PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Numerical and experimental investigation of the turbofan first
booster stage tone noise
Victor Mileshin, Sergey Pankov, Anton Rossikhin1
Central Institute of Aviation Motors, Russia

ABSTRACTAn important problem of turbofan design is maintenance of low noise emission. One of the components ofturbofan noise is the tone noise arising due to inter-row interactions in the booster stages. For modern
turbofans this component of noise can be significant at approach certification point. The investigations of thetone noise of the first booster stage have been carried out in CIAM during last several years. They were
performed using a frequency domain method for simulation of tone noise of multistage turbomachinesdeveloped in CIAM. The advantage of the method is the high speed of calculations; however it relies on somead hoc assumptions about the process of interactions between flow field disturbances and blade rows. This
work presents the results of the tone noise calculation of the first booster stage using time domain approachbased on multirate Runge-Kutta scheme. The result, obtained in the calculation, is independent from any
assumptions about the interactions in the multistage turbomachine. This allows verifying the results ofprevious calculations. In general satisfactory correspondence were found between the results ofcomputations in time and in frequency domains and between them and experimental data obtained in the
CIAM C-3A acoustic test facility.
Keywords: tone noise, booster stages, multirate Runge-Kutta scheme

1. INTRODUCTION
The main source of modern turbofan noise is the fan. However, such sources of tone noise , as low
pressure turbines and boosters, can also contribute to the tone noise at some operational conditions[1,2]. The advances in the reduction of a fan tone noise can make the contribution of these sources
more significant. The increase of the low pressure cascades shaft rotation speed in the modern
turbofans can also lead to the same result. The feature of low pressure turbines and boosters is that they
are usually multistage turbomachines. The necessity of taking into account of interactions between
different stages significantly complicates simulations of their tone noi se [2,3,4] in comparison withordinary fans.
A method of multistage turbomachines tone noise calculation, formulated in the frequency domain,
was developed in CIAM as a part of activities devoted to the calculation of tone noise of the booster
stages. It is based on the kinematic relations, describing a dependence of flow field in a turbomachinefrom time and azimuth angle. The flow field in each row is represented according to the method as a
combination of harmonic fragments in one blade channel. The notion of harmonic fragment – anassembly of disturbances having equal frequency and phase shift between the boundaries of one blade
channel was introduced in [5] (see also references therein). As it was shown in the paper [5], a flow
field in a row can be decomposed into fields of harmonics of some base frequency, and each field ofharmonic can be decomposed into a finite number of fields of harmonic fragments. The number
depends on the turbomachine geometry only. For example for single stage turboimachine the flow field
for each harmonic can be described by only one harmonic fragment, which reduces the method
introduced in [5] to the well-known method of rotor-stator interaction calculation based onphase-lagged boundary conditions [6]. The method, developed in CIAM, makes possible a significant
reduction of computational resources, needed for a turbomachine tone noise simulation, by taking into
account only those harmonic fragments, which contain the strongest disturbances, generated by theturbomachine.
However the specification of the set of harmonic fragments, needed for the calculation of a1

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aarossikhin@ciam.ru

turnomachine tone noise with sufficient prediction, is impossible without some heuristic
considerations [5]. Though the results of the calculations with the method, showed satisfactory
correspondence with experimental data, it seems important to compare these results with thoseobtained by a method of calculation being less sensitive to the assumptions, made at the statement of
the calculations.
Such method of calculation is the method of calculation in the time domain. According to thismethod the discretized equations are solved for specified number of time steps. The calculation is
performed for whole spatial period (a sector of full annulus equal to 2π/P, where P is the greatest
common divisor of numbers of blades in blade rows). In such a calculation all disturbances , generated
by turbomachine, which can be resolved on the computational grid with the specified time step aretaken into account. Unfortunately the method is extremely expensive in terms of computational
resources.
The paper is devoted to the application of the time domain method of calculation for the problem of
the first booster stage tone noise, which was earlier considered in [5] with the usage of the frequency
domain method. Both methods of calculation are implemented in CIAM 3DAS solver [7,8]. The solver
was designed for the calculation of tone noise of the conventional and counter-rotating fans, open
rotors, low pressure turbines and compressors.

