3. Singularities in solutions of three-dimensional boundary layer equations

The laminar boundary layer problem on a thin round cone with the half apex angle <sup>δ</sup><sup>c</sup> , , 1 at the angle of attack <sup>α</sup>\* depends on the parameter <sup>k</sup> <sup>¼</sup> <sup>4</sup>α<sup>∗</sup>=ð Þ <sup>3</sup>δ<sup>c</sup> only. Firstly, analytical results about singularities were obtained for outer BL part for a such cone. It is understood from previous works [15–18, 20], the singularity can arise when two subcharacteristic (streamlines) families collided —this is a necessary condition. Such situation arises usually in the leeward symmetry (runoff) plane over a body of revolution at an angle of attack. Unusual properties in numerical solutions of self-similar equations in this plane for a round slender cone in supersonic freestreams were studied in many works due to the practical interest of the heat exchange on flying vehicles head parts [16–18, 20]. In this case, one parameter defines the flow. Two solutions were found in the windward symmetry (attachment) plane and at small angles of attack (k≤kc) in the leeward symmetry plane. In this plane, no solutions were

#### 3D Boundary Layer Theory DOI: http://dx.doi.org/10.5772/intechopen.83519

obtained at moderate angles of attack (kc ≤k≤ 2=3) and many solutions at larger incidences up to BL separation (2=3≤k , 1). Full BL equation solutions with initial conditions in the windward symmetry plane fixed the violation of symmetry conditions in the runoff plane, a velocity jump through this plane in the angle of attack diapason, when the self-similar solution has been absent [10, 21]. The task for the cone was solved numerically on the base of parabolized Navier-Stokes equations, without the streamwise viscous diffusion [20]. However the problem is retained since the flow structure and reasons of unusual BL properties have not been explained.

Analytical solutions of full equations for the outer BL part on the slender round cone with initial conditions in the windward symmetry plane showed the singularity presence in the leeward symmetry plane of the logarithmic type at k ¼ 1=3 and of a power type at k . 1=3 [10, 21]. It had been shown numerical solutions provided incorrect results near the singularity due to the accuracy loss. Similar but more complex results were obtained for arbitrary cones; they allow defining the sufficient conditions of the singularity arising [10, 22]. The asymptotic flow structure at large Reynolds number near the singularity on the base of Navier-Stokes equations was constructed, and analytical solutions in different asymptotic regions were obtained, which were matched with BL solutions. The analysis of the viscous-inviscid interaction region, in particular, revealed that the singularity can arise not only in selfsimilar but in full 3D BL equations [10, 22]. The theory showed that the singularity appearance relates with eigensolutions of the BL equations appearing near the runoff plane; it also explained numerical modeling results on the base of parabolized Navier-Stokes equations.

In the outer BL part, the theory gives the critical angle of attack for the singularity appearance kc ¼ 1=3. However calculations showed that this parameter is a function on numbers of Mach M and Prandtl Pr and the wall temperature hw, kc ¼ kcð Þ M∞; Pr; hw [10]. This indicates that a singularity can arise in the near-wall region. The series decomposition of the near-wall solution in the runoff plane showed the presence of a parameter α, the linear combination of skin friction components, and the sign change of which leads to the change of the physical flow

Figure 3. Solutions of boundary layer equations (dotted lines) and parabolized Navier-Stokes equations (solid lines).

These figures demonstrate that the asymptotic solution very well reproduce numerical results as for the skin friction and for displacement thicknesses in the

, Re <sup>¼</sup> <sup>2</sup>:<sup>246</sup> � <sup>10</sup><sup>6</sup>

Displacement thickness distributions on the model of supercritical wing; <sup>M</sup> <sup>¼</sup> <sup>0</sup>:99, Re <sup>¼</sup> <sup>2</sup>:<sup>246</sup> � <sup>10</sup><sup>6</sup>,

Boundary Layer Flows - Theory, Applications and Numerical Methods

The laminar boundary layer problem on a thin round cone with the half apex angle <sup>δ</sup><sup>c</sup> , , 1 at the angle of attack <sup>α</sup>\* depends on the parameter <sup>k</sup> <sup>¼</sup> <sup>4</sup>α<sup>∗</sup>=ð Þ <sup>3</sup>δ<sup>c</sup> only. Firstly, analytical results about singularities were obtained for outer BL part for a such cone. It is understood from previous works [15–18, 20], the singularity can arise when two subcharacteristic (streamlines) families collided —this is a necessary condition. Such situation arises usually in the leeward symmetry (runoff) plane over a body of revolution at an angle of attack. Unusual properties in numerical solutions of self-similar equations in this plane for a round slender cone in supersonic freestreams were studied in many works due to the practical interest of the heat exchange on flying vehicles head parts [16–18, 20]. In this case, one parameter defines the flow. Two solutions were found in the windward symmetry (attachment) plane and at small angles of attack (k≤kc) in the leeward symmetry plane. In this plane, no solutions were

3. Singularities in solutions of three-dimensional boundary layer

Skin friction distributions on the model of supercritical wing; <sup>M</sup> <sup>¼</sup> <sup>0</sup>:5, <sup>α</sup><sup>∗</sup> <sup>¼</sup> <sup>12</sup>:09°

large parameter diapason.

(solid lines), and Re <sup>¼</sup> <sup>2</sup>:<sup>58</sup> � 107 (dotted lines).

equations

6

Figure 1.

Figure 2.

and <sup>α</sup><sup>∗</sup> <sup>¼</sup> <sup>3</sup>:12°

.

topology near this plane [24]. The analysis of BL equations in the near-wall region showed that α ¼ 0 corresponds to the critical value kc, and it was confirmed by all published numerical calculations [16–18, 20]. In the runoff plane, the new power type singularity in solutions of full BL equations revealed that it is related with the eigensolutions appearing near this plane. Calculation results for BL on delta wing confirm the singularity presence.

<sup>h</sup> <sup>¼</sup> hw <sup>þ</sup> hru � <sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83519

3D Boundary Layer Theory

Me <sup>¼</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>M</sup><sup>2</sup>

α<sup>∗</sup>, simple expressions for outer functions are

uyy ¼ Awu<sup>φ</sup> þ vuy,

wyy ¼ Aww<sup>φ</sup> þ vwy þ w

3.2 Singularities in the outer boundary layer region

to the wall), flow functions are represented as [17, 36]

the first-order approximation satisfy to equations [10, 21]

<sup>h</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>H</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

<sup>U</sup>ηη <sup>þ</sup> <sup>η</sup>U<sup>η</sup> � aAU<sup>φ</sup> <sup>¼</sup> <sup>0</sup>, Wηη <sup>þ</sup> <sup>η</sup>W<sup>η</sup> � <sup>2</sup>

<sup>U</sup>ð Þ¼ <sup>η</sup>; <sup>φ</sup> <sup>C</sup>1erfc <sup>η</sup><sup>=</sup> ffiffi

These equations have solutions:

which satisfy to equations [10, 21, 22]

9

web<sup>φ</sup> þ 2 1ð Þ þ M weφb ¼ 2pMweφ, pð Þ¼ φ 1 þ 1 þ

wea<sup>φ</sup> <sup>þ</sup> <sup>2</sup>ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> weφ<sup>a</sup> <sup>¼</sup> <sup>2</sup>Nweφ, Nð Þ¼ <sup>φ</sup> <sup>3</sup>Mð Þ¼ <sup>φ</sup> <sup>K</sup>�<sup>1</sup>

2 Meu<sup>2</sup>

, hr ¼ 1 � hw þ

� �,

<sup>u</sup> <sup>þ</sup> Kw � � � <sup>h</sup> <sup>2</sup>

2

AW<sup>φ</sup> þ ð Þ 1 þ 3K W � �

3 2 K � � 1

:

2

Me þ hw � 1 � �,

, v ¼ � f þ Kg þ Ag<sup>φ</sup>

2 3

For the slender round cone with the apex half angle δc< < 1 at the angle of attack

we <sup>¼</sup> <sup>2</sup>α<sup>∗</sup> sin <sup>φ</sup>, Kð Þ¼ <sup>φ</sup> <sup>k</sup> cos <sup>φ</sup>, Að Þ¼ <sup>φ</sup> <sup>k</sup> sin <sup>φ</sup>, k <sup>¼</sup> <sup>4</sup>α<sup>∗</sup>

In the outer boundary layer region, y ≫ 1, (y is the Dorodnitsyn variable normal

<sup>u</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>U</sup>ð Þ <sup>η</sup>; <sup>φ</sup> , w <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>W</sup>ð Þ <sup>η</sup>; <sup>φ</sup> , <sup>η</sup> <sup>¼</sup> ð Þ <sup>y</sup> � <sup>δ</sup> <sup>=</sup> ffiffiffiffiffiffiffiffiffi

Me þ hw � 1 � �<sup>U</sup> � <sup>1</sup>

Here δ φð Þ is the displacement thickness defined by the equation of F. Moore [6], the function að Þ φ is found from the local self-similarity condition, and U , , 1 and W , , 1 are velocity perturbations with respect to boundary conditions, which in

Wð Þ¼� η; φ bð Þ φ U, W1ð Þ¼� η; φ bð Þ φ U þ B1ð Þk Vð Þ η; φ

Constants C<sup>1</sup> and B<sup>1</sup> are calculated from matching condition with a numerical solution inside the boundary layer. These solutions satisfy to initial conditions in the attachment plane and must tend to zero at η ! ∞. The function Vð Þ η; φ is the solution of the homogeneous equation for the cross-velocity perturbation, when the right-hand side equals to zero; it is expressed by Veber-Hermite functions [21]. The coefficient B<sup>1</sup> � 1=Kð Þ 0 has the singularity at Kð Þ! 0 0. For the round cone this limit corresponds to zero angle of attack; in this case, the analytical expression for W1ð Þ η; φ shows the presence of the power type singularity in the leeward plane φ ¼ φ<sup>1</sup> [10, 21]. The first solution Wð Þ η; φ is regular in this limit, and its behavior is defined by functions að Þ φ and bð Þ φ ,

2

<sup>2</sup> � � <sup>p</sup>

1 2 Me,

> <sup>3</sup> <sup>þ</sup> <sup>K</sup> � �:

> > 3δc

<sup>a</sup>ð Þ <sup>φ</sup> <sup>p</sup> ,

MeU<sup>2</sup> (8)

¼ 2 3

apð Þ φ U:

(9)

(10)

(11)

(6)

(7)

