Mass Conservation Errors Research Articles

A split-step finite element method for the space-fractional Schrödinger equation in two dimensions

In this article, we propose a split-step finite element method (FEM) for the two-dimensional nonlinear Schrödinger equation (NLS) with Riesz fractional derivatives in space. The space-fractional NLS is first spatially discretized by finite element scheme and the semi-discrete variational scheme is obtained. We prove that it maintains the mass and energy conservation laws. Then, we establish a fully discrete split-step finite element scheme for the considered problem, which avoids the iteration at each time layer, thereby significantly reducing computational cost. The discrete mass conservation property and error estimate of this split-step finite element scheme is derived. Finally, illustrative tests and the numerical simulation of dynamic of wave solutions are included to confirm its effectiveness and capability.

Non-homogeneous anisotropic bulk viscosity for acoustic wave attenuation in weakly compressible methods

A major limitation of the weakly compressible approaches to simulate incompressible flows is the appearance of artificial acoustic waves that introduce mass conservation errors and lead to spurious oscillations in the force coefficients. In this work, we propose a non-homogeneous anisotropic bulk viscosity term to effectively damp the acoustic waves. By implementing this term in a computational framework based on the recently proposed general pressure equation, we demonstrate that the non-homogeneous and anisotropic nature of the term makes it significantly more effective than the isotropic homogeneous version widely used in the literature. Moreover, it is computationally more efficient than the pressure (or mass) diffusion term, which is an alternative mechanism used to suppress acoustic waves. We simulate a range of benchmark problems to comprehensively investigate the performance of the bulk viscosity on the effective suppression of acoustic waves, mass conservation error, order of convergence of the solver, and computational efficiency. The proposed form of the bulk viscosity enables fairly accurate modelling of the initial transients of unsteady simulations, which is highly challenging for weakly compressible approaches, and to the best of our knowledge, existing approaches can't provide an accurate prediction of such transients.

A weighted shifted boundary method for immersed moving boundary simulations of Stokes' flow

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. The surrogate domain is constructed so as to avoid cut cells and the associated problematic implementation and numerical integration issues. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions: hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Stokes flow, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach allows to drastically reduce spurious pressure oscillations in time, which are produced if the total volume of active fluid were to change abruptly over a time step. The proposed Weighted SBM (W-SBM) exactly preserves states of hydrostatic equilibrium, and induces small mass and momentum conservation errors, which converge as the grid is refined. This is in analogy to cutFEMs and related unfitted approaches, which rely on an affine representation of cut boundaries. We demonstrate the robustness and accuracy of the proposed method with an extensive suite of two-dimensional tests.

Modeling liquid transport in the Earth's mantle as two-phase flow: effect of an enforced positive porosity on liquid flow and mass conservation

Abstract. Fluid and melt transport in the solid mantle can be modeled as a two-phase flow in which the liquid flow is resisted by the compaction of the viscously deforming solid mantle. Given the wide impact of liquid transport on the geodynamical and geochemical evolution of the Earth, the so-called “compaction equations” are increasingly being incorporated into geodynamical modeling studies. When implementing these equations, it is common to use a regularization technique to handle the porosity singularity in the dry mantle. Moreover, it is also common to enforce a positive porosity (liquid fraction) to avoid unphysical negative values of porosity. However, the effects of this “capped” porosity on the liquid flow and mass conservation have not been quantitatively evaluated. Here, we investigate these effects using a series of 1- and 2-dimensional numerical models implemented using the commercial finite-element package COMSOL Multiphysics®. The results of benchmarking experiments against a semi-analytical solution for 1- and 2-D solitary waves illustrate the successful implementation of the compaction equations. We show that the solutions are accurate when the element size is smaller than half of the compaction length. Furthermore, in time-evolving experiments where the solid is stationary (immobile), we show that the mass balance errors are similarly low for both the capped and uncapped (i.e., allowing negative porosity) experiments. When Couette flow, convective flow, or subduction corner flow of the solid mantle is assumed, the capped porosity leads to overestimations of the mass of liquid in the model domain and the mass flux of liquid across the model boundaries, resulting in intrinsic errors in mass conservation even if a high mesh resolution is used. Despite the errors in mass balance, however, the distributions of the positive porosity and peaks (largest positive liquid fractions) in both the uncapped and capped experiments are similar. Hence, the capping of porosity in the compaction equations can be reasonably used to assess the main pathways and first-order distribution of fluids and melts in the mantle.

