Hyperbolic Tangent Profile Research Articles

Cuscuta-Galileon braneworlds

We investigate braneworlds modeled by topological solutions that arise from the so-called Cuscuta-Galileon model. We develop a first order framework and illustrate our procedure with the scalar field having the well-known hyperbolic tangent profile. We find conditions that must be imposed to the parameters of the model in order to have solutions connecting minima of the potential, with the brane constrained to interpolate Minkowski and anti de Sitter geometries. We also find solutions where the brane only interpolates anti de Sitter geometry. In both cases, the gravity sector of the brane is stable against small fluctuations of the metric.

Disk wakes in nonlinear stratification

Nonlinearity of density stratification modulates buoyancy effects. We report results from a body-inclusive large eddy simulation of a wake in nonlinear stratification, specifically for a circular disk at diameter-based Reynolds number ( ${\it Re} $ ) of $5000$ . Five density profiles are considered; the benchmark has linear stratification and the other four have hyperbolic tangent profiles of the same thickness to model a pycnocline. The disk moves inside the central core of the pycnocline in two of those four cases and, in the other two cases with a shifted density profile, the disk moves partially/completely outside the pycnocline. The maximum buoyancy frequency ( $N_{max}$ ) for all the profiles is the same. The first part of the study investigates the centred cases. Non-uniform stratification results in increasing wake turbulence relative to the benchmark owing to reduced suppression of turbulence production as well as wave trapping in the pycnocline. Steady lee waves are also quantified to understand the limitations of linear theory. The second part pays attention to the effect of a relative shift between the pycnocline and the disk. The wake defect velocity decays substantially faster in the cases with a shift and the wake has higher turbulence level. The effect of disk location on the Kelvin wake waves (a family of steady waves within the pycnocline) and its modal form is obtained and explained by solving the Taylor–Goldstein equation. The family of unsteady internal gravity waves that are generated by the wake is also studied and the effect of disk shift is quantified.

Longitudinal velocity profile of flows in open channel with double-layered rigid vegetation

Aquatic vegetation of different heights is widely scattered in natural rivers and is conducive to their environmental function while affecting the flow hydrodynamic conditions. A semi-analytical velocity model is constructed and used to study the longitudinal velocity profile in open channel flow through double-layered rigid vegetation. The double-layered vegetation flow is separated into three zones according to the velocity profile: 1) nearly uniform distributed velocity zone 1A in the lower region of the short vegetation layer, 2) a mixing layer zone B, 3) uniform distributed velocity zone 2A in the upper region of the tall vegetation layer. Two force equilibrium equations about the gravity-driving and vegetation drag are solved to obtain the uniform velocity distribution equations in zone 1A and 2A. The velocity of zone 1A and B is further modeled as a linear superposition of two concepts: the uniform velocity distribution term of zone 1A and a hyperbolic tangent profile. Meanwhile, longitudinal velocity and the lateral vorticity profiles of open channel flow through double-layered rigid vegetation are studied by laboratory flume tests of different vegetation arrangements exposed to two water depths and three slopes. The experimental results show that the longitudinal velocity increases with the slope increase. The verification of the velocity model is based on the instantaneous velocity measured by Acoustic Doppler Velocimetry (ADV), which shows acceptable agreement, indicating that the model can give a reference to the longitudinal velocity of multi-layered vegetation flow in some cases. The effects of wake vortices and boundary friction on the model are further explored in the discussions. The results presented in this study could contribute to the management of aquatic vegetation configurations and the restoration of freshwater ecology.

