Recent Numerical Solutions Research Articles

Bridging High-density Electron Beam Coronal Transport and Deep Chromospheric Heating in Stellar Flares

The optical and near-ultraviolet (NUV) continuum radiation in M-dwarf flares is thought to be the impulsive response of the lower stellar atmosphere to magnetic energy release and electron acceleration at coronal altitudes. This radiation is sometimes interpreted as evidence of a thermal photospheric spectrum with T ≈ 104 K. However, calculations show that standard solar flare coronal electron beams lose their energy in a thick target of gas in the upper and middle chromosphere (log10 column mass/[g cm−2] ≲ −3). At larger beam injection fluxes, electric fields and instabilities are expected to further inhibit propagation to low altitudes. We show that recent numerical solutions of the time-dependent equations governing the power-law electrons and background coronal plasma (Langmuir and ion-acoustic) waves from Kontar et al. produce order-of-magnitude larger heating rates than those that occur in the deep chromosphere through standard solar flare electron beam power-law distributions. We demonstrate that the redistribution of beam energy above E ≳ 100 keV in this theory results in a local heating maximum that is similar to a radiative-hydrodynamic model with a large, low-energy cutoff and a hard power-law index. We use this semiempirical forward-modeling approach to produce opaque NUV and optical continua at gas temperatures T ≳ 12,000 K over the deep chromosphere with log10 column mass/[g cm−2] of −1.2 to −2.3. These models explain the color temperatures and Balmer jump strengths in high-cadence M-dwarf flare observations, and they clarify the relation among atmospheric, radiation, and optical color temperatures in stellar flares.

The high-speed submerged hydrofoil

This paper gives, in the limit of large Froude number, a closed-form, analytical solution for steady, two-dimensional, inviscid, free-surface attached flow over a submerged planar hydrofoil for arbitrary angles of attack and depths of submergence. The doubly connected flow domain is conformally mapped to a concentric annulus in an auxiliary plane. The complex flow potential and its derivative, the complex velocity, are obtained in the auxiliary plane by considering their form at known special points in the flow, and the required conformal mapping is determined by explicit integration. The four real solution parameters are determined as the simultaneous roots of four real nonlinear algebraic equations arising from the flow normalisation. The explicit form allows accurate evaluation of various flow quantities, including the lift on the foil, and these are related to the large-Froude-number results in recent numerical solutions.

Particle dispersion and deposition in wall-bounded turbulent flow

Particle dispersion and deposition in a fully developed turbulent channel flow at a friction Reynolds number of Reτ=180 for a wide range of Stokes numbers (St = 1-130) are investigated using the direct numerical simulation (DNS). The combined effect of gravity and the Saffman lift force on the particle deposition velocities in upward and downward flows are considered. The results are compared with recent experimental measurements and numerical benchmark solutions. For particle–wall collisions, two scenarios of fully elastic and no-rebound (trap-wall) collisions are considered. Due to the turbophoresis effect and the steady migration of particles toward the walls, a stationary state cannot be reached for particle concentration with the fully elastic assumption. To alleviate this problem, for every particle that impacts the wall a new particle is added to the computational domain with the trap-wall assumption. The stationary particle field is further time-averaged to present the average particle velocities, particle concentrations, and particle velocity fluctuations profiles. It is shown that at lower Stokes numbers, the deposition velocity of downward channel flow is higher than the upward flow, whereas at higher Stokes numbers the upward flow has a higher deposition rate but a lower near-wall particle concentration. This higher deposition rate is attributed to the combined effect of gravity and the positive Saffman lift force that brings the inactive particles trapped in the viscous sublayer to the buffer layer, resulting in an increase of their wall-normal turbulent fluctuations and their wall-collision rates.