Experimental and CFD modeling of a progressive cavity pump (PCP) using overset unstructured mesh part 1: Single-phase flow

Progressive Cavity Pumps (PCP) are integral in industries for transporting high-viscosity fluids due to their efficacy. Nevertheless, their computational modeling via Computational Fluid Dynamics (CFD) poses challenges, primarily attributed to meshing algorithms and computational demands. This research underscores that structured meshes are not imperative for assessing PCP efficiency, given that operational parameters reflect a 10% accuracy when juxtaposed with experimental data. An overset mass conservation model was harnessed to rectify mass conservation errors and the computational burden from interpolation. The overset grid methodology emerged as a favorable technique for PCP modeling, with its applicability critically analyzed. Experimental data comparison revealed that variables like volumetric flow rate, power consumption, and efficiency were accurately depicted with a marginal error of less than 10%. Interestingly, while high-viscosity oils resonated with the Hagen-Poiseuille flow equation in clearance zones, water showcased elevated velocities, inducing a non-linear pressure profile. Analysis of slip components indicated that high-viscosity oils offered superior efficiencies due to augmented slip resistance. Thus, CFD modeling of PCPs across different viscosity spectrums is feasible without mandating operations to enhance mesh quality. This study pioneers in offering insights into PCP dynamics, emphasizing the accuracy and optimization potential of the overset unstructured mesh approach.

A One-Dimensional Numerical Model for High-Performance Two-Stroke Engines

Computer software that simulates the thermodynamic and gas dynamic properties of internal combustion engines can play a significant role in the design and optimization of internal combustion engines. In the present work, a quasi-dimensional numerical model for two-stroke engines is presented. Particular attention was paid to reporting in-cylinder models, combustion (turbulent with flame development and flame–wall interaction), and turbulence (K-k-ϵ model), with the addition of tumble- and squish-generated turbulence that is quite common in such engines. The aim was to reduce the role of the calibration constants, which are fundamental for correlating the models with the experiments, and relations for calculating the tumble ratio and turbulent scales were reported. A one-dimensional model for manifolds is also presented (solving the Euler equations), using the second-order Roe Riemann solver with some improvements, paying particular attention to the source terms, such as area variation. Additionally, a new approach to the end-pipe boundaries, which would reduce the mass conservation error, is reported. The engines tested were two kart two-stroke engines, used for racing purposes: the IAME X30 engine and the IAME Screamer III KZ engine. A comparison between the model results and the experimental data was made, and good accordance was observed, with a root mean square error of about 0.5 kW and providing good accuracy in evaluating changes, such as the combustion chamber squish area and the exhaust pipe length.

On-the-fly accuracy evaluation of artificial neural networks and hybrid method to improve the robustness of neural network accelerated chemistry solving

A chemistry acceleration strategy based on the coupling of artificial neural networks (ANNs) and direct integration (DI) is proposed and evaluated in the context of turbulent combustion. The main novelty of this study is its focus on ANNs robustness assessment. A hybrid DI/ANN strategy is proposed, which allows for a direct control of the prediction errors. This control is achieved by evaluating the ANN prediction error after each inference. To this end, a simple yet novel criterion based on mass conservation is proposed and compared to a criterion based on the distance between the inferred state and the training database, as done previously in the literature in the context of on-the-fly learning. A two-dimensional turbulent premixed H2 ignition case is used to assess the performance of the strategy and challenge the two criteria. An a priori study demonstrates that the state space-based criterion cannot correctly describe the ANN error, while the mass conservation-based one provides a good match with the ANN error. An a posteriori evaluation, involving actual simulations of the turbulent case, shows the ability of the hybrid DI/ANN model based on mass conservation error to improve the quality of the predictions and thus the robustness of ANNs. The increase in computational cost due to the hybrid model is acceptable as the DI is only applied in very localized regions in space and time.