#### 3.1 Self-similar boundary layer on a cone

The 3D laminar boundary layer on a conical surface in the orthogonal coordinate system xyφ (Figure 3) is described by following self-similar equations and boundary conditions [10, 22]:

$$\begin{aligned} u\_{\gamma\gamma} &= Awu\_{\sigma} + vu\_{\gamma} + A\_1 w(u - w), \\ w\_{\gamma\gamma} &= Aww\_{\sigma} + vw\_{\gamma} + w\left(\frac{2}{3}u + Aw\right) - h\left(\frac{2}{3} + K\right), \\ h\_{\gamma\gamma} &= Awh\_{\sigma} + vh\_{\gamma} - M\_{\epsilon}\left(u\_{\gamma}^2 + \frac{3}{2}A\_1 w\_{\gamma}^2\right), \rho h = 1, \\ \gamma &= \epsilon \sqrt{\frac{3\rho\_{\epsilon}\mu\_{\epsilon}}{2\pi\mu\_{\epsilon}}} \left[\rho d \frac{y}{l}\right.\left.\text{Re}\left.e^{-2}\right| = \frac{\rho\_{\infty}u\_{\infty}l}{\mu\_{\infty}}\right.\end{aligned} \tag{4}$$
 
$$\begin{aligned} f\_{\gamma} = u, \,\,g\_{\gamma} = w, \,\,v = -f - \left[K - \frac{1}{2}A(\ln\left(\rho\_{\epsilon}\mu\_{\epsilon}/\mu\_{\epsilon}\right))\_{\varphi}\right] \text{g} - A\varrho\_{\varphi}, \\ y = 0: \,\,u = v = w = 0, \,\,h = h\_{w}\left(h\_{\gamma} = 0\right); \,\,y = \infty: \,u = w = h = 1. \end{aligned}$$

Equation coefficients are defined by expressions

$$\begin{aligned} M\_{\epsilon}(\rho) &= (\gamma - 1)M\_{\infty}^2 \frac{u\_{\epsilon}^2}{h\_{\epsilon}}, \; K(\rho) = \frac{2w\_{eq}}{3Ru\_{\epsilon}}, \\ A(\rho) &= \frac{2w\_{\epsilon}}{3Ru\_{\epsilon}}, \; A\_1(\rho) = \frac{2}{3} \left(\frac{w\_{\epsilon}}{u\_{\epsilon}}\right)^2 \end{aligned} \tag{5}$$

In these equations, to reduce formulas, Pr ¼ 1 and the linear dependence of the viscosity on the temperature (ρμ ¼ 1) are assumed. Indexes y and φ denote derivatives with respect to the corresponding variables; x is the distance from the body nose along the generator referenced to the body length l; y is the Dorodnitsyn variable; y<sup>∗</sup> is normal to the body surface; φ is the transversal coordinate, and it can be the polar angle for a round cone; f y ð Þ ; φ and g y ð Þ ; φ are longitudinal and transverse stream functions; v y ð Þ ; φ is transformed normal velocity; and Rð Þ φ is the metric coefficient. The density ρ, the enthalpy h, the viscosity μ, the longitudinal u, and transversal w velocities are referenced to the values at the outer boundary indexed by e, which are normalized to their freestream values indexed by ∞; they are functions of φ only. The transversal velocity on the outer boundary layer edge we ¼ 0 in the initial value plane (the attachment plane) φ ¼ 0, in which Kð Þ 0 . 0, and in the runoff plane φ ¼ φ1, in which K φ<sup>1</sup> ð Þ¼�k , 0, and two boundary layer parts that came from different sides of the attachment plane collided. For the round cone, φ<sup>1</sup> ¼ π.

Eq. (4) is simplified for slender bodies since in this case, ue ¼ ρ<sup>e</sup> ¼ μ<sup>e</sup> ¼ 1, A<sup>1</sup> , , 1. Neglecting proportional to A<sup>1</sup> terms in (4), we obtain the Crocco integral for the enthalpy and momentum equations in the form

3D Boundary Layer Theory DOI: http://dx.doi.org/10.5772/intechopen.83519

topology near this plane [24]. The analysis of BL equations in the near-wall region showed that α ¼ 0 corresponds to the critical value kc, and it was confirmed by all published numerical calculations [16–18, 20]. In the runoff plane, the new power type singularity in solutions of full BL equations revealed that it is related with the eigensolutions appearing near this plane. Calculation results for BL on delta wing

Boundary Layer Flows - Theory, Applications and Numerical Methods

The 3D laminar boundary layer on a conical surface in the orthogonal coordinate system xyφ (Figure 3) is described by following self-similar equations and bound-

> u þ Kw � �

� �

<sup>l</sup> , Re <sup>¼</sup> <sup>ε</sup>�<sup>2</sup> <sup>¼</sup> <sup>ρ</sup>∞u∞<sup>l</sup>

Að Þ ln ð Þ ρeμe=ue <sup>φ</sup> � �

, Kð Þ¼ φ

we ue

2 3

<sup>y</sup> þ 3 2 A1w<sup>2</sup> y

2

∞ u2 e he

, A1ð Þ¼ φ

Eq. (4) is simplified for slender bodies since in this case, ue ¼ ρ<sup>e</sup> ¼ μ<sup>e</sup> ¼ 1, A<sup>1</sup> , , 1. Neglecting proportional to A<sup>1</sup> terms in (4), we obtain the Crocco integral

In these equations, to reduce formulas, Pr ¼ 1 and the linear dependence of the viscosity on the temperature (ρμ ¼ 1) are assumed. Indexes y and φ denote derivatives with respect to the corresponding variables; x is the distance from the body nose along the generator referenced to the body length l; y is the Dorodnitsyn variable; y<sup>∗</sup> is normal to the body surface; φ is the transversal coordinate, and it can be the polar angle for a round cone; f y ð Þ ; φ and g y ð Þ ; φ are longitudinal and transverse stream functions; v y ð Þ ; φ is transformed normal velocity; and Rð Þ φ is the metric coefficient. The density ρ, the enthalpy h, the viscosity μ, the longitudinal u, and transversal w velocities are referenced to the values at the outer boundary indexed by e, which are normalized to their freestream values indexed by ∞; they are functions of φ only. The transversal velocity on the outer boundary layer edge we ¼ 0 in the initial value plane (the attachment plane) φ ¼ 0, in which Kð Þ 0 . 0, and in the runoff plane φ ¼ φ1, in which K φ<sup>1</sup> ð Þ¼�k , 0, and two boundary layer parts that came from different sides of the attachment plane collided. For the round

<sup>y</sup> <sup>¼</sup> <sup>0</sup> : <sup>u</sup> <sup>¼</sup> <sup>v</sup> <sup>¼</sup> <sup>w</sup> <sup>¼</sup> <sup>0</sup>, h <sup>¼</sup> hw hy <sup>¼</sup> <sup>0</sup> � �; y <sup>¼</sup> <sup>∞</sup> : <sup>u</sup> <sup>¼</sup> <sup>w</sup> <sup>¼</sup> <sup>h</sup> <sup>¼</sup> <sup>1</sup>:

� <sup>h</sup> <sup>2</sup>

<sup>3</sup> <sup>þ</sup> <sup>K</sup> � �

, ρh ¼ 1,

μ∞ ,

> 2we<sup>φ</sup> 3Rue ,

,

g � Agφ,

� �<sup>2</sup> (5)

(4)

uyy ¼ Awu<sup>φ</sup> þ vuy þ A1w uð Þ � w ,

2 3

confirm the singularity presence.

ary conditions [10, 22]:

cone, φ<sup>1</sup> ¼ π.

8

3.1 Self-similar boundary layer on a cone

wyy ¼ Aww<sup>φ</sup> þ vwy þ w

y ¼ ε

hyy <sup>¼</sup> Awh<sup>φ</sup> <sup>þ</sup> vhy � Me <sup>u</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffi 3ρeue 2xμ<sup>e</sup>

 ð y∗

0

Með Þ¼ <sup>φ</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>M</sup><sup>2</sup>

2we 3Rue

<sup>ρ</sup><sup>d</sup> <sup>y</sup><sup>∗</sup>

r

<sup>f</sup> <sup>y</sup> <sup>¼</sup> u, gy <sup>¼</sup> w, v ¼ �<sup>f</sup> � <sup>K</sup> � <sup>1</sup>

Equation coefficients are defined by expressions

Að Þ¼ φ

for the enthalpy and momentum equations in the form

$$\begin{aligned} h &= h\_w + h\_r u - \frac{1}{2} M\_\epsilon u^2, \; h\_r = 1 - h\_w + \frac{1}{2} M\_\epsilon, \\ M\_\epsilon &= (\chi - 1) M^2, \; v = -\left( f + Kg + A g\_\varphi \right), \\ u\_{\mathcal{yy}} &= A u u\_\varphi + v u\_\mathcal{yy} \\ w\_{\mathcal{yy}} &= A u w u\_\varphi + v u w\_\mathcal{y} + w \left( \frac{2}{3} u + K w \right) - h \left( \frac{2}{3} + K \right). \end{aligned} \tag{6}$$

For the slender round cone with the apex half angle δc< < 1 at the angle of attack α<sup>∗</sup>, simple expressions for outer functions are

$$w\_{\varepsilon} = 2a^\* \sin \varrho,\ K(\varrho) = k \cos \varrho,\ A(\varrho) = k \sin \varrho,\ k = \frac{4a^\*}{3\delta\_c} \tag{7}$$

#### 3.2 Singularities in the outer boundary layer region

In the outer boundary layer region, y ≫ 1, (y is the Dorodnitsyn variable normal to the wall), flow functions are represented as [17, 36]

$$\begin{aligned} \mu &= \mathbf{1} + U(\boldsymbol{\eta}, \boldsymbol{\rho}), \; w = \mathbf{1} + W(\boldsymbol{\eta}, \boldsymbol{\rho}), \; \eta = (\boldsymbol{\eta} - \boldsymbol{\delta}) / \sqrt{a(\boldsymbol{\rho})}, \\\ h &= \mathbf{1} + H = \mathbf{1} - \left(\frac{\mathbf{1}}{2} M\_{\varepsilon} + h\_{w} - \mathbf{1}\right) U - \frac{\mathbf{1}}{2} M\_{\varepsilon} U^{2} \end{aligned} \tag{8}$$

Here δ φð Þ is the displacement thickness defined by the equation of F. Moore [6], the function að Þ φ is found from the local self-similarity condition, and U , , 1 and W , , 1 are velocity perturbations with respect to boundary conditions, which in the first-order approximation satisfy to equations [10, 21]

$$dU\_{\eta\eta} + \eta U\_{\eta} - aAU\_{\eta} = 0,\\ \mathcal{W}\_{\eta\eta} + \eta \mathcal{W}\_{\eta} - \frac{2}{3}a\left[\frac{3}{2}AW\_{\eta} + (1+3K)W\right] = \frac{2}{3}ap(\eta)U. \tag{9}$$

These equations have solutions:

$$\begin{aligned} U(\eta,\rho) &= \mathbf{C\_1} \text{erfc}(\eta/\sqrt{2}) \\ W(\eta,\rho) &= -b(\rho)U, \; W\_1(\eta,\rho) = -b(\rho)U + B\_1(k)V(\eta,\rho) \end{aligned} \tag{10}$$

