P o = P ox .After the shock, at position y, there is the stagnation pressure P oy , whose magnitude is the same up to the nozzle outlet.The exhaust area A e , the static pressure P e and the Mach number M e at subsonic velocity are found at the nozzle exit [1,3].
The trajectory of the Mach number curve is shown in Fig. 1b.The divergent shows the behavior of the Mach number pattern before and after the shock, where the flow goes from supersonic velocity to subsonic velocity; for these conditions, the flow is classified as over-expanded flow [2,3].In the gas dynamics for compressible flow, the value of k is variable and is a thermodynamic parameter of the gas.It is known as the specific heat ratio; this parameter is a positive value, and it is in the range of 1 < k ≤ 1.67.For air, the value is k = 1.4; for other gases, such as those generated by the combustion of solid, liquid, and hybrid fuels used in rocket engines, the value of k is less than 1.4 [3].
For an over-expanded isentropic flow, the calculations begin with the pressure ratio P e /P 0 , whose value is previously determined under the condition that the shock is present in the divergent at an arbitrary position A x /A * (Figure 1a), being P e less than atmospheric pressure.In addition, we have the area ratio A e /A * for the design condition.The use of analytical equations is mentioned below.
At the nozzle exit, the subsonic Mach number M e is determined by equation ( 1) [1]: The ratio of stagnation pressures P oy /P ox is determined by equation (2) [1], which is a function of M e : P 0y /P 0x is substituted in equation ( 3) [1], and by iterative methods, the supersonic Mach number M x is determined at the position of the shock, at reference x: With equation ( 4) [1] is calculated A x /A * at the position of the shock, for M = M x : The subsonic Mach number M y at the shock position, at the y reference, is calculated with equation ( 5) [1]: The static pressure ratio P y /P x at the shock position is determined with equation (6) [1], which is useful to determine the minimum pressure drop P x at the supersonic speed position M x .Where P y is the pressure reached when the flow slows down after the shock at position M y : The calculations for equations ( 4), (5), and ( 6) are stopped since M x is determined by iterative methods from equation (3).Equation ( 4) is known as the Stodola equation.Calculating the Mach number as a function of the area ratio is useful.To obtain the solution, iterative methods are applied; this equation has two solutions, one for subsonic flow and one for supersonic flow.
The analytical equations (3) and ( 4) cannot be inverted by algebraic procedures for solving the Mach number and are nonlinear problems.Therefore, some known method or other mechanisms must be applied to obtain empirical equations that yield approximate solutions for these two analytical equations.
As a scientific methodology, heuristics is a set of methods and techniques that allow finding a satisfactory solution to a problem, and Polya [21] explains how to approach it.In engineering sciences, empirical equations are very common to obtain approximate solutions to certain problems, such as fluid mechanics, heat transfer, and deformations of solid materials, among others.The complexity of constructing empirical equations lies in the methods and tools used to obtain them.
Lagrange's inversion theorem is reported in the literature, which allows inverting analytic equations by means of the Taylor series expansion of the inverse function of an analytic function.This theorem was generalized by Bürmann [22].
The Stodola equation ( 4) was approached by Abu-Irshaid [23], who applied Bürmann's theorem to obtain a mathematical expression for an approximate solution; for values close to Mach 1, the series required more terms to reduce the error.Authors such as Majdalani and Maicke [24,25] also applied Bürmann's theorem to the Stodola equation ( 4) and obtained results similar to those reported in [23] for values close to Mach 1.
For further studies of compressible flow, Ferrari [26] applied the Abel equation to obtain new analytical solutions for the steady-state one-dimensional viscous adiabatic flow of an ideal gas through a conical nozzle or diffuser.
On the other hand, for the case of incompressible flow, such as the case of obtaining the explicit solution of the nonlinear Colebrook equation, different researchers applied other methods, such as the application of infinite recursion [27] and the Wright-ω function [28].
From the above, the application of different methods to obtain approximate solutions to implicit equations in fluid mechanics is observed.In that sense, since the solution of the analytical equation ( 3) is of great importance to obtain direct calculations of the Mach number M x for supersonic flow at the shock position at A x /A * , it has been proposed to obtain the solution by applying a method different from the mentioned methods.
In the present work, the objective is to construct an empirical equation for the Mach number as a function of the stagnation pressure ratio for a quasi-one-dimensional compressible flow, which satisfies as an approximate solution the inverse of the analytical equation (3).
The methodology presents the applied steps that allowed for constructing the empirical equation.The coefficients and exponents during construction of the empirical equation are calibrated with the Solver iterative simulation analysis tool in Excel 2019.The results of four case studies and the conclusions of the analysis are presented.

