Analysis and Characterization of Flow-Generated Sound

Flow-generated sound and velocity distributions of free flows are characterized by frequency spectra (FFT and several spectra from the octave spectrum over the one-third octave spectrum till the narrow band spectrum), by the PSD (power spectrum density) and wavelets. Whereas previously, flow processes are characterized by the average velocity, by the local velocity, if needed also by the three-dimensional velocity distribution and the degree of turbulence, also the flows and particular velocity distributions could be described by the acoustic pressure and level, by the power spectrum density and by wavelets, especially by the Gabor-Wavelet, as shown below.


Introduction
Velocities of air and gas flows and their frequency of the sound pressure oscillation are coupled together by the Strouhal number Sr = fd/c.The range of the Strouhal number, in dependence of the Reynolds number Re = cd/ν, can be divided into a subcritical section Sr = 0.10 to 0.18, in the Reynolds number range Re = 60 to 3 × 10 5 into a critical range from Re = 2 × 10 5 bis 5 × 10 6 and into a supercritical range with values of Sr = 0.28 at Reynolds numbers of Re = 5 × 10 6 .This context is successfully used in the manufacture of musical instruments and plucked instruments industry.It can be used successfully in other technical areas, in particular in the field of flow measurement as described below.In a gas and liquid flow, nothing is constant except for the postulated mean values of the velocity and pressure.The sound pressure and the sound pressure level as well as the velocity change with high frequency from f = 10 Hz to 100 kHz, but also into lower ultrasonic range.The frequencies are caused by the turbulence of the flow, which is characterized by the degree of turbulence.The degree of turbulence of a flow is defined by the speed of variations in the three coordinate directions of a Cartesian coordinate system.

Flow Velocity and Frequency of the Sound Pressure Oscillation
So far, flows and leakage flows as shown in Figure 1 and velocity distributions by the local velocities are described by the average values and the turbulent fluctuations in addition by the degree of freedom [1,2].For the description of the energy distribution of a turbulent flow, which is a function of the wavelength or the frequency of the turbulent vortices, the energy spectrum with the energy spectral density E in dependence on the wavelength λ or wave number k [3][4][5].The wave number or vortex number k = ω/c = 2π/λ is a ordinal number for the transfer of energy from large to small turbulence numbers.The flow of energy in turbulent flows is directed to larger wave numbers k (Figure 2).This form of energy transfer is the reason why the frictional resistance in turbulent flows and the distribution of the average velocity is only weakly dependent on the Reynolds number.The pressure losses in turbulent flows are mainly caused by the vortex viscosity.Going one step further in the description of the flow, we come to the spectral analysis of the flow.Spec-  tral analysis of the flow means the separation of the input signal of the flow into a series of sinusoidal and cosinusoidal oscillations.Each oscillation has its own frequency, amplitude and phase.The sum of sine and cosine waves is called the spectrum of which there are many.Well known are time signals (Figure 3) and the frequency spectra, which can be determined using the Fourier transform.Figure 4 shows the frequency spectra of three very similar time signals from Figure 3 for the times t = 1s with 200,000 measured values, t = 0.1 s with 20,000 measured values and for t = 10 ms with 2000 measured values.The following results outline the acoustic data such as the acoustic pressure, the acoustic pressure spectra, specific spectra and the sound pressure level.

Sound Pressure Spectra of Air Outflow
In the following the acoustic effects of air outflow form a pressure vessel through defined outlet openings according to Figure 1 are analyzed and outlined.The outlet openings are d = 1.6 mm to d = 18.4 mm with the exit areas A = 2.0 mm 2 to A = 265.9mm 2 .The outflow from the pressure vessel is carried out at p = 0.25 bar to p = 5bar with outlet velocities of c = 12 m/s up to c = 48 m/s.In Figure 5 the frequency spectra (FFT) for two outlet openings of d = 10 mm and d = 18.4 mm for the tank pressures of p = 1.5 bar to 5.0 bar shown.In Figure 5 is seen that the amplitudes at pressures of p = 4.0 bar and p = 4.5 bar with L p = 0.74 mPa und L p = 0.68 mPa take maximum values and that also the outlet opening has a strong influence on the spectral distribution.The amplitudes of the frequency spectra at the pressures p = 4.0 bar and p = 4.5 bar at the frequencies of f = 5 kHz, 8 kHz, 13 kHz und f = 24 kHz are relative maximum values.The frequency spectra of the flow-generated sound extend to the ultrasonic range to about f = 100 kHz, but the significant amplitudes for the pressures from p = 3.0 bar up to p = 5.0 bar extend to the frequency f = 60 kHz.This distribution of the spectra is also set for the other pressures and other outlets.
Examining the octave spectra, the third octave spectra as 1/3-octave spectrum, as well as narrow-band spectra,   is called Lighthill tensor.The pressure oscillations per unit volume described by the Lighthill tensor represents a field of stress fluctuations.
If these four spectra of the sound pressure are shown as sound pressure level in the same Figure 6, this results in a different form.In the representation of sound pressure level, which extends from L p = −45 dB to 52 dB, the sound pressure levels in the lower frequency range of f = 10 bis 1000 Hz are greatly increased.Thus, the amplitudes of the pressure level obtained in the frequency range from f = 1 Hz to 1000 Hz far greater importance, while the level values of the octave and third-octave spectrum are zero and only from the frequency f = 150 Hz take positive values.On the other hand the peak values of sound pressure level in the range of f = 5.5 to 100 kHz are strongly attenuated.They are easily highlighted only in the narrow-band spectra.In spectral analysis so both forms of representation, the sound pressure spectra as well the level representation should be used.
Comparing the narrow band spectra (1/24 octave spectrum) for the outlet openings with the opening widths d = 1 to 5 mm and pressure values of p = 2 bar and p = 3 bar in Figure 7, we obtain the following results.At the pressure of p = 2 bar for the opening d = 5 mm is a obvious maximum amplitude of L p = 0.83 mPa at the frequency f = 6.2 kHz.The maximum amplitude of all other openings from d = 1 mm up to d = 3 mm does not exceed the amplitude value of L p = 0.52 mPa, while the frequencies reach the range between f = 3 kHz and 24 kHz.The results at the pressure of p = 3.0 bar behave similar.The frequency band extend from f = 3 kHz to 60 kHz, but the

