**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 **V** 2

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

*M*∞2 1 � *u*2 � *v*2 � � � � 1

velocity, *U*∞ is free-stream flow velocity at infinity.

*κ* � 1 2

*κ*�1, *a*0

The speed of sound is defined by formula *a* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

*M*2 � � 1

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

*ρ* ¼ ½ � *E M*ð Þ 1

the case of an isentropic air flow is *a* ¼ ffiffiffiffiffiffiffiffiffiffi

*ρ*0

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

Mach number.

**Figure 1.**

**173**

*The function σ vs. Mach number.*

*ρ* ¼ 1 þ

2 þ *ρκ*�1

ð Þ *κ* � 1 *ρκ*�1 ∞ *M*2

*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 , *a a*∞

here *u* ¼ *u=U*∞, *v* ¼ *v=U*∞ are the dimensionless components of the local

*κ*�1 , *a*0

*a* ¼ ½ � *E M*ð Þ 1

*a* ¼ *a*∞

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*∞ � �*κ*�1 2

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

1 ð Þ *κ* � 1 *M*2

¼ 1 þ

*a* ¼ 1 þ

2, *E M*ð Þ¼ 1 þ

∞

*κ* � 1 2

*κ* � 1 2

� �1

*M*2

2

*κ* � 1 2

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

*M*2

*dp=dρ* p . The speed of sound in

*:* (11)

, (9)

*:* (10)

*M*∞2 1 � *u*2 � *v*2 � � � �1

, (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/m3* ; 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*0 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*∞ 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*∞ are the dimensionless components of the local velocity, *U*∞ 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 *a* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi *dp=dρ* p . The speed of sound in the case of an isentropic air flow is *a* ¼ 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**

*∂x* ¼ 0, *∂ rn* ð Þ *ρu*

*n* ¼ 0 � 2D body;

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

¼ 0, *∂ ∂x*

> *ρ ρ*∞

¼ *p p*∞ � �1 *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

*h* ¼ *ρκ*�1

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 *κ* � 1 *ρκ*�1 ∞ *M*2

∞

*dp*

�

*τ* ¼ 2 *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*:*

; *σ* ¼ *ρ*0*=ρ* � 1 *ρ*0*=ρ* þ 1

> *rnτu* 1 þ *σ* � �

þ *∂ ∂r*

*rnτv* 1 þ *σ* � �

, (4)

*ρ :* (5)

, (6)

*ρ*0 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

*∂ rn* ð Þ *ρv*

; parameter *n* equal

*∂r* ¼ 0, (1)

, (2)

¼ 0, (3)

**characteristics**

*∂u ∂r* � *∂v*

are coordinates; *ρ* is local density of gas, *kg/m3*

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*∞ 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*o and *v*o 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*c 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

*u*2 þ *v*2 þ *κ*�1

incompressible flow. The critical Mach number *M*ð Þ1

calculating the critical Mach number

*M*∗ ¼

**175**

*M* ¼ *M*∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2 þ *v*2

*u*ð Þ1 ¼ *u*0*=U*∞; *v*ð Þ1 ¼ *v*0*=U*∞*:* (20)

*:* (19)

1 þ *κ*�1 2 *M*2 ∞ 1 � *u*2 � *v*2 � � s

*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 Þ2 , *v*ð Þ2 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 Þ3 , *v*ð Þ3 , and so on. The approximation process must be continued

> ð Þ *n c* � *η*

Putting in the formula (Eq. (21)) *M* = 1 and *M*∞ = *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 Þ1 and *v*ð Þ1 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 Þ2 and *v*ð Þ2 . 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 *u*2 þ *v*2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ *κ*�1 2 *M*2 ∞ q ¼ *M* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*M*∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2 *u*2 þ *v*2 � 1 � � s

ð Þ *n*�1 *c*

1 þ *κ*�1 2 *M*2

> *U*2 þ *κ*�1 2 *U*2

� � � � *η*

� � �

ð Þ *n*�1 *c* � *η*

q *:* (21)

