Non-Synchronous Vibrations of Turbomachinery Airfoils 600 500 NSV Frequency,!, hz 400 300 200 F.R. Flutter 100 SFV 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Rotor Speed,!, RPM Kenneth C. Hall, Jeffrey P. Thomas, Meredith Spiker & Robert E. Kielb Department of Mechanical Engineering and Materials Science Edmund T. Pratt, Jr. School of Engineering Duke University 9th National Turbine Engine High Cycle Fatigue Conference Pinehurst, North Carolina
Outline Objectives of the present work. Description of non-synchronous vibration (NSV), review. Some preliminary results of a conventional time-marching simulation of NSV. 1. 3D front stage compressor The harmonic balance method a nonlinear eigenvalue formulation. Computational results. 1. 2D vortex shedding. 2. 2D compressor instability. Conclusions and future work.
Objectives of Present Study Objectives: To develop an understanding of the most significant types of NSV, with emphasis on fan & compressor blades & vanes. To develop an efficient computational tool to predict NSV frequencies (campbell diagram) and modal force. To develop a design approach. Existing capability Time domain simulations can capture NSV phenomena, but at a high computational cost. Our approach: Frequency domain (harmonic balance) methods to model nonlinear fluid mechanics instabilities. Novel search techniques to find nonlinear eigenvalues (frequencies) of NSV drivers.
Classical Aeroelastic Phenomena: What is NSV? Forced Response Synchronous with engine order excitations. Flutter Non-Synchronous vibrations at low to moderate reduced frequencies. 600 500 NSV Frequency,!, hz 400 300 200 F.R. Flutter 100 SFV 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Rotor Speed,!, RPM Non-synchronous vibration (NSV) Coherent flow instability. Separated flow vibration (SFV) Broadband flow instability.
Non-Synchronous Vibration Characteristics of NSV: Blades excited by a coherent fluid dynamic instability (e.g. Strouhal shedding). High amplitude response possible, especially when the excitation frequency is near the blade natural frequencies. Blade motion is frequency and phase locked. Flutter design parameters are well within the stable region not flutter. Occurs in blades & vanes of fans, compressors and turbines and can cause high cycle fatigue failures. NSV is missing line on Campbell diagram. Although NSV frequencies are influenced by blade motion, our initial research will emphasize the role of fluid dynamic instabilities only.
Experimental Evidence of NSV Airfoil strain gauge Casing pressure measurement
Fluid Dynamic Instabilities A number of potential phenomena may potentially contribute to NSV, including; dynamic boundary-layer separation, shock/boundary-layer dynamics, vortex shedding, tip flow/vortices, hub vortices, rotating stall, combustion instabilities. Fluid dynamic instabilities are main driver. Blade dynamics play a secondary role, with fluid instability locking on to blade natural frequency.
Time-Marching Simulation of NSV Numerically modeled five passages of C1 compressor using TURBO time marching simulation. TURBO simulation included tip clearance and turbulence model. (Model also included wakes from upstream inlet guide vane) Blades modeled as rigid (no aeroelastic coupling). } {{ } Near Midspan } {{ } Near Tip
C1 Compressor TURBO simulation shows fluid dynamic instability involves tip leakage vortex from one blade interacting with neighboring suction side blade. Unsteady fluid dynamic eigenmode dominated by unsteadiness near the tip. Numerical simulation provided useful insight into physical mechanisms of NSV, but required significant computer resources (turnaround time for one case was months).
Previous Studies for Cascades Mailach et al. (1999, 2000 & 2001) 4 Stage LSRC & Linear Cascade Tip Flow Instability Multi-Cell Circumferentially Traveling Wave Near Stall Line with Large Tip Clearance (> 2%) Strouhal-type Number Proposed Marz et al. (1999) Low Speed Fan Rig Tip Flow Instability Near Stall Line with Large Tip Clearance CFD Frequency Prediction 8% Higher Than That Measured Camp (1999)
Previous Studies for Cascades Inoue et al. (1999) Lenglin & Tan (1999) Vo (2001)
Derivation of Harmonic Balance Euler Equations For the moment, consider two-dimensional Euler equations. U t + F(U) + G(U) = 0 x y where the vector of conservation variables U and the flux vector F are given by ρ ρu ρu ρu U = and F = 2 + p ρv ρuv ρe ρuh For an ideal gas with constant specific heats, the pressure and enthalpy may be expressed in terms of the conservation variables, i.e. h = ρe + p ρ and { p = (γ 1) ρe 1 [(ρu) 2 + (ρv) 2]} 2ρ The flux vector G can be similarly expressed.
