Mass Conservation Errors Research Articles

Assessment of numerical schemes for solving the advection–diffusion equation on unstructured grids: case study of the Guaíba River, Brazil

Abstract. In this work, a first-order upwind and a high-order flux-limiter schemes for solving the advection–diffusion equation on unstructured grids were evaluated. The numerical schemes were implemented as a module of an unstructured two-dimensional depth-averaged circulation model for shallow lakes (IPH-UnTRIM2D), and they were applied to the Guaíba River in Brazil. Their performances were evaluated by comparing mass conservation balance errors for two scenarios of a passive tracer released into the Guaíba River. The circulation model showed good agreement with observed data collected at four water level stations along the Guaíba River, where correlation coefficients achieved values up to 0.93. In addition, volume conservation errors were lower than 1% of the total volume of the Guaíba River. For all scenarios, the higher order flux-limiter scheme has been shown to be less diffusive than a first-order upwind scheme. Accumulated conservation mass balance errors calculated for the flux limiter reached 8%, whereas for a first-order upwind scheme, they were close to 18% over a 15-day period. Although both schemes have presented mass conservation errors, these errors are assumed negligible compared with kinetic processes such as erosion, sedimentation or decay rates.

An Eulerian–Lagrangian moving immersed interface method for simulating burning solids

This study concerns the development of a numerical methodology to conduct conjugate heat and mass transfer simulations of burning and moving solids. The flow is described using an Eulerian representation and the solid described using a Lagrangian finite element (FE) method. The fluid–solid interface is defined using a level set function that is initialized by a surface mesh representation that balances accuracy with the computational cost of re-initializing moving interfaces. A ghost–fluid methodology is implemented that results in low errors in mass and energy conservation. Mass transfer from solid ablation is computed explicitly using surface integration over the solid surface mesh to guarantee enforcement of conservation principles. The introduced methodologies are applied to the study of the burning of carbon-epoxy composites.

Flow Characteristics According to Velocity Conditions of Cylinder Boundary Under Low Reynolds Number

기존의 천수흐름 해석 상용모형에서는 내부 경계조건을 단순히 완전활동조건으로 가정하여 유체의 흐름을 해석함으로써 구조물 주위에서의 유속, 와도, 수위, 전단력의 분포, 항력 및 양력의 시간에 따른 변화 등을 올바르게 해석하지 못하였다. 본 연구에서는 구조물 주위에서의 흐름특성을 정확하게 예측할 수 있는 유한요소모형을 개발하고, 구조물에서의 경계조건을 활동길이를 이용한 부분활동조건으로 묘사하여 내부경계조건에 따른 원형 실린더 후면에서의 층류 흐름특성을 분석하였다. 종횡방향 유속 및 와도의 시간에 따른 변화, 후류길이, 활동길이에 따른 와류열의 변화와 질량보존율을 비교한 결과 완전활동조건을 부여한 경우에는 와류열이 전혀 형성되지 않고 완전한 층류흐름이 발생하였다. 부분활동조건을 입력한 경우 실린더 표면에서의 유속분포가 변화되어 전단력의 크기와 와도의 발생에 영향을 미치므로 무활조건을 부여한 경우에 비해 와류열의 발생 주기가 짧아졌다. 최대 질량보존 오차는 무활조건을 적용한 경우 0.73%로 나타났으며, 무활조건에 비해 부분활동조건을 부여한 경우의 오차율이 최대 0.21% 감소하였다. Existing conventional model for analysis of shallow water flow just assumed the internal boundary condition as free-slip, which resulted in the wrong prediction about the velocity, vorticity, water level, shear stress distribution, and time variation of drag and lift force around a structure. In this study, a finite element model that can predict flow characteristics around the structure accurately was developed and internal boundary conditions were generalized as partial slip condition using slip length concept. Laminar flow characteristics behind circular cylinder were analyzed by varying the internal boundary conditions. The simulation results of (1) time variations of longitudinal and transverse velocities, and vorticity; (2) wake length; (3) vortex shedding phenomena by slip length; (4) and mass conservation showed that the vortex shedding had never observed and laminar flow like creeping motion was occurred under free-slip condition. Assignment of partial slip condition changed the velocity distribution on the cylinder surface and influenced the magnitude of the shear stress and the occurrence of vorticity so that the period of vortex shedding was reduced compared with the case of no slip condition. The maximum mass conservation error occurred in the case of no slip condition, which had the value of 0.73%, and there was 0.21 % reduction in the maximum mass conservation error by changing the internal boundary condition from no slip to partial slip condition.

