**Table 1.**

*Comparison Burago's and Kárman-Tsien methods.*

method is observed. For the values of the relative coordinate along the major axis of the ellipse 0*:*1≤*x*≤ 0*:*5, there is a significant discrepancy between the Kárman-Tsien results and the Burago method. However, it can be noted that the calculations of other authors are in good agreement with the calculations by the Kárman-Tsien formula (34). The calculation according to this method approaches the value of the pressure coefficient, indicated in **Figure 4** as *C*<sup>∗</sup> *<sup>p</sup>* , which corresponds to the critical Mach number. As can be seen from **Figure 4**, the values determined in the article [16] and by the author practically coincide. **Figure 4** also shows the distribution of the pressure-drop coefficient *C*<sup>0</sup> *<sup>p</sup>* for an incompressible fluid. **Figure 4** shows the strong influence of the compressibility factor. Of greatest interest to researchers is the value of the critical Mach number for a cylinder in a transverse flow. In **Figure 5 (a, b)**, the results of calculations of the relative velocity on the surface of the cylinder, obtained using the proposed approximate technique, are compared with the calculations of other authors performed by other methods.

Pai [6]) are compared in **Figure 5**. The results of Simisaki's calculations were taken from [17]. The calculations were carried out for the Mach number almost equal to the critical Mach number *M*<sup>∞</sup> ≈ 0.372 (**Table 2**). It can be noted that in **Figure 5**, the author's calculation is in good agreement with the data of other authors for velocities

0.390 Calculations by I. Imei. Z. Hashimoto 5 0.37170¤0.00001 author's calculation —

*M***\* Method and authors of calculation results** *δ* **%** 0.415 C. Jacob's calculation [7] 10 0.409 Poggi method. Calculations by S. Kaplan [12] 9 0.404 I. Imei's calculation 8 0.400 Rayleigh-Janzen method. T. Simisaki's calculations 7

*Comparison of the calculated values of the pressure drop coefficient on the surface of the ellipse (relative*

*Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies*

*DOI: http://dx.doi.org/10.5772/intechopen.94981*

*thickness 20%) (a); for a biconvex airfoil (relative thickness 6%) (b).*

0.3983¤0.0002 The Rayleigh-Janzen method. Hoffman's calculations 7

calculations by P.I. Chushkin

Dorodnitsyn. calculations by R. Melnik and D. C. Ives

G. A. Dombrovsky

A. Dombrovsky

Calculations by M. Holt and B. Messon

7

7

6

5

5

0.399 The method of integral relations by A.A. Dorodnitsyn.

0.39853¤0.00002 The multi-layer method of integrated relations by A.A.

0.396 The method of approximation of adiabat. Approximation A4.

0.390 The method of approximation of adiabat. A3 approximation. G.

*Comparison of the calculated values for the critical Mach number on the surface of the cylinder.*

0.390 The method of integral relations by A.A. Dorodnitsyn.

Let us consider the numerical values of the critical Mach numbers *M*\* for a flow around a circular cylinder obtained by different authors. **Table 1** shows a comparison of the results of calculating the critical Mach numbers by different methods. The value *δ* denotes the relative error of comparing the values of the critical Mach

on the cylinder surface, taking into account the compressibility factor.

numbers calculated by different methods and by this method.

**Table 2.**

**181**

**Figure 4.**

Integral relationship method [17], method of Legendre transformations [10], method approximation of adiabat [7], Rayleigh-Janzen method (Simisaki [17]; Shih-I *Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies DOI: http://dx.doi.org/10.5772/intechopen.94981*

#### **Figure 4.**

*Comparison of the calculated values of the pressure drop coefficient on the surface of the ellipse (relative thickness 20%) (a); for a biconvex airfoil (relative thickness 6%) (b).*