Constants C<sup>1</sup> and B<sup>1</sup> are calculated from matching condition with a numerical solution inside the boundary layer. These solutions satisfy to initial conditions in the attachment plane and must tend to zero at η ! ∞. The function Vð Þ η; φ is the solution of the homogeneous equation for the cross-velocity perturbation, when the right-hand side equals to zero; it is expressed by Veber-Hermite functions [21]. The coefficient B<sup>1</sup> � 1=Kð Þ 0 has the singularity at Kð Þ! 0 0. For the round cone this limit corresponds to zero angle of attack; in this case, the analytical expression for W1ð Þ η; φ shows the presence of the power type singularity in the leeward plane φ ¼ φ<sup>1</sup> [10, 21]. The first solution Wð Þ η; φ is regular in this limit, and its behavior is defined by functions að Þ φ and bð Þ φ , which satisfy to equations [10, 21, 22]

$$\begin{aligned} w\_{\varepsilon}b\_{\varphi} + 2(1+M)w\_{\varepsilon\varphi}b &= 2pMw\_{\varepsilon\varphi}, \; p(\varphi) = 1 + \left(1 + \frac{3}{2}K\right)\left(\frac{1}{2}M\_{\varepsilon} + h\_{w} - 1\right), \\ w\_{\varepsilon}a\_{\varphi} + 2(N+1)w\_{\varepsilon\varphi}a &= 2Nw\_{\varepsilon\varphi}, \; N(\varphi) = 3M(\varphi) = K^{-1}. \end{aligned} \tag{11}$$

Solutions of these equations with initial conditions in the attachment plane are represented in integral forms in the general case and have analytical expressions for the round cone [10, 21]. Their properties near the leeward plane, at ζ ¼ φ<sup>1</sup> � φ , , 1, are represented by expressions

Uyy þ kUzz þ ð Þ 1 � k yUy þ kzUz ¼ 0,

DOI: http://dx.doi.org/10.5772/intechopen.83519

2 z

ð Þ <sup>1</sup> � <sup>k</sup> <sup>=</sup><sup>2</sup> � � <sup>p</sup> erf z<sup>=</sup> ffiffi

� �Bz � <sup>2</sup>ð Þ <sup>m</sup> � <sup>1</sup> <sup>B</sup> ¼ �2mp1F zð Þ,Fzð Þ¼ erf <sup>z</sup><sup>=</sup> ffiffi

3 2 ; � 1 2 z2 � �, Bm <sup>¼</sup> bm <sup>R</sup> ffiffiffiffiffi

ζ ! 0 is the viscous-inviscid interaction. This effect is important in the region, where the inviscid and induced cross velocities have same orders; this condition defines the transverse dimension of the region Δφ and the

<sup>ε</sup> <sup>p</sup> <sup>x</sup>�<sup>1</sup>

<sup>Δ</sup><sup>φ</sup> � ffiffi

The function B zð Þ is expressed by Kummer's function Φð Þ a; b; x [10, 22, 23]:

B0ð Þz is a particular solution of the inhomogeneous equation; the coefficient Bm

In Figure 4, comparisons of solutions of BL (dotted lines) and Navier-Stokes (solid lines) equations for m ¼ 1=2 (curves 1 and 2) and m ¼ 1 (curves 3 and 4) are presented. It is seen that regular solutions of Navier-Stokes equations are converged

Another effect generated by the singularity at k≥1=3 due to the BL growth at

<sup>4</sup>, we � kRue

In this region, the flow has the two-layer structure. Assuming the potential flow in the outer inviscid region, the solution here is presented by the improper integral

ffiffi <sup>ε</sup> <sup>p</sup> <sup>x</sup>�<sup>1</sup>

<sup>þ</sup> kz � �Wz <sup>þ</sup> <sup>2</sup>k mð Þ � <sup>1</sup> <sup>W</sup> <sup>þ</sup>

For k , 1 these equations have the solution corresponding to the regular at k ! 0

2 3

<sup>2</sup> � � <sup>p</sup> , W ¼ �B zð ÞC1erfc <sup>y</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � � <sup>p</sup>

kx <sup>p</sup> <sup>=</sup>ε<sup>1</sup> � �2 1ð Þ �<sup>m</sup>

<sup>p</sup>1<sup>U</sup> <sup>¼</sup> <sup>0</sup> (14)

ð Þ <sup>1</sup> � <sup>k</sup> <sup>=</sup><sup>2</sup> � � <sup>p</sup> ,

(15)

: (16)

<sup>4</sup>: (17)

Wyy þ kWzz þ ð Þ 1 � k yWy þ

U y ð Þ¼ ; <sup>z</sup> <sup>C</sup><sup>1</sup> erfc y ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B ¼ mp1B0ð Þþ z BmΦ 1 � m;

is determined from matching condition.

quickly to singular solutions of BL equations.

The general flow scheme and the coordinate system for a cone.

solution of BL equations:

3D Boundary Layer Theory

2 z þ z

Bzz þ

velocity scale:

Figure 4.

11

$$m\_{\epsilon} = \frac{3}{2} k R \zeta, \; k = -K(\rho\_1), \; R = R(\rho\_1), \; p\_1 = p(\rho\_1), \; n = 3m = -K^{-1}(\rho\_1),$$

$$m \neq \mathbf{1}: \; b = \frac{m p\_1}{m - 1} - b\_m \zeta^{2(m - 1)}, \; m = \mathbf{1}: \; b = -2p\_1 \ln \zeta + b\_1. \tag{12}$$

$$n \neq \mathbf{1}: \; a = \frac{n}{n - 1} + a\_n \zeta^{2(n - 1)}, \; n = \mathbf{1}: \; a = -2 \ln \zeta + a\_1$$

Here an and bm are known coefficients [10, 21]. These formulas are true for nonslender bodies also [10, 22, 23].

These results show the presence in the outer BL part of two singularity types in the leeward plane related with properties of functions að Þ φ and bð Þ φ . For k , 1 the function Uð Þ η; ζ exists at ζ ¼ 0 but reaches this limit irregularly; its behavior is studied analytically in details for the slender round cone [10, 21]. For k≥1 the function Uð Þ η; ζ is singular at ζ ! 0 since að Þ!ζ ∞ and the BL thickness tend to infinity as ffiffiffiffiffiffiffiffiffi <sup>a</sup>ð Þ<sup>ζ</sup> <sup>p</sup> : the logarithmic singularity type takes place at k ¼ 1, and it is of the power type at k . 1. At k≥1 the flow separation is observed in experimental and numerical studies; this phenomenon changes not only the outer part but also the inner boundary layer structure. It should be noted that such behavior of velocity viscous perturbations near the BL outer part at the separation development is a new property in the comparison with the 2D flows.

The function Wð Þ η; ζ has irregular but finite limit in the leeward plane for ζ ! 0 at k , 1=3. This limit is singular at k≥ 1=3: the singularity has the logarithmic or power type, if k ¼ 1=3 or k . 1=3. At 1=3≤k , 1 the singularity is related with the behavior of cross-flow velocity only. This singularity leads to the longitudinal vortex component strengthening in the outer part of the viscous region. The singularity takes place, if the pressure gradient is negative (k≤2=3) or positive (k . 2=3). It is formed by BL proper solutions, which have homogeneous conditions on both boundaries and arise near the runoff plane. The critical value kc ¼ 1=3 for the outer BL part is undependable on the wall temperature and Mach and Prandtl numbers; however the considered singularities define the real flow structure near the leeward plane at k≥1=3 [17, 36, 37].

#### 3.3 Asymptotic flow structure near the singularity

Due to the irregularity of solutions already at k≥1=6 (m ≤2), the vortex boundary region near the runoff plane is formed with transverse dimension <sup>ζ</sup> � <sup>ε</sup> <sup>1</sup> <sup>2</sup>�<sup>m</sup>; at m � 1 this value is of the order of the BL thickness ε. In this region, the transverse diffusion is the effect of the first order, and to describe it we introduce the following variables:

$$\begin{aligned} \varepsilon\_1 &= \left[ \frac{3}{2} \text{Re}\rho\_\epsilon(\rho\_1) \mathfrak{u}\_\epsilon(\rho\_1) / \mu\_\epsilon(\rho\_1) \right]^{-\frac{1}{2}}. \\ \varepsilon &= \sqrt{\text{k}\text{xR}} \zeta/e\_1, \; \mathfrak{u} = \mathfrak{u}(\mathfrak{y}, \mathfrak{z}), \; \mathfrak{h} = \mathfrak{h}(\mathfrak{y}, \mathfrak{z}), \; \mathfrak{w} = \mathfrak{w}(\mathfrak{y}, \mathfrak{z}) \end{aligned} \tag{13}$$

Using these variables from Navier-Stokes equations at ζ � ε<sup>1</sup> , , 1 for this region, we derive self-similar equations, which in its outer part, at y ≫ 1, reduce to the form

3D Boundary Layer Theory DOI: http://dx.doi.org/10.5772/intechopen.83519

Solutions of these equations with initial conditions in the attachment plane are represented in integral forms in the general case and have analytical expressions for

kRζ, k ¼ �<sup>K</sup> <sup>φ</sup><sup>1</sup> ð Þ, R <sup>¼</sup> <sup>R</sup> <sup>φ</sup><sup>1</sup> ð Þ, p<sup>1</sup> <sup>¼</sup> <sup>p</sup> <sup>φ</sup><sup>1</sup> ð Þ, n <sup>¼</sup> <sup>3</sup><sup>m</sup> ¼ �K�<sup>1</sup> <sup>φ</sup><sup>1</sup> ð Þ,

� bmζ2ð Þ <sup>m</sup>�<sup>1</sup> , m <sup>¼</sup> <sup>1</sup> : <sup>b</sup> ¼ �2p1ln <sup>ζ</sup> <sup>þ</sup> <sup>b</sup>1,

Here an and bm are known coefficients [10, 21]. These formulas are true for non-

These results show the presence in the outer BL part of two singularity types

The function Wð Þ η; ζ has irregular but finite limit in the leeward plane for ζ ! 0 at k , 1=3. This limit is singular at k≥ 1=3: the singularity has the logarithmic or power type, if k ¼ 1=3 or k . 1=3. At 1=3≤k , 1 the singularity is related with the behavior of cross-flow velocity only. This singularity leads to the longitudinal vortex component strengthening in the outer part of the viscous region. The singularity takes place, if the pressure gradient is negative (k≤2=3) or positive (k . 2=3). It is formed by BL proper solutions, which have homogeneous conditions on both boundaries and arise near the runoff plane. The critical value kc ¼ 1=3 for the outer BL part is undependable on the wall temperature and Mach and Prandtl numbers; however the considered singularities define the real flow structure near the leeward