A comparative study of turbulent stratified shear layers: effect of density gradient distribution

Direct numerical simulations are performed to compare the evolution of turbulent stratified shear layers with different density gradient profiles at a high Reynolds number. The density profiles include uniform stratification, two-layer hyperbolic tangent profile and a composite of these two profiles. All profiles have the same initial bulk Richardson number (Ri_{b,0}); however, the minimum gradient Richardson number and the distribution of density gradient across the shear layer are varied among the cases. The objective of the study is to provide a comparative analysis of the evolution of the shear layers in term of shear layer growth, turbulent kinetic energy as well as the mixing efficiency and its parameterization. The evolution of the shear layers in all cases shows the development of Kelvin–Helmholtz billows, the transition to turbulence by secondary instabilities followed by the decay of turbulence. Comparison among the cases reveals that the amount of turbulent mixing varies with the density gradient distribution inside the shear layer. The minimum gradient Richardson number and the initial bulk Richardson number do not correlate well with the integrated TKE production, dissipation and buoyancy flux. The bulk mixing efficiency for fixed Ri_{b,0} is found to be highest in the case with two-layer density profile and lowest in the case with uniform stratification. However, the parameterizations of the flux coefficient based on buoyancy Reynolds number and the ratio of Ozmidov and Ellison scales show similar scaling in all cases.

Modeling and simulation of multi-component immiscible flows based on a modified Cahn–Hilliard equation

In this study, an efficient method will be developed for the phase-field model of multi-component immiscible phases. The formulation of surface tension requires the interfaces to satisfy the hyperbolic tangent property. However, the interfacial transitions between different phases are not hyperbolic tangent profiles. The enclosed area is not preserved although the total mass is conserved by the original Cahn–Hilliard equation. This study is an extended research based on our previous study (Li et al., 2016). This work aims to apply the modified Cahn–Hilliard model into the multi-phase system. The interface is forced to be hyperbolic tangent by the modified Cahn–Hilliard system. The computational accuracy of the surface tension is improved under our multiphase framework. The mass loss of each phase can be reduced and the enclosed area can be preserved by the proposed method. We show various numerical results to demonstrate the robustness of the proposed modified model.

An asymptotic analysis and numerical simulation of a prostate tumor growth model via the generalized moving least squares approximation combined with semi-implicit time integration

This paper focuses on presenting an asymptotic analysis and employing simulations of a model describing the growth of local prostate tumor (known as a moving interface problem) in two-dimensional spaces. We first demonstrate that the proposed mathematical model produces the diffuse interfaces with the hyperbolic tangent profile using an asymptotic analysis. Next, we apply a meshless method, namely the generalized moving least squares approximation for discretizing the model in the space variables. The main benefits of the developed meshless method are: First, it does not need a background mesh (or triangulation) for constructing the approximation. Second, the obtained numerical results suggest that the proposed meshless method does not need to be combined with any adaptive technique for capturing the evolving interface between the tumor and the neighboring healthy tissue with proper accuracy. A semi-implicit backward differentiation formula of order 1 is used to deal with the time variable. The resulting fully discrete scheme obtained here is a linear system of algebraic equations per time step solved by an iterative scheme based on a Krylov subspace, namely the generalized minimal residual method with zero–fill incomplete lower–upper preconditioner. Finally, some simulation results based on estimated and experimental data taken from the literature are presented to show the process of prostate tumor growth in two-dimensional tissues, i.e., a square, a circle and non-convex domains using both uniform and quasi-uniform points.

Consistent scalar transport with front capturing methods: application to two-phase heat transfer

Accurate numerical simulations of heat transfers in 3D liquid-gas flows are of first importance in multiple industrial applications such as droplet evaporation in combustion chambers, spray cooling or propellant behavior in cryogenic tanks. Liquid-gas flows are characterized by the discontinuity of properties (e.g. viscosity, thermal diffusivities, ...) and flow variables (e.g. pressure, energy, chemical composition) across the interface. Robust and accurate algorithms are then necessary to transport the flow variables consistently with the interface. This work presents an algebraic interface capturing method which enables a consistent transport of scalars with the interface. This two-fluid approach ensures conservation of the transported scalars while controlling accurately the flux at the interface. The interface represented by a hyperbolic tangent profile is transported and reinitialized without geometric reconstruction. The scalars are transported separately in each phase and a reinitialization step is performed in order to impose the correct flux at the interface. The method is implemented in the YALES2 low-Mach number flow solver and takes advantage of adaptive unstructured grids to handle complex geometries. It has been assessed on various test cases and compared to analytical solutions.

