**4.4 Combining method of interior and exterior asymptotics for boundary layer of supersonic flow in compressed viscous gas by TPPA**

We consider the boundary layer in hypersonic flow of viscous gas and solve a model problem which reduces to ordinary differential equations with appropriate boundary conditions. The TPPAs parameters are calculated and relevant questions *Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer… DOI: http://dx.doi.org/10.5772/intechopen.93084*

are discussed. The equations of laminar boundary layer near a semi-infinite plate in the supersonic flow of viscous perfect gas, as it is known [2, 7], can be reduced to the form:

$$\left(\boldsymbol{\varrho}^{\prime\prime}\frac{\boldsymbol{\mu}}{T}\right)^{\prime} + \boldsymbol{\varrho}\boldsymbol{\varrho}^{\prime\prime} = \mathbf{0},\tag{40}$$

$$\left(\mu \frac{T'}{T}\right)' + \sigma \rho T' + a\sigma \frac{\mu}{T} \rho''^2 = 0\tag{41}$$

where

Species of TPPA taking into account four nontrivial parameters:

*Mathematical Theorems - Boundary Value Problems and Approximations*

Therefore,

<sup>2</sup>*a*2*<sup>ζ</sup>* � *<sup>a</sup>*<sup>2</sup> 2

second local equality:

*φ*0

**88**

<sup>2</sup> � <sup>2</sup>*a*2*<sup>D</sup>*

*φ*0

<sup>6</sup> *<sup>ζ</sup>*<sup>4</sup> <sup>¼</sup> 2 1 �

Therefore, the TPPA has the form:

internal domain, we will write down the local equality:

*α*3, *β*1, *β*2, *β*<sup>4</sup>

*<sup>a</sup>*ð Þ¼ *<sup>ζ</sup>* 2 1 � <sup>1</sup> <sup>þ</sup> *<sup>α</sup>*3*ζ*<sup>3</sup> � � exp �*ζ*<sup>2</sup> <sup>þ</sup> *<sup>c</sup><sup>ζ</sup>* � �

Parameter values are determined using local asymptotic and TPPA in the respective domain. Taking into account the decomposition of the exponent in the

<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*<sup>ζ</sup>* <sup>þ</sup> *<sup>β</sup>*2*ζ*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*4*ζ*<sup>4</sup>

� �

<sup>2</sup> þ …

(35)

(36)

(37)

(38)

(39)

" #

<sup>1</sup> <sup>þ</sup> *<sup>α</sup>*3*ζ*<sup>3</sup> � � <sup>1</sup> <sup>þ</sup> *<sup>c</sup><sup>ζ</sup>* � *<sup>ζ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>c</sup>ζ*�*ζ*<sup>2</sup> ð Þ<sup>2</sup>

Taking into account Eq. (33) in the external domain, we will write down the

Equalizing the coefficients in Eqs. (36) and (37) at the same degrees *ζ*, we get

" #

*a*<sup>2</sup> ¼ 0*:*6641, *c* ¼ 1*:*7308, *D* ¼ 0*:*3357

By substituting (39) in (38), we get an explicit expression for the TPPA.

**of supersonic flow in compressed viscous gas by TPPA**

**4.4 Combining method of interior and exterior asymptotics for boundary layer**

We consider the boundary layer in hypersonic flow of viscous gas and solve a model problem which reduces to ordinary differential equations with appropriate boundary conditions. The TPPAs parameters are calculated and relevant questions

<sup>2</sup>*<sup>ζ</sup>* � *<sup>c</sup>* exp �*ζ*<sup>2</sup> <sup>þ</sup> *<sup>c</sup><sup>ζ</sup>* � � <sup>¼</sup> 2 1 � <sup>1</sup> <sup>þ</sup> *<sup>α</sup>*3*ζ*<sup>3</sup> � � exp �*ζ*<sup>2</sup> <sup>þ</sup> *<sup>c</sup><sup>ζ</sup>* � �

*<sup>α</sup>*<sup>3</sup> <sup>¼</sup> *<sup>a</sup>*2*D*, *<sup>β</sup>*<sup>1</sup> <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>c</sup>*, *<sup>β</sup>*<sup>2</sup> ¼ � *<sup>c</sup>*<sup>2</sup>