Due to the irregularity of solutions already at k≥1=6 (m ≤2), the vortex bound-

<sup>p</sup> <sup>R</sup>ζ=ε1, u <sup>¼</sup> u yð Þ ; <sup>z</sup> , h <sup>¼</sup> h yð Þ ; <sup>z</sup> , w <sup>¼</sup> w yð Þ ; <sup>z</sup>

Using these variables from Navier-Stokes equations at ζ � ε<sup>1</sup> , , 1 for this region, we derive self-similar equations, which in its outer part, at y ≫ 1, reduce to

2 :

ary region near the runoff plane is formed with transverse dimension <sup>ζ</sup> � <sup>ε</sup> <sup>1</sup>

<sup>2</sup> Reρ<sup>e</sup> <sup>φ</sup><sup>1</sup> ð Þue <sup>φ</sup><sup>1</sup> ð Þ=μ<sup>e</sup> <sup>φ</sup><sup>1</sup> ð Þ � ��<sup>1</sup>

m � 1 this value is of the order of the BL thickness ε. In this region, the transverse diffusion is the effect of the first order, and to describe it we introduce the following

<sup>a</sup>ð Þ<sup>ζ</sup> <sup>p</sup> : the logarithmic singularity type takes place

in the leeward plane related with properties of functions að Þ φ and bð Þ φ . For k , 1 the function Uð Þ η; ζ exists at ζ ¼ 0 but reaches this limit irregularly; its behavior is studied analytically in details for the slender round cone [10, 21]. For k≥1 the function Uð Þ η; ζ is singular at ζ ! 0 since að Þ!ζ ∞ and the BL

at k ¼ 1, and it is of the power type at k . 1. At k≥1 the flow separation is observed in experimental and numerical studies; this phenomenon changes not only the outer part but also the inner boundary layer structure. It should be noted that such behavior of velocity viscous perturbations near the BL outer part at the separation development is a new property in the comparison with

(12)

<sup>2</sup>�<sup>m</sup>; at

(13)

<sup>n</sup> � <sup>1</sup> <sup>þ</sup> anζ2ð Þ <sup>n</sup>�<sup>1</sup> , n <sup>¼</sup> <sup>1</sup> : <sup>a</sup> ¼ �2ln <sup>ζ</sup> <sup>þ</sup> <sup>a</sup><sup>1</sup>

the round cone [10, 21]. Their properties near the leeward plane, at

Boundary Layer Flows - Theory, Applications and Numerical Methods

ζ ¼ φ<sup>1</sup> � φ , , 1, are represented by expressions

we <sup>¼</sup> <sup>3</sup> 2

the 2D flows.

variables:

the form

10

plane at k≥1=3 [17, 36, 37].

3.3 Asymptotic flow structure near the singularity

<sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>3</sup>

<sup>z</sup> <sup>¼</sup> ffiffiffiffiffi kx

<sup>m</sup> 6¼ <sup>1</sup> : <sup>b</sup> <sup>¼</sup> mp<sup>1</sup>

<sup>n</sup> 6¼ <sup>1</sup> : <sup>a</sup> <sup>¼</sup> <sup>n</sup>

m � 1

slender bodies also [10, 22, 23].

thickness tend to infinity as ffiffiffiffiffiffiffiffiffi

$$\begin{aligned} U\_{\mathcal{W}} + kU\_{\mathcal{xx}} + (1 - k)yU\_{\mathcal{y}} + kzU\_{\mathcal{z}} &= 0, \\ W\_{\mathcal{W}} + kW\_{\mathcal{xz}} + (1 - k)yW\_{\mathcal{y}} + \left(\frac{2}{\mathfrak{z}} + kz\right)W\_{\mathfrak{z}} + 2k(m - 1)W + \frac{2}{3}p\_1U &= 0 \end{aligned} \tag{14}$$

For k , 1 these equations have the solution corresponding to the regular at k ! 0 solution of BL equations:

$$\begin{aligned} U(y, z) &= \mathbf{C}\_1 \operatorname{erfc}\left(y\sqrt{(1-k)/2}\right) \operatorname{erf}\left(z/\sqrt{2}\right), \; W = -B(z)\mathbf{C}\_1 \operatorname{erfc}\left(y\sqrt{(1-k)/2}\right), \\ B\_{xx} + \left(\frac{2}{z} + z\right)B\_x - 2(m-1)B &= -2mp\_1F(z), \; F(z) = \operatorname{erf}\left(z/\sqrt{2}\right) \end{aligned} \tag{15}$$

The function B zð Þ is expressed by Kummer's function Φð Þ a; b; x [10, 22, 23]:

$$B = m p\_1 B\_0(z) + B\_m \Phi\left(1 - m, \frac{3}{2}, -\frac{1}{2}z^2\right), \ B\_m = b\_m \left(R\sqrt{k\varkappa}/e\_1\right)^{2(1-m)}.\tag{16}$$

B0ð Þz is a particular solution of the inhomogeneous equation; the coefficient Bm is determined from matching condition.

In Figure 4, comparisons of solutions of BL (dotted lines) and Navier-Stokes (solid lines) equations for m ¼ 1=2 (curves 1 and 2) and m ¼ 1 (curves 3 and 4) are presented. It is seen that regular solutions of Navier-Stokes equations are converged quickly to singular solutions of BL equations.

Another effect generated by the singularity at k≥1=3 due to the BL growth at ζ ! 0 is the viscous-inviscid interaction. This effect is important in the region, where the inviscid and induced cross velocities have same orders; this condition defines the transverse dimension of the region Δφ and the velocity scale:

$$
\Delta\rho \sim \sqrt{\varepsilon} \mathbf{x}^{-\frac{1}{4}}, \ w\_{\varepsilon} \sim k R u\_{\varepsilon} \sqrt{\varepsilon} \mathbf{x}^{-\frac{1}{4}}.\tag{17}
$$

In this region, the flow has the two-layer structure. Assuming the potential flow in the outer inviscid region, the solution here is presented by the improper integral

Figure 4. The general flow scheme and the coordinate system for a cone.

from the displacement thickness δð Þ x; s . In the boundary layer, the flow is described by full 3D equations:

solution behavior of Eq. (6) at y , , 1 in the runoff plane φ ¼ φ<sup>1</sup> where the solution

2

0

ð Þ τ<sup>0</sup> � kθ<sup>0</sup>

(21)

(22)

(23)

(24)

(25)

<sup>u</sup><sup>0</sup> <sup>¼</sup> <sup>τ</sup>0<sup>y</sup> <sup>þ</sup> <sup>U</sup>0ð Þ<sup>y</sup> , w<sup>0</sup> <sup>¼</sup> <sup>θ</sup>0<sup>y</sup> <sup>þ</sup> <sup>W</sup>0ð Þ<sup>y</sup> , v<sup>0</sup> ¼ �αy<sup>2</sup> � <sup>F</sup><sup>0</sup> <sup>þ</sup> kG0, <sup>α</sup> <sup>¼</sup> <sup>1</sup>

Second terms of these decompositions can be presented by series

αiy<sup>i</sup>þ<sup>4</sup> ð Þ i þ 4 !

> βiy<sup>i</sup>þ<sup>2</sup> ð Þ i þ 2 !

> > 3

<sup>6</sup> <sup>k</sup>β0y<sup>3</sup> 

In the plane ζ ¼ 0, the cross-flow velocity w ¼ 0 due to symmetry conditions. Here two critical points, in which v ¼ 0, can be. The first point locates on the cone surface y ¼ 0, and the second one y ¼ �yc appears in the physical space at α , 0, if p . 0 (k , 2=3), that corresponds to small angles of attack for the round cone and at α . 0, if p , 0. Commonly, the critical value of the cross-flow velocity gradient kc ≤1=3 corresponds to the negative cross-flow pressure gradient p . 0, the trans-

Using these expressions, the equation for the subcharacteristics is obtained in

; <sup>α</sup> 6¼ <sup>0</sup> : <sup>y</sup> <sup>¼</sup> ycy0<sup>s</sup>

The subcharacteristic behavior is shown in Figure 5a and b for p . 0. At α . 0 velocities v , 0 and w , 0; the only critical point node is in the coordinate origin, and subcharacteristics go to it from the region ζ 6¼ 0 (Figure 5a). At α ¼ 0 yc ¼ 0 and the point ζ ¼ y ¼ 0 is double critical point of the type saddle node: the saddle is

Using these decompositions we can study qualitatively a dependence of the flow structure near the runoff plane from parameters by analyzing the subcharacteristic behavior. The transformed normal to the body surface v and transverse w velocities at ζ , , 1 and y , , 1 in the first-order approximation are represented in the form

, F<sup>0</sup> ¼ ∑ i¼0

, G<sup>0</sup> ¼ ∑ i¼0

¼ � <sup>1</sup>

, w ¼ �w<sup>0</sup> ¼ �kθ0ζy

αiyiþ<sup>5</sup> ð Þ i þ 5 ! ,

> βiy<sup>i</sup>þ<sup>3</sup> ð Þ i þ 3 ! :

ð Þ <sup>τ</sup><sup>0</sup> � <sup>3</sup>kθ<sup>0</sup> <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>2</sup>pMeτ<sup>2</sup>

<sup>6</sup> <sup>k</sup>β0y<sup>2</sup> <sup>y</sup> <sup>þ</sup> yc 

β yc <sup>þ</sup> <sup>y</sup><sup>0</sup> <sup>1</sup> � <sup>s</sup><sup>β</sup> ð Þ, s <sup>¼</sup> <sup>ζ</sup>

ζ0 

 ,

i¼0

i¼0

First three coefficients of these series are defined by relations

is presented in the form

3D Boundary Layer Theory

dy , <sup>θ</sup><sup>0</sup> <sup>¼</sup> dw0ð Þ <sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83519

dy ,

U<sup>0</sup> ¼ F0<sup>y</sup> ¼ ∑

W<sup>0</sup> ¼ G0<sup>y</sup> ¼ ∑

α<sup>0</sup> ¼ �2τ0α, α<sup>1</sup> ¼ kτ0β0, α<sup>2</sup> ¼ kτ0β<sup>1</sup> <sup>β</sup><sup>0</sup> ¼ �phw, <sup>β</sup><sup>1</sup> ¼ �pτ0hr, <sup>β</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>

<sup>v</sup> <sup>¼</sup> <sup>v</sup><sup>0</sup> ¼ � <sup>α</sup>y<sup>2</sup> � <sup>1</sup>

<sup>¼</sup> <sup>6</sup><sup>α</sup> kphw

yc ¼ � <sup>6</sup><sup>α</sup> kβ<sup>0</sup>

verse skin friction in this region θ<sup>0</sup> . 0.

<sup>ζ</sup> , <sup>β</sup> <sup>¼</sup> <sup>α</sup>

1 � y0dln s

kθ<sup>0</sup>

Here y<sup>0</sup> and z<sup>0</sup> define the initial point in the cross-plane.

