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The development of an analysis to predict the unsteady compressible flows in blade boundary layers and wakes is presented. The equations that govern the flows in these regions are transformed using an unsteady turbulent generalization of the Levy–Lees transformation. The transformed equations are solved using a finite difference technique in which the solution proceeds by marching in time and in the streamwise direction. Both laminar and turbulent flows are studied, the latter using algebraic turbulence and transition models. Laminar solutions for a flat plate are shown to approach classical asymptotic results for both high and low-frequency unsteady motions. Turbulent flat-plate results are in qualitative agreement with previous predictions and measurements. Finally, the numerical technique is also applied to the stator and rotor of a low-speed turbine stage to determine unsteady effects on surface heating. The results compare reasonably well with measured heat transfer data and indicate that nonlinear effects have minimal impact on the mean and unsteady components of the flow.

This paper describes two new developments in the numerical analysis of linearized unsteady cascade flows, which have been motivated by the need for an accurate analytical procedure for predicting the onset of flutter in highly loaded compressors. In previous work, results were determined using a two-step or single-pass procedure in which a solution was first determined on a rectilinear-type cascade mesh to determine the unsteady flow over an extended blade-passage solution domain and then on a polar-type local mesh to resolve the unsteady flow in high-gradient regions. In the present effort a composite procedure has been developed in which the cascade-and local-mesh equations are solved simultaneously. This allows the detailed features of the flow within the local mesh region to impact the unsteady solution over the entire domain. In addition, a new transfinite local mesh has been introduced to permit a more accurate modeling of unsteady shock phenomena. Numerical results are presented for a two dimensional compressor-type cascade operating at high subsonic inlet Mach number and high mean incidence to demonstrate the impact of the new composite- and local-mesh analyses on unsteady flow predictions.

An experiment was conducted on a heavily instrumented isolated model compressor rotor to study the unsteady aerodynamic response of the blade row to a controlled pitching oscillation of all blades in an undistorted flow, and to a circumferential inlet flow distortion with nonoscillating blades. To accomplish this, miniature pressure transducers were embedded in the blades and the unsteady pressure time histories were recorded. Both phases of the experiment were performed over a wide range of flow coefficient, from Cx/Um = 0.6 to 0.95 in 0.05 steps, and data were taken at each condition for sinusoidal disturbances characterized by one, two, and four per revolution waves. Steady-state data were acquired for flow coefficients from 0.55 to 0.99 in 0.05 steps. In this paper the steady and unsteady results of the portion of this experiment dealing with oscillating blades are compared with analytical predictions, and the steady results are compared with experimental data from previous work. Although the model blades were instrumented at five spanwise stations, only the midspan measurements will be presented herein. The measured pressures for nonoscillating blades were in good agreement with the steady potential flow predictions (and with previous steady experimental data) when the measured exit angle was imposed as the downstream boundary condition for the analysis. It was found that a quasi-steady approach yielded marginally acceptable agreement with the experimental results for the lowest frequency tested. For the higher reduced frequencies, the experimental data could not be modeled in this manner. In contrast, a comparison of the measurements with the Verdon–Caspar unsteady potential flow theory produced generally good agreement except near the leading edge at high mean incidence (i.e., at low flow coefficient). At high incidence the blades in this experiment had very high steady pressure gradients near the leading edge and it is suspected that this may be responsible for the lack of agreement. The agreement was somewhat better at the higher frequencies.

A time-accurate Navier–Stokes analysis is needed for understanding the relative importance of nonlinear and viscous effects on the unsteady flows associated with turbomachinery blade vibration and blade-row noise generation. For this purpose an existing multi-blade-row Navier–Stokes analysis has been modified and applied to predict unsteady flows excited by entropic, vortical, and acoustic disturbances through isolated, two-dimensional blade rows. In particular, time-accurate, non-reflecting inflow and outflow conditions have been implemented to allow specification of vortical, entropic, and acoustic excitations at the inlet, and acoustic excitations at the exit, of a cascade. To evaluate the nonlinear analysis, inviscid and viscous numerical simulations were performed for benchmark unsteady flows and the predicted results were compared with analytical and numerical results based on linearized inviscid flow theory. For small-amplitude unsteady excitations, the unsteady pressure responses predicted with the nonlinear analysis show very good agreement, both in the field and along the blade surfaces, with linearized inviscid solutions. Based on a limited range of parametric studies, it was also found that the unsteady responses to inlet vortical and acoustic excitations are linear over a surprisingly wide range of excitation amplitudes, but acoustic excitations from downstream produce responses with significant nonlinear content.

A linearized potential flow analysis, which accounts for the effects of nonuniform steady flow phenomena on the unsteady response to prescribed blade motions, has been applied to five two-dimensional cascade configurations. These include a flat-plate cascade and three cascades which are representative of the tip sections of current fan designs. Here the blades are closely spaced, highly staggered, and operate at low mean incidence. The fifth configuration is a NASA Lewis cascade of symmetric biconvex airfoils for which experimental measurements are available. Numerical solutions are presented that clearly illustrate the effects and importance of blade geometry and mean blade loading on the linearized unsteady response at high subsonic inlet Mach number and high blade-vibrational frequency. In addition, a good qualitative agreement is shown between the analytical predictions and experimental measurements for the cascade of symmetric biconvex airfoils. Finally, recommendations on the research needed to extend the range of application of linearized unsteady aerodynamic analyses are provided.

