Steady-state Numerical Solutions Research Articles

Analytical and Numerical Models of a New Hybrid System of Earth Air Tunnel Integrated Solar Air Heater With Sensible Storage

Abstract A novel combination of an earth air tunnel (EAT) and a sensible thermal storage-aided solar air heater has been proposed when the sensible storage medium is taken to be rocks filled within the air flow passage. Both time-dependent and steady-state models are reported. The former describes the exact thermal performance of the system for a two-month wintertime for the city of Baghdad, Iraq. The latter describes average performance of the system. The implicit finite difference method and method of separation of variables have been used to solve the pertinent equations in the respective models. Acceptable level of accuracy was obtained between the steady-state numerical and analytical solutions as well as between the numerical and the published data. It is revealed that preheating by a sufficiently long EAT module improves the effective power as well as the output temperature from the rock bed solar air heater by about 35% and 9% respectively for the present set of parameters considered. It is also observed that for the present set of parameters considered, the temperature gradients in the direction normal to air flow in the solar air heater are insignificant and may be ignored. A parametric study is also carried out that assesses the impact of system parameters on the quality as well as quantity of the energy extracted from the system. This work is hence the first to couple an EAT with a sensible thermal storage equipped solar air heater and may pave the way for future studies.

Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.

Modeling inductive radio frequency coupling in powerful negative hydrogen ion sources: validating a self-consistent fluid model

Radio frequency (RF) negative hydrogen ion sources utilized in fusion and for accelerators use inductively coupled plasmas, which are operated at a low driving frequency, high power densities and gas pressures in the order of 1 MHz, 10 W cm−3 and 1 Pa, respectively. In this work a numerical fluid model is developed for a self-consistent description of the RF power coupling in these discharges. After validating the RF power coupling mechanism, such a model is a valuable tool for the optimization of RF power coupling and hence can help to increase the efficiency and reliability of RF ion sources. The model validation is achieved using measurements from the ITER RF prototype ion source. Steady state numerical solutions are obtained for the first time, where all modeled trends fit well. Remaining systematic quantitative differences could be caused by 3D effects such as highly non-uniform magnetic fields that cannot be captured in the current model formulation, which is 2D cylindrically symmetric. The coupling between the RF fields and the electrons is realized in the electron momentum transport equation, where approximations consistent with the operating regime of RF ion sources are applied. Here large magnetic RF fields lead to a plasma compression by the nonlinear RF Lorentz force. Using a local approximation for the electron viscosity, it is found that increased diffusion of the RF current density mitigates the compression. Navier–Stokes equations for the neutral atoms and molecules are used to capture neutral depletion. In this way it is shown that at high powers neutral depletion has a large impact on the power coupling via the viscosity of the electrons. The application of the self-consistent model for optimization of the RF power coupling will be described in a forthcoming paper.

Investigating Neutral and Stable Atmospheric Surface Layers Using Computational Fluid Dynamics

Computational fluid dynamics (CFD) is an effective technique for investigating atmospheric processes at a local scale. For example, in near-source atmospheric dispersion applications, the effects of meteorology, air-pollutant sources, and buildings can be included. A prerequisite is to establish realistic atmospheric conditions throughout the computational domain. This work investigates the modeling of the atmospheric surface layer under neutral and stable boundary-layer conditions, respectively. Steady-state numerical solutions of the Reynolds averaged Navier–Stokes equations were used, including the k-ε turbulence model. Atmospheric profiles derived from the Cooperative Atmosphere–Surface Exchange Study-99 (Kansas, USA) were used as reference data. The results indicate that the observed profiles of velocity and potential temperature can be reproduced with CFD, while turbulent kinetic energy showed less agreement with the observations. For the stable boundary layer, reasonable agreement of the numerical results with the observations was also obtained for surface layer depth, shear stress, and heat-flux profiles, respectively. The results are discussed in relation to the boundary conditions and sources, and the observation data.

A Simplified Way to Evaluate the Effect of Temperature on a Circular Tunnel

Temperature differences between the surrounding ground and the tunnel lining will cause a variation of the tunnel lining forces. The hyperstatic reaction method (HRM) could be efficiently and simply used to investigate the impact of thermal load on tunnel linings. First, the steady state numerical solution is derived for a shallow circular tunnel to estimate the internal forces and displacements of tunnel lining caused by thermal load. The effective strain coefficient βTl is deduced to calculate the thermal stresses in tunnel lining. Secondly, the influence of the temperature difference on the internal forces of tunnel lining is investigated using the HRM method considering different lining elastic modulus, lining thickness and ground coefficient of thermal expansion. Lastly, the impact of fires which will be able to modify the elastic modulus of tunnel lining is investigated, which makes it possible to predict the damage of tunnel lining caused by fires.

