A Newton/upwind method and numerical study of shock wave/boundary layer interactions

The objective of the paper is twofold. First we describe an upwind/central differencing method for solving the steady Navier–Stokes equations. The symmetric line relaxation method is used to solve the resulting algebraic system to achieve high computational efficiency. The grid spacings used in the calculations are determined from the triple-deck theory, in terms of Mach and Reynolds numbers and other flow parameters. Thus the accuracy of the numerical solutions is improved by comparing them with experimental, analytical and other computational results. Secondly we proceed to study numerically the shock wave/boundary layer interactions in detail, with special attention given to the flow separation. The concept of free interaction is confirmed. Although the separated region varies with Mach and Reynolds numbers, we find that the transverse velocity component behind the incident shock, which has not been identified heretofore, is also an important parameter. A small change of this quantity is sufficient to eliminate the flow separation entirely.     

The cumulant method for computational kinetic theory

We propose a new method for numerical simulation of gas dynamics based on kinetic theory. The method is based on a cumulant-expansion-ansatz for the phase space density, which leads to a set of quasi-linear, hyperbolic partial differential equations. The method is compared to the moment method of Grad. Both methods agree for low-order approximations but the method proposed shows additional non-linear terms for high order approximations. Boundary conditions on the cumulants for an ideally reflecting and an ideally rough boundary surface are derived from conditions on the phase space density. A Lax-method is used for numerical analysis of a 2d-BGK fluid, which results in an easy-to-implement algorithm well suited for implementation on massivly parallel computers. The results are found to agree qualitatively with predictions from moment theories. Received August 10, 2000     

The three-dimensional boundary layer on a rotating flat plate as influenced by the containing end-walls

Growth of the turbulent boundary layer over a flat plate rotating about an axis parallel to the leading edge is considered in which the axial length (or span) is contained between rotating radial end-plates (the hub and shroud, in effect, of a centrifugal impeller). The problem of the influence of the cross-flows in the boundary layers on the end-plates as they affect the blade boundary layer is considered. The latter is treated as a three-dimensional problem and the dependence of the solution on the boundary conditions is discussed. The integral equations of this boundary layer reduce to a pair of quasi-linear partial differential equations which are weakly elliptic, parabolic, or weakly hyperbolic according to the rotation number. When the equations are exactly parabolic and the boundary layers remain thin it is shown that the end-plate boundary layers can have no influence upon the blade boundary layer if the flow is initially radial; separation of the end-plate cross flows takes place in the corners.     

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

Formulation of boundary integral equations for three-dimensional elasto-plastic flow

A general theory of elasto-plastic flow is formulated for work-hardening materials that may be both anisotropic and compressible. Because the theory is quasi-linear, it may be cast in terms of integral equations and the result is an extended form of Somigliana's identity. When these relations are evaluated on the boundary of a solid, their dimensionality is reduced. Previous experience with simpler materials shows that arbitrary problems may be solved in a direct manner.     

Global dynamics of a harmonically excited oscillator with a play: Numerical studies

In this paper a harmonically excited linear oscillator with a play is investigated. Direct numerical simulation and numerical continuation techniques were employed to study the system behaviour. To conduct the numerical analysis, the system differential equations were transformed into the autonomous form and were then solved using our newly developed in-house Matlab-based computational suite ABESPOL [1]. The results are presented in form of trajectories and Poincaré maps on the phase plane, bifurcation diagrams and basins of attraction. The bifurcation analysis was supported by a path following method. The influence of each system parameter (except gap) on the system dynamics was studied in detail. The bifurcations known as interior crisis and boundary crisis were observed and discussed in this work. Notably, the parameter regions where various types of grazing induced bifurcations occurred were detected and investigated.     

考虑法向压力梯度的三维旋转边界层计算

本文以量级分析为基础,建立了一般曲线坐标系上的三维旋转边界层方程。对旋转在边界层中的影响进行分析之后,提出了一个能够处理壁面法向压力梯度不为零问题的压力梯度迭代方法。在传统的Box法的基础上发展了一套完整的求解三维旋转边界层的方法和程序,并对螺旋面、压气机转子叶面以及圆柱面上的旋转边界层进行了计算,与他人的计算和实验的对比分析表明,该方法和程序是正确的,可用于求解任意几何物面上的三维旋转边界层。     

