Magnetohydrodynamic (MHD) flow of a micropolar fluid towards a stagnation point on a vertical surface

The steady MHD mixed convection stagnation point flow towards a vertical surface immersed in an incompressible micropolar fluid is investigated. The external velocity impinges normal to the wall and the wall temperature is assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a system of ordinary differential equations, which is then solved numerically by a finite-difference method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Both assisting and opposing flows are considered. It is found that dual solutions exist for the assisting flow, besides that usually reported in the literature for the opposing flow.     

Closed Form Solution of Local Shape from Shading at Critical Points

In the theory of shape from shading, behaviours of the local solution around a critical point of the image play an important role. This paper shows that the second derivatives of the object surface can be locally determined at these image critical points. Closed form expressions of the surface second derivatives in terms of the second derivatives of the image brightness and of the reflectance map are shown. They are derived as follows: By differentiating the image irradiance equation twice at an image critical point, a set of polynomial equations is obtained that contains the second derivatives of the surface, of the image brightness and of the reflectance map. Regarding these equations as simultaneous equations for unknown surface second derivatives, they are algebraically solved and their explicit expressions are derived. Such a derivation is possible only at image critical points and is impossible at any other image point. The applicability of the derived expressions to noisy images is tested using synthetic images.     

Orthotropic cylindrical shells under line load along a generator

A technique for elastic analysis of an orthotropic cylindrical shell subjected to a uniform line load along a generator is developed. An accurate form of governing differential equations is derived and a mathematically discrete element method is used for its solution. The shell is divided into a finite number of longitudinal strips and the derivatives with respect to the circumferential coordinate in the governing equation are replaced by their finite difference relationships. The solution of the resulting equations is written in closed form. A computer program to implement this technique is developed and the computed results are compared with published experimental and analytical results. An excellent agreement is obtained. Some new results for a shell with fixed end boundary conditions are also presented.     

A numerical method for predicting natural convection in horizontal cylinders with asymmetric boundary conditions

A numerical technique is developed to predict the two-dimensional transient natural convection heat transfer within a horizontal cylinder. Finite difference analogs of the Navier-Stokes and thermal energy equations are solved in the stream function-vorticity framework. The solution method, which is a modification of an alternating-direction implicit (ADI) scheme wherein the convective terms are evaluated explicitly, is found to be computationally more efficient than either an ADI or an explicit method. Unlike previous work, the present technique will accommodate completely arbitrary temperature boundary conditions. Thus, rather than considering an annular space or half of a symmetric cylinder, the solutions are determined for a full cylinder. A Cartesian form of the governing equations is employed at the point $\text{r= 0}$ where the polar coordinate equations become singular. The computed results are found to be in good agreement with previously published experimental data.     

Linear eigenvalue code for edge plasma in full tokamak X-point geometry

A new code is presented for solving linear eigenvalue problems from fluid models of the edge plasma of tokamaks. The 2DX code solves linearized fluid equations in a 2D cross-section of the plasma, with toroidal mode number resolving the third dimension. Geometry capabilities include both closed and open field lines, allowing solution of X-point problems as well as a variety of other toroidal and cylindrical systems. The code generates a pair of sparse matrices forming a generalized eigenvalue problem which is then solved using a standard sparse eigensolver package. Use of a specialized equation parser permits a high degree of flexibility in both equations and coordinate systems. Both analytic and full geometry benchmark cases are presented.     

Modified algorithms for nonconforming taylor discretization

Algorithms for solving partial differential equations which extend previous applications of the nonconforming Taylor discretization method (NTDM) are presented. In one modification the number of interrelated grid points is variable, thus enabling additional geometric flexibility. Another modification is the approximation of the governing differential equation using the method of weighted residuals. A simple one-dimensional test case with a known analytic solution is solved using this code. The results demonstrate that precision is enhanced when using the method of weighted residuals with an increased number of interrelated points. The algorithm is applied as a general purpose two-dimensional code for nonlinear steady state heat-conduction. Two-dimensional examples with complex geometry and boundary conditions are then solved both by the NTDM and by the finite elements method (FEM). The results obtained by the two methods are compared.     

Numerical studies of the flow around a circular cylinder by a finite element method

Numerical solutions of the steady, incompressible, viscous flow past a circular cylinder are presented for Reynolds numbers R ranging from 1 to 100. The governing Navier-Stokes equations in the form of a single, fourth order differential equation for stream function and the boundary conditions are replaced by an equivalent variational principle. The numerical method is based on a finite element approximation of this principle. The resulting non-linear system is solved by the Newton-Raphson process. The pressure field is obtained from a finite element solution of the Poisson equation once the stream function is known. The results are compared with those determined by other numerical techniques and experiments. In particular, the discussion is concerned with the development of the closed wake with Reynolds number, and the tendency of R ≥ 40 flow toward instability.     

