In turbomachines, the relative motion of fixed and rotating blades gives rise to deterministic unsteady interactions at frequencies termed BPFs (Blade Passing Frequencies). In a multi-stage turbomachine, a row sandwiched between two other rows is submitted to (at least) two BPFs, hence the need for multiple frequency methods. Initially developed for single frequency problems, harmonic methods have been extended to account for multiple frequencies. All the variations of the Harmonic Balance (HB) technique proposed in the literature rely on a uniform time sampling of the longest period of interest (though the number of samples can differ). This can compromise the efficiency of the method, as too many time samples are computed. Besides, as demonstrated in the present contribution, uniform time sampling can also raise stability issues. To overcome these computational limitations, a new approach using non-uniform time sampling is proposed in the present contribution. This paper will be organized as follows: first, the multi-frequency HB methods is presented, and the impact of time sampling on numerical stability is discussed. Then, algorithms for an automatic choice of the time samples are presented and compared. The proposed non-uniform sampling is assessed for a model problem (i.e. a pulsating channel). Finally, a section is dedicated to the application to a turbomachinery configuration, with emphasis on the choice of frequencie

GOMAR, Adrien

GUÉDENEY, Thomas

SICOT, Frédéric

I-Revues

21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
Multi-frequential harmonic balance approach for the
computation of unsteadiness in multi-stage
turbomachines
T. Gue´deneya, A. Gomara and F. Sicota
a. CERFACS, CFD Team, 42 avenue Gaspard Coriolis 31057 Toulouse cedex 1, France
Re´sume´ :
Dans les turbomachines, la vitesse relative entre les parties fixes et les parties mobiles ge´ne`re dans
l’e´coulement des interactions instationnaires de´terministes a` une fre´quence appele´e fre´quence de pas-
sage des aubes. Dans une turbomachine multi-e´tage´e, une roue aubage´e comprise entre deux autres est
soumises a` (au moins) deux fre´quences de passages des aubes, justifiant la ne´cessite´ d’une me´thode
capable de prendre en compte ces instationnarite´s. Initialement de´veloppe´ pour des proble`mes mono-
fre´quentiels, les me´thodes d’e´quilibrage harmonique ont e´te´ e´tendues afin de prendre en compte des
fre´quences multiples. Applique´e a` une turbomachine multi-e´tage´e, elles permettent de capturer les ins-
tationnarite´s de l’e´coulement a` un couˆt 70 fois infe´rieur par rapport aux me´thodes instationnaires
classiques.
Abstract :
In turbomachines, the relative motion of fixed and rotating blades gives rise to deterministic unsteady
interactions at frequencies termed BPFs (Blade Passing Frequencies). In a multi-stage turbomachine,
a row sandwiched between two other rows is submitted to (at least) two BPFs, hence the need for me-
thods able to capture these unsteady effects. Initially developed for single frequency problems, harmonic
methods have been extended to account for multiple frequencies. Applied to turbomachines, these me-
thods allows to capture unsteady phenomena of the flow field with a cost four seventy cheaper compared
to classical unsteady methods.
Mots clefs : Harmonic Balance ; Turbomachines ; Row interactions
1 Introduction
With the need to ever increase performance, aggressive design choices foster unsteady phenomenon,
such as : separated flows at or close to stable operability limits, aeroelastic phenomenon, or blade
interactions in compact turbo-engines, to name but a few. In such a context, engineers need tools
to account for these effects as early as possible in the design cycle. For this reason, efficient and/or
accurate unsteady approaches are receiving a lot of attention.
One approach is to improve the time-integration algorithm to reduce the computational cost as compa-
red to standard techniques. To achieve this, Fourier-based methods for periodic flows have undergone
major developments in the last decade (see He [10] for a recent review, or the special issue of the In-
ternational J. of CFD [11]). In this context, the present paper focuses on a time-domain Fourier-based
method, namely the Harmonic Balance (HB) method, which transforms an unsteady time-marching
problem into the coupled resolution of several steady computations at different time samples of the
period of interest.
Initially developed for single (fundamental) frequency problems [9, 6], the HB method has been ex-
tended to account for multiple frequencies [7, 2, 3].
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21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
The multi-frequential formulation of the HB is first presented, an extension of the chorochronic boun-
dary condition is then described to solve for only one blade passage per row even in a multi-stage
turbomachinery. Finally, a multistage compressor case is presented to demonstrate the accuracy and
efficiency of the proposed approach for industrial applications.
2 Time-domain almost-periodic harmonic balance technique
The semi-discrete finite-volume form of the Unsteady Reynolds-Averaged Navier-Stokes (U-RANS)
equations is given by
d
dt
(VW ) +R (W ) = 0, (1)
where W is the vector of the conservative unknowns (conservative variables and turbulent variables),
V the volume of the cell and R the residual resulting from the discretization of the fluxes and the
source terms (including the turbulent equations).
If the flow variables are composed of non-harmonically related frequencies (i.e. the flow spectrum has
high-energy discrete-frequency modes), the flow regime can be termed as almost-periodic [1]. Instead
of a regular Fourier series, the U-RANS equations are projected on a set of complex exponentials with
arbitrary angular frequencies ωk. The conservative variables and the residuals are then approximated
by
W (t) ≈
N∑
k=−N
Ŵke
iωkt, R(t) ≈
N∑
k=−N
R̂ke
iωkt, (2)
where Ŵk and R̂k are the coefficients of the almost-periodic Fourier series for the frequency fk = ωk/2pi.
Injecting this decomposition in Eq. (1) yields
N∑
k=−N
(
iωkV Ŵk + R̂k
)
eiωkt = 0. (3)
Sampling in time onto a set of 2N + 1 time levels to solve Eq. (3), the following matrix formulation
is obtained :
A−1 ·
(
iV PŴ ? + R̂?
)
= 0, (4)
where the almost-periodic inverse discrete Fourier transform (IDFT) matrix reads :
A−1 =

