Mass Conservative Scheme Research Articles

Numerical simulation of 3D sloshing flow in partially filled LNG tank using a coupled level-set and volume-of-fluid method

The motion induced 3D sloshing flow in partially filled LNG tank was investigated using a new coupled Level-Set and Volume-of-Fluid (CLSVOF) method incorporated into Finite-Analytic Navier–Stokes (FANS) method. In addition to accurately simulating the sloshing pattern inside LNG tank, impact pressures acting upon tank wall were also predicted by the CLSVOF method. Furthermore, the predicted impact pressures were compared with the corresponding experimental data and another numerical data by the pure level set method with global mass conservation scheme. The comparisons showed the CLSVOF method is capable of predicting the accurate sloshing induced impact and capturing the violent sloshing flow including wave breaking, jet formation, gas entrapping and liquid-gas interactions. The CLSVOF method also demonstrated the prominent feature in local and stable mass conservation.

A Lattice Boltzmann Method for the Advection-Diffusion Equation with Neumann Boundary Conditions

AbstractIn this paper, we study a lattice Boltzmann method for the advection-diffusion equation with Neumann boundary conditions on general boundaries. A novel mass conservative scheme is introduced for implementing such boundary conditions, and is analyzed both theoretically and numerically.Second order convergence is predicted by the theoretical analysis, and numerical investigations show that the convergence is at or close to the predicted rate. The numerical investigations include time-dependent problems and a steady-state diffusion problem for computation of effective diffusion coefficients.

Mixing properties of SLICE and other mass-conservative semi-Lagrangian schemes

Recently, new test cases and associated measures have been proposed in the literature to evaluate the mixing and filament preservation properties of transport schemes on the sphere. In this Note, we evaluate the performance of SLICE and a semi-Lagrangian scheme with a new conservative correction using these tests and diagnostics. The results show that, based on these measures, both schemes considered here perform well and produce results comparable to some results in the literature. However, SLICE outperforms other schemes for some measures.

A mass-conserving and multi-tracer efficient transport scheme in the online integrated Enviro-HIRLAM model

Abstract. In this paper a new advection scheme for the online coupled chemical–weather prediction model Enviro-HIRLAM is presented. The new scheme is based on the locally mass-conserving semi-Lagrangian method (LMCSL), where the original two-dimensional scheme has been extended to a fully three-dimensional version. This means that the three-dimensional semi-implicit semi-Lagrangian scheme which is currently used in Enviro-HIRLAM is largely unchanged. The HIRLAM model is a computationally efficient hydrostatic operational short-term numerical weather prediction model, which is used as the base for the online integrated Enviro-HIRLAM. The new scheme is shown to be efficient, mass conserving, and shape preserving, while only requiring minor alterations to the original code. It still retains the stability at long time steps, which the semi-Lagrangian schemes are known for, while handling the emissions of chemical species accurately. Several mass-conserving filters have been tested to assess the optimal balance of accuracy vs. efficiency.

A semi‐implicit numerical model for urban drainage systems

SUMMARYIn this paper, a semi‐implicit numerical model for one‐dimensional urban drainage networks is formulated in such a fashion as to intrinsically account for arbitrary cross sections, for the occurrence of dry areas, for free surface, and for pressurized flows. The governing differential equations are discretized with a consistent mass conservative scheme that naturally applies to all flow regimes. The resulting mildly nonlinear system, at every time step, is efficiently solved with a converging, properly devised, nested Newton‐type algorithm. It will be shown that with the proposed semi‐implicit model, high accuracy can be achieved at a moderate computational cost. Copyright © 2013 John Wiley & Sons, Ltd.

A mixed finite element discretization scheme for a concrete carbonation model with concentration-dependent porosity

We investigate a prototypical reaction–diffusion-flow problem in saturated/unsaturated porous media. The special features of our problem are twofold: the reaction produces water and therefore the flow and transport are coupled in both directions and moreover, the reaction may alter the microstructure. This means we have a variable porosity in our model. For the spatial discretization we propose a mass conservative scheme based on the mixed finite element method (MFEM). The scheme is semi-implicit in time. Error estimates are obtained for some particular cases. We apply our finite element methodology for the case of concrete carbonation—one of the most important physico-chemical processes affecting the durability of concrete.

A well-balanced finite volume scheme for 1D hemodynamic simulations

We are interested in simulating blood flow in arteries with variable elasticity with a one dimensional model. We present a well-balanced finite volume scheme based on the recent developments in shallow water equations context. We thus get a mass conservative scheme which also preserves equilibria of Q=0. This numerical method is tested on analytical tests.

