1 Introduction

Acoustic mirrors and phased microphone arrays are standard tools used to localize and quantify aeroacoustic sources. However, as with all imaging methods, these acoustic techniques also have a limited resolution. On the one hand, measurements in the far-field do not give any direct information about the source processes occurring in the near-field, whereas measurements in the near-field, although able to deliver information about the structures and the dynamics of the involved flow, are nevertheless very difficult to analyze in terms of an estimation of the radiated sound from near-field data. One way around this is the simultaneous measurement of the acoustic pressure in the far-field together with some other near-field quantity. In this manner, the far-field pressure can be correlated with the near-field quantity to identify aeroacoustic sources. This approach was used by many researchers in the past, and different near-field quantities were considered for the correlation.

For example Clark and Ribner (1969) correlated the fluctuating lift of an airfoil with the acoustic pressure. Siddon (1973) measured the fluctuating surface pressure on a circular plate and correlated it with the acoustic pressure to obtain the surface dipole strength on the plate. Lee and Ribner (1972) used a hot film probe to measure the flow velocity in a jet, in which they correlated v 2 r with the acoustic pressure, where v r is the component of the flow velocity in the direction of the far-field microphone. Sunyach et al. (1974) followed a similar approach using a hot-wire probe in the wake of an airfoil. The expression v 2 r was used as near-field quantity, because it corresponds to the source strength in Proudman’s form of Lighthill’s integral for aerodynamic noise (see Lee and Ribner 1972). Siddon (1974) and Rackl and Siddon (1979), who introduced the name ”causality correlation” for the correlation approach, identified a possible difficulty with this method when the near-field quantity is measured using a probe. If a hot-wire or a microphone with a nose cone is brought into the flow-field, extra sound is generated by the probe itself. This sound can lead to a contamination of the correlation function. Hence, this effect was called ”probe contamination.” The ”probe contamination” can be avoided completely by using a nonintrusive technique to measure the nearfield quantity. For example, Schaffar (1979) used laser-Doppler velocimetry to measure flow velocity fluctuations in a jet. Then, he correlated them and their square values with the acoustic pressure. Panda and Seasholtz (2002) used the density fluctuations in a high-speed jet as near-field quantity. They obtained the density using a Raleigh-scattering technique. Later, Panda et al. (2005) extended their nonintrusive technique to measure two velocity components and the density simultaneously at one point. In this way, they were able to correlate the fluctuating stress ρ v i v j , as it occurs in the source term of Lighthill’s acoustic analogy, with the acoustic pressure in the far-field. In all aforementioned references, a correlation was calculated between the acoustic pressure in the far-field and a single near-field quantity, which could be selected by considering the source terms in the aeroacoustic equations.

The correlation with other quantities was also performed in the past. For example, Hileman et al. (2005) presented an approach where the far-field pressure was correlated with flow visualization patterns in the near-field. They investigated the flow in a high-speed jet, where turbulent structures in the shear layer were visible due to condensation effects. These structures were captured on an ultrahigh-speed camera, while, simultaneously, the far-field pressures were recorded by an array of microphones. The flow images were kept and stored only if the microphone array recognized a strong sound wave originating from the observed region. Then proper orthogonal decomposition (POD) was used to analyze the images. The POD was first introduced by Lumley (1967) for characterizing coherent structures in turbulent flows. The concept of correlating any physical quantity in any domain with the projection of the velocity field on POD modes is described by Borée (2003). He introduced this method to analyze correlated events in turbulent flows and referred to it as extended POD.

In this paper, a correlation approach is presented where particle image velocimetry (PIV) is used to measure the flow velocity in the near-field. The PIV technique combines the advantages of several of the methods mentioned above. It is a nonintrusive technique of instantaneously measuring the flow velocity over an extended region. The main goal of this paper is to ascertain whether PIV can be used to obtain the cross-correlation function between a near-field quantity and the acoustic pressure in the far-field. For that purpose, two test cases are considered.

First, the flow around a circular cylinder at a Reynolds number of Re = 19,000 is considered. The PIV measurement plane is perpendicular to the cylinder axis and parallel to the flow, so that the wake behind the cylinder is probed by this near-field measurement. A strong tone is generated by the flow in this case, which clearly dominates the background noise in the wind tunnel. Etkin et al. (1957) stated that this strong tone radiates most strongly in a direction perpendicular to the flow, while a higher harmonic at double that frequency radiates most strongly in a streamwise direction.

The second example considered here is the flow about a swept wing with extracted slat. The noise generated by the slat flow has broadband character, but for the parameters selected here a weaker tonal component is present as well. The PIV measurement covers an area in the slat cove including the slat cusp, and the measurement plane is perpendicular to the leading edge. The details of both experimental setups are explained in Sect. 3.

In the present work, a standard PIV system is used with a recording rate of 2.5 measurements per second. Clearly, this rate is too slow to enable a capturing of the unsteady flow-field in the considered test cases. The cross-correlation between a near-field quantity and the acoustic pressure is calculated by an ensemble average over a certain number of PIV snapshots. The adopted procedure is described in Sect. 2. A high number of snapshots is desired, so that the contribution of uncorrelated noise components and random errors can be reduced via the averaging process. In this regard, a compromise has to be found, which takes into account the required overall recording time as well as the numerical effort for the evaluation of the PIV recordings. One objective of this work is to determine whether, for the considered test cases, it is possible to obtain a useful correlation result with an acceptable amount of effort. In the first step, the acoustic pressure is correlated with the fluctuations of the velocity components and of the vorticity in the PIV measurement plane. The motivation for the selection of these near-field quantities is given in Sect. 2. Additionally, a snapshot POD is computed for the measured velocity fields and the acoustic pressure is correlated with the amplitude of the POD modes. Then the obtained correlation coefficients for the local near-field quantities and the POD modes, which represent the flow-field in an extended region, are compared. The respective results are presented in Sect. 4.

