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.     

Self-similar solutions to Lin-Reissner-Tsien equation

The Lin-Reissner-Tsien equation describes unsteady transonic flows under the transonic approximation. In the present paper, the equation is reduced to an ordinary differential equation via a similarity transformation. The resulting equation is then solved analytically and even exactly in some cases. Numerical simulations are provided for the cases in which there is no exact solution. Travelling wave solutions are also obtained.     

Far nonlinear field of a nonlifting airfoil for a generalised equation of transonic flow

The transonic flow equation [1] for plane unsteady irrotational idealgas flows is extended to the case of subsonic, transonic or supersonic flows in a region with an almost constant value of the velocity using orthogonal flow coordinates (family of equipotential lines and streamlines). A solution for the nonlinear far field of steady transonic flow past an airfoil has been obtained for the transonic equation [2]. In this paper it is obtained for a generalized transonic equation and its asymptotic expansion is given. In using difference methods of calculating the flow past an airfoil in the transonic regime a knowledge of the nonlinear field makes it possible to reduce the dimensions of the calculation region (near field) as compared with the region determined by the far field of the linear theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–91, January–February, 1986.     

Engine Modeling for Small-Disturbance-CFD Related to Aircraft Flutter Investigations

Aeroelastic behavior of aircraft is significantly affected by the presence of engines mounted under the wings. Powered engines influence the unsteady aerodynamics on the one hand and lead to additional unsteady forces due to thrust vector oscillations on the other hand. This work focuses on the incorporation of aerodynamic engine effects into a small disturbance CFD framework to enhance the modeling accuracy of unsteady aerodynamics of aircraft. The effects are numerically modeled by mimicking physically reasonable flow conditions at the intake and nozzle planes of the engine nacelle. Subsequently, the influence of the engine effects on the flutter behavior of an aircraft is studied employing the small-disturbance-CFD-based flutter analysis. The basis for the investigation is the Common Research Model, which represents a modern transonic commercial airliner with a cruise Mach number of 0.85. Two configurations are considered: aircraft with passive engines represented by flow-through nacelles and aircraft with powered engines, where the novel small disturbance engine model is applied. The results are compared in terms of the flutter trends and the predicted flutter boundary. Furthermore, the impact of the modal induced thrust oscillations on the aeroelastic behavior of the aircraft is discussed.     

Computation of three-dimensional flow using the Euler equations and a multiple-grid scheme

The unsteady Euler equations are numerically solved using the finite volume one-step scheme recently developed by Ron-Ho Ni. The multiple-grid procedure of Ni is also implemented. The flows are assumed to be homo-enthalpic; the energy equation is eliminated and the static pressure is determined by the steady Bernoulli equation; a local time-step technique is used. Inflow and outflow boundaries are treated with the compatibility relations method of ONERA. The efficiency of the multiple-grid scheme is demonstrated by a two-dimensional calculation (transonic flow past the NACA 12 aerofoil) and also by a three-dimensional one (transonic lifting flow past the M6 wing). The third application presented shows the ability of the method to compute the vortical flow around a delta wing with leading-edge separation. No condition is applied at the leading-edge; the vortex sheets are captured in the same sense as shock waves. Results indicate that the Euler equations method is well suited for the prediction of flows with shock waves and contact discontinuities, the multiple-grid procedure allowing a substantial reduction of the computational time.     

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.     

Implicit schemes for unsteady Euler equations on triangular meshes

An implicit finite element method is presented for the solution of steady and unsteady inviscid compressible flows on triangular meshes under transonic conditions. The method involves a first-order time-stepping scheme with a finite element discretization that reduces to central differencing on a rectangular mesh. On a solid wall the slip condition is prescribed and the pressure is obtained from an approximation of the normal momentum equation. With this solver no artificial viscosity is added to ensure the success of the calculation. Numerical examples are given for steady and unsteady cases.     

