Direct numerical simulation of the turbulent Ekman layer over a smooth wall is used to investigate bulk properties of a planetary boundary layer under stable stratification. Our simplified configuration depends on two non-dimensional parameters: a Richardson number characterizing the stratification and a Reynolds number characterizing the turbulence scale separation. This simplified configuration is sufficient to reproduce global intermittency, a turbulence collapse, and the decoupling of the surface from the outer region of the boundary layer. Global intermittency appears even in the absence of local perturbations at the surface; the only requirement is that large-scale structures several times wider than the boundary-layer height have enough space to develop. Analysis of the mean velocity, turbulence kinetic energy, and external intermittency is used to investigate the large-scale structures and corresponding differences between stably stratified Ekman flow and channel flow. Both configurations show a similar transition to the turbulence collapse, overshoot of turbulence kinetic energy, and spectral properties. Differences in the outer region resulting from the rotation of the system lead, however, to the generation of enstrophy in the non-turbulent patches of the Ekman flow. The coefficient of the stability correction function from Monin-Obukhov similarity theory is estimated as β ≈ 5.7 in agreement with atmospheric observations, theoretical considerations, and results from stably stratified channel flows. Our results demonstrate the applicability of this setup to atmospheric problems despite the intermediate Reynolds number achieved in our simulations.
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.
Existence of non-turbulent flow patches in the vicinity of the wall of a turbulent flow is known as global intermittency. Global intermittency challenges the conventional statistics approach when describing turbulence in the inner layer and calls for the use of conditional statistics. We extend the vorticity-based conditioning of a flow to turbulent and non-turbulent sub-volumes by a high-pass filter operation. This modified method consistently detects non-turbulent flow patches in the outer and inner layers for stratifications ranging from the neutral limit to extreme stability, where the flow is close to a complete laminarization. When applying this conditioning method to direct numerical simulation data of stably stratified Ekman flow, we find the following. First, external intermittency has a strong effect on the logarithmic law for the mean velocity in Ekman flow under neutral stratification. If instead of the full field, only turbulent sub-volumes are considered, the data fit an idealized logarithmic velocity profile much better; in particular, a problematic dip in the von Kármán measure κ in the surface layer is decreased by approximately 50 % and our data only support the reduced range 0.41 κ 0.43. Second, order-one changes in turbulent quantities under strong stratification can be explained by a modulation of the turbulent volume fraction rather than by a structural change of individual turbulence events; within the turbulent fraction of the flow, the character of individual turbulence events measured in terms of turbulence dissipation rate or variance of velocity fluctuations is similar to that under neutral stratification.
The zero-divergence constraint on the velocity field in the numerical simulation of incompressible flows can be reduced, in certain cases, to a set of one-dimensional linear difference equations for the pressure. These equations involve the secondorder derivative δxδxp expressed in terms of twice the first-order derivative. When implicit finite-difference schemes are used, those equations lead to full linear systems, which are computationally prohibitive. Hence, it is a common practice to substitute δxδxp by a different discretization δxxp. However, it is well known that this step results in a non-zero divergence in the velocity field. This paper presents a factorization of the original equation that allows to satisfy the discrete solenoidal constraint exactly while maintaining a linear relation between the number of operations and the grid size. As an example, the method is particularized to compact schemes often found in the literature.
We perform direct numerical simulation of the Couette flow as a model for the stable boundary layer. The flow evolution is investigated for combinations of the (bulk) Reynolds number and the imposed surface buoyancy flux. First, we establish what the similarities and differences are between applying a fixed buoyancy difference (Dirichlet) and a fixed buoyancy flux (Neumann) as boundary conditions. Moreover, two distinct parameters were recently proposed for the turbulent-to-laminar transition: the Reynolds number based on the Obukhov length and the “shear capacity,” a velocity-scale ratio based on the buoyancy flux maximum. We study how these parameters relate to each other and to the atmospheric boundary layer. The results show that in a weakly stratified equilibrium state, the flow statistics are virtually the same between the different types of boundary conditions. However, at stronger stratification and, more generally, in nonequilibrium conditions, the flow statistics do depend on the type of boundary condition imposed. In the case of Neumann boundary conditions, a clear sensitivity to the initial stratification strength is observed because of the existence of multiple equilibriums, while for Dirichlet boundary conditions, only one statistically steady turbulent equilibrium exists for a particular set of boundary conditions. As in previous studies, we find that when the imposed surface flux is larger than the maximum buoyancy flux, no turbulent steady state occurs. Analytical investigation and simulation data indicate that this maximum buoyancy flux converges for increasing Reynolds numbers, which suggests a possible extrapolation to the atmospheric case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.