Numerical Simulation of Laminar and Turbulent Two-Phase Flow in Pressure-Swirl Atomizers

This paper has developed an axisymmetric laminar and turbulent two-phase flow solver to simulate pressure swirl atomizer. Equations include explicit algebraic Reynolds stress model, Reynolds averaged Navier Stokes and level set equation. Applying high order compact upwind finite difference scheme with the level set equation being culminated to capture the interface between air-liquid two-phase flow and decreasing the mass conservation error in the level set equation. The results show that close to swirl chamber wall and to axis, some recirculation zones are observed. Converting Rankin vortex from the swirl chamber to forced vortex in orifice section can be predicted by this model. The proposal model shows a considerable improvement in the numerical results especially in laminar flow so that the discharge coefficient, film thickness, and spray cone angle are satisfactory with the previous experimental data. In addition, the error is less than 10 percent.

Incorporating metamorphism in geodynamic models: the mass conservation problem

SUMMARY Geodynamic models incorporating metamorphic phase transformations almost invariably assume the validity of the Boussinesq approximation that violates conservation of mass. In such models metamorphic density changes take place without volumetric effects. We assess the impact of the Boussinesq approximation by comparing models of orogeny accompanied by lower crustal eclogitization both with and without the approximation. Our results demonstrate that the approximation may cause errors approaching 100 per cent in characteristic measures of orogenic shape. Mass conservation errors in Boussinesq models amplify with model time. Mass conservative models of metamorphism are therefore essential to understand long-term tectonic evolution and to assess the importance of the different geodynamic processes.

Validation of a mesoscale weather prediction model using subdomain budgets

The monitoring of conservation properties is essential for model development and for the investigation of the hydrological cycle. This is especially relevant for models that do not solve equations in flux form and do not apply a finite volume discretization. The conservation properties of the mesoscale model COSMO are evaluated by using a finite volume diagnostic approach. That is the subdomain budget of energy, water mass and total mass are diagnosed in a control volume that can be placed at each site in the model domain and is independent of the grid size. Thus, this diagnostic method has the major advantage that it can be applied to realistic simulations. The application of the diagnostic method to the COSMO model reveals a good preservation of the water mass, but large errors in energy and total mass conservation. The analysis shows to which extent errors in the treatment of thermodynamical processes, numerical filters and moisture advection schemes contaminate the subdomain budgets. In this paper we will show that the application of a saturation adjustment scheme under a fixed volume condition is required for models, which use the non-hydrostatic equations and height-based coordinates. Also, a further extension of the model physics will be introduced and discussed for a realistic test case.

Compressible Streamline-Based Simulation With Changes in Oil Composition

Summary In Brazil, there are several offshore oil fields with horizontal oil-density variations. API tracking, which is available in some commercial finite difference simulators, can deal with such cases by allowing the definition of an initial oil-gravity distribution and tracking variations of oil density owing to the movement of oil. Streamline-based simulators can be much faster than conventional finite-difference simulators when applied to large and heterogeneous models. However, this approach is most accurate and efficient when it is assumed that the rock and fluids are incompressible. In previous work, Beraldo et al. (2007) presented an incompressible formulation for streamline simulation with an API tracking option using two components in the oleic phase. This paper presents a compressible formulation for streamlines that also considers API tracking. It has an approach similar to the one presented by Cheng et al. (2006) and Osako and Datta-Gupta (2007). However, cumulative volume is used in the streamline solution, instead of time of flight, which reduces mass-conservation errors. The method was implemented in a 3D two-phase streamline-based simulator. Tests based on a Brazilian oilfield model demonstrate that the implementation can reproduce the results of a conventional simulator, even when significant liquid compressibilities are considered. Another test, using the SPE 10th Comparative Case, indicates that it can be substantially faster for finely resolved models.

J0601-3-3 フラクショナルステップ法による多孔質内流動解析スキームの開発(PEFC内の凝縮水挙動)

Solid matrix drag term usually results in a significant pressure drop across the porous medium. The general momentum conservation equation reduces to the extended Darcy's law for the flow in porous media. So Darcy resistance force is very important for the numerical simulation inside the porous media. A new implicit fractional step algorithm is used for the numerical treatment of Darcy drag term and its effect on the residual error of continuity is studied in the present investigation. Faster convergence and reduced mass conservation error is found by using this algorithm.