Improving estimated soil moisture fields through assimilation of AMSR-E soil moisture retrievals with an ensemble Kalman filter and a mass conservation constraint

Abstract. Model simulated soil moisture fields are often biased due to errors in input parameters and deficiencies in model physics. Satellite derived soil moisture estimates, if retrieved appropriately, represent the spatial mean of near surface soil moisture in a footprint area, and can be used to reduce bias of model estimates (at locations near the surface) through data assimilation techniques. While assimilating the retrievals can reduce bias, it can also destroy the mass balance enforced by the model governing equation because water is removed from or added to the soil by the assimilation algorithm. In addition, studies have shown that assimilation of surface observations can adversely impact soil moisture estimates in the lower soil layers due to imperfect model physics, even though the bias near the surface is decreased. In this study, an ensemble Kalman filter (EnKF) with a mass conservation updating scheme was developed to assimilate Advanced Microwave Scanning Radiometer (AMSR-E) soil moisture retrievals, as they are without any scaling or pre-processing, to improve the estimated soil moisture fields by the Noah land surface model. Assimilation results using the conventional and the mass conservation updating scheme in the Little Washita watershed of Oklahoma showed that, while both updating schemes reduced the bias in the shallow root zone, the mass conservation scheme provided better estimates in the deeper profile. The mass conservation scheme also yielded physically consistent estimates of fluxes and maintained the water budget. Impacts of model physics on the assimilation results are discussed.

Mixed finite element discretization and Newton iteration for a reactive contaminant transport model with nonequilibrium sorption: convergence analysis and error estimates

We present a numerical scheme for reactive contaminant transport with nonequilibrium sorption in porous media. The mass conservative scheme is based on Euler implicit, mixed finite elements, and Newton method. We consider the case of a Freundlich-type sorption. In this case, the sorption isotherm is not Lipschitz but just Holder continuous. To deal with this, we perform a regularization step. The convergence of the scheme is analyzed. An explicit order of convergence depending only on the regularization parameter, the time step, and the mesh size is derived. We give also a sufficient condition for the quadratic convergence of the Newton method. Finally, relevant numerical results are presented.

Newton method for reactive solute transport with equilibrium sorption in porous media

We present a mass conservative numerical scheme for reactive solute transport in porous media. The transport is modeled by a convection–diffusion–reaction equation, including equilibrium sorption. The scheme is based on the mixed finite element method (MFEM), more precisely the lowest-order Raviart–Thomas elements and one-step Euler implicit. The underlying fluid flow is described by the Richards equation, a possibly degenerate parabolic equation, which is also discretized by MFEM. This work is a continuation of Radu et al. (2008) and Radu et al. (2009) [1,2] where the algorithmic aspects of the scheme and the analysis of the discretization method are presented, respectively. Here we consider the Newton method for solving the fully discrete nonlinear systems arising on each time step after discretization. The convergence of the scheme is analyzed. In the case when the solute undergoes equilibrium sorption (of Freundlich type), the problem becomes degenerate and a regularization step is necessary. We derive sufficient conditions for the quadratic convergence of the Newton scheme.

Improving the Numerical Solution of Soil Moisture–Based Richards Equation for Land Models with a Deep or Shallow Water Table

Abstract The soil moisture–based Richards equation is widely used in land models for weather and climate studies, but its numerical solution using the mass-conservative scheme in the Community Land Model is found to be deficient when the water table is within the model domain. Furthermore, these deficiencies cannot be reduced by using a smaller grid spacing. The numerical errors are much smaller when the water table is below the model domain. These deficiencies were overlooked in the past, most likely because of the more dominant influence of the free drainage bottom boundary condition used by many land models. They are fixed here by explicitly subtracting the hydrostatic equilibrium soil moisture distribution from the Richards equation. This equilibrium distribution can be derived at each time step from a constant hydraulic (i.e., capillary plus gravitational) potential above the water table, representing a steady-state solution of the Richards equation. Furthermore, because the free drainage condition has serious deficiencies, a new bottom boundary condition based on the equilibrium soil moisture distribution at each time step is proposed that also provides an effective and direct coupling between groundwater and surface water.

