The Lie-group shooting method for boundary-layer problems with suction/injection/reverse flow conditions for power-law fluids

For power-law fluids we propose a Lie-group shooting method to tackle the boundary-layer problems under a suction/injection as well as a reverse flow boundary conditions. The Crocco transformation is employed to reduce the third-order equation to a second-order ordinary differential equation, and then through a symmetric extension to a boundary value problem with constant boundary conditions, which can be solved numerically by the Lie-group shooting method. However, the resulting equation is singular, which might be difficult to solve, and we propose a technique to overcome the initial impulse caused by the singularity using a very small time-step integration at the first few time steps. Because we can express the missing initial condition through a closed-form formula in terms of the weighting factor r∈(0,1), the Lie-group shooting method is very effective for searching the multiple-solutions of a reverse flow boundary condition.     

A PIECEWISE–LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO–POINT BOUNDARY–VALUE PROBLEMS

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton–Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.     

Thermal stresses induced by a point heat source in a circular plate by quasi-static approach

The present paper deals with the determination of quasi-static thermal stresses due to an instantaneous point heat source of strength g_pi situated at certain circle along the radial direction of the circular plate and releasing its heat spontaneously at time t = τ. A circular plate is considered having arbitrary initial temperature and subjected to time dependent heat flux at the fixed circular boundary of r = b. The governing heat conduction equation is solved by using the integral transform method, and results are obtained in series form in terms of Bessel functions. The mathematical model has been constructed for copper material and the thermal stresses are discussed graphically.     

A non-iterative transformation method for Newton's free boundary problem

In book II of Newton's Principia Mathematica of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in the literature as a transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in the literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height.     

非惯性坐标系中的边界层方程

本文用匹配渐近展开法导出二维翼型在非惯性坐标系中的一阶和二阶边界层方程,消去压强项后,一阶边界层方程与经典边界层方稃相同;惯性力的作用在二阶边界层中才出现。     

Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration

In this article we derive semi-analytical/numerical solutions for transport phenomena (momentum, heat and mass transfer) in a nanofluid regime adjacent to a nonlinearly porous stretching sheet by means of the Homotopy analysis method (HAM). The governing equations are reduced to a nonlinear, coupled, non-similar, ordinary differential equation system via appropriate similarity transformations. This system is solved under physically realistic boundary conditions to compute stream function, velocity, temperature and concentration function distributions. The results of the present study are compared with numerical quadrature solutions employing a shooting technique with excellent correlation. Furthermore the current HAM solutions demonstrate very good correlation with the non-transpiring finite element solutions of Rana and Bhargava (Commun. Nonlinear Sci. Numer. Simul. 17:212–226, 2012). The influence of stretching parameter, transpiration (wall suction/injection) Prandtl number, Brownian motion parameter, thermophoresis parameter and Lewis number on velocity, temperature and concentration functions is illustrated graphically. Transpiration is shown to exert a substantial influence on flow characteristics. Applications of the study include industrial nanotechnological fabrication processes.     

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.     

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.     

Incompressible laminar 2D steady thermal boundary layers with temperature-dependent kinematic viscosity and thermal diffusivity

We prove that the incompressible 2D steady thermal boundary layer equations with temperature-dependent kinematic viscosity ν and thermal diffusivity α is maximally symmetric provided the Prantl number Pr=ν/α is constant and or ν=K₂(AT+B)^K₁ if we neglect energy dissipation and if we take into account dissipation. This result corroborates assumptions often made in applications. When we disregard dissipation, the symmetry Lie algebra assumes the forms L^r⊕L^∞, where L^∞ is an infinite-dimensional Lie algebra and L^r is an r-dimensional Lie algebra with r∈{3,4,5,6}. If we include dissipation, r∈{2,3}. We notice that dissipation has a symmetry breaking effect.We also show how the symmetries can be employed for the calculation of invariant solutions.     

Improvement for expansion of parabolized stability equation method in boundary layer stability analysis

An improved expansion of the parabolized stability equation(iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber α and its streamwise gradient dα/dx are unknown variables. This eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation used in the PSE. The i EPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the i EPSE is significantly higher than that of the ESPE, and it is in excellent agreement with the PSE which is regarded as the baseline for comparison. In addition, the unphysical multiple eigenmode problem in the EPSE is solved by using the i EPSE. As a local non-parallel stability analysis tool, the i EPSE has great potential application in the eNtransition prediction in general three-dimensional boundary layers.     

