Complex Source Terms Research Articles

High-order compact gas-kinetic scheme for two-layer shallow water equations on unstructured mesh

For the two-layer shallow water equations, a high-order compact gas-kinetic scheme (GKS) on triangular mesh is proposed. The two-layer shallow water equations have complex source terms in comparison with the single layer equations. The main focus of this study is to construct a time-accurate evolution solution at a cell interface and to design a well-balanced scheme. The evolution model at a cell interface provides not only the numerical fluxes, but also the flow variables. In the development of the well-balanced scheme, the time-dependent flow variables at the cell interfaces surrounding the cell can be utilized to update the cell-averaged gradients. These time-evolving gradients are subsequently applied in the discretization of source terms inside each control volume. Based on the cell-averaged flow variables and their gradients, high-order initial data reconstruction can be achieved with compact stencils. The compact high-order GKS has advantages to simulate the flow evolution in complex domain covered by unstructured mesh. Many test cases are used to validate the accuracy and robustness of the scheme for the two-layer shallow water equations.

Revisiting the second-order convergence of the lattice Boltzmann method with reaction-type source terms

This study analyses an approach to consistently recover the second-order convergence of the lattice Boltzmann method (LBM), which is frequently degraded by an improper discretisation of the required source terms. The current work focuses on advection-diffusion models, in which the source terms are dependent on the intensity of transported fields. Such terms can be observed in reaction-type equations used in heat and mass transfer problems or multiphase flows. The investigated scheme is applicable to a wide range of formulations within the LBM framework. All considered source terms are interpreted as contributions to the zeroth-moment of the distribution function. These account for sources in a scalar field, such as density, concentration, temperature or a phase field. Further application of this work can be found in the method of manufactured solutions or in the immersed boundary method. This paper is dedicated to three aspects regarding proper inclusion of the source term in LBM schemes. Firstly, it identifies the differences observed between the ways in which source terms are included in the LBM schemes present in the literature. The algebraic manipulations are explicitly presented in this paper to clarify the observed differences, and to identify their origin. Secondly, it analyses in full detail, the implicit relation between the value of the transported macroscopic field, and the sum of the LBM densities. This relation is valid for any source term discretization scheme. It is a crucial ingredient for preserving the second-order convergence in the case of complex source terms. Moreover, three equivalent forms of the second-order accurate collision operator are presented. Finally, closed form solutions of this implicit relation are shown for a variety of common models, including general linear and second order terms; population growth models, such as the Logistic or Gompertz model and the Allen-Cahn equation. The second-order convergence of the proposed LBM schemes is verified on both linear and non-linear source terms. The pitfalls of the commonly used acoustic and diffusive scalings are identified and discussed. Furthermore, for a simplified case, the competing errors are shown visually with isolines of error in the space of spatial and temporal resolutions.

Validation of the neural network for 3D photon radiation field reconstruction under various source distributions

Introduction: This paper proposes a five-layer fully connected neural network for predicting radiation parameters in a radiation space based on detector readings.Methods: The network is trained and tested using gamma flux values from individual detector positions as input, and is used to predict the gamma radiation field in 3D space under different source term distributions. The method is evaluated using the mean percentage change error (PCT) for the test set under different source term distributions.Results: The results show that the neural network method can accurately predict radiation parameters with an average PCT error range of 0.53% to 3.11%, within the given measurement input error range of ± 10%. The method also demonstrates its ability to directly reconstruct the 3D radiation field with some simple source terms.Discussion: The proposed method has practical value in real operations within radiation spaces, and can be used to improve the accuracy and efficiency of predicting radiation parameters. Further research could explore the use of more complex source term distributions and the integration of other types of sensors for improved accuracy.