Medium-induced radiative kernel with the Improved Opacity Expansion

We calculate the fully differential medium-induced radiative spectrum at next-to-leading order (NLO) accuracy within the Improved Opacity Expansion (IOE) framework. This scheme allows us to gain analytical control of the radiative spectrum at low and high gluon frequencies simultaneously. The high frequency regime can be obtained in the standard opacity expansion framework in which the resulting power series diverges at the characteristic frequency ωc ∼ hat{q} L2. In the IOE, all orders in opacity are resumed systematically below ωc yielding an asymptotic series controlled by logarithmically suppressed remainders down to the thermal scale T « ωc, while matching the opacity expansion at high frequency. Furthermore, we demonstrate that the IOE at NLO accuracy reproduces the characteristic Coulomb tail of the single hard scattering contribution as well as the Gaussian distribution resulting from multiple soft momentum exchanges. Finally, we compare our analytic scheme with a recent numerical solution, that includes a full resummation of multiple scatterings, for LHC-inspired medium parameters. We find a very good agreement both at low and high frequencies showcasing the performance of the IOE which provides for the first time accurate analytic formulas for radiative energy loss in the relevant perturbative kinematic regimes for dense media.

Importance of on-time decision making in energy sector based on perspectives: Case study new Stavalj project

Expression "Development conditions and perspectives" became important for numerous analyses in various industrial and social areas. Several strategic documents and studies in last two decades analysed projects with topics on perspectives of future development in Serbia. Various tools are used for development of such documents, based on recent scientific and numerical solutions, thus providing reliable assessment for strategic decision making. Almost all analyses tried to implement the theories and practical experiences through the prism of "sustainable development", which included establishment of most important sustainability parameters. Analysis and ranking presented in this paper considered the potential of the Stavalj coal deposit, near city of Sjenica in Serbia, and feasibility of construction project of new mine and thermal power plant. Basis for analysis was a hybrid assessment model which takes into account principles of sustainable development. The model incorporates quantified SWOT analysis, which applies to active underground mines in Serbia. Special attention was given to the parameters describing potential for development.

Further Results on Guderley Mach Reflection and the Triple Point Paradox

Recent numerical solutions and shock tube experiments have shown the existence of a complex reflection pattern, known as Guderley Mach reflection, which provides a resolution of the von Neumann paradox of weak shock reflection. In this pattern, there is a sequence of tiny supersonic patches, reflected shocks and expansion waves behind the triple point, with a discontinuous transition from supersonic to subsonic flow across a shock at the rear of each supersonic patch. In some experiments, however, and in some numerical computations, a distinctly different structure which has been termed Guderley reflection has been found. In this structure, there appears to be a single expansion fan at the triple point, a single supersonic patch, and a smooth transition from supersonic to subsonic flow at the rear of the patch. In this work, we present numerical solutions of the compressible Euler equations written in self-similar coordinates at a set of parameter values that were used in previous computations which found the simple single patch structure described above. Our solutions are more finely resolved than these previous solutions, and they show that Guderley Mach reflection occurs at this set of parameter values. These solutions lead one to conjecture that the two patterns are not distinct: rather, Guderley reflection is actually underresolved Guderley Mach reflection.

The dome and the ring: Verification of an old mathematical model for the design of a stiffened shell roof

We study an old mathematical model, developed before the computer era, for analyzing the strength of a stiffened shell roof. The specific problem considered is a textbook example presented in K. Girkmann: Flächentragwerke, 3rd edition, 1954. Here the roof consists of a spherical dome and a stiffening ring of rectangular cross section attached to the edge of the dome. The problem is to compute the resultant force and moment acting at the junction of the dome and the ring. We approach the old model for solving the problem in two different ways. First we carry out a historical study, where we look for possible improvements of the old model while limiting ourselves to manual computations only. We find a variant of the model which, despite being about as simple as the original one, is considerably more accurate in comparison with recent numerical solutions based on FEM and axisymmetric 3D elastic formulation of the problem. The second approach in our study is to carry out an a posteriori error analysis of our refined old model. The analysis is based on variational methods and on the Hypercircle theorem of the linear theory of elasticity. The error analysis confirms, and largely also explains, the observed–rather high–accuracy of the refined old mathematical model.

Large amplitude internal solitary waves in a two-layer system of piecewise linear stratification

We study large amplitude internal solitary waves in a two-layer system where each layer has a constant buoyancy frequency (or Brunt–Väisälä frequency). The strongly nonlinear model originally derived by Voronovich [J. Fluid Mech. 474, 85 (2003)] under the long wave assumption for a density profile discontinuous across the interface is modified for continuous density stratification. For a wide range of depth and buoyancy frequency ratios, the solitary wave solutions of the first two modes are described in detail for both linear-constant and linear-linear density profiles using a dynamical system approach. It is found that both mode-1 and mode-2 solitary waves always point into the layer of smaller buoyancy frequency. The width of mode-1 solitary waves is found to increase with wave amplitude while that of mode-2 solitary waves could decrease. Mode-1 solitary wave of maximum amplitude reaches the upper or lower wall depending on its polarity. On the other hand, mode-2 solitary wave of maximum amplitude can reach the upper or lower wall only when the interface is displaced toward the shallower layer; otherwise, the maximum wave amplitude is smaller than the thickness of the deeper layer. Streamlines and various physical quantities including the horizontal velocity and the Richardson number are computed and discussed in comparison with the recent numerical solutions of the Euler equations by Grue et al. [J. Fluid Mech. 413, 181 (2000)].

Convective mass transfer across fluid interfaces in straight angular pores

Steady convective mass transfer to or from fluid interfaces in pores of angular cross-section is theoretically investigated. This situation is relevant to a variety of mass transport process in porous media, including the fate of residual non-aqueous phase liquid ganglia and gas bubbles. The model incorporates the essential physics of capillarity and solute mass transfer by convection and diffusion in corner fluid filaments. The geometry of the corner filaments, characterized by the fluid–fluid contact angle, the corner half-angle and the interface meniscus curvature, is accounted for. Boundary conditions of zero surface shear (‘perfect-slip’) and infinite surface shear (‘no-slip’) at the fluid–fluid interface are considered. The governing equations for laminar flow within the corner filament and convective diffusion to or from the fluid–fluid interface are solved using finite-element methods. Flow computations are verified by comparing the dimensionless resistance factor and hydraulic conductance of corner filaments against recent numerical solutions by Patzek and Kristensen (J. Colloid Interface Sci 236, 305–317 2001). Novel results are obtained for the average effluent concentration as a function of flow geometry and pore-scale Peclet number. These results are correlated to a characteristic corner length and local pore-scale Peclet number using empirical equations appropriate for implementation in pore network models. Finally, a previously published “2D-slit” approximation to the problem at hand is checked and found to be in considerable error.

A computational procedure for finding multiple solutions of convective heat transfer equations

In recent years numerical solutions of the convective heat transfer equations have provided significant insight into the complex materials processing operations. However, these computational methods suffer from two major shortcomings. First, these procedures are designed to calculate temperature fields and cooling rates as output and the unidirectional structure of these solutions preclude specification of these variables as input even when their desired values are known. Second, and more important, these procedures cannot determine multiple pathways or multiple sets of input variables to achieve a particular output from the convective heat transfer equations.Here we propose a new method that overcomes the aforementioned shortcomings of the commonly used solutions of the convective heat transfer equations. The procedure combines the conventional numerical solution methods with a real number based genetic algorithm (GA) to achieve bi-directionality, i.e. the ability to calculate the required input variables to achieve a specific output such as temperature field or cooling rate. More important, the ability of the GA to find a population of solutions enables this procedure to search for and find multiple sets of input variables, all of which can lead to the desired specific output. The proposed computational procedure has been applied to convective heat transfer in a liquid layer locally heated on its free surface by an electric arc, where various sets of input variables are computed to achieve a specific fusion zone geometry defined by an equilibrium temperature. Good agreement is achieved between the model predictions and the independent experimental results, indicating significant promise for the application of this procedure in finding multiple solutions of convective heat transfer equations.

SUPERSYMMETRIC YANG-MILLS QUANTUM MECHANICS IN VARIOUS DIMENSIONS

Recent analytical and numerical solutions of above systems are reviewed. Discussed results include: a) exact construction of the supersymmetric vacua in two space-time dimensions, and b) precise numerical calculations of the coexisting, continuous and discrete, spectra in the four-dimensional system, together with the identification of dynamical supermultiplets and SUSY vacua. New construction of the gluinoless SO (9) singlet state, which is vastly different from the empty state, in the ten-dimensional model is also briefly summarized.