Interactions fluide–paroi

The object of this work is the modelling and the analysis of fluid–wall interactions in a cylindrical cavity with a unique orifice serving both as inflow and outflow of an incompressible fluid. The analysis shows that the system's behavior is governed by the values of five dimensionless characteristic parameters. The coupled equations system is then approximated by using a staggered scheme which enables to integrate the structure and the fluid parts separately during each time step. A numerical solution is given with the respective influence of each introduced parameter. We conclude by giving an approximation of the viscous friction effects allowing to uncouple the system of equations.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Finite elasto-plastic deformation—I: Theory and numerical examples

It is demonstrated that the problem of elasto-plastic finite deformation is governed by a quasi-linear model irrespective of deformation magnitude. This feature follows from the adoption of a rate viewpoint toward finite deformation analysis in an Eulerian reference frame. Objectivity of the formulation is preserved by introduction of a frame-invariant stress rate.Equations for piece-wise linear incremental finite element analysis are developed by application of the Galerkin method to the instantaneously linear governing differential equations of the quasi-linear model. Numerical solution capability has been established for problems of plane strain and plane stress. The accuracy of the numerical analysis is demonstrated by consideration of a number of problems of homogeneous finite deformation admiting comparative analytic solution. It is shown that accurate, objective numerical solutions can be obtained for problems involving dimensional changes of an order of magnitude and rotations of a full radian.     

Numerical simulation of vortical ideal fluid flow through curved channel

A numerical algorithm to study the boundary‐value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co‐ordinate system. The convergence of the finite‐difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka–Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two‐dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented. Copyright © 2003 John Wiley & Sons, Ltd.     

THE GENERALIZED GALERKIN’S EQUATION OF THE FINITE ELEMENT,THE BOUNDARY VARIATIONAL EQUATIONS AND THE BOUNDARY INTEGRAL EQUATIONS

Based on[1 ],we have further applied the variational princi-ple of the variable boundary to investigate the discretizationanalysis of the solid system and derived the generalized Ga-lerkin’s equations of the finite element.the boundary varia-tional equations and the boundary integral equations.These e-quations indicate that the unknown functions of the solid sys-tem must satisfy the conditions in the element S_a or on the boun-dariesГ_a.These equations are applied to establishing the discreti-zation equations in order to obtain the numerical solution ofthe unknown functions.At a time these equations can be usedas the basis for the simplified calculation in various corres-ponding cases.In this paper,the results of boundary integral equationsshow that the calculation of integration is not accurate alongthe surfaceГ_a of interior element S_a by J-integral suggestedby Rice[2].     

非线性流体-刚体结构相互作用问题的一种数值模拟方法

给出了一种模拟非线性流体-刚体结构相互作用问题的数值方法．文中假定结构承受大的刚体运动,流体流动受非线性有粘或无粘的场方程支配并满足自由表面和两相耦合界面上的非线性边界条件,利用任意拉氏-欧氏（ALE）网格系统构造了数值模型．采用所探讨的多块数值格式,允许可动重造网格间有独立的相对运动,从而克服了流体网格与固体大运动匹配的困难．通过数值离散化,导出了描述非线性流固耦合动力学的数值方程并应用耦合迭代过程对其作了求解．通过算例,说明了所提出数值模型的应用．     

A NUMERICAL METHOD FOR SIMULATING NONLINEAR FLUID-RIGID STRUCTURE INTERACTION PROBLEMS

A numerical method for simulating nonlinear fluid-rigid structure interaction problems is developed. The structure is assumed to undergo large rigid body motions and the fluid flow is governed by nonlinear, viscous or non-viscous, field equations with nonlinear boundary conditions applied to the free surface and fluid-solid interaction interfaces. An Arbitrary-Lagrangian-Eulerian (ALE) mesh system is used to construct the numerical model. A multi-block numerical scheme of study is adopted allowing for the relative motion between moving overset grids, which are independent of one another. This provides a convenient method to overcome the difficulties in matching fluid meshes with large solid motions. Nonlinear numerical equations describing nonlinear fluid-solid interaction dynamics are derived through a numerical discretization scheme of study. A coupling iteration process is used to solve these numerical equations. Numerical examples are presented to demonstrate applications of the model developed.     

Turbulent boundary layer heat transfer for a constant property particle-laden gas flow

Solution of a turbulent boundary layer for a constant property, particle-laden gas flow is obtained by a differential method. A dimensionless analysis shows importance of an interaction parameter in increasing heat flux. Boundary layer analysis is done in usual manner by transforming partial differential equations and solution is started at the leading edge by Runge-Kutta method. Velocity and temperature profiles at downstream planes for gas and particles are calculated by an implicit finite-difference iterative procedure, and numerical results are compared with available experimental data.     

