Radiative heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary

The heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary over a flat plate, over a 90-deg wedge, and in stagnation flow is solved by application of the Galerkin method with the particular solution boundary condition I _p (τ₀,ξ,?μ) of the equation of radiative transfer for an inhomogeneous term and the Box method. The exact integral expressions for the radiation part of this problem are developed. The coupling between convective and radiative heat transfer in boundary-layer flows is described by a set of nonlinear simultaneous equations including differential equations and integrodifferential equations. The Galerkin method and the particular solution boundary condition I _p (τ₀,ξ,?μ) are used to analyze the radiation part of the problem. The nonsimilar boundary-layer equations are solved by the Box method. The present numerical procedure solutions are compared in tables with the other exact treating results, the P-3, and P-1 approximation methods for the case of isotropically scattering boundary-layer flows. The effects of linearly anistropically scattering and reflecting surface are taken into account. It is found that the present method is a reliable and efficient numerical procedure and scattering leads to a reduction in the total heat flux. The influence of the forward-backward scattering parameter on the total heat flux decreases with the increase of the surface reflectivity.     

Three-dimensional computer model of nonequilibrium aerophysics of the spacecraft entering in the Martian atmosphere

A three-dimensional computer model of nonequilibrium aerophysics of the spacecraft entering in the Martian atmosphere and landing on the planet surface is presented. The model is based on the system of Navier-Stokes equations, the energy conservation equation in the form of the Fourier-Kirchhoff equation for the translational temperature, the system of equations of vibrational kinetics for six vibrational modes of N₂, O₂, CO₂ and CO molecules, and the equation of selective thermal radiation transfer in the multigroup spectral approximation. The results of two-dimensional and threedimensional calculations of the nonequilibrium flowfield and the convective and radiative heating of the EXOMARS spacecraft surface are presented. The calculations are performed on regular multiblock curvilinear grids using the NERAT-2D and NERAT-3D computer codes developed in the Institute for Problems in Mechanics of the Russian Academy of Sciences.     

Monte Carlo simulation of radiative transfer in a refractive layered medium

In this work, we adapted the Monte Carlo method to simulate radiative transfer in a two-layer scattering slab with continuously varying refractive index in each of the two layers and a jump of refractive index at the interface between the two layers. The hemispherical reflectance (R _h ) and transmittance (T _h ) of the slab are obtained by tracing photon bundles propagating along curved trajectories. There is a very satisfying correspondence between the present results and those obtained by numerical solution of integral radiative transfer equation for the special cases with constant refractive index in each of the layers. The magnitude of numerical uncertainty decreases with the increase of bundles. The results show that the R _h decreases with the increase of the positive gradient of the refractive index considered. For the cases with constant total thickness, the R _h and the T _h increase with the increase of the ratio of upper-layer thickness to lower-layer thickness.     

Transient radiative and conductive heat transfer in non-gray semitransparent two-dimensional media with mixed boundary conditions

This paper is devoted to transient heat transfer involving radiation and conduction. Considering a non-gray purely absorbing media, the radiative heat transfer equation (RTE) is solved iteratively with the Discrete Ordinates Method (DOM) using an exponential differencing scheme. The energy balance equation is used to compute temperature at each time step with the Crank–Nicholson technique. Energy equation is coupled to the RTE through the radiative source term. Both equations are discretized with finite differencing schemes. The energy conservation leads to the sparse system of linear equations A× T=B which is solved with a bi-conjugate stabilized gradient technique (BCSG). Validation of the model with different test cases is achieved and application to transient heating of glass is also studied.     

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T _a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.     

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic solution of the equations of a laminar multicomponent boundary layer with intensive suction

An asymptotic solution is obtained for the equations of the laminar multicomponent boundary layer encountered in the plane-parallel and axially symmetrical flow of a gas with large values of the suction parameter. It is shown that the roots of the characteristic equation to which the solution of the diffusion equations reduce in the first approximation may be found in the form of radicals when the external gas flow contains chemical components capable of being combined into r5 groups as regards their diffusion properties. The number of components in the groups and the number of components in the boundary layer may be arbitrary. Asymptotic equations are obtained for the coefficient of friction, the temperature and concentration gradients, and the diffusion flows of the components on the surface of the body. By way of example, formulas are given for the thermal flux passing to a body during the flow of dissociated air or a dissociated mixture of N₂ and CO₂. A numerical solution is given for the equations of the boundary layer in the case of the flow of dissociated air. The asymptotic solution is compared with the numerical result, and the range of applicability of the asymptotic equations is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 66–74, November–December, 1970.The author wishes to thank G. A. Tirskii for discussion of this analysis.     

