Transient Analysis of Mixed Free-Surface-Pressurized Flows With Modified Slot Model: Part 2 — Similarity Law and Eigenvalue Problem

2003 ◽  
Author(s):  
Kazutoshi Arai ◽  
Katsuhiro Yamamoto ◽  
Mitsuhiro Saitou ◽  
Shigeru Fukumoto
Keyword(s):  
Free Surface ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Transient Flow ◽  
Pressure Rise ◽  
Complex Eigenvalue ◽  
Velocity Difference ◽  
Model Calculations ◽  
Pressurized Flows ◽  
Entrapped Air

To analyze transient free-surface-pressurized flows with entrapped air in a drainage system, a virtual slot model with ceiling has been introduced. Using this model, calculations for the reduced model and the actual size model are carried out to examine a scale effect on the transient flow. The results show that up-scaling the system causes the increase in compressibility of entrapped air so that the rate of pressure rise decreases. Next, it is confirmed that the partial differential equations for the modified slot model have real eigenvalues and then the initial value problem is well-posed when the velocity difference between air and water is not so large. The increase in the velocity difference yields a system having complex eigenvalue, however the well behaved numerical solutions can be obtained since the friction terms in the differential equations suppress the numerical instability.

Transient Analysis of Mixed Free-Surface-Pressurized Flows With Modified Slot Model: Part 1 — Computational Model and Experiment

2003 ◽  
Author(s):  
Kazutoshi Arai ◽  
Katsuhiro Yamamoto
Keyword(s):  
Free Surface ◽  
Computational Model ◽  
Drainage System ◽  
Scale Model ◽  
Small Scale ◽  
Horizontal Pipe ◽  
Pressurized Flows ◽  
Calculation Results ◽  
Gas Liquid Flow ◽  
Entrapped Air

A new computational model is developed here to analyze the influence of entrapped air on free-surface-pressurized flows in a drainage system. A virtual slot with ceiling on the top of the pipe is introduced to treat a separated gas-liquid flow. This model is a modified model of Preissmann’s and is applicable not only to open channel flow and closed conduit flow but also pressurized flow with entrapped air. Compared to experimental results using the model of 1/50 scale of actual drainage system, the calculation results show that the entrapped air in a horizontal pipe advances the time of pressure rising and makes the maximum value of pressure higher. The escape flow of entrapped air at a dropshaft is caused by long waves pushing the air in the horizontal pipe, and then the pipe slope affects the flow rate of air. The air compressibility has less effect on the transient separated air-water flow in the small-scale model.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

scholarly journals A numerical frame work of magnetically driven Powell-Eyring nanofluid using single phase model

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Wasim Jamshed ◽  
Mohamed R. Eid ◽  
Kottakkaran Sooppy Nisar ◽  
Nor Ain Azeany Mohd Nasir ◽  
Abhilash Edacherian ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Dissipative Flow ◽  
Uniform Medium ◽  
Heat Transport Properties ◽  
Flow Heat ◽  
Frame Work ◽  
Slippery Surface ◽  
Nanoparticle Shapes

AbstractThe current investigation aims to examine heat transfer as well as entropy generation analysis of Powell-Eyring nanofluid moving over a linearly expandable non-uniform medium. The nanofluid is investigated in terms of heat transport properties subjected to a convectively heated slippery surface. The effect of a magnetic field, porous medium, radiative flux, nanoparticle shapes, viscous dissipative flow, heat source, and Joule heating are also included in this analysis. The modeled equations regarding flow phenomenon are presented in the form of partial-differential equations (PDEs). Keller-box technique is utilized to detect the numerical solutions of modeled equations transformed into ordinary-differential equations (ODEs) via suitable similarity conversions. Two different nanofluids, Copper-methanol (Cu-MeOH) as well as Graphene oxide-methanol (GO-MeOH) have been taken for our study. Substantial results in terms of sundry variables against heat, frictional force, Nusselt number, and entropy production are elaborate graphically. This work’s noteworthy conclusion is that the thermal conductivity in Powell-Eyring phenomena steadily increases in contrast to classical liquid. The system’s entropy escalates in the case of volume fraction of nanoparticles, material parameters, and thermal radiation. The shape factor is more significant and it has a very clear effect on entropy rate in the case of GO-MeOH nanofluid than Cu-MeOH nanofluid.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

A New Numerical Method for Solving Nonlinear Fractional Fokker–Planck Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
BeiBei Guo ◽  
Wei Jiang ◽  
ChiPing Zhang
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Simple Algorithm ◽  
High Accuracy ◽  
Computational Work ◽  
Reproducing Kernel Theory ◽  
Transport Problems ◽  
Fokker Planck

The nonlinear fractional-order Fokker–Planck differential equations have been used in many physical transport problems which take place under the influence of an external force filed. Therefore, high-accuracy numerical solutions are always needed. In this article, reproducing kernel theory is used to solve a class of nonlinear fractional Fokker–Planck differential equations. The main characteristic of this approach is that it induces a simple algorithm to get the approximate solution of the equation. At the same time, an effective method for obtaining the approximate solution is established. In addition, some numerical examples are given to demonstrate that our method has lesser computational work and higher precision.

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