An Approximate Three-Dimensional Solution for Melting or Freezing Around a Buried Pipe Beneath a Free Surface

1986 ◽  
Vol 108 (4) ◽  
pp. 900-906 ◽  
Author(s):  
G.-P. Zhang ◽  
S. Weinbaum ◽  
L. M. Jiji
Keyword(s):  
Approximate Solution ◽  
Free Surface ◽  
Phase Change ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Computational Time ◽  
Buried Pipe ◽  
Dimensional Solution ◽  
Pipe Surface

This paper presents a quasi-steady-state approximate solution for small Stefan number for the three-dimensional melting or freezing around a fluid-carrying pipe buried in a semi-infinite phase change medium (PCM). The two-dimensional quasi-steady approximate solution method, the virtual free surface technique [18], has been extended to three dimensions where axial thermal interaction between the moving fluid and the PCM is considered. Of particular interest in the motion of the phase change interface and the time variation of the axial temperature distribution in the fluid. Due to the singularities of the differential equations along the pipe surface, an axisymmetric analytic solution is provided for the region near the pipe wall. Solutions are presented for several representative dimensionless pipe burial depths and initial conditions. The computational time to predict the three-dimensional interface location up to 10 years is several minutes on an IBM 4341 computer.

Solving linear systems from dynamical energy analysis - using and reusing preconditioners

2021 ◽  
Vol 263 (5) ◽  
pp. 1041-1052
Author(s):  
Martin Richter ◽  
Gregor Tanner ◽  
Bruno Carpentieri ◽  
David J. Chappell
Keyword(s):  
Wave Equations ◽  
Energy Analysis ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Iterative Solvers ◽  
Computational Method ◽  
Computational Time ◽  
Shell Elements ◽  
Finite Dimensional ◽  
Large Matrix

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.

An Investigation Into Nonlinear Growth Rate of Two-Dimensional and Three-Dimensional Single-Mode Richtmyer–Meshkov Instability Using an Arbitrary-Lagrangian–Eulerian Algorithm

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Mike Probyn ◽  
Ben Thornber ◽  
Dimitris Drikakis ◽  
David Youngs ◽  
Robin Williams
Keyword(s):  
Growth Rate ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Numerical Study ◽  
Three Dimensional ◽  
Single Mode ◽  
Computational Time ◽  
Initial Growth ◽  
Arbitrary Lagrangian Eulerian ◽  
Layer Width

This paper presents an investigation into the use of a moving mesh algorithm for solving unsteady turbulent mixing problems. The growth of a shock induced mixing zone following reshock, using an initial setup comparable to that of existing experimental work, is used to evaluate the behavior of the numerical scheme for single-mode Richtmyer–Meshkov instability (SM-RMI). Subsequently the code is used to evaluate the growth rate for a range of different initial conditions. The initial growth rate for three-dimensional (3D) SM Richtmyer–Meshkov is also presented for a number of different initial conditions. This numerical study details the development of the mixing layer width both prior to and after reshock. The numerical scheme used includes an arbitrary Lagrangian–Eulerian grid motion which is successfully used to reduce the mesh size and computational time while retaining the accuracy of the simulation results. Varying initial conditions shows that the growth rate after reshock is independent of the initial conditions for a SM provided that the initial growth remains in the linear regime.

Two- and Three-Dimensional EBSD Measurement of Dislocation Density in Deformed Structures

2010 ◽  
Vol 160 ◽  
pp. 17-22 ◽  
Author(s):  
David P. Field ◽  
Colin C. Merriman ◽  
Ioannis N. Mastorakos
Keyword(s):  
Free Surface ◽  
Dislocation Density ◽  
Bulk Material ◽  
Ion Beam ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Dislocation Dynamics ◽  
Three Dimensions ◽  
Lattice Curvature ◽  
Orientation Data

