**2. The approximate method for calculating flow compressibility characteristics**

The equations for the existence of the velocity potential and continuity are written for two cases: two-dimensional and axisymmetric for compressible irrotational gas flow

$$\frac{\partial u}{\partial r} - \frac{\partial v}{\partial \mathbf{x}} = \mathbf{0}, \quad \frac{\partial (r^{\mathbf{n}} \rho u)}{\partial \mathbf{x}} + \frac{\partial (r^{\mathbf{n}} \rho v)}{\partial r} = \mathbf{0}, \tag{1}$$

j j **<sup>V</sup>** <sup>2</sup>

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

*<sup>M</sup>*<sup>∞</sup><sup>2</sup> <sup>1</sup> � *<sup>u</sup>*<sup>2</sup> � *<sup>v</sup>*<sup>2</sup> � � � � <sup>1</sup>

velocity, *U*<sup>∞</sup> is free-stream flow velocity at infinity.

*κ* � 1 2

*<sup>κ</sup>*�1, *<sup>a</sup>*<sup>0</sup>

The speed of sound is defined by formula *<sup>a</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

*M*<sup>2</sup> � � <sup>1</sup>

here lower index "0" specifies parameters in stagnation point.

*ρ ρ*∞

¼ 1 þ

*κ* � 1 2

*ρ*0

Formulas (9) can be written as

*<sup>ρ</sup>* <sup>¼</sup> ½ � *E M*ð Þ <sup>1</sup>

the case of an isentropic air flow is *<sup>a</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

*ρ*0

form (Eq. (4)), we can write

Mach number.

**Figure 1.**

**173**

*The function σ vs. Mach number.*

*<sup>ρ</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

<sup>2</sup> <sup>þ</sup> *<sup>ρ</sup><sup>κ</sup>*�<sup>1</sup>

ð Þ *<sup>κ</sup>* � <sup>1</sup> *ρκ*�<sup>1</sup> <sup>∞</sup> *<sup>M</sup>*<sup>2</sup>

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

∞ ¼ 1 2 þ

where j j **V** is the modulus of the vector of the total local velocity of flow. Based on formula (Eq. (7)), the following equalities can be written:

> *κ*�1 , *<sup>a</sup> a*<sup>∞</sup>

here *u* ¼ *u=U*∞, *v* ¼ *v=U*<sup>∞</sup> are the dimensionless components of the local

*κ*�1 , *<sup>a</sup>*<sup>0</sup>

*<sup>a</sup>* <sup>¼</sup> ½ � *E M*ð Þ <sup>1</sup>

*a* ¼ *a*<sup>∞</sup>

Based on the isentropic relationship for density (Eq. (9)), the function *σ* (Eq. (2)) can be calculated. **Figure 1** shows a graph of the function *σ* for air versus

*ρ ρ*∞ � �*<sup>κ</sup>*�<sup>1</sup> 2

Let us write the well-known isentropic relations for density and sound speed

1 ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>M</sup>*<sup>2</sup>

¼ 1 þ

*<sup>a</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

2, *E M*ð Þ¼ 1 þ

∞

*κ* � 1 2

*κ* � 1 2

� �<sup>1</sup>

*M*<sup>2</sup>

2

*κ* � 1 2

*κp=ρ* p . Based on the equation of state in the

*M*<sup>2</sup>

*dp=dρ* p . The speed of sound in

*:* (11)

, (9)

*:* (10)

*<sup>M</sup>*<sup>∞</sup><sup>2</sup> <sup>1</sup> � *<sup>u</sup>*<sup>2</sup> � *<sup>v</sup>*<sup>2</sup> � � � �<sup>1</sup>

, (7)

2 ,

(8)

where *u*, *v* are components of the velocity along axis *x* и *r* accordingly, *m/s*;, *r, x* are coordinates; *ρ* is local density of gas, *kg/m<sup>3</sup>* ; parameter *n* equal

$$n = \begin{cases} 0 & -2\text{D body;} \\ 1 & - \text{ axisymmetric body.} \end{cases}$$

Here *x*, *r* are coordinates in the meridional plane for the axisymmetric case (*n* = 1). Let us introduce the special functions proposed by Burago [20]

$$
\sigma = \frac{2}{\rho\_0/\rho + 1}; \quad \sigma = \frac{\rho\_0/\rho - 1}{\rho\_0/\rho + 1}, \tag{2}
$$

*ρ*<sup>0</sup> is gas density at the stagnation point. Taking into account equation (Eq. (2)), the equation (Eq. (1)) can be rewritten as

$$
\frac{\partial}{\partial r} \left( \frac{\pi u}{1 - \sigma} \right) - \frac{\partial}{\partial \mathbf{x}} \left( \frac{\pi u}{1 - \sigma} \right) = \mathbf{0}, \quad \frac{\partial}{\partial \mathbf{x}} \left( \frac{r^{\mathbf{n}} \pi u}{1 + \sigma} \right) + \frac{\partial}{\partial r} \left( \frac{r^{\mathbf{n}} \pi v}{1 + \sigma} \right) = \mathbf{0}, \tag{3}
$$

For barotropic model of compressible air we have

$$\frac{\rho}{\rho\_{\infty}} = \left(\frac{p}{p\_{\infty}}\right)^{\frac{1}{\kappa}},\tag{4}$$

where *p* is static pressure, *Pa*; *κ* is ratio of specific heats (for air *κ* = 1.4); ∞ subscript indicates flow parameters at infinity. The equation (Eq. (4)) refers to as Poisson's adiabatic curve.

According to the accepted model of a barotropic gas, the enthalpy and pressure function differ only by a constant value *h* = *P*(*p*) + const. The pressure function is

$$P(p) = \int\_{p\_{\approx}}^{p} \frac{dp}{\rho} \,. \tag{5}$$

If we carry out the integration in formula (Eq. (5)) taking into account formula (Eq. (4)), then we get (Eq. (6))

$$h = \frac{\rho^{\kappa - 1}}{(\kappa - 1)\rho\_{\infty}^{\kappa - 1}M\_{\infty}^2},\tag{6}$$

here *M*<sup>∞</sup> is Mach number at infinity.

