Fundamental Conservation Laws Research Articles

Numerical modeling of propagation of breaking nonlinear acoustic-gravity waves from the lower to the upper atmosphere

Acoustic-gravity waves (AGWs) observed in the upper atmosphere may be generated near the Earth’s surface due to a variety of meteorological sources. Two-dimensional simulations of vertical propagation and breaking of nonlinear AGWs in the atmosphere are performed. Forcing near the Earth’s surface is used as the AGW source in the model. We use a numerical method based on finite-difference analogues of fundamental conservation laws for solving atmospheric hydrodynamic equations. This approach selects physically correct generalized solutions of the wave hydrodynamic equations. Numerical simulations are performed in a representative region of the Earth’s atmosphere up to altitude 500km. Vertical profiles of temperature, density, molecular viscosity and heat conductivity were taken from the standard atmosphere model MSIS-90 for January. Calculations were made for different amplitudes and frequencies of lower boundary wave forcing. It is shown that after activating the tropospheric wave forcing, the initial pulse of AGWs may very quickly propagate to altitudes of 100km and above and relatively slowly dissipate due to molecular viscosity and heat conduction. This may increase the role of transient nonstationary waves in effective energy transport and variations of atmospheric parameters and gas admixtures in a broad altitude range.

Study of Fukushima Daiichi Nuclear Power Station Unit 4 Spent-Fuel Pool

A study on the Fukushima Daiichi nuclear power station spent-fuel pool (SFP) at Unit 4 (SFP4) is presented in this paper. We discuss the design characteristics of SFP4 and its decay heat load in detail and provide a model that we developed to estimate the SFP evaporation rate based on the SFP temperature. The SFP level of SFP4 following the March 11, 2011, accident is predicted based on the fundamental conservation laws of mass and energy. Our predicted SFP level and temperatures are in good agreement with measured data and are consistent with Tokyo Electric Power Company evaluation results.

Modeling wave action effects in internal combustion engine air path systems: comparison of numerical and system dynamics approaches

The challenge of continuously improving engine fuel economy and emissions has pushed the automotive industry to adopt more efficient procedures for modeling, system simulation and model-based control. Such procedures require accurate and computationally efficient engine system simulation models that are able to predict the cylinder charge composition and the thermodynamic conditions. To this extent, reduced-order models for unsteady compressible flow systems, namely models derived from the fundamental conservation laws without over-simplifying the topology of the engine air path, are receiving considerable interest for performance simulation and control design. This paper presents a comparative study on modeling approaches for the prediction of one-dimensional unsteady compressible flow in the internal combustion engine air path system. Specifically, the paper compares a well-known second-order, shock-capturing finite-difference scheme with two novel solution methods, namely a finite-volume method and a model-order reduction method. The study aims at evaluating the ability of each method to trade-off the prediction accuracy and robustness with the computation time, in light of potential applications to real-time simulation. This is achieved by increasing the discretization length of each method, and evaluating the accuracy of the predicted response in the time and frequency domain. The characterization of the intake and exhaust flows in the manifolds of a single-cylinder engine is considered as a case study. The results are compared for the three solution methods at different discretization lengths to evaluate the ability of each model to retain accuracy and robustness. The computation times of the different methods are also evaluated. The results presented in this paper establish a clear trade-off between accuracy, stability and computation time for each solution method. This allows one to formulate considerations on the advantages and disadvantages of each method in light of potential applications to engine performance prediction, optimization and control design.

Ultrashort pulses and short-pulse equations in 2+1 dimensions

In this paper, we derive and study two versions of the short pulse equation (SPE) in $(2+1)$ dimensions. Using Maxwell's equations as a starting point, and suitable Kramers-Kronig formulas for the permittivity and permeability of the medium, which are relevant, e.g., to left-handed metamaterials and dielectric slab wave guides, we employ a multiple scales technique to obtain the relevant models. General properties of the resulting $(2+1)$-dimensional SPEs, including fundamental conservation laws, as well as the Lagrangian and Hamiltonian structure and numerical simulations for one- and two-dimensional initial data, are presented. Ultrashort one-dimensional breathers appear to be fairly robust, while rather general two-dimensional localized initial conditions are transformed into quasi-one-dimensional dispersing wave forms.

