Incompressible multiphase flow and encapsulation simulations using the moment‐of‐fluid method

A moment‐of‐fluid method is presented for computing solutions to incompressible multiphase flows in which the number of materials can be greater than two. In this work, the multimaterial moment‐of‐fluid interface representation technique is applied to simulating surface tension effects at points where three materials meet. The advection terms are solved using a directionally split cell integrated semi‐Lagrangian algorithm, and the projection method is used to evaluate the pressure gradient force term. The underlying computational grid is a dynamic block‐structured adaptive grid. The new method is applied to multiphase problems illustrating contact‐line dynamics, triple junctions, and encapsulation in order to demonstrate its capabilities. Examples are given in two‐dimensional, three‐dimensional axisymmetric (R–Z), and three‐dimensional (X–Y–Z) coordinate systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Two‐dimensional dispersion analyses of finite element approximations to the shallow water equations

Dispersion analysis of discrete solutions to the shallow water equations has been extensively used as a tool to define the relationships between frequency and wave number and to determine if an algorithm leads to a dual wave number response and near 2Δx oscillations. In this paper, we explore the application of two‐dimensional dispersion analysis to cluster based and Galerkin finite element‐based discretizations of the primitive shallow water equations and the generalized wave continuity equation (GWCE) reformulation of the harmonic shallow water equations on a number of grid configurations. It is demonstrated that for various algorithms and grid configurations, contradictions exist between the results of one‐dimensional and two‐dimensional dispersion analysis as a result of subtle changes in the mass matrix. Numerical experiments indicate that the two‐dimensional dispersion analysis correctly predicts the existence and onset of near 2Δx noise in the solution. Copyright © 2004 John Wiley & Sons, Ltd.     

A finite element method for the two‐dimensional extended Boussinesq equations

A new numerical method for Nwogu's (ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 1993; 119 :618)two‐dimensional extended Boussinesq equations is presented using a linear triangular finite element spatial discretization coupled with a sophisticated adaptive time integration package. The authors have previously presented a finite element method for the one‐dimensional form of these equations (M. Walkley and M. Berzins (International Journal for Numerical Methods in Fluids 1999; 29 (2):143)) and this paper describes the extension of these ideas to the two‐dimensional equations and the application of the method to complex geometries using unstructured triangular grids. Computational results are presented for two standard test problems and a realistic harbour model. Copyright © 2002 John Wiley & Sons, Ltd.     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

Flow past a cylinder: shear layer instability and drag crisis

Flow past a circular cylinder for Re=100 to 10⁷ is studied numerically by solving the unsteady incompressible two‐dimensional Navier–Stokes equations via a stabilized finite element formulation. It is well known that beyond Re ～ 200 the flow develops significant three‐dimensional features. Therefore, two‐dimensional computations are expected to fall well short of predicting the flow accurately at high Re. It is fairly well accepted that the shear layer instability is primarily a two‐dimensional phenomenon. The frequency of the shear layer vortices, from the present computations, agree quite well with the Re^0.67 variation observed by other researchers from experimental measurements. The main objective of this paper is to investigate a possible relationship between the drag crisis (sudden loss of drag at Re ～ 2 × 10⁵) and the instability of the separated shear layer. As Re is increased the transition point of shear layer, beyond which it is unstable, moves upstream. At the critical Reynolds number the transition point is located very close to the point of flow separation. As a result, the shear layer eddies cause mixing of the flow in the boundary layer. This energizes the boundary layer and leads to its reattachment. The delay in flow separation is associated with narrowing of wake, increase in Reynolds shear stress near the shoulder of the cylinder and a significant reduction in the drag and base suction coefficients. The spatial and temporal power spectra for the kinetic energy of the Re=10⁶ flow are computed. As in two‐dimensional isotropic turbulence, E(k) varies as k^?5/3 for wavenumbers higher than energy injection scale and as k^?3 for lower wavenumbers. The present computations suggest that the shear layer vortices play a major role in the transition of boundary layer from laminar to turbulent state. Copyright © 2004 John Wiley & Sons, Ltd.     

Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems

This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called ‘‘time-2τ² map’’ of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T¹ (Hopf invariant circles), tori 2T¹ and tori 2T² depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms’ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.The project supported by the National Natural Science Foundation of China (10472096)The English text was polished by Ron Marshall.     

Exact Relations for Effective Tensors of Polycrystals. II. Applications to Elasticity and Piezoelectricity

An important necessary condition for an exact relation for effective moduli of polycrystals to hold is stability of that relation under lamination. This requirement is so restrictive that it is possible (if not always feasible) to find all such relations explicitly. In order to do this one needs to combine the results developed in Part I of this paper and the representation theory of the rotation groups SO(2) and SO(3). More precisely, one needs to know all rotationally invariant subspaces of the space of material moduli. This paper presents an algorithm for finding all such subspaces. We illustrate the workings of the algorithm on the examples of 3‐dimensional elasticity, where we get all the exact relations, and the examples of 2‐dimensional and 3‐dimensional piezoelectricity, where we get some (possibly all) of them. (Accepted September 24, 1997)     

On dimensionless numbers for dynamic plastic response of structural members

Summary A dimensional analysis is reported for the dynamic plastic response and failure of structural members, which includes material strain hardening, strain rate and temperature effects. Critical shear failure conditions are also discussed based on the dimensional analysis results. It is shown that the response number R _n proposed in [3], is an important independent dimensionless number for the dynamic plastic bending and membrane response of structural members. However, additional dimensionless numbers are necessary when transverse shear, strain hardening, strain rate, and temperature effects are important. Received 22 February 1999; accepted for publication 15 June 1999     

Model reduction using L1‐norm minimization as an application to nonlinear hyperbolic problems

We are interested in the model reduction techniques for hyperbolic problems, particularly in fluids. This paper, which is a continuation of an earlier paper of Abgrall et al, proposes a dictionary approach coupled with an L¹ minimization approach. We develop the method and analyze it in simplified 1‐dimensional cases. We show in this case that error bounds with the full model can be obtained provided that a suitable minimization approach is chosen. The capability of the algorithm is then shown on nonlinear scalar problems, 1‐dimensional unsteady fluid problems, and 2‐dimensional steady compressible problems. A short discussion on the cost of the method is also included in this paper.     

Ergodicity of the Infinite Dimensional Fractional Brownian Motion

In this short note we prove that an infinite dimensional fractional Brownian motion B _H of any Hurst parameter ${H \in (0, 1)}In this short note we prove that an infinite dimensional fractional Brownian motion B _H of any Hurst parameter H ? (0, 1){H \in (0, 1)} forms an ergodic metric dynamical system. For the proof we mainly use the fundamental theorems of Kolmogorov.     

Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type

Methods based on exponential finite difference approximations of h⁴ accuracy are developed to solve one and two‐dimensional convection–diffusion type differential equations with constant and variable convection coefficients. In the one‐dimensional case, the numerical scheme developed uses three points. For the two‐dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri‐diagonal systems. The methods are applied on a number of linear and non‐linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non‐uniform mesh is also illustrated. The h⁴ accuracy of the schemes is also computationally demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.     

A second order solution for the distortion of large scale turbulence around a 2-D obstacle

The problem of large scale (d/L « 1, where d is the dimension of the dimension of the body and L is the integral length scale of turbulence) three dimensional turbulence approaching an arbitrary two dimensional body is analysed following Hunt's (1973) theory. The analysis, based on Rapid Distortion Theory, incorporates both blocking and weak distortion of the velocity field caused by the obstacle. Besides obvious engineering applications, the solution illustrates the application of RDT for inhomogeneous turbulence problems.     