#### **Table 2.**

method is observed. For the values of the relative coordinate along the major axis of the ellipse 0*:*1≤*x*≤ 0*:*5, there is a significant discrepancy between the Kárman-Tsien results and the Burago method. However, it can be noted that the calculations of other authors are in good agreement with the calculations by the Kárman-Tsien formula (34). The calculation according to this method approaches the value of the

*<sup>p</sup>* **Mach number** *M***<sup>∞</sup> Cp Kárman-Tsien Cp Burago** *δ* **%** 0.1 0.3 0.1051 0.1048 0.29

0.5 0.3 0.5305 0.5315 0.18

1.0 0.3 1.0742 1.0796 0.5

1.5 0.3 1.6315 1.6474 1.0

2.0 0.3 2.2029 2.2389 1.6

2.5 0.3 2.7890 2.8618 2.5

3.0 0.3 3.3904 3.5221 3.7

3.5 0.3 4.0076 4.2352 5

4.0 0.3 4.6414 5.0245 8

4.5 0.3 5.2922 5.9404 11

0.6 0.1266 0.1269 0.24 M\* = 0.886 0.2289 0.2165 6

0.6 0.6667 0.7042 5.0 M\* = 0.679 0.7489 0.8696 14

0.4 1.1432 1.1659 2.0 M\* = 0.558 1.3427 1.5874 15

0.4 1.7566 1.8320 4 M\* = 0.486 1.9245 2.2790 16

0.4 2.4009 2.6067 8 M\* = 0.437 2.5034 2.9683 16

M\* = 0.400 3.0782 3.6432 16

M\* = 0.371 3.6515 4.3127 15

M\* = 0.348 4.2265 4.9949 15

M\* = 0.329 4.8020 5.6830 15

M\* = 0.312 5.3714 6.3339 15

Mach number. As can be seen from **Figure 4**, the values determined in the article [16] and by the author practically coincide. **Figure 4** also shows the distribution of

strong influence of the compressibility factor. Of greatest interest to researchers is the value of the critical Mach number for a cylinder in a transverse flow. In **Figure 5 (a, b)**, the results of calculations of the relative velocity on the surface of the cylinder, obtained using the proposed approximate technique, are compared with

Integral relationship method [17], method of Legendre transformations [10], method approximation of adiabat [7], Rayleigh-Janzen method (Simisaki [17]; Shih-I

*<sup>p</sup>* , which corresponds to the critical

*<sup>p</sup>* for an incompressible fluid. **Figure 4** shows the

pressure coefficient, indicated in **Figure 4** as *C*<sup>∗</sup>

*Comparison Burago's and Kárman-Tsien methods.*

the calculations of other authors performed by other methods.

the pressure-drop coefficient *C*<sup>0</sup>

**Table 1.**

**180**

*C***0**

*Aerodynamics*

*Comparison of the calculated values for the critical Mach number on the surface of the cylinder.*

Pai [6]) are compared in **Figure 5**. The results of Simisaki's calculations were taken from [17]. The calculations were carried out for the Mach number almost equal to the critical Mach number *M*<sup>∞</sup> ≈ 0.372 (**Table 2**). It can be noted that in **Figure 5**, the author's calculation is in good agreement with the data of other authors for velocities on the cylinder surface, taking into account the compressibility factor.

Let us consider the numerical values of the critical Mach numbers *M*\* for a flow around a circular cylinder obtained by different authors. **Table 1** shows a comparison of the results of calculating the critical Mach numbers by different methods. The value *δ* denotes the relative error of comparing the values of the critical Mach numbers calculated by different methods and by this method.

to other methods. Holt and Messon [17] point out that "symmetric flows (in which a shock does not occur) were calculated only up to the Mach number of the freestream flow *M*∞=0.37". This circumstance indirectly confirms that the exact value of the critical Mach number is closer to that determined by the Burago method than

*Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies*

An important question is the question: how does the viscosity of the medium affect the value of the critical Mach number? **Figure 6** shows a comparison of the calculated data for the flow around the biconvex airfoil using the ideal gas and viscous models. **Figure 6** also shows the experimental results that can be observed

Calculations for viscous gas are performed in a computational package Flow Simulation (SolidWorks). The calculations in the Flow Simulation software are based on solving the Navier–Stokes equations. The calculation domain and the grid used in the calculations Flow Simulation are shown in **Figure 7(a)** and **(b)**. The flow velocity was calculated at a distance equal to the thickness of the boundary

by other methods.

