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Procedía Engineering 6 (2010) 303-311

Procedía Engineering

www.elsevier.com/locate/procedia

IUTAM Symposium on Computational Aero-Acoustics for Aircraft Noise Prediction

Wall pressure fluctuations in transonic shock/boundary layer

interaction

Sergio Pirozzoli and Matteo Bernardini

Universita di Roma 'La Sapienza', Dipartimento di Meccanica e Aeronáutica Via Eudossiana 18, 00184, Roma (Italy)

Abstract

The structure of wall pressure fluctuations beneath a turbulent boundary layer interacting with a normal shock wave is investigated through direct numerical simulation (DNS). In the zero-pressure-gradient (ZPG) region upstream of the interaction pressure statistics well compare with canonical boundary layers in terms of fluctuation intensities and frequency spectra. Across the interaction zone, the r.m.s. wall pressure fluctuations attain large values (in excess of » 162 dB), with an increase of about 7 dB from the upstream level. The main effect of the interaction on the frequency spectra is to enhance of the low-frequency Fourier modes, while inhibiting the high-frequency ones. Excellent collapse of frequency spectra is observed past the interaction zone when data are scaled with the local boundary layer units. In this region an extended w-1'3 power-law behavior is observed, which is associated with the suppression of mean shear caused by the imposed adverse pressure gradient. © 2010 Published by Elsevier Ltd.

Keywords: pressure fluctuations, shock/boundary layer interaction, direct numerical simulation

1. Introduction

The importance of interactions between shock waves and turbulent boundary layers (SBLI) in aeronautical and aerospace applications is widely recognized, since the occurrence of localized fluctuating pressure loads across the interaction region can negatively affect the lifetime of aircraft structures [1]. Especially relevant is the research on interactions occurring in the transonic regime, having an impact on the design of high-speed wings and diffuser, ad well as of turbo-machinery components. Despite its relevance in practical applications, the unsteady wall pressure signature of transonic SBLI has not been investigated in detail, and most studies have

Email address: sergio.pirozzoli@uniroma1.it (Sergio Pirozzoli and Matteo Bernardini)

1877-7058 © 2010 Published by Elsevier Ltd. doi:10.1016/j.proeng.2010.09.032

focused on the unsteady pressure signature for flows with global, low-frequency motion of the impinging shock [2, 3], that is relevant for the prediction of transonic buffet on airfoils [4, 5].

At present, most of the available information on the effect of adverse pressure gradients on the wall pressure stems from investigations of low-speed boundary layers. Measurements of surface pressure fluctuation spectra for a separated turbulent boundary layer under adverse pressure gradient were reported by [6], who found that pressure fluctuations increase monotonically through the APG region, and showed that the maximum turbulent Reynolds shear stress is the proper scale for normalization. A similar conclusion was also reached by [7], who investigated the structure of wall pressure fluctuations from a DNS database of turbulent boundary layer over a flat plate. In the case of extended flow separation, the frequency spectra were found to exhibit distinct power-law scalings in different regions of the flow (upstream, within and past the separation bubble).

The aim of the present work is to investigate the structure of the wall pressure field induced by a transonic shock/boundary layer interaction, providing description of the surface pressure field in terms of the frequency spectra, that are useful to predict the structural dynamical response [8]. For this purpose we interrogate a DNS database of a canonical flow case, whereby a normal shock wave is made to impinge on a turbulent boundary layer developing over a flat plate. The focus of the study is on the characterization of pressure fluctuations associated with fine-grain turbulence, and no attempt is made to investigate the possible presence of low-frequency unsteadiness that may result from self-sustained oscillations.

2. DNS database

The pressure field is analyzed exploiting the DNS database of [9], consisting of a turbulent boundary layer that develops over a flat plate with Mach number M„ = 1.3, Reynolds number Reg = 1200 (based on the momentum thickness of the upstream boundary layer), and made to interact with a normal shock wave. The convective terms in the Navier-Stokes equations are discretized using a hybrid approach, whereby sixth-order central discretization of the skew-symmetric split form is used in smooth regions, and shock waves are captured through a seventh-

Figure 2: Instantaneous density field in x - y plane. 32 contour levels, 0.77 < p/pM < 1.53.

tion # x* Pw/Pm Pe/PM Ue /Mm 6/60 6*/6*0 6'0/6v A1/60 A3/6*0

0 -0.2 1 1 1 4.58 1 73.67 1.83 1.94

1 0.6 1.27 1.16 0.89 5.87 1.70 46.90 2.18 4.68

2 2.0 1.61 1.40 0.74 9.84 4.63 26.92 6.04 11.71

3 4.0 1.74 1.47 0.69 11.88 4.83 46.88 8.49 15.61

Table 1: Boundary layer properties at various streamwise stations. Subscripts: e indicates properties at the edge of the boundary layer, w indicates wall properties, 0 indicates properties taken at station |~0], and m free-stream properties.

order WENO scheme, the switch being based on the Ducros sensor. Viscous fluxes are computed by means of standard sixth-order central formulas, and a fourth-order Runge-Kutta method is used to perform time integration. Inlet conditions for the turbulent boundary layer are based on the rescaling-recycling procedure developed by [10] and extended to the compressible case by [11]. The mean field is kept constant at the inflow station, and fluctuations are recycled from a cross-stream slice, after suitable rescaling.

