Frequency Domain Techniques for Complexand Nonlinear Flows in Turbomachinery

Kenneth C. Hall∗, Jeffrey P. Thomas†,Kivanc Ekici‡ and Dmytro M. Voytovych§

Duke University, Durham, NC 27708–0300

In this paper, we review frequency domain techniques for computing unsteady flowsin turbomachinery. The frequency domain approaches, which can be used to model bothsmall-disturbance and nonlinear time periodic flows, are computationally very efficient,typically one to two orders of magnitude more efficient than time domain techniques.Furthermore, they can be used to analyze a wide variety of aeromechanic, performance,unsteady heat transfer, and flow stability problems in turbomachinery, including theimportant problem of unsteady flows in multistage machines.

1 IntroductionIn this paper, we review the state of the field of com-

putational unsteady aerodynamics, particularly fre-quency domain methods used for the calculation ofthe unsteady aerodynamic forces arising in turboma-chinery aeroelasticity problems. While the emphasishere is on turbomachinery aerodynamics, the methodsdescribed in this paper have analogues for the analysisof isolated airfoils, wings, and even whole aircraft.

The two main aeromechanics problems in turboma-chinery are flutter and gust response. In the flutterproblem, the unsteady aerodynamic loads acting on acascade of turbomachinery airfoils arise from the mo-tion of the airfoils themselves. In the gust responseproblem, the original excitation arises away from theblade row in question. Typical sources of excitationare wakes or potential fields from neighboring bladerows or support structures (struts), inlet distortions,and hot streaks from the combustor.

More recently, a third class of aeromechanical prob-lems has been identified.1 This class of problems isakin to galloping of power lines or buffeting of air-craft wings. Examples include so-called separated flowvibrations and non-synchronous vibrations. In sepa-rated flow vibrations, the flow over a row of airfoilsis separated, or nearly so. The flow itself is un-stable, producing unsteady air loads with broadbandfrequency distributions that excite the airfoil at allfrequencies producing a large response at the naturalstructural frequencies of the airfoil. Non-synchronous

∗Professor and Chair, Department of Mechanical Engineeringand Materials Science, Associate Fellow AIAA.

†Research Assistant Professor, Department of MechanicalEngineering and Materials Science, Senior Member AIAA.

‡Research Associate, Department of Mechanical Engineeringand Materials Science, Member AIAA.

§Graduate Research Assistant, Department of MechanicalEngineering and Materials Science, Member AIAA.

Copyright c© 2003 by Kenneth C. Hall, Jeffrey P. Thomas, Ki-vanc Ekici, Dmytro Voytovych. Published by the American Instituteof Aeronautics and Astronautics, Inc. with permission.

flow vibrations, on the other hand, are similar to sep-arated flow vibrations in that the source of the exci-tation is thought to be a fluid dynamics instability(rather than an aeroelastic instability), except thatnon-synchronous vibrations can also occur well awayfrom a stalled condition and the response tends to beat a single frequency. If this fluid dynamic naturalfrequency happens to be close to a structural naturalfrequency, then the fluid dynamic instability frequencycan “lock on” to the structural frequency of the airfoilsresulting in a large-amplitude vibration.

Most unsteady flows of interest in turbomachineryaeromechanics are periodic in time, and for many ofthese flows, the unsteadiness in the flow is also rela-tively small. Thus, the flow may be decomposed intotwo parts: a nonlinear steady or mean flow, plus adynamically linear small perturbation flow. In gen-eral, the steady flow is described by a set of nonlin-ear partial differential equation, whereas the unsteadysmall-perturbation flow is described by a set of linearvariable coefficient partial differential equations thatare hyperbolic in time. Because the unsteady per-turbation flow is periodic in time, we may, withoutloss of generality, represent the unsteady flow as aFourier series in time with coefficients that vary spa-tially. Each Fourier coefficient is described by a set ofpartial differential equations in which time does notappear explicitly. Such equations are called “time-linearized” equations, and their solution is the one ofthe main topics of this paper.

Working in the frequency domain rather than thetime domain has a number of distinct advantages.The available frequency solution techniques tend to bemuch more computationally efficient than the equiv-alent time domain techniques. Additionally, certainparts of the unsteady flow problem – for instance, thedescription of the behavior of acoustic, vortical, andentropic waves – are simplified in the frequency do-main.

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American Institute of Aeronautics and Astronautics Paper 2003-3998

33rd AIAA Fluid Dynamics Conference and Exhibit23-26 June 2003, Orlando, Florida

AIAA 2003-3998

Copyright © 2003 by Kenneth C. Hall, Jeffrey P. Thomas, Kivanc Ekici, and Dmytro M. Voytovych. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

In the case where the unsteadiness in the flow islarge, the time-linearized equations are not valid. For-tunately, however, the unsteady flow may still becalculated in the frequency domain using a recentlydeveloped harmonic balance technique.

Finally, although the emphasis in this paper is un-steady aerodynamics associated with aeromechanicalproblems, frequency domain techniques can be usedto solve of wide variety of problems, including perfor-mance, unsteady heat transfer, aeroacoustic, and flowstability problems.

