Finite Difference Method For Solution Research Articles

Parametric study of a solar pond for Northern Jordan

A mathematical model is developed to determine the various parameters affecting the performance of a salt-gradient solar pond (SGSP). We use the finite difference method for solution. Monthly average meteorological data for the Irbid region are used. Our results indicate that the thickness of the non-convection zone (NCZ) has a significant effect on the storage-zone temperature. The optimum value of this thickness is found to be one meter. Storage-temperature fluctuations due to weather changes may be minimized by increasing the thickness of the storage zone. This result may also be achieved by assuming that the rate of heat extraction is proportional to the intensity of the incoming solar radiation.

A One-dimensional Heat Transfer Model for Laser Surface Alloying of Chromium on Copper Substrate

A one-dimensional unsteady-state heat transfer model based on explicit finite difference method of solution has been developed to simulate the thermal profile in the course of laser surface alloying of a Cr-deposited Cu substrate with a continuous-wave CO 2 laser. In this model, a suitable temperature/phase dependence of the material properties has been considered. The temperature profile and depth of melting are found to be strong functions of the incident power density and interaction time. The mechanism of transient melting and solidification following laser irradiation is discussed in the light of the present model

The three-dimensional boundary layer on a yawed body of revolution

A three-dimensional boundary layer calculation is carried out for the flow over a semi-infinite circular cylinder which is placed at a small angle of incidence to an oncoming uniform stream. To elucidate the details of the flow both a spectral and a finite difference method of solution has been employed. The results show that separation of ‘open’ type on a body of revolution is characterized by a line of Goldstein singularities that originates, spontaneously, at a finite distance from the nose of the body.

FINITE-DIFFERENCE METHOD OF INCOMPRESSIBLE FLOWS WITH ROTATION AND MOVING BOUNDARY IN A NONSTAGGERED GRID

Abstract In this article a finite-difference solution method for simulating incompressible two-dimensional and axisymmetric flow problems with rotation and moving boundary is developed. In this method conservative governing equations are approximated by a vertex-based control-volume discretization in a nonstaggered grid. The solutions for velocity field and pressure are coupled in a similar manner to the SIMPLE family. Moving boundaries are solved step by step from the kinematic boundary condition, and the changing flow domain is fitted by a curvilinear coordinate system with an updated grid. To avoid unrealistic distributions of velocity components and pressure, the Rhie-Chow interpolation is employed and the boundary value of pressure on solid surfaces is determined from the local mass conservation. The validation is carried out in simulating two-dimensional lid-driven square cavity flows and rotating coaxial disk flows with and without free surfaces.

Electron degradation and thermalization in SiH 4 gas

The time-dependent Boltzmann equation is applied to the study of the degradation and thermalization of a nonequilibrium distribution of subexcitation electrons in SiH 4. For the small electron to SiH 4 mass ratio, the collision operators in the Boltzmann equation reduce to the differential Fokker-Planck operator for elastic collisions and the difference operator for inelastic collisions. A finite difference method of solution of the Boltzmann equation is used. The time evolution of the electron energy distribution and the thermalization time of subexcitation electrons are calculated for initial electron energies of 1, 3 and 5 eV. It is shown that super elastic collisions due to vibrational de-excitation play an important role in the thermalization process.

Calculation of Turbulent Flow for an Enclosed Rotating Cone

Prediction of the flow in the cavity between a rotating cone and an outer stationary cone with and without throughflow is considered. A momentum-integral method and a finite difference method for solution of the Reynolds-averaged Navier–Stokes equations with a mixing-length model of turbulence are applied. These two methods have previously been validated for flow between corotating and rotor–stator disk systems, but have not been properly tested for conical systems. Both methods have been evaluated by comparing predictions with the experimental measurements of other workers. There is good agreement for cone half-angles greater than or equal to 60 deg but discrepancies are evident for smaller angles. “Taylor-type” vortices, the existence of which has been postulated by other workers and which are not captured by the present steady, axisymmetric models, may contribute to these discrepancies.

Free convection in a saturated porous medium beyond the similarity solution

The non-similar problems associated with a non-isothermal vertical flat plate embedded in a fluid-saturated porous medium were considered to assess the performances of the two distinctive boundary layer solution methods, namely, the local similarity solution and the integral method. The results generated from these two approximate solution methods are compared against the results from a two-point finite difference and those based on a Merk-type series expansion. Comparison of the results reveals that both integral and local similarity methods perform excellently. Especially, the accuracy acquired by the local similarity solution is so high that the difference between the results from the local similarity solution and those from the two-point finite difference and local non-similarity solution methods is hardly discernible for the case of monotonic increasing wall temperature.

