Abstract: The mathematical formulation, basic concept and numerical implementation of a meshless method for solving three dimensional fluid flow and related heat transfer problems are presented in this paper. A second order moving least squares approximation is used for the spatial discretization together with an implicit scheme for time marching. The vorticity and vector potential formulation of NavierâStokes equations is employed to avoid the difficulties associated with pressureâvelocity coupling. Two three-dimensional examples of natural convection in a differentially heated cubic cavity and in the annular space between a sphere and a cube are considered and steady-state solutions are obtained for Rayleigh numbers in the range of 103â106. Results show the flexibility of the method and demonstrate its accuracy.
Notes: A meshless method using a vorticity and vector potential formulation is presented. ⺠Convection problems in a cubic cavity and in the space between a sphere and a cube are treated. ⺠Calculations show the good accuracy of the method as compared to available results.
Abstract: A diffuse approximation meshless (DAM) method is presented to solve the radiative transfer equation (RTE) in a graded index medium. The meshless method can solve the equation directly without using an upwind scheme. Absorbing, emitting and scattering media with different kinds of graded indices in 1D and 2D geometries are tested. Prediction results obtained by the proposed method are compared with different references in order to illustrate the performance of this solution method.
Notes: Refers to
Vital Le Dez, Hamou Sadat, Radiative transfer in a semi-transparent medium enclosed in a cylindrical annulus, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 113, Issue 1, January 2012, Pages 96-116
Abstract: A meshless method DAM is employed to solve the coupled radiative and conductive heat transfer problem in a semi-transparent medium enclosed in complex 2D and 3D geometries. The meshless method for radiative transfer is based on the even parity formulation of the discrete ordinates method. Cases of combined conductionâradiation are presented and the results are compared with other benchmark approximate solutions.
Abstract: An exact analytical description of the internal radiative field inside an emittingâabsorbing grey semi-transparent medium at radiative equilibrium, enclosed in a cylindrical annular cavity bounded by hot and diffusely reflecting surfaces is proposed. It is shown that the radiative field can be completely described by purely geometrical weighting coefficients which allow determining the temperature field for any combined heat transfer with radiation, so as the two-dimensional vectorial radiative flux field. It is worth noting that no ray effect occurs since the only numerical discretization is purely spatial. Some examples show that the proposed method gives perfectly smooth results of good accuracy for both the temperature and flux fields inside the medium.
Abstract: This paper reports on experimental measurements of the gas temperature in a dielectric barrier discharge reactor powered by a high voltage pulsed signal. It is shown that the thermal behavior of the reactor follows a first order model. However, an unexpected runaway phenomenon was observed at a frequency of 300 Hz. A sudden increase in the power source and consequently in reactor temperature which reaches 170 °C is observed. This behavior is discussed in terms of input power variation with temperature, possibly due to a resonance phenomenon.
Notes: ⺠Temperature measurements in a Pulsed DBD reactor have been carried out. ⺠The electrode and gas temperatures were monitored until a steady state was reached. ⺠Temperature runaway has been observed at a frequency of 300 Hz.
Abstract:
In this paper, a perturbation method is used to solve a two-dimensional unsteady heat conduction problem. Low-order transfer functions are defined. Step responses are obtained and compared to the complete numerical solutions given by a previous termmeshlessnext term method. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the used method.
Abstract: Magnetic convection of air around a hot circular cylinder placed in a square cavity under the influence of an
external magnetic field and in absence of gravity is considered. Fluid motion occurs due to the gradients of the magnetic and temperature fields. A collocation meshless method is used for the numerical solution of the governing equations. The results are presented through temperature contours and velocity vectors at steady state. Nusselt numbers are also given for magnetic Rayleigh numbers of 105 and 106.
Abstract: In the present study, we investigate the flow of a paramagnetic fluid in a two dimensional heated channel when an external magnetic gradient is imposed. In the fully developed regime, an analytical solution shows that a flow reversal may occur; the condition of this is given n terms of the Reynolds number. Numerical simulations are then carried out for more general situations. It is shown that the analytical model gives good qualitative predictions.