Identification of the influencing input parameters and their effects on the split dynamics of liquid–liquid two-phase mixture through T-junction: a numerical analysis

Split dynamics of the liquid–liquid two-phase mixture through horizontal T-junction is numerically investigated by varying the parameters like inlet-volume-fraction, inlet-mixture-velocity, liquid pair, and conduit-diameter. Ultimately, the effects of these parameters on the said dynamics are extracted. 3D steady-state numerical solutions are achieved by adopting a Finite Volume-based Eulerian Multi-Fluid VOF model. To simulate the turbulence, SST k–w model is used. Before producing the results, the adopted numerical methodology is successfully validated with the results available in the literature. Composition of the liquid–liquid two-phase mixture obtained along the branch and run arms differ from that along inlet arm. Again, that composition along the branch and run arm are also different from each other. Inertia imbalance is found as the most vital factor to influence the dynamics of phase split. More split is obtained for the mixture with high density-difference between the component liquids. No significant phase split is found for a liquid pair when any one of them has high viscosity. Depending on the conduit-diameter, liquid pair, and inlet-mixture-velocity, a critical inlet-volume-fraction (> 0.5) is found where no split occurs, and the phenomenon of phase split is reversed at the critical inlet-volume-fraction. As the inlet-velocity increases, critical inlet-volume-fraction increases for T-junction having a smaller conduit-diameter, but it increases very slowly for a larger conduit-diameter. With the increment in inlet-velocity, effective phase split reduces for smaller conduit-diameter, and it slightly increases for a larger conduit-diameter. The intensity of the split decreases with the increment in conduit-diameter. Larger critical inlet-volume-fraction is found for a larger conduit-diameter.

A mathematical analysis of an isothermal tube drawing process

The aim of this paper is to investigate stability of an isothermal glass tube drawing process through control parameters by the method of linear stability analysis. We want to see how the process parameters effect stability of the physical system being considered. For this purpose, we not only prove the existence and uniqueness of the solutions of steady state isothermal tube drawing model but also determine its numerical solution. To perform linear stability analysis, steady state numerical solution is incorporated in the eigenvalue problem, formulated by linearizing the isothermal model. The eigenvalue problem is then solved numerically to determine the critical draw ratio which indicates the onset of instabilities. To the end, stability of the process is analyzed using three different values of space step size. We also observe and discuss the effect of density, viscosity and pressure on stability of the isothermal tube drawing model.

Prediction of mechanical hemolysis in medical devices via a Lagrangian strain-based multiscale model.

This work introduces a new Lagrangian strain-based model to predict the shear-induced hemolysis in biomedical devices. Current computational models for device-induced hemolysis usually utilize empirical fitting of the released free hemoglobin (Hb) in plasma from damaged red blood cells (RBCs). These empirical correlations contain parameters that depend on specific device and operating conditions, thus cannot be used to predict hemolysis in a general device. The proposed algorithm does not have any empirical parameters, thus can presumably be used for hemolysis prediction in various blood-wetting medical devices. In contrast to empirical correlations in which the Hb release is related to the shear stress and exposure time without considering the physical processes, the proposed model links flow-induced deformation of the RBC membrane to membrane permeabilization and Hb release. In this approach, once the steady-state numerical solution of blood flow in the device is obtained under a prescribed operating condition, sample path lines are traced from the inlet of the device to the outlet to calculate the history of the shear stress tensor. In solving the fluid flow, it is assumed that RBCs do not have any influence on the flow pattern. Along each path line, shear stress tensor will be input into a coarse-grained (CG) RBC model to calculate the RBC deformation. Then the correlations obtained from molecular dynamics (MD) simulations are applied to relate the local areal RBC deformation to the perforated area on the RBC membrane. Finally, Hb released out of transient pores is calculated over each path line via a diffusion equation considering the effects of the steric hindrance and increased hydrodynamic drag due to the size of the Hb molecule. The total index of hemolysis (IH) is calculated by integration of released Hb over all the path lines in the computational domain. Hemolysis generated in the Food and Drug Administration (FDA) nozzle and two blood pumps, that is, a CentriMag blood pump (a centrifugal pump) and HeartMate II (an axial pump), for different flow regimes including the laminar and turbulent flows are calculated via the proposed algorithm. In all the simulations, the numerical predicted IH is close to the range of experimental data. The results promisingly indicate that this multiscale approach can be used as a tool for predicting hemolysis and optimizing the hematologic design of other types of blood-wetting devices.