exp(iω−N t0) · · · exp(iω0t0) · · · exp(iωN t0)
...
...
...
exp(iω−N tk) · · · exp(iω0tk) · · · exp(iωN tk)
...
...
...
exp(iω−N t2N ) · · · exp(iω0t2N ) · · · exp(iωN t2N )
 , (5)
with ω0 = 0, t0 = 0, ω−N = −ωN and
P = diag (−ωN , . . . , ω0, . . . , ωN ) ,
Ŵ ? =
[
Ŵ−N , . . . , Ŵ0, . . . , ŴN
]>
,
R̂? =
[
R̂−N , . . . , R̂0, . . . , R̂N
]>
.
(6)
Knowing a time sampling that allows A−1 to be invertible, the almost-periodic Fourier coefficients can
be approximated thanks to{
Ŵ ? = AW ?, with W ? = [W (t0) , . . . ,W (ti) , . . . ,W (t2N )]
> ,
R̂? = AR?, with R? = [R (t0) , . . . , R (ti) , . . . , R (t2N )]
> .
(7)
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21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
Equation (4) thus becomes
iV A−1PA+R? = V Dt[W ?] +R? = 0, (8)
where the multiple-frequency HB time-derivative operator Dt[.] = iA
−1PA, the HB source term, can
not be easily derived analytically, and has to be numerically computed.
3 Turbomachinery Application
Under the assumption that all unsteady phenomena in a blade row during stable operation are periodic
and can be correlated with the rotation rate Ω of the shaft, the dominant frequencies are those created
by the passage of the neighboring blades. In a multi-row turbomachine, a blade row sandwiched
between the upstream and downstream rows is subjected to wake and potential effects. In practical
turbomachines, the blade counts of neighboring rows are generally different and coprime. Consequently,
a sandwiched blade row resolves various combinations of the frequencies, which are additions and/or
subtractions of multiples of the blade passing frequencies : according to Tyler and Sofrin [16], the kth
frequency in the blade row j is given by
ωrowjk =
nRows∑
i=1
nk,iBi(Ωi − Ωj). (9)
Here, Bi and Ωi are respectively the blade count and the rotation rate of the i
th blade row, nk,i is the
kth set of nRows integers driving the frequency combinations. It must be noted that only the blade
rows that are mobile relative to the considered j one contribute to its temporal frequencies and that
every blade row solves its own set of frequencies and thus its own set of time levels. To set up a HB
computation for a multistage configuration, it is of course impossible to use each and every possible
nk,i, and the user has to choose which frequency combinations will appear in the computation of each
row.
3.1 Phase-lagged azimuthal boundary conditions
In a single blade passage computation of a multi-row configuration, the phase-lag condition [4] needs
to be used to take the space-time periodicity into account. It states that the flow in one blade passage θ
is the same as next blade passage θ + ∆θ but at another time t+ δt :
W (θ + ∆θ, t) = W (θ, t+ δt) , (10)
where ∆θ is the pitch of the considered row. Assuming that every temporal lag is associated with a
rotating wave of rotational speed ωk, the constant time lag can be expressed as
δt =
βk
ωk
, ∀k, (11)
where
βk = 2pisign(ωk)
1− 1
Bj
∑
i 6=j
nk,iBi
 , (12)
the nk,i being the integers specified for the computation of the frequencies from Eq. (9), Bi the number
of blades in row i and subscript j denoting the current row.
The phase-lag condition was adapted to the time-domain HB by Gopinath et al. [6]. The derivation
starts with the almost-periodic Fourier transform of Eq. (10) :
N∑
k=−N
Ŵk (θ + ∆θ, t) e
iωkt =
N∑
k=−N
Ŵk (θ, t) e
iωkδteiωkt. (13)
