Vibrations of cascade profiles in nonuniform incompressible flow

The influence of flow nonuniformity on the aerodynamic characteristics of profiles in a cascade has been studied in [1, 2, 3]. In these studies the general problem was separated into two independent problems: first, uniform flow of a fluid about a cascade of oscillating profiles disturbed only by the cascade and second, flow of a nonuniform stream past a cascade of stationary profiles. This separation is possible within the framework of linear theory, in which the nonuniformity of the flow approaching the cascade and the profile vibration amplitudes are sufficiently small. However, the order of smallness of these two factors is different, which often leads to consideration of the influence of flow nonuniformity on the unsteady aerodynamic characteristics of the oscillating profiles. This investigation concerns that problem. In particular, certain conditions of flow nonuniformity, giving rise to parametric resonance of turbomachinery blades, are discussed.     

Unsteady incompressible fluid flow past a cascade of thin curved plates

A large number of papers have been devoted to the study of unsteady flow past airfoil cascades. The majority of authors solve the problem for slightly curved profiles oscillating at low angles of attack.Among other work, we note that of Söhngen [1] on the flow past a dense cascade of plates oscillating synchronously and in phase in a potential fluid flow at a high angle of attack. Samoilovich [2] studied the flow past a cascade of plates of arbitrary shape oscillating with an arbitrary phase shift between neighboring plates. He presents the solution for the case of variable circulation in the quasisteady formulation. Stepanov [3] studied the same question with a linear approach to the flow behind the cascade. Musatov [4] examined the problem of the flow past a cascade of plates oscillating with an arbitrary phase shift between neighboring plates in a fluid flow, again at a high angle of attack, and considered the variation of the relative position of the plates durilng the oscillation process.The present paper considers the flow of a perfect incompressible fluid past a cascade of thin curved oscillating plates with account for the relative displacements of the plates during oscillation. To determine the intensity of the bound vortices per unit length, a linear integral equation is obtained. This represents a generalization of the Birnbaum equation to the case considered (see [5]). Equations are presented for calculating the unsteady aerodynamic forces and moments acting on the plates. As an example, the aerodynamic forces and moments are calculated for the quasistationary formulation of the problem.     

Plate cascade in subsonic unsteady gas flow

Accounting for fluid compressibility creates serious difficulties in solving the problem of oscillations of a grid of thin, slightly curved profiles in a subsonic stream. The problem has been solved in [1–3] for a widely-spaced cascade without stagger whose profiles oscillate in phase opposition. The phenomenon of aerodynamic (acoustic) resonance, which may arise in a grid in the direction transverse to the stream for definite values of the stream velocity and profile oscillation frequency, was discovered in [2]. An approximate solution of the problem in which account is not taken of the effect of the vortex trails on the gas flow has been obtained in [4]. In [5, 6] Meister studied in the exact linear formulation the problem of unsteady gas motion through an unstaggered cascade of semi-infinite plates. In [7] Meister considered a grid of profiles with finite chords, but the problem solution was not completed. The problem of subsonic gas flow through a staggered lattice whose profiles oscillate following a single law with constant phase shift was solved most completely in the studies of Kurzin [8, 9] using the method of integral equations. A method of solving the problem for the case of arbitrary harmonic oscillation laws for the lattice profiles was indicated in [10]. The results of the calculation of the unsteady aerodynamic forces for the particular case of a plate cascade without stagger are presented in [9,11], and the possibility of the occurrence of aerodynamic resonance in the cascade in the directions transverse to and along the stream is indicated.Another method of solving the problem is given in [12], in which the more general problem of unsteady subsonic gas flow through a three-dimensional cascade of plates is solved. In the present study this method is applied to the solution of the problem of oscillations of staggered plate cascades in a two-dimensional subsonic gas flow. The results are presented of an electronic computer calculation of the unsteady aerodynamic characteristics of the cascade profiles, which show the essential influence of fluid compressibility on these characteristics. In particular, a sharp decrease of the aerodynamic damping in the acoustic resonance regimes is obtained.     

