The Low Reduced Frequency Limit of Vibrating Airfoils—Part II: Numerical Experiments

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Mode Shape ◽  
Numerical Experiments ◽  
Order Approximation ◽  
Unsteady Aerodynamics ◽  
Direct Consequence ◽  
Operating Conditions ◽  
Influence Coefficient ◽  
Frequency Limit ◽  
Unsteady Pressure ◽  
Reduced Frequency

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis on its impact on the scaling of the work-per-cycle curves by means of numerical experiments. Simulations using a frequency domain linearized Navier–Stokes solver have been carried out on rows of a low-pressure turbine (LPT) airfoil section, the NACA0012 and NACA65 profiles, and a flat-plate cascade operating at different flow conditions. Both the traveling wave (TW) and the influence coefficient (IC) formulations of the problem are used in combination to investigate the nature of the unsteady pressure perturbations. All the theoretical conclusions derived in Part I of the paper have been confirmed, and it is shown that the behavior of the unsteady pressure modulus and phase, as well as the work-per-cycle curves, are fairly independent of the geometry of the airfoil, the operating conditions, and the mode-shape in first-order approximation in the reduced frequency. The second major conclusion is that the airfoil loading and the symmetry of the cascade play an essential role in this trend. Simulations performed at reduced frequency ranges beyond the low reduced frequency limit reveal that, in this regimen, the ICs modulus varies linearly with the reduced frequency, while the phase is always π/2, and then, the classical sinusoidal antisymmetric shape of work-per-cycle curves in the low reduced frequency limit turns into a cosinusoidal symmetric shape. It is then concluded that the classical cosinusoidal shape of compressor airfoils is not neither a geometric nor a flow effect, but a direct consequence of the fact that the natural frequencies of the lowest modes of compressors are higher than that of high aspect ratio cantilever LPT rotor blades. Numerical simulations have also confirmed that the actual mode-shape of the airfoil motion does not alter the conclusions derived in Part I of the paper.

Quantification of the Influence of Unsteady Aerodynamic Loading on the Damping Characteristics of Airfoils Oscillating at Low Reduced Frequency—Part II: Numerical Verification

2016 ◽  
Vol 139 (3) ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Unsteady Aerodynamics ◽  
Operating Conditions ◽  
Mode Shapes ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Numerical Verification ◽  
Low Pressure Turbine ◽  
Reduced Frequency ◽  
Unsteady Loading ◽  
Loading Parameter

This paper numerically investigates the correlation between the so-called unsteady loading parameter (ULP), derived in Part I of the corresponding paper, and the unsteady aerodynamics of oscillating airfoils at low reduced frequency with special emphasis on the work-per-cycle curves. Simulations using a frequency-domain linearized Navier–Stokes solver have been carried out on rows of a low-pressure turbine airfoil section, the NACA65 section, and a flat plate, to show the correlation between the actual value of the ULP and the flutter characteristics, for different airfoils, operating conditions, and mode shapes. Both the traveling wave and influence coefficient formulations of the problem are used in combination to increase the understanding of the ULP influence in different aspects of the unsteady flow field. It is concluded that, for a blade vibrating in a prescribed motion at design conditions, the ULP can quantitatively predict the effect of unsteady loading variations due to changes in both the incidence and the mode shape on the work-per-cycle curves. It is also proved that the unsteady loading parameter can be used to qualitatively compare the flutter characteristics of different airfoils.

The Low Reduced Frequency Limit of Vibrating Airfoils: Part II — Numerical Experiments

2015 ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Mean Value ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Navier Stokes Equations ◽  
Unsteady Pressure ◽  
Reduced Frequency ◽  
Influence Coefficients ◽  
Frequency Regime

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis in its impact on the scaling of the work per cycle curves using an asymptotic approach (Part I) and numerical simulations. A perturbation analysis of the linearized Navier-Stokes equations at low reduced frequency is presented and some conclusions are drawn (Part I of the corresponding paper). The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure field caused by the vibration of the airfoil. For lightly loaded airfoils the unsteady pressure and the influence coefficients scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence the work-per-cycle is proportional to the reduced frequency for any inter-blade phase angle and it is independent of its sign. For highly loaded airfoils the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero varies linearly with it. In this case only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and in first approximation of the mode-shape. For symmetric cascades the work-per-cycle scales linearly with the reduced frequency irrespectively of whether the airfoil is loaded or not. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on a low-pressure turbine airfoil section, the NACA0012 and NACA65 profiles and a flat plate operating at different flow conditions to show the generality and correctness of the analytical conclusions. Both the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding and explore the nature of the unsteady pressure perturbations.

