Pure Advection Research Articles

Compressive VOF method with skewness correction to capture sharp interfaces on arbitrary meshes

The accurate and efficient modelling of two-phase flows is at present mostly limited to structured, unskewed meshes, due to the additional topological and numerical complexity of arbitrary, unstructured meshes. Compressive VOF methods which discretize the interface advection with algebraic differencing schemes are computationally efficient and inherently applicable to arbitrary meshes. However, compressive VOF methods evidently suffer severely from numerical diffusion on meshes with topological skewness. In this paper we present a compressive VOF method using a state-of-the-art donor–acceptor advection scheme which includes novel modifications to substantially reduce numerical diffusion on arbitrary meshes without adding computational complexity. The new methodology accurately captures evolving interfaces on any arbitrary, non-overlapping mesh and conserves mass within the limits of the applied solver tolerance. A thorough validation of the presented methods is conducted, examining the pure advection of the interface indicator function as well as the application to evolving interfaces with surface tension. Crucially, the results on equidistant Cartesian and arbitrary tetrahedral meshes are shown to be comparable and accurate.

Interface control domain decomposition methods for heterogeneous problems

SUMMARYThis paper is concerned with the solution of heterogeneous problems by the interface control domain decomposition (ICDD) method, a strategy introduced for the solution of partial differential equations in computational domains partitioned into subdomains that overlap. After reformulating the original boundary value problem by introducing new additional control variables, the unknown traces of the solution at internal subdomain interfaces; the latter are determined by requiring that the (a priori) independent solutions in each subdomain undergo the minimization of a suitable cost functional.We provide an abstract formulation for coupled heterogeneous problems and a general theorem of well‐posedness for the associated ICDD problem. Then, we illustrate and validate an efficient algorithm based on the solution of the Schur‐complement system restricted solely to the interface control variables by considering two kinds of heterogeneous boundary value problems: the coupling between pure advection and advection–diffusion equations and the coupling between Stokes and Darcy equations. In the latter case, we also compare the ICDD method with a classical approach based on the Beavers–Joseph–Saffman conditions. Copyright © 2014 John Wiley & Sons, Ltd.

A linear systems approach to watershed transport simulation

AbstractModels that simulate loadings of pollutants from agricultural landscapes to surface waters often operate at time scales that are relatively coarse (e.g. daily) compared with how fast water moves in streams, suggesting a commensurate physical scale that is substantially larger than typical agricultural fields. In general, as pollutants enter water and move downstream, longitudinal dispersive effects and travel time de‐synchronization tend to cause flattening and broadening of concentration peaks—an effect with implications for potential impacts on ecological and human health, and for which adequate representation is thus important for risk assessment. In‐stream transport is often approximated in practice using numerical implementation of the one‐dimensional advection–dispersion equation (ADE), with streams discretized into linked homogeneous segments. However, when a daily time step is employed, limitations inherent in the finite difference methodology may constrain simulated dispersion in lotic waters to unrepresentative or unrealistic magnitudes. In this paper, a convolution‐based approach to surface water transport is suggested as an alternative to the ADE, for use in combination with daily input loading models. This approach offers the advantage of greater flexibility in representing longitudinal mixing by using impulse response functions (IRF) to represent inter‐segment transport. Networks of stream segments are represented using nested convolutions, implemented using forward and inverse discrete Fourier transform to simplify calculations. Enhanced representational flexibility arises from the freedom afforded the modeller in selecting each segment's IRF, which may be chosen to represent dispersive regimes ranging from pure advection (plug flow) to compete mixing, and beyond to the sort of long‐tailed mixing characterized by fractal inverse frequency power‐law scaling. The approach is explored in proof‐of‐concept exercises that make use of atrazine monitoring data sets collected over common time periods from upstream and downstream locations within the same watersheds. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.

