Experimental Setup

A sketch of the experimental setup was already shown in figure 4. In a steel pressure/vacuum vessel a jet is released from a (cylindrical) nozzle with a subsonic velocity. The ambient pressure in the vessel can be varied between $0.01$ and $10$ bar. This allows us to vary the Reynolds number of the flow without changing the Mach number or the physical dimensions of the setup. The Reynolds and Mach numbers are defined as

$$Re={\frac{\rho Ud}{\mu }}\,\,\,\,\,Ma={\frac{U}{c}}$$

In a gas the speed of sound $c$ is only a function of the temperature and can be kept approximately constant $\simeq$ 340 m/s for the temperature equal to the ambient value. With a jet nozzle $d$ of say 10mm and an exit velocity $U$ of 40 m/s the Reynolds number can in principle be varied between, 400 and 400000 while the Mach number is constant $\approx 40/340=0.12$. The lower value of Re can be easily calculated with help of DNS and the higher value will be a challenge for LES. From experiments listed in Table 1 it is known that the dominant frequency of the sound $F$ is equal to

$$F\propto StU/d$$

where $St$ is the Strouhal number (around 0.2-0.6 for a jet, depending on the shape of the jet nozzle) , $U$ is again the jet exit velocity and $d$ the jet diameter. With the numbers given above we expect a frequency of $%% F=0.4\cdot 40/0.01=1600Hz$ (note that this frequency is in principle independent of the value of $\rho$). The acoustic pressure fluctuations and the sound frequency $f$ (not equal to $F$ due to the Doppler shift) of the jet are according to Goldstein (1974)

$$p' \propto {\frac{K\rho^2 u^{\prime }{}^{4}d^{3}}{16\pi ^{2... ...{5}r^{2}}},\,\,\,\,f(\theta ))={\frac{F}{1-M_{c}\cos \theta }}$$

where $r$ is the position of the observer, $K$ a non-dimensional constant, $%% u^{\prime }$ is the velocity fluctuation, $M_{c}$ the convective Mach number, roughly half of the jet Mach number, $\theta$ the angle with the main jet axis and $F$ the source frequency of the sound. Most microphones measure $p'^2$ while $p'$ is believed to be the quantity most closely related to the sensation of loudness (see Goldstein 1974). >From turbulence theory we expect that $u^{\prime }$ scales with the jet velocity $U$ and will more or less be independent of the Reynolds number. So in our experiment we expect that the microphone output $p'^2$ is mainly a function of $\rho$ and the angle $\theta$ between the jet centerline and the observer. To measure the acoustic field of the jet we will place ($\approx 20$) acoustic pressure transducers (microphone) in the vessel. We will do this at various different angles $\theta$. Due to the small diameter of the jet nozzle and the wave length of the acoustic wave ( $\lambda = c/f =340/1600=0.2m$) we need rather small microphones with physical dimensions which are much smaller than $\lambda$. Such microphones are commercially available.

The acoustic field can be visualized with the Schlieren method.⁵ This method can be used to study the directivity pattern of the sound and acoustic reflection on the sides of the vessel. Detailed information about the structures in the flow, like vortices secondary motion, etc. can be obtained with Particle Image Velocimetry (PIV).⁶

Figure: A cross section of a turbulent jet flow obtained from a PIV experiment.The arrows denote the velocity in the plane perpendicular to the main jet axis. The color of the arrow is a measure for the axial velocity. (courtesy of ir. C. van Doorne, Laboratory for Aero and Hydrodynamics.

The statistics of the flow field can be obtained by PIV measurements or by Laser Doppler Anemometry (LDA) measurements.⁷Furthermore, we have to measure the temperature distribution in the vessel. We do not expect that at low velocities the temperature will play an important role but we also can not rule this effect out at forehand. For higher Mach numbers, which we want to study in a later stage, temperature effects will be important. The spatial temperature field can be measured with infrared thermography.

Another problem, put forward by Prof. M. Hirschberg of the TUE, will be the reflection of the acoustic waves on the wall off the low pressure vessel. To minimize this reflections we have to place acoustic damping material on the wall of the vessel. Such material is commercially available. If the damping of the acoustic reflections is not sufficient we have to incorporate the acoustic reflections in our simulation model.