Showing papers on "Volume of fluid method published in 1997"

Volume‐tracking methods for interfacial flow calculations

Abstract: SUMMARY A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced. It is applicable to incompressible 2D flow simulations on finite volume and difference meshes. The method requires no explicit interface reconstruction, is direction-split and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner. A comparison of the new scheme against well-known existing 2D finite volume techniques is undertaken. A series of progressively more difficult advection tests is used to test the accuracy of each scheme and it is seen that simple advection tests are inadequate indicators of the performance of volume-tracking methods. A straightforward methodology is presented that allows more rigorous estimates to be made of the error in volume advection and coupled volume and momentum advection in real flow situations. The volume advection schemes are put to a final test in the case of Rayleigh‐Taylor instability. 1997 by CSIRO. In the numerical computation of multifluid problems such as density currents or Rayleigh‐Taylor instability there is a need for an accurate representation of the interface separating two immiscible fluids. Free surface flows such as water waves and splashing droplets are an approximation to the multifluid problem in which one of the fluids (usually a gas) is neglected as having an insignificant influence on the dynamics of the system. In a general free surface flow problem, fluid coalescence and detachment may occur and deforming meshes cannot be used. In this case the need of an accurate and sharp interface is even greater than in true multifluid computations. Although a slightly diffuse interface may be acceptable in a problem where the continuity, momentum and energy equations are solved throughout the entire mesh, in a free surface simulation the location of the interface determines the size and shape of the computational domain and specifies where boundary conditions must be applied. In this case a diffuse interface cannot be tolerated. On finite volume (or difference) meshes, standard advection techniques can be used in multifluid problems to advect either the density or a material indicator function, however these methods are either diffusive (e.g. first order upwinding) or unstable (higher order schemes in which unphysical oscillations appear in the vicinity of the interface). Numerous techniques have been devised to limit the diffusiveness of low order schemes and to minimize the instability of high order schemes (see e.g.

A PV-Based Shallow-Water Model on a Hexagonal-Icosahedral Grid

Abstract: A new global shallow-water model has been developed. It uses a hexagonal–icosahedral grid, potential vorticity as a prognostic variable, and a conservative, shape-preserving scheme for advection of mass, potential vorticity, and tracers. A semi-implicit time scheme is used so that the maximum time step for stable integrations is limited by the advection speed rather than the gravity wave phase speed. This combination of numerical methods avoids some of the major problems of more traditional numerical methods, such as pole problems, and spurious oscillations and negatives in advected quantities. Sample results from a standard set of test cases are presented to illustrate the model’s performance. In a pure advection test case the model’s advection scheme shows good isotropy and phase-speed properties, but it is a little diffusive. In the remaining test cases the model’s overall accuracy is comparable to that of other gridpoint models for which results are available. Two sources of error are noted. ...

Coupled Fluid-Structural Characteristics of Actuators for Flow Control

Abstract: The characteristics of a typical flow control actuator design are discussed. The device is based on a resonating structure that interacts with a closed volume of fluid to create a concentrated jet through an exit orifice. The resulting unsteady flow through the orifice introduces viscous effects that are characterized by the Stokes parameter based on the orifice diameter. An optimum operating Stokes parameter is then computed by matching this viscous dominated solution to an ideal, inviscid result. The actuator is modeled with a system of coupled equations that describe its fluid-structural behavior.

Method and apparatus for measuring flow rate and controlling delivered volume of fluid flow through a valve aperture

Abstract: An apparatus for measuring flow rate, and controlling delivered volume of fluid flow in a fluid conduit, is disclosed including a valve (14) having a fixed aperture disposed in the fluid conduit for selectively opening and closing the aperture between an open position and a closed position The valve (14) is selectively closed when a predetermined volume has passed through the valve Differential pressure is measured across the valve, and the differential pressure is accumulated with respect to time to provide a value corresponding to volume of fluid flow passing through the valve when in the open position A method for measuring flow rate, and controlling delivered volume of fluid flow through a fluid conduit, includes the steps of selectively opening and closing a valve having a fixed aperture disposed in the fluid conduit between an open position and a closed position, measuring a differential pressure between a first pressure on an upstream side of the valve, and a second pressure on a downstream side of the valve, and when the valve is in the open position, accumulating the differential pressure with respect to time to obtain a value corresponding to volume of fluid flow passing through the valve

