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DOE OSTI2015

Particle tracking approach for transport in three-dimensional discrete fracture networks: Particle tracking in 3-D DFNs

The discrete fracture network (DFN) model is a method to mimic discrete pathways for fluid flow through a fractured low-permeable rock mass, and may be combined with particle tracking simulations to address solute transport. However, experience has shown that it is challenging to obtain accurate transport results in three-dimensional DFNs because of the high computational burden and difficulty in constructing a high-quality unstructured computational mesh on simulated fractures. We present a new particle tracking capability, which is adapted to control volume (Voronoi polygons) flow solutions on unstructured grids (Delaunay triangulations) on three-dimensional DFNs. The locally mass-conserving finite-volume approach eliminates mass balance-related problems during particle tracking. The scalar fluxes calculated for each control volume face by the flow solver are used to reconstruct a Darcy velocity at each control volume centroid. The groundwater velocities can then be continuously interpolated to any point in the domain of interest. The control volumes at fracture intersections are split into four pieces, and the velocity is reconstructed independently on each piece, which results in multiple groundwater velocities at the intersection, one for each fracture on each side of the intersection line. This technique enables detailed particle transport representation through a complex DFN structure. Verified for small DFNs, the new simulation capability enables numerical experiments on advective transport in large DFNs to be performed. As a result, we demonstrate this particle transport approach on a DFN model using parameters similar to those of crystalline rock at a proposed geologic repository for spent nuclear fuel in Forsmark, Sweden.

58 GEOSCIENCES↗
DOE OSTI2020

Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. Finally, the convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes

We study the convergence properties of the recently developed Dynamically Orthogonal (DO) field equations [1] in comparison with the Polynomial Chaos (PC) method. To this end, we consider a series of one-dimensional prototype SPDEs, whose solution can be expressed analytically, and which are associated with both linear (advection equation) and nonlinear (Burgers equation) problems with excitations that lead to unimodal and strongly bi-modal distributions. We also propose a hybrid approach to tackle the singular limit of the DO equations for the case of deterministic initial conditions. The results reveal that the DO method converges exponentially fast with respect to the number of modes (for the problems considered) giving same levels of computational accuracy comparable with the PC method but (in many cases) with substantially smaller computational cost compared to stochastic collocation, especially when the involved parametric space is high-dimensional.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

A monolithic mass tracking formulation for bubbles in incompressible flow

We devise a novel method for treating bubbles in incompressible flow that relies on the conservative advection of bubble mass and an associated equation of state in order to determine pressure boundary conditions inside each bubble. We show that executing this algorithm in a traditional manner leads to stability issues similar to those seen for partitioned methods for solid–fluid coupling. Therefore, we reformulate the problem monolithically. This is accomplished by first proposing a new fully monolithic approach to coupling incompressible flow to fully nonlinear compressible flow including the effects of shocks and rarefactions, and then subsequently making a number of simplifying assumptions on the air flow removing not only the nonlinearities but also the spatial variations of both the density and the pressure. The resulting algorithm is quite robust, has been shown to converge to known solutions for test problems, and has been shown to be quite effective on more realistic problems including those with multiple bubbles, merging and pinching, etc. Notably, this approach departs from a standard two-phase incompressible flow model where the air flow preserves its volume despite potentially large forces and pressure differentials in the surrounding incompressible fluid that should change its volume. Our bubbles readily change volume according to an isothermal equation of state.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

A multiphase electrokinetic flow model for electrolytes with liquid/liquid interfaces

A numerical model for electrokinetic flow of multiphase systems with deformable interfaces is presented, based on a combined level set-volume of fluid technique. A new feature is a multiphase formulation of the Nernst–Planck transport equation for advection, diffusion and conduction of individual charge carrier species that ensures their conservation in each fluid phase. The numerical model is validated against the analytical results of Zholkovskij et al. (2002) [1], and results for the problem of two drops coalescing in the presence of mobile charge carriers are presented. The time taken for two drops containing ions to coalesce decreases with increasing ion concentration.

42 ENGINEERING↗
DOE OSTI2013

Subcell resolution in simplex stochastic collocation for spatial discontinuities

Subcell resolution has been used in the Finite Volume Method (FVM) to obtain accurate approximations of discontinuities in the physical space. Stochastic methods are usually based on local adaptivity for resolving discontinuities in the stochastic dimensions. However, the adaptive refinement in the probability space is ineffective in the non-intrusive uncertainty quantification framework, if the stochastic discontinuity is caused by a discontinuity in the physical space with a random location. The dependence of the discontinuity location in the probability space on the spatial coordinates then results in a staircase approximation of the statistics, which leads to first-order error convergence and an underprediction of the maximum standard deviation. To avoid these problems, we introduce subcell resolution into the Simplex Stochastic Collocation (SSC) method for obtaining a truly discontinuous representation of random spatial discontinuities in the interior of the cells discretizing the probability space. The presented SSC–SR method is based on resolving the discontinuity location in the probability space explicitly as function of the spatial coordinates and extending the stochastic response surface approximations up to the predicted discontinuity location. The applications to a linear advection problem, the inviscid Burgers’ equation, a shock tube problem, and the transonic flow over the RAE 2822 airfoil show that SSC–SR resolves random spatial discontinuities with multiple stochastic and spatial dimensions accurately using a minimal number of samples.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Computation of probabilistic hazard maps and source parameter estimation for volcanic ash transport and dispersion