Ballistic thermal rectification in asymmetric homojunctions.

Ballistic thermal rectification is of significance for the management of thermal transport at the nanoscale since the size of thermal devices shrinks down to the phonon mean free path. By using the single-particle Lorentz gas model, the ballistic thermal transport in asymmetric homojunctions is investigated. The ballistic thermal rectification of the asymmetric rectangular homojunction is enhanced by the increasing structural asymmetry. A hyperbolic tangent profile is introduced to the interface to study the effect of interface steepness on thermal transport. We find that the thermal rectification ratio increases with the decreasing interface steepness, indicating that a gradual interface is of benefit to increase the thermal rectification. Moreover, the thermal rectification of the asymmetric homojunction can be improved by either increasing the temperature gradient or decreasing the average temperature of two heat sources.

Horizontal shear instabilities in rotating stellar radiation zones

Context. Stellar interiors are the seat of efficient transport of angular momentum all along their evolution. In this context, understanding the dependence of the turbulent transport triggered by the instabilities of the vertical and horizontal shears of the differential rotation in stellar radiation zones as a function of their rotation, stratification, and thermal diffusivity is mandatory. Indeed, it constitutes one of the cornerstones of the rotational transport and mixing theory, which is implemented in stellar evolution codes to predict the rotational and chemical evolutions of stars. Aims. We investigate horizontal shear instabilities in rotating stellar radiation zones by considering the full Coriolis acceleration with both the dimensionless horizontal Coriolis component f̃ and the vertical component f. Methods. We performed a linear stability analysis using linearized equations derived from the Navier-Stokes and heat transport equations in the rotating nontraditional f-plane. We considered a horizontal shear flow with a hyperbolic tangent profile as the base flow. The linear stability was analyzed numerically in wide ranges of parameters, and we performed an asymptotic analysis for large vertical wavenumbers using the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation for nondiffusive and highly-diffusive fluids. Results. As in the traditional f-plane approximation, we identify two types of instabilities: the inflectional and inertial instabilities. The inflectional instability is destabilized as f̃ increases and its maximum growth rate increases significantly, while the thermal diffusivity stabilizes the inflectional instability similarly to the traditional case. The inertial instability is also strongly affected; for instance, the inertially unstable regime is also extended in the nondiffusive limit as 0 < f < 1 + f̃ 2/N2, where N is the dimensionless Brunt-Väisälä frequency. More strikingly, in the high thermal diffusivity limit, it is always inertially unstable at any colatitude θ except at the poles (i.e., 0° < θ < 180°). We also derived the critical Reynolds numbers for the inertial instability using the asymptotic dispersion relations obtained from the WKBJ analysis. Using the asymptotic and numerical results, we propose a prescription for the effective turbulent viscosities induced by the inertial and inflectional instabilities that can be possibly used in stellar evolution models. The characteristic time of this turbulence is short enough so that it is efficient to redistribute angular momentum and to mix chemicals in stellar radiation zones.