Mathematical Modelling Time Dependent for Heat Transfer Process in Homogenous Composite ZnO/Activated CNF

A theoretical study of heat transfer on composite Zinc Oxide (ZnO)/Activated Carbon Nanofibers (CNF) by using mathematical model from diffusion equation for stationary composite with various heating time was studied. The equation and boundary conditions are interpreted by using Taylor series expansion to obtain a steady-state numerical solution from the Jacobi iteration algorithm. The composites where put inside the furnace model by filaments on the two sides of the walls as the source of heat (general model for furnace). By using this model, the position of sample to receive the homogeneous temperature determined. We compared the steady-state and unsteady-state conditions to identify the temperature distribution at real-time. The results show that the heat distribution will reach an asymptotic state by performing a numerical analysis is 200 K and 300 K for 25 minute time of heating. In this study, shows the efficient way for annealing homogenous materials to identify the position inside the furnace which can be used as a guide to receive the homogenous temperature during the annealing processes.

Steady-State Analytical and Numerical Solutions of Confined and Unconfined Flows in Aquifers with Discontinuous Aquiclude

Solution to combined confined and unconfined flows in multiple aquifer systems has been a challenging issue for groundwater flow modelers. In this study, steady-state solutions to four different problems are presented using analytical and numerical methods. For analytical solution, the concept of discharge potential is used. In this method, the governing equation becomes the same for both confined and unconfined flows. Moreover, discharge potential satisfies Laplace’s Equation and hence the potential theory becomes applicable which makes easier to reach solutions to complex problems. The problem consisting of an aquifer with a discontinuous aquiclude was solved by using comprehensive potential which gives the solution in an entirely unconfined aquifer with equivalent boundary conditions. This facilitates the solutions to confined and unconfined regions above and below the aquiclude. For the numerical solutions, two different versions of MODFLOW, namely, MODFLOW-2005 and MODFLOW-NWT were used. Initially, two different resolution schemes were tested. One of the schemes yielded good predictions of heads for different types of input in which the flow direction and aquifer type varied.

Steady state numerical solutions for determining the location of MEMS on projectile

This paper is devoted to compare the numerical solutions for the steady and unsteady state heat distribution model on projectile. Here, the best location for installing of the MEMS on the projectile based on the surface temperature is investigated. Numerical iteration methods, Jacobi and Gauss-Seidel have been elaborated to solve the steady state heat distribution model on projectile. The results using Jacobi and Gauss-Seidel are shown identical but the discrepancy iteration cost for each methods is gained. Using Jacobi’s method, the iteration cost is 350 iterations. Meanwhile, using Gauss-Seidel 188 iterations are obtained, faster than the Jacobi’s method. The comparison of the simulation by steady state model and the unsteady state model by a reference is shown satisfying. Moreover, the best candidate for installing MEMS on projectile is observed at pointT(10, 0) which has the lowest temperature for the other points. The temperature using Jacobi and Gauss-Seidel for scenario 1 and 2 atT(10, 0) are 307 and 309 Kelvin respectively.

Comparison Between Equivalent Continuum and Discrete Crack Models of Transient Radon Transport at the Soil–Building Foundation Crack Interface Using TOUGH2/EOS7Rn

