Numerical Diffusion Research Articles

A simple FORCE-type centred scheme accurate for contact discontinuities: Application to compressible Euler flows

The FORCE-type centred schemes are simple and efficient and do not explicitly require the wave propagation information of the system to calculate the numerical flux. However, their poor resolution for contact discontinuities seriously affects their applications. In the present work, a simple FORCE-type centred scheme accurate for contact discontinuities is proposed and applied to calculations of compressible Euler flows. The missing contact wave of the original FORCE centred scheme is restored with an algebraic method and the resolution for contact discontinuities is further improved by using the boundary variation diminishing (BVD) algorithm to minimize the density jump in the numerical diffusion term of the original FORCE centred scheme. Numerical results of several one- and two-dimensional benchmark test problems fully demonstrate that the proposed centred scheme is capable of capturing contact discontinuities more sharply than the complete-wave HLLC upwind scheme. In addition, another advantage of the proposed centred scheme is that it is free from the carbuncle phenomenon which afflicts many contact-capturing upwind schemes (e.g. Roe and HLLC) in calculations of multidimensional flow problems involving strong shock waves.

A Method to Solve Hamilton–Jacobi Type Equation on Unstructured Meshes

A new method is developed to approximate a first-order Hamilton–Jacobi equation. The constant motion of an interface in the normal direction is of interest. The interface is captured with the help of a “Level-Set” function approximated through a finite-volume Godunov-type scheme. Contrarily to most computational approaches that consider smooth Level-Set functions, the present one considers sharp “Level-Set”, the numerical diffusion being controlled with the help of the Overbee limiter (Chiapolino et al. in J Comput Phys 340:389–417, 2017). The method requires gradient computation that is addressed through the least squares approximation. Multidimensional results on fixed unstructured meshes are provided and checked against analytical solutions. Geometrical properties such as interfacial area and volume computation are addressed as well. Results show excellent agreement with the exact solutions.

Development and evaluation of a new particle tracking solver for hydrodynamic and mass transport characterization of porous media – A case study on periodic open cellular structures

The detailed understanding of local mass transport characteristics in porous catalyst supports is essential for the evaluation of their performance in reaction engineering applications. This applies particularly for the development and assessment of novel concepts such as additively manufactured periodic open cellular structures (POCS). A major drawback of the frequently used Eulerian based mass transport simulation methods is their tendency to overestimate molecular diffusion due to artificial numerical diffusion, especially in the range of high Péclet numbers (boundary-layer dispersion and pure mechanical dispersion regime) that is often encountered in industrial applications.To overcome this problem, we present a new efficient solver implementation for the OpenFOAM® simulation toolbox based on the Lagrangian Random Walk Particle Tracking method to calculate the mass transport via a deterministic convective and a stochastic diffusion term. Besides the numerical calculation of effective diffusion as well as dispersion coefficients, the new solver is also capable of analyzing residence time behavior, flow tortuosity and contact patterns with catalytically active surfaces of catalyst support structures in detail. Following a thorough validation study, we demonstrate the evaluation options provided by the results of this new solver named disTrackFoam on the example of three POCS with distinct unit cell types.

Comparison of well-posedness criteria of two-fluid models for numerical simulation of gas-liquid two-phase flows in vertical pipes

The purpose of the present study is to compare the well-posedness criteria of the free-pressure two-fluid model, single-pressure two-fluid model, and two-pressure two-fluid model in a vertical pipe. Two-fluid models were solved using the conservative shock capturing method. A water faucet case is used to compare two-fluid models. The free pressure two-fluid model can accurately predict discontinuities in the solution field if the problem's initial condition satisfies the Kelvin Helmholtz instability conditions. The single-pressure two-fluid model can accurately predict the behavior of flows in which the two phases are poorly coupled. The two-pressure two-fluid model is an unconditionally well-posed one. If in the free-pressure two-fluid model and single-pressure two-fluid model, the range of velocity difference of two phases exceeds certain limits, the models will be ill-posed. The two-pressure two-fluid model produces more numerical diffusion than the free-pressure two-fluid and single-pressure two-fluid models in the solution field. High numerical diffusion of two-pressure two-fluid models leads to failure to better comply with the problem's analytical solution. Results show that a single-pressure model is a powerful model for numerical modeling of gas-liquid two-fluid-flows in the vertical pipe due to a broader range of well-posed than free-pressure models and less numerical diffusion than the two-pressure mode.