2 Linearized Analysis of UnsteadyFlows

Recently, a number of investigators have developedlinearized Euler and Navier–Stokes solvers to predictunsteady flows in turbomachinery. To motivate thesemethods, consider the two-dimensional Euler equa-tions, given by

∂U∂t

+∂F∂x

+∂G∂y

= 0 (1)

where U is the vector of conservation variables, and Fand G are so-called flux vectors. These are given by

U =

ρρuρve

, F =

ρu

ρu2 + pρuvρuh0

, G =

ρvρuv

ρv2 + pρvh0

(2)

where ρ is the density, u and v are the x and y compo-nents of velocity, p is the static pressure, e is the totalinternal energy, and h0 is the total enthalpy.

The flow is assumed to be composed of a mean flowplus an unsteady perturbation flow that is harmonicin time, so that

U(x, y, t) = U(x, y) + u(x, y)ejωt (3)

where the perturbation amplitude u(x, y) is small com-pared to the mean flow U(x, y), and ω is the frequencyof the unsteadiness. This assumption is substitutedinto the Euler equations, Eq. (1), and the result isexpressed as a perturbation series in the small param-eter. The collected zeroth-order terms result in thesteady Euler equations, which are solved using nowconventional computational fluid dynamic techniques.The collected first-order terms give the linearized Eu-ler equations, i.e.,

jωu +∂

∂x

(∂F∂U

u)

+∂

∂y

(∂G∂U

u)

= 0 (4)

where ∂F/∂U and ∂G/∂U are steady flow Jacobians.These equations are solved subject to appropriate so-called “upwash” boundary conditions at the airfoil

surface, far-field boundary conditions, and periodic-ity conditions that allow the computational domain tobe reduced to a single blade passage.

One of the earliest linearized Euler solvers was de-veloped by Ni and Sisto.2 So that they could make useof more traditional time marching CFD algorithms, Niand Sisto introduced a pseudo-time term into the lin-earized Euler equations, i.e.

∂u∂τ

+ jωu +∂

∂x

(∂F∂U

u)

+∂

∂y

(∂G∂U

u)

= 0 (5)

where now u = u(x, y, τ). This additional term makesthe equations hyperbolic in pseudo time, and hencethey may be marched in time using traditional CFDtechniques. As time advances, the solution reachesa steady state and the additional term goes to zero.Thus the solution to the original linearized Euler equa-tions [Eq. (4)] is obtained. The results of this earlyanalysis, however, were limited to flat plate cascadeswith homentropic flows.

Hall and Crawley3 discretized the linearized Eulerequations using a finite volume scheme, and assembledthe resulting equations into a large sparse set of matrixequations which were then solved using LU decompo-sition. Shock fitting and shock capturing were usedto explicitly model the motion of wakes and shocks.However, due to the complexity of the shock fittingalgorithm, only a few model transonic flows problemswere computed. Nevertheless, this work demonstratedthe feasibility of using the linearized Euler equationsto model transonic flows, at least for cases where themotion of the shock is not too large.

In recent years, a number of investigators4–8 havecontinued to develop the linearized Euler techniqueusing pseudo time marching. The time-linearized tech-nique has been extended to three-dimensional inviscid(Euler) flows9,10 and to two and three-dimensional vis-cous (Navier–Stokes) flows11–13 in turbomachinery.

The time-linearized frequency-domain approach hasa number of distinct advantages over conventionaltime-domain approach. By using pseudo time march-ing, conventional steady CFD techniques may be usedto solve the time-linearized equations. Thus, accel-eration techniques such as local time stepping andmultiple grid acceleration techniques may be used toaccelerate convergence. Furthermore, by using com-plex periodicity conditions, the computational domaincan be reduced to a single blade passage. The result isthat time-linearized solutions can be obtained in com-putational times comparable to the time required toobtain a steady flow solution.

One feature of most linearized Euler and Navier-Stokes solvers is the use of harmonically deformingcomputational meshes.7 The use of the deforminggrid eliminates the need for a troublesome extrapola-tion term from the upwash boundary condition appliedat the airfoil surface, but also produces an inhomoge-

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American Institute of Aeronautics and Astronautics Paper 2003-3998

Fig. 1 Trajectory of shock in a channel or on airfoilsurface and corresponding effect on unsteady flowquantities.

neous term in the linearized Euler equation, Eq. (4),that depends on the mean flow and the prescribed gridmotion.

One important aspect of a linearized Euler analy-sis is how shocks are treated. Two approaches havebeen used. Hall and Crawley3 used shock fitting tomodel shocks within a linearized framework. Shockfitting, while providing explicit descriptions of theshock motion, is somewhat difficult to implement forgeneral cascade flows which may have rather compli-cated shock features. For this reason, shock captur-ing is favored. However, until recently it was notknown whether shock capturing correctly predicts theunsteady aerodynamic lads induced by the unsteadyshock motion.