Squared error conditioning based on fixed data and variable data
First, in general, the squared error e 2 is conditioned so that it can be applied to a standard data set that is considered fixed data for the position (x i ,y i ), and to a data set that is considered variable data for the position (x i , y´i), but only y´i varies since x i is the same parameter for both fixed and variable data.Fig. 2 illustrates the positions of the fixed data starting at position (x 1 ,y 1 ) and ending at position (x n ,y n ), and of the variable data starting at position (x 1 ,y 1 ) and ending at position (x i , y´n).
The squared error is expressed as: It should be noted that fixed data is associated with experiments, implicit analytic functions, and inverted data from analytic equations.The variable data are associated with an empirical equation under construction y´i = h(x i ), composed of trigonometric, polynomial, logarithmic, and exponential functions.The empirical equation's purpose is to be well structured to model the curve's path on the fixed data faithfully.

Figure 2. Fixed data and variable data positions
The empirical equation y´i = h(x i ) has coefficients a i and exponents n j , and for its calibration, computational tools are required, and one of them is the Excel Solver.
In the Excel Solver, the sum of the squared error Σe 2 is assigned as the objective function in order to minimize, and the coefficients and exponents of the empirical equation under construction are assigned as variable cells.The purpose is for y´i to be as close as possible to y´i to reduce the squared error to the minimum so that the trajectory of the data of the empirical equation is superimposed on the trajectory of the fixed data.
There are other tools, such as Lingo, R, Maple, Matlab, Fortran, and Python, among other similar, for which code must be structured to calculate the minimum of the objective function.On the other hand, for the Excel Solver, it is enough to enter the necessary data.

Data conditioning of the analytical equation
The analytical equation ( 3) was conditioned to generate the discrete Mach number data for two phases: the first phase was set for the Mach number range from 1 to 10, and the second phase for the Mach number range from 1 to 5 is expressed as 67 are presented in Table 1.Discrete Mach number data was generated for 1.
6 and k = 1.67, in the range of 1 ≤ M x ≤ 10 and 297x10 -9 < P oy /P ox ≤ 1; the Mach number is discretized with an increment ΔM x = 0.01 for the range of 1 ≤ M x ≤ 2 and ΔM x = 0.1 for 2 ≤ M x ≤ 10.The discretization for ΔM x = 0.01 is in order to smooth the curve patch for values close to the position P oy /P ox = 1 and M x = 1.In total, n = 190 was discretized for the range of 1 ≤ M x ≤ 10, used in the first phase, and n = 170 for 1 ≤ M x ≤ 5, used in the second phase.The discrete data generated by the analytical equation ( 3) were plotted for 1 ≤ M x ≤ 10, as shown in Fig. 3a in the initial position.And for a second graph, the position was reversed, becoming the data of the Mach number M x of the abscissa to be the ordinate and the P oy /P ox data of the ordinate to be the abscissa, as shown in Fig. 3b in the inverted position.The same figure shows that curve k = 1.4 is located in the center of the group of analytic curves, both in its initial and inverted position.The curves shown in Fig. 3b are very important since the Mach number data of those curves are the fixed data, which belong to the exact solution of the analytical equation ( 3), and all the curves have a convergence point at position M x = 1 and P oy /P ox = 1.And with the support of the standard data, the empirical equation will be constructed.

