Explicit Finite Difference Technique Research Articles

A model for steam bubble formation at a submerged orifice in a flowing liquid.

A model for describing steam bubble formation at a submerged orifice in flowing liquid is developed. It is assumed that the bubble grows continuously in the radial direction, translates continuously in the vertical direction, and is surrounded by a thin liquid layer unaffected by the bulk liquid motion. The model allows fluctuation of pressure, temperature, and density of the vapor in a bubble in equilibrium with the liquid at the bubble wall. The temperature profile in the boundary layer is not limited to a quadratic function as assumed in Denekamp’s model6) but varies in accordance with the propagation of heat. The detachment criterion assumed is that the bubble neck is equal to zero for the first bubble of the pairing and equal to half of the bubble radius for the bubble containing the second bubble of the pairing.The prediction values from the present model using an explicit finite-difference technique are compared with experimental data in the literature. The results are in good agreement and show a significant improvement over Denekamp’s model.

Effect of grain refining on fatigue-creep interaction behaviour in a nickel-base superalloy: Yang, W., Zhang, J., Chen, G., Tian, S. and Geng, Q Acta Metall. Sin. (China) Aug. 1988 24, (4), A266–A269 (in Chinese)

A model for simulating the constrained thermal fatigue of an aluminium casting alloy is presented in this work. The specimen being tested has its ends fixed to a given length while its centre is subjected to thermal cycling. The thermal gradients developed within the test are calculated by an explicit finite difference technique, and with them, the strain and stress gradients are calculated. The agreement between the predicted and measured values of stress is good enough to consider that the model reflects the behaviour of the material. It is found that the predicted life of the specimen is higher than that experimentally observed, an effect that might be due to overageing of the material as the specimen is being tested.

Non-newtonian secondary flows in ducts of rectangular cross-section

A numerical study of the secondary flows which occur in the laminar pressure-driven motion of dilute polymer solutions in ducts of rectangular cross-section is presented. The full nonlinear equations of motion for a Maxwell fluid with shear thinning are solved by an explicit finite-difference technique. Results are presented in an inertial framing which indicate the existence of the previously observed eight-vortex secondary flow in ducts of various aspect ratios. In addition, the effect of spanwise rotations on the strength and the structure of the secondary flow is examined. The results indicate that even the mere presence of the diurnal rotation of the earth can give rise to a double-vortex secondary flow which is of a comparable order of magnitude to the viscoelastic secondary flows in dilute polymer solutions. Specific computations are presented which illustrate the distortional effect that the imposition of a spanwise rotation has on the viscoelastic secondary flow structure in square ducts. The implications that these results can have on laboratory experiments are discussed briefly along with other possible applications to rotating and curved flows.

STEADY AND TRANSIENT MIXED CONVECTION NEAR A VERTICAL UNIFORMLY HEATED SURFACE EXPOSED TO HORIZONTAL FLUID FLOW

The numerical solution of flow and heat transfer for steady and transient laminar mixed convection near a vertical uniformly heated surface exposed to a horizontal cross-flow is presented. The transients considered include the simultaneous initiation of flow and heating and the time-varying flow generated by starting or stopping forced convection with the surface heating condition unchanged. The partial differential equations describing the conservation of mass, momentum, and energy were solved in their time-dependent forms by an explicit finite-difference technique. Calculations were performed for fluids with Prandtl numbers of 0.733 and 6.7, nominally air and water. During both steady and transient circumstances, the effects of the horizontal forced flow were dominant near the vertical leading edge, whereas natural convection dictated the flow at large values of the horizontal coordinate. The heat transfer coefficient during mixed convection was significantly higher than that with either forced or free convection atone. During simultaneous starting of flow and heating, the time to reach steady state decreased with increase in cross-flow velocity. For the transients initiated by suddenly imposing a horizontal forced flow on an existing natural convection flow, the local temperature and vertical velocity were found to undershoot before reaching their respective steady-stale values. The overshoot in heat transfer coefficient in such situations was significant for fluids with smaller Prandtl number. The transient initiated by stopping the horizontal flow during mixed convection was associated with overshoot in both vertical component of velocity and local fluid temperature.

Development of Axisymmetric Laminar to Turbulent Free Jets From Initially Parabolic Profiles

Experiments were carried out on air in air axisymmetric free jets having parabolic profiles at the nozzle exit. The range of Reynolds numbers investigated was from 2342 to 11,000 and transition from laminar to turbulent flow was observed. The development of the laminar mean velocity profiles agrees well with that predicted from solution of the continuity and momentum equations using an explicit finite-difference technique of the Dufort-Frankel type. Some observations are made of the transition from laminar to turbulent flow using hot-wire turbulence measurements, and an empirical equation for the location of the transition is given.

