3D Boundary Layer Theory

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 εcharacterizes the gradient of the pressure ptxzwith respect to transverse nondimensional coordinate z; tand xhere are dimensionless time and longitudinal coordinate, λis body span, H2is metric coefficient, k1is longitudinal coordinate line curvature, and θis the angle between coordinate lines on the body surface. Using these conditions flow parameters in the 3D boundary layer are presented by asymptotic expansions:

Here, stxzis a dimensionless length of the coordinate line z=constmeasured from the critical point xctz; nis normal coordinate transformed with Dorodnitsyn transformation; vwtxzand hwtxzare blow (suction) velocity and the surface temperature; uand v, h, ρ, κand μare dimensionless longitudinal and normal velocities, enthalpy, density, thermal conductivity, and viscosity. The parameter ε1<<1is 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 ε1=ε/λ2, for slightly asymmetric bodies εcharacterizes the asymmetry and ε1=α∗/λ, where λis relative body thickness [4, 5, 6, 7, 8, 9, 10].

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 (Re=2.246⋅106); dotted lines on Figure 4 are results for full scale wing (Re=2.58⋅107); symbols present solutions of full 3D boundary layer equations [11].

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 δc<<1at the angle of attack α* depends on the parameter k=4α∗/3δconly. Firstly, analytical results about singularities were obtained for outer BL part for a such cone. It is understood from previous works [15, 16, 17, 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, 17, 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 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=1/3and 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 self-similar 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=1/3. However calculations showed that this parameter is a function on numbers of Mach Mand Prandtl Prand the wall temperature hw, kc=kcM∞Prhw[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 topology near this plane [24]. The analysis of BL equations in the near-wall region showed that α=0corresponds to the critical value kc, and it was confirmed by all published numerical calculations [16, 17, 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.

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]:

uyy=Awuφ+vuy+A1wu−w,wyy=Awwφ+vwy+w23u+Kw−h23+K,hyy=Awhφ+vhy−Meuy2+32A1wy2,ρh=1,y=ε3ρeue2xμe  ∫0y∗ρd y∗l,Re=ε−2=ρ∞u∞lμ∞,fy=u,gy=w,v=−f−K−12Alnρeμe/ueφg−Agφ,y=0:u=v=w=0,h=hwhy=0;y=∞:u=w=h=1.E4

Equation coefficients are defined by expressions

Meφ=γ−1M∞2ue2he,Kφ=2weφ3Rue,Aφ=2we3Rue,A1φ=23weue2E5

In these equations, to reduce formulas, Pr=1and the linear dependence of the viscosity on the temperature (ρμ=1) are assumed. Indexes yand φdenote derivatives with respect to the corresponding variables; xis the distance from the body nose along the generator referenced to the body length l; yis the Dorodnitsyn variable; y∗is normal to the body surface; φis the transversal coordinate, and it can be the polar angle for a round cone; fyφand gyφare longitudinal and transverse stream functions; vyφis transformed normal velocity; and Rφis the metric coefficient. The density ρ, the enthalpy h, the viscosity μ, the longitudinal u, and transversal wvelocities 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 φonly. The transversal velocity on the outer boundary layer edge we=0in the initial value plane (the attachment plane) φ=0, in which K0>0, and in the runoff plane φ=φ1, in which Kφ1=−k<0, and two boundary layer parts that came from different sides of the attachment plane collided. For the round cone, φ1=π.

Eq. (4) is simplified for slender bodies since in this case, ue=ρe=μe=1, A1<<1. Neglecting proportional to A1terms in (4), we obtain the Crocco integral for the enthalpy and momentum equations in the form

h=hw+hru−12Meu2,hr=1−hw+12Me,Me=γ−1M2,v=−f+Kg+Agφ,uyy=Awuφ+vuy,wyy=Awwφ+vwy+w23u+Kw−h23+K.E6

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

we=2α∗sinφ,Kφ=kcosφ,Aφ=ksinφ,k=4α∗3δcE7

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 ζ∼ε12−m; at m∼1this 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:

ε1=32Reρeφ1ueφ1/μeφ1−12.z=kxRζ/ε1,u=uyz,h=hyz,w=wyzE13

Using these variables from Navier-Stokes equations at ζ∼ε1<<1for this region, we derive self-similar equations, which in its outer part, at y≫1, reduce to the form

Uyy+kUzz+1−kyUy+kzUz=0,Wyy+kWzz+1−kyWy+2z+kzWz+2km−1W+23p1U=0E14

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

Uyz=C1erfcy1−k/2erfz/2,W=−BzC1erfcy1−k/2,Bzz+2z+zBz−2m−1B=−2mp1Fz,Fz=erfz/2E15

The function Bzis expressed by Kummer’s function Φabx[10, 22, 23]:

B=mp1B0z+BmΦ1−m32−12z2,Bm=bmRkx/ε121−m.E16

B0zis a particular solution of the inhomogeneous equation; the coefficient Bmis determined from matching condition.