In the presentation of the results the origin of the longitudinal coordinate is set at the beginning of the interaction (x0), corresponding to the point where the mean wall pressure distribution pw, reported in Fig. 1, starts to rise (we assume pw = 1.005pM). Coordinates are normalized by the interaction length scale L, that in the transonic case [12] corresponds to the distance between the sonic point location (i.e. the streamwise station where the mean wall pressure is equal to the critical value), and the origin of the interaction (indicated with x0). Scaled coordinates are therefore denoted as x* = (x - X0)/L, y* = y/L and z* = z/L. The computational region is ideally divided in three zones: i) the zero-pressure gradient region (ZPG) upstream of the interaction (for x* < 0); ii) the supersonic adverse-pressure-gradient (APG) region (for 0 < x* < 1) and a subsonic adverse-pressure-gradient region (for x* > 1).

The interaction pattern (see Fig. 2) consists of a fan of compression waves originating upstream of the nominal impingement point, and of a nearly normal shock that drives the incoming flow to subsonic conditions. Past the interaction zone the flow is characterized by the the formation of a turbulent mixing layer, with unsteady release of large eddies. The analysis of the flow recovery process past the interaction zone shows that the boundary layer reacts to the impose adverse pressure gradient by attaining a new equilibrium state over a distance of O(L) past x* = 1,

Figure 3: Distribution of the r.m.s. wall pressure fluctuations scaled by the reference dynamic pressure

which is characterized by self-similarity of the mean velocity field in the scaling of [13]. The DNS data were validated through comparison with experimental measurements in a transonic channel [12].

In the following the pressure statistics are reported at various streamwise stations, listed in Table 1, together with the corresponding boundary layer parameters.

3. Results

The distribution of the r.m.s. wall pressure fluctuations (prms) is reported in Fig. 3, normalized by the free-stream dynamic pressure q„ = 1/2p„ u^. The distribution of prms at the wall, reported in Fig. 3, exhibits a nearly flat distribution in the ZPG region, where prms « 0.010 q„, corresponding to a sound pressure level (SPL) of about 155 dB (assuming free-stream atmospheric pressure). In terms of wall units, the pressure fluctuation intensity is very nearly prms = 2.50 tw, in good agreement with the findings of [14], who reported prms = 2.55 tw for low-speed turbulent boundary layers at ReT < 333, and the recent DNS data of [15]. Pressure fluctuations experience strong amplification in the supersonic APG region, attaining a peak value prms « 0.022 q„ (corresponding to approximately 162.5 dB) at x* « 0.6, and relax towards a nearly constant value (0.0145 q„) in the subsonic APG region. Amplification of pressure fluctuations in the presence of shear layers was also reported for separated low-speed turbulent boundary layers [6, 7].

The structure of the pressure fluctuations is analyzed looking at the one-point frequency spectra, x, y), whose integral over all frequencies yields the r.m.s. pressure. The shape of the frequency spectrum in the ZPG region, reported in Fig. 4 well conforms to that found in incompressible boundary layer experiments [17], whereas differences are found from previous DNS at lower Reynolds number [7]. As theoretically predicted [8], the spectra drop off as approximately a-5 at high frequency. The incompressible low-frequency scaling a2 is not observed here, either due to the limited duration of the time sample (this is true of all DNS studies published so far), or to the effect of finite compressibility, that should imply a flat spectrum at low frequencies [19]. The intermediate a-1 scaling associated with turbulent activity in the log layer [20], and expected for ReT > 333 [14] is also absent from the present data.

In Fig. 5 the frequency spectra across the interaction zone are reported in outer scaling. Note that, to compare data from different stations, the units are referred to the reference state

30 : 20 10 0 -10 -20

-40 -50

2 10-1 10° w vwlu2T

Figure 4: Wall pressure frequency spectrum at station | 0 | (solid line) scaled in inner variables, compared with the experimental data of [17] at Ree = 1577 (circles) and the DNS data of [7] at Ree = 300.

1-1 -20

o M 8 -30

8 s -40

Og _o -70

o -800

10-1 100 0}5*0lurXl

Figure 5: Wall pressure frequency spectra at various stations scaled in outer variables taken upstream of the interaction. The arrows denote the direction of increasing x*. Data are reported at x* = -0.2; 0.13; 0.37; 0.6; 0.86; 1.6; 2.0; 2.3; 2.5; 3.0; 3.5; 4.0.

