Irrotational Research Articles

A flow force reformulation of steady periodic fixed-depth irrotational equatorial flows

In this paper, we study steady two-dimensional periodic equatorial water waves of small amplitude which propagate over a flat bed and with a specified fixed mean-depth. In particular, we focus on irrotational flows and develop an equivalent formulation using flow force functions. The analysis is based on the local bifurcation theory due to Crandall-Rabinowitz.

Almost $$\eta $$-Ricci solitons on Kenmotsu manifolds

We characterize the Einstein metrics in such broad classes of metrics as almost \(\eta \)-Ricci solitons and \(\eta \)-Ricci solitons on Kenmotsu manifolds, and generalize some known results. First, we prove that a Kenmotsu metric as an \(\eta \)-Ricci soliton is Einstein metric if either it is \(\eta \)-Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost \(\eta \)-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of \(\eta \)-Ricci solitons and gradient \(\eta \)-Ricci solitons, which illustrate our results.

Classification of dark solitons via topological vector potentials.

Dark solitons are some of the most interesting nonlinear excitations and are considered to be the one-dimensional topological analogs of vortices. However, in contrast to their two-dimensional vortex counterparts, the topological characteristics of a dark soliton are far from fully understood because the topological charge defined according to the phase jump cannot reflect its essential property. Here, similar to the complex extension used in the exploration of the partition-function zeros to depict thermodynamic states, we extend the complex coordinate space to explore the density zeros of dark solitons. Surprisingly we find that these zeros constitute some pointlike magnetic fields, each of which has a quantized magnetic flux of elementary π. The corresponding vector potential fields demonstrate the topology of the Wess-Zumino term and can depict the essential characteristics of dark solitons. Then we classify the dark solitons according to the Euler characteristic of the topological manifold of the vector potential fields. Our study not only reveals the topological features of dark solitons but can also be applied to explore and identify new dark solitons with high topological complexity.

A rigorous bound on the scaling of dissipation with velocity amplitude in flow past a sphere

Abstract

The Numerical Prediction of The Flow Pattern under a Stationary Hovercraft.(Dept.M)

This paper presents a numerical solution procedure for the prediction of the flow pattern created under a stationary hovercraft using the irrotational flow model. A finite difference computer program using a rectangular grid was employed in this study. Three flow patterns are investigated. These flow patterns are the flow pattern at constant mass flow rate M, flow pattern at constant skirt height and the flow pattern at constant mean pressure along the underside of the hovercraft floor. It, also, determines the dependence of the pressure distribution on the relative height of the skirt and the mass flow rate. The relation between the mass flow rate and the skirt height, at constant average pressure along the underside of the hovercraft, was also obtained. The results show that the flow pattern changes with the variation the skirt height, while there is no change in the flow pattern with the increase of the mass flow rate, if the skirt . height is kept constant. It also explains the practical limitation of the unability of a hovercraft to operate over very rough seas.

Existence of global steady subsonic Euler flows with collision through 2D infinitely long nozzles

In this paper, we study the global existence of steady subsonic flows with collision, where the collision is caused by a confluence of two semi‐infinitely incoming flows that are nonmiscible steady subsonic irrotational Euler flows come from two different infinitely nozzles. First, we prove that when the total flux of two incoming flows is less that the critical mass flux, there exists a unique global smooth subsonic flow with collision. Meanwhile, the interface between two flows is a smooth free interface, which is determined uniquely by the mass fluxes of incoming flows. Second, by using the blowup argument, we establish the asymptotic behaviors for the stream function. Finally, we prove that the position of free interface can be determined uniquely by the mass fluxes of incoming flows. Moreover, we establish the monotonicity of the relation between the position of free interface and the mass fluxes of incoming flows.

Remarks on almost Riemann solitons with gradient or torse-forming vector field

We consider almost Riemann solitons \((V,\lambda )\) in a Riemannian manifold and underline their relation to almost Ricci solitons. When V is of gradient type, using Bochner formula, we explicitly express the function \(\lambda \) by means of the gradient vector field V and illustrate the result with suitable examples. Moreover, we deduce some properties for the particular cases when the potential vector field of the soliton is solenoidal or torse-forming, with a special view towards curvature.

$$\zeta $$-Ricci Soliton on Real Hypersurfaces of Nearly Kaehler 6-Sphere with SSMC

The main intention of this paper is to study the real hypersurfaces in nearly Kaehler $$\mathbb {S}^{6}$$ endowed with a semi-symmetric metric connection. We characterize the real hypersurfaces of the nearly Kaehler $$\mathbb {S}^{6}$$ admitting semi-symmetric metric connection, and investigate the curvature properties of these submanifolds. Moreover, it is shown that a real hypersurface is congruent to an open segment of a totally-geodesic hypersphere or a tube over an almost complex curve in $$\mathbb {S}^{6}$$ if such a connected real hypersurface of nearly Kaehler $$\mathbb {S}^{6}$$ is an $$\zeta $$ -Ricci soliton with the potential vector field $$\xi $$ .

