## Abstract

Some new analytical results in 3D boundary layer theory are reviewed and discussed. It includes the perturbation theory for 3D flows, analyses of 3D boundary layer equation singularities and corresponding real flow structures, investigations of 3D boundary layer distinctive features for hypersonic flows for flat blunted bodies including the heat transfer and the laminar-turbulent transition and influences of these phenomena on flows, and the new approach to the analysis of the symmetric flow instability over thin bodies and studies of the control possibility with the electrical discharge using new model of this phenomenon interaction with the 3D boundary layer. Some new analytical solutions of boundary layer and Navier-Stokes equations are presented. Applications of these results to analyze viscous flow characteristics of real objects such as aircraft wings, fuselages, and other bodies are considered.

### Keywords

- 3D boundary layer
- asymptotic perturbation theory
- singularities
- flow structures
- applications

## 1. Introduction

Despite the intensive development of computer technologies and numerical methods for the Navier-Stokes and Reynolds equations, problems of the three-dimensional 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, 4, 5, 6, 7, 8, 9, 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.

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, 17, 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, 22, 23, 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, 27, 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, 31, 32, 33, 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

The small parameter

Here,

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

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, 5, 6, 7, 8, 9, 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 (

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.

## 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 *α** depends on the parameter

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

In the outer BL part, the theory gives the critical angle of attack for the singularity appearance

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

The 3D laminar boundary layer on a conical surface in the orthogonal coordinate system

Equation coefficients are defined by expressions

In these equations, to reduce formulas,

Eq. (4) is simplified for slender bodies since in this case,

For the slender round cone with the apex half angle

### 3.2 Singularities in the outer boundary layer region

In the outer boundary layer region,

Here

These equations have solutions:

Constants

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

Here

These results show the presence in the outer BL part of two singularity types in the leeward plane related with properties of functions

The function

### 3.3 Asymptotic flow structure near the singularity

Due to the irregularity of solutions already at

Using these variables from Navier-Stokes equations at

For

The function

In Figure 4, comparisons of solutions of BL (dotted lines) and Navier-Stokes (solid lines) equations for

Another effect generated by the singularity at

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 from the displacement thickness

For these equations boundary conditions have the form (1). A solution of these equations will be matched with the boundary layer solution at

The solution in the outer boundary layer part, at

Along characteristics

Coefficients

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

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

The singularity in the outer BL part gives the critical value

Second terms of these decompositions can be presented by series

First three coefficients of these series are defined by relations

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

In the plane

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

Here

The subcharacteristic behavior is shown in Figure 5a and b for

This analysis shows that at the parameter

To support this hypothesis, equations for functions

At

At the limit

Solutions of these equations can be represented as

First terms of these expressions are solutions of homogeneous equations, with zero right-hand sides;

Solutions grow exponentially at

Then we consider the solution behavior of full BL equations in the near-wall region beside the runoff plane at

The function

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-wall region beside the runoff plane:

Here

The constant *C* is found from a comparison with numerical calculations. It follows from this relation at

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

Here

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 *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.

## 4. Studies of the symmetric flow instability over thin bodies and the control possibility on the base of the interaction model of 3D boundary layer with the electrical discharge

The electric discharge is considered as one of effective methods for control of the flow asymmetry over bodies [23, 24, 25, 26, 27]. However, to select optimal control parameters, it needs to have a reasonable criterion for the asymmetry origin and a possibility for fast estimation of the control effect. For the second problem, the model of the boundary layer and discharge interaction is proposed. The scheme of this model is shown in Figure 8 [28, 29, 30, 31, 37].

It is assumed the plasma discharge effect can be modeled by the heat source in the boundary layer. The effect of gas ionization is neglected since the ionization coefficient is of the order of 10^{−5}. This source in the energy equation is presented by formulas:

Here

Calculations of the turbulent boundary layer characteristics were conducted using the method [10] for a slender cone of half-apex angle

In Figure 9, the dimensionless enthalpy (Figure 9a) and circumferential velocity (Figure 9b) profiles across the boundary layer are shown as functions of

Figure 10a demonstrates the plasma discharge effect on the separation point. As the heat source intensity increases from 0 to 400, the separation angle,

Figure 10b illustrates feasibility of the vortex structure control using a local boundary-layer heating on the base of the developed criterion of symmetric flow stability (solid line). Due to the heat release, the flow configuration changes from the initial asymmetric state (

The method of the global flow stability was developed [27, 28, 29, 30, 31] using the asymptotic approach for the flow over slender cones, the separated inviscid flow model [34] and the stability theory of autonomous dynamical systems [35]. Comparison of the calculated criteria for different elliptic slender cones with experimental data for laminar and turbulent boundary layers sowed its efficiency.