� 1 � � vuut , (22)

∗ attained in the first

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

ð Þ *n*�2 *c*

� �

�≤*ε*, where *ε*

The coefficient *η*c 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*0 , *v*0 ). 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*0 , *v*0 ). 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*0 þ *ρ* � �

*ρ*

*=* 1 þ *ρ*0 *ρ*∞ � �

þ

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

*∂ rnv*0 ð Þ

� *∂ ∂x*

*ρ* þ *ρ*0 ð Þ

*τu ρ*0 þ *ρ* � �

*ρ*∞ þ *ρ*0 ð Þ *v*0 ¼ 1 þ *ρ*0

¼ 0*:* (12)

*v*0 ¼ *ηcv*0*:*

(13)

*ρ* 1 þ *ρ*0 *ρ*∞

*:* (14)

*∂r* ¼ 0*:* (15)

*ρ* þ *ρ*0 ð Þ *vn* ¼ 0*:* (16)

, *v*0 ).

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

> ¼ 0, *∂ ∂r*

*ηc* ¼ 1 þ *ρ*0

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

*∂x* ¼ 0, *∂ rnu*0 ð Þ

incompressible liquid (*ρ* ≈ *const*) with a velocity having components (*u*0

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

isentropic relationships for the local velocity of a flow and density

¼ *M* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ *κ*�1 2 *M*2 q , *ρ*0

¼ *U*∞*a*∞

j j **V** *a*0

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

formula (Eq. 17) can be written as

*v*0 *n* ¼ *ρ ρ*∞

If we introduce new variables *u*o and *v*o associated with u and *v* as follows

*u*0 ¼ *ηcu*0; *v* ¼ *ρ*∞

*ρ* � �

The coefficient *η*c 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*∞ þ *ρ*0 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 *u*2 þ *v*2 p *a*∞*a*0

¼ 1 þ

¼ *M*∞

*κ* � 1 2

� � 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2 þ *v*2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ *κ*�1 2 *M*2 ∞

*M*2 ∞ *κ*�1

q *:* (18)

, (17)

approximate

*Aerodynamics*

*∂ ∂x*

*u* ¼ *ρ*∞ *ρ*

nents (*u*0

**174**

, *v*0

*rnτu ρ*0 þ *ρ* � �

*ρ* þ *ρ*0 ð Þ

where *η*c is introduced

þ *∂ ∂r*

*ρ*∞ þ *ρ*0 ð Þ *u*0 ¼ 1 þ *ρ*0

*∂u*0 *∂r* � *∂v*0

*rnτv ρ*0 þ *ρ* � �

> *ρ* 1 þ *ρ*0 *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 Þ2 , *v*ð Þ2 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 Þ3 , *v*ð Þ3 , and so on. The approximation process must be continued until the following inequality is satisfied *η* ð Þ *n c* � *η* ð Þ *n*�1 *c* � � � � � � � *η* ð Þ *n*�1 *c* � *η* ð Þ *n*�2 *c* � � � � � �≤*ε*, where *ε* 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*∞ = *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 Þ1 and *v*ð Þ1 calculated for an incompressible flow. The critical Mach number *M*ð Þ1 ∗ 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 Þ2 and *v*ð Þ2 . 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*i* ∗ � *M*ð Þ *i*�1 ∗ ≤ *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*∞ ≤ *M*∗ is valid. This condition determines the absence of transonic and supersonic zones in the flow field. To calculate the critical Mach number *M*∗ and the velocity field (*u, v*) in a compressible flow, it is sufficient to calculate the velocity field (*u*0 , *v*0 ) 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 ∞ *E M*ð Þ *:*

*E M*ð Þ ∞

*κ*�1, *p*0*=p*∞ ¼ ½ � *E M*ð Þ ∞

*κ*�1 � 1

0 *p*

> 0 *p*

*E M*ð Þ � 1 þ ½ � *E M*ð Þ 1

( )

