REDUCED-ORDER MODELS OF UNSTEADY TRANSONIC VISCOUS FLOWS IN TURBOMACHINERY

The proper orthogonal decomposition (POD) technique is applied in the frequency domain to obtain a reduced-order model of the unsteady flow in a transonic turbomachinery cascade of oscillating blades. The flow is described by a inviscid—viscous model, i.e. a full potential equation outer flow model and an integral equation boundary layer model. The nonlinear transonic steady flow is computed first and then the unsteady flow is determined by a small perturbation linearization about the nonlinear steady solution. Solutions are determined for a full range of frequencies and validated. The full model results and the POD method are used to construct a reduced-order model in the frequency domain. A cascade of airfoils forming the Tenth Standard Configuration is investigated to show that the reduced-order model with only 15–75 degrees of freedom accurately predicts the unsteady response of the full system with approximately 15 000 degrees of freedom.     

Cascade aerodynamic gust response including steady loading effects

To predict the unsteady convected gust aerodynamic response of a cascade comprised of arbitrary thick and cambered aerofoils in an incompressible, inviscid, flow field, a complete first-order model is formulated. The flow is analysed by considering a periodic flow channel. The velocity potential is separated into steady and unsteady harmonic components, each described by a Laplace equation. The strong dependence of the unsteady aerodynamics on the steady effects of aerofoil and cascade geometry and incidence angle is manifested in the coupling of the unsteady and steady flow fields through the unsteady boundary conditions. Analytical solutions in individual grid elements of a body-fitted computational grid are then determined, with the complete solution obtained by assembly of these local solutions. The validity and capabilities of this model and solution technique are then demonstrated by analysing the steady and unsteady aerodynamics of both theoretical and experimental cascade configurations.     

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.     

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.     

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.     

International journal of computational fluid dynamics real-time prediction of unsteady flow based on POD reduced-order model and particle filter

An integrated method consisting of a proper orthogonal decomposition (POD)-based reduced-order model (ROM) and a particle filter (PF) is proposed for real-time prediction of an unsteady flow field. The proposed method is validated using identical twin experiments of an unsteady flow field around a circular cylinder for Reynolds numbers of 100 and 1000. In this study, a PF is employed (ROM-PF) to modify the temporal coefficient of the ROM based on observation data because the prediction capability of the ROM alone is limited due to the stability issue. The proposed method reproduces the unsteady flow field several orders faster than a reference numerical simulation based on Navier–Stokes equations. Furthermore, the effects of parameters, related to observation and simulation, on the prediction accuracy are studied. Most of the energy modes of the unsteady flow field are captured, and it is possible to stably predict the long-term evolution with ROM-PF.     

Numerical and experimental investigations on the cavitating flow in a cascade of hydrofoils

The cavitating flow in a cascade of three hydrofoils was investigated by experimental means and numerical simulation. Experiments on the 2D-hydrofoils cascade were carried out at Darmstadt University of Technology in a rectangular test section of a cavitation tunnel. A numerical model developed at LEGI (Grenoble) to describe the unsteady behaviour of cavitation, including the shedding of vapour structures, was applied to the hydrofoils cascade geometry. Results of both experimental and numerical studies show a strong interaction between the cavities of each flow channel besides the typical self-oscillation of cloud cavitation. A detailed comparison of the results allows an interpretation of the interaction mechanisms to be proposed.     

REDUCED-ORDER MODELS OF UNSTEADY VISCOUS FLOWS IN TURBOMACHINERY USING VISCOUS–INVISCID COUPLING

The proper orthogonal decomposition technique is applied in the frequency domain to obtain a reduced-order model of the flow in a turbomachinery cascade. The flow is described by an inviscid–viscous interaction model where the inviscid part is described by the full potential equation and the viscous part is described by an integral boundary layer model. The fully nonlinear steady flow is computed and the unsteady flow is linearized about the steady solution. A frequency-domain model is constructed and validated, showing to provide similar results when compared with previous computational and experimental data presented in the literature. The full model is used to obtain a reduced-order model in the frequency domain. A cascade of airfoils forming a slightly modified Tenth Standard Configuration is investigated to show that the reduced-order model with only 25 degrees of freedom accurately predicts the unsteady response of the full system with approximately 10 000 degrees of freedom.     