## 5. Investigations of abnormal features of the heat transfer and the laminar-turbulent transition for hypersonic flows around flat delta wing with blunted leading edges

Although found in the experimental zones of abnormal high heat fluxes on the windward flat surface of the half cone with blunted nose and delta wings with blunted leading edges, the phenomenon of the early laminar-turbulent transition [38, 39, 40, 41, 42, 43, 44, 45, 46] cannot be explained in frameworks of the boundary layer theory and on the base of solutions of parabolized Navier-Stocks equations. Only detailed flow simulations using full Navier-Stocks equations allowed to find reasons of such anomalies [46, 47, 48].

Figure 8 shows the comparison of calculated (the upper part) and experimental (the lower part) heat flux distributions on the delta wing with the leading edge sweep angle χ = 75°, the bluntness radius of cylindrical edges and the spherical nose R = 8 at the angles of attack α = 0°, M = 6, unit Reynolds numbers Re_{1} = 1.1556 × 10^{6} m^{−1} [47, 48]. Similar patterns were obtained in numerical simulations for different Reynolds numbers and Mach numbers up to 10.5 [46]. At moderate Mach numbers, a flow on such simple surface outside the nose and leading edge regions is described very well by the flat plate approximation and has no anomalies.

At hypersonic speeds, high heat flux regions, which is present in Figure 11, are observed in the middle wing span and near the symmetry plane. It is seen that the experimental middle high heat flux streak is finished by the turbulent wedge. Calculations were conducted only for the laminar flow.

To understand the reason for the heat flux anomaly, the cross-flow pattern helps (Figure 9). Three longitudinal vortexes are in this flow. The largest vortex is in the inviscid region above shock (the dark layer) and boundary (the light layer) layers. Vortex near the symmetry plane and in the middle of the span occupies both layers. Its mutual location depends on the blunt radius, Mach, and Reynolds numbers [43, 46]. For the considered case, the middle vortex is above the high heat flux region that is shown below the cross-flow pattern (Figure 12).

The analysis shows that high heat flux streaks are formed by the convective transfer of heat gas from the shock layer to the wing surface by the gas rotation inside the vortex. In the considered case, the middle vortex is formed before the symmetry plane vortex near the nose in the narrowing flow region between the head shock and the leading edge due to the cross-flow acceleration near the leading edge and the induced pressure gradient related with the domed flow structure near the symmetry plane.

In considered conditions, the middle vortex also is the reason for the laminar-turbulent transition. Formed along the vortex center, streamwise velocity profiles have inflection points that lead to the Rayleigh instability development. Transverse velocity profiles along this line have the S-shaped form that leads to the cross-flow instability. Both these processes result to the more early transition than Tollmien-Schlichting wave evolution.

## 6. Conclusions

In this work, the short review of researches on the study of BL equation singularities, which are formed when two streamline families are collided, is presented. This phenomenon can arise only in unsteady and 3D problems and has no analogue in 2D flows. A typical example of such problem is the flow around a slender cone in the vicinity of the runoff plane. In this case, solutions are found in the analytical form that allows to analyze explicitly the singularity character.

The analysis of solutions for the outer flow part revealed two singularity types. One type is in streamwise and cross-velocity viscous perturbations; it arises at values of relative cross pressure gradient

To find the dependence of the critical parameter of the singularity appearance

Presented research allows concluding that the flow in symmetry planes, for example, on wings, has the complex structure, which is needed to take into account the numerical modeling in order to eliminate the accuracy loss. Regular flow function decompositions commonly used at solutions of BL equations are not applied near this plane, and it cannot be considered as a boundary condition plane due to a possible solution disappearance.