A constrained transport scheme for MHD on unstructured static and moving meshes

Magnetic fields play an important role in many astrophysical systems and a detailed understanding of their impact on the gas dynamics requires robust numerical simulations. Here we present a new method to evolve the ideal magnetohydrodynamic (MHD) equations on unstructured static and moving meshes that preserves the magnetic field divergence-free constraint to machine precision. The method overcomes the major problems of using a cleaning scheme on the magnetic fields instead, which is non-conservative, not fully Galilean invariant, does not eliminate divergence errors completely, and may produce incorrect jumps across shocks. Our new method is a generalization of the constrained transport (CT) algorithm used to enforce the $\nabla\cdot \mathbf{B}=0$ condition on fixed Cartesian grids. Preserving $\nabla\cdot \mathbf{B}=0$ at the discretized level is necessary to maintain the orthogonality between the Lorentz force and $\mathbf{B}$. The possibility of performing CT on a moving mesh provides several advantages over static mesh methods due to the quasi-Lagrangian nature of the former (i.e., the mesh generating points move with the flow), such as making the simulation automatically adaptive and significantly reducing advection errors. Our method preserves magnetic fields and fluid quantities in pure advection exactly.

Development of a Dispersion Relation Equation - Preserving Pure Advection Scheme for Solving the Navier-Stokes Equations with/Without Free Surface

In this article we develop a dispersively more accurate pure advection scheme and apply it to properly simulate incompressible free surface flow equations. According to Clebsch velocity decomposition, the incompressible flow equations are decomposed into the equations accounting respectively for the flow potential, rotation, and dissipation. For the Euler equation, the midpoint implicit symplectic time integrator is applied to approximate the temporal derivative term. For the sake of reducing numerical dispersion error, the upwinding scheme is developed to minimize the difference between the numerical and exact dispersion relation equations for the time-dependent pure advection equation in wavenumber space.

Fractional Ensemble Average Governing Equations of Transport by Time-Space Nonstationary Stochastic Fractional Advective Velocity and Fractional Dispersion. I: Theory

AbstractIn this study, starting from a time-space nonstationary general random walk formulation, the pure advection and advection-dispersion forms of the fractional ensemble average governing equations of solute transport by time-space nonstationary stochastic flow fields were developed. In the case of the purely advective fractional ensemble average equation of transport, the advection coefficient is a fractional ensemble average advective flow velocity in fractional time and space that is dependent on both space and time. As such, in this case, the time-space nonstationarity of the stochastic advective flow velocity is directly reflected in terms of its mean behavior in the fractional ensemble average transport equation. In fact, the derived purely advective form represents the Lagrangian derivation of the ensemble average mass conservation equation for solute transport in fractional time-space. In the case of the fractional ensemble average advection-dispersion transport equation, the moment and cumula...

GPGPU implementation of the BFECC algorithm for pure advection equations

In the present work an implementation of the Back and Forth Error Compensation and Correction (BFECC) algorithm specially suited for running on General-Purpose Graphics Processing Units (GPGPUs) th...

Different particle-accumulation structures arising from particle–boundary interactions in a liquid bridge

The formation of particle-accumulation structures in the flow in a cylindrical liquid bridge driven by the thermocapillary effect is studied. The problem is modeled as an incompressible fluid seeded with a low concentration of small spherical particles which are assumed to have a negligible effect on the flow. The particle motion is modeled by pure advection except for small regions at the free surface and walls where the particle interaction with the free surface and walls is modeled by a hard-wall potential which restricts the motion of the particle within some interaction length from the boundaries. The model yields a wide range of particle-accumulation structures when varying the relative interaction length from 5×10-3 to 3×10-2. It is found that in many cases each particle need undergo no more than a handful of returns to the free surface for particle-accumulation structures to arise. It is also found that the shape of the particle-accumulation structures changes qualitatively at certain critical interaction lengths.

Local Discontinuous Galerkin methods for fractional diffusion equations

We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems in one space dimension, characterized by having fractional derivatives, parameterized by beta in [1,2]. After demonstrating that a classic approach fails to deliver optimal order of convergence, we introduce a modified local numerical flux which exhibits optimal order of convergence O(h^(k+1)) uniformly across the continuous range between pure advection (beta=1) and pure diffusion (beta=2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.

On the application of SUPG/θ-method in 2D advection–diffusion-reaction simulation

The SUPG/θ-method for applications in 2D advection–diffusion transport simulation is reviewed and it is presented in details. Spatial discretization is done with finite elements method with stabilized streamline upwind Petrov–Galerkin method, and time discretization was done with θ-stable finite difference operator. Numerical tests showed that this algorithm is general and robust and it can be applied to simulate simple problems of mass transport; for example, pure advection phenomenon in structured mesh, as well as reactive advection–diffusion transport with an unstructured mesh, and not well-behaved velocity field. The method was applied in a simulation of the transport of total phosphorus in the Lake Agua Preta, and the results were validated by comparison with experimental data. The computational cost is directly associated with the value of the safety factor, introduced to assure the stability of the method.