2 Theoretical background

2.1 Acoustic analogy

The acoustic radiation from an unsteady flow-field can be described by Lighthills acoustic analogy (Lighthill 1952), which has the form

$$\left(\frac{\partial {^2}}{\partial {t^2}} - c_0^2 \Updelta \right) (\rho - \rho_0) = \frac{\partial^2} {\partial {x_i} \partial {x_j}} T_{ij} $$
(1)

where

$$T_{ij} = \rho v_i v_j + \tau_{ij} - \delta_{ij} \left\{p - c_0^2 (\rho - \rho_0) \right\} $$
(2)

represents a stress system. Here, ρ is the density, p is the pressure, v i are the velocity components, τ ij is the viscous stress tensor, and c 0 and ρ 0 are selected values for the speed of sound and a reference density. The term on the right hand side of Eq. 1 can be interpreted as equivalent sources, which act on an acoustic medium at rest and which generate in this medium the same density fluctuations as they are generated by the unsteady flow.

Howe (1975) reformulated the acoustic analogy of Lighthill and introduced a different formulation of the equivalent sources. He expressed the source strength using the Lamb vector \({\varvec{\omega}} \times {\bf v},\) where \({\varvec{\omega}} = \hbox{curl} {\bf v}\) denotes the vorticity. In the low Mach-number limit, following Howe, the acoustic pressure generated by an unsteady flow-field around a rigid body is approximately given by

$$p^{\prime}({{\mathbf{x}}},t) = - \rho_0 \int \int \limits_{V} ({\varvec{\omega}} \times {{\mathbf{v}}})_i \frac{\partial {G}} {\partial {y_i}} \; \hbox{d}^3 {{\mathbf{y}}} \hbox{d} \tau , $$
(3)

where G(x, t; y, τ) is a Greens function. The integration is performed over all source points y in the volume V outside the body and over all possible source times τ. In this approach, the effect of the solid surface is treated as diffraction problem. To fulfill the boundary condition at the surface, the derivative of G in normal direction n has to vanish: n·∇ x G = 0 at the surface. The approximate solution (3) shows that, at low Mach numbers, the sound generation by flows around rigid bodies can be described using a source distribution, which is determined by the velocity field alone. To characterize which part of the source strength contributes to the acoustic pressure in the far-field, the cross-correlation between \({\varvec{\omega}} \times {\bf v}\) and p′ is considered. It is reasonable to split the vorticity and the velocity into mean and fluctuating parts: \({\varvec{\omega}} = {\bar{\varvec{\omega}}} + {\varvec{\omega}}^{\prime}\) and \({\bf v} = {\bar{{\mathbf{v}}}} + {\bf v}^{\prime}.\) One then obtains the following relationship for the cross-correlation:

$$ \begin{aligned} \langle ({\varvec{\omega}} \times {{\mathbf{v}}})_i \; p^{\prime} \rangle &= (\bar{\varvec{\omega}} \times \bar{{\mathbf{v}}})_i \; \langle p^{\prime} \rangle+ \left( \bar{\varvec{\omega}} \times \langle {{\mathbf{v}}}^{\prime} p^{\prime} \rangle \right)_i \\ &\quad + \left( \langle {\varvec{\omega}}^{\prime} p^{\prime} \rangle \times \bar{{\mathbf{v}}} \right)_i + \langle ({\varvec{\omega}}^{\prime} \times {{\mathbf{v}}}^{\prime} )_i \; p^{\prime} \rangle . \\ \end{aligned} $$
(4)

Since the acoustic pressure has a zero mean value, the first term on the right hand side vanishes. The second term contains an effect of fluctuating velocity and the third of fluctuating vorticity. In the fourth term, both fluctuations are combined. The different parts can be considered separately and the correlations to be calculated are \(\langle {{\mathbf{v}}}^{\prime} p^{\prime} \rangle, \langle {\varvec{\omega}}^{\prime} p^{\prime} \rangle\), and \(\langle ({\varvec{\omega}}^{\prime} \times {{\mathbf{v}}}^{\prime})_i \; p^{\prime} \rangle.\) The 2D-PIV method used in the present study measures the velocity field in a plane, from which only the out-of-plane component of the vorticity and the two in-plane components of the velocity are determined. Hence, although the complete vector \({\varvec{\omega}} \times {\bf v}\) cannot be obtained from the experimental data, it is nevertheless possible to calculate the in-plane components of the correlation \(\langle {{\mathbf{v}}}^{\prime} p^{\prime} \rangle\) and the out-of-plane component of \(\langle {\varvec{\omega}}^{\prime} p^{\prime} \rangle.\) This obviously does not give a complete correlation of the source expression in Eq. 3 with the acoustic pressure, but these partial correlations are a reasonable starting point to test the feasibility of the proposed method. The investigated unsteady flow-fields involve flow separation, shear layer instabilities, vortex production and convection, interaction between vortices and geometry, vortex shedding, and also a possible pressure feedback on the flow. Therefore, one can expect a larger correlation length in the flow-field. Since the sound is not generated by individually radiating source elements, which are compact, it is not possible in this case to calculate directly the local source efficiency from the cross-correlation results. Nevertheless, the cross-correlation with fluctuations v′ and \({\varvec{\omega}}^{\prime}\) can indicate which processes in the near-field are responsible for the main part of the radiated sound.