Wetting of a High-Energy Fiber Surface

Abstract: The measurement of the equilibrium contact angle of a small droplet of fluid partially wetting a flat solid surface provides information on the solid–liquid interfacial energy. However, if the spreading power,S= γSV− (γSL+ γLV), of the surface is positive the liquid spreads completely, no equilibrium contact angle exists, and the resulting thin film has an ultimate thickness determined by Van der Waal's forces. On a chemically identical solid surface with only the geometry changed to a cylinder the same droplet of fluid which completely wets the flat surface can provide an equilibrium conformation. The indefinite spreading tendency is inhibited and the equilibrium is not necessarily a thin sheathing film about the fiber, but can have a macroscopic profile. On a high energy cylindrical surface a barrelling type droplet is only approximately spherical in cross section. Near the three phase contact line the curvature can change sign and measurement of the contact angle becomes difficult. In this work we consider the theoretical profile for such droplets and calculate the extent to which decreasing the fiber radius changes the surface energy and the maximum slope of the profile. We suggest that measurements of the inflection angle in addition to the reduced thickness and reduced length of the droplet provide an improved means of characterizing droplet on fiber systems. Experiments are reported showing the changes in contact length, droplet height, and inflection angle for poly(dimethyl)siloxane oils on copper cylinders of different diameters. These cylinders are produced from the same initial copper wire by etching in sodium hydroxide to produce controlled diameters ranging from 0.07 to 0.49 mm. As the curvature increases with reducing diameter the influence of gravity diminishes and the shape increasingly conforms to a symmetric barreling droplet type. Furthermore, as the reduced volume of fluid increases the inflection angle increases from 7° to 30° while the contact angle remains at 0°. Consistency between measured values of equilibrium parameters are compared to the theoretical values which we compute numerically and the suggested radius and volume dependence of the inflection angle is confirmed.

Generation and maintenance of pore pressure excess in a dehydrating system 1. Experimental and microstructural observations

Abstract: The generation and maintenance of excess pore pressures in dehydrating gypsum aggregates were investigated using experiments and microstructural analyses. A triaxial deformation apparatus, was equipped with a pore fluid system connected directly to the dehydrating sample. This system was operated in constant fluid volume mode to monitor pore pressure increase under undrained conditions, and in constant pore pressure mode to monitor fluid expulsion under drained conditions. X ray diffraction and backscatter scanning electron microscopy were used to characterize the spatial relationship among gypsum, the product phase bassanite, and the pores. In addition, we measured the permeability and pore compressibility of the starting material and explored the influence of effective and pore pressures, temperature, and axial load on fluid expulsion. Three stages of fluid expulsion and microstructural evolution during dehydration of an initially low-porosity, low-permeability gypsum aggregate are defined: (1) Initially, fluid released by the reaction is trapped in isolated or discontinuous pore networks and high pore pressures are possible. (2) An interconnected pore network eventually develops and fluid readily escapes. (3) Fluid expulsion slows down drastically as the reaction nears completion. As a result of coupling between dehydration and porosity production, both the cumulative volume of fluid expelled and the expulsion rate increase with increasing temperature, effective pressure, and axial load and with decreasing pore pressure. Our hydrological and microstructural data, combined with previous mechanical data, provide a better understanding of the relationships among changes in fluid volume, porosity, and pore pressure excess, and the deformation behavior of a dehydrating system where drainage evolves with time.

Modelling the volume of expandable body fluid spaces during i.v. fluid therapy.

Abstract: We have developed mathematical models to represent the changes in volume of fluid spaces associated with i.v. administration of a crystalloid solution. Input data for parameter estimations were dilution of blood, measured as reduction of blood haemoglobin concentration. The models were based on the assumption that the body strives to maintain volume homeostasis of fluid spaces and that the rate of restoration is a function of deviation from resting volume. Two models were derived; the first had a single fluid space into which fluid was administered and from which fluid left, the other model had a second fluid space communicating with the first. These models may be useful in the description and analysis of the effects of i.v. fluid therapy.