Using the equation for enthalpy (Eq. (6)), the Bernoulli equation for a compressible gas can be written as

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

$$\frac{\left|\mathbf{V}\right|^{2}}{2} + \frac{\rho^{\kappa-1}}{(\kappa-1)\rho\_{\infty}^{\kappa-1}M\_{\infty}^{2}} = \frac{1}{2} + \frac{1}{(\kappa-1)M\_{\infty}^{2}},\tag{7}$$

where j j **V** is the modulus of the vector of the total local velocity of flow. Based on formula (Eq. (7)), the following equalities can be written:

$$\frac{\rho}{\rho\_{\infty}} = \left[\mathbf{1} + \frac{\kappa - 1}{2} M\_{\infty 2} \left(\mathbf{1} - \overline{u}^2 - \overline{v}^2\right)\right]^{\frac{1}{\kappa - 1}}, \quad \frac{a}{a\_{\infty}} = \left[\mathbf{1} + \frac{\kappa - 1}{2} M\_{\infty 2} \left(\mathbf{1} - \overline{u}^2 - \overline{v}^2\right)\right]^{\frac{1}{2}},\tag{8}$$

here *u* ¼ *u=U*∞, *v* ¼ *v=U*<sup>∞</sup> are the dimensionless components of the local velocity, *U*<sup>∞</sup> is free-stream flow velocity at infinity.

Let us write the well-known isentropic relations for density and sound speed

$$\frac{\rho\_0}{\rho} = \left[1 + \frac{\kappa - 1}{2}M^2\right]^{\frac{1}{\kappa - 1}}, \quad \frac{a\_0}{a} = \left[1 + \frac{\kappa - 1}{2}M^2\right]^{\frac{1}{2}},\tag{9}$$

here lower index "0" specifies parameters in stagnation point. Formulas (9) can be written as

$$\frac{\rho\_0}{\rho} = [E(M)]^{\frac{1}{\kappa - 1}}, \quad \frac{a\_0}{a} = [E(M)]^{\frac{1}{\kappa}}, \quad E(M) = \mathbf{1} + \frac{\kappa - 1}{2}M^2. \tag{10}$$

The speed of sound is defined by formula *<sup>a</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi *dp=dρ* p . The speed of sound in the case of an isentropic air flow is *<sup>a</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi *κp=ρ* p . Based on the equation of state in the form (Eq. (4)), we can write

$$\mathfrak{a} = \mathfrak{a}\_{\infty} \left( \frac{\rho}{\rho\_{\infty}} \right)^{\frac{\kappa - 1}{2}}. \tag{11}$$

Based on the isentropic relationship for density (Eq. (9)), the function *σ* (Eq. (2)) can be calculated. **Figure 1** shows a graph of the function *σ* for air versus Mach number.

**Figure 1.** *The function σ vs. Mach number.*

**2. The approximate method for calculating flow compressibility**

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> 0, *<sup>∂</sup> <sup>r</sup><sup>n</sup>* ð Þ *<sup>ρ</sup><sup>u</sup>*

*<sup>n</sup>* <sup>¼</sup> <sup>0</sup> � 2D body;

(*n* = 1). Let us introduce the special functions proposed by Burago [20]

<sup>¼</sup> 0, *<sup>∂</sup> ∂x*

> *ρ ρ*∞

<sup>¼</sup> *<sup>p</sup> p*∞ � �<sup>1</sup> *κ*

where *p* is static pressure, *Pa*; *κ* is ratio of specific heats (for air *κ* = 1.4); ∞ subscript indicates flow parameters at infinity. The equation (Eq. (4)) refers to as

According to the accepted model of a barotropic gas, the enthalpy and pressure function differ only by a constant value *h* = *P*(*p*) + const. The pressure

*P p*ð Þ¼

*<sup>h</sup>* <sup>¼</sup> *<sup>ρ</sup><sup>κ</sup>*�<sup>1</sup>

Using the equation for enthalpy (Eq. (6)), the Bernoulli equation for a

ð*p p*∞

If we carry out the integration in formula (Eq. (5)) taking into account formula

ð Þ *<sup>κ</sup>* � <sup>1</sup> *ρκ*�<sup>1</sup> <sup>∞</sup> *<sup>M</sup>*<sup>2</sup>

∞

*dp*

�

*<sup>τ</sup>* <sup>¼</sup> <sup>2</sup> *ρ*0*=ρ* þ 1

*τu* 1 � *σ* � �

For barotropic model of compressible air we have

The equations for the existence of the velocity potential and continuity are written for two cases: two-dimensional and axisymmetric for compressible irrota-

*∂x*

where *u*, *v* are components of the velocity along axis *x* и *r* accordingly, *m/s*;, *r, x*

1 � axisymmetric body*:*

; *<sup>σ</sup>* <sup>¼</sup> *<sup>ρ</sup>*0*=<sup>ρ</sup>* � <sup>1</sup> *ρ*0*=ρ* þ 1

> *rnτu* 1 þ *σ* � �

þ *∂ ∂r*

*r<sup>n</sup>τv* 1 þ *σ* � �

, (4)

*<sup>ρ</sup> :* (5)

, (6)

*ρ*<sup>0</sup> is gas density at the stagnation point. Taking into account equation (Eq. (2)),

Here *x*, *r* are coordinates in the meridional plane for the axisymmetric case

þ

*<sup>∂</sup> <sup>r</sup><sup>n</sup>* ð Þ *<sup>ρ</sup><sup>v</sup>*

; parameter *n* equal

*<sup>∂</sup><sup>r</sup>* <sup>¼</sup> 0, (1)

, (2)

¼ 0, (3)

**characteristics**

*∂u <sup>∂</sup><sup>r</sup>* � *<sup>∂</sup><sup>v</sup>*

are coordinates; *ρ* is local density of gas, *kg/m<sup>3</sup>*

the equation (Eq. (1)) can be rewritten as

� *∂ ∂x*

tional gas flow

*Aerodynamics*

*∂ ∂r*

*τu* 1 � *σ* � �

Poisson's adiabatic curve.

(Eq. (4)), then we get (Eq. (6))

compressible gas can be written as

here *M*<sup>∞</sup> is Mach number at infinity.

function is

**172**

#### *Aerodynamics*

In **Figure 1** it is seen that the values *σ* lie in a narrow range [0; 0.22] for air when changing the Mach number from 0 to 1.0. This narrow range allows us to take σ ≈ const, and then using formulas (Eq. (2)), we can replace (Eq. (3)) with an approximate

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{r^n \pi u}{\rho\_0 + \rho} \right) + \frac{\partial}{\partial r} \left( \frac{r^n \pi v}{\rho\_0 + \rho} \right) = \mathbf{0}, \quad \frac{\partial}{\partial r} \left( \frac{\pi u}{\rho\_0 + \rho} \right) - \frac{\partial}{\partial \mathbf{x}} \left( \frac{\pi u}{\rho\_0 + \rho} \right) = \mathbf{0}. \tag{12}$$

