scholarly journals Ill posedness in modelling two-dimensional morphodynamic problems: effects of bed slope and secondary flow

2019 ◽  
Vol 868 ◽  
pp. 461-500 ◽  
Author(s):  
Víctor Chavarrías ◽  
Ralph Schielen ◽  
Willem Ottevanger ◽  
Astrid Blom
Keyword(s):  
Sediment Transport ◽  
Numerical Simulations ◽  
Secondary Flow ◽  
Vertical Mixing ◽  
Well Posedness ◽  
Two Dimensional ◽  
Transport Direction ◽  
Flow Curvature ◽  
Ill Posed ◽  
Well Posed

A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.

scholarly journals Fluid velocity profile reconstruction for non-Newtonian shear dispersive flow

2001 ◽  
Vol 5 (2) ◽  
pp. 87-104 ◽  
Author(s):  
Paul R. Shorten ◽  
David J. N. Wall
Keyword(s):  
Inverse Problem ◽  
Velocity Profile ◽  
Fluid Velocity ◽  
Average Concentration ◽  
Two Dimensional ◽  
Cross Sectional ◽  
Dispersive Flow ◽  
Ill Posed ◽  
Profile Reconstruction ◽  
Well Posed

An inverse problem associated with the mass transport of a material concentration down a pipe where the flowing non-Newtonian medium has a two-dimensional velocity profile is examined. The problem of determining the two-dimensional fluid velocity profile from temporally varying cross-sectional average concentration measurements at upstream and downstream locations is considered. The special case of a known input upstream concentration with a time zero step, and a strictly decreasing velocity profile is shown to be a well-posed problem. This inverse problem is in general ill-posed and mollification is used to obtain a well conditioned problem.

A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

2015 ◽  
Vol 7 (1) ◽  
pp. 31-42 ◽  
Author(s):  
Jingjun Zhao ◽  
Songshu Liu ◽  
Tao Liu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Kernel Method ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Ill Posed ◽  
Well Posed ◽  
Regularized Solution

AbstractIn this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.

scholarly journals A Well-posed Model for Mixed-Sediment River Morphodynamics

2018 ◽  
Vol 40 ◽  
pp. 05060
Author(s):  
Víctor Chavarrías ◽  
Guglielmo Stecca ◽  
Robert Jan Labeur
Keyword(s):  
Active Layer ◽  
Measured Data ◽  
Additional Parameter ◽  
Well Posedness ◽  
Layer Model ◽  
Ill Posed ◽  
Basic Concepts ◽  
Physical Phenomena ◽  
Well Posed ◽  
Two Alternatives

The mixed-size character of sediment is a necessary property to ex- plain physical phenomena such as downstream fining or the presence of armor layers. The active layer model was developed to model mixed-size sediment in river morphodynamics. This model assumes that the topmost part of the bed, the active layer, has no vertical stratification and interacts with the flow. The substrate, below the active layer, only interacts with the active layer in case of aggradation or degradation. The active layer model has been used in morphody- namic modelling for more than four decades but under certain conditions it may become mathematically ill-posed. When a model becomes ill-posed, the solu- tion presents unphysical oscillations and its predictive capabilities are lost. We present two alternatives to the active layer model. The first one retains the basic concepts and guarantees well-posedness by means of an additional parameter controlling the celerity of mixed-size sediment processes. The second solution yields a well-posed model by means of considering the sediment transport rate as a stochastic process rather than to adapt instantaneously to the flow. Both models provide reasonable results when compared to measured data from a lab- oratory experiment conducted under conditions in which the active layer model is ill-posed.

scholarly journals The Cauchy problem for a two-dimensional generalized Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces

2019 ◽  
Vol 18 (03) ◽  
pp. 469-522
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Jianhua Huang ◽  
Jinqiao Duan
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
Well Posedness ◽  
Two Dimensional ◽  
Anisotropic Sobolev Spaces ◽  
The Cauchy Problem ◽  
Kp Equations ◽  
Fifth Order ◽  
Well Posed

