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Research Papers

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

[+] Author and Article Information
Almudena Vega

School of Aeronautics and Space,
Universidad Politécnica de Madrid,
Plaza Cardenal Cisneros, 3,
Madrid 28040, Spain
e-mail: almudena.vega@upm.es

Roque Corral

Department of Fluid Dynamics and
Aerospace Propulsion,
School of Aeronautics and Space,
Universidad Politecnica de Madrid,
Madrid 28040, Spain;
Advanced Engineering Direction,
Industria de TurboPropulsores S.A.,
Francisca Delgado, 9,
Alcobendas, Madrid 28108, Spain
e-mail: roque.corral@itp.es

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received June 29, 2015; final manuscript received October 6, 2015; published online November 3, 2015. Editor: Kenneth C. Hall.

J. Turbomach 138(2), 021005 (Nov 03, 2015) (9 pages) Paper No: TURBO-15-1130; doi: 10.1115/1.4031777 History: Received June 29, 2015; Revised October 06, 2015

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.

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References

Corral, R. , and Vega, A. , 2015, “ The Low Reduced Frequency Limit of Vibrating Airfoils—Part I: Theoretical Analysis,” ASME J. Turbomach. (in press).
Corral, R. , Escribano, A. , Gisbert, F. , Serrano, A. , and Vasco, C. , 2003, “ Validation of a Linear Multigrid Accelerated Unstructured Navier–Stokes Solver for the Computation of Turbine Blades on Hybrid Grids,” AIAA Paper No. 2003-3326.
Corral, R. , and Gisbert, F. , 2003, “ A Numerical Investigation on the Influence of Lateral Boundaries in Linear Vibrating Cascades,” ASME J. Turbomach., 125(3), pp. 433–441. [CrossRef]
Vega, A. , Corral., R. , Zanker, A. , and Ott, P. , 2014, “ Experimental and Numerical Assessment of the Aeroelastic Stability of Blade Pair Packages,” ASME Paper No. GT2014-25607.
Burgos, M. , Corral, R. , and Contreras, J. , 2011, “ Efficient Edge Based Rotor/Stator Interaction Method,” AIAA J., 41(1), pp. 19–31. [CrossRef]
Jameson, A. , Schmidt, W. , and Turkel, E. , 1981, “ Numerical Solution of the Euler Equations by Finite Volume Techniques Using Runge–Kutta Time Stepping Schemes,” AIAA Paper No. 81-1259.
Roe, P. , 1981, “ Approximate Riemann Solvers, Parameters, Vectors and Difference Schemes,” J. Comput. Phys., 43(2), pp. 357–372. [CrossRef]
Swanson, R. C. , and Turkel, E. , 1992, “ On Central-Difference and Upwinding Schemes,” J. Comput. Phys., 101(2), pp. 292–306. [CrossRef]
Corral, R. , Burgos, M. A. , and Garcia, A. , 2000, “ Influence of the Artificial Dissipation Model on the Propagation of Acoustic and Entropy Waves,” ASME Paper No. 2000-GT-563.
Giles, M. B. , 1990, “ Non-Reflecting Boundary Conditions for Euler Equation Calculations,” AIAA J., 28(12), pp. 2050–2057. [CrossRef]
Vega, A. , and Corral, R. , 2013, “ Physics of Vibrating Airfoils at Low Reduced Frequency,” ASME Paper No. GT2013-94906.
Bölcs, A. , and Fransson, T. H. , 1986, “ Aeroelasticity in Turbomachines: Comparison of Theoretical and Experimental Cascade Results,” Laboratoire de Thermique Appliquee et de Turbomachines, EPFL, Lausanne, Switzerland, Technical Report No. 13.
Fransson, T. H. , and Verdon, J. M. , 1991, “ Updated Report on Standard Configurations for Unsteady Flow Through Vibrating Axial-Flow Turbomachine Cascades,” KTH, Stockholm, Sweden.

Figures

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Fig. 1

Isentropic Mach number distributions along the airfoil chord

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Fig. 2

(a) Real and (b) imaginary components of the unsteady pressure for an asymmetric cascade of flat plates vibrating in bending (Min = 0.45, θ = 45 deg, s/c = 0.5, and k = 0.1)

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Fig. 3

(a) Nondimensional IC modulus and (b) phase as a function of the reduced frequency for an asymmetric cascade of flat plates vibrating in bending (Min = 0.45, θ = 45 deg, and s/c = 0.5)

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Fig. 4

Nondimensional damping as a function of the IBPA and the reduced frequency for an asymmetric cascade of flat plates vibrating in bending (M = 0.45, θ = 45 deg, and s/c = 0.5)

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Fig. 5

(a) Nondimensional IC modulus of the 0th and ±1st airfoils as a function of the reduced frequency and (b) nondimensional damping as a function of the IBPA and the reduced frequency for the NACA65 compressor (Min = 0.7)

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Fig. 6

Phases of the 0th and ± 1 airfoils as a function of the reduced frequency for the NACA65 compressor (Min = 0.7)

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Fig. 7

Real (left) and imaginary (right) part of the unsteady pressure of the NACA65 compressor (k = 0.1)

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Fig. 8

Phase of the ICs of the 0th and + 1st airfoils as a function of the reduced frequency and Mach numbers for an LPT airfoil

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Fig. 9

Variation of the nondimensional IC modulus of the central airfoil with the square of the Mach number and the reduced frequency

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Fig. 10

Nondimensional damping as a function of the IBPA and the reduced frequency for the LPT case (Mexit = 0.75)

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Fig. 11

IC phases of the 0th and ± 1st airfoils as a function of the reduced frequency for the NACA 0012 symmetric (filled symbols) and asymmetric (open symbols) cases

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Fig. 12

Nondimensional IC modulus of the 0th and +1st airfoils as a function of the reduced frequency for the NACA 0012 symmetric (filled symbols) and asymmetric (open symbols) cases

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Fig. 13

Nondimensional damping as a function of the IBPA and the reduced frequency for the NACA 0012 symmetric case

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Fig. 14

Nondimensional damping as a function of the IBPA and the reduced frequency for the NACA 0012 asymmetric case

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Fig. 15

Mean (filled symbols) and minimum (open symbols) damping as a function of the reduced frequency of all the configurations

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Fig. 16

Mean value, W¯cycle-mean (solid symbols) and first harmonic, maxσ Wcycle′(k) (filled symbols) of the nondimensional aerodynamic damping for the flap, edge, and torsion modes of an LPT airfoil as a function of the reduced frequency

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