The situation is shown graphically in Fig. 1. Shownare the mean and unsteady shock trajectories on thesurface of an airfoil or channel. Also shown are typ-ical unsteady flow quantities, e.g., pressure. In alinearized analysis, the motion of the shock appearsas an impulse in pressure. At this impulse, the pres-sure is not small (in fact it is of the same order as themean flow pressure), and therefore, one might con-clude that the linearization would break down at this

0.0 0.2 0.4 0.6 0.8 1.0

1.0

0.5

0.0

-0.5

-1.0

-1.5

Distance Along Chord, /c

Stea

dy C

oeffi

cient

of P

ress

ure,

Cp

x

Suction Surface

Pressure Surface

MethodSymbolPresent Steady EulerNonlinear Euler (Huff)

Fig. 2 Steady surface pressure for the Tenth Stan-dard Configuration transonic compressor cascade.

point. Lindquist and Giles14 argued that to obtainthe proper shock impulse, one must faithfully linearizea conservative discretization of the Euler equations.Hall et al.8 demonstrated that the correct shock im-pulse is produced if a conservative discretization of theEuler equations is used, and further, that the shock issmoothed so that it is smeared over several grid points.

A typical transonic analysis is shown in Figs. 2and 3. Shown in Fig. 2 is the computed steady dis-tribution for the Tenth Standard Configuration,15,16one of a series of Standard Configurations used to val-idate unsteady aerodynamic models. Shown in Fig. 3is the computed unsteady pressure distribution on theairfoils of this cascade for the case where the airfoilsvibrate in plunge with a reduced frequency ω of 1.287with an interblade phase angle σ of -90 deg. Shownare two solutions, one computed with a time-linearizedflow solver, and one computed using a conventionaltime-domain flow solver.17 Note the very good agree-ment between the two solutions everywhere except atthe shock impulse. However, even though the details ofthe shock impulse differ between the two solutions, theintegrated force is very nearly equal. The integratedunsteady lift differs by just 2 percent.

Although we have only considered two-dimensionalinviscid flows, the time-linearized approach can be ap-plied equally well to viscous and/or three-dimensionalflows. As an example, consider the case of the frontstage of a modern front stage transonic compressorrotor shown in Figure 4. Shown are the blades ofthe rotor together with the computed streamlines as-sociated with the steady flow at a particular operat-ing condition. These results were computed using asteady Navier-Stokes flow solver that uses the Spalart-Allmaras18 turbulence model to predict turbulent vis-cosity. Note the clearly visible tip vortex due to leak-age flow through the tip clearance.

Next, the rotor blades are assumed to vibrate in

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American Institute of Aeronautics and Astronautics Paper 2003-3998

0.0 0.2 0.4 0.6 0.8 1.0

-10

010

20

30

Real U

nst

eady

Pre

ssure

, R

e [P

]

Suction Surface

Pressure Surface

MethodSymbol

Linearized Euler

Nonlinear Euler (Huff)

0.0 0.2 0.4 0.6 0.8 1.0

-20

-10

010

20

Scheme Integrated Lift

Lin. Euler (193x49)

Huff (121x41)

3.1627 , -86.7 deg

3.2343 , -83.8 degImagin

ary

Unst

eady

Pre

ssure

, Im

[P

]

Suction Surface

Pressure Surface

Fig. 3 Real and imaginary surface pressure for theTenth Standard Configuration cascade vibrating inplunge with a reduced frequency ω of 1.287 withan interblade phase angle σ of -90 deg.

Fig. 4 Computed steady streamline patterns fora modern front stage compressor rotor.

0.0 0.2 0.4 0.6 0.8 1.0

Surface Location, x/c

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

Re

al P

art

Un

ste

ad

y P

ress

ure

, R

e(p−

1)/

pre

f

0.0 0.2 0.4 0.6 0.8 1.0

Surface Location, x/c

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

Ima

g P

art

Un

ste

ad

y P

ress

ure

, Im

(p− 1)/

pre

f

Fig. 5 Unsteady pressure distribution near themidspan station of a front stage compressor rotor.The airfoil vibrates in its first bending mode withan interblade phase angle of zero.

their first bending mode (and at the natural frequencyof that mode) with zero interblade phase angle. Fig-ure 5 shows the computed steady and unsteady pres-sure distributions for for this compressor rotor nearthe midspan of the blade computed using the time-linearized technique. A shock impulse is seen in theunsteady pressure distribution on the surface of theairfoil at about 35 percent of the chord.