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation [Formula: see text] is locally well-posed in the anisotropic Sobolev spaces [Formula: see text] with [Formula: see text] and [Formula: see text]. Second, we prove that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text]. Finally, we show that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text] Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. 79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl. 90 (2008) 338–352].

scholarly journals NUMERICAL SIMULATIONS OF SEDIMENT TRANSPORT INDUCED BY THE 2004 INDIAN OCEAN TSUNAMI NEAR KIRINDA PORT IN SRI LANKA

2011 ◽  
Vol 1 (32) ◽  
pp. 12 ◽  
Author(s):  
Naoto Kihara ◽  
Masafumi Matsuyama
Keyword(s):  
Sediment Transport ◽  
Sri Lanka ◽  
Indian Ocean ◽  
Numerical Simulations ◽  
Secondary Flow ◽  
Suspended Sediment ◽  
Indian Ocean Tsunami ◽  
2004 Indian Ocean Tsunami ◽  
Flow And Sediment Transport ◽  
Induced Flow

Numerical simulations of sediment transport induced by the 2004 Indian Ocean tsunami near Kirinda port in southeastern Sri Lanka is carried out and the relationships between the tsunami-induced flow and sediment transport are discussed. The results show two clear relationships. The first one is that the local scour occurs and a large amount of suspended sediment is generated around the head of breakwaters when the tsunami-induced flow passes through there. The second one is that the suspended sediment is deposited near vortex centers due to a secondary flow.

WAVE EQUATION WITH SECOND-ORDER NON-STANDARD DYNAMICAL BOUNDARY CONDITIONS

2008 ◽  
Vol 18 (12) ◽  
pp. 2019-2054 ◽  
Author(s):  
JUAN LUIS VAZQUEZ ◽  
ENZO VITILLARO
Keyword(s):  
Space Variable ◽  
Regularity Theory ◽  
Laplacian Operator ◽  
Well Posedness ◽  
Open Domain ◽  
One Dimensional ◽  
Ill Posed ◽  
Dynamical Boundary Conditions ◽  
The One ◽  
Well Posed

The paper deals with the well-posedness of the problem [Formula: see text] where u = u(t, x), t ∈ ℝ, x ∈ Ω, Δ = Δx denotes the Laplacian operator with respect to the space variable, Ω is a bounded regular (C∞) open domain of ℝN (N ≥ 1), Γ = ∂Ω, ν is the outward normal to Ω, k is a constant. We prove that it is ill-posed if N ≥ 2, while it is well-posed when N = 1. In the one-dimensional case, we give a complete existence, uniqueness and regularity theory. We also give some existence result for regular initial data when N ≥ 2 and Ω is a ball.

Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements

2002 ◽  
Vol 459 ◽  
pp. 93-102 ◽  
Author(s):  
GRETCHEN L. HOLTZER ◽  
LANCE R. COLLINS
Keyword(s):  
Inverse Problem ◽  
Distribution Function ◽  
Numerical Simulations ◽  
Power Law ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Two Dimensional ◽  
Dimensional Measurements ◽  
Well Posed ◽  
Lower Dimensional

In this paper, we present relationships between the intrinsic radial distribution function (RDF) for a three-dimensional, isotropic system of particles and the lower-dimensional RDFs obtained experimentally from either two-dimensional or one-dimensional sampling of the data. The lower-dimensional RDFs are shown to be equivalent to integrals of the three-dimensional function, and as such contain less information than their three-dimensional counterpart. An important consequence is that the lower-dimensional RDFs are attenuated at separation distances below the characteristic length scale of the measurement. In addition, the inverse problem (calculating the three-dimensional RDF from the lower-dimensional measurements) is not well posed. However, recent results from direct numerical simulations (Reade & Collins 2000) showed that the three-dimensional RDF for aerosol particles in a turbulent flow field obeys a power-law dependence on r for r [Lt ] η, where η is the Kolmogorov scale of the turbulence. In this case, the inverse problem is well posed and it is possible to obtain the prefactor and exponent of the power law from one- or two-dimensional measurements. A procedure for inverting the data is given. All of the relationships derived in this paper have been validated by data derived from direct numerical simulations.

Sign in / Sign up

Export Citation Format

Share Document