Coefficient of determination and percentage error
The coefficient of determination R 2 is calculated with equation ( 8) [29]: The sum of the squared error is expressed as: 1 where n is the number of discrete Mach number data, the same quantity for the analytical and empirical equations, it should be noted that during the construction of the empirical equation, the sum of the squared error must be considered as the minimum of the objective function in the Excel Solver.
For calculation purposes, the Mach number of the analytical equation ( 3) is represented as M x,a which is fixed standard data, and of the empirical equation as M x,e , which is variable data, which is based on the squared error conditioning principle e 2 = (M x,a -M x,e ) 2 .
The variance of the dependent variable is expressed as: The average of discrete M x,a is expressed as The following expression determines the percentage error e of the Mach number: , , , 100

Solver iterative simulation analysis tool
Excel Solver version 2019 was used to calibrate the coefficients and exponents of the empirical equation.And its use was based on the fact that it is a versatile and easy-to-use tool to optimize the minimum of the sum of the squared error set as the objective function.In addition to including within the options the generalized reduced gradient method (GRG) [30] for nonlinear optimization solutions, it facilitates the search for solutions for the calibration of coefficients and exponents without restrictions during the construction of the empirical equations.Table 2 presents the parameters and options in ExcelSolver that are applied to calibrate the coefficients and exponents of the empirical equation.Solver options considered it for time 10000 seconds and 10000 iterations.As Solver considers evaluating between zero and one, the range of accuracy and convergence, as well as tolerance equal to or greater than zero; for these three options, the decimals were set to 10 -12 .The closer to zero, the accuracy, tolerance, and convergence, the better the results; however, more time is required to get the minimum objective function's solution during optimization.It should be noted the decimal numbers that Excel takes into account in its spreadsheets are up to 10 -15 .