Steady and Transient Mixed Convection Near a Vertical Uniformly Heated Surface Exposed to Horizontal Fluid Flow

The numerical solution of flow and heat transfer for steady and transient laminar mixed convection near a vertical uniformly heated surface exposed to a horizontal cross-flow is presented. The transients considered include the simultaneous initiation of flow and heating and the time-varying flow generated by starting or stopping forced convection with the surface heating condition unchanged. The partial differential equations describing the conservation of mass, momentum, and energy were solved in their time-dependent forms by an explicit finite-difference technique. Calculations were performed for fluids with Prandtl numbers of 0.733 and 6.7, nominally air and water. During both steady and transient circumstances, the effects of the horizontal forced flow were dominant near the vertical leading edge, whereas natural convection dictated the flow at large values of the horizontal coordinate. The heat transfer coefficient during mixed convection was significantly higher than that with either forced or free ...

A two-dimensional dispersion analysis of selected methods for solving the linearized shallow water equations

The accuracy and cost of three finite element methods for solving the linearized, two-dimensional shallow water equations are compared with a traditional explicit finite difference technique. Accuracy is determined by comparing numerical and analytic plane wave solutions. Cost is measured as the number of computations per unit of real time and per unit of model area. Two of the finite element methods are shown to be cost competitive, and as accurate as the chosen explicit finite difference technique. Though not comprehensive, the finite element analyses also suggest that meshes composed of equilateral triangles most accurately represent phase and group velocity.

Numerical modelling of electric current induced growth layers generated during Czochralski pulling

A one-dimensional mathematical model is presented which accounts for discontinuous absorption of heat at the crystal/melt interface (Peltier effect) associated with the formation of electric current induced growth layers during Czochralski pulling. The energy transport equations are solved using an explicit finite difference technique. Simulations were performed using crystal growth and heat transfer parameters which reflect actual experimental conditions encountered during growth of doped InSb. Assumptions made in order to calculate the Peltier coefficient from the dimensions of contiguous current induced growth layers are verified in simulation. The results are found to be in excellent agreement with the experiment.

Soil-structure interaction in explosive testing of model containments

The SIMQUAKE series of field tests on model containment structures provides a data base which may be used to validate analytic models of soil-structure interaction (SSI). In addition, the test produced significant evidence of nonlinear rocking response of 1/4-embedded model structures due to debonding and rebonding at the soil-structure interface. This paper describes the SIMQUAKE II test and an analytic method for soil-structure interaction based on explicit finite difference techniques, which consider both nonlinear behavior at the soil-structure interface and nonlinear constitutive behavior of the site. The analytic method is applied to two-dimensional analyses of both 1/8- and 1/12-size structures and results are compared with test measurements. Structural rocking response is shown to be sensitive to the inelastic compaction characteristic of the soil adjacent to the structure. A three-dimensional pretest analysis of the SIMQUAKE II test is also briefly reviewed.

Nonlinear truncation error analysis of finite difference schemes forthe Euler equations

It is pointed out that, in general, dissipative finite difference integration schemes have been found to be quite robust when applied to the Euler equations of gas dynamics. The present investigation considers a modified equation analysis of both implicit and explicit finite difference techniques as applied to the Euler equations. The analysis is used to identify those error terms which contribute most to the observed solution errors. A technique for analytically removing the dominant error terms is demonstrated, resulting in a greatly improved solution for the explicit Lax-Wendroff schemes. It is shown that the nonlinear truncation errors are quite large and distributed quite differently for each of the three conservation equations as applied to a one-dimensional shock tube problem.

Ensembles of microelectrodes: A digital- simulation

The rate of diffusion to ensembles of microelectrodes is calculated by digital simulation, based on the explicit finite-difference technique. Two limiting cases are shown in the current-time relationship at short times, when the diffusion layer is small in comparison to the radius of each microelectrode, the current behaves as in the case of semi-infinite planar diffusion to the active area on the surface. At long times, when the diffusion layers of adjacent electrodes overlap significantly, the current approaches the case of planar diffusion to the total area of the surface, including the non-active regions. The results are compared with previous calculations based on analytical solutions, which were derived on the basis of different approximations. Good agreement between reported experimental data and results based on simulation is observed. Edge effects arising as a result of contributions of radial components to the total diffusion rate are shown to be significant, even on relatively large electrodes. The errors which arise due to such effects are estimated as a function of the size of the electrode and the duration of measurement.

An alternative conceptual model of groundwater flow

By means of examples from finite mathematics and probability theory the relation between analytic and numerical methods in groundwater flow is investigated; as a result a conceptual model of groundwater flow is developed, based on probability and a particulate description of the interstitial water. The model seems to merge with the standard model of groundwater flow in the finite-difference approximation to the diffusion equation and to be capable of explaining some results of the standard model in a direct way. The model gives an interesting insight into the nature of the stability criterion for the explicit finite-difference technique.

TIDE-INDUCED RESIDUAL FLOW IN SHALLOW BAYS

For a square bight (= square basin with two neighboring open boundaries) residual flows are computed 1) by (semi-analytically) solving the time averaged tidal equations, 2) by numerically solving the tidal equations and time averaging the tidal flows. For a semi-enclosed basin, residual flows are computed by numerically solving the tidal equations. An explicit finite difference technique is used for the numerical solutions. The numerical computations for the square bight are carried out using three different approaches (approximations) to accomodate the cross-differentiated terms at the open boundaries. When restricting attention to the interior, for all three approximations good agreement is found between the residual flows computed with the methods 1 and 2. When inconsistent with the true (= semi-analytical solution) the approximations lead to an erroneous residual flow pattern adjacent to the boundary. Numerically computed residual flow patterns for the semi-enclosed basin show large eddies. Eddies are...

