Reynolds-number dependence of wall-pressure fluctuations in a pressure-induced turbulent separation bubble

Abstract

Direct numerical simulations are used to examine the behaviour of wall-pressure fluctuations$p_{w}$in a flat-plate turbulent boundary layer with large adverse and favourable pressure gradients, involving separation and reattachment. The Reynolds number$Re_{\unicode[STIX]{x1D703}}$based on momentum thickness is equal to 300, 600 and 900. Particular attention is given to effects of Reynolds number on root-mean-square (r.m.s.) values, frequency/power spectra and instantaneous fields. The possible scaling laws are also examined as compared with the existing direct numerical simulation and experimental data. The r.m.s. value of$p_{w}$normalized by the local maximum Reynolds shear stress$-\unicode[STIX]{x1D70C}\overline{uv}_{max}$(Simpson et al. J. Fluid Mech.vol. 177, 1987, pp. 167–186; Na & MoinJ. Fluid Mech.vol. 377, 1998b, pp. 347–373) leads to near plateau (i.e.$p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}=2.5\sim 3$) in the adverse pressure gradient and separated regions in which the frequency spectra exhibit good collapse at low frequencies. The magnitude of$p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{uv}_{max}$is however reduced down to 1.8 near reattachment where good collapse is also obtained with normalization by the local maximum wall-normal Reynolds stress$\unicode[STIX]{x1D70C}\overline{vv}_{max}$. Near reattachment,$p_{w\,rms}/-\unicode[STIX]{x1D70C}\overline{vv}_{max}=1.2$is attained unambiguously independently of the Reynolds number and pressure gradient. The present magnitude (1.2) is smaller than (1.35) obtained for step-induced separation by Ji & Wang (J. Fluid Mech.vol. 712, 2012, pp. 471–504). The reason for this difference is intrinsically associated with convective nature of a pressure-induced separation bubble near reattachment where the magnitude of$p_{w\,rms}$depends essentially on the favourable pressure gradient. The resulting mean flow acceleration leads to delay of the r.m.s. peak after reattachment. Attention is also given to structures of$p_{w}$. It is shown that large-scale spanwise rollers of low pressure fluctuations are formed above the bubble, whilst changing to large-scale streamwise elongated structures after reattachment. These large-scale structures become more prominent with increasing$Re_{\unicode[STIX]{x1D703}}$and affect$p_{w}$significantly.