Simulation of flow noise generated by the interaction of inflow turbulence withthe leading edge of a foil
Mattias LIEFVENDAHL(1)(1) Swedish Defence Research Agency (FOI), Sweden, mattias.liefvendahl@foi.se

Abstract
Noise generated by the interaction of inflow turbulence with a lifting surface may be of interest for a numberof hydrodynamic applications, including a propeller operating in non-cavitating condition. In the present work,
simulation-based methods are applied for the prediction of the radiated noise from a wing in strong inflow
turbulence, at a chord-based Reynolds number, Rec = 256 000, and a Mach number of 0.09. The case has
also been investigated experimentally, and a comparison is made between measurement and simulation results.
Large-eddy simulation is employed to predict the flow, with synthetic turbulence fluctuations applied at the
inflow boundary. The LES-results are then used to compute the acoustic source terms of an acoustic analogy.
The sensitivity of the results to inflow turbulence level and turbulent length scale is investigated. The analysis isfocused on the unsteady flow features associated with flow noise generation.
Keywords: Large-eddy simulation, Flow noise, Hydroacoustics

1

INTRODUCTION

Flow noise generated by a lifting surface is an important model problem for a number of hydrodynamic applications, including a propeller operating in non-cavitating condition. For a lifting surface, several flow mechanisms
may induce significant flow-generated sound, including the trailing edge flow, the tip vortex and the interaction
between inflow turbulence and the leading edge, [2]. In certain conditions, including strong inflow turbulence,
the leading edge contribution may be dominant. In the present study, such a test case is investigated, primarily
by simulation-based methods.
Pioneering studies of leadinge-edge noise was carried out in the 1970:ies. Amiet derived an analytical model ofthe radiated sound from a thin foil, [1], by analysis of scattering from a semi-infinite plane. See [13] for a later
review of developments and refinements of this approach which is widely used. An aeroacoustic measurement
campaign of leadinge-edge noise in an open jet anechoic wind tunnel, [11], was made in connection withthese original investigations. This type of models require a characterization of the inflow turbulence in terms ofintensity, in particular of the fluctuations in the upwash velocity component, and various length scales associatedwith the turbulence.
The development of computational technology has lead to the possibility of employing CFD, in particular scaleresolving simulations, in order to directly simulate the flow mechanisms generating sound. Such is the approachof the present paper, where LES is employed to simulate the flow around the foil, including the inflow turbulence which was generated by a synthetic method, [7]. The use of such high-fidelity methods has the potential
to improve predictions as compared to classical techniques, e.g. concerning the effect of wing thickness andthe deformation of turbulence in the flow around the wing. This LES approach, and its use to compute soundsources in an acoustic analogy, was applied in, [9], to investigate aeolian tones.
The simulated case consists of isotropic and homogeneous turbulence impinging on a wing with a NACA0012 profile at zero angle of attack. Both radiated sound and the inflow turbulence has been investigatedexperimentally for this case, [15]. The flow is simulated at conditions corresponding to the experimental set-up,
and the results are analyzed with a particular focus on to flow around the leading edge with a view to soundgeneration. Furthermore, the influence of intensity and length scale of the incoming turbulence is investigated.

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1.1 Case description
The main physical parameters of the simulated case are given in table 1, and the relatively thick wing profileis shown in figure 1. The turbulence characteristics have been inferred from the results reported in [15]. The
inflow turbulence of the simulations is characterized by a turbulence intensity, k = 4.86 m2 /s2 , and a turbulent
length scale of, lt = 15 mm, which are input parameters to the synthetic turbulence generation method.
Table 1. Main physical parameters, corresponding to the experimental measurements, [15], leading to a Machnumber, Ma = V0 /c = 0.09, and a chord-based Reynolds-number, Rec = cV0 /ν = 256 000.
Quantity
Wing length
Wing chord
Max. wing thickness
Velocity
Kin.visc.
Sound speed

Notation
L
c
t
V0
ν
c0

Value
0.2
0.1
11.9
30
1.17·10−5
333

Unit
m
m
mm
m/s
m2 /s
m/s

Figure 1. The NACA-0012 wing profile.
The simulation campaign, see table 2, is designed to investigate the sensitivity of the flow and the radiatedsound to changes in turbulence intensity and length scale. The first set of four simulations (Case 1–4) tests
independent variation of k and lt . The second set of three simulations (Case 5–7) tests variation of the turbulent
length scale lt around the value measured in [15]. The simulated time intervals are described in section 2, inconnection with a description of time step size and other numerical parameters.
Table 2. Simulation campaign. Turbulent kinetic energy, k, and turbulent length scale, lt , is varied between
simulation cases. For the remaining physical parameters, see table 1.
Case

1

2

3

4

5

6

7

k (m2 /s2 )

1.50
10

1.50
15

3.00
10

3.00
15

4.86
10

4.86
15

4.86
20

ly (mm)Throughout this paper, the following two coordinate systems are used. A Cartesian coordinate system with thex-axis pointing in the main flow direction, with, x = 0, at the wing leading edge, the y-axis in the spanwise
direction, with, y = 0, at mid-span, and the z-axis in order to have a right-handed coordinate system, with, z = 0,in the plane of the wing. A polar coordinate system in the xz–plane, with radial coordinate, r = (x2 + z2 )1/2 ,and the polar coordinate defined with, θ = 0, in the upstream direction.