Travel time distributions under convergent radial flow in heterogeneous formations: Insight from the analytical solution of a stratified model

We analyze conservative solute transport under convergent flow to a well in perfectly stratified porous media, in which the hydraulic conductivity is treated as a random spatial function along the vertical direction (K(z)). The stratified model provides a rare exception of an exact analytical solution of travel time distributions in the proximity of pumping wells, and it is used here to obtain insights about ergodic and nonergodic transport conditions under nonuniform flow conditions. In addition, it provides a benchmark for numerical models aiming to correctly reproduce convergent flow transport in heterogeneous media, such as indicating the minimum number of layers required to obtain ergodic travel time distributions using only one model realization. The model provides important insights about the shape of the depth-integrated concentrations over time measured at the well (breakthrough curves, BTCs), which are usually applied to obtain transport parameters of the subsurface. It can be applied to any degree of system’s heterogeneity and using either resident or flux-weighted injection modes. It can be built using different probabilistic distributions of K. In our analysis, we consider a log-normal K distribution, and the results indicate that, especially for highly heterogeneous systems, described by the log-K variance (σY2), the minimum number of layers required for from one model simulation to reproduce ergodic travel time distributions can be prohibitively high, e.g., above 106 for σY2=8 considering flux-weighted injections. This issue poses serious concerns for numerical applications aiming to simulate transport in the proximity of pumping wells. In addition, this simple solution confirms that stratification can lead BTCs to display strong preferential flow and persistent, power-law-like late-time tailing. Since the latter are common phenomenological macroscale evidences of other microscale hydrodynamic processes than pure advection (e.g., mass-transfer), caution must be taken when inferring aquifer properties controlling the anomalous transport dynamics in heterogeneous media from BTCs fitting.

Energy stable flux reconstruction schemes for advection–diffusion problems on triangles

The Flux Reconstruction (FR) approach unifies several well-known high-order schemes for unstructured grids, including a collocation-based nodal discontinuous Galerkin (DG) method and all types of Spectral Difference (SD) methods, at least for linear problems. The FR approach also allows for the formulation of new families of schemes. Of particular interest are the energy stable FR schemes, also referred to as the Vincent–Castonguay–Jameson–Huynh (VCJH) schemes, which are an infinite family of high-order schemes parameterized by a single scalar. VCJH schemes are of practical importance because they provide a stable formulation on triangular elements which are often required for numerical simulations over complex geometries. In particular, VCJH schemes are provably stable for linear advection problems on triangles, and include the collocation-based nodal DG scheme on triangles as a special case. Furthermore, certain VCJH schemes have Courant–Friedrichs–Lewy (CFL) limits which are approximately twice those of the collocation-based nodal DG scheme. Thus far, these schemes have been analyzed primarily in the context of pure advection problems on triangles. For the first time, this paper constructs VCJH schemes for advection–diffusion problems on triangles, and proves the stability of these schemes for linear advection–diffusion problems for all orders of accuracy. In addition, this paper uses numerical experiments on triangular grids to verify the stability and accuracy of VCJH schemes for linear advection–diffusion problems and the nonlinear Navier–Stokes equations.

Stabilized finite element discretization applied to an operator-splitting method of population balance equations

An operator-splitting method is applied to transform the population balance equation into two subproblems: a transient transport problem with pure advection and a time-dependent convection–diffusion problem. For discretizing the two subproblems the discontinuous Galerkin method and the streamline upwind Petrov–Galerkin method combined with a backward Euler scheme in time are considered. Standard energy arguments lead to error estimates with a lower bound on the time step length. The stabilization vanishes in the time-continuous limit case. For this reason, we follow a new technique proposed by John and Novo for transient convection–diffusion–reaction equations and extend it to the case of population balance equations. We also compare numerically the streamline upwind Petrov–Galerkin method and the local projection stabilization method.

