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Non-Synchronous Vibrations of Turbomachinery Airfoils
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NSV
Flutter
SFV
F.R.
Kenneth C. Hall, Jeffrey P. Thomas, Meredith Spiker & Robert
E. KielbDepartment of Mechanical Engineering and Materials
Science
Edmund T. Pratt, Jr. School of EngineeringDuke University
9th National Turbine Engine High Cycle Fatigue
ConferencePinehurst, North Carolina
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Outline
• Objectives of the present work.
• Description of non-synchronous vibration (NSV), review.
• Some preliminary results of a conventional time-marching
simula-tion of NSV.
1. 3D front stage compressor
• The harmonic balance method – a nonlinear eigenvalue
formula-tion.
• Computational results.
1. 2D vortex shedding.
2. 2D compressor instability.
• Conclusions and future work.
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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
frequen-cies (campbell diagram) and modal force.
• To develop a design approach.
Existing capability
• Time domain simulations can capture NSV phenomena, but at
ahigh computational cost.
Our approach:
• Frequency domain (harmonic balance) methods to model
nonlinearfluid mechanics instabilities.
• Novel search techniques to find nonlinear eigenvalues
(frequencies)of NSV drivers.
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What is NSV?
Classical Aeroelastic Phenomena:
• Forced Response – Synchronous with engine order
excitations.
• Flutter – Non-Synchronous vibrations at low to moderate
reducedfrequencies.
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NSV
Flutter
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• Non-synchronous vibration (NSV) – Coherent flow
instability.
• Separated flow vibration (SFV) – Broadband flow
instability.
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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
excitationfrequency is near the blade natural frequencies.
• Blade motion is frequency and phase locked.
• Flutter design parameters are well within the stable region –
notflutter.
• Occurs in blades & vanes of fans, compressors and turbines
andcan cause high cycle fatigue failures.
NSV is “missing line” on Campbell diagram. Although NSV
frequenciesare influenced by blade motion, our initial research
will emphasize therole of fluid dynamic instabilities only.
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Experimental Evidence of NSV
Airfoil strain gauge Casing pressure measurement
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Fluid Dynamic Instabilities
• A number of potential phenomena may potentially con-tribute 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
“lockingon” to blade natural frequency.
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Time-Marching Simulation of NSV
• Numerically modeled five passages of C1 compressor using
TURBOtime 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
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C1 Compressor
• TURBO simulation shows fluid dynamic instability involves
tipleakage vortex from one blade interacting with neighboring
suc-tion side blade.
• Unsteady fluid dynamic “eigenmode” dominated by
unsteadinessnear the tip.
• Numerical simulation provided useful insight into physical
mech-anisms of NSV, but required significant computer
resources(turnaround time for one case was months).
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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)
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Previous Studies for Cascades
• Inoue et al. (1999)
• Lenglin & Tan (1999)
• Vo (2001)
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Derivation of Harmonic Balance Euler Equations
For the moment, consider two-dimensional Euler equations.
∂U
∂t+
∂F(U)
∂x+
∂G(U)
∂y= 0
where the vector of conservation variables U and the flux vector
F aregiven by
U =
ρρuρvρe
and F =
ρu
ρu2 + pρuvρuh
For an ideal gas with constant specific heats, the pressure and
enthalpymay be expressed in terms of the conservation variables,
i.e.
h =ρe + p
ρ
and
p = (γ − 1)
{
ρe −1
2ρ
[
(ρu)2 + (ρv)2]}
The flux vector G can be similarly expressed.
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Solution of Harmonic Balance Euler Equations
In harmonic balance approach, assume unsteady periodic flow may
berepresented by Fourier series in time, i.e.
ρ(x, y, t) =∑
n
Rn(x, y)ejωnt
Harmonic balance equations then take the form
∂Ũ
∂τ+
∂F̃(Ũ)
∂x+
∂G̃(Ũ)
∂y+ S̃(Ũ) = 0
If n harmonics are kept in solution, then 2n + 1 coefficients
are storedfor each flow variable (1 for mean flow, 2n for real and
imaginary partsof unsteady harmonics).
Note harmonics are coupled via nonlinearities in governing
equations.
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“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 + 1equally spaced points in time over one temporal
period.
Ũ = EU∗
U∗ = E−1Ũ
Where the matrices E and E−1 are discrete Fourier transform and
in-verse Fourier transform operators. Thus, pseudo-time harmonic
balanceequations become
∂EU∗
∂τ+
∂EF∗
∂x+
∂EG∗
∂y+ jωNEU∗ = 0
Pre-multiplying by E−1 gives
∂U∗
∂τ+
∂F∗
∂x+
∂G∗
∂y+ jωE−1NE
︸ ︷︷ ︸
∂/∂t
U∗ = 0
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“Simultaneous Dual Time-Step” Form of Harmonic Balance
∂U∗
∂τ+
∂F∗
∂x+
∂G∗
∂y+ S∗ = 0
where
S∗ = jω[E]−1[N][E]
︸ ︷︷ ︸
∂/∂t
U∗≈
∂U∗
∂t
Here we use spectral operator to compute time derivative. Using
finitedifference does not work well. Use of spectral difference
operator allowsfor very coarse temporal discretization.
• Note that since only “steady-state” solution is desired, can
use lo-cal time stepping, multiple-grid acceleration techniques,
and resid-ual 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
non-reflecting boundary conditions are available.
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Harmonic Balance
• For NSV problem, frequency of limit cycle oscillation ω is
unknowna priori. Must determine frequency as part of the solution
proce-dure.
• When discretized, HB equations are of the form
jωMU∗︸ ︷︷ ︸
Linear
+ N(U∗)︸ ︷︷ ︸
Nonlinear
= 0
• This equation may be thought of as a nonlinear eigenvalue
prob-lem for the unknown frequency ω and mode shape (including
theamplitude) U∗.
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Cylinder in Cross Flow – HB Solution
• Computational time is on the order of a single steady
calculation(times about 20).
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Cylinder in Cross Flow – HB Solution
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Reynolds Number, Re
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Str
ou
ha
l N
um
be
r, S
t
Williamson, 1996
HB Method
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• Cylinder shedding occurs at frequency close to experimentally
mea-sured frequency.
• Method predicts both frequency and amplitude of unsteady
load-ing.
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2D C1 Compressor – “Steady” Flow Computation
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• Steady computation uses pseudo time marching to obtain
con-verged solution.
• Unsteady residual evidence of physical periodic
unsteadiness.
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C1 Compressor
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Search for zero residual
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Search for zero phase error
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Possible Design Strategy for NSV Avoidance
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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Rotor Speed, !, RPM
Freq
uen
cy, !
, h
z
NSV
Flutter
SFV
F.R.
• Compute eigenfrequencies of NSV and plot on Campbell
diagram.
• Where possible, avoid crossings with blade frequencies within
op-erating range.
• For unavoidable crossings, compute LCO amplitude using
harmonicbalance technique. Only accept crossings within acceptable
HCFlimits.
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Conclusions
• NSV is a recurring design problem in modern
turbomachinery.
• Have demonstrated using a time-marching technique the
feasibilityof predicting NSV in a compressor.
• Frequency finding HB method has been applied to model
two-dimensional periodic flow instability problems with
success.
– Phase error search method more reliable and efficient thanzero
residual search.
– Currently applying HB technique to 3D flow geometry.
– Working on methods to reduce time required for iterativesearch
of nonlinear eigenfrequency.
– HB method is potentially orders of magnitude more
efficientthan time marching simulation.
• Eigenfrequencies of fluid alone (uncoupled) provides important
in-formation for Campbell diagram based aeromechanical design
ofrotors.