Water wave transients in an ice-covered channel

2011 ◽  
Vol 38 (4) ◽  
pp. 404-414 ◽  
Author(s):  
François Nzokou ◽  
Brian Morse ◽  
Jean-Loup Robert ◽  
Martin Richard ◽  
Edmond Tossou
Keyword(s):  
Analytical Solutions ◽  
Water Waves ◽  
Numerical Solutions ◽  
Wave Attenuation ◽  
Ice Sheet ◽  
Ice Cover ◽  
Beam Equation ◽  
Transverse Cracks ◽  
Time Step ◽  
The Impact

Many studies show that the propagation of a breakup water surge in impeded rivers (ice cover present) differs from the unimpeded case. Some of the differences are due to ice sheet breaking into pieces as the wave travels downstream while others are due to the effect of a fissured but otherwise intact ice cover’s resistance to motion. This is the subject of this paper: water waves that are sufficiently strong to break the cover away from the banks but not strong enough to create transverse cracks. Although some analytical solutions exist for the propagation of these transients for simple cases, for the first time in the literature, this paper introduces numerical solutions using a FEM model (HYDROBEAM) that simulates this interaction using the one-dimensional Saint-Venant equations appropriately written for rivers having an intact fissured floating ice cover coupled with a classic beam equation subject to hydrostatic loads (often referred to as a beam on an elastic foundation). The governing equations are numerically expressed and are solved using a finite element method (FEM) for the hydrodynamic and ice beam equations separately. A coupling technique is used to converge to a unique solution at each time step (for more information on the numerical characteristics of the model, see companion paper presented by the authors in this issue). The coupled model, gives a first and unique opportunity to compare the simplified analytical solutions to the full numerical solutions. A parametric analysis is herein presented that quantifies the impact of the ice cover's presence and stiffness on wave attenuation and wave celerity as well as to quantify tensile stresses generated in the ice sheet as a function of ice properties (thickness and strength) and channel shape (rectangular and trapezoidal). In general, for rectangular channels, it was found that the simplified analytical solutions are quite representative of the phenomenon namely that short wave transients are affected by the cover’s stiffness but long waves (>400 m) are not.

SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL

2017 ◽  
Vol 59 (2) ◽  
pp. 167-182 ◽  
Author(s):  
H. Y. ALFIFI
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Partial Differential ◽  
Analytical Results ◽  
Transient Solutions ◽  
The Impact

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

scholarly journals Mathematical modeling of groundwater contamination with varying velocity field

2017 ◽  
Vol 65 (2) ◽  
pp. 192-204 ◽  
Author(s):  
Pintu Das ◽  
Sultana Begam ◽  
Mritunjay Kumar Singh
Keyword(s):  
Groundwater Contamination ◽  
Analytical Solutions ◽  
Diffusion Coefficients ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Time Dependent ◽  
Analytical Models ◽  
Retardation Factor ◽  
Hydrodynamic Dispersion ◽  
The Impact

Abstract In this study, analytical models for predicting groundwater contamination in isotropic and homogeneous porous formations are derived. The impact of dispersion and diffusion coefficients is included in the solution of the advection-dispersion equation (ADE), subjected to transient (time-dependent) boundary conditions at the origin. A retardation factor and zero-order production terms are included in the ADE. Analytical solutions are obtained using the Laplace Integral Transform Technique (LITT) and the concept of linear isotherm. For illustration, analytical solutions for linearly space- and time-dependent hydrodynamic dispersion coefficients along with molecular diffusion coefficients are presented. Analytical solutions are explored for the Peclet number. Numerical solutions are obtained by explicit finite difference methods and are compared with analytical solutions. Numerical results are analysed for different types of geological porous formations i.e., aquifer and aquitard. The accuracy of results is evaluated by the root mean square error (RMSE).

scholarly journals Water movement through plant roots – exact solutions of the water flow equation in roots with linear or exponential piecewise hydraulic properties

