On separating plumes from boundary layers in turbulent convection

Abstract

We present a simple, novel kinematic criterion – that uses only the horizontal velocity fields and is free of arbitrary thresholds – to separate line plumes from local boundary layers in a plane close to the hot plate in turbulent convection. We first show that the horizontal divergence of the horizontal velocity field ( $\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$ ) has negative and positive values in two-dimensional (2D), laminar similarity solutions of plumes and boundary layers, respectively. Following this observation, based on the understanding that fluid elements predominantly undergo horizontal shear in the boundary layers and vertical shear in the plumes, we propose that the dominant eigenvalue ( $\lambda _D$ ) of the 2D strain rate tensor is negative inside the plumes and positive inside the boundary layers. Using velocity fields from our experiments, we then show that plumes can indeed be extracted as regions of negative $\lambda _D$ , which are identical to the regions with negative $\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$ . Exploring the connection of these plume structures to Lagrangian coherent structures (LCS) in the instantaneous limit, we show that the centrelines of such plume regions are captured by attracting LCS that do not have dominant repelling LCS in their vicinity. Classifying the flow near the hot plate based on the distribution of eigenvalues of the 2D strain rate tensor, we then show that the effect of shear due to the large-scale flow is felt more in regions close to where the local boundary layers turn into plumes. The lengths and areas of the plume regions, detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion applied to our experimental and computational velocity fields, are then shown to agree with our theoretical estimates from scaling arguments. Using velocity fields from numerical simulations, we then show that the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion detects all the upwellings, while the available criteria based on temperature and flux thresholds miss some of these upwellings. The plumes detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion are also shown to be thicker at Prandtl numbers ( $Pr$ ) greater than one, expectedly so, due to the thicker velocity boundary layers of the plumes at $Pr>1$ .

Stability analysis of sheared thermal boundary layers and its implication for modelling turbulent Rayleigh–Bénard convection

“…The relation between the wind and V r.m.s. is very likely non trivial and V w should probably be described in a statistical fashion [35]. Our approach could nonetheless remain valid and useful as an "elementary brick" upon which to build a more complex model.…”

“…While two-and three-dimensional DNS and experiments with aspect ratios close to unity exhibit a single large scale circulation in the bulk, several cells separated by plumes have been reported in large aspect ratio two-dimensional DNS [19] and a complex network of plumes circumscribing regions of obvious sheared ther-mal boundary layers is observed experimentally [34].…”

Effect of shear on local boundary layers in turbulent convection

“…More importantly, the observed vertical distributions of fluctuations of velocities and temperature near the hot plate have been predicted well by a model that assumes laminar boundary layers giving rise to laminar plumes (Theerthan & Arakeri 1998). This means that the observed fluctuations in velocity and temperature near the hot plate are created by the spatial averaging of a spatially non-uniform field consisting of many laminar local boundary layers giving rise to many plumes, as well as by the lateral motion of such plumes (Shevkar & Puthenveettil 2022). Also, as an a posteriori justification, the scaling laws obtained in the present study using two-dimensional steady laminar local boundary layer equations, forced by shear, match well with our measurements in turbulent RBC shown later, in figure 10.…”