This work provides an accurate and efficient numerical method for turbomachinery flutter. The unsteady Euler or Reynolds-averaged Navier-Stokes equations are solved in integral form, the blade passages being discretised using a background fixed C-grid and a body-fitted C-grid moving with the blade. In the overlapping region data are exchanged between the two grids at every time step, using bilinear interpolation. The method employs Roe’s second-order-accurate flux difference splitting scheme for the inviscid fluxes, a standard second-order discretisation of the viscous terms, and a three-level backward difference formula for the time derivatives. The dual-time-stepping technique is used to evaluate the nonlinear residual at each time step. The state-of-the-art second-order accuracy of unsteady transonic flow solvers is thus carried over to flutter computations. The code is proven to be accurate and efficient by computing the 4th Aeroelastic Standard Configuration, namely, the subsonic flow through a turbine cascade with flutter instability in the first bending mode, where viscous effect are found practically negligible. Then, for the very severe 11th Aeroelastic Standard Configuration, namely, transonic flow through a turbine cascade at off-design conditions, benchmark solutions are provided for various values of the inter-blade phase angle.

The aeroelastic behavior of vibrating blade assemblies is usually investigated in the frequency domain where the determination of aeroelastic stability boundaries is separated from the computation of linearized unsteady aerodynamic forces. However, nonlinear fluid-structure interaction caused by oscillating shocks or strong flow separation may significantly influence the aerodynamic damping and hence effect a shift of stability boundaries. In order to investigate such aeroelastic phenomena, the governing equations of structural and fluid motion have to be simultaneously integrated in time. In this paper a technique is presented which analyzes the aeroelastic behavior of an oscillating compressor cascade in the time domain. The structural part of the governing aeroelastic equations is time-integrated according to the algorithm of Newmark, while the unsteady airloads are computed at every time step by an Euler upwind code. The link between the two time integrations is an automatic grid generation in which the used mesh is dynamically deformed as such that it conforms with the deflected blades at every time step. The computed time series of the aeroelastic simulation of an assembly of twenty compressor blades performing torsional vibrations in transonic flow are presented. For subsonic flow, the differences between time domain and frequency domain results are of negligible order. For transonic flow, however, where vibrating shocks and a temporarily choked flow in the blade channel dominate the unsteady flow, the energy transfer between fluid and structure is no longer comparable to that of a linear system. It is demonstrated that the application of the time domain method leads to a significantly different aeroelastic behavior of the blade assembly including a shift of the stability boundary.

The three-dimensional, multistage, unsteady, turbomachinery analysis, TURBO, has been extended to predict the aeroelastic response of a blade row operating within a cylindrical annular duct. In particular, a blade vibration capability has been incorporated, so that the TURBO analysis can be applied over a solution domain that deforms with a vibratory blade motion. Also, unsteady far-field conditions have been implemented to render the computational inlet and exit boundaries transparent to outgoing unsteady disturbances and to allow for the prescription of incoming aerodynamic excitations. The modified TURBO analysis has been applied to predict unsteady subsonic and transonic flows. The intent is to validate this nonlinear analysis partially for blade flutter applications via numerical results for benchmark unsteady flows, and to demonstrate this analysis for a realistic fan rotor. For these purposes, we have considered unsteady subsonic flows through a three-dimensional version of the 10th Standard Cascade and unsteady transonic flows through the first-stage rotor of the NASA Lewis Rotor 67 fan. Some general correlations between aeromechanical stabilities and fan operating characteristics will be presented.

