Effect of adverse pressure gradient and separation
An overall effect of adverse pressure gradients onto the wall pressure field statistics is an increase of the wall pressure fluctuations and a reduction of the convection velocity. This behavior was first observed by Schloemer (1967) through an experimental study devoted to the investigation of the influence of a mild adverse pressure gradient on wall pressure fluctuations. Owing to changes in the streamwise turbulent intensity, Schloemer also noticed an increase in the wall pressure spectral densities at low-frequencies (in outer scaling), whereas little effect was observed in the high-frequency range. This result has been later confirmed [see e. g. Lim (1971)] and seems to suggest that the pressure gradient influences the outer layer region which, as described above, is directly correlated to the mid and low frequency range of the wall pressure frequency spectra.
Na & Moin (1998) performed a Direct Numerical Simulation (DNS) of a turbulent boundary layer developing over a flat plate, under both mild and strong imposed adverse pressure gradient. In the latter case (involving extensive separation) the frequency spectra in the separation bubble were found to exhibit a ш-4 decay, whereas a u-2 behavior at high frequencies was observed for the spectra downstream of the reattachment position. The analysis of two-point correlations of wall pressure fluctuations also revealed strong coherence in the spanwise direction, that was attributed to the occurrence of large two-dimensional roller-type vortical structures. These authors also showed that the presence of flow separations, re-circulations and re-attachments lead to the generation of wall pressure fluctuations whose overall level might be significantly larger (up to 30dB) than that observed in equilibrium turbulent boundary layer with no separations.
Measurements of surface pressure fluctuations for a separated turbulent boundary layer under adverse pressure gradient were reported by Simpson, Ghodbane & McGrath (1987). Those authors found that pressure fluctuations increase monotonically through the adverse pressure gradient region, and showed that the maximum turbulent shear stress in the wall-normal direction can be used as a scaling variable since it yields good collapse of the normalized spectra at various streamwise stations.
Several studies have been conducted to characterize the fluid dynamic structure of flows whose separation is induced by a surface discontinuity. Detailed results have been obtained for several geometries, including backward facing steps [see Simpson (1989), and the literature cited therein for a comprehensive review in the field], sharp edges [as in Kiya, Sasaki & Arie (1982), Kiya & Sasaki (1985), and Hudy, Naguib & Humphreys (2003)], inclined surfaces [e. g. Song, DeGraaff & Eaton (2000)] and surface bumps [e. g. Kim & Sung (2006)]. Most of these studies have shown that the wall pressure fluctuations are driven by a low frequency excitation linked to the expansion and contraction of the separation bubble, a phenomenon usually designated as flapping motion. Besides, the vortical structures within the shear layer have been identified as the source of higher frequency peaks normally observed close to the reattachment position.
Stiier, Gyr & Kinzelbach (1999) analyzed the separation bubble upstream of a Forward Facing Step (FFS) in laminar flow conditions through flow visualizations and particle tracking velocimetry measurements. They demonstrated that the laminar re-circulating region upstream of the step is an open separation bubble characterized by spanwise quasi-periodic unsteadiness. The flow topology and the pressure field upstream and downstream of an FFS at much higher Reynolds numbers have been recently studied by Largeau & Moriniere (2007). The effect of the relevant length-scales has been underlined in this work and the influence of the flapping motion upon the pressure field at the reattachment point has been demonstrated by means of pressure-velocity cross-correlations obtained from simultaneous wall microphones and hot wire anemometry measurements. Fourier pressure spectra upstream and downstream of a FFS have been presented also by Efimtzov et al. (1999) who showed that the region downstream of the step is the most significant in terms of pressure level. On the other hand, Leclercq et al. (2001) considered the acoustic field induced by a forward – backward step sequence and suggested that the most effective region in terms of noise emission is located just upstream of the FFS. The experimental results reported in Leclercq et al. (2001) have been successfully reproduced in a large eddy simulation performed by the same group, Addad et al. (2003). It was confirmed that the largest acoustic source is located in the separated region upstream of the wall discontinuity. Camussi, Guj & Ragni (2006) and Camussi et al. (2006) measured the pressure fluctuations at the wall of a shallow cavity representing a backward-forward step sequence. The authors again showed that the region close to the FFS is the most effective in terms of wall pressure fluctuations level even though the origin of the observed acoustic field was not clarified. In a recent study of the incompressible flow past a forward-facing step, Camussi et al. (2008) also observed the increase of energy at low-frequencies and a decrease at higher ones.
A flow separation can be induced also by the effect of a shockwave interacting with the boundary layer, a situation that can typically be encountered in transonic flow conditions. The prediction of pressure fluctuations in the transonic regime is particularly important in the vibro-acoustic design of aerospace launch vehicles. As a matter of fact, vibrations induced in the interior of the vehicle can exceed design specifications, and cause payload damage, as well as structural damage due to fatigue problems.
The presence of a shockwave and the consequent separation, causes an adverse pressure gradient that modifies significantly the boundary layer dynamics and causes substantial modification of the wall pressure signature. The Mach number effect in attached boundary layers has been taken into account in a few literature models [see e. g. the one proposed by Efimtsov (1982) and cited above]. On the other hand, the effect of the shockwave induced separation on the wavenumber-frequency spectrum is the subject of quite a few literature papers. We remind the numerical studies conducted by Pirozzoli and co-workers [Pirozzoli, Bernardini & Grasso (2010) and Bernardini, Pirozzoli & Grasso (2011)] based on a DNS approach used to simulate the shockwave induced separation on a flat plate at a transonic Mach number (M = 1.3). They show that the shape of the frequency wall pressure spectra is qualitatively modified by the interaction with the shock wave. In the region with zero pressure gradient, the shape of the spectra is similar to that observed in low-speed boundary layers. When the pressure gradient is relevant, the low-frequency components of the spectrum are enhanced while the higher ones are attenuated. This observation is in agreement with results obtained in low-speed boundary layers in adverse pressure gradient and it is the signature of the greater importance of large – scale, low-frequency dynamics past the interacting shock, with respect to the fine scale effects. According to observations in low speed flows upstream an FFS by Camussi et al. (2008), in the separated region downstream of the shock, a self-similar structure of the pressure spectra is observed exhibiting the -7/3 inertial scaling at intermediate frequencies and a -5 decay law at high frequencies.
Similar scalings were observed in transonic and supersonic flow conditions by Camussi et al. (2007). They analyzed the statistics of the wall pressure fluctuations on a scaled model of an aerospace launcher that has been investigated in transonic and supersonic wind tunnels. Even though qualitatively, the -1 and -7/3 scalings were documented at several stations along the surface of the model.
The determination of a general predictive model for the wavenumber – frequency spectrum in the presence of shockwaves is however still far and, to the authors’ opinion, this topic merits to be the task for future extensive research.