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

*E M*ð Þ ∞ *E M*ð Þ � � *κ*

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 ∞ *F M*ð Þ � 1

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

*κ*�1, *G M*ð Þ¼ *M*2

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

*G M*ð Þ� *G M*ð Þ ∞ 1 � *c*

*cp* ¼ 2 *κM*2 ∞

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 ∗ 1 � *c*

*E M*ð Þ *:* (27)

*:* (28)

*:* (29)

*κ*�1 n o�2

� � ¼ 0, (30)

� � ¼ 0*:* (32)

� �*:* (31)

*κ κ*�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 *p* ¼ 1 � *η*2 *c M*2 *M*2 ∞ �

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

*p*0*=p* ¼ ½ � *E M*ð Þ *κ*

*cp* ¼ 2 *κM*2 ∞

Using isentropic formulas for pressure ratios in the form

¼ *M*2 *M*2 ∞ �

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 *κ*

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*i*

pressible flow.

*Aerodynamics*

equalities

**176**

∗ � *M*ð Þ *i*�1 ∗

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*∞ ≤ *M*∗ is valid. This condition determines the absence of transonic and supersonic zones in the flow field. To calculate the critical Mach number *M*∗ and the velocity field (*u, v*) in a compressible flow, it is

, *v*0

¼ 1 � *V*0

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

*ηcV*∞ 2

*p* ¼ 1 � *V*

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

¼ 2 *p*∞ *ρ*∞*V*2 ∞

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

> ¼ 1 *κM*2 ∞ ,

¼ 2 *κM*2 ∞

*V*∞ 2

> *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*0

pressure-drop coefficient for an incompressible flow

*c* 0

ible flow can be represented through the gas flow velocity

*c* 0

*cp* ¼ *p* � *p*∞ 0*:*5*ρ*∞*V*2 ∞

> *p*∞ *ρ*∞*V*2 ∞ 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 ∞ ¼ *M*2 *M*2 ∞

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

*V*2 *V*2 ∞ ¼ *V*2 *a*2 *a*2 *a*2 0

*cp* ¼ 2 *κM*2 ∞

represented as the identity

*p* ¼ *p*0 � *p*∞ 0, 5*ρV*2 ∞

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*0

*Comparison of different methods account compressibility.*

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

increases with decreasing value *C*0

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

*p* by the values of the

*p* and at *M*∞ ! *M*∗ . 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*∞ ≥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*0 *p* ¼ �0*:*6 and *C*0 *p* 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*0 *p* 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*0 *p* and at *M*∞ ! *M*∗ . 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* ¼ *C*0

*Cp* ¼ *C*0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � *M*2 ∞

q

*p* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � *M*2 ∞

*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*∞ ≥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-

*p* ¼ �0*:*6 and *C*0

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.*

q , (33)

*C*0 *p*

*p* ¼ �0*:*73) it is inferior in

*:* (34)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � *M*2 ∞

� � q

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

*Aerodynamics*

and the Kárman-Tsien formula is

sidered pressure-drop coefficients (*C*0

**Figure 2.**

**178**


**1. Introduction**

Okara is a by-product of soybean processing. Over 20 million tons of wet soybean dregs (okara) are produced each year in China [1], which is one of the world-leading producers and consumers of soybean. This huge sum of okara produced during soybean processing is evidenced to be packed with a significant amount of nutritional and nonnutritional constituents. However, these are usually used as animal feeds, fertilizers, landfills, or discarded as waste due to their high susceptibility to spoilage, which exerts serious economic and socio-environmental problems [2]. Hence, its valorization will be important to help utilize the untapped nutrients and minimize the environmental problems caused by this waste disposal. Okara is high in nutrients, with insoluble fiber fraction accounting for 70% of the total dietary fiber [1]. Besides, the functional characteristics of insoluble dietary fiber have recently been discovered and gained recognition. Dietary fiber is tagged as "The Seventh Nutrient" [3], and is necessary for a balanced diet [4]. Hence, can be utilized in shaping the gut microbiota [5]. The consumption of a high dietary fiber diet is evidenced by decreases in the bioavailability of some important nutritional components, such as some vitamins and minerals [6], and may also impact the rate of digestion of food