2017 ◽  
Vol 21 (12) ◽  
pp. 6519-6540 ◽  
Author(s):  
Félicien Meunier ◽  
Valentin Couvreur ◽  
Xavier Draye ◽  
Mohsen Zarebanadkouki ◽  
Jan Vanderborght ◽  
...  
Keyword(s):  
Water Flow ◽  
Water Uptake ◽  
Analytical Solutions ◽  
Hydraulic Properties ◽  
Numerical Solutions ◽  
Water Movement ◽  
Axial Flow ◽  
Flow Equation ◽  
Root Tip ◽  
The Impact

Abstract. In 1978, Landsberg and Fowkes presented a solution of the water flow equation inside a root with uniform hydraulic properties. These properties are root radial conductivity and axial conductance, which control, respectively, the radial water flow between the root surface and xylem and the axial flow within the xylem. From the solution for the xylem water potential, functions that describe the radial and axial flow along the root axis were derived. These solutions can also be used to derive root macroscopic parameters that are potential input parameters of hydrological and crop models. In this paper, novel analytical solutions of the water flow equation are developed for roots whose hydraulic properties vary along their axis, which is the case for most plants. We derived solutions for single roots with linear or exponential variations of hydraulic properties with distance to root tip. These solutions were subsequently combined to construct single roots with complex hydraulic property profiles. The analytical solutions allow one to verify numerical solutions and to get a generalization of the hydric behaviour with the main influencing parameters of the solutions. The resulting flow distributions in heterogeneous roots differed from those in uniform roots and simulations led to more regular, less abrupt variations of xylem suction or radial flux along root axes. The model could successfully be applied to maize effective root conductance measurements to derive radial and axial hydraulic properties. We also show that very contrasted root water uptake patterns arise when using either uniform or heterogeneous root hydraulic properties in a soil–root model. The optimal root radius that maximizes water uptake under a carbon cost constraint was also studied. The optimal radius was shown to be highly dependent on the root hydraulic properties and close to observed properties in maize roots. We finally used the obtained functions for evaluating the impact of root maturation versus root growth on water uptake. Very diverse uptake strategies arise from the analysis. These solutions open new avenues to investigate for optimal genotype–environment–management interactions by optimization, for example, of plant-scale macroscopic hydraulic parameters used in ecohydrogolocial models.

scholarly journals Generation of solitary waves by transcritical flow over a step

2007 ◽  
Vol 587 ◽  
pp. 235-254 ◽  
Author(s):  
R. H. J. GRIMSHAW ◽  
D.-H. ZHANG ◽  
K. W. CHOW
Keyword(s):  
Analytical Solutions ◽  
Water Waves ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Undular Bore ◽  
Downstream Side ◽  
Upstream Side ◽  
Transcritical Flow ◽  
De Vries ◽  
Positive Step

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. Inthispaper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg–de Vries equation, together with numerical solutions of the full Eulerequations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore.

Verification of ANSYS and Matlab Conduction Results Using Analytical Solutions

2020 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
Giampaolo D'Alessandro ◽  
James V. Beck
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Early Time ◽  
The Body ◽  
Numerical Code ◽  
Transient Temperature ◽  
Dimensionless Time ◽  
Time Step ◽  
First Case

Abstract A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element method, are compared against an analytical solution. Various different grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ANSYS software, including coarse, medium and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. The errors found in the numerical solutions by comparing them directly with the analytical solution vary depending primarily on the time step size used. The errors are much larger if calculated using the analytical solution at a given time as a basis of the comparison between the two solutions as opposed to using the steady-state temperature as a basis. The largest errors appear in the early time steps of the problem, which is typically the regime wherein the largest errors occur in mathematical solutions to transient conduction problems. Conversely, errors at larger values of dimensionless time are extremely small and the numerical solutions agree within one tenth of one percent of the analytical solutions at even the worst locations. In addition to difficulties during the early time values of the problem, temperatures calculated on convective boundaries or prescribed-heat-flux boundaries are locations generating larger-magnitude errors. Corners are particularly difficult locations and require finer gridding and finer time steps in order to generate a very precise solution from a numerical code. These regions are compared, using several grid densities, against the analytical solutions. The analytical solutions are, in turn, intrinsically verified to eight significant digits by comparing similar analytical solutions against one another at very small values of dimensionless time. The solution developed using the Matlab differential equation solver was found to have errors of a similar magnitude to those generated using ANSYS. Two different test cases are examined for the various numerical solutions using the selected grid densities. The first case involves steady heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case involves constant heating for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times were extremely small, the errors found within the short duration test were more significant.

