Using Small Silicon Pressure Transducers to Determine the Longitudinal Space-Time Correlation Coefficient of the Wall Pressure
Paper in proceedings, 1994
Wall-Pressure fluctuations beneath a turbulent boundary layer are a key issue in studies of the boundary layer dynamics. The pressure fluctuations are coupled to the velocity fluctuations via complex interactions with the mean shear as well as with the velocity fluctuations themselves. These phenomena have been studied theoretically as well as experimentally, and reviews of earlier work can be found in e.g. Blake (1986).
From an experimental point of view the pressure fluctuations in turbulence flow is far from being as comprehensive as that of the velocity fluctuations. In case of all wall-pressure fluctuations, where a pressure transducer is mounted flush on the surface, some general facts have been proposed like the order of magnitude of the pressure fluctuation and a general form of the power spectrum. However, one main criticism may be raised about these experiments, namely that the size of the pressure the transducer was employed. A large transducer attenuates the small scale high frequency components of the pressure signals due to spatial and temporal averaging.
The introduction of silicon technology into fluid dynamics offers new possibilities in the design of extremely small sensors for the study of turbulent phenomena. A first generation of small silicon based pressure transducers was developed and tested by Löfdahl et al. (1993). Important issues were to find a reliable method to detect the diaphragm deflection and to take into consideration the handling of a sensor with a diaphragm thickness of less than one micron. The present paper is based on further development second generation pressure transducers, which are fabricated employing more complex and sophisticated methods, yielding even smaller sensors and thereby an increased spatial and temporal resolution.
A classic way to detect pressure fluctuations is to use cavity covered with a diaphragm, where a stable static pressure is maintained on one side of the diaphragm, while the other side is exposed to the fluctuations. The deflection of a diaphragm is a measure of the fluctuations, and can be detected in many ways. In accordance with the results of Löfdahl et al. (1993), the diaphragm deflection is detected by two resistors located on the diaphragm. Two additional resistors are integrated close to the diaphragm making it possible to connect the four resistors into a temperature insensitive Wheatstone bridge. All sensors are equipped with a vent channel starting at the cavity and emerging at the opposite side of the chip. Figure 1 shows a principal drawing of the second generation of pressure transducers. The different parts such as the diaphragm, polysilicon, piezoresistive gauges, conductors, vent channel and bonding pads are clearly indicated in the figure. The side lengths of the diaphragms are 100 and 300 μm, respectively and the thickness is 0.4 μm.
All measurements were carried out in a closed low-speed wind tunnel. In the horizontal symmetry plane of the test section (1.25 m x 1.80 m x 3.00 m), a flat plate of 2.5 m lengths was positioned. The present experiments were all carried out in a cross-section of the flat plate at a distance of approximately 1.5 m from the leading edge. In this cross-section, the boundary layer thickness, δ, was in the range of 27 to 39 mm and the friction velocity, ___, was determined to be 0.235 to 1.390 m/s, respectively.
In Figure 2, the probability density distribution of the pressure amplitudes are shown for the diaphragm side lengths of 100 and 300 μm. The distribution is normalized so that the area underneath the curve is unity. As is shown, a decreasing diaphragm size is associated with a shrinking peak and a continuous expansion in the probability distribution, i.e. a wider distribution capturing more of the larger pressure amplitudes. The 300 μm transducer revealed a slightly more peaked distribution.
To elucidate the influence of the diaphragms size of the measured fluctuating pressure, the rms value is plotted as a function of the wall unit, ___. This is shown in Figure 3. The quantity, ___ is again interpreted as an intensity of the fluctuating pressure, and it is clearly shown in Figure 3 that the 100 μm sensor is capable of capturing higher values of the fluctuating pressures. Thus the 100 μm sensor resolves the smallest eddies, spatially and temporally, better than the 300 μm sensor. A physical interpretation of this might be that the larger sensor attenuates and integrates eddies with a length scale less than a certain threshold value, while the smaller transducer is capable of resolving these eddies, see Blake (1986), and Ligrani & Bradshaw (1987) for a corresponding discussion on hot-wires. An estimate of this threshold value might be obtained for the present experimental set up. As seen in Figure 3, the fitted curves are converging at approximately 10 ___. implying that pressure fluctuations associated with a length scale less than this value can not be separated. Interpreting this a measure of the threshold value, it is interesting to know that 10 wall units is approximately 125 μm for the present flat plate flow case, and the determined threshold value hence is about two times the estimated Kolmogorov scale for the flat plate boundary layer.
In the full paper, the wall pressure correlation coefficient as a function of dimensionless time delay and longitudinal spatial separation will be shown. Four transducers have been used in a squared array to accomplish a contour map of the longitudinal space time correlation, similar to the map of Willmarth & Woolridge (1962), and the local advection speed of the pressure producing eddies for various frequency bands is determined. Here it is worthwhile to point out that the ratio of the transducer diameter to boundary layer thickness of Willmarth & Woolridge (1962) was of the order 1/30, while this ratio is 1/300 for present work.