DEVELOPMENT OF RIGOROUS METHODS IN FLUID MECHANICS AND THEORY OF WATER WAVES

The thesis consists of five chapters where the following problems are considered - the transformation of long linear waves in an ocean with a variable depth; long wave scattering in a canal with a rapidly varying cross-section; long linear surface waves on stationary currents in a canal of constant depth and variable width; and a resonant interaction of a solitary wave with external pulse-type perturbations within the framework of forced Korteweg-de Vries equation. Chapter 1 reviews the history of these problems, and notes previous literature and research in this area. In Chapter 2, the transformation of long linear waves in an ocean of a variable depth is studied. The transformation coeffcients are considered as functions of frequency and the total depth drop for three typical models of bottom profile variation: piecewise linear, piecewise quadratic and hyperbolic tangent profiles. The results obtained for all these profiles are analysed and compared from the physical point of view. In Chapter 3, long wave scattering in a canal with a rapidly varying cross-section is studied. The scattering coeffcients are calculated for all possible incident wave orientations (background current downstream and upstream with respect to the background flow); and current types subcritical, transcritical, and supercritical. It is shown that when the over-reflection or over-transmission occurs, negative energy waves can appear in the flow. A spontaneous wave generation can happen in a transcritical accelerating flow, resembling a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect in astrophysics. In Chapter 4, long wave transformation is studied in the canal with abrupt variation of width and depth. Again, all possible wave orientation with respect to the background current is considered, and all types of a current is studied (subcritical, transcritical, and supercritical). The transformation coefficients are calculated and analysed as functions of wave frequency, Froud number, and depth drop. In Chapter 5, we revise the solutions of the forced Korteweg{de Vries equation for the resonant interaction of a solitary wave with the various external pulse-type perturbations. In contrast to the previous studies, we consider an arbitrary relationship between the width of a solitary wave and external forcing. In many cases, exact solutions of the forced Korteweg{de Vries equation can be obtained for the specific forcings of arbitrary amplitude. The theoretical outcomes obtained by asymptotic method are in good agreement with the results of direct numerical modelling within the framework of forced Korteweg-de Vries equation. In Conclusion, we summarise the results obtained within the various models and equations; discuss the future applications and innovation of the results; and suggest further perspectives for the future research.

Numerical study of incompressible interfacial flows by an one-step level set method

We develop a two-phase flow model for solving the nonlinear level set (LS) equation and Navier-Stokes equations in two dimension. In the proposed one-step LS method, the sharp interface is captured by implicitly representing the zero LS contour. The nonlinear LS equation is solved by upwinding Combined Compact Difference (CCD) scheme for the convection terms and center difference scheme for the other terms in space for predicting propagation of interface. Within the framework of Navier-Stokes solver, the semi-implicit Gear algorithm on a semi-staggered fixed grid is used for velocity-pressure coupling. The interface normal and curvature are calculated in terms of a hyperbolic tangent profile across the interface. To verify the proposed LS method, we consider reversed single vortex and Zalesak’s cases which are amenable to exact solutions. The proposed two-phase model is also investigated by solving the dam-break, Rayleigh-Taylor instability and bubble rising problems.

Internal wave boluses as coherent structures in a continuously stratified fluid

Internal waves shoaling on the continental slope can break and form materially coherent vortices called boluses. These boluses are able to trap and transport material up the continental slope, yet the global extent of bolus transport is unknown. Previous studies of bolus formation primarily focused on systems consisting of two layers of uniform density, which do not account for the presence of ocean pycnoclines of finite thickness. We use hyperbolic tangent profiles to model the density stratification in our simulations and demonstrate the impact of the pycnocline on the bolus. A spectral clustering method is used to objectively identify the bolus as a Lagrangian coherent structure that contains the material advected upslope. The bolus size and displacement upslope are examined as a function of the pycnocline thickness, incoming wave energy, density change across the pycnocline and topographic slope. The dependence of bolus transport on the pycnocline thickness demonstrates that boluses in continuous stratifications tend to be larger and transport material further than in corresponding two-layer stratifications.

A Novel Hyperbolic Tangent Profile for Optical Fiber for Next Generation OAM-MDM Systems

In this work, we propose and theoretically/numerically investigate a novel parametric refractive index profile for few mode fibers, referred to as a hyperbolic tangent few mode fiber (HTAN-FMF). To the best of our knowledge, the HTAN-FMF is adopted for the first time, to simultaneously maximize the inter-channels separation and OAM purity. The proposed FMF achieves a wide separation between cylindrical vector modes ( $\geq 3\times 10\,\,^{\mathrm{ -4}}$ between TE0,1, HE2,1, and TM0,1). This enables low-level crosstalk and clearly overcomes recently reported levels in literature. On the other hand, simulations show that the generated and evaluated OAM modes are mostly high pure (≥ 99.9%) leading to intrinsic crosstalk as minimum as -30 dB during propagation through HTAN-FMF. Other propagation performance metrics of the proposed fiber including chromatic dispersion and differential mode delay are also evaluated over the C ITU-T band showing that HTAN-FMF has the potential to achieve stable OAM modes transmission. Bending insensitivity is numerically proved showing that the proposed FMF has the potential to enhance the transmission capacity and the spectral efficiency in next generation OAM mode division Multiplexing (OAM-MDM) systems.