Psychophysiology,Cortical Arousal and Dynamical Complexity (DCX)

Recent studies in brain dynamics have utilized a dependent variable calculated from the electroencephalogram (EEG) known as dimensional complexity (DC _x ), where variables such as scalp locus, cognitive task difficulty, or cortical arousal, are manipulated to test quantitative hypotheses regarding brain-state changes. The technique has been criticised on technical and theoretical grounds, yet its application to experimental time series in many domains has succeeded in yielding information about cortical activity which either complements or surpasses spectral band analysis, and other linear-stochastic techniques. The aim of this paper is to provide a pedagogical review of the contribution of dimensional complexity studies in understanding the psychophysiology of cortical arousal by outlining strategies for the successful estimation of DC _x as an empirical measure.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical model for run‐up of breaking waves

In the present paper, the numerical method for the three‐dimensional run‐up, given in Johnsgard and Pedersen [‘A numerical model for three‐dimensional run‐up’, Int. J. Numer. Methods Fluids, 24 , 913–931 (1997)], is extended to include wave breaking. In the fundamental problem of run‐up of a uniform bore, the present model is compared with analytical solutions from the literature. The numerical solutions converge, but very slowly. This is not due to the numerical model, but rather to the structure of the solutions themselves. Numerical results for two realistic but simplified tsunami cases are also presented. In the first case, two‐dimensional simulations are performed concerning the run‐up of a tsunami in Portugal, in the second case, the three dimensional wave pattern generated after a slide in Tafjord, Norway in 1931, is studied. A discussion of different aspects of the model is summarized at the end of the paper. Copyright © 1999 John Wiley & Sons, Ltd.     

Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

In this article, we present two improved third‐order weighted essentially nonoscillatory (WENO) schemes for recovering their design‐order near first‐order critical points. The schemes are constructed in the framework of third‐order WENO‐Z scheme. Two new global smoothness indicators, τ_L3 and τ_L4, are devised by a nonlinear combination of local smoothness indicators (IS_k) and reference values (IS_G) based on Lagrangian interpolation polynomial. The performances of the proposed schemes are evaluated on several numerical tests governed by one‐dimensional linear advection equation or one‐ and two‐dimensional Euler equations. Numerical results indicate that the presented schemes provide less dissipation and higher resolution than the original WENO3‐JS and subsequent WENO3‐N scheme.     

Natural convection in a partitioned enclosure heated from below

Cubic spline collection numerical method has been developed to determine two dimensional natural convection in a partitioned enclosure heated from below. The both sides of impermeable partition are considered to have continuity in heat flux and temperatures. The governing equations are solved with aid of the SADI procedure. Parametric studies of the effects of the partition and Rayleigh number on the fluid flow and temperature fields have been performed. Results show that the location of the partition and Rayleigh number have a significant influence on the flow and heat transfer characteristics.

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Freie Konvektion in einem von unten beheizten, unterteiltem Hohlraum

Zusammenfassung Eine numerische dreidimensionale SplineMethode zur Berechnung der zweidimensionalen Naturkonvektion in einem von unten beheizten, unterteiltem Hohlraum wird vorgestellt. Der Wärmestrom und die Temperatur auf beiden Seiten der undurchlässigen Trennwand werden als konstant betrachtet. Mit Hilfe der SADI-Prozedur werden die beschreibenden Gleichungen gelöst. Über den Einfluß der Unterteilung und der Rayleigh-Zahl auf die Strömung des Fluids und das Temperaturfeld wird eine Parameter-Studie durchgeführt. Die Ergebnisse zeigen, daß die Anordnung der Unterteilung und die Rayleigh-Zahl einen entscheidenden Einfluß auf das Wärmeübertragungsverhalten haben.

