Non-flat Bottom Research Articles

Two‐dimensional stationary fluids with gravity and nonflat bottom

AbstractWe establish well‐posedness theory for the two‐dimensional stationary fluids with gravity and a nonflat bottom. The major purpose in this paper is to study free streamline theory in hydrodynamics of the fluids with gravity in rotational flows. More precisely, given any positive horizontal velocity of incoming flow in the inlet of the nozzle, there exists a smooth fluids with gravity. In addition, there exists a smooth free surface, which goes to infinity. Furthermore, we obtain the asymptotic behavior of the fluids with gravity in upstream and downstream by means of a blow‐up argument. As byproducts, the monotone relationship between the location of the free boundary and the asymptotic height in the inlet are also established.

Moving water equilibria preserving nonstaggered central scheme for open‐channel flows

In this paper, we investigate a well‐balanced and positive‐preserving nonstaggered central scheme, which has second‐order accuracy on both time and spatial scales, for open‐channel flows with the variable channel width and the nonflat bottom. We perform piecewise linear reconstructions of the conserved variables and energy as well as discretize the source term using the property that the energy remains constant, so that the complex source term and the flux can be precisely balanced so as to maintain the steady state. The scheme also ensures that the cross‐sectional wet area is positive by introducing a draining time‐step technique. Numerical experiments demonstrate that the scheme is capable of accurately maintaining both the still steady‐state solutions and the moving steady‐state solutions, simultaneously. Moreover, the scheme has the ability to accurately capture small perturbations of the moving steady‐state solution and avoids generating spurious oscillations. It is also capable of showing that the scheme is positive‐preserving and robust in solving the dam‐break problem.

Well-Balanced and Positivity-Preserving Surface Reconstruction Schemes Solving Ripa Systems With Nonflat Bottom Topography

Well-Balanced and Positivity-Preserving Surface Reconstruction Schemes Solving Ripa Systems With Nonflat Bottom Topography

A Well-Balanced Asymptotic Preserving Scheme for the Two-Dimensional Rotating Shallow Water Equations with Nonflat Bottom Topography

A Well-Balanced Asymptotic Preserving Scheme for the Two-Dimensional Rotating Shallow Water Equations with Nonflat Bottom Topography

A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms. Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservative finite element method whose approximate solutions are discontinuous across the inter-element boundaries. The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-LobattoLegendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as a high-order version of upwind finite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standard finite element methods fail. To deal with the source terms we also implement a high-order splitting operator for the time integration. The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surface flows over both flat and non-flat beds. The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method. The proposed method is also applied to solve a recirculation flow problem in the Strait of Gibraltar.

A High-Order Well-Balanced Positivity-Preserving Moving Mesh DG Method for the Shallow Water Equations With Non-Flat Bottom Topography

A rezoning-type adaptive moving mesh discontinuous Galerkin method is proposed for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the simulation of perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance and positivity-preserving properties, strategies are discussed in the use of slope limiting, positivity-preservation limiting, and data transferring between meshes. Particularly, it is suggested that a DG-interpolation scheme be used for the interpolation of both the flow variables and bottom topography from the old mesh to the new one and after each application of the positivity-preservation limiting on the water depth, a high-order correction be made to the approximation of the bottom topography according to the modifications in the water depth. Moreover, mesh adaptation based on the equilibrium variable and water depth is shown to give more desirable results than that based on the commonly used entropy function. Numerical examples in one and two spatial dimensions are presented to demonstrate the well-balance and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state.

Micro-ECM of the Aerodynamic Surface Understating

This paper demonstrates that 18 μm deep surface understating with a tolerance of 4.5 μm and a roughness Ra=0.4 μm can be produced to the required accuracy by electrochemical die-sinking if configured appropriately.Theoretical analysis shows that the bottom profile error can be presented as a superposition of the errors of surface alignment (non-parallel bottom) and shape (non-flat bottom). In this case, the alignment error accounts for a greater part of the size tolerance (3 μm out of 4.5 μm). This is why the attainment of desired accuracy revolves around the development, analysis, and assessment of ways to reduce this error.

A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies

We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only first order derivatives, thus allowing to overcome the numerical difficulties arising from higher order derivative terms, especially in the context of high order discontinuous Galerkin finite element schemes. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an extra conservation law from which the energy-type conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. In order to provide a theoretical basis for the proposed model, a derivation from the vertical average of the compressible Euler equations has been proposed. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation recently proposed in [1].