New Strategies for Mitigating the Gray Area in Delayed-Detached Eddy Simulation Models

This paper presents a new approach for mitigating the unphysical delay in the Reynolds-averaged Navier–Stokes (RANS) to large-eddy simulation (LES) transition, often referred to as the gray area, which is a common issue for hybrid RANS–LES turbulence models such as delayed-detached eddy simulation. An existing methodology designed for improving the LES performance in complex flows is adapted and tested. This is based on reducing the numerical diffusion in critical areas for permitting a more accurate development of turbulence. The new formulation comprises both a two-dimensional sensitive velocity gradient model and an alternative definition of the subgrid length scale, which are tested both individually and in tandem, and compared with the other formulations commonly used for addressing the gray area. Four test cases are examined, a flat plate, two variants of the incompressible backward-facing step, and an open jet compressible case, all of which are selected to expose the adverse impact of numerical diffusion that this study seeks to address. Furthermore, the proposed changes are implemented in two different codes for the purpose of cross-validation. Encouraging results are observed, supporting the suitability of the new approach as a candidate for addressing the gray area issue in flows of this kind.

Altered White Matter Microstructures in Type 2 Diabetes Mellitus: A Coordinate-Based Meta-Analysis of Diffusion Tensor Imaging Studies.

ObjectiveType 2 diabetes mellitus (T2DM) is often accompanied by cognitive decline and depressive symptoms. Numerous diffusion tensor imaging (DTI) studies revealed microstructural white matter (WM) abnormalities in T2DM but the findings were inconsistent. The present study aimed to conduct a coordinate‐based meta‐analysis (CBMA) to identify statistical consensus of DTI studies in T2DM.MethodsWe performed a systematic search on relevant studies that reported fractional anisotropy (FA) differences between T2DM patients and healthy controls (HC). The anisotropic effect size seed‐based d mapping (AES-SDM) approach was used to explore WM alterations in T2DM. A meta‐regression was then used to analyze potential influences of sample characteristics on regional FA changes.ResultsA total of eight studies that comprised 245 patients and 200 HC, along with 52 coordinates were extracted. The meta‐analysis identified FA reductions in three clusters including the left inferior network, the corpus callosum (CC), and the left olfactory cortex. Besides, FA in the CC was negatively correlated with body mass index (BMI) in the patients group.ConclusionsT2DM could lead to subtle WM microstructural alterations, which might be associated with cognitive deficits or emotional distress symptoms. This provides a better understanding of the pathophysiology of neurodegeneration and complications in T2DM.Systematic Review RegistrationRegistered at PROSPERO (http://www.crd.york.ac.uk/PROSPERO), registration number: CRD42020218737.

Using the Després and Lagoutière (1999) antidiffusive transport scheme: a promising and novel method against excessive vertical diffusion in chemistry-transport models

Abstract. The potential of the antidiffusive transport scheme proposed by Després and Lagoutière (1999) for resolving vertical transport in chemistry-transport models is investigated in an idealized framework with very encouraging results. We show that, compared to classical higher-order schemes, the Després and Lagoutière (1999) scheme reduces numerical diffusion and improves accuracy in idealized cases that are typical of atmospheric transport of tracers in chemistry-transport models. The increase in accuracy and the reduction in diffusion are substantial when, and only when, vertical resolution is insufficient to properly resolve vertical gradients, which is very frequent in chemistry-transport models. Therefore, we think that this scheme is an extremely promising solution for reducing numerical diffusion in chemistry-transport models.

Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media

In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the L-scheme developed in finite element/volume approaches. The resulting GRW L-schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards’ equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.

A comparison of semi-Lagrangian vortex method and lattice Boltzmann method for incompressible flows

The semi-Lagrangian Vortex method (VM) and the Lattice Boltzmann method (LBM) are used to investigate flows simulations in the incompressible regime. In this study, a proven version of each method is used and compared on different three dimensional benchmarks in terms of numerical accuracy, convergence, numerical diffusion and dissipation. The first comparisons are made on a convected vortex to study and compare the numerical dissipation of LBM and VM. Then the Taylor–Green vortex is investigated to compare the dissipation rate of the kinetic energy of each method. It is shown that both methods converge to the same solution but in a different way. The VM performs better than the LBM for the lowest resolution whereas LBM appears to be more accurate for the growing resolutions. These results are confirmed on 3D simulations with wall boundaries for the stiff test case of the wake behind a 3D cube at Re=290 and Re=570.

