Nonlinear calculation of arbitrarily shaped supercavitating hydrofoils near a free surface

A nonlinear exact solution to the problem of two-dimensional gravity-free incompressible potential flow around an arbitrarily shaped supercavitating hydrofoil near a free surface is obtained. A combination of Newton’s method with a functional iterative procedure is used to solve the nonlinear integral and algebraic equations of this problem. Fast and stable convergence results by starting the iteration with a readily chosen initial solution. Some representative numerical computations are made for practical hydrofoils having both generally shaped camber and leading-edge thickness distributions. The force coefficients, pressure distribution and free-streamline shapes of the cavity are calculated for each case with an execution time on an IBM 370-158 of 200-530s depending upon the initial trial solution

Three-dimensional theory on supercavitating hydrofoils near a free surface

Supercavitating hydrofoils of large aspect ratio operating near a free surface are investigated, assuming an inviscid and irrotational flow with the effects of gravity and surface tension neglected. The flow near the foil, treated as two-dimensional, is solved by a nonlinear free-streamline theory, then a three-dimensional 'downwash' correction is made using Prandtl's lifting-line theory. The strength of the lifting-line vortex is determined by information from the two-dimensional solution through a matching procedure, in which the inverse of aspect ratio is used as a small parameter for asymptotic expansions. The analysis incorporates a free-surface reference level to determine the submergence depth of the foil. The present method can be applied to any type of foil having an arbitrary planform or profile shape, including a rounded leading edge, a twist and even a small dihedral angle, within the assumption of large aspect ratio. Numerical computations made on rectangular flat-plate hydrofoils show excellent agreement of results with existing experimental data, even for large angles of attack and relatively low aspect ratios. The pressure distributions, shapes of the cavity and free surface are also calculated as a function of spanwise position

Infall, outflow, and rotation in the G19.61-0.23 hot molecular core

Aims: The main goal of this study is to perform a sub-arcsecond resolution analysis of the high-mass star formation region G19.61-0.23, both in the continuum and molecular line emission. While the centimeter continuum images will be discussed in detail in a forthcoming paper, here we focus on the (sub)mm emission, devoting special attention to the hot molecular core. Results: Our observations resolve the HMC into three cores whose masses are on the order of 10^1-10^3 Msun. No submm core presents detectable free-free emission in the centimeter regime, but they appear to be associated with masers and thermal line emission from complex organic molecules. Towards the most massive core, SMA1, the CH3CN (18_K-17_K) lines reveal hints of rotation about the axis of a jet/outflow traced by H2O maser and H13CO+ (1--0) line emission. Inverse P-Cygni profiles of the 13CO (3--2) and C18O (3--2) lines seen towards SMA1 indicate that the central high-mass (proto)star(s) is (are) still gaining mass with an accretion rate $ge 3 ~10^{-3}$ Msun/yr. Due to the linear scales and the large values of the accretion rate, we hypothesize that we are observing an accretion flow towards a cluster in the making, rather than towards a single massive star.Comment: A&A accepted; 18 pages; Preprint with full-resolution figures is available at http://subarutelescope.org/staff/rsf/publication.htm

Warm Extended Dense Gas Lurking At The Heart Of A Cold Collapsing Dense Core

In order to investigate when and how the birth of a protostellar core occurs, we made survey observations of four well-studied dense cores in the Taurus molecular cloud using CO transitions in submillimeter bands. We report here the detection of unexpectedly warm (~ 30 - 70 K), extended (radius of ~ 2400 AU), dense (a few times 10^{5} cm^{-3}) gas at the heart of one of the dense cores, L1521F (MC27), within the cold dynamically collapsing components. We argue that the detected warm, extended, dense gas may originate from shock regions caused by collisions between the dynamically collapsing components and outflowing/rotating components within the dense core. We propose a new stage of star formation, "warm-in-cold core stage (WICCS)", i.e., the cold collapsing envelope encases the warm extended dense gas at the center due to the formation of a protostellar core. WICCS would constitutes a missing link in evolution between a cold quiescent starless core and a young protostar in class 0 stage that has a large-scale bipolar outflow.Comment: Accepted for publication in The Astrophysical Journal Letter

On the kk-free divisor problem

Let $\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\geq 2$. In this paper we establish an asymptotic formula of the integral $\int_1^T|\Delta^{(k)}(x)|^2dx$ for each $k\geq 4.

Path-factors involving paths of order seven and nine

In this paper, we show the following two theorems (here $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$): (i)~If a graph $G$ satisfies $c_{1}(G-X)+\frac{1}{3}c_{3}(G-X)+\frac{1}{3}c_{5}(G-X)\leq \frac{2}{3}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{7}\}$-factor. (ii)~If a graph $G$ satisfies $c_{1}(G-X)+c_{3}(G-X)+\frac{2}{3}c_{5}(G-X)+\frac{1}{3}c_{7}(G-X)\leq \frac{2}{3}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{9}\}$-factor.Comment: 29 pages, 4 figure