After systems (31)–(33) are solved, we will obtain

*<sup>a</sup>*ð Þ¼ *<sup>ζ</sup>* 2 1 � <sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2*Dζ*<sup>3</sup> � � exp �*ζ*<sup>2</sup> <sup>þ</sup> *<sup>c</sup><sup>ζ</sup>* � � <sup>1</sup> <sup>þ</sup> ð Þ *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>c</sup> <sup>ζ</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*<sup>ζ</sup>* <sup>þ</sup> *<sup>β</sup>*2*ζ*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*4*ζ*<sup>4</sup>

<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*<sup>ζ</sup>* <sup>þ</sup> *<sup>β</sup>*2*ζ*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*4*ζ*<sup>4</sup>

) *<sup>a</sup>*2*<sup>D</sup>* <sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*<sup>ζ</sup>* <sup>þ</sup> *<sup>β</sup>*2*ζ*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*4*ζ*<sup>4</sup> � � <sup>¼</sup> ð Þ <sup>2</sup>*<sup>ζ</sup>* � *<sup>c</sup>* <sup>1</sup> <sup>þ</sup> *<sup>α</sup>*3*ζ*<sup>3</sup> � �

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2ð Þ *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>c</sup>* , *<sup>β</sup>*<sup>4</sup> <sup>¼</sup> <sup>2</sup>*:*

" #

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*<sup>c</sup>* <sup>þ</sup> *<sup>c</sup>*<sup>2</sup>

<sup>2</sup> � <sup>1</sup> � �*ζ*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ζ*<sup>4</sup>

$$\rho = \frac{\mathcal{W}}{\sqrt{\mathcal{X}}} = \rho(\zeta), \ T = T(\zeta), \ \zeta = \frac{\eta}{2\sqrt{\mathcal{X}}}, \ \eta = \int\_0^\mathcal{Y} \frac{dy}{T}, \ \ a = \frac{1}{4}M^2(\kappa - 1)$$

*M* is the Mach number, *σ* is the Prandtl number**,** *κ* is the adiabatic index, *ψ* is the stream function, *T* is the temperature, *μ* is the viscosity coefficient, and *x* and *y* are the Cartesian coordinates.

The boundary conditions at the wall are

$$
\rho(\mathbf{0}) = \rho'(\mathbf{0}) = \mathbf{0}, \ T(\mathbf{0}) = T\_s \tag{42}
$$

At external boundary of layer is

$$
\rho'(\lnot) = \mathbf{2}, \ T(\lnot) = \mathbf{1}.\tag{43}
$$

Interior asymptotic expansions are for *<sup>μ</sup>* <sup>¼</sup> *<sup>T</sup><sup>n</sup>*

$$\begin{aligned} q' &= 2a\_2\zeta - (n-1)a\_2 \frac{T\_1}{T\_s} \zeta^2 + O(\zeta^3), \\ T &= T\_s + T\_1\zeta - \left(2a\sigma a\_2^2 + \frac{(n-1)}{2} \frac{T\_1^2}{T\_s}\right) \zeta^2 + O(\zeta^3) \end{aligned} \tag{44}$$

where two constants *a*<sup>2</sup> and *T*<sup>1</sup> remain undefined. Exterior asymptotics for *ς* ! ∞

$$\begin{aligned} \ln \varrho'' &= c^2 + c\zeta + \ln A + o(\mathbf{1}), \\ \ln \left(-T'\right) &= -\sigma \zeta^2 + \sigma c\zeta + \ln B + o(\mathbf{1}) \end{aligned} \tag{45}$$

where three constants are unknown: *c*, *A*, and *B.*

We solve boundary problems (40) and (41) approximately by connecting asymptotics (44) and (45) TPPA

$$\rho\_a'(\zeta) = 2 \left[ 1 - \frac{\left( 1 + A\zeta^3 \right) \exp \left( -\zeta^2 + c\zeta \right)}{1 + a\_1\zeta + a\_2\zeta^2 + a\_4\zeta^4} \right] \tag{46}$$

$$T\_a'(\zeta) = \frac{\zeta\_m - \zeta}{\beta\_0 + \beta\_1 \zeta} \exp\left(\sigma(-\zeta^2 + c\zeta)\right) \tag{47}$$