A comparative study of split advection algorithms on the moment-of-fluid (MOF) method for incompressible flow

The moment-of-fluid method (MOF) is known as an extension of the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). In this study, several directional splitting advection algorithms are extended from the VOF-PLIC method to the MOF method. These methods, along with some existing directional splitting algorithms, are presented in detail, especially on the geometrical and non-geometrical nature of the advection algorithm. In addition, a simple and efficient analytic form to calculate the volume and centroid of the intersection polyhedron in 3D is proposed. Several numerical tests, including both 2D and 3D tests, are carried out to investigate mass conservation and geometrical error. Numerical results suggest that the mixed EI and LE scheme (EILE2D) (Aulisa et al., 2003) has the overall best performance in 2D, and Weymouth and Yue’s scheme (WY) has the overall best performance in 3D.

Remedy for ill-posedness and mass conservation error of 1D incompressible two-fluid model with artificial viscosities

The two-fluid model is widely used to describe two-phase flows in complex systems such as nuclear reactors. Although the two-phase flow was successfully simulated, the standard two-fluid model suffers from an ill-posed nature. There are several remedies for the ill-posedness of the one-dimensional (1D) two-fluid model; among those, artificial viscosity is the focus of this study. Some previous works added artificial diffusion terms to both mass and momentum equations to render the two-fluid model well-posed and demonstrated that this method provided a numerically converging model. However, they did not consider mass conservation, which is crucial for analyzing a closed reactor system. In fact, the total mass is not conserved in the previous models. This study improves the artificial viscosity model such that the 1D incompressible two-fluid model is well-posed, and the total mass is conserved. The water faucet and Kelvin-Helmholtz instability flows were simulated to test the effect of the proposed artificial viscosity model. The results indicate that the proposed artificial viscosity model effectively remedies the ill-posedness of the two-fluid model while maintaining a negligible total mass error.

A conservative and consistent implicit Cartesian cut-cell method for moving geometries with reduced spurious pressure oscillations

A conservative and consistent three-dimensional Cartesian cut-cell method is presented for reducing the spurious pressure oscillations often observed in moving body simulations in sharp-interface Cartesian grid methods. By analysing the potential sources of the oscillation in the cut-cell framework, an improved moving body algorithm is proposed for the cut-cell method for the temporal discontinuity of the solid volume change. Strict conservation of mass and momentum for both fluid and cut cells is enforced through pressure-velocity coupling to reduce local mass conservation errors. A consistent mass and momentum flux computation is employed in the finite volume method. In contrary to the commonly cut-cell methods, an implicit time integration scheme is employed in the present method, which prevents numerical instability without any additional small cut-cell treatment. The effectiveness of the present cut-cell method for reducing spurious pressure oscillations is demonstrated by simulating various two- and three-dimensional benchmark cases (in-line and transversely oscillating cylinder, oscillating and free-falling sphere), with good agreement with previous experimental measurements and other numerical methods available in the literature.

