1. Introduction

Despite the intensive development of computer technologies and numerical methods for the Navier-Stokes and Reynolds equations, problems of the threedimensional boundary layer are of significant interest in the fluid dynamics. So far these problems have been little studied as a result of objective difficulties related with the large dimensionality and complexity of equations. Therefore, analytic results in this field can play an important role in the depth understanding of fluid dynamics phenomena and their study. In this part, some modern results in the three-dimensional boundary layer theory are discussed.

The small perturbation theory for inviscid flows is well developed and widely applied to estimate aerodynamic characteristics of real flight apparatus. Also it has been attempted to develop such theory for the boundary layer [1]. However, the zero approximation ("flat plate" approximation, zero cross-flow approximation) only given a rational contribution and were used in calculations. Equations for perturbations were complex. They required a numerical solution that was not much simpler than the full equation system. Father investigations of three-dimensional effects in the boundary layer theory became possible only after developments of computers with the enough power, numerical methods, and turbulence models [2].

Another approach was developed on the base of the rational perturbation theory including the first-order approximation [3–10] for some class of flows, such as flows over aircraft wings and fuselages at small angle of attack, which have high importance as for the theory and the practice. In this case, zero-order approximation functions do not depend on the cross coordinate. Equations of the first-order

approximation reduce to a two-dimensional system by introducing a new variable. The cross coordinate is included to this system as a parameter. This property of the self-similarity simplifies the solution procedures allowing to apply two-dimensional numerical methods and to reduce computing resources.

vw <sup>¼</sup> vw0ð Þþ <sup>t</sup>; <sup>s</sup> <sup>ε</sup>vw1ð Þ <sup>t</sup>; <sup>s</sup>; <sup>z</sup> , hw <sup>¼</sup> hw0ð Þþ <sup>t</sup>; <sup>s</sup> <sup>ε</sup>hw1ð Þ <sup>t</sup>; <sup>s</sup>; <sup>z</sup> , w <sup>¼</sup> <sup>ε</sup>

<sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>∗</sup>=λ, where <sup>λ</sup> is relative body thickness [4–10].

λH<sup>2</sup> � � ∂u

� �

λH<sup>2</sup> � � ∂h

∂t

� �

∂p <sup>∂</sup><sup>z</sup> � εβ

∂p1 ∂s

� �

� �

∂s þ v ∂u ∂n

∂s þ v ∂h ∂n

<sup>þ</sup> <sup>u</sup> � <sup>β</sup><sup>w</sup>

∂p1 ∂s

λH<sup>2</sup> � � ∂p

þ cos θ

� �<sup>2</sup> " #,

<sup>þ</sup> <sup>u</sup> � <sup>β</sup><sup>w</sup>

<sup>þ</sup> <sup>u</sup> � <sup>β</sup><sup>w</sup>

þð Þ <sup>γ</sup> � <sup>1</sup> <sup>M</sup><sup>2</sup> <sup>∂</sup><sup>p</sup>

∂p <sup>∂</sup><sup>z</sup> � εβ

1 λH<sup>2</sup>

wings <sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>ε</sup>=λ<sup>2</sup>

ρ ∂u ∂t

3D Boundary Layer Theory

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

ρ ∂h ∂t

ρ ∂w ∂t þ u ∂w ∂s þ v ∂w ∂n <sup>þ</sup> <sup>k</sup>1u<sup>2</sup> � <sup>k</sup>2uw � � <sup>¼</sup> <sup>∂</sup>

ρ ∂q ∂t þ u ∂q ∂s þ v ∂q ∂n þ ∂k1 ∂z <sup>u</sup><sup>2</sup> � <sup>k</sup>2uq � � <sup>¼</sup> <sup>∂</sup>

� ∂ ∂z

∂ρ ∂t þ ∂ρu ∂s þ ∂ρv ∂n þ 1 λH<sup>2</sup>

wing (Re <sup>¼</sup> <sup>2</sup>:<sup>58</sup> � <sup>10</sup><sup>7</sup>

equations [11].

