Showing papers on "Volume of fluid method published in 1995"

An Introduction to Computational Fluid Dynamics: The Finite Volume Method

Abstract: *Introduction. *Conservation Laws of Fluid Motion and Boundary Conditions. *Turbulence and its Modelling. *The Finite Volume Method for Diffusion Problems. *The Finite Volume Method for Convection-Diffusion Problems. *Solution Algorithms for Pressure-Velocity Coupling in Steady Flows. *Solution of Discretised Equations. *The Finite Volume Method for Unsteady Flows. *Implementation of Boundary Conditions. *Advanced topics and applications. Appendices. References. Index.

An algorithm for analysis of polymer filling of molds

Abstract: A numerical implementation of the Volume of Fluid (VOF) free surface technique, applied to a filling process governed by a potential flow, is introduced. The implementation is based on a finite element control volume space discretization and uses an implicit time integration. This allows for the accurate tracking of the filling front using large simulation time steps in molds with complex geometries. The approach is validated on comparision with the analytical solution for filling a one-dimensional tube and with two-dimensional results obtained with previously-presented filling algorithms. Examples of the application of the approach in the analysis of structural resin transfer molding (SRTM) are presented. The capability of the method is demonstrated on predicting “race tracking” and “dry spotting” phenomena. The CPU requirement for a typical analysis is on the order of 1 to 20 s on pesonal computers.

Numerical simulation and validation of plunging breakers using a 2d navier-stokes model

Abstract: The numerical model SKYLLA, developed for simulation of breaking waves on coastal structures is described. The model is based on the Volume Of Fluid method and solves the two-dimensional (2DV) Navier-Stokes equations. Weakly reflecting boundary conditions allow waves to enter and leave the computational domain. Impermeable boundaries can be introduced to simulate a structure. A two-model approach can be used to simulate overtopping over a low crested structure. Results obtained with the model are compared with those obtained with physical model tests for waves on a 1:20 slope of a submerged structure.

Wave Action On and In Permeable Structures

Abstract: A numerical model that can simulate plunging waves on permeable structures is described. The 'Volume Of Fluid' method is used to solve the two-dimensional (2D-V) incompressible Navier-Stokes equations. After implementation of porous media flow for applications with permeable structures, the model has been verified by using several analytical solutions and by comparisons with physical model tests to study breaking waves on and inside permeable structures.

A hybrid finite‐element–volume‐of‐fluid method for simulating free surface flows and interfaces

Abstract: SUMMARY A numerical technique is developed for the simulation of free surface flows and interfaces. This technique combines the strength of the finite element method (FEM) in calculating the field variables for a deforming boundary and the versatility of the volume-of-fluid (VOF) technique in advection of the fluid interfaces. The advantage of the VOF technique is that it allows the simulation of interfaces with large deformations, including surface merging and breaking. However, its disadvantage is that in solving the flow equations, it cannot resolve interfaces smaller than the cell size, since information on the subgrid scale is lost. Therefore the accuracy of the interface reconstruction and the treatment of the boundary conditions (i.e. viscous stresses and surface tension forces) become grid-size-dependent. On the other hand, the FEM with deforming interface mesh allows accurate implementation of the boundary conditions, but it cannot handle large surface deformations occurring in breaking and merging of liquid regions. Combining the two methods into a hybrid FEM-VOF method eliminates the major shortcomings of both. The outcome is a technique which can handle large surface deformations with accurate treatment of the boundary conditions. For illustration, two computational examples are presented, namely the instability and break-up of a capillary jet and the coalescence collision of two liquid drops. Free surface flows and interfaces between two immiscible fluids or materials with different phases are observed in many natural and industrial processes. Various numerical techniques have been developed to simulate these flows. However, owing to the complexity of the problem, each technique is tailored to a particular category of flows. For instance, boundary integral techniques'" are mainly used for simulating inviscid irrotational flows and the limiting case of zero Reynolds number. Finite element methods (FEMs) and finite difference methods (FDMs) are potentially applicable to generalized Navier-Stokes equations; however, they have to be coupled with a technique to track the advecting fluid boundaries and interfaces. The difficulty in the interface tracking is inherently related to the complexity of its topology. Therefore techniques which can handle small surface deformations fail when applied to large interface distortions. For simulation of the former category of flows (small surface deformations) the FEM is more popular. Here the fluid boundary is described by a set of fixed4 or def~nning~-'~ meshes, the location of which is obtained by either an iterative procedure or the Lagrangian movement of the interface nodes. This results in the simultaneous calculation of the position of the free surface and the field variables at the new nodal positions. Boundary-fitted orthogonal c~-ordinates*~-'~ and Lagrangian technique^'^>'^ are also used to follow the advecting liquid interfaces. These techniques are confronted with difficulties when applied to large surface deformations, surface breaking and merging. CCC 0271-2091/95/121363-18 0 1995 by John Wiley & Sons, Ltd.

