Closure Relations Research Articles

Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions

One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.

Prediction of mean radical concentrations in lean hydrogen-air turbulent flames at different Karlovitz numbers adopting a newly extended flamelet-based presumed PDF

A recent analysis (Lipatnikov et al., 2020) of complex-chemistry direct numerical simulation (DNS) data obtained from lean hydrogen-air flames associated with corrugated-flame (case A), thin-reaction-zone (case B), and broken-reaction-zone (case C) regimes of turbulent burning has shown that the flamelet concept (i) can predict mean concentrations of various species in those flames if the probability density function (PDF) for the fuel-based combustion progress variable c is extracted from the DNS data, but (ii) poorly performs for the mean rate W¯c of product creation. These results suggest applying the concept to evaluation of mean species concentration (but not the mean rate) in combination with another closure relation for W¯c whose predictive capabilities are better. This proposal is developed in the present paper whose focus is placed on studying a new flamelet-based presumed PDF P(c) for predictions of mean concentration of radicals in engineering computational fluid dynamics (CFD) applications. Analysis of the DNS data shows that (i) the flamelet PDF performs well at intermediate values of c in cases A and B, but should be truncated at small and large c, (ii) modeling P(c) in the radical recombination zone (i.e., at large c) is of importance for predicting mean concentrations of H,O, and OH. Accordingly, the flamelet PDF is truncated and combined with a uniform P(c) at large c. Moreover, the mean rate W¯c extracted from the DNS data is used to calibrate the PDF (the rate is considered to be given by another model). Assessment of the approach against the DNS data shows that it well predicts mean density, temperature, and concentrations of reactants, product, and the aforementioned radicals in cases A and B. In case C, the approach performs worse for OandOH at large c¯ and moderately underestimates the mean concentration of H in the entire flame brush.

GRB Fermi-LAT Afterglows: Explaining Flares, Breaks, and Energetic Photons

Abstract The Fermi-LAT collaboration presented the second gamma-ray burst (GRB) catalog covering its first 10 years of operations. A significant fraction of afterglow-phase light curves in this catalog cannot be explained by the closure relations of the standard synchrotron forward-shock model, suggesting that there could be an important contribution from another process. In view of the above, we derive the synchrotron self-Compton (SSC) light curves from the reverse shock in the thick- and thin-shell regime for a uniform-density medium. We show that this emission could explain the GeV flares exhibited in some LAT light curves. Additionally, we demonstrate that the passage of the forward shock synchrotron cooling break through the LAT band from jets expanding in a uniform-density environment may be responsible for the late time (≈102 s) steepening of LAT GRB afterglow light curves. As a particular case, we model the LAT light curve of GRB 160509A that exhibited a GeV flare together with a break in the long-lasting emission, and also two very high energy photons with energies of 51.9 and 41.5 GeV observed 76.5 and 242 s after the onset of the burst, respectively. Constraining the microphysical parameters and the circumburst density from the afterglow observations, we show that the GeV flare is consistent with an SSC reverse-shock model, the break in the long-lasting emission with the passage of the synchrotron cooling break through the Fermi-LAT band, and the very energetic photons with SSC emission from the forward shock, when the outflow carries a significant magnetic field (R B ≃ 30) and it decelerates in a uniform-density medium with a very low density ( ).

Drift Flux Model Parameters Estimation based on Numerical Simulation of Slug Flow Regime with High-Viscous Liquids in Pipelines

Multiphase flow models involve more variables than available equations, thus constitutive equations are required to solve the system's governing equations. In the Drift Flux Model, mean drift velocity of the gas phase is estimated by using a Dispersed-Phase Distribution Coefficient and Gas Drift Velocity closure relationships. This study investigates the hydrodynamic of the slug flow regime with high-viscous liquids, and develops an equation for the Distribution Coefficient. Simulation cases were run with inclination angles varying from 0° (horizontal direction) to 90° (vertical upward) and a 0.051m-ID pipe. OpenFOAM, an open-source CFD software, was used to numerically solve the two-phase flow problem. A detailed methodology to estimate drift flux parameters from the data stored at each grid block is presented. This step-by-step methodology allows researchers to further develop the constitutive equations for Distribution Coefficient and the Drift Velocity for other cases. Simulated results were compared with experimental data from the literature and other published models for horizontal and vertical pipes. The results showed that this CFD modeling approach is suitable for representing actual slug flow in pipes. The newly developed Distribution Coefficient constitutive equation has less than 10% absolute average relative error for the viscosity range of 0.14 to 1.120Pa·s, and pipe inclinations from 0° to 90°.