If we introduce new variables *u*<sup>o</sup> and *v*<sup>o</sup> associated with u and *v* as follows

$$u = \frac{\rho\_{\text{os}}}{\rho} \frac{(\rho + \rho\_0)}{(\rho\_{\text{os}} + \rho\_0)} u^0 = \frac{\mathbf{1} + \frac{\rho\_0}{\rho}}{\mathbf{1} + \frac{\rho\_0}{\rho\_{\text{os}}}} u^0 = \eta\_c u^0; \quad v = \frac{\rho\_{\text{os}}}{\rho} \frac{(\rho + \rho\_0)}{(\rho\_{\text{os}} + \rho\_0)} v^0 = \frac{\mathbf{1} + \frac{\rho\_0}{\rho}}{\mathbf{1} + \frac{\rho\_0}{\rho\_{\text{os}}}} v^0 = \eta\_c v^0. \tag{13}$$

where *η*<sup>c</sup> is introduced

$$\eta\_c = \left(\mathbf{1} + \frac{\rho\_0}{\rho}\right) / \left(\mathbf{1} + \frac{\rho\_0}{\rho\_\infty}\right). \tag{14}$$

From (Eq. (17)) and (Eq. (18)) it follows

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

until the following inequality is satisfied *η*

From (Eq. (18) and (19)) we can write

*<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>2</sup> <sup>þ</sup> *<sup>κ</sup>*�<sup>1</sup>

incompressible flow. The critical Mach number *M*ð Þ<sup>1</sup>

calculating the critical Mach number

*M*<sup>∗</sup> ¼

**175**

*M* ¼ *M*<sup>∞</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>u</sup>*ð Þ<sup>1</sup> <sup>¼</sup> *<sup>u</sup>*0*=U*∞; *<sup>v</sup>*ð Þ<sup>1</sup> <sup>¼</sup> *<sup>v</sup>*0*=U*∞*:* (20)

*:* (19)

<sup>1</sup> <sup>þ</sup> *<sup>κ</sup>*�<sup>1</sup> <sup>2</sup> *<sup>M</sup>*<sup>2</sup> <sup>∞</sup> <sup>1</sup> � *<sup>u</sup>*<sup>2</sup> � *<sup>v</sup>*<sup>2</sup> � � <sup>s</sup>

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

which we will substitute in (Eq. (19)) at the first stage of approximation

Equation (Eq. 19) shows that the local Mach number *M* is determined by the local dimensionless velocities *u* and *v*, which, according equations (Eq. 13) and (Eq. 10) depends on local Mach number. To account for this, we will use successive approximations. We will use the velocities calculated for an incompressible flow,

Thus, based on these values, the local Mach number *M*(1) for the first approximation is determined. Thereafter, the computed first approximation Mach number *M*(1) is used to calculate the second approximation velocity components from *u*ð Þ<sup>2</sup> , *v*ð Þ<sup>2</sup> from (Eqs. (13) and (14)), and isentropic formula for density (Eq. (10)). The obtained values of the velocity components make it possible to calculate on the basis (Eq. (19)), the local Mach number for the second approximation *M*(2), which, in turn, is used to calculate the local velocity components at the stage of the third approximation *u*ð Þ<sup>3</sup> , *v*ð Þ<sup>3</sup> , and so on. The approximation process must be continued

> ð Þ *n <sup>c</sup>* � *η*

Putting in the formula (Eq. (21)) *M* = 1 and *M*<sup>∞</sup> = *M*\*, we can get the formula for

�

where the notation *U* is introduced for the local total relative velocity. From (Eq. (22)) it can be seen that the minimal value *M*\* corresponds to the maximal value *U*. Therefore, to calculate the critical Mach number for a flow near a body, it is necessary to determine the maximum local velocity on the body surface. The dimensionless velocities *u* and *v* (Eq. (22)) also should correspond to the critical free-stream Mach number. For this, it is necessary to apply a method of successive approximations, similar to that used to calculate the local Mach number (Eq. (19)). As a first approximation, we will use the velocities *u*ð Þ<sup>1</sup> and *v*ð Þ<sup>1</sup> calculated for an

approximation should be considered as the free-stream Mach number for calculating the local relative velocities for the second approximation *u*ð Þ<sup>2</sup> and *v*ð Þ<sup>2</sup> . The approximation process must be continued until the calculated critical Mach number for subsequent approximations changes by a given error *ε*, i.e., until the

� � �

is a given calculation error. Preliminary calculations using the described algorithm showed that the number of approximation steps increases as *M* ! 1.0, but usually

the maximum number of approximations does not exceed 30.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>2</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>κ</sup>*�<sup>1</sup> <sup>2</sup> *<sup>M</sup>*<sup>2</sup> ∞ <sup>q</sup> <sup>¼</sup> *<sup>M</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*M*<sup>∞</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

<sup>2</sup> *<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>2</sup> � <sup>1</sup> � � <sup>s</sup>

ð Þ *<sup>n</sup>*�<sup>1</sup> *<sup>c</sup>*

<sup>1</sup> <sup>þ</sup> *<sup>κ</sup>*�<sup>1</sup> <sup>2</sup> *<sup>M</sup>*<sup>2</sup>

> *U*2 <sup>þ</sup> *<sup>κ</sup>*�<sup>1</sup> <sup>2</sup> *<sup>U</sup>*<sup>2</sup>

� � � � *<sup>η</sup>*

� � �

ð Þ *<sup>n</sup>*�<sup>1</sup> *<sup>c</sup>* � *<sup>η</sup>*

<sup>q</sup> *:* (21)

� 1 � � vuut , (22)

<sup>∗</sup> attained in the first

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>

ð Þ *<sup>n</sup>*�<sup>2</sup> *<sup>c</sup>*

� �

�≤*ε*, where *<sup>ε</sup>*

The coefficient *η*<sup>c</sup> will be called the coefficient of compressibility. The new speeds (Eq. (13)) allow equations (Eq. (12)) to be written in the form

$$\frac{\partial u^{0}}{\partial r} - \frac{\partial v^{0}}{\partial \mathbf{x}} = \mathbf{0}, \quad \frac{\partial (r^{n} u^{0})}{\partial \mathbf{x}} + \frac{\partial (r^{n} v^{0})}{\partial r} = \mathbf{0}. \tag{15}$$

Equations (Eq. (15)) repeat equations (Eq. (1)) describing potential flow of an incompressible liquid (*ρ* ≈ *const*) with a velocity having components (*u*<sup>0</sup> , *v*<sup>0</sup> ). Equations (Eq. (15)) allow us to assert that the boundary conditions at infinity for a compressible gas flow with velocity components (*u*, *v*) will be identical to the corresponding conditions for an incompressible fluid flow with velocity components (*u*<sup>0</sup> , *v*<sup>0</sup> ). The boundary conditions on the body surface will also be the same. Indeed, in a compressible flow on the surface of a body *vn* = 0. On the body surface in an incompressible fluid flow, based on (Eq. (13)), we can write

$$
\upsilon\_n^0 = \frac{\rho}{\rho\_\infty} \frac{(\rho\_\infty + \rho\_0)}{(\rho + \rho\_0)} \upsilon\_n = \mathbf{0}.\tag{16}
$$