Energy Stable Semi-implicit Schemes for Allen–Cahn–Ohta–Kawasaki Model in Binary System

In this paper, we propose a first order energy stable linear semi-implicit method for solving the Allen-Cahn-Ohta-Kawasaki equation. By introducing a new nonlinear term in the Ohta-Kawasaki free energy functional, all the system forces in the dynamics are localized near the interfaces which results in the desire hyperbolic tangent profile. In our numerical method, the time discretization is done by some stabilization technique in which some extra nonlocal but linear term is introduced and treated explicitly together with other linear terms, while other nonlinear and nonlocal terms are treated implicitly. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are proved for this method in both semi-discretization and full discretization levels. Numerical experiments indicate the force localization and desire hyperbolic tangent profile due to the new nonlinear term. We test the first order temporal convergence rate of the proposed scheme. We also present hexagonal bubble assembly as one type of equilibrium for the Ohta-Kawasaki model. Additionally, the two-third law between the number of bubbles and the strength of long-range interaction is verified which agrees with the theoretical studies.

Transformation of Long Surface and Tsunami-Like Waves in the Ocean with a Variable Bathymetry

We consider a transformation of long linear waves in an ocean with a variable depth. We calculate the transformation coefficients (the coefficients of transmission and reflection) as the functions of frequency and the total depth drop for three typical models of bottom profile variation: (i) piecewise-linear, (ii) piecewise-quadratic, and (iii) hyperbolic tangent profiles. For all these cases, exact solutions are obtained, analysed and graphically illustrated. This allows us to derive the transformation coefficients in the analytic form for the subsequent comparison with the results of approximate, numerical, or experimental study. We show that the results obtained are in agreement with both the energy flux conservation and Lamb’s formulae in the limiting case of zero frequency. We also study wave transformation on the underwater barriers and trenches of different shapes and compare the results obtained.

A fractional step lattice Boltzmann model for two-phase flow with large density differences

In this paper, a fractional step lattice Boltzmann method is proposed to model two-phase flow with large density differences by solving a modified Cahn-Hilliard phase-field equation and the incompressible Navier-Stokes equations. In order to maintain the interface profile and conserve the volume, the normal flux in the original Cahn-Hilliard equation is removed and a profile-corrected term to enforce the interface to be a hyperbolic tangent profile is added. The modified Cahn-Hilliard equation is split to two sub-equations. One is the original CH equation and the other is the correction equation. Both the original CH and the hydrodynamic equations are solved by the lattice Boltzmann method. The correction equation is solved by the finite difference method. With the multi-relaxation-time collision model and a high-order compact selective filter operation employed in the lattice Boltzmann method, numerical stability is much improved. Compared with the previous lattice Boltzmann methods, the proposed method is able to maintain the order parameter within a physically meaningful range, which is capable of tracking the interface accurately. The proposed method can also simulate two-phase fluid flows with the density ratio up to 1000. Several benchmark problems, including single vortex deform of a circle, translation of a drop, stationary droplet, capillary wave and rising bubble with large density ratios, are presented to validate the accuracy and efficiency of the present method.

Generation of bright-dark soliton trains with a central wide dip in optical fibers

The nonlinear propagation dynamics of the dark soliton pulse with hyperbolic tangent profile in the anomalous dispersion regime of an optical fiber is investigated numerically. The corresponding temporal evolutions are provided for different integral or non-integral soliton order and loss perturbations. The results show that, irrespective of the integral or non-integral soliton order, with increase of distance, the width of the dark soliton broadens and more and more bright and dark pulses appear symmetrically on the two sides of the central black soliton. Correspondingly, bright-dark soliton trains with a central wide black soliton can gradually form. The higher the soliton order, the more the bright-dark pulses at the same distance. With increase of distance, the peak values of the bright pulses exhibit oscillation behavior. The loss perturbation only influences the evolutions slightly, which means that the generated soliton trains can resist loss perturbations to some extent. Moreover, this work provides us an alternative approach to generate bright-dark soliton trains by using the dark soliton pulse propagating in the anomalous dispersion regime of the passive optical fiber besides directly using the fiber laser.