Construction of the empirical equation
This section presents the steps of constructing the empirical equation for the Mach number range from 1 to 10.
As a strategy, as an initial phase, the empirical equation for the curve k = 1.4 is built since it is located in the center of the family of curves, later with the empirical equation obtained, it is approached for the other curves to calibrate their coefficients and exponents as discrete data; and as a final phase, the coefficients and exponents are calibrated as a function of k.
The steps to construct the empirical equation M x = f(P oy /P ox ) for the curve k = 1.4,are as follows: Step 1: It was taken as a reference that the empirical curve begins at P oy /P ox = 1 and M x = 1, the convergence point for all analytical and empirical curves.It is known that the ratio of stagnation pressures is in the range of 297x10 -9 < P oy /P ox ≤ 1 on the abscissa axis and of Mach number in the range of 1 ≤ M x ≤ 10 on the ordinate axis.
Within the family of functions, the natural logarithm function M x = ln(P oy /P ox ) has the virtue of evaluating the argument for P oy /P ox = 1, where M x = ln(1) = 0. Therefore, the beginning of the construction of the empirical equation is given by equation ( 12): The graph of the curves of the analytical and empirical equations is shown in Fig. 4a.Where the empirical curve registers negative Mach number values and is out of phase, and starts at the position P oy /P ox = 1 and M x = 0; therefore, equation ( 12) must be manipulated to adjust its behavior on the analytical curve.
Step 2: To translate the curve on the ordinate axis, the unit was added to equation (12); in addition, the argument P oy /P ox of the logarithm became the deno-minator to flip the curve, for which a new equation ( 13) was obtained.The graph of the empirical curve is shown in Fig. 4b, where the Mach number has positive values, and the curve begins at the position P oy /P ox = 1 and M x = 1: Step 3: This step adds coefficients and exponents to equation (13) to fit the curve.As coefficients and exponents are added in each step, the iteration is done in Solver, which is routinely repeated.
The logarithm was raised to the power p, as shown in the new equation (14).As an initial condition, p = 1 was established to start the iteration.By minimizing the sum of the squared error, which is the objective function, Solver gave p = 1.253343.The graph of the curves in Fig. 4c shows that the empirical curve is closer to the analytical curve: Step 4: The above equation was raised to the power n = 1 and p = 1.253343 as shown in equation (15); Solver reported n = 1.491785 and p = 0.712427.In Fig. 4d, it is shown that the empirical curve is more in line with the analytical curve: Step 5: The term (P oy /P ox ) m was added to divide the previous equation and the coefficient a to multiply; as shown in equation ( 16), m = 0.1 and a = 1 were assigned, and the values n = 1.491785 and p = 0.712427.When optimizing, Solver reported: m = 0.150776, a = 0.947419, p = 0.389651, n = 1.349928.In this step, the empirical curve was superimposed on the analytical curve, as shown in Fig. 4e: For equation ( 16), the percentage error reached a maximum peak e < 0.9% for the region of Mach number close to 1; for other regions, the error was below e < 0.35%, as shown in Fig. 4f.In addition, the coefficient of determination R 2 = 0.999996 was obtained, close to R 2 ≈ 1; and the sum of the squared error yielded Σe 2 = 0.075.
It should be noted that Fig. 4 illustrates the evolution of the empirical curves on the inverse analytical equation (3) curve for stages 1, 2, 3, 3, 3, 4, and 5 performed.
The error peak close to 0.9% is high despite having R 2 ≈ 1.Therefore, the empirical equation ( 16) must be adapted, and another term must be added to reduce the error.
This consideration of reducing the error allows better numerical precision; in this way, the results of the empirical equation will be more adjusted to the numerical data of the analytical equation (3).Therefore, a new empirical equation has to be obtained, which is presented in step 6.
