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.

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 IIIA — Theoretical Quantification and Influence of Unsteady Loading

2016 ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Vibration Mode ◽  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Airfoil Surface ◽  
Navier Stokes Equations ◽  
Reduced Frequency ◽  
Frequency Regime ◽  
Unsteady Loading

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. In Part I of the corresponding paper, a perturbation analysis of the linearized Navier-Stokes equations for real modes at low reduced frequency was presented and some conclusions were drawn. It was discovered that a new parameter, the unsteady loading, plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the oscillation of the airfoil. In this third (a) part, the theory is extended in order to quantify the new parameter. It is shown that this parameter depends solely on the steady flowfield on the airfoil surface and the vibration mode-shape. As a consequence, the effect of changing the design operating conditions or the vibration mode onto the work-per-cycle curves (and therefore in the stability) can be easily predicted and, what is more important, quantified without conducting additional flutter analysis. The relevance of the parameter has been numerically confirmed in the Part IIIb of the paper.

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 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.

Quantification of the Influence of Unsteady Aerodynamic Loading on the Damping Characteristics of Airfoils Oscillating at Low-Reduced Frequency—Part I: Theoretical Support

2016 ◽  
Vol 139 (3) ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Vibration Mode ◽  
Stokes Equations ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Airfoil Surface ◽  
Navier Stokes Equations ◽  
Aerodynamic Loading ◽  
Unsteady Aerodynamic ◽  
Reduced Frequency ◽  
Frequency Regime

The effect of the unsteady aerodynamic loading of oscillating airfoils in the low-reduced frequency regime on the work per cycle curves is investigated. The theoretical analysis is based on a perturbation analysis of the linearized Navier–Stokes equations for real modes at low-reduced frequency. It was discovered that a new parameter, the unsteady loading, plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the airfoil oscillation. Here, the theory is extended in order to quantify this new parameter. It is shown that this parameter depends solely on the steady flow-field on the airfoil surface and the vibration mode-shape. As a consequence, the effect of changing the design operating conditions or the vibration mode onto the work-per-cycle curves (and therefore in the stability) can be easily predicted and, what is more important, quantified without conducting additional flutter analysis. The relevance of the parameter has been numerically confirmed in the Part II of the paper.

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 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.

scholarly journals Physics of Vibrating Airfoils at Low Reduced Frequency

2013 ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Mach Number ◽  
Acoustic Waves ◽  
Unsteady Aerodynamics ◽  
Low Pressure ◽  
Travelling Wave ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Reduced Frequency ◽  
Influence Coefficients ◽  
Key Aspects

The unsteady aerodynamics of low pressure turbine vibrating airfoils in flap mode is studied in detail using a frequency domain linearized Navier-Stokes solver. Both the travelling-wave and influence coefficient formulations of the problem are used to highlight key aspects of the physics and understand different trends such as the effect of reduced frequency and Mach number. The study is focused in the low-reduced frequency regime which is of paramount relevance for the design of aeronautical low-pressure turbines and compressors. It is concluded that the effect of the Mach number on the unsteady pressure phase can be neglected in first approximation and that the unsteadiness of the vibrating and adjacent airfoils is driven by vortex shedding mechanisms. Finally a simple model to estimate the work-per-cycle as a function of the reduced frequency and Mach Number is provided. The edge-wise and torsion modes are presented in less detail but it is shown that acoustic waves are essential to explain its behaviour. The non-dimensional work-per-cycle of the edge-wise mode shows a large dependence with the Mach number while in the torsion mode a large number of airfoils is needed to reconstruct the work-per-cycle departing from the influence coefficients.

Flutter of Low Pressure Turbine Blades With Cyclic Symmetric Modes: A Preliminary Design Method

2004 ◽  
Vol 126 (2) ◽  
pp. 306-309 ◽  
Author(s):  
Robert Kielb ◽  
Jack Barter ◽  
Olga Chernycheva ◽  
Torsten Fransson
Keyword(s):  
Mode Shape ◽  
Design Method ◽  
Turbine Blades ◽  
Current Method ◽  
Low Pressure ◽  
Mode Shapes ◽  
Preliminary Design ◽  
Complex Mode ◽  
Low Pressure Turbine ◽  
Reduced Frequency

A current preliminary design method for flutter of low pressure turbine blades and vanes only requires knowledge of the reduced frequency and mode shape (real). However, many low pressure turbine (LPT) blade designs include a tip shroud that mechanically connects the blades together in a structure exhibiting cyclic symmetry. A proper vibration analysis produces a frequency and complex mode shape that represents two real modes phase shifted by 90 deg. This paper describes an extension to the current design method to consider these complex mode shapes. As in the current method, baseline unsteady aerodynamic analyses must be performed for the three fundamental motions, two translations and a rotation. Unlike the current method work matrices must be saved for a range of reduced frequencies and interblade phase angles. These work matrices are used to generate the total work for the complex mode shape. Since it still only requires knowledge of the reduced frequency and mode shape (complex), this new method is still very quick and easy to use. Theory and an example application are presented.

scholarly journals Calculation and Correlation of the Unsteady Flowfield in a High Pressure Turbine

2002 ◽  
Author(s):  
Milind A. Bakhle ◽  
Jong S. Liu ◽  
Josef Panovsky ◽  
Theo G. Keith ◽  
Oral Mehmed
Keyword(s):  
High Pressure ◽  
Three Dimensional ◽  
Unsteady Aerodynamics ◽  
Forced Vibrations ◽  
Forced Response ◽  
Operating Conditions ◽  
Test Facility ◽  
Navier Stokes ◽  
Turbine Stage ◽  
High Pressure Turbine

Forced vibrations in turbomachinery components can cause blades to crack or fail due to high-cycle fatigue. Such forced response problems will become more pronounced in newer engines with higher pressure ratios and smaller axial gap between blade rows. An accurate numerical prediction of the unsteady aerodynamics phenomena that cause resonant forced vibrations is increasingly important to designers. Validation of the computational fluid dynamics (CFD) codes used to model the unsteady aerodynamic excitations is necessary before these codes can be used with confidence. Recently published benchmark data, including unsteady pressures and vibratory strains, for a high-pressure turbine stage makes such code validation possible. In the present work, a three dimensional, unsteady, multi blade-row, Reynolds-Averaged Navier Stokes code is applied to a turbine stage that was recently tested in a short duration test facility. Two configurations with three operating conditions corresponding to modes 2, 3, and 4 crossings on the Campbell diagram are analyzed. Unsteady pressures on the rotor surface are compared with data.

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