Transient analysis of solar air heaters using a finite difference technique

The transient heat transfer model of a solar air heater is developed and the resulting system is solved by using an explicit finite difference technique. the solar radiation data and the ambient temperature have been represented by a Fourier series in the numerical calculation. the effects of various design parameters of the heater on its performance are studied.

The formation and growth of bubbles at a submerged orifice

A model has been developed to describe the formation of single bubbles at a submerged orifice. The model is based on a modified Rayleigh equation for bubble growth and describes the effect of gas momentum by assuming that the flow field inside the growing bubble is in the form of a circulating toroidal vortex. The equations describing the bubbling system are solved numerically using an explicit finite-difference technique. The model shows that bubble growth is characterised by an initial outward movement of the base of the bubble along the plate floor followed by an inward movement back towards the orifice which leads to a severing of the bubble from the orifice and termination of the growth cycle. Computed bubble growth rates, formation times and chamber pressure fluctuations are shown to be in good agreement with available experimental data for a wide range of system pressure (0–1.37 MN/m 2) and computed bubble shapes are similar to those observed experimentally.

Heat flow in controlled directional solidification of metals: II. Mathematical model

This paper outlines the development of a mathematical model to describe heat flow during directional solidification of a cylindrical specimen by traversing from a furnace into a cooling device. Thermal profiles were set up along the specimen and its environment and an explicit finite difference technique used to describe radial and axial heat exchange. The validity of the model was verified by comparison with experimental data for the growth behaviour and thermal characteristics during solidification of aluminium, obtained in the manner outlined in the first of this pair of papers. It was confirmed that, for a typical experimental set-up, the interface velocity during the period of stabilized growth can differ markedly from the crucible traverse speed. The model was then used to examine the conditions under which this departure from ideal behaviour would be expected to be significant, and some practical steps are suggested for such cases.

Determination of surface heat-transfer coefficients during quenching of steel plates

AbstractThe experimentally determined relationships between temperature and time in quenched steel plates have been used in conjunction with an explicit finite-difference technique to determine the effect of surface temperature on the magnitude of the surface heat-transfer coefficient in water, polymer, and oil quenchants. In the case of the water quenchant the results obtained were very sensitive to the degree of oxidation of the specimen surface; this effect was associated with the degree of stability of the vapour blanket produced during the early stages of quenching. The polymer quenchant did not possess a clear advantage over water except where the concentration of the latter was as high as 25%. In comparison, the values of the surface heat-transfer coefficients obtainedfrom the oil quenchant were very low; this was associated with a relatively short nucleate boiling stage.

Numerical Model of Estuarine Circulation

A real-time numerical model is developed to predict the dynamics of partially mixed estuaries. An explicit finite difference technique which conserves mass, salt, and momentum was used to solve the governing equations. A consistent set of vertical mixing coefficients is proposed such that an application of the model to the Potomac River Estuary produces simulations of velocity, salinity, and tidal heights comparable to field observations. The model also permits an investigation of the time variability in the magnitude of the various terms composing the salt balance equation.

An analytical and experimental investigation ofgravity effects upon laminar gas jet-diffusion flames

A mathematical model is presented for the description of axisymmetric laminar jet-diffusionflames. The analysis includes the effects of inertia, viscosity, diffusion, gravity, and combustion. These mechanisms are coupled in a boundary-layer-type formulation, and solutions are obtained by an explicit finite-difference technique. A dimensional analysis shows that the maximum flame-width radius, velocity, and thermodynamic state characterize the flame structure. Comparisons with NASA-Lewis drop-tower data showed excellent agreement for normal gravity flames and fair agreement for steady-state low Reynolds Number zero-gravity flames. Kinetics effects are shown to be the primary mechanism responsible for this discrepancy, while additional factors are discussed, including ellipticity, radiation, and transient effects associated with drop-tower testing.

Explicit Numerical Solution of the Three-Dimensional Incompressible Turbulent Boundary-Layer Equations

An explicit finite-difference technique has been developed and applied to solve the governing partial differential equations for the three-dimensional incompressible turbulent boundary layer. The method uses a mixing-length concept suggested by Prandtl for representing the turbulent shear stress tensor. The only empirical input is a mathematical model for the mixing length taken after Maise and McDonald which incorporates the van Driest damping factor in the near-wall flow region. The predicted boundary-layer parameters and the velocity profiles agree well with the experimental data for two geometries considered, bom of which contained a plane of symmetry with large transverse gradients in flow properties to provide a severe test of the method of solution and to a lesser extent the shear model. The analysis does not indicate a collateral near-wall layer as suggested by several experiments.