Thus, the equations (Eq. (13)) determine the approximate relationship between the components of the velocities in compressible and incompressible flows around the same body under the same conditions at infinity and on the surface of the body. So, to determine velocities of compressible gas flow, it is necessary to use the known isentropic relationships for the local velocity of a flow and density

$$\frac{|\mathbf{V}|}{a\_0} = \frac{M}{\sqrt{1 + \frac{\kappa - 1}{2}M^2}}, \quad \frac{\rho\_0}{\rho\_\infty} = \left[1 + \frac{\kappa - 1}{2}M\_\infty^2\right]^\frac{1}{\kappa - 1} \tag{17}$$

here M is the local Mach number at an arbitrary point. The left side of the first formula (Eq. 17) can be written as

$$\frac{|\mathbf{V}|}{a\_0} = \frac{U\_{\infty}a\_{\infty}\sqrt{\overline{u}^2 + \overline{v}^2}}{a\_{\infty}a\_0} = \frac{M\_{\infty}\sqrt{\overline{u}^2 + \overline{v}^2}}{\sqrt{1 + \frac{\kappa - 1}{2}M\_{\infty}^2}}.\tag{18}$$

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

From (Eq. (17)) and (Eq. (18)) it follows

In **Figure 1** it is seen that the values *σ* lie in a narrow range [0; 0.22] for air when

*τu ρ*<sup>0</sup> þ *ρ* � �

*ρ*

*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup> *ρ*∞ � �

þ

). The boundary conditions on the body surface will also be the same.

*<sup>∂</sup> <sup>r</sup>nv*<sup>0</sup> ð Þ

� *∂ ∂x*

*ρ* þ *ρ*<sup>0</sup> ð Þ

*τu ρ*<sup>0</sup> þ *ρ* � �

*<sup>ρ</sup>*<sup>∞</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup> ð Þ *<sup>v</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup>

¼ 0*:* (12)

*<sup>v</sup>*<sup>0</sup> <sup>¼</sup> *<sup>η</sup>cv*<sup>0</sup>*:*

(13)

*ρ* <sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup> *ρ*∞

*:* (14)

*<sup>∂</sup><sup>r</sup>* <sup>¼</sup> <sup>0</sup>*:* (15)

*<sup>ρ</sup>* <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup> ð Þ *vn* <sup>¼</sup> <sup>0</sup>*:* (16)

, *v*<sup>0</sup> ).

changing the Mach number from 0 to 1.0. This narrow range allows us to take σ ≈ const, and then using formulas (Eq. (2)), we can replace (Eq. (3)) with an

> <sup>¼</sup> 0, *<sup>∂</sup> ∂r*

*<sup>η</sup><sup>c</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup>

speeds (Eq. (13)) allow equations (Eq. (12)) to be written in the form

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> 0, *<sup>∂</sup> rnu*<sup>0</sup> ð Þ

incompressible liquid (*ρ* ≈ *const*) with a velocity having components (*u*<sup>0</sup>

in an incompressible fluid flow, based on (Eq. (13)), we can write

isentropic relationships for the local velocity of a flow and density

<sup>¼</sup> *<sup>M</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>κ</sup>*�<sup>1</sup> <sup>2</sup> *<sup>M</sup>*<sup>2</sup> <sup>q</sup> , *<sup>ρ</sup>*<sup>0</sup>

<sup>¼</sup> *<sup>U</sup>*∞*a*<sup>∞</sup>

j j **V** *a*0

> j j **V** *a*0

formula (Eq. 17) can be written as

*v*0 *<sup>n</sup>* <sup>¼</sup> *<sup>ρ</sup> ρ*∞

If we introduce new variables *u*<sup>o</sup> and *v*<sup>o</sup> associated with u and *v* as follows

*<sup>u</sup>*<sup>0</sup> <sup>¼</sup> *<sup>η</sup>cu*0; *<sup>v</sup>* <sup>¼</sup> *<sup>ρ</sup>*<sup>∞</sup>

*ρ* � �

The coefficient *η*<sup>c</sup> will be called the coefficient of compressibility. The new

*∂x*

Equations (Eq. (15)) repeat equations (Eq. (1)) describing potential flow of an

Equations (Eq. (15)) allow us to assert that the boundary conditions at infinity for a compressible gas flow with velocity components (*u*, *v*) will be identical to the corresponding conditions for an incompressible fluid flow with velocity compo-

Indeed, in a compressible flow on the surface of a body *vn* = 0. On the body surface

*ρ*<sup>∞</sup> þ *ρ*<sup>0</sup> ð Þ

Thus, the equations (Eq. (13)) determine the approximate relationship between the components of the velocities in compressible and incompressible flows around the same body under the same conditions at infinity and on the surface of the body. So, to determine velocities of compressible gas flow, it is necessary to use the known

*ρ*∞

here M is the local Mach number at an arbitrary point. The left side of the first

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>2</sup> <sup>p</sup> *a*∞*a*<sup>0</sup>

¼ 1 þ

<sup>¼</sup> *<sup>M</sup>*<sup>∞</sup>

*κ* � 1 2

� � <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>2</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>κ</sup>*�<sup>1</sup> <sup>2</sup> *<sup>M</sup>*<sup>2</sup> ∞

*M*<sup>2</sup> ∞ *κ*�1

<sup>q</sup> *:* (18)

, (17)

approximate

*Aerodynamics*

*∂ ∂x*

*<sup>u</sup>* <sup>¼</sup> *<sup>ρ</sup>*<sup>∞</sup> *ρ*

nents (*u*<sup>0</sup>

**174**

, *v*<sup>0</sup>

*rnτu ρ*<sup>0</sup> þ *ρ* � �

*ρ* þ *ρ*<sup>0</sup> ð Þ

where *η*<sup>c</sup> is introduced

þ *∂ ∂r*

*<sup>ρ</sup>*<sup>∞</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup> ð Þ *<sup>u</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup>

*∂u*<sup>0</sup> *<sup>∂</sup><sup>r</sup>* � *<sup>∂</sup>v*<sup>0</sup>

*rnτv ρ*<sup>0</sup> þ *ρ* � �

> *ρ* <sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup> *ρ*∞

$$M = M\_{\infty} \sqrt{\frac{\overline{u}^2 + \overline{v}^2}{1 + \frac{\kappa - 1}{2} M\_{\infty}^2 \left(1 - \overline{u}^2 - \overline{v}^2\right)}}. \tag{19}$$