A Comparison of Closures for Stochastic Advection-Diffusion Equations

Perturbation-based moment equations for advection-diffusion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to the same equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches, using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, and prove that the resulting moment equations are in fact hyperbolic for the special case of pure advection b...

Dynamics of mixing and bimolecular reaction kinetics in aquifers

Descriptions of chemical transformation kinetics and hydrologic transport need to be coupled to understand the composition of flowing waters. The coupling of a bimolecular transformation reaction (reactant 1 + reactant 2 → product; rate \( r = \kappa c_{1} c_{2} \)) with spatially heterogeneous subsurface flows is addressed here. The flow microstructure—that controls the spreading rate of solutes and the mean reactant concentrations (C 1, C 2)—creates concentration microstructure whose intensity is characterized by the variances \( \sigma_{{c_{1} }}^{2} \), \( \sigma_{{c_{2} }}^{2} \), and the cross-covariance \( \overline{{c_{1} c_{2} }} \). In addition to the macroscopic overlap of the reactants, that is quantified by the product of the mean concentrations that are routinely modeled using effective dispersion coefficients \( D_{ij} \), the concentration microstructure plays an important role in determining reaction macro-kinetics as the mean reaction rate is \( \overline{r} = \kappa (C_{1} C_{2} + \overline{{c_{1} c_{2} }} ) \). For initially non-overlapping reactants \( \overline{{c_{1} c_{2} }} + C_{1} C_{2} = 0 \) and \( \overline{r} = 0 \) under pure advection. It is shown that due to the action of local dispersion, at large time, the \( \overline{{c_{1} c_{2} }} \) budget is characterized by a balance between its rate of production and dissipation, which results in \( \overline{{c_{1} c_{2} }} \approx 2\tau D_{ij} ({{\partial C_{1} } \mathord{\left/ {\vphantom {{\partial C_{1} } {\partial x_{i} )({{\partial C_{2} } \mathord{\left/ {\vphantom {{\partial C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}} \right. \kern-0pt} {\partial x_{i} )({{\partial C_{2} } \mathord{\left/ {\vphantom {{\partial C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}} \), where τ is the dissipation time-scale characteristic to the destruction of concentration fluctuation by local dispersion. This results in \( \overline{r} = \kappa_{\text{eff}} C_{1} C_{2} \), where \( \kappa_{\text{eff}} \approx \kappa [1 + 2\tau D_{ij} ({{\partial { \ln }C_{1} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{1} } {\partial x_{i} )({{\partial { \ln }C_{2} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}} \right. \kern-0pt} {\partial x_{i} )({{\partial { \ln }C_{2} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}] \), which accounts for the influence of concentration microstructure and small-scale mixing on the macroscopic bimolecular kinetics. The effective rate parameter κeff is greater than the intrinsic rate constant κ measured under well-mixed conditions if the macroscopic concentration gradients have the same sign (initially overlapping reactants). For the initially non-overlapping reactants which result in macroscopic gradients having opposite signs, κ eff < κ. The macroscopic reactant concentration gradients, effective dispersion coefficients, and the dissipation time-scale control the reaction macro-kinetics, in addition to the intrinsic rate constant κ and the mean reactant concentrations. The formulation for reaction macro-kinetics developed here helps explain previously reported disparities between laboratory and field-scale transformation rates and also provides a way to represent the influence of reactant concentration microstructure in large-scale descriptions of reactive transport.

High-Order Preserving Residual Distribution Schemes for Advection-Diffusion Scalar Problems on Arbitrary Grids

This paper deals with the construction of a class of high-order accurate residual distribution schemes for advection-diffusion problems using conformal meshes. The problems considered range from pure diffusion to pure advection. The approximation of the solution is obtained using standard Lagrangian finite elements and the total residual of the problem is constructed taking into account both the advective and the diffusive terms in order to discretize with the same scheme both parts of the governing equation. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is reconstructed at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. Linear and nonlinear schemes are constructed and their accuracy is tested with the discretization of advection-diffusion and anisotropic diffusion problems.

