A layerwise boundary integral equation model for layers and layered media

A hybrid method is presented for the analysis of layers, plates, and multilayered systems consisting of isotropic and linear elastic materials. The problem is formulated for the general case of a multilayered system using a total potential energy formulation. The layerwise laminate theory of Reddy is employed to develop a layerwise, two-dimensional, displacement-based, hybrid boundary element model that assumes piecewise continuous distribution of the displacement components through the system's thickness. A one-dimensional finite element model is used for the analysis of the multilayered system through its thickness, and integral Fourier transforms are used to obtain the exact solution for the in-plane problem. Explicit expressions are obtained for the fundamental solution of a typical infinite layer (element) assuming linear displacement distribution through its thickness. This fundamental solution is given in a closed form in the cartesian space, and it can be applied in the two-dimensional boundary integral equation model to analyze layered structures with finite dimensions. The proposed method provides a simple, efficient, and versatile model for a three-dimensional analysis of thick plates or multilayered systems.     

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。     

Investigation of three-dimensional boundary layers on blunt bodies with permeable surface

Numerical and approximate analytic methods are used to investigate three-dimensional laminar boundary layers on blunt bodies with permeable surface in a supersonic gas stream. In the first approximation of the integral method of successive approximation an analytic solution is obtained to the problem for an impermeable surface, small values of the blowing parameter, and arbitrary suction. For large parameters of the blowing (or suction), whose velocity vector in the general case is directed at a certain angle to the vector of the outer normal to the body, asymptotic expressions are derived for the components of the frictional stress and the heat flux. A numerical solution is obtained to the equations of the three-dimensional boundary layer in a wide range of variation of the blowing (or suction) parameter. The accuracy and region of applicability of the analytic solutions is estimated by comparison with the numerical solutions. On the basis of the solutions obtained in the present paper and the work of other authors an expression is proposed for calculating the heat fluxes to a perfectly catalytic surface of a body in a three-dimensional supersonic flow of dissociated or ionized air. The present paper continues earlier work of the authors [1, 2] on boundary layers in the neighborhood of a symmetry plane and on sweptback wings of infinite span.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 49–58, May–June, 1982.     

多层复合壳体三维振动分析的谱——微分求积混合法

对于较厚的多层复合壳体,其振动位移沿厚度方向呈锯齿形变化且层间剪切和拉、压应力呈三维耦合状态,采用传统的等效单层理论分析已不能满足精度要求. 建立不受结构厚度、铺层材料性质和铺层方式限制的三维分析方法具有重要的研究价值. 本文以独立铺层为建模对象,结合广义谱方法与微分求积技术建立了一种适用一般边界条件和铺层方式的多层复合壳体三维分析新方法——谱--微分求积混合法. 该方法应用三维弹性理论对独立铺层进行精确建模,有效克服了二维简化理论对横向变形以及层间应力估计不确切的缺点;引入微分求积技术对铺层进行数值离散,将三维偏微分问题转化为二维偏微分问题,降低了求解维度和难度;应用广义谱方法近似地表述离散计算面上的场变量,将获取的二维偏微分方程转化为以场变量谱展开系数为未知量的线性代数方程组,避免了对超越方程的求解. 数值验证结果表明该方法收敛性好,计算精度高.      

Multi-directional and multi-time step absorbing layer for unbounded domain

Variant techniques are proposed for reproducing the elastic wave propagation in an unbounded medium such as the infinite elements, the absorbing boundary conditions or the perfect matched layers. Here, a simplified approach is adopted by considering absorbing layers characterized by the viscous Rayleigh matrix as studied by Semblat et al. [16] and Rajagopal et al. [14]. Here, further improvements to this procedure are provided. First, we start by establishing the strong form for the elastic wave propagation in a medium characterized by the Rayleigh matrix. This strong form will be used for deriving optimal conditions for damping out in the most efficient way the incident waves while minimizing the spurious reflected waves at the interface between the domain of interest and the Rayleigh damping layer. A procedure for designing the absorbing layer is proposed by targeting a performance criterion expressed in terms of logarithmic decrement of the wave amplitude in the layer thickness. Second, the GC subdomain coupling method, proposed by Combescure and Gravouil [9], is introduced for enabling the choice of any Newmark time integration schemes associated with different time steps depending on subdomains. When wave propagation is predicted by an explicit time integrator, the subdomain strategy is of great interest because it enables a different time integrator for the absorbing layer to be adopted. An external coupling software, based on the GC method, is used to carry out multi=time step explicit/implicit co-computations, making interact in time an explicit FE code (Europlexus) for the domain of interest, with an implicit FE code (Cast3m) handling the absorbing boundary layers. The efficiency of the approach is shown in 1D and 2D elastic wave propagation problems.     