Equation (Eq. 19) shows that the local Mach number *M* is determined by the local dimensionless velocities *u* and *v*, which, according equations (Eq. 13) and (Eq. 10) depends on local Mach number. To account for this, we will use successive approximations. We will use the velocities calculated for an incompressible flow, which we will substitute in (Eq. (19)) at the first stage of approximation

$$
\overline{u}^{(1)} = u^0 / U\_{\infty}; \quad \overline{v}^{(1)} = v^0 / U\_{\infty}. \tag{20}
$$

Thus, based on these values, the local Mach number *M*(1) for the first approximation is determined. Thereafter, the computed first approximation Mach number *M*(1) is used to calculate the second approximation velocity components from *u*ð Þ<sup>2</sup> , *v*ð Þ<sup>2</sup> from (Eqs. (13) and (14)), and isentropic formula for density (Eq. (10)). The obtained values of the velocity components make it possible to calculate on the basis (Eq. (19)), the local Mach number for the second approximation *M*(2), which, in turn, is used to calculate the local velocity components at the stage of the third approximation *u*ð Þ<sup>3</sup> , *v*ð Þ<sup>3</sup> , and so on. The approximation process must be continued until the following inequality is satisfied *η* ð Þ *n <sup>c</sup>* � *η* ð Þ *<sup>n</sup>*�<sup>1</sup> *<sup>c</sup>* � � � � � � � *<sup>η</sup>* ð Þ *<sup>n</sup>*�<sup>1</sup> *<sup>c</sup>* � *<sup>η</sup>* ð Þ *<sup>n</sup>*�<sup>2</sup> *<sup>c</sup>* � � � � � �≤*ε*, where *<sup>ε</sup>* is a given calculation error. Preliminary calculations using the described algorithm showed that the number of approximation steps increases as *M* ! 1.0, but usually the maximum number of approximations does not exceed 30.

From (Eq. (18) and (19)) we can write

$$\frac{M\_{\infty}\sqrt{\overline{u}^{2}+\overline{v}^{2}}}{\sqrt{1+\frac{\kappa-1}{2}M\_{\infty}^{2}}}=\frac{M}{\sqrt{1+\frac{\kappa-1}{2}M^{2}}}.\tag{21}$$

Putting in the formula (Eq. (21)) *M* = 1 and *M*<sup>∞</sup> = *M*\*, we can get the formula for calculating the critical Mach number

$$M\_{\*} = \sqrt{\frac{1}{\overline{u}^{2} + \overline{v}^{2} + \frac{\kappa - 1}{2} \left(\overline{u}^{2} + \overline{v}^{2} - 1\right)}} \equiv \sqrt{\frac{1}{\overline{U}^{2} + \frac{\kappa - 1}{2} \left(\overline{U}^{2} - 1\right)}},\tag{22}$$

where the notation *U* is introduced for the local total relative velocity. From (Eq. (22)) it can be seen that the minimal value *M*\* corresponds to the maximal value *U*. Therefore, to calculate the critical Mach number for a flow near a body, it is necessary to determine the maximum local velocity on the body surface. The dimensionless velocities *u* and *v* (Eq. (22)) also should correspond to the critical free-stream Mach number. For this, it is necessary to apply a method of successive approximations, similar to that used to calculate the local Mach number (Eq. (19)). As a first approximation, we will use the velocities *u*ð Þ<sup>1</sup> and *v*ð Þ<sup>1</sup> calculated for an incompressible flow. The critical Mach number *M*ð Þ<sup>1</sup> <sup>∗</sup> attained in the first approximation should be considered as the free-stream Mach number for calculating the local relative velocities for the second approximation *u*ð Þ<sup>2</sup> and *v*ð Þ<sup>2</sup> . The approximation process must be continued until the calculated critical Mach number for subsequent approximations changes by a given error *ε*, i.e., until the

condition is satisfied *M*ð Þ*<sup>i</sup>* <sup>∗</sup> � *<sup>M</sup>*ð Þ *<sup>i</sup>*�<sup>1</sup> <sup>∗</sup> ≤ *ε*. It turns out that the critical Mach number can be calculated using only the values of the incompressible flow speeds. It should be noted that the method described above is applicable for compressible flows for which the inequality *M*<sup>∞</sup> ≤ *M*<sup>∗</sup> is valid. This condition determines the absence of transonic and supersonic zones in the flow field. To calculate the critical Mach number *M*<sup>∗</sup> and the velocity field (*u, v*) in a compressible flow, it is sufficient to calculate the velocity field (*u*<sup>0</sup> , *v*<sup>0</sup> ) around the body for an incompressible flow.

It follows from the described method that the critical Mach number for a given 2-D body can be calculated using only the values of the incompressible flow speeds. The method is applicable for compressible flows for which the free-stream Mach numbers are less than the critical Mach number. So, to calculate the critical Mach number, for an incompressible flow, it is sufficient to calculate the velocity field near the streamlined body, but in practical aerodynamics, the pressure distribution on the body surface is obtained experimentally more often than the velocity field. To compare the results of calculations and experimental data, it is convenient to use the pressure-drop coefficient. This circumstance leads to the necessity of obtaining a connection between the pressure field in a compressible flow and the pressure field in an incompressible flow. To establish this connection we write down the pressure-drop coefficient for an incompressible flow

$$c\_p^0 = \frac{p^0 - p\_{\infty}}{0, 5\rho V\_{\infty}^2} = 1 - \left(\frac{V^0}{V\_{\infty}}\right)^2. \tag{23}$$

The use of isentropic formula (Eq. (13)) allows us to write the last equation in

Substitution of the found ratio of the squares of the velocities into the formula

*E M*ð Þ <sup>∞</sup> *E M*ð Þ *:*

*E M*ð Þ <sup>∞</sup>

*<sup>κ</sup>*�1, *p*0*=p*<sup>∞</sup> ¼ ½ � *E M*ð Þ <sup>∞</sup>

*κ*�1 � 1

0 *p*

> 0 *p*

*E M*ð Þ � <sup>1</sup> <sup>þ</sup> ½ � *E M*ð Þ <sup>1</sup>

( )

we write down the pressure-drop coefficient of the compressible gas (Eq. (26))

*E M*ð Þ <sup>∞</sup> *E M*ð Þ � � *<sup>κ</sup>*

In order to calculate the pressure coefficient for a compressible gas from the known value of the incompressible flow coefficient, we introduce auxiliary func-

Using the new functions, the formulas for the pressure-drop coefficients of an

*F M*ð Þ <sup>∞</sup> *F M*ð Þ � <sup>1</sup>

In order to recalculate the values of the pressure-drop coefficient of the incompressible flow to the pressure-drop coefficient of the compressible gas for a given value of the Mach number of the free-stream flow, it is necessary to solve the transcendental Eq. (30) with respect to the local Mach number, and then use the formulas (31) and (29) to calculate the pressure-drop coefficient of the compressible gas.