*DOI: http://dx.doi.org/10.5772/intechopen.94981*

also in **Figure 4(b)**.

**Figure 7.**

**183**

*The calculation domain and grid used in package flow simulation.*

**Figure 5.** *The velocity on the cylinder surface vs. Mach number.*

If we exclude from the comparison the early work of Kaplan [12] in which the adiabatic exponent for air was taken equal *κ* = 1.408 which affected the value of the calculated critical Mach number and the work of Jacob (see [7]) then based on the data in **Table 1** it can be argued that the accuracy of calculating the critical Mach number on the surface of the circular cylinder by the Burago's method in comparison with other methods on average approximately corresponds to a relative error of 5%. From **Table 1** it can be seen that the critical Mach number on the surface of the circular cylinder calculated by Burago's method is the smallest of the data presented in the **Table 1**. This does not mean that this value is the most inaccurate compared

**Figure 6.** *Critical Mach number vs. relative thickness of the biconvex airfoil.*

## *Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies DOI: http://dx.doi.org/10.5772/intechopen.94981*

to other methods. Holt and Messon [17] point out that "symmetric flows (in which a shock does not occur) were calculated only up to the Mach number of the freestream flow *M*∞=0.37". This circumstance indirectly confirms that the exact value of the critical Mach number is closer to that determined by the Burago method than by other methods.

An important question is the question: how does the viscosity of the medium affect the value of the critical Mach number? **Figure 6** shows a comparison of the calculated data for the flow around the biconvex airfoil using the ideal gas and viscous models. **Figure 6** also shows the experimental results that can be observed also in **Figure 4(b)**.

Calculations for viscous gas are performed in a computational package Flow Simulation (SolidWorks). The calculations in the Flow Simulation software are based on solving the Navier–Stokes equations. The calculation domain and the grid used in the calculations Flow Simulation are shown in **Figure 7(a)** and **(b)**. The flow velocity was calculated at a distance equal to the thickness of the boundary

**Figure 7.** *The calculation domain and grid used in package flow simulation.*

If we exclude from the comparison the early work of Kaplan [12] in which the adiabatic exponent for air was taken equal *κ* = 1.408 which affected the value of the calculated critical Mach number and the work of Jacob (see [7]) then based on the data in **Table 1** it can be argued that the accuracy of calculating the critical Mach number on the surface of the circular cylinder by the Burago's method in comparison with other methods on average approximately corresponds to a relative error of 5%. From **Table 1** it can be seen that the critical Mach number on the surface of the circular cylinder calculated by Burago's method is the smallest of the data presented in the **Table 1**. This does not mean that this value is the most inaccurate compared

**Figure 5.**

*Aerodynamics*

**Figure 6.**

**182**

*Critical Mach number vs. relative thickness of the biconvex airfoil.*

*The velocity on the cylinder surface vs. Mach number.*

layer. In the calculations, the free-stream velocity varied, and as soon as the local flow velocity at the boundary of the layer boundary became equal to the sound velocity, the critical Mach number was calculated.

Based on **Figure 6**, it can be concluded that viscosity has practically no effect on the value of the critical Mach number.

Another two-dimensional body, the flow around which is well studied is an ellipse. **Figure 8** shows the calculated results for the critical Mach number depending on the degree of compression (relative thickness, *p* = *a*/*b*) of the ellipse. **Figure 8** also shows the value of the critical Mach number for a two-dimensional body formed by two contacting cylinders. Calculations of the critical Mach number for such a 2-D body were first performed by the author [1].