This paper is the second of a two-part report on a theoretical analysis of the aerodynamic response to an oscillating supersonic cascade in subsonic axial flow. Supersonic resonance criteria are discussed and lead to the distinction between subresonant and superresonant cascade motions. Numerical predictions, based on the unsteady solution reported in Part 1, are presented for two typical cascade configurations. These reveal the possibility of both subresonant and superresonant single-degree-of-freedom torsional instabilities. Subresonant instabilities occur over a broad range of frequencies and interblade phase angles, whereas superresonant instabilities occur only over a narrow range of such cascade parameter values. For a given blade motion frequency and free-stream Mach number, it appears that the least stable condition will usually lie in the subresonant region.

This paper presents, in two parts, a theoretical investigation of the aerodynamic response produced by an oscillating cascade placed in a supersonic stream with subsonic axial velocity component. Predictions are based on the successive solution of two linear boundary value problems which treat the velocity potential and the pressure, respectively, as basic dependent variables. A solution for the potential has been reported earlier and is used here to provide upper surface blade pressure distributions. This information serves as a boundary condition for the second problem. The solution for the unsteady pressure field, described in Part 1, is obtained by a construction procedure which parallels that used earlier to determine the potential. With the present procedure, blade pressure difference distributions and aerodynamic coefficients are accurately and efficiently determined for both subresonant and superresonant blade motions. Supersonic resonance phenomena and selected numerical results are discussed in Part 2 of the paper.

A time-accurate Navier–Stokes analysis is needed for understanding the relative importance of nonlinear and viscous effects on the unsteady flows associated with turbomachinery blade vibration and blade-row noise generation. For this purpose an existing multi-blade-row Navier–Stokes analysis has been modified and applied to predict unsteady flows excited by entropic, vortical, and acoustic disturbances through isolated, two-dimensional blade rows. In particular, time-accurate, non-reflecting inflow and outflow conditions have been implemented to allow specification of vortical, entropic, and acoustic excitations at the inlet, and acoustic excitations at the exit, of a cascade. To evaluate the nonlinear analysis, inviscid and viscous numerical simulations were performed for benchmark unsteady flows and the predicted results were compared with analytical and numerical results based on linearized inviscid flow theory. For small-amplitude unsteady excitations, the unsteady pressure responses predicted with the nonlinear analysis show very good agreement, both in the field and along the blade surfaces, with linearized inviscid solutions. Based on a limited range of parametric studies, it was also found that the unsteady responses to inlet vortical and acoustic excitations are linear over a surprisingly wide range of excitation amplitudes, but acoustic excitations from downstream produce responses with significant nonlinear content.

The development of an analysis to predict the unsteady compressible flows in blade boundary layers and wakes is presented. The equations that govern the flows in these regions are transformed using an unsteady turbulent generalization of the Levy–Lees transformation. The transformed equations are solved using a finite difference technique in which the solution proceeds by marching in time and in the streamwise direction. Both laminar and turbulent flows are studied, the latter using algebraic turbulence and transition models. Laminar solutions for a flat plate are shown to approach classical asymptotic results for both high and low-frequency unsteady motions. Turbulent flat-plate results are in qualitative agreement with previous predictions and measurements. Finally, the numerical technique is also applied to the stator and rotor of a low-speed turbine stage to determine unsteady effects on surface heating. The results compare reasonably well with measured heat transfer data and indicate that nonlinear effects have minimal impact on the mean and unsteady components of the flow.

This paper describes two new developments in the numerical analysis of linearized unsteady cascade flows, which have been motivated by the need for an accurate analytical procedure for predicting the onset of flutter in highly loaded compressors. In previous work, results were determined using a two-step or single-pass procedure in which a solution was first determined on a rectilinear-type cascade mesh to determine the unsteady flow over an extended blade-passage solution domain and then on a polar-type local mesh to resolve the unsteady flow in high-gradient regions. In the present effort a composite procedure has been developed in which the cascade-and local-mesh equations are solved simultaneously. This allows the detailed features of the flow within the local mesh region to impact the unsteady solution over the entire domain. In addition, a new transfinite local mesh has been introduced to permit a more accurate modeling of unsteady shock phenomena. Numerical results are presented for a two dimensional compressor-type cascade operating at high subsonic inlet Mach number and high mean incidence to demonstrate the impact of the new composite- and local-mesh analyses on unsteady flow predictions.

A method is presented for determining the unsteady flow field and the aerodynamic response which occurs when a finite oscillating cascade is placed in a supersonic stream, which has a subsonic velocity component normal to the cascade. Solutions are obtained through the combined use of closed-form and numerical procedures. Computed results indicate that the finite cascade analysis should provide a reasonable indication of the influence of the cascade parameters on the response of the infinite array. A brief parametric study for a typical configuration reveals possible aerodynamic instabilities when the blades perform single-degree-of-freedom pitching oscillations over a broad range of frequencies and interblade phase angles.