Mean velocity profile in stably stratified turbulent channel flow

TL;DR

The paper addresses extending Monin-Obukhov Similarity Theory (MOST) to stably stratified, wall-bounded channel flows. It develops a region-based, MOST-inspired framework with a local Obukhov length $\Lambda(z)$ and a closure for $Ri_w=h/L$ derived from $Ri_{\tau}$ and $Re_{\tau}$, enabling velocity-profile reconstruction across the channel. Validation against DNS over a broad range of $Re_{\tau}$ and $Ri_{\tau}$ demonstrates accurate mean-velocity predictions and skin-friction estimates, with typical errors in the a few percent range. The approach provides a practical tool for predicting pressure losses and designing stratified-flow systems where buoyancy suppresses turbulence, while maintaining consistency with atmospheric MOST concepts.

Abstract

The Monin-Obukhov Similarity Theory (MOST) is a cornerstone of atmospheric science for describing turbulence in stable boundary layers. Extending MOST to stably stratified turbulent channel flows, however, is non-trivial due to confinement by solid walls and the much smaller turbulent length scales involved. In this study, we investigate the applicability of MOST in closed channels and identify where and to what extent the theory remains valid. A key finding is that the ratio of the half-channel height to the Obukhov length serves as a governing parameter for identifying distinct flow regions and determining the scaling of the mean velocity within them. Hence, we propose a closure relation to estimate this ratio directly from the governing input parameters: friction Reynolds and friction Richardson numbers ($Re_τ$ and $Ri_τ$). The framework is tested against a series of direct numerical simulations (DNS) across a range of $Re_τ$ and $Ri_τ$. The reconstructed velocity profiles enable accurate prediction of the skin friction coefficient crucial for quantifying pressure losses in stratified flows in engineering applications.

Figures (8)

• Figure 1: