Identification of triadic phase coupling in wall-bounded turbulence using the bispectrum

TL;DR

This work addresses how external forcing perturbs energy transfer in wall-bounded turbulence by characterizing triadic phase coupling with the bispectrum $B(f_1,f_2)$, its normalized form $b(f_1,f_2)$, and the phase $\beta(f_1,f_2)$. The authors apply these tools to canonical and forced turbulent boundary-layer data, augmenting with a skewness/asymmetry decomposition and a spectral-energy-transfer function $\hat{T}(f)$ to quantify interscale dynamics without arbitrary filtering. Key findings show that forcing at $f=35$ and/or $f=50$ Hz induces strong triads, with both forward and reverse cascades depending on wall-normal position, and that two-frequency forcing broadens and strengthens interscale coupling, concentrating energy transfer near the near-wall critical layers. This framework provides a robust, filter-free method to diagnose perturbation-induced energy redistribution in turbulence, with potential implications for flow control and drag-reduction strategies, by directly linking triadic phase relationships to energy transfer processes. All quantities are expressed in the Fourier domain via $B(f_1,f_2)$, $b(f_1,f_2)$, $\beta(f_1,f_2)$, and related transfer metrics $\hat{T}(f)$, $\hat{T}_S(f)$, and $\hat{T}_D(f)$.

Abstract

The direction and magnitude of energy transfer between turbulence scale brought about by external forcing on a turbulent boundary layer are uncovered through the bispectrum, bicoherence, and biphase. The bispectrum is a third-order, complex-valued spectrum of the streamwise velocity that preserves the phase information between triadically consistent scales. Normalized bispectrum is the bicoherence, a measure of the relative amount of energy at a higher frequency that results from quadratic phase coupling of two lower frequencies. The phase of the bispectrum, the biphase, measures the phase lag between the high frequency and two lower frequencies that add to it, unveiling whether a triadic interaction produces a forward or reverse cascade of energy. Summing the bispectrum over triadically consistent frequencies allows a spectral decomposition of the velocity skewness and asymmetry, unveiling the triadically active scales in the energy transfer processes. An average sense of energy transfer is inferred from the phase of this skewness spectrum, which shows that scales smaller than the boundary layer thickness contribute to a forward cascade on average, while those larger than the boundary layer thickness have a mix of forward and reverse events. These measures show that the forced scales in the perturbed boundary layer have a mixture of forward and reverse energy transfer processes for different sets of triadic scales and wall-normal locations, providing a method of quantifying the effects of external perturbations on turbulent flows without any need for artificial filtering.

Figures (13)

• Figure 1: Schematic of the experimental setup. Figure reproduced from duvvuri2016nonlinear.
• Figure 2: Statistics of the three experimental cases. (a) The streamwise variance as a function of wall position, normalized in outer units: canonical flow (- -); single forcing frequency ($\triangle$); two forcing frequencies ($\square$). (b) The skewness ($S_u$) with the same symbols representing the three cases. Inset shows the region of maximal deviation from the canonical flow.
• Figure 3: Pre-multiplied velocity spectra for the three flow configurations, with the canonical (unforced) boundary layer in (a), the single forcing case in (b), and the two-frequency forcing in (c).
• Figure 4: Magnitude of the bispectrum at $y/\delta =0.2$ in the low frequency region for (a) the canonical boundary layer, (b) the single forcing frequency case, and (c) two-frequency forcing case.
• Figure 5: All plots shown at $y/\delta =0.20$. (a) Bicoherence over the entire frequency domain (no statistical masking). (b) Bicoherence shown only in the low-frequency range. Data has been masked for statistical significance following elgar1989statistics. (c) Biphase over the same frequency region in (b) for statistically significant values corresponding to the bicoherence. The left column is the canonical boundary layer, middle column is the single forcing frequency case, and the right column is the two-frequency forcing case.