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Nomenclature A aspect ratio=L/H - g gravitational acceleration - H enclosure height - H₁ distance between the top wall of enclosure and the partition - H₂ distance between the bottom wall of enclosure and the partition - k thermal conductivity of fluid - L enclosure length - m number of vertical grid lines - n number of horizontal grid lines - Nu Nusselt number - P pressure - Pr Prandtl number - Q heat transfer across enclosure - Ra Rayleigh number based onH - t time - T dimensional temperature - T _H temperature of warm horizontal wall - T _L temperature of cold horizontal wall - T ₀ average temperature=T(_H+T_L)/2 - T temperature difference between the hot and cold wall =T _H–T_L - u, U dimensional and dimensionless horizontal velocity - , V dimensional and dimensionless vertical velocity - x, X dimensional and dimensionless horizontal coordinate - y, Y dimensional and dimensionless vertical coordinate - fluid thermal diffusivity - coefficient of thermal expansion - viscosity - kinematic viscosity=/g9 - density - , dimensional and dimensionless stream function - dimensionless temperature - , dimensional and dimensionless vorticity - dimensionless time     

Numerical modelling of 3D stratified free surface flows: a case study of sediment dumping

A three‐dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds‐averaged Navier–Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ‐coordinate system with the use of operator‐splitting method (Int. J. Numer. Meth. Fluids 2002; 38 :1045–1068). The numerical model is validated against the one‐dimensional diffusion problem and the two‐dimensional density‐gradient flow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratified flow. For the two‐dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium‐size sediments. The model is also employed to study the dumping of sediments in a three‐dimensional environment with the presence of free surface. It is found that during the descending process an annulus‐like cloud is formed for fine sediments whereas a plate‐like cloud for medium‐size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright © 2005 John Wiley & Sons, Ltd.     

A physically based flux limiter for QUICK calculations of advective scalar transport

Transient, advective transport of a contaminant into a clean domain will exhibit a moving sharp front that separates contaminated and clean regions. Due to ‘numerical diffusion’—the combined effects of ‘cross‐wind diffusion’ and ‘artificial dispersion’—a numerical solution based on a first‐order (upwind) treatment will smear out the sharp front. The use of higher‐order schemes, e.g. QUICK (quadratic upwinding) reduces the smearing but can introduce non‐physical oscillations in the solution. A common approach to reduce numerical diffusion without oscillations is to use a scheme that blends low‐order and high‐order approximations of the advective transport. Typically, the blending is based on a parameter that measures the local monotonicity in the predicted scalar field. In this paper, an alternative approach is proposed for use in scalar transport problems where physical bounds C_Low?C?C_High on the scalar are known a priori. For this class of problems, the proposed scheme switches from a QUICK approximation to an upwind approximation whenever the predicted upwind nodal value falls outside of the physical range [C_Low, C_High]. On two‐dimensional steady‐state and one‐dimensional transient test problems predictions obtained with the proposed scheme are essentially indistinguishable from those obtained with monotonic flux‐limiter schemes. An analysis of the modified equation explains the observed performance of first‐ and second‐order time‐stepping schemes in predicting the advective transport of a step. In application to the transient two‐dimensional problem of contaminate transport into a streambed, predictions obtained with the proposed flux‐limiter scheme agree with those obtained with a scheme from the literature. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical simulations of the steady Navier–Stokes equations using adaptive meshing schemes

In this paper, we consider an adaptive meshing scheme for solution of the steady incompressible Navier–Stokes equations by finite element discretization. The mesh refinement and optimization are performed based on an algorithm that combines the so‐called conforming centroidal Voronoi Delaunay triangulations (CfCVDTs) and residual‐type local a posteriori error estimators. Numerical experiments in the two‐dimensional space for various examples are presented with quadratic finite elements used for the velocity field and linear finite elements for the pressure. The results show that our meshing scheme can equally distribute the errors over all elements in some optimal way and keep the triangles very well shaped as well at all levels of refinement. In addition, the convergence rates achieved are close to the best obtainable. Extension of this approach to three‐dimensional cases is also discussed and the main challenge is the efficient implementation of three‐dimensional CfCVDT generation that is still under development. Copyright © 2007 John Wiley & Sons, Ltd.