用扰动涡方法模拟叶轮机内动静叶的相互作用

发展了一种研究叶轮机内动、静叶间的相互作用的新方法———扰动涡方法，它利用全三维的定常解为基础解，并由此给出非定常扰动场的初始解．为计算叶片对扰动场的响应过程，采用拉格朗日方法追踪扰动涡团的对流流动过程，用确定性涡方法来描述流体的粘性扩散过程．发展了代数湍流模型（Ｂａｌｄｗｉｎ Ｌｏｍａｘ湍流模型）在尾迹中的应用方法，克服了其它数值方法中无法准确捕捉尾迹中心线的运动轨迹，以及计算出的边界层外的湍流涡粘性系数偏大的缺陷．利用该方法计算轴流叶轮机内由于动、静叶间的相互作用而引起的非定常流动过程，与实验的对照表明，模拟结果与实验数据吻合得相当好，从而说明本文发展的方法是可信的，为更直观地描述尾迹等非定常因素的流动及叶轮机内的掺混问题提供依据．     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Disturbance evolution in the leading-edge region of a blunt flat plate in supersonic flow

The spatio-temporal dynamics of small disturbances in viscous supersonic flow over a blunt flat plate at freestream Mach number M_∞=2.5 is numerically simulated using a spectral approximation to the Navier–Stokes equations. The unsteady solutions are computed by imposing weak acoustic waves onto the steady base flow. In addition, the unsteady response of the flow to velocity perturbations introduced by local suction and blowing through a slot in the body surface is investigated. The results indicate distinct disturbance/shock-wave interactions in the subsonic region around the leading edge for both types of forcing. While the disturbance amplitudes on the wall retain a constant level for the acoustic perturbation, those generated by local suction and blowing experience a strong decay downstream of the slot. Furthermore, the results prove the importance of the shock in the distribution of perturbations, which have their origin in the leading-edge region. These disturbance waves may enter the boundary layer further downstream to excite instability modes.     

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.     

Time marching integral equation method for unsteady transonic flows

In this paper, we have proposed a time marching intregral equation method which does not have the limitation of the time linearized integral equation method in that the latter method can not satisfactorily simulate the shock-wave motions. Firstly, a model problem—one dimensional initial and boundary value wave problem is treated to clarify the basic idea of the new method. Then the method is implemented for 2-D and 3-D unsteady transonic flow problems. The introduction of the concept of a quasi-velocity-potential simplifies the time marching integral equations and the treatment of trailing vortex sheet condition. The numerical calculations show that the method is reasonable and reliable.     

Enhanced performance of fast-response 3-hole wedge probes for transonic flows in axial turbomachinery

The paper presents the development and application of a three-sensor wedge probe to measure unsteady aerodynamics in a transonic turbine. CFD has been used to perform a detailed uncertainty analysis related to probe-induced perturbations, in particular the separation zones appearing on the wedge apex. The effects of the Reynolds and Mach numbers are studied using both experimental data together with CFD simulations. The angular range of the probe and linearity of the calibration maps are enhanced with a novel zonal calibration technique, used for the first time in compressible flows. The data reduction methodology is explained and demonstrated with measurements performed in a single-stage high-pressure turbine mounted in the compression tube facility of the von Karman Institute. The turbine was operated at subsonic and transonic pressure ratios (2.4 and 5.1) for a Reynolds number of 10⁶, representative of modern engine conditions. Complete maps of the unsteady flow angle and rotor outlet Mach number are documented. These data allow the study of secondary flows and rotor trailing edge shocks.     

Study of modeling unsteady blade row interaction in a transonic compressor stage part 1: code development and deterministic correlation analysis

The average-passage equation system (APES) provides a rigorous mathematical framework for accounting for the unsteady blade row interaction through multistage compressors in steady state environment by introducing deterministic correlations (DC) that need to be modeled to close the equation system.The primary purpose of this study is to provide insight into the DC characteristics and the influence of DC on the time-averaged flow field of the APES.In Part 1 of this two-part paper,firstly a 3D viscous unsteady and time-averaging flow CFD solver is developed to investigate the APES technique.Then steady and unsteady simulations are conducted in a transonic compressor stage.The results from both simulations are compared to highlight the significance of the unsteady interactions.Furthermore,the distribution characteristics of DC are studied and the DC at the rotor/stator interface are compared with their spatial correlations (SC).Lastly,steady and time-averaging (employing APES with DC) simulations for the downstream stator alone are conducted employing DC derived from the unsteady results.The results from steady and time-averaging simulations are compared with the time-averaged unsteady results.The comparisons demonstrate that the simulation employing APES with DC can reproduce the time-averaged field and the 3D viscous time-averaging flow solver is validated.     

