Unsteady Aerodynamic Response Research Articles

Review of unsteady aerodynamic methods for turbomachinery aeroelastic and aeroacoustic applications

Introduction T HE unsteady aerodynamic analyses intended for turbomachinery aeroelastic and aeroacoustic predictions must be applicable over wide ranges of blade-row geometries and operating conditions and unsteady excitation modes and frequencies. Also, because of the large number of controlling parameters involved, there is a stringent requirement for computational efficiency. To date these requirements have been met only to a limited extent. As a result, aeroelastic and aeroacoustic design predictions are, for the most part, still based on the classical linearized unsteady aerodynamic analyses developed in the early 1970s. During the past decade, significant advances in unsteady aerodynamic prediction capabilities have been achieved. In particular, researchers have developed efficient linearized analyses that account for the effects of important design features, such as real blade geometry, mean blade loading, and operation at transonic Mach numbers, on the unsteady aerodynamic response of the blading to imposed structural and external aerodynamic excitations. The improvements in physical modeling that such linearizations allow are motivating their current implementation into aeroelastic and aeroacoustic design prediction systems. Also, considerable progress has been made on developing time-accurate Euler and Navier-Stokes simulations of nonlinear unsteady flows through blade rows. Although not yet suitable for design use, such analyses offer opportunities for an improved understanding of the unsteady aerodynamic processes associated with blade vibration and noise generation. These recent advances in the theoretical and computational modeling of turbomachinery unsteady flows are reviewed in the present survey.

Double Linearization Theory Applied to Three-Dimensional Cascades Oscillating under Supersonic Axial Flow Condition.

The double linearization theory is extended to a three-dimensional straight cascade between parallel-plane end walls operating at supersonic axial velocity. Numerical examples are presented to show unsteady aerodynamic responses and flutter boundaries in correlation with cascade parameters. The three-dimensional effects are evaluated by comparing them with the strip theory predictions. To attain a large steady loading on the stability boundary of the first bending vibration, it is advantageous to design blades with angle of attack and camber decreasing from hub to tip. The three-dimensional effects are small, and the strip theory was found to be a useful method for estimation of the unsteady aerodynamic work of the three-dimensional straight cascade.

A New Technique for Analysis of Unsteady Aerodynamic Responses of Cascade Airfoils with Blunt Leading Edge. Unsteady Aerodynamic Responses of the Cascade in Incompressible Flow.

Unsteady aerodynamic responses of cascade airfoils with a blunt leading edge, which are subjected to incident rotational and irrotational fluctuations, are analyzed by use of the method developed in the previous paper of Funazaki and Kakudate. Since some difficulties in the imposition of the boundary condition have been pointed out in that paper, most of the efforts in this study are devoted to refinement of the method of the boundary condition imposition on the inlet/outlet and periodic boundaries by employing the analytical solutions in the upstream/downstream far-fields. Although the method presented in this paper could be applied to compressible flow problems, numerical examples employed this time are limited to incompressible flow cases. Obtained results then show the usefulness of the method, and also reveal some of the notable features of the unsteady flow field induced within the blade-to-blade passage vividly.

The Effect of Steady Aerodynamic Loading on the Flutter Stability of Turbomachinery Blading

An aeroelastic analysis is presented that accounts for the effect of steady aerodynamic loading on the aeroelastic stability of a cascade of compressor blades. The aeroelastic model is a two-degree-of-freedom model having bending and torsional displacements. A linearized unsteady potential flow theory is used to determine the unsteady aerodynamic response coefficients for the aeroelastic analysis. The steady aerodynamic loading was caused by the addition of (1) airfoil thickness and camber and (2) steady flow incidence. The importance of steady loading on the airfoil unsteady pressure distribution is demonstrated. Additionally, the effect of the steady loading on the tuned flutter behavior and flutter boundaries indicates that neglecting either airfoil thickness, camber, or incidence could result in nonconservative estimates of flutter behavior.

Unsteady Aerodynamic Responses of Mistuned Cascades to Incoming Wakes. Mistuning of Stagger Angle.