An Optimal control least-squares method for the solution of advection dispersion problems

This paper discusses the numerical solution of advection dispersion equations using an Optimal control,H ¹, least-squares formulation, associated with a quasi-Newton conjugate gradient algorithm. The suggested algorithm represents an extension of the method proposed by Bristeauxet al., for the solution of nonlinear fluid flow problems.At each time step, the discretized differential equation is transformed into an optimal control problem. This problem is then stated as an equivalent minimization one, whose objective function allows the capture of the advective behavior of the equation for high values of the P_e number.A general presentation is made of the optimization algorithm. Validation runs, for a one-dimensional example, show fairly accurate results for a wide range of Péclet and Courant numbers. Comparisons with several numerical schemes are also presented.     

Numerical modelling of heat and mass transfer in adsorption solar reactor of ammonia on active carbon

 In this paper, we present a modelling of the performance of a reactor of a solar cooling machine based carbon–ammonia activated bed. Hence, for a solar radiation, measured in the Energetic Laboratory of the Faculty of Sciences in Tetouan (northern Morocco), the proposed model computes the temperature distribution, the pressure and the ammonia concentration within the activated carbon bed. The Dubinin–Radushkevich formula is used to compute the ammonia concentration distribution and the daily cycled mass necessary to produce a cooling effect for an ideal machine. The reactor is heated at a maximum temperature during the day and cool at the night. A numerical simulation is carried out employing the recorded solar radiation data measured locally and the daily ambient temperature for the typical clear days. Initially the reactor is at ambient temperature, evaporating pressure; P _ev =P _st (T _ev =0 ^∘C) and maintained at uniform concentration. It is heated successively until the threshold temperature corresponding to the condensing pressure; P _cond =P _st (T _am ) (saturation pressure at ambient temperature; in the condenser) and until a maximum temperature at a constant pressure; P _cond . The cooling of the reactor is characterised by a fall of temperature to the minimal values at night corresponding to the end of a daily cycle. We use the mass balance equations as well as energy equation to describe heat and mass transfer inside the medium of three phases. A numerical solution of the obtained non linear equations system based on the implicit finite difference method allows to know all parameters characteristic of the thermodynamic cycle and consider principally the daily evolution of temperature, ammonia concentration for divers positions inside the reactor. The tube diameter of the reactor shows the dependence of the optimum value on meteorological parameters for 1 m² of collector surface. Received on 10 January 2001     

The Effects of Blowing and Suction on Double Diffusion by Mixed Convection over Inclined Permeable Surfaces

A boundary layer analysis is used to investigate the effect of lateral mass flux on mixed convection heat and mass transfer over inclined permeable surfaces in porous media. The conservation equations that govern the problem are reduced to a system of non-linear ordinary differential equations and then the resulting equations is solved by numerical method. The numerical results for heat and mass transfer in terms of Nusselt and Sherwood number are presented in x-y plots for the buoyancy ratio (N) and Lewis number (Le) with mass flux pammeter (Fw). The obtained results are validated against previously published results with on special case of the problem.     

THE THREE-DIMENSIONAL UNSTEADY BOUNDARY LAYER OVER AN IMPULSIVELY STARTED ROTATING DISK

The unsteady boundary layer over an impulsively started rotating disk isstudied,a complete solution describing the smooth transition from vortex diffusionat ωt=0 to Kármán’s steady solution is obtained by series expansion and itsnumerical continuation. The angle of body streamlines,together withexperimental values, are given as the function of time t as well as the momentcoefficient C_M and on-coming velocity w(∞).     

Flow and heat transfer in a moving fluid over a moving flat surface

In this paper, a numerical analysis of the momentum and heat transfer of an incompressible fluid past a parallel moving sheet based on composite reference velocity U is carried out. A single set of equations has been formulated for both momentum and thermal boundary layer problems containing the following parameters: r the ratio of the free stream velocity to the composite reference velocity, σ (Prandtl number) the ratio of the momentum diffusivity of the fluid to its thermal diffusivity, and E _c (E _ck ) (Eckert number). The present study has been carried out in the domain 0 ≤ r ≤ 1. It is found that the direction of the wall shear changes in such an interval and an increase of the parameter r yields an increase in temperature.      

