Explicit Finite Difference Technique Research Articles

Spherical vessel subjected to explosive detonation loading

The following work utilizes a calculation method for the design of spherical containment vessels for pressure gases likely subjected to internal detonation. A centrally initiated explosion within the vessel is taken into account and the history of pressures on the internal vessel wall is investigated by means of the fundamental compressible fluid dynamics, for which a non-viscous perfect gas is assumed. The obtained differential equations are integrated with explicit finite difference techniques, introducing artificial viscosity. The response of vessel is decoupled from the fluid and it is determined by a dynamic finite element code of explicit type; the investigated material is considered to have an elastic-plastic behaviour. Comparison with data found in references on PBX-9404 high-explosive induced explosions confirms the calculation method developed.

TEMPERATURE DISTRIBUTION AND MELT POOL SIZE IN A SEMI-INFINITE BODY DUE TO A MOVING LASER HEAT SOURCE

Heating and melting of a semi-infinite body due to volumetric absorption of a moving laser radiation were studied. The problem is essentially a transient three-dimensional conduction problem with a moving heat source and a moving phase boundary. An explicit finite difference technique was used to solve for the temperature distribution and the melt pool size. In order to save computing time the solution domain was divided into inner and outer regions. The inner region was sized to contain the maximum pool size. A fine grid pattern was used for the inner region, where the temperature gradient is large and where the accurate location of the solid-liquid interface is desired. In order to validate the numerical results, comparisons were made with experimental data. The results show that the heat-affected zone and the melt pool size decrease as the translational speed of the beam increases.

Synchronization-Free Parallel Implementation of Factored-Implicit and Explicit Techniques

This work is motivated by the desire to assess the adaptability of alternating direction implicit and explicit finite difference techniques, used in computationl fluid dynamics, to parallelization on tightly-coupled, shared-memory parallel computers. The aim of the work is to achieve synchronization-free parallel implementations by the introduction and use of load balancing and processor routing schemes. Two representative and popular finite difference techniques, the implicit Beam and Warming and the explicit MacCormack techniques, are parallelized and tested on a 20-processor computer. Several parallel implementations of increasing complexity are investigated in the context of the simple heat and the more computationally intensive curvilinear grid mapping equations. In attaining near-theoretical speedups, the need for processor synchronization has to be eliminated completely. To achieve a synchronization-free parallel implementation, three stratagems are introduced and evaluated. First, each finite diff...

Influence of adsorption on the measurement of diffusion coefficients by Taylor dispersion

The effect of adsorption on the measurement of diffusion coefficients by the Taylor dispersion technique is investigated by modifying the governing equation to account for reversible, nonequilibrium adsorption. The resulting two-dimensional equations are solved by an explicit finite-difference technique. Experimental data for the acridine carbon dioxide system indicated that acridine adsorbs on the walls on the tubing and these data were investigated with this model. The influence of carious parameters including the number of sites and the rates of adsorption desorption was investigated by conducting a parametric sensitivity analysis on the model. It was found that adsorption of the solute on the wall of the tubing could produce an error as high as 35% on the measured diffusion coefficient compared to the actual diffusion coellicient. Examination of the influence of each of the parameters will enable Inure investigators to reduce the effect of adsorption in the measurement of diffusion coefficients by Taylor dispersion.

SIMULTANEOUS HEAT and MASS TRANSFER DURING the DEEP FAT FRYING of TORTILLA CHIPS

The heating and water loss of a single tortilla chip during deep fat frying was investigated experimentally and the data were analyzed by writing two heat and mass transfer balances. the equations were solved using explicit finite difference technique. Oil uptake as a function of frying time was described by a first order exponential equation. the empirical and theoretical heat and mass transfer agreed well.The effect of oil temperature on the moisture loss and oil uptake as a function of frying time was analyzed. Moisture loss rate increased as temperature increased. the effects of temperature on oil uptake were not significant during the first 15 s of frying, however, the final oil content was higher for the tortilla chip fried at 190C than at 150C for 60 s.

Transient conjugate heat transfer between a laminar stagnation zone and a solid disk

An explicit finite difference technique is used to solve the case of transient, forced convective conjugate heat transfer between an axisymmetric incompressible laminar impinging jet and a solid disk. The solution to the governing equations is based on the assumptions of constant physical properties and no viscous dissipation. The hydrodynamics of the flow are considered to be steady state, whereas thermal transients take place with a step temperature change at the jet upstream boundary condition. The unexposed bottom surface of the disk is treated as an isothermal boundary, and the radial boundary of the disk is modeled as adiabatic. In this work, the effect of the ratios of thermal conductivity and thermal diffusivity between fluid and solid on transient heat transfer characteristics of the system have been investigated.

