Parameters for the Collapse of Turbulence in the Stratified Plane Couette Flow

Abstract

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.

Stable channel flow with spanwise heterogeneous surface temperature

“…Numerical studies that investigated stably stratified flows have reported that with increasing stratification, the turbulence is dampened so much that it becomes intermittent in time and laminar-turbulent patches coexist near the wall, until a critical stratification level is reached where all turbulence is extinguished and the flow becomes laminar (e.g. Nieuwstadt 2005;Flores & Riley 2011;García-Villalba & del Álamo 2011;He & Basu 2015;van Hooijdonk et al 2018;Atoufi, Scott & Waite 2019). Long before, Gage & Reid (1968) had already developed a linear stability theory for homogeneous stratified plane Poiseuille flow, and found an explicit relation between the Reynolds number of the flow and the critical Richardson number at which the flow becomes stable.…”

“…Although first attempts of numerical simulations with stably stratified plane channel flow showed turbulence collapse already at much lower Richardson numbers then those predicted by linear theory (Garg et al 2000;Iida, Kasagi & Nagano 2002), more recent DNS and LES results were consistent with this theory. It was demonstrated that a large enough domain size is crucial to maintain the turbulence (Flores & Riley 2011;García-Villalba & del Álamo 2011), while other authors suggest that the laminarization is also sensitive to the choice of boundary and initial conditions and the type of forcing to the flow (Brethouwer, Duguet & Schlatter 2012;van Hooijdonk et al 2018). Numerical studies in the ABL community have focused on finding a stability threshold for turbulence collapse expressed in a different parameter than the Richardson number, such as a critical Obukhov length (Nieuwstadt 2005), Obukhov-Reynolds number (Flores & Riley 2011;Deusebio et al 2014;Zhou, Taylor & Caulfield 2017) or 'shear capacity' (Donda et al 2016;van Hooijdonk et al 2018).…”

“…It was demonstrated that a large enough domain size is crucial to maintain the turbulence (Flores & Riley 2011;García-Villalba & del Álamo 2011), while other authors suggest that the laminarization is also sensitive to the choice of boundary and initial conditions and the type of forcing to the flow (Brethouwer, Duguet & Schlatter 2012;van Hooijdonk et al 2018). Numerical studies in the ABL community have focused on finding a stability threshold for turbulence collapse expressed in a different parameter than the Richardson number, such as a critical Obukhov length (Nieuwstadt 2005), Obukhov-Reynolds number (Flores & Riley 2011;Deusebio et al 2014;Zhou, Taylor & Caulfield 2017) or 'shear capacity' (Donda et al 2016;van Hooijdonk et al 2018). Moreover, Donda et al (2015) argue that turbulence collapse is only temporary, and turbulence is regenerated due to increased shear in the laminar state in combination with small perturbations.…”

Geostrophic drag law for conventionally neutral atmospheric boundary layers revisited

“…Two parameters govern the physics in stratified boundary layers: one that characterizes the size, speed, or scale separation, and another that describes the degree of stratification. Therefore, the dynamics can be described by different parameters such as a flux or gradient Richardson number or a buoyancy Reynolds number (Coleman et al ., 1992; Flores and Riley, 2011; van Hooijdonk et al ., 2018; Sun et al ., 2020). However, in this study, we use the Rossby number Ro = G /( z 0 | f |) and the Zilitinkevich number , as they were used in the GDL for CNBLs introduced by Zilitinkevich and Esau (2005).…”

Scale Dependence of Atmosphere–Surface Coupling Through Similarity Theory

“…Intermittent turbulence is not appropriately characterized by bulk and conventional statistics; instead, conditional sampling can be employed to gain insight into the complex dynamics of such intermittent flows (Ansorge and Mellado 2016;Ansorge 2016). Even in critical conditions (at intermediate stratification with very little or no global intermittency), L alone does not characterize the flow state appropriately (Hooijdonk et al 2018). In these cases, the buoyancy flux (and thus L) is limited while the bulk stratification and also the wind speed correction increase further, which is consistently observed for both the stability correction of buoyancy and momentum and thus does not appear to impact on the turbulent Prandtl number.…”