Volcanic ash advisory centers are charged with forecasting the movement of volcanic ash plumes, for aviation, health and safety preparation. Deterministic mathematical equations model the advection and dispersion of these plumes. However initial plume conditions – height, profile of particle location, volcanic vent parameters – are known only approximately at best, and other features of the governing system such as the windfield are stochastic. These uncertainties make forecasting plume motion difficult. As a result of these uncertainties, ash advisories based on a deterministic approach tend to be conservative, and many times over/under estimate the extent of a plume. This paper presents an end-to-end framework for generating a probabilistic approach to ash plume forecasting. This framework uses an ensemble of solutions, guided by Conjugate Unscented Transform (CUT) method for evaluating expectation integrals. This ensemble is used to construct a polynomial chaos expansion that can be sampled cheaply, to provide a probabilistic model forecast. The CUT method is then combined with a minimum variance condition, to provide a full posterior pdf of the uncertain source parameters, based on observed satellite imagery. The April 2010 eruption of the Eyjafjallajökull volcano in Iceland is employed as a test example. The puff advection/dispersion model is used to hindcast the motion of the ash plume through time, concentrating on the period 14–16 April 2010. Variability in the height and particle loading of that eruption is introduced through a volcano column model called bent. Output uncertainty due to the assumed uncertain input parameter probability distributions, and a probabilistic spatial-temporal estimate of ash presence are computed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes

We present a novel numerical scheme for the deterministic solution of the Wigner transport equation, especially suited to deal with situations in which strong quantum effects are present. The unique feature of the algorithm is the expansion of the Wigner function in local basis functions, similar to finite element or finite volume methods. This procedure yields a discretization of the pseudo-differential operator that conserves the particle density on arbitrarily chosen grids. The high flexibility in refining the grid spacing together with the weighted essentially non-oscillatory (WENO) scheme for the advection term allows for an accurate and well-resolved simulation of the phase space dynamics. A resonant tunneling diode is considered as test case and a detailed convergence study is given by comparing the results to a non-equilibrium Green's functions calculation. The impact of the considered domain size and of the grid spacing is analyzed. The obtained convergence of the results towards a quasi-exact agreement of the steady state Wigner and Green's functions computations demonstrates the accuracy of the scheme, as well as the high flexibility to adjust to different physical situations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

An adjoint view on flux consistency and strong wall boundary conditions to the Navier–Stokes equations

Inconsistent discrete expressions in the boundary treatment of Navier–Stokes solvers and in the definition of force objective functionals can lead to discrete-adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations. The underlying problem is studied for an elementary 1D advection–diffusion problem first using a node-centred finite-volume discretisation. The defect of the boundary operators in the inconsistently defined discrete-adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous-adjoint approach. A homogenisation of the discretisations for the primal boundary treatment and the force objective functional yields second-order functional accuracy and eliminates the defect in the discrete-adjoint boundary treatment. Subsequently, the issue is studied for aerodynamic Reynolds-averaged Navier–Stokes problems in conjunction with a standard finite-volume discretisation on median-dual grids and a strong implementation of noslip walls, found in many unstructured general-purpose flow solvers. Going out from a base-line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver, two improved flux-consistent schemes are presented; based on either body wall-defined or farfield-defined control-volumes they resolve the dual inconsistency. The behaviour of the schemes is investigated on a sequence of grids in 2D and 3D.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

A flux-splitting method for hyperbolic-equation system of magnetized electron fluids in quasi-neutral plasmas

A flux-splitting method is proposed for the hyperbolic-equation system (HES) of magnetized electron fluids in quasi-neutral plasmas. The numerical fluxes are split into four categories, which are computed by using an upwind method which incorporates a flux-vector splitting (FVS) and advection upstream splitting method (AUSM). The method is applied to a test calculation condition of uniformly distributed and angled magnetic lines of force. All of the pseudo-time advancement terms converge monotonically and the conservation laws are strictly satisfied in the steady state. The calculation results are compared with those computed by using the elliptic–parabolic-equation system (EPES) approach using a magnetic-field-aligned mesh (MFAM). Both qualitative and quantitative comparisons yield good agreements of results, indicating that the HES approach with the flux-splitting method attains a high computational accuracy.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