The dependence of the critical Mach number on the pressure-drop coefficient of the incompressible flow can be obtained. For this, it is necessary to solve Eq. (30)

The presented method is approximate; therefore, it is necessary to demonstrate the consistency of this method with other calculation methods in order to analyze

*<sup>κ</sup>*�1, *G M*ð Þ¼ *<sup>M</sup>*<sup>2</sup>

incompressible flow and a compressible gas can be represented in the form

*G M*ð Þ� *G M*ð Þ <sup>∞</sup> 1 � *c*

*cp* <sup>¼</sup> <sup>2</sup> *κM*<sup>2</sup> ∞

with respect to the condition *M* = 1.0, i.e. solve an equation of the form

*G*ð Þ� 1*:*0 *G M*ð Þ <sup>∗</sup> 1 � *c*

*E M*ð Þ *:* (27)

*:* (28)

*:* (29)

*κ*�1 n o�<sup>2</sup>

� � <sup>¼</sup> 0, (30)

� � <sup>¼</sup> <sup>0</sup>*:* (32)

� �*:* (31)

*κ <sup>κ</sup>*�1,

*V*2 *V*2 ∞

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

(Eq. (24)) for the pressure coefficient of the incompressible flow gives

*c* 0 *<sup>p</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>η</sup>*<sup>2</sup> *c M*<sup>2</sup> *M*<sup>2</sup> ∞ �

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

*<sup>p</sup>*0*=<sup>p</sup>* <sup>¼</sup> ½ � *E M*ð Þ *<sup>κ</sup>*

*cp* <sup>¼</sup> <sup>2</sup> *κM*<sup>2</sup> ∞

Using isentropic formulas for pressure ratios in the form

<sup>¼</sup> *<sup>M</sup>*<sup>2</sup> *M*<sup>2</sup> ∞ �

the form

as follows

tions proposed by G. Burago [20].

**3. Results and discussions**

the accuracy of the calculations.

**177**

*F M*ð Þ¼ ½ � *E M*ð Þ *<sup>κ</sup>*

Based on formulas (Eq. (13)), the pressure-drop coefficient for an incompressible flow can be represented through the gas flow velocity

$$c\_p^0 = 1 - \left(\frac{V}{\eta\_c V\_\infty}\right)^2. \tag{24}$$

The pressure-drop coefficient for the compressible gas flow can be written

$$\mathcal{L}\_p = \frac{p - p\_{\text{os}}}{0.5 \rho\_{\text{os}} V\_{\text{os}}^2} = 2 \frac{p\_{\text{os}}}{\rho\_{\text{os}} V\_{\text{os}}^2} \left(\frac{p}{p\_{\text{os}}} - 1\right) \tag{25}$$

Let us rewrite the last formula in terms of the Mach number of the free-stream flow. Using the barotropic model of compressed air, we can write the following equalities

$$\frac{p\_{\infty}}{\rho\_{\infty}V\_{\infty}^{2}} = \frac{1}{\kappa} \frac{a\_{\infty}^{2}}{V\_{\infty}^{2}} = \frac{1}{\kappa M\_{\infty}^{2}},$$

then we rewrite formula (Eq. (25)) in the form

$$c\_p = \frac{2}{\kappa M\_{\infty}^2} \left(\frac{p}{p\_{\infty}} - 1\right) = \frac{2}{\kappa M\_{\infty}^2} \left(\frac{p}{p\_0} \frac{p\_0}{p\_{\infty}} - 1\right). \tag{26}$$

In formula (Eq. (24)), the ratio of the squares of the velocities can be represented as the identity

$$\frac{V^2}{V\_{\infty}^2} = \frac{V^2}{a^2} \frac{a^2}{a\_0^2} \frac{a\_0^2}{a\_{\infty}^2} \frac{a\_{\infty}^2}{V\_{\infty}^2} = \frac{M^2}{M\_{\infty}^2} \frac{a^2}{a\_0^2} \frac{a\_0^2}{a\_{\infty}^2}.$$

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

The use of isentropic formula (Eq. (13)) allows us to write the last equation in the form

$$\frac{V^2}{V\_{\infty}^2} = \frac{M^2}{M\_{\infty}^2} \cdot \frac{E(M\_{\infty})}{E(M)} \dots$$

Substitution of the found ratio of the squares of the velocities into the formula (Eq. (24)) for the pressure coefficient of the incompressible flow gives

$$c\_p^0 = 1 - \eta\_c^2 \frac{M^2}{M\_{\infty}^2} \cdot \frac{E(M\_{\infty})}{E(M)}.\tag{27}$$

Using isentropic formulas for pressure ratios in the form

$$p\_0/p = [E(\mathcal{M})]^{\frac{\kappa}{\kappa - 1}}, \quad p\_0/p\_\infty = [E(\mathcal{M}\_\infty)]^{\frac{\kappa}{\kappa - 1}},$$

we write down the pressure-drop coefficient of the compressible gas (Eq. (26)) as follows

$$c\_p = \frac{2}{\kappa M\_{\infty}^2} \left\{ \left[ \frac{E(M\_{\infty})}{E(M)} \right]^{\frac{\kappa}{\kappa - 1}} - 1 \right\}. \tag{28}$$

In order to calculate the pressure coefficient for a compressible gas from the known value of the incompressible flow coefficient, we introduce auxiliary functions proposed by G. Burago [20].

$$F(M) = [E(M)]^{\frac{r}{r-1}}, \quad G(M) = \frac{M^2}{E(M)} \cdot \left\{ \mathbf{1} + [E(M)]^{\frac{1}{r-1}} \right\}^{-2}.\tag{29}$$

Using the new functions, the formulas for the pressure-drop coefficients of an incompressible flow and a compressible gas can be represented in the form

$$\mathbf{G(M)} - \mathbf{G(M\_{\Leftrightarrow})} \left(\mathbf{1} - c\_p^0 \right) = \mathbf{0},\tag{30}$$

$$c\_p = \frac{2}{\kappa \mathcal{M}\_{\infty}^2} \left[ \frac{F(\mathcal{M}\_{\infty})}{F(\mathcal{M})} - 1 \right]. \tag{31}$$

In order to recalculate the values of the pressure-drop coefficient of the incompressible flow to the pressure-drop coefficient of the compressible gas for a given value of the Mach number of the free-stream flow, it is necessary to solve the transcendental Eq. (30) with respect to the local Mach number, and then use the formulas (31) and (29) to calculate the pressure-drop coefficient of the compressible gas.

