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

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.

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.

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

SFV

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• 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 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.

Experimental Evidence of NSV

Airfoil strain gauge Casing pressure measurement

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.

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

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).

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)

∂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.

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.

“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

“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.

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∗.

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

40 60 80 100 120 140 160 180

Reynolds Number, Re

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Str

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be

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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.

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.

C1 Compressor

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Search for zero residual

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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.

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.