Simulations of the Early Postbounce Phase of Core-collapse Supernovae in Three-dimensional Space with Full Boltzmann Neutrino Transport

Abstract We report on the core-collapse supernova simulation we conducted for a 11.2M ⊙ progenitor model in three-dimensional space up to 20 ms after bounce, using a radiation-hydrodynamics code with full Boltzmann neutrino transport. We solve the six-dimensional Boltzmann equations for three neutrino species and the three-dimensional compressible Euler equations with Furusawa and Togashi's nuclear equation of state. We focus on the prompt convection at ∼10 ms after bounce and investigate how neutrinos are transported in the convective matter. We apply a new analysis based on the eigenvalues and eigenvectors of the Eddington tensor and make a comparison between the Boltzmann transport results and the M1 closure approximation in the transition regime between the optically thick and thin limits. We visualize the eigenvalues and eigenvectors using an ellipsoid, in which each principal axis is parallel to one of the eigenvectors and has a length proportional to the corresponding eigenvalue. This approach enables us to understand the difference between the Eddington tensor derived directly from the Boltzmann simulation and the one given by the M1 prescription from a new perspective. We find that the longest principal axis of the ellipsoid is almost always nearly parallel to the energy flux in the M1 closure approximation, whereas in the Boltzmann simulation it becomes perpendicular in some transition regions, where the mean free path is ∼0.1 times the radius. In three spatial dimensions, the convective motions make it difficult to predict where this happens and to possibly improve the closure relation there.

On the Investigation of the Closure Relations for Gamma-Ray Bursts Observed by Swift in the Post-plateau Phase and the GRB Fundamental Plane

Abstract Gamma-ray Bursts (GRBs) are the most explosive phenomena in the universe after the big bang. A large fraction of GRB lightcurves (LCs) shows X-ray plateaus. We perform the most comprehensive analysis of all GRBs (with known and unknown redshifts) with plateau emission observed by The Neil Gehrels Swift Observatory from its launch until 2019 August. We fit 455 LCs showing a plateau and explore whether these LCs follow closure relations, relations between the temporal and spectral indices of the afterglow, corresponding to two distinct astrophysical environments and cooling regimes within the external forward shock (ES) model, and find that the ES model works for the majority of cases. The most favored environments are a constant-density interstellar or wind medium with slow cooling. We also confirm the existence of the fundamental plane relation between the rest-frame time and luminosity at the end of the plateau emission and the peak prompt luminosity for this enlarged sample, and test this relation on groups corresponding to the astrophysical environments of our known redshift sample. The plane becomes a crucial discriminant corresponding to these environments in terms of the best-fitting parameters and dispersions. Most GRBs for which the closure relations are fulfilled with respect to astrophysical environments have an intrinsic scatter σ compatible within 1σ of that of the “Gold” GRBs, a subset of long GRBs with relatively flat plateaus. We also find that GRBs satisfying closure relations indicating a fast cooling regime have a lower σ than ever previously found in literature.

Averaging over spatiotemporal heterogeneity substantially biases evapotranspiration rates in a mechanistic large-scale land evaporation model