A Numerical Model for Breaking Waves: The Volume of Fluid Method.

Abstract: : This report reviews a numerical model for calculating the evolution of a breaking wave. The model is the combination of a modified version of RIPPLE which was originally developed at Los Alamos National Laboratory (Kothe et al., 1991) and the kappa - epsilon turbulence model. In the model, finite difference solutions to the incompressible Reynolds equations for the mean flow field and the kappa - epsilon equations for the turbulent field are obtained on a nonuniform mesh. The free surface locations are represented by the volume of fluid (VOF) data on the mesh. A two-step projection method is used for the mean flow solutions, aided by the incomplete Cholesky conjugate gradient technique solving the Poisson equation for the mean pressure field. Advections of momentum in Reynolds equations and turbulent kinetic energy and dissipation rate in the kappa - epsilon equations are estimated by the combination of the upwind method and the central difference method. Several numerical examples, including the runup and rundown of nonbreaking and breaking solitary waves, are given. Agreement between the experimental data and the numerical results is very good.

Non-convex profile evolution in two dimensions using volume of fluids

Abstract: A new Volume of Fluid (VoF) method is applied to the problem of surface evolution in two dimensions (2D). The VoF technique is applied to problems that are representative of those that arise in semiconductor manufacturing, specifically photolithography and ion-milling. The types of surface motion considered are those whose etch rates vary as a function of both surface position and orientation. Functionality is demonstrated for etch rates that are non-convex in regard to surface orientation. A new method of computing surface curvature using divided differences of the volume fractions is also introduced, and applied to the advancement of surfaces as a vanishing diffusive term.

Refined Numerical Scheme for Advective Transport in Diffusion Simulation

Abstract: We propose an accurate numerical scheme for calculating advection in the simulations of mass, heat, and momentum transport. The second-order spatial derivatives of the advection-diffusion equation can be discretized more accurately by the usual finite-difference approximation than the first-order spatial derivatives. By taking this feature into account, we can expect that the second-order wave equation is more available for numerical calculation of pure advection than the first-order advection equation. However, the second-order wave equation has two types of propagating wave solutions, one of which is unnecessary. To get a unique solution that shows the downstream advection only, the concept of characteristics method is applied. By minimizing truncation errors in the scheme, the parameters involved could be determined as functions of the Courant number. Comparison of this scheme with several others in model calculations proves its superior accuracy and stability. The proposed scheme uses only three grid points in space in case of one-dimensional problems. In addition, this can be easily applied to multidimensional practical problems.

Monitoring cleaning effectiveness of a cleaning system

Abstract: The invention can be used to monitor the cleaning effectiveness of a cleaning system in which vibrations (such as, for example, vibrations having a megasonic or ultrasonic frequency) are applied to a volume of fluid (e.g., water) in which an object to be cleaned (e.g., a substrate, such as semiconductor wafer or other semiconductor substrate) is at least partially immersed. Generally, the invention monitors the state of a physical characteristic of the fluid during the application of the vibrations, thus providing a more direct and accurate measure of the effectiveness of the cleaning than has heretofore been obtainable. For example, the magnitude of the acceleration of a pressure wave produced in the volume of fluid by the vibrations can be monitored, thus enabling the amplitude and/or frequency of the pressure wave to be determined. Or, the formation of bubbles on a surface that is immersed in the fluid can be detected in a manner that enables the size and/or the frequency of formation of such surface bubbles to be determined. These quantities can be used to evaluate the effectiveness of the cleaning. The invention can be used, for example, during installation of a cleaning system, during qualification of a cleaning method, or as part of scheduled or diagnostic maintenance of a cleaning system. Further, the invention can be implemented with a cleaning system or method so as to enable control of the system or method in response to the evaluation of cleaning effectiveness, thus enabling real-time control of the cleaning system or method so as to increase the cleaning effectiveness of the system or method.

