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 [38–46] 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 [46–48].

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 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>1</sup> [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.

In considered conditions, the middle vortex also is the reason for the laminarturbulent 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-

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

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≥1 and leads to the exponential disturbance growth as the runoff plane is approached. At k ¼ 1 the singularity is logarithmic and at k . 1 it is power; its appearance is correlated with the BL separation appearance. Another singularity type at smaller values of k≥1=3 in 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 ¼ 1=3 the singularity is logarithmic, and at k . 1=3 it 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

form that allows to analyze explicitly the singularity character.

Schlichting wave evolution.

3D Boundary Layer Theory

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

The cross-flow structure above the wing in the section Х = 0.1 m.

6. Conclusions

21

Figure 12.

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.

#### Figure 11.

Comparison of numerical (the upper part) and experimental (the lower part) specific heat flux distribution on the wing surface.

5. Investigations of abnormal features of the heat transfer and the laminar-turbulent transition for hypersonic flows around flat delta

Boundary Layer Flows - Theory, Applications and Numerical Methods

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 [38–46] 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

Figure 8 shows the comparison of calculated (the upper part) and experimental

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.

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

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

Comparison of numerical (the upper part) and experimental (the lower part) specific heat flux distribution on

Calculations were conducted only for the laminar flow.

that is shown below the cross-flow pattern (Figure 12).

(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 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>1</sup> [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

wing with blunted leading edges

[46–48].

no anomalies.

the symmetry plane.

Figure 11.

20

the wing surface.

Figure 12. The cross-flow structure above the wing in the section Х = 0.1 m.

In considered conditions, the middle vortex also is the reason for the laminarturbulent 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.