In Figure 4, comparisons of solutions of BL (dotted lines) and Navier-Stokes (solid lines) equations for m=1/2(curves 1 and 2) and m=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≥1/3due to the BL growth at ζ→0is 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 Δφand the velocity scale:

Δφ∼εx−14,we∼kRueεx−14.E17

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 δxs. In the boundary layer, the flow is described by full 3D equations:

s=Rζε,we=32ueεWexsWexs=−ks1+r,r=4mπ∂∂x∫0∞δxtdts2−t2v=f+Kg+Ags+23xfx,h=hw+hru−12Meφ1u2uyy=Wewus+vuy+23xuuxwyy=Wewws+vwy+w23u+Wesw−h23+Wes+23xuwxE18

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=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

t=y/dxs,u=1+Uxts,w=1−cxsUU=C1erfct/2,p0=3212M0+hw−11+rsds−2mxdx−2n−1−rsd=−2n1+rscs−2mxcx−2m−1−rsc=−2mp1−qp0E19

Along characteristics ξxs=const, which are streamlines of the inviscid flow, the equations for functions d=dξsand c=cξsare integrated. At s→0these functions are represented in the form

c=CsL+mp1+p0rsm−1−r,Lξs=m−1−rs1+r;d=DsI+nn−1−rs,Iξs=n−1−rs1+r,C=bmεm−1,D=anεn−1.E20

Coefficients Cand Dare obtained by matching dξsand cξsat s→∞in relation to aζand bζat ζ→0[17, 36]. The logarithmic singularity appears in these functions at I=0or L=0. At Lξ0<0or Iξ0<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=1/3, although calculations show kc=kcM∞Prhw. This indicates on the possibility of singularity arising in the near-wall region. To study this possibility, at the first, we study the solution behavior of Eq. (6) at y<<1in the runoff plane φ=φ1where the solution is presented in the form

τ0=du00dy,θ0=dw00dy,u0=τ0y+U0y,w0=θ0y+W0y,v0=−αy2−F0+kG0,α=12τ0−kθ0E21

Second terms of these decompositions can be presented by series

U0=F0y=∑i=0αiyi+4i+4!,F0=∑i=0αiyi+5i+5!,W0=G0y=∑i=0βiyi+2i+2!,G0=∑i=0βiyi+3i+3!.E22

First three coefficients of these series are defined by relations

α0=−2τ0α,α1=kτ0β0,α2=kτ0β1β0=−phw,β1=−pτ0hr,β2=13τ0−3kθ0θ0+2pMeτ02E23

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 vand transverse wvelocities at ζ<<1and y<<1in the first-order approximation are represented in the form

v=v0=−αy2−16kβ0y3=−16kβ0y2y+ycyc=−6αkβ0=6αkphw,w=−w0=−kθ0ζyE24

In the plane ζ=0, the cross-flow velocity w=0due to symmetry conditions. Here two critical points, in which v=0, can be. The first point locates on the cone surface y=0, and the second one y=−ycappears 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≤1/3corresponds to the negative cross-flow pressure gradient p>0, the transverse skin friction in this region θ0>0.

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

ycdyyy+yc=βdζζ,β=αkθ0;α≠0:y=ycy0sβyc+y01−sβ,s=ζζ0,α=0:y=y01−y0dlns,d=phw6θ0E25

Here y0and z0define the initial point in the cross-plane.