Abstract: An explicit simple analytical method is presented for periodic heat conduction transfer in solids by using a perturbation method. Low order models are developed and their accuracy was compared to that of the complete numerical model. It is shown that first and second order models can be used efficiently for relatively low frequencies. An improvement of the method is then proposed by using a convergence acceleration of the series. This allows the use of the method at much higher frequencies.
Abstract: An exact analytical description of the internal radiative field inside an emitting-absorbing gray semi-transparent medium at radiative equilibrium, enclosed in a square cavity bounded by hot and black surfaces is proposed. It is shown that purely geometrical coefficients (in the case of optical constants non-depending on the temperature) completely describe the radiative field inside the cavity. The temperature field and two-dimensional vectorial radiative flux field are then easily deduced. It is worth noting that no angular discretization has to be done (no ray effect), the only numerical discretization being purely spatial. Some examples show that the proposed method gives perfectly smooth results of good accuracy for both the temperature and flux fields inside the medium.
Abstract: A lumped thermal model was developed in order to analyse the transient gas temperature rise and fall in a dielectric barrier discharge reactor. A laboratory reactor was employed to validate the model by means of the experimental data. It has been shown that this simple model is of sufficient accuracy for engineering applications
Abstract: A meshless method is presented to solve the radiative transfer equation in complex 2D and 3D geometries. In order to avoid numerical oscillations, the even parity formulation of the discrete ordinates method is used. A moving least squares approximation meshless method is used to solve the second order partial differential equations. Prediction results of radiative heat transfer problems obtained by the proposed method are compared with reference in order to assess the correctness of the present method.
Abstract: We show in this paper that the Adomian decomposition method is closely related to a perturbation method used to deal with some heat conduction problems. Three examples demonstrate that the obtained series are strictly identical.
Abstract: A coherent definition of the reflection and transmission factors at a plane interface separating two uniaxial crystals is proposed, from the photons impulsion-energy 4-vectors conservation. This definition, different from the classical electromagnetic one, is compatible with the completely resolved extended Fermat's principle of the geometric optics for extraordinary luminous rays inside uniaxial media, and allows the exact calculation of the transmission factors at the plane interface for any practical configuration, combining all possible optical axes and anisotropy factors variations. Furthermore, this particular technique points out the existence of quasi-particles strongly associated to the photons, whose behaviour is highly correlated to the photons transmission/reflection possibilities.
Abstract: A perturbation method is used to solve an unsteady one-dimensional heat conduction problem
in a cylinder. A simple second order explicit solution is obtained. It is shown that this solution
is accurate even for high values of the Biot number in a region surrounding the center of the
cylinder.
Abstract: The purpose of this paper is to obtain the temperature field inside a cylinder filled in with a dense nonscattering semitransparent medium from directional intensity data by solving the inverse radiative transfer equation. This equation is solved in a first approach with the help of a discrete scheme, and the solution is then exactly obtained by separating the physical set on two disjoint domains on which a Laplace transform is applied, followed by the resolution of a first kind Fredholm equation.
Abstract: A lumped thermal model was developed in order to analyse the transient gas temperature rise and fall in a dielectric barrier discharge reactor. A laboratory reactor was employed to validate the model by means of the experimental data. It has been shown that this simple model is of sufficient accuracy for engineering applications.
Abstract: A second-order model to solve the transient heat conduction problems in the case of a spherical solid is developed. The solid is placed in a medium at variable temperature f(t) and a constant convection coefficient h is considered at the external surface of the sphere. When the fluid temperature is constant [F(t) = 1T], the analytical solution can be calculated by employing the separation of variables techniques. A first-order is used to calculate the first-order step response. To show the accuracy of the second-order model, the obtained step responses to the analytical solution given by the infinite series are compared. The comparison showed that the second-order model gives accurate results and is close to the analytical solution even for an infinite Biot number. The accuracy of the second-order model can ensures that the model can be used safely for engineering calculations.
