Scale interactions driving the nonlinear forcing and response in turbulent channel flow

In turbulence, non-linearity is essential for the transfer of energy between different scales, yet it remains a challenging part of our understanding and modeling efforts. This quadratic non-linearity is treated as forcing to the linear resolvent operator (McKeon and Sharma, JFM, 2010), and is studied in the Fourier domain, in which it becomes a convolution sum over all triadically compatible wavenumber-frequency triplets. These triadic interactions are dissected in this work to reveal the spatio-temporal nature of the interactions, by applying the linear resolvent operators to each pair of interacting triplets and quantifying the resulting contributions to the forcing and velocity fields.

The results show the importance of interactions involving streamwise large scales, consistent with the interactions permitted under the quasi-linear (QL) and generalized quasi-linear (GQL) assumptions (Marston et al., PRL, 2016). Additionally, sparsity of the non-linear interactions is observed, where most active triadic interactions concentrate around a region with similar wavespeeds for all three modes in the triad. Finally, the effect of the linear resolvent operator is studied, where the low-rank linear amplification of the resolvent enables the generation of strong velocity responses even from weak forcing.