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21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
Thus, the flow spectrum from one blade passage is equal to that of the next blade passage modulated
by the inter-blade phase angle βk :
Ŵk (θ + ∆θ, t) = Ŵk (θ, t) e
iωkδt = Ŵk (θ, t) e
iβk . (14)
Using the same notation as previously, the following matrix formulation is obtained :
W ? = A−1MAW ? (θ) , (15)
where
M = diag (−βN , . . . , β0, . . . , βN ) , (16)
and A−1 is given by Eq. (5).
3.2 Application to a subsonic compressor
In order to validate the non-uniform HB method on a turbomachinery test case, a subsonic compressor
case is studied. It is the mid-span slice of the inlet guide vanes (IGV) and the first stage of the axial
compressor CREATE [8], located in Lyon (France) at the Laboratoire de Me´canique des Fluides et
Acoustique (LMFA). This configuration is composed of 32 IGV blades, 64 rotor (RM1) blades and
96 stator (RD1) blades. The full 3D 3.5-stage computation is presented in Ref. [15].
3.2.1 Mesh and numerical parameters
The blade passages are meshed with a block-structured topology. It is composed of five grid points in
the radial direction, 33 in the azimuthal direction and 100 in the axial direction for both rows. This
leads to a total number of approximately 50,000 mesh cells.
The IGV blade is not actually meshed but taken into account through a non-uniform injection boun-
dary condition that represents the wake of the IGV entering the RM1 domain. This injection follows
the self-similarity law of Lakshminarayana and Davino [12], which states that the spatial evolution of
a wake can be described by a Gaussian function. As BRD1 = 3 ·BIGV , the frequency content remains
mono-frequential in the rotor (i.e., the BPF of the downstream rotor is just an harmonic of the IGV’s
BPF). Therefore the number of blades composing the IGV has been changed from 32 to 80 so that
the configuration still presents a 2pi/16 periodicity but the frequency content is now multi-frequential
in the rotor.
The outlet duct is modeled by a valve condition coupled with a simplified radial equilibrium equation.
At the blades surfaces, wall laws [5] are imposed. The lower and upper radial conditions are slip walls.
The convective fluxes are discretized using the second-order Jameson scheme with added artificial
viscosity, or a second-order Roe scheme. For this study, the turbulent viscosity is computed with the
one-equation model proposed by Spalart and Allmaras.
The DTS scheme is used to get a numerical reference solution. The periodicity of the different blade
passages is such that a 2pi/16 periodicity is enough to perform the unsteady computations. To reach
an established periodic state, 67 passages (using 400 instants per azimuthal period) of the periodic
sector are necessary.
3.2.2 Results
Two sets of frequencies (one of four and one of six frequencies), are chosen to run the harmonic balance
computations. They are summarized Tab. 1.
Figure 1 shows the instantaneous entropy flow field for the maximum-efficiency operating point η∗is = 1.
For the HB computations, the computed passage is duplicated using phase-lag to check that the
azimuthal phase-lag boundary conditions ensure the continuity of the flow field between the original
blade passage and the duplicated ones. All the wakes are correctly convected downstream and very few