, d <sup>¼</sup> phw 6θ<sup>0</sup>

the form

13

ycdy y y þ yc

<sup>¼</sup> <sup>β</sup> <sup>d</sup><sup>ζ</sup>

<sup>α</sup> <sup>¼</sup> <sup>0</sup> : <sup>y</sup> <sup>¼</sup> <sup>y</sup><sup>0</sup>

<sup>τ</sup><sup>0</sup> <sup>¼</sup> du0ð Þ <sup>0</sup>

$$\begin{aligned} s &= \frac{R\zeta}{\sqrt{\varepsilon}}, \ w\_{\varepsilon} = \frac{3}{2} u\_{\varepsilon} \sqrt{\varepsilon} W\_{\varepsilon}(\mathbf{x}, s) \ W\_{\varepsilon}(\mathbf{x}, s) = -k\varepsilon [1 + r], \ r = \frac{4m}{\pi} \frac{\partial}{\partial x} \left[ \frac{\delta(\mathbf{x}, t) dt}{\delta^2 - t^2} \right] \\\\ v &= f + \mathcal{H} + \mathcal{A}\mathfrak{g}\_{\varepsilon} + \frac{2}{3} \mathfrak{x} \mathcal{f}\_{\text{x}}, \ h = h\_{w} + h\_{r} u - \frac{1}{2} M\_{\varepsilon}(\rho\_{1}) u^{2} \\\\ u\_{\mathcal{\mathcal{H}}} &= W\_{\varepsilon} u u\_{\delta} + \nu u\_{\mathcal{\mathcal{H}}} + \frac{2}{3} \mathfrak{x} u u\_{\mathcal{\mathcal{E}}} \\\\ w\_{\mathcal{\mathcal{H}}} &= W\_{\varepsilon} u w\_{\delta} + \nu u\_{\mathcal{\mathcal{H}}} + w \left( \frac{2}{3} u + W\_{\varepsilon} w \right) - h \left( \frac{2}{3} + W\_{\alpha} \right) + \frac{2}{3} \mathfrak{x} u u\_{\mathcal{\mathcal{E}}} \end{aligned} \tag{18}$$

For these equations boundary conditions have the form (1). A solution of these equations will be matched with the boundary layer solution at s ! ∞. Initial conditions are needed at some streamwise location x ¼ x0, which can be obtained from a solution of Navier-Stokes equations near the body nose; this feature does the problem more complicated. Obtained equations allow a self-similar solution for hypersonic flows at some additional assumptions.

The solution in the outer boundary layer part, at y ≫ 1, is described by formulas

$$\begin{aligned} t &= y / \sqrt{d(\mathbf{x}, s)}, \; u = \mathbf{1} + U(\mathbf{x}, t, s), \; w = \mathbf{1} - c(\mathbf{x}, s)U \\ U &= \mathbf{C}\_1 \operatorname{erfc}(t/\sqrt{2}), \; p\_0 = \frac{3}{2} \left( \frac{1}{2} M\_0 + h\_w - \mathbf{1} \right) \\ (1 + r)s d\_i &- 2m \mathbf{x} d\_x - 2(n - \mathbf{1} - r\_i)d = -2n \\ (1 + r)s c\_i &- 2m \mathbf{x} c\_x - 2(m - \mathbf{1} - r\_i)c = -2m(p\_1 - q p\_0) \end{aligned} \tag{19}$$

Along characteristics ξð Þ¼ x; s const, which are streamlines of the inviscid flow, the equations for functions d ¼ dð Þ ξ; s and c ¼ cð Þ ξ; s are integrated. At s ! 0 these functions are represented in the form

$$\begin{aligned} c &= \mathcal{G}^L + \frac{m\left(p\_1 + p\_0 r\_s\right)}{m - 1 - r}, & L(\xi, s) &= \frac{m - 1 - r\_s}{1 + r}; \; d = D\mathcal{s}^I + \frac{n}{n - 1 - r\_s}, \\ I(\xi, s) &= \frac{n - 1 - r\_s}{1 + r}, & C = b\_m \mathcal{e}^{m - 1}, \; D = a\_n \mathcal{e}^{n - 1}. \end{aligned} \tag{20}$$

Coefficients C and D are obtained by matching dð Þ ξ; s and cð Þ ξ; s at s ! ∞ in relation to að Þζ and bð Þζ at ζ ! 0 [17, 36]. The logarithmic singularity appears in these functions at I ¼ 0 or L ¼ 0. At Lð Þ ξ; 0 , 0 or Ið Þ ξ; 0 , 0, the singularity is of the power type.

Following from presented results, in contrast with the 2D separation, the viscous-inviscid interaction does not eliminate the singularity in 3D boundary layer; this effect moves only the critical value of kc.

#### 3.4 Singularities in the boundary layer near-wall region

The singularity in the outer BL part gives the critical value kc ¼ 1=3, although calculations show kc ¼ kcð Þ M∞; Pr; hw . This indicates on the possibility of singularity arising in the near-wall region. To study this possibility, at the first, we study the

from the displacement thickness δð Þ x; s . In the boundary layer, the flow is described

<sup>ε</sup> <sup>p</sup> Weð Þ <sup>x</sup>; <sup>s</sup> Weð Þ¼� <sup>x</sup>; <sup>s</sup> ks½ � <sup>1</sup> <sup>þ</sup> <sup>r</sup> , r <sup>¼</sup> <sup>4</sup><sup>m</sup>

2

� <sup>h</sup> <sup>2</sup>

Me <sup>φ</sup><sup>1</sup> ð Þu<sup>2</sup>

<sup>3</sup> <sup>þ</sup> Wes � �

M<sup>0</sup> þ hw � 1 � �

; d <sup>¼</sup> Ds<sup>I</sup> <sup>þ</sup>

� �

n n � 1 � rs

,

xf <sup>x</sup>, h <sup>¼</sup> hw <sup>þ</sup> hru � <sup>1</sup>

u þ Wesw � �

For these equations boundary conditions have the form (1). A solution of these equations will be matched with the boundary layer solution at s ! ∞. Initial conditions are needed at some streamwise location x ¼ x0, which can be obtained from a solution of Navier-Stokes equations near the body nose; this feature does the problem more complicated. Obtained equations allow a self-similar

The solution in the outer boundary layer part, at y ≫ 1, is described by formulas

d xð Þ ; <sup>s</sup> <sup>p</sup> , u <sup>¼</sup> <sup>1</sup> <sup>þ</sup> U xð Þ ; <sup>t</sup>; <sup>s</sup> , w <sup>¼</sup> <sup>1</sup> � c xð Þ ; <sup>s</sup> <sup>U</sup>

2 1 2

ð Þ 1 þ r scs � 2mxcx � 2ð Þ m � 1 � rs c ¼ �2m p<sup>1</sup> � qp<sup>0</sup>

Along characteristics ξð Þ¼ x; s const, which are streamlines of the inviscid flow, the equations for functions d ¼ dð Þ ξ; s and c ¼ cð Þ ξ; s are integrated. At s ! 0 these

> m � 1 � rs 1 þ r

, D <sup>¼</sup> anε<sup>n</sup>�<sup>1</sup>

Coefficients C and D are obtained by matching dð Þ ξ; s and cð Þ ξ; s at s ! ∞ in relation to að Þζ and bð Þζ at ζ ! 0 [17, 36]. The logarithmic singularity appears in these functions at I ¼ 0 or L ¼ 0. At Lð Þ ξ; 0 , 0 or Ið Þ ξ; 0 , 0, the singularity is of

Following from presented results, in contrast with the 2D separation, the viscous-inviscid interaction does not eliminate the singularity in 3D boundary layer;

The singularity in the outer BL part gives the critical value kc ¼ 1=3, although calculations show kc ¼ kcð Þ M∞; Pr; hw . This indicates on the possibility of singularity arising in the near-wall region. To study this possibility, at the first, we study the

:

π

þ 2 3 xuwx

∂ ∂x ∞ð

0

δð Þ x; t dt s<sup>2</sup> � t<sup>2</sup>

(18)

(19)

(20)

by full 3D equations:

v ¼ f þ Kg þ Ags þ

uyy ¼ Wewus þ vuy þ

wyy ¼ Wewws þ vwy þ w

2 ue ffiffi

> 2 3

<sup>t</sup> <sup>¼</sup> <sup>y</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

functions are represented in the form

� � m � 1 � r

this effect moves only the critical value of kc.

<sup>c</sup> <sup>¼</sup> CsL <sup>þ</sup> m p<sup>1</sup> <sup>þ</sup> <sup>p</sup>0rs

n � 1 � rs 1 þ r

Ið Þ¼ ξ; s

the power type.

12

<sup>U</sup> <sup>¼</sup> <sup>C</sup><sup>1</sup> erfc t<sup>=</sup> ffiffi

2 3 xuux

> 2 3

Boundary Layer Flows - Theory, Applications and Numerical Methods

solution for hypersonic flows at some additional assumptions.

<sup>2</sup> � � <sup>p</sup> , p<sup>0</sup> <sup>¼</sup> <sup>3</sup>

, Lð Þ¼ ξ; s

, C <sup>¼</sup> bmε<sup>m</sup>�<sup>1</sup>

3.4 Singularities in the boundary layer near-wall region

ð Þ 1 þ r sds � 2mxdx � 2ð Þ n � 1 � rs d ¼ �2n

<sup>s</sup> <sup>¼</sup> <sup>R</sup><sup>ζ</sup> ffiffi <sup>ε</sup> <sup>p</sup> , we <sup>¼</sup> <sup>3</sup> solution behavior of Eq. (6) at y , , 1 in the runoff plane φ ¼ φ<sup>1</sup> where the solution is presented in the form

$$\begin{aligned} \tau\_0 &= \frac{du\_0(0)}{dy}, \; \theta\_0 = \frac{dw\_0(0)}{dy}, \\\\ u\_0 &= \tau\_0 y + U\_0(y), \; w\_0 = \theta\_0 y + W\_0(y), \; \nu\_0 = -a\eta^2 - F\_0 + kG\_0, \; a = \frac{1}{2}(\tau\_0 - k\theta\_0) \end{aligned} \tag{21}$$

Second terms of these decompositions can be presented by series

$$\begin{split} U\_{0} = F\_{0\text{y}} &= \sum\_{i=0} \frac{a\_{i} y^{i+4}}{(i+4)!}, \; F\_{0} = \sum\_{i=0} \frac{a\_{i} y^{i+5}}{(i+5)!}, \\ W\_{0} = G\_{0\text{y}} &= \sum\_{i=0} \frac{\beta\_{i} y^{i+2}}{(i+2)!}, \; G\_{0} = \sum\_{i=0} \frac{\beta\_{i} y^{i+3}}{(i+3)!}. \end{split} \tag{22}$$

