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Internal Shocks in Supersonic Flows past the Contours of Optimal Bodies and Nozzles

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Abstract—

The problems of designing the contours of two-dimensional and axisymmetric bodies of minimum wave drag and maximum-thrust nozzles, such that flow past them can contain internal shocks, are considered within the ideal (inviscid and non-heat-conducting) gas model. These shocks may occur in flows past internal breaks in the optimal contours or on intersection of characteristics of the same family proceeding from these breaks. The submerged shocks occurring in the latter case are named internal, if their initial points lie within the domain of determinacy of the flowed-over optimal contour region. Further downstream the domain of determinacy is bounded by the characteristic coming to the end point of this contour. The internal submerged shocks can appear, when the optimal contour includes an interval of an isobaric streamline. The appearance of internal breaks in the optimal contours is possible due to the reflection of the pressure waves induced by them from the bow shock (for body noses) or, as might be expected, from tangential discontinuities (for afterbodies). In the case of supersonic freestreams having a tangential discontinuity the search for optimal two-dimensional afterbodies with an internal concave break flowed-over with an oblique shock is performed using the direct method with the representation of the required contours by the Bernstein—Bézier curves, a genetic algorithm, and a rapid and accurate marching scheme for calculating steady supersonic flows. While the optimal two-dimensional afterbodies with a convex internal break designed earlier using the general method of Lagrange multipliers could be reliably reproduced using the above-mentioned method, the expected optimal configurations with a concave break and an internal shock were not found. Instead, optimal contours with centered compression and expansion waves with the common focus on a tangential discontinuity and an “external” oblique shock proceeding from it were discovered. Then the new “discontinuous shock-free” solutions were constructed in the exact formulation, within the framework of the method of indefinite control contours and the method of characteristics. The solutions are generalized to include the contours of the afterbodies of revolution and supersonic parts of nozzles. In the case of a uniform freestream the flow past optimal afterbodies never includes internal shocks.

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Funding

The study was carried out with the support of the Russian Foundation of Basic Research (projects 19-01-00671 and 20-01-00100).

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Authors and Affiliations

  1. Baranov Central Institute of Aviation Motors (CIAM), ul. Aviamotornaya 2, 111116, Moscow, Russia

    A. N. Kraiko, K. S. P’yankov, N. I. Tillyaeva & V. A. Shapovalov

  2. Moscow Institute of Physics and Technology, Institutski per. 9, 141701, Dolgoprudny, Moscow region, Russia

    A. N. Kraiko & V. A. Shapovalov

Authors
  1. A. N. Kraiko
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  2. K. S. P’yankov
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  3. N. I. Tillyaeva
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  4. V. A. Shapovalov
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Correspondence to A. N. Kraiko, K. S. P’yankov, N. I. Tillyaeva or V. A. Shapovalov.

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Translated by M. Lebedev

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Kraiko, A.N., P’yankov, K.S., Tillyaeva, N.I. et al. Internal Shocks in Supersonic Flows past the Contours of Optimal Bodies and Nozzles. Fluid Dyn 55, 840–857 (2020). https://doi.org/10.1134/S0015462820060075

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