**Citation:** Fan, H.; Zhang, Y.; Swallah, M.S.; Wang, S.; Zhang, J.; Fang, J.; Lu, J.; Yu, H. Structural Characteristics of Insoluble Dietary Fiber from Okara with Different Particle Sizes and Their Prebiotic Effects in Rats Fed High-Fat Diet. *Foods* **2022**, *11*, 1298. https://doi.org/10.3390/ foods11091298

Academic Editor: Antonella Pasqualone

Received: 1 April 2022 Accepted: 26 April 2022 Published: 29 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

substances as well as energy metabolism. The intake of a high-fiber diet can also improve the activities of the gut microbial composition [7].

The gut ecosystem is a home for over 1014 microbial cells, and collectively comprise of at least 150 times more genes than their host [8]. In both humans and animals, the gastrointestinal tract is host to a complex community of varied microorganisms, whose activities considerably impact host nutrition and health [9]. The composition of the gut microbiota is said to vary based on genetic characteristics, sex, age, and diet [10]. The association between the host gut-microbiota are dynamic and are highly susceptible to several environmental conditions, mainly diet [8], and the interactive relationship between prebiotics and dietary fiber is reliant on variations in the gut microflora as well as their colonic degradation [7]. Numerous studies indicates that the gut microorganisms can directly impact the physiological conditions of the host organism by encouraging the immune system through stimulating the defensive mechanisms against pathogens and inflammatory bowel diseases, as well as improving the roles of the intestinal barrier, regulating autoimmunity, producing biological metabolites, destroying cancer cells, regulating diabetes and preventing obesity. The host gut-microbiota interactions are highly influenced by several environmental conditions, buy mainly by diet [3,10]. Dietary fiber intake is reported to impact gastrointestinal health via encouraging the gastrointestinal barrier function, nutrient absorption, as well as selectively stimulating the composition and activities of the gut microbes, that can confer health benefits to the host, including lowering the risk of cardiovascular disease, cancer, diabetes, and other ailments [11,12]. For instance, dietary fiber alters the gastrointestinal barrier function through increasing cells and the mucins that produce the "goblet cells" [11]. Mucins are large glycoproteins that, along with antibodies, bacteria, lipids, proteins, ions, antimicrobial peptides, and water, form what is known as mucus. Mucus shields the gut epithelium from mechanical stress, to inhibit the translocation of toxic substances and to lubricate the intestine as well as encourage smooth transportation of digested material [3,12]. In addition, dietary fiber may absorb and excrete toxic chemicals in the intestines, promote intestinal flora, and offer energy and nutrients for probiotic growth [13]. The majority of studies on okara's insoluble dietary fiber focus on changing their fundamental physical and chemical characteristics [14,15], and only a few studies have been conducted on the functional qualities of okara due to their unpleasant taste [16]. The prolonged intake of a high-fat diet poses a major hazard to human/animal health. Emerging evidence has indicated that high-fat-diet can alter the composition of the microbiota, by inducing dysbiosis or imbalance of the gut microbial ecosystem. Dysbiosis is associated with diseases and metabolic health ranging from insulin resistance, glucose intolerance and to metabolic syndrome [17]. Furthermore, the gut microbiota can influence lipid metabolism via short-chain fatty acid (SCFA) metabolites [18]. SCFAs are primarily generated in the large intestine by anaerobic bacteria fermenting indigestible carbohydrates. They provide the material foundation for the gut flora's "conversation" with the host [13]. They have the ability to stimulate cell development, enhance intestinal function, and have an impact on cardiovascular metabolism, as well as anti-inflammatory, anti-tumor, and immunological regulatory activities [19].