5

� 1 λH<sup>2</sup>

V ¼ ð Þ¼ u; v; h; ρ; μ; κ V0ð Þþ t; s; n εV10ð Þþ t; s; n; z ε1V11ð Þ t; s; n; z :

Here, s tð Þ ; x; z is a dimensionless length of the coordinate line z ¼ const measured from the critical point xcð Þ t; z ; n is normal coordinate transformed with Dorodnitsyn transformation; vwð Þ t; x; z and hwð Þ t; x; z are blow (suction) velocity and the surface temperature; u and v, h, ρ, κ and μ are dimensionless longitudinal and normal velocities, enthalpy, density, thermal conductivity, and viscosity. The parameter ε<sup>1</sup> , , 1 is not known a priori, it describes own flow perturbations inside the boundary layer. The following is found from the analysis of equations: for thin

To calculate boundary layer characteristics, the equation system for the composite solution incorporated in all terms of asymptotic expansion (2) was derived:

> þ ∂p <sup>∂</sup><sup>s</sup> <sup>¼</sup> <sup>∂</sup> ∂n μ ∂u ∂n ,

∂s þ μ

∂p ∂s ,

� cos θ

<sup>ρ</sup><sup>q</sup> � <sup>β</sup> <sup>∂</sup>ρ<sup>w</sup> ∂s

� �

� �,q tð Þ¼ ; <sup>s</sup>; <sup>n</sup>; <sup>z</sup>

n ¼ 0 : u ¼ w ¼ 0, v ¼ vwð Þ t; x; z , h ¼ hwð Þ t; x; z

n ! ∞ : u ¼ ueð Þ t; x; z , w ¼ weð Þ t; x; z , h ¼ heð Þ t; x; z :

Eq. (3) is not true in the vicinity of the wing leading edge, where the pressure perturbation has the singularity. Using the asymptotic theory, singular regions near blunted and sharp leading edges were analyzed. It was found that the boundary layer in these regions is described by equations for the boundary layer on the sweep parabola or wedge. On a body the boundary layer begins in the critical point.

The system (2) was applied to the solution of different problems for wings and bodies [4–10]. To illustrate the developed approach in Figures 1 and 2, calculations of displacement thicknesses (Figure 1) and skin frictions (Figure 2) on the wind tunnel model of the US Air Force fighter TF-8A supercritical wing at Mach numbers M = 0.99 and 0.5 are presented. Solid lines correspond to solutions of Eq. (3) for the wing model (Re <sup>¼</sup> <sup>2</sup>:<sup>246</sup> � <sup>10</sup>6); dotted lines on Figure 4 are results for full scale

); symbols present solutions of full 3D boundary layer

∂p ∂s

¼ ∂ ∂n κ Pr ∂h ∂n

, for slightly asymmetric bodies ε characterizes the asymmetry and

∂u ∂n

> ∂n μ ∂w ∂n

∂n μ ∂q ∂n �

� k2ρu ¼ 0,

∂w ∂z ,

∂h <sup>∂</sup><sup>n</sup> <sup>¼</sup> <sup>0</sup> � �,

λ

w1ð Þ t; s; n; z ,

(2)

(3)

The singularity in the solution of 2D steady boundary layer equation is well known as the separation. Singularities arising in solutions of unsteady or 3D laminar boundary layer equations are not related directly with the flow separation and are slightly studied due to difficulties of analytical investigations of complex equations and uncertainty of numerical result treatments. However, this task is of interest for the mathematical physics and for numerical modeling of aerodynamic applications. For the first time, a singularity was found in the solution of 2D unsteady BL equations for the flow around the flat plate impulsively set into motion [12]. The singularity of the similar type was discovered on the side edge of a quarter flat plate in a uniform freestream [13] and at a collision of two jets [14]. In Ref. [15], necessary conditions were formulated for a singularity formation in self-similar solutions of the unsteady model and 3D incompressible laminar boundary layers on a flat surface with pressure gradients. Sufficient conditions and singularity types were not studied, and real flow conditions were not considered. Singularities of numerical solutions (the nonuniqueness or the absence of a solution) were found for the laminar boundary layer in the leeward symmetry plane on a round cone at incidence [16–18]. Similar results were obtained inside the computation region of the 3D turbulent boundary layer on the swept wing [19]. The singular behavior of boundary layer characteristics (the skin friction tends to the infinity in the symmetry plan) was found for the boundary layer on the small span delta wing [8, 10]. The explanation of these phenomena was found on the base of analytical solutions of laminar boundary layer equations on conical surfaces [10, 21–24]. The asymptotic flow structure on the base of Navier-Stocks equations in the singularity vicinity is constructed.

The problem of the flow separation control using plasma actuators on the base of the electrical discharge is assumed as a perspective aerodynamic instrument [26–28]. It is considered as a one method for the control of the separated flow asymmetry near the nose part of aircrafts. The problem was complicated by the absence of an adequate model for the boundary layer-discharge interaction and a criterion for flow asymmetry arising. The use as a criterion numerical results and experimental data is restricted as a result of the high sensitivity of the asymmetry origin to different parameters [29]. Solution of these problems was obtained with the development of new models [30–34].