进口受控非定常来流同振荡叶栅相互作用的Euler方程数值解

将 Dcnton 的四角点有限面积格式用于求解变换坐标系下的非定常 Euler 方程组,在振荡叶片附近局部子域构造了一种按等差级数分布的动态网格,从而节省了计算时间,对于本方法的正确性进行了初步检验并与相应的实验进行了对比,在进口总压简谐振荡的束流条件下,计算了零安装角平板叶栅和53°安装角双凸叶栅扭转振荡的非定常流场,结果表明,进口受控扰动同振荡叶栅之间的相互作用在有压力梯度的情况下可以定性地改变叶栅的气动阻尼。     

Calculation of the unsteady aerodynamic parameters of a rotating cascade of oscillating blades in incompressible flow

The unsteady aerodynamic parameters of 3D blade cascades oscillating in incompressible flow are determined with account for blade geometry and the influence of the steady hydrodynamic loads acting on the blades. On the assumption of separationless flow and harmonic blade oscillations, the corresponding boundary-value problem for the amplitude function of the unsteady velocity potential component is solved in the linear formulation, using a finite-element method. Test calculation results are presented and an example of calculating the unsteady aerodynamic parameters of a hydro-turbine model is given.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 40–52. Original Russian Text Copyright © 2005 by Kurzin and Tolstukha.     

Influence of boundary layer on unsteady aerodynamic characteristics of blunt cones in a supersonic flow

A study is made of the influence of the boundary layer on the unsteady aerodynamic characteristics of blunt cones oscillating in a supersonic gas stream about zero angle of attack. A solution to the problem is constructed in the framework of the linear theory of bodies of finite thickness. Such an approach has been used [1–3] in the case of the equations of motion of an ideal gas to calculate the unsteady aerodynamic characteristics of sharp and blunt bodies of various configurations. The influence on these characteristics of the viscosity effects due to the presence on the surface of the body of a laminar boundary layer has been taken into account [4–6] for bodies of the simplest shapes (wedge, cone). The present paper considers the unsteady aerodynamic characteristics of cones and investigates the influence of rounding of the tips and laminar and turbulent flow regimes in the boundary layer.     

Oscillations of a plate cascade in a transonic gas flow

The transonic unsteady flow of a gas through a cascade of thin, slightly curved plates is quite complex and has received little study. The main difficulties are associated with the nonlinear dependence of the aerodynamic characteristics on the plate thickness. In [1] it is shown that, for a single thin plate performing high-frequency oscillations in a transonic gas stream, the variation of the unsteady aerodynamic characteristics with plate thickness may be neglected. For a plate cascade, the flow pattern is complicated by the aerodynamic interference between the plates, which may depend significantly on their shape. A solution of the problem of transonic flow past a cascade without account for the plate thickness has been obtained by Hamamoto [2].The objective of the present study is the clarification of the dependence of the aerodynamic characteristics of a plate cascade on plate thickness in transonic unsteady flow regimes. The nonlinear equation for the velocity potential is linearized under the assumption that the motionless plate causes significantly greater disturbances in the stream than those due to the oscillations. A similar linearization was carried out for a single plate in [3]. The aerodynamic interference between the plates is determined by the method presented in [4]. As an example, the aerodynamic forces acting on a plate oscillating in a duct and in a free jet are calculated.     

Lattice of plates in an unsteady supersonic flow

The supersonic unsteady flow of a gas past a lattice of thin, slightly curved profiles has been investigated in several studies. The paper [1] is devoted to an evaluation of the effect of wind tunnel walls on the unsteady aerodynamic characteristics of a profile, and [2] investigates the effects of the boundaries of a free jet. These cases are equivalent respectively to the anti-phase and in-phase oscillations of the profiles of an unstaggered grid. In [3] consideration is given to a more general case of gas flow past a profile in a wind tunnel with perforated walls. Flow past a lattice of profiles with stagger is studied in [4], where the magnitude of the stagger angle is limited by the condition that the lattice leading edge is located in the undisturbed stream.In the present paper we present a method of calculation of the unsteady supersonic flow of a gas past a lattice of profiles with arbitrary stagger. As an example the results are presented of the calculation of the aerodynamic forces and moments acting on an oscillating profile in a wind tunnel with solid walls and in a free jet.     

