System Of Flow Equations Research Articles

Lagrangian particle model for multiphase flows

A Lagrangian particle model for multiphase multicomponent fluid flow, based on smoothed particle hydrodynamics (SPH), was developed and used to simulate the flow of an emulsion consisting of bubbles of a non-wetting liquid surrounded by a wetting liquid. In SPH simulations, fluids are represented by sets of particles that are used as discretization points to solve the Navier–Stokes fluid dynamics equations. In the multiphase multicomponent SPH model, a modified van der Waals equation of state is used to close the system of flow equations. The combination of the momentum conservation equation with the van der Waals equation of state results in a particle equation of motion in which the total force acting on each particle consists of many-body repulsive and viscous forces, two-body (particle–particle) attractive forces, and body forces such as gravitational forces. Similar to molecular dynamics, for a given fluid component the combination of repulsive and attractive forces causes phase separation. The surface tension at liquid–liquid interfaces is imposed through component dependent attractive forces. The wetting behavior of the fluids is controlled by phase dependent attractive interactions between the fluid particles and stationary particles that represent the solid phase. The dynamics of fluids away from the interface is governed by purely hydrodynamic forces. Comparison with analytical solutions for static conditions and relatively simple flows demonstrates the accuracy of the SPH model.

Iterative coupling reservoir simulation on high performance computers

In this paper, the iterative coupling approach is proposed for applications to solving multiphase flow equation systems in reservoir simulation, as it provides a more flexible time-stepping strategy than existing approaches. The iterative method decouples the whole equation systems into pressure and saturation/concentration equations, and then solves them in sequence, implicitly and semi-implicitly. At each time step, a series of iterations are computed, which involve solving linearized equations using specifi c tolerances that are iteration dependent. Following convergence of subproblems, material balance is checked. Convergence of time steps is based on material balance errors. Key components of the iterative method include phase scaling for deriving a pressure equation and use of several advanced numerical techniques. The iterative model is implemented for parallel computing platforms and shows high parallel effi ciency and scalability.

Analysis of semi-staggered finite-difference method with application to Bingham flows

In this paper, a finite-difference scheme for incompressible flow problems is treated. The scheme uses non-staggered grid for velocity approximation. A special stabilization is introduced to ensure the well-posedness and optimal approximation properties of the scheme. The stability estimate is proved in the form of a mesh-independent bound for the norm of discrete operator inverse. The finite-difference method is particularly suitable for problems in which velocity vector is involved in additional quantities that enter the system of flow equations as, for example, in the Bingham problem. We describe this application in the paper in some detail. Results of numerical experiments are included that confirm the applicability and optimality of the method.

Unsteady RANS simulation of oscillating mould flows

AbstractMould flow oscillations are of major importance for the performance of the continuous casting process. They are suspected to promote entrainment of slag and other unwanted secondary phases into the melt pool. These oscillating turbulent flows are investigated by means of numerical simulations. The numerical model is based on the equation of continuity and the unsteady Reynolds averaged Navier–Stokes equations. The system of flow equations is closed by a Reynolds stress turbulence model in combination with non‐equilibrium wall functions.The unsteady simulation resolves low‐frequency oscillations of the flow field. These frequencies and numerically resolved mean values are in agreement with results of corresponding model experiments.The proposed model should be advantageous in order to investigate the mechanisms of the oscillations and the process of slag entrainment in more detail. Copyright © 2006 John Wiley & Sons, Ltd.

Recent advances in finite element methods for multi-phase flow processes in porous media

In this paper, we will focus on three essentially important issues considering the modeling of multi-phase flow in heterogeneous porous media. Depending on the governing processes, the multi-phase flow equation system can show either predominant advection or diffusion/dispersion. Numerical methods have to be able to capture both cases. Furthermore, realistic porous media are always heterogeneous. The physically correct description of the interfacial behaviour is numerically very challenging, as discontinuities in primary variables may occur. In this paper, we review these issues and present the box scheme, a control volume finite element method, including an interface condition as one numerical method capable of tackling the three problems. In the end, we will give the results of a 3D simulation.

Hubble slow roll expansion for multifield inflation

We examine the dynamics of inflation driven by multiple, interacting scalar fields and derive a multi field version of the Hubble slow roll expansion. We show that the properties of this expansion naturally generalize those of the single field case. We present an analogous hierarchy of slow roll parameters, and derive the system of ``flow'' equations that describes their evolution, and show that when this system is truncated at finite order, it can always be solved exactly. Lastly, we express the scalar and tensor perturbation spectra in terms of the slow roll parameters.

