Jump Conditions for Hyperbolic Systems of Forced Conservation Laws with an Application to Gravity Currents

Weak solutions to systems of nonlinear hyperbolic conservation laws admit discontinuities that result from either an initial value or as part of the temporally developing solution itself. The propagation of such shocks or jumps is affected by forcing terms for the nonlinear system in a way that has not been investigated fully in standard references. Jump conditions for systems of conservation laws with discontinuous forcing terms are derived herein, following the method used to derive the Rankine–Hugoniot jump conditions, and the generalized results are illustrated for the one-dimensional inviscid Burger's equation with discontinuous forcing. The main application of this type of jump condition, and the primary motivation for its study, is its application to a shallow-water model of gravity currents previously described by the authors. Specifically, a new result relation between the front and height at a gravity current front is obtained by using the existing model. Front speeds for gravity currents resulting from instantaneous release are calculated numerically and used to determine the suitability of the jump conditions, which are then compared with existing theoretical expressions and experimental observations. New numerical results are portrayed for the gravity current model, suggesting that the standard method of modeling shallow-water gravity currents with a simple Froude number front condition may tend to suppress some of the finer details of the flow resolved by the numerical scheme used by the authors.     

A Study of Multiple Solutions for the Navier-Stokes Equations by a Finite Element Method

In this paper, a finite element method is proposed to investigate multiple solutions of the Navier-Stokes equations for an unsteady, laminar, incompressible flow in a porous expanding channel. Dual or triple solutions for the fixed values of the wall suction Reynolds number $R$ and the expansion ratio $α$ are obtained numerically. The computed multiple solutions for the symmetric flow are validated by comparing them with approximate analytic solutions obtained by the similarity transformation and homotopy analysis method. Unlike previous works, our method deals with the Navier-Stokes equations directly and thus has no similarity and other restrictions as in previous works. Finally we use the method to study multiple solutions for three cases of the asymmetric flow (which has not been studied before using the similarity-type techniques).     

动脉血管流动计算的伽辽金有限元法研究

得到大动脉三维模型的过二重分叉的二维截定常流的NS方程有限元解,采用了物理坐标系统换到曲线边界贴休坐标系的数学技巧,以支流至主动脉流率为参数,计算了雷诺数为1000的壁面切应力,所得结果与前人的工作（包括实验数据）进行了比较,发现与他们的结果非常接近,改进了Sharma和Kapoor(1995)的工作,相比之下,所用的数值方法上更经济,适用的雷诺数更大。     

Development of the collocations and least squares method

We propose and implement new, more general versions of the method of collocations and least squares (the CLS method) and, for a system of linear algebraic equations, an orthogonal method for accelerating the convergence of an iterative solution. The use of the latter method and the proper choice of values of control parameters, based on the results of investigating the dependence of the properties of the CLS method on these parameters, as well as some other improvements of the CLS method suggested in this paper, allow one to solve numerically problems for Navier-Stokes equations in a reasonable time using a single-processor computer even for grids as large as 1280 × 1280. In this case, the total number of unknown variables is ～ 25 · 10⁶. The numerical results for the problem of the lid-driven cavity flow of a viscous fluid are in good agreement with known results of other authors, including those obtained by means of schemes of higher approximation order with a small artificial viscosity. This and some other facts prove that the new versions of the CLS method make it possible to obtain an approximate solution with high accuracy.     

A finite element method for heat transfer of power‐law flow in channels with a transverse magnetic field

Heat transfer of a power‐law non‐Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power‐law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power‐law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Annular liquid jets and other axisymmetric free-surface flows at high Reynolds numbers

A perturbation method based on a long wavelength approximation is used to obtain the leading order equations governing the fluid dynamics of laminar, annular, round and compound liquid jets and liquid films on convex and concave cylindrical surfaces. An approximate, integral balance method is also used to determine the inviscid core and the thickness of the boundary layers of annular liquid jets near the nozzle exit. The steady state equations are transformed into parabolic ones by means of the von Mises transformation and solved in an adaptive, staggered grid to determine the axial velocity distribution and the location of the free surfaces. It is shown that, for free surface flows subject to inertia, gravity and surface tension, there is a contraction near the nozzle which increases as the Reynolds and Froude numbers are decreased, and is nearly independent of the Weber number for Weber numbers larger than about one hundred. It is also shown that this contraction depends on the flow considered, and is larger for films on convex surfaces. It is also shown that, for round jets, the acceleration of the jet's free surface is larger than that of the jet's centerline, although, sufficiently far from the nozzle exit, the axial velocity is uniform across the jet.     