scholarly journals The Framework For Ice Sheet–Ocean Coupling (FISOC) V1.1

2021 ◽  
Vol 14 (2) ◽  
pp. 889-905
Author(s):  
Rupert Gladstone ◽  
Benjamin Galton-Fenzi ◽  
David Gwyther ◽  
Qin Zhou ◽  
Tore Hattermann ◽  
...  
Keyword(s):  
Dynamic Model ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Coupled Processes ◽  
Earth System ◽  
Ice Shelf ◽  
Time Step ◽  
Ice Dynamics ◽  
Ocean Coupling ◽  
The Impact

Abstract. A number of important questions concern processes at the margins of ice sheets where multiple components of the Earth system, most crucially ice sheets and oceans, interact. Such processes include thermodynamic interaction at the ice–ocean interface, the impact of meltwater on ice shelf cavity circulation, the impact of basal melting of ice shelves on grounded ice dynamics and ocean controls on iceberg calving. These include fundamentally coupled processes in which feedback mechanisms between ice and ocean play an important role. Some of these mechanisms have major implications for humanity, most notably the impact of retreating marine ice sheets on the global sea level. In order to better quantify these mechanisms using computer models, feedbacks need to be incorporated into the modelling system. To achieve this, ocean and ice dynamic models must be coupled, allowing runtime information sharing between components. We have developed a flexible coupling framework based on existing Earth system coupling technologies. The open-source Framework for Ice Sheet–Ocean Coupling (FISOC) provides a modular approach to coupling, facilitating switching between different ice dynamic and ocean components. FISOC allows fully synchronous coupling, in which both ice and ocean run on the same time step, or semi-synchronous coupling in which the ice dynamic model uses a longer time step. Multiple regridding options are available, and there are multiple methods for coupling the sub-ice-shelf cavity geometry. Thermodynamic coupling may also be activated. We present idealized simulations using FISOC with a Stokes flow ice dynamic model coupled to a regional ocean model. We demonstrate the modularity of FISOC by switching between two different regional ocean models and presenting outputs for both. We demonstrate conservation of mass and other verification steps during evolution of an idealized coupled ice–ocean system, both with and without grounding line movement.

Comparing Numerical and Analytical Solutions of Solitary Water Waves Over Finite and Variable Depth

2021 ◽  
Author(s):  
Thiago S. Hallak ◽  
Hafizul Islam ◽  
Sarat Chandra Mohapatra ◽  
C. Guedes Soares
Keyword(s):  
Analytical Solutions ◽  
Potential Flow ◽  
Water Waves ◽  
Numerical Solutions ◽  
Turbulence Models ◽  
Boundary Element Methods ◽  
Linear Potential ◽  
Variable Depth ◽  
Viscous Effects ◽  
Non Linear

Abstract In this paper, three methods are used in order to obtain the solution for the propagation of water solitons over finite and variable depth. First, the exact analytical solitary wave solutions of the one-dimensional non-linear Boussinesq equations under shallow water condition are described for constant and variable depth. Second, the three-dimensional Fully Non-linear Potential Flow code OceanWave3D is used in order to obtain the numerical solutions for the solitary waves’ propagation over same depth ranges, providing robust solutions for the potential flow problem. Third, Computational Fluid Dynamics’ tool OpenFOAM is used in order to obtain the viscous solution for the same problem, however, without the accounts of turbulence models. The free-surface profiles are drawn and compared; and the stability of the numerical solutions are assessed. Since the approximations of Boussinesq-type equations depend mainly on the orders of magnitude of amplitude and depth, the numerical-analytical comparison will draw the limits for the validity of the analytical solutions. On the other hand, the comparison will provide the limits where viscous effects start playing an important role, whereas the CFD simulations predict the occurrence of wave breaking. These benchmark cases are compared with past references. After all, results regarding the same phenomena have been described in the literature according to, e.g. Fully Non-linear Boussinesq Models, and Fully Nonlinear Potential Flow schemes solved by Boundary Element Methods. Last but not least, the open source Fully Non-linear Potential Flow code is used in order to provide the potential flow solution for some extra cases of water soliton propagation, in order to capture the trends in weak shoaling scenarios.