In this work, high-pressure microfluidic technology was employed to extract three insoluble dietary fibers (IDFs) with varying particle sizes from okara (purity > 90%) to investigate the effects of insoluble dietary fiber from okara (OIDF) on blood lipid levels and intestinal flora in rats fed a high-fat diet. Okara can obtain high-purity OIDF and increase its value addition.

#### **2. Materials and Methods**

#### *2.1. Preparation of OIDFs with Three Different Particle Sizes*

Okara, a by-product of producing soybean protein isolate with a protein level of 10–15% (Liaocheng, China), was generously provided by Shandong Jiahua Health Care Products Co., Ltd. The IDF from okara was further processed into three different particle sizes using High Pressure Microfluidic Technology to obtain the OIDFs. The steps comprise of α-amylase treatment at 95 ◦C for 35 min, being starch glucosidase treated at 60 ◦C for 30 min, being neutral protease treated 60 ◦C for 30 min, 70 ◦C water precipitation for 1.5 h, centrifugation at 3500 rpm for 10 min, alcohol precipitation until colorless, and being freeze-dried for standby. According to the Chinese national standard GB 5009.88-2014, OIDF has a purity of 90.50 percent [20]. The OIDF (purity higher than 90.50%) extracted by biological enzymatic method was processed by dynamic high pressure microfluidization (DHPM). OIDF weighing 20 g was mixed with 800 mL of deionized water, and treated at 10 MPa, 50 MPa, and 150 MPa for 2 min, respectively. Three distinct particle sizes of OIDFs were named OIDF-10 (μm), OIDF-50 (μm), and OIDF-100 (μm).

#### *2.2. Determination of the Basic Components of the Three OIDFs*

The protein content was determine according to the Kjeldahl's method in GB5009.5-2016 "Determination of Protein in Food"; Ash content was determine based on GB5009.4-2016 "Determination of ash in food; Moisture content was determined by MB35 Ohaus moisture analyzer; SDF (soluble dietary fiber), IDF and TDF content were determined using GB5009.88-2014 "Determination of Dietary Fiber in Food".


**3.1 Microcontact printing (μCP)**

μCP is a direct method for pattern transfer, generating a non-structured, chemically modified surface. The process of μCP is shown in figure 2. Photolithography was used for the fabrication of silicon-based masters in preparing PDMS stamps. Multi-layer molds were made of thick photoresist like SU-8 on silicon or glass wafers by standard lithography techniques. It was subsequently placed at least 30min in an oven at 160°C to make the photoresist adhere to the substrate closely. Release agent DC20 or OTS were always spincoated and drying on the master before pouring PDMS. Liquid PDMS (Sylgard-184 from Dow Corning) was poured onto the mold and clamped by the foil, so that the shape of the mold microstructure was transferred to PDMS membrane. It was subsequently placed at least 2 h in an oven at 80°C. The molded PDMS slab was then peeled off and placed onto a glass slide for handling. After curing, PDMS stamps are soaked in a protein "ink", such as PEI, PLL, or LN. 20 minutes later, the "ink" was blew off by using nitrogen gas. Then the raised regions were brought into conformal contact with a substrate in order to print the ink onto the substrate surface. The material of interest was transferred from the PDMS stamp onto the substrate surface. The microscopy of the PDMS stamps were shown in figure 3.

biocompatibility, chemical stability, optical transparency, air permeability, elasticity. Moreover, the polymer precursors can be aggregated into a mold by UV radiation. PDMS

To cure the PDMS prepolymer in general, a mixture of silicon elastomer and a curing agent (10:1, Sylgard 184 silicone elastomer kit, Dow Corning Corp.) is poured onto the master and placed at 70-80°C for 1 h. The character of the PDMS is closely related to the mixture ratio, curing temperature, and vacuum. Silicon, quartz or glass, and some photoresist are the most common materials to fabricate the masters by standard lithography, transferring the

**3. Soft lithography fabrication methods and it's application in patterning**

Various soft lithographic technologies have been applied to fabricate high-quality microstructures and nanostructures including micro contact printing (μCP), replica molding (REM), microtransfer molding (μTM), micromolding in capillaries (MIMIC), and solventassisted micromolding (SAMIM). Here, three soft lithographic methods are introduced to fabricate micropatterns onto a surface or MEA: μCP, microfluidic patterning technique and microstencil. The former two can be achieved using the same PDMS stamps and molds. A novel technology was applied to get the high depth-to-width ratio silicon-based mold to fabricate the topographic PDMS microstencil with microfluidic channel (Y.Nam, et al, 2006). Finally, microchannels on MEA with polyimide (PI) guiding the cell growing were also

μCP is a direct method for pattern transfer, generating a non-structured, chemically modified surface. The process of μCP is shown in figure 2. Photolithography was used for the fabrication of silicon-based masters in preparing PDMS stamps. Multi-layer molds were made of thick photoresist like SU-8 on silicon or glass wafers by standard lithography techniques. It was subsequently placed at least 30min in an oven at 160°C to make the photoresist adhere to the substrate closely. Release agent DC20 or OTS were always spincoated and drying on the master before pouring PDMS. Liquid PDMS (Sylgard-184 from Dow Corning) was poured onto the mold and clamped by the foil, so that the shape of the mold microstructure was transferred to PDMS membrane. It was subsequently placed at least 2 h in an oven at 80°C. The molded PDMS slab was then peeled off and placed onto a glass slide for handling. After curing, PDMS stamps are soaked in a protein "ink", such as PEI, PLL, or LN. 20 minutes later, the "ink" was blew off by using nitrogen gas. Then the raised regions were brought into conformal contact with a substrate in order to print the ink onto the substrate surface. The material of interest was transferred from the PDMS stamp onto the substrate surface. The microscopy of the PDMS stamps were shown in figure 3.

polymerization is shown in Figure 1.

Fig. 1. PDMS polymerization

patterns to the PDMS stamp.

**3.1 Microcontact printing (μCP)**

introduced.

Fig. 2. Schematics of the processes of μCP

Application of Soft Lithography and Micro-Fabrication on Neurobiology 21

MEA was a cell-based biosensor for extracellular electrophysiological investigations of neuronal networks. PDMS microstencil was designed to pattern adhesion molecules at the surface of MEA guiding cultured cells grow along the patterns. PDMS microstencil mold was fabricated by a complex photomask aligning method, shown in figure 4.

HF etching

Spin-coat photoresist

DRIE

DRIE

Spin-coat photoresist

SiO2

thermal oxidation

Si

Si

Si

Si

Si

Si

Fig. 4. Schematics of the fabrication of microstencil mold

d

e

f

a

b

c

work.

525μm silicon was thermal oxidated with 4000 Ǻ SiO2 (see Fig. 4a). The substrate was rst coated with a thin photoresist (AZ AZ9912) for 30 s at 3000 rpm (see Fig. 4b). 8 × 8 SiO2 arrays were fabricated by wet etching as the RIE mask which was the same as the MEA structure (see Fig. 4c). Photoresist was spin-coated to the silicon with SiO2 mask again (see Fig. 4d) and selectively exposed to UV under a chromium photomask. The silicon was etched 30μm by deep reactive ion etching (DRIE) with photoresist in order to construct the microchannel , (see Fig. 5a). Then photoresist was removed by ultrasonication in acetone. The silicon was selectively etched 70μm by DRIE with SiO2 resist, forming the topographic PDMS microstencil mold (see Fig. 5b). The high depth-to-width ratio silicon-based mold was designed to penetrate through the PDMS membrane on the MEA to exposure the electrodes and form the microchannel between the electrodes so that the MEA could also

Fig. 3. Microscopy of the PDMS stamps

The bare areas of substrate surface that the PDMS stamp has not touched can be exposed to another coating material. μCP provides the patterning of self-assembled monolayers(SAMs) of alkanethiols on gold, and the resulting control over the adsorption of adhesive proteins facilitates the patterning of cells on substrates.

μCP enables easy stamp replication, fast printing using parallelization, and low-cost batch production. A conformal contact between the stamp and the surface of the substrate is the key to its success. The polymer stamps also minimize the problems of sample carry-over and cross contamination. Printing has the advantage of simplicity and convenience: Once the stamp is available, multiple copies of the pattern can be produced using straightforward experimental techniques. Printing is an additive process; the waste of material is minimized. Printing also has the potential to be used for patterning large areas.

However, μCP has some limitations that are mainly caused by the use of a soft polymer stamp. The swelling of a stamp during inking often results in an increase in the pattern size by diffusion of the excessive printed molecules on the substrate.

## **3.2 Microfluidic patterning using microchannels**

The difference between μCP and microfluidic patterning is that PDMS stamps are soaked in the "ink" in the former usage, but the stamps contact the substrate forming microchannels delivering the materials for cell adhesion or cell suspension to the desired area in the latter usage because of the elastic nature and hydrophobicity of PDMS. The substrate tilted 45 degree, drop of liquids were injected to the PDMS microchannels by the pipette. Then the substrate was put on the test tube rack vertically for 25 to 30 min. Patterns were formed of after the liquid dried.

While this method has been used primarily for surface attachment of cells, it may be possible to adapt this method to three-dimensional tissue constructs. In many cases three dimensional tissue constructs promote cellular differentiation and more authentic cellular morphology and metabolism.

#### **3.3 Microstencil on MEA**

The former two methods enable patterning adhesion molecules and guiding cultured cells grow physically. But the cells' communication and interactions in co-cultures are difficult to be detected, which is important to research the function of the cells network.

The bare areas of substrate surface that the PDMS stamp has not touched can be exposed to another coating material. μCP provides the patterning of self-assembled monolayers(SAMs) of alkanethiols on gold, and the resulting control over the adsorption of adhesive proteins

μCP enables easy stamp replication, fast printing using parallelization, and low-cost batch production. A conformal contact between the stamp and the surface of the substrate is the key to its success. The polymer stamps also minimize the problems of sample carry-over and cross contamination. Printing has the advantage of simplicity and convenience: Once the stamp is available, multiple copies of the pattern can be produced using straightforward experimental techniques. Printing is an additive process; the waste of material is minimized.

However, μCP has some limitations that are mainly caused by the use of a soft polymer stamp. The swelling of a stamp during inking often results in an increase in the pattern size

The difference between μCP and microfluidic patterning is that PDMS stamps are soaked in the "ink" in the former usage, but the stamps contact the substrate forming microchannels delivering the materials for cell adhesion or cell suspension to the desired area in the latter usage because of the elastic nature and hydrophobicity of PDMS. The substrate tilted 45 degree, drop of liquids were injected to the PDMS microchannels by the pipette. Then the substrate was put on the test tube rack vertically for 25 to 30 min. Patterns were formed of

While this method has been used primarily for surface attachment of cells, it may be possible to adapt this method to three-dimensional tissue constructs. In many cases three dimensional tissue constructs promote cellular differentiation and more authentic cellular

The former two methods enable patterning adhesion molecules and guiding cultured cells grow physically. But the cells' communication and interactions in co-cultures are difficult to be detected, which is important to research the function of the cells network.

Fig. 3. Microscopy of the PDMS stamps

facilitates the patterning of cells on substrates.

Printing also has the potential to be used for patterning large areas.

by diffusion of the excessive printed molecules on the substrate.

**3.2 Microfluidic patterning using microchannels**

after the liquid dried.

morphology and metabolism.

**3.3 Microstencil on MEA**

MEA was a cell-based biosensor for extracellular electrophysiological investigations of neuronal networks. PDMS microstencil was designed to pattern adhesion molecules at the surface of MEA guiding cultured cells grow along the patterns. PDMS microstencil mold was fabricated by a complex photomask aligning method, shown in figure 4.