A mass and momentum conserving unsplit semi-Lagrangian framework for simulating multiphase flows

In this work, we present a computational methodology for convection and advection that handles discontinuities with second order accuracy and maintains conservation to machine precision. This method can transport a variety of discontinuous quantities and is used in the context of an incompressible gas–liquid flow to transport the phase interface, momentum, and scalars. The proposed method provides a modification to the three-dimensional, unsplit, second-order semi-Lagrangian flux method of Owkes & Desjardins (JCP, 2014). The modification adds a refined grid that provides consistent fluxes of mass and momentum defined on a staggered grid and discrete conservation of mass and momentum, even for flows with large density ratios. Additionally, the refined grid doubles the resolution of the interface without significantly increasing the computational cost over previous non-conservative schemes. This is possible due to a novel partitioning of the semi-Lagrangian fluxes into a small number of simplices. The proposed scheme is tested using canonical verification tests, rising bubbles, and an atomizing liquid jet.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2017

On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry-standard tools

First- and second-order accurate numerical methods, implemented for CPUs, underpin the majority of industrial CFD solvers. Whilst this technology has proven very successful at solving steady-state problems via a Reynolds Averaged Navier–Stokes approach, its utility for undertaking scale-resolving simulations of unsteady flows is less clear. High-order methods for unstructured grids and GPU accelerators have been proposed as an enabling technology for unsteady scale-resolving simulations of flow over complex geometries. In this study we systematically compare accuracy and cost of the high-order Flux Reconstruction solver PyFR running on GPUs and the industry-standard solver STAR-CCM+ running on CPUs when applied to a range of unsteady flow problems. Specifically, we perform comparisons of accuracy and cost for isentropic vortex advection (EV), decay of the Taylor–Green vortex (TGV), turbulent flow over a circular cylinder, and turbulent flow over an SD7003 aerofoil. We consider two configurations of STAR-CCM+: a second-order configuration, and a third-order configuration, where the latter was recommended by CD-adapco for more effective computation of unsteady flow problems. Results from both PyFR and STAR-CCM+ demonstrate that third-order schemes can be more accurate than second-order schemes for a given cost e.g. going from second- to third-order, the PyFR simulations of the EV and TGV achieve 75× and 3× error reduction respectively for the same or reduced cost, and STAR-CCM+ simulations of the cylinder recovered wake statistics significantly more accurately for only twice the cost. Moreover, advancing to higher-order schemes on GPUs with PyFR was found to offer even further accuracy vs. cost benefits relative to industry-standard tools.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

1-D coupled surface flow and transport equations revisited via the physics-informed neural network approach

The de Saint-Venant equation (SVE) and advection–diffusion equation (ADE) are commonly employed to solve solute transport problems in surface water. In this work, we propose a mesh-free method based on the physics-informed neural network (PINN) to solve the one dimensional (1-D) SVE, ADE, and the coupled SVE and ADE (SVE-ADE) under various initial and boundary conditions. The PINN model extends the architecture of deep neural networks (DNNs) with implementation of loss function, which are additionally subject to constraints imposed by the physical laws of SVE and ADE, along with their initial and boundary conditions. In such a manner, PINNs can be quickly steered to the true solution while obeying the physical laws. The results of PINN model are compared with the analytical and/or numerical solutions under various conditions to investigate its accuracy and efficiency in solving the SVE, ADE, and SVE-ADE. Our results indicate PINN can accurately simulate the shock wave morphology and avoid numerical dissipation in unsteady flow condition. The PINN method outweighed traditional numerical methods in several aspects, including its ability to function with small amounts of data, no grid discretization, and random selection of sampling points, etc. Additionally, the PINN method is also suitable for solving inverse problems with sparse and noisy data. With 1% noise and 2000 initial and boundary condition points (N u ), the errors of the estimated flow rate (v) and diffusion coefficient (D) are 0.003% and 0.105%, respectively, which indicate the accuracy and robustness of the proposed method. Finally, our results indicate the capability and robustness of the proposed PINN methodology for solving multi-physics problems, irrespective of the presence of sparse and noisy data in the training dataset.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Coupled neutronics and species transport simulation of the Molten Salt Reactor Experiment

This paper presents the development of coupling between the molten salt reactor species transport code Mole and the reactor physics code Griffin. Here, in this study, tracking of delayed neutron precursors was investigated in the Molten-Salt Reactor Experiment (MSRE), accounting for changes in fuel flow velocity as a function of position in the primary loop. The neutron transport calculations in Griffin were performed using 11 energy groups, and the species advection calculations in Mole used 6 delayed neutron precursor groups to predict spatial distribution of the neutron flux and neutron precursors in the MSRE. Mole–Griffin was used to calculate $k$ eff and $β$ eff in the reactor as a function of different volumetric flow rates.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2017