2. 3DAS SOLVER
2.1 The Description of the Solver
The method of calculation used in the 3DAS solver is based on the decomposition of the unsteadyviscous flow into two parts - inhomogeneous viscous steady flow field and unsteady inviscid
disturbances. For turbomachinery tone noise applications the steady mean flow field in rows can becalculated by any steady aerodynamic solver using Reynolds-Averaged Navier–Stokes (RANS)
equations, structured grid, finite volume spatial approximation and "mixing-plane" interfaces between
rows. Calculation of unsteady inviscid equations for disturbances (linear or nonlinear dependent on the
problem) over the mean steady flow field in a row in the reference frame, rotating with the row, is
performed by 3DAS solver.
The equations are solved using the numerical methods of computational aeroacoustics. The finite
volume approach is used for spatial discretization. The fluxes are approximated using fourth order
DRP (Dispersion Relation Preserving) scheme [9]. The details of the DRP scheme implementation canbe found in [7] and references therein.
Different Runge-Kutta schemes or dual time step method can be used in 3DAS solver as advancing
schemes. Usually second-order four stages Runge-Kutta LDDRK (Low Dissipation and Dispersion
Runge-Kutta Scheme) scheme [10] was used for calculations in the time domain. Also a multirate
Runge-Kutta scheme, developed by authors on the bases of LDDRK scheme can be applied when thecomputational domain can be divided on several subdomains with significantly different cell sizes.
3DAS solver can be used for calculations either in the time domain or in the frequency domain. In
the first case the direct evaluation of the discretized equations solution for the prespecified number ofsteps is performed. The interaction between blade rows is provided using sliding grid interfaces. In thesecond case the solution is searched in the form of a finite set of fields of harmonic fragments,
sufficient for the description of flow field evolution in time. It is obtained using the pseudotime
relaxation method with local time step. The interaction between blade rows is provided by special
interfaces, which preserve continuity of flow parameters for prespecified set of harmonic functions oftime and circumferential angle (circumferential modes).
Other important details about 3DAS solver can be found in [8].
2.2 Multirate Runge-Kutta Scheme
The second-order four stages Runge-Kutta LDDRK scheme is the usual choice for calculations in
the time domain with 3DAS solver. The examples of calculations with the scheme can be found in [11]and references therein. However the explicit scheme can be used only on the grids with relatively large
minimal cell sizes. The situation is different for frequency domain methods, where there are no
limitations on the cell sizes. The problem under consideration was first formulated in the frequency
domain, and it was found out that the calculation on the same grid in the time domain using explicit
scheme demanded too much time steps on one period of rotation. As alternatives two meth ods ofcalculations were tested – dual time step method and multirate Runge-Kutta scheme. The latter1609

approach showed significantly higher calculation speed.
Multirate Runge-Kutta schemes are an attractive approach for the solution of ordinary and partial
differential equations. In the formulation of our multirate Runge-Kutta method we followed theapproach for construction of such methods presented in [12]. The paper [12] is in turn based on thewell-established theory of partitioned Runge-Kutta methods [13].
Let’s briefly describe the method. Assume that we have a system of ordinary differential equations:y(t ) f (t , y), y  Rn , y(t0 ) y0 ,t ! t0
(1)
In the following we will assume that the solution of equation (1) can be split into two componentsy L (t) and y A(t), where the first of the components (latent) varies relatively small in time in comparison
with the second component (active). For the components we can write out:yA (t ) f A ( yA , yL )), yL (t ) f L ( yA , yL )), yA  RnA , yL  RnL ,nA  nL n
(2)
The latent component can be integrated with the time step H, while active with the time step h=H/p(p – integer). Thus following the standard representation of the Runge-Kutta scheme we can express
the solution for the latent and the active components in the formf A ( y A,O  h¦ j 1 aij k AO, j , YLO,i ),) i 1,
1 2,s

k AO,iy A,O  h¦ i 1 bi k AO,i | y A (t0  (O  1)h),s

y A,O 1O

Here ci

¦

s

, s,

0,1, 2,(3), p  1,
1,YLO,i | yL (t0  (O  ci )h).

k L ,if L (YA,i , yL ,00  H ¦ j 1 aij k L , j ),i 1, 2,yL,1yL,0  H ¦ i 1 bi k L ,i | yL (t0  H ),YA,i | y A (t0  ci h)a , а ci

j 1 ij

s

,s,

(4)

s

¦

s

a .