Lattice-Boltzmann coupled models for advection–diffusion flow on a wide range of Péclet numbers

Traditional Lattice-Boltzmann modelling of advection–diffusion flow is affected by numerical instability if the advective term becomes dominant over the diffusive (i.e., high-Péclet flow). To overcome the problem, two 3D one-way coupled models are proposed. In a traditional model, a Lattice-Boltzmann Navier–Stokes solver is coupled to a Lattice-Boltzmann advection–diffusion model. In a novel model, the Lattice-Boltzmann Navier–Stokes solver is coupled to an explicit finite-difference algorithm for advection–diffusion. The finite-difference algorithm also includes a novel approach to mitigate the numerical diffusivity connected with the upwind differentiation scheme.The models are validated using two non-trivial benchmarks, which includes discontinuous initial conditions and the case Peg→∞ for the first time, where Peg is the grid Péclet number. The evaluation of Peg alongside Pe is discussed. Accuracy, stability and the order of convergence are assessed for a wide range of Péclet numbers. Recommendations are then given as to which model to select depending on the value Peg—in particular, it is shown that the coupled finite-difference/Lattice-Boltzmann provide stable solutions in the case Pe→∞, Peg→∞.

Is There Anything Better than Pt for HER?

The search for cheap and abundant alternatives to Pt for the hydrogen evolution reaction (HER) has led to many efforts to develop new catalysts. Although the discovery of promising catalysts is often reported, none can compete with Pt in intrinsic activity. To enable true progress, a rigorous assessment of intrinsic catalytic activity is needed, in addition to minimizing mass-transport limitations and following best practices for measurements. Herein, we underline the importance of measuring intrinsic catalytic activities, e.g., turnover frequencies (TOFs). Using mass-selected, identical Pt nanoparticles at a range of loadings, we show the pervasive impact of mass-transport limitations on the observed activity of Pt in acid. We present the highest TOF measured for Pt at room temperature. Since our measurements are still limited by mass transport, the true intrinsic HER activity for Pt in acid is still unknown. Using a numerical diffusion model, we suggest that hysteresis in cyclic voltammograms arises from H2 oversaturation, which is another indicator of mass-transport limitations.

Cloud droplet diffusional growth in homogeneous isotropic turbulence: bin microphysics versus Lagrangian super-droplet simulations

Abstract. The increase in the spectral width of an initially monodisperse population of cloud droplets in homogeneous isotropic turbulence is investigated by applying a finite-difference fluid flow model combined with either Eulerian bin microphysics or a Lagrangian particle-based scheme. The turbulence is forced applying a variant of the so-called linear forcing method that maintains the mean turbulent kinetic energy (TKE) and the TKE partitioning between velocity components. The latter is important for maintaining the quasi-steady forcing of the supersaturation fluctuations that drive the increase in the spectral width. We apply a large computational domain (643 m3), one of the domains considered in Thomas et al. (2020). The simulations apply 1 m grid length and are in the spirit of the implicit large eddy simulation (ILES), that is, with small-scale dissipation provided by the model numerics. This is in contrast to the scaled-up direct numerical simulation (DNS) applied in Thomas et al. (2020). Two TKE intensities and three different droplet concentrations are considered. Analytic solutions derived in Sardina et al. (2015), valid for the case when the turbulence integral timescale is much larger than the droplet phase relaxation timescale, are used to guide the comparison between the two microphysics simulation techniques. The Lagrangian approach reproduces the scalings relatively well. Representing the spectral width increase in time is more challenging for the bin microphysics because appropriately high resolution in the bin space is needed. The bin width of 0.5 µm is only sufficient for the lowest droplet concentration (26 cm−3). For the highest droplet concentration (650 cm−3), an order of magnitude smaller bin size is barely sufficient. The scalings are not expected to be valid for the lowest droplet concentration and the high-TKE case, and the two microphysics schemes represent similar departures. Finally, because the fluid flow is the same for all simulations featuring either low or high TKE, one can compare point-by-point simulation results. Such a comparison shows very close temperature and water vapor point-by-point values across the computational domain and larger differences between simulated mean droplet radii and spectral width. The latter are explained by fundamental differences in the two simulation methodologies, numerical diffusion in the Eulerian bin approach and a relatively small number of Lagrangian particles that are used in the particle-based microphysics.