Numerical studies of solute transport in a fracture surrounded by rock matrix: Effect of lateral diffusion and chemical reactions on the overall dispersion

In some approximations to the transport of radionuclides in geological media, it is assumed that the radionuclides are transported by advection-dispersion in a single fracture and diffusion in the surrounding rock matrix. The concentration in the main fracture is assumed to be constant in the direction transverse to groundwater flow, and the effect of this approximation is accounted for by use of the Taylor dispersion coefficient. Since in the calculations carried out to date this dispersion coefficient is based on Taylor's original analysis which had not considered interactions with a surrounding rock, the adequacy of the approximation needs to be examined carefully. For this purpose, we carry out a detailed numerical analysis of the problem by using an explicit finite difference technique. We find that indeed in the presence of the interactions with the surrounding rock, it would be necessary to modify the standard dispersion coefficient so that the solution of the laterally averaged equation produces agreement with the more detailed analysis.

Nonlinear hydrodynamics of cosmological sheets. 1: Numerical techniques and tests

We present the numerical techniques and tests used to construct and validate a computer code designed to study the multidimensional nonlinear hydrodynamics of large-scale sheet structures in the universe, especially the fragmentation of such structures under various instabilities. This code is composed of two codes, the hydrodynamical code ZEUS-2D and a particle-mesh code. The ZEUS-2D code solves the hydrodynamical equations in two dimensions using explicit Eulerian finite-difference techniques, with modifications made to incorporate the expansion of the universe and the gas cooling due to Compton scattering, bremsstrahlung, and hydrogen and helium cooling. The particle-mesh code solves the equation of motion for the collisionless dark matter. The code uses two-dimensional Cartesian coordinates with a nonuniform grid in one direction to provide high resolution for the sheet structures. A series of one-dimensional and two-dimensional linear perturbation tests are presented which are designed to test the hydro solver and the Poisson solver with and without the expansion of the universe. We also present a radiative shock wave test which is designed to ensure the code's capability to handle radiative cooling properly. And finally a series of one-dimensional Zel'dovich pancake tests used to test the dark matter code and the hydro solver in the nonlinear regime are discussed and compared with the results of Bond et al. (1984) and Shapiro & Struck-Marcell (1985). Overall, the code is shown to produce accurate and stable results, which provide us a powerful tool to further our studies.

Analysis of transient nonlinear heat conduction in wood using finite-difference solutions

This paper describes three-dimensional, transient heat conduction in a rectangular piece of wood with crossgrain, and in an orthotropic wooden cylinder. Computerized solutions of a generalized, nonlinear heat equation are derived by discretizing the space and the time domains, using explicit and implicit finite-difference techniques. A simplified example of linear heat conduction in cylindrical coordinates illustrates how to apply the finite-difference solutions. The accuracy of the solutions for this special case is evaluated via comparing them with a well-known exact solution.

Performance analysis of a solar concrete collector

This article presents a transient analysis of solar concrete collectors used for providing domestic hot water. A mathematical model has been developed to estimate the performance of such collectors. This involves the solution of the two-dimensional time-dependent heat conduction equation with appropriate initial and boundary conditions. An explicit finite difference technique has been used for the purpose, and calculations have been carried out to validate the model by comparing its predictions with several sets of experimental data. Further, a parametric calculation and sensitivity analysis have been done to study the effect of various values of the governing parameters on the collector performance.

Three‐dimensional Characteristics Model of Wind‐Generated Turbulent Flow

A three‐dimensional finite difference model is presented for the calculation of the turbulent flow generated by the action of the blowing wind over a water surface. Initially, the free‐surface elevation is calculated by means of an accurate two‐dimensional, depth‐averaged model based on the method of characteristics. The method has the advantage of transforming the partial differential equations describing hydrodynamic flow to a set of characteristic conditions holding along specific lines of the (x, y, t) domain. These contain only total derivatives and, thus, they allow a better representation of the nonlinear convective terms. Subsequently, the calculated free‐surface gradient values are introduced in the three‐dimensional momentum equations with the view of computing both the wind‐induced currents u, v, and the vertical velocity w by means of an explicit finite difference technique. This combined two‐dimensional and three‐dimensional model is tested and validated by applying to several simple flow cases for which Ekman's classical solution is also available for comparison.