2.2 Computation of the cross-correlation

The normalized cross-correlation \(R_{\phi p^{\prime}}({\bf x},{\bf y},\tau)\) is defined as

$$ \begin{aligned} R_{\phi,p^{\prime}}({{\mathbf{x}}},{{\mathbf{y}}},\tau) &=\frac{S_{\phi,p^{\prime}} ({{\mathbf{x}}},{{\mathbf{y}}},\tau)} {\sigma_{\phi}({{\mathbf{y}}}) \sigma_{p^{\prime}}({{\mathbf{x}}})}\\ &= \frac{ \langle \phi({{\mathbf{y}}},t) p^{\prime}({{\mathbf{x}}},t + \tau) \rangle}{\sqrt{ \langle \phi^2({{\mathbf{y}}},t) \rangle \langle p^{\prime 2} ({{\mathbf{x}}}, t) \rangle}}\\ \end{aligned} $$
(5)

where ϕ(y, t) represents a near-field quantity measured at position y and time t. The variable τ is the time shift between the pressure signal and ϕ. The cross-correlation coefficient is normalized by the root-mean-square (RMS) values of ϕ and p′, which are denoted by σ ϕ(y) and \(\sigma_{p^{\prime}}({\bf x}).\) In the present experiments, the flow-field is recorded by the PIV system at discrete times t n and the near-field quantity is evaluated from these recordings. The time between the PIV measurements is large enough so that the individual images can be considered as statistically independent. The far-field pressure is recorded at discrete times simultaneously to the PIV flow-field measurements, but using a much higher sampling rate than with PIV. The system clock of the PIV system must be synchronized with the data-acquisition system used for the acoustic pressure. Then the cross-correlation between the near-field quantity and the acoustic pressure can be calculated in a discrete manner using

$$ S_{\phi,p^{\prime}}({{\mathbf{x}}},{{\mathbf{y}}},\tau)= \frac{1}{N} \sum^{N}_{n=1} \left[ \phi({{\mathbf{y}}},t_n) p^{\prime}({{\mathbf{x}}},t_n + \tau) \right],$$
(6)

where N is the number of PIV measurements. Note that the time delay τ is given only in discrete steps, where the step size Δτ is determined by the sampling rate of the acoustic pressure. The RMS-value of the near-field quantity, which is required for normalization, can be calculated from the measured data by

$$ \sigma_{\phi}({{\mathbf{y}}}) = \sqrt{ \frac{1} {N}\sum^{N}_{n=1} \phi^2({{\mathbf{y}}},t_n) } . $$
(7)

In an analogous manner, the standard deviation \(\sigma_{p^{\prime}}({\bf x})\) of the acoustic pressure is computed, except that there the averaging is carried out over all recorded pressure samples. The number of these samples is typically several orders of magnitude higher than N. It is to be noted here that the critical parameter is the number of PIV recordings N. If N is too low, uncorrelated parts of the measured quantities cannot be sufficiently suppressed. This results in a high amount of noise in the resulting \(S_{\phi,p^{\prime}}({\bf x},{\bf y},\tau).\) The definition of a sufficient number of recordings may of course depend strongly on the particular case under consideration.

2.3 Cross-correlation using POD modes

A section of a measured flow-field, which consists of K fluctuating velocity vectors

$$ \left( u^{\prime}({{\mathbf{y}}}_k, t_n), v^{\prime}({{\mathbf{y}}}_k, t_n) \right) \qquad \hbox{with} \; k=1,2,\ldots,K $$
(8)

can be considered as a point in a 2K-dimensional vector space. Here u′ and v′ denote the fluctuations of the velocity components in a local coordinate system, which is aligned with the PIV-measurement plane. From N flow-field measurements, one can construct the vectors

$${{\mathbf{U}}}(t_n)= \left( u^{\prime}({{\mathbf{y}}}_1,t_n) \ldots u^{\prime}({{\mathbf{y}}}_K,t_n), v^{\prime}({{\mathbf{y}}}_1,t_n) \ldots v^{\prime}({{\mathbf{y}}}_K,t_n) \right) $$
(9)

with n = 1,2,...,N. By searching for a deterministic basis set \({\varvec{\psi}} = (\psi_1,...,\psi_{2K})\) that spans this vector space, where a minimum number of components captures a maximum of the energy, which is present in the flow-field in a mean square sense, this then leads to the POD. It can be shown (e.g., Berkooz et al. 1993; Sirovich 1987) that, to find the optimal choice of \({\varvec{\psi}}\), the following eigenvalue problem has to be solved:

$${{\mathbf{R}}} {\varvec{\psi}} = {\varvec{\lambda}} {\varvec{\psi}}.$$
(10)

Here the kernel of Eq. 10 is defined as

$$ {{\mathbf{R}}} \equiv \frac{1}{N} \sum_{n=1}^N \left( {{\mathbf{U}}}(t_n) \otimes {{\mathbf{U}}}(t_n) \right).$$
(11)