j 1 ij

The main question in the problem of construction of multirate advancing scheme is how to find thecontribution of passive components in the time evolution of active and vice versa. Following [12] weare using interpolation formulas of the form

YLL,O,iyL,00  h¦ j 1 (J ij  K j (O ))k Lj ,i 1, 2,

YA,iy A,0  H ¦ j 1 J ij k A0 , j ,i 1, 2,ss, s,O0,1,

p  1,(5),s

The constants in the formulas (5) should obey special assumptions in order to provide the desiredorder of the scheme. Using the theory of partitioned Runge-Kutta methods it is possible to show, thatin order to preserve the second order the constants should satisfy the following conditions [12]:¦ K (O ) O,¦ J
s

j 1

s

j

j 1 ijci ,i 1, 2,, s,¦ j 1J ijsci ,i 1, 2,

,s,

(6)

In this paper we will use the previously mentioned second order LDDRK scheme as a base for theconstruction of multirate scheme. The constants for the LDDRK scheme are:s=4; b1 = b2 = b3 = 0; b4 = 1.0; a2,1 = 0.234570947137; a3,1 = 0; a3,2 = 0.333333333333333;
a4,1 = a4,2 = 0; a4,3 = 0.5.
The constants in our multirate scheme are the same for both active and latent components. As forthe formulas (5) we assumed that the values for γ i,j are the same as for ai,j (again both for active andlatent components):
γ2,1 = 0.234570947137; γ3,1 = 0; γ3,2 = 0.333333333333333; γ4,1 = γ4,2 = 0; γ4,3 = 0.5,
Also it was assumed that η 1 = η 2 = η 3 = 0; η4 = λ. This choice of the constants satisfies conditions (6).
If we have several levels of variables activity we have to introduce several time scales. For the levelof activity l, l=0..k, it is reasonable to take time step in the form h l =H/p l so that to preserve the integer
ratio between time steps corresponding to different levels of activity. In our calculations it is usuallyassumed that p=2.
The scheme under consideration can provide relatively high computational speed. However it has
some significant disadvantages. First of all it is not conservative, being applied to partial differential1610

equations. Also in some cases there is a lack of stability.

3. THE OBJECT OF INVESTIGATION
The fan under consideration is a 700 mm diameter high bypass ratio wide chord fan model withswept stator vanes, designed for an advanced turbofan of civil aircraft. It has three booster stages. The
description of the model can be found in [5] and in references therein.

a)Figure 1. - Scheme of the fan model: a) general view; b) magnified fragment
The model has 18 blades in the fan rotor and 41 vanes in the stator of bypass duct. It also has 71
vanes in the inlet guide vanes (IGV) of the booster stages, 86 blades in the rotor of the first booster
stage and 100 blades in the stator of the first stages. The scheme of a fan model can be seen in the
fig. 1.
The calculation was performed at approach operational conditions. At these conditions the tonenoise of the booster stages dominates in the forward hemisphere over the tone noise of the fanrotor-stator interaction [14]. Design parameters of the fan model at approach and designed mode arerepresented in Table 1.
Table 1. Parameters of the fan model
DesignApproach

Relative shaft speed

100%

53.9%

Fan rotor tip speed

395 m/s

212.9 m/s

First stage rotor tip speed

221.5 m/s

119.4 m/s

Bypass ratio

8.5

9.7

The aim of the calculation is to resolve the main tones connected with the first booster stage of the
fan model. The noise generation in the first booster stage reveals itself by the tones with frequencies
being the combinations of the blade passing frequency of the first booster stage rotor f 2=86N, where N
is the shaft rotational frequency and some harmonic of the fan rotor blade passing frequency f 1=18N.
The experience obtained in the previous calculation showed [5,15] that the tones arise as a result of
the scattering of the disturbance with the frequency equal to f2 on the fan rotor. Therefore, first of all,there was a task in view to correctly describe mechanisms of the generation of the tone with thisfrequency in the booster stage. Also it was assumed beneficial to the calculation precision to take into
account the mechanisms of tones generation at frequencies f 2 -f 1 and f 2+f 1. These considerations wereused both in the processes of grid generation and computation setup.
The test campaign for the fan in the rig setup was performed in the C -3A test facility [14]. It is
designed for acoustic, aerodynamic, and mechanical investigations of counter rotating andconventional fan models.