A differential quadrature analysis of unsteady open channel flow

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation of unsteady open channel flow. To the best of authors’ knowledge, this is the first attempt to use the DQM in open channel hydraulics. The Saint-Venant equations and the related nonhomogenous, time dependent boundary conditions are discretized in spatial and temporal domain by DQ rules. The unknowns in the entire domain are computed by satisfying governing equations, boundary and initial conditions simultaneously. By employing DQM, accurate results can be obtained using dramatically less grid points in spatial and time domain. The stability of DQM solution is not sensitive to choosing time step or Courant number unlike other methods. Although numerical problems such as instability, oscillation and underestimation near critical depth can be seen by using other methods but DQM solution is smooth and accurate in this case. The results are sensitive to grid distribution in time domain. In light of this, Chebyshev–Gauss–Lobatto distribution performance is excellent. To validate the DQM solutions, the obtained results are compared with those of the characteristic method. In conclusion, DQM is a potential powerful method with minimum computational effort for unsteady flow simulation.     

小波分析方法在金融股票数据预测中的应用

利用小波分析预测方法对金融数据—股票收盘价这一典型的非平稳时间序列进行预测.使用M a llat小波分解算法对数据进行分解,对分解后的数据进行平滑处理,然后再进行重构,而重构之后的数据就成为近似意义的平稳时间序列,这样就得到了原始数据的近似信号,再应用传统时间序列预测方法对重构后的数据进行预测,将预测结果与实际值,以及和传统预测方法预测结果比较,小波分析方法预测效果更为理想.     

带有长期趋势的近似周期时间序列的刻度变换估计

近似周期时间序列具有近似的周期趋势,即近似周期性.所谓近似周期性是指它看起来有周期性,但是每个周期的长度不是常数,比如太阳黑子数序列.近似周期时间序列在社会经济现象建模中有着广泛的应用前景.对于近似周期时间序列,关键在于刻画它的近似周期趋势,因为一旦近似周期趋势被刻画出来它就可以作为一个普通的时间序列来处理.然而,关于近似周期趋势刻画的研究却很少. 本文首先建立一些必要的理论,特别地,提出了带长度压缩的保形变换概念,并且得到了带长度压缩的线性保形变换的充分必要,然后基于此理论作者提出了一种估计尺度变换的方法,该方法可以很好地估计出近似周期趋势.最后,对一个仿真实例进行了分析.结果表明,本文所提出的方法强力有效.     

Existence and a priori bounds for steady stagnation flow toward a stretching cylinder

We investigate the nonlinear boundary value problem (BVP) that is derived from a similarity transformation of the Navier-Stokes equations governing fluid flow toward a stretching permeable cylinder. Existence of a solution is proven for all values of the Reynolds number and for both suction and injection, and uniqueness results are obtained in the case of a monotonic solution. A priori bounds on the skin friction coefficient are also obtained. These bounds achieve any desired order of accuracy as the injection parameter tends to negative infinity.     

Application of the differential transformation method to vibration analysis of pipes conveying fluid

In this paper, a relatively new semi-analytical method, called differential transformation method (DTM), is generalized to analyze the free vibration problem of pipes conveying fluid with several typical boundary conditions. The natural frequencies and critical flow velocities are obtained using DTM. The results are compared with those predicted by the differential quadrature method (DQM) and with other results reported in the literature. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of pipes conveying fluid.     