The flow induced in a three layer stratified fluid by a submerged sink or source with stagnation points on the free surfaces

A boundary integral technique is developed to study the free surface flow of a steady, two-dimensional, incompressible, irrotational and inviscid fluid which is induced in both two and three layer stratified fluids in the presence of gravity by a submerged sink or source with stagnation points on the free surfaces. A special form of the Riemann–Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing non-linear boundary integral–differential equations which have been solved for the fluid velocity on the free surfaces and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various Froude numbers and sink heights in both two and three layer fluids. Further, we have also studied the critical Froude numbers for which no convergent solutions are possible for any larger values of the Froude number. We have found that the free surfaces are dependent on two parameters, namely the Froude number and the ratio of sink height to the thickness of either the middle layer in a three layer system and the bottom layer in a two layer system.     

Large deflection of clamped skew plates

A theoretical analysis is presented for the large deflection elastic behaviour of clamped, uniformly loaded orthotropic skew plates. The governing nonlinear partial differential equations are transformed into a set of nonlinear algebraic equations. For this purpose a network of discrete points over the domain is considered, and the integral equation of beams along the skew directions is used with appropriate boundary conditions. The resulting nonlinear algebraic equations are then solved by a Newton-Raphson procedure.A convergence study of the solution has been made, and numerical results for a few cases of orthotropy and skew angles are presented in graphical form. The results obtained by this method are compared with available results of other investigators.     

Theoretical research on naturally curved and twisted beams under complicated loads

This paper presents an analytical method for the study of naturally curved and twisted beams under complicated loads, with special attention devoted to the solving process of governing equations which take into account the effects of torsion-related warping as well as transverse shear deformations. The solutions derived in this paper can be used for the analysis of the beams, including the calculation of various internal forces, stresses, strains and displacements. These governing equations, in special cases, can be readily solved and yield the solutions to the problem. A generalized warping coordinate for a curved coplanar beam subjected to the action of vertical distributed loads is given for verification.     

Least square finite element solution of stagnation point flow

A detailed analysis of the least square finite element solution of nonlinear boundary value problems is presented with reference to a particular example of nonlinear coupled differential equations governing the flow of an incompressible viscous fluid in the vicinity of a forward stagnation point at a blunt body. The numerical solutions are presented for different cases. The results obtained by the least square finite element method are in very good agreement with the results available in the literature confirming the versatility and usefulness of the application of the method to nonlinear boundary value problems governing the fluid flow problems.     

Numerical investigation of meniscus deformation and flow in an isothermal liquid bridge subject to high-frequency vibrations under zero gravity conditions

This paper deals with meniscus deformation and flow in an isothermal liquid bridge maintained between two circular rods, when one rod is subject to axial monochromatic vibrations. It concerns a fundamental aspect of the problem of crystal growth from melt by the floating-zone technique which is often considered in weightlessness conditions. In the absence of vibrations the bridge is cylindrical; but due to vibration the mean shape of the meniscus is no more cylindrical and the meniscus oscillates around this mean shape. Two models are developed. First, we take into account the pulsating deformations of the meniscus (free surface), but we assume that the mean shape of meniscus remains cylindrical (i.e., we neglect the influence of vibration on this mean shape). For this simple case, a solution of the problem for the pulsating meniscus deformations and the pulsating velocity field is found in explicit form. For the mean flow, the problem is solved numerically by a finite-difference method. The calculations demonstrate the contribution of two basic mechanisms of mean flow generation due to vibrations, related to the generation of mean vorticity in the viscous boundary layer near the rigid boundaries and surface-wave propagation at a free surface. The intensity of the mean flow induced by surface waves is found to be sharply increasing when the vibration frequency approaches the resonance values that are determined from the explicit form of the solution of pulsation problem. In the second model, we take into account both pulsating and mean deformations of the meniscus. The governing equations for the potential of pulsating velocity and mean velocity, and for the pressure, are solved by using a finite-difference method and a boundary-fitted curvilinear coordinate system fitting the free surface.     

Similarity solutions for axisymmetric plane radial power law fluid flows through a porous medium

This paper investigates the unsteady flow of non-Newtonian fluids of power low behavior through a porous medium in a plane radial geometry. The equation governing the flow is a nonlinear parabolic partial differential equation with a source term whose solution satisfies certain fixed and moving boundary conditions. The attention is focused on the finding of similarity solution when the fixed boundary condition and the source term satisfy certain restrictions. In this case similarity transformations are determined and the resulting ordinary differential equations are deduced. For shear thinning fluids the existence of a pressure disturbance front moving with finite velocity is shown and expression for its location as a function of time is determined. The solutions in closed form have been given for certain particular cases where the resulting differential equations can be analytically solved. A numerical procedure has also been presented.     