Capillary driven flow along interior corners formed by planar walls of varying wettability

Closed-form analytic solutions useful for the design of capillary flows in a variety of containers possessing interior corners were recently collected and reviewed. Low-g drop tower and aircraft experiments performed at NASA to date show excellent agreement between theory and experiment for perfectly wetting fluids. The analytical expressions are general in terms of contact angle, but do not account for variations in contact angle between the various surfaces within the system. Such conditions may be desirable for capillary containment or to compute the behavior of capillary corner flows in containers consisting of different materials with widely varying wetting characteristics. A simple coordinate rotation is employed to recast the governing system of equations for flows in containers with interior corners with differing contact angles on the faces of the corner. The result is that a large number of capillary driven corner flows may be predicted with only slightly modified geometric functions dependent on corner angle and the two (or more) contact angles of the system. A numerical solution is employed to verify the new problem formulation. The benchmarked computations support the use of the existing theoretical approach to geometries with variable wettability. Simple experiments may be performed to confirm the theoretical findings. Favorable agreement between such experiments and the present theory may argue well for the extension of the analytic results to predict fluid performance in future large length scale capillary fluid systems for spacecraft as well as for small scale capillary systems on Earth.

Development of low-diffusion flux-splitting methods for dense gas–solid flows

The development of a class of low-diffusion upwinding methods for computing dense gas–solid flows is presented in this work. An artificial compressibility/low-Mach preconditioning strategy is developed for a hyperbolic two-phase flow equation system consisting of separate solids and gas momentum and continuity equations. The eigenvalues of this system are used to devise extensions of the AUSM+ [1] and LDFSS [2] flux-splitting methods that provide high resolution capturing of bubble growth and collapse in gas–solid fluidized beds. Applications to several problems in fluidization are presented.

Modeling multiphase non-isothermal fluid flow and reactive geochemical transport in variably saturated fractured rocks: 1. Methodology

Reactive fluid flow and geochemical transport in unsaturated fractured rocks have received increasing attention for studies of contaminant transport, ground- water quality, waste disposal, acid mine drainage remediation, mineral deposits, sedimentary diagenesis, and fluid-rock interactions in hydrothermal systems. This paper presents methods for modeling geochemical systems that emphasize: (1) involvement of the gas phase in addition to liquid and solid phases in fluid flow, mass transport, and chemical reactions; (2) treatment of physically and chemically heterogeneous and fractured rocks, (3) the effect of heat on fluid flow and reaction properties and processes, and (4) the kinetics of fluid-rock interaction. The physical and chemical process model is embodied in a system of partial differential equations for flow and transport, coupled to algebraic equations and ordinary differential equations for chemical interactions. For numerical solution, the continuum equations are discretized in space and time. Space discretization is based on a flexible integral finite difference approach that can use irregular gridding to model geologic structure; time is discretized fully implicitly as a first-order finite difference. Heterogeneous and fractured media are treated with a general multiple interacting continua method that includes double-porosity, dual-permeability, and multi-region models as special cases. A sequential iteration approach is used to treat the coupling between fluid flow and mass transport on the one hand, chemical reactions on the other. Applications of the methods developed here to variably saturated geochemical systems are presented in a companion paper (part 2, this issue).

Stability and Newton's Law of Cooling in Double Diffusive Flow

We study the system of equations for flow of a viscous heat conducting fluid which has dissolved in a salt field. Under the approximation, often employed in the fluid dynamical literature, that the Prandtl number be infinite, we developa prioribounds and we show the solution depends continuously on changes in the Newton cooling coefficient.

Hamiltonian flow equations for the Lipkin model

We derive Hamiltonian flow equations giving the evolution of the Lipkin Hamiltonian to a diagonal form using continuous unitary transformations. To close the system of flow equations, we present two different schemes. First we linearize an operator with three pairs of creation and destruction operators by reducing it to the z component of the quasi spin. We obtain the well known RPA-result in the limit of large particle number. In the second scheme we introduce a new operator which improves the resulting spectrum considerably especially for few particles.