A comparison of mass conservation methods for air quality models

Mass inconsistency in air quality models (AQMs) can be severe particularly over complex terrain. The vertical winds adjustment is a remedy tool that can maintain the mass conservation in AQMs. We implemented this approach in the Community Multiscale Air Quality model (CMAQ) with two different vertical advection schemes. With the first-order upwind scheme, vertical winds adjustment made mass strictly conserved, while with the Bott's scheme, a higher-order scheme, there were very small errors in the total mass because of the iterative solution for adjustment. Bott's scheme results in better ozone performance than the upwind scheme. However, both vertical winds adjustment methods introduce large trajectory deviations compared to the unadjusted wind fields generated by meteorological models. We then developed a third method with mixed use of vertical winds adjustment and concentration renormalization under different conditions. This method limits the trajectory deviations, introduces some mass conservation errors but at a lesser degree than the renormalization approach.

Conservative Wetting and Drying Methodology for Quadrilateral Grid Finite-Volume Models

Algebraic equations relating fluid volume and the free surface elevation in partially wetted quadrilateral computational cells are derived and incorporated into a Godunov-type, finite-volume, shallow-water model. These equations make it straightforward to reconstruct the free surface elevation based on the volume of fluid in a computational cell, the dependent variable tracked by finite volume models for conservation purposes, regardless of whether the cell is fully or partially wetted. Improvements to the variable reconstruction process streamline the computation of mass and momentum fluxes with approximate Riemann solvers, yielding a model that simulates sub-, super-, and transcritical flows over irregular topography with wetting and drying fronts. Furthermore, the model is free from fluid and scalar mass conservation errors and it eliminates nonphysical distributions of scalars by avoiding artificial concentration and/or dilution at wet/dry interfaces. Use of this wetting and drying methodology adds roughly 10% to the execution time of flow simulations.

High-fidelity interface tracking in compressible flows: Unlimited anchored adaptive level set

The interface-capturing-fidelity issue of the level set method is addressed wholly within the Eulerian framework. Our aim is for a practical and efficient way to realize the expected benefits of grid resolution and high order schemes. Based on a combination of structured adaptive mesh refinement (SAMR), rather than quad/octrees, and on high-order spatial discretization, rather than the use of Lagrangian particles, our method is tailored to compressible flows, while it provides a potentially useful alternative to the particle level set (PLS) for incompressible flows. Interesting salient features of our method include (a) avoidance of limiting (in treating the Hamiltonian of the level set equation), (b) anchoring the level set in a manner that ensures no drift and no spurious oscillations of the zero level during PDE-reinitialization, and (c) a non-linear tagging procedure for defining the neighborhood of the interface subject to mesh refinement. Numerous computational results on a set of benchmark problems (strongly deforming, stretching and tearing interfaces) demonstrate that with this approach, implemented up to 11th order accuracy, the level set method becomes essentially free of mass conservation errors and also free of parasitic interfacial oscillations, while it is still highly efficient, and convenient for 3D parallel implementation. In addition, demonstration of performance in fully-coupled simulations is presented for multimode Rayleigh–Taylor instability (low-Mach number regime) and shock-induced, bubble-collapse (highly compressible regime).

A note on least-squares mixed finite elements in relation to standard and mixed finite elements

The least-squares mixed finite-element method for second-order elliptic problems yields an approximation u h ∈ V h C H 1 0 (Q) of the potential u together with an approximation p h ∈ r h ⊂ H(div; Q) of the vector field p = -AVu. Comparing u h with the standard finite-element approximation of u in V h , and p h with the mixed finite-element approximation of p, it turns out that they are higher-order perturbations of each other. In other words, they are 'superclose'. Refined a priori bounds and superconvergence results can now be proved. Also, the local mass conservation error is of higher order than could be concluded from the standard a priori analysis.

Improvements in mass conservation using alternative boundary implementations for a quasi-bubble finite element shallow water model

Finite element approaches generally do not guarantee exact satisfaction of conservation laws especially when Dirichlet-type boundary conditions are imposed. This article discusses improvement of the global mass conservation property of quasi-bubble finite element solutions for the shallow water equations, focusing on implementations of the surface-elevation boundary conditions. We propose two alternative implementations, which are shown by numerical verification to be effective in improving the smoothness of solutions near the boundary and in reducing the mass conservation error. The improvement of the mass conservation property contributes to augmenting the reliability and robustness of long-term time integrations. Copyright © 2006 John Wiley & Sons, Ltd.

Mass conservation in the Community Multiscale Air Quality model

Mass conservation errors were discovered when the Community Multiscale Air Quality (CMAQ) model (version 4.3) was coupled with the Fifth-Generation PSU/NCAR Mesoscale Model (MM5). These errors can be severe particularly over complex terrain. A recent correction made to the vertical advection module (in version 4.4) reduces the mass conservation errors, but does not completely eliminate them. This means that the method of renormalizing concentrations according to air density, which is currently applied to address the inconsistency problem, is not an ideal solution. Reconstruction of the wind fields by adjusting the vertical wind is proposed as a simple remedy to make CMAQ strictly mass conservative.