In the framework of radon risk management in France, it is necessary to enhance knowledge on radon transfer from its source to exposure areas (e.g., buildings) by developing simple, accurate, numerical models for transient radon transport in three-dimensional (3D) unsaturated porous materials. The equivalent continuum model (ECM) of flow and transport at the interface between the soil and cracks (fissures) in a building foundation (e.g., slab on grade, basement) is attractive, since equivalent (effective) continuum properties assigned to model cells can represent the combined effect of individual cracks and solid matrix of the cracked concrete of the foundation (slab and blocks walls). Although the ECM approach based on the volume averaging method has been used to model flow and transport through cracks at the soil–building interface, it has never been verified numerically. Thus, the goal of the present work is to develop an ECM using this averaging method and to quantify its uncertainties based on its comparison to an accurate numerical discrete crack model (DCM) for flow and transport in the crack. As a first step, the DCM implemented in the TOUGH2/EOS7Rn module has been verified numerically through a comparison to a reference 3D steady-state numerical solution for radon transport into a house with basement under constant negative pressure. Then, 3D results of the DCM and ECM approaches were compared, under time-dependent indoor–outdoor pressure differentials conditions, for two crack line configurations in the basement slab floor and two different soil configurations with different soil permeability and radium \(^{226}\)Ra mass content values. Results of this comparison show that, for a homogeneous soil configuration, discrepancies between ECM and DCM simulated indoor radon activity concentrations decrease with the increase in soil permeability, regardless crack line configuration in the slab floor and soil radium mass content. However, ECM uncertainties were not within the range of absolute errors on measured radon concentration for the higher soil permeability \((1\times 10^{-9}, 1\times 10 ^{-8} \hbox { m}^{2})\) and the higher \(^{226}\hbox {Ra}\) mass content values (4500 \(\hbox {Bq\;kg}^{-1})\), especially for high radon pics induced by sudden increase in indoor air pressure drop. Regardless soil \(^{226}\hbox {Ra}\) mass content and crack line configuration in the slab floor, the ECM showed to be conservative for the two-layered soil configuration with the presence of aggregates beneath the slab foundation, generally practiced in buildings constructions.

2-D unsteady free convective heat and mass transfer Newtonian Hartmann flow with thermal diffusion and Soret effects: Network model and finite differences

In this paper we present the numerical solution of the problem 2-D unsteady free convective heat and mass transfer flow over a moving semi-infinite vertical porous plate with thermal diffusion in presence of magnetic field, dissipative heat and Soret effects taking into account the induced magnetic field. The steady-state numerical solutions for the velocity field, induced magnetic field, temperature distribution and concentration distribution are obtained by the explicit finite difference method and the network simulation method. The obtained results have been shown graphically. The obtained numerical results were compared with other author data to demonstrate the efficiency of the method used, moreover all the cases the convergence is always quickly reached. An increase in Soret or Eckert number is found to strongly enhance the fluid velocity and temperature values, and this effect is reversed for the induced magnetic field. It is found that the flow velocity decreases with the increase in Hartmann and magnetic Prandtl number, and an opposite behaviour is observed for the fluid temperature. Applications of the study arise in the thermal plasma reactor modelling, the electromagnetic induction, MHD transport phenomena in chromatographic systems, and the magnetic field control of materials processing. The results show that both methods provide excellent approximations to the solution of this nonlinear system with high accuracy.

Two Approaches for Simulating the Burning Surface in Gas Dynamics

Two approaches for simulating the burning surface in gas dynamics by means boundary conditions and right sides in the equations involving Dirac delta function are discussed. A comparison of numerical steady-state solutions and the exact ones in one-dimensional approximation is performed for two approaches. It is shown that the numerical solutions obtained with the finite-difference scheme of first order accuracy on the base of two considered approaches converge to each other when the mesh refinement is applied. The numerical solution for the steady state problem coincides with the analytical one, if the pressure at the boundary cell face is set equal to the pressure in the center of the boundary cell.

A two-dimensional depth-averaged -rheology for dense granular avalanches

Steady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged ${\it\mu}(I)$-rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional ${\it\mu}(I)$-rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory.

MODIFIED METHOD OF CHARACTERISTICS VARIATIONAL MULTISCALE FINITE ELEMENT METHOD FOR TIME DEPENDENT NAVIER-STOKES PROBLEMS

In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent NavierStokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient.