Electron backscatter diffraction (EBSD) techniques have been used to measure the dislocation density tensor for various materials. Orientation data are typically obtained over a planar array of measurement positions and the minimum dislocation content required to produce the observed lattice curvature is calculated as the geometrically necessary (or excess) dislocation density. The present work shows a comparison of measurements in two-dimensions and three-dimensions using a dual beam instrument (focused ion beam, electron beam) to obtain the data. The two-dimensional estimate is obviously lower than that obtained from three-dimensional data since the 2D analysis necessarily assumes that the third dimension has no curvature in the lattice. Effects of the free-surface on EBSD measurements are discussed in conjunction with comparisons against X-ray microdiffraction experiments and a discrete dislocation dynamics model. It is observed that the EBSD measurements are sensitive to free-surface effects that may yield dislocation density observations that are not consistent with that of the bulk material.

scholarly journals The Diffraction of Free-Surface Waves by a Slender Ship

1984 ◽  
Vol 28 (01) ◽  
pp. 29-47
Author(s):  
P. D. Sclavounos
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Three Dimensional ◽  
Practical Interest ◽  
Prolate Spheroid ◽  
Exciting Force ◽  
Dimensional Solution ◽  
Free Surface Waves ◽  
Dimensional Approximation ◽  
Inner Region

A linear theory is presented for the scattering of small-amplitude monochromatic and unidirectional free-surface waves by a ship fixed at its mean advancing position. In an inner region close to the ship the hull geometrical slenderness is used to justify a quasi-two-dimensional approximation of the flow. The method of matched asymptotic expansions is then introduced to enforce the compatibility of the inner solution with the three-dimensional solution in the far field. The theory is shown to be uniformly valid for all wavelengths of practical interest and all angles of wave incidence. In the short-wavelength limit, existing theories are recovered and the singularity that is present in the limit from oblique to head incidence is removed. Computations are included for the pressure and the sectional exciting force distributions, the wave elevation, and the vertical exciting force and moment in head and bow waves on a prolate spheroid.

scholarly journals Dependence of Enstrophy Transport and Mixed Mass on Dimensionality and Initial Conditions in the Richtmyer–Meshkov Instability Induced Flows1

2020 ◽  
Vol 142 (12) ◽  
Author(s):  
Ye Zhou ◽  
Michael Groom ◽  
Ben Thornber
Keyword(s):  
Inertial Confinement Fusion ◽  
Initial Conditions ◽  
Broad Band ◽  
Initial Perturbation ◽  
Density Ratio ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Late Time ◽  
Production Term

Abstract This paper presents a comparative study of the enstrophy budget and mixed mass between two- and three-dimensional flows induced by Richtmyer–Meshkov instability (RMI). Specifically, the individual contributions to the enstrophy budget due to the production from baroclinicity and from vortex stretching (which vanishes in two-dimensional (2D) flow) are delineated. This is enabled by a set of two- and three-dimensional computations at Atwood 0.5 having both narrow- and broad-band perturbations. A further three-dimensional (3D) computation is conducted at Atwood 0.9 using an identical narrowband perturbation to the Atwood 0.5 case to examine the sensitivity to density ratio. The mixed mass is also considered with the goal to obtain insight on how faithfully a simplified calculation performed in two dimensions can capture the mixed mass for an inertial confinement fusion (ICF) or other practical application. It is shown that the late time power law decay of variable density enstrophy is substantially different in two and three dimensions for the narrowband initial perturbation. The baroclinic production term is negligible in three dimensions (aside from the initial shock interaction), as vortex stretching is larger by two orders of magnitude. The lack of vortex stretching considerably reduces the decay rate in both narrowband and broadband perturbations in two dimensions. In terms of mixed mass, the lack of vortex stretching reduces the mixed mass in two dimensions compared to three in all cases. In the broadband cases, the spectral bandwidth in the 2D case is wider; hence, there is a longer time period of sustained linear growth which reduces the normalized mixed mass further.

scholarly journals Akbari–Ganji Method for Solving Equations of Euler–Bernoulli Beam with Quintic Nonlinearity

Acoustics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 337-353
Author(s):  
Iman Khatami ◽  
Mohsen Zahedi ◽  
S. Abolfazl Zahedi ◽  
Mohammad Yaghoub Abdollahzadeh Jamalabadi
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Computational Time ◽  
Nonlinear Frequency ◽  
Response Curves ◽  
High Dependency ◽  
System Variables