We have looked at comparing calculations for two-dimensional bodies. The high accuracy of the calculation of the critical Mach numbers by the Burago method is shown.

Let us now consider the applicability of the Burago method for axisymmetric flows. It is also of interest to compare the critical Mach numbers for two-dimensional and axisymmetric flows. Ellipses and ellipsoids of revolution (spheroids) with various factor of compression *δ* are chosen. For calculations the well-known potential models for ellipses and spheroids [21] were used. It is obvious that the maximal error will correspond to flow around thick bodies (*δ* ! 1.0) and for the case *M*<sup>∞</sup> = *M*\*. In **Table 3** and in **Figure 9(a)** results of calculation of critical Mach number for ellipses and spheroids with an offered method and with Dorodnitsyn method of integral relations executed by Chushkin [14, 15] are compared.

elliptical contour to the velocity at infinity as a function of the Mach number (symmetric flow). **Figure 9(b)** uses the relative ellipse thickness *δ* = 0.1. Here, along with the calculations by the proposed method, the results of calculations obtained by the theory of small perturbations and by the Hantz and Wendt second approximation (see the Book [7]), as well as the results of calculations obtained by the Dombrovsky method [7] are shown. It can be noted that the results of calcula-

*δ* **Ellipses Ellipsoids**

*Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies*

**Relative difference %**

0.05 0.103 0.869 0.884 1.8 0.014 0.984 0.980 0.4 0.10 0.210 0.803 0.807 0.5 0.042 0.957 0.945 1.3 0.15 0.323 0.752 0.748 0.5 0.080 0.929 0.905 2.6 0.20 0.440 0.709 0.700 1.3 0.122 0.899 0.868 4.0 0.40 0.960 0.588 0.566 3.7 0.337 0.783 0.742 5 0.60 1.560 0.506 0.480 5 0.602 0.692 0.648 6 0.80 2.240 0.447 0.418 6 0.908 0.620 0.576 7 1.00 3.000 0.399 0.372 7 1.250 0.563 0.519 8

*C***0**

*<sup>p</sup>* **min Chushkin [14]** *M***\***

**Calculation** *M***\***

**Relative difference %**

**Calculation** *M***\***

*Critical Mach number at a flow around ellipses and spheroids.*

tions using the proposed method practically coincided with the results of Dombrovsky. The critical Mach number equal to *M\** ≈ 0.807 (**Figure 9(b)**) was obtained by the Dombrovsky method of approximation of the adiabat. This is in very good agreement with Lighthill's data (see Book [7]) and Dombrovsky's result [7] *M*\* ≈ 0.81 and Chushkin's result *M\** ≈ 0.803 (**Table 2**). Kaplan's result [12] *M\** ≈ 0.857 should be considered less accurate, but the relative error does not

Also, it follows from **Figure 9(a)** that critical Mach number for the twodimensional case is always less than for axisymmetric case for bodies with the same

*(a) Critical Mach number vs. relative thickness of an ellipse and ellipsoid; (b) relative maximal velocity of the*

exceed 7%.

**Table 3.**

*C***0**

*<sup>p</sup>* **min Chushkin [14]***M***\***

*DOI: http://dx.doi.org/10.5772/intechopen.94981*

cross-section.

**Figure 9.**

**185**

*flow around ellipse vs. Mach number.*

**Table 3** and **Figure 9(a)** shows a comparison of the results of calculating the critical Mach numbers by two different methods. **Table 3** and **Figure 9(a)** shows a good agreement. It is necessary to notice that calculations in a range of ð Þ 0*:*4≤*δ*≤0*:*8 were executed by Chushkin only for the second approximation and each subsequent approximation resulted in reduction the value of critical Mach number. Therefore, some additional error can be explained by this fact. It can be noted that the relative difference in values of the critical Mach numbers increases with increasing *δ* for both ellipses and spheroids.

