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

Influence of the Mesh Size on the Aerodynamic and Aeroacoustics
of a Centrifugal Fan using the Lattice-Boltzmann Method
Rebecca SCHAEFER1; Martin BOEHLE2
1

University of Kaiserslautern, Germany

2

University of Kaiserslautern, Germany

ABSTRACTThe present paper deals with the direct and coupled simulation of the aerodynamics and aeroacoustics of acentrifugal fan connected to a suction-side test bench using the commercial Lattice-Boltzmann CodePowerFLOW of the Exa Corporation. The focus is on the variation of the mesh size in the finest grid
resolution, which influence the time step immensely. The aim is to determine the smallest grid size in such a
way that both the flow and acoustics of the centrifugal fan can be accurately predicted with an acceptabletotal computing time. In this study, three different mesh sizes for two different geometries of the finest
resolution area will be investigated and compared.
The results of the Lattice-Boltzmann simulation are compared and validated with measurement results
obtained according to an ISO standard procedure at the Institute of Fluid Mechanics and Fluid Machinery at
the University of Kaiserslautern. Flow information in term of the pressure increase of the centrifugal fan iswell captured by the simulation. In addition, the noise is well predicted with respect to blade passing
frequency and the general level of the spectrum at the considered microphone positions.
Keywords: Lattice-Boltzmann Method, Centrifugal Fan, Aeroacoustics

1. INTRODUCTION
Due to market requirements and ever stricter guidelines, noise emissions of fans are becoming
increasingly important in addition to energy efficiency. This applies in particular when people haveto work near running fans. For this reason, the development of fans aims not only at more efficientrunning fans but also ever quieter fans. In order to keep the development process cost -effective andas short as possible, numerical investigations are used to day. Classic numerical methods do not allow
to calculate aerodynamics coupled with aeroacoustics. The Lattice - Boltzmann method is a relatively
new method that promises to close this gap. It always solves for the unsteady flow field using
relatively small time steps which allow for an accurate resolution of the alternating pressures that are
responsible for the acoustics. This allows to solve the sound pressures directly without using an
acoustic model. Also, the method is known to be low dissipative, which could be shown by
Marié et al. (1). Therefore, the LBM is ideally suited for the simulation of aeroacoustics. A big
advantage of the Lattice-Boltzmann method is also that it is more or less an immersed boundary
method. Flow areas of complex geometry can be meshed without effortful geometry preparation,
which leads to a huge time saving.
Perot et al. (2) could show that the LBM can be used to determine the performance and frequency
spectrum of an axial fan. Also Zhu et al. (3) and Sturm et al. (4) could successfully show that the
aerodynamics and aeroacoustics for an axial fan with different head gaps and under different inflow
conditions can be calculated with the commercial LBM code PowerFLOW of the company Exa. There
are also some studies on centrifugal fans where the performance and acoustics have been successfully
calculated using PowerFLOW. The fans were connected on the pressure -side and the investigation of
the influence of so-called obstructions was in focus. (5, 6)
In this paper, the commercial Lattice-Boltzmann code PowerFLOW is used to simulate the flow
and acoustics in a centrifugal fan without a volute casing connected to a suction-side test bench. A
detailed mesh study is in focus. Two different geometries are selected for refinement because it has
1
2

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Rebecca.schaefer@mv.uni-kl.de
Martin.boehle@mv.uni-kl.debeen shown that the choice of the refinement zone has an immense influence on the results especiallyregarding the acoustics. In the first case an offset of the impeller geometry is chosen for refinement,in the second case an addition donut-like ring in areas of high velocity is selected to set a finer
resolution.
For the validation of the LBM simulation, aerodynamic and aeroacoustics measurements are
performed on the institute’s own fan test bench, which was developed according to DIN ISO 5801.
The results of the LBM simulation are compared to the experimental results.

