Critical Mach Numbers of Flow around Two-Dimensional and Axisymmetric Bodies

The paper presents the calculated results obtained by the author for critical Mach numbers of the flow around two-dimensional and axisymmetric bodies. Although the previously proposed method was applied by the author for two media, air and water, this chapter is devoted only to air. The main goal of the work is to show the high accuracy of the method. For this purpose, the work presents numerous comparisons with the data of other authors. This method showed acceptable accuracy in comparison with the Dorodnitsyn method of integral relations and other methods. In the method under consideration, the parameters of the compressible flow are calculated from the parameters of the flow of an incompressible fluid up to the Mach number of the incoming flow equal to the critical Mach number. This method does not depend on the means determination parameters of the incompressible flow. The calculation in software Flow Simulation was shown that the viscosity factor does not affect the value critical Mach number. It was found that with an increase in the relative thickness of the body, the value of the critical Mach number decreases. It was also found that the value of the critical Mach number for the two-dimensional case is always less than for the axisymmetric case for bodies with the same cross-section.

Relationship of the method of obtaining the original billet with the accuracy of manufacturing of the extended axisymmetric bodies

A comparative description of the technology for producing blanks for the artillery shell cases or the rocket shell bodies using flat rolled stock or rolled tubular products is given. It is noted that the lower cost of flat rolled stock leads to its wider application for stamping blanks of cartridge caps or bodies, including cutting a circle from flat rolled stock, convolution and several transitions of drawing, alternating with heat treatment. It is noted that the disadvantage of this technology are the errors in the shape of the blanks, amounting to 0.75÷1.5 of the tolerance for diametrical dimensions. When machining, these errors decrease according to the laws of copying the errors, however, the anisotropy of the material of the part blank leads to the appearance of errors in the shape-ovality and curvature of the pipe, which lead to the formation of a large radial runout of the prefabricated rocket body. It has been found that the scattering fields of all output quality parameters exceed the existing tolerances: radial runout of the assembled body by 1.3 times; ovality of the middle centering bulge by 2.15 times. It is shown that the technological process of manufacturing prefabricated housings using initial blanks from flat rolled stock does not provide the required precision technology reliability. For the manufacture of monolithic extended axisymmetric bodies, hot-rolled thickwalled pipes are used as an initial billet. The operations are performed in the following sequence: cutting pipes into dimensional workpieces; machining (turning, boring); heat treatment (quenching, tempering); machining (fine turning, boring); rotary drawing (first and second transitions); crimping the thickening; low temperature annealing. Statistical studies have found a regression equation showing at a confidence level β = 0.95, dependence of the diameter of the base hole of the workpiece after drawing in from the current value of the diameter before drawing.

Ring dynamics around the Centaur Chariklo and the dwarf planet Haumea: effects of high-order resonances

<p>Narrow and dense rings have been detected around the small Centaur body Chariklo (Braga-Ribas et al. 2014), as well as around the dwarf planet Haumea (Ortiz et al. 2017).</p> <p>Both objects have non-axisymmetric shapes that induce strong resonant effects between the rotating central body with spin rate <em>Ω </em>and the radial epicyclic motion of the ring particles, <em>κ</em>. These resonances include the classical Eccentric Lindblad Resonances (ELR), where <em>κ = m(n-Ω),</em> with <em>m</em> integer, <em>n </em>being the particle mean motion. These resonances create an exchange of angular momentum between the body and the collisional ring, clearing the corotation zone, pushing the inner disk onto the body and repelling the outer part outside of the outermost 1/2 ELR, where the particles complete one orbital revolution while the body executes two rotations, i.e. <em>n/Ω ~ </em>1/2 (Sicardy et al. 2019)</p> <p>Here I will focus on higher-order resonances. They may appear either by considering other resonances such as <em>n/Ω ~ </em>1/3, or the same resonance as above (<em>n/Ω ~ </em>1/2), but with a body that has a triaxial shape. In this case, the invariance of the potential under a rotation of<em> π</em> radians transforms the 1st-order 1/2 Lindblad Resonance into a 2nd order 2/4 resonance.</p> <p>Second-order resonances are of particular interest because they force a strong response of the particles near the resonance radius, in spite of the intrinsically small strength of their forcing terms. This stems from the topography of the associated resonant Hamiltonian, which possesses an unstable hyperbolic point at its origin.</p> <p>The width of the region where this strong response is expected will be discussed for both Chariklo's and Haumea's rings. The special case of the second-order 1/3 resonance will be discussed, as it appears that both ring systems are close to that resonance.</p> <p>This work is intended, among others, to pave the way for future collisional simulations of rings around non-axisymmetric bodies.</p> <div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p>Braga-Ribas et al., 2014, <em>Nature</em> <strong>508</strong>, 72<br />Ortiz et al., 2017, <em>Nature</em> <strong>550</strong>, 219<br />Sicardy et al., 2019, <em>Nature Astronomy</em> <strong>3</strong>, 146</p> <p>The work leading to these results has received funding from the European Research Council under the European Community's H2020 2014-2020 ERC Grant Agreement n°669416 "Lucky Star"</p> </div> </div> </div>