However, the maximum relative difference in values of the critical Mach numbers for the elliptic cylinder does not exceed 7% and for the sphere 8%. **Figure 9(b)** shows a comparison of the ratio of the maximum velocity at the surface of an

**Figure 8.** *Critical Mach number vs. relative thickness of the ellipse.*

*Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies DOI: http://dx.doi.org/10.5772/intechopen.94981*


#### **Table 3.**

layer. In the calculations, the free-stream velocity varied, and as soon as the local flow velocity at the boundary of the layer boundary became equal to the sound

Another two-dimensional body, the flow around which is well studied is an

depending on the degree of compression (relative thickness, *p* = *a*/*b*) of the ellipse. **Figure 8** also shows the value of the critical Mach number for a two-dimensional body formed by two contacting cylinders. Calculations of the critical Mach number

We have looked at comparing calculations for two-dimensional bodies. The high accuracy of the calculation of the critical Mach numbers by the Burago method is

Let us now consider the applicability of the Burago method for axisymmetric flows. It is also of interest to compare the critical Mach numbers for two-dimensional and axisymmetric flows. Ellipses and ellipsoids of revolution (spheroids) with various factor of compression *δ* are chosen. For calculations the well-known potential models for ellipses and spheroids [21] were used. It is obvious that the maximal error will correspond to flow around thick bodies (*δ* ! 1.0) and for the case *M*<sup>∞</sup> = *M*\*. In **Table 3** and in **Figure 9(a)** results of calculation of critical Mach number for ellipses and spheroids with an offered method and with Dorodnitsyn method of integral

**Table 3** and **Figure 9(a)** shows a comparison of the results of calculating the critical Mach numbers by two different methods. **Table 3** and **Figure 9(a)** shows a

ð Þ 0*:*4≤*δ*≤0*:*8 were executed by Chushkin only for the second approximation and each subsequent approximation resulted in reduction the value of critical Mach number. Therefore, some additional error can be explained by this fact. It can be noted that the relative difference in values of the critical Mach numbers increases

However, the maximum relative difference in values of the critical Mach numbers for the elliptic cylinder does not exceed 7% and for the sphere 8%. **Figure 9(b)** shows a comparison of the ratio of the maximum velocity at the surface of an

good agreement. It is necessary to notice that calculations in a range of

ellipse. **Figure 8** shows the calculated results for the critical Mach number

Based on **Figure 6**, it can be concluded that viscosity has practically no effect on

velocity, the critical Mach number was calculated.

for such a 2-D body were first performed by the author [1].

relations executed by Chushkin [14, 15] are compared.

with increasing *δ* for both ellipses and spheroids.

*Critical Mach number vs. relative thickness of the ellipse.*

the value of the critical Mach number.

shown.

*Aerodynamics*

**Figure 8.**

**184**

*Critical Mach number at a flow around ellipses and spheroids.*

elliptical contour to the velocity at infinity as a function of the Mach number (symmetric flow). **Figure 9(b)** uses the relative ellipse thickness *δ* = 0.1. Here, along with the calculations by the proposed method, the results of calculations obtained by the theory of small perturbations and by the Hantz and Wendt second approximation (see the Book [7]), as well as the results of calculations obtained by the Dombrovsky method [7] are shown. It can be noted that the results of calculations using the proposed method practically coincided with the results of Dombrovsky. The critical Mach number equal to *M\** ≈ 0.807 (**Figure 9(b)**) was obtained by the Dombrovsky method of approximation of the adiabat. This is in very good agreement with Lighthill's data (see Book [7]) and Dombrovsky's result [7] *M*\* ≈ 0.81 and Chushkin's result *M\** ≈ 0.803 (**Table 2**). Kaplan's result [12] *M\** ≈ 0.857 should be considered less accurate, but the relative error does not exceed 7%.

Also, it follows from **Figure 9(a)** that critical Mach number for the twodimensional case is always less than for axisymmetric case for bodies with the same cross-section.

