Numerical investigation of the wake bi-stability behind a notchback Ahmed body

Large-eddy simulations are used to investigate the origin of the wake asymmetry and symmetry behind notchback Ahmed bodies. Two different effective backlight angles, ${\beta _1} = 17.8\mathrm{^\circ }$ and ${\beta _2} = 21.0\mathrm{^\circ }$ , are simulated resulting in wake asymmetry and symmetry in flows without external perturbations, in agreement with previous experimental observations. In particular, the asymmetric case presents a bi-stable nature showing, in a random fashion, two stable mirrored states characterized by a left or right asymmetry for long periods. A random switch and several attempts to switch between the bi-stability are observed. The asymmetry of the flow is ascribed to the asymmetric separations and reattachments in the wake. The deflection of the near-wall flow structures behind the slant counteracting the asymmetry drives the wake to be temporarily symmetric, triggering the switching process of the bi-stable wake. The consequence of deflection that forces the flow structure to form on the opposite side of the slant is the decisive factor for a successful switch. Modal analysis applying proper orthogonal decomposition is used for the exploration of the wake dynamics of the bi-stable nature observed.

Symmetric and Non-Symmetric Flows of Burgers’ Fluids through Porous Media between Parallel Plates

Unidirectional unsteady flows of the incompressible Burgers’ fluids between two infinite horizontal parallel plates are analytically studied when the magnetic and porous effects are taken into consideration. The fluid motion is induced by the two plates, which move in their planes with time-dependent velocities. Exact general expressions are established both for the dimensionless velocity and shear stress fields as well as the corresponding Darcy’s resistance in the channel using the Laplace transform. If both plates move with equal velocities in the same direction, the fluid motion becomes symmetric with respect to the mid-plane between them. Otherwise, its motion is non-symmetric. To bring to light the behavior of the fluid, the dimensionless velocity profiles versus the spatial variable as well as its time evolution are presented both for the symmetric and asymmetric case. Finally, for comparison, similar graphical representations are presented together for the velocities of the incompressible Oldroyd-B and Burgers’ fluids. For large values of the time t, as expected, the behavior of the two different fluids is almost identical. The Darcy’s resistance against y is also graphically represented for the symmetric flow at different values of the time t. The influence of the magnetic field on the fluid motion is graphically revealed and discussed.

On a Generalization of One-Dimensional Kinetics

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.

Caustics in Gravitational Lensing by Mixed Binary Systems

We investigated binary lenses with 1/rn potentials in the asymmetric case with two lenses with different indexes n and m. These kinds of potentials have been widely used in several contexts, ranging from galaxies with halos described by different power laws to lensing by wormholes or exotic matter. In this paper, we present a complete atlas of critical curves and caustics for mixed binaries, starting from the equal-strength case, and then exploring unequal-strength systems. We also calculate the transitions between all different topology regimes. Finally we find some useful analytic approximations for the wide binary case and for the extreme unequal-strength case.

Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. The numerical results suggest that there exist three solutions designated as type $I$, type $II$ and type $III$ for the asymmetric case, type $I$ solution exists for all non-negative Reynolds number and types $II$ and $III$ solutions appear simultaneously at a common Reynolds number that depends on the value of asymmetric parameter $a$ and with the increase of $a$ the common Reynolds numbers are decreasing. We also theoretically show that there exist three solutions. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. These asymptotic solutions are all verified by their corresponding numerical solutions.

FTA: A Stumbling Bloc towards Global Free Trade

This paper examines the impact of free trade areas (FTAs) on the world trading system. It uses a partial equilibrium framework among three countries where Cobb-Douglas utility function is applied. It is found that FTAs are primarily detrimental to multilateral trading system and are stumbling blocs to global free trade (GFT), whether in a symmetric or an asymmetric case. This conclusion is based, firstly, upon the findings that the utility of member countries in a FTA are higher than status quo, so they are interested in forming a FTA, and secondly, after forming a FTA, they don't have incentive to let the third country join in since their utility would be lowered. In addition, this paper shows that when there are two large countries and a small one, it is more profitable for one large country to form a FTA with the small one instead of the other large one.