Moving water equilibria preserving nonstaggered central scheme for open‐channel flows

In this paper, we investigate a well‐balanced and positive‐preserving nonstaggered central scheme, which has second‐order accuracy on both time and spatial scales, for open‐channel flows with the variable channel width and the nonflat bottom. We perform piecewise linear reconstructions of the conserved variables and energy as well as discretize the source term using the property that the energy remains constant, so that the complex source term and the flux can be precisely balanced so as to maintain the steady state. The scheme also ensures that the cross‐sectional wet area is positive by introducing a draining time‐step technique. Numerical experiments demonstrate that the scheme is capable of accurately maintaining both the still steady‐state solutions and the moving steady‐state solutions, simultaneously. Moreover, the scheme has the ability to accurately capture small perturbations of the moving steady‐state solution and avoids generating spurious oscillations. It is also capable of showing that the scheme is positive‐preserving and robust in solving the dam‐break problem.

Extension of the WRF-Chem volcanic emission preprocessor to integrate complex source terms and evaluation for different emission scenarios of the Grimsvötn 2011 eruption

Abstract. Volcanic eruptions may generate volcanic ash and sulfur dioxide (SO2) plumes with strong temporal and vertical variations. When simulating these changing volcanic plumes and the afar dispersion of emissions, it is important to provide the best available information on the temporal and vertical emission distribution during the eruption. The volcanic emission preprocessor of the chemical transport model WRF-Chem has been extended to allow the integration of detailed temporally and vertically resolved input data from volcanic eruptions. The new emission preprocessor is tested and evaluated for the eruption of the Grimsvötn volcano in Iceland 2011. The initial ash plumes of the Grimsvötn eruption differed significantly from the SO2 plumes, posing challenges to simulate plume dynamics within existing modelling environments: observations of the Grimsvötn plumes revealed strong vertical wind shear that led to different transport directions of the respective ash and SO2 clouds. Three source terms, each of them based on different assumptions and observational data, are applied in the model simulations. The emission scenarios range from (i) a simple approach, which assumes constant emission fluxes and a predefined vertical emission profile, to (ii) a more complex approach, which integrates temporarily varying observed plume-top heights and estimated emissions based on them, to (iii) the most complex method that calculates temporal and vertical variability of the emission fluxes based on satellite observations and inversion techniques. Comparisons between model results and independent observations from satellites, lidar, and surface air quality measurements reveal the best performance of the most complex source term.

Monte Carlo Simulation and Experimental Validation for Radiation Protection with Multiple Complex Source Terms and Deep Penetration for a Radioactive Liquid Waste Cementation Facility

A new radioactive liquid waste cementation facility was under commissioning recently in the Institute of Nuclear and New Energy Technology of Tsinghua University, which is designed to simultaneously process multiple intermediate-level radioactive waste drums. Therefore, the multiple volume sources and the scattering effect becomes a key issue in its radiation protection. For this purpose, the Monte Carlo program FLUKA code and experimental measurement were both adopted. In the FLUKA simulation, five different scenarios were considered, i.e., one drum, two drums, four drums, six drums, and eight drums. For the multiple volume sources, the source subroutine code of FLUKA was rewritten to realize the sampling. The complex shielding also leads to a deep penetration problem; hence, the optimization algorithm and variance reduction techniques were adopted. During the measurement, two scenarios, outdoor and indoor, were carried out separately representing the dose field when only one drum is considered and when the scattering effect is considered. A comparison between the experiments and calculations shows very good agreement. From both of the Monte Carlo simulation and the experimental measurement, it can be drawn that, in the horizontal direction, with the increase of the drum number, the dose rate increases very little, while in the vertical direction, the increase of the dose rate is very obvious with the increase of the drum number. The complicated source term sampling methods, the optimization algorithm and variance reduction techniques, and the experimental verification can provide valuable references for the similar scattering problem in radiation protection and shielding design.