4. THE CALCULATION OF TONE NOISE
4.1 Calculation of Steady Flow Field and Computation Setup
The first stage of the simulation was the computation of mean steady flow fields using RANS
equations, the semi-empirical model of turbulence, and "mixing-plane" interfaces between blade rows.
It was performed with CIAM in-house aerodynamic solvers 3D-IMP_MULTI [7]. The computational
domain included one blade passage for rotor, stator and each row of booster stages. The boundaries of1611

the computational domain and boundaries between blocks for individual rows are shown by black
curves in the fig. 2. The size of the computational grid was 5.6 million cells. The grid had H-type andwas generated by 3D-IMP_MULTI. The results of the computation were compared with experimental
data. Satisfactory correspondence for the fan integral characteristics was obtained for all operation
range.

Figure 2. - Scheme of the computational domain: I) Low pressure compressor; II) Booster. Here: a - theboundaries of the fine grid region in the computational domain; b - the boundaries of buffer blocks (shown
only partially); c - the surface for modal analysis upstream of the fan
Next step was the generation of the grid for unsteady calculations. Corresponding computational
domain included the domain, containing fan rotor blade, the domain containing booster IGV vane, thedomains containing rotor and stator blades of the first booster stage and the domain, containing
entrance of the bypass duct. We excluded fan stator from the calculation, because our attention was
attracted to the forward hemisphere, where booster noise in the experiment was more noticeable. The
process of reflection of noise from the stator has been decided to neglect. Also in our calculation weneglected interaction between the noise of the first stage and the second and third stages.
The computational grid for unsteady simulations was derived from the grid for the steady flow fieldcalculations. The grid was resized in X direction in order to provide the propagation of thehigh-frequency disturbances generated by the first booster stage without significant error. We tried toprovide resolution approximately equal to 15 points per wavelength for disturbances which
participates in the generation of tones at frequencies f 2 -f 1 , f 2 and f 2+f 1.
Finally, to avoid unphysical reflections from the boundaries we added to the grid buffer zones inwhich grid cells are stretched near the outer boundaries. In these blocks outgoing waves are absorbed
by numerical viscosity. The boundaries of the fine grid region in the computational domain for
unsteady calculations are shown in the fig. 2 by green lines. The boundaries of buffer blocks (partially)
are shown in the fig. 2 by the blue lines. The size of the grid was approximately 4.2 million cells.
4.2 Brief Description of the Calculation in the Frequency Domain
Though the article is devoted to the calculation in the time domain, some brief description of thecalculation in the frequency domain is needed for a better understanding of the comparison between
the results of different computations. The description follows the paper [15], from which we will takethe results for comparison (slightly different from those presented in [5] due to higher number of
harmonic fragments involved in the calculation).
We started the calculation setup from the determination of those circumferential modes for each of
the frequencies of interest (f 1, f 2 -f 1 and f 2+f 1 ), which are cut-on and probably arises in the strongest
interactions in the booster stage (the corresponding theory can be found in [5]). It is modes withcircumferential numbers m=-43; -14; 15; 28; 44 for f2, modes with m=-32; -3; 10; 26; 39 for f2 -f 1 andmodes with m=-38; -25; 4; 33; 46; 62 for f 2+f 1. Most of these circumferential modes arise in theinteractions between two or three blade rows. To resolve the modes, in the calculation were involved
67 harmonic fragments: 9 fragments in the first row, 13 fragments in the second row, 23 fragments inthe third row and 22 fragments in the fourth row. The size of the problem was equivalent to the one forthe calculation in the time domain with 94 million grid cells.
In spite of the large number of variables the calculation converged quite fast. It took approximately
10000 steps for satisfactory and 20000 steps for fine convergence. The results of the calculations will
be discussed in the next section.