On the application of en-methods to three-dimensional boundary-layer flows

Extension of the eⁿ-method from two-dimensional to three-dimensional boundary-layer flows has not been straightforward. Confusion has centred on whether to use temporal or spatial stability theories, conversion between the two approaches, and the choice of integration path. The aim of this study is to clarify the confusion about the direction and magnitude of maximum growth in convectively unstable three-dimensional non-parallel boundary layers. To this end, the time-asymptotic response of the boundary layer to an impulsive point excitation is considered. Since all frequencies and all wavenumbers are excited by an impulsive point source, the most amplified component of the response is equivalent to the result of maximizing the growth over arbitrary choices of harmonic point excitation; the standard eⁿ-approach. The impulse response is calculated using a spatial steepest-descent method, which is distinct from the earlier Cebeci–Stewartson method. It is necessary to allow both time and spanwise distance to become complex during integration, but with the constraint that both are real at the end point. This method has been applied to the two-dimensional Blasius boundary layer, for which validation of the method is more straightforward, and also to a three-dimensional Falkner–Skan–Cooke (with non-zero pressure gradient and sweep) boundary layer. Dimensional frequencies and spanwise wavenumbers of propagating components are kept constant (although not necessarily real), as is physically relevant to steady flows with spatial inhomogeneity in the chordwise direction only. With this method a spatial approach is taken without having to make a priori choices about the value of disturbance frequency or wavenumber. Further, purely by choosing a downstream observation point, it is possible to find the maximum-amplitude component directly without having to calculate the entire impulse response (or wave packet). If the flow is susceptible to more than one convective instability mode, provided the modes are separated in the frequency–wavenumber space, separate n-factors can be calculated for each mode. Wave-packet propagation in the Ekman layer (a strictly parallel three-dimensional boundary layer) is also discussed to draw comparisons between the conditions for maximum growth in parallel and non-parallel boundary layers.     

On the boundary layer methods

In this paper, the defect of the traditionary boundary layer methods (including the method of matched asymptotic expansions and the method of Visik-Lyusternik) is noted, from those methods we can not construct the asymptotic expansion of boundary layer term substantially. So the method of multiple scales is proposed for constructing the asymptotic expansion of boundary layer term, the reasonable result is obtained. Furthermore, we compare this method with the method used by Levinson, and find that both methods give the same asymptotic expansion of boundary layer term, but our method is simpler.Again, we apply this method to study some known works on singular perturbations. The limitations of those works have been noted, and the asymptotic expansion of solution is constructed in general condition.     

Growth of artificially induced disturbances in a supersonic boundary layer

This work proposes a method of inducing artificial disturbances of adjustable amplitude in a supersonic boundary layer. Using the proposed method, an experimental study is made of the development of a three-dimensional wave packet of low intensity at a frequency of 20 kHz in the boundary layer of a flat plate at Mach number M = 2.0. The Fourier components of the wave packet are determined. The data obtained are compared with the results of calculating the linear stability of the supersonic boundary layer in a plane-parallel flow approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–43, September–October, 1984.     

The flow structure of dilute gas-particle suspensions behind a shock wave moving along a flat surface

The asymptotic and numerical investigations of shock-induced boundary layers in gas-particle mixtures are presented.The Saffman lift force acting on a particle in a shear flow istaken into account.It is shown that particle migration across the boundary layer leads tointersections of particle trajectories.The corresponding modification of dusty gas model isproposed in this paper.The equations of two-phase sidewall boundary layer behind a shock wave moving at aconstant speed are obtained by using the method of matched asymptotic expansions.Themethod of the calculation of particle phase parameters in Lagrangian coordinates isdescribed in detail.Some numerical results for the case of small particle concentration aregiven.     