A massively parallel accurate conservative level set algorithm for simulating turbulent atomization on adaptive unstructured grids

This article presents a massively parallel and robust strategy to perform the simulation of turbulent incompressible two-phase flows on unstructured grids in complex geometries. This strategy relies on a combination of a narrow-band accurate conservative level set (ACLS)/ghost-fluid framework with isotropic adaptive mesh refinement. This combination enables to accurately capture interface dynamics and topology, and the small physical scales at the liquid-gas interface are resolved at an affordable cost. The ACLS method, even if not strictly mass-conserving, ensures the exact conservation of a smoothed phase indicator, which minimizes the liquid mass conservation errors. In the accurate conservative level set framework, presented first in Desjardins et al. (2008) [25], the interface is defined as the iso-contour of a hyperbolic tangent function, which is advected by the fluid, and then reshaped using a reinitialization equation. Several forms of this reinitialization exist: the original ACLS form proposed by Desjardins et al. involves numerical estimation of the hyperbolic tangent gradient, which is difficult to compute accurately on unstructured meshes. It is thus susceptible to induce artificial deformation of the interface. A new form has been recently proposed in Chiodi and Desjardins (2017) [29], which takes advantage of a mapping onto a classical distance level set while much better preserving the interface shape. Nevertheless, the implementation of this new form on unstructured grids requires special care. In this work, a robust implementation of this new form on unstructured meshes is proposed and implemented in the YALES2 low-Mach flow solver. In order to compute interface normals and curvature, the signed-distance function is reconstructed in parallel at nodes in the narrow band around the interface using a Geometric-Projection Marker Method (GPMM). This method relies on the triangulation of the level set iso-contour and exact geometric projection to the closest surface elements. Spatial convergence, robustness and efficiency of the overall procedure are firstly demonstrated through classical interface transport test cases and two-phase flow examples. Eventually, to emphasize the significant computational gain using adaptive mesh refinement and the ability to compute complex turbulent flows with large density ratios, two Large-Eddy Simulations (LES) of atomizing liquid jets in air are presented, each one at various resolutions. The first one is a low-pressure water jet in quiescent air from a compound nozzle with full computation of the internal injector flow, while the latter is a high-pressure kerosene jet in crossflow. Both simulations are validated against experiments, demonstrating the potential of the method to access a deep numerical insight into jet instabilities and internal flow dynamics with 3D unstructured meshes.

Re-examining the partially saturated-cells method for incompressible flows with stationary and moving bodies

We revisit the Partially-Saturated-Cells method (PSC) in the context of incompressible single-phase flows past stationary and moving bodies. This methodology adds a solid collision operator to the Lattice Boltzmann Equation, resulting in a unified evolution equation everywhere in the domain that suitably accounts for the boundary conditions. The unified evolution equation, which also allows for a smooth transition from solid cells to fluid cells, is dictated by the solid volume fraction, obtained using a simple geometric approach. We prove that the PSC approach exhibits nominal second-order accuracy in presence of straight and curved boundaries for pressure, although spatial order of accuracy in velocity degrades to first-order. Our analysis also reveals that PSC is approximately discretely conservative in the incompressible limit with the mass conservation errors being dependent on the lattice size and relaxation time. For simulations involving moving body problems, the method is found to introduce acceptable level of spurious force oscillations, which rapidly decrease with increasing grid resolution. To the best of the authors' knowledge, the present investigations pertaining to discrete conservation and spurious force oscillations in the PSC approach have never been discussed in literature. Extensive numerical experiments are performed involving both stationary and moving bodies, with imposed and induced motion, to assess the applicability of the PSC approach. Our findings illustrate that the inclusion of added mass effects are important for accurate force and torque computations in case of accelerating solids. Numerical investigations, that include unsteady flows featuring three degrees-of-freedom fluid-particle interactions clearly highlight the ability of the PSC approach in resolving complex hydrodynamical phenomena. This in-depth evaluation of the PSC approach conclusively demonstrates its efficacy as a promising computational framework for efficient and robust computations of incompressible fluid flows over a range of flow scenarios.

