ISSN 10283358, Doklady Physics, 2011, Vol. 56, No. 12, pp. 618–621. © Pleiades Publishing, Ltd., 2011. Original Russian Text © I.A. Bedarev, A.V. Fedorov, V.M. Fomin, 2011, published in Doklady Akademii Nauk, 2011, Vol. 441, No. 5, pp. 621–624.
MECHANICS
Numerical Simulation of Flow around a Body’s System beyond a Transmitted Shock Wave I. A. Bedarev, A. V. Fedorov, and Academician V. M. Fomin Received August 4, 2011
DOI: 10.1134/S1028335811120056
INTRODUCTION For constructing mathematical models in mechan ics of reacting and inert heterogeneous media, infor mation is important on local characteristics of flow fields arising as a result of gasphase interaction with small particles of the discrete phase. In order to describe the penetration of a shock wave through a cloud of particles, we need to know characteristics of flow around them and of chemical transformations, such as damping and heat exchange, between a particle and the environment, as well as inflammation and combustion times for small metal particles, etc. These characteristics, without a doubt, depend on whether or not the flow around the particles is subsonic or super sonic. In other words, we should distinguish which mode of flow is realized. In turn, the mode of flow around the particles is dependent on whether the col lective jump is formed ahead of the particle’s cloud or individual flows around the particles take place [1–3]. Indeed, the presence of an individual shock wave in the flow of interacting continua, e.g., in the case of carbon particles, was taken into account in [4]. There, the existence of the effect of the flow mode on the inflammationdelay time was proved. It is well known that a collective shock wave or a wave structure exhib iting either the Mach interaction or the regular inter action can arise ahead of bodies. Of course, this pro cess is dependent on both the supersonicflow velocity and the distance between the bodies. In the present study, we numerically investigate the interaction of a transmitted shock wave with a system of immobile bodies. Our goal is to determine maps for possible modes of supersonic flows around this system
Christianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia email:
[email protected];
[email protected];
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beyond the transmitted shock wave. As examples, the system of bodies is represented by either cylinders or spheres. CALCULATING THE SHOCKWAVE TRANSMISSION THROUGH A GRID OF TRANSVERSELY LOCATED CYLINDERS IN A TWODIMENSIONAL SETTING We now study initially unsteady problem on the incidence of a shock wave onto a system of transversely located cylinders. In the twodimensional unsteady calculations, we make use of the mathematical model for Favraveraged Navier–Stokes equations. The model is complemented by the SST modification of the k–ω turbulence model, by the ANSYS Fluent pro gram package, and by the quadrangular net. The net became denser towards the body’s surface and was dynamically adapted to flow gasdynamic features in accordance with the density gradient (i.e., to shock waves, contact discontinuities, and rarefaction waves). We here pay attention to certain data of numerical cal culations, which were obtained in the case of the cor responding boundary conditions for the mathematical viscousgas model. Figure 1 demonstrates dynamics of the shockwave transmission through the grid of cylinders. The shock wave is characterized by the Mach number M0 = 3 (beyond the shock wave, the local Mach number of flow is M1 = 1.36). The grid step is λ = l/d = 10. The staticpressure fields are shown at different instants of time. The upper and lower boundaries of the region are symmetry lines. The wave pattern obtained for the interaction shows that, in the case of the body’s location being close, the flow mode is sequentially transformed from the regular shockwave interaction (Fig. 1b) to the Mach interaction (Fig. 1c) and then to collective flow around the bodies (Fig. 1d). The appearance of the collective wave under the conditions of flow around the infinite grid of cylinders
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Fig. 1. Pressure fields at various instants of time for M0 = 3 and λ = 10.
makes impossible the formation of a steadystate flow pattern. At the subsequent instants of time, the collec tive shock wave permanently detaches from the grid upstream to the left. If the bodies are located at a great distance from each other, then only one or two initial stages of the interaction (regular interaction or Mach reflection) can be observed once the shock wave has been formed. In this case, a steadystate pattern of flow around the grid with individual shock waves ahead of the cylinders is formed. For the parametric analysis of flow, it is convenient to consider steadystate supersonic flow around the system of cylinders beyond the shock wave.
symmetry line. The cylinders are 5 mm in diameter. Figure 2 illustrates various modes of flow around the system for variable distances between the bodies. For distances λ = 10 between the cylinders (Fig. 2a), we can observe the collective shock wave. For λ = 20, there are individual flows around the cylinders, and the Mach interaction of shock waves is observed (Fig. 2b). Finally, for λ = 50 (Fig. 2c), the interaction is regular. As was noted in [1], the collective shock wave is produced as a result of combining transonic zones beyond shock waves. This can be seen from the sound line plotted with the solid curve in the figure. With approaching the bodies, merging subsonic zones occurs, and a common shock wave is formed ahead of the bodies.
