Unstable turbulent channel flow response to spanwise-heterogeneous heat fluxes: Prandtl's secondary flow of the third kind

Abstract

Turbulent secondary flows are defined as Prandtl's secondary flow of the first or second kind, the former produced by stretching and/or tilting of vorticity, the latter produced via spatial heterogeneity of Reynolds stresses. Both mechanisms are instantaneously active within inertia-dominated wall turbulence; Reynolds stress spatial heterogeneity is required for Reynolds-averaged secondary flows. Spanwise-variable surface roughness can induce turbulent stress spatial heterogeneity in the spanwise–wall-normal plane and provide sustenance for streamwise-aligned mean secondary flows. Herein, we demonstrate that turbulent secondary flows can also be sustained by spanwise variability in the surface heat flux in unstably stratified turbulent channels, defined hereafter as Prandtl's secondary flow of the third kind. Support for this mechanism is established with scaling arguments, while large-eddy simulation is used to model inertia-dominated channel turbulence responding to a lower boundary with uniform aerodynamic/hydrodynamic roughness but spanwise-variable surface heat flux. Transport equations for streamwise vorticity and turbulent kinetic energy, $k$ , outline the conditions needed for third-kind production: shear and buoyancy production over the elevated heat flux regions necessitates lateral entrainment of low- $k$ fluid, inducing mean counter-rotating secondary cells aligned such that upwelling and downwelling occur over the high and low heat flux regions, respectively. Buoyancy-driven production of $k$ alters aggregate flow response and thus is a distinctly different mechanism responsible for sustenance of secondary flows.