2. FUNDAMENTALS OF THE LATTICE-BOLTZMANN METHOD
2.1 Velocity Distribution Function
In contrast to the Navier-Stokes equation, which is based on a macroscopic view, the Lattic eBoltzmann Method uses a mesoscopic approach. A molecular consideration is taken in which themolecular movements are described statistically. The molecules themselves are characteriz ed by a socalled velocity distribution function, which is given in Equation (1).
(1)
It describes the probability of finding dN molecules with a certain velocity d[ at a certain point inspace dV.
2.2 Boltzmann-Equation
The Boltzmann equation is a generally valid determination equation for the distribution function f.
Due to the fact that it is a complex integro-differential equation, it cannot be solved even with modern
methods. The derivation of the Boltzmann equation is based on the preservation of the particles in thephase space. The basis form in tensor notation is presented as(2)
where f = f (x,[,t) is the above described velocity distribution function. The left side of Equation (2)contains the time and convective change of the distribution function as well as the change due toexternal forces Fi. The right side of Equation (2) describes the change of the distribution function due
to interaction of the particles and is called the collision term. Since the collision term is composed of
a complex integral, it is approximated by a much simpler mathematical approach. Often the so -called
BGK model according to Bhatnagar, Gross and Krook (7) is used. This assumes that every nonequilibrium condition leads to an equilibrium condition, cf. Equation (3)(3)
Z describes the molecular collision frequency and indicates the number of transitions from non equilibrium to equilibrium per time interval t. f stands for the non -equilibrium distribution and f eq for
the local equilibrium distribution. The equilibrium distribution is equal to the Maxwell distribution.
(8, 9) Using the BGK model (7) for the collision modeling and a Chapman-Enskog development (8)the Boltzmann Equation in Equation (2) recovers the Navier-Stokes equation.
2.3 Lattice-BGK-Equation
In order to solve the BGK-Boltzmann equation numerically, a discretization in time and phasespace has to be applied. In the latter case, a spatial discretization of the Cartesian space into an
equidistant mesh is performed, cf. Figure1. Each mesh cell is called a lattice.
In addition to the spatial discretization, the velocity is discretized to predefined velocity directions.
In the three-dimensional case, model D3Q19 is often used. This is also implemented in PowerFLOW.
The model works with i = 19 defined directions, which are indicated by [i in Figure 1.
The discrete BGK-Boltzmann equation presented in Equation (4) is called the Lattice-BGK or the
LBGK equation.
(4)
The equation indicates that a calculation step consists of a transport step ݂௜ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ and a collision௘௤
step ߱ ή ߲‫ ݐ‬ή ݂௜ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ െ ݂௜ ሺ‫ݔ‬ǡ ‫ݐ‬ሻ. Algorithmically, these two steps are usually calculated separately for1883

each velocity [i in each lattice per time step. In an explicit procedure the new distribution functions
f i(x,t) for the next time step ‫ ݐ‬൅ ߲‫ ݐ‬can be determined.

Figure 1 – Spatial discretization (left) and model of D3Q19 of the velocity discretization (right)
With the known distribution function f (x,[t) the macroscopic quantities of interest can becalculated by determining the so-called moments. It is shown in Equation (5) and (6) exemplary forthe calculation of macroscopic density and velocity.
(5)(6)
2.4 Selection of Timestep
The timestep selection is based on the fact that a particle moves with the velocity [0 in one timestep
exactly one lattice length 'x. The particle velocity ߦ଴ ൌ ξ͵ ή ܿ௦ is determined by the so-calledܿ
simulation Mach number ‫ܽܯ‬௦௜௠ ൌ ௖௛௔௥ൗܿ௦ with the characteristic velocity c char and the isothermal
speed of sound c s. The timestep 't can thus be determined from(7)
whereby the simulation Mach number ‫ܽܯ‬௦௜௠ in only a scaling factor with which ensures that thecharacteristic velocity c char is one scale order smaller than the speed of sound. (8) In the case of
acoustic problems, the simulation Mach number ‫ܽܯ‬௦௜௠ is set equal to the physical Mach number‫ܽܯ‬௥௘௔௟ ൌ ‫ݑ‬ൗܽ௦ , so that the time step is ultimately only dependent on the grid size 'x.

3. EXPERIMANTAL SETUP
3.1 Test Rig
The experimental investigations were performed on the fan test rig in the anechoic room of theInstitute of Fluid Mechanics and Fluid Machinery of the University of Kaisers lautern. It is a suctionside chamber test bench, which is developed according to DIN EN ISO 5801 and is schematicallyshown in Figure 2.
The test bench consists of two main components: the inlet section and the suction chamber. The
inlet section includes an inlet nozzle, a diffusor, a throttle and a pipe section. All components of theinlet section with the exceptions of the pipe section, which connects the inlet section with the suction
chamber, are located outside the anechoic room. This should ensure that the noise, which is generated
by the throttle, is decoupled to the acoustic measurements of the fan. In addition, perforated discs
according DIN EN ISO 5136 are used, which allow low-noise throttling due to the uniform distribution1884

of the holes. The inlet nozzle, at which the differential pressure 'p 0 is measured, is used to determine
the volume flow. The differential pressure 'p 1 is acquired to determine the pressure increase of thefan. Furthermore, the temperature and the ambient pressure in the anechoic room, which are used forthe physical conditions in the simulation, are recorded. The sound pressure is measured on ahemisphere behind the fan connected to the suction chamber. As in the simulation, the sound pressure
and aerodynamic parameters are measured simultaneously.

