In a flow past a blunt-fin body mounted on a plate, a complex vortex structure arises, characterized by a different configuration of horseshoe-shaped vortices. A large number of studies are devoted to investigation of the topology of the vortex structure depending on the flow parameters for the case of a subsonic flow [1, 2]. In the case of a supersonic flow around a body, the resulting shock waves lead to complex effects of viscous–inviscid interaction, which significantly affects the vortex structure. Such problems, which are important both theoretically and for practical applications, for example, in the aerospace industry, have also been studied by many researchers (see, for example, [3–7] and references therein). The model problem of supersonic viscous gas flow around a blunt body in the form of a stream-oriented thick plate with a rounded leading edge is most often considered. In particular, in the calculation and experimental work [4], for this model configuration, the structure of the laminar flow in front of the streamlined body was studied for three values of the Reynolds number constructed by diameter: ReD = 1.25 × 104, 2.50 × 104, 3.75 × 104. For all ReD values, the authors of [4] interpreted the obtained numerical solutions as stationary and unique. Later, in a computational study [5], it was shown that the solution even at ReD = 2.50 × 104 is unsteady.

The purpose of this work is to clarify the possibility of the duality and/or unsteadiness of the numerical solution of the viscous–inviscid interaction problem for the same model configuration for values of the ReD number in the range from 1.25 × 104  to 2.50 × 104. To a certain extent, this work is a continuation of our studies [8], in which, at lower Reynolds numbers, the influence of the Mach number, plate length, and temperature factor on the flow structure and heat transfer was studied.

The calculations were carried out in two problem formulations: for half of the considered configuration (the computational domain is shown in Fig. 1) and in the complete computational domain without the condition of symmetry in the median plane XZ. According to the used model of a viscous perfect gas, the flow is determined by the following set of dimensionless parameters: free-stream Mach number M, Reynolds number ReD constructed from blunt diameter D, Prandtl number Pr, temperature factor Tw/Tin (Tin is the free-stream temperature), and adiabatic exponent γ, as well as length of the plate Lplate to the junction. Numerical solutions to the problem were found for a set of parameters corresponding to the data of [4]: M = 6.7, Tw/Tin = 4.75, Pr = 0.7, γ = 1.4, and Lplate = 145 mm. The calculations were performed at two Reynolds numbers: ReD = 1.25 × 104 and 1.56 × 104 (varied due to a change in D from 2.5 to 3.125 mm). A uniform flow was set at the inlet boundary of the computational domain, and the no-slip condition was set on the body surface and plate. A constant temperature Tw was maintained for the surface of the body and plate.

Fig. 1.
figure 1

Computational domain.

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For calculations (based on nonstationary formulation), the finite-volume “unstructured” SINF/Flag-S program code developed at the Institute of Applied Mathematics and Mechanics of Peter the Great St. Petersburg Polytechnic University was used. Details of the numerical scheme are given in [9].

In the case of the computational domain with the imposed symmetry condition, a quasi-structured grid consisting of 10 × 106 cells was first used. Then, in order to check the grid convergence, the calculations were also performed on a grid with 25 × 106 cells. Calculations for the “full” region were performed on a grid consisting of 20 × 106 cells. For calculations, the  computing resources of the Peter the Great St. Petersburg Polytechnic University supercomputer center were used [10].

Previous studies (see, for example, [3–5]) showed that in a supersonic flow past a blunt-fin body mounted on a plate, the opposing pressure gradient leads to separation of the boundary layer. A recirculation flow with supersonic velocities arises inside the separation region (Mach numbers make up about a third of the incoming flow Mach number), as well as local compression waves, which leads to re-separation of the near-wall flow, and, as a result, an extended separation region is formed in front of the body containing a set of horseshoe-shaped vortex.

As a result of calculations starting from different initial conditions, it was found that, at ReD = 1.25 × 104, there are two stable stationary solutions that correspond to two metastable flow states with different configurations of the vortex structure (Solution 1 and Solution 2 in Fig. 2a).

Fig. 2.
figure 2

The calculation results at ReD = 1.25 × 104 for two solutions: (a) the field of the modulus of the density gradient and streamlines in the plane of symmetry; (b) the heat flux distribution along the line of symmetry.

