Some explicit solutions of the three-dimensional Euler equations with a free surface

We present a family of radial solutions (given in Eulerian coordinates) to the three-dimensional Euler equations in a fluid domain with a free surface and having finite depth. The solutions that we find exhibit vertical structure and a non-constant vorticity vector. Moreover, the flows described by these solutions display a density that depends on the depth. While the velocity field and the pressure function corresponding to these solutions are given explicitly through (relatively) simple formulas, the free surface defining function is specified (in general) implicitly by a functional equation which is analysed by functional analytic methods. The elaborate nature of the latter functional equation becomes simpler when the density function has a particular form leading to an explicit formula of the free surface. We subject these solutions to a stability analysis by means of a Wentzel–Kramers–Brillouin (WKB) ansatz.

Transition of Near-Ground Vorticity Dynamics During Tornadogenesis

Although much is known about the environmental conditions necessary for tornadogenesis, the near-ground vorticity dynamics during the tornadogenesis process itself are still somewhat poorly understood. For instance, seemingly contradicting mechanisms responsible for large near-ground vertical vorticity can be found in the literature. Broadly, these mechanisms can be sorted into two classes, one being based on upward tilting of mainly baroclinically produced horizontal vorticity in descending air (here called downdraft mechanism), while in the other the horizontal vorticity vector is abruptly tilted upward practically at the surface by a strong updraft gradient (referred to as in-and-up mechanism). In this study, full-physics supercell simulations and highly idealized simulations show that both mechanisms play important roles during tornadogenesis. Pretornadic vertical vorticity maxima are generated via the downdraft mechanism, while the dynamics of a fully developed vortex are dominated by the in-and-up mechanism. Consequently, a transition between the two mechanisms occurs during tornadogenesis. This transition is a result of axisymmetrization of the pretornadic vortex patch and intensification via vertical stretching. These processes facilitate the development of the corner flow, which enables production of vertical vorticity by upward tilting of horizontal vorticity practically at the surface, i.e. the in-and-up mechanism. The transition of mechanisms found here suggests that early stages of tornado formation rely on the downdraft mechanism, which is often limited to a small vertical component of baroclinically generated vorticity. Subsequently, a larger supply of horizontal vorticity (produced baroclinically or via surface drag, or even imported from the environment) may be utilized, which marks a considerable change in the vortex dynamics.

Three-Dimensional Numerical Study of the Effect of Protective Barrier on the Dispersion of the Contaminant in a Building

The finite volume method and potential-vorticity vector formalism in their three-dimensional form were used to numerically study the impact of an adiabatic and impermeable vertical barrier on the dispersion of a local aero-contaminant due to the double-diffusive Rayleigh–Benard convection inside a cubic container. Different governing parameters such as the Rayleigh number, buoyancy ratio and barrier height were analyzed for Le = 1.2 and Pr = 0.7, representing an air-contaminant mixture. The potential-vector-vorticity formalism in the three-dimensional form allowed the elimination of the pressure terms appearing in the Navier–Stokes equations. It was found that the heat and mass transfer as well as the effectiveness of the barrier in reducing contaminant dispersion are strongly influenced by the buoyancy ratio, the barrier size and the Rayleigh number. In addition, the barrier effectiveness is more than 70% for a height of half the building height.

The role of curvature in modifying frontal instabilities, part 1

In this study, we examine the role of curvature in modifying frontal stability. We first evaluate the classical criterion that the Coriolis parameter, f, multiplied by the Ertel potential vorticity (PV), q, is positive for stable flow and that instability is possible when this quantity is negative. The first portion of this statement can be deduced from Ertel’s PV theorem, assuming an initially positive fq. Moreover, the full statement is implicit in the governing equation for the mean flow, as the discriminant, fq, changes sign. However, for curved fronts in cyclo-geostrophic or gradient wind balance (GWB), an additional term enters the discriminant owing to conservation of absolute angular momentum, L. The resulting expression, Lq < 0, simultaneously generalizes Rayleigh’s (1917) criterion by accounting for baroclinicity and Hoskins’ (1974) criterion by accounting for centrifugal effects. In particular, changes in the front’s vertical shear and stratification owing to curvature tilt the absolute vorticity vector away from its thermal wind state; in an effort to conserve the product of absolute angular momentum and Ertel PV, this modifies gradient Rossby and Richardson numbers permitted for stable flow. This forms the basis of a non-dimensional expression valid for inviscid, curved fronts on the f -plane, which can be used to classify frontal instabilities. In conclusion, the classical criterion, fq < 0, should be replaced by the more general criterion, Lq < 0, for studies involving gravitational, centrifugal, and symmetric instabilities at curved density fronts. In Part 2 of the study, we examine interesting outcomes of the criterion applied to low-Richardson number fronts and vortices in GWB.