SUPERSONIC FLOW AROUND TWO TRANSVERSELY LOCATED CYLINDERS BY SUPERSONIC FLOW BEYOND THE SHOCK WAVE
Figure 3 demonstrates in the phase plane the gener alized calculation data in the form of mode maps for flows around the system of transversely located cylinders for different Mach numbers and relative distances λ between the bodies. Figure 3 allows us to reveal the position of neutral curves separating the boundaries of the region for various interaction modes. As is clearly seen, for small Mach numbers (M < 1.3), individual flows around the cylinders are observed only when they are significantly removed from each other
We now consider supersonic flow around a system of two transversely located cylinders at both different Mach numbers and different distances between the cylinders. Figure 2 presents staticpressure fields for two cylinders at M1 = 1.5 and various distances between them. The lower boundary of the region is the DOKLADY PHYSICS
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Fig. 2. Staticpressure fields for a cylinder at M = 1.5. The transition is seen from the collective flow around the cylinder for (a) λ = 10 to (b) individual flow mode with the Mach interaction for λ = 20, and to (c) the regular interaction for λ = 50.
(λ > 40). An attempt was made to compare the calcu lation data for flow around cylinders and the experi mental data of [3], where the modes of supersonic flows around two spheres were studied. It was shown in the calculation that the transitions from the collective mode to the individual mode of flow around the cylin ders occurred at significantly larger (approximately by an order of magnitude) distances between the bodies. λ 120 1 100
SUPERSONIC FLOW AROUND A SYSTEM OF SPHERES IN THE THREEDIMENSIONAL SETTING AT DIFFERENT MACH NUMBERS AND DISTANCES BETWEEN SPHERES
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Evidently, this fact is explained by the difference in supersonic flow near a transversely located cylinder or sphere. The comparison of supersonic flow around a sphere or a cylinder shows the substantial difference in the wave pattern for flows around these bodies. In the first case, the shock wave detaches from the body at a shorter distance and is characterized by a significantly smaller inclination angle. Thus, in order to adequately simulate the modes of flows around spherical parti cles, it is necessary to perform the threedimensional calculation of the problem.
2.0 M
Fig. 3. Calculated modes of flow around two cylinders as functions of the Mach number and of the distance between the cylinders: (1) collective wave; (2) Mach interaction of shock waves; and (3) regular interaction of shock waves.
The calculation of threedimensional flow around two spheres was performed on the basis of tetrahedral nets in the main part of the calculation region. The hexagonal net that became denser towards the sphere surface was constructed only in the vicinity of the sphere. In accordance with the density gradient, the net was also adapted to flow gasdynamic features. Figure 4 makes it possible to compare calculated and experimental flows around two spheres as func tions of both the Mach number and the distance between the spheres. Dots 1 and 4 correspond, respec tively, to the calculated and experimental modes with DOKLADY PHYSICS
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1 2 3 4 5 6
9 7 5 3
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 M Fig. 4. Calculated and experimental modes of flow around two spheres as functions of the Mach number and of the distance between the spheres.
the unified wave front. Dots 2 and 5 correspond to the Mach interaction, and dots 3 and 6 correspond to the regular interference of the shock wave. Figure 4 con firms the consistency of the calculated and experi mental data. CONCLUSIONS The results presented allow us to make the follow ing conclusions. In order to reveal the interaction type (collective, regular, or Mach interaction) for detached shock waves ahead of bodies, we have managed to solve the problem of flow around a system of bodies (of both cylinders and spheres).
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The wave pattern for the transmitted shockwave interaction with transversely located cylinders at dif ferent instants of time testifies to the fact that in the case of a close location of bodies, flow around a system of bodies sequentially transforms from regular reflec tion to the Mach interaction of the shock wave and then to the collective shock wave. With an increase in the distance between the bod ies, the transition occurs from the collective flow around particles to individual flow. In this case, we ini tially deal with the Mach interaction, and then with the regular interaction between the shock waves. The mode maps for flows around a system of trans verse cylinders or spheres as functions of the Mach number and the distance between the bodies describe transitions between different flow types. The compar ison of the calculated mode maps and the experimen tal data shows their good consistency. REFERENCES 1. V. I. Blagosklonov, V. M. Kuznetsov, A. N. Minailos, et al., Prikl. Mekh. Tekh. Fiz., No. 5, 59 (1979). 2. A. V. Fedorov and N. N. Fedorova, Prikl. Mekh. Tekh. Fiz., No. 4, 10 (1992). 3. V. M. Boiko, K. V. Klinkov, and S. V. Poplavskii, Izv. Akad. Nauk, Ser. Mekh. Zhidk. Gaza, No. 2, 183 (2004). 4. A. V. Fedorov and T. A. Khmel’, Fiz. Goreniya Vzryva, No. 1, 89 (2005).
Translated by G. Merzon