The Low Reduced Frequency Limit of Vibrating Airfoils: Part IIIB — Numerical Quantification and Influence of Unsteady Loading

2016 ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Unsteady Aerodynamics ◽  
Operating Conditions ◽  
Mode Shapes ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Low Pressure Turbine ◽  
Reduced Frequency ◽  
Frequency Regime ◽  
Unsteady Loading ◽  
Loading Parameter

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis on its impact on the work per cycle curves. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on rows of a low-pressure turbine airfoil section, the NACA65 section and a flat plate, to show the correlation between the actual value of the unsteady loading parameter (ULP), theoretically derived in Part IIIa, and the flutter characteristics, for different airfoils, operating conditions and mode-shapes. Both, the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding of the ULP influence in different aspects of the unsteady flowfield. It is concluded that, for a blade vibrating in a prescribed motion at design conditions, the ULP can quantitatively predict the effect of loading variations due to changes in the incidence, and also in the mode shape. It is also proved that the unsteady loading parameter can be used to compare the flutter characteristics of different airfoils.

The Low Reduced Frequency Limit of Vibrating Airfoils—Part I: Theoretical Analysis

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Stokes Equations ◽  
Order Approximation ◽  
Unsteady Aerodynamics ◽  
Mean Value ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Unsteady Pressure ◽  
Reduced Frequency ◽  
Influence Coefficients ◽  
Frequency Regime

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis on its impact on the scaling of the work-per-cycle curves, using an asymptotic approach. A perturbation analysis of the linearized Navier–Stokes equations for real modes at low reduced frequency is presented and some conclusions are drawn. The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the vibration of the airfoil. For lightly loaded airfoils, the unsteady pressure and the influence coefficients (ICs) scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence, the work-per-cycle scales linearly with the reduced frequency for any interblade phase angle (IBPA), and it is independent of its sign. For highly loaded airfoils, the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero and varies linearly with it. In this case, only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and the mode shape in first-order approximation in the reduced frequency. For symmetric cascades, the work-per-cycle scales linearly with the reduced frequency irrespective of whether the airfoil is loaded or not. These conclusions have been numerically confirmed in Part II of the paper.

The Low Reduced Frequency Limit of Vibrating Airfoils: Part I — Theoretical Analysis

2015 ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Mean Value ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Navier Stokes Equations ◽  
Unsteady Pressure ◽  
Reduced Frequency ◽  
Influence Coefficients ◽  
Frequency Regime

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis in its impact on the scaling of the work per cycle curves using an asymptotic approach (Part I) and numerical simulations (Part II). A perturbation analysis of the linearized Navier-Stokes equations for real modes at low reduced frequency is presented and some conclusions are drawn. The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the vibration of the airfoil. For lightly loaded airfoils the unsteady pressure and the influence coefficients scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence the work-per-cycle scales linearly with the reduced frequency for any inter-blade phase angle and it is independent of its sign. For highly loaded airfoils the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero and varies linearly with it. In this case only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and the modeshape in first approximation. For symmetric cascades the work-per-cycle scales linearly with the reduced frequency irrespectively of whether the airfoil is loaded or not. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on a low-pressure turbine airfoil section, the NACA0012 and NACA65 profiles and a flat plate to show the generality and correctness of the analytical conclusions (Part II of the corresponding paper). Both, the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding and explore the nature of the unsteady pressure perturbations.