Unsteady aerodynamic responses of a mistuned cascade subjected to incoming wakes from an upstream cascade are investigated in this study, where the extended Nishiyama-Funazaki method is employed. Numerical calculations are made, focussing on the effects of stagger angle mistuning of the cascade. It is accordingly found that there is some possibility of reducing wake-induced unsteady forces by controlling stagger angles without the severe expense of cascade performance deterioration. In addition, intense correlation between unsteady lifts and the corresponding steady lifts in the case of in-phase incoming wake conditions is observed.

Unsteady Aerodynamic Response of a Cascade to Nonuniform Inflow

A solution method is developed for the small perturbation approximation to the inviscid, unsteady fluid equations for a two-dimensional cascade of airfoils. By introducing the unsteady field as a locally small perturbation from a nonuniform steady field, a first-order coupling between the unsteady fluid properties and the gradients of the steady field properties results. Results of this analysis procedure are compared with the experimentally determined unsteady pressure distribution on the vane of a low-speed research compressor produced by the wakes from the upstream rotor. Correlation between experimental and analytical results is good. Similar comparisons of the data with an analysis method that approximates the vane as a two-dimensional cascade of flat plates at zero mean incidence did not exhibit acceptable correlation. The lack of correlation was particularly evident for the chordwise phase angle distribution.

Analysis of forced response of detuned blade rows

A mathematical model is developed and utilized to demonstrate the enhanced forced response behavior associated with aerodynamic, structural, and combined aerodynamic-structural detuning of a loaded rotor operating in an incompressible flow field. The unsteady aerodynamic gust response and oscillating cascade aerodynamics are determined by developing both a complete first-order unsteady aerodynamic analysis and a locally analytical solution in individual grid elements of a body fitted computational grid. The aerodynamic detuning is accomplished by means of alternate circumferential airfoil spacing, with alternate blade structural detuning also considered. The beneficial forced response effects of these detuning techniques are then demonstrated by applying this model to various detuned rotor configurations.

Modal Analysis of Unsteady Aerodynamic Response of Subsonic Annular Cascade with Steady Loading under Elastic Vibration.

Aerodynamic analysis based on the double linearization theory has been combined with a structural analysis by a finite element method to study the unsteady aerodynamic response of a rotating subsonic annular cascade composed of twisted blades undergoing three-dimensional elastic vibrations with chordwise displacements and flexible deformations. Numerical results are presented to demonstrate the unsteady pressure difference distributions on blade surfaces, aerodynamic works, three-dimensional effects and flutter boundaries for two kinds of cascade blades. In particular, it is found that the second elastic vibration mode can be most unstable for a turbine-type cascade with small camber operating at a subsonic Mach number.

A New Technique for Analysis of Unsteady Aerodynamic Responses of Cascade Airfoils with Blunt Leading Edge. 1st Report. Theory.

A new technique has been developed, based on the technique proposed by Goldstein (1978), in order to analyze unsteady aerodynamic responses of cascade airfoils subjected to incident wakes. One of the important features of this technique is the introduction of the additional perturbation potential for the purpose of preventing the solution being indeterminate due to the singular behavior of the unsteady perturbed velocities around the blunt leading edge. In this paper, the main focus is placed on the theoretical development of the technique, and some simple cases of numerical calculations with no stagger angle are made in order to show the validity of our method.

Gust response analysis for cascades operating in nonuniform mean flows

The linearized unsteady aerodynamic response of a cascade of airfoils subjected to entropic, vortical, and acoustic gusts is analyzed. Field equations for the first-order unsteady perturbation flow are obtained by linearizing the full time-dependent mass, momentum, and energy conservation equations about a nonlinear, isentropic, and irrotational mean or steady flow. A splitting technique is then used to decompose the unsteady velocity field into irrotational and rotational parts leading to field equations for the unsteady entropy, rotational velocity, and irrotational velocity fluctuations that are coupled only sequentially. The entropic and rotational velocity fluctuations can be described in terms of the mean-flow drift and stream functions which can be computed numerically. The irrotational unsteady velocity is described by an inhomogeneous linearized potential equation which contains a source term that depends on the rotational velocity field. This equation is solved via a finite difference technique. Results are presented to indicate the status of the numerical solution procedure and to demonstrate the impact of blade geometry and mean blade loading on the aerodynamic response of cascades to vortical gust excitations. The analysis described leads to very efficient predictions of cascade unsteady aerodynamics phenomena making it useful for turbomachinery aeroelastic and aeroacoustic design applications.