The subcharacteristic behavior is shown in Figure 5a and b for p>0. At α>0velocities v<0and w<0; the only critical point node is in the coordinate origin, and subcharacteristics go to it from the region ζ≠0(Figure 5a). At α=0yc=0and the point ζ=y=0is double critical point of the type saddle node: the saddle is in the lower half-plane, i.e., out of the physical space. The node is in the upper half-plane, and the subcharacteristic pattern retains the same as at α>0. At α<0the node drifts in the point ζ=0, y=yc>0, and the coordinate origin becomes by the saddle point (Figure 5b). In this case, at y>ycthe normal velocity v<0and at 0<y<ycv>0; v=0on the line y=yc.

This analysis shows that at the parameter αsign change, the physical flow structure varies qualitatively, and the value α=0is 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=y=0in 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 kchwMis defined by the relation

2αkc=τ0kc−kcθ0kc=0E26

To support this hypothesis, equations for functions U0yand W0yare analyzed by substituting near-wall decompositions to Eq. (6). Considering functions U0yand W0yas perturbations, we can linearize resulting equations and obtain in the first-order approximation:

U0yy+αy2U0y+τ0F0−kG0=−ατ0y2,W0yy+αy2W0y−23τ0−3θ0yW0+θ0F0−kG0=β0+β1y+12β2y2+23θ0y−phr−2Meτ0yU0E27

At y→0U0yand W0yare 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≠0, we introduce the new variable:

ξ=−αy3/3,y=−3ξ/α13.E28

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

ξ∂2U0∂ξ2+23−ξ∂U0∂ξ=−τ033ξα13,c=2τ0−3kθ09αξ∂2W0∂ξ2+23−ξ∂W0∂ξ+cW0=−β13α+β26α3ξα13−29αθ0+3Meτ0U0E29

Solutions of these equations can be represented as

U0=A00∫0ye−13αs3ds+τ03ξα13,W0=B00Φ−c23ξ+B01ξ13Φ13−c43ξ−3β12τ0−3kθ0−3β2τ0−9kθ03ξα13−θ0+3Meτ0τ0−3kθ0U00E30

First terms of these expressions are solutions of homogeneous equations, with zero right-hand sides; A00, B00, and B01are constants; Φabxis Kummer’s degenerate hypergeometric function, which has asymptotes at ξ→∞:

α>0,ξ<0:Φ∼−ξc;α<0,ξ>0:Φ∼eξξc−23E31

Solutions grow exponentially at α<0and 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, 26, 27, 28, 29, 30, 31, 32, 37], a part of which is presented in Figure 6. In this figure, symbols correspond to calculations of limit values αkcfor 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<1/3data are grouped near the value α=0in 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 kcand 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 τζand cross-flow θζfriction stresses and the self-similar variable ηat ζ<<1will be defined by expressions

τζ=τ0aζ,θz=θ0aζ,η=yaζE32

The function azat α≥0will satisfy to the condition a0=1. In this case, flow functions in the boundary layer near the wall can be represented in the form

fηζ=aζτ0η22+Fηζ,uηζ=fη=τ0η+Uηζgηζ=aζθ0η22+Gηζ,wηζ=gη=θ0η+Wηζv=aα−12θ0kζaζaη2+F−kG1+kζaζa−kζGζ−kζηζWE33

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:

Uηη+αη2Uη+a2kθ0ζηUζ+τ0F−kG1+aζa−kζGζ=−ατ0η2Wηη+αη2Wη+a2kθ0ζηWζ+θ0F−kG1+ζaζa−kζGζ−3αcηW=−αθ0η2+a2β0+β1η+12β3η2+23θ0+3pMeτ0η−phrUE34

Here β3=23τ0θ0−kθ02+pMeτ02. Due to local self-similarity at α≥0, we define the function aζas

αa2−12kθ0ζaaζ=α,a2=1+Cζq,q=4αkθ0E35

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

Fηζ=F0η+ζqFqη+…,Uηζ=U0η+ζqUqη+…,Gηζ=G0η+ζqGqη+…,Wηζ=W0η+ζqWqη+…,E36

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]

ξ=−αη33,Uqξ=Aq0Φ4323ξ+Aq1ξ13Φ5343ξWqξ=Bq0Φ43−c23ξ+Bq1ξ13Φ53−c43ξ+9β12τ0+3β2113τ0−kθ03ξα13+θ0+3Meτ02τ0U0−θ0+3Meτ0τ0−3kθ0Uq.E37

Here Aq0, Aq1, Bq0, and Bq1are constants. These relations show that the proper solution in near-wall BL region near the runoff plane is nonzero. It is irregular at α≥0and 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 kcin 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 f1ʹʹzand g1ʹʹz, induced by the second order BL approximation (Figure 7a) and the angle of attack (Figure 7b) are presented in dependence on transverse coordinate z=1−Z/X, where Xand Zare Cartesian streamwise and transverse coordinates. By approaching the symmetry plane (z=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 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 Q∗is a dimensional source intensity; Q0is a maximum of dimensionless heat-release intensity; σcharacterizes the discharge width; ycφis a centerline of the discharge that is approximated by the parabola; y0is a maximum distance from the discharge centerline to the wall; and the angles φ1and φ2determine the electrode locations.