The eigenvectors in the columns of the matrix \({\varvec{\psi}}\) build a new orthogonal basis. The eigenvalues in the components of the vector \({\varvec{\lambda}}\) provide information about the amount of energy contained in the corresponding eigenvector. In the present study, the method is applied, for example, to a field of 40 × 40 two-component velocity vectors to obtain 2K = 3,200 eigenvectors. The kernel R is obtained by averaging the cross-correlation over N recordings. After having the new orthonormal basis \(\varvec{\psi},\) we can calculate coefficient vectors \({\user2{a}}(t_n) = \left( a(1,t_n),\ldots,a(2K,t_n) \right)\) by back-projecting the basis onto the original velocity field using

$${\user2{a}}(t_n) = {\varvec{\psi}}^{-1} {\user2{U}}(t_n) $$
(12)

where the vector U(t n ) represents the fluctuating velocity field in the nth measurement. a(m, t n ) is the amplitude of the mth mode in the nth measurement. Finally, the normalized cross-correlation between the amplitude of mode m and the acoustic pressure at x is defined by

$$ R_{a,p^{\prime}} ({{\mathbf{x}}},m,\tau) =\frac{S_{a,p^{\prime}} ({{\mathbf{x}}},m,\tau)}{\sigma_{a}(m) \sigma_{p^{\prime}}({{\mathbf{x}}})}, $$
(13)

where the cross-correlation is given by

$$S_{a,p^{\prime}} ({{\mathbf{x}}},m,\tau) =\frac{1} {N} \sum^{N}_{n=1} \left[ a(m,t_n) p^{\prime} ({{\mathbf{x}}},t_n + \tau) \right] $$
(14)

and the RMS-value of the amplitude by

$$ \sigma_{a}(m) = \sqrt{ \frac{1}{N}\sum^{N}_{n=1} a^2(m,t_n)}$$
(15)

The resulting \(R_{a,p^{\prime}}({\bf x},m,\tau)\) can provide information about the contribution of each eigenmode to the generated noise.

3 Experimental setup and methods

3.1 Facility

Experiments are conducted in the 2.0 × 1.4 m2 wind tunnel facility at the Technical University of Berlin. It is a Göttingen type wind tunnel with a closed test section and reverberant side walls [see Fig. 1 (bottom left)]. The turbulence level in the core region of the test section is less than 0.23% at a mean free stream velocity of 20 m/s. Sound absorbers upstream and downstream of the driving fan are installed to reduce the background noise level.

Fig. 1
figure 1

Left top Schematic setup of the experiment. Left bottom Wind tunnel facility of the Technical University of Berlin. Right Experimental setup of PIV and microphone measurement of the cylinder wake flow

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3.2 PIV-system

Velocity data are acquired with a 2D PIV system. It consists of a Nd:YAG pulsed laser that generates the light sheet and a CCD camera that records the light scattered by tracer particles. The frequency-doubled laser (Type: Quantel Twins, Brilliant W) emits laser pulses with a maximum energy of 120 mJ per pulse. It is operated at a repetition rate of 10 Hz. The CCD camera (PCO Sensicam) is mounted on the wind tunnel ceiling and records particle images via an upstream mirror. The CCD camera has a resolution of 1,280 pixels ×  1,024 pixels. It is operated at a frame rate of 2.5 Hz, which therefore is the sampling rate of the whole PIV system. The flow is seeded with diethylhexylsebacate (DEHS) tracer particles of approximately 1 μm in diameter (see Kaehler et al. 2002). They are generated by a seeding generator with 40 Laskin atomizer nozzles and distributed via a rake, which is installed inside the settling chamber. The PIV data are processed using the PivView Software (see DLR contribution to Stanislas et al. 2005). A multipass algorithm with an additional image deformation correction is used. For the evaluation, in the present cases, the interrogation window size is set to 32 pixels ×  32 pixels with an overlap of 50%, which leads to fields of 79 × 63 vectors. The vorticity component normal to the measurement plane is estimated using a center difference scheme (Raffel et al. 1998). Five-thousand pairs of images (snapshots) are analyzed. To synchronize the PIV and the microphone measurements, the TTL signal from the Q-switch of the first laser is recorded in parallel with the microphone signal: its leading edge corresponds to the firing of the laser pulse and hence provides the time of the captured velocity field snapshot. For the results of the calculations presented in the following sections, the zero point (τ = 0) is defined as the moment of the first laser pulse. The Q-switch signal is acquired with the same data-acquisition system as the microphone signal, also sampled with a rate of 102.4 kHz. Note that this results in a maximum jitter of approximately ±5 μs between the PIV timing and the pressure data acquisition, which, considering the much lower frequency (<10 kHz) of the processes being examined here, leads to a negligible error and uncertainty.

3.3 Microphone

A single 1/4 inch Brüel & Kjær microphone (Type 4919) records the radiated sound simultaneously with the PIV measurement. The microphone is covered with a nose cone to reduce the interference from aerodynamically induced noise, while the microphone cable is wound around the supporting rod to avoid an additional aeolian tone (see Fig. 1). Sound pressure data are acquired with a sampling rate of 102.4 kHz. An antialiasing filter is used at a frequency of 40 kHz.