Double linearization theory of three-dimensional cascades with vibrating blades under spanwise-nonuniform mean loading, I: Subsonic flow

The improved double linearization theory is extended to three-dimensional subsonic cascades in order to study three-dimensional effects upon the unsteady aerodynamic forces on vibrating blades under spanwise-nonuniform mean loading. Two kinds of unsteady spanwise aerodynamic forces arise due to non-zero mean loading. One is proportional to the blade inclination times the steady pressure difference between upper and lower blade surfaces. Another is the aerodynamic force which acts on the streamwise component of steady bound vortices moving with the blades relative to the undisturbed fluid. Numerical examples which illustrate the comparison between the present theory and the strip theory are presented, and some features of the three-dimensional effects of nonuniform mean loading on the unsteady aerodynamic response are made clear.

Double linearization theory of three-dimensional cascades with vibrating blades under spanwise-nonuniform mean loading, II: Supersonic flow

The double linearization theory is extended to a supersonic straight cascade placed between parallel plane end walls, and the effect of spanwise-nonuniform mean loading upon the unsteady aerodynamic force for vibrating blades is studied. The three-dimensional version of the theory takes account of the unsteady spanwise aerodynamic force component due to spanwise nonuniformities of the mean blade loading and the vibration amplitude. The unsteady shock displacement effect on the unsteady aerodynamic force is found to be quasi-two-dimensional. Numerical examples are presented to demonstrate several features of the unsteady aerodynamic response ascribable to the effects of the spanwise-nonuniform angle of attack, camber and thickness of blades. The effect of nonuniformity of camber or thickness is larger than that of angle of attack. To attain a large steady loading on the stability boundary of the first mode bending vibration, it is advantageous to design blades with the angle of attack decreasing and the chamber and thickness increasing from hub to tip.

Double linearization theory for prediction of mean loading effect on cascade flutter, II: Two-dimensional supersonic cascade

In this paper an extension of the improved double linearization theory to two-dimensional supersonic cascades is presented, to predict the mean loading effect on unsteady aerodynamic force on blades in vibrating motion. In the case of supersonic cascades, the aerodynamic instability of the blade motion is significantly affected by the unsteady displacement of shock reflection points, which appears in the linearized formulation as fluctuating concentrated forces acting at mean reflection points. Sample results are presented to demonstrate the effects of blade angle of attack, camber, thickness, Mach number and pitch-chord ratio on the unsteady aerodynamic response. The results show that single-degree-of-freedom bending flutter is more likely to occur for supersonic cascades than for subsonic cascades.

Unsteady Aerodynamic Responses of Mistuned Cascade to Incoming Wakes. Mistuning of Stagger Angle.

Unsteady Aerodynamic Responses of Mistuned Cascade to Incoming Wakes. Mistuning of Stagger Angle.

Unsteady Aerodynamic Responses of Mistuned Cascade to Incoming Wakes. Mistuning Effects of Blade Pitch.

Wake-induced unsteady aerodynamic responses of mistuned compressor cascades, in which blade pitch is purposefully made nonuniform, are investigated by means of the cascade superposition method developed in the previous paper. Numerical calculations are performed in focus, mainly on the effect of the blade pitch nonuniformity which is defined by the off-set level of the cascade superposition. It is consequently found that in most cases, unsteady lift on the compressor blade tends to increase in magnitude according to the off-set level, and it is also found that the reduction effect of the unsteady lift cannot be obtained by the simple mistuning of the blade pitch considered in this paper.