Mass Conservative Transport Scheme for the Application of the ELCIRC Model to Water Quality Computation

An attempt was made to couple the water quality model of Danshuei River to the three-dimensional unstructured-grid hydrodynamic model [Eulerian–Lagrangian circulation model (ELCIRC)]. The Eulerian–Lagrangian scheme for the solution of the transport equations of salt in ELCIRC was demonstrated to be not mass conservative. The scheme was replaced with a finite-volume/finite-difference upwind scheme to ensure mass conservation both locally and globally. The same scheme was also used for the scalar transport equation in the water quality model. The representation of mass flux in the scalar transport equation is carefully formulated to be consistent with that of volume flux used in the continuity equations of ELCIRC. It was demonstrated that the newly revised scheme (1) conserved mass locally and globally; (2) conserved mass for both conservative and nonconservative substances subjected to biogeochemical transformation; and (3) preserved the integrity of the wetting-and-drying scheme. Further, the baroclinic s...

Numerical simulation of Richards equation in partially saturated porous media: under-relaxation and mass balance

Numerical simulation of Richards equation in unsaturated soil is known to be difficult because of the highly non-linear material properties involved. An extensive study was conducted to clarify the role of mass balance and under-relaxation on the rate in which the correct solution is approached. Both finite element and finite difference techniques were considered. For the former, the h-based formulation was compared with a mass-conservative mixed form. The conductivity function (K) was under-relaxed in two ways while the capacity function (C) is computed following the standard mass or non-mass conservative schemes recommended in literature. For fairly coarse discretisation, it was found that large errors were produced when K was under-relaxed by evaluating it at the average of heads from current and previous time step (UR1), regardless of the numerical scheme used. Maintaining global mass balance is found to have little impact on the accuracy. All numerical schemes that under-relaxed K by computing it at the average of two most recent iterations in the current time step (UR2) converged quicker to the correct solution with increasing discretisation, although more iterations per time step than UR1 is needed to achieve a stable solution. An important practical ramification is that it appears to be possible to achieve reasonably accurate and oscillation-free results using fairly coarse discretisation by making only minor modifications (namely, using UR2 for conductivity function) to the h-based finite element formulation and applying some minimum time step criteria.

Fundamental parameters and granulation properties of Alpha Centauri A and B obtained from inversions of their spectra

Properties of stellar granulation are obtained by inverting spectra of the late-type stars α Centauri A and B. Our inversions are based on a multi-component model of the stellar photosphere and take into account the center-to-limb variation and rotational broadening. The different atmospheric components describe the areas harboring up-, down- and horizontal flows. The inversions are constrained by fitting not only the flux profiles, but also their line bisectors, and by using a simple mass conservation scheme. The inversions return the properties of convection at the stellar surface, including the stratification of the thermodynamic parameters, as well as fundamental parameters such as the gravitational acceleration, v sini and the element abundances. For α Cen A (G2V) the derived stratifications of the temperature and convective velocity are very similar to the Sun, while for α Cen B (K1V) we find similar up- and downflow velocities, but lower horizontal speeds and a reduced overshoot. The latter is consistent with the smaller scale height of the atmosphere, while mass conservation arguments taken with the lower horizontal speed imply that the granules on α Cen B are smaller than on the Sun. Both these properties are in good agreement with the hydrodynamic simulation of Nordlund & Dravins (1990, A&A, 228, 155). The inversions also return the fundamental parameters (Teff ,l ogg, abundances, v sini, etc.) of the two stars. These values are on the whole in good agreement with literature values. Also, most of them do not strongly depend on the details of the inversion. However, importantly, the element abundances are 0.1 to 0.15 dex lower when a 2- or 3-component inversion is carried out than with a 1-component inversion.