A Lagrangian level‐set approach for the simulation of incompressible two‐fluid flows

AbstractA Lagrangian level‐set method to solve incompressible two‐dimensional two‐fluid flows is presented. The Navier–Stokes equations are discretized by a Galerkin finite element method. A projection method is employed to decouple the system of non‐linear equations. The interface between fluids is represented by the zero level set of a function ϕ plus additional marker points of the computational mesh. In the standard Eulerian level‐set method, this function is advected through the domain by solving a pure advection equation. To reduce mass conservation errors that can arise from this step, our method employs a Lagrangian technique which moves the nodes of the finite element mesh, and consequently, the information stored in each node. The quality of the mesh is controlled by a remeshing procedure, avoiding bad triangles by flipping edges, inserting or removing vertices from the triangulation. Results of numerical simulations are presented, illustrating the improvements in mass conservation and accuracy of this new methodology. Copyright © 2005 John Wiley & Sons, Ltd.

High-order surface tension VOF-model for 3D bubble flows with high density ratio

An improved Volume of Fluid (VOF) method is presented which is applicable to high density ratio 3D flows for a large range of bubble Reynolds number ( Re). The method is based on the Navier–Stokes equations for incompressible multi-phase flows which are discretized on a Cartesian staggered grid. The multi-grid technique together with the pressure–velocity coupling scheme for multi-phase flows have resulted in an efficient solver which nearly exponentially converge with the number of iterations. The convergence speed also shows negligible dependence on density ratio, viscosity ratio and Re. A second-order accurate, non-diffusive, mass conservative phase transport model is presented which does not suffer from unphysical over- or under-shoots of the phase variable. The high accurate normal, curvature and surface tension force model in combination with the high-order defect-correction scheme for multi-phase flows shows second-order global accuracy when applied to the transient bubble rise where the viscosity ratio is equal to one. In contrast, the commonly used viscosity model for VOF introduces a first order error for the same problem. The VOF method has been tested for different types of bubble flows at low Re and for path-oscillating and wobbling air bubbles (in water) with a diameter range of 1.82< D<6 mm. The numerical results agree quantitatively with the available experimental data. The investigations show that the proposed high accurate surface tension model can be used successfully for wobbling flows with bubble deformation while maintaining the mass of the phases. The error in mass conservation is directly proportional to the residual in solving the discrete problem.

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

AbstractA wetting–drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting–drying condition based on steady‐state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting–drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications. Copyright © 2004 John Wiley &amp; Sons, Ltd.

A general paradigm to model reaction‐based biogeochemical processes in batch systems

This paper presents the development and illustration of a numerical model of reaction‐based geochemical and biochemical processes with mixed equilibrium and kinetic reactions. The objective is to provide a general paradigm for modeling reactive chemicals in batch systems, with expectations that it is applicable to reactive chemical transport problems. The unique aspects of the paradigm are to simultaneously (1) facilitate the segregation (isolation) of linearly independent kinetic reactions and thus enable the formulation and parameterization of individual rates one reaction by one reaction when linearly dependent kinetic reactions are absent, (2) enable the inclusion of virtually any type of equilibrium expressions and kinetic rates users want to specify, (3) reduce problem stiffness by eliminating all fast reactions from the set of ordinary differential equations governing the evolution of kinetic variables, (4) perform systematic operations to remove redundant fast reactions and irrelevant kinetic reactions, (5) systematically define chemical components and explicitly enforce mass conservation, (6) accomplish automation in decoupling fast reactions from slow reactions, and (7) increase the robustness of numerical integration of the governing equations with species switching schemes. None of the existing models to our knowledge has included these scopes simultaneously. This model (BIOGEOCHEM) is a general computer code to simulate biogeochemical processes in batch systems from a reaction‐based mechanistic standpoint, and is designed to be easily coupled with transport models. To make the model applicable to a wide range of problems, programmed reaction types include aqueous complexation, adsorption‐desorption, ion‐exchange, oxidation‐reduction, precipitation‐dissolution, acid‐base reactions, and microbial mediated reactions. In addition, user‐specified reaction types can be programmed into the model. Any reaction can be treated as fast/equilibrium or slow/kinetic reaction. An equilibrium reaction is modeled with an infinite rate governed by a mass action equilibrium equation or by a user‐specified algebraic equation. Programmed kinetic reaction rates include multiple Monod kinetics, nth order empirical, and elementary formulations. In addition, user‐specified rate formulations can be programmed into the model. No existing models to our knowledge offer these simultaneous features. Furthermore, most available reaction‐based models assume chemical components a priori so that reactions can be written in basic (canonical) forms and implicitly assume that fast equilibrium reactions occur only for homogeneous reactions. The decoupling of fast reactions from slow reactions lessens the stiffness typical of these systems. The explicit enforcement of mass conservation overcomes the mass conservation error due to numerical integration errors. The removal of redundant fast reactions alleviates the problem of singularity. The exclusion of irrelevant slow reactions eliminates the issue of exporting their problematic rate formulations/parameter estimations to different environment conditions. Taking the advantage of the nonuniqueness of components, a dynamic basis‐species switching strategy is employed to make the model numerically robust. Backward basis switching allows components to freely change in the simulation of the chemistry module, while being recovered for transport simulation. Three example problems were selected to demonstrate the versatility and robustness of the model.