Map of fluid flow in fractal porous medium into fractal continuum flow

This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided.

Stability of algebraic multigrid for Stokes problems

SUMMARYAn investigation is made of the performance of algebraic multigrid (AMG) solvers for the discrete Stokes problem. The saddle‐point formulations are based on the direct enforcement of the fundamental conservation laws in discrete spaces and subsequently stabilised with the aid of a regular splitting of the diffusion operator. AMG solvers based on an independent coarsening of the fields (the unknown approach) and also on a common coarsening (the point approach) are investigated. Both mixed‐order and equal‐order interpolations are considered. The dependence of convergence on the ‘degree of coarsening’ is investigated by studying the ‘convergence versus coarsening’ characteristics and their variation with mesh resolution. They show a consistency in shape, which reveals two distinct performance zones, one convergent the other divergent. The transition from the convergent to the divergent zones is discontinuous and occurs at a critical coarsening factor that is largely mesh independent. It signals a breakdown in the stability of the smoothing at the coarser levels of coarse grid approximation. It is shown that the previously observed, mesh‐dependent, scaling of convergence factors, which had suggested inconsistencies in the coarse grid approximation, is not a reliable marker of inconsistency. It is an indirect consequence of the breakdown in the stability of smoothing. For stable smoothing, reduction factors are shown to be largely mesh independent. The ability of mixed‐order interpolation to permit stable smoothing and therefore to deliver mesh‐independent convergence is explained. Two expedient options are suggested for obtaining mesh‐independent convergence for those AMG codes that are based on an equal‐order interpolation. Copyright © 2012 John Wiley & Sons, Ltd.

Drug eluting sutures: A model for in vivo estimations

This work is focused on the development of a transient 1-dimension model to describe drug release from a bioresorbable suture thread in a living tissue and the pharmacologic behavior of the active substance being delivered from the device into the tissue. The model is based on fundamental conservation laws, represented by mass balances, being the thread degradation described through population balances and involving detailed hydrolysis kinetics. Monomer, water and drug diffusion are assumed as Fickian, and the increasing of diffusion coefficient is expressed with the “free volume” theory. Drug behavior in tissue is described with a “diffusion and reaction” approach. The model leads to a system of partial differential equations solved by applying the method of lines and then numerically integrated. Simulations allowed to estimate release dynamics and drug behavior in tissue and to obtain spatial and temporal profiles of drug in tissue. Moreover, phase diagrams, which show drug effect in time and space, are here introduced for the first time.

Conservation law of operator current in open quantum systems

We derive a fundamental conservation law of operator current for master equations describing reduced quantum systems. If this law is broken, the temporal integral of the current operator of an arbitrary system observable does not yield in general the change of that observable in the evolution. We study Lindblad-type master equations as examples and prove that the application of the secular approximation during their derivation results in a violation of the conservation law. We show that generally any violation of the law leads to artificial corrections to the complete quantum dynamics, thus questioning the accuracy of the particular master equation.

Hydrodynamics of fractal continuum flow

A model of fractal continuum flow employing local fractional differential operators is suggested. The generalizations of the Green-Gauss divergence and Reynolds transport theorems for a fractal continuum are suggested. The fundamental conservation laws and hydrodynamic equations for an anisotropic fractal continuum flow are derived. Some physical implications of the long-range correlations in the fractal continuum flow are briefly discussed. It is noteworthy to point out that the fractal (quasi)metric defined in this paper implies that the flow of an isotropic fractal continuum obeying the Mandelbrot rule of thumb for intersection is governed by conventional hydrodynamic equations.