Calculation of unsteady supersonic flow past a two-dimensional cascade of plates ween vorticity inhomogeneities of the flow act on it

In the framework of the linear theory of small perturbations the problem of unsteady subsonic flow past a two-dimensional cascade of plates has been considered in a number of papers. Thus, the unsteady aerodynamic characteristics of a cascade of vibrating plates were calculated in [1] by the method of integral equations, while the same method was used in [2, 3] to calculate the sound fields that are excited when sound waves Coming from outside or vorticity inhomogeneities of the oncoming flow act on the cascade. The problem of a two-dimensional cascade of vibrating plates in a supersonic flow was solved in [4, 5]. In [4] the solution was constructed on the basis of the well-known solution of the problem of vibrations of a single plate, while in [5] a variant of the method of integral equations was used which differed slightly from the usual formulation of this method [1–3]. The approach proposed in [5] is used below to calculate the unsteady flow past a two-dimensional cascade of plates in the case when vorticity inhomogeneities of a supersonic oncoming flow act on it. Equations are obtained for the strength of the unsteady pressure jumps arising in such a flow and the vortex wakes shed from the trailing edges of the plates. Examples of the calculations illustrating the accuracy of the method and its possibilities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 152–160, May–June, 1986.     

Unsteady aerodynamic interaction of two annular blade rows of thin lightly loaded blades rotating one relative to the other in a subsonic flow

The problem of subsonic unsteady ideal-gas flow over two annular blade rows of thin lightly loaded blades rotating one relative to the other is solved within the framework of linear small perturbation theory. As in the case of the interaction of two-dimensional cascades [1], the problem reduces to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on one blade of each row. The system of integral equations for a finite number of harmonics is solved numerically by the collocation method. The kernels of the integral equations are regularized on the basis of the method proposed in [2].Translated from Izvestiya Akademii Nauk.SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 165–174, May–June, 1991.The author is grateful to A. A. Osipov and K. K. Butenko for their considerable assistance in the preparation of this paper.     

Plane incompressible fluid flow past an array of arbitrary profiles vibrating with arbitrary phase shift

We consider the problem of the vibration of an array of arbitrary profiles with arbitrary phase shift. Account is taken of the influence of the vortex wakes. The vibration amplitude is assumed to be small. The problem reduces to a system of two integral Fredholm equations of the second kind, which are solved on a digital computer. An example calculation is made for an array of arbitrary form.A large number of studies have considered unsteady flow past an array of profiles. Most authors either solve the problem for thin and slightly curved profiles or they consider the flow past arrays of thin curvilinear profiles [1].In [2] a study is made of the flow past an array of profiles of arbitrary form oscillating with arbitrary phase shift in the quasi-stationary formulation. The results are reduced to numerical values. Other approaches to the solution of the problem of unsteady flow past an array of profiles of finite thickness are presented in [3–5] (the absence of numerical calculations in [3, 4] makes it impossible to evaluate the effectiveness of these methods, while in [5] the calculation is made for a symmetric profile in the quasi-stationary formulation).     

气固耦合振动叶栅非定常流动分析研究

引入结构动力学方程建立了二维N-S/结构振动耦合方程组,采用双时间法建立了气固耦合方程组的非定常数值求解体系,研究了叶栅间的二维非定常粘性流动及叶栅振动特性。对两种叶型分别计算了不同折合振动频率下的流场,振动叶栅位移随时间变化的曲线表明,采用气固耦合得到的叶栅振动频率与非耦合自振频率相比均有所下降;振动位移-时间曲线在不同振动折合频率下有显著差别。在气固耦合情况下叶栅振动规律及其稳定性与非耦合情形差异较大,因此研究叶栅振动稳定性应当考虑气动/结构的耦合。     