A comparison of numerical models for evaporative two-phase flow in a self-heated porous medium

Two different numerical models using the finite difference method (FDM) for one-component time-dependent two-phase flows in a porous medium are investigated: the iterative four-variable model (I4VM) and the direct three-variable model (D3VM). The former includes the pressure gradient and uses the iterative method to solve a system of flow equations, whereas for the latter, the formulation without the pressure gradient is simultaneously solved using the algorithm for tri-tridiagonal equations of three dependent variables. The steady-state solution as well as the unsteady results obtained by two models are compared only for the low heat generation rate below the dryout limit. For the high heat generation rate the effects of two numerical models on the time-dependent flow and dryout behavior up to incipient dryout are discussed in terms of liquid volumetric fraction and liquid superficial velocity distributions. It was found that the I4VM is numerically more stable for the case of strongly nonlinear physical models (e.g. the Ergun constants model of Fand, R. M., Kim, B. Y. K., Lam, A. C. and Phan, R. T., Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. J. Fluids Engng, 1987, 109, 268–274) and enables us to analyze those, whereas the D3VM is advantageous for fast analysis of the weakly nonlinear model (e.g. the Ergun constants model of Macdonald, I. F., El-Sayed, M. S., Mow, K. and Dullien, F. A. L., flow through porous media—the Ergun equation revisited. Ind. Engng Chem. Fundam., 1979, 18, 199–208). Finally, a comparative evaluation of both numerical models is presented.

Critical conditions for detonation propagation through mixture with decreasing reaction rate

A problem of detonation propagation through decreasing reaction rate gradient is examined. To estimate corresponding critical conditions, a simplified problem is considered where the width of reaction zone increases linearly with distance. The stationary solution for this self-similar problem exists only if the gradient of reactivity is below a certain critical value. The critical conditions are found analytically for a one-step Arrhenius kinetic model with larger activation energy. The critical gradient is expressed through the characteristic reaction zone width, its dependence on temperatue, and thermodynamic properties of the mixture. In the cases of expanding waves (cylindrical or spherical symmetry), superposition of limitations imposed by the gradient and by the wave curvature is observed. The critical gradient decreases with an increase of E a /T ooN (E a , activation energy; T OoN , von Neumann temperature for stationary detonation wave) and with the change of the wave symmetry from planar to spherical. Numerical integration of the self-similakr system of flow equations for a wide range of E a /T OoN is presented. the results show that the analytical solution can be applied for moderate values of effective activation energy as well. The critical conditions are also examined by time-dependent numerical simulation. The results are in agreement with the analytical solution if E a /T OoN is less than 6–8. For higher E a / T ooN , longitudinal instability of the detonation wave becomes the most important factor.

Collocation Finite-Element Simulation of Dam-Break Flows

The one-dimensional hydrodynamic equations based on shallow-water theory are used to compute the flood wave resulting from the total instantaneous collapse of a dam. The collocation method is used in conjunction with quintic Hermite elements to solve the system of flow equations. Quintic Hermite elements are used to provide the high resolution required in the solution of discontinuities, thereby producing satisfactorily stable solutions. The model can simulate both subcritical and supercritical flows in different parts of the channel or in a sequence in time. The model is stable, although there are no second-order pseudoviscosity terms in it. No upwinding of the basis function is required. Linear stability analysis reveals the selective damping of short wavelengths and excellent phase accuracy of the model and insensitivity to parameter variation. The computed results are compared with the test results of the U.S. Corps of Engineers, Waterways Experiment Station. Close agreement between computed and measu...

Stability of explicit navier-stokes procedures using k- ε and k - ε/algebraic reynolds stress turbulence models

A three-dimensional explicit Navier-Stokes procedure has been developed for application to compressible turbulent flows, including rotation effects. In the present work, a numerical stability analysis of the discrete, coupled system of seven governing equations is presented. Order of magnitude arguments are presented for flow and geometric properties typical of internal flows, including turbomachinery applications, to ascertain the relative importance of grid stretching, rotation and turbulence source terms, and effective diffusivity on the stability of the scheme. It is demonstrated through both analysis and corroborative numerical experiments that: (1 ) It is quite feasible to incorporate, efficiently, a two-equation k- ϵ turbulence model in an explicit time marching scheme, provided certain numerical stability constraints are enforced. (2) The role of source terms due to system rotation on the stability of the numerical scheme is not significant when appropriate grids are used and realistic rotor angular velocities are specified. (3) The direct role of source terms in the turbulence transport equations on the stability of the numerical scheme is not significant when appropriate grids are used and realistic freestream turbulence quantities are specified, except in the earliest stages of iteration (a result which is contrary to that generally perceived). (4) There is no advantage to numerically coupling the two-equation model system to the mean flow equation system, in regard to convergence or accuracy. (5) For some flow configurations, including turbomachinery blade rows, it is useful to incorporate the influence of artificial dissipation in the prescription of a local timestep. (6) Explicit implementation of an algebraic Reynolds stress model (ARSM) is intrinsically stable provided that the discrete two-equation transport model which provides the necessary values of k and ϵ is itself stable.