On the incremental singular value decomposition method to support unsteady adjoint-based optimization

This paper proposes and evaluates an approximation model based on an incremental Singular Value Decomposition (iSVD) algorithm, for unsteady flow field reconstructions, needed for integrating the unsteady adjoint equations backward in time, within a gradient-based optimization loop. Due to the iSVD algorithm, the computational cost of solving the unsteady adjoint equations is reduced considerably, without practically affecting the accuracy of the computed gradient. Approximations to the unsteady flow fields are constructed while solving the time-varying flow equations (moving forward in time) and used to reconstruct these fields during the backward-in-time integration of the continuous adjoint equations. Optimization results obtained using the proposed method are compared to those computed using the binomial checkpointing technique, which acts as the reference method. Test cases for both flow control and shape optimization problems are presented.     

基于边界元方法的气动弹性降阶模型及仿真

传统气动弹性的时域计算耗费了大量时间,为了提高计算效率,本文发展了基于边界元方法的降阶模型技术。首先基于边界元方法建立非定常流场的求解模型,结合特征值分析技术建立了非定常气动力的低阶模型;然后,利用边界元方法建立了气动网格和结构网格之间的信息转换矩阵;最后将非定常气动力降阶模型和结构动力学方程联合,建立了气动弹性系统的低阶状态空间模型。将所发展的降阶模型方法应用于NACA0012翼型的非定常气动力求解中,结果表明降阶模型可以在保证原系统计算精度的同时提高了计算效率;将降阶模型技术应用到三维机翼的气动弹性响应计算中,在系统阶数仅为12阶的情况下可以得到与原系统一致的极限环响应,说明降阶模型技术在求解气动弹性问题中的巨大优势。     

Numerical Investigation of Unsteady Transitional Flows in Turbomachinery Components Based on a RANS Approach

This paper presents results of the numerical simulation of periodically unsteady flows with focus on turbomachinery applications. The unsteady CFD solver used for the simulations is based on the Reynolds averaged Navier–Stokes equations. The numerical scheme applies an extended version of the Spalart–Allmaras one-equation turbulence model coupled with a transition correlation. The first example of validation consists of boundary layer flow with separation bubble on a flat plate, both under steady and periodically unsteady main flow conditions. The investigation includes a variation of the major parameters Strouhal number, amplitude, and Reynolds number. The second, more complex test case consists of the flow through a cascade of turbine blades which is influenced by wakes periodically passing over the cascade. The computations were carried out for two different blade loadings. The results of the numerical simulations are discussed and compared with experimental data in detail. Special emphasis is given to the investigation of boundary layers with regard to transition, separation and reattachment under the influence of main flow unsteadiness. This revised version was published online in July 2006 with corrections to the Cover Date.     

A check on the energy method of predicting blade transonic stall flutter

An improved structural dynamic model of an oscillating blade in two degrees of freedom is combined with an unsteady aerodynamic model for the transonic flow about a cascade with separation, which results in a coupled system. The system is solved in an iterative way between the two models. As a check on the current energy methods, the stall flutter boundaries for two real rotors are predicted by using the present method and the results are compared with the experiments and those predicted by using an energy method.     

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。     

Fluidelastic instability study in a rotated triangular tube array subject to two-phase cross-flow. Part I: Fluid force measurements and time delay extraction

Fluidelastic instability is a key issue in steam generator tube bundles subjected to cross-flow. The extension to two-phase flow of the existing theoretical models, developed and tested mostly for single phase flow, is investigated in this paper. The time delay is one of the key parameter for modeling fluidelastic instability, especially the damping controlled mechanism. The direct measurement of the time delay between the tube motion and the fluid force faces certain difficulties in two-phase flow since the high turbulence due to the interaction of the two components of the flow may increase the randomness of the measured force. To overcome this difficulty, an innovative method for extracting the time delay inherent to the quasi-steady model for fluidelastic instability is proposed in this study.Firstly, experimental measurements of unsteady and quasi-static fluid forces (in the lift direction) acting on a tube subjected to air–water two-phase flow were conducted. The unsteady fluid forces were measured by exciting the tube using a linear motor. These forces were measured for a wide range of void fractions, flow velocities and excitation frequencies. The experimental results showed that the unsteady fluid forces could be represented as single valued function of the reduced flow velocity. It was also found that for a given frequency, the unsteady fluid force phase was weakly dependent on the void fraction for the range of flow velocities considered.The time delay was determined by equating the unsteady fluid forces with the quasi-steady forces. The results given by this innovative method of measuring the time delay in two-phase flow were consistent with theoretical expectations. The time delay could be expressed as a linear function of the convection time and the time delay parameter was determined for void fractions ranging from 60% to 90%.     

