Validity Of Numerical Solutions Research Articles

Impact of heat transfer in a duct composed of anisotropic porous material: A non-linear Brinkman-Forchheimer extended Darcy's model: A computational study

Forced convective heat transport in a rectangular porous channel with anisotropic permeability is investigated in this study. A fully developed flow field is assumed, and the Brinkman-Forchheimer extended Darcy's equation is adopted to model the fluid flow. The thermal field is developing. A weak formulation of the momentum equation has been obtained. The existence and uniqueness of the solution for the weak momentum equation are proved using the Browder-Minty theorem. The finite difference approach has been utilized to solve these coupled differential equations. Non-equidistant grids are implemented in the axial direction to reduce computational time and space. The effect of anisotropic parameters, such as permeability ratio and orientation angle, on the hydrodynamic and thermal characteristics of the flow has been discussed. An increase in the permeability ratio (K) causes an increase in the local Nusselt number. Anisotropy has been identified as crucial in heat transmission for the Darcy numbers less than0.75. The anisotropic phenomenon of the channel contributed to a more than 13 % increase in the heat transmission rate compared to the isotropic scenario. Also, as the anisotropic parameters increase, the amount of heat transferred from the walls to the fluid up to any desired axial distance increases. Validation of numerical solutions is done using existing literature.

Numerical study on the effect of viscosity on a multistage pump running in reverse mode

Utilization of pump as turbine (PAT) systems is known as an economically beneficial method to harvest hydropower energy, especially in a remote area. Despite the extensive investigations on the operation and parametric study of the PAT components, the turbine mode of multistage PATs in case of a viscous working fluid is found as a literature gap. In this paper, a numerical simulation method is employed as a predicting tool to investigate the consequences of changing the fluid viscosity affecting the flow patterns within the multistage PATs. The commercial solver ANSYS CFX 16 was used for simulation. Validation of numerical solution was done for the single-stage pump considering water and 48 cSt viscous fluid. The numerical results in the single-stage PAT showed that increasing the viscosity causes an efficiency drop mainly at the part-load condition and translates the maximum efficiency point to higher flow rates, where the vortices at impeller passage and draft tube are weaker. Increasing the viscosity up to 48 cSt, results in 12.5% reduction of efficiency at the best efficiency point (BEP) in two stage PAT. Based on the two-stage PAT results, the total hydraulic efficiency at the BEP has been increased by 1% and 1.5% for water and the 48 cSt viscous fluid, respectively, in comparison with the single-stage PAT. According to the post-processed streamlines at different stages of the two-stage PAT, the undesired vortical structures were caused by the diffuser and the return channel, which were mainly designed for the pump mode. The hydraulic analysis of two-stage PAT shows that using multistage pumps in reverse mode is reasonable for power generation in high head and flow rate sites.

Hydromagnetic flow of a variable viscosity nanofluid in a rotating permeable channel with hall effects

The flow, heat and mass transfer of water-based nanofluid are examined between two horizontal parallel plates in a rotating system. The effects of Brownian motion, thermophoresis, viscosity and Hall current parameters are considered. The governing partial differential equations are reduced to ordinary differential equations that are then solved numerically using the Runge–Kutta–Fehlberg method. Validation of numerical solution is achieved with an exact solution of primary velocity and found to be in good agreement. Results show that both surfaces experience opposite behavior regarding skin friction, Nusselt and Sherwood numbers in both primary and secondary flows. These physical quantities depend upon dimensionless parameters and numbers.

The Equivalent Thévenin-Network Representation of a Pulse-Excited Power-Ground Structure

The $N$ -port equivalent Thevenin representation of a power-ground structure is constructed using the Lorentz reciprocity theorem of the time-convolution type. It is shown that the introduced multiport Kirchhoff-circuit representation offers a number of alternative routes to determining the pulsed electromagnetic radiated emissions of a power-ground structure from its response to an electromagnetic disturbance in the receiving situation. Moreover, the result may serve as an important check on the validity of numerical solutions to related electromagnetic radiation problems. An illustrative example that validates the equivalent-circuit representation is given.

