Pressure power spectrum in high-Reynolds number wall-bounded flows

We study the behaviors of pressure fluctuations in high Reynolds number wall-bounded flows. Pressure fluctuations are small-scale quantities compared to velocity fluctuations in a wall-bounded flow (Tsuji, Marusic, & Johansson, Int. J. Heat Fluid Flow, vol. 61, 2016, pp. 2–11.): at a given wall-normal distance y, the premultiplied velocity spectrum peaks at a streamwise wavelength on the order of the boundary layer thickness (λ_x=O(δ)), whereas the premultiplied pressure spectrum peaks at λ_x < O(y). The differing scales of pressure and velocity pose a challenge to modeling, and the scaling of the pressure spectrum in wall-bounded flows remains an unsolved issue from both a theoretical and measurement standpoint. To address this unresolved issue, we incorporate Kolmogorov's theory (K41) within the framework of Townsend's attached eddy hypothesis to account for the small scale nature of pressure fluctuations, leading to the first derivation that is consistent with both theories. Our main result is that at a wall-normal distance in the logarithmic layer the premultiplied pressure power spectrum scales as [k_xE_pp]∼λ_x ⁿ⁻¹y^−(3+n)/4, for λ_x < y/tan (θ), and as [k_xE_pp]∼λ_x ^(3n−7)/4, for λ_x > y/tan (θ). Here, θ is the attached-eddy inclination angle, k_x is the streamwise wavenumber, the velocity spectrum follows a k⁻¹ scaling for 1/k_x > y/tan(θ) and a k^−5/3 scaling for 1/k_x < y/tan (θ), and n is a Reynolds-number-dependent constant. This result conforms to Kolmogorov's theory of small scale turbulence, i.e., it yields a −7/3 scaling for the small scales at high Reynolds numbers, and also yields the anticipated −1 scaling for the logarithmic layer scales. Detailed analysis shows that pressure and spanwise velocity have differently statistical properties: while an outer peak emerges in the premultiplied spanwise velocity spectrum at high Reynolds numbers, no outer peak is expected in the premultiplied pressure spectrum. The derived scalings are confirmed using data from a direct numerical simulation of a channel flow at friction Reynolds number Re_τ=5200.