A flux correction approach for the pressure equation in incompressible flows on overset meshes in OpenFOAM

Standard interpolation methods between overset meshes in finite-volume methods such as distance-weighted or least squares based methods do not respect the conservation property of the finite-volume method. As the resultant fluxes exchanged between meshes do not vanish on a global scale, there is either a source or sink in the mass, momentum or energy. This then manifests itself as reduced order of accuracy in the flow variables such as velocity and pressure or high amplitude oscillations in the pressure field when the mesh moves. For incompressible flows, the pressure Poisson equation unanimously decides the extent to which mass is conserved. In this paper, the pressure Poisson equation is isolated and analyzed independently to understand the source of the conservation error. Based on the discretized pressure Poisson equation, a novel flux correction method applicable to moving overset meshes is derived. Using exact solutions for the pressure Poisson equation, it is shown that this method improves the order of accuracy and the discretization error and reduces the mass conservation error. An order of magnitude reduction in the oscillations of the pressure field is also observed when the mesh moves.

A conservative level set method for liquid-gas flows with application in liquid jet atomisation

In this paper, a methodology for modelling two-phase flows based on a conservative level set method in the framework of finite volume method is presented. The novelty of the interface capturing method used here lies on the advection of level set which is solved with a WENO scheme and corrected with a novel re-initialisation method for retaining its signed distance function character. The coupling with the volume of fluid method is done with a simple algebraic approach, and with the new algorithm the accumulated mass conservation errors remain reasonably low. The paper presents a unique coupling between the level set method and the Eulerian-Lagrangian Spray Atomisation approach for modelling spray dispersion in liquid atomisation systems. The method is shown to have good accuracy providing similar results to other numerical codes for the classical tests presented. Preliminary results are also shown for three-dimensional simulations of the primary break-up of a turbulent liquid jet obtaining results comparable to direct numerical simulations. Consequently, the coupled method can be used for simulating various two-phase flow applications offering an accurate representation of the interface dynamics.

On element mass conservation in Eulerian stochastic fields modeling of turbulent combustion

Eulerian stochastic fields (ESF) based transported PDF method has evolved as one of the general purpose methods for the numerical simulation of complex chemically reactive turbulent flows in which the effect of turbulence on the chemical reaction rates can be computed directly without the need of modeling. It is found that ESF method may suffer from violation of the element mass conservation law due to the stochastic Wiener process and the highly non-linear dependence of the chemical reaction rates on the stochastic fields. This paper presents a modified ESF method to remove the error in the element mass conservation. The method is evaluated in numerical simulation of two turbulent flames, (i) Reynolds averaged Navier-Stokes (RANS) simulation of a turbulent non-premixed methane/air counterflow flame under stationary burning and transient flame extinction conditions, (ii) large eddy simulation of swirling turbulent methane/air non-premixed flames under local extinction and re-ignition conditions. The original ESF method violates the element mass conservation in both flames, and the element mass error would not disappear even if a large number of stochastic fields were used. The new method yields a satisfactory prediction of the element mass conservation even with a small number of stochastic fields. The new and original methods predict similar stationary flame structures but the results under flame extinction and re-ignition conditions are rather different.

A generalized Fourier\u2013Hermite method for the Vlasov\u2013Poisson system

A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the symmetrically-weighted and asymmetrically-weighted Fourier–Hermite methods from the literature. The numerical scheme is formulated as a weighted Galerkin method with two separate scaling parameters for the Hermite polynomial and the exponential part of the new basis functions. Exact formulas for the error in mass, momentum, and energy conservation of the method depending on the parameters are devised and L^2 stability is discussed. The numerical experiments show that an optimal choice of the additional parameter in the generalized method can yield improved accuracy compared to the existing methods, but also reveal the distinct stability properties of the symmetrically-weighted method.

A control volume method based interface movement equation for one-dimensional Stefan problem achieving mass conservation

For heat-transfer-controlled or diffusion-controlled phase transformation simulations, the movement of the interface between different phases needs to be determined, which is commonly called the Stefan problem. Various numerical methods have been developed for this and the movement of the interface is conventionally calculated by a flux balance equation, which is usually called the Stefan condition equation. However, mass conservation errors have been reported in the literature for some of the numerical methods. In this research, a control volume method based interface movement equation was proposed to achieve mass conservation during calculation. The obtained maximum relative mass conservation error for the new interface movement equation is less than 10−13 and the total mass is constant to the machine precision. This is much smaller than that of the Stefan condition discrete equation, which is in the order of 10−2. It was also found that the mass conservation accuracy of the numerical methods partly depends on the choice of the interface movement equation. The mass change term in the Stefan condition discrete equation is only a part of the whole mass change during interface movement, leading to the mass non-conservation found in the literature. In contrast, the piecewise integration treatment in the proposed interface movement equation in this research gives a more accurate mass change description during the interface movement. The correctness of the proposed interface movement equation was also proved by a comparison with the analytical solutions.