Abstract. Evapotranspiration (ET) influences land–climate interactions, regulates the hydrological cycle, and contributes to the Earth's energy balance. Due to its feedback to large-scale hydrological processes and its impact on atmospheric dynamics, ET is one of the drivers of droughts and heatwaves. Existing land surface models differ substantially, both in their estimates of current ET fluxes and in their projections of how ET will evolve in the future. Any bias in estimated ET fluxes will affect the partitioning between sensible and latent heat and thus alter model predictions of temperature and precipitation. One potential source of bias is the so-called “aggregation bias” that arises whenever nonlinear processes, such as those that regulate ET fluxes, are modeled using averages of heterogeneous inputs. Here we demonstrate a general mathematical approach to quantifying and correcting for this aggregation bias, using the GLEAM land evaporation model as a relatively simple example. We demonstrate that this aggregation bias can lead to substantial overestimates in ET fluxes in a typical large-scale land surface model when sub-grid heterogeneities in land surface properties are averaged out. Using Switzerland as a test case, we examine the scale dependence of this aggregation bias and show that it can lead to an average overestimation of daily ET fluxes by as much as 10 % across the whole country (calculated as the median of the daily bias over the growing season). We show how our approach can be used to identify the dominant drivers of aggregation bias and to estimate sub-grid closure relationships that can correct for aggregation biases in ET estimates, without explicitly representing sub-grid heterogeneities in large-scale land surface models.

Development of a four-fluid model for condensing steam flows

A new four-fluid model is proposed for condensing water-steam flows applicable to a wide variety of different flow conditions relevant for nuclear power plants. The model expands upon current state-of-the-art three-fluid models by adding a second droplet fluid phase to account for homogeneous nucleation of droplets from the bulk vapor field. Consistent with prior three-fluid models, liquid wall film and entrained droplet fluids share a common temperature and energy equation, resulting in a system of eleven conservation equations. An additional governing PDE is required for the nucleated droplet number density, for twelve total governing equations. A full set of closure relations for the interphase and wall interactions is given. An initial test of the model formulation in a quasi-one-dimsensional converging-diverging nozzle with condensing steam was carried out using a simple in-house solver as a proof of concept. Results showed qualitatively correct trends compared with experimental data.

Nonhyperbolicity of Conservation Equations of RELAP5 Two-Fluid Model in Nuclear Reactor Safety Results (Investigation and Eigenvalue Analysis)

Abstract The RELAP5 code simulates the thermal-hydraulic characteristics of nuclear reactors by the use of a two-fluid one-dimensional, nonequilibrium, nonhomogeneous two-phase flow model. This model consists of six governing equations to describe the mass, energy, and momentum of the two fluids. The scope of this work comprises the study of the mathematical nature of the code model and to predict the accuracy of the model in the nuclear reactor safety analysis. The method of characteristics (MOC) is applied to check the nonhyperbolic nature of conservation equations for all normal and accident conditions of light water reactors (LWRs). The analysis also gives information about the soundness of the model and to identify the regions where the solutions obtained from it will be numerically convergent. The characteristics of equations of nonhyperbolic nature are complex. It implies that results thus obtained (by finite difference method) have to be interpreted very carefully in view of the sensitive nature of reactor safety analysis. The present analysis shows that governing equations of the code exhibit complex characteristics for some operating conditions thus implying nonhyperbolicity under those conditions. Results are less accurate under such conditions, so sensitivity analysis plays an important role. The sensitivity of closure relationship on the conservation equation's stability is also checked. The analysis is performed in matlab environment for three different systems, (a) pressurized water reactor (PWR), boiling water reactor (BWR), and (c) natural circulation reactor or advance heavy water reactor (AHWR). These results can also be extended to other thermal-hydraulic systems. The different values of the coefficient of closure relationship are taken for different flow regimes. It is observed that the coefficient of virtual mass (for momentum equation) has a significant effect on the hyperbolicity of the system. It is recommended that further development of the RELAP5 model be performed to identify changes that would reduce the region of complex characteristics. The importance of MOC (in nuclear reactor thermal-hydraulic safety analysis) is evident here. In addition, a detailed analysis for operating pressures range of 0.1–22.5 MPa is also performed to find out the nonhyperbolic regions of code model and realistic data of the different type of reactors is used as input of the code. It is also observed here that RELAP5 results are less accurate when system pressure exceeds 19.5 MPa.