Determining unsteady aerodynamic forces for three-dimensional plate cascade in subsonic gas flow

The general solution for the problem of three-dimensional gas flow through a cascade of plates was obtained in [1], It is shown below that in the considered direct and inverse aerodynamic problems there may exist nontrivial solutions for uniform boundary conditions. The flow regimes for which such solutions exist may be treated as aerodynamic resonance.The examples presented of the calculation for a plate cascade illustrate the behavior of the unsteady aerodynamic forces near the aerodynamic resonance regimes and make it possible to evaluate the limits of applicability of the hypothesis of plane sections. In addition, the calculation results show the possibility of self-induced vibrations of plate cascades with a single degree of freedom in a subsonic gas stream.     

Mathematical model of aeroelastic vibrations of cascades of axial turbomachine blades,caused by circumferential nonuniformity of the flow

A system of linear differential equations with time-dependent coefficients, which describes aeroelastic vibrations of blade cascades in a nonuniform flow, is derived. With the use of the model of an ideal incompressible fluid and the hypothesis of cylindrical sections, determination of aerodynamic forces acting on the blades is reduced to solving problems by methods fairly well developed in the theory of cascades in unsteady flow. The possibility of the emergence of a parametric resonance is analyzed. It is demonstrated that circumferential nonuniformity of the flow in the turbomachine duct can substantially reduce the critical velocity of the cascade flutter.     

叶轮机中振动叶片非定常气动力的研究

采用“单一叶片振动方法”代替传统的“全部叶片振动方法”，用线性化的方法推导出计算振动叶片非定常气动力（引起叶片弯曲振动）及力矩（引起叶片扭转振动）方程。通过输入叶型及气流参数，它可以方便有效地估计出作用在振动叶片上的非定常气动力和力矩，以及振动叶片不稳定工况发生的条件和范围。并在氟里昂超音速风洞上进行了实验测量，结果表明，理论计算结果与实验测试结果符合较好。     

Fluid flow in an arbitrary cascade on an axisymmetric stream surface in a variable thickness layer

A solution is given for the problem of flow past a cascade on an axisymmetric stream surface in a layer of variable thickness, which is a component part of the approximate solution of the three-dimensional problem for a three-dimensional cascade. Generalized analytic functions are used to obtain the integral equation for the potential function, which is solved via iteration method by reduction to a system of linear algebraic equations. An algorithm and a program for the Minsk-2 computer are formulated. The precision of the algorithm is evaluated and results are presented of the calculation of an example cascade.In the formulation of [1, 3] the problem of flow past a three-dimensional turbomachine cascade is reduced approximately to the joint solution of two-dimensional problems of the averaged axisymmetric flow and the flow on an axisymmetric stream surface in an elementary layer of variable thickness.In the following we solve the second problem for an arbitrary cascade with finite thickness rotating with constant angular velocity in ideal fluid flow: the solution is carried out on a Minsk-2 computer.Many studies have been devoted to this problem. A method for solving the direct problem for a cascade of flat plates in a hyperbolic layer was presented in [2]. Methods were developed in [1, 3] for constructing the flow for the case of a channel with variable thickness; these methods are approximately applicable for dense cascades but yield considerable error for small-load turbomachine cascades. The solution developed in [4], somewhat reminiscent of that of [2], is applicable for thin, slightly curved profiles in a layer with monotonically varying thickness. A solution has been given for a circular cascade for layers varying logarithmically [5] and linearly [6]. Approximate methods for slightly curved profiles in a monotonically varying layer with account for layer variability only in the discharge component were examined in [7–9]. A solution is given in [10] for an arbitrary layer by means of the relaxation method, which yields a roughly approximate flow pattern. The general solution of the problem by means of potential theory and the method of singularities presented in [11] is in error because of neglect of the crossflow through the skeletal line. The computer solution of [12] contains an unassessed error for the calculations in an arbitrary layer. The finite difference method is used in [13] to solve the differential equation of flow, which is illustrated by numerical examples for monotonie layers of axial turbomachines. The numerical solution of [13] is very complex.The solution presented below is found in the general formulation with respect to the geometric parameters of the cascade and the axisymmetric surface and also in terms of the layer thickness variation law.The numerical solution requires about 15 minutes of machine time on the Minsk-2 computer.     