Characterization and Control for the Laminar Flow of Liquid Polyurethane System in a Wide Angle Diffuser with Transversely Arrayed Obstacles

In the manufacturing process of hard-board poly-urethane foams, the uniformity is a very important issue for the raw compound of the liquid poly-urethane system flow for the quality control of such products. One of the universal methods to generate more uniform flow is that some obstacles are located inside the diffuser at the end of injector. For the regime of non-Newtonian laminar flow, better flow uniformity can be achieved with the enhancement of mixing in the wake after the resistive obstacles. In this research, the parametric study is made for the gap interval between adjacent obstacle components as well as the cross-sectional shape with a computational fluid dynamics (CFD) technique. The flow fields around circular and elliptic cylinders are visualized for flow velocity and vorticity with the comparison of root-mean-square (RMS) error for the deviation of velocity at the outlet as a lumped parameter to estimate flow uniformity and mixing. When the blockage ratio is fixed 0.3 for the pipe of Reynolds number 58.5 based on its diameter, eliminating the effect of wall boundary ratio with the classical Blasius velocity profile, the RMS error is reduced 77% to 92% from the baseline case in the case of 60%-diameter gaps for the figure of circles and 2:1 longitudinal ellipse, respectively. The flow is visualized around obstacle components with vorticity as well as flow velocity where the three-dimensional components of vorticity vector are also elucidated in physics for the evolution of complex multi-dimensional flow wake.

Anisotropy of the vorticity tensor as a magnetic indicator of aromaticity

The vorticity vector of the current density JB, induced in the electron cloud of a molecule by a magnetic field B, is defined by VB = ∇ × JB.

Gravitational Radiation, Vorticity And Super–Energy: A Conspicuous Threesome

We elaborate on the link relating gravitational radiation, vorticity and a flux of super–energy on the plane orthogonal to the vorticity vector. We examine the vorticity appearing in the congruence of observers at the outside of the source, as well as the vorticity of the fluid distribution, the source of the gravitational radiation is made of. The information provided by the study of the physical aspects of the source poses new questions which could, in principle, be solved by the observational evidence. Besides the study of the theoretical issues associated to such relationship, we also stress the new observational possibilities to detect gravitational radiation, appearing as consequence of the above mentioned link. The high degree of development achieved in the gyroscope technology, as well as recent proposals to detect rotations by means of ring lasers, atom interferometers, atom lasers and anomalous spin–precession experiments, lead us to believe that an alternative to the laser interferometers used so far to detect gravitational waves, may be implemented based on the detection of the vorticity associated with gravitational radiation. Additionally, this kind of detectors might be able to elucidate the open question about the physical properties of the tail of the waves appearing as the consequence of the violation of the Huygens’s principle in general relativity.

A Spatial Pattern Analysis of Land Surface Roughness Heterogeneity and its Relationship to the Initiation of Weak Tornadoes

North Alabama is among the most tornado-prone regions in the United States and is composed of more spatially variable terrain and land cover than the frequently studied North American Great Plains region. Because of the high tornado frequency observed across north Alabama, there is a need to understand how land surface roughness heterogeneity influences tornadogenesis, particularly for weak-intensity tornadoes. This study investigates whether horizontal gradients in land surface roughness exist surrounding locations of tornadogenesis for weak (EF0–EF1) tornadoes. The existence of the horizontal gradients could lead to the generation of positive values of the vertical components of the 3D vorticity vector near the surface that may aid in the tornadogenesis process. In this study, surface roughness was estimated using parameterizations from the Noah land surface model with inputs from MODIS 500-m and Landsat 30-m data. Spatial variations in the parameterized roughness lengths were assessed using GIS-based grid and quadrant pattern analyses to quantify observed variation of land surface features surrounding tornadogenesis locations across spatial scales. This analysis determined that statistically significant horizontal gradients in surface roughness exist surrounding tornadogenesis locations.