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

Abstract

We generate predictions for the fluctuating pressure field in turbulent pipe flow by reformulating the resolvent analysis of McKeon and Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336-382) in terms of the so-called primitive variables. Under this analysis, the nonlinear convective terms in the Fourier-transformed Navier-Stokes equations (NSE) are treated as a forcing that is mapped to a velocity and pressure response by the resolvent of the linearized Navier-Stokes operator. At each wavenumber-frequency combination, the turbulent velocity and pressure field are represented by the most-amplified (rank-1) response modes, identified via a singular value decomposition of the resolvent. We show that these rank-1 response modes reconcile many of the key relationships among the velocity field, coherent structure (i.e. hairpin vortices), and the high-amplitude wall-pressure events observed in previous experiments and direct numerical simulations (DNS). A Green's function representation shows that the pressure fields obtained under this analysis correspond primarily to the fast pressure contribution arising from the linear interaction between the mean shear and the turbulent wall-normal velocity. Recovering the slow pressure requires an explicit treatment of the nonlinear interactions between the Fourier response modes. By considering the velocity and pressure fields associated with the triadically consistent mode combination studied by Sharma and McKeon (J. Fluid Mech., vol. 728, 2013, pp. 196-238), we identify the possibility of an apparent amplitude modulation effect in the pressure field, similar to that observed for the streamwise velocity field. However, unlike the streamwise velocity, for which the large scales of the flow are in phase with the envelope of the small-scale activity close to the wall, we expect there to be a π/2 phase difference between the large-scale wall-pressure and the envelope of the small-scale activity. Finally, we generate spectral predictions based on a rank-1 model assuming broadband forcing across all wavenumber-frequency combinations. Despite the significant simplifying assumptions, this approach reproduces trends observed in previous DNS for the wavenumber spectra of velocity and pressure, and for the scale-dependence of wall-pressure propagation speed.

An input–output based analysis of convective velocity in turbulent channels

“…In figure 4, the convective velocities of structures in the intermediate scale range λ x ≈ 2 show a discontinuity as the streamwise wavelength λ x varies. This phenomenon was also observed for the scale-dependent convective velocity of wallpressure in pipe flow predicted using resolvent analysis with broadband forcing; see figure 12(a) in Luhar et al (2014). In the term-by-term analysis in section 7, we will further confirm that the convective velocity of structures associated with these scales is highly influenced by the pressure.…”

“…As shown in figure 11(a), the pressure plays an important role for the intermediate scale structures (λ x ≈ 2 and λ z > λ x ), which supports our conjecture that the discontinuity in these scales shown in figure 4 is related to the pressure. Luhar et al's (2014) figure 12(a) also showed a discontinuity of the scale-dependent convective velocity of wall-pressure computed using resolvent analysis and the maximum of the PSD to define the convective velocity. As discussed in Section 5, using the center of gravity of the PSD to define the convective velocity eliminates the discontinuity.…”

Amplitude modulation of pressure in turbulent boundary layer

“…Small-scale wall pressure associates with turbulent structures, which do not locate just above the wall sensor but exist slightly downstream at a distance of the order boundary layer thickness. A similar physical description was suggested by Luhara et al (2014) . Moreover, Ahn et al (2010) proposed the structurebased model for wall pressure ( Ahn et al, 2010 ).…”

Wall-based identification of coherent structures in wall-bounded turbulence

“…Since then, several other authors have analysed the spectrum of the pressure fluctuations, using DNS using numerical [17,18] or experimental data [19,20], corroborating the scaling arguments mentioned above. Also, models for the wall-pressure have been formulated using structures based on Townsend's attached eddy hypothesis [21], and some authors have linked specific features of the pressure fluctuations at the wall to coherent structures in the flow [22,23].…”