3 Analysis of Unsteady Flows inMultistage Machines

Unsteady fluid motion is essential to gas turbine en-gine operation. Only through unsteady flow processescan a machine do work on a fluid to increase its totalenthalpy. This unsteadiness is provided in compressorsand turbines by relative motion of adjacent stators androtors. Unfortunately, this motion also produces un-desirable aeroacoustic and aeroelastic phenomena, i.e.tonal noise and forced blade vibrations induced by ro-tor/stator interactions. Furthermore, the aeroelastic(flutter) stability of a rotor can be profoundly influ-

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American Institute of Aeronautics and Astronautics Paper 2003-3998

enced by the presence of nearby stators and rotors.Most current unsteady aerodynamic theories model

a single blade row in an infinitely long duct, ignor-ing potentially important multistage effects. However,unsteady flows are made up of acoustic, vortical, andentropic waves. These waves provide a mechanism forthe rotors and stators of multistage machines to com-municate with one another. For example, consider thecase of a row of vibrating rotor blades (see Fig. 6).The blades will respond aerodynamically, producingacoustic, vortical, and entropic waves which propa-gate away from the rotor. Some of these waves willthen impinge on the neighboring stators. The statorswill in turn respond aerodynamically, again producingwaves, some of which will impinge upon the originalrotor, and so on. In other words, wave behavior makesunsteady flow in turbomachinery fundamentally a mul-tistage phenomenon.

The basic time-linearized Euler and Navier-Stokesapproach can be extended to the case of multistageflows.19 For a given multistage fan or compressor, onefirst generates a computational mesh for each bladerow. Unlike time-domain multistage analyses, thecomputational mesh for each blade row need only in-clude a single passage.

Having generated a computational mesh, the steadyflow is computed using the steady Euler or Navier-Stokes equations and conventional computational fluiddynamic (CFD) techniques, with so-called “mixingplanes” (the inter-row computational boundaries ofthe computational grid) used to couple together thesolutions computed in the individual blade rows. Thesolution in each blade row is computed independently,except that at each time step in the pseudo timemarching, the circumferentially averaged flow at theexit of one blade row is made to match the circum-ferentially averaged flow at the inlet of the next bladerow.

For the unsteady flow solution, the process is sim-ilar. However, instead of a single solution in eachpassage, several linearized unsteady flow calculationsare performed simultaneously in each blade row, onecorresponding to each spinning mode retained in themodel. Each spinning mode is identified by a set ofintegers that describe the scattering process that cre-ates the mode (n and k in Fig. 7). These integers,along with the freqquency and interblade phase angleof the initial disturbance and the blade counts in eachblade row, determine the interblade phase angle andfrequency of the mode. The only coupling among thevarious spinning modes is at the inter-row boundaries.

The advantages of this approach are several. First,unlike time-domain techniques where many blade pas-sages are required to model each blade row, only asingle passage is required. Second, as in the singleblade row computations, a pseudo time marching tech-nique is used to march the solution to steady state,

and acceleration techniques such as multigrid may beused. Finally, as a practical matter, only a handfulof spinning modes are usually need to obtain accu-rate solutions. These three factors mean that mem-ory requirements and computational time required tocompute the unsteady multistage solution are greatlyreduced compared to time domain solvers. A typicalunsteady multistage flow calculation might take on theorder of just ten times the computational time requiredfor a single steady blade row flow computation.

As an example, consider the case of the front stage ofa modern compressor. This three-dimensional configu-ration consists of three blade rows (IGV/rotor/stator).Figure 7 shows the static pressure distribution nearthe midspan location computed using a steady Eulermultistage flow solver. Note, that the contours at theinter-row boundaries are not continuous since the onlythe circumferential averages of the flows in each bladerow match at the boundaries.

Next, the middle blade row (rotor) is assumed to vi-brate in its first bending mode and frequency. Figure 8shows the real and the imaginary parts of the unsteadypressure on the rotor row for an interblade phase angleof -30. deg. The multistage solution was computedusing one and eight spinning modes in the couplingprocedure. One can see that there is significant dif-ference between the isolated and coupled (multistage)computations.

The unsteady pressure distribution can be inte-grated to obtain the unsteady modal force acting onthe rotor blades due to their vibration. Figure 9 showsthe real and imaginary parts of the computed gen-eralized forces on the rotor blades as a function ofinterblade phase angle. Note the significant differencebetween the forces computed using a single blade row(the rotor) and three blade rows (the rotor and theupstream IGV and downstream stator). Clearly, mul-tistage effects are important. However, the generalizedforce computed using just one spinning mode is in verygood agreement with the force computed using eightspinning modes, indicating the dominant coupling isin the fundamental mode.

These results, and earlier two-dimensional resultsproduced by Hall and Silkowski,20–22 confirm that theaerodynamic damping of a blade row that is part of amultistage machine can be significantly different thanthat predicted using an isolated blade row model. Thisis an important result since most unsteady aerody-namic theories currently used in industry assume thatthe blade row can be modeled as isolated in an in-finitely long duct. However, a good estimate of theaerodynamic damping can be obtained using just afew spinning modes in the model. In fact, most of theunsteady aerodynamic coupling between blade rowsoccurs in the fundamental spinning mode, that is,the spinning mode associated with the original distur-bance. Scattered modes are relatively less important.