Method of coating a thin film on a substrate

Abstract: A method for coating a substrate with a relatively thin, uniform film of a fluid with a minimum of waste. A volume of fluid is produced and the size and velocity of the volume of fluid are selected such that the volumes of fluid break up upon impact with the substrate without splashing or rippling. The apparatus and method are ideal for coating semiconductor wafers with a photoresist solution. The kinetic energy of the volume of fluid is adjusted to overcome the free energy associated with surface tension on impact. The collision of the fluid thus results in a uniform, thin coating of photoresist or other coating solution which may then be further processed by conventional techniques.

Free surface flow and heat transfer in cavities: the sea algorithm

Abstract: This study concerns the mathematical modeling of heat transfer and free surface motion under gravity, in cavities partially filled with a liquid. This two-phase flow problem is solved using a single-phase technique that assumes the air and liquid occupying the volume of the cavity can be treated as a single fluid with a sharp property discontinuity at the interface. A two-valued scalar advection equation is solved to mark the extent of each fluid. This idea is simple in concept, but requires careful application for two reasons: (1) The interface must remain sharp throughout the simulation; and (2) the equations of motion have to be expressed in a way that prevents the numerical “smothering” of the lighter fluid by the heavy one during the iteration process. To satisfy (1), the Van Leer TVD differencing scheme is adopted for the scalar advection equation, with appropriate flux corrections in the momentum and enthalpy equations. To satisfy (2), the continuity equation is expressed in volumetric form. The te...

An adaptive multifluid interface-capturing method for compressible flow in complex geometries

Abstract: We present a numerical method for solving the multifluid equations of gas dynamics using an operator-split second-order Godunov method for flow in complex geometries in two and three dimensions. The multifluid system treats the fluid components as thermodynamically distinct entities and correctly models fluids with different compressibilities. This treatment allows a general equation-of-state (EOS) specification and the method is implemented so that the EOS references are minimized. The current method is complementary to volume-of-fluid (VOF) methods in the sense that a VOF representation is used, but no interface reconstruction is performed. The Godunov integrator captures the interface during the solution process. The basic multifluid integrator is coupled to a Cartesian grid algorithm that also uses a VOF representation of the fluid-body interface. This representation of the fluid-body interface allows the algorithm to easily accommodate arbitrarily complex geometries. The resulting single grid multifluid-Cartesian grid integration scheme is coupled to a local adaptive mesh refinement algorithm that dynamically refines selected regions of the computational grid to achieve a desired level of accuracy. The overall method is fully conservative with respect to the total mixture. The method will be used for a simple nozzle problem in two-dimensional axisymmetric coordinates.

Advection of axisymmetric interfaces by the volume‐of‐fluid method

Abstract: SUMMARY A criterion is proposed for the advection of axisymmetric interfaces. The location of an interface is followed by a volume-tracking technique wherein a volume fraction parameter is assigned to each of the cells in a Eulerian grid system. The interface is discretized into a set of line segments fitted at the boundary of every pair of neighburing computational cells. The orientation of a line segment is obtained by inspecting the volume fractions of two neighbouring cells. The volume fractions are then advected using the velocity components at the boundary of the two cells. The following advection criterion is proposed: for advection in the axial direction the axial velocity I( is assumed constant in the vicinity of each cell face; for advection in the radial direction the radial velocity v times the radial distance r is assumed constant in the vicinity of each cell face, i.e. gv = const., where fi = 0 for Cartesian and fi = 1 for axisymmetric systems. The above criterion is used to develop an algorithm for the advection of axisymmetric interfaces which is referred to as the ‘axisymmetric flux line segment model for advection and interface reconstruction’ or A-FLAIR.

Numerical modelling of breaking wave impacts on a vertical wall

Abstract: The impact processes on a vertical wall resulting from breaking waves are numerical simulated. Two dimensional incompressible viscous flow which is governed by the Navier-Stokes Equations and the continuity equation is solved by a finite difference scheme based on the Volume of Fluid(VOF) concept. Some comparisons with experimental results reveal that the present model is able to simulate the impact process with negligible air entrappment not only qualitatively but also quantitatively well. Although the impact pressure of a plunging breaker with nonnegligible air entrapment can not be quantitatively well simulated by this model due to the restriction of the incompressible flow, the wave kinematics is still well simulated.