Spline Interpolations for Water Hammer Analysis

The method of characteristics (MOC) with spline polynomials for interpolations was investigated for numerical water hammer analysis for a frictionless horizontal pipe. A new analysis quantifies numerical errors in a systematic and dimensionless way and is used to evaluate the performance of three spline polynomials and to compare them with other numerical schemes for water hammer analysis, as well as to assess the effect of required additional interpolation boundary conditions on the overall accuracy of spline and two-point fourth-order Hermite schemes. For the sudden valve closure test case investigated, the overall accuracy with splines is significantly improved compared to the MOC with linear interpolations or to second-order explicit finite difference techniques. Compared to the Hermite method, the spline scheme has about the same overall accuracy and has the advantage of offering a choice between several spline polynomials with different interpolating characteristics and of being unconditionally stable.

Steam bubble formation at a submerged orifice in quiescent water

The formation of steam bubbles at a submerged orifice in quiescent water was studied theoretically and experimentally. The experiments were performed for liquid temperature from 80 to 98°C, for liquid height above the orifice from 2.5 to 10 cm, for orifice diameter of 1.48 mm and for constant pressure difference of 245 Pa between the vapor pressure in the vapor chamber and the hydrostatic pressure of the orifice. The flow rate of steam entering through the orifice, the detachment time of the bubble, and the bubble size at detachment were increased as the temperature of the water in the liquid column was raised. Sixty-two per cent of the steam was condensed at a liquid temperature of 80°C and 11% at 98°C while the flow rate of the steam entering through the orifice was almost the same. The liquid depth had a negligible effect on the vapor bubble formation in this experimental region. A model for describing the vapor bubble formation in the single-bubble region is presented. The predicted values from the present model using an explicit finite difference technique are compared with experimental data, and the results are in good agreement.

A Poroelastic PKN Hydraulic Fracture Model Based on an Explicit Moving Mesh Algorithm

This paper describes the mathematical formulation of a Perkins-Kern-Nordgren (PKN) fracture model, that accounts for the existence of poroelastic effects in the reservoir. The poroelastic effects, induced by leak-off of the fracturing fluid, are treated in a manner consistent with the basic assumptions of the PKN model, by means of a transient influence function. The fracture model is formulated in a moving coordinates system and solved using an explicit finite difference technique. The numerical algorithm has the following features: fixed mesh, adaptive control of the time step, and unconstrained fracture length during shut-in. Numerical simulation with this model indicates that poroelastic processes could be responsible for a significant increase of the treatment pressure, but that they have virtually no influence on the fracture length and fracture width.

Parallel computation with adaptive methods for elliptic and hyperbolic systems

We consider the solution of two-dimensional vector systems of elliptic and hyperbolic partial differential equations on a shared memory parallel computer. For elliptic problems, the spatial domain is discretized using a finite quadtree mesh generation procedure and the differential system is discretized by a finite element-Galerkin technique with a piecewise linear polynomial basis. Resulting linear algebraic systems are solved using the conjugate gradient technique with element-by-element and symmetric successive over-relaxation preconditioners. Stiffness matrix assembly and linear system solutions are processed in parallel with computations schedules on noncontiguous quadrants of the tree in order to minimize process synchronization. Determining noncontiguous regions by coloring the regular finite quadtree structure is far simpler than coloring elements of the unstructured mesh that the finite quadtree procedure generates. We describe linear-time complexity coloring procedures that use six and eight colors. For hyperbolic problems, the rectangular spatial domain is discretized into a grid of rectangular cells, and the differential system is discretized by an explicit finite difference technique. Recursive local refinement of the time steps and spatial cells of a coarse base mesh is performed in regions where a refinement indicator exceeds a prescribed tolerance. Data management involves the use of a tree of grids with finer grids regarded as offspring of coarser ones. Computational procedures that sequentially traverse the tree structure while processing solutions on each grid in parallel and that process solutions at the same tree level in parallel have been developed. Computational results using the sequential tree traversal scheme are presented and compared with results using a non-adaptive strategy. Heuristic processor load balancing techniques are suggested for the parallel tree traversal procedure.

