Unbounded wall turbulence induced by inverse cascade*

While seeking the ultimate statistical invariance of turbulence, classical boundary layer theory is unable to distinguish between the averaged states of wall flows with two- (2D) and three-dimensional (3D) fluctuations. Here we demonstrate glaring differences between 2D and 3D Poiseuille channel flows and present theoretical explanations for the differences in their Reynolds numbers ($Re_ au$) asymptotic behaviors. In particular, due to the peculiar inverse cascade, large-scale wavy structures (LSWS) are developed in 2D flows which inject high energy flux toward the wall and cause extreme wall dissipation. The latter follows a distinct $Re^{1/3}_ au$ scaling in our direct numerical simulation domain ($130<Re_ au<8100$), the trend observed also for the root mean square of the pressure and velocity fluctuations (as well as for the bulk velocity). Rationale for the scaling is further given through an LSWS-induced dissipative time scale (provided with the 2D friction law), which is unlike the viscous time scale in 3D flows due to the absence of LSWS. As a counterpart to the classical boundary layer of bounded 3D fluctuations (Chen & Sreenivasan J. Fluid Mech. 908, 2021; 933 2022), the results here reveal an unprecedented asymptotic state of wall flows in which the inverse cascade induces unbounded 2D fluctuations.