Application of the Volume-of-Fluid Method to the Advection–Condensation Problem

Abstract: The authors demonstrate the application of the volume of fluid (VOF) method, a specialized grid refinement technique, to the numerical simulation of clouds. In particular, it is shown that VOF eliminates most of the well-recognized numerical difficulties (spurious oscillations and/or diffusion in vicinity of a cloud–environment interface) associated with finite-difference Eulerian advection of cloud boundaries. In essence, VOF is a subgrid-scale advection parameterization that accounts for the transport of material interfaces. VOF is an Eulerian approach, as it does not track explicitly material interfaces. Instead, it reconstructs such interfaces using auxiliary dependent variables—the partial volume fractions of immiscible materials within computational cells. A feature of VOF particularly important for cloud modeling is its ability to identify cells with a subgrid-scale cloud–environment interface. Consequently, relevant parameterizations of microphysical processes can be applied consistently ...

Motion of interacting gas bubbles in a viscous liquid including wall effects and evaporation

Abstract: The motion of single and multiple gas bubbles in an otherwise stationary liquid contained in a closed right vertical cylinder is investigated using a modified volume-of-fluid (VOF) method incorporating surface tension stresses. An isolated bubble was considered in a separate paper [4], where the initial bubble radius was small in comparison with that of the cylinder and watt effects were negligible. In this work the focus is on the interference effects during the motion of two initially spherical bubbles in a gravitational field, as well as the influence of the container wall on the bubble motion: the initial bubble diameter in the present study is more than half the cylinder diameter. The bubble size is also much larger than that required to satisfy the condition in which the gas can be treated as incompressible. In addition, the effect on bubble motion of the inclusion of evaporation at the gas-liquid interface as well as the bursting of a bubble through a five surface are considered. The modif...

Method for separating solids from drilling fluids

Abstract: A solids separation system may be used to separate solids, such as cuttings from drilling fluids used in well drilling operations. The system includes a settling tank having transverse baffles defining a fluid receiving chamber, a fluid output chamber and one or more intermediate chambers. Fluid introduced into the fluid receiving chamber can flow in a sinuous path through apertures in the baffles to the fluid output chamber. Solids settle to the bottom of the settling tank. A material conveyor, preferably an auger, extends along a bottom surface of the settling tank to an outlet port in the fluid receiving chamber. A centrifuge is connected to the output port to receive fluid in which solids have been concentrated. Fluid output from the centrifuge is reintroduced into the settling tank. The apparatus and method of the invention permit a single centrifuge to be used to handle a higher volume of fluid than is possible with conventional methods and apparatus. This provides significant cost savings.

Intrusive Gravity Currents

Abstract: Intrusive gravity currents arise when a fluid of intermediate density intrudes into an ambient fluid. These intrusions may occur in both natural and human-made settings and may be the result of a sudden release of a fixed volume of fluid or the steady or time-dependent injection of such a fluid. In this article we analytically and numerically analyze intrusive gravity currents arising both from the sudden release of a fixed volume and the steady injection of fluid having a density that is intermediate between the densities of an upper layer bounded by a free surface and a heavier lower layer resting on a flat bottom. For the physical problems of interest we assume that the dynamics of the flow are dominated by a balance between inertial and buoyancy forces with viscous forces being negligible. The three-layer shallow-water equations used to model the two-dimensional flow regime include the effects of the surrounding fluid on the intrusive gravity current. These effects become more pronounced as the fraction of the total depth occupied by the intrusive current increases. To obtain some analytical information concerning the factors effecting bore formation we further reduce the complexity of our three-layer model by assuming small density differences among the different layers. This reduces the model equations from a 6×6 to a 4×4 system. The limit of applicability of this weakly stratified model for various ranges of density differences is examined numerically. Numerical results, in most instances, are obtained using MacCormack's method. It is found that the intrusive gravity current displays a wide range of flow behavior and that this behavior is a strong function of the fractional depth occupied by the release volume and any asymmetries in the density differences among the various layers. For example, in the initially symmetric sudden release problem it is found that an interior bore does not form when the fractional depth of the release volume is equal to or less than 50% of the total depth. The numerical simulations of fixed-volume releases of the intermediate layer for various density and initial depth ratios demonstrate that the intermediate layer quickly slumps from any isostatically uncompensated state to its Archimedean level thereby creating a wave of opposite sign ahead of the intrusion on the interface between the upper and lower layers. Similarity solutions are obtained for several cases that include both steady injection and sudden releases and these are in agreement with the numerical solutions of the shallow-water equations. The 4×4 weak stratification system is also subjected to a wavefront analysis to determine conditions for the initiation of leading-edge bores. These results also appear to be in agreement with numerical solutions of the shallow-water equations.

