Contouring the nozzles producing a uniform supersonic flow or a thrust maximum in the presence of a curvilinear sonic line

Kraiko, A. A.; Kraiko, A. N.; P’yankov, K. S.; Tillyaeva, N. I.

doi:10.1134/s001546281202010x

Cited by 3 publications

(2 citation statements)

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“…The conditions (2.12) are used only for designing the long section. As was established in [8], the smooth contours d + f possessing a decelerating and equalizating short initial section, as well as a long accelerating section are inferior in thrust to contours with two bend points only by hundredth fractions of per cent.…”

Section: Is a Region Of Boundary Extremum Under The Conditionsmentioning

confidence: 96%

“…By [8], in the presence of an abrupt contraction, the optimal contour of the supersonic nozzle part consists of not one (as in Figure 1), but two sections: a short section that contracts after the lower bend point of the abrupt contraction and a long extending section adjacent to the first one also with a bend. The short section decelerates and approaches the supersonic flow accelerated in the expansion fan to the axial direction.…”

Section: Is a Region Of Boundary Extremum Under The Conditionsmentioning

confidence: 98%

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Conjugate Problem for Lagrange Multipliers and Consequences for Partial Differential Equations of Mixed Type

Kraiko

Tillyaeva

2015

J Math Sci

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UDC 517.9We formulate and solve the conjugate problem for Lagrange multipliers connected with designing a Laval nozzle optimal contour, including its subsonic part. In approximation of an ideal (inviscid and nonheat-conducting) gas, the sought contour provides a thrust maximum under a number of constraints; in particular, given total nozzle length and gas mass flow. For a contour of a contracting (subsonic) part of the nozzle (which is suspected to be optimal) we take an abrupt contraction. Because of the constraint on the nozzle length, the abrupt contraction can be a region of boundary extremum with positive permissible variations of the longitudinal ("axial") coordinate of the contour. To clarify whether this is true, we use the method of Lagrange multipliers and formulate the conjugate problem for finding the Lagrange multipliers. As in the case of quasilinear Euler equations governing a flow, the linear equations in the conjugate problem are elliptic (hyperbolic) in the subsonic (supersonic) flow domain. The requirement that the conjugate problem be solvable for any contour highlights some features that can be of interest not only for this special problem, but, possibly for the general theory of mixed type partial differential equations. Bibliography: 8 titles. Illustrations: 8 figures.

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Section: Is a Region Of Boundary Extremum Under The Conditionsmentioning

confidence: 96%

Section: Is a Region Of Boundary Extremum Under The Conditionsmentioning

confidence: 98%