Numerical simulation of turbulent flow over a moving wavy boundary: Norris and Reynolds extended

A numerical model of fully developed turbulent channel flow over a moving wavy boundary is presented. The model is of the linear-stability type, employs a transformed coordinate system, and uses the finite difference method for solution. Both the time-independent mean flow and the time-dependent perturbation to the mean flow caused by the traveling wavy boundary are computed. A low-turbulence-Reynolds-number form of the k-epsilon turbulence closure model is employed and three levels of sophistication in modeling the wave-induced Reynolds stresses (in the transformed coordinate system) are introduced. These models are the viscous quasilaminar (VQL) model, the mean eddy viscosity (EV) model, and the full perturbation (FP) model. The model is applied to simulate previous experimental investigations of the flow over wavy boundaries (including a water wave) under a variety of conditions. The solutions generated using both the EV and FP Reynolds stress closure models generally demonstrate very good agreement with the available pressure, stress, velocity, and Reynolds stress experimental data. In contrast, due to its neglect of the perturbation Reynolds stresses, the simple VQL closure model has limited predictive ability, especially for downstream running waves whose wave speed exceeds about half the channel centerline velocity and for cases with low inverse Reynolds numbers (based on the shear velocity and wavelength). For the water wave case, the component of the wave growth rate due to the pressure mechanism predicted by the EV and FP models can be up to three times those of the VQL model in the gravity wave growth range.

Description and Evaluation of Program: CARIMA

This paper is one of a series resulting from the ASCE Task Committee on Irrigation Canal System Hydraulic Modeling, including presentation and evaluation of several programs, one of which is CARIMA. The CARIMA code simulates unsteady free‐surface flow in simple or multiply connected systems of rivers or canals. The simulation uses the Preissmann implicit finite difference method for solution of the complete de St. Venant equations and appropriate equations for hydraulic structures. Special features of the code include automatic topology recognition, a user interface for fully implicit incorporation of specialized supervisory and global control schemes, and the lack of any preset limits on network complexity or size of data tables. CARIMA was used to evaluate a canal automation algorithm at the California Polytechnic Model Canal facility. This investigation provided a useful comparison of CARIMA modeling results with actual results recorded on the Model Canal, and helped to set the specifications for the i...

Curvature corrections to Reynolds stress model for computation of turbulent recirculating flows

References ^rabha, S., and Jain, A. C., On the Use of Compatibility Conditions in the Solution of Gas Paniculate Boundary Layer Equations, Applied Scientific Research, Vol. 36, No. 1, 1980, pp. 81-91. Osiptsov, A. N., Structure of the Laminar Boundary Layer of a Disperse Medium on a Flat Plate, Fluid Dynamics, Vol. 15, No. 4, 1980, pp. 512-517. Marble, F. E., of Dusty Gases, Annual Review of Fluid Mechanics, Vol. 2, No. 1, 1970, pp. 397-446. Blottner, F. G., Finite Difference Methods of Solution of the Boundary-Layer Equations, AIAA Journal, Vol. 8, No. 2, 1970, pp. 193-205. Spo, S. L., Fluid Dynamics of Multiphase Systems, Blaisdell, Waltham, MA, 1967, Chap. 8.

Parallel Computing Solution of Coupled Flow Processes in Soil

A new parallel computing solution of problems of coupled flows in soils is presented. Such phenomena occur when more than one transport process exists below ground as is the case, for example, for coupled chemical, electrical, heat, or moisture flow. A numerical solution of the governing differential equations is achieved via a new parallel computing algorithm, programmed using new parallel software and operated on new parallel hardware. An attempt is made to exploit a high degree of parallelism based on the use of a finite difference solution method. The specific problem considered in detail in the paper, as an example of one particular case that may be addressed, is that of coupled heat and moisture transfer in unsaturated soil. The validity of the results achieved are first discussed in relation to physically observed behavior. Qualitatively and quantitatively correct results are shown to have been obtained. The performance of the approach as an efficient parallel computing solution is then assessed. The results are found to be very encouraging.

Viscoelastic Simulation of Flow of Rubber Compounds

Abstract A hybrid scheme has been developed for numerical simulation of a nonisothermal viscoelastic flow in an arbitrary planar geometry of uniform thickness during extrusion and injection molding. This formulation is based on the control-volume finite-element method for solution of the continuity and momentum equations and finite difference method for solution of the energy equation. Application of this numerical technique for simulation of an extrusion and injection molding process was performed in a slit die and a quarter of a circular disk cavity, respectively. Development of pressures, shear and normal stresses, velocities, and temperature fields were calculated during nonisothermal flow of the rubber compound. Furthermore, the relaxation of stresses was calculated after cessation of the flow. The location of the meltfront was predicted during the cavity filling process. The contribution of normal stresses was studied by comparing results following from the viscoelastic and inelastic simulations.