2

METHODS

The computational approach consists of incompressible LES of the flow around the foil, and then the application of an acoustic analogy for the far-field radiated sound. The source terms in the acoustic analogy areapproximated using the simulated flow. Spatial filtering of the incompressible and viscous momentum equation

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of the flow leads to the following equation, see [14].
∂ τi j
∂ vi
∂ vi
1 ∂p
∂ 2 vi
+vj
=−
+ν
−
∂t
∂xj
ρ0 ∂ xi∂ x j x j ∂ xi x j
Here, τi j = ui u j − vi v j , is the subgrid stress tensor and overbar indicated filtered quantities. LES subgrid modelling consists of providing a computable expression for τi j . In the simulations of this paper, an implicit LES
approach was used, where the subgrid dissipation is considered to be represented by the numerical diffusion,see e.g. [6]. See, [14], for a general reference on LES and, [9], and the references therein for more information
about the particular approach and implementation used here. From now on, the overbar notation is dropped
and use is made of the notation, u = v1 , v = v2 , and w = v3 for the velocity components. The acoustic analogy
employed was proposed in, [5], and also employed in [8, 9]. The formulation supports so-called porous acoustic
surfaces for the sampling of flow quantities for source computation, but here the acoustic surface is chosen tocoincide with the foil, which reduces the method to that proposed by Curle, [3]. The results analysis below is
however strongly focused on the flow around the leading edge.
The flow solver and the acoustic analogy have been implemented in the open source software library OpenFOAM, see [10] and the references therein. The finite volume method was used to discretize the LES equations,with cell-centered storage of the unknown quantities. All approximations are second-order accuracte. For theconvective scheme, a blending is used with 25% second-order upwind and 75% linear scheme. This ensures anadequate level of numerical dissipation for ILES, [12]. The computational domain is the following rectangular
block.
−24 cm < x < 110 cm− 10 cm < y < 10 cm− 50 cm < z < 50 cm
Recall that the wing chord is, c = 10 cm, and the leading edge is located at, (x, z) = (0, 0). At the inflow,
x = −24 cm, a uniform inflow, with turbulence characterized by k and lt , is prescribed. This boundary condition
for synthetic turbulence, [7], is implemented in an openly available library to OpenFOAM.
The computational grid is generated as a block-structured grid with hexahedral cells. Care must be taken to
make the grid sufficiently fine to sustain the convection of the inflow turbulence and also to allow a proper
representation of the boundary layer over the wing. The total number of cells in the grid is 46 · 106 . Between
the inflow and the leading edge, the cells are roughly rectangular blocks with maximum cell edge length in thethree coordinate directions of, ∆x ≈ 1.0 mm, ∆y ≈ 1.0 mm, and ∆z ≈ 0.5 mm. This is to be compared with thesimulated turbulent length scales, 10 mm < lt < 20 mm. The Re-number is sufficiently low so that there is notransition to turbulence in the boundary layer over the wing. Hence the wing is simulated as laminar and nowall model is employed. This implies that very thin cells (aiming for y+ ≈ 1) must be used adjacent to thewing in order to resolve the boundary layer profile. The typical height of the first cell layer over the wing is,∆h ≈ 10 µm.
The simulations are time-resolved and CFL-number limited. The time step, ∆t = 1 µs, was used in all simulations. This time step can be compared to the wing flow-pass time, T f p = c/V0 ≈ 3.33 ms. The simulations werestarted from a uniform flow state and simulated during 20 ms (= 6T f p ) before the flow was considered to haveassumed a statistical steady state. After that, flow statistics was gathered during at least 10 ms for all simulation
cases, and up to 90 ms for the most interesting cases.

3

RESULTS

An illustration of the simulated flow around the wing, for Case 6, is shown in figure 2. From the stagnationline along the leading edge, the flow is accelerated over the wing until it reaches the adverse pressure gradientnear the trailing edge. The scales of the inflow turbulent structures, lt = 15 mm, can be seen relative to the wing
thickness, t = 11.9 mm. The radius of curvature at the leading edge is, rc = 1.6 mm. Additional unsteadinessdevelops in the flow near the trailing edge, as can be seen in the surface pressure distribution. The structure of
the turbulence in the near wake of the wing can also be seen in the figure.