Figure 2 – Schematic of the fan test rig (left) and investigated centrifugal fan (cf. (10)) (right)
3.2 Measurement Technique
The aerodynamic of a fan is quantified by the pressure increase and the efficiency. The values arecalculated according to DIN EN ISO 5801. In Equation (8) the formula for the static pressure increaseof the fan is given. The indices 1 and 2 indicate the position of the evaluation position upstream anddownstream of the fan, whereby the ambient pressure p amb is be assumed for the pressure p st2.
(8)
The aeroacoustics of a fan is determined by the sound pressure level L p as well the total soundpressure ‫ܮ‬ത௣ obtained from the frequency analysis. The quantities are defined according DIN 45635 1 and are given in Equation (9) and (10) with p as the alternating pressures and p 0 as the reference
sound pressure level.
(9)
(10)

4. NUMERICAL SETUP AND PARAMETER
The commercial LBM-Code PowerFLOW 6-2019 is used to compute the unsteady flow field andthe acoustic of a centrifugal fan without the volute casing. Because the pressure loss of the screens
and the honeycomb inside the suction chamber are not known until now the simulation area starts at
the pressure measurement location 'p1 , cf. Figure 2 and 3. The overall simulation area includes theanechoic room; foams at the walls are modelled with so-called sponge zones, where the viscosity is
increased over three layers. A velocity boundary is selected as the inlet boundary condition. The
velocity corresponds to the mean velocity resulting from the measured volumetric flow. At theboundaries of the overall simulation area a pressure outlet is set. The rotation of the fan is modelled
by a local reference frame (LRF) with the sliding mesh approach using the rotating speed obtained by
the measurements.
Several variable resolution (VR) regions are used to refine the computational domain, some ofthem are shown in grey in Figure 3. Two different studies of the geometry for the finest resolution arecompared: In the first case the refinement for the grid study is done by an offset of the impeller

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geometry. The offset is chosen so that the gap between the impeller and the impeller inlet nozzle isalso refined and the flow can be dissolved. In the following, this method is r eferred to as “Offset”
refinement.

Figure 3 – Simulation model (left) and measurement locations (right)
In the second case the refinement is done by a ring which resolves the high velocity regions at the
impeller’s outlet and the gap between the impeller and the impeller inlet nozzle. In the ensuing, thiscase is named as “Donut” refinement. Three different meshes will be studied for the two different
refinement cases - one fine mesh with finest cell size of 'x =0.2 mm, one medium with 'x = 0.4 mm
and one coarse with 'x =0.8 mm.
In Figure 3 the positions of the microphone locations are shown on the right side. The grid
resolution of 'x 5 = 12.8mm is determined in the area of the microphone positions. According toBrès (11) et al., at least 12 – 16 grid points per wavelength should be selected for the propagation ofsound waves, which results to a maximum cutoff frequency of around f max = 1.7 kHz – 2.2kHz.

Figure 4 – Mesh with offset refinement (left) and with donut refinement (right)
In Figure 4 a section of the discretization of the computational domain in a cross-sectional area
through the impeller main axis for the two different refinement cases representative with the medium
mesh are shown. In Table 1 the total number of voxel of the different meshes is presented.
For the simulation Mach number, the real physical Mach number is used, which leads to a Mach
number of approximately Masim = 0.1.
Table 1 – Total number of voxels of the different meshes
Mesh size

Offset Refinement

Donut Refinement

Coarse (0.8 mm)

26.6 x 106

9.7 x 106

Medium (0.4 mm)

35.0 x 106

31.3 x 106

Fine (0.2 mm)

106.1 x 106

183.7 x 106

The simulation is executed over a physical time of approximately 0.85 s. Convergence is archived

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after approximately 0.2 s physical time, which corresponds to approximately 12.0 impeller revolutions.
The evaluations are made over a signal time of 0.65 s. The simulations are performed onIntel XEON E5-2670 processors in case of the offset refinement and on Intel XEON SP 6126 in caseof the donut refinement. In Table 2 the CPU-h per impeller revolution are listed for the different
meshes.
Table 2 – CPU-h per impeller Revolution for the different meshes
Mesh size

Offset Refinement

Donut Refinement

Coarse (0.8 mm)

99.2

71.3

Medium (0.4 mm)

291.4

298.4

Fine (0.2 mm)

2103.62642.1

5. RESULTS
In order to make the results comparable to the measurements, the same settings for the evaluationof the power spectrum are used for simulation and measurement, cf. Table 3.
Table 3 – Settings for evaluation of the power spectrum
Parameter

Settings

Frequency resolution

6.25 Hz

Window

HanningSignal length

0.65 sWindow overlapping

50 %AveraginglinearIn Figure 5 on the left side the fan’s pressure increase is presented over the different mesh sizes
for the two refinement approaches “Offset” and “Donut”. In both approaches the simulation results
converges to the experimental values with finer mesh. The deviation between the fine mesh and the
medium mesh as well between the two refinement approaches are very small, but better results are
obtained by the donut refinement. With that refinement case and the fine mesh an absolute deviationof approximate 8 Pa is achieved which correspond to a relative deviation of 2 %. The greatest
deviation is around 10 % with the coarse mesh.

