3D transonic flow over swept and unswept wings with an J-78 airfoil at spanwise sections is studied numerically at negative and vanishing angles of attack. Solutions of the unsteady Reynolds-averaged Navier-Stokes equations are obtained with a finite-volume solver on unstructured meshes. The numerical simulation shows that adverse Mach numbers, at which the lift coefficient is highly sensitive to small perturbations, are larger than those obtained earlier for 2D flow. Due to the larger Mach numbers, there is an onset of self-exciting oscillations of shock waves on the wings. The swept wing exhibits a higher sensitivity to variations of the Mach number than the unswept one.

In the 1960s through 1990s, numerical and experimental studies of transonic flow demonstrated formation of double supersonic regions on supercritical airfoils that comprise a flat or nearly flat arc [

If a double supersonic region (S-region) forms on an airfoil, then it consists of a bow S-region, which expands as the free-stream Mach number

Lift coefficient

This paper addresses the 3D transonic flow over wings with J-78 airfoil at spanwise sections. The study is focused on the determination of Mach numbers and angles of attack that admit a high sensitivity of the flow field to small perturbations. Such adverse free-stream conditions may occur in practice at a vertical gust of wind in cruise flight of civil or transport aircraft.

We consider the fully turbulent flow past (semi) wings which extend spanwise from the root plane

The wing is inclined at an angle of attack ^{2} is given. The no-slip condition and vanishing heat flux are used on the wing, which is normally assumed to be smooth. In the end of Section

Sketch of the computational domain.

The air is treated as a perfect gas whose specific heat at constant pressure is 1004.4 J/(kg K) and the ratio of specific heats is 1.4. Then the temperature

Solutions of the unsteady Reynolds-averaged Navier-Stokes equations were obtained with an ANSYS-15 CFX finite-volume solver of the second order accuracy, which is based on a high-resolution scheme for convective terms [

Hybrid computational meshes were constituted by several millions prisms in 35 layers on the wing and over 10^{7} tetrahedrons in the remaining region. The nondimensional thickness^{+} of the first mesh layer on the wing was less than 1. Apart from the boundary layer region, mesh nodes were clustered in vicinities of shock waves and in the wake.

Test computations of transonic flow over the unswept J-78 wing on uniformly refined meshes of approximately 6 × 10^{6}, 14 × 10^{6}, and 20 × 10^{6} cells showed that differences in the lift coefficient obtained on the second and third meshes do not exceed 5%. Global time steps of 2 × 10^{−5} s and 4 × 10^{−5} s yielded undistinguishable solutions. That is why we employed meshes of 14 × 10^{6} cells and the time step of 2 × 10^{−5} s for the study of aerodynamic characteristics of the J-78 wings at various

The solver was verified by computation of transonic flow for a commonly used test case of the ONERA M6 wing at

A comparison of the pressure coefficient contours on the upper surface of ONERA M6 wing at

Also verifications of the solver were performed by the simulation of an oscillatory transonic flow past a 18% thick circular-arc airfoil at ^{6}. The obtained amplitude of lift coefficient oscillations was 0.35. This agrees well with the value of 0.37 obtained numerically in [

When the angle of attack is −0.6 deg, two-dimensional flow over the J-78 airfoil exhibits a lift coefficient jump at ^{2}, Re = 5.2 × 10^{6} showed that S-regions reside at a considerable distance from each other. With the increase of

Sonic surfaces on the J-78 unswept wing at ^{6}. (a)

Due to the larger Mach numbers, the terminating shock waves on the upper and lower surfaces of the wing are stronger than those in 2D flow. As a consequence, there is an onset of self-exciting oscillations of the shock waves and separated boundary layer at the rear of the wing. A dependence of the lift coefficient on time, for example, at ^{6}, is depicted in Figure

Lift coefficient as a function of time for the unswept wing at ^{6}.

Figure ^{6} the amplitude of lift coefficient oscillations is larger than that at Re = 5.2 × 10^{6}, though a jump of

Lift coefficient versus ^{6}, 2—Re = 10.4 × 10^{6}.

The leading-edge sweep angle of 10 deg provides a nearly uniform spacing between the bow and rear S-regions on the upper surface of the wing at various spanwise sections. Figure

Sonic surfaces on the J-78 swept wing at ^{2}, Re = 5.1 × 10^{6}: (a) tip view, (b) side view.

It can be seen from Figure

With an increase of

Lift coefficient

Pressure coefficient ^{6}: 1—

When the Mach number increases from 0.857 to 0.861, the lift coefficient declines due to the flow stabilization on the upper surface of the wing and an expansion of the supersonic region beneath the wing. The expansion yields a pressure drop on the lower surface and, hence, the drop of lift coefficient.

At a smaller angle of attack,

Dashed curves in Figure ^{−6} m. As seen, there is a good agreement of the results obtained for the smooth and rough surfaces.

For the swept and unswept wings with J-78 airfoil at spanwise sections and a span of two chord lengths, the numerical study revealed adverse bands of free-stream parameters

The author declares that there is no conflict of interests regarding the publication of this paper.

This research was performed using computational resources provided by the Computational Center of St. Petersburg State University (