Fig. 4. Schematics of the fabrication of microstencil mold

525μm silicon was thermal oxidated with 4000 Ǻ SiO2 (see Fig. 4a). The substrate was rst coated with a thin photoresist (AZ AZ9912) for 30 s at 3000 rpm (see Fig. 4b). 8 × 8 SiO2 arrays were fabricated by wet etching as the RIE mask which was the same as the MEA structure (see Fig. 4c). Photoresist was spin-coated to the silicon with SiO2 mask again (see Fig. 4d) and selectively exposed to UV under a chromium photomask. The silicon was etched 30μm by deep reactive ion etching (DRIE) with photoresist in order to construct the microchannel , (see Fig. 5a). Then photoresist was removed by ultrasonication in acetone. The silicon was selectively etched 70μm by DRIE with SiO2 resist, forming the topographic PDMS microstencil mold (see Fig. 5b). The high depth-to-width ratio silicon-based mold was designed to penetrate through the PDMS membrane on the MEA to exposure the electrodes and form the microchannel between the electrodes so that the MEA could also work.

Application of Soft Lithography and Micro-Fabrication on Neurobiology 23

However, PDMS microstencil is difficult to practise because it is hard to align with MEA

Planar MEA are developed to study electrogenic tissues such as dissociated neuronal cultures (Hiroaki Oka,et al. 1999). They have been widely used with dissociated cultures for a variety of neuroscience investigation including learning and memory and cell-based biosensors for the detection of neurotoxins (Conrad D. James, et al. 2004). But the neurons grow disorderly and cannot form a network so that the function is not the same as the cells in vitro. Combining the patterning technology and MEA forming neuronal networks is the

μCP, microfluidic patterning technique and microstencil are difficult to operate because the space is too small and the PDMS stamp or stencil can hardly align with MEA. The best and easiest way to forming neuronal networks is to fabricate the microchannels on MEA with

MEA were fabricated using a conventional semiconductor process (Guangxin Xiang, et al. 2007). After cleaning the polished quartz glass wafer, the conductive layer of Au/Ti lm (Au 3000 Ǻ and Ti 700 Ǻ) was sputtered. 8 × 8 electrode arrays were left with the photomask protection by standard photolithography. Then, a combination of SiO2/Si3N4/SiO2 (3000 Ǻ /4000 Ǻ /3000 Ǻ) passivation layers was deposited onto the substrate using plasma enhanced chemical vapor deposition (PECVD), and the insulating layers on the electrodes and the bonding-pads were removed by inductively coupled plasma (ICP) (see Fig. 7a). Finally, Negative photosensitive polyimide (AP2210B, Fujifilm Electronic Materials Inc) was spin-coated to form microchannels having a thickness of 3~4μm and photo-etched by the

standard procedure to expose the microelectrodes and the terminals (see Fig. 7b).

(a) (b)

GABAergic neurons in the striatum and PC12 cells were cultured on MEA with PI microchannels which were coated with poly-l-lysine (PLL) to promote cell adhesion, (see Fig. 8a, 8b). PI microchannels could be seen between the electrodes and the neural cell can grow along the microchannels. However the nerve cell synapse could not formed along the microchannels. Because the depth of microchannels could not match the neurons and the

Fig. 7. Microscopy of the MEA with PI microchannels

and the silicon mold is easy to fracture when lifting off.

**3.4 MEA with microchannels for patterning**

efficient method to research the neurobiology.

polyimide (PI) guiding the cell growing.

Fig. 5. SEM images of PDMS microstencil mold

The mixture of PDMS prepolymer and curing agent was spin-coated on the mold for 40 s at 4000 rpm. The coated mold was cured for 2 hours at 110◦C in a convection oven. The fullycured PDMS-coated mold was soaked in an acetone ultra-sonication bath until the PDMS layer released from the mold. The detached microstencil was rinsed with IPA and DI water. The upside and downside of PDMS microstencil with microholes and microchannels SEM images was shown in figure 6.

Fig. 6. SEM images of PDMS microstencil

However, PDMS microstencil is difficult to practise because it is hard to align with MEA and the silicon mold is easy to fracture when lifting off.