Step 6: The new empirical equation ( 17) is presented in this step.The procedure to obtain equation ( 17) M x = f(P oy /P ox ) is as follows: the nomenclature of the coefficients was adapted as a i and that of the exponents as n j .Equation ( 16), the section of the first term, was taken as the basis.The second term was carefully constructed to satisfy the decrease of the error percentage and increase the exact decimal digits of the Mach number.
For a prior adjustment of the coefficients and exponents of equation ( 17), the Solver was not applied; therefore, the calibration was manual.To avoid divergence of the sum of the squared error, the assignment of the coefficient and exponent values was controlled so that the sum of the squared error was less than unity.A series of numerical tests were performed, and the most satisfactory results, which are four tests, are presented in Table 3.It should be noted that the values of a 1 , n 1 , n 2 , n 3 , and n 4 in tests 1 and 2 correspond to the data generated in step 5 for equation (16).From Table 3, test 4 was the most satisfactory since it has the percentage error of Mach number less than 1%, and the sum of the squared error Σe 2 = 0.05.Therefore, the data from test 4 were taken as the initial condition to calibrate the coefficients and exponents with the Solver.
To optimize the minimum of the sum of the squared error, as an objective function, the following was taken into account changing variable cells: a1, a2, n 1 , n 2 , n 3 , n 4 , n 5 , n 6 and n 7 , and without restrictions.Consecutive repetitions of runs were performed until numerical convergence was obtained, the calibrated values of coefficients a 1 = 0.476086 and a 2 = 1.245852, and the exponents n 1 = 0.108771, n 2 = 0.091835, n 3 =0.328732,n 4 = 1.821138, n 5 + 0,083785, n 6 = 0.176509, and n 7 = 0.618819, respectively.Fig. 5a illustrates the empirical curve superimposed on the analytical curve, and Fig. 5b shows the percentage error curve of the Mach number.The error reached the maximum peak e < 0.012% for the region of Mach number close to 1, the smallest error concerning equation ( 16) which has a peak of e < 0.9%.In addition, the coefficient of determination R 2 = 0.999996 was obtained, close to R 2 ≈ 1, and the sum of the squared error yielded Σe 2 = 9.9864x10 -7 .
With equation ( 17) already constructed in its final stage, the coefficients and exponents of the curves of the extremes were determined for for k = 1.1, k = 1.2, k = 1.3, k = 1.4,k = 1.5, k = 1.6 and k = 1.67.For example, to calibrate the a i and n j for k = 1.3, we took as initial data the a i and n j obtained for k = 1.4,applied the Solver, and obtained the new values of a i and n j for k = 1.3.Similarly, we calibrated the a i and n j for the other k values.The values of the calibrated a i and n j for the Mach number range from 1 to 10 are presented in the resultsin Tables 4 and 5. Similarly, with equation ( 17), a i and n j were calibrated for the Mach number range from 1 to 5. The values are presented in the resultsin Tables 7 and 8. To obtain equation ( 17) M x = f(P oy /P ox ,k) as a function of two variables, with the discrete data of the coefficients and exponents for each value of k.For example, for each data of k = 1.1, k = 1.2, k = 1.3, k = 1.4,k = 1.5, k = 1.6 and k = 1.67 we have the values of the coefficients a i and the exponent n j , so that with these data we construct the polynomials a i = f(k) and n j = f(k), which are the initial data to apply the Solver.
The data in Tables 4 and 5 were used for the Mach number in the range 1 to 10, and Tables 7 and 8 were used for the Mach number in the range 1 to 5.
This phase of polynomial calibration started with the smoothest curve that reported the discrete data and was anchored as fixed and constant data.The remainder of the discrete data was taken as variables in the changing cell and re-optimized to obtain new values of coef-ficients and exponents.Then, the smoothest curve trajectory was searched again, anchored as a constant value, and optimized until new values of the remaining discrete data were recorded.The procedure was repeated until the last polynomial was obtained.It was very complicated because it was trial and error.The poly-nomials obtained are shown in the results; the Mach number range from 1 to 1 is presented in Table 6, and for Mach number range from 1 to 5 is presented in Table 9.

