Velocity dip in turbulent mixed convection of an open Poiseuille-Rayleigh-Bénard channel

Abstract

We study the emergence of a velocity-dip phenomenon in turbulent mixed convection in open Poiseuille-Rayleigh-Bénard (PRB) channels with a free-slip upper boundary. Three-dimensional direct numerical simulations (DNS) are performed for Rayleigh numbers in the range $10^5 \leq Ra \leq 10^8$, at a fixed Prandtl number $Pr = 0.71$ and a bulk Reynolds number $Re_b = 2850$. In the shear-dominated regime, the flow is characterised by small-scale structures such as near-wall streaks. As buoyancy becomes comparable to shear, streamwise-oriented large-scale rolls emerge and span the full channel height. At higher Rayleigh numbers, buoyancy dominates and the rolls fragment, giving rise to a convection-cell-dominated regime. Short-time-averaged flow fields show that streamwise rolls transport low-speed fluid from the bottom wall towards the upper boundary, forming laterally extended low-speed regions, while roll fragmentation induces upstream low-speed regions near the upper boundary. Both mechanisms locally reduce the near-surface mean velocity, leading to a velocity dip in which the maximum mean streamwise velocity is located below the upper boundary. Consistent with the mean momentum budget, the near-surface region exhibits a large-scale Reynolds shear stress that exceeds the local total shear stress, implying a negative viscous contribution and a reversal of the mean velocity gradient. To model this behaviour, we propose a model based on a balance between buoyancy and shear production with dissipation, incorporating a linear wall-normal profile for the Reynolds shear stress, a wall-normal-independent buoyancy-production term, and a decomposition of the dissipation into shear-induced and buoyancy-induced contributions. Our model accurately reproduces the DNS mean velocity profiles across the explored $Ra$ range.

Figures (28)

• Figure 1: Schematic of the open Poiseuille--Rayleigh--Bénard (PRB) channel. The bottom wall is no-slip and maintained at a constant hot temperature $T_{\mathrm{bottom}}=T_{\mathrm{hot}}$, while the top boundary is free-slip and held at a constant cold temperature $T_{\mathrm{top}}=T_{\mathrm{cold}}$.
• Figure 2: Instantaneous volume renderings of the temperature field at a fixed bulk Reynolds number $Re_b=2850$. (a) $Ra=10^5$, (b) $Ra=10^6$, (c) $Ra=10^7$ and (d) $Ra=10^8$. The corresponding bulk Richardson numbers are $1.7\times10^{-2}$, $1.7\times10^{-1}$, $1.7\times10^{0}$ and $1.7\times10^{1}$, respectively. Colour denotes the normalised temperature $(T-T_0)/\varDelta_T$. Opacity increases monotonically with temperature to emphasise hot buoyant structures, while cooler regions are rendered transparent.
• Figure 3: Instantaneous volume renderings of the streamwise velocity field at a fixed bulk Reynolds number $Re_b=2850$. (a) $Ra=10^5$, (b) $Ra=10^6$, (c) $Ra=10^7$ and (d) $Ra=10^8$. The corresponding bulk Richardson numbers are $1.7\times10^{-2}$, $1.7\times10^{-1}$, $1.7\times10^{0}$ and $1.7\times10^{1}$, respectively. Colour denotes the normalised streamwise velocity $u/u_b$. Opacity decreases monotonically with streamwise velocity to emphasise low-speed structures, while higher-speed regions are rendered transparent.
• Figure 4: (a--c) The most energetic POD mode, shown as isosurfaces of the wall-normal velocity $v$ (red denotes positive values and blue denotes negative values), and (d--f) the energy contained in each mode. Panels (a,d), (b,e) and (c,f) correspond to $Ra=10^6$, $10^7$ and $10^8$, respectively, at a fixed bulk Reynolds number $Re_b=2850$. The corresponding bulk Richardson numbers are $1.7\times10^{-1}$, $1.7\times10^{0}$ and $1.7\times10^{1}$, respectively.
• Figure 5: Non-dimensional Monin--Obukhov length $H/|L_{MO}|$ as a function of the bulk Richardson number $Ri_b$. Circles denote the present open PRB simulations, while triangles and squares correspond to closed PRB and CRB configurations taken from pirozzoli2017mixed, schafer2022effect and blass2020flow, respectively. The dashed line indicates a unit-slope power-law scaling, $H/|L_{MO}| \sim Ri_b$.