Triads of Domination in the Non-linear Forcing

Analyzing dominant triadic interactions in turbulent channel flow to improve modeling of non-linearities.

The beauty and complexity of turbulent fluid flow arise largely from the non-linear convective term in the Navier-Stokes equations (NSE), and it is also one of the greatest obstacles in understanding, effectively simulating, and modeling turbulent flows. One approach to reduce the cost and complexity of the nonlinear terms is to employ a quasi-linear (QL) or generalized quasi-linear (GQL) model, which restricts the resolved nonlinear interactions to those involving the large streamwise structures (Farrell and Ioannou 2007, Marston et al. 2016). When viewed from the Fourier domain, which naturally decomposes the flowfields into structures of different sizes, the quadratic non-linear term manifests itself as a convolution between triadically compatible interactions. Our work focuses on analyzing the non-linear terms to find the dominating triadic interactions. To do this, we develop interaction coefficients to quantify the contribution to the forcing by each pair of wavenumber-frequency triplets, and we compute the coefficients using data from direct numerical simulations of a turbulent channel. The coefficients show the importance of interactions involving streamwise large-scale structures, which is consistent with previous studies such as the superposition and modulation mechanisms observed in Marusic et al. 2010. Furthermore, using the coefficients, we are able to show that the regions of non-linear interactions permitted under the QL and GQL assumptions are significant contributors to the forcing, providing a possible reason for the success of QL/GQL. The tools we developed are expected to be useful for improving modeling of the nonlinearity, especially in QL, GQL, and resolvent analyses, and we would like to further improve it by closing the loop with the linear resolvent operator.

Figure: The interaction coefficient in the streamwise direction, showing large contributions from triadic interactions involving streamwise wavenumbers of 0 (the streamwise large structures).

Yuting Huang and Beverley McKeon

Funding: The support of the U.S. Office of Naval Research under grant numbers N00014-17-1-2960 and N00014-17-1-3022 is gratefully acknowledged.