A numerical modelling of stator–rotor interaction in a turbine stage with oscillating blades

In real flows unsteady phenomena connected with the circumferential non-uniformity of the main flow and those caused by oscillations of blades are observed only jointly. An understanding of the physics of the mutual interaction between gas flow and oscillating blades and the development of predictive capabilities are essential for improved overall efficiency, durability and reliability. In the study presented, the algorithm proposed involves the coupled solution of 3D unsteady flow through a turbine stage and the dynamics problem for rotor-blade motion by the action of aerodynamic forces, without separating the outer and inner flow fluctuations. The partially integrated method involves the solution of the fluid and structural equations separately, but information is exchanged at each time step, so that solution from one domain is used as a boundary condition for the other domain. 3-D transonic gas flow through the stator and rotor blades in relative motion with periodicity on the whole annulus is described by the unsteady Euler conservation equations, which are integrated using the explicit monotonous finite volume difference scheme of Godunov–Kolgan. The structural analysis uses the modal approach and a 3-D finite element model of a blade. The blade motion is assumed to be constituted as a linear combination of the first natural modes of blade oscillations, with the modal coefficients depending on time. A calculation has been done for the last stage of the steam turbine, under design and off-design regimes. The numerical results for unsteady aerodynamic forces due to stator–rotor interaction are compared with results obtained while taking into account blade oscillations. The mutual influence of both outer flow non-uniformity and blade oscillations has been investigated. It is shown that the amplitude-frequency spectrum of blade oscillations contains the high-frequency harmonics, corresponding to the rotor moving past one stator blade pitch, and low-frequency harmonics caused by blade oscillations and flow non-uniformity downstream from the blade row; moreover, the spectrum involves the harmonics which are not multiples of the rotation frequency.     

Aerodynamic interaction of two plane cascades of thin lightly loaded blades in relative motion in a subsonic gas flow

The problem of subsonic ideal-gas flow over two plane cascades of thin lightly loaded blades in relative motion is solved within the framework of the linear theory of small perturbations. By means of the method of integral equations [1] the problem is reduced to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on the blades. The regularized system of integral equations for a finite number of harmonics is solved numerically by a collocation method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 168–175, May–June, 1987.     

Computation of cascade flutter by uncoupled and coupled methods

A computational method for flutter prediction of turbomachinery cascades is presented. The flow through multiple blade passages is calculated using a time-domain approach with coupled aerodynamic and structural models. The unsteady Euler/Navier-Stokes equations are solved in quasi-three-dimensions using a second-order implicit scheme with dual time-stepping and a multigrid method. A structural model for the blades with bending and torsion degrees of freedom is integrated in time together with the flow field. Information between structural and aerodynamic models is exchanged until convergence in each real-time step. Computational results for a cascade are presented and compared with those obtained by the conventional energy method and with experimental and numerical data by other authors. Significant differences are found between the coupled and uncoupled methods at low mass ratios. A transonic test case with strong nonlinear phenomena is investigated with the fluid-structure coupled method. Results for inviscid flow are compared with results of Navier-Stokes computations.     

Mathematical modeling of the sound induced in a subsonic flow around cascades in relative motion

A method of mathematical modeling of the tonal sound induced by the unsteady aerodynamic interaction of two plane airfoil cascades in a subsonic flow and in uniform relative motion in the direction of their fronts is developed. The method is based on the numerical integration of the unsteady flow equations using a simplified model for the periodic system of edge wakes shed from the airfoils of the first (leading) cascade in the viscous flow and acting on the second (trailing) cascade. An analysis of the distinctive features of the flow under consideration demonstrates the efficiency of the model proposed.