Application of a shape-preserving advection scheme to the moisture equation in an E-grid regional forecast model

Abstract: This paper presents a methodology which is very useful to design shape-preserving advection finite difference scheme on general E-grid horizontal arrangement of variables through introducing a two-step shape-preserving positive definite advection scheme in the moisture equation of the LASG-REM (LASG regional E-grid eta-coordinate forecast model) By trial-forecasting six local heavy raincases, the efficiency of the shape-preserving advection scheme in practical application has been examined The LASG-REM with the shape-preserving advection scheme has a good forecasting ability for local precipitation

A comparison of three dimensional photolithography simulators

Abstract: Methods for negative volume artifact removal and mesh regularization are introduced and applied to triangle based surface advancement simulation in the area of photolithography dissolution in three dimensions. These methods were then compared against the popular cell and advection (level-set) techniques for solving the Hamilton-Jacobi equation for a surface etching front advancing through an inhomogeneous exposed photoresist media. An order of magnitude increase in speed and a significant improvement in robustness have been achieved simultaneously for large ($>$10,000) triangle ray-based representation of dissolution. This has been achieved by applying spatial decomposition in a locally refined octtree to store the triangles and graph theory to identify loops to be removed. A general approach for removing all loops for display and a swallow-tail removal for preserving shock lines for advancement were implemented. Heuristic techniques were introduced for removing thin triangles from a triangular mesh. These techniques were based on recognizing small altitudes. A look ahead method was used to judge between removal or a 'Delaunay like' flip. A single pass using a 28 degree bending angle criterion was found to be adequate for triangles with a 0.5 minimum segment length altitude. Heuristics for removing crenulations from a triangular surface are also described for addressing ray scattering. Three first order methods of simulating photoresist dissolution were compared using a benchmark case that is representative of typical photolithography problems. These methods were ray-trace triangle advancement, advection contour advancement, and a cell based volume removal method. The speed and memory comparisons were normalized for equivalent levels of accuracy. Ray-trace was found to be 2x faster than cells and $>$10x faster than advection. Ray-trace was found to have equivalent memory consumption to cells and $<$1/10th the amount required for advection. Ray-trace and cells were found to have greater accuracy than advection along the coordinate axes for similar grid sizes. The cell method demonstrated anisotropic behavior that was not exhibited by the other two methods. A new technique for improving the accuracy of the advection method is also introduced.

On geostrophic adjustment of a two-layer, uniformly rotating fluid in the presence of a step escarpment

Abstract: This paper addresses the Rossby adjustment problem for an inviscid uniformly rotating two-layer fluid in the presence of a step escarpment of infinite length. The problem can be solved analytically for the case when the ratio of the step height to the average depth of the lower layer is small. In this case two well-separated adjustment time scales emerge; the rapid, inertial and the slow, topographic vortex-stretching time scales.The fluid is assumed to be at rest initially with imposed step discontinuities in the free surface and interfacial displacements oriented perpendicular to the escarpment. A two time-scale approach shows that during the rapid inertial adjustment the fluid is not influenced by the topography. On the slow vortex-stretching time scale the fluid adjusts via the propagation of topographic Rossby waves, modified by stratification, along the step. A steady state solution is established in which the flow is geostrophically balanced in both layers. Therefore, in this steady state no fluid in the lower layer crosses the escarpment. However, cross-escarpment flow occurs in the upper layer. The volume of fluid in the upper layer that crosses the escarpment, rather than being deflected parallel to the topography, is calculated.

Condensate flow inside paper dryer cylinders

Abstract: The rimming flow of condensate in horizontal rotating dryer cylinders has been studied computationally by solving the full Navier-Stokes equations coupled with a volume of fluid method for tracking the free surface. It was shown that significant variations in both condensate velocity and thickness exist at moderate dryer speeds, whereas at higher speeds the variations are of less significance. Regardless of dryer speed or condensate film thickness, the film can be divided into two distinct regions: a viscous sub-layer adjacent to the cylinder wall and an inviscid, oscillating layer close to the free surface. The thickness of the viscous layer decreases as the dryer speed increases, whereas, for a certain speed, it is independent of the total film thickness. The computational results are compared with measurements of both the film thickness and the pressure normal to the cylinder wall. In both cases the agreement is excellent. Some implications for heat transfer through the condensate film are briefly discussed.