The dependence of the critical Mach number on the pressure-drop coefficient of the incompressible flow can be obtained. For this, it is necessary to solve Eq. (30) with respect to the condition *M* = 1.0, i.e. solve an equation of the form

$$(G(\mathbf{1}.\mathbf{0}) - G(M\_\* \,) \left(\mathbf{1} - \mathbf{c}\_p^0\right) = \mathbf{0}.\tag{32}$$

## **3. Results and discussions**

The presented method is approximate; therefore, it is necessary to demonstrate the consistency of this method with other calculation methods in order to analyze the accuracy of the calculations.

condition is satisfied *M*ð Þ*<sup>i</sup>*

pressible flow.

*Aerodynamics*

equalities

**176**

<sup>∗</sup> � *<sup>M</sup>*ð Þ *<sup>i</sup>*�<sup>1</sup> <sup>∗</sup>

number can be calculated using only the values of the incompressible flow speeds. It should be noted that the method described above is applicable for compressible flows for which the inequality *M*<sup>∞</sup> ≤ *M*<sup>∗</sup> is valid. This condition determines the absence of transonic and supersonic zones in the flow field. To calculate the critical Mach number *M*<sup>∗</sup> and the velocity field (*u, v*) in a compressible flow, it is

, *v*<sup>0</sup>

<sup>¼</sup> <sup>1</sup> � *<sup>V</sup>*<sup>0</sup>

Based on formulas (Eq. (13)), the pressure-drop coefficient for an incompress-

*ηcV*<sup>∞</sup> <sup>2</sup>

*<sup>p</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>V</sup>*

The pressure-drop coefficient for the compressible gas flow can be written

<sup>¼</sup> <sup>2</sup> *<sup>p</sup>*<sup>∞</sup> *ρ*∞*V*<sup>2</sup> ∞

Let us rewrite the last formula in terms of the Mach number of the free-stream flow. Using the barotropic model of compressed air, we can write the following

> <sup>¼</sup> <sup>1</sup> *κM*<sup>2</sup> ∞ ,

<sup>¼</sup> <sup>2</sup> *κM*<sup>2</sup> ∞

*V*<sup>∞</sup> <sup>2</sup>

> *p p*∞ � 1

*p p*0 *p*0 *p*∞ � 1 

> *a*2 *a*2 0

*a*2 0 *a*2 ∞ *:*

It follows from the described method that the critical Mach number for a given 2-D body can be calculated using only the values of the incompressible flow speeds. The method is applicable for compressible flows for which the free-stream Mach numbers are less than the critical Mach number. So, to calculate the critical Mach number, for an incompressible flow, it is sufficient to calculate the velocity field near the streamlined body, but in practical aerodynamics, the pressure distribution on the body surface is obtained experimentally more often than the velocity field. To compare the results of calculations and experimental data, it is convenient to use the pressure-drop coefficient. This circumstance leads to the necessity of obtaining a connection between the pressure field in a compressible flow and the pressure field in an incompressible flow. To establish this connection we write down the

≤ *ε*. It turns out that the critical Mach

) around the body for an incom-

*:* (23)

*:* (24)

(25)

*:* (26)

sufficient to calculate the velocity field (*u*<sup>0</sup>

pressure-drop coefficient for an incompressible flow

*c* 0

ible flow can be represented through the gas flow velocity

*c* 0

*cp* <sup>¼</sup> *<sup>p</sup>* � *<sup>p</sup>*<sup>∞</sup> 0*:*5*ρ*∞*V*<sup>2</sup> ∞

> *p*∞ *ρ*∞*V*<sup>2</sup> ∞ ¼ 1 *κ a*2 ∞ *V*2 ∞

*p p*∞ � 1 

In formula (Eq. (24)), the ratio of the squares of the velocities can be

*a*2 0 *a*2 ∞

*a*2 ∞ *V*2 ∞ <sup>¼</sup> *<sup>M</sup>*<sup>2</sup> *M*<sup>2</sup> ∞

then we rewrite formula (Eq. (25)) in the form

*V*2 *V*2 ∞ ¼ *V*2 *a*2 *a*2 *a*2 0

*cp* <sup>¼</sup> <sup>2</sup> *κM*<sup>2</sup> ∞

represented as the identity

*<sup>p</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>0</sup> � *<sup>p</sup>*<sup>∞</sup> 0, 5*ρV*<sup>2</sup> ∞

The calculation of the dependence of the critical Mach number on the pressuredrop coefficient of an incompressible flow using Eq. (30) is shown in **Figure 2**. **Figure 2** also shows the Khristianovich's curve [13].

Comparison of the two calculations shows good agreement. After the publication of the work of Burago [20] in 1949, a number of approximate methods based on the Chaplygin gas model were developed. Burago's method refers to the approximate mathematical models for accounting for the compressibility of the flow. The effectiveness of approximate mathematical models is always shown by comparison with experimental or calculated data for various bodies. Calculations carried out according to the method described above showed that this method is not inferior in accuracy to more rigorous methods, at the same time the advantage of this method over other methods of accounting for the flow compressibility in terms of speeds is undeniable.

Here are some comparisons. In **Figure 3** shows a comparison of the calculation by the method described above with the calculated data of L. Sedov [5] and by the Glauert formulas ([4], p. 311) and Kárman-Tsien ([4], p. 311). Glauert formula is

$$\mathbf{C}\_{p} = \frac{\mathbf{C}\_{p}^{0}}{\sqrt{\mathbf{1} - \mathbf{M}\_{\infty}^{2}}},\tag{33}$$

of the chapter by the method outlined above [1]. It should be noted that the calculated data by Burago's method for all three considered cases are closer to the experimental data of G. Stack [19] than the results of the calculation by the Sedov's

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

the Kárman-Tsien method [see. Formula (35)] is based on recalculating the

pressure-drop coefficients of an incompressible liquid *C*<sup>0</sup>

*Comparison of different methods account compressibility.*

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

increases with decreasing value *C*<sup>0</sup>

pressure coefficients reaches 15–16%.