THREE-FLUID, THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC SOLAR WIND MODEL WITH EDDY VISCOSITY AND TURBULENT RESISTIVITY

We have developed a three-fluid, three-dimensional magnetohydrodynamic solar wind model that incorporates turbulence transport, eddy viscosity, turbulent resistivity, and turbulent heating. The solar wind plasma is described as a system of co-moving solar wind protons, electrons, and interstellar pickup protons, with separate energy equations for each species. Numerical steady-state solutions of Reynolds-averaged solar wind equations coupled with turbulence transport equations for turbulence energy, cross helicity, and correlation length are obtained by the time relaxation method in the corotating with the Sun frame of reference in the region from 0.3 to 100 AU (but still inside the termination shock). The model equations include the effects of electron heat conduction, Coulomb collisions, photoionization of interstellar hydrogen atoms and their charge exchange with the solar wind protons, turbulence energy generation by pickup protons, and turbulent heating of solar wind protons and electrons. The turbulence transport model is based on the Reynolds decomposition and turbulence phenomenologies that describe the conversion of fluctuation energy into heat due to a turbulent cascade. In addition to using separate energy equations for the solar wind protons and electrons, a significant improvement over our previous work is that the turbulence model now uses an eddy viscosity approximation for themore » Reynolds stress tensor and the mean turbulent electric field. The approximation allows the turbulence model to account for driving of turbulence by large-scale velocity gradients. Using either a dipole approximation for the solar magnetic field or synoptic solar magnetograms from the Wilcox Solar Observatory for assigning boundary conditions at the coronal base, we apply the model to study the global structure of the solar wind and its three-dimensional properties, including embedded turbulence, heating, and acceleration throughout the heliosphere. The model results are compared with plasma and magnetic field observations on WIND, Ulysses, and Voyager 2 spacecraft.« less

Derivation of reference distribution functions for Tokamak-plasmas by statistical thermodynamics

A general approach for deriving the expression of reference distribution functions by statistical thermodynamics is illustrated, and applied to the case of a magnetically confined plasma. The local equilibrium is defined by imposing the minimum entropy production, which applies only to the linear regime near a stationary thermodynamically non-equilibrium state and the maximum entropy principle under the scale invariance restrictions. This procedure may be adopted for a system subject to an arbitrary number of thermodynamic forces, however, for concreteness, we analyze, afterwords, a magnetically confined plasma subject to three thermodynamic forces, and three energy sources: i) the total Ohmic heat, supplied by the transformer coil, ii) the energy supplied by Neutral Beam Injection (NBI), and iii) the RF energy supplied by Ion Cyclotron Resonant Heating (ICRH) system which heats the minority population. In this limit case, we show that the derived expression of the distribution function is more general than that one, which is currently used for fitting the numerical steady-state solutions obtained by simulating the plasma by gyro-kinetic codes. An application to a simple model of fully ionized plasmas submitted to an external source is discussed. Through kinetic theory, we fixed the values of the free parameters linking them with the external power supplies. The singularity at low energy in the proposed distribution function is related to the intermittency in the turbulent plasma.

Transient mixed convective flow of reactive viscous fluid in vertical tube

The problem of fully-developed transient mixed convection flow of viscous reactive fluid in a vertical tube is analysed both analytically and numerically. The study reports the effect of several operating parameters on the flow hydrodynamics and thermal characteristics. The solutions for transient velocity and temperature fields are obtained by implicit finite difference method. To check the accuracy of the numerical solution steady-state solutions for temperature and velocity field are obtained by using perturbation series method. Skin-friction and Nusselt number at the surface of tube are determined. A selected set of graphical results illustrating the effects of the various parameters involved in the problem including mixed convection parameter, pressure gradient, reactant consumption parameter on the velocity and temperature profiles as well as flow reversal situations and Nusselt numbers are presented and discussed.

Dynamics of the solar tachocline – III. Numerical solutions of the Gough and McIntyre model

We present the first numerical simulations of the solar interior to exhibit a tachocline consistent with the Gough and McIntyre (1998) model. We find nonlinear, axisymmetric, steady-state numerical solutions in which: (1) a large-scale primordial field is confined within the radiation zone by downwelling meridional flows that are gyroscopically pumped in the convection zone (2) the radiation zone is in almost-uniform rotation, with a rotation rate consistent with observations (3) the bulk of the tachocline is magnetic free, in thermal-wind balance and in thermal equilibrium and (4) the interaction between the field and the flows takes place within a very thin magnetic boundary layer, the tachopause, located at the bottom of the tachocline. We show that the thickness of the tachocline scales with the amplitude of the meridional flows exactly as predicted by Gough and McIntyre. We also determine the parameter conditions under which such solutions can be obtained, and provide a simple explanation for the failure of previous numerical attempts at reproducing the Gough and McIntyre model. Finally, we discuss the implications of our findings for future numerical models of the solar interior, and for future observations of the Sun and other stars.