Numerical modelling of twin‐roll casting by the coupled fluid flow and heat transfer model

Abstract: The twin-roll process is modelled by a coupled fluid flow and phase change model by means of a versatile finite element method. Here, a simple numerical scheme is proposed to solve the problem of determining the interface shape under the thermal equilibrium condition. The procedure is based on a finite element method using a transform technique. The simple numerical method provides an efficient and accurate way to find the interface position and shape with arbitrary boundary geometry. This method can easily be implemented on the existing finite element program, and provides a simple and efficient tool to simulate the solidification as well as the fluid flow problem of the twin-roll casting process. © 1997 by John Wiley & Sons, Ltd.

a Wave Equation Model to Solve the Multidimensional Transport Equation

Abstract: SUMMARY The wave equation model, originally developed to solve the advection‐diffusion equation, is extended to the multidimensional transport equation in which the advection velocities vary in space and time. The size of the advection term with respect to the diffusion term is arbitrary. An operator-splitting method is adopted to solve the transport equation. The advection and diffusion equations are solved separately at each time step. During the advection phase the advection equation is solved using the wave equation model. Consistency of the first-order advection equation and the second-order wave equation is established. A finite element method with mass lumping is employed to calculate the three-dimensional advection of both a Gaussian cylinder and sphere in both translational and rotational flow fields. The numerical solutions are accurate in comparison with the exact solutions. The numerical results indicate that (i) the wave equation model introduces minimal numerical oscillation, (ii) mass lumping reduces the computational costs and does not significantly degrade the numerical solutions and (iii) the solution accuracy is relatively independent of the Courant number provided that a stability constraint is satisfied. The transport equation arises naturally in several areas of environmental science and energy engineering. In many cases advection plays a dominant role in transport process. The numerical solution of the advection-dominated transport equation has been one of the most difficult problems in computational fluid dynamics. It is known that the advection component is the main source of difficulties in solving the transport equation. Many numerical papers have been published over the past decade. An accurate and effective numerical method for solving the transport equation is of great interest. In physics the advection equation represents the conservation laws of fluid mechanics. The mass, energy and momentum carried by fluid particles remain conservative in a flow field. The general solution of the advection equation is similar to a progressive wave in the flow direction where physical information is transferred from upstream to downstream and the advection equation is nonsymmetric. Symmetric numerical methods such as central finite differences and the Galerkin finite element method are not likely to solve the advection equation accurately. It is preferable to adopt a numerical method consistent with the physical nature of the problem being considered. The characteristic method 1 and upwind method 2 are typical non-symmetric numerical methods which have been widely used. The characteristic method with linear interpolation gives a smooth but seriously damped solution. The characteristic method with high-order interpolation, developed by

Wave dynamics at coastal structures: development of a numerical model for free surface flow

Abstract: The development of third generation wave models is needed for a detailed study of wave dynamics and impact at coastal structures. This would require the modelling of wave flows with high distortion of the free surface at confined boundaries. In our opinion, the Volume of fluid method, which uses concepts of local advection of fluid in free surface flow modelling, is the prime candidate for simulating realistic flows at sea defences and walls. In this paper, the numerical techniques for the simulation of waves with highly distorted water/air interfaces at a slope, using the volume of fluid method, are considered.

Non-convex profile evolution in two dimensions using volume of fluids

Abstract: A new Volume of Fluid (VoF) method is applied to the problem surface evolution in two dimensions. The VoF technique is applied to problems that are representative of those that arise in semiconductor manufacturing, specifically photolithography and ion milling. The types of surface motion considered are those whose etch rates vary as a function of both surface position and orientation. Functionality is demonstrated for etch rates that are (non?)-convex in regard to surface orientation. A new method of computing surface curvature using divided differences of the volume fractions is also introduced and applied to the advancement of surfaces as a vanishing diffusive term.