An evaluation of size-resolved cloud microphysics scheme numerics for use with radar observations Part II: Condensation and evaporation

AbstractAccurate numerical modeling of clouds and precipitation is essential for weather forecasting and climate change research. While size-resolved (bin) cloud microphysics models predict particle size distributions without imposing shapes, results are subject to artificial size distribution broadening owing to numerical diffusion associated with various processes. Whereas Part 1 addressed collision-coalescence, here we investigate numerical diffusion that occurs in solving condensation and evaporation. In a parcel model framework, all of the numerical schemes examined converge to one solution of condensation and evaporation as the mass grid is refined, and the advection-based schemes are recommended over the reassigning schemes. Including Eulerian vertical advection in a column limits the convergence to some extent, but that limitation occurs at a sufficiently fine mass grid, and the number of iterations in solving vertical advection should be minimized to reduce numerical diffusion. Insubstantial numerical diffusion in solving condensation can be amplified if collision-coalescence is also active, which in turn can be substantially diminished if turbulence effects on collision are incorporated. Large-eddy simulations of a drizzling stratocumulus field reveal that changes in moments of Doppler spectra obtained using different mass grids are consistent with those obtained from the simplified framework, and that spectral moments obtained using a mass grid designed to effectively reduce numerical diffusion are generally closer to observations. Notable differences between the simulations and observations still exist, and our results suggest a need to consider whether factors other than numerical diffusion in the fundamental process schemes employed can cause such differences.

Multicollision source method for discrete ordinates neutron transport calculation

Discretization errors like ray effects and numerical diffusion are major shortcomings of SN calculation in shielding problems involving strong anisotropy. Especially in isolated point source shielding problems with cavities and ducts, the angular flux distribution in spatial-angular space may become unsmooth and discontinuous because of heterogeneity effects in the first few iterations. We have developed the multicollision sources (MCS) algorithm via spatial decomposition. The method semi-analytically calculates the scattering source for particles that have suffered several collisions in a selected region, and the SN transport sweeping calculates the scattering for particles in the remaining regions. The calculation parameters of the MCS method is selected according to the distribution of flux in different regions and the scattering effect of shielding materials. For practical complex multi-group problems, the MCS method can be carried out independently in several groups considering the specific problems. We implemented the MCS method in the 3D, parallel, multigroup discrete ordinates code ARES, and numerical results and evaluations are given for a self-designed problem and several general benchmarks. All these results can verify the ability of the MCS method to reduce discretization errors and demonstrate that the MCS method can improve calculation accuracy obviously with an acceptable additional computation cost.

Importance of parameter optimization in a nonlinear stabilized method adding a crosswind diffusion

Numerical solutions of convection-dominated problems are known to exhibit spurious oscillations whose suppression requires the use of numerical stabilization. Stabilized methods which involve heuristic parameters are often applied. The parameters influence the quality of the solution but their optimal values are unknown.In this paper, we consider a stabilization method which adds numerical diffusion adaptively, based on minimizing a functional. The novelty of our approach consists in combining an error indicator with reduced residuals with a nonlinear SOLD method adding artificial diffusion in the crosswind direction. We demonstrate that this approach can lead to more physically meaningful solutions than techniques considered before.

Time variation of concentration-dependent interdiffusion coefficient obtained by numerical simulation analysis

A newly developed numerical diffusion model that conserves solute is coupled with a recently reported forward simulation technique to address the key flaws in conventional methods, including the Boltzmann-Matano, Sauer-Freise and Hall procedures, which are used to study the dependence of interdiffusion coefficient on concentration. The new model involves a combination of fully explicit finite difference analysis with the Leapfrog / Dufort-Frankel scheme to effectively eliminate non-trivial simplification assumptions that degrade accuracy. Numerical analyses of experimental data obtained in the present work and those reported by other authors in the literature show that despite not being generally acknowledged, the concentration dependency of interdiffusion coefficient can significantly vary with time isothermally, even in binary systems. In these situations, the use of a time-independent function, through which interdiffusion coefficient changes with concentration, as commonly derived by the conventional methods, to compute diffusion-controlled kinetics or solute distribution, can be grossly unreliable. Furthermore, unawareness or disregard of this vital concept can lead to significant misinterpretation of experimental data, such as misidentification of the underlying mechanism of microstructural changes by phase transformation reactions.