Evaluating the performance of the two-phase flow solver interFoam

The performance of the open source multiphase flow solver, interFoam, is evaluated in this work. The solver is based on a modified volume of fluid (VoF) approach, which incorporates an interfacial compression flux term to mitigate the effects of numerical smearing of the interface. It forms a part of the C + + libraries and utilities of OpenFOAM and is gaining popularity in the multiphase flow research community. However, to the best of our knowledge, the evaluation of this solver is confined to the validation tests of specific interest to the users of the code and the extent of its applicability to a wide range of multiphase flow situations remains to be explored. In this work, we have performed a thorough investigation of the solver performance using a variety of verification and validation test cases, which include (i) verification tests for pure advection (kinematics), (ii) dynamics in the high Weber number limit and (iii) dynamics of surface tension-dominated flows. With respect to (i), the kinematics tests show that the performance of interFoam is generally comparable with the recent algebraic VoF algorithms; however, it is noticeably worse than the geometric reconstruction schemes. For (ii), the simulations of inertia-dominated flows with large density ratios yielded excellent agreement with analytical and experimental results. In regime (iii), where surface tension is important, consistency of pressure–surface tension formulation and accuracy of curvature are important, as established by Francois et al (2006 J. Comput. Phys. 213 141–73). Several verification tests were performed along these lines and the main findings are: (a) the algorithm of interFoam ensures a consistent formulation of pressure and surface tension; (b) the curvatures computed by the solver converge to a value slightly (10%) different from the analytical value and a scope for improvement exists in this respect. To reduce the disruptive effects of spurious currents, we followed the analysis of Galusinski and Vigneaux (2008 J. Comput. Phys. 227 6140–64) and arrived at the following criterion for stable capillary simulations for interFoam: where . Finally, some capillary flows relevant to atomization were simulated, resulting in good agreement with the results from the literature.

An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.

Numerical Simulation of Free Surface by an Area-Preserving Level Set Method

AbstractWe apply in this study an area preserving level set method to simulate gas/water interface flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the fifth-order accurate upwinding combined compact difference (UCCD) scheme. This scheme development employs two coupled equations to calculate the first- and second-order derivative terms in the momentum equations. For accurately predicting the level set value, the interface tracking scheme is also developed to minimize phase error of the first-order derivative term shown in the pure advection equation. For the purpose of retaining the long-term accurate Hamiltonian in the advection equation for the level set function, the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. Also, to keep as a distance function for ensuring the front having a finite thickness for all time, the re-initialization equation is used. For the verification of the optimized UCCD scheme for the pure advection equation, two benchmark problems have been chosen to investigate in this study. The level set method with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break, Rayleigh-Taylor instability, two-bubble rising in water, and droplet falling problems.

Transient Schemes for Capturing Interfaces of Free-Surface Flows

This article presents a new methodology for the development of Transient Interpolation for Capturing of Surfaces schemes suitable for the simulation of free-surface flows, which is given the acronym TICS. The newly developed approach is based on a switching strategy that combines a bounded high-order transient scheme with a bounded compressive transient scheme. Bounded high-order and compressive transient schemes are constructed by discretizing the transient term in the volume-of-fluid (r) equation over a temporal control-volume in a way similar to the discretization of the convection term over a spatial control-volume, allowing advances in building convective schemes to be exploited in the development of bounded high-order and compressive transient schemes. Following that approach, a bounded version of the second-order upwind Euler scheme is constructed (B-SOUE). The B-SOUE is used to develop a family of temporal compressive schemes that is denoted by the B-CEm family, where “m” refers to the slope of the scheme on a temporal normalized variable diagram. The TICS methodology is then applied to the B-SOUE scheme and the B-CEm family of schemes to create a new family of transient interface-capturing schemes that is designated by TICSm. The virtues of the TICSm family, in producing a steep interface for the volume-of-fluid (r) field that defines the volume fraction occupied by the different fluids in a computational domain, are demonstrated through results generated using two schemes of the family (TICS1.75 and TICS2.5). The accuracy of the new transient TICS schemes is compared to the first-order Euler scheme, the Crank-Nicolson scheme, and the B-SOUE scheme by solving four pure advection test problems (advection of hollow shapes in an oblique flow field and advection of a solid body in a rotational flow field) and one flow problem (the break of a dam) using both the SMART and the STACS convective schemes. Results, displayed in the form of interface contours, demonstrate that predictions obtained with TICS1.75 and TICS2.5 are far more accurate and less diffusive, preserving interface sharpness and boundedness at all Courant number values considered.