Simulation of foaming in reaction injection molding

AbstractBlowing agents are often used in the reaction injection molding process to compensate for the shrinkage that occurs upon polymerization, thereby creating a structural foam part. Controlling the thickness of the solid skin and foamed core is essential to achieve the intended mechanical properties of the molded part. A numerical model was developed to predict the foaming behavior in reaction injection molding. This algorithm employs a novel primitive cell construction to enable it to analyze complex rectangular geometries, including inserts, with a two‐dimensional, finite difference solution method. The analysis was applied to foaming of polyurethane in a rectangular cavity. The predicted skin thickness was found to be in good agreement with actual structural foam parts. Foaming as a function of cavity thickness was also treated. The algorithm Is useful for understanding and interpreting nonlinear phenomena of rapid, exothermic polymerizations such as foam formation adjacent to the mold wall or around a metal insert. The results can be used to formulate design guidelines for achieving desired skin/core thicknesses as a function of design, material, and process parameters.

Difference solution of a viscous flow with free-surface wave about an advancing ship

A finite-difference solution method is described for a 31) viscous flow with free-surface waves about an advancing ship in steady course. A Navier-Stokes equation of finite-difference form is solved by a time-marching scheme in a boundary-fitted curvilinear coordinate system which is deformed to fit the moving free-surface at each time step. A sub-grid-scale turbulence model is introduced to achieve computations at higher Reynolds numbers. It is demonstrated that the computations by this method simulate fairly well free-surface waves, viscous flows and their interactions including three-dimensional separation under the influence of the free-surface.

A Full Potential Rotor Analysis With Wake Influence Using an Inner‐Outer Domain Technique

A three-dimensional, quasi-steady, full potential flow solver was adapted to include realistic rotor wake influence for the aerodynamic analysis of helicopter rotors. The method uses an inner/outer domain technique to accommodate wake effects. Nonlinear flow is computed in the inner domain using a finite difference solution method. The wake is modeled using prescribed wake techniques to allow for the inclusion of realistic wake geometries. Portions of the wake passing inside the inner domain are treated using an embedded vortex technique. The procedure couples the wake influence with the inner domain solution in a consistent and efficient solution process. Correlation with measured lifting transonic data in hover and forward flight is shown which demonstrates the merits of the approach.

Determination of the Interface Heat Transfer Coefficient for Non-Isothermal Bulk-Forming Processes

Experimental and analytical techniques have been developed for the determination of the interface heat transfer coefficient for nonisothermal bulk-forming processes. A fixture consisting of two flat IN-100 alloy dies was instrumented with high-response thermocouples. With this tooling, heat-transfer experiments were conducted in which (1) the two dies were heated to different temperatures and brought together under varying pressure levels and (2) the two dies were heated to the same temperature and were used to upset an aluminum alloy 2024-0 ring specimen heated to a higher temperature. Data from both sets of tests were analyzed to determine heat-transfer coefficients by using calibration curves derived from analytical and finite-difference method solutions. By this means, the effects of interface pressure, deformation, and deformation rate on the heat-transfer coefficient were established.

Numerical Simulation of a Viscous Flow with a Free Surface around a Ship Model

A new, finite difference solution method for the Navier-Stokes equations is developed to simulate viscous flow with a free surface around a ship. A body-fitted coordinates system that does not fit to the free surface is used for easy application to ships of arbitrary form. The nonlinear free surface condition is implemented by a special method. Simulated results for free surface flows around a Wigley hull at low Reynolds number are presented and compared with experimental data.

Stress analysis of ribbed flanges

In this paper an attempt is made to study the role of stiffening ribs on the behaviour of flanges. Shell theory is resorted to in developing the ribbed shell equations. A shell element with stiffening ribs in the meridional direction is obtained by superposing the stress resultants of a grid shell with the stress resultants of a isotropic shell element. A case of unequal bolt loads is considered by expressing them as Fourier series in the circumferential direction. The effect of varying the rib parameters like rib height, number, etc. is studied. The finite-difference method of solution is adopted. The behaviour of a ribbed flange is found to be similar to that of a taper hub flange.

Simulations of a Viscous Flow around a Circular Cylinder by a Higher-Order Accurate Difference Method

A higher-order accurate finite-difference solution method is developed for three-dimensional incompressible viscous flows. The Navier-Stokes equations in rotational form are solved in boundary-fitted coordinates. Fourth-order compact scheme is used for the space differencing in order to have adequate degree of accuracy for viscous flow problems. A viscous flow around a circular cylinder is solved at R=100 and 1000. The simulated twin vortices and secondary twin vortices show fairly good agreement with experimental results.

New Finite-Difference Solution Methods for Wave Instability Problems

New Finite-Difference Solution Methods for Wave Instability Problems