An alternative method to implement contact angle boundary condition and its application in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces.

We propose an alternative method to implement the contact angle boundary condition on a solid wall and apply it in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces in which the flow equations are solved by the lattice-Boltzmann method and the interface equations are solved by the finite-difference method. Using the hyperbolic tangent profile of the order parameter across an interface in phase-field theory, we were able to obtain its unknown value at a ghost point from the information at only one point in the fluid domain. This is in contrast with existing approaches relying on interpolations involving several points. The special feature allows it to be more easily implemented on immersed surfaces cutting through the grid lines. It was properly incorporated into the framework of the hybrid lattice-Boltzmann finite-difference simulation, and applied successfully for several problems with different levels of complexity. First, the equilibrium shapes of a droplet on a sphere with different contact angles and radii were studied under cylindrical geometry and a good agreement with theoretical predictions was found. Preliminary studies on a three-dimensional droplet spreading on a sphere were also performed and the results agreed well with the corresponding axisymmetric results. Second, the spreading of a two-dimensional drop on an embedded inclined wall with a given contact angle was investigated and the results matched those on a flat wall at the domain boundary under the same condition. Third, capillary filling in a cylindrical tube was studied and the speed of the interface in the tube was found to follow Washburn's law. Fourth, a droplet impacting on a sphere was investigated and several different outcomes were captured depending on the Reynolds number, the viscosity ratio, and the wettability and radius of the sphere. Finally, the proposed method was shown to be capable of studying even more complicated problems involving the interaction between a droplet and multiple objects of different sizes and contact angles.

Extreme nonlinear waves in external gravitational-like potentials: Possible applications for the optical soliton supercontinuum generation and the ocean coast line protection

In view of the fact that the nonlinear Schrödinger equation model has a wide range of applicability, from nonlinear femtosecond optics, Bose–Einstein condensates and plasmas, to ocean extreme (monster) waves and hurricanes, we conclude that existing mathematical analogies open advantageous possibility to study optical, matter-wave and hydrodynamic nonlinear waves and solitons in parallel and due to the evident complexity of experiments with extreme nonlinear waves in oceans, offer remarkable potentialities in studies of these so complex systems by performing “optic hydrodynamics” experiments in the nonlinear fiber-optics systems. Specifically, we present the results of computer experiments of soliton interactions with different external gravitational-like potentials including that with varying levels of inclinations. We demonstrate novel effective mechanism of the supercontinuum generation both in nonlinear optics and in hydrodynamics based on the higher order N-soliton wave packets decay induced by external potentials of the hyperbolic tangent profiles with varying inclinations similar to the real ocean coast line profiles.

Enhancement of Low Illumination Images based on an Optimal Hyperbolic Tangent Profile

Contrast enhancement is a critical pre-processing stage for many image based applications. It is frequently encountered that the illumination condition, while capturing the image, is imperfect. Specific algorithms have to be applied to restore these images from, for instance, the degradation due to low illumination. An adaptive enhancement method is developed here that tackles the image quality enhancement problem from an optimization perspective. In particular, the input image intensity is mapped to the output based on a weighted hybrid of a hyperbolic tangent and a linear profile. The mapping parameters are optimized, with regard to maximizing the image global entropy, by using the Golden Section Search algorithm for its implementation efficiency. Moreover, user interventions are not necessary. Better qualitative and comparable quantitative performances are obtained from experiments, with regard to the increase of brightness, information content and suppression of unwanted artifacts, as compared to recent profile mapping based methods.