### 2. Small perturbation theory for three-dimensional boundary layer

As follows from the cross-flow impulse equation in biorthogonal coordinates [2], the necessary conditions for a small cross velocity (|w| < < 1) are the relations

$$\frac{1}{\lambda H\_2} \frac{\partial p}{\partial \mathbf{z}} \sim \cos \theta \sim k\_1 \sim \frac{\varepsilon}{\lambda} < \varepsilon \mathbf{1}.\tag{1}$$

The small parameter ε characterizes the gradient of the pressure p tð Þ ; x; z with respect to transverse nondimensional coordinate z; t and x here are dimensionless time and longitudinal coordinate, λ is body span, H<sup>2</sup> is metric coefficient, k<sup>1</sup> is longitudinal coordinate line curvature, and θ is the angle between coordinate lines on the body surface. Using these conditions flow parameters in the 3D boundary layer are presented by asymptotic expansions:

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

approximation reduce to a two-dimensional system by introducing a new variable. The cross coordinate is included to this system as a parameter. This property of the self-similarity simplifies the solution procedures allowing to apply two-dimensional

The singularity in the solution of 2D steady boundary layer equation is well known as the separation. Singularities arising in solutions of unsteady or 3D laminar boundary layer equations are not related directly with the flow separation and are slightly studied due to difficulties of analytical investigations of complex equations and uncertainty of numerical result treatments. However, this task is of interest for the mathematical physics and for numerical modeling of aerodynamic applications. For the first time, a singularity was found in the solution of 2D unsteady BL equations for the flow around the flat plate impulsively set into motion [12]. The singularity of the similar type was discovered on the side edge of a quarter flat plate

in a uniform freestream [13] and at a collision of two jets [14]. In Ref. [15], necessary conditions were formulated for a singularity formation in self-similar solutions of the unsteady model and 3D incompressible laminar boundary layers on a flat surface with pressure gradients. Sufficient conditions and singularity types were not studied, and real flow conditions were not considered. Singularities of numerical solutions (the nonuniqueness or the absence of a solution) were found for the laminar boundary layer in the leeward symmetry plane on a round cone at incidence [16–18]. Similar results were obtained inside the computation region of the 3D turbulent boundary layer on the swept wing [19]. The singular behavior of boundary layer characteristics (the skin friction tends to the infinity in the symmetry plan) was found for the boundary layer on the small span delta wing [8, 10]. The explanation of these phenomena was found on the base of analytical solutions of laminar boundary layer equations on conical surfaces [10, 21–24]. The asymptotic flow structure on the base of Navier-Stocks equations in the singularity vicinity is

The problem of the flow separation control using plasma actuators on the base of

the electrical discharge is assumed as a perspective aerodynamic instrument [26–28]. It is considered as a one method for the control of the separated flow asymmetry near the nose part of aircrafts. The problem was complicated by the absence of an adequate model for the boundary layer-discharge interaction and a criterion for flow asymmetry arising. The use as a criterion numerical results and experimental data is restricted as a result of the high sensitivity of the asymmetry origin to different parameters [29]. Solution of these problems was obtained with

2. Small perturbation theory for three-dimensional boundary layer

<sup>∂</sup><sup>z</sup> � cos <sup>θ</sup> � <sup>k</sup><sup>1</sup> � <sup>ε</sup>

As follows from the cross-flow impulse equation in biorthogonal coordinates [2], the necessary conditions for a small cross velocity (|w| < < 1) are the relations

The small parameter ε characterizes the gradient of the pressure p tð Þ ; x; z with respect to transverse nondimensional coordinate z; t and x here are dimensionless time and longitudinal coordinate, λ is body span, H<sup>2</sup> is metric coefficient, k<sup>1</sup> is longitudinal coordinate line curvature, and θ is the angle between coordinate lines on the body surface. Using these conditions flow parameters in the 3D boundary

λ

, , 1: (1)

numerical methods and to reduce computing resources.

Boundary Layer Flows - Theory, Applications and Numerical Methods

constructed.

4

the development of new models [30–34].

1 λH<sup>2</sup>

layer are presented by asymptotic expansions:

∂p

$$\begin{aligned} \boldsymbol{w}\_{w} &= \boldsymbol{v}\_{w0}(t,s) + \varepsilon \boldsymbol{v}\_{w1}(t,s,z), \; h\_{w} = h\_{w0}(t,s) + \varepsilon h\_{w1}(t,s,z), \; w = \frac{\varepsilon}{\lambda} \boldsymbol{w}\_{1}(t,s,n,z), \\ \mathbf{V} &= (\boldsymbol{u}, \boldsymbol{v}, h, \boldsymbol{\rho}, \boldsymbol{\mu}, \kappa) = \mathbf{V}\_{0}(t,s,n) + \varepsilon \mathbf{V}\_{10}(t,s,n,z) + \varepsilon\_{1} \mathbf{V}\_{11}(t,s,n,z). \end{aligned} \tag{2}$$