This paper presents the numerical results of a code for computing the unsteady transonic viscous flow in a two-dimensional cascade of harmonically oscillating blades. The flow field is calculated by a Navier–Stokes code, the basic features of which are the use of an upwind flux vector splitting scheme for the convective terms (Advection Upstream Splitting Method), an implicit time integration, and the implementation of a mixing length turbulence model. For the present investigations, two experimentally investigated test cases have been selected, in which the blades had performed tuned harmonic bending vibrations. The results obtained by the Navier–Stokes code are compared with experimental data, as well as with the results of an Euler method. The first test case, which is a steam turbine cascade with entirely subsonic flow at nominal operating conditions, is the fourth standard configuration of the “Workshop on Aeroelasticity in Turbomachines.” Here the application of an Euler method already leads to acceptable results for unsteady pressure and damping coefficients and hence this cascade is very appropriate for a first validation of any Navier–Stokes code. The second test case is a highly loaded gas turbine cascade operating in transonic flow at design and off-design conditions. This case is characterized by a normal shock appearing on the rear part of the blades’s suction surface, and is very sensitive to small changes in flow conditions. When comparing experimental and Euler results, differences are observed in the steady and unsteady pressure coefficients. The computation of this test case with the Navier–Stokes method improves to some extent the agreement between the experiment and numerical simulation.

The purpose of this work is to validate a time-nonlinear three-dimensional Euler solver for vibrating cascades aerodynamics by comparison with available theoretical semi-analytical results from flat-plate cascades. First the method is validated with respect to the purely two-dimensional theory of Verdon (for supersonic flow) by computing two-dimensional vibration (spanwise constant) in linear three-dimensional cascades. Then the method is validated by comparison with the theoretical results of Namba and the computational results of He and Denton, for subsonic flow in a linear three-dimensional cascade with three-dimensional vibratory mode. Finally the method is compared with results of Chi from two subsonic rotating annular cascades of helicoi¨dal flat plates. Quite satisfactory agreement is obtained for all the cases studied. A first code-to-code comparison is also presented.

New blading concepts as used in modern transonic axial-flow compressors require improved loss and deviation angle correlations. The new model presented in this paper incorporates several elements and treats blade-row flows having subsonic and supersonic inlet conditions separately. In the first part of this paper two proved and well-established profile loss correlations for subsonic flows are extended to quasi-two-dimensional conditions and to custom-tailored blade designs. Instead of a deviation angle correlation, a simple method based on singularities is utilized. The comparison between the new model and a recently published model demonstrates the improved accuracy in prediction of cascade performance achieved by the new model.

The aerodynamics of a biconvex airfoil cascade oscillating in torsion is investigated using the unsteady aerodynamic influence coefficient technique. For subsonic flow and reduced frequencies as large as 0.9, airfoil surface unsteady pressures resulting from oscillation of one of the airfoils are measured using flush-mounted high-frequency-response pressure transducers. The influence coefficient data are examined in detail and then used to predict the unsteady aerodynamics of a cascade oscillating at various interblade phase angles. These results are correlated with experimental data obtained in the traveling-wave mode of oscillation and linearized analysis predictions. It is found that the unsteady pressure disturbances created by an oscillating airfoil excite wind tunnel acoustic modes, which have detrimental effects on the experimental results. Acoustic treatment is proposed to rectify this problem.

Experiments in a linear oscillating cascade reveal that the wind tunnel walls enclosing the airfoils have, in some cases, a detrimental effect on the oscillating cascade aerodynamics. In a subsonic flow field, biconvex airfoils are driven simultaneously in harmonic, torsion-mode oscillations for a range of interblade phase angle values. It is found that the cascade dynamic periodicity—the airfoil-to-airfoil variation in unsteady surface pressure—is good for some values of interblade phase angle but poor for others. Correlation of the unsteady pressure data with oscillating flat plate cascade predictions is generally good for conditions where the periodicity is good and poor where the periodicity is poor. Calculations based upon linearized unsteady aerodynamic theory indicate that pressure waves reflected from the wind tunnel walls are responsible for the cases where there is poor periodicity and poor correlation with the predictions.