Exact piezothermoelastic axisymmetric solution of a finite transversely isotropic cylindrical shell

This work presents an exact piezothermoelastic solution of a finite transversely isotropic piezoelectric cylindrical shell under axisymmetric thermal, pressure and electrostatic excitation. The general solution of the governing equations is obtained in terms of potential functions which satisfy the boundary conditions at the ends. The axisymmetric loadings are expanded as a Fourier series in the axial coordinate. The coefficients in the infinite set of potential functions are obtained by solving sets of six algebraic equations resulting from the satisfaction of boundary conditions at the inner and outer surfaces of the shell for each Fourier component. The inverse problem of inferring the applied temperature and pressure fields from the given measured distribution of electrical potential difference between the inner and outer surfaces of the shell has also been solved. Numerical results are presented for typical pressure, thermal and electrostatic loadings.     

Numerical analysis of hydraulically-driven fractures

A general method-of-lines numerical approach for modeling hydraulically-driven fractures is developed and tested. The methodology employs several novel features: a straining coordinate system that elongates as the fracture grows, an evolutionary equation to describe growth of the fracture length, direct treatment of the fluid/elastic-solid coupling, and a control volume equation which governs fluid motion near the tip and thus circumvents local degeneracy of the differential equations. Spatial discretization of the governing equations leads to a nonsparse system of implicit, coupled ordinary differential equations that is solved for time derivatives that are then integrated with a Runge-Kutta algorithm. The numerical solutions agree very well with known similarity solutions for laminar and for turbulent flow. New solutions for nonsimilar flows are also presented and these converge to proper limits as the fracture becomes very long. Acceptable accuracy, in all cases, is obtained using a very few numerical grid points and with modest execution times.     

Simulation of runaway electrons during tokamak disruptions

When modeling the generation of runaway electrons in tokamak disruptions, it is essential to account for the evolution of the electric field in a self-consistent way. This is achieved by the ARENA code, which is described in the present paper. In this code, the relativistic electron kinetic equation is solved by the Monte Carlo method, supplemented with a weighting scheme to enhance the accuracy of the simulated fast-electron dynamics. Finite elements are employed to solve Maxwell's equations governing the electric field, and this solution is coupled to the Monte Carlo solution of the kinetic equation in a semi-implicit way in order to maintain numerical stability. This numerical scheme thus makes it possible, for the first time, to simulate runaway avalanche kinetics in a disruption self-consistently, accounting both for the acceleration of runaway electrons by the electric field and for the change in the electric field induced by the runaway current. The first results of such a simulation of a JET-like disruption are presented.     

Symmetric laminar incompressible flow past sharp wedges

The symmetric laminar incompressible flow part wedges with a sharp leading edge has been determined using the Navier-Strokes equations written in a coordinate system that is optimal to second order. Because of a singularity that develops in the skin friction at such a sharp leading edge, the flow at this singular point is calculated from the Stokes approximation to the full equations. The free constant in the Stokes solution is determined by equating the skin friction function from the two solutions at a point, on the stagnation streamline, in the immediate vicinity of the sharp tip. Good agreement is obtained with the available results for the flow past the two limiting cases: the semi-finite flat plate and the infinite vertical wall.An implicit alternating direction method with high accuracy and rapid convergence is used to obtain the numerical solutions.     

The d’Alembert–Lagrange equation exploited on a Riemannian manifold

In this paper, we present a geometric exploitation of the d’Alembert–Lagrange equation (or alternatively, Lagrange form of the d’Alembert’s principle) on a Riemannian manifold. We develop the d’Alembert–Lagrange equation in a geometric form, as well as an explicit analytic form with respect to an arbitrary frame in a coordinate neighborhood on the configuration manifold. We provide a procedure to determine the governing dynamic equations of motion. Examples are given to illustrate the new formulation of dynamic equations and their relations to alternative ones. The objective is to provide a generalized perspective of governing equations of motion and its suitability for studying complex dynamic systems subject to nonholonomic constraints.     

再入系统的热流分析

针对再入过程中返回舱体存在严重的气动热问题,提出一种新型再入系统.通过建立减速系统的阻力估算模型和求解动力学方程,分析再入过程的速度特性和驻点热流密度,计算结果与文献算例数据吻合,验证飞行器再入过程的热流特性.采用Euler数值计算与边界层内工程算法相结合的方法计算充气阻力罩表面热流密度,结果表明热流密度在驻点附近较大,远离驻点后迅速减小.     

A finite difference model for air pollution simulation

A 3-D model for atmospheric pollutant transport is proposed considering a set of coupled convection–diffusion–reaction equations. The convective phenomenon is mainly produced by a wind field obtained from a 3-D mass consistent model. In particular, the modelling of oxidation and hydrolysis of sulphur and nitrogen oxides released to the surface layer is carried out by using a linear module of chemical reactions. The dry deposition process, represented by the so-called deposition velocity, is introduced as a boundary condition. Moreover, the wet deposition is included in the source term of the governing equations using the washout coefficient. Before obtaining a numerical solution, the problem is transformed using a terrain conformal coordinate system. This allows to work with a simpler domain in order to build a mesh that provides finite difference schemes with high spatial accuracy. The convection–diffusion–reaction equations are solved with a high order accurate time-stepping discretization scheme which is constructed following the technique of Lax and Wendroff. Finally, the model is tested with a numerical experiment in La Palma Island (Canary Islands).