RESULTS AND DISCUSSION
This section presents the results for four case studies, calculated with the empirical equation ( 17): two cases for 1 ≤ M x ≤ 10, one for a i and n j , and another for a i = f(k) and n j = f(k).And two cases for 1 ≤ M x ≤ 5, one for a i and n j , and another for a i = f(k) and n j = f(k).

Case studies
For case 1, in Tables 4 and 5, the results are presented; it was evaluated for discrete data, where the coefficients a 1 and a 2 and of the exponents n 1 , n 2 , n 3 , n 4 , n 5 , n 6 and n 7 were calibrated for each value of k, for M x = f(P oy /P ox ) in the range of 1 ≤ M x ≤ 10.For case 2, Table 6 presents the results; it was evaluated for polynomials, where the coefficients a i = f(k), and of the exponents n j = f(k), for M x = f(P oy /P ox ,k) in the range of 1 ≤ M x ≤ 10.
All the analytical and empirical curves converge at the position P oy /P ox = 1 and M x = 1.It should be noted that the graphs of case 2. Cases 1, 3, and 4 are not included since the trajectories of the curves are similar.In addition, it is indicated that cases 3 and 4 only cover the range of 1 ≤ M x ≤ 5.

Comparative analysis of cases
The Mach number values obtained with the empirical equation ( 17 In Tables 10,11, 12, and 13, the proximity to the exact solution is evidenced when comparing the results of the empirical equation.How the decimals fluctuate by excess and by default indicates that the empirical curves oscillate and intercept with analytical curves.For case 1 (Table 10), the Mach number error is less than 0.018%; for case 2 (Table 11), the error is less than 0.063%; for case 3 (Table 12), the error is less than 0.004%, finally, for case 4 (Table 13) the error is less than 0.00988%.4 and 5 Specific heats ratio, k 1.1 The most accurate values are for discrete data; however, they have the disadvantage that they are individual curves for each k value.
While the empirical equation with the data of coefficients and exponents as a function of k allows dealing with a family of curves within the range of 1.1 ≤ k ≤ 1.67, the advantage is much greater regarding individual curves.The results of the Mach numbers presented in Tables 10-13 have to do with the calibrations of the coefficients and exponents.The solution search engine by iterative methods in Solver was based on the GRG.The unrestricted condition was applied to the variables, where the objective function was the minimum sum of the squared error.
The calibration phase to obtain the polynomials a i = f(k) and n j = f(k) presented difficulty and was very laborious because the discrete data for each value of k did not only present curves but also fluctuations.In addition, this was compounded by the fact that the errors during calibration increased rather than remained the same or decreased.Therefore, to avoid sharp increases in the errors, the polynomials of orders 2, 3, and 4 were evaluated repeatedly, and the one with the lowest error was selected, and this was done for each coefficient and exponent.It should be noted that sometimes the errors increased above 2%, which is why we tried to obtain the errors as close as possible to the discrete data values.
It is summarized, in Table 14 shows the magnitudes of the percentage errors of the Mach number and the coefficients of determination R 2 ≈ 1 for the four cases of study.Figure 7 shows the behavior of the Mach number errors for the four cases.The empirical equation ( 17) is a mathematical representation of the solution of the inverse of equation ( 3).It significantly contributes to the area of gas dynamics for quasi-one-dimensional isentropic compressible flow.However, it can be improved by modifying or adding more terms to reduce the error further.
The analytical equations ( 3) and ( 4), among other mathematical expressions, are widely used to simulate the compressible flow in one dimension for an ideal gas or real gas, for which different computational iterative calculation procedures are applied to obtain the solutions [31][32][33][34][35][36].
Applying the empirical equation ( 17), the following are considered for the design of a nozzle at Mach 3, flow k = 1.4 and A x /A * =4.234.The inlet pressure of the nozzle P 0 =3.5547 atm is considered a constant value, and the pressure at outlet Pe is variable.For the divergent shock to occur,the pressure ratio rp = P e /P 0 is 0.2183 < rp < 0.9866, and the Mach number at the outlet is 0.1384 < M e < 0.4751.For the shock to occur in the divergent section, the outlet pressure P e is 1 < P e < 3.5075; for this range, four cases are established: rp = 0.4; rp = 0.5; rp = 0.6 and rp = 0.7.
The behavior patterns of the trajectories of the curves for the Mach number and how the normal shocks are distributed in the local positions in A x /A * for the supersonic flow M x and subsonic flow M y are shown in Fig. 8.As the magnitude of rp decreases, the position of the shock is translated toward the nozzle outlet.
It should be noted that the curve span for the supersonic flow up to the shock position and for the subsonic flow after the shock is determined with equation ( 4

Figure 1 .
Figure 1.A simple scheme of a convergent-divergent nozzle.(a) The divergent section refers to the position of a normal shock wave.(b) Flow rate pattern in terms of the Mach number.Adapted from Anderson [2] Being the expressions for the exponents n = k/(k-1) and m = 1/(k-1) and for the coefficients a = 2/(k+1), b = (k-1)/(k+1) and c = 2k/(k+1).The numerical values of the coefficients and exponents for k

Figure 3 .
Figure 3.The behavior of the trajectories of the analytical curves.(a) Initial position.(b) Inverted position

Table 2 .
Parameters and options in the Solver tool Parameter Objective function cell: sum of square error Objective cell condition: minimum Changing variable cells: coefficients and exponents Subject to restrictions: no restrictions Option Estimate: squared Derivatives: central Iterative method: GRG Nonlinear

Figure 4 .
Figure 4.The behavior of the evolution and adjustment of the empirical curve on the analytical curve for k = 1.4.(a) Step 1, (b) step 2, (c) Step 3, (d) Step 4, (e) Step 5, and (f) Mach number percentage error for step 5.

Figure 5 .
Figure 5.The behavior of the evolution and adjustment of the empirical curve on the analytical curve for k = 1.4(a) Step 6, and (b) Mach number percentage error for step 6.
) are compared with the exact values of the analytical equation (3).They are only considered for k = 1.1, k = 1.3 k = 1.5 and k = 1.67; the other values of for k = 1.2, k = 1.4,k = 1.6, the results are similar.

Figure7.
Figure7.Percentage errors of Mach number.(a) Case 1, (b) Case 2, (c) Case 3, and (d) Case 4 ) for each position of A/A*.The subsonic Mach number, after the shock, from M y to M e , is calculated with the area ratio A , the area * x A is the critical area in the throat, the value of A is variable in the range from