(Multi)wavelets increase both accuracy and efficiency of standard Godunov-type hydrodynamic models: Robust 2D approaches

Multiwavelets (MW) enable the compression, analysis and assembly of model data on a multiresolution grid within Godunov-type solvers based on second-order discontinuous Galerkin (DG2) and first-order finite volume (FV1) methods. Multiwavelet adaptivity has been studied extensively with one-dimensional (1D) hydrodynamic models (Kesserwani et al., 2019), revealing that MWDG2 can be 20 times faster than uniform DG2 and 2 times faster than uniform FV1, while preserving the accuracy and robustness of the underlying formulation. The potential of the MWDG2 scheme has yet to be studied for two-dimensional (2D) modelling, but this requires a design that robustly and efficiently solves the 2D shallow water equations (SWE) with complex source terms and wetting and drying. This paper presents a two-dimensional MWDG2 scheme that: (1) adopts a slope-decoupled DG2 solver as a reference scheme, for its ability to deliver well-balanced piecewise-planar solutions shaped by a simplified 3-component basis; and, (2) adapts the multiresolution analysis of multiwavelets for compatibility with the slope-decoupled DG2 basis. A scaled reformulation of slope-decoupled DG2 is presented alongside two multiwavelet approaches that yield MWDG2 schemes with similar properties, and a Haar wavelet FV1 (HFV1) variant for adapting piecewise-constant model data. The performance of the adaptive HFV1 and MWDG2 solvers is explored alongside their uniform counterparts, while analysing their accuracy, efficiency, grid-coarsening ability, reliability in handling wet-dry fronts across steep bed-slopes, and ability to capture features relevant to practical hydraulic modelling. The results indicate a particular multiwavelet approach that allows the MWDG2 scheme to exploit its grid-coarsening ability for the widest range of flow types. Results also indicate that the proposed (multi)wavelet-based adaptive schemes are even more efficient for the 2D case. Accompanying model software is openly available online.

Fully Coupled Discontinuous Galerkin Modeling of Dam-Break Flows over Movable Bed with Sediment Transport

A one-dimensional (1D) discontinuous Galerkin morphodynamic model has been devised with application to simulate of dam-break flows over erodible beds with suspended sediment transport. The morphodynamic equations adopt the shallow-water equations (SWE) considering the interaction of sediment transport and bed changes on the flow. A local second-order Runge-Kutta discontinuous Galerkin (RKDG2) model has been reformulated to numerically solve the morphodynamic equations in a fully coupled manner and with a noncapacity sediment model. The model’s implementation is thoroughly detailed with focus on the discretization of the complex source terms, the treatment of wetting and drying, and other stabilizing issues pertaining to high-solution gradients and the transient character of the topography. The model has been favorably applied to replicate experimental dam-break flow over erodible sediment beds.

Closure to “Discussion of ‘Flow-Excited Acoustic Resonance: Excitation Mechanism, Design Guidelines and Counter Measures,’” (Ziada, S., and Lafon, P., 2014, Appl. Mech. Rev., 66(1), p. 010802)

Closure to “Discussion of ‘Flow-Excited Acoustic Resonance: Excitation Mechanism, Design Guidelines and Counter Measures,’” (Ziada, S., and Lafon, P., 2014, Appl. Mech. Rev., 66(1), p. 010802)

An inverse modeling method to assess the source term of the Fukushima Nuclear Power Plant accident using gamma dose rate observations

