Finite element method solution of electrically driven magnetohydrodynamic flow

The magnetohydrodynamic (MHD) flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes, placed one in each of the walls of the duct where the applied magnetic field is perpendicular. The flow is steady, laminar and the fluid is incompressible, viscous and electrically conducting. A stabilized finite element with the residual-free bubble (RFB) functions is used for solving the governing equations. The finite element method employing the RFB functions is capable of resolving high gradients near the layer regions without refining the mesh. Thus, it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the Hartmann number. Before employing the bubble functions in the global problem, we have to find them inside each element by means of a local problem. This is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem. Equivelocity and current lines are drawn to show the well-known behaviours of the MHD flow. Those are the boundary layer formation close to the insulated walls for increasing values of the Hartmann number and the layers emanating from the endpoints of the electrodes. The changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field.     

High Hartmann number flow through a rectangular duct with unequally and finitely conducting walls

Nonsymmetric Hartmann flow through a rectangular duct is investigated for thin duct walls with, generally, unequal but finite conductivities. A high Hartmann number is adopted. Consistent with known phenomena, both Hartmann layers transverse to the applied magnetic field are assumed to be separated from the two side boundary layers by four corner regions plus four inner corner regions. The method of singular perturbations and matched asymptotic expansions is applied to the coupled system. The equations governing the core and Hartmann layers are first partially resolved for leading terms. This is then followed by tackling equations governing one side layer and two adjacent corner regions. The latters' incorporation secures, for the former, only those boundary conditions that are compatible along the transverse walls. Both corner regions are denied access to non-required boundary conditions along the neighbouring side wall by the adjoining inner corner regions. However, the latters' boundary value problems need not be tackled for the acquirement of only dominant terms beyond all four inner corner regions. The complementary side layer and associated corners are accounted for by a non-symmetric reflection principle. Results reveal that a difference between conductivities in the transverse walls together with at least one finitely conducting side wall impart to disturbances within the core and Hartmann layers (i) a nontrivial dependence on the transverse coordinate relative to the magnetic field and flow in addition to the (usual) dependence on the field aligned coordinate, (ii) a dependence on side wall parameters in addition to the dependence on transverse wall parameters. Applications to related situations are considered. These include the case for a perfectly conducting lower wall, a finitely conducting upper wall, and equally and finitely conducting side walls.     

Fundamental solution for coupled magnetohydrodynamic flow equations

In this paper, a fundamental solution for the coupled convection–diffusion type equations is derived. The boundary element method (BEM) application then, is established with this fundamental solution, for solving the coupled equations of steady magnetohydrodynamic (MHD) duct flow in the presence of an external oblique magnetic field. Thus, it is possible to solve MHD duct flow problems with the most general form of wall conductivities and for large values of Hartmann number. The results for velocity and induced magnetic field is visualized in terms of graphics for values of Hartmann number M?300$M?300$.     

The application of the boundary element method to the magnetohydrodynamic duct flow

A boundary element method is developed in order to investigate the flow of viscous, incompressible, electrically conducting fluids in cylindrical ducts having arbitrary cross-sections. The numerical results obtained for the circular duct with insulating wall are very close to the results obtained by means of analytical formulas.     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers     

Distortion of liquid metal flow in a square duct due to the influence of a magnetic point dipole

We consider liquid metal flow in a square duct with electrically insulating walls under the influence of a magnetic point dipole using three-dimensional direct numerical simulations with a finite-difference method. The dipole acts as a magnetic obstacle. The Lorentz force on the magnet is sensitive to the velocity distribution that is influenced by the magnetic field. The flow transformation by an inhomogeneous local magnetic field is essential for obtaining velocity information from the measured forces. In this paper we present a numerical simulation of a spatially developing flow in a duct with laminar inflow and periodic boundary conditions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Design of axisymmetric ducts for incompressible flow with vorticity.

