Holes in ordered sets

We pursue the technique of “holes” to study the retracts of an ordered set. This is applied to establish a close connection between the class of absolute retracts and the class of dismantlable ordered sets.     

The rank of a distributive lattice

The rank of a partial ordering P is the maximum size of an irredundant family of linear extensions of P whose intersection is P. A simple relationship is established between the rank of a finite distributive lattice and its subset of join irreducible elements.     

Certain partially ordered sets of dimension three

Some new 3-irreducible posets are constructed.     

On the stabilization of leading-edge vortices with spanwise flow

The influence of spanwise flow on the development and stabilization of leading-edge vortices (LEVs) on a foil (without rotational acceleration) has been investigated. The plunging swept wing/fin geometry used in this study, characteristic of fish-like tails, has been found to be insufficient in the stabilization of LEVs. Direct force measurements and visualizations, including Particle Image Velocimetry and lead precipitation, show that despite the presence of a strong spanwise flow at higher sweepback angles, the vortex breaks off and convects downstream at the same relative time as found for low sweepback angles, which experience little spanwise contribution. Although the LEV stabilization is insensitive to bulk spanwise flow, the LEV and tip vortex have been observed to maintain a stronger connection with one another at higher sweepback angles. This result implies that despite similar forces developing for low and high sweepback angles alike, the resulting vortex-wake topologies can vary significantly from one another.     

Measurement of parallel blade–vortex interaction at low Reynolds numbers

In this study parallel blade–vortex interaction for a Schmidt-propeller configuration has been examined using particle image velocimetry (PIV). This tandem configuration consists of a leading airfoil (forefoil), used to generate a vortical wake of leading-edge vortices (LEVs) and trailing-edge vortices (TEVs) through a pitching or plunging motion, and a trailing airfoil (hindfoil), held fixed with a specified angle of attack and vertical spacing in its wake. The hindfoil incidence (loading) and not the vertical spacing to the incoming vortical wake has been found to dictate the nature of the interaction (inviscid vs. viscous). For cases where the vortex–blade offset is small and the hindfoil is loaded, vortex distortion and vortex-induced separations are observed. By tracking the circulation of the LEV and TEV, it has been found that the vortices are strengthened for the tandem arrangement and in certain cases dissipate quicker in the wake when interacting with the hindfoil. Time-averaged forces obtained using a standard control-volume analysis are then obtained and used to evaluate these vortex-interaction cases. A subsequent analysis of the varying pressure distribution over the suction side of the hindfoil is performed by integrating the Navier–Stokes equations through the velocity field. This allows for a direct comparison of the vortex-induced loading for the various configurations.     

The Complexity of Upward Drawings on Spheres

Although there is a linear time algorithm to decide whether an ordered set has an upward drawing on a surface topologically equivalence to a sphere, we shall prove that the decision problem whether an ordered set has an upward drawing on a sphere is NP-complete. The proof involves the investigation of the surface topology of ordered sets highlighting especially their saddle points. It echoes the recent, important result due to A. Garg and R. Tamassia (1995) that upward planarity testing is NP-complete, for which we give a new proof.