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American Institute of Aeronautics and Astronautics Paper 2003-3998

1

4

3 2

N = k0 + kB2N = k0 + kB2

N = k0 + nB1 + kB2 N = k0 + nB1 + kB2

w ' = w0 + (k0 + kB2 )W

w ' = w0 + (k0 + kB2 )W w = w0

w = w0

N = k0

STATOR ROTOR

w = w0 - nB1W

5WR

x

q¢q

¢x

Fig. 6 Kinematics of mode scattering and frequency shifting in a multistage machine.

Fig. 7 Pressure contours at the midspan of a front stage compressor.

Although not shown here, Hall and Silkowski20–22have shown that the two neighboring stator blade rowsadjacent to a rotor have the strongest influence on theunsteady aerodynamic response of the rotor. The nextnearest blade rows are less important, but can have amodest influence. As a practical matter, this meansthat only three blade rows need to be included in mostunsteady flow computations.

4 The Harmonic Balance Method forNonlinear Unsteady Aerodynamics

A number of investigators have developed frequencydomain analyses of nonlinear unsteady flows.23–31While the methods differ somewhat in detail, most canbe viewed as a form of harmonic balance. To motivatethe development of the harmonic balance analysis, andfor simplicity, we assume for the moment that the flowin a blade row is two-dimensional, inviscid, and non-heat-conducting, with constant specific heats. Thus,the flow may be modeled by the two-dimensional Eulerequations, Eq. (1).

In this paper, we consider unsteady flows that aretemporally and spatially periodic. In particular, tem-

poral periodicity requires that

U(x, y, t) = U(x, y, t + T ) (6)

where T is the temporal period of the unsteadiness.Similarly, for cascade flow problems arising from vibra-tion of the airfoils with fixed interblade phase anglesσ, or incident gusts that are spatially periodic, spatialperiodicity requires that

U(x, y + G, t) = U(x, y, t + ∆T ) (7)

where G is the blade-to-blade gap, and ∆T is the timelag associated with the interblade phase lag. As anexample, consider a cascade of airfoils where the sourceof aerodynamic excitation is blade vibration with aprescribed interblade phase angle σ and frequency ω.Then T = 2π/ω and ∆T = σ/ω.

Because the flow is temporally periodic, the flowvariables may be represented as a Fourier series in timewith spatially varying coefficients. For example, theconservation variables may be expressed as

ρ(x, y, t) =∑

n

Rn(x, y)ejωnt

ρu(x, y, t) =∑

n

Un(x, y)ejωnt

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American Institute of Aeronautics and Astronautics Paper 2003-3998

0.0 0.2 0.4 0.6 0.8 1.0Surface Location, x/c

-3.0

-2.0

-1.0

0.0

1.0

2.0

Real

Par

t Uns

tead

y Pr

essu

re R

e(p 1)/p

ref

IsolatedCoupled (1 Mode)Coupled (8 Modes)

0.0 0.2 0.4 0.6 0.8 1.0Surface Location, x/c

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

Imag

Par

t Uns

tead

y Pr

essu

re Im

(p1)/p

ref

IsolatedCoupled (1 Mode)Coupled (8 Modes)

Fig. 8 Unsteady pressure distribution on the sur-face of rotor blades vibrating in first bending withσ = −30.o

-180.0 -90.0 0.0 90.0 180.0Interblade Phase Angle

-1.5

-1.0

-0.5

0.0

0.5

Gen

eral

ized

Forc

e

IsolatedCoupled (1 Mode)Coupled (8 Modes)Real

Imaginary

Fig. 9 Generalized force acting on rotor bladesvibrating in first bending.

ρv(x, y, t) =∑

n

Vn(x, y)ejωnt

ρe(x, y, t) =∑

n

En(x, y)ejωnt (8)

where in principle the summations are taken over allinteger values of n. In practice, these series are trun-cated to a finite number of terms, −N ≤ n ≤ +N .

Next, at least conceptually, we substitute the seriesexpansions for ρ, ρu, ρv, and ρe into the Euler equa-tions, Eq. (1). The result is expanded in a Fourierseries, with terms grouped by frequency. Using a tra-ditional harmonic balance approach, each frequencycomponent must vanish. Collecting the resulting har-monic balance equations together into one vector equa-tion gives

∂F(U)∂x

+∂G(U)

∂y+ S(U) = 0 (9)

where

U =

R0

U0

V0

E0

R+1

U+1

V+1

E+1...

, S = jω

0 ·R0

0 · U0

0 · V0

0 · E0

+1 ·R+1

+1 · U+1

+1 · V+1

+1 · E+1...

(10)

The vectors F and G are much more complicated, butcan be written as a nonlinear functiosn of the vectorof Fourier coefficients of the conservation variables U.

Finally, we note the conservation variables are realquantities, so that

U−n = Un (11)

where Un is the complex conjugate of Un. Thus, weonly need to store Fourier coefficients for non-negativen. If N harmonics are retained in the Fourier seriesrepresentation of the flow, then 2N +1 coefficients arestored for each flow variable (one for the zeroth har-monic or mean flow, and 2N for the real and imaginaryparts of the remaining harmonics).

Computation of the harmonic fluxes is difficult andcomputationally expensive; on the order of N3 oper-ations are required, so that the cost of the harmonicbalance analysis grows rapidly with the number of har-monics. Also, this approach is not readily applicableto viscous flows, because turbulence models tend tobe quite complex, and not readily expressed in simplealgebraic forms.

To alleviate these problems, we note that alterna-tively, one can reconstruct the Fourier coefficients ofthe conservation variables U and the flux vectors F

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American Institute of Aeronautics and Astronautics Paper 2003-3998

and G from a knowledge of the temporal behavior ofU, F, and G at 2N +1 equally spaced points over onetemporal period. In other words,

U = EU∗ (12)

where U∗ is the vector of conservation variables at2N+1 equally spaced points in time over one temporalperiod, and E is matrix that is the discrete Fouriertransform operator. Conversely,

U∗ = E−1U (13)

where E−1 is the corresponding inverse Fourier trans-form operator. Similar expressions hold for the fluxvectors.

Substitution of Eq. (12) into Eq. (8) gives

∂EF∗

∂x+

∂EG∗

∂y+ jωNEU∗ = 0 (14)

where N is a diagonal matrix with n in the entriescorresponding to the nth harmonic. Pre-multiplyingEq. (14) by E−1 gives

∂F∗

∂x+

∂G∗

∂y+ S∗ = 0 (15)

whereS∗ = jωE−1NEU∗ ≈ ∂U∗

∂t(16)

The product jωE−1NE is just the spectral operatorthat approximates ∂/∂t. The advantage of Eq. (15)over the original form of the harmonic balance equa-tions, Eq. (9), is that the fluxes in Eq. (15) are mucheasier to compute. The fluxes are simply computed ateach of the 2N + 1 time levels in the usual way, usingEq. (2). Also, the alternate form of the harmonic bal-ance equations can easily be applied to more complexflow equations, such as the Navier-Stokes equations,whereas the original form, Eq. (9), cannot.

To solve the harmonic balance equations, we intro-duce a “pseudo-time” term so that the equations maybe marched to a steady state condition using a con-ventional computational fluid dynamic scheme. Usingthe harmonic balance form of the Euler equations asan example, we let

∂U∗

∂τ+

∂F∗

∂x+

∂G∗

∂y+ S∗ = 0 (17)

where τ is a fictitious time, used only to march Eq. (17)to steady state, driving the pseudo time term to zero.Note that pseudo-time harmonic balance equations,Eq. (17), are similar in form to the original time-domain form of the Euler equations, Eq. (12).

In Eq. (17), we use a spectral operator to computethe time S∗ term, as described in the previous section.As presented, this operator requires O(N2) operations

to compute. However, the calculation of the flux vec-tor terms is greatly simplified, requiring only O(N)computations. As a practical matter, the flux calcula-tions, and other calculations requiring O(N) compu-tations, require much more computational time thanthe relatively simple time derivative term. Thus, thecomputational time scales like the number of Fourierterms retained in the solution.

Although the harmonic balance technique has beendescribed using the Euler equations, the method canbe readily applied to the Navier-Stokes equations. Asan example, we apply the harmonic balance techniqueto a representative flutter problem. We consider atwo-dimensional section near the tip of the front stagetransonic rotor of a modern high pressure compressor.At this spanwise station, the inflow Mach number Mis 1.27, the inflow angle Θ, measured from the axial di-rection, is 59.3◦, and the Reynolds number Re is about1.35×106. The computational grid used is an H-O-Hgrid, which has good resolution near the airfoil surfacefor resolving viscous boundary layers, as well as goodresolution in the far field for modeling outgoing waves.

Shown in Fig. 10 is the steady flow (i.e., no unsteadydisturbances) in the blade row computed using a gridwith 193×33 nodes in the O-grid section. Note thefairly complex shock structure, with a shock extend-ing from the leading edge both above and below theairfoil. This shock impinges on the suction surface ofthe airfoil, causing a local strong adverse pressure gra-dient, which causes the boundary layer to separate.The rapid growth of the boundary layer results in anoblique shock forming just upstream of the separationpoint. Also, the flow accelerates over the front por-tion of the pressure surface resulting in a weak normalshock at about 40 percent of the chord on the pressuresurface.

Next, we consider the unsteady aerodynamic re-sponse of the rotor for the case where the airfoilsvibrate harmonically in pitch about their midchordswith a reduced frequency ω equal to 1.0 (based onchord and upstream velocity), an interblade phaseangle σ equal to 30◦, and amplitude α. Shown inFig. 11 is the mean pressure distribution (the zerothFourier component) computed for a pitching ampli-tude α = 1.0◦. The harmonic balance solution wascomputed using one, three, five, and seven harmon-ics (N = 1, 3, 5, and 7). Note that the mean pressuredistributions computed with various numbers of har-monics are different. However, the solutions convergerapidly as the number of harmonics is increased.