Flow in a fractured medium with fractal fracture geometry

A model of a fractured porous medium in which the fracture system forms a fractal with Hausdorff-Bezikovich dimension d is proposed. The fractal is immersed in a saturated porous medium with the dimension D (D ≥ d, D=2, 3). The rock skeleton is assumed to be nondeformable. The system of flow equations is written out for cylindrically (D=2) and spherically (D=3) symmetrical flows. When D=d the model reduces to the well-known Barenblatt-Zheltov model. Certain particular solutions, which make it possible to determine the phenomenological parameters of the model experimentally, are obtained.

1‐D Dam‐Break Flood‐Wave Propagation on Dry Bed

The complete hydrodynamic equations of the shallow‐water theory in one dimension are employed to compute the flood wave resulting from the total and instantaneous collapse of a dam in a broad rectangular channel. The MacCormack numerical scheme is employed to solve the system of flow equations in a conservation‐law form. The actual upstream and downstream boundary conditions of the flow field, i.e., a depth of flow equal to zero at these boundaries and beyond, are used for the solution of the flow equations. The results obtained are compared with computational and experimental data obtained by other investigators. The broad rectangular channel was employed for the purpose of comparing results. The method can easily be applied to channels of arbitrary cross‐sectional shapes, as well as to almost any other unsteady open‐channel flow problem by incorporating the appropriate initial and boundary conditions.

An Efficient Finite-Difference Method for Simulating Phase Segregation in the Matrix Blocks in Double-Porosity Reservoirs

Summary An efficient finite-difference method is presented for simulating unsteady-state multi phase flow in the matrix blocks (primary porosity) of naturally fractured systems or other reservoirs that have two different porosities. The scheme obtains pressure and saturation distributions in the matrix blocks (rather than average values) by dividing the matrix into subdomains. The method is an extension of the double-porosity concept in which fractures are the continuum for fluid flow and the matrix rock is the primary storage medium that acts as a source term to the fractures. Examples show that this subdivision results in different recoveries from the conventional double-porosity models when phase segregation in the matrix blocks is important to the recovery mechanism. In this formulation, the fracture medium is the major flow path to the wellbore with the matrix subdomains as source terms. The matrix subdomains can be connected to each other or to the fractures in any way desired to conform to the physics of the reservoir. This approach lends itself to an efficient scheme for reducing the system of flow equations to the same structure as conventional single-porosity simulators. The method is presented for a fully implicit, one-dimensional (1D), two-phase (oil/water) simulator.

Grid Generation for Fluid Mechanics Computations

Fluid mechanics is understood in a descriptive way through experimental observation, mathematical analysis, and numerical simulation. When the understanding is required for flows with complex internal structure and with complicated regional boundaries, insight is gained primarily from experiments and simulations. Numerical simulations are motivated by the prospect of economically obtaining a detailed flow-field description. In each instance, a governing system of flow equations is analytically formulated over the region and is solved in an approximate form on a computer. Various numerical methods have been devised for such ap­ proximations, and most depend upon some representation of the flow-field region by means of finitely many points and the interconnections between them. When each regional boundary is given by a sequence of connected points, the efficiency and accuracy of the various methods are enhanced, since boundary conditions can be applied without interpolation. Further enhancement comes when the connectivity pattern is regular. The most regular and thus preferable patterns are those that result from coordinate transformations. With the application of transformations, regions with topological complexity are consistently treated even when points are in motion. During the course of simulation, motion can be advantageously used to adaptively resolve the significantly varying solution quantities. From a geometric viewpoint, the quantities determine a surface over the physical region. The resolution of the surface as it evolves then determines the adaptive movement.

Numerical calculations for two-phase flow analyses in pin bundles

Two-phase flow equation systems, in which equations are defined for each phase, are discussed for use in analyzing coolant behaviors in LMFBR pin bundles. These equation systems have not yet consolidated, because of theoretical and experimental difficulties and complexities. One of the problems is the equation systems' stability. This paper shows the stability for the low Reynolds number (O(1)) system, using a one dimensional linear equation system. Based on this fact, a two-phase flow equation system is numerically solved by using the subchannel method for 19- and 37-pin bundles. The calculational examples are LOF and TOP conditions with/without the blockage, and fission gas release.