EXPERIMENTAL EVIDENCE ON THE STRUCTURE AND EVOLUTION OF THE PENTA–HEPTA DEFECT IN HEXAGONAL LATTICES DUE TO BÉNARD–MARANGONI CONVECTION

Evidence is provided to show that the penta–hepta defect in a Bénard convective hexagonal lattice arises as a result of the attraction of two dislocations having opposite topological charges and belonging to different modes. The dislocations approach each other along a corridor connecting those topological charges. Fields of long–living penta–hepta defects have been analyzed. The comparison of experimental results with recent numerical calculations and solutions obtained analytically shows good qualitative agreement.

Accurate approximate equations for intensive sub-sonic evaporation

The paper describes an approximate solution of the problem of intensive evaporation of a monatomic substance at a plane surface and with condensation coefficient equal to unity. The result is in the form of a set of algebraic equations which may readily be solved and is in excellent agreement with recent numerical solutions.

Analytical solutions for the heat and mass transfer in a falling film absorber

In this paper, fully analytical solutions for the temperature and mass fraction are obtained for a falling film absorber in an absorption heat pump. The analytical solutions described here lead to an analytical expression for the local and total mass absorbed. This situation is due to the fact that the vast majority of mass transfer takes place in a thin layer near the liquid-vapor interface and that the temperature distribution is conduction-dominated. A design procedure based on the analytical solutions is developed and results for a libr-water solution are compared with recent experiments and previous numerical solutions for the temperature and the agreement with each is found to be very good. Heat and mass transfer coefficients are outputs of the problem and it is shown that the heat transfer coefficient increases with decreasing film thickness, whereas the mass transfer coefficient decreases. This situation makes it difficult to evaluate design procedures based solely on increasing heat transfer coefficients gleaned from experiments or the open literature. It is shown that for given inlet bulk temperature, solutions may be obtained only for a limited range of mass fraction.

Thermal conductivity of a-Si:H thin films.

The thermal conductivity of sputtered a-Si:H thin films for a hydrogen content of 1--20 % and a film thickness of 0.2--1.5 \ensuremath{\mu}m is determined in the temperature range 80--400 K using an extension of the 3\ensuremath{\omega} measurement technique. The reliability of the method is demonstrated on 1-\ensuremath{\mu}m-thick a-${\mathrm{SiO}}_{2}$ thermally grown on Si. Scattering of phonons at the interface between the a-Si:H film and the substrate places a simple upper limit on the heat transport by long-wavelength phonons and facilitates the comparison of the experimental data to recent numerical solutions of a Kubo formula using harmonic vibrations.