A second-order numerical scheme for the porous shallow water equations based on a DOT ADER augmented Riemann solver

In the present work, a novel DOT ADER numerical solver capable of handling porosity and bottom discontinuities in the framework of the 1D porous Shallow Water Equations (SWEs) is presented. In order to ensure the preservation of the water at rest condition, a new set of well-balanced governing equations based on the isotropic porosity parameter is derived. The effects exerted by the bed slope and porosity variation source terms are accurately accounted for inside the Riemann solver: to this purpose, an augmented Riemann problem is created by adding two fictitious equations stating the invariance of porosity and bottom in time to the SWEs system. With the aim of computing the non-conservative fluxes, which in the augmented system replace the original source terms, meanwhile ensuring robustness, stability and accuracy, a novel approximate numerical scheme, based on the entropy-satisfying DOT family, is introduced. The extension of the novel Riemann solver, which strictly conserves mass, to a second order of accuracy in both space and time is addressed in the ADER framework. The fulfillment of the C-property condition (i.e. the exact preservation of an initial quiescent flow) in the presence of a discontinuous porosity field and over a non-flat bottom with abrupt variation is theoretically proved and numerically verified. The capability of the proposed numerical scheme to simulate some Riemann problems developing across porosity discontinuities and bed steps is finally assessed.

Modification of the Endohedral Potential after Instant Ionization of an Inner Atom

We investigate here the variation of endohedral A@CN potential due to instant ionization of an inner atom A. Using a reasonable model to describe the fullerenes shell, we managed to calculate this variation that is a consequence of the monopole polarization of CN shell. We demonstrate that the phenomenological potentials that properly simulate the CN shell should belong to a family of potentials with a non-flat bottom, instead of very often used square well potential. As concrete example, we use the Lorentz-bubble model potential. By varying the shape of this potential, we describe the various degrees of the monopole polarization of the CN shell by positive electric charge at the center of the shell. We calculated the photoionization cross sections of He, Ar, and Xe atoms located at the center of C60 shell with and without taking into account accompanying this process monopole polarization of the fullerenes shell. Unexpectedly, we found that the monopole polarization do not affect the photoionization cross sections of these endohedral atoms, thus presenting a very seldom example of potential conspiracy.

Existence of a regular solution for 1D Green\u2013Naghdi equations with surface tension at a large time instant

In this paper the model 1D-GNσ is considered, which concerns the 1D Green–Naghdi equations with non-flat bottom and under the influence of surface tension, to be widely used in coastal oceanography to describe the propagation of large-wave amplitudes. The purpose of this paper is to show that the solution of 1D-GNσ can be made by the Picard iterative scheme, which proves that there is no loss of regularity of the solution relative to the initial condition.

Using the AIM for solving the non-relativistic wave equation for a new class of infinite one-dimensional well with non-flat bottom

The main goal of this work is to solve the non-relativistic wave equation for a new potential configuration that describes the quantum states of a particle that lies within a one-dimensional infinite well of width L using the asymptotic iteration method (AIM). This potential was introduced recently by Alhaidari to be added to the class of exactly solvable potentials in the tridiagonal representation approach (TRA). We have obtained the energy eigenvalues for different choices of the potential parameters. A good match between our results and the ones obtained by the TRA together with new results for the energy spectrum have been presented.

The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom

The goal of this work is to investigate, analytically and numerically, the dynamics of gravity water waves with the effects of the small surface tension and the bottom topography taken into account. Using a third-order perturbative approach of the Boussinesq equation, we obtain a new third-order perturbed Korteweg-de Vries (KdV) equation which includes nonlinear, dispersive, nonlocal and mixed nonlinear-dispersive terms, describing shallow water waves with a non-flat bottom and the surface tension. We show by numerical simulations, for various bottom shapes, that this new third-order perturbed KdV equation can support the propagation of solitary waves, whose profiles strongly depend on the surface tension. In particular, we show that the instability observed in the numerical simulation can be suppressed by the inclusion of small surface tension.

Nonlinear boundary layers for rotating fluids

We investigate the behaviour of rotating incompressible flows near a nonflat horizontal bottom. In the flat case, the velocity profile is given explicitly by a simple linear ODE. When bottom variations are taken into account, it is governed by a nonlinear PDE system, with far less obvious mathematical properties. We establish the well-posedness of this system and the asymptotic behaviour of the solution away from the boundary. In the course of the proof, we investigate in particular the action of pseudodifferential operators in nonlocalized Sobolev spaces. Our results extend an older paper of Gerard-Varet (J. Math. Pures Appl. (9) 82:11 (2003), 1453–1498), restricted to periodic variations of the bottom, using the recent linear analysis of Dalibard and Prange (Anal. & PDE 7:6 (2014), 1253–1315).

Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system

A lot of well-balanced schemes have been proposed for discretizing the classical Saint-Venant system for shallow water flows with nonflat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example, Godunov, Roe, kinetic, etc.). If this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semi-discrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We also prove that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term.

