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1 Introduction

In the last three decades, the rapid increase in computational power has led researchers in computational fluid dynamics to attempt to fully couple unsteady flows to flexible structures for simple geometries. Frendi (1997) showed the presence of a strong coupling between a supersonic turbulent boundary layer and a flexible structure. Handler et al. (1984) obtained the wall pressure fluctuations in a channel flow using a coarse direct numerical simulation (DNS). The coarse grid led to a lack of resolution of the high frequencies. Choi and Moin (1990) studied the structure of the wall pressure fluctuations in a turbulent channel flow using a DNS database of Kim et al. (1987). Choi and Moin (1990) computed the wave number/frequency spectra and the convection velocities of the wall pressure fluctuations and found that small structures convected at a slower speed than large structures. Na and Moin (1998) studied the effect of an adverse pressure gradient on the structure of the wall pressure fluctuations using DNS. They found that adverse pressure gradients led to elongated two-point correlation maps in the spanwise direction and decreased convection velocities. Viazzo et al. (2001) studied the spectral features of the wall pressure fluctuations using a large eddy simulation database of a plane channel flow. They reported that a sinusoidal wall perturbation had a very weak effect in terms of time mean values. Wall pressure fluctuations and flow-induced noise from a turbulent boundary layer over a bump were investigated by Kim and Sung (2006) using DNS. They found that wall pressure fluctuations increased near the trailing edge of the bump along with the presence of large structures that convected rapidly downstream. The method used in this paper is referred to as a hybrid RANS-LES method, namely detached eddy simulation (DES) (Spalart et al. 1998) which has been used extensively and has shown promising results.

2 Numerical Approach

The equations solved are derived from the baseline RANS model of Menter et al. (1994) modified by Nichols and Nelson (2003) to become a hybrid RANS-LES. The numerical procedure followed is as follows: Once a RANS mean flow is obtained, several hybrid RANS/LES computations are carried out. The parameters investigated are the protuberance height to boundary layer thickness and the surface curvature. The flow parameters used are as follows: free stream Mach number 1.6, free stream air velocity 463.3 m/s, free stream air temperature 422 K and a Reynolds number per meter of 4.92 million. Protuberance height to boundary layer thickness ratios, \( h/\delta \), studied are 0.5, 1.0 and 2.0, and the radius of curvature of the surfaces studied are as follows: 9.6, 4.8 cm and a flat surface. The 9.6 cm curved surface corresponds to the one used in the experiments (Hahn and Frendi 2013). All protuberances were cylindrical with a diameter, D, of 0.315 cm. When running a hybrid RANS/LES computation with a high Reynolds number, one needs to make sure the grid resolution is sufficient to capture the relevant flow physics of the problem. In the present computations, a hybrid unstructured grid is used with cell densities varying from 55 to 75 million depending on protuberance height. Near viscous walls, \( y^{ + } < 1 \), were enforced.

3 Results and Discussion

3.1 Overall Sound Pressure Level on the Surface

Figure 1 shows the OASPL map on a flat surface for a small protuberance, Fig. 1a, and a tall protuberance, Fig. 1b. It is clear from the figure that the tall protuberance, Fig. 1b, has a huge impact on the surface in terms of high acoustic loads over a large area surrounding the protuberance. Figure 1c shows the OASPL contours on a curved surface for a protuberance having a height to boundary layer thickness ratio of 2.0. The curved surface has a radius of curvature of 9.6 cm. The figure shows that the presence of the curvature, Fig. 1c, leads to the spread of the high dB-levels area in the spanwise direction, i.e., y-direction, both upstream and downstream of the protuberance. Downstream of the protuberance, Fig. 1c, shows lower dB-levels over a larger area than that on a flat surface, Fig. 1b.

Fig. 1
figure 1

OASPL in dB on a flat surface for a protuberance height to boundary layer thickness of (i) 0.5 (ii) 2.0 and for a curved surface for a protuberance height to boundary layer thickness of 2.0

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3.2 Surface Pressure Spectra, Two-Point and Space–Time Correlation

Figure 2a shows that the wall pressure spectra at various distances downstream of the protuberance collapse well at low frequencies using the outer scaling but not so well at high frequencies. The ratio of protuberance height to boundary layer thickness in this case is 2.0. The two-point space correlation three-diameter downstream of the protuberance shows the presence of large structures with stretching in the spanwise direction, Fig. 2b. The space–time correlation, Fig. 2c, shows a short-lived coherence in both space and time. One can also deduce the convection velocity of the flow structures based on the slope of the curves; in this case, it is found to be between 0.6 and 0.77 of the free stream.

Fig. 2
figure 2

(a) Wall pressure spectra at various downstream locations from the protuberance, (b) two-point space correlation three-diameter downstream of the protuberance (c) two-point space–time correlation three-diameter downstream of the protuberance. The protuberance height to boundary layer thickness is 2.0, and for both the two-point and the space–time correlations, the contours levels are between 0.1 and 0.9

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4 Conclusions

Results from an extensive numerical investigation on the effects of cylindrical protuberances on the wall pressure fluctuations in a supersonic turbulent boundary layer have been presented. The effects of protuberance height to boundary layer thickness and surface curvature on wall pressure fluctuations were studied. Our CFD results compared well to the experimental data for pressure coefficient and the fluctuating pressure coefficient. For tall protuberances, our results show an increase in overall sound pressure levels on the surface surrounding the protuberance. Surface curvature is found to increase the area affected by high overall sound pressure levels ahead of the protuberance but slightly decreases it downstream. Additional results can be found in (Hahn and Frendi 2013).