A Review of Water Hammer Theory and Practice

2005 ◽  
Vol 58 (1) ◽  
pp. 49-76 ◽  
Author(s):  
Mohamed S. Ghidaoui ◽  
Ming Zhao ◽  
Duncan A. McInnis ◽  
David H. Axworthy
Keyword(s):  
Numerical Solutions ◽  
Practical Interest ◽  
Water Hammer ◽  
Theory And Practice ◽  
Full Description ◽  
Turbulence Structure ◽  
Two Dimensional ◽  
Hydraulic Transients ◽  
Research And Practice ◽  
One Dimensional

Hydraulic transients in closed conduits have been a subject of both theoretical study and intense practical interest for more than one hundred years. While straightforward in terms of the one-dimensional nature of pipe networks, the full description of transient fluid flows pose interesting problems in fluid dynamics. For example, the response of the turbulence structure and strength to transient waves in pipes and the loss of flow axisymmetry in pipes due to hydrodynamic instabilities are currently not understood. Yet, such understanding is important for modeling energy dissipation and water quality in transient pipe flows. This paper presents an overview of both historic developments and present day research and practice in the field of hydraulic transients. In particular, the paper discusses mass and momentum equations for one-dimensional Flows, wavespeed, numerical solutions for one-dimensional problems, wall shear stress models; two-dimensional mass and momentum equations, turbulence models, numerical solutions for two-dimensional problems, boundary conditions, transient analysis software, and future practical and research needs in water hammer. The presentation emphasizes the assumptions and restrictions involved in various governing equations so as to illuminate the range of applicability as well as the limitations of these equations. Understanding the limitations of current models is essential for (i) interpreting their results, (ii) judging the reliability of the data obtained from them, (iii) minimizing misuse of water-hammer models in both research and practice, and (iv) delineating the contribution of physical processes from the contribution of numerical artifacts to the results of waterhammer models. There are 134 refrences cited in this review article.

A Two-Dimensional Cylindrical Transient Conduction Solution Using Green’s Functions

2013 ◽  
Author(s):  
Robert L. McMasters ◽  
James V. Beck
Keyword(s):  
Parameter Estimation ◽  
Green's Functions ◽  
Bessel Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Two Dimensional ◽  
Trigonometric Functions ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
One Dimensional

There are many applications for problems involving thermal conduction in two-dimensional cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green’s functions where the principle of superposition is employed in combining a one-dimensional cylindrical case with a one-dimensional Cartesian case to provide a temperature solution for a two-dimensional cylindrical. Green’s functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

Weakly nonlinear internal waves in a two-fluid system

1996 ◽  
Vol 313 ◽  
pp. 83-103 ◽  
Author(s):  
Wooyoung Choi ◽  
Roberto Camassa
Keyword(s):  
Internal Waves ◽  
Numerical Solutions ◽  
Lower Layer ◽  
Bottom Topography ◽  
Ocean Dynamics ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Characteristic Wavelength ◽  
Upper Boundary

We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin–Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak.

The effect of a fissure on storage in a porous medium

2009 ◽  
Vol 639 ◽  
pp. 239-259 ◽  
Author(s):  
JEROME A. NEUFELD ◽  
DOMINIC VELLA ◽  
HERBERT E. HUPPERT
Keyword(s):  
Porous Medium ◽  
Numerical Solutions ◽  
Gravity Current ◽  
Practical Interest ◽  
Two Dimensional ◽  
Geological Sequestration ◽  
Long Term Storage ◽  
Cap Rock ◽  
Two Parameters

We consider the two-dimensional buoyancy driven flow of a fluid injected into a saturated semi-infinite porous medium bounded by a horizontal barrier in which a single line sink, representing a fissure some distance from the point of injection, allows leakage of buoyant fluid. Our studies are motivated by the geological sequestration of carbon dioxide (CO2) and the possibility that fissures in the cap rock may compromise the safe long-term storage of CO2. A theoretical model is presented that accounts for leakage through the fissure using two parameters, which characterize leakage driven both by the hydrostatic pressure within the overriding fluid and by the buoyancy of the fluid within the fissure. We determine numerical solutions for the evolution of both the gravity current within the porous medium and the volume of fluid that has escaped through the fissure as a function of time. A quantity of considerable practical interest is the efficiency of storage, which we define as the amount of fluid remaining in the porous medium relative to the amount injected. This efficiency scales like t−1/2 at late times, indicating that the efficiency of storage ultimately tends to zero. We confirm the results of our model by comparison with an analogue laboratory experiment and discuss the implications of our two-dimensional model of leakage from a fissure for the geological sequestration of CO2.

Numerical Solution for Two-Dimensional Elastic Wave Propagation in Finite Bars

1967 ◽  
Vol 34 (3) ◽  
pp. 725-734 ◽  
Author(s):  
L. D. Bertholf
Keyword(s):  
Experimental Data ◽  
Wave Propagation ◽  
Elastic Wave ◽  
Numerical Solutions ◽  
Step Change ◽  
Two Dimensional ◽  
One Dimensional ◽  
Elastic Bar ◽  
The One ◽  
The Impact

Numerical solutions of the exact equations for axisymmetric wave propagation are obtained with continuous and discontinuous loadings at the impact end of an elastic bar. The solution for a step change in stress agrees with experimental data near the end of the bar and exhibits a region that agrees with the one-dimensional strain approximation. The solution for an applied harmonic displacement closely approaches the Pochhammer-Chree solution at distances removed from the point of application. Reflections from free and rigid-lubricated ends are studied. The solutions after reflection are compared with the elementary one-dimensional stress approximation.