Next, we consider the first harmonic of the unsteadypressure distribution on the airfoil surface. This com-ponent is important, because it is the only componentthat contributes to aerodynamic damping for har-monic pitching motion of the airfoil. Shown in Fig. 12is the first harmonic of the unsteady pressure on theairfoil surface, scaled by the amplitude of the pitching

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American Institute of Aeronautics and Astronautics Paper 2003-3998

Fig. 10 Computed steady pressure (top) and Machnumber (bottom) contours for transonic viscousflow through front stage compressor rotor.

amplitude. As in the case of the mean flow, the un-steady pressure distributions computed with variousnumbers of harmonics are different, but again the so-lutions converge rapidly as the number of harmonicsis increased.

To demonstrate the influence of nonlinearities on theunsteady flow, we again plot the zeroth and first har-monics of the unsteady flow in Fig. 13. In these results,the larger amplitude motion solutions are computedusing five harmonics so that the results are convergedin the harmonic balance sense. The pressure distri-butions are plotted for several pitching amplitudes.The pressure distributions associated with the largeramplitude pitching motion is seen to be substantiallydifferent from the small amplitude case. In the small-amplitude case, the mean pressure distribution showssigns of very sharp shocks. For the larger amplitudemotion, the shocks get “smeared” out. Physically,

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

0.3

0.4

0.5

0.6

0.7

0.8

Mean F

low

Pre

ssure

, p/(

qin

letc

)

1 Harmonic

3 Harmonics

5 Harmonics

7 Harmonics

Fig. 11 Zeroth harmonic (mean flow) of unsteadypressure distribution for front stage compressor ro-tor airfoils vibrating in pitch with ω = 1.0 andσ = 30◦, and α = 1.0◦.

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

−10

−8

−6

−4

−2

0

2

4

6

8R

eal U

nst

eady

Pre

ssure

, R

e(p

1)/

(qin

letc

a −)

1 Harmonic

3 Harmonics

5 Harmonics

7 Harmonics

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

−6

−4

−2

0

2

4

6

8

Imag U

nst

eady

Pre

ssure

, Im

(p1)/

(qin

letc

a −)

1 Harmonic

3 Harmonics

5 Harmonics

7 Harmonics

Fig. 12 First harmonic of unsteady pressure distri-bution for front stage compressor rotor airfoils vi-brating in pitch with ω = 1.0, σ = 30◦, and α = 1.0◦.

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American Institute of Aeronautics and Astronautics Paper 2003-3998

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

0.3

0.4

0.5

0.6

0.7

0.8

Mean F

low

Pre

ssure

, p/(

qin

letc

)

a − = 0.01 (deg)

a − = 0.1 (deg)

a − = 0.5 (deg)

a − = 1.0 (deg)

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

−8

−6

−4

−2

0

2

4

6

8

10

Real U

nst

eady

Pre

ssure

, R

e(p

1)/

(qin

letc

a −)

a − = 0.01 (deg)

a − = 0.1 (deg)

a − = 0.5 (deg)

a − = 1.0 (deg)

0.0 0.2 0.4 0.6 0.8 1.0

Airfoil Surface Location, x/c

−6

−4

−2

0

2

4

6

8

Imag U

nst

eady

Pre

ssure

, Im

(p1)/

(qin

letc

a −)

a − = 0.01 (deg)

a − = 0.1 (deg)

a − = 0.5 (deg)

a − = 1.0 (deg)

Fig. 13 Unsteady pressure distribution for frontstage compressor rotor airfoils vibrating in pitchwith ω = 1.0 and σ = 30◦. Top: zeroth harmonic.Middle and bottom: first harmonic.

−180 −120 −60 0 60 120 180

Interblade Phase Angle, s (deg)

−2.4

−2.0

−1.6

−1.2

−0.8

−0.4

0.0

0.4

0.8

Real M

om

ent, R

e(M

1e

.a.)/

(qin

letc

a −)

a −=0.01 (deg) 1 Harmonic

a −=0.1 (deg) 5 Harmonics

a −=0.5 (deg) 5 Harmonics

a −=1.0 (deg) 5 Harmonics

−180 −120 −60 0 60 120 180

Interblade Phase Angle, s (deg)

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

Imag M

om

ent, Im

(M1e

.a.)/

(qin

letc

a −)

a −=0.01 (deg) 1 Harmonic

a −=0.1 (deg) 5 Harmonics

a −=0.5 (deg) 5 Harmonics

a −=1.0 (deg) 5 Harmonics

Fig. 14 First harmonic of unsteady pitching mo-ment for front stage compressor rotor airfoils vi-brating in pitch with ω = 1.0.

this is because the shocks oscillate, and when tempo-rally averaged, the shocks appear smeared. Of course,when viewed at any instant in time, the shocks aresharp. Also shown are the real and imaginary partsof the first harmonic of the unsteady pressure. In thesmall-amplitude case, very large and narrow peaks ofpressure are seen. These are the so-called “shock im-pulses” associated with the unsteady motion of theshock. As the amplitude of the pitching vibrationis increased, these peaks are reduced and spread out,because the shock motion is larger and the resultingshock impulse is spread over a larger chordwise extent.