Here, s tð Þ ; x; z is a dimensionless length of the coordinate line z ¼ const measured from the critical point xcð Þ t; z ; n is normal coordinate transformed with Dorodnitsyn transformation; vwð Þ t; x; z and hwð Þ t; x; z are blow (suction) velocity and the surface temperature; u and v, h, ρ, κ and μ are dimensionless longitudinal and normal velocities, enthalpy, density, thermal conductivity, and viscosity. The parameter ε<sup>1</sup> , , 1 is not known a priori, it describes own flow perturbations inside the boundary layer. The following is found from the analysis of equations: for thin wings <sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>ε</sup>=λ<sup>2</sup> , for slightly asymmetric bodies ε characterizes the asymmetry and <sup>ε</sup><sup>1</sup> <sup>¼</sup> <sup>α</sup><sup>∗</sup>=λ, where <sup>λ</sup> is relative body thickness [4–10].

To calculate boundary layer characteristics, the equation system for the composite solution incorporated in all terms of asymptotic expansion (2) was derived:

$$\begin{aligned} &\rho \left[ \frac{\partial u}{\partial t} + \left( u - \frac{\partial w}{\partial H\_2} \right) \frac{\partial u}{\partial t} + \nu \frac{\partial u}{\partial m} \right] + \frac{\partial p}{\partial x} = \frac{\partial}{\partial u} \mu \frac{\partial u}{\partial n}, \\ &\rho \left[ \frac{\partial u}{\partial t} + \left( u - \frac{\partial w}{\partial H\_2} \right) \frac{\partial h}{\partial t} + \nu \frac{\partial h}{\partial m} \right] = \frac{\partial}{\partial u} \frac{\kappa}{\mathrm{Pr}} \frac{\partial h}{\partial n} \\ &+ (\gamma - 1) \mathsf{M}^2 \left[ \frac{\partial p}{\partial t} + \left( u - \frac{\partial w}{\partial H\_2} \right) \frac{\partial p}{\partial s} + \mu \left( \frac{\partial u}{\partial n} \right)^2 \right], \\ &\rho \left[ \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial \nu} + \nu \frac{\partial w}{\partial m} + k\_1 u^2 - k\_2 \mu w \right] = \frac{\partial}{\partial u} \mu \frac{\partial w}{\partial n} \\ &- \frac{1}{\lambda H\_2} \left( \frac{\partial p}{\partial x} - \epsilon \rho \frac{\partial p}{\partial x} \right) + \cos \theta \frac{\partial p}{\partial s}, \\ &\rho \left[ \frac{\partial q}{\partial t} + u \frac{\partial q}{\partial s} + \nu \frac{\partial q}{\partial n} + \frac{\partial k\_1}{\partial x} u^2 - k\_2 uq \right] = \frac{\partial}{\partial u} \mu \frac{\partial q}{\partial n} - \\ &- \frac{\partial}{\partial x} \left[ \frac{1}{\lambda H\_2} \left( \frac{\partial p}{\partial x} - \epsilon \rho \frac{\partial p$$

Eq. (3) is not true in the vicinity of the wing leading edge, where the pressure perturbation has the singularity. Using the asymptotic theory, singular regions near blunted and sharp leading edges were analyzed. It was found that the boundary layer in these regions is described by equations for the boundary layer on the sweep parabola or wedge. On a body the boundary layer begins in the critical point.

The system (2) was applied to the solution of different problems for wings and bodies [4–10]. To illustrate the developed approach in Figures 1 and 2, calculations of displacement thicknesses (Figure 1) and skin frictions (Figure 2) on the wind tunnel model of the US Air Force fighter TF-8A supercritical wing at Mach numbers M = 0.99 and 0.5 are presented. Solid lines correspond to solutions of Eq. (3) for the wing model (Re <sup>¼</sup> <sup>2</sup>:<sup>246</sup> � <sup>10</sup>6); dotted lines on Figure 4 are results for full scale wing (Re <sup>¼</sup> <sup>2</sup>:<sup>58</sup> � <sup>10</sup><sup>7</sup> ); symbols present solutions of full 3D boundary layer equations [11].

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

parabolized Navier-Stokes equations, without the streamwise viscous diffusion [20]. However the problem is retained since the flow structure and reasons of

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

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

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

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

unusual BL properties have not been explained.

3D Boundary Layer Theory

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

parabolized Navier-Stokes equations.

Figure 3.

7

Figure 1.

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

Figure 2.

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° , Re <sup>¼</sup> <sup>2</sup>:<sup>246</sup> � <sup>10</sup><sup>6</sup> (solid lines), and Re <sup>¼</sup> <sup>2</sup>:<sup>58</sup> � 107 (dotted lines).

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