Fluidic oscillator and method for measuring the volume of a fluid flowing therethrough

Abstract: A fluidic oscillator symmetrical with respect to a longitudinal plane of symmetry (P) in which the longitudinal direction of a fluid flow lies. Said oscillator includes a component (26b) for generating a two-dimensional fluid jet oscillating transversely to said longitudinal plane of symmetry (P); two ultrasonic transducers (52, 54); components (62-72) for outputting an ultrasonic signal from one transducer through the fluid flow towards the other transducer, and receiving the ultrasonic signal modulated by the oscillation of the fluid jet; and components (100) for processing the received signal to determine the volume of fluid that has flowed through said fluidic oscillator, said ultrasonic transducers (52, 54) being substantially in alignment along the longitudinal plane of symmetry (P).

Anti-siphon valve

Abstract: A method of valving a flow of pumped fluid, to prevent siphoning of the flow when the pump is stopped, including the steps of filling a chamber with fluid, directing a small flow of pumped fluid into the chamber to increase its volume, opening a main flow valve by the increase in chamber volume, providing a leak in the chamber to reduce the volume of fluid therein when the small flow of pumped fluid is terminated, and closing the main flow valve by the decrease in chamber volume.

Erosion of a fluid mud layer due to entrainment: Numerical modelling

Abstract: A continuous transport cycle of mud material can be noticed in a natural water environment. Aggregation, settling, deposition, consolidation and erosion are typically interlinked. These processes are influenced by the cohesive properties of the mud and by the characteristics of its environment. Fluid mud is a highly concentrated near-bed sediment suspension with a sediment concentration between about 10 and 300 g/l, and can be formed by hindered settling or by the fluidization of the bed. Once formed, the fluid mud can be transported due to • horizontal pressure gradients, frictional and gravitational forces. • turbulence and instability of the interface between the fluid mud layer and the water layer above, resulting in mass transport from a non-turbulent layer to a turbulent layer. This process is defined as entrainment. This report concentrates on the process of entrainment by turbulent water flow. A quantitative measure for entrainment is the dimensionless entrainment rate E, which is the ratio of the entrainment rate ue (i.e. the entrained volume of fluid mud per unit area and per unit time) to a characteristic flow velocity. Dimensional analysis indicated that £ is a function of an overall Richardson number. From the literature it followed that the entrainment of fluid mud resembles the fresh/saline water entrainment process, though properties of the cohesive sediments may greatly influence the entrainment behaviour. Two numerical models have been used to predict the entrainment of fluid mud: an entrainment model describing the small scale behaviour (1) and the two-layer fluid mud model which considers mud transport on a larger scale (2). The results have been compared with experimental data or observations. (1) From the analysis of the integral entrainment model of Kranenburg (1994), it resulted that the values taken for the empirical coefficients involved and the assumptions made for the effects of viscous drag and side wall friction are satisfactory. The effect of consolidation and the related change from entrainment to floc erosion becomes apparent for large times. (2) The two-layer fluid mud model, developed by Delft Hydraulics, showed the importance of entrainment for mud transport. When applying a settling velocity, which varies with the sediment concentration, upward transports (due to entrainment) and downward transports (caused by settling) are much larger than in the case of a constant settling velocity. Also the results agree better with observations then. The major limitation of this model originated from the fact that in the model no differences in bed material were made and only neap tide was simulated instead of a neap tide - spring tide cycle. The incorporation of the integral entrainment model of Kranenburg into the two-layer fluid mud model will only be one step forward and further improvements are needed.

HDR reservoir analysis incorporating acoustic emission data

Abstract: A set of models of HDR systems is presented which attempts to explain the formation and operation of HDR systems using only the in-situ properties of the fractured rock mass, the earth stress field, the engineering intervention applied by way of stimulation and the relative positions and pressures of the well(s). A statistical and rock mechanics description of fractures in low permeability rocks provides the basis for modeling of stimulation, circulation and water loss in HDR systems. The model uses a large number of parameters, chiefly simple directly measurable quantities, describing the rock mass and fracture system. The effect of stimulation (raised fluid pressure allowing slip) on fracture apertures is calculated, and the volume of rock affected per volume of fluid pumped estimated. The total rock volume affected by stimulation is equated with the rock volume containing the associated AE (microseismicity). The aperture and compliance properties of the stimulated fractures are used to estimate impedance and flow within the reservoir. Fluid loss from the boundary of the stimulated volume is treated using radial leak-off with pressure-dependent permeability.