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Figure 2. Illustration of the simulated flow around the wing (Case 6). The instantaneous axial velocity, u, isshown on a streamwise plane (y constant) and on a cross-plane (x constant) in the near wake. The range of thecolor scale is, 10 m/s < u < 40 m/s. The instantaneous pressure distribution on the wing is shown in grey scalewith range, −600 Pa < p < 200 Pa.

Figure 3. Flow statistics for Case 6. The normalized mean axial velocity, hui/V0 , is shown in the top graph,with color scale in the range, 0.0 – 1.2. The second graph shows the standard deviation of axial velocity
fluctuations, normalized by the free-stream velocity, σ (u0 )/V0 , with color scale in the range, 0 – 10%. The
third graph shows, σ (v0 )/V0 , and the bottom graph shows, σ (w0 )/V0 , both with the same color scale range, 0 –
10%.

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The flow statistics, for Case 6, is shown in figure 3. The mean flow is not significantly affected by the strong
inflow turbulence, but shows the usual behaviour for a wing at no angle of attack. Near the leading edge,
the level of fluctuations behaves quite differently for the three components of velocity. The inflow value, k =4.86 m2 /s2 , corresponds to, σ (u0 )/V0 = 6%, and the same for the other two velocity components. This level isroughly maintained as the inflow turbulence is convected from the inflow to the leading edge. In the vicinity of
the leading edge, the axial fluctuations, σ (u0 ), decreases sharply while the upwash fluctuations, σ (w0 ), displays arapid increase. The spanwise fluctuations, σ (v0 ), exhibits a mild increase near the whole wing surface, including
at the leading edge.

Figure 4. Time series and spectrum of velocity components in a point 10 cm upstream of the leading edge,at (y, z) = 0, 0, for Case 6. The left graph shows the time series of the three velocity components during thecomplete simulation interval. The right graph shows the power spectral density of the corresponding time series.
The vertical lines in the right graph indicates the turbulence frequency, ft = 2 kHz, and the grid frequency,f∆x = 30 kHz.
The inflow turbulence, for Case 6, is shown in figure 4, in terms of the velocity signals (both time series
and spectrum) in the indicated point location 10 cm upstream of the leading edge. The initial flat part in thetime series, up to t ≈ 5 ms, indicates the time it takes for the inflow turbulence to be convected from the
inflow boundary to the point. After that, the curves show similar level of turbulent fluctuations around the
local mean value. The influence of the wing is negligible at this point. The power spectral densities of thevelocity fluctuations are also shown in figure 4. The length scale of the inflow turbulence, lt = 15 mm, whichis convected with the free-stream velocity, corresponds to a frequency, ft = V0 /lt = 2 kHz, which is indicated in
the graph. Also of interest are the finest scales which are possible to resolve in the LES computation. To get
an idea of this, the grid size, ∆x ≈ 1.0 mm, can likewise be converted to a frequency, f∆x = V0 /∆x ≈ 30 kHz,
which is also indicated in the graph. The power spectral densities decrease with a factor ∼ 109 between these
two frequencies.
The turbulence deformation around the wing was shown in figure 3. In figure 5, the same quantities are plottedalong a streamwise line upstream of the leading edge of the wing. Figure 5 also compares results from Cases
5–7 and measurement data from [15]. The focus is on the levels of fluctuations, but the mean velocity curve is
included to show the upstream influence of the leading edge, which is noticeable up to ≈ 5 mm upstream of theleading edge. The level of fluctuations in the velocity components show very significant discrepancies between
the simulations and the measurements. The measurement data is taken from figure 12b in [15]. It should bepointed out that this is a difficult aspect of the flow to measure, and also to simulate to some extent. First it isobserved that there is a problem with the measurement data for σ (u0 ), since this quantity is required to decay