Solution of Harmonic Balance Euler Equations In harmonic balance approach, assume unsteady periodic flow may be represented by Fourier series in time, i.e. ρ(x, y, t) = n R n (x, y)e jωnt Harmonic balance equations then take the form Ũ τ + F(Ũ) + G(Ũ) + S(Ũ) = 0 x y If n harmonics are kept in solution, then 2n + 1 coefficients are stored for each flow variable (1 for mean flow, 2n for real and imaginary parts of unsteady harmonics). Note harmonics are coupled via nonlinearities in governing equations.
Simultaneous Dual Time-Step Form of Harmonic Balance Computation of harmonic fluxes difficult and computationally expensive, especially for viscous flows. Alternatively, could store solution at 2n + 1 equally spaced points in time over one temporal period. Ũ = EU U = E 1 Ũ Where the matrices E and E 1 are discrete Fourier transform and inverse Fourier transform operators. Thus, pseudo-time harmonic balance equations become EU τ + EF x + EG y + jωneu = 0 Pre-multiplying by E 1 gives U τ + F x + G y + jωe 1 NE U = 0 }{{} / t
Simultaneous Dual Time-Step Form of Harmonic Balance U τ + F x + G y + S = 0 where S = jω[e] 1 [N][E] U U }{{} t / t Here we use spectral operator to compute time derivative. Using finite difference does not work well. Use of spectral difference operator allows for very coarse temporal discretization. Note that since only steady-state solution is desired, can use local time stepping, multiple-grid acceleration techniques, and residual smoothing to speed convergence. For 2D and 3D cascades, only a single blade passage is required, with complex periodicity conditions along periodic boundaries. Because we work in the frequency domain, essentially exact nonreflecting boundary conditions are available.
Harmonic Balance For NSV problem, frequency of limit cycle oscillation ω is unknown a priori. Must determine frequency as part of the solution procedure. When discretized, HB equations are of the form jωmu + N(U ) = 0 }{{}}{{} Linear Nonlinear This equation may be thought of as a nonlinear eigenvalue problem for the unknown frequency ω and mode shape (including the amplitude) U.
Cylinder in Cross Flow HB Solution Computational time is on the order of a single steady calculation (times about 20).
Cylinder in Cross Flow HB Solution Strouhal Number, St 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 Williamson, 1996 HB Method 40 60 80 100 120 140 160 180 Reynolds Number, Re 9:;*<=1-(/+>/?*.=(:-)/@<>=5/AB!/, & A "6$ "68 "6# "67 "6! "6& @CD/D*.(=/'(E#F "6"!" #" $" %" &"" &!" &#" &$" &%"!"" '()*+,-./0123(45/'( Cylinder shedding occurs at frequency close to experimentally measured frequency. Method predicts both frequency and amplitude of unsteady loading.
2D C1 Compressor Steady Flow Computation!#1! 3,45'+,-/6(7+85*4./4,9 :! ;< " ;N;Q===!#12!21!!212!$1!!$12! "!!! #!!! $!!! %!!! &'()*'+,-./0 Steady computation uses pseudo time marching to obtain converged solution. Unsteady residual evidence of physical periodic unsteadiness.
C1 Compressor 47*(89/:;80*<-=;>/?93*@8A!" B@ # BNBQCCC!6!!"!!6 D*4?,E80;1= 6F4?,E80;1=!#"!"""!#""!$""!%""!&"" #""" '()*+,-./-0123*45 } {{ } Search for zero residual 7-89:*;0<9-:=2*>?@9*AB:<-*C-,*D9-,:9?E03*=-F #6" #6! #6#!#6!!#6" &*4:,GE0?1< $*4:,GE0?1<!#6$!"##!$##!%##!&## '()*+,-./-0123*45 } {{ } Search for zero phase error
Possible Design Strategy for NSV Avoidance 600 500 NSV Frequency,!, hz 400 300 200 F.R. Flutter 100 SFV 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Rotor Speed,!, RPM Compute eigenfrequencies of NSV and plot on Campbell diagram. Where possible, avoid crossings with blade frequencies within operating range. For unavoidable crossings, compute LCO amplitude using harmonic balance technique. Only accept crossings within acceptable HCF limits.
Conclusions NSV is a recurring design problem in modern turbomachinery. Have demonstrated using a time-marching technique the feasibility of predicting NSV in a compressor. Frequency finding HB method has been applied to model twodimensional periodic flow instability problems with success. Phase error search method more reliable and efficient than zero residual search. Currently applying HB technique to 3D flow geometry. Working on methods to reduce time required for iterative search of nonlinear eigenfrequency. HB method is potentially orders of magnitude more efficient than time marching simulation. Eigenfrequencies of fluid alone (uncoupled) provides important information for Campbell diagram based aeromechanical design of rotors.