^ -40 u

1 -50 lJ^ -60

tg -70 O

2 -80 -90

To5 101

o)S'0/ue

Figure 6: Wall pressure frequency spectra at various stations in the subsonic APG region. Outer variable scaling with pressure scaled by q2 (a) and r2m (b). Data are reported at x* = 1.6; 2.0; 2.3; 2.5; 3.0; 3.5; 4.0.

upstream of the interaction (station |"o]). As found for low-speed boundary layer flows in adverse pressure gradient [21, 7], as well as for the flow past a forward-facing step [22], the shock wave enhances the lower frequencies and inhibits the higher ones, indicating the occurrence of large-scale dynamics past the interacting shock. The frequency spectra in the downstream subsonic-APG region are reported in Fig. 6, where data are normalized using the local free-stream velocity (ue) and dynamic pressure (qe). Excellent collapse of data is observed using this scaling, thus confirming the self-similar structure of the boundary layer recovery region as far as the mean flow properties are concerned [9]. In this zone the spectra still exhibit a w-5 high-frequency scaling, but an extended w-7/3 power-law scaling also appears at intermediate frequencies, followed by a spectral bump associated with the change of slope of the PSD. A w-7/3 spectral scaling was first theoretically predicted by [25] for locally isotropic turbulence, as the counterpart of the Kolmogorov k-5/3 energy spectrum scaling. Such inertial pressure scaling has been occasionally observed in experiments of turbulent jets [26] and forward-facing step flows [22], and in DNS of isotropic turbulence at large Reynolds number [27, 28]. Note that a w-7/3 is not observed in ZPG boundary layers (even at very large Reynolds numbers), owing to the influence of the mean shear [29]. To our knowledge, the w-7/3 has never been observed in DNS of wall-bounded turbulence, even though a narrow power-law spectral scaling (with exponent close to -2) was reported by [7]. As seen in Fig. 7, the power-law spectral scaling is lost moving away from the wall, entering the mixing layer area.

To understand the physical significance of the observed spectral scalings, an analysis has been carried out to investigate the sources of pressure fluctuations. For that purpose, we consider Lighthill's equation for the instantaneous pressure under the assumption of weak compressibility effects, upon neglect of both entropy fluctuations and viscous effects, that reads [8]

1 d2P d2P 52 , v

=2 Of - j = Ql-Qlj UiUj>' (1)

where c is a reference speed of sound. Introducing Favre decomposition into (1) yields

1 d2p' d2p' _ d2

1 UL- a p = _(TS-T + TT-n (2)

-2 dt2 dxjXj dxidxj V ij ij >' ()

Figure 8: Distribution of r.m.s. of pressure sources in (a) the ZPG region (x* = -0.2) and in (b) the mixing layer region (x* = 4). Solid line, total source; dashed line, S-T source, dotted line, T-T source.

Tj- = p (üiu'j + /), (3)

accounts for the interaction between mean velocity gradients and turbulence, and

j = puiu'j) > (4)

accounts for turbulence-turbulence interactions. The distributions of the r.m.s. source terms JS-T = Tj—T, JT-T = TT-, are reported in Fig. 8 at the stations [0 and [2], as a function of the wall distance. In the ZPG region (station [0) the S-T and the T-T source terms have comparable magnitude throughout the boundary layer, the former attaining a peak at y+ x 36, and the latter peaking at y+ x 18, and being dominant in the near-wall region. In the subsonic APG region (station |~2]) the T-T source term is dominant, and it peaks near the mid-line of the mixing layer. The S-T term stays much smaller that the T-T term through the inner part of the boundary layer, and it only becomes similar in magnitude in the outer part of the mixing layer. Taking into account these evidences, we argue that the power-law spectral scaling of wall pressure observed past the shock is related to the reduction of the mean shear caused by the adverse pressure gradient, that makes dominant the contribution of the turbulence-turbulence interaction. However, moving away from the wall, the importance of the S-T source terms again becomes non negligible, and the to-7'3 scaling is not observed, as seen in Fig. 7).

4. Conclusions

The wall pressure signature of a transonic shock/boundary layer interaction has been analyzed upon interrogation of a DNS database. The structure of the pressure field upstream of the interaction is found to conform well with available experimental and DNS data, with a clear a— scaling at high frequency. The main effect of the interaction with the impinging shock is the enhancement of low frequencies, and suppression of the higher ones, with an overall increase, mostly limited to the supersonic part of the interaction. In the downstream recovery region the pressure spectra exhibit self-similarity when plotted in local boundary layer units, and a distinct to-7'3 spectral range emerges. The analysis of the pressure source terms has shown that such scaling is due to reduction of the mean shear caused by the imposed adverse pressure gradient, which makes the turbulence-turbulence source term dominant throughout the recovery region.

Acknowledgments

The support of the CASPUR computing consortium through the 2009 Competitive HPC Grant "Extremely large scale simulation of transonic turbulent flows" is gratefully acknowledged.

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