Abstract. The Chernobyl nuclear accident, and more recently the Fukushima accident, highlighted that the largest source of error on consequences assessment is the source term, including the time evolution of the release rate and its distribution between radioisotopes. Inverse modeling methods, which combine environmental measurements and atmospheric dispersion models, have proven efficient in assessing source term due to an accidental situation (Gudiksen, 1989; Krysta and Bocquet, 2007; Stohl et al., 2012a; Winiarek et al., 2012). Most existing approaches are designed to use air sampling measurements (Winiarek et al., 2012) and some of them also use deposition measurements (Stohl et al., 2012a; Winiarek et al., 2014). Some studies have been performed to use dose rate measurements (Duranova et al., 1999; Astrup et al., 2004; Drews et al., 2004; Tsiouri et al., 2012) but none of the developed methods were carried out to assess the complex source term of a real accident situation like the Fukushima accident. However, dose rate measurements are generated by the most widespread measurement system, and in the event of a nuclear accident, these data constitute the main source of measurements of the plume and radioactive fallout during releases. This paper proposes a method to use dose rate measurements as part of an inverse modeling approach to assess source terms. The method is proven efficient and reliable when applied to the accident at the Fukushima Daiichi Nuclear Power Plant (FD-NPP). The emissions for the eight main isotopes 133Xe, 134Cs, 136Cs, 137Cs, 137mBa, 131I, 132I and 132Te have been assessed. Accordingly, 105.9 PBq of 131I, 35.8 PBq of 132I, 15.5 PBq of 137Cs and 12 134 PBq of noble gases were released. The events at FD-NPP (such as venting, explosions, etc.) known to have caused atmospheric releases are well identified in the retrieved source term. The estimated source term is validated by comparing simulations of atmospheric dispersion and deposition with environmental observations. In total, it was found that for 80% of the measurements, simulated and observed dose rates agreed within a factor of 2. Changes in dose rates over time have been overall properly reconstructed, especially in the most contaminated areas to the northwest and south of the FD-NPP. A comparison with observed atmospheric activity concentration and surface deposition shows that the emissions of caesiums and 131I are realistic but that 132I and 132Te are probably underestimated and noble gases are likely overestimated. Finally, an important outcome of this study is that the method proved to be perfectly suited to emergency management and could contribute to improve emergency response in the event of a nuclear accident.

Review of Airborne Emissions from Agricultural Fumigants: Design and Uncertainty Considerations for the Use of the Integrated Horizontal Flux Method

Ground-level area sources, such as those associated with the use of agricultural fumigants, waste disposal sites, wastewater lagoons, and other applications, present a challenge in terms of characterizing atmospheric flux as a function of time. Studies are costly in terms of field activities and laboratory analysis. The optimization of field study design, therefore, is essential to conduct cost-effective research. The collection of on-field profile data for airborne concentration, wind speed, and wind direction can be used in conjunction with the integrated horizontal flux (IHF) method to empirically compute complex source terms as a function of time. This paper focuses on complicating factors and field study design issues for the use of the IHF method. Insights and examples are drawn from five field research studies. The methods and results of characterizing the uncertainty and method precision in the emission fitting for the IHF method also are presented.

Concurrent use of finite element and finite volume methods for shallow water flows in locally 1-D channel networks

SUMMARY A numerical model is developed for shallow water equation in locally 1-D channel networks. The model concurrently uses the standard Galerkin finite element method for the continuity equation and the finite volume method with an upwind scheme for the momentum equation. The surface gradient method is consistently employed. A minimum treatment is given for channel junctions so that application to multiply connected channels do not require any special consideration The model is capable of computing different types of transcritical flows, wet and dry flows, and flows with complex source terms. Standardized test problems and laboratory experimental data are used for verifying the model. Applicability of the models is validated in a multiply connected channel network draining hydromorphic farmlands located in a West African savanna, and Manning's roughness coefficient is identified, so that the steady solution is consistent with field observations. Unsteady simulation demonstrates that the model is capable of stably reproducing shifts of hydraulic jumps in flows of sub-millimeter water depths. Copyright © 2011 John Wiley & Sons, Ltd.

A Godunov-type scheme for modelling 1D channel flow with varying width and topography

One-dimensional (1D) numerical models have long been used in simulating fluvial hydrodynamics. While most of these models are based on the solutions to some approximate forms of the fully 1D St. Venant equations, it is desirable to have a 1D code that can deal with those highly dynamic and complex flows under certain flood conditions, with full consideration of the convective and source terms. This paper therefore presents a Godunov-type alternative for solving the 1D inhomogeneous shallow water equations with complex source terms. The model is also implemented with a wetting and drying condition to avoid producing negative water depth. The proposed model is validated by a selection of steady and transient hydraulic problems with reference solutions.

