Constructing the External Contour of the Frankl Nozzle Using S-Shaped Curves with Quadratic Distribution of the Curvatur
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CONSTRUCTING THE EXTERNAL CONTOUR OF THE FRANKL NOZZLE USING S-SHAPED CURVES WITH QUADRATIC DISTRIBUTION OF THE CURVATURE* P. I. Stetsyuk,1† O. V. Tkachenko,2† O. M. Khomyak,1‡ and O. L. Gritsay2‡
UDC 519.85
Abstract. A mathematical model, an algorithm, and a software are developed for the problem of constructing an S-shaped curve passing through two given points with specified tangent inclination angles and providing a tangent inclination angle at a point with a given abscissa. To control the inflection point of the S-shaped curve with quadratic distribution of curvature in natural parameterization, the tangent inclination angle at the point with the known abscissa is used. The algorithm is based on the modification of the method with space dilation in the direction of the difference of two successive generalized gradients. Computational experiments have shown the efficiency of the developed algorithm for constructing the external contour of a Frankl-type nozzle. Keywords: external nozzle contour, S-shaped curve, natural parameterization, quadratic curvature, nonsmooth optimization, r-algorithm. INTRODUCTION One of the main elements of any jet engine is a nozzle, which is necessary to obtain the maximum possible gas flow rate, i.e., create jet power [1, 2]. To obtain supersonic flow rates of gas flow, G. Laval, T. Stanton, and F. I. Frankl [3] developed supersonic nozzles, in which the gas flow is accelerated to speeds greater than the speed of sound. Nozzles (of Laval, Stanton, and Frankl) differ, in particular, in the shape of the outer contour (Fig. 1). A supersonic jet nozzle consists of the following three main parts: the part where the nozzle narrows (subsonic), the part of the critical section, and the part where the nozzle expands (supersonic) [4]. The profiling of the nozzle and the construction of the outer contour of each of its parts are important elements of the design of aircraft and rocket engines. It is necessary to combine the design and calculation problems where the parameters of gas dynamics, heat and mass transfer, and nozzle strength can be used. This requires solving different types of complex gas dynamics problems for each of the three parts of the nozzle, during which the traction and energy characteristics are determined with the required accuracy and the limitations on the dimensions and the weight of the nozzle [4] are taken into account. Methods of solving these gas dynamics problems require a lot of time. Design tasks often require rapid approximate methods of a nozzle contour construction with high traction characteristics. This imposes additional restrictions onto the geometric properties of the contour, which can be provided by methods of applied geometry, in particular, by the methods of geometric modeling of flat curves [5]. These methods *
The present paper is published with the financial support of the NAS of Ukraine (Project No. 0120U002085).
1 V. M. Glushkov Institute of Cybernetics, National Academy † [email protected]; ‡[email protected]. 2Ivchenko-Progress ZMKB,
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