Evaluation of Advection Schemes for Estuary Transport

The quest for a conservative, accurate, and physically realistic yet computationally efficient numerical method for advective transport continues to pervade the physical sciences. The authors are to be commended for a paper contributing to this topic of seemingly never-ending interest. The aim of this discussion is to comment on some aspects of advection modeling in the hope of providing clarification for fellow modelers. One of the advection schemes used by the authors is a nonconservative Eulerian-Lagrangian (EL) method. Not surprisingly, in their practical application the authors have found (as have other workers) that this type of method introduces significant errors into solute transport computations. The authors indicate that these errors are mass conservation errors. Are the authors aware that there is a relatively simple way of eliminating these errors to create conservative EL methods? All that is required is to cast the EL method in an appropriately constructed mass balance form. Although this may be accomplished in a number of apparently different ways, the essential ingredients are the tracking of fluid control volumes and the spatial interpolation of solute mass, a general approach for one-dimensional unsteady non-uniform advection-diffusion, as described in Manson and Wallis (1999) and earlier papers referenced therein. Note that nonconservative EL methods involve the spatial interpolation of solute concentrations in the absence of a strict mass balance. In non-uniform flows, such methods do not conserve mass. The discussers support the authors in expounding the good aspects of the QUICKEST scheme. The discussers are only surprised that it seems to have taken almost twenty years for water industry modelers to recognize the merits of the scheme. Readers should be aware, however, that there is some confusion over the nature of the scheme. It is most often thought of as a Eulerian scheme, since in the original paper (Leonard 1979) it appears as an extension of the Eulerian QUICK scheme, an extension specifically for unsteady state transport. Strictly speaking, however, the scheme is more closely allied to EL methods because in Leonard’s paper, the advective transport terms are derived in exactly the same way as is used in one method of constructing conservative EL methods, i.e., by using a control volume mass balance with Lagrangian integrals. In this, spatial interpolation of solute concentrations is used to estimate the solute fluxes at the control volume faces. However, QUICKEST is a special, i.e., restricted, EL scheme because the particular Lagrangian integrals account for material being swept downstream only from the nearest upstream cell, whereas a general EL scheme is unlimited in from how far upstream material can be tracked. This is why QUICKEST has a Courant number‐based stability limit while general EL methods do not. In particular, for 1D pure advection with QUICKEST the upper limit for the Courant number is one, but for 1D advection-diffusion the upper limit apa

Quasi-steady friction in transient polytropic flow

An accurate method of integrating quasi-steady friction in transient polytropic flows is introduced and is justified both physically and analytically. When applied to subsonic flows, it is found to be especially informative at higher Mach numbers where existing methods of integration can lead to serious errors in mass conservation. The method is used herein to assess the accuracy of widely used finite difference representations of quasi-steady friction, attention focusing principally on solutions obtained using the method of characteristics. The new method is exact in the special case of steady flows. Of the approximate methods considered, the popular approximation V t ¦ V t − Δ t ¦Δ t is found to perform well. A modification introduced by Karney and McInnis ( Journal of Hydology Engineering, ACSE, 1992, 118(7), 1014–1030 [1]) improves the correlation for downstream characteristic lines, but has the opposite effect for upstream characteristics. A simple cubic scheme, loosely related to a more complex one used successfully by Murray ( Proceedings of the International Conference on Unsteady Flow and Fluid Transients, 29 September–1 October. Balkema, Amsterdam, 1992, pp. 143–157 [2]), illustrates difficulties in higher-order schemes. The new method is not exact in transient flows, but it gives good accuracy in slowly varying flows and it is plausible in rapidly varying flows. Using it as a benchmark, the approximation V t ¦ V t − Δ t ¦Δ t is again shown to perform well. All of the schemes considered are unreliable when the flow is about to become choked within the region of integration