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4.3 Calculation in the Time Domain
The calculation in the time domain was performed in full annulus setup. The computational grid
was obtained from the previously described grid for one blade channel. The grid blocks for n-th blade
channel were generated by duplication of the blocks in the first blade channel and subsequent turn on
to corresponding angle. The size of the final grid was 135 million cells.
The calculation was performed with the usage of multirate Runge-Kutta scheme, described earlier.
The number of time steps for one period of rotation was 25250. There were at least 170 steps per period
for any of disturbance among those which were resolved in the framework of the frequency domain
method. This gave significant reserve in time resolution for the calculation.
It took approximately one and the half periods of rotation for a solution to converge. In fig. 3 areshown the results of calculation as the instantaneous fields of static pressure (pulsations over steadyflow field) on the sections of constant diameter d=0.5D, where D is the diameter of the fan. We see
quite complex structure of the flow field.
The accumulation of data for modal analyses was started when the solutions had converged (and it
took whole period of rotation). The surface, on which modal analysis was performed, is shown in fig. 2
by red curve. The results of the modal analysis in the form of sound power levels of separate
circumferential modes are shown in fig. 4. The modes are designated by the pair of integers (h,m),where h is the number of harmonic of shaft rotation frequency and m – circumferential order. The
modes which power levels are lower than those of the strongest modes on more than 15 dB are notshown. Also only the modes with frequency less or equal to 122 harmonic of shaft rotation frequency
are shown in fig. 4, as the grid had insufficient resolution for the modes with higher frequency.
The results of the calculation in the frequency domain are also plotted in fig. 4. We see that both
calculations predict the same set of the strongest modes, and the sound power levels of the modes,
obtained in the different calculations are close to each other.

Figure 3. – Unsteady pulsations of static pressure field on the section of constant diameter d=0.5D

Figure 4. – Results of the modal analysis upstream of the fan. The modes are designated by the pair of
integers (h,m).
In general the calculation in the time domain took much more time than the calculation in the1613frequency domain. One time step in the frequency domain calculation is 4 times shorter than in thetime domain calculation, and also the number of time steps needed for the calculation is smaller.
The results of modal analysis were used to set sources for noise propagation problem in t he inlet.
The method of calculation and the geometry of the computational domain are completely equivalent tothose described in the paper [5]. For the calculation of noise propagations we used linearized Euler
equations for circumferential modes in the frequency domain, specified on the longitudinal section of
the inlet [8]. As a result the flow fields for each of the modes, presented in fig. 4 in the near field of theinlet were obtained.
To compute tone noise radiation in the far field Ffowcs Williams and Hawkings method was used.
Finally pressure pulsations were calculated in the points placed along the arc with radius equal to 4 mwith the center in the point of intersection between the surface, containing tips of the leading edges of
fan rotor blades, and the axis of shaft rotation. The points had uniform angular distribution within 1 -90
degrees with 1 degrees spacing.

5. THE COMPARISON WITH EXPERIMENTAL DATA
The results of calculations were compared with the results of the experiment for the fan model in
the CIAM test rig C-3A. Comparison was performed for 12 microphones placed at the same distancefrom the fan, as in the computation. As an initial data for comparison we used narrowband spectra forthese microphones obtained in the experiment. The frequency bands of the spectra were equal to 19.5
Hz. On the basis of these spectra the directivity diagrams for the harmonics of tone noise underconsideration were calculated.
The results of comparison are presented in the fig. 5. In the figure the directivity diagrams for boththe time domain and the frequency domain calculations and the experimental data for frequencies f 2 -f 1,
f 2 , f 2+f1 and f 2+2f 1 are shown. In general we see good correspondence between the calculations andbetween them and the experiment.

Figure 5. – Comparison between computational and experimental directivity diagrams in a forward
hemisphere

6. CONCLUSIONS
The results of the numerical investigation of the first booster stage tone noise at approach1614

operational conditions are presented. The investigation was performed using the time domain methodof calculation, based on multirate Runge-Kutta scheme. The method was developed in CIAM and
implemented in CIAM 3DAS solver. The results were compared with the results of the frequency
domain calculations of the tone noise of the stage and also with the experimental data obtained in the
CIAM C-3A acoustic test facility.
In general the results of the calculations in the time and in the frequency domains are close to eachother. Both the powers of different circumferential modes in the inlet and the directivity diagrams in
the far field are similar. Especially close results are observed for the directivity diagrams
corresponding to the blade passing frequency of the first stage rotor. Also for both calculations there issatisfactory correspondence with experiment. The results can be treated as a strong argument for thevalidity of the frequency domain method of the multistage turbomachinery tone noise calculations
which was presented in our previous works.

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