An investigation of the “additional” total pressure losses occurring in combining flow through several sharp-edged three-leg junctions has been made. Experimental results covering a wide speed range up to choking are presented for three flow geometries of a lateral branch off a straight duct using dry air as the working fluid. A new theoretical flow model provided results in fairly good agreement with the experimental data obtained. Flow visualization of the high-speed flow using the Schlieren method revealed the presence of normal shock waves in the combined flow about one duct diameter downstream of the junction. The highest attainable Mach number (M 3 ) of the averaged downstream (combined) flow was 0.66 for several of the flow geometries. This value of M 3 appears to be the maximum possible and is the result of a combination of flow separation and local flow choking.

An aerodynamic design method is described and used to implement a parametric study of radial turbomachinery blade design in three-dimensional subsonic flow. Given the impeller hub and shroud, the number of blades and their stacking position, the design method gives the detailed blade shape, flow, and pressure fields that would produce a prescribed tangentially averaged swirl schedule. The results from that study show that decreasing the number of blades increases the blade wrap, and that the blade loading is strongly affected by the rate of change of mean swirl along the mean streamlines. The results also show that the blade shape and the pressure field are rather sensitive to the prescribed mean swirl schedule, which suggests that, by carefully tailoring the swirl schedule, one might be able to control the blade shape and the pressure field and hence secondary flow.

The study presents a numerical method, based on the flux vector splitting approach, to the problem of unsteady one-dimensional and two-dimensional inviscid transonic flows, with emphasis on the numerical determination of the shock position, through nozzles with time-varying back pressure. The model is first validated by comparison with exact (one dimension) and numerical (two dimensions) steady-state solutions. It is thereafter applied to the problem of time-fluctuating back pressure in quasi-one-dimensional and two-dimensional nozzles. The one-dimensional results are validated by comparison with a small perturbation analytical unsteady solution, whereafter a few sample cases are presented with the objective of understanding fundamental aspects of unsteady transonic flows. It is concluded that both the amplitude and frequency of the imposed fluctuating exit pressure are important parameters for the location of the unsteady shock. It is also shown that the average unsteady shock position is not necessarily identical with the steady-state position, and that the unsteady shock may, under certain circumstances, propagate upstream into the subsonic flow domain. The pressure jump over the shock, as well as the unsteady post-shock pressure, is different for identical shock positions during the cycle of fluctuation, which implies that an unsteady shock movement, imposed by oscillating back pressure, may introduce a significant unsteady lift and moment. This may be of importance for flutter predictions. It is also noted that, although the sonic velocity is obtained in the throat of steady-state, quasi-one-dimensional flow, this is not necessarily true for the unsteady solution. During part of the period with fluctuating back pressure, the flow velocity may be subsonic at the throat and still reach a supersonic value later in the nozzle. This phenomenon depends on the frequency and amplitude of the imposed fluctuation, as well as on the nozzle geometry.

An experimental study on a gas turbine cascade operating under transonic flow conditions is presented. The flow is compared for airfoil shapes corresponding to the design geometry and the geometry taken from a rotor blade, in an industrial gas turbine burning heavy oil, after a few thousand hours of operation. Steady-state data have been obtained in a linear cascade over a range of isentropic exit Mach numbers from 0.6 to 1.6. The flow field was determined by static pressure measurements on the side walls up- and downstream of the cascade, on one side wall in the blade passage, and on the blade surface at midspan. Furthermore, the flow was visualized by the methods of Schlieren and laser holography. The results show that the choked flow conditions are reached at different steady-state isentropic outlet Mach numbers for the two blade shapes. The deposit, typical for a gas turbine, does not however significantly modify the boundary layer separation point. The flow visualization indicates that the shock wave fluctuations have not been significantly influenced by the important roughness and thickness of the deposit. The experimental results on the two cascades are also compared with two-dimensional time-marching calculations after Denton. In the subsonic regime, good agreement was found for the “clean” blade. For the profile with deposit, the flow cannot be correctly predicted by the time-marching calculation, even in subsonic flow condition. The sonic line calculated by the numerical model under transonic outlet conditions (0.9 < M 2S < 1.20) does not agree with the laser holography measurements for either of the two cascades.