Analytical solution to transient Richards' equation with realistic water profiles for vertical infiltration and parameter estimation

AbstractA general analytical model for one‐dimensional transient vertical infiltration is presented. The model is based on a combination of the Brooks and Corey soil water retention function and a generalized hydraulic conductivity function. This leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index. Accordingly, the proposed analytical solution covers many existing realistic models in the literature. The general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time. It can be used to model infiltration through semi‐infinite dry soils with prescribed water content or flux boundary conditions. Some mathematical expressions of practical importance are also derived. The general form solution is useful for comparison between models, validation of numerical solutions and for better understanding the effect of some hydraulic parameters. Based on the analytical expression, a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented.

Discontinuous Solutions Using the Method of Manufactured Solutions on Finite Volume Solvers

When applying the method of manufactured solutions on computational fluid dynamic software, it is a requirement that all solutions be continuous on the computational domain. This stipulation is limiting for the verification and validation of numerical solutions where discontinuities are frequent. In an effort to adapt the standard method of manufactured solutions procedure, we propose a piecewise approach for modeling solutions with discontinuities. Linearly and quadratically exact transformations are used for determining the exact solutions and source terms for cells split by discontinuities. Upwind manufactured solutions are initialized and a least squares fit is used to solve for solutions downwind of the discontinuity such that the Rankine–Hugoniot conditions are satisfied. The codes used throughout this research are finite volume, 1st and 2nd order, inviscid schemes combined with uniform structured grids. This study shows that a modified method of manufactured solutions procedure can be performed with fully general discontinuous solutions yielding 1st order convergences typically associated with shocks.

Experimental Validation of Numerical Solutions Using the Explicit Green's Approach to Simulate Transient Heat Conduction in Multilayer Systems

This article presents an experimental validation of numerical solutions using the explicit Green's approach (ExGA) for transient heat conduction in multilayer systems. The ExGA is an efficient recurrence relationship for the temperature in the time domain, based on the Green's matrix, which allows explicit time marching with a larger time step than is required by other methods found in the literature without losing accuracy. The multilayer system used in the experimental validation is built by overlaying different materials. The systems were subjected to heat variations that were recorded over time using thermocouples, and these results were used for comparison.

A solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation

This letter presents an analytical solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation. The solution is found by solving the Reynolds equation with Reynolds boundary condition for cavitation. The cavitation boundary is shown to be straight lines directed 108.4° against the sliding direction. The result is experimentally verified in the limit of large values of viscosity, sliding velocity and radius of a spherical ball. The solution raises questions about the coupling between cavitation and film rupture and can be used as an independent check on the validity of numerical solutions.

An Analytical Framework for the Solution of Autofrettaged Tubes Under Constant Axial Strain Condition

Autofrettage is a technique for introducing beneficial residual stresses into cylinders. Both analytical and numerical methods are used for the analysis of the autofrettage process. Analytical methods have been presented only for special cases of autofrettage. In this work, an analytical framework for the solution of autofrettaged tubes with constant axial strain conditions is developed. Material behavior is assumed to be incompressible, and two different quadratic polynomials are used for strain hardening in loading and unloading. Clearly, elastic perfectly plastic and linear hardening materials are the special cases of this general model. This quadratic material model is convenient for the description of the behavior of a class of pressure vessel steels such as A723. The Bauschinger effect is assumed fixed, and the total deformation theory based on the von Mises yield criterion is used. An explicit solution for the constant axial strain conditions and its special cases such as plane strain and closed-end conditions is obtained. For an open-end condition, for which the axial force is zero, the presented analytical method leads to a simple numerical solution. Finally, results of the new method are compared with those obtained from other analytical and numerical methods, and excellent agreement is observed. Since the presented method is a general analytical method, it could be used for validation of numerical solutions or analytical solutions for special loading cases.

Extension of the Bissell–Johnson plasma-sheath model for application to fusion-relevant and general plasmas

This article presents an approach to solving a special Fredholm-type integral equation of the first kind with a particular kernel containing a modified Bessel function for applications in plasma physics. From the physical point of view, the problem was defined by Bissell and Johnson (B&J) [Phys. Fluids 30, 779 (1987)] as a task to find the potential profile and the ion velocity distribution function in a plane-parallel discharge with a Maxwellian ion source. The B&J model is a generalization of the well-known Tonks–Langmuir (T&L) [Phys. Rev. 34, 876 (1929)] discharge model characterized by a “cold” ion source. Unlike the T&L model, which can be readily solved analytically, attempts to solve the B&J model with a “warm” ion source have been done only numerically. However, the validity of numerical solutions up to date remains constrained to a rather limited range of a crucial independent parameter of the B&J integral equation, which mathematically is the width of a Gaussian distribution and physically represents the ion temperature. It was solved only for moderately warm ion sources. This paper presents the exact numerical solution of the B&J model, which is valid without any restriction regarding the above-mentioned parameter. It is shown that the ion temperature is very different from the temperature of the ion source. The new results with high-temperature ion sources are not only of particular importance for understanding and describing the plasma-sheath boundary in fusion plasmas, but are of considerable interest for discharge problems in general. The eigenvalue of the problem, found analytically by Harrison and Thompson [Proc. Phys. Soc. 74, 145 (1959)] for the particular case of a cold ion source, is here extended to arbitrary ion-source temperatures.

