Upper bounds for centerlines

In 2008, Bukh, Matousek, and Nivasch conjectured that for every n-point set S in R^d and every k, 0 <= k <= d-1, there exists a k-flat f in R^d (a "centerflat") that lies at "depth" (k+1) n / (k+d+1) - O(1) in S, in the sense that every halfspace that contains f contains at least that many points of S. This claim is true and tight for k=0 (this is Rado's centerpoint theorem), as well as for k = d-1 (trivial). Bukh et al. showed the existence of a (d-2)-flat at depth (d-1) n / (2d-1) - O(1) (the case k = d-2). In this paper we concentrate on the case k=1 (the case of "centerlines"), in which the conjectured value for the leading constant is 2/(d+2). We prove that 2/(d+2) is an *upper bound* for the leading constant. Specifically, we show that for every fixed d and every n there exists an n-point set in R^d for which no line in R^d lies at depth larger than 2n/(d+2) + o(n). This point set is the "stretched grid"---a set which has been previously used by Bukh et al. for other related purposes. Hence, in particular, the conjecture is now settled for R^3.Comment: This paper (without the appendix) has been published in Journal of Computational Geometry 3:20--30, 2012. 17 pages; 10 figure

CFD Mixing Analysis of Jets Injected from Straight and Slanted Slots into Confined Crossflow in Rectangular Ducts

A CFD study was performed to analyze the mixing potential of opposed rows of staggered jets injected into confined crossflow in a rectangular duct. Three jet configurations were numerically tested: (1) straight (0 deg) slots; (2) perpendicular slanted (45 deg) slots angled in opposite directions on top and bottom walls; and (3) parallel slanted (45 deg) slots angled in the same direction on top and bottom walls. All three configurations were tested at slot spacing-to-duct height ratios (S/H) of 0.5, 0.75, and 1.0; a jet-to-mainstream momentum flux ratio (J) of 100; and a jet-to-mainstream mass flow ratio of 0.383. Each configuration had its best mixing performance at S/H of 0.75. Asymmetric flow patterns were expected and predicted for all slanted slot configurations. The parallel slanted slot configuration was the best overall configuration at x/H of 1.0 for S/H of 0.75