Diffusion with nonlinear adsorption: an eigenfunction expansion solution

An eigenfunction expansion (spectral) method is used to transform a partial differential equation for diffusion with nonlinear adsorption into a system of first-order, nonlinear, ordinary differential equations that are readily solved with standard numerical routines. The nonlinearity is expressed as a polynomial in the dependent variable, which restricts the solution to a high-capacity adsorbent. The technique can be used easily to generate results for solid- and fluid-phase uptake. The addition of terms to the expansion increases the accuracy of the method but also increases the computing time. Nevertheless, this method produces accurate results more efficiently than does the explicit finite difference technique used for comparison

Numerical Simulation of Crevice Corrosion of Stainless Steels and Nickel Alloys in Chloride Solutions

Abstract This paper describes the development of a new mathematical model of the incubation period of crevice corrosion. The model uses a new treatment of the transport processes, which includes both ionic migration and diffusion. An explicit finite-difference technique was applied to a uni-dimensional crevice along the crevice depth, with appropriate boundary conditions at both the crevice tip and bulk solution. Transient concentration results demonstrate that the major reason for the pH decrease in crevice solution is the production of soluble metal hydroxides, particularly CrOH2+. No solid monomer metal hydroxides (ex. Cr(OH)3 〈s〉) were formed in any of the simulations, which suggests that metal oligomeric complexes will be formed in the crevice solution. In addition, a comparison of the simulation results and experimental literature data (for alloys 304 [UNS S30400], 316L [UNS S31600], 904L [UNS N08904], and Inconel 625 [UNS N06625]) are discussed. The numerical simulation results, using Inconel 625, ...

Resonance scattering in waveguides: Acoustic scatterers

In this paper, acoustic scattering in shallow and deep inhomogeneous waveguides is analyzed. The full acoustic wave equation that describes the scattered and reflected wave fields, as well as all multipathing within the scatterer and the waveguide, is employed. The explicit finite difference time integration and the k–w transform in the space domain (pseudospectral method) were utilized. Properly chosen boundary conditions enabled the authors to model both shallow and deep oceans. The pseudospectral method was compared with the explicit finite difference technique (see Appendix). The pseudospectral method can be valuable for modeling different underwater wave phenomena: It is characterized by much smaller numerical dispersion than the conventional finite difference method. The results show that in shallow ocean, strong resonance coupling between the underwater scatterer and the waveguide may occur. An important conclusion of this paper is that in a limited aperture experiment when the acoustic reflections are beyond the recording aperture (due to the finite length of the recording cable), the measured data are mainly represented by the primary diffraction arrivals and ‘‘diffraction resonances.’’ In this paper, a detailed discussion will be given on the acoustic scattering in an inhomogeneous waveguide. (For the sake of simplicity, an acoustic scatterer of the rectangular shape was considered.) Different complexities (elastic scatterers, more complicated structures of the index of refraction in the water, etc.) will be compounded in the original model and reported elsewhere.

Investigation of thermal instabilities in a forced convection upward boiling system

This work covers investigations of thermal instabilities in forced convection boiling in a vertical single channel system, with Freon-11 as the working fluid. Experiments with various heat inputs and inlet fluid temperatures, as well as for different heater tube sizes, have been conducted to investigate the effects of these parameters on the instabilities. Under the experimental conditions of this study, pressure-drop type and thermal oscillations have been observed. In the theoretical study, the one-dimensional governing equations have been solved using the explicit finite difference technique to determine the wall conditions, fluid properties, and flow conditions at any point along the test section. The model is based on the assumption of homogeneous two-phase flow, with thermodynamic equilibrium between the phases. The conditions under which thermal oscillations occur are predicted. The periods and amplitudes predicted by the model are in good agreement with experimental results.

Transient free convection flow of water at 4°C past a semi-infinite vertical plate

Transient free convection flow of water at 4°C past an isothermal semi-infinite vertical plate, when its density is maximum, is analysed by employing an explicit finite-difference technique. The transient velocity and temperature, the average skin-friction and the average Nusselt number are shown graphically for Prandtl number Pr=11.4 (4°C) and 7.0 (20°C). It has ben observed that when the density of water is maximum, it takes more time to reach its steady state. Also the average skin-friction decreases considerably when the water is at 4°C, as compared to that at 20°C. The average Nusselt number increases at small values of time and decreases at large values of time when the water is at 4°C.