By appropriate integration of the first harmonic ofthe unsteady pressure distribution, one can obtain thefirst harmonic of the pitching moment. The imagi-nary part determines the aeroelastic stability of therotor. In the absence of mechanical damping, therotor is stable only if the imaginary moment is lessthan zero for all interblade phase angles. Shown inFig. 14 is the pitching moment as a function of in-terblade phase angle for several pitching amplitudes.For small amplitude motions, the rotor is unstable forinterblade phase angles σ between −10◦ and +60◦.Thus, the amplitude of an initially infinitesimal mo-tion will grow. As the motion grows, however, theaerodynamic damping of the least stable interbladephase angle goes to zero. This is seen more clearly in

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American Institute of Aeronautics and Astronautics Paper 2003-3998

0.0 0.2 0.4 0.6 0.8 1.0

Pitch Amplitude, a − (deg)

−1.4

−1.2

−1.0

−0.8

−0.6

−0.4

Re

al M

om

en

t, R

e(M

1e

.a.)/

(qin

letc

a −)

1 Harmonic

3 Harmonics

5 Harmonics

7 Harmonics

0.0 0.2 0.4 0.6 0.8 1.0

Pitch Amplitude, a − (deg)

−0.2

0.0

0.2

0.4

0.6

0.8

Ima

g M

om

en

t, I

m(M

1e

.a.)/

(qin

letc

a −)

1 Harmonic

3 Harmonics

5 Harmonics

7 Harmonics

Fig. 15 First harmonic of unsteady pitching mo-ment for front stage compressor rotor airfoils vi-brating in pitch with σ = 30◦.

Fig. 15. Shown is the pitching moment for σ = +30◦as a function of pitch amplitude computed using one,three, five and seven harmonics. Clearly, the solutioncomputed with just one harmonic is not converged (ex-cept at very small amplitudes), and gives erroneousresults. However, with three or five harmonics, the so-lution is converged to engineering accuracy. Note thatthe imaginary moment is positive (unstable) for smallamplitude motions, but goes to zero at a pitching am-plitude of about 0.7◦. Thus, the blade will vibrate ina stable limit cycle with with this pitch amplitude. Itis also remarkable that the nonlinear fluid dynamicseffects are important at such a small geometric dis-placement.

We next consider the computational efficiency of thepresent method. Shown in Fig. 16 are the convergencehistories for the steady flow and harmonic balance cal-culations. Note that except for N = 7, the steady flowsolver and harmonic balance flow solver converge inabout the same number of iterations. For the N = 7case, the harmonic balance solution does not converge.Fortunately, we have found that three to five harmon-ics are more than adequate to obtain mode convergedsolutions of the zeroth and first harmonic componentsof the unsteady flow, and the harmonic balance solverusually converges for this number of harmonics. Fi-

0 2000 4000 6000 8000 10000

Iteration Number

−10

−9

−8

−7

−6

−5

−4

So

lutio

n R

esi

du

al,

Lo

g1

0(L

2(R

esi

du

al))

0 2000 4000 6000 8000 10000

Iteration Number

−10

−9

−8

−7

−6

−5

−4

So

lutio

n R

esi

du

al,

Lo

g1

0(L

2(R

esi

du

al))

1 Harmonic

3 Harmonics

5 Harmonics

7 Harmonics

Fig. 16 Convergence history for steady flow solver(top) and harmonic balance flow solver (bottom)with σ = 30◦, ω = 1.0, and α = 1.0◦.

nally, the CPU time per iteration of the harmonicbalance flow solver for one, three, five, and seven har-monics was found to be 2.15, 4.62, 7.45, and 10.29times the cost per iteration of the steady flow solver.Even using seven harmonics, the cost to compute thefully nonlinear, viscous, transonic flow about a vibrat-ing blade row is only about ten times the cost of acomparable steady flow calculation.

5 ConclusionsFrequency domain techniques for computing un-

steady flows in turbomachinery have evolved consid-erably over the past two decades. These techniquesare capable of computing both small-disturbance un-steady flows (using the time-linearized approach) andlarge amplitude flows (using the harmonic balance ap-proach). In addition, the time-linearized approachhas been applied to the difficult but important prob-lem of computing unsteady flows in multistage ma-chines. In all cases, the frequency domain solvershave a number of distinct advantages over their time-domain equivalents. Using complex periodicity condi-tions, the computational domain can be reduced to asingle blade passage (in each row). Furthermore, us-ing the pseudo time marching technique, the governingEuler and Navier-Stokes equations can be solved us-ing steady-state acceleration techniques such as local

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American Institute of Aeronautics and Astronautics Paper 2003-3998

time stepping and multigrid acceleration. The result isthat computing these very complex unsteady flows inturbomachinery is only slightly more expensive thansolving a steady flow problem.

AcknowledgmentsPortions of this review paper are excerpted from a

previous publication30 written by the two of the au-thors (Hall and Thomas) and Dr. William S. Clark,whose contribution is gratefully acknowledged.

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