Very high-order asymptotic-preserving schemes for hyperbolic systems of conservation laws with parabolic degeneracy on unstructured meshes

In this paper, we consider the numerical approximation of hyperbolic systems of conservation laws with stiff source terms and parabolic degeneracy in the asymptotic limit. We are more precisely interested in the design of high-order asymptotic-preserving schemes on unstructured meshes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme and boils down to a sharp control of the underlying numerical diffusion. The strategy allows to capture the correct asymptotic parabolic behaviour and to preserve the high-order accuracy also in the asymptotic limit. Numerical experiments are proposed to illustrate these properties.

Computational general relativistic force-free electrodynamics

Scientific codes are an indispensable link between theory and experiment; in (astro-)plasma physics, such numerical tools are one window into the universe’s most extreme flows of energy. The discretization of Maxwell’s equations – needed to make highly magnetized (astro)physical plasma amenable to its numerical modeling – introduces numerical diffusion. It acts as a source of dissipation independent of the system’s physical constituents. Understanding the numerical diffusion of scientific codes is the key to classifying their reliability. It gives specific limits in which the results of numerical experiments are physical. We aim at quantifying and characterizing the numerical diffusion properties of our recently developed numerical tool for the simulation of general relativistic force-free electrodynamics by calibrating and comparing it with other strategies found in the literature. Our code correctly models smooth waves of highly magnetized plasma. We evaluate the limits of general relativistic force-free electrodynamics in the context of current sheets and tearing mode instabilities. We identify that the current parallel to the magnetic field (j∥), in combination with the breakdown of general relativistic force-free electrodynamics across current sheets, impairs the physical modeling of resistive instabilities. We find that at least eight numerical cells per characteristic size of interest (e.g., the wavelength in plasma waves or the transverse width of a current sheet) are needed to find consistency between resistivity of numerical and of physical origins. High-order discretization of the force-free current allows us to provide almost ideal orders of convergence for (smooth) plasma wave dynamics. The physical modeling of resistive layers requires suitable current prescriptions or a sub-grid modeling for the evolution of j∥.

Development of an Advection-diffusion Model Using Depth-integrated Equations Based on GPU Acceleration

A scalar transport model is developed by adding a depth-averaged advection-diffusion equation to Celeris Advent, which is a Boussinesq-type numerical model that utilizes GPU acceleration. The hybrid finite volume-finite difference method is used to guarantee numerical stability along with the high accuracy of the Boussinesq equation. The advective and diffusive terms are numerically discretized using the finite volume and finite difference methods, respectively. Results of a one-dimensional scalar advection benchmark test showed that the scalar advection by the proposed model was very close to the analytical solution without any remarkable numerical diffusion. In addition, two benchmark tests using experimental data from different hydraulic experiments were numerically reproduced, and the computed results and observed data for scalar transport were found to be in good agreement. The developed model is expected to contribute to real-time disaster prediction for contaminant spills and can assist in preparing countermeasures for these types of disasters.

Effect of non-condensable gas on heat transfer within the hollow fin condenser of metallic and polymer vapor chambers

• Effect of non-condensable gas (NCG) in condensers with hollow fins is investigated. • Thermal behavior and performance of polymer and aluminum condensers are compared. • Predictions of the numerical diffusion model agree well with experimental data. • Performance of air-cooled polymer and aluminum condensers is comparable. • Aluminum condensers are more sensitive to NCG than polymer ones. A vapor chamber with hollow fins was manufactured and tested with condensers made of polymer and aluminum. The fins are 5.3 mm in inner diameter and cooled by forced air convection. The use of polymer instead of aluminum has little influence on the performance: it is of interest for weight reduction when the cooling medium is air. The impact of non-condensable gas (NCG) is studied experimentally using IR temperature measurements on the fins and performance monitoring over time. To complete the study, a numerical model of film condensation inside a vertical tube in the presence of NCG is developed. The model combines hydrodynamic and axial mass diffusion phenomena with 2D heat conduction in the wall. A good agreement is found between experimental results and model predictions. The effect of the NCG mass, heat load, external cooling coefficient and wall thermal conductivity is studied though a parametric analysis. When the wall thermal conductivity increases, so does the volume occupied by NCG inside the fins and the performance degradation. In other words, aluminum condensers are more sensitive to NCG than polymer ones, and the difference is all the more important as external cooling is efficient.