A New Analytical Approach to Solve Magnetohydrodynamics Flow Over a Nonlinear Porous Stretching Sheet by Laplace Padé Decomposition Method

The aspire of this article is to bring in a new approximate method, that is to say the Laplace Padé decomposition method which is a mixture of Laplace decomposition and Padé approximation to offer an analytical approximate way out to magnetohydrodynamics flow over a nonlinear porous stretching sheet. This new iteration approach provides us with a convenient way to approximate solution. A closed agreement between the obtained solution and some well-known results has been established. The proposed procedure can be applied to handle other nonlinear problems.     

A variational iteration method for solving Troesch’s problem

Troesch’s problem is an inherently unstable two-point boundary value problem. A new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve Troesch’s problem. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. Only the second-order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods.     

A fast and accurate algorithm for computing radial transonic flows

An efficient algorithm is described for calculating stationary one-dimensional transonic outflow solutions of the compressible Euler equations with gravity and heat source terms. The stationary equations are solved directly by exploiting their dynamical system form. Transonic expansions are the stable manifolds of saddle-point-type critical points, and can be obtained efficiently and accurately by adaptive integration outward from the critical points. The particular transonic solution and critical point that match the inflow boundary conditions are obtained by a two-by-two Newton iteration which allows the critical point to vary within the manifold of possible critical points. The proposed Newton Critical Point (NCP) method typically converges in a small number of Newton steps, and the adaptively calculated solution trajectories are highly accurate. A sample application area for this method is the calculation of transonic hydrodynamic escape flows from extrasolar planets and the early Earth. The method is also illustrated for an example flow problem that models accretion onto a black hole with a shock.     

Fluid flow through a vertical slot under gravity

The flow of fluid from a horizontal uniform channel of finite depth into a vertical slot under the influence of gravity is considered. The shapes of a top and bottom free surface are computed for a range of parameter values. Solutions in which the top free surface attaches smoothly to the vertical wall were only found for Froude numbers greater than or equal to one, though solutions were found for all gap sizes.     

DISCRETE GALERKIN METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

In this article, we develop a fully Discrete Galerkin(DG) method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.     

The study of Heat and Mass Transfer in a Visco elastic fluid due to a continuous stretching surface using Homotopy Analysis Method

In this paper, an approximate analytical solution is derived for the flow velocity and temperature due to the laminar, two-dimensional flow of non-Newtonian incompressible visco elastic fluid due to a continuous stretching surface. The surface is stretched with a velocity proportional to the distance $x$ along the surface. The surface is assumed to have either power-law heat flux or power-law temperature distribution. The presence of source/sink and the effect of uniform suction and injection on the flow are considered for analysis. An approximate analytical solution has been obtained using Homotopy Analysis Method(HAM) for various values of visco elastic parameter, suction and injection rates. Optimal values of the convergence control parameters are computed for the flow variables. It was found that the computational time required for averaged residual error calculation is very very small compared to the computation time of exact squared residual errors. The effect of mass transfer parameter, visco elastic parameter, source/sink parameter and the power law index on flow variables such as velocity, temperature profiles, shear stress, heat and mass transfer rates are discussed.     

Modeling and control of surface gravity waves in a model of a copper converter

Undesirable splashing appears in copper converters when air is injected into the molten matte to trigger the conversion process. We consider here a cylindrical container horizontally placed and containing water, where gravity waves on the liquid surface are generated due to water injection through a lateral submerged nozzle. The fluid dynamics in a transversal section of the converter is modeled by a 2-D inviscid potential flow involving a gravity wave equation with local damping on the liquid surface. Once the model is established, using a finite element method, the corresponding natural frequencies and normal modes are numerically computed in the absence of injection, and the solution of the system with injection is obtained using the spectrum. If a finite number of modes is considered, this approximation leads to a system of ordinary differential equations where the input is represented by the fluid injection. The dynamics is simulated as perturbations around a constant fluid injection solution, which is the desired operating state of the system, considering that the conversion process does not have to be stopped or seriously affected by the control. The solution is naturally unstable without control and the resulting increase of amplitude of the surface waves are assimilable to the splashing inside the converter. We show numerically that a variable flow around the operating injection is able to sensibly reduce these waves. This control is obtained by a LQG feedback law by measuring the elevation of the free surface at the point corresponding to the opposite extreme to where the nozzle injection is placed.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.