Analytical solutions of linear finite- and small-strain one-dimensional consolidation

Analytical solutions are presented for linear finite-strain one-dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one- and two-way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small-strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small-strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non-linear finite-strain consolidation. They also clarify a formerly perplexing aspect of finite-strain solution charts first noted in numerical solutions. Copyright (C) 2004 John Wiley Sons, Ltd.

Reply to the ‘Comment on “Return stroke transmission line model for stroke speed near and equal that of light” by R. Thottappillil, J. Schoene, and M.A. Uman’ by B. Kordi, R. Moini, and V.A. Rakov

[1] The ideal and exact solution for a vertical wire antenna of infinite length above a ground plane excited at the bottom, all conductors being perfect, is spherical TEM. Currents along such an antenna do not suffer from attenuation and dispersion. The energy lost due to radiation from such an antenna appears to originate from the source at the bottom. Practical systems are different from this ideal case because there are no ideal sources and no ideal conductors. However, ideal analytical solutions should be used, when possible, as a validation of numerical solutions in the appropriate limit.

Almost analytic solutions to equilibrium sequences of irrotational binary polytropic stars for n = 1

A solution to an equilibrium of irrotational binary polytropic stars in Newtonian gravity is expanded in a power of epsilon = a(0)/R, where R and a(0) are the separation of the binary system and the radius of each star for R = infinity. For the polytropic index n = 1, the solutions are given almost analytically up to order epsilon(6). We have found that in general an equilibrium solution should have the velocity component along the orbital axis and that the central density should decrease when R decreases. Our almost analytic solutions can be used to check the validity of numerical solutions.

An Investigation into the Validation of Numerical Solutions of Complex Flowfields

Three cases, including a two-dimensional, an axisymmetric, and a three-dimensional flowfield, were studied to demonstrate the effectiveness and reliability of a method proposed for validation of numerical solutions of complex flowfields. Images of these flowfields were first constructed from numerical solutions based on the principle of experimental flow visualization, and then compared directly with experimental interferograms. Because both experimental and numerical results are of identical physical representation, agreement between them can be evaluated effectively by examining characteristic flow structures of the flowfields as well as comparing differences in density. An efficient algorithm for three-dimensional density integration was also proposed to replace the conventional one that is computationally expensive. The study shows that reliable validation can be achieved in this way because it allows a direct comparison between numerical and experiment results without any loss of accuracy in either of them. The validation method is highly recommended for three-dimensional flowfields where quantification of images from experimental flow visualization is very difficult or impossible.

Numerical Solution and Sensitivity Analysis of Pore Liquid Pressure and Cake Porosity to Model Parameters

Filter cake formation is important in groundwater and oil wells where drilling contains suspended mud particles. The accumulation of these mud particles on the borehole wall creates a pressure drop in the well. Furthermore, the migration of colloidal particles into adjacent porous rock could damage the formation and cause productivity decline. In this study, numerical solutions for pore liquid pressure variation across the cake with variable total stress and associated porosity variation are obtained. Mass equations for captured and suspended particles are averaged along the mud cake thickness, taking into account conditions on the cake surface and at the filter septum. The variability of total stress in soil consolidation’s problem is considered to determine the pore liquid pressure along the mud cake thickness. Then, the relation between porosity and pressure is studied to determine the mud cake porosity. Experimental data obtained by various researchers is used to compare and test the validity of numerical solutions to develop guidelines for model applications. Results show that the pore liquid pressure increases with the decrease of membrane impedance value (i.e. less pervious membrane). Also, the pressure profile has a cubic function of dimensionless cake thickness. The conclusions from the sensitivity analysis conducted in this study agree with earlier conclusions.