First three coefficients of these series are defined by relations

$$\begin{aligned} a\_0 &= -2\tau\_0 a, \ a\_1 = k\tau\_0 \beta\_0, \ a\_2 = k\tau\_0 \beta\_1\\ \beta\_0 &= -p h\_w, \ \beta\_1 = -p \tau\_0 h\_r, \ \beta\_2 = \frac{1}{3}(\tau\_0 - 3k\theta\_0)\theta\_0 + 2p\mathsf{M}\_\varepsilon \tau\_0^2 \end{aligned} \tag{23}$$

Using these decompositions we can study qualitatively a dependence of the flow structure near the runoff plane from parameters by analyzing the subcharacteristic behavior. The transformed normal to the body surface v and transverse w velocities at ζ , , 1 and y , , 1 in the first-order approximation are represented in the form

$$\begin{aligned} v = v\_0 &= -\left( a\eta^2 - \frac{1}{6} k\beta\_0 \eta^3 \right) = -\frac{1}{6} k\beta\_0 \eta^2 (y + y\_c) \\\ y\_c = -\frac{6a}{k\beta\_0} &= \frac{6a}{kph\_w}, \; w = -w\_0 = -k\theta\_0 \zeta y \end{aligned} \tag{24}$$

In the plane ζ ¼ 0, the cross-flow velocity w ¼ 0 due to symmetry conditions. Here two critical points, in which v ¼ 0, can be. The first point locates on the cone surface y ¼ 0, and the second one y ¼ �yc appears in the physical space at α , 0, if p . 0 (k , 2=3), that corresponds to small angles of attack for the round cone and at α . 0, if p , 0. Commonly, the critical value of the cross-flow velocity gradient kc ≤1=3 corresponds to the negative cross-flow pressure gradient p . 0, the transverse skin friction in this region θ<sup>0</sup> . 0.

Using these expressions, the equation for the subcharacteristics is obtained in the form

$$\begin{aligned} \frac{y\_c dy}{y(y+y\_c)} &= \beta \frac{d\zeta}{\zeta}, \; \beta = \frac{a}{k\theta\_0}; \; a \neq 0: y = \frac{y\_c y\_0 s^{\beta}}{y\_c + y\_0(1-s^{\beta})}, \; s = \left| \frac{\zeta}{\zeta\_0} \right|, \\\ a = 0: y = \frac{y\_0}{1 - y\_0 d \ln s}, \; d = \frac{p h\_w}{6\theta\_0} \end{aligned} \tag{25}$$

Here y<sup>0</sup> and z<sup>0</sup> define the initial point in the cross-plane.

The subcharacteristic behavior is shown in Figure 5a and b for p . 0. At α . 0 velocities v , 0 and w , 0; the only critical point node is in the coordinate origin, and subcharacteristics go to it from the region ζ 6¼ 0 (Figure 5a). At α ¼ 0 yc ¼ 0 and the point ζ ¼ y ¼ 0 is double critical point of the type saddle node: the saddle is

ξ ∂2 U<sup>0</sup> ∂ξ<sup>2</sup> þ

ξ ∂2 W<sup>0</sup> ∂ξ<sup>2</sup> þ

ξ ! ∞:

Figure 6.

15

2 <sup>3</sup> � <sup>ξ</sup> � � ∂U<sup>0</sup>

3D Boundary Layer Theory

2 <sup>3</sup> � <sup>ξ</sup> � � ∂W<sup>0</sup>

<sup>U</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup>00<sup>ð</sup>

y

DOI: http://dx.doi.org/10.5772/intechopen.83519

0 e �1 3αs 3 ds þ τ<sup>0</sup>

� <sup>3</sup>β<sup>2</sup> τ<sup>0</sup> � 9kθ<sup>0</sup>

W<sup>0</sup> ¼ B00Φ �c;

<sup>∂</sup><sup>ξ</sup> ¼ � <sup>τ</sup><sup>0</sup> 3

3ξ α � �<sup>1</sup> 3 , c ¼ 2

> <sup>3</sup><sup>α</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> 6α

<sup>∂</sup><sup>ξ</sup> <sup>þ</sup> cW<sup>0</sup> ¼ � <sup>β</sup><sup>1</sup>

3ξ α � �<sup>1</sup> 3 ,

þ B01ξ

<sup>1</sup>=3Φ 1 <sup>3</sup> � <sup>c</sup>; 4 3 ; ξ � �

� <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>3</sup>Meτ<sup>0</sup> τ<sup>0</sup> � 3kθ<sup>0</sup>

with zero right-hand sides; A00, B00, and B<sup>01</sup> are constants; Φð Þ a; b; x is Kummer's degenerate hypergeometric function, which has asymptotes at

First terms of these expressions are solutions of homogeneous equations,

Solutions grow exponentially at α , 0 and p . 0; they cannot be matched with the solution in the main BL part. Therefore, at these conditions a solution of BL equations cannot exist. This conclusion and also the criterion (26) for the boundary of the existing leeward symmetry plane solution are confirmed by numerical calculations for the slender round cone at an angle of attack [25–32, 37], a part of which is presented in Figure 6. In this figure, symbols correspond to calculations of limit

The boundary of the solution existing in the leeward symmetry plane of the slender round cone at the angle of

attack and Pr ¼ 1 in the dependence of the critical value kc: ▲, [28]; ■, [29]; and ○, [37].

Solutions of these equations can be represented as

2 3 ; ξ � �

> 3ξ α � �<sup>1</sup> 3

<sup>α</sup> . <sup>0</sup>, <sup>ξ</sup> , <sup>0</sup> : <sup>Φ</sup> � �ð Þ<sup>ξ</sup> <sup>c</sup>

τ<sup>0</sup> � 3kθ<sup>0</sup> 9α

> 3ξ α � �<sup>1</sup> 3 � 2

U<sup>00</sup>

; α , 0, ξ . 0 : Φ � e

<sup>9</sup><sup>α</sup> ð Þ <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>3</sup>Meτ<sup>0</sup> <sup>U</sup><sup>0</sup>

� <sup>3</sup>β<sup>1</sup> 2ð Þ τ<sup>0</sup> � 3kθ<sup>0</sup>

ξ

ξ<sup>c</sup>�2=<sup>3</sup> (31)

(29)

(30)

Figure 5. Subcharacteristics in the cross-plane at α ≥0 (a) and α , 0 (b); p . 0.

in the lower half-plane, i.e., out of the physical space. The node is in the upper halfplane, and the subcharacteristic pattern retains the same as at α . 0. At α , 0 the node drifts in the point ζ ¼ 0, y ¼ yc . 0, and the coordinate origin becomes by the saddle point (Figure 5b). In this case, at y . yc the normal velocity v , 0 and at 0 , y , yc v . 0; v ¼ 0 on the line y ¼ yc.

This analysis shows that at the parameter α sign change, the physical flow structure varies qualitatively, and the value α ¼ 0 is a criterion of the new flow property appearance. It should be noted that in solutions of Navier-Stokes equations for similar problems near the coordinate origin z ¼ y ¼ 0 in the leeward symmetry plane, the streamwise-oriented vortex arises, and the flow is not described by the BL theory since the viscous diffusion inside the vortex is distributed along the radius from its axis, but not along the normal to the body surface. On the base of this qualitative analysis, it is supposed that the critical value kcð Þ hw; M is defined by the relation

$$
\Delta a(k\_{\varepsilon}) = \tau\_0(k\_{\varepsilon}) - k\_{\varepsilon}\theta\_0(k\_{\varepsilon}) = \mathbf{0} \tag{26}
$$

To support this hypothesis, equations for functions U0ð Þy and W0ð Þy are analyzed by substituting near-wall decompositions to Eq. (6). Considering functions U0ð Þy and W0ð Þy as perturbations, we can linearize resulting equations and obtain in the first-order approximation:

$$\begin{aligned} U\_{0\eta\gamma} + a\eta^2 U\_{0\eta} + \tau\_0 (F\_0 - kG\_0) &= -a\tau\_0 \eta^2, \\ W\_{0\eta\gamma} + a\eta^2 W\_{0\eta} - \frac{2}{3} (\tau\_0 - 3\theta\_0) \eta W\_0 + \theta\_0 (F\_0 - kG\_0) &= \\ \beta\_0 + \beta\_1 \eta + \frac{1}{2} \beta\_2 \eta^2 + \left[\frac{2}{3} \theta\_0 \eta - p(h\_r - 2M\_\epsilon \tau\_0 \eta)\right] U\_0 \end{aligned} \tag{27}$$

At y ! 0 U0ð Þy and W0ð Þy are expressed by above series, and in order to match them with the solution of full Eq. (6) in the main BL part, it is required that these functions will grow at y ! ∞ not faster than a power function. To study their solution behavior at y ! ∞ and α 6¼ 0, we introduce the new variable:

$$\xi = -ay^3/3, \ y = -(\mathfrak{Z}\xi/a)^{\frac{1}{5}}.\tag{28}$$

At the limit ξ ! ∞, previous equations are reduced in the first-order approximation to the form

3D Boundary Layer Theory DOI: http://dx.doi.org/10.5772/intechopen.83519

$$\begin{aligned} \xi \frac{\partial^2 U\_0}{\partial \xi^2} + \left(\frac{2}{3} - \xi\right) \frac{\partial U\_0}{\partial \xi} &= -\frac{\tau\_0}{3} \left(\frac{3\xi}{a}\right)^{\frac{1}{3}}, c = 2\frac{\tau\_0 - 3k\theta\_0}{9a} \\ \xi \frac{\partial^2 W\_0}{\partial \xi^2} + \left(\frac{2}{3} - \xi\right) \frac{\partial W\_0}{\partial \xi} + cW\_0 &= -\frac{\beta\_1}{3a} + \frac{\beta\_2}{6a} \left(\frac{3\xi}{a}\right)^{\frac{1}{3}} - \frac{2}{9a} (\theta\_0 + 3\mathsf{M}\_\ell \tau\_0) U\_0 \end{aligned} \tag{29}$$

Solutions of these equations can be represented as

$$\begin{split} U\_{0} &= A\_{00} \Big| \Big[ e^{-\frac{1}{3}\mu^{3}} ds + \tau\_{0} \left( \frac{3\xi}{\alpha} \right)^{\frac{1}{3}}, \\ W\_{0} &= B\_{00} \Phi \Big( -\varepsilon, \frac{2}{3}, \xi \Big) + B\_{01} \xi^{\circ \flat} \Phi \Big( \frac{1}{3} - \varepsilon, \frac{4}{3}, \xi \Big) - \frac{3\theta\_{1}}{2(\tau\_{0} - 3k\theta\_{0})} \\ & \qquad - \frac{3\theta\_{2}}{\tau\_{0} - 9k\theta\_{0}} \left( \frac{3\xi}{\alpha} \right)^{\frac{1}{3}} - \frac{\theta\_{0} + 3\mathcal{M}\_{\varepsilon}\tau\_{0}}{\tau\_{0} - 3k\theta\_{0}} U\_{00} \end{split} \tag{30}$$

First terms of these expressions are solutions of homogeneous equations, with zero right-hand sides; A00, B00, and B<sup>01</sup> are constants; Φð Þ a; b; x is Kummer's degenerate hypergeometric function, which has asymptotes at ξ ! ∞:

$$a \ge 0, \xi \le 0 \;:\ \Phi \sim (-\xi)^{\epsilon}; \; a \le 0, \xi \ge 0 \;:\ \Phi \sim e^{\xi} \xi^{\epsilon - 2\zeta} \tag{31}$$

Solutions grow exponentially at α , 0 and p . 0; they cannot be matched with the solution in the main BL part. Therefore, at these conditions a solution of BL equations cannot exist. This conclusion and also the criterion (26) for the boundary of the existing leeward symmetry plane solution are confirmed by numerical calculations for the slender round cone at an angle of attack [25–32, 37], a part of which is presented in Figure 6. In this figure, symbols correspond to calculations of limit

Figure 6.