Numerical Analysis of 2-dimensional Fluid Flow with Free Surface using VOF technique

Abstract: A computer code for analyzing 2-dimensional fluid flow with free surface has been developed. The main features of this computer code are (1) the modifica-tion of a VOF technique with improved numerical accuracy of stress boundary condition and the mass conservation law on free surface, and (2) the introduce of a k-e model in which an attenuating behavior of turbulent intensity near free surface can be expressed.The results of verification show that the numerical solutions agree well with the experimental data for free surface shape and velocity profiles.

A method and a control system for the production of fluid from a well

Abstract: A method of controlling the production of fluid from a well by gas lift injection and a control system (35) for controlling the production of fluid from a well by gas lift injection are described. The control system (35) comprises a cycle monitoring means for monitoring the number of cycles performed by a plunger in the well and volume metering means to meter the produced volume of fluid. A processing device (36) is also provided and is coupled to the cycle monitoring means and the volume metering means to receive output signals from the cycle monitoring means and the volume metering means. The processing device (36) controls the number of cycles and the amount of gas injected in response to the output signals received to control the volume of fluid produced from the well.

Numerical and Experimental Study of Porosity Evolution during Plasma Spray Deposition of W

Abstract: The porosity that is commonly associated with discrete droplet processes, such as plasma spraying and spray deposition, effectively degrades the quality of the sprayed material. In the present paper, numerical and experimental studies on porosity evolution in plasma spray deposition of W are undertaken to provide insight into the formation and evolution of porosity. In the numerical study, deformation, interaction and solidification of molten droplets impinging onto a flat and non-flat substrate during plasma spraying are investigated. The full Navier-Stokes equations coupled with the Volume Of Fluid (VOF) function are solved to determine the exact movement and interaction of droplets. A 2-domain method is employed for the treatment of the thermal field and solidification problem within the flattening droplet to track the moving solid/liquid interface. A two-phase flow continuum model is employed for the simulation of the flow problem with a growing solid layer during droplet impingement. On the basis of the VOF function and the two-phase flow continuum model the micro-porosity is quantitatively calculated. In the experimental study, a W deposit of 2 - 3 mm in thickness is prepared using low pressure plasma spraying (LPPS). The microstructure of the deposit is characterized in detail, paying particular attention to the presence of porosity. The mechanisms that govern the formation of porosity during LPPS are proposed in light of numerical and experimental results. On the basis of the mechanisms, some fundamental trends and effects of important processing parameters on micro-porosity may be reasonably explained and optimal processing conditions for reducing microporosity may be determined.

Extension of the volume‐of‐fluid method for analysis of free surface viscous flow in an ideal gas

Abstract: The volume-of-fluid (VOF) method is a simple and robust technique for simulating free surface flows with large deformations and intersecting free surfaces. Earlier implementations used Laplace's formula for the normal stress boundary condition at the interface between the liquid and vapour phases. We have expanded the interfacial boundary conditions to include the viscous component of the normal stress in the liquid phase and, in a limited manner, to allow the pressure in the vapour phase to vary. Included are sample computations that show the accuracy of added third-order-accurate differencing schemes for the convective terms in the Navier-Stokes equation (NSE), the viscous terms in the normal stress at the interface and the solution of potential flow in the vapour phase coupled with the solution of the NSE in the liquid phase. With these modifications we show that the VOF method can accurately predict the instability of a thin viscous sheet flowing through a stagnant vapour phase.

1 – Fluids in motion

Abstract: Mass, length, and time are commonly used primary units, other units being derived from them. Their dimensions are written as M, L, and T respectively. Although gases and liquids consist of molecules, it is possible in most cases to treat them as continuous media for the purposes of fluid-flow calculations. On a length scale comparable to the mean free-path among collisions, large rapid fluctuations of properties such as the velocity and density occur. However, fluid flow is concerned with the macroscopic “scale” ; the typical length scale of the equipment is many orders of magnitude greater than the mean free path. Even when an instrument is placed in the fluid to measure some property such as the pressure, the measurement is not made at a point; rather, the instrument is sensitive to the properties of a small volume of fluid around its measuring element. Although, this measurement volume may be minute compared with the volume of fluid in the equipment, it will generally contain millions of molecules and consequently the instrument measures an average value of the property. In almost all fluid-flow problems it is possible to select a measurement volume that is very small compared with the flow field, yet it contains so many molecules that the properties of individual molecules are averaged out. The total energy of a fluid in motion consists of the following components: internal, potential, pressure, and kinetic energies. Each of these energies may be considered with reference to an arbitrary base level. It is also convenient to make calculations on unit mass of fluid.