Dimension reduction numerical closure method for advection–diffusion-reaction systems

Many natural physical processes allow different mathematical descriptions on different scales. The microscale description is usually based on fundamental conservation laws that form a closed system of ordinary differential equations (ODEs) or partial differential equations (PDEs) but the numerical discretization of these equations may produce a system of ODEs with an enormous number of unknowns. Furthermore, time integration of the microscale equations usually requires time steps that are smaller than the observation time by many orders of magnitude. A direct solution of these ODEs can be extremely expensive. Often, we are only interested in the average behavior of the microscale system rather than the exact solution of the ODEs. Here we propose a novel dimension reduction computational closure (DRNC) method that gives an approximate solution of the ODEs and provides an accurate prediction of the average behavior. The DRNC method consists of two main elements. First, effective ODEs for evolution of average variables (e.g. average velocity, concentration and mass of a mineral precipitate) are obtained by averaging the micro-scale ODEs over the entire micro-scale domain. These effective ODEs contain non-local terms in the form of volume integrals of functions of the micro-scale variables. Second, a numerical closure is used to close the system of the effective equations. The numerical closure is achieved via short bursts of the microscale model. The DRNC method is used to simulate flow and transport with mixing controlled reactions and mineral precipitation by reducing porescale (microscale) Navier–Stokes and advection–diffusion-reaction equations to ODEs for averaged velocity and concentrations. Good agreement between direct solutions of the microscale equations and DRNC solutions for different boundary conditions and Damkohler numbers confirms the accuracy and computational efficiency of DRNC method. The DRNC method significantly accelerates microscale simulations, while providing accurate approximation of the microscale solution and accurate prediction of the average behavior of the system.

The role of self‐coherence in correlations of bosons and fermions in linear counting experiments. Notes on the wave‐particle duality

AbstractCorrelations of detection events in two detectors are studied in case of linear excitations of the measuring apparatus. On the basis of classical probability theory and fundamental conservation laws, a general formula is derived for the two‐point correlation functions for both bosons and fermions. The results obtained coincide with that derivable from quantum theory which uses quantized field amplitudes. By applying both the particle and the wave picture at the same time, the phenomena of photon bunching and antibunching, photon anticorrelation and fermion antibunching measured in beam experiments are interpreted in the frame of an intuitively clear description.

Study of species, temperature distributions and the solid oxide fuel cells performance in a 2-D model

A two-dimensional mathematical model for a planar SOFC (solid oxide fuel cell) is constructed. The distribution of the chemical species, the temperature and the performance (power) and the current density were calculated using a single-unit model with double channels of co-flow pattern. The finite volume method was employed for the calculation. The method was based on the fundamental conservation laws of continuity, momentum, energy and mass. The equations are implemented in FORTRAN language. The effects of several heat sources and flow rates on the calculated results were also investigated. The reference SOFC polarization curve has been calculated by imposing a uniform temperature of 800 K, a pressure equal to 1 atm; H2 and O2 molar fractions equal to 0.97 and 0.21 respectively. Results of temperature, chemical species distributions, performance and efficiency under several heat sources are shown and discussed. At a current density of about 23500 A/m2, the power densities under all sources and chemical sources reached their maximums of 12965 W/m2 and 16209 W/m2 (i.e. 25% lower) respectively. However the temperature increment in the anode is analyzed toward all sources and chemical reaction. The temperature maximum values for each heat source type reached 1005.81 K and 984.69 K respectively.

Study on the Coupled Neutronic and Thermal-Hydraulic Characteristics of the New Concept Molten Salt Reactor