Excitation source of a side-branch shear layer

The excitation source of flow-induced acoustic resonances in closed side-branches is characterized experimentally for circular pipes excited by turbulent flow in the main pipe. The shear layer at the branch junction is modeled by an unsteady complex source which is dependent on the Strouhal number and the acoustic particle velocity at the shear layer. The amplitude and phase of this source are determined experimentally and presented in the form of a dimensionless complex source term. This determined shear layer source term and the acoustic description of the piping system are then combined in a semi-empirical model to predict the frequency and pulsation amplitude of flow-excited acoustic resonance. The model results exemplify important experimental observations of flow excited side-branch resonances; including the occurrence of the lock-in phenomenon, the excitation of resonance by the single and double vortex modes of the shear layer, and nonlinear saturation at large pulsation amplitude due to vortex damping. The dependence of the pulsation amplitude on the Strouhal number, the static test pressure and on friction and radiation losses is also reproduced by the model. Finally, the effect of the acoustic particle velocity distribution at the branch junction on the shear layer source term is quantified.

Numerical resolution of well-balanced shallow water equations with complex source terms

This paper presents a well-balanced numerical scheme for simulating frictional shallow flows over complex domains involving wetting and drying. The proposed scheme solves, in a finite volume Godunov-type framework, a set of pre-balanced shallow water equations derived by considering pressure balancing. Non-negative reconstruction of Riemann states and compatible discretization of slope source term produce stable and well-balanced solutions to shallow flow hydrodynamics over complex topography. The friction source term is discretized using a splitting implicit scheme. Limiting value of the friction force is derived to ensure stability. This new numerical scheme is validated against four theoretical benchmark tests and then applied to reproduce a laboratory dam break over a domain with irregular bed profile.

Consistent hybrid finite volume/element formulations: model and complex viscoelastic flows

AbstractThe accuracy and consistency of a new cell‐vertex hybrid finite element/volume scheme are investigated for viscoelastic flows. Finite element (FE) discretization is employed for the momentum and continuity equation, with finite volume (FV) applied to the constitutive law for stress. Here, the interest is to explore the consequences of utilizing conventional cell‐vertex methodology for an Oldroyd‐B model and to demonstrate resulting drawbacks in the presence of complex source terms on structured and unstructured grids. Alternative strategies worthy of consideration are presented. It is demonstrated how high‐order accuracy may be achieved in steady state by respecting consistency in the formulation. Both FE and FV spatial discretizations are embedded in the scheme, with FV triangular sub‐cells referenced within parent triangular finite elements. Both model and complex flow problems are selected to quantify and assess accuracy, appealing to analysis and experimental validation. The test problem is that of steady sink flow, a pure extensional flow, which reflects some of the numerical difficulties involved in solving more generalized viscoelastic flows, where both source and flux terms may contribute equally to stress propagation. In addition, a complex transient filament‐stretching flow is chosen to compute the evolution of stress fields within liquid bridges. Shortcomings of the various stress upwinding schemes are discussed in this context, whilst dealing with such free‐surface type problems. Here, stress fluctuation distribution alone is advocated, and a Lax‐scheme is found to deliver accuracy and stability to the computational results, comparing well with the literature. Copyright © 2004 John Wiley & Sons, Ltd.

Adaptive implicit/explicit finite element methods for axisymmetric viscous turbulent flows with moving boundaries

The extension of the two-dimensional adaptive implicit/explicit finite element method to axisymmetric turbulent flow problems is presented. Difficulties involved in the selection of proper flow variables for the finite element interpolation to avoid singularities and the treatment of complex source terms resulting from the axisymmetric formulation are covered in this study. A special data structure is proposed for efficiently implementing the Prandtl-Van Driest turbulence model with an implicit/explicit algorithm for adapted unstructured grids. In addition, a moving-grid/eroding-boundary algorithm is implemented (specifically, for modeling the burning of a solid propellant) for internal flow analysis in solid rocket motors. A series of benchmark problems are solved to demonstrate the effectiveness of the methodology.