Abstract: The thermal behaviour of a dielectric barrier discharge reactor is studied. The experimental tests are performed on a laboratory reactor with two working fluids: helium and air. A simple heat conduction model for calculating the heat loss is developed. By using temperature measurements in the internal and external electrodes, a thermal resistance of the reactor is defined. Finally, the percentage of the input power that is dissipated to the environment is given.
Abstract: The present paper deals with the influence of the Lorentz force associated with an applied radial magnetic field on the axisymmetric stagnation flow impinging obliquely onto a uniformly rotating circular cylinder. It is found that the boundary layer flow is described by an exact solution of the NavierâStokes equations when Hall effects are ignored. The stability of this basic solution is then considered in the framework of GörtlerâHammerlin assumption according to which linear disturbances inherit the underlying symmetry of the basic flow. The resulting eigenvalue problem is solved numerically by means of a pseudospectral method using Laguerre's polynomials. The scheme is specifically designed to solve boundary layer equations. The numerical experiments indicate that the effect of cylinder rotation is to reduce the stability of the basic flow and most importantly the magnetic field acts to either increase or decrease it, depending on whether the rotation rate is smaller or greater than some critical value that changes with the Hartmann number. At criticality, the basic flow undergoes Hopf bifurcations leading to branching off solutions in the form of azimuthally travelling waves. In the case of axisymmetric disturbances the bifurcation remains of Hopf type provided that the Hartmann number is small enough, a saddle-node bifurcation is encountered in the opposite case
Abstract: A high order version of a diffuse approximate meshless method has been implemented and tested on two problems, namely diffusion and transport diffusion two-dimensional equations. It is shown that the fourth-order approximation greatly improves the numerical solution.
Abstract: The determination of the optical trajectories on which the radiative energy is transported has been achieved by solving the anisotropic Fermat's principle in a sphere of uniaxial semi-transparent material when the dielectric perpendicular and parallel permittivities are space dependant; first established for the general case of an optical axis function of the location inside the medium, it has completely been solved for the particular case of a constant optical axis with a constant anisotropy factor when the perpendicular permittivity is a function of the radius; the calculation shows a lack of the spherical symmetry due to the presence of the optical axis which induces non planar trajectories contrarily to the isotropic case.
Abstract: The present analysis deals with the onset of instability in an axisymmetric stagnation flow obliquely impinging on a uniformly rotating circular cylinder. The basic flow is described by an exact solution of the Navier-Stokes equations, discovered by Weidmann and Putkaradze [Eur. J. Mech. B/Fluids 22, 123 (2003)]. An eigenvalue problem for the linear stability is formulated, regardless of the free stream obliqueness, and then solved numerically by means of a collocation method using Laguerre's polynomials. It is established that the basic stagnation flow is stable for sufficiently high Reynolds numbers. This is in conformity with the unconditional linear stability of two-dimensional Hiemenz stagnation flow. Instability occurs for Reynolds numbers smaller than some threshold value that increases with the rotation rate of the cylinder. At criticality, the flow undergoes a Hopf bifurcation, leading then to an oscillatory secondary motion.
Abstract: In this paper we present a second order model for transient heat conduction in a slab obtained by using a perturbation method. We show that this simple model is accurate even for high values of the Biot number in a region surrounding the center of the slab.
Abstract: A diffuse approximation-based collocation method is used to solve a two-dimensional heterogeneous heat conduction problem. It is shown that one has to take into account the neighbors of each neighbor of the calculation node in order to achieve sufficient accuracy in the case of strong heterogeneity.
Abstract: A meshless method is presented for solving the radiative transfer equation in the discrete ordinates approach. It is shown that the primitive variables formulation is unstable for low values of the absorption coefficient while the even parity formulation is always stable and accurate.