Glassy dynamics from generalized mode-coupling theory: existence and uniqueness of solutions for hierarchically coupled integro-differential equations

Generalized mode-coupling theory (GMCT) is a first-principles-based and systematically correctable framework to predict the complex relaxation dynamics of glass-forming materials. The formal theory amounts to a hierarchy of infinitely many coupled integro-differential equations, which may be approximated using a suitable finite-order closure relation. Although previous studies have suggested that finite-order GMCT leads to well-defined solutions, and that the hierarchy converges as the closure level increases, no rigorous and general result in this direction is known. Here we unambiguously establish the existence and uniqueness of solutions to generic, schematic GMCT hierarchies that are closed at arbitrary finite order. We consider two types of commonly invoked closure approximations, namely mean-field and exponential closures. We also distinguish explicitly between overdamped and underdamped glassy dynamics, corresponding to hierarchies of first-order and second-order integro-differential equations, respectively. We find that truncated GMCT hierarchies closed under an exponential closure conform to previously developed mathematical theories, both in the overdamped and underdamped case, such that the existence of a unique solution can be readily inferred. Self-consistent mean-field closures, however, of which the well-known standard-MCT closure approximation is a special case, warrant additional arguments for mathematical rigor. We demonstrate that the existence of a priori bounds on the solution is sufficient to also prove that unique solutions exist for such self-consistent hierarchies. To complete our analysis, we present simple arguments to show that these a priori bounds must exist, motivated by the physical interpretation of the GMCT solutions as density correlation functions. Overall, our work contributes to the theoretical justification of GMCT for studies of the glass transition, placing this hierarchical framework on a firmer mathematical footing.

Modeling of the ECCD injection effect on the Heliotron J and LHD plasma stability

The aim of the study is to analyze the stability of the energetic particle modes (EPM) and Alfven Eigenmodes (AE) in Helitron J and LHD plasma if the electron cyclotron current drive (ECCD) is applied. The analysis is performed using the code FAR3d that solves the reduced MHD equations describing the linear evolution of the poloidal flux and the toroidal component of the vorticity in a full 3D system, coupled with equations of density and parallel velocity moments for the energetic particle (EP) species, including the effect of the acoustic modes. The Landau damping and resonant destabilization effects are added via the closure relation. The simulation results show that the n = 1 EPM and n = 2 global AE (GAE) in Heliotron J plasma can be stabilized if the magnetic shear is enhanced at the plasma periphery by an increase (co-ECCD injection) or decrease (ctr-ECCD injection) of the rotational transform at the magnetic axis (). In the ctr-ECCD simulations, the EPM/AE growth rate decreases only below a given , similar to the ECCD intensity threshold observed in the experiments. In addition, ctr-ECCD simulations show an enhancement of the continuum damping. The simulations of the LHD discharges with ctr-ECCD injection indicate the stabilization of the n = 1 EPM, n = 2 toroidal AE (TAE) and n = 3 TAE, caused by an enhancement of the continuum damping in the inner plasma leading to a higher EP β threshold with respect to the co- and no-ECCD simulations.

Experimental study of a single elongated bubble in liquid in under 10-degree upwardly inclined pipes

Two phase flow is of great interest in chemical and petroleum industries, and multiphase pipe flow models with closure relationships require experimental data for their development and validation. However, only little experimental information is available for slightly upward inclined pipes. Experimental investigations of single elongated bubble in marginally upwardly inclined pipes less than 10° have therefore been performed. Observations of the bubble drift velocity along the pipe has been highlighted. The drift velocity data presented here can contribute to improve knowledge of pipe inclination and viscosity dependency in drift velocity correlations. The new data on the bubble characteristics - shape, length, fraction and drift velocity may also provide useful information for the development and validation of numerical models. The measured drift velocity data have therefore been compared with some recently developed bubble velocity correlations.

The extension of the one‐dimensional two‐fluid slug capturing method to simulate slug flow in vertical pipes