Tensor Universal Serendipity Elements and unsteady Taylor-Galerkin finite element method

A new kind of Universal Serendipity Element (USE)— the Tensor Universal Serendipity Element (TUSE) is constructed by using both tensor force finite elements and the basic idea of USE. The formulation of shape functions and their derivatives for TUSE is presented. TUSE can be used to study steady and unsteady transonic flow fields when combined with Taylor-Galerkin Finite Element Methods, the NND scheme in FDM, and four-stage Runge-Kutta methods. As numerical examples the transonic flow in cascades and one kind of complex unsteady transonic axisymmetric flow in engineering are studied. It is shown that the algorithm presented in this paper is efficient and robust. The project supported by the National Natural Science Foundation of China     

计算含动边界非定常流动的无网格算法

在无网格算法中考虑了含动边界的流动问题,研究了可以计算处理包含一定位移及扭转动边界非定常流动的算法.创建了无网格算法的动点法则,并引入抗扭方法对弹簧方法进行改进来处理离散点运动,提高了方法的可用度及精度.发展了求解基于无网格的ALE方程组的算法,在点云离散的基础上采用曲面逼近计算空间导数及HLLC格式计算数值通量,运用四步龙格-库塔法进行时间推进.在跨、超音速条件下,计算模拟了典型翼型简谐振动流场,计算结果与实验结果及文献对比吻合,验证了该算法的正确性.     

Resonant phenomena in a transonic flow around an airfoil with pulsed periodic energy supply

Interaction of a pulsed periodic source of energy with a closing shock wave arising near airfoils in transonic flight is studied. The evolution of the shock-wave structure of the flow around a symmetric airfoil is examined by solving two-dimensional unsteady gas-dynamic equations, and a resonant mechanism of interaction is found, which leads to considerable (by an order of magnitude) reduction of the wave drag of the airfoil.     

Computation of unsteady transonic flow over a fighter wing using a zonal Navier–Stokes/full‐potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier–Stokes equations are solved using a diagonal form of an alternating‐direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full‐potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic‐based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier–Stokes CPU times for unsteady flows in non‐vector processing machines. Applications of the method are presented for a F‐5 wing in steady and unsteady transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to the full Navier–Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier–Stokes equations can be retained with significant savings in computational time. Copyright © 1999 John Wiley & Sons, Ltd.     

Stability of flow past a cylinder: Energy budget of eigenmodes

Global linear stability analysis of the flow past a circular cylinder at the onset of primary wake instability is carried out. The real and imaginary parts of the most unstable eigenmode, responsible for vortex shedding, are very similar but associated with a spatial shift in the vortex structures. This shift results in the convection of vortices that are observed in the unsteady flow, which is actually a consequence of global absolute instability. The kinetic energy density, associated with the most unstable eigenmode, is studied. At the onset of the instability the energy density of the disturbance field is found to be stronger in the far wake compared with the near wake. With increase in Re the region where the disturbance is strong moves upstream closer to the cylinder. However, the maximum value of the kinetic energy density of the disturbance lies outside the recirculation zone even for Re upto 100. A linearized mechanical energy equation for the time evolution of the kinetic energy density of the disturbance is utilized to examine the energy budget of the most unstable eigenmode at various Re. It is found that the most significant contribution to the growth rate of the disturbance arises from the transfer of the energy due to the strain rate of the base flow to the perturbation. The stabilizing effect of the viscous dissipation increases with increase in Re, but saturates for Re beyond ～70. Copyright © 2009 John Wiley & Sons, Ltd.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.