Recent developments in “added-mass” planing theory

The results of several recent isolated investigations in planing theory are consolidated in this paper, together with new insights generated by a recent numerical solution of the vertically impacting wedge problem by Zhao and Faltinsen [(1992), Water entry of two-dimensional bodies. J. Fluid Mech. 246, 593–612]. As a result, in contrast to some earlier studies, it is found that the “wetted width” associated with the added mass is not that of the intersection of the wedge with the undisturbed water surface, but the wetted width of the splashed-up water, as originally proposed by Wagner [(1932), Uber Stoss-und Gleitvorgange an der Oberflache von Flussig-Keiten, Zeitschrift für Angewandte Mathematik und Mechanik, Band 12, Heft 4 (August)]. However, the splash-up ratio is not the value of (π/2–1) which he proposed, but a value which decreases with increasing deadrise, originally proposed in the late-1940s by Pierson (“Pierson's hypothesis” in the paper). For 30° deadrise, for example, Pierson's splash-up ratio is two-thirds that of Wagner's. The new equations are employed to determine the increase in the “added mass” of prismatic hull sections due to chine immersion, using experimental data. If m o ′ is the added amss of the hull section whose chines are just wetted, Payne [(1988), Design of High-speed Boats. Volume 1: Planing. Fishergate, Inc., Annapolis, Maryland, U.S.A.] postulated that the increase in added mass due to a chine submergence ( z c ) would be Δm ′ =km ′ o c b where b is the chine beam and k is a constant which Payne [(1988), Design of High-speed Boats. Volume 1: Planing. Fishergate, Inc., Annapolis, Maryland, U.S.A.] gave as k = 4 3 . The present analysis includes the “one-sided flow” correction introduced in Payne [(1990), Planing and impacting forces at large trim angels. Ocean Engng 17, 201–234]. Partly for that reason and partly because of the more precise analysis of the experimental data, the present paper revises the value to k = 2 for wetted length to beam ratios normally employed. For deadrise angles in excess of 40° and wetted keel to beam ratios in excess of 2.0, there is some evidence that k < 2.0. The revised theoretical formulation is compared with eight different sets of experimental data for flat plate and prismatic hull forms and is found to be in excellent agreement when the speed is high enough for “dynamic suction” (a loss of buoyancy at low speeds and low wetted lenghts) to be unimportant. This is true for “chines-dry” operation with deadrise angles up to 50° and chines-wet operation at length to beam ratios far in excess of the most extreme conventional practice. The research involved in performing this analysis led to the realization that different towing tanks measure different wetted chine lengths for the same hulls and test conditions. Some consistently measure more splash-up than “theory” (based on Pierson's splash-up hypothesis) predicts and others measure somewhat less than the theory. Some examples are given in Appendix B. The reason for this is not understood.

A direct theory of viscous fluid flow in pipes II. Flow of incompressible viscous fluid in curved pipes

This paper is a companion to Part I under the same title and is concerned with the development of an analytical solution (via a direct formulation) for flow of an incompressible viscous fluid in curved pipes. Although a major portion of the analysis is presented for circular pipes of elliptical cross section, detailed calculations are carried out only for pipes of circular cross section. These calculations include the friction loss factor and the velocity contours, both of which are presented over a range of Dean number th a t significantly goes beyond the corresponding range of recent numerical solutions and the available experimental data. The results obtained show very favourable agreements with existing experimental data and several recent numerical solutions of the same problem based on the Navier—Stokes equations. Also included (for pipes of circular cross section) is a comparison of the multiple solutions and bifurcation points, as predicted by the present analytical solution, with corresponding available inform ation from several recent num erical solutions of the problem.

Multiple equilibria in two-dimensional thermohaline circulation

As a model of the thermohaline circulation of the ocean we study the two-dimensional Boussinesq equations forced by prescribing the surface temperature and the surface salinity flux. We simplify the equations of motion using an expansion based on the small aspect ratio of the domain. The result is an amplitude equation governing the evolution of the depth averaged salinity field. This amplitude equation has multiple, linearly stable equilibria. The simplified dynamics has a Lyapunov functional and this variational structure permits a simple characterization of the relative stability of the alternative steady solutions.Even when the thermal and salinity surface forcing functions are symmetric about the equator there are asymmetric solutions, representing pole to pole circulations. These asymmetric solutions are stable to small perturbations and are always found in conjunction with symmetric solutions, also stable to small perturbations. Recent numerical solutions of the full two-dimensional equations have shown very similar flow patterns.

Onset of periodic convection in a vertical slot

New observations of the convective states in a vertical slot in the neighborhood of the transition to multicellular motion are reported. Experiments in large Prandtl number fluids reveal that the departure from unicellular flow develops gradually in the central region of the slot with an initial closure of the distorted primary flow. Subsequently, as the Rayleigh number increases, a multiplicity of transverse stationary eddies occur in the same vicinity within the primary flow. Similar observations in an air-filled slot display a different response. In this case the initial perturbation of the base flow, resulting in closure of the unicellular motion takes place near the end walls. These and other observations illustrate that differences arise because of the Prandtl number. Measurements of the thermal stratification in the core of the oil-filled cavity are in good agreement with recent numerical solutions, based on the boundary layer approximation. The results are also found to be in reasonable accord with the neutral curve predicted from a linear stability analysis.

Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers

Abstract Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.