The new concept molten salt reactor is the only liquid-fuel reactor of the six Generation IV advanced nuclear energy systems. The liquid molten salt serves as the fuel and coolant simultaneously and causes one important feature: the delayed neutron precursors are drifted by the fuel flow, which leads the spread of delayed neutrons’ distribution to noncore parts of the primary circuit, and it also results in reactivity variation depending on the flow condition of the fuel salt. Therefore, the neutronic and thermal-hydraulic characteristics of the molten salt reactor are quite different from the conventional nuclear reactors using solid fissile materials. Besides, there is no other reactor design theory and safety analysis methodologies can be used for reference. The neutronic model is derived based on the conservation of particles considering the flow effect of the fuel salt in the molten salt reactor, while the thermal-hydraulic model applies the fundamental conservation laws: the mass, momentum, and energy conservation equations. Then, the neutronic and thermal-hydraulic calculations are coupled and the influences of inflow temperature and flow velocity on the reactor physical properties are obtained. The calculated results show that the flow effect on the distributions of thermal and fast neutron fluxes is very weak, as well as on the effective multiplication factor keff, while the flow effect on the distribution of delayed neutron precursors is much stronger. The inflow temperature influences the distribution of neutron fluxes and delayed neutron precursors slightly, and makes a significant negative reactivity. Coupled calculation also reveals that the flow velocity of molten salt has little effect on the distribution of neutron fluxes in the steady-state, but affects the delayed neutron precursors’ distribution significantly.

Charge Conservation, Klein’s Paradox and the Concept of Paulions in the Dirac Electron Theory

An algebraic block-diagonalization of the Dirac Hamiltonian in a time-independent external field reveals a charge-index conservation law which forbids the physical phenomena of the Klein paradox type and guarantees a single-particle nature of the Dirac equation in strong external fields. Simultaneously, the method defines simpler quantum-mechanical objects—paulions and antipaulions, whose 2-component wave functions determine the Dirac electron states through exact operator relations. Based on algebraic symmetry, the presented theory leads to a new understanding of the Dirac equation physics, including new insight into the Dirac measurements and a consistent scheme of relativistic quantum mechanics of electron in the paulion representation. Along with analysis of the mathematical anatomy of the Klein paradox falsity, a complete set of paradox-free eigenfunctions for the Klein problem is obtained and investigated via stationary solutions of the Pauli-like equations with respective paulion Hamiltonians. It is shown that the physically correct Dirac states in the Klein zone are characterized by the total particle reflection from the potential step and satisfy the fundamental charge-index conservation law.

Ballistic heat transport of quantum spin excitations as seen inSrCuO2

Fundamental conservation laws predict ballistic, i.e., dissipationless transport behaviour in one-dimensional quantum magnets. Experimental evidence, however, for such anomalous transport has been lacking ever since. Here we provide experimental evidence for ballistic heat transport in a S=1/2 Heisenberg chain. In particular, we investigate high purity samples of the chain cuprate SrCuO2 and observe a huge magnetic heat conductivity $\kappa_{mag}$. An extremely large spinon mean free path of more than a micrometer demonstrates that $\kappa_{mag}$ is only limited by extrinsic scattering processes which is a clear signature of ballistic transport in the underlying spin model.

Dynamic finite deformation viscoelasticity and energy‐consistent time integration

AbstractIn the present work we deal with the conserving integration of viscoelastic bodies undergoing finite deformations.Isotropic viscoelastic materials can be described by using a symmetric viscous internal variable ${\cal C}_i$ for measuring the inelastic strains, and the right Cauchy‐Green tensor $\cal C$ as measure of the total strain (see Reese & Govindjee [1]). Then, by using the unsymmetric product tensor ${\cal C} {\cal C}_i^{-1}$, the purely elastic strains enter the isotropic free energy function. Alternatively, the i application of the symmetric right stretch tensor ${\cal U}_i$ as internal variable allows to define a symmetric elastic strain tensor ${\cal C}_e$ which enters the free energy (see also Miehe [2]). These two different approaches lead to different evolution equations for the viscous internal variable.In this lecture, both evolution equations are discretised by an ordinary midpoint rule at each Gauss point of a standard nonlinear displacement‐based finite element in space. For discretising the semidiscrete Hamilton's equations of motion in time, we use numerical time integrators which preserve the fundamental conservation laws of the underlying system. In particular, we make use of a modified midpoint rule according to the discrete gradient method, proposed in Gonzalez [3]. Numerical examples demonstrate the difference between both formulations. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A two-fluid model of abrasive waterjet