3.4 Cylinder-wake experiment

Figure 1 (right) shows the experimental setup for the test case with the circular cylinder. The Reynolds number is Re = 19,000 based on a cylinder diameter of d = 14 mm and a free stream velocity U ∞ = 20.55 m/s. The blockage and aspect ratios, defined as d/W and H/d, are 0.007 and 100. The laser light sheet is oriented perpendicular to the cylinder axis and directed through a window into the test section. The thickness of the light sheet is 1.5 mm in the investigation area, which is located directly downstream of the cylinder. The camera of the PIV system is equipped with a 60 mm lens, which results in a field of view of 121.5 × 96.6 mm2 and a spatial resolution of 1.6 mm (=0.11d) for the resulting vector field.

The microphone is located at a position downstream of the PIV-measurement area. It is laterally shifted, so that it is far outside the cylinder wake; the distance between microphone and cylinder being about 0.65 m. Assuming a dipole character of the generated sound field where the maximum emission is in lateral direction, the microphone observes from a direction where the received sound waves are relatively strong but not maximal.

3.5 Slat-cove experiment

The unsteady flow-field inside a leading edge slat is investigated. The experiments are carried out on a generic swept constant cord half (SCCH) model. It is equipped with a three-element high lift device. The cord length is c = 0.45 m for the cruise configuration with a half span of 1.2 m. The flap is retracted for the measurements carried out here. A cross sectional view of the SCCH airfoil is shown in Fig. 2, where the box indicates the area observed by the camera. The present paper describes the results for a configuration with α = 16° and U ∞ = 20 m/s. The Reynolds number based on the slat thickness of 2.5 cm is Re = 33,000. The camera of the PIV system is equipped with a 135 mm lens, which enables the observation of an area of approximately 54 × 43 mm2 and leads to a spatial resolution of 0.68 mm for the chosen parameters of the PIV evaluation (see Sect. 3.2). The measurement error is estimated to be approximately 0.32 m/s based on an uncertainty of 0.1 pixel in the detection of the correlation maximum. For a detailed description of the experimental setup and results for various α and U ∞ combinations, see Kaepernick et al. (2005).

Fig. 2
figure 2

Cross sectional view of the airfoil of the SCCH model. The box indicates the area observed by the camera

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4 Results and discussion

4.1 Cylinder wake

Figure 3 shows an example of the instantaneous flow-field of the cylinder wake. All quantities and axes are made dimensionless using the cylinder diameter d and the free stream velocity U ∞. x and y are local coordinates in the measurement plane, where the x-axis points in the mean flow direction. The vorticity component ω z normal to the measurement plane is coded by color. Velocity vectors are superimposed to illustrate the flow-field. Vortices can be seen to emanate from the upper and lower shear layer and form a vortex street in the wake of the cylinder with alternating sign.

Fig. 3
figure 3

Instantaneous flow-field behind the cylinder (U ∞ = 20.55 m/s)

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4.1.1 Acoustic measurement results

A time sequence of the recorded sound pressure and the corresponding acoustic power spectrum are shown in Fig. 4. The spectrum is calculated using a Hanning window with averaging occurring over 30 s. The sound pressure level (SPL) is given in dB with a reference pressure of p ref = 2 × 10−5 Pa. A predominant oscillation frequency can be identified clearly in the time series, which appears as the resulting peak in the averaged power spectrum at 265 Hz. Such an aeolian tone produced by translating cylindrical rods through air has already been observed by Strouhal (1878). The sound spectrum of such a configuration consists of one strong tone at the Strouhal frequency and a weaker peak at double the frequency. The tone at the fundamental frequency has a dipole character, radiating most strongly perpendicular to the stream, while the higher harmonic radiates most strongly in the direction of the stream, as observed and theoretically explained by Etkin et al. (1957). Because of the microphone position in the experimental setup used here, only the fundamental frequency is observed. Since the measurement is conducted in a closed test section with a reverberant build-up, reflections at the side-walls lead to an even more dominant tone. The normalized frequency S t = f d /U ∞ is 0.186, which is in good agreement with values found in the literature (Williamson 1996). Acoustic measurements in the empty test section at the free stream velocity U ∞ = 20 m/s have shown that the second peak at around 90 Hz is due to the background noise of the wind tunnel.

Fig. 4
figure 4

Cylinder wake experiment. Time sequence of the recorded sound pressure (left) and the corresponding acoustic power spectrum (right)

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4.1.2 Cross-correlation results

To illustrate the statistical significance of the calculated correlation coefficients \(R_{\phi p^{\prime}},\) a Student’s t test against zero is conducted (van der Waerden 1971) and the t value

$$t_N = \frac{\sqrt{N-2}}{\sqrt{1-R_{\phi p^{\prime}}^2}} R_{\phi p^{\prime}} $$
(16)

is calculated. The evolution of the t value for \(R_{v^{\prime}p^{\prime}}\) with increasing N at [x/d;y/d] = [1.65;0.00] for selected values of τ is depicted in Fig. 5 (left). The selected values are at local peak levels of \(R_{v^{\prime}p^{\prime}}.\) The dashed lines show constant values of t N for four different confidence intervals (taken from van der Waerden 1971). In case of τ = 0.0126 s, the correlation coefficient is different from zero with 99.9% confidence for N ≥ 400, and from thereon, t N increases with N as one would expect. The same trend is observed for the other depicted t values, although more samples are needed to obtain significance in case of smaller final values of \(R_{v^{\prime}p^{\prime}}.\) Note that the final error margin for \(R_{\phi p^{\prime}}\) with 5,000 samples is approximately ±0.028 (95% probability).