Figure 5 – Pressure increase (left) and total sound pressure (right) over the mesh size
The total sound pressure level is also shown in Figure 5 on the right side. Here, the choice ofrefinement geometry has a greater influence on the results. Best results are received by the fine andmedium mesh with the donut refinement where the relative deviation is around 1 - 2 %. The deviationsof the offset refinement are up to 6 % compared to the experiment.

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To better understand where the deviations are exactly, the sound pressure above the frequency isshown in Figure 6. On the left side the power spectrum of the offset refinement is presented and on
the right of the donut refinement. Areas above the cutoff frequency, which results from the grid
resolution in the area of the microphone positions, are marked in grey. Also the blade passing
frequencies (BPF) is marked as grey dotted line.
In both cases, the coarse mesh does not reproduce the acoustics of the fan well. A similar shape aswith the coarse mesh can also be achieved with the medium mesh with offset refinement (cf. Figure 6,left). There is a drop of the spectrum noticeable starting at approximately 800 Hz with those threemeshes. This indicates that the flow in high velocity regions, which is discretized with 0.8 mm cells,
is not completely resolved. No cutoff frequency in the range of 800 Hz is detectable with the finemesh and the medium mesh with donut refinement, which seems that a cell size of 0.4 mm is sufficient
to resolve the high velocity areas.
With all meshes except the fine mesh with offset refinement, the blade passing frequency is
captured very well. The best agreement can be obtained by the medium mesh. The coarse mesh slightly
overestimates the amplitude of the blade passing frequency and with the fine mesh it is slightly
underestimated. So in total, the results show that a simple offset refinement is not sufficient to depict
the acoustics correctly.

Figure 6 – FFT of offset refinement (left) and donut refinement (right) with different mesh sizes
If the accuracy of the simulation results is considered together with the computation time, which
is given in Table 2, it turns out that the best compromise between accuracy and computation time isavailable with the medium mesh and the donut refinement. The flow in the form of the pressure
increase of the fan is still represented with an accuracy of 3 % and the acoustics in the form of thetotal sound level with approximately 2 % deviation in comparison to the experiment. The blade
passing frequency and the general sound power level except in the range of 500 Hz and 800 Hz arevery well achieved. It is assumed that the overestimation can be explained by the fact that the boundary
layer of the impeller surface with 0.4 mm is not as well resolved as with 0.2 mm. However, this needs
further investigations.

6. CONCLUSION AND OUTLOOKIn this paper a direct and coupled simulation of the aerodynamic and aeroacoustics of a centrifugal
fan with the commercial Lattice-Boltzmann Code PowerFLOW was executed. A mesh size study,
where the finest cell size was varied, was performed. Two different geometries to set the finest cellsize was analyzed and compared. It could be shown that the shape of the refinement geometry has an
influence on the results, especially on the acoustics. A simple offset of the impeller geomet ry is not
sufficient for the refinement to reproduce the acoustics by the simulation, but improvements inperformance can be observed. Very good results can be achieved if, in addition, a refinement in theform of a donut-like ring geometry in the range of high velocity areas is taken into account. Best result
regarding accuracy and computational effort can be obtained with a medium mesh with a cell size of
0.4 mm with the donut-like ring refinement.
In future investigation additional centrifugal fan’s operating points could be analyzed and the casesetup could be extended to include also the volute casing. In addition, geometry changes can be made1888

to the volute casing in order to determine whether these can also be reproduced with the Lattice Boltzmann simulation. In addition to these points, the simulation and the measurement are compared
under visual aspects in order to gain a deeper understanding of the noise generation mechanisms.

ACKNOWLEDGEMENTS
The study was funded by the German Federal Ministry of Economics and Energy (BMWi) via the“Arbeitsgemeinschaft industrieller Forschungsvereinigungen Otto von Guericke e.V. “ (AiF) and the“Forschungsvereinigung für Luft und Trocknungstechnik e.V.” (FLT). The numerical simulations were
executed on the high performance cluster “Elwetritsch”, which is part of the “Alliance of High
Performance Computing Rhineland-Palatinate” (AHRP) and is managed by the regional computer
center of the University of Kaiserslautern (RHRK).

NOMENCLATURE
Symbols

Subscripts and Greek Symbolsc

Velocity1

InflowfFrequency

2

Outflowf

Velocity distribution function

amb

Ambient

Lp

Sound pressure

charCharacteristic

Ma

MACH NumbereqEquilibriump

PressurefaFree blowingpoReference sound pressure level'x

Lattice size

Q

Volume flowUDensity

t

Time

[

Velocity of particlesuCircumferential velocity

Z

Collision frequency

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