AbstractWhile many studies have been conducted on the application of the one‐dimensional (1D) two‐fluid model in horizontal and nearly horizontal pipes under slug flow condition, very few works have been carried out on the mechanistic simulation of the slug flow regime in vertical pipe configurations.This article presents the extension of a novel numerical scheme for slug capturing in pipes in vertical slug flows. The original numerical scheme includes an 1D transient, hyperbolic five‐equation two‐fluid model, and it has been embodied in a computer code called 5ESCARGOTS (5 Equations for Slug Capturing Analysis by a Roe solver in Gas and Oil Transient Simulation).This article demonstrates that the model is able to properly automatically capture and simulate the slug generation even in vertical flow. It shows that when appropriate closure relations are used for interfacial shear forces, the model is able to properly characterize the vertical slug flow.Numerous simulations have been carried out for different pipe configurations and flow conditions leading to slug flow. First, a qualitative discussion of the numerical results is proposed underlining that the model is able to properly simulate the evolution of the slug flow in vertical pipes. Experimental flow pattern maps have also been numerically computed and it has been demonstrated that the model correctly simulates slug conditions. Moreover, slug frequency, velocity, and average liquid holdup, computed numerically, have been compared with experimental data in order to quantitatively validate the model. It is shown that the numerical results agree fairly well with the experimental data. A sensitivity analysis on the mesh size and on the inlet boundary conditions has also been carried out.

Interface learning in fluid dynamics: Statistical inference of closures within micro–macro-coupling models

Many complex multiphysics systems in fluid dynamics involve using solvers with varied levels of approximations in different regions of the computational domain to resolve multiple spatiotemporal scales present in the flow. The accuracy of the solution is governed by how the information is exchanged between these solvers at the interface, and several methods have been devised for such coupling problems. In this Letter, we construct a data-driven model by spatially coupling a microscale lattice Boltzmann method (LBM) solver and macroscale finite difference method (FDM) solver for reaction–diffusion systems. The coupling between the micro–macro-solvers has one to many mapping at the interface leading to the interface closure problem, and we propose a statistical inference method based on neural networks to learn this closure relation. The performance of the proposed framework in a bifidelity setting partitioned between the FDM and LBM domains shows its promise for complex systems where analytical relations between micro–macro-solvers are not available.

Indirect Solution of Ornstein-Zernike Equation Using the Hopfield Neural Network Method

Microscopic information, such as the pair distribution and direct correlation functions, can be obtained from experimental data. From these correlation functions, thermodynamical quantities and the potential interaction function can be recovered. Derivations of Ornstein-Zernike equation and Hopfield Neural Network method are given first, as a theoretical background to follow the present work. From these two frameworks, structural information, such as the radial distribution (g(r)) and direct correlation (C(r)) functions, were retrieved from neutron scattering experimental data. The problem was solved considering simple initial conditions, which does not require any previous information about the system, making it clear the robustness of the Hopfield Neural Network method. The pair interaction potential was estimated in the Percus-Yevick (PY) and hypernetted chain (HNC) approximations and a poor agreement, compared with the Lennard-Jones 6-12 potential, was observed for both cases, suggesting the necessity of a more accurate closure relation to describe the system. In this sense, the Hopfield Neural Network together with experimental information provides an alternative approach to solve the Ornstein-Zernike equations, avoiding the limitations imposed by the closure relation.

Structure Factor of Charged Microemulsion Nanodroplets Decorated with Telechelic Polymers: Experimental and Numerical Study

We investigate thermodynamic and structural properties of positively charged O/W microemulsion spherical nanodroplets, suspended in salt water and decorated with telechelic polymers PEO-m and PEO-2m at low and high volume fraction F (6.98%, 26.5%) by using Small Angle Neutron Scattering (SANS). We describe and propose an effective pair potential interaction between charged colloidal particles in solution both with and without added telechelic polymers PEO-m or PEO-2m.We solve Ornstein-Zernicke (OZ) integral equation in the Hypernetted Chain (HNC) closure relation to obtain the pair correlation function g(r) and the structure factor S(q). A good agreement is found between the experimental and numerical spectra assuming the interaction potentials to be the sum of attractive and repulsive contributions in terms of decoration or bridging of the nanodroplets by the PEO polymer chains.

