On the role of return to isotropy in wall-bounded turbulent flows with buoyancy

Abstract

High Reynolds number wall-bounded turbulent flows subject to buoyancy forces are fraught with complex dynamics originating from the interplay between shear generation of turbulence ($S$) and its production or destruction by density gradients ($B$). For horizontal walls, $S$ augments the energy budget of the streamwise fluctuations, while $B$ influences the energy contained in the vertical fluctuations. Yet, return to isotropy remains a tendency of such flows where pressure–strain interaction redistributes turbulent energy among all three velocity components and thus limits, but cannot fully eliminate, the anisotropy of the velocity fluctuations. A reduced model of this energy redistribution in the inertial (logarithmic) sublayer, with no tuneable constants, is introduced and tested against large eddy and direct numerical simulations under both stable ($B<0$) and unstable ($B>0$) conditions. The model links key transitions in turbulence statistics with flux Richardson number (at $Ri_{f}=-B/S\approx$$-2$, $-1$ and $-0.5$) to shifts in the direction of energy redistribution. Furthermore, when coupled to a linear Rotta-type closure, an extended version of the model can predict individual variance components, as well as the degree of turbulence anisotropy. The extended model indicates a regime transition under stable conditions when $Ri_{f}$ approaches $Ri_{f,max}\approx +0.21$. Buoyant destruction $B$ increases with increasing stabilizing density gradients when $Ri_{f}<Ri_{f,max}$, while at $Ri_{f}\geqslant Ri_{f,max}$ limitations on the redistribution into the vertical component throttle the highest attainable rate of buoyant destruction, explaining the ‘self-preservation’ of turbulence at large positive gradient Richardson numbers. Despite adopting a ‘framework of maximum simplicity’, the model results in novel and insightful findings on how the interacting roles of energy redistribution and buoyancy modulate the variance budgets and the energy exchange among the components.

The nocturnal boundary layer transition from weakly to very stable. Part II: Numerical simulation with a second‐order model

“…Bou‐Zeid et al . () found a theoretical maximum attainable Ri f of 0.21, associated with the value of the Rotta constant in the return‐to‐isotropy term of the TKE budget equation. The present result indicates that a similar limit occurs in the second‐order model used.…”

Critical flux Richardson number for Kolmogorov turbulence enabled by TKE transport

“…Using velocity variance budget equations with the simplest linear Rotta closure model for return‐to‐isotropy terms (Rotta, ), and assuming isotropy for the velocity variance dissipation terms, Bou‐Zeid et al . () obtained Ri f , c = 0.21. This is exactly equal to the value obtained by Mellor and Yamada () and Yamada () using analytical models for the full Reynolds stress tensor and temperature variance budgets, along with more detailed redistribution models.…”

“…As proposed by Bou‐Zeid et al . (), by summing Equations and , assuming an approximately isotropic dissipation rate to write and using Equation to replace ϵ w , all three variance equations can be combined into one equation given by …”

“…Following Mellor and Yamada () and Bou‐Zeid et al . (), a linear Rotta‐type closure is adopted for the pressure redistribution term (Rotta, ; Davidson, ), namely …”

“…Extending the approach presented by Bou‐Zeid et al . (), we derive a new Ri f , c that includes the TKE transport term. In the reduced TKE phase space proposed by Chamecki et al .…”