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The first solution is characterized by a shorter separation region where the center of the main horseshoe-shaped vortex is shifted from the front edge of the body by a distance of about one caliber (D). In the second solution, the separation region is more extended, and the center of the main vortex is located almost twice as far from the streamlined body. In addition, the number of vortices in the second solution is one less (four vs. five).

The first solution with a given value of ReD = 1.25 × 104 is obtained using uniform initial fields of the sought quantities, evaluated from the values ​​of the flow parameters at the inlet of the computational domain. This regime is also established when specifying the initial conditions of the solution obtained with a significantly lower Reynolds number—in particular, for a fivefold decrease in the Reynolds number (ReD = 0.25 × 104, starting from homogeneous fields). We also note that in the solution obtained at ReD = 0.25 × 104 (not shown in the figures), the center of the main vortex is located at approximately the same distance from the body as in the first solution (Solution 1), shown in Fig. 2a. The solution of the second type is established when the instaneous fields of the nonstationary solution that develops from homogeneous fields with a noticeably higher Reynolds number are specified as initial conditions. Specifically, in this work, this “auxiliary” nonstationary solution was obtained for ReD = 2.50 × 104, and the structure of the averaged flow field for this solution is similar to that shown in Fig. 2a for the second (stationary) solution in the case of ReD = 1.25 × 104. The problem of finding possible bifurcation points or turning points of stationary solutions (stable and unstable) was not posed in this paper. The study of these issues, including clarification of the possibility of hysteresis, requires a very large amount of computations and will be the subject of a separate work.

Returning to the consideration of the features of the obtained solutions, it should be noted that the distribution of the relative heat flux along the line of symmetry on the plate (Fig. 2b), in addition to the global maximum (located outside the field of Fig. 2b), contains one more pronounced local maximum in the first solution and two local maxima of a lower level in the second solution. The latter is in good agreement with the numerical solution and experimental data from [4] (it should be noted here that in [4] there is no mention of the possibility of a dual solution).

Both solutions remain stable even in calculations in the full formulation (without imposing the symmetry condition). The refinement of the grid had practically no effect on the calculated flow fields (Fig. 2b). In particular, for the first solution, the maximum values of the Mach number of the recirculation flow (in the lower part of the main vortex) in the calculations on the initial and refined grids were 33.1 and 33.5% of free-stream Mach number M = 6.7, respectively.

When the Reynolds number is increased to a value of 1.56 × 104, one of the solutions (Solution 1) remains stationary, while the second (Solution 2) is unsteady with stable quasi-periodic flow oscillations, which are especially strong inside the separation region (Fig. 3a, for the second solution, the averaged time field). Earlier [8], we showed that an increase in the length of the separation region can be accompanied by a transition to an unsteady flow regime. Since, in the case considered here, the second solution corresponds to a separation region of greater length, it is the first to become unsteady with an increase in the Reynolds number.

Fig. 3.
figure 3

The calculation results at ReD = 1.56 × 104 for two solutions: (a) the field of the modulus of the density gradient and streamlines in the plane of symmetry (the nonstationary solution is averaged over time); (b) temporary changes in temperature at points near the surface of the plate.

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In Fig. 3b, the value of relative temperature T/Tin at several monitoring points near the plate surface is presented (for comparison, the temperature values for the first stationary solution are also shown). Oscillations with the largest amplitude are observed near the streamlined body and in the region of the global maximum of the heat flux, and the oscillations are insignificant at X/D < −0.8. The configuration of the separation zone and its length and thickness practically do not change in time.

Three-dimensional numerical simulation of a supersonic flow of a viscous gas during its interaction with a blunt body mounted on a streamlined surface in the form of a flow-oriented thick plate with a rounded leading edge was carried out. Numerical solutions were obtained with a Mach number of 6.7 and two Reynolds numbers.

It was found that, for the Reynolds number ReD = 1.25 × 104, there are two stable stationary solutions that correspond to two metastable states of the flow with different configurations of the vortex structure. The first solution is characterized by a less extended separation region and the location of the center of the main horseshoe-shaped vortex at a distance of about one caliber from the front edge of the body; in the second solution, the center of the main vortex is located significantly further from the streamlined body. The second solution is in good agreement with the experimental and calculated data of [4].

It was also shown that, with an increase in the Reynolds number to 1.56 × 104, one of the two solutions remains stationary, while the second is unsteady and is characterized by a stable quasi-periodic oscillations of the flow, which is especially strong inside the separation region.