Nonlinear boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

In this paper, we consider the boundary control problem of the unforced generalized Korteweg–de Vries–Burgers (GKdVB) equation when the spatial domain is [0,1]. Three control laws are derived for this equation and the L ²-global exponential stability of the solution is proved analytically. Numerical results using the finite element method (FEM) are presented to illustrate the developed control schemes.     

Reynolds方程在纹理表面动压润滑计算中的有效性评价

从理论上阐述了纹理表面动压润滑计算中决定Reynolds方程有效性的两个关键因素为油膜厚度与纹理特征长度的比值h/L和缩减的雷诺数re;只有当h/L和re同时趋近于零时Reynolds方程才能够获得准确的结果,并由此在h/L-Re平面上标注了Reynolds方程的适用范围.继而以二维矩形沟槽为实例,采用数值方法计算了h/L和re对Reynolds方程误差的影响规律;分析了Reynolds方程在不同条件下的失效机制;分析了矩形沟槽纹理表面Reynolds方程有效性的评价标准:当缩减的雷诺数re小于0.20,并且h/L小于0.015时能够保证Reynolds方程的误差在10%以下.     

Flow characteristics around a circular cylinder subjected to vortex-induced vibration near a plane boundary

Although vortex-induced vibration (VIV) has been extensively studied, much of existing literature deals with uniform flow in the absence of a boundary. The VIV flow field of a structure close to a boundary generally remains unexplored, but it can have important engineering implications, such as pipeline scour if the boundary is an erodible seabed. In this paper, laboratory experiments are performed to investigate the flow characteristics of an elastically mounted circular cylinder undergoing VIV, and a rigid plane boundary is considered to simplify the problem. The initial gap-to-diameter ratio is fixed at 0.8, and six different reduced velocities are considered. The velocity field is measured using a high resolution particle image velocimetry (PIV) system, which has several advantages over traditional PIV systems, including high sampling rate and the ability to mitigate scatter of laser light near the boundary, allowing accurate measurements at the viscous sublayer. This paper presents the vibration amplitude and oscillation frequency for different V_r; in addition, the mean velocity field, turbulence characteristics, vortex behavior, gap flow velocity, and normal/shear stresses on the boundary were measured/calculated, leading to new insights on the flow field behavior.     

A second-order approximation to natural convection for large Rayleigh numbers and small Prandtl numbers

The problem under investigation is that of fluid flow within an enclosed rectangular cavity. It is assumed that one wall is maintained at a constant temperature T₁ (hot wall) and the other wall is maintained at a constant temperature T₀ (cold wall). At the remaining walls, two separate cases are studied. In the first, an adiabatic boundary condition is assumed. That is, the normal derivative of the temperature function is assumed to be 0. In the second, it is assumed the temperature varies linearly from T₀ to T₁. The purpose of this paper is the application of a second order numerical technique to the problem of fluid flow within a heated closed cavity. The method is a modification of a method developed by Shay¹ and applied to the driven cavity problem. In order to test the viability of this technique, it was decided to extend the technique to the problem of natural convection in a square. Jones² proposed that this problem is suitable for testing techniques that may be applied to a wide range of practical problems such as reactor insulation, cooling of radioactive waste containers, solar energy collection and others.³ The technique makes use of second-order finite difference approximations to all derivatives in the governing equations. Furthermore, second-order approximations are also used to determine boundary vorticities and, when the adiabatic boundary condition is used, for the boundary temperatures as well. In some works, where second-order approximations are used at interior points, second-order boundary approximations have been sacrificed in favour of a more stable, but first-order boundary approximation. The current approximations are generated by writing the unknown value of a function at a given interior node as a linear combination of unknown function values at all of the neighbouring nodes. Then the function values at these neighbouring nodes are expanded in a Taylor series about the given node. Through appropriate regrouping of terms and the use of the equations to the solved, constraints are imposed on the coefficients of the linear combination to yield a second-order approximation. As it turns out, there are more unknowns than constraints and, as a result, we are left with some freedom in choosing coefficients. In this work this freedom was used to choose coefficients in such a way as to maximize stability of the resulting system of equations. In other words, the approximations to the governing partial differential equation are individually determined at each point dependent on the direction of flow in order to generate the best possible stability. This idea is analogous to that used in the derivation of the upwind method. However, the current method is second-order accurate where the upwind method is only first-order accurate. Thus, what is generated is an easily implemented second-order method that yields a system of equations that has proved easy to solve. The system of equations is solved via the method of successive overrelaxation. The stability of the method is shown in the convergence for a wide range of Rayleigh numbers, Prandtl numbers and mesh sizes. Level curves of the stream, vorticity and temperature functions are provided for Rayleigh numbers (Ra) as large as 100,000, Prandtl numbers (Pr) as small as 0.0001, and mesh sizes as small as 0.0125. Values of the Nusselt number have also been calculated through the use of Simpson's rule, and a second order approximation to the normal derivative of the temperature along the cold wall. Comparisons are made with other current works to aid in the verification of this methods' accuracy and also with the first-order upwind method to demonstrate superiority over the first-order method.     