DYNAMICAL ARTIFICIAL BOUNDARY FOR FLUID MEDIUM IN WAVE MOTION PROBLEMS

无限域流体介质的波动辐射效应是影响海域工程动力反应的重要因素,人工边界是实现此类开放系统近场波动问题数值分析的有效方法.基于位移格式的流体波动理论推导开放域流体介质的人工边界,分别给出一维、二维和三维空间中平面波、柱面波和球面波的流体介质动力人工边界条件,其中一维平面波动人工边界为经典的黏性边界,二维柱面波、三维球面波的人工边界处节点应力与节点速度和加速度成正比,可等效为由阻尼与质量系统构成的人工边界条件.讨论相应的数值模拟技术,给出流体介质动力人工边界在ANSYS软件平台的具体实现方法.近场流体介质动力反应问题的算例表明,所发展的流体动力人工边界对于轴对称波动与非轴对称波动在近场有限域截断处的透射吸收作用的模拟计算精度均较为良好,说明此流体介质人工边界具有较高的可靠性与实用性.所发展的流体介质动力人工边界可较为方便地与大型商用有限元软件结合,可为包括海域地形和海岛在内的海域工程的动力分析提供一定的方法借鉴.     

Approximate solution of the problem of the three-dimensional boundary layer in an incompressible fluid

A method of successive approximations is proposed for the solution of the equations of the three-dimensional incompressible boundary layer on bodies of arbitrary shape. A coordinate system connected with the streamlines of the external nonviscous flow is used. It is assumed that the velocity across the external streamlines is small. When the intensity of secondary flow is low the equations describing the boundary layer in an incompressible fluid are reduced to a form analogous to the equations for the boundary layer on axially symmetrical bodies. An approximate analytical solution is obtained for the velocity and for the friction in the form of equations which can be used for any problems of a three-dimensional incompressible boundary layer. The method developed was applied to the problem of the three-dimensional boundary layer at a plate with a cylindrical obstacle in the presence of a slip angle.     

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.     

Near-wall damping in model predictions of separated flows

Different near-wall scalings are reviewed by the use of data from direct numerical simulations (DNS) of attached and separated adverse pressure gradient turbulent boundary layers. The turbulent boundary layer equation is analysed in order to extend the validity of existing wall damping functions to turbulent boundary layers under severe adverse pressure gradients. A proposed near-wall scaling is based on local quantities and the wall distance, which makes it applicable for general computational fluid dynamics (CFD) methods. It was found to have a similar behaviour as the pressure-gradient corrected analytical y^* scaling and avoids the inconsistencies present in the y⁺ scaling. The performance of the model is illustrated by model computations using explicit algebraic Reynolds stress models with near-wall damping based on different scalings.     

Shock-induced turbulent boundary layers

Experimental data for an incompressible turbulent moving surface boundary layer are reviewed and a theoretical extension of their predictions is suggested for the case of finite free stream velocities. It is argued that such a boundary layer provides an incompressible analogue for shock-induced turbulent boundary layers. Coles's transformation is used to predict the behaviour of the shock-induced case from the incompressible analogue. These predictions are used to attempt to correlate the available experimental shock-induced turbulent boundary layer data. It is felt that the correlations are reasonably successful for some of the data. It is suggested that the remaining data have been affected by the premature arrival of the contact region and reflected rarefaction wave.     

Applications of EPSE method for predicting crossflow instability in swept-wing boundary layers

The nth-order expansion of the parabolized stability equation(EPSEn) is obtained from the Taylor expansion of the linear parabolized stability equation(LPSE) in the streamwise direction. The EPSE together with the homogeneous boundary conditions forms a local eigenvalue problem, in which the streamwise variations of the mean flow and the disturbance shape function are considered. The first-order EPSE(EPSE1) and the second-order EPSE(EPSE2) are used to study the crossflow instability in the swept NLF(2)-0415 wing boundary layer. The non-parallelism degree of the boundary layer is strong. Compared with the growth rates predicted by the linear stability theory(LST),the results given by the EPSE1 and EPSE2 agree well with those given by the LPSE.In particular, the results given by the EPSE2 are almost the same as those given by the LPSE. The prediction of the EPSE1 is more accurate than the prediction of the LST, and is more efficient than the predictions of the EPSE2 and LPSE. Therefore, the EPSE1 is an efficient e~N prediction tool for the crossflow instability in swept-wing boundary-layer flows.     