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function and the function as independent variables where for irrotational flow can be recognized as the velocity potential function, for rotational flow ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences on a uniform mesh is employed. The technique described is capable of tackling the so–called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct wall shapes. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

服从OldroydB型微分模型的粘弹性流体问题的数值解法

本文对服从OldroydB型微分模型的粘弹性流体问题给出了一种数值逼近算法．该算法对压力方程采用标准混合有限元方法,对速度方程采用并行非重叠区域分解方法和特征线法．这种并行算法在子区域上用Galerkin方法,通过积分平均方法显式地给出内边界的数值流．在本文最后还给出了该算法的最优L^2。一误差估计．     

The Propagation of Sound in Cylindrical Ducts with Mean Flow and Bulk-reacting Lining: III. Step Discontinuities

We consider the transmission and reflexion of sound at suddenarea changes in a cylindrical waveguide, which consists of twosemi-infinite ducts. One duct is composed of two coaxial cylinders;the outer is sound-hard and the inner is perforated, and thespace between the cylinders is filled with a sound-absorbingmaterial. The other duct is a sound-hard cylinder with the sameradius and axis as the perforated tube. A uniform gas flow isassumed in this duct and inside the perfrated tube. Transmissionand reflexion matrices are calculated for the causal solution.Causality and the boundary conditions imply a convective instabilitywave (Helmholtz instability) when the perforate is downstreamthe junction. (We have chosen the parameter values in orderto avoid so-called absolute instabilities.) Calculations ofthe reflexion coefficient for the abrupt area expansion showgood agreement with experiments.     

Large Eddy Simulation of turbulent confined coaxial swirling jets

Large eddy simulations of swirling flow in a coaxial-jet combustor are reported. Two experimental test cases have been chosen from the literature. In both cases the configuration consists of two coaxial jets which enter into an expansion duct with the annular jet being swirled, the inner jet unswirled. The main features of the flow are well predicted in the simulations. The mean velocities and the turbulent fluctuations are compared with the experimental data and show good agreement. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of gas–solid flow in a cyclone separator

This paper presents a numerical study of the gas–powder flow in a typical Lapple cyclone. The turbulence of gas flow is obtained by the use of the Reynolds stress model. The resulting pressure and flow fields are verified by comparing with those measured and then used in the determination of powder flow that is simulated by the use of a stochastic Lagrangian model. The separation efficiency and trajectory of particles from simulation are shown to be comparable to those observed experimentally. The effects of particle size and gas velocity on separation efficiency are quantified and the results agree well with experiments. Some factors which affect the performance of cyclone were identified. It is shown that the collision between gas streams after running about a circle and that just entering occurred around the junction of the inlet duct and the cylinder of the cyclone, resulting in a short-circuiting flow. The combination of flow source and sink was distributed near the axis of cyclone, forming a flow dipole at axial section. Particles entering at different positions gave different separation efficiency. A particle with size exceeding a critical diameter, which was condition-dependant, would stagnate on the wall of cyclone cone. This was regarded as one of the main reasons for the deposition on the inner conical surface in such cyclones used in the cement industry.     

Laminar magnetohydrodynamic duct flow in the presence of a magnetic dipole

We study magnetohydrodynamic flow of a liquid metal in a straight duct. The magnetic field is produced by an exterior magnetic dipole. This basic configuration is of fundamental interest for Lorentz force velocimetry (LFV), where the Lorentz force opposing the relative motion of conducting medium and magnetic field is measured to determine the flow velocity. The Lorentz force acts in equal strength but opposite direction on the flow as well as on the dipole. We are interested in the dependence of the velocity on the flow rate and on strength of the magnetic field as well as on geometric parameters such as distance and position of the dipole relative to the duct. To this end, we perform numerical simulations with an accurate finite-difference method in the limit of small magnetic Reynolds number, whereby the induced magnetic field is assumed to be small compared with the external applied field. The hydrodynamic Reynolds number is also assumed to be small so that the flow remains laminar. The simulations allow us to quantify the magnetic obstacle effect as a potential complication for local flow measurement with LFV. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pressure and flow in two dimensional numerical bifurcation models

The finite element method has been used to solve the Navier-Strokes equations for steady flow conditions in bifurcations. The results are presented as pressure, velocity and streamline plots at different Reynolds number. The three bifurcations considered have rigid walls and bifurcation angles of 0°, 20° and 180°. For the bifurcation with branch angles 0° and 20° there is flow separation along the inner wall of the outlet branches and large spatial pressure variations, these phenomena being more pronounced at the higher Reynolds numbers. For the bifurcation with a branch angle of 180° the high pressure gradients occured at the outer corner and for the high Reynolds number a vortex formation developed downstream of this corner.     