Jellium model potentials for the C60 molecule and the photoionization of endohedral atoms, A@C60

Approximating the C60 shell as a collection of carbon atoms, the potential experienced by a confined atom has been calculated within the framework of the self-consistent spherical jellium model. It has been found that the potential well in this model has a cusp-shaped Lorentz-like profile. The parameters of the model Lorentz-bubble potential (depth and thickness) have been selected so that in the potential well there would be an electronic level corresponding to the experimental electron affinity of the C60 molecule. The spatial distribution of the positive charge of the C-atomic nuclei and the negative charge of the electron clouds forming the electrostatic potential of C60, as a whole, has been analyzed using the Poisson equation. It is demonstrated that the often used radial square-well potential to approximate the C60 corresponds to a non-physical charge density for the C60 molecule. This analysis demonstrates that the phenomenological potentials simulating the C60 shell potential should belong to a family of potentials with a non-flat bottom and non-parallel potential walls similar to the Lorentz-bubble potential. The photoionization cross-sections of a hydrogen atom localized at the center of the C60 shell have been calculated as well. It is found that confinement oscillations in the cross-sections are exhibited within the framework of the cusp-shaped potential model and these oscillations are essentially the same as those in the case of the potential wells with well-defined borders (parallel walls), thereby demonstrating that the inherent characteristic distances of the potential, e.g., radii of the potential walls, or the distances between potential walls, are not necessary to produce confinement resonances; this should be a general result for atoms or molecules confined in near-spherical fullerenes.

3D analytical solution to the tidally induced Lagrangian residual current equations in a narrow bay

The 3D first-order Lagrangian residual velocity (LRV) equation is established, and its analytical solution is obtained in a narrow bay. The results show clearly the 3D structure of the first-order LRV. When the exponential bottom profile is assumed, the upper half layer of the water flows in through the deep channel from the open boundary directly to the head of the bay. Then the water will return to the area surrounding the lower half of the inflow area. The downwelling area is located mainly at the deep channel, while the upwelling area occupies both sides of the bay. The inter-tidal water transport, obtained by integrating the 3D first-order LRV through the water column, has a pattern similar to the previous study in which the 2D depth-averaged Lagrangian residual current equations were solved. The inter-tidal water transport is used to analyze the water exchange, and it is found that the water exchange at different cross sections increases smoothly with the distance between the cross sections and the head of the bay until about one wavelength. It is also found that the pattern of the breadth-averaged Lagrangian residual current varies with the length of the bay if a non-flat bottom profile is used. The depth-integrated LRV and the breadth-averaged LRV are mainly determined by the different terms of the tidal body force, with the former determined by the bottom friction related term and the latter by the eddy viscosity related term. When the bay is longer than one wavelength, different results in the outer bay can be observed.

Effect of stratification on the frequency of bounded Rossby modes over a non-flat bottom

This work attempts to express and analyze the challenges, induced by stratification, affecting the Rossby-topographic eigenmodes of a closed domain with a general uneven bottom of arbitrary shape filled with a uniform fluid in the unperturbed configuration. The modified eigenmodes have been computed analytically: stratification is introduced in the mathematical form of a perturbation of a homogeneous fluid over a non-flat bottom. The eigenmodes lose their barotropic character and differences appear in the dynamical fields (velocity and pressure) from upper to lower layer, as expected. Expressions for the baroclinic and ageostrophic velocity components due to the perturbation are given. The analysis is carried out in the frame of linear shallow water approximation. All terms have been retained apart from nonlinear advection in the governing equations. We find that the frequencies of the eigenmodes change; an analytical expression of frequency correction as a function of layer density difference and interface depth is found. Initial results for some elementary geometrical settings with a waveguide bottom are determined and expressed in a concise, easily readable closed form. The results obtained in the shallow water approximation are expanded in series with respect to the Rossby number. Next, they are compared with the frequency correction obtained in an alternative framework in which the quasi-geostrophic approximation is used, and a purely baroclinic perturbation is imposed from the outset as the result of the introduction of stratification in the otherwise homogeneous fluid. In this scenario, reduced gravity and the ratio of upper to lower layer depth are, in turn, used as the expansion parameters in lieu of the Rossby number.

Paralic Confinement: Models and Simulations

This paper deals with the mathematical modelling of confinement of paralic ecosystems. It is based on the recent paper (Frenod and Goubert in Ecol. Model. 200(1---2):139---148, 2007) that presents a modelling procedure in order to compute the confinement field of a lagoon. Here, we improve the existing model in order to account for tide oscillations in any kind of geometry such as non-rectangular lagoons with a non-flat bottom. The new model, that relies on PDEs rather than ODEs, is then implemented thanks to the finite element method. Numerical results confirm the feasibility of confinement studies thanks to the introduced model.

Validity of the Reynolds equation for miscible fluids in microchannels

In this paper, we consider asymptotic models for miscible flows in microchannels. The characteristics of the flows in microfluidics imply that usually the Hele-Shaw approximation is valid. We present asymptotic models in the Hele-Shaw regime for flows of miscible fluids in a channel in the case where the bottom and the top of the channels have been modified in two different ways. The first case concerns a flat bottom with slip boundary conditions obtained by chemical patterning. The second one is a non-flat bottom with a non-slipping surface. We derive in both cases 2.5D and 2D asymptotic models. We prove global well-posedness of the 2D model. We also prove that both approaches are asymptotically equivalent in the Hele-Shaw regime and we present direct 3D simulations showing that for passive mixing strategy, the Hele-Shaw approximation is not valid anymore.