On the Formation of Discontinuities of Wave Fronts in a Saturated Granular Body

The paper deals with propagation of plane wave fronts in a solid with a non-linear relation between stress and deformation. The objective is to calculate the distance that a wave front covers before it loses continuity. The formulae derived for a general quasi-linear system of two equations are applied to the propagation of plane compression waves in dry and partially saturated granular bodies. In the case of a saturated body with gas bubbles, the influence of gas and capillary pressure on the stiffness of the body is taken into account. Numerical calculations relevant to soil mechanics are presented. For the numerical calculations a constitutive equation of the hypoplasticity theory for granular materials has been used. Received November 1, 1997     

Numerical and experimental investigation of the laminar boundary layer over riblets

One of the main aims of this work is to show to what extent drag reduction in a turbulent boundary layer can be ascribed to a purely viscous effect. A numerical and experimental study is performed in a laminar boundary layer over triangular riblets. The 2-D parabolic equations of motion are integrated using an x marching method and the discretised system is solved with the MSI algorithm. The influence of the riblet geometrical parameters and of the number of grid points is studied. Measurements are carried out in a water tunnel with forward scatter and backscatter laser-Doppler velocimetry extending within the riblets. The longitudinal velocity component measurements and computations are practically identical. Numerical results presented herein show that a slight drag reduction is obtained for s/h=1.2. It appears that, as far as friction is concerned, the wetted area is not the surface to be considered. Thus, the boundary layer over riblets would behave like a boundary layer on an equivalent smooth plate located beneath the crest plane. The numerical study in terms of the riblet height h shows best results are for h tending to zero, with the ratio s/h being equal to 1.2.     

The Absolute Instability of Joukowski-Type Airfoils

In this paper an investigation is undertaken to explore the nature of the flow in the vicinity of the trailing edge of Joukowski-type airfoil configurations. Making use of the asymptotic interactive boundary layer theory, the basic flow profiles in the attached and detached flow regions are computed numerically through integrating the interactive boundary layer equations governing the flow motion for sufficiently large Reynolds numbers. Employing a Spectral Chebyshev collocation numerical integration scheme, boundary layer features corresponding to a number of thickness-to-chord ratio parameter cases are produced. The analysis carried out over the interaction region of the trailing edge shows that flow separation always takes place beyond certain critical value of the thickness-to-chord ratio parameter under the action of a self-induced pressure gradient. In addition, reversed flow regions of a sufficiently large size are found to be absolutely unstable, within the framework of linear spatio-temporal stability analysis in combination with the Briggs--Bers branch point criterion. Received 10 January 2001 and accepted 15 August 2001     

地形和土-结相互作用效应对三维跨峡谷桥梁地震响应的影响分析

评估跨峡谷桥梁的地震性态需要考虑地形效应、行波效应以及土-结相互作用效应.将峡谷-桥梁系统在地震波输入下的反应分析看作波动散射问题,即桥梁及其邻近非规则区域对峡谷场地"自由场"的扰动. 基于此思想,本文发展了一套跨峡谷桥梁地震反应分析方法,通过二维模型分析得到峡谷场地的"自由场", 结合人工边界输入到峡谷-桥梁体系,采用土-结相互作用分区并行方法对其进行分析, 并编制了相应的分析程序.该方法可在自由场分析时考虑非垂直入射地震波, 计入行波效应,因此可综合考虑行波效应、地形效应和土-结相互作用效应. 通过峡谷场地分析算例,验证了自由场和人工边界实施的正确性; 并以马水河大桥为对象,通过5种计算模型结果的比较,分析了地形效应和土-结相互作用效应对跨峡谷桥梁地震反应的影响, 算例结果表明,地形效应对墩底剪力、弯矩和轴力有明显影响,对位移的影响要比对剪力、弯矩的影响小; 土-结相互作用对桥梁反应的影响较大,较大地减小了桥梁反应.      

On Disturbance Propagation in Internal-Flow Boundary Layers

The unsteady interaction of plane-channel wall boundary layers with a supersonic inviscid flow is investigated. The flow regimes in which disturbances introduced by the boundary layer developing on one wall influence the boundary layer on the other wall are considered. The regime of relatively large pressure disturbance amplitudes generated near the nozzle outlet or by deforming the channel walls is studied. In these conditions, the interaction process is described by a system of Burgers equations with retarded arguments. Numerical solutions of this system are obtained for symmetric and antisymmetric perturbations of the channel walls.