Abstract: The aim of the present paper is to examine the effects of a constant magnetic field on the thermal instability of a two-dimensional stagnation point flow. First, it is shown that a basic flow, described by an exact solution of the full NavierâStokes equations exists under some conditions relating the orientation of the magnetic field in the plane of motion to the obliqueness of free stream. The stability of the basic flow is then investigated in the usual fashion by making use of the normal mode decomposition. The resulting eigenvalue problem is solved numerically by means of a pseudo spectral collocation method based upon Laguerreâs functions. The use of this procedure is warranted by the exponential damping of disturbances far from the boundary layer and the appropriate distribution of the roots of Laguerreâs polynomials to treat boundary layer problems. It is found through the calculation of neutral stability curves that magnetic field acts to increase the stability of the basic flow.
Abstract: Unsteady one-dimensional heat conduction problem is analysed within the framework of a perturbation method. It is shown that simple first order models for the centre, the surface and the average temperature can be developed. The step responses are given for the slab, the infinite cylinder and the sphere.
Abstract: A diffuse approximation method (DAM) for three-dimensional, incompressible, viscous fluid flow is presented. The method works directly with primitive variables. The discretized equations are solved using a first-order-in-time, implicit projection algorithm. The proposed method is verified by applying it to the steady three-dimensional differentially heated cubic cavity. The results are compared to those of numerical and experimental investigations in the literature.
Abstract: A diffuse approximation method for the solution of time-dependent Navier-Stokes equations is presented. Different preconditioned iterative methods for solving the pressure correction equation are tested. Sample results are presented for the window cavity problem and the fluid flow around a circular cylinder.
Abstract: The main objective of this article is to assess the performance of a meshless method called the diffuse approximation method (DAM). This numerical method is presented for the calculation of two-dimensional, incompressible, laminar thermal flows. The two-dimensional natural-convection problems in a square cavity and in an eccentric annulus demonstrate the accuracy and capabilities of the proposed technique for high Rayleigh numbers.
Abstract: The aim of this work is to solve an inverse boundary convective and diffusive problem. A flow of an incompressible Newtonian fluid between two plane slabs, when the dynamical regime is established, is considered. In order to retrieve an unknown parietal flux from measurements collected within a channel, an asymptotic technique coupled with a constraint optimisation method is developed to avoid the regularisation process. Two numerical examples are given to show the efficiency of this approach.
Abstract: This article presents a comparison for several test cases between the diffuse approximation method (DAM) and the control-volume finite element method (CVFEM) The diffuse approximation method is a relatively new numerical approach that does not need a finite-element mesh to solve fluid flow and heat transfer problems in complex geometries. It appears that the two methods provide similar accurate results. However, when some manipulations of the grids are needed during the calculation process, such as a local refinement, the use of a simple set of nodes appears to be less time consuming, as shown in the last example.
Notes: this paper is a comparison between a meshless method and a control volume based finite element method
Abstract: An adaptative diffuse approximation method for convection diffusion problems has been developed. The grid refinement method is first described and two examples are reported to illustrate the validity of the proposed method. The first one is a two-dimensionnal diffusion problem and the second one is a one-dimensionnal convection-diffusion problem
Abstract: The diffuse approximation is presented and applied to natural convection problems in porous media. A comparison with the control volume-based finite-element method shows that, overall, the diffuse approximation appears to be fairly attractive.
Abstract: This paper deals with the numerical solution of laminar natural convection in a trapezoidal enclosure by a control volume based finite element method and the vorticity-stream function formulation. Steady state solutions have been obtained for aspect ratios of 3 and 6 and for Rayleigh numbers ranging from 103 to 2 Ã 105. Numerical results are presented for the flow patterns and for the average heat transfer characteristics.
Abstract: he development of meshless approaches may lead to a new generation of computational methods in engineering and applied science. The contributors focus on fundamental ideas (Moving Least Squares, Smooth Particle Hydrodynamics and Generalized Finite Differences) illustrated with applications in acoustics, fluid and solid mechanics, as well as numerical and experimental data smoothing. Testifying to the vitality of this research area in Europe, these papers represent state-of-the-art contributions from researchers in Poland, Belgium, United Kingdom, France, and Spain.
