Triadic orthogonal decomposition reveals nonlinearity in fluid flows

Energy transfer across scales is fundamental in fluid dynamics, linking large-scale flow motions to small-scale turbulent structures in engineering and natural environments. Triadic interactions among three wave components form complex networks across scales, challenging understanding and model reduction. We introduce triadic orthogonal decomposition (TOD), a method that identifies coherent flow structures optimally capturing spectral momentum transfer, quantifies their coupling and energy exchange in an energy-budget bispectrum and reveals the regions where they interact. Triadic orthogonal decomposition distinguishes three components – a momentum recipient, donor and catalyst – and recovers laws governing pairwise, six-triad and global triad conservation. We apply TOD to three examples: the classical cylinder wake, experimental wind turbine wake data and a direct numerical simulation of isotropic turbulence. Energy transfer can be spatially distributed but vanish upon integration or spatially localised but facilitate net interscale exchange, so a complete characterisation of nonlinearity requires examination of both integral and local transfers. In the cylinder wake, we link backscatter of energy from high to low frequencies to a compact attenuation region downstream of the cylinder. In the turbine wake, we confirm the known association between energy amplification and decay and vortex tilting, but observe more complex secondary mechanisms in suboptimal modes. For isotropic turbulence, we derive and confirm inertial-range frequency scaling for convective–recipient covariances, then demonstrate self-similar energy transfer at each rank.