Unsteady aerodynamic gust response including steady flow separation

A series of experiments are performed to investigate and quantify the unsteady aerodynamic response of an airfoil to a high reduced frequency gust including the effects of the gust forcing function waveform, airfoil loading and steady flow separation. this is accomplished by using an axial flow research compressor to experimentally model the high reduced frequency gust forcing function and replacing the last stage stator row with isolated instrumented airfoils. Appropriate data are correlated with predictions from flat plate and chambered airfoil convected gust models. The airfoil surface steady loading is shown to have a large effect on the unsteady aerodynamic response. Also the steady flow separation has a significant influence on the gust response, particularly upstream of the separation point and in the airfoil trailing-edge region.

Cascade aerodynamic gust response including steady loading effects

AbstractTo 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.

The unsteady aerodynamic response to arbitrary modes of blade motion

Analytical expressions are derived for predicting the unsteady aerodynamic response associated with small-amplitude, harmonic vibrations of the blades of a two-dimensional cascade. These expressions can be applied to predict the aeroelastic behavior of blades undergoing arbitrary, i.e. rigid and/or flexible, motions. In addition, a new local response function, termed the pressure-displacement function, has been identified. This function reveals the positions on a moving blade surface at which the local unsteady loads tend to support or suppress a prescribed blade motion and, hence, contribute to the stability or instability of that motion. The pressure-displacement function is continuous along a blade surface except at mean shock locations, where it exhibits a delta function behavior. The integral of this function over the entire blade surface, i.e. the aerodynamic work per cycle, determines the rate at which energy is transferred from an airstream to a moving blade. Example solutions are presented to illustrate the behavior of the global aerodynamic work per cycle and the local pressure-displacement function for a vibrating compressor-type cascade operating at subsonic and transonic Mach numbers.

Double linearization theory applied to unsteady aerodynamic analysis of three-dimensional supersonic cascade vibrating with spanwisely nonuniform mean loading.

This paper extends the double linearization theory to three-dimensional supersonic cascades to study the spa wisely nonuniform mean loading effects upon unsteady aerodynamic forces of vibrating blades. The theory takes account of the unsteady shock displacement effect and the two kinds of unsteady spanwise aerodynamic forces due to nonuniform mean loadings and nonuniform vibration amplitudes in the three-dimensional flow field. Numerical examples show several features of unsteady aerodynamic responses which are ascribable to the effects of the spanwise nonuniform angle of attack, camber and thickness. The nonuniform effects of the camber and thickness are greater than those of the angle of attack. In order to obtain a large total steady lift coefficient on a stability boundary for the first bending mode, the cascade configuration with the angle of attack decreasing and the camber and thickness increasing from hub to tip is advantageous.

Airfoil dynamic stall at constant pitch rate and high Reynolds number

An experiment has been performed to study the aerodynamics of dynamic stall penetration at constant pitch rate and high Reynolds number, in an attempt to model more accurately conditions during aircraft poststall maneuvers and during helicopter high-speed forward flight. An airfoil was oscillated at pitch rates, A = ac/2U between 0.001 and 0.020, Mach numbers between 0.2 and 0.4, and Reynolds numbers between 2-4 x 10. Surface pressures were measured using 72 miniature transducers, and the locations of transition and separation were determined using 8 surface hot-film gages. The results demonstrate the influence of the leading-edge vorticity on the unsteady aerodynamic response during and after stall. The vortex is strengthened by increasing the pitch rate and is weakened by increasing the Mach number and by starting the motion close to the steady-state stall angle. A periodic pressure oscillation occurred after stall at high pitch angle and moderate Reynolds number; the oscillation frequency was close to that predicted for a von Karman vortex street. A small supersonic zone near the leading edge at M = 0.4 was found to reduce significantly the peak suction pressures and the unsteady increments to the airloads. These results provide the first known data base of constant-pitch-rate aerodynamic information at realistic combinations of Reynolds and Mach numbers.