Fig. 5
figure 5

Left Cylinder wake experiment: the evolution of the t value t N for \(R_{v^{\prime}p^{\prime}}\) with increasing N at [x/d;y/d] = [1.65;0.00] for τ = −0.113 s, −0.0368 s, 0.0126 s, 0.1147  s. The final values of the cross-correlation coefficient for N = 5,000 are \(R_{v^{\prime}p^{\prime}} = 0.0586, 0.1002, 0.2160, 0.1195\), respectively. Right Slat cove experiment: t N for \(R_{v^{\prime}p^{\prime}}\) with increasing N at [x/c;y/c] = [0.03;0.02] for τ = −0.0913 s, −0.0128s. The final values of the cross-correlation coefficient for N = 5,000 are \(R_{v^{\prime}p^{\prime}} = 0.0304, 0.0323\), respectively

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Figure 6 shows the instantaneous distribution of the normalized cross-correlation coefficient behind the cylinder for τ = 1 ms. \(R_{\omega_z^{\prime}p^{\prime}}\) is color-coded (left hand side of figure) and \((R_{u^{\prime} p^{\prime}}, R_{v^{\prime} p^{\prime}})\) is depicted as a vector plot (right hand side). \(R_{\omega_z^{\prime} p^{\prime}}\) becomes close to zero for regions outside of the wake. The sign of the correlation coefficient alternates between positive and negative values downstream of the cylinder, and the distance between two neighboring local maxima increases further downstream, which can be explained by the effect of accelerating vortices. The cross-correlation coefficient reaches a maximum approximately 1.6 diameters downstream of the cylinder and then decreases further downstream. In this area of maximal correlation, a rapid acceleration of the released vortices takes place. As explained by Howe (1998), most of the sound is generated during the initial period of acceleration. It is therefore plausible that the largest correlation coefficients are to be found in this region.

Fig. 6
figure 6

Left Instantaneous distribution of the normalized cross-correlation coefficient \(R_{\omega_z^{\prime} p^{\prime}}\) behind the cylinder for τ = 1 ms. Right Instantaneous distribution of \((R_{u^{\prime} p^{\prime}}, R_{v^{\prime} p^{\prime}})\) as a vector plot for τ = 1 ms

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The temporal evolution of the cross-correlation coefficient \(R_{v^{\prime} p^{\prime}}\) and the corresponding cross power spectrum at position [x/d;y/d] = [1.65;0.00] are plotted in Fig. 7. The correlation becomes significant at τ ≈ −0.04 s. The cross-correlation coefficient fluctuates over time in a sine-type oscillation with maximum values of about \(|R_{v^{\prime}p^{\prime}}| = 0.2\) at τ = 0.025 s. From thereon, the amplitude of \(R_{v^{\prime}p^{\prime}}\) decreases with τ. The background noise at frequencies below 100 Hz [see Fig. 4 (right)] is suppressed, but the peak at around 90 Hz is still visible, even though its amplitude has been reduced by at least 10 dB. This filtering of uncorrelated noise is brought about by the averaging process during the cross-correlation calculation. The predominant oscillation with the shedding frequency of the cylinder wake can be seen clearly in the time plot as well as in the cross-power spectrum. \(R_{u^{\prime}p^{\prime}}\) and \(R_{v^{\prime}p^{\prime}}\) show this periodic behavior with a phase shift of π/2 with respect to each other.

Fig. 7
figure 7

Cylinder wake experiment. Left Temporal evolution of the normalized cross-correlation coefficient at [x/d;y/d] = [1.65;0.00]. Right Power spectrum of the shown correlation coefficient

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The observed result is very different from those that have been presented in the literature for the correlation function between the acoustic pressure and the velocity fluctuation in a jet (see, for example, Lee and Ribner 1972; Schaffar 1979; Panda et al. 2005). In the case of a jet, typically the correlation function shows only a relatively short event, which consists mainly of a single positive and negative deflection. The regular oscillations of \(R_{v^{\prime}p^{\prime}}\) in the present result can be explained by coherent periodic structures in the flow-field. These structures generate a sound field with the same periodicity, which is perceived as tone. Hence, the correlation between the velocity fluctuations and the acoustic pressure shows the same oscillations as the input signals. The larger temporal coherence of both signals leads also to a significant correlation at negative τ values. This is also observed by Chatellier and Fitzpatrick (2005) for the correlation between the velocity at two different positions in the cylinder wake. The shape of the envelope of the plot \(R_{v^{\prime}p^{\prime}}\) as a function of τ can be explained by reflections at the side walls of the test section. Without reflections, one would expect the maximum amplitude of the correlation function at a value τ 0, which matches with the travel time directly from the cylinder to the microphone, but in the present experiment, multiple reflected signals are also received by the microphone and contribute to the correlation function at larger positive delay times τ, which correspond to travel distances of several times the channel width. A complicated interference of the directly emitted and the reflected waves takes place in the test section, and the superimposed waves interact constructively and destructively. Depending on the coherence length in the flow and the position of the microphone, the maximum amplitude of the correlation function can be shifted towards larger τ values, as is observed here. The delay time of τ = 0.025 s is equivalent to a travel distance of about 8 m. Thus, in the present case, the correlation function is clearly dominated by reflected waves.