An Experimental Study on the Variation of Atmospheric Magnetic-Field Intensity Related to Dust, Haze, Rain, Snow, and Thunderstorms

Simultaneous measurements of the vertical magnetic field (VMF), three-dimensional electric field, ambient temperature, ambient relative humidity, particle mass concentration, and three-dimensional velocity are conducted during dust, haze, rain, snow, and thunderstorms. The hourly VMF variation (the rate of VMF increment and time increment with a 1-h time interval) is used to evaluate the disturbance level of the atmospheric magnetic field during the abovementioned weather conditions. There is no significant difference in the hourly VMF variations between hazy days and fair weather. However, the hourly VMF variations on dusty, rainy, snowy, and thunderstorm days differ from those in fair weather, implying that these weather conditions significantly affect the atmospheric magnetic field. On hazy days, although the haze particles are charged, the VMF does not change compared with in fair weather, which suggests that the horizontal electric field generated by haze particles is an irrotational field. On dusty and snowy days, the heterogeneity of the charge distribution forms a rotational horizontal electric field, which can induce the VMF to disturb the atmospheric magnetic field. During rain and thunderstorms, the hourly variation in the VMF is larger than in dust and snow, which can be attributed to the rotational horizontal electric field generated by raindrops and clouds. In addition, the hourly variation in the VMF in thunderstorms is one order of magnitude larger than that during days without thunderstorms, indicating that the VMF induced by the cumulonimbus cloud electric field is more significant than that in other weather conditions.

On non-gradient $$(m,\rho )$$-quasi-Einstein contact metric manifolds

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $$(m,\rho )$$ -quasi-Einstein structure on a contact metric manifold. First, we prove that if a K-contact or Sasakian manifold $$M^{2n+1}$$ admits a closed $$(m,\rho )$$ -quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $$2n(2n+1)$$ , and for the particular case—a non-Sasakian $$(k,\mu )$$ -contact structure—it is locally isometric to the product of a Euclidean space $${\mathbb {R}}^{n+1}$$ and a sphere $$S^n$$ of constant curvature 4. Next, we prove that if a compact contact or H-contact metric manifold admits an $$(m,\rho )$$ -quasi-Einstein structure, whose potential vector field V is collinear to the Reeb vector field, then it is a K-contact $$\eta $$ -Einstein manifold.

Divergence-Free Fitting-Based Incompressible Deformation Quantification of Liver.

Liver is an incompressible organ that maintains its volume during the respiration-induced deformation. Quantifying this deformation with the incompressible constraint is significant for liver tracking. The constraint can be accomplished with retaining the divergence-free field obtained by the deformation decomposition. However, the decomposition process is time-consuming, and the removal of non-divergence-free field weakens the deformation. In this study, a divergence-free fitting-based registration method is proposed to quantify the incompressible deformation rapidly and accurately. First, the deformation to be estimated is mapped to the velocity in a diffeomorphic space. Then, this velocity is decomposed by a fast Fourier-based Hodge-Helmholtz decomposition to obtain the divergence-free, curl-free, and harmonic fields. The curl-free field is replaced and fitted by the obtained harmonic field with a translation field to generate a new divergence-free velocity. By optimizing this velocity, the final incompressible deformation is obtained. Moreover, a deep learning framework (DLF) is constructed to accelerate the incompressible deformation quantification. An incompressible respiratory motion model is built for the DLF by using the proposed registration method and is then used to augment the training data. An encoder-decoder network is introduced to learn appearance-velocity correlation at patch scale. In the experiment, we compare the proposed registration with three state-of-the-art methods. The results show that the proposed method can accurately achieve the incompressible registration of liver with a mean liver overlap ratio of 95.33%. Moreover, the time consumed by DLF is nearly 15 times shorter than that by other methods.

Mechanisms of entrainment in a turbulent boundary layer

Data from direct numerical simulation of a zero-pressure-gradient incompressible turbulent boundary layer (TBL) [You and Zaki, “Conditional statistics and flow structures in turbulent boundary layers buffeted by free-stream disturbances,” J. Fluid Mech. 866, 526 (2019)] are analyzed to examine the entrainment process. The two mechanisms by which the outer irrotational flow can be entrained into the turbulent region and their relative contribution to the growth of the spatially developing boundary layer are evaluated: (i) nibbling is the enstrophy transport across the turbulent/non-turbulent interface (TNTI), and (ii) engulfment is the entrapment of pockets of irrotational flow inside the TBL prior to finally breaking apart. The relative importance of the two mechanisms depends on the normalized vorticity threshold adopted to identify the TNTI. Our choice of this threshold highlights the structure of the TNTI and entrainment within this layer by engulfment of irrotational pockets. The sizes of the engulfed pockets are of the same order as the heads of the hairpin vortices underneath the TNTI. The vortices straddle larger streaky structures of internal layers and cause handle shaped deformations on the TNTI, which leads to engulfment as they fold onto themselves and entrap the external potential flow. Three dynamical regions are distinguished: a TNTI region (interface layer), an adjustment region, and the turbulent core. The first of these is further sub-divided into a viscous superlayer and a turbulent sublayer. It is shown as the irrotational fluid elements cross the interface layer toward the turbulent core, a smooth transition from the non-focal topology to the well-known primarily focal topology of fully developed turbulence occur. The viscous superlayer is similar to previously studied flow configurations, such as jets and mixing layers. In contrast, vorticity stretching in the turbulent sublayer is significantly weaker in the boundary layer relative to free-shear flows, which results in a smaller rate of entrainment by nibbling.

Self-Adaptive Asymmetrical Artificial Potential Field Approach Dedicated to the Problem of Position Tracking by Nonholonomic UAVs in Windy Enivroments

Abstract Artificial potential fields (APFs) are a popular method of planning and controlling the path of robot movement, including unmanned aerial vehicles (UAVs). However, in the case of nonholonomic robots such as fixed-wing UAVs, the distribution of velocity vectors should be adapted to their limited manoeuvrability to ensure stable and precise position tracking. The previously proposed local asymmetrical potential field resolves this issue, but it is not effective in the case of windy environments, where the UAV is unable to maintain the desired position and drifts due to the wind drift effect. This is reflected in the growth of position error, which, similar to the steady-state error in the best case, is constant. To compensate for it, the asymmetrical potential field approach is modified by extending definitions of potential function gradient and velocity vector field (VVF) with elements based on the integral of position tracking error. In the case of wind drift, the value of this integral increases over time, and lengths and orientations of velocity vectors will also be changed. The work proves that redefining gradient and velocity vector as a function of position tracking error integrals allows for minimisation of the position tracking error caused by wind drift.

Sasakian 3-metric as a generalized Ricci-Yamabe soliton

In the present paper, we first investigate a Sasakian 3-metric as a quasi-Yamabe gradient soliton. In the sequel, extending the notions of quasi-Yamabe soliton and Ricci-Yamabe soliton, the notion of generalized Ricci-Yamabe soliton is introduced. It is shown that if (g, V, λ, α, β, γ) is a generalized gradient Ricci-Yamabe soliton on a complete Sasakian 3-manifold M with potential function f , then M is compact Einstein and locally isometric to a unit sphere. Moreover, the potential vector field V is an infinitesimal contact transformation and pointwise collinear with the characteristic vector field ξ. Further, if h is the Hodge-de Rham potential for V, then, upto a constant, f = h.

The $k$-almost Yamabe solitons and contact metric manifolds

We introduce the concept of a k-almost Yamabe soliton which extends naturally from Yamabe solitons. Our aim is to study the k-almost Yamabe soliton (g,V,k,λ) on a contact metric manifold M2n+1. Firstly, for a general contact metric manifold, it is proved that V is Killing if the potential vector field V is a contact vector field and that M is K-contact if V is collinear with Reeb vector field. Secondly, we prove that a compact K-contact manifold, admitting a k-almost Yamabe gradient soliton, is isometric to a standard unit sphere. Moreover, for a complete Sasakian manifold admitting a k-almost Yamabe soliton, we show that it is isometric to a standard unit sphere 𝕊2n+1(1) when n>1 and for n=1, M is also isometric to a standard unit sphere if it admits a closed k-almost Yamabe soliton. Finally, we consider a contact metric (κ,μ)-manifold with a nontrivial k-almost Yamabe gradient soliton and show that it is flat in dimension 3 and in higher dimension M is locally isometric to En+1×𝕊n(4). In the end, we construct two examples of contact metric manifolds with a k-almost Yamabe soliton.

Critical homoclinics in a restricted four-body problem: numerical continuation and center manifold computations

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for computing normal forms and center stable/unstable manifold parameterizations. After careful numerical study of a number of special cases we formulate several conjectures about the global bifurcations of the homoclinic families.

Natural Ricci Solitons on Tangent and Unit Tangent Bundles

Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on a flat Riemannian manifold. When unit tangent bundles over a constant curvature Riemannian manifold are endowed with pseudo-Riemannian Kaluza-Klein type metric, we give a classification of Ricci soliton structures whose potential vector fields are fiber-preserving, inferring the existence of some of them which are non Einstein.

A Partition of Unity Method for Divergence-Free or Curl-Free Radial Basis Function Approximation

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications ...

On Ricci solitons with a semi-symmetric metric connection

We find some properties of Ricci solitons with a semi-symmetric metric connection. When the potential vector field is torse-forming, we obtain some characterizations. Applications to submanifolds are also given.

Einstein solitons with unit geodesic potential vector field

We obtain some results on almost Einstein solitons with unit geodesic potential vector field and provide necessary and sufficient conditions for the soliton to be trivial.