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.

Methodology for developing a high-precision ultrasound flow meter and fluid velocity profile reconstruction

2008 ◽  
Vol 55 (1) ◽  
pp. 161-172 ◽  
Author(s):  
Emmanuelle Mandard ◽  
Denis Kouame ◽  
Rodolphe Battault ◽  
Jean-Pierre Remenieras ◽  
Frederic Patat
Keyword(s):  
Velocity Profile ◽  
High Precision ◽  
Fluid Velocity ◽  
Flow Meter ◽  
Profile Reconstruction

scholarly journals A Hölder-logarithmic stability estimate for an inverse problem in two dimensions

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Matteo Santacesaria
Keyword(s):  
Inverse Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stability Estimate ◽  
Two Dimensions ◽  
Positive Energy ◽  
Two Dimensional ◽  
Ill Posed ◽  
Dirichlet To Neumann Map ◽  
Logarithmic Stability

AbstractThe problem of the recovery of a real-valued potential in the two-dimensional Schrödinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient

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.

2-D CFD Model for Free Surface River Flow with Tilt Rigid Leave of Submerged Vegetation

2012 ◽  
Vol 212-213 ◽  
pp. 332-335 ◽  
Author(s):  
Yan Hong Li ◽  
Li Quan Xie
Keyword(s):  
Free Surface ◽  
Velocity Profile ◽  
River Flow ◽  
Longitudinal Velocity ◽  
Submerged Vegetation ◽  
Two Dimensional ◽  
Cfd Model ◽  
Cross Sectional ◽  
Vertical Velocity Profile ◽  
Flow Through

Keywords: river flow; two-dimensional CFD model; velocity profile; submerged vegetation leave Abstract. River flow with submerged foliage vegetation in straight and rectangular cross-sectional channel is numerically simulated through a vertical two-dimensional CFD model. Tilt thin strips are assigned in river flow to mimic the configuration of vegetation leave. The free surface line and the vertical profiles of longitudinal velocity are presented. The vertical velocity profile differs from the well acknowledged logarithmic or semi-logarithmic law. The submerged leave canopy resist the flow through it and pilots the flow upward over it, resulting in a decreased velocity within the canopy and an increased velocity above the canopy. The velocity profiles within the leave canopy are impacted by the configurations of the leave.

scholarly journals ANALYSIS OF AN INVERSE PROBLEM ARISING IN PHOTOLITHOGRAPHY

2012 ◽  
Vol 22 (05) ◽  
pp. 1150026 ◽  
Author(s):  
LUCA RONDI ◽  
FADIL SANTOSA
Keyword(s):  
Inverse Problem ◽  
Variational Formulation ◽  
Diffractive Optics ◽  
Shape Design ◽  
Two Dimensional ◽  
Convergence Properties ◽  
Approximate Problem ◽  
The Relationship ◽  
Well Posed ◽  
Dimensional Domain

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods, much like the Ambrosio–Tortorelli's approximation of the Mumford–Shah functional in image processing.

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.

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.

scholarly journals Generalised unsteady plume theory

2016 ◽  
Vol 792 ◽  
pp. 1013-1052 ◽  
Author(s):  
John Craske ◽  
Maarten van Reeuwijk
Keyword(s):  
Velocity Profile ◽  
Integral Equations ◽  
Longitudinal Velocity ◽  
Travelling Wave ◽  
Radial Dependence ◽  
Large Set ◽  
Ill Posed ◽  
Velocity Turbulence ◽  
Limiting Case ◽  
Well Posed

We develop a generalised unsteady plume theory and compare it with a new direct numerical simulation (DNS) dataset for an ensemble of statistically unsteady turbulent plumes. The theoretical framework described in this paper generalises previous models and exposes several fundamental aspects of the physics of unsteady plumes. The framework allows one to understand how the structure of the governing integral equations depends on the assumptions one makes about the radial dependence of the longitudinal velocity, turbulence and pressure. Consequently, the ill-posed models identified by Scase & Hewitt (J. Fluid Mech., vol. 697, 2012, pp. 455–480) are shown to be the result of a non-physical assumption regarding the velocity profile. The framework reveals that these ill-posed unsteady plume models are degenerate cases amongst a comparatively large set of well-posed models that can be derived from the generalised unsteady plume equations that we obtain. Drawing on the results of DNS of a plume subjected to an instantaneous step change in its source buoyancy flux, we use the framework in a diagnostic capacity to investigate the properties of the resulting travelling wave. In general, the governing integral equations are hyperbolic, becoming parabolic in the limiting case of a ‘top-hat’ model, and the travelling wave can be classified as lazy, pure or forced according to the particular assumptions that are invoked to close the integral equations. Guided by observations from the DNS data, we use the framework in a prognostic capacity to develop a relatively simple, accurate and well-posed model of unsteady plumes that is based on the assumption of a Gaussian velocity profile. An analytical solution is presented for a pure straight-sided plume that is consistent with the key features observed from the DNS.

Sign in / Sign up

Export Citation Format

Share Document