4.1.3 Cross-correlation results using POD modes

Figure 8 (left) shows the energy distribution of the 40 most energetic eigenmodes. The dots characterize the fraction of the fluctuating kinetic energy (energy per mode) and the solid line represents the accumulated fractions over all modes. The first two eigenmodes alone encompass already 64% of the fluctuating kinetic energy. In Fig. 8 (right), the vector representation of the first eigenmode is depicted. It is interesting that this plot shows a similar pattern of vortical structures such as that seen in Fig. 6. Thus, the correlation technique described in the previous section functions as a filter of the most energetic structures in the flow. This is at least true for the cylinder wake flow, where temporal and spatial structures are strongly coupled.

Fig. 8
figure 8

Cylinder wake experiment. Left Energy distribution of the eigenmodes; dots stand for the fraction of the total fluctuating kinetic energy in each mode; line stands for the accumulated values. Right First eigenmode represented as a vector field in the original physical domain

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The temporal evolution of \(R_{ap^{\prime}}\) is shown for m = 1, 3, and 6 in Fig. 9. The first two most energetic modes (m = 2 is not presented here) are the major contributors to the radiated sound. In case of m = 6, the correlation \(R_{ap^{\prime}}\) reaches values of approximately 0.1 at its maximum. As seen in the plot for m = 3, other modes do not correlate significantly with the pressure signal. \(R_{ap^{\prime}}\) for m = 1 and m = 2 both show a similar time-dependent behavior, such as the one shown for \(R_{v^{\prime}p^{\prime}}\) in Fig. 7 (left). The cross power spectrum of \(R_{ap^{\prime}}\) for m = 1 [Fig. 9 (bottom right)] has a peak for the dominant Strouhal frequency at 265 Hz. The peak is narrower than the one in the spectrum of \(R_{v^{\prime}p^{\prime}}\) [Fig. 7 (right)] and the background noise has been suppressed more efficiently.

Fig. 9
figure 9

Cylinder wake experiment: the temporal evolution of \(R_{ap^{\prime}}\) for m = 1 (top left), 3 (top right), and 6 (bottom left). Frequency spectrum of \(R_{ap^{\prime}}\) for m = 1 (bottom right)

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No significant increase in the maximum value of the correlation coefficient is observed when compared with the values of \(R_{v^{\prime}p^{\prime}}\), which was presented in the last section.

4.2 Slat cove

An instantaneous velocity field is shown in Fig. 10. It shows the normalized vorticity \(\omega_z c/U_{\infty}\) as a color plot with scale given by the color bar at right. Again, all quantities and axes are made dimensionless by normalizing with the cord length c and the free stream velocity U ∞. The strong spatial gradient of the velocity magnitude clearly identifies the shear layer, which becomes unstable approximately 0.02 chord lengths downstream of the slat cusp and thereafter breaks up into discrete vortices, as described by Takeda et al. (2001). As can be seen, the shear layer impinges on the inner slat surface; the vortices were either ejected through the slat gap or became trapped inside the recirculation area. One can again refer to Kaepernick et al. (2005) for a very detailed description of this complex and interesting flow-field.

Fig. 10
figure 10

Instantaneous slat flow-field for α = 16° and U ∞ = 20 m/s

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4.2.1 Acoustic measurement results

The measured acoustic spectrum, calculated in the same manner as in Sect. 4.1.1, is depicted in Fig. 11. There are several peaks between 2.3 and 3.4 kHz. One dominant tone is found with a sound pressure level of 76.5 dB at 3 kHz, which is approximately 8 dB above the ambient noise level (see plot “no model” in Fig. 11). The intermittent character of the tone becomes obvious in a time-dependent spectrum, as has been described by Kaepernick et al. (2005), where the tonal character and an observed mode switching has been ascribed by them to a resonance process. This has also been described by Hein et al. (2007).

Fig. 11
figure 11

Slat cove experiment. Acoustic spectrum for the α = 16° configuration at U ∞ = 20 m/s

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4.2.2 Cross-correlation results

As for the case of the cylinder wake, the evolution of the t value for \(R_{v^{\prime}p^{\prime}}\) with increasing N at [x/c;y/c] = [0.03;0.02] for selected values of τ is depicted in Fig. 5 (right). The selected values are at local peak levels of \(R_{v^{\prime}p^{\prime}}\) and the dashed lines show constant values of t N for four different confidence intervals. Only for N > 4,000, the correlation coefficients are different from zero with 95% confidence. Again, the final error margin for \(R_{v^{\prime} p^{\prime}}\) with 5,000 samples is approximately ±0.028 (95% probability).

The coefficients of the correlation between the flow structures and the far-field pressure fluctuations for the slat cove are small in comparison to the values obtained for the cylinder wake flow. They do not exceed 3%, which is in the range of the remaining error margin. The interpretation of the results must therefore be done very carefully. Figure 12 shows the distribution of \(\sigma_{v^{\prime}}\) and the spatial distribution of \(\langle R_{v^{\prime}p}^2\rangle ^{1/2}\) time-averaged over the interval of −10 ms < τ < 10 ms. The investigated flow-field in the slat cusp region contains areas of strong fluctuating velocities (shear layer and recirculation region) and areas with \(\sigma_{v^{\prime}}\) values less than the measurement error (downstream of the shear layer). The fluctuating quantities in the latter regions cannot be considered as reliable, so that those regions with \(\sigma_{v^{\prime}}\) values below the measurement error have been spatially filtered out in the calculation of the cross-correlation coefficient. This is done by normalizing the pressure–velocity correlation by \(\max(\sigma_{v^{\prime}},\varsigma)\) instead of \(\sigma_{v^{\prime}}\) in Eq. 5, where \({\varsigma}=0.25\, \rm m/s.\) Maxima of the temporally averaged correlation can be identified at different positions around the shear layer and inside the recirculation region. The location of spots with high correlation values depends on the time interval chosen. Note that the distribution of \(R_{v^{\prime}p}\) for a single delay time τ shows only an irregular pattern.