Waterjet flow added with abrasive solid particles, the so called abrasive waterjet, can enhance its performance for cutting various materials from soft food products to very hard titanium alloys. In this study, a theoretical analysis is conducted in order to develop a flow model for the abrasive waterjet. The main concern is whether the abrasive particles can be treated as a pseudo-fluid phase. A two-fluid model is developed based on the fundamental laws of conservation. A control volume method is used to discretize the equations, and a phase-coupled SIMPLE algorithm is adopted to solve the pressure–velocity coupling equations. The quasi two-dimensional flow field outside a conventional nozzle used in abrasive waterjet is analyzed and computed to validate the model. Good agreement is observed comparing the numerical results with the experimental measurements.

Dynamic model for control system design and simulation of a low temperature nuclear reactor

A lumped parameter dynamic model for the primary-loop and the U-tube steam generator of a low temperature power reactor is developed based on the fundamental conservation laws of fluid mass, energy and momentum. The dynamic model is formulated by coupling the point kinetics with reactivity feedback and the thermal-hydraulics of the reactor. The developed dynamic model is implemented on a personal computer using MATLAB/SIMULINK. Numerical simulation results for steady-state and transient responses are then presented, which show that the steady-state precision of the newly developed dynamic model is acceptable and the trend of the transient responses is correct. In addition, the “swell and shrink” behavior of the U-tube steam generator is also verified by numerical simulation. This newly established model can be utilized to control system design and simulation for the low temperature power reactor.

A Simple Mathematical Approach to Data Reconciliation in a Single-Shaft Combined Cycle System

The ultimate proof of the soundness and viability of a novel technology is a full-scale demonstration test in which actual components are run successfully over the entire operating envelope. Consequently, the collection of accurate and meaningful test data is of utmost importance to the success of the test. An analysis of such data will validate the original design concepts and will lead to paths of further improvement for the next generations thereof. Statistical fundamentals to determine the accuracy and precision of measured data are amply documented and readily available in well-established standards. The yardstick that should be used for the “meaningfulness” of the measured test data is the satisfaction of the fundamental laws of conservation. While it is known that the “true” values of the sensor data when inserted into the governing equations for the tested component will result in perfect balances, “actual” measured values will always result in “imbalances.” Therefore, reconciliation of the individual measurements with the governing conservation equations is a must prior to the actual analysis of the data. Reconciliation in this context is an estimation of the true values of the sensor data from the actual sensor data by using statistical concepts. This paper describes the development of a data reconciliation concept that is universally applicable to any process or power plant system where sensor data are used. The usefulness and power of the technique are demonstrated by its application to a single-shaft combined cycle with both gas turbine and steam turbine driving a common generator. In the absence of a reliable and accurate measuring system to individually determine gas and steam turbine shaft outputs, data reconciliation is vital to an accurate analysis of the data.

Material models in principal stretches for elastodynamics

AbstractIn the present work we deal with the conserving integration of elastic bodies undergoing finite deformations. In particular, we make use of constitutive laws formulated in terms of principal stretches.Most material models for hyperelastic isotropic materials are described in terms of principal stretches (Simo and Taylor [1]), like the Neo–Hooke material which is a special case of the Ogden material, or in invariants. The main advantage of principal stretches is the fact that they can be measured directly, which means that the numerical results can be compared easily with experimental ones, see for example, Ogden [2]. Moreover, it is advantageous to describe viscoelastic material behaviour (e.g. rubberlike materials) in terms of principal stretches.Concerning the discretization in space we apply the enhanced assumed strain (EAS) method, see Simo and Armero [3]. For the discretization in time we aim at numerical integrators which inherit fundamental conservation laws from the underlying continuous system. In particular, we propose an energy and momentum conserving time–stepping scheme which relies on the notion of a discrete gradient (or derivative) in the sense of Gonzalez [4]. The proposed approach starts from our previous developments in [5]. Numerical examples demonstrate the advantageous properties of the present formulation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)