An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes,stability analysis,and energy conservation

We set up a numerical strategy for the simulation of the Euler equations, in the framework of finite volume staggered discretizations where numerical densities, energies, and velocities are stored on different locations. The main difficulty relies on the treatment of the total energy, which mixes quantities stored on different grids. The proposed method is strongly inspired, on the one hand, from the kinetic framework for the definition of the numerical fluxes, and, on the other hand, from the discrete duality finite volume (DDFV) framework, which has been designed for the simulation of elliptic equations on complex meshes. The time discretization is explicit and we exhibit stability conditions that guaranty the positivity of the discrete densities and internal energies. Moreover, while the scheme works on the internal energy equation, we can define a discrete total energy which satisfies a local conservation equation. We provide a set of numerical simulations to illustrate the behavior of the scheme.     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

Numerical simulation of 3D free surface flows,with multiple incompressible immiscible phases. Applications to impulse waves

A numerical method for the solution to the density‐dependent incompressible Navier–Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume‐of‐fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density‐dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide‐generated impulse wave in a reservoir. Copyright © 2014 John Wiley & Sons, Ltd.     

Effects of surface radiation on natural convection in a rayleigh-benard square enclosure: steady and unsteady conditions

Coupled laminar natural convection with radiation in air-filled square enclosure heated from below and cooled from above is studied numerically for a wide variety of radiative boundary conditions at the sidewalls. A numerical model based on the finite difference method was used for the solution of mass, momentum and energy equations. The surface-to-surface method was used to calculate the radiative heat transfer. Simulations were performed for two values of the emissivities of the active and insulated walls (ɛ₁=0.05 or 0.85, ɛ₂=0.05 or 0.85) and Rayleigh numbers ranging from 10³ to 2.3×10⁶ . The influence of those parameters on the flow and temperature patterns and heat transfer rates are analyzed and discussed for different steady-state solutions. The existing ranges of these solutions are reported for the four different cases considered. It is founded that, for a fixed Ra, the global heat transfer across the enclosure depends only on the magnitude of the emissivity of the active walls. The oscillatory behavior, characterizing the unsteady-state solutions during the transitions from bicellular flows to the unicellular flow are observed and discussed.     

Radiative heat transfer calculations in real gases

A general formulation for radiative heat transfer calculations is presented, based on integrated quantities such as total emissivities and absorptivities. The procedure is intended particularly for combustion chamber applications of varying degree of complexity, the radiative active medium consisting of gases such as H₂O and CO₂ and of soot. First, some preliminary calculations are given for the often treated radiative equilibrium cases of plane parallel plates and infinite concentric cylinders. Then an example of a combustion chamber calculation is studied where the radiative heat transfer calculation is included in a system of partial differential equations describing momentum, heat and mass transfer with combustion.     

Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

The evolution of heat in crystalline solids is described at low temperatures by the Boltzmann-Peierls equation, which is a kinetic equation for the phase density of phonons.In this study, we solve initial value problems for the Boltzmann-Peierls equation in relation to the following issues: In thermodynamics, a given kinetic equation is usually replaced by a truncated moment system, which in turn is supplemented by a closure principle so that a system of PDEs results for some moments as thermodynamic variables. A very popular closure principle is the maximum entropy principle, which yields a symmetric hyperbolic system. In recent times, this strategy has led to serious studies on two problems that might arise: 1. Do solutions of the maximum entropy principle exist? 2. Is the physics that is embodied by the kinetic equation more or less equivalently displayed by the truncated moment system? It was Junk who proved for the BOLTZMANN equation of gases that maximum entropy solutions do not exist. The same failure appears for the Fokker-Planck equation, which was proved by means of explicit solutions by Dreyer, Junk, and Kunik.This study has two main objectives:1. We give a positive existence result for the maximum entropy principle if the underlying kinetic equation is the Boltzmann-Peierls equation. In other words we show that the maximum entropy principle can be used here to establish a closed hyperbolic moment system of PDEs. However, the intent of the paper is by no means a general justification of the maximum entropy principle.2. We develop an approximative method that allows the solutions of the kinetic equations to be compared with the solutions of the hyperbolic moment systems. To this end we introduce kinetic schemes that consists of free flight periods and two classes of update rules. The first class of rules is the same for the kinetic equation as well as for the maximum entropy system, while the second class of update rules contains additional rules for the maximum entropy system. It is shown that if a sufficient number of moments are taken into account, the two solutions converge to each other. However, in terms of numerical effort, the presented solver for the kinetic equation clearly outperforms the one for the maximum entropy principle.Received: 15 August 2003, Accepted: 8 November 2003, Published online: 11 February 2004PACS: 02.30.Jr, 02.60.Cb, 05.30.Jp, 44.10. + i, 63.20.-e, 66.70. + f, 65.40.Gr Correspondence to: M. Herrmann     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.