differences can be seen in the different flow fields. Some numerical wiggles can be observed downstream
the RM1/RD1 interface, but the higher the number of frequencies, the better the solution, as already
shown by Sicot et al. [14].
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21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
n IGV n RM1 n RD1 Initialization
1 1 0 Restart from steady computation
N = 4 0 2 1 15,000 it. in Roe second order scheme
2 3 0 then 10,000 it. in Jameson scheme
0 4 2
1 1 0
0 2 1 Restart from N = 4
2 3 0 5,000 it. in Jameson scheme k4=0.064
N = 6 0 4 2 then 5,000 it. in Jameson scheme k4=0.032
3 5 0
0 6 3
Table 1 – Frequency combination coefficients to compute only harmonics of the fundamental blade-
passing frequencies.
(a) DTS (b) HB N = 4
(c) HB N = 6
Figure 1 – Comparison of the entropy flow field at η∗is = 1.0 and t = 0.
3.2.3 Computational gain
The HB computations allow a reduction of the CPU cost by a factor 4.5 for four frequencies and 2
for the six frequencies set, the gain being higher with fewer harmonics. However, it should be kept
in mind that the reference DTS simulations are done on a 2pi/16 periodic sector, whereas practical
turbomachinery configurations usually do not have such periodicity, thus requiring simulations on the
whole 360˚ machine. In this case, an additional factor 16 in gain can thus be estimated, suggesting a
gain of almost two orders of magnitude.
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21e`me Congre`s Franc¸ais de Me´canique Bordeaux, 26 au 30 aouˆt 2013
4 Conclusions
A multi-frequential harmonic balance approach is presented. The flow in a multi-stage axial compressor
is computed to prove the maturity of the method. It is shown that nonlinear flows can be modeled
to engineering accuracy with only four frequencies. In the present case, the HB method is about 70
(4.5× 16) times faster than a classical time-marching computation over the whole annulus, thanks to
the efficient spectral-integration scheme and to the generalized phase-lag boundary conditions. The
conclusions obtained for the present quasi-2D case have been extended to 3D geometries without any
new assumption [13].
Re´fe´rences
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[3] K. Ekici and K. C. Hall. Nonlinear Frequency-Domain Analysis of Unsteady Flows in Turboma-
chinery with Multiple Excitation Frequencies. AIAA Journal, 46(8) :1912–1920, 2008.
[4] J. I. Erdos, E. Alznert, and W. McNally. Numerical Solution of Periodic Transonic Flow through
a Fan Stage. AIAA Journal, 15(11) :1559–1568, November 1977.
[5] E. Goncalves and R. Houdeville. Reassessment of the Wall Function Approach for RANS Com-
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[8] N. Gourdain, X. Ottavy, and A. Vouillarmet. Experimental and Numerical Investigation of Uns-
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[9] K. C. Hall, J. P. Thomas, and W. S Clark. Computation of Unsteady Nonlinear Flows in Cascades
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[12] B. Lakshminarayana and R. Davino. Mean Velocity and Decay Characteristics of the Guide Vane
and Stator Blade Wake of an Axial Flow Compressor. In Gaz Turbine Conference and Exhibit
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Multi-frequential harmonic balance approach for the computation of unsteadiness in multi-stage turbomachines

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