Approximation of the third two-point moments of the velocity field in isotropic turbulence

Different methods for approximating the third two-point moments of the velocity fields entering in the von Kármán-Howarth equation are compared using the known experimental results of A. Townsend and R. Stewart, together with the model form of the energy spectrum of turbulence, which makes it possible to approximate the experimentally determined second moments of the velocity field. The latter moments are used for calculating the third two-point moments following the methods proposed by various authors. The calculated results are compared with the experimental data which makes it possible to quantitatively estimate the approximation accuracy. For several models the second-order structure function is calculated from the Kolmogorov equation for the inertial interval. In this case, for the models of K. Hasselmann and Yu.M. Lytkin the power-law dependence D _LL (r) ～ r ^2/3 expected for the inertial interval is obtained for the structure function D _LL (r).     

A penalization method applied to the wave equation

In this Note we investigate the mathematical properties of the volume penalization method applied to the one-dimensional wave equation. Generally speaking, the penalization method allows one to handle complex geometries by simply adding a term to the equation to impose the boundary conditions. We study the convergence of the method with regards to the penalization parameter and we present error and stability analyses for the wave equation. Numerical simulations using a finite difference scheme illustrate the results. To cite this article: A. Paccou et al., C. R. Mecanique 333 (2005).     

极坐标系下Fourier-Legendre谱元方法与有限差分法数值扩散的比较

提出一种Fourier-Legendre谱元方法用于求解极坐标系下的Navier-Stokes方程,其中极点所在单元的径向采用Gauss-Radau积分点,避免了r=0处的1/r坐标奇异性。时间离散采用时间分裂法,引入数值同位素模型跟踪同位素的输运过程验证数值模拟的精度,分别利用谱元法和有限差分法的迎风差分格式求解匀速和加速坩埚旋转流动中的同位素方程。计算结果表明,有限差分法中的一阶迎风差分格式存在严重的数值假扩散,二阶迎风差分格式的数值结果较精确,增加节点可以有效地缓解数值扩散。然而,谱元法具有以较少节点得到高精度解的优势。     

A uniqueness criterion for fractional differential equations with Caputo derivative

We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order α∈(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann–Liouville derivative of this nonlinearity verifies a special inequality.     

Theoretical study of the flow field inside a hydrocyclone with vortex finder diameter greater than that of apex opening—I. Laminar case

The axi-symmetric laminar boundary layers, formed on the conical surface and under the cyclone roof, have been calculated by Pohlhausen's method assuming that the tangential velocity outside boundary layers varies as r ^?n up to the point where boundary layers meet solid body rotating liquid column and thereafter as r—as found in experiments—where r is the distance from the axis of the cone and n is flow pattern constant. Point of interaction of the boundary layers with solid body rotating liquid column has been taken from experimental results obtained with a hydrocyclone having vortex finder diameter greater than that of apex opening. Results show that there is no possibility of separation of boundary layers from cyclone wall.     

Exact analytic solutions for the damped Duffing nonlinear oscillatorLes solutions analytiques exactes pour le amortissable non linéaire oscillateur de Duffing

We prove that the second-order damped nonlinear Duffing oscillator is reduced to an equivalent equation of the normal Abel form of the second kind. Based on a recently developed mathematical methodology for the construction of exact analytic solutions of Abel's equation, exact analytic solutions are obtained for the nonlinear damped Duffing oscillator obeying the initial conditions adapted to the physical problem. To improve the general developed methodology an application concerning the nonlinear Van der Pol free oscillator is briefly discussed. To cite this article: D.E. Panayotounakos et al., C. R. Mecanique 334 (2006).