The boundary of the solution existing in the leeward symmetry plane of the slender round cone at the angle of attack and Pr ¼ 1 in the dependence of the critical value kc: ▲, [28]; ■, [29]; and ○, [37].

in the lower half-plane, i.e., out of the physical space. The node is in the upper halfplane, and the subcharacteristic pattern retains the same as at α . 0. At α , 0 the node drifts in the point ζ ¼ 0, y ¼ yc . 0, and the coordinate origin becomes by the saddle point (Figure 5b). In this case, at y . yc the normal velocity v , 0 and at

This analysis shows that at the parameter α sign change, the physical flow structure varies qualitatively, and the value α ¼ 0 is a criterion of the new flow property appearance. It should be noted that in solutions of Navier-Stokes equations for similar problems near the coordinate origin z ¼ y ¼ 0 in the leeward symmetry plane, the streamwise-oriented vortex arises, and the flow is not described by the BL theory since the viscous diffusion inside the vortex is distributed along the radius from its axis, but not along the normal to the body surface. On the base of this qualitative analysis, it is supposed that the critical value kcð Þ hw; M is defined by

To support this hypothesis, equations for functions U0ð Þy and W0ð Þy are analyzed by substituting near-wall decompositions to Eq. (6). Considering functions U0ð Þy and W0ð Þy as perturbations, we can linearize resulting equations and obtain

At y ! 0 U0ð Þy and W0ð Þy are expressed by above series, and in order to match them with the solution of full Eq. (6) in the main BL part, it is required that these functions will grow at y ! ∞ not faster than a power function. To study their solution behavior at y ! ∞ and α 6¼ 0, we introduce the new

<sup>U</sup>0yy <sup>þ</sup> <sup>α</sup>y<sup>2</sup>U0<sup>y</sup> <sup>þ</sup> <sup>τ</sup>0ð Þ¼� <sup>F</sup><sup>0</sup> � kG<sup>0</sup> ατ0y<sup>2</sup>,

2 3

3

<sup>ξ</sup> ¼ �αy<sup>3</sup>

2αð Þ¼ kc τ0ð Þ� kc kcθ0ð Þ¼ kc 0 (26)

ð Þ τ<sup>0</sup> � 3θ<sup>0</sup> yW<sup>0</sup> þ θ0ð Þ¼ F<sup>0</sup> � kG<sup>0</sup>

U<sup>0</sup>

<sup>3</sup>: (28)

(27)

θ0y � p hð Þ <sup>r</sup> � 2Meτ0y 

<sup>=</sup>3, y ¼ �ð Þ <sup>3</sup>ξ=<sup>α</sup> <sup>1</sup>

At the limit ξ ! ∞, previous equations are reduced in the first-order approxi-

0 , y , yc v . 0; v ¼ 0 on the line y ¼ yc.

Subcharacteristics in the cross-plane at α ≥0 (a) and α , 0 (b); p . 0.

Boundary Layer Flows - Theory, Applications and Numerical Methods

in the first-order approximation:

<sup>W</sup>0yy <sup>þ</sup> <sup>α</sup>y<sup>2</sup>W0<sup>y</sup> � <sup>2</sup>

1 2 <sup>β</sup>2y<sup>2</sup> <sup>þ</sup>

β<sup>0</sup> þ β1y þ

the relation

Figure 5.

variable:

14

mation to the form

values αð Þ kc for the solution existing at different boundary conditions in the diapason of Mach numbers from 2 to ∞ at the Prandtl number 1 for different surface temperatures. At k , 1=3 data are grouped near the value α ¼ 0 in accordance with the criterion (26). The data scatter is, apparently, due to the decrease of the calculation accuracy at the approach to the critical value kc and also with errors of data copying from papers. At k . 1=3, all calculations are finished with α . 0, since the solution existing in this region is determined by singularities in the outer BL part, but not in the near-wall region.

Then we consider the solution behavior of full BL equations in the near-wall region beside the runoff plane at ζ , , 1. 3D BL equations have the parabolic type, and their solution before the runoff plane knows nothing about the solution in this plane; however, in order for the first solution to move smoothly into the last one at α . 0, the first will be locally self-similar. Due to this condition, the streamwise τ ζð Þ and cross-flow θ ζð Þ friction stresses and the self-similar variable η at ζ , , 1 will be defined by expressions

$$\pi(\zeta) = \frac{\tau\_0}{a(\zeta)}, \ \theta(z) = \frac{\theta\_0}{a(\zeta)}, \ \eta = \frac{\mathcal{Y}}{a(\zeta)}\tag{32}$$

<sup>F</sup>ð Þ¼ <sup>η</sup>; <sup>ζ</sup> <sup>F</sup>0ð Þþ <sup>η</sup> <sup>ζ</sup><sup>q</sup>

3D Boundary Layer Theory

<sup>G</sup>ð Þ¼ <sup>η</sup>; <sup>ζ</sup> <sup>G</sup>0ð Þþ <sup>η</sup> <sup>ζ</sup><sup>q</sup>

<sup>ξ</sup> ¼ � αη<sup>3</sup>

region.

tributions f

Figure 7.

17

approximation and (b) angle of attack.

ʹʹ

<sup>1</sup> ð Þ<sup>z</sup> and <sup>g</sup>ʹʹ

Wqð Þ¼ ξ Bq0Φ

þ

ζ , , 1, which at η ! ∞, has the form [37]

DOI: http://dx.doi.org/10.5772/intechopen.83519

4 3 ; 2 3 ; ξ 

þ Bq1ξ

<sup>U</sup><sup>0</sup> � <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>3</sup>Meτ<sup>0</sup> τ<sup>0</sup> � 3kθ<sup>0</sup>

<sup>3</sup> , Uqð Þ¼ <sup>ξ</sup> Aq0<sup>Φ</sup>

4 <sup>3</sup> � <sup>c</sup>; 2 3 ; ξ 

θ<sup>0</sup> þ 3Meτ<sup>0</sup> 2τ<sup>0</sup>

Fqð Þþ <sup>η</sup> …, Uð Þ¼ <sup>η</sup>; <sup>ζ</sup> <sup>U</sup>0ð Þþ <sup>η</sup> <sup>ζ</sup><sup>q</sup>

Gqð Þþ <sup>η</sup> …, Wð Þ¼ <sup>η</sup>; <sup>ζ</sup> <sup>W</sup>0ð Þþ <sup>η</sup> <sup>ζ</sup><sup>q</sup>

The first term of this expansion is the solution for the runoff plane but depends on the self-similar variable. Second terms define the proper solution of BL Eq. (6) at

> þ Aq1ξ 1 <sup>3</sup><sup>Φ</sup> <sup>5</sup> 3 ; 4 3 ; ξ

> > <sup>3</sup> � <sup>c</sup>; 4 3 ; ξ

> > > Uq:

Here Aq0, Aq1, Bq0, and Bq<sup>1</sup> are constants. These relations show that the proper solution in near-wall BL region near the runoff plane is nonzero. It is irregular at α≥ 0 and it is singular at α , 0. The logarithmic singularity is not in this case, and the solution of BL equations exists at the critical value kc in contrast to the outer

In the work of [15], at the analysis of perturbations in the boundary layer related with the angle of attack, it was found that they lead to infinite disturbances in the symmetry plane, although equations have no visible singularities contained. In this case, the first-order approximation is described by the Blasius solution for the delta flat plate. In Figure 7, dimensionless longitudinal and transverse skin friction dis-

(Figure 7a) and the angle of attack (Figure 7b) are presented in dependence on transverse coordinate z ¼ 1 � Z=X, where X and Z are Cartesian streamwise and transverse coordinates. By approaching the symmetry plane (z ¼ 1), skin friction perturbations infinitely grow. Detailed investigation of equations for these functions showed that in these cases singularities take place as in the near-wall and outer BL parts. In the outer part, the singularity corresponds to values of the parameter

Skin friction distributions on the small aspect ratio delta wing at М = 2 related with (a) second BL

<sup>1</sup> ð Þz , induced by the second order BL approximation

þ 9β<sup>1</sup> 2τ<sup>0</sup> þ

<sup>1</sup>=3<sup>Φ</sup> <sup>5</sup>

Uqð Þþ η …,

Wqð Þþ <sup>η</sup> …, (36)

3β<sup>2</sup>

3ξ α <sup>1</sup> 3

(37)

<sup>3</sup> <sup>τ</sup><sup>0</sup> � <sup>k</sup>θ<sup>0</sup>

11

The function a zð Þ at α ≥0 will satisfy to the condition að Þ¼ 0 1. In this case, flow functions in the boundary layer near the wall can be represented in the form

$$f(\eta,\zeta) = a(\zeta)\left[\tau\_0 \frac{\eta^2}{2} + F(\eta,\zeta)\right], \ u(\eta,\zeta) = f\_\eta = \tau\_0 \eta + U(\eta,\zeta)$$

$$g(\eta,\zeta) = a(\zeta)\left[\theta\_0 \frac{\eta^2}{2} + G(\eta,\zeta)\right], \ w(\eta,\zeta) = \mathfrak{g}\_\eta = \theta\_0 \eta + W(\eta,\zeta) \tag{33}$$

$$\nu = a\left[\left(a - \frac{1}{2}\theta\_0 k\zeta \frac{a\_\zeta}{a}\right)\eta^2 + F - kG\left(1 + k\zeta \frac{a\_\zeta}{a}\right) - k\zeta G\_\zeta - k\zeta \eta\_\zeta W\right]$$