#### **Figure 9.**

*(a) Critical Mach number vs. relative thickness of an ellipse and ellipsoid; (b) relative maximal velocity of the flow around ellipse vs. Mach number.*

proposed method is shown. The method allows one to determine the parameters of a compressible flow from the values of the flow of an incompressible fluid up to a speed corresponding to the critical Mach number. This method does not depend on the means determination parameters of the incompressible flow. The calculation in software Flow Simulation was shown that the viscosity factor does not affect the value critical Mach number. It was found that with an increase in the relative thickness of the body, the value of the critical Mach number decreases. It was also found that the value of the critical Mach number for the two-dimensional case is always less than for the axisymmetric case for bodies with the same cross-section.

*Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies*

**Conflict of interest**

**Acronyms and abbreviations**

TM Technical Memorandum

**Appendices and nomenclature**

HSH High Speed Hydrodynamics

TR Technical Report

The author declares no conflict of interest.

*DOI: http://dx.doi.org/10.5772/intechopen.94981*

AMM Applied Mechanics and Materials

TsAGI Central Aerohydrodynamic Institute

*ρ* local density of gas; *τ*, *σ* special functions (Eq. (2));

*p* the static pressure;

*h* the enthalpy;

*c*0

**187**

*κ* the ratio of specific heats; *P(p)* the pressure function;

*M*<sup>∞</sup> the Mach number at infinity; *M* the local Mach number; *u* ¼ *u=U*∞, *v* ¼ *v=U*<sup>∞</sup> dimensionless velocities; *E(M)* the function (Eq. (11)); *a* the sound speed;

*ε* given accuracy of calculation;

*G(M), F(M)* the functions (Eq. (30)).

*U* local total relative velocity (Eq. (23));

(Eq. (28));

NACA National Advisory Committee for Aeronautics

*u*, *v* velocity components along axis *x* и *r*;

**V** the vector of full local velocity of flow;

∞ subscript indicates flow parameters at infinity;

*η<sup>c</sup>* the compressibility factor of the flow (Eq. (15));

*<sup>p</sup>* the pressure-drop coefficient for an incompressible flow

0 lower index, specifies parameters of the flow in the stag-

the flow of the incompressible liquid;

nation point or upper index and specifies parameters of

#### **Figure 10.**

*Maximal velocities on surface of the ellipse and spheroid vs. Mach number.*

#### **Figure 11.**

*Influence of Mach number of compressible flow on pressure-drop coefficient on the surface of the an ellipse and spheroid.*

The effect of compressibility is shown in **Figure 10**, which shows the ratio of the relative maximum velocities *U*max ¼ *U*max*=U*<sup>∞</sup> for compressible and incompressible flows on the surface of 2D ellipses and 3D spheroids.

It follows from **Figure 10** that the effect of compressibility for two-dimensional ellipses at the same Mach number is greater than for 3D-spheroids. **Figure 11** shows the results of calculating the pressure-drop coefficient for compressible and incompressible flows on the surfaces of two- and three-dimensional bodies.

As seen in **Figure 11**, the compressibility effect has a stronger effect on 2D bodies than on axisymmetric bodies.

## **4. Conclusions**

This paper presents the results of calculating the critical Mach numbers of the flow around two-dimensional and axisymmetric bodies. A sufficiently high accuracy of calculating the critical Mach numbers for engineering calculations using the *Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies DOI: http://dx.doi.org/10.5772/intechopen.94981*

proposed method is shown. The method allows one to determine the parameters of a compressible flow from the values of the flow of an incompressible fluid up to a speed corresponding to the critical Mach number. This method does not depend on the means determination parameters of the incompressible flow. The calculation in software Flow Simulation was shown that the viscosity factor does not affect the value critical Mach number. It was found that with an increase in the relative thickness of the body, the value of the critical Mach number decreases. It was also found that the value of the critical Mach number for the two-dimensional case is always less than for the axisymmetric case for bodies with the same cross-section.