Spectrogram and Power Spectrum Density (PSD)
In

Wavelets of the Discrete Wavelet Transform
The predetermined functions in a continuous or discrete wavelet transformation are called wavelets.They generalize the short-time Fourier transforms.In contrast to sine and cosine functions of the Fourier transform, the wavelets have not only a location in the frequency spectrum, but also in the time

Summary
The advanced new forms of Fourier transform as the spectrograms, the power spectral density and the wavelet transforms provide a deeper insight into the energy transport of the turbulent flow and in the spectra of the vortex turbulence of gas and air flows.

Figure 2 .
Figure 2. Energy spectrum as function of the wave number.

Figure 3 .Figure 4 .Figure 5 .Figure 6 .
Figure 3.Time signals of a measurement for the three measurement times from t = 1 s to t = 10 ms.

Figure 7 .
Figure 7. Comparing the 1/24 octave spectra of the sound pressure for openings of d = 1 mm up to 5 mm and pressures of p = 2.0 bar (a) and 3.0 bar (b).
addition to the FFT and the spectra, also the spectrograms provide a good insight into the spectral distribution of the flow acoustics and into the turbulence of the flow.The spectrogram is a color-scaled representation of the 200-Hz narrow-band spectrum as Figure 8 shows.In the Figure 8(a) is the spectrogram for the opening of d = 1mm and L = 12 mm and the pressure p = 3 bar and in Figure 8(b) is the spectrogram for d = 2 mm and L = 12 mm for the expansion pressure of p = 5 bar shown, where the four amplitude lines extend over the entire period of t = 1 s.Only the fifth spectral line at f = 34 kHz is discontinuous and shifts in time from f = 27 kHz to 34 kHz.The spectrogram for p = 3 bar in Figure 8(a) is much more homogeneous, with five dark brown spectral lines between f = 1.5 kHz and f = 34 kHz visible.The amplitude level is determined by the color scale a i = 0 bis 120.In Figure9the power spectrum density plots are shown in linear and logarithmic scale for the opening diameter of d = 2 mm and p = 5 bar (Figure9).In Figure9you can see the outstanding amplitudes at frequencies of f = 5.7 kHz, f = 8.7 kHz, f = 14.4 kHz and f = 23.9kHz.Similar to the representation of the sound pressure level in logarithmic scale, also the amplitudes in the frequency range f = 0.01 kHz to 10 kHz in the logarithmic representation of the power spectrum density are greatly extended and so the lower frequency range is very clear.To make the power spectrum density over the entire frequency range significantly, both forms of representation are recommended.

Figure 10 .
Figure 10.Plane and three-dimensional Gabor wavelet for the sound pressure at d = 1.6 mm and p = 5 bar.

Figure 11 .
Figure 11.Plane and three-dimensional Gabor wavelet for the sound pressure at d = 2.0 mm and p = 5 bar.

Figure 12 .
Figure 12.Plane and three-dimensional Gabor wavelet for the sound pressure at d = 3 mm and p = 4.25 bar.

Figure 13 .
Figure 13.Plane and three-dimensional Gabor wavelet for the sound pressure at d = 4 mm and p = 5 bar.

= 4 .
25 bar are shown.Here again the traces of the two maximum values in the planar representation is visible.The 3-D display shows the large number of maximum values in the two frequency tracks.The frequency range extends from f = 3 kHz to about 32 kHz, so the results of the octave and narrow-band spectra have been confirmed.In Figure13are the plane and the 3-D wavelets for the opening diameter of d = 4 mm and p = 5.0 bar visible with two distinct frequency tracks.