Substituting these expressions to Eq. (6) and linearizing the result with respect to disturbances, we obtain the first-order approximation for the flow in the nearwall region beside the runoff plane:

$$\begin{aligned} &U\_{\eta\eta} + a\eta^2 U\_{\eta} + a^2 \left\{ k\theta\_0 \zeta \eta U\_{\zeta} + \tau\_0 \left[ F - kG \left( 1 + \frac{a\zeta}{a} \right) - k\zeta \mathcal{G}\_{\zeta} \right] \right\} = -a\tau\_0 \eta^2 \\ &W\_{\eta\eta} + a\eta^2 W\_{\eta} + a^2 \left\{ k\theta\_0 \zeta \eta W\_{\zeta} + \theta\_0 \left[ F - kG \left( 1 + \frac{\zeta a\_{\zeta}}{a} \right) - k\zeta \mathcal{G}\_{\zeta} \right] - 3a\alpha \eta W \right\} = \\ &- a\theta\_0 \eta^2 + a^2 \left\{ \beta\_0 + \beta\_1 \eta + \frac{1}{2} \theta\_3 \eta^2 + \left[ \frac{2}{3} \left( \theta\_0 + 3pM\_{\varepsilon} \varepsilon\_0 \right) \eta - ph\_r \right] U \right\} \end{aligned} \tag{34}$$

Here <sup>β</sup><sup>3</sup> <sup>¼</sup> <sup>2</sup> <sup>3</sup> <sup>τ</sup>0θ<sup>0</sup> � <sup>k</sup>θ<sup>2</sup> <sup>0</sup> <sup>þ</sup> pMeτ<sup>2</sup> 0. Due to local self-similarity at α ≥0, we define the function að Þζ as

$$
tau^2 - \frac{1}{2}k\theta\_0\zeta a a\_\zeta = a,\ a^2 = 1 + C\zeta^q,\ q = \frac{4a}{k\theta\_0} \tag{35}$$

The constant C is found from a comparison with numerical calculations. It follows from this relation at α≥0 and q , 2 the solution of Eq. (6) in the near-wall region at ζ , , 1 can find in the form of the series:

3D Boundary Layer Theory DOI: http://dx.doi.org/10.5772/intechopen.83519

values αð Þ kc for the solution existing at different boundary conditions in the diapason of Mach numbers from 2 to ∞ at the Prandtl number 1 for different surface temperatures. At k , 1=3 data are grouped near the value α ¼ 0 in accordance with the criterion (26). The data scatter is, apparently, due to the decrease of the calculation accuracy at the approach to the critical value kc and also with errors of data copying from papers. At k . 1=3, all calculations are finished with α . 0, since the solution existing in this region is determined by singularities in the outer BL part,

Then we consider the solution behavior of full BL equations in the near-wall region beside the runoff plane at ζ , , 1. 3D BL equations have the parabolic type, and their solution before the runoff plane knows nothing about the solution in this plane; however, in order for the first solution to move smoothly into the last one at α . 0, the first will be locally self-similar. Due to this condition, the streamwise τ ζð Þ and cross-flow θ ζð Þ friction stresses and the self-similar variable η at ζ , , 1 will be

τ ζð Þ¼ <sup>τ</sup><sup>0</sup>

Boundary Layer Flows - Theory, Applications and Numerical Methods

<sup>2</sup> <sup>þ</sup> <sup>F</sup>ð Þ <sup>η</sup>; <sup>ζ</sup> � �

<sup>2</sup> <sup>þ</sup> <sup>G</sup>ð Þ <sup>η</sup>; <sup>ζ</sup> � �

η2

η2

<sup>U</sup>ηη <sup>þ</sup> αη<sup>2</sup>U<sup>η</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>k</sup>θ0ζηU<sup>ζ</sup> <sup>þ</sup> <sup>τ</sup><sup>0</sup> <sup>F</sup> � kG <sup>1</sup> <sup>þ</sup>

<sup>W</sup>ηη <sup>þ</sup> αη<sup>2</sup>W<sup>η</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>k</sup>θ0ζηW<sup>ζ</sup> <sup>þ</sup> <sup>θ</sup><sup>0</sup> <sup>F</sup> � kG <sup>1</sup> <sup>þ</sup> <sup>ζ</sup>a<sup>ζ</sup>

1 2 <sup>β</sup>3η<sup>2</sup> <sup>þ</sup>

<sup>0</sup> <sup>þ</sup> pMeτ<sup>2</sup>

<sup>a</sup>ð Þ<sup>ζ</sup> , <sup>θ</sup>ð Þ¼ <sup>z</sup>

<sup>η</sup><sup>2</sup> <sup>þ</sup> <sup>F</sup> � kG <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>ζ</sup>

� �

Substituting these expressions to Eq. (6) and linearizing the result with respect to disturbances, we obtain the first-order approximation for the flow in the near-

n o h i

2

� �

<sup>k</sup>θ0ζaa<sup>ζ</sup> <sup>¼</sup> <sup>α</sup>, a<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>C</sup>ζ<sup>q</sup>

The constant C is found from a comparison with numerical calculations. It follows from this relation at α≥0 and q , 2 the solution of Eq. (6) in the near-wall

θ0

The function a zð Þ at α ≥0 will satisfy to the condition að Þ¼ 0 1. In this case, flow functions in the boundary layer near the wall can be represented in the

<sup>a</sup>ð Þ<sup>ζ</sup> , <sup>η</sup> <sup>¼</sup> <sup>y</sup>

, uð Þ¼ η; ζ f <sup>η</sup> ¼ τ0η þ Uð Þ η; ζ

, wð Þ¼ η; ζ g<sup>η</sup> ¼ θ0η þ Wð Þ η; ζ

� kζG<sup>ζ</sup> � kζηζW

� kζG<sup>ζ</sup>

U

aζ a � �

> aζ a � �

� �

� �<sup>η</sup> � phr � �

<sup>3</sup> <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>3</sup>pMeτ<sup>0</sup>

� kζG<sup>ζ</sup>

0. Due to local self-similarity at α ≥0, we define

, q <sup>¼</sup> <sup>4</sup><sup>α</sup> kθ<sup>0</sup>

a � �

� �

<sup>a</sup>ð Þ<sup>ζ</sup> (32)

¼ �ατ0η<sup>2</sup>

� 3αcηW

(33)

¼

(34)

(35)

but not in the near-wall region.

defined by expressions

fð Þ¼ η; ζ að Þζ τ<sup>0</sup>

gð Þ¼ η; ζ að Þζ θ<sup>0</sup>

2 θ0kζ aζ a

wall region beside the runoff plane:

�αθ0η<sup>2</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>β</sup><sup>0</sup> <sup>þ</sup> <sup>β</sup>1<sup>η</sup> <sup>þ</sup>

<sup>3</sup> <sup>τ</sup>0θ<sup>0</sup> � <sup>k</sup>θ<sup>2</sup>

<sup>α</sup>a<sup>2</sup> � <sup>1</sup> 2

region at ζ , , 1 can find in the form of the series:

Here <sup>β</sup><sup>3</sup> <sup>¼</sup> <sup>2</sup>

16

the function að Þζ as

� �

<sup>v</sup> <sup>¼</sup> <sup>a</sup> <sup>α</sup> � <sup>1</sup>

form

$$\begin{aligned} F(\eta,\zeta) &= F\_0(\eta) + \zeta^q F\_q(\eta) + \dots, \ U(\eta,\zeta) = U\_0(\eta) + \zeta^q U\_q(\eta) + \dots \\ G(\eta,\zeta) &= G\_0(\eta) + \zeta^q G\_q(\eta) + \dots, \ W(\eta,\zeta) = W\_0(\eta) + \zeta^q W\_q(\eta) + \dots \end{aligned} \tag{36}$$

The first term of this expansion is the solution for the runoff plane but depends on the self-similar variable. Second terms define the proper solution of BL Eq. (6) at ζ , , 1, which at η ! ∞, has the form [37]

$$\begin{split} \xi &= -\frac{a\eta^{3}}{3}, \ U\_{q}(\xi) = A\_{q0}\Phi\left(\frac{4}{3}, \frac{2}{3}, \xi\right) + A\_{q1}\xi^{\sharp}\Phi\left(\frac{5}{3}, \frac{4}{3}, \xi\right) \\ \mathcal{H}\_{q}(\xi) &= B\_{q0}\Phi\left(\frac{4}{3} - c, \frac{2}{3}, \xi\right) + B\_{q1}\xi^{\sharp\prime}\Phi\left(\frac{5}{3} - c, \frac{4}{3}, \xi\right) + \frac{9\beta\_{1}}{2\tau\_{0}} + \frac{3\beta\_{2}}{\frac{11}{3}\tau\_{0} - k\theta\_{0}} \left(\frac{3\xi}{a}\right)^{\sharp} \\ &+ \frac{\theta\_{0} + 3M\_{\varepsilon}\tau\_{0}}{2\tau\_{0}}U\_{0} - \frac{\theta\_{0} + 3M\_{\varepsilon}\tau\_{0}}{\tau\_{0} - 3k\theta\_{0}}U\_{q} . \end{split} \tag{37}$$

Here Aq0, Aq1, Bq0, and Bq<sup>1</sup> are constants. These relations show that the proper solution in near-wall BL region near the runoff plane is nonzero. It is irregular at α≥ 0 and it is singular at α , 0. The logarithmic singularity is not in this case, and the solution of BL equations exists at the critical value kc in contrast to the outer region.

In the work of [15], at the analysis of perturbations in the boundary layer related with the angle of attack, it was found that they lead to infinite disturbances in the symmetry plane, although equations have no visible singularities contained. In this case, the first-order approximation is described by the Blasius solution for the delta flat plate. In Figure 7, dimensionless longitudinal and transverse skin friction distributions f ʹʹ <sup>1</sup> ð Þ<sup>z</sup> and <sup>g</sup>ʹʹ <sup>1</sup> ð Þz , induced by the second order BL approximation (Figure 7a) and the angle of attack (Figure 7b) are presented in dependence on transverse coordinate z ¼ 1 � Z=X, where X and Z are Cartesian streamwise and transverse coordinates. By approaching the symmetry plane (z ¼ 1), skin friction perturbations infinitely grow. Detailed investigation of equations for these functions showed that in these cases singularities take place as in the near-wall and outer BL parts. In the outer part, the singularity corresponds to values of the parameter

Figure 7.

Skin friction distributions on the small aspect ratio delta wing at М = 2 related with (a) second BL approximation and (b) angle of attack.

m = 3/4 and 7/8 in relation to cases а and b, respectively. The longitudinal velocity perturbation singularity is related only with the near-wall singularity.

Near-wall singularities generate the flow structure including three asymptotic sublayers describing the viscous-inviscid interaction similar as near the 2D separation point. However, the viscous-inviscid interaction is not enough to remove the singularity of the obtained type. Near the wall sublayer close to the symmetry plane the fourth region is formed, in which the flow is described by the parabolized Navier-Stocks equations similar to the above case of the outer singularity.