Self-consistent parabolized stability equation (PSE) method for compressible boundary layer

The parabolized stability equation (PSE) method has been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulation, for nonlinear problems, the complex wave number of each Fourier mode is determined by the so-called phase-locked rule, which results in non-self-consistency in the wave numbers. In this paper, a modification is proposed to make it self-consistent. The main idea is that, instead of allowing wave numbers to be complex, all wave numbers are kept real, and the growth or decay of each mode is simply manifested in the growth or decay of the modulus of its shape function. The validity of the new formulation is illustrated by comparing the results with those from the corresponding direct numerical simulation (DNS) as applied to a problem of compressible boundary layer with Mach number 6.     

The Multi-point Optimization of Shock Control Bump with Constant-Lift Constraint Enhanced with Suction and Blowing for a Supercritical Airfoil

Both shock control bump (SCB) and suction and blowing are flow control methods used to control the shock wave/boundary layer interaction (SWBLI) in order to reduce the resulting wave drag in transonic flows. A SCB uses a small local surface deformation to reduce the shock-wave strength, while suction decreases the boundary-layer thickness and blowing delays the flow separation. Here a multi-point optimization method under a constant-lift-coefficient constraint is used to find the optimum design of SCB and suction and blowing. These flow control methods are used separately or together on a RAE-2822 supercritical airfoil for a wide range of off-design transonic Mach numbers. The RANS flow equations are solved using Roe’s averages scheme and a gradient-based adjoint algorithm is used to find the optimum location and shape of all devices. It is shown that the simultaneous application of blowing and SCB (hybrid blowing/SCB) improves the average aerodynamic efficiency at off-design conditions by 18.2 % in comparison with the clean airfoil, while this increase is only 16.9 % for the hybrid suction/SCB. We have also studied the SWBLI and how the optimization algorithm makes the flow wave structure and interactions of the shock wave with the boundary layer favorable.     

波动方程的精细逐步积分法

应用现有的波动方程求解方法解决工程实际问题尚存在一定的局限性。本文在结构动力方程精细逐步积分的基础上,提出了波动方程初边值问题的精细逐步积分法,并分别给出了不同边界条件下的精细逐步积分格式。此数值方法虽然是显式积分方法,却是无条件稳定的。分别用精细逐步积分法和其它已有的方法对两个算例进行了计算,一个是有解析解的例子,该例验证了此方法的准确性,另一个例子是求解由波动方程及初始条件和边界条件组成的有杆抽油系统预测模型,此例验证了精细逐步积分法的高效性。     

Development of three-dimensional disturbances in a boundary layer

In the region of transition from a two-dimensional laminar boundary layer to a turbulent one, three-dimensional flow occurs [1–3]. It has been proposed that this flow is formed as the result of nonlinear interaction of two-dimensional and three-dimensional disturbances predicted by linear hydrodynamic stability theory. Using many simplifications, [4, 5] performed a calculation of this interaction for a free boundary layer and a boundary layer on a wall with a very coarse approximation of the velocity profile. The results showed some argreement with experiment. On the other hand, it is known that disturbances of the Tollmin—Schlichting wave type can be observed at sufficiently high amplitude. This present study will use the method of successive linearization to calculate the primary two- and three-dimensional disturbances, and also the average secondary flow occurring because of nonlinear interaction of the primary disturbances. The method of calculation used is close to that of [4, 5], the disturbance parameters being calculated on the basis of a Blazius velocity profile. A detailed comparison of results with experimental data [1] is made. It developed that at large disturbance amplitude the amplitude growth rate differs from that of linear theory, while the spatial distribution of disturbances agree s well with the distribution given by the natural functions and their nonlinear interaction. In calculating the secondary flow an experimental correction was made to the amplitude growth rate.