The Impact of Blade Loading and Unsteady Pressure Bifurcations on Low-Pressure Turbine Flutter Boundaries

2015 ◽  
Author(s):  
Joshua J. Waite ◽  
Robert E. Kielb
Keyword(s):  
Mode Shape ◽  
Pressure Ratio ◽  
Low Pressure ◽  
Forced Response ◽  
Blade Loading ◽  
Unsteady Pressure ◽  
Aerodynamic Damping ◽  
Low Pressure Turbine ◽  
Reduced Frequency ◽  
The Impact

The three major aeroelastic issues in the turbomachinery blades of jet engines and power turbines are forced response, non-synchronous vibrations, and flutter. Flutter primarily affects high-aspect ratio blades found in the fan, fore high-pressure compressor stages, and aft low-pressure turbine (LPT) stages as low natural frequencies and high axial velocities create smaller reduced frequencies. Often with LPT flutter analyses, physical insights are lost in the exhaustive quest for determining whether the aerodynamic damping is positive or negative. This paper underlines some well known causes of low-pressure turbine flutter in addition to one novel catalyst. In particular, an emphasis is placed on revealing how local aerodynamic damping contributions change as a function of unsteady (e.g. mode shape, reduced frequency) and steady (e.g. blade torque, pressure ratio) parameters. To this end, frequency domain RANS CFD analyses are used as computational wind tunnels to investigate how aerodynamic loading variations affect flutter boundaries. Preliminary results show clear trends between the aerodynamic work influence coefficients and variations in exit Mach number and back pressure, especially for torsional mode shapes affecting the passage throat. Additionally, visualizations of qualitative bifurcations in the unsteady pressure phases around the airfoil shed light on how local damping contributions evolve with steady loading. Final results indicate a sharp drop in aeroelastic stability near specific regions of the pressure ratio indicating a strong correlation between blade loading and flutter. Passage throat shock behavior is shown to be a controlling factor near the trailing edge, and like critical reduced frequency, this phenomenon is shown to be highly dependent on the vibratory mode shape.

Three-Dimensional Unsteady Flow for an Oscillating Turbine Blade and the Influence of Tip Leakage

1998 ◽  
Vol 122 (1) ◽  
pp. 93-101 ◽  
Author(s):  
D. L. Bell ◽  
L. He
Keyword(s):  
Unsteady Flow ◽  
Turbine Blade ◽  
Three Dimensional ◽  
Unsteady Aerodynamics ◽  
Tip Clearance ◽  
Computational Results ◽  
Blade Loading ◽  
Tip Leakage ◽  
Unsteady Pressure ◽  
Reduced Frequency

The results of two investigations, concerning the aerodynamic response of a turbine blade oscillating in a three-dimensional bending mode, are presented in this paper. The first is an experimental and computational study, designed to produce detailed three-dimensional test cases for aeroelastic applications and examine the ability of a three-dimensional time-marching Euler method to predict the relevant unsteady aerodynamics. Extensive blade surface unsteady pressure measurements were obtained over a range of reduced frequency from a test facility with clearly defined boundary conditions (Bell and He, 1997, ASME Paper No. 97-GT-105). The test data indicate a significant three-dimensional effect, whereby the amplitude of the unsteady pressure response at different spanwise locations is largely insensitive to the local bending amplitude. The computational results, which are the first to be supported by detailed three-dimensional test data, demonstrate the ability of the inviscid method to capture the three-dimensional behavior exhibited by the experimental measurements and a good level of quantitative agreement is achieved throughout the range of reduced frequency. Additional computational solutions, obtained through application of the strip methodology, reveal inadequacies in the conventional quasi-three-dimensional approach to the prediction of oscillating blade flows. The issue of linearity is also considered, and both experimental and computational results indicate a linear behavior of the unsteady aerodynamics. The second, an experimental investigation, addresses the influence of tip leakage upon the unsteady aerodynamic response of an oscillating turbine blade. Results are provided for three settings of tip clearance. The steady flow measurements show marked increases in the size and strength of the tip leakage vortex for the larger settings of tip clearance and deviations are present in the blade loading toward the tip section. The changes in tip clearance also caused distinct trends in the amplitude of the unsteady pressure at 90 percent span, which are observed to correspond with localized regions where the tip leakage flow had a discernible impact on the steady flow blade loading characteristic. The existence of these trends in the unsteady pressure response warrants further investigation into the influence of tip leakage on the local unsteady flow and aerodynamic damping. [S0889-504X(00)01101-6]

scholarly journals A Design Method to Prevent Low Pressure Turbine Blade Flutter

1998 ◽  
Author(s):  
Josef Panovsky ◽  
Robert E. Kielb
Keyword(s):  
Mode Shape ◽  
Design Method ◽  
Turbine Blades ◽  
Low Pressure ◽  
Stability Parameter ◽  
Influence Coefficient ◽  
Low Pressure Turbine ◽  
Reduced Frequency ◽  
Blade Flutter ◽  
The Stability