In many real word applications, beam has nonlinear transversely vibrations. Solving nonlinear beam systems is complicated because of the high dependency of the system variables and boundary conditions. It is important to have an accurate parametric analysis for understanding the nonlinear vibration characteristics. This paper presents an approximate solution of a nonlinear transversely vibrating beam with odd and even nonlinear terms using the Akbari–Ganji Method (AGM). This method is an effective approach to solve nonlinear differential equations. AGM is already used in the heat transfer science for solving differential equations, and in this research for the first time, it is applied to find the approximate solution of a nonlinear transversely vibrating beam. The advantage of creating new boundary conditions in this method in additional to predefined boundary conditions is checked for the proposed nonlinear case. To illustrate the applicability and accuracy of the AGM, the governing equation of transversely vibrating nonlinear beams is treated with different initial conditions. Since simply supported and clamped-clamped structures can be encountered in many engineering applications, these two boundary conditions are considered. The periodic response curves and the natural frequency are obtained by AGM and contrasted with the energy balance method (EBM) and the numerical solution. The results show that the present method has excellent agreements in contrast with numerical and EBM calculations. In most cases, AGM is applied straightforwardly to obtain the nonlinear frequency– amplitude relationship for dynamic behaviour of vibrating beams. The natural frequencies tested for various values of amplitude are clearly stated the AGM is an applicable method for the proposed nonlinear system. It is demonstrated that this technique saves computational time without compromising the accuracy of the solution. This approach can be easily extended to other nonlinear systems and is therefore widely applicable in engineering and other sciences.

Simulation of Turbulent Flows in River Confluences and Meandering Channels with a Cartesian 3D Free Surface Hydrodynamic Model

2015 ◽  
Vol 12 (06) ◽  
pp. 1550035 ◽  
Author(s):  
C. L. Ramón ◽  
J. Prats ◽  
F. J. Rueda
Keyword(s):  
Free Surface ◽  
Turbulent Flows ◽  
Three Dimensional ◽  
Computational Time ◽  
Natural Environments ◽  
Computational Domain ◽  
Meandering Channels ◽  
Lateral Cavity ◽  
Reasonable Approach ◽  
Simple Flows

Three-dimensional primitive equations (3DPE) become a reasonable approach in hydrodynamics in terms of computational costs when the length of the computational domain and/or computational time scales increases. However, given the simplified set of equations used in the analysis, results with 3DPE-based models are expected to be approximate and before attempting to reproduce complex natural flows they first need to be validated against more simple flows observed in laboratory settings. Here, the validity of Cartesian free-surface hydrodynamic models to reproduce three turbulent flows characteristic of river environments is tested: (1) the development of shallow mixing layers, (2) flow pass a lateral cavity and (3) flow in open channel with mild curvature. Errors between measured and modeled values were generally less than 10%, proving their validity to reproduce such turbulent flows and their potential for simulations in more complex natural environments, such it is the case of the confluence between the Ebro and Segre rivers into Ribarroja reservoir.

The Influence of a Free Surface on the Hydroelastic Stability of a Flat Panel

1972 ◽  
Vol 39 (1) ◽  
pp. 53-58 ◽  
Author(s):  
D. S. Weaver ◽  
T. E. Unny
Keyword(s):  
Free Surface ◽  
Three Dimensional ◽  
Surface Displacement ◽  
Two Dimensional ◽  
Flat Panel ◽  
Dimensional Solution ◽  
Dimensional Approximation ◽  
Free Surface Displacement

This paper examines the influence of a parallel free surface on the hydroelastic stability of a flat panel. A quasi-two-dimensional approximation is made for the free surface displacement and the results compared with the more general but cumbersome three-dimensional solution. This comparison shows that the former approach is quite reasonable as well as being considerably simpler and more instructive. It is found that the free surface has no effect for depth ratios greater than about one half and is stabilizing for smaller depth ratios.

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