A Hamiltonian gyrofluid model based on a quasi-static closure

A Hamiltonian six-field gyrofluid model is constructed, based on closure relations derived from the so-called ‘quasi-static’ gyrokinetic linear theory where the fields are assumed to propagate with a parallel phase velocity much smaller than the parallel particle thermal velocities. The main properties captured by this model, primarily aimed at exploring fundamental problems of interest for space plasmas such as the solar wind, are its ability to provide a reasonable agreement with kinetic theory for linear low-frequency modes, and at the same time to ensure a Hamiltonian structure in the absence of explicit dissipation. The model accounts for equilibrium temperature anisotropy, ion and electron finite Larmor radius corrections, electron inertia, magnetic fluctuations along the direction of a strong guide field and parallel Landau damping, introduced through a Landau-fluid modelling of the parallel heat transfers for both gyrocentre species. Remarkably, the quasi-static closure leads to exact and simple expressions for the nonlinear terms involving gyroaveraged electromagnetic fields and potentials. One of the consequences is that a rather natural identification of the Hamiltonian structure of the model becomes possible when Landau damping is neglected. A slight variant of the model consists of a four-field Hamiltonian reduction of the original six-field model, which is also used for the subsequent linear analysis. In the latter, the dispersion relations of kinetic Alfvén waves and the firehose instability are shown to be correctly reproduced, relatively far in the sub-ion range (depending on the plasma parameters), while the spectral range where the slow-wave dispersion relation and the field-swelling instabilities are precisely described is less extended. This loss of accuracy originates from the breaking of the condition of small phase velocity, relative to the parallel thermal velocity of the electrons (for kinetic Alfvén waves and firehose instability) or of the ions (in the case of the field-swelling instabilities).

Four field flow modeling of forced convective boiling

A two-phase, four-field computational fluid dynamics (CFD) model was implemented in ANSYS CFX and subsequent three-dimensional calculations were compared with experimental data collected from an R134a experimental test facility. These calculations considered the following four fields for mass, momentum, and energy continuity – continuous liquid, dispersed vapor (bubbles), continuous vapor, and dispersed liquid (droplets). This four-field approach enabled CFD calculations to transition smoothly between the flow regimes present in heated, multiphase flows without the use of a-priori heuristics or flow-regime mapping. Such flow regimes included subcooled nucleate boiling, transition flow, annular flow, and dryout in three dimensions. This two-fluid, four-field model and related closure relationships predicted a variety of experimental data for multiple combinations of inlet temperatures, pressures, test section powers, and inlet velocities in a vertically oriented R134a flow loop.

The Bruhat order on abelian ideals of Borel subalgebras

Let G G be a quasi-simple algebraic group over an algebraically closed field k \mathsf {k} whose characteristic is not very bad for G G , and let B B be a Borel subgroup of G G with Lie algebra b \mathfrak {b} . Given a B B -stable abelian subalgebra a \mathfrak {a} of the nilradical of b \mathfrak {b} , we parametrize the B B -orbits in a \mathfrak {a} and we describe their closure relations.

A prediction model for the critical liquid-carrying velocity of gas–liquid stratified flow in micro-tilting line pipes with low liquid contents

Wet gas can form liquid loading at the lower line pipe sections, so the transportation efficiency will be impacted and the line pipes will be corroded and even blocked. Therefore, to accurately predict the critical liquid-carrying velocity of gas is of great significance to preventing the liquid loading in wet gas line pipes. In view of the gas–liquid two-phase stratified flow in micro-tilting line pipes with low liquid content, this paper newly established a critical liquid-carrying velocity prediction model considering droplet entrainment according to the momentum balance equation of a gas–liquid two-phase flow and the closure relationship of a new gas–liquid interface shape. Then, based on the experimental data, the new model, FLAT model, ARS model, double-circle model and MARS model were verified and their prediction results were compared. Finally, the new model was applied to analyze the effects of pipe dip, operational pressure, liquid density and gas component on the critical liquid-carrying velocity and critical liquid content of natural gas–water and natural gas–60% glycerine with water stratified flow in a micro-tilting line pipe. And the following research results were obtained. First, with the increase of pipe dip and liquid density, the critical liquid-carrying velocity increases continuously and the critical liquid content decreases gradually. Second, with the increase of operational pressure and heavy component content, the critical liquid-carrying velocity decreases continuously and the critical liquid content increases gradually. In conclusion, the new model is higher in prediction accuracy and its prediction result is better accordant with the experimental value, so it can be used to predict the critical liquid-carrying velocity in wet gas line pipes.