Fully Implicit Dynamic Pore‐Network Modeling of Two‐Phase Flow and Phase Change in Porous Media

AbstractDynamic pore‐network model (PNM) has been widely used to model pore‐scale two‐phase flow. Numerical algorithms commonly used for dynamic PNM including IMPES (implicit pressure explicit saturation) and IMP‐SIMS (implicit pressure semi‐implicit saturation) can be numerically unstable or inaccurate for challenging flow regimes such as low capillary number (Ca) flow and unfavorable displacements. We perform comprehensive analyses of IMPES and IMP‐SIMS for a wide range of flow regimes under drainage conditions and develop a novel fully implicit (FI) algorithm to address their limitations. Our simulations show the following: (1) While IMPES was reported to be numerically unstable for low Ca flow, using a smoothed local pore‐body capillary pressure curve appears to produce stable simulations. (2) Due to an approximation for the capillary driving force, IMP‐SIMS can deviate from quasi‐static solutions at equilibrium states especially in heterogeneous networks. (3) Both IMPES and IMP‐SIMS introduce mass conservation errors. The errors are small for networks with cubic pore bodies (less than 1.4% for IMPES and 1.2% for IMP‐SIMS). They become much greater for networks with square‐tube pore bodies (up to 45% for IMPES and 46% for IMP‐SIMS). Conversely, the new FI algorithm is numerically stable and mass conservative regardless of the flow regimes and pore geometries. It also precisely recovers the quasi‐static solutions at equilibrium states. The FI framework has been extended to include compressible two‐phase flow, multicomponent transport, and phase change dynamics. Example simulations of two‐phase displacements accounting for phase change show that evaporation and condensation can suppress fingering patterns generated during invasion.

A weighted Shifted Boundary Method for free surface flow problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods and was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Navier-Stokes flows with moving free-surfaces, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach prevents spurious pressure oscillations in time, which would otherwise be produced if the total active fluid volume were to change abruptly over a time step. In fact, the proposed weighted SBM method induces small mass (i.e., volume) conservation errors, which converge quadratically in the case of piecewise-linear finite element interpolations, as the grid is refined. We present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method.

A nearly-conservative high-order Lagrange–Galerkin method for the resolution of scalar convection-dominated equations in non-divergence-free velocity fields

In this work, we present a novel Lagrange–Galerkin method for the resolution of the scalar pure-convection and convection-dominated diffusion equations in non-divergence-free velocity fields. The scheme has been formulated so that (i) time-integration is carried out by means of an arbitrary order backward differentiation formula, (ii) finite element space discretizations of any order can be employed, and (iii) it satisfies a discrete mass balance equation at each simulation instant and preserves mass in those cases in which no fluid enters nor leaves the control domain, as long as the integrals involved are computed exactly. These properties are achieved by posing a conservation law for a weighted mass –from which the standard mass can be seen as a particular case– that can be easily discretized in time and in space with any order of accuracy. Numerical experiments –which include a three-dimensional test– show that the method is as accurate in time as the employed time-marching formula, as accurate in space as the finite element discretization, and generally more accurate –especially in terms of mass conservation errors– and efficient than the classic, non-conservative Lagrange–Galerkin method. To the best of our knowledge, this is the first conservative Lagrange–Galerkin method for convection-dominated equations that considers both non-divergence-free velocity fields and time and space discretizations of any order.