Accounting for gas compressibility according to the method described above and

pressure-drop coefficients of a compressible gas *Cp* for the given free-stream Mach numbers and ratio of specific heats*κ*. An interesting question is: how much differ these two methods in a wide range of variation of Mach numbers and pressure coefficients for an incompressible fluid? **Table 1** compares these two methods based on the data in **Figure 3**, from which it follows that the maximum discrepancy

of the two methods in the considered range of variation of Mach numbers and

The advantage of the Burago method over the Kárman-Tsien method is demonstrated in **Figure 4**, which presents the calculated data and experimental data of various authors for two bodies: an ellipse (**Figure 4a**) and a biconvex airfoil

(**Figure 4b**) with relative thicknesses of 20% and 6% respectively. **Figure 4a** shows a comparison of the results of calculating the pressure drop coefficient on the surface of an ellipse of relative thickness *δ* = 0.2 in a compressible flow. The calculations are performed for the critical Mach number *M*\* = 0.7 (**Table 2**). The calculated data using the Sells finite difference method (1968) are taken from [16]. It can be noted that the results of the Burago method and the Kárman-Tsien (34) are in good agreement to within approximate values *x*≤0*:*1. For values of relative coordinate along the big axis of an ellipse 0*:*1≤*x*≤ 0*:*5 the significant divergence between the results of the Kárman-Tsien formula (34) and Burago's

*<sup>p</sup>* by the values of the

*<sup>p</sup>* and at *M*<sup>∞</sup> ! *M*<sup>∗</sup> . The maximum relative error

method [5].

**179**

**Figure 3.**

and the Kárman-Tsien formula is

$$\mathcal{C}\_{p} = \frac{\mathcal{C}\_{p}^{0}}{\sqrt{\mathbf{1} - \boldsymbol{\mathcal{M}}\_{\text{os}}^{2}} + \mathbf{0}, \mathbf{5} \left[\mathbf{1} - \sqrt{\mathbf{1} - \boldsymbol{\mathcal{M}}\_{\text{os}}^{2}}\right] \mathcal{C}\_{p}^{0}}.\tag{34}$$

In **Figure 3**, for the pressure coefficient, the experimental data are presented as points ([4], Figure 121) and a curve that is an approximation of these points ([5], p.388). Comparison analysis indicates that the calculation by the Glauert formula (34) is approximate and for the Mach number *M*<sup>∞</sup> ≥0, 4 it can be argued that the Glauert formula should not be used. The Kárman-Tsien formula (35) gives a better agreement with the experimental data than the Glauert formula, but for the considered pressure-drop coefficients (*C*<sup>0</sup> *<sup>p</sup>* ¼ �0*:*6 and *<sup>C</sup>*<sup>0</sup> *<sup>p</sup>* ¼ �0*:*73) it is inferior in accuracy to the calculated data of L. Sedov [5] and the data obtained by the author

**Figure 2.** *Comparison of calculation results for critical Mach number based on Burago and Khristianovich methods.*

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

**Figure 3.** *Comparison of different methods account compressibility.*

of the chapter by the method outlined above [1]. It should be noted that the calculated data by Burago's method for all three considered cases are closer to the experimental data of G. Stack [19] than the results of the calculation by the Sedov's method [5].

Accounting for gas compressibility according to the method described above and the Kárman-Tsien method [see. Formula (35)] is based on recalculating the pressure-drop coefficients of an incompressible liquid *C*<sup>0</sup> *<sup>p</sup>* by the values of the pressure-drop coefficients of a compressible gas *Cp* for the given free-stream Mach numbers and ratio of specific heats*κ*. An interesting question is: how much differ these two methods in a wide range of variation of Mach numbers and pressure coefficients for an incompressible fluid? **Table 1** compares these two methods based on the data in **Figure 3**, from which it follows that the maximum discrepancy increases with decreasing value *C*<sup>0</sup> *<sup>p</sup>* and at *M*<sup>∞</sup> ! *M*<sup>∗</sup> . The maximum relative error of the two methods in the considered range of variation of Mach numbers and pressure coefficients reaches 15–16%.

The advantage of the Burago method over the Kárman-Tsien method is demonstrated in **Figure 4**, which presents the calculated data and experimental data of various authors for two bodies: an ellipse (**Figure 4a**) and a biconvex airfoil (**Figure 4b**) with relative thicknesses of 20% and 6% respectively. **Figure 4a** shows a comparison of the results of calculating the pressure drop coefficient on the surface of an ellipse of relative thickness *δ* = 0.2 in a compressible flow. The calculations are performed for the critical Mach number *M*\* = 0.7 (**Table 2**).

The calculated data using the Sells finite difference method (1968) are taken from [16]. It can be noted that the results of the Burago method and the Kárman-Tsien (34) are in good agreement to within approximate values *x*≤0*:*1. For values of relative coordinate along the big axis of an ellipse 0*:*1≤*x*≤ 0*:*5 the significant divergence between the results of the Kárman-Tsien formula (34) and Burago's

The calculation of the dependence of the critical Mach number on the pressuredrop coefficient of an incompressible flow using Eq. (30) is shown in **Figure 2**.

Comparison of the two calculations shows good agreement. After the publication of the work of Burago [20] in 1949, a number of approximate methods based on the Chaplygin gas model were developed. Burago's method refers to the approximate mathematical models for accounting for the compressibility of the flow. The effectiveness of approximate mathematical models is always shown by comparison with experimental or calculated data for various bodies. Calculations carried out according to the method described above showed that this method is not inferior in accuracy to more rigorous methods, at the same time the advantage of this method over other methods of accounting for the flow compressibility in terms of speeds is undeniable. Here are some comparisons. In **Figure 3** shows a comparison of the calculation by the method described above with the calculated data of L. Sedov [5] and by the Glauert formulas ([4], p. 311) and Kárman-Tsien ([4], p. 311). Glauert formula is

*Cp* <sup>¼</sup> *<sup>C</sup>*<sup>0</sup>

*Cp* <sup>¼</sup> *<sup>C</sup>*<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>M</sup>*<sup>2</sup> ∞

q

*p* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>M</sup>*<sup>2</sup> ∞

*p*

þ 0, 5 1 �

In **Figure 3**, for the pressure coefficient, the experimental data are presented as points ([4], Figure 121) and a curve that is an approximation of these points ([5], p.388). Comparison analysis indicates that the calculation by the Glauert formula (34) is approximate and for the Mach number *M*<sup>∞</sup> ≥0, 4 it can be argued that the Glauert formula should not be used. The Kárman-Tsien formula (35) gives a better agreement with the experimental data than the Glauert formula, but for the con-

*<sup>p</sup>* ¼ �0*:*6 and *<sup>C</sup>*<sup>0</sup>

accuracy to the calculated data of L. Sedov [5] and the data obtained by the author

*Comparison of calculation results for critical Mach number based on Burago and Khristianovich methods.*

<sup>q</sup> , (33)

*C*0 *p*

*<sup>p</sup>* ¼ �0*:*73) it is inferior in

*:* (34)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>M</sup>*<sup>2</sup> ∞

� � q

**Figure 2** also shows the Khristianovich's curve [13].

*Aerodynamics*

and the Kárman-Tsien formula is

sidered pressure-drop coefficients (*C*<sup>0</sup>

**Figure 2.**

**178**