Arctic Tropospheric Winds Derived from TOVS Satellite Retrievals

Abstract Accurate three-dimensional wind fields are essential for diagnosing a variety of important climate processes in the Arctic, such as the advection and deposition of heat and moisture, changes in circulation features, and transport of trace constituents. In light of recent studies revealing significant biases in upper-level winds over the Arctic Ocean from reanalyses, new daily wind fields are generated from 22.5 yr of satellite-retrieved thermal-wind profiles, corrected with a recently developed mass-conservation scheme. Compared to wind measurements from rawinsondes during the Surface Heat Budget of the Arctic (SHEBA) experiment, biases in satellite-retrieved winds are near zero in the meridional direction, versus biases of over 50% for reanalyses. Errors in the zonal component are smaller than those observed in reanalysis winds in the upper troposphere, while in the lower troposphere the effects of Greenland introduce uncertainty in the mass-conservation calculation. Further reduction in error may be achieved by incorporating winds retrieved from feature-tracking techniques using satellite imagers. Overall, satellite-retrieved winds are superior to reanalysis products over the data-sparse Arctic Ocean and provide increased accuracy for analyses requiring wind information. Trends and anomalies for the 22.5-yr record are calculated for both meridional and zonal winds at eight levels between the surface and 300 hPa. Annual mean trends are similar at varying levels, reflecting the relatively barotropic nature of the Arctic troposphere. Zonal winds are more westerly over Eurasia and the western Arctic Ocean, while westerlies have weakened over northern Canada. Combined with the corresponding pattern in meridional winds, these results suggest that the polar vortex has, on average, shifted toward northern Canada. Seasonal trends show that some changes persist throughout the year while others vary in magnitude and sign. Most striking are spring patterns, which differ markedly from the other seasons. Changes in meridional winds are consistent with observed trends in melt-onset date and sea ice concentration in the marginal seas. Anomalies in zonal wind profiles exhibit decadal-scale cyclicity in the eastern Arctic Ocean, while overall shifts in anomaly signs are evident and vary by region. The winter North Atlantic Oscillation (NAO) index correlates moderately with meridional wind anomalies in the Atlantic sector of the Arctic Ocean: positively (0.48) in the Barents Sea and negatively (−0.59) in the Lincoln Sea. These observed trends and anomalies are expected to translate to changes in advected heat and moisture into the Arctic basin, which are likely linked to trends in sea ice extent, melt onset, cloud properties, and surface temperature.

A mass conservative scheme for simulating shallow flows over variable topographies using unstructured grid

Most available numerical methods face problems, in the presence of variable topographies, due to the imbalance between the source and flux terms. Treatments for this problem generally work well for structured grids, but most of them are not directly applicable for unstructured grids. On the other hand, despite of their good performance for discontinuous flows, most available numerical schemes (such as HLL flux and ENO schemes) induce a high level of numerical diffusion in simulating recirculating flows. A numerical method for simulating shallow recirculating flows over a variable topography on unstructured grids is presented. This mass conservative approach can simulate different flow conditions including recirculating, transcritical and discontinuous flows over variable topographies without upwinding of source terms and with a low level of numerical diffusion. Different numerical tests cases are presented to show the performance of the scheme for some challenging problems.

Development of a technique for modelling clay liner desiccation

AbstractThis paper presents a model for the analysis of clay liner desiccation in a landfill barrier system due to temperature effects. The model incorporates consideration of fully coupled heat‐moisture‐air flow, a non‐linear constitutive relationship, the dependence of void ratio and volumetric water content on stress, capillary pressure and temperature, and the effect of mechanical deformation on all governing equations. Mass conservative numerical schemes are proposed to improve the accuracy of the finite element solution to the governing equations. The application of the model is then demonstrated by examining three test problems, including isothermal infiltration, heat conduction and non‐isothermal water and heat transport. Comparisons are made with results from literature, and good agreement is observed. Copyright © 2003 John Wiley & Sons, Ltd.

A cascadic conjugate gradient algorithm for mass conservative, semi‐implicit discretization of the shallow water equations on locally refined structured grids

AbstractA semi‐implicit, mass conservative discretization scheme is applied to the two‐dimensional shallow water equations on a hierarchy of structured, locally refined Cartesian grids. Different resolution grids are fully interacting and the discrete Helmholtz equation obtained from the semi‐implicit discretization is solved by the cascadic conjugate gradient method. A flux correction is applied at the interface between the coarser and finer discretization grids, so as to ensure discrete mass conservation, along with symmetry and diagonal dominance of the resulting matrix. Two‐dimensional idealized simulations are presented, showing the accuracy and the efficiency of the resulting method. Copyright © 2002 John Wiley & Sons, Ltd.

Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow

We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.

A mass conservative flow field visualization method

This paper introduces a new technique for computer visualization of three-dimensional flow fields. The most powerful feature of this technique is that the streamlines and stream surface are generated by mass conservative interpolation schemes. Interpolation is an important topic in flow visualization because CFD velocity fields are defined at a discrete location in space. Interpolation errors are more significant than those arising from numerical integration. The main draw-back of conventional trilinear interpolation of velocity is that it is not mass conservative. Failure to conserve mass can produce errors which can not be eliminated by reducing the integration step. A significant feature of the relationship between the velocity field and the stream functions is that it implies conservation of mass. So a mass conservative interpolation scheme is developed using a stream function, which is obtained by solving the partial differential equation in the local cell and approximated by a cluster of stream surfaces. Then the streamline can be traced using numerical techniques with mass conservative interpolation and the stream surface is directly calculated by slicing the stream function. The result is more accurate because we replace the polygoned tiling of streamlines by mass conservative stream surface generation. Results presented here compare the performance of the new method to the trilinear interpolation scheme and demonstrate its effectiveness.