Short-term solar irradiance forecasting via satellite/model coupling

Here, the short-term (0-3 h) prediction of solar insolation for renewable energy production is a problem well-suited to satellite-based techniques. The spatial, spectral, temporal and radiometric resolution of instrumentation hosted on the geostationary platform allows these satellites to describe the current cloud spatial distribution and optical properties. These properties relate directly to the transient properties of the downwelling solar irradiance at the surface, which come in the form of 'ramps' that pose a central challenge to energy load balancing in a spatially distributed network of solar farms. The short-term evolution of the cloud field may be approximated to first order simply as translational, but care must be taken in how the advection is handled and where the impacts are assigned. In this research, we describe how geostationary satellite observations are used with operational cloud masking and retrieval algorithms, wind field data from Numerical Weather Prediction (NWP), and radiative transfer calculations to produce short-term forecasts of solar insolation for applications in solar power generation. The scheme utilizes retrieved cloud properties to group pixels into contiguous cloud objects whose future positions are predicted using four-dimensional (space + time) model wind fields, selecting steering levels corresponding to the cloud height properties of each cloud group. The shadows associated with these clouds are adjusted for sensor viewing parallax displacement and combined with solar geometry and terrain height to determine the actual location of cloud shadows. For mid/high-level clouds at mid-latitudes and high solar zenith angles, the combined displacements from these geometric considerations are non-negligible. The cloud information is used to initialize a radiative transfer model that computes the direct and diffuse-sky solar insolation at both shadow locations and intervening clear-sky regions. Here, we describe the formulation of the algorithm and validate its performance against Surface Radiation (SURFRAD; Augustine et al., 2000, 2005) network observations. Typical errors range from 8.5% to 17.2% depending on the complexity of cloud regimes, and an operational demonstration outperformed persistence-based forecasting of Global Horizontal Irradiance (GHI) under all conditions by ~10 W/m 2 .

14 SOLAR ENERGY↗
DOE OSTI2016

Dispersion controlled by permeable surfaces: surface properties and scaling

Permeable and porous surfaces are common in natural and engineered systems. Flow and transport above such surfaces are significantly affected by the surface properties, e.g. matrix porosity and permeability. However, the relationship between such properties and macroscopic solute transport is largely unknown. In this work, we focus on mass transport in a two-dimensional channel with permeable porous walls under fully developed laminar flow conditions. By means of perturbation theory and asymptotic analysis, we derive the set of upscaled equations describing mass transport in the coupled channel–porous-matrix system and an analytical expression relating the dispersion coefficient with the properties of the surface, namely porosity and permeability. Our analysis shows that their impact on the dispersion coefficient strongly depends on the magnitude of the Péclet number, i.e. on the interplay between diffusive and advective mass transport. Additionally, we demonstrate different scaling behaviours of the dispersion coefficient for thin or thick porous matrices. Our analysis shows the possibility of controlling the dispersion coefficient, i.e. transverse mixing, by either active (i.e. changing the operating conditions) or passive mechanisms (i.e. controlling matrix effective properties) for a given Péclet number. By elucidating the impact of matrix porosity and permeability on solute transport, our upscaled model lays the foundation for the improved understanding, control and design of microporous coatings with targeted macroscopic transport features.

36 MATERIALS SCIENCE↗
DOE OSTI2013

The microphysics and macrophysics of cosmic rays

This review paper commemorates a century of cosmic ray research, with emphasis on the plasma physics aspects. Cosmic rays comprise only ∼10{sup −9} of interstellar particles by number, but collectively their energy density is about equal to that of the thermal particles. They are confined by the Galactic magnetic field and well scattered by small scale magnetic fluctuations, which couple them to the local rest frame of the thermal fluid. Scattering isotropizes the cosmic rays and allows them to exchange momentum and energy with the background medium. I will review a theory for how the fluctuations which scatter the cosmic rays can be generated by the cosmic rays themselves through a microinstability excited by their streaming. A quasilinear treatment of the cosmic ray–wave interaction then leads to a fluid model of cosmic rays with both advection and diffusion by the background medium and momentum and energy deposition by the cosmic rays. This fluid model admits cosmic ray modified shocks, large scale cosmic ray driven instabilities, cosmic ray heating of the thermal gas, and cosmic ray driven galactic winds. If the fluctuations were extrinsic turbulence driven by some other mechanism, the cosmic ray background coupling would be entirely different. Which picture holds depends largely on the nature of turbulence in the background medium.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