scholarly journals New solutions for the propagation of long water waves over variable depth

1994 ◽  
Vol 278 ◽  
pp. 391-406 ◽  
Author(s):  
Yinglong Zhang ◽  
Songping Zhu
Keyword(s):  
Wave Propagation ◽  
Gravity Waves ◽  
Analytical Solutions ◽  
Water Waves ◽  
Numerical Solutions ◽  
Practical Significance ◽  
Variable Depth ◽  
Surface Gravity Waves ◽  
Long Wave ◽  
Variable Water

Based on the linearized long-wave equation, two new analytical solutions are obtained respectively for the propagation of long surface gravity waves around a conical island and over a paraboloidal shoal. Having been intensively studied during the last two decades, these two problems have practical significance and are physically revealing for wave propagation over variable water depth. The newly derived analytical solutions are compared with several previously obtained numerical solutions and the accuracy of those numerical solutions is discussed. The analytical method has the potential to be used to find solutions for wave propagation over more natural bottom topographies.

scholarly journals A Framework for Ice Sheet – Ocean Coupling (FISOC) V1.1

2020 ◽  
Author(s):  
Rupert Gladstone ◽  
Benjamin Galton-Fenzi ◽  
David Gwyther ◽  
Qin Zhou ◽  
Tore Hattermann ◽  
...  
Keyword(s):  
Dynamic Model ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Coupled Processes ◽  
Earth System ◽  
Ice Shelf ◽  
Time Step ◽  
Ice Dynamics ◽  
Ocean Coupling ◽  
The Impact

Abstract. A number of important questions concern processes at the margins of ice sheets where multiple components of the Earth System, most crucially ice sheets and oceans, interact. Such processes include thermodynamic interaction at the ice-ocean interface, the impact of melt water on ice shelf cavity circulation, the impact of basal melting of ice shelves on grounded ice dynamics, and ocean controls on iceberg calving. These include fundamentally coupled processes in which feedback mechanisms between ice and ocean play an important role. Some of these mechanisms have major implications for humanity, most notably the impact of retreating marine ice sheets on global sea level. In order to better quantify these mechanisms using computer models, feedbacks need to be incorporated into the modelling system. To achieve this ocean and ice dynamic models must be coupled, allowing run time information sharing between components. We have developed a flexible coupling framework based on existing Earth System coupling technologies. The open-source Framework for Ice Sheet – Ocean Coupling (FISOC) provides a modular approach to online coupling, facilitating switching between different ice dynamic and ocean components. FISOC allows fully synchronous coupling, in which both ice and ocean run on the same time-step, or semi-synchronous coupling in which the ice dynamic model uses a longer time step. Multiple regridding options are available, and multiple methods for coupling the sub ice shelf cavity geometry. Thermodynamic coupling may also be activated. We present idealised simulations using FISOC with a Stokes flow ice dynamic model coupled to a regional ocean model. We demonstrate the modularity of FISOC by switching between two different regional ocean models and presenting outputs for both. We demonstrate conservation of mass and other verification steps during evolution of an idealised coupled ice – ocean system, both with and without grounding line movement.

Instability and Collapse in Discrete Wave Equations

2005 ◽  
Vol 5 (3) ◽  
pp. 223-241
Author(s):  
A. Carpio ◽  
G. Duro
Keyword(s):  
Continuum Limit ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Blow Up ◽  
Theoretical Predictions ◽  
Initial States ◽  
The Impact ◽  
The Continuum

AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

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