Calculations of the turbulent boundary layer characteristics were conducted using the method [10] for a slender cone of half-apex angle δc=5∘at the angle of attack α=α∗/δc=3.15. Other parameters are: l=1 m, T∞=288K, u∞=10m/s, σ=1, and y0=1; the center between electrodes is located at ϕ0=0.5ϕ1+ϕ2=1.714rad98.25°, ϕ1=ϕ0−3Δϕ, and ϕ2=ϕ0+3Δϕ, where Δϕ=0.0314159is the integration step of the finite-difference approximation.

In Figure 9, the dimensionless enthalpy (Figure 9a) and circumferential velocity (Figure 9b) profiles across the boundary layer are shown as functions of ηfor Q0=200and for different polar angles φ. These profiles are similar to the source heat intensity distribution across the boundary layer. The temperature reaches the maximum value near the rear electrode, φ=φ2=1.809. Behind the heat source region, the temperature maximum decreases and moves toward the upper boundary-layer edge due to the heat diffusion. The station φ=1.87is located just after the separation point

Figure 10a demonstrates the plasma discharge effect on the separation point. As the heat source intensity increases from 0 to 400, the separation angle, φs, decreases from 133° to about 105°. It is seen that the plasma heating is more effective in the range Q0<100, where the slope dφs/dQ0is relatively large.

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 (φs≈133°, symbol 1) to the symmetric state with θs≈120°(symbol 2). This requires a nondimensional heat source intensity Q0≈30that corresponds to the total power which is approximately equal to 480 W. This example indicates that the method is feasible for practical applications of the global flow structure control.

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 Re₁ = 1.1556 × 10⁶ 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 k≥1and leads to the exponential disturbance growth as the runoff plane is approached. At k=1the singularity is logarithmic and at k>1it is power; its appearance is correlated with the BL separation appearance. Another singularity type at smaller values of k≥1/3in the first-order approximation leads to the infinite growth of transverse velocity perturbations only and is not related directly with the flow separation; at k=1/3the singularity is logarithmic, and at k>1/3it is power. These BL singularities correspond to some asymptotic flow structure at Re≫1. This structure includes the boundary region with the dimension of the order of the BL thickness, in which the viscous transverse diffusion effect smoothes the singularity. The comparison of obtained parabolized Navier-Stokes equation solutions describing the flow in the boundary region with BL equations solutions confirms this conclusion. Second region induced by the viscous-inviscid interaction effect has the transverse dimension of the order of square root from the BL thickness and the two-layer structure. For the potential flow in the outer inviscid subregion, the integral solution representation is found on the base of the slender wing theory. The inner subregion is described by full 3D BL equations, the solution of which is obtained for the outer viscous subregion part. It was shown that the viscous-inviscid interaction does not eliminate the singularity but drifts it in the parametric space. To eliminate the irregularity, the boundary region is needed.

To find the dependence of the critical parameter of the singularity appearance kcon Mach and Prandtl numbers and the wall temperature BL equations, solutions are studied in the near-wall region beside the runoff plane. Equation subcharacteristic (streamlines) analysis showed the presence of one parameter α, the sing of which defines the qualitative change of the streamline topology and, consequently, the physical flow structure. It is shown and confirmed by comparison with all available calculations that the boundary of the solution which exists in the runoff plane corresponds to the criterion αkc=0. The analysis of BL equation solutions near the runoff plane revealed the presence at α≥0of irregular and at α<0singular proper solutions. This is confirmed by numerical calculations of the flow around slender delta wing with the small aspect ratio. Singularities in the near-wall region generate the some flow structure in its vicinity, the study of which is out of this paper framework. Presented results do not depend on outer boundary conditions and are true for the full freestream velocity diapason including hypersonic flows.

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.