Fig. 12
figure 12

Slat cove experiment: left spatial distribution of σ v′; right cross-correlation coefficient \(\langle R_{v^{\prime}p^{\prime}}^2\rangle ^{1/2}\) time-averaged over −10 ms < τ < 10 ms

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4.2.3 Cross-correlation results using POD modes

As for the case of the cylinder wake, a POD is also applied to the slat cove flow-field. The energy distribution for the eigenmodes is plotted in Fig. 13 (top left). Significant more eigenmodes are needed to represent most of the fluctuating kinetic energy compared to the cylinder wake flow (see Fig. 8). This is reasonable, because in contrast to the cylinder wake the flow in the slat cove of a swept wing is highly three-dimensional. From hot-wire measurements (not presented here) and from the obtained instantaneous velocity fields, it can be seen that, in contrast to the cylinder wake, there is no dominant frequency associated with dominant large-scale structure in the flow. Only small-scale vortices emanate from the shear layer downstream of the slat cusp. They were either ejected through the slat gap or became trapped inside a recirculation area, thus forming a highly irregular pattern. The small scale of the convected structures together with the absence of a strong periodicity leads to an energy distribution over a wider range of eigenmodes. Ausseur et al. (2006) proposed a convection-based POD approach to represent the vortical structures in such a flow-field with fewer modes. Such an approach was not applied in the present work.

Fig. 13
figure 13

Slat cove experiment. Top left Energy distribution for the eigenmodes; dots stand for the fraction of the total fluctuating kinetic energy in each mode; line stands for the accumulated values. Top right Second eigenmode represented as a vector field in the original physical domain. Bottom Temporal evolution of the normalized cross-correlation coefficient \(R_{ap^{\prime}}\) for m = 2 (left) and m = 7 (right)

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Figure 13 (top right) shows the second eigenmode represented as a vector field in the original physical domain. The dominant vortical structure of the recirculation region inside the slat cove is represented by this mode. The correlation coefficients \(R_{ap^{\prime}}\) are calculated in the same manner as described in Sect. 2.3 by using Eq. 13. The temporal evolution of the normalized cross-correlation coefficient \(R_{ap^{\prime}}\) for both m = 2 and 7 are shown in Fig. 13 (bottom) as an example. A difference in the frequency distribution can be seen in both plots, but the results do not identify any specific eigenmodes (such as in the cylinder wake flow), which could act as sources of the far-field pressure fluctuations. The amplitude of \(|R_{ap^{\prime}}|\) is only 0.04, and it does not decrease with τ as one would expect from a correlation of the eigenmode with the far-field pressure. Thus, the POD approach could not give an improvement of the signal-to-noise ratio compared to the direct correlation method with v′.

5 Conclusion

The presented tests show that the PIV method can be used to obtain a correlation function between near-field quantities and the generated acoustic pressure. Five-thousand PIV recordings are sufficient in the case of the flow around a cylinder to identify the flow structures, which are involved in the generation of sound. In contrast, the correlation values for the slat-flow problem remain in the range of the noise level. In both cases, the correlation with the eigenmodes of a POD gives no larger correlation coefficients than the correlation with a local near-field quantity. In the case of the cylinder, the shedded vortices and the concomitant fluctuations of the wake are responsible for the generation of the strong tonal noise. If the vortices extend far in the direction of the cylinder axis, a sound emission takes place, which is coherent in a larger axial section of the cylinder. The frequency of the sound and the vortex shedding frequency are the same. The observed shedding of each vortex requires the development of a counter circulation around the cylinder and therefore a lift. The produced tone has a dipole character and it is linked to alternating lift forces acting on the cylinder, radiating most strongly in the direction perpendicular to the stream (Etkin et al. 1957). Under these circumstances, one can expect a relatively high correlation coefficient between a locally measured near-field quantity and the acoustic pressure. This was the reason why the example of the cylinder was selected for the first test case. The SCCH model features a 30° swept-back wing, resulting in a flow-field, which has far great complexity and is highly three-dimensional. Measurements using two hot-wire probes positioned parallel to the leading edge in the slat cove showed that the spatial correlation of the flow velocity decays rapidly in a spanwise direction. Hence, the PIV measurement here, carried out in one plane only, represents only the flow in one small spanwise section of the slat. The microphone receives the sound from many uncorrelated sources at the slat and additionally the background noise. This explains the observed weak correlation between the measured near-field quantity and the acoustic pressure in this case. One can assume that by using more microphones and focusing on the area of the velocity measurement, the signal-to-noise ratio of the technique presented here could be significantly increased. An improvement should also be possible by taking a higher number of flow-field snapshots into account. To achieve that, future investigations will concentrate on measurements with a high-speed PIV system and a phased microphone array.