On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis

摘要

We generate predictions for the fluctuating pressure field in turbulent pipe flow byudreformulating the resolvent analysis of McKeon and Sharma (J. Fluid Mech., vol. 658,ud2010, pp. 336–382) in terms of the so-called primitive variables. Under this analysis,udthe nonlinear convective terms in the Fourier-transformed Navier–Stokes equationsud(NSE) are treated as a forcing that is mapped to a velocity and pressure response byudthe resolvent of the linearized Navier–Stokes operator. At each wavenumber–frequencyudcombination, the turbulent velocity and pressure field are represented by theudmost-amplified (rank-1) response modes, identified via a singular value decompositionudof the resolvent. We show that these rank-1 response modes reconcile many of theudkey relationships among the velocity field, coherent structure (i.e. hairpin vortices),udand the high-amplitude wall-pressure events observed in previous experiments anduddirect numerical simulations (DNS). A Green’s function representation shows thatudthe pressure fields obtained under this analysis correspond primarily to the fastudpressure contribution arising from the linear interaction between the mean shear andudthe turbulent wall-normal velocity. Recovering the slow pressure requires an explicitudtreatment of the nonlinear interactions between the Fourier response modes. Byudconsidering the velocity and pressure fields associated with the triadically consistentudmode combination studied by Sharma and McKeon (J. Fluid Mech., vol. 728, 2013,udpp. 196–238), we identify the possibility of an apparent amplitude modulation effect inudthe pressure field, similar to that observed for the streamwise velocity field. However,udunlike the streamwise velocity, for which the large scales of the flow are in phaseudwith the envelope of the small-scale activity close to the wall, we expect there to beuda π/2 phase difference between the large-scale wall-pressure and the envelope of theudsmall-scale activity. Finally, we generate spectral predictions based on a rank-1 modeludassuming broadband forcing across all wavenumber–frequency combinations. Despiteudthe significant simplifying assumptions, this approach reproduces trends observedudin previous DNS for the wavenumber spectra of velocity and pressure, and for theudscale-dependence of wall-pressure propagation speed.