Spatially fourth-order-accurate scheme for unsteady-convection problems

Abstract: This work introduces a method for solving advection and advection-diffusion equations which is fourth-order-accurate in space and second-order-accurate in time. MacCormack (MC) time-splitting schemes are used with various differencing schemes in discretizing the advection terms. By using various differencing schemes, it is found that not all differencing schemes produce accurate results even though the truncation orders are same. The suggested method is compared with an analytical solution and conventional MC for pure advection and advection-diffusionh equations. For advection problems the method is more accurate than the MC method. Also, two-dimensional Navier-Stokes equations are solved for forced (Re = 1,000, 3,200, and 5000) and natural convection (Ra = 1 X 10 4 and 1 X 10 5 ) in a closed cavity. The results are compared with benchmark solutions.

A Lagrangian Advection Scheme Using Tracer Points

Abstract: A new quasi-Lagrangian advection scheme with similarities to the so-called Particle-In-Cell methods bas been introduced with the purpose of describing advection processes in atmospheric modelling in an accurate way. The new Full Particle In Cell (FPIC) scheme has been tested and compared with other advection schemes commonly used in atmospheric modelling. The first tests deal with plane passive advection in a non-deforming as well as in a strongly deforming flow. In these cases, the FPIC scheme is compared with a traditional semi-Lagrangian advection method. The new scheme has the advantage of being exact for linear advection and in the case of non-linear advection, much less damping is seen than with the semi-Lagrangian scheme. In order to investigate if the FPIC scheme behaves reasonably in the atmospheric dynamical environment, it has been fully implemented in a shallow water model including orography and with semi-implicit treatment of gravity wave terms. Also in this case, the scheme behaves...

Apparatus and method for separating solids from drilling fluids

Abstract: A solids separation system may be used to separate solids, such as cuttings from drilling fluids used in well drilling operations. The system includes a settling tank having transverse baffles defining a fluid receiving chamber, a fluid output chamber and one or more intermediate chambers. Fluid introduced into the fluid receiving chamber can flow in a sinuous path through apertures in the baffles to the fluid output chamber. Solids settle to the bottom of the settling tank. A material conveyor, preferably an auger, extends along a bottom surface of the settling tank to an outlet port in the fluid receiving chamber. A centrifuge is connected to the output port to receive fluid in which solids have been concentrated. Fluid output from the centrifuge is reintroduced into the settling tank. The apparatus and method of the invention permit a single centrifuge to be used to handle a higher volume of fluid than is possible with conventional methods and apparatus. This provides significant cost savings.

Finite element formulation for filling a thin section cavity

Abstract: Presents a quasi three‐dimensional formulation for filling a thin section cavity which is derived under the assumption that no transverse flow occurs in the gap. A no‐slip condition was applied on all surfaces occupied by the fluid and a slip condition on all air‐filled (empty) surfaces. The formulation was developed to analyse the sections which lie in the xy‐plane or may be oriented arbitrarily in three‐dimensional space. Solves the discretized thickness‐integrated finite element flow equations by using the implicit mixed velocity‐pressure formulation, and uses the volume of fluid (VOF) method to track the free surfaces. Presents numerical examples which confirm the accuracy of the formulation and demonstrate how it can be used to model the filling of planar and three‐dimensional thin section cavities of irregular shape.

Equivalence of Fluctuation Splitting and Finite Volume for One-Dimensional Gas Dynamics

Abstract: The equivalence of the discretized equations resulting from both fluctuation splitting and finite volume schemes is demonstrated in one dimension. Scalar equations are considered for advection, diffusion, and combined advection/diffusion. Analysis of systems is performed for the Euler and Navier-Stokes equations of gas dynamics. Non-uniform mesh-point distributions are included in the analyses.

Finite element analysis of flow with moving free surface by volume of fluid method

Abstract: A numerical technique for simulating incompressible viscous flow with free surface is presented. The flow field is obtained by penalty finite element formulation. In this work, a modified volume of fluid (VOF) method which is compatible with 4-node element is proposed to track the moving free surface. This scheme can be applied to irregular mesh system, and can be easily extended to three dimensional geometries. Numerical analyses were done for two benchmark examples, namely the broken dam problem and the solitary wave propagation problem. The numerical results were in close agreement with the existing data. Illustrative examples were studied to show the effectiveness of the proposed numerical scheme.