Quasi-one-dimensional model equations for plasma flows in high-pressure discharges in ablative capillaries

1992 ◽  
Vol 48 (2) ◽  
pp. 215-227 ◽  
Author(s):  
D. Zoler ◽  
S. Cuperman
Keyword(s):  
High Pressure ◽  
Numerical Solutions ◽  
Experimental Information ◽  
Dimensional System ◽  
System Of Equations ◽  
Two Dimensional ◽  
One Dimensional ◽  
Pressure Discharge ◽  
Model Equations ◽  
Energy Equations

Quasi-one dimensional hydrodynamic continuity, momentum and energy equations describing the plasma flow in high-pressure-discharge ablative capillaries are derived. To overcome the formidable difficulties arising in the solution of a fully two-dimensional system of equations, experimental information on the structure (geometry) of the generated plasma is used. Thus the two-dimensional hydrodynamic equations are averaged over the cross-section of the capillary to obtain a quasi-one-dimensional system of equations in which, however, the essential two-dimensional features are present. These include the radial outwards radiative transfer of energy and the radial inwards ablative mass flow. Some particular cases, including their thermodynamical aspects, are discussed. Illustrative analytical and numerical solutions of the equations are also presented.

scholarly journals Determination of load distribution in a given medium according to the values of the loads at certain points

2021 ◽  
pp. 167-175
Author(s):  
Oleksandr Mostovenko ◽  
Serhii Kovalov ◽  
Svitlana Botvinovska
Keyword(s):  
Boundary Conditions ◽  
Load Distribution ◽  
Numerical Solutions ◽  
Dimensional Space ◽  
Geometric Model ◽  
Three Dimensional ◽  
The Body ◽  
Three Dimensions ◽  
Two Dimensional ◽  
One Dimensional

Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2]. In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7]. From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space [8], where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes. Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions. If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid. If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes. The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells. As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.

Experiments on Foucault Imaging of Superconducting Fluxons

1997 ◽  
Vol 3 (S2) ◽  
pp. 507-508
Author(s):  
T. Yoshida ◽  
J. Endo ◽  
K. Harada ◽  
H. Kasai ◽  
T. Matsuda ◽  
...  
Keyword(s):  
Phase Contrast ◽  
Theory And Practice ◽  
Structural Defects ◽  
Two Dimensional ◽  
Transmitted Beam ◽  
Standard Methods ◽  
Easy Task ◽  
One Dimensional ◽  
Lorentz Microscopy ◽  
Theoretical Results

The out-of-focus method has been successfully employed in the dynamical observation of superconducting fluxons. However, owing to the large defocus distance needed to image fluxons with enough contrast it turns out that their correlation with structural defects, better imaged at focus, is troublesome. Among the standard methods of Lorentz microscopy the Foucault technique is a good candidate for removing this drawback, since it generates phase contrast in the focused image by masking part of the transmitted beam by means of an aperture.Therefore, simulations have been carried out both for one-dimensional fluxon models and more realistic two dimensional ones, with the result that enough contrast can be generated in the focused image in order to detect them. However, although these theoretical results suggest the feasibility of Foucault experiments, filling the gap between theory and practice is not an easy task, especially when considering the small angular deflections involved, of the order of 10−5 rad.

scholarly journals Calculation of the pressure on the valves of a sluice

1997 ◽  
Vol 19 (3) ◽  
pp. 25-34
Author(s):  
Tran Gia Lich ◽  
Le Kim Luat ◽  
Han Quoc Trinh
Keyword(s):  
Numerical Method ◽  
Method Of Characteristics ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Sweep Method ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Interior Points

This paper is devoted to a numerical method for calculating the pressure on the vertical two-dimensional valve basing on Navier-Stokes equations. Numerical solutions at interior points are established by splitting Navie-Stokes unsteady two-dimensional equations into two unsteady one-dimensional equations. An implicit scheme is obtained and the solution for these equations is established by the double sweep method. The values at the boundary points are calculated by the method of characteristics.

Temperature Depression Under Heat Sources With Cylindrical Contact Thermocouples

2015 ◽  
Author(s):  
Kayvan Abbasi ◽  
Sukhvinder Kang
Keyword(s):  
Heat Sink ◽  
Numerical Solutions ◽  
Heat Sources ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
One Dimensional ◽  
Dimensionless Groups ◽  
Dimensional Approximation ◽  
The One ◽  
Temperature Depression

The thermal performance of heat sinks is commonly measured using heat sources with spring loaded thermocouples contained within plastic poppets that press against the heat sink to measure its surface temperature where the heat is applied. However, when the thickness of the heat sink base is small or the effective heat transfer coefficient on the fin side is large, the temperature at the thermocouple contact point is less than the nearby temperature where the heat source contacts the heat sink. This temperature depression under the contact thermocouples has been studied. The heat conduction equation is solved analytically to determine the temperature distribution around the contact thermocouple using a one-dimensional approximation and also a detailed two-dimensional approach. Two dimensionless groups are identified that characterize the detailed two-dimensional solution. The combination of the two dimensionless groups also appears in the one dimensional solution. The temperature distributions are validated using finite difference numerical solutions. It is shown that the one dimensional solution is the limit of the detailed solution when one of the dimensionless groups tends to infinity. A simple equation is provided to estimate the temperature measurement error on the heat sink surface.

Sign in / Sign up

Export Citation Format

Share Document