A design approach to avoid flutter of low pressure turbine blades in aircraft engines is described. A linearized Euler analysis, previously validated using experimental data, is used for a series of parameter studies. The influence of mode shape and reduced frequency are investigated. Mode shape is identified as the most important contributor to determining the stability of a blade design. A new stability parameter is introduced to gain additional insight into the key contributors to flutter. This stability parameter is derived from the influence coefficient representation of the cascade, and includes only contributions from the reference blade and its immediate neighbors. This has the effect of retaining the most important contributions to aerodynamic damping while filtering out terms of less significance. This parameter is utilized to develop a stability map, which provides the critical reduced frequency as a function of torsion axis location. Rules for preliminary design and procedures for detailed design analysis are defined.

scholarly journals Three Dimensional Unsteady Flow for an Oscillating Turbine Blade and the Influence of Tip Leakage

1998 ◽  
Author(s):  
D. L. Bell ◽  
L. He
Keyword(s):  
Unsteady Flow ◽  
Steady Flow ◽  
Turbine Blade ◽  
Test Data ◽  
Three Dimensional ◽  
Unsteady Aerodynamics ◽  
Blade Loading ◽  
Tip Leakage ◽  
Unsteady Pressure ◽  
Reduced Frequency

The results of two investigations, conducted on the aerodynamic response of a turbine blade oscillating in a three dimensional bending mode, are presented in this paper. The first is an experimental and computational study, designed to produce detailed three dimensional test cases for aeroelastic applications and examine the ability of a 3D time-marching Euler method to predict the relevant unsteady aerodynamics. Extensive blade surface unsteady pressure measurements were obtained for a range of reduced frequency, from a test facility with clearly defined boundary conditions, Bell & He (1997). The test data exhibits a significant three dimensional effect, whereby the amplitude of the unsteady pressure response at different spanwise positions is largely insensitive to the local bending amplitude. The inviscid numerical scheme successfully captured this behaviour, and a good qualitative and quantitative agreement with the test data was achieved for the full range of reduced frequency. In addition, the issue of linearity is addressed and both experimental and numerical tests demonstrate a linear behaviour of the unsteady aerodynamics. The second, an experimental investigation, considers the influence of tip leakage on the unsteady pressure response of an oscillating turbine blade. Results are provided for three tip clearances. The steady flow measurements show marked increases in the size and strength of the tip leakage vortex for the larger tip gaps and deviations in the blade loading towards the tip section. The changes in tip gap also caused distinct trends in the amplitude of the unsteady pressure at 90% span, which were consistent with those observed for steady flow blade loading. It is the authors opinion, that the existence of these trends in unsteady pressure warrants further investigation into the influence of tip leakage upon the local unsteady flow and aerodynamic damping.

A Design Method to Prevent Low Pressure Turbine Blade Flutter

1999 ◽  
Vol 122 (1) ◽  
pp. 89-98 ◽  
Author(s):  
J. Panovsky ◽  
R. E. Kielb
Keyword(s):  
Mode Shape ◽  
Design Method ◽  
Turbine Blades ◽  
Low Pressure ◽  
Stability Parameter ◽  
Influence Coefficient ◽  
Low Pressure Turbine ◽  
Reduced Frequency ◽  
Blade Flutter ◽  
The Stability

A design approach to avoid flutter of low pressure turbine blades in aircraft engines is described. A linearized Euler analysis, previously validated using experimental data, is used for a series of parameter studies. The influence of mode shape and reduced frequency are investigated. Mode shape is identified as the most important contributor to determining the stability of a blade design. A new stability parameter is introduced to gain additional insight into the key contributors to flutter. This stability parameter is derived from the influence coefficient representation of the cascade, and includes only contributions from the reference blade and its immediate neighbors. This has the effect of retaining the most important contributions to aerodynamic damping while filtering out terms of less significance. This parameter is utilized to develop a stability map, which provides the critical reduced frequency as a function of torsion axis location. Rules for preliminary design and procedures for detailed design analysis are defined. [S0742-4795(00)01401-0]

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