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Figure 5. Flow statistics along a streamwise line at, (y, z) = (0, 0), upstream of the wing leading edge. Comparison of results from Case 5–7 (line style indicated in the figure) with measurements from figure 12b in [15],shown with the full line in the three graphs (for fluctuation levels). The top graph (contains no measurementdata) shows the normalized axial mean velocity, hui/V0 . The remaining three graphs show the standard deviationof the velocity fluctuations in percent, for u0 , v0 and w0 respectively.
to zero at the leading edge. As is clear from figure 3, only a small deviation in the measurement point implies
that this decay is missed. For the other two components, spanwise (v0 ) and upwash (w0 ), both simulations andmeasurements show a significant increase close to the leading edge, at a distance of less than 2 mm, say. The
measured increase is however very much larger than that predicted by simulations. E.g. σ (v0 ) increases fromthe inflow value of 6% to 17% in the measurements, whereas the maximum value in the simulations of thisquantity is only around 9%. Similar discrepancy is also noted for σ (w0 ). Based on the simulation results, theeffect of turbulent length scale can also be investigated. The only quantity in figure 5 which is affected by thisis the upwash fluctuations, σ (w0 ). An increase in turbulent length scale leads to an increase in peak σ (w0 ). The
maximum for Case 5 (lt = 10 mm) is 8%, the maximum for Case 6 (lt = 15 mm) is 11%, and the maximum forCase 7 (lt = 20 mm) is 11%,
Next, the influence of the turbulent length scale, lt , on the radiated noise is checked using the model ofAmiet, [1]. In figure 6, these predictions are compared with the measurement data from [15]. Results arecompared in a microphone placed 1 m above the leading edge of the wing, roughly in the main radiation direction. According to Amiet’s model, an increase in lt leads to a shift of the spectrum towards lower frequencies,
and has a minor effect on the level. The agreement with measurement data is reasonable, but not sufficiently
good to determine which lt corresponds best to the experimental set-up. It is also noticeable that the measurement resolution for low frequencies is not sufficient to identify whether a peak exists in the spectrum. A
further interesting uncertainty which concerns the model is how large the error is in the thin foil approximation. The assumption in the model is that of scattering from a plane (zero thickness) whereas the thicknessof the NACA-0012 foil is comparable to the turbulent length scale. Furthermore, as was discussed for fluctuations in velocity components in connection with figures 3 and 5, the turbulence is significantly deformed at
the leading edge. Hence, the inflow values (k, lt ) are hardly accurate at the sound source (the leading edge).
These are well-known issues, see e.g. [4, 13], and various techniques exist to improve the models, such as

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Figure 6. Sound-pressure spectrum in a microphone located 1 m above the wing leading edge, (x, y, z) =(0, 0, 1) m. The curves indicate spectra obtained with Amiet’s model, [1], with k = 4.86 m2 /s2 and varying
turbulence length scale, lt , as indicated in the graph (10, 15 and 20 mm, respectively). The measurement data
is from figure 14 in [15].
Rapid-distortion theory as applied in, [15]. It is however an area where LES-based predictions may provide asignificant improvement over existing techniques due to the potential to accurately simulate the interaction of
the inflow turbulence with the foil leading edge. At the time of writing of this paper however, the processing of
the results from the acoustic analogy has not been fully carried out, but will be included in future publications.

4

CONCLUSIONS

Large-eddy simulation was applied to investigate a foil in strong inflow turbulence, corresponding to the experimental set-up of [15]. The influence of turbulence level and turbulent length scale was investigated in thesimulation campaign. The turbulent length scale, 10–20 mm, is comparable to the thickness of the foil, 11.9 mm.
The analysis was focused on the flow around the leading edge, and the deformation of the turbulence in thisregion, including comparison with measurements.
Significant modification of the level of fluctuations in the velocity components occurs only quite near to theleading edge, at a distance of less than 2 mm. Hence it is challenging to characterize it with measurements,
and it is also believed that many RANS turbulence models have difficulties prediction the components of the
Reynolds stress in this region. At the same time, these flow features are critical for the flow-generated soundat the leading edge.
The results in figure 5 show large discrepancies between the LES results and the measurements of [15]. In
the author’s opinion, the LES results are more accurate. This is supported by the unphysical behaviour of themeasurement curve for σ (u0 ) which should decay to zero at the leading edge, and also by the fact that the grid
resolution for LES should be sufficiently fine to accurately capture this part of the flow. This opinion should
be substantiated by a mesh refinement study, which would provide error estimates for the LES prediction. It
is noted that the measurements of this difficult flow quantity are not provided with uncertainty quantificationin [15]. The simulation results show an increase in the fluctuation level of the upwash velocity component near5318

the leading edge as the turbulent length scale, lt , is decreased. The fluctuation level of the other two velocity
components is rather insensitive to lt .
In conclusion, further work is necessary to clarify the exact behaviour of the turbulence near the leading edge.
It is believed that LES is a tool which can provide accurate prediction of the relevant flow quantities. These
can then be used either as input to simplified noise models of the type described in, [1, 13], or directly input
into an acoustic analogy as described in section 2. At the time of writing of this paper, the processing of theresults from the acoustic analogy has not not been fully carried out, but will be included in future publications.

5

ACKNOWLEDGEMENTS

Paul Croaker is gratefully acknowledged for interesting discussions and valuable insights concerning different
modelling techniques for flow-generated sound. His assistance in producing the results of the acoustic model infigure 6 is also much appreciated.

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