Experimental study on the effect of unsteadiness on boundary layer development on a linear turbine cascade

 The results from an experimental investigation of unsteady boundary layer behavior on a linear turbine cascade are presented in this paper. To perform a detailed study on unsteady cascade aerodynamics and heat transfer, a new large-scale, high-subsonic research facility for simulating the periodic unsteady flow has been developed. It is capable of sequentially generating up to four different unsteady inlet flow conditions that lead to four different passing frequencies, wake structures, and freestream turbulence intensities. For a given Reynolds number, two different unsteady wake formations are utilized. Detailed unsteady boundary layer velocity. turbulence intensity, and pressure measurements are performed along the suction and pressure surfaces of one blade. The results display the transition and development of the boundary layer, ensemble-averaged velocity, and turbulence intensity. Received: 23 September 1996/Accepted: 19 February 1997     

Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows

In this paper, a high‐order DG method coupled with a modified extended backward differentiation formulae (MEBDF) time integration scheme is proposed for the solution of unsteady compressible flows. The objective is to assess the performance and the potential of the temporal scheme and to investigate its advantages with respect to the second‐order BDF. Furthermore, a strategy to adapt the time step and the order of the temporal scheme based on the local truncation error is considered. The proposed DG‐MEBDF method has been evaluated for three unsteady test cases: (i) the convection of an inviscid isentropic vortex; (ii) the laminar flow around a cylinder; and (iii) the subsonic turbulent flow through a turbine cascade. Copyright © 2014 John Wiley & Sons, Ltd.     

Buffeting in transonic flow prediction using time‐dependent turbulence model

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and flow separations on wing upper surface induce flow instabilities, ‘buffet’, and then the buffeting (structure vibrations). This phenomenon can greatly influence the aerodynamic performance. These flow excitations are self‐sustained and lead to a surface effort due to pressure fluctuations. They can produce enough energy to excite the structure. The objective of the present work is to predict this unsteady phenomenon correctly by using unsteady Navier–Stokes‐averaged equations with a time‐dependent turbulence model based on the suitable (k–ε) turbulent eddy viscosity model. The model used is based on the turbulent viscosity concept where the turbulent viscosity coefficient (Cμ) is related to local deformation and rotation rates. To validate this model, flow over a flat plate at Mach number of 0.6 is first computed, then the flow around a NACA0012 airfoil. The comparison with the analytical and experimental results shows a good agreement. The ONERA OAT15A transonic airfoil was chosen to describe buffeting phenomena. Numerical simulations are done by using a Navier–Stokes SUPG (streamline upwind Petrov–Galerkin) finite‐element solver. Computational results show the ability of the present model to predict physical phenomena of the flow oscillations. The unsteady shock wave/boundary layer interaction is described. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

PANS模型在空化湍流数值计算中的应用

采用一种基于标准k-ε模型改进的局部时均化模型（Partially-Averaged Navier-Stokes Model,PANS）,并应用于空化流动计算。控制不同的模型参数,分别对绕平头轴对称回转体和Clark-Y型水翼的空化流动进行模拟,并与实验结果进行对比。结果表明：PANS模型中未分解湍动能比率fk的取值对预测空化流动的数值计算精度有重要影响,改变fk的取值可实现对不同滤波尺度范围内的求解;随着fk值的减小PANS的预测精度逐步提高,能在相对较大范围内求解较小尺度的湍流运动过程中,预测到湍流运动中强烈的非定常特性;同时可以比较准确地预测空化流场结构和动力特性。