Boundary conditions (45) and (46) are satisfied if to put

$$\rho\_a(\zeta) = \bigcap\_{0}^{\infty} \rho\_a'(\zeta) d\zeta, \quad T\_a(\zeta) = T\_s + \bigcap\_{0}^{\infty} T\_a'(\zeta) d\zeta \tag{48}$$

We complement the last equalities (50) and (51) with a normalizing condition:

$$\mathbf{1} = T\_s + \bigcap\_{a=1}^{\infty} T'\_a(\zeta)d\zeta \tag{49}$$

and integrating from 0 to ∞ taking into account Eq. (48):

*Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer…*

*<sup>φ</sup>* � *<sup>μ</sup>*ð Þ *<sup>T</sup>*

þ *a*<sup>2</sup>

Thus, the integral relations (52) and (55)–(47) form a nonlinear system of

*T*1, *a*2, *c*, *A*, *B:*

Integrals of the systems (37) and (42)–(44) solution were approximated using Simpson quadrature formulas. The behavior of magnitude *B* proved to be highly dependent on the behavior of the exponent at large, so the integral relation had to be replaced by the local condition (52), besides controlling the behavior of the TPPA near the maximum is more important than the weight of the exponent away from the wall. Thus, instead of the value of *B*, we include the value among the parameters

*<sup>T</sup>* ð Þ <sup>2</sup>*<sup>ζ</sup>* � *<sup>c</sup>*

*μ*ð Þ *T <sup>T</sup> <sup>φ</sup>*00<sup>2</sup> � � exp *σ ζ*<sup>2</sup> � *<sup>c</sup><sup>ζ</sup>* � � � � *<sup>d</sup><sup>ζ</sup>* (56)

� �*φ*<sup>00</sup> exp *<sup>ζ</sup>*<sup>2</sup> � *<sup>c</sup><sup>ζ</sup>* � �*d<sup>ζ</sup>* (55)

*<sup>x</sup>*€ <sup>þ</sup> *<sup>μ</sup>x x* \_ <sup>2</sup> � <sup>1</sup> � � <sup>þ</sup> *<sup>x</sup>* <sup>¼</sup> <sup>0</sup> (57)

<sup>18432</sup> *<sup>μ</sup>*<sup>5</sup> <sup>þ</sup> … , *<sup>μ</sup>* ! 0, (58)

2 3ð Þ � 2 ln 2 *<sup>μ</sup>* <sup>þ</sup> … , *<sup>μ</sup>* ! <sup>∞</sup>*:* (59)

0

*<sup>T</sup>* ð Þ <sup>2</sup>*<sup>ζ</sup>* � *<sup>c</sup>* � �

sought, and the value of *B* is expressed from Eqs. (50) and (51).

*<sup>λ</sup>*<sup>2</sup> ¼ �*<sup>μ</sup>* � <sup>1</sup>

<sup>16</sup> *<sup>μ</sup>*<sup>3</sup> <sup>þ</sup>

*<sup>λ</sup>*<sup>2</sup> ¼ � <sup>3</sup> <sup>þ</sup> 4 ln 2

As an example of TPPA (see Section 3.2) used for matching of limiting asymptotics, consider the paper by Grasman et al. [33]. They dealt with Lyapunov exponents which characterize the dynamics of a system near its attractor. For the

Similar to the asymptotic approximation of amplitude and period, expressions are derived for the nonzero Lyapunov exponent *λ*<sup>2</sup> for both small and large param-

263

The overlap of these series does not take place. The authors of [33] remark: "Such an overlap comes within reach if in the regular expansion a large number of terms is included." This is not correct, because the obtained series is asymptotic; so, with increasing of number of terms, the results will be worst. So, one needs a summation procedure. Some authors [34] proposed to use PAs, but in this case one needs hundreds of perturbation series terms. That is why we use TPPA. Using two

terms from expansion (58) and one term from expansion (59), one obtains

equations for determining the following parameters:

*A* ¼ 2*a*<sup>2</sup>

*B* ¼ *σ*

**5. Results**

eter *μ* values:

**91**

Van der Pol oscillator:

∞ð

0

Similarly, from Eq. (41), we get

*DOI: http://dx.doi.org/10.5772/intechopen.93084*

*<sup>T</sup>*<sup>0</sup> *<sup>φ</sup>* � *<sup>μ</sup>*ð Þ *<sup>T</sup>*

*μ*ð Þ *Ts Ts* � ∞ð

Following the procedure of the previous section, we will calculate the coefficients at *ζ* and *ζ*<sup>2</sup> in asymptotic expansions (44) and, equating them with the corresponding expressions from Eqs. (46) and (47), we will obtain equalities, from which values *α*1, *α*2, *α*4, *β*0, *β*<sup>1</sup> are expressed through *a*1,*c*, *T*1, *ζm*:

$$\begin{aligned} a\_1 &= a\_2 + c, \\ a\_2 &= -\frac{1}{2}(n-1)a\_2 \frac{T\_1}{T\_s} - \mathbf{1} + \frac{c^2}{2} + a\_2(a\_2 + c), \ a\_4 = 4, \\ \beta\_0 &= \frac{\zeta\_m}{T\_1}, \ \beta\_1 = \sigma c \frac{\zeta\_m}{T\_1} - \frac{1}{T\_1} + (n-1)\frac{\zeta\_m}{T\_1^2} \end{aligned} \tag{50}$$

Three parameters in asymptotics (44) are defined in the outer region if the following condition is met:

$$
\beta\_1 = -\mathbb{1}\_{\mathsf{B}} \tag{51}
$$

A priori at large *M* numbers, it is known that the temperature profile is non-monotonic and has a maximum within the layer at point *ς<sup>m</sup>* at which, as can be seen from the second equation of the systems (40) and (41), the following condition is used:

$$T''(\zeta\_m) = -a\sigma\rho^{\prime\prime 2}(\zeta\_m) \tag{52}$$

From the convexity condition of the temperature profile in the vicinity of the point *ςm*, the following equality is used:

$$(\beta\_0 + \beta\_1 \zeta\_m) a \sigma \rho^{\prime 2}(\zeta\_m) = \exp \left( -\sigma (\zeta\_m^2 - \zeta\_m) \right) \tag{53}$$

Let us add the received equations with the integrated ratios received on the basis of coincidence of TPPAs (46) and (47); in this case, three members in asymptotic decompositions (50) and (51), the initial system of Eqs. (40) and (41), with boundary conditions (42) and (43), by using the technique stated in the previous sections.

$$\begin{aligned} a\_2 &= \frac{1}{2} \int\_0^\infty \rho\_a'(2 - \rho') d\zeta \\\\ c &= \int\_0^\infty (2 - \rho') d\zeta \end{aligned} \tag{54}$$

The integral relation for parameter *A* is obtained by multiplying Eq. (40) by

$$\exp\left(\zeta^2 - c\zeta\right)$$

*Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer… DOI: http://dx.doi.org/10.5772/intechopen.93084*

and integrating from 0 to ∞ taking into account Eq. (48):

$$A = 2a\_2 \frac{\mu(T\_s)}{T\_s} - \int\_0^\infty \left(\rho - \frac{\mu(T)}{T}(2\zeta - c)\right) \rho'' \exp\left(\zeta^2 - c\zeta\right) d\zeta \tag{55}$$

Similarly, from Eq. (41), we get

$$B = \sigma \int\_0^\infty \left( T' \left( \rho - \frac{\mu(T)}{T} (2\zeta - c) \right) + a\_2 \frac{\mu(T)}{T} \rho''^2 \right) \exp\left( \sigma (\zeta^2 - c\zeta) \right) d\zeta \tag{56}$$

Thus, the integral relations (52) and (55)–(47) form a nonlinear system of equations for determining the following parameters:

$$T\_1, \ a\_2, \ c, \ A, \ B.$$

Integrals of the systems (37) and (42)–(44) solution were approximated using Simpson quadrature formulas. The behavior of magnitude *B* proved to be highly dependent on the behavior of the exponent at large, so the integral relation had to be replaced by the local condition (52), besides controlling the behavior of the TPPA near the maximum is more important than the weight of the exponent away from the wall. Thus, instead of the value of *B*, we include the value among the parameters sought, and the value of *B* is expressed from Eqs. (50) and (51).