The dynamic behaviour of elastic coaxial cylindrical shells conveying fluid

The dynamic behaviour of elastic coaxial cylindrica shells, which interact with a flow of compressible fluid in the inner shell or in the annular gap between the shells when both flows are present, is investigated by the finite element method. A number of test problems is considered in the case of cantilevered coaxial shells. The effect on the stability limit of the gap between the two shells is investigated for different values of the stiffness parameters of the outer shell and the fluid flow. An important difference is found with existing solutions in cases when the loss of stability of the coaxial shells occurs at higher oscillation modes. It is established that, for a certain value of the gap between the shells, the elasticity of the outer shell may have a stabilizing effect. It is shown that the presence of internal and annular flows simultaneously has a considerable stabilizing effect, while a loss of stability when the flow rates increase occurs at extremely high oscillation modes.     

Haar wavelet approximation for magnetohydrodynamic flow equations

This study proposes Haar wavelet (HW) approximation method for solving magnetohydrodynamic flow equations in a rectangular duct in presence of transverse external oblique magnetic field. The method is based on approximating the truncated double Haar wavelets series. Numerical solution of velocity and induced magnetic field is obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The calculations show that the accuracy of the Haar wavelet solutions is quite good even in the case of a small number of grid points. The HW approximation method may be used in a wide variety of high-order linear partial differential equations. Application of the HW approximation method showed that it is reliable, simple, fast, least computation at costs and flexible.     

Direct solution of Navier–Stokes equations by radial basis functions

The pressure–velocity formulation of the Navier–Stokes (N–S) equation is solved using the radial basis functions (RBF) collocation method. The non-linear collocated equations are solved using the Levenberg–Marquardt method. The primary novelty of this approach is that the N–S equation is solved directly, instead of using an iterative algorithm for the primitive variables. Two flow situations are considered: Couette flow with and without pressure gradient, and 2D laminar flow in a duct with and without flow obstruction. The approach is validated by comparing the Couette flow results with the analytical solution and the 2D results with those obtained using the well-validated CFD-ACE™ commercial package.     

The Propagation of Sound in Cylindrical Ducts with Mean Flow and Bulk-reacting Lining II. Bifurcated Ducts

The transmission and reflexion of sound in a bifurcated coaxialcylindrical duct is investigated. The inner tube carries a uniformflow and consists of two semi-infinite tubes: one is hard andthe other is perforated. The space between the coaxial cylindersis filled with a sound-absorbing material. Transmission andreflexion matrices are calculated for the causal solution. Itis found that causality and the boundary conditions requirean instability wave when the perforated tube is downstream itsjunction to the hard tube. When the perforate is situated upstreamthe junction the analysis permits incident waves that are unstable.This is important for applications to multiple reflexions. Itfollows from the analysis that, in addition to giving rise tothe instability wave, the gas flow has several other importanteffects on the acoustic properties of the junction.     

二维分岔槽道内非牛顿流体流动的有限元分析

本文介绍二维分岔槽道内非牛顿流体流动的有限元分析.采用Galerkin法及混合有限元法,流体看作不可压缩的非牛顿流体,满足Oldyord微分型本构方程.由有限元法形成的非线性代数方程组用连续微分法求解.结果表明有限元法适于分析复杂流场中非牛顿流体的流动.     

多重尺度法在椭圆管道磁流体动力流中的应用

本文研究不可压缩可导流体沿一均匀磁场中的椭圆管道的流动.在Hartmann数是充分大的情形下,应用多重尺度法,作出解的准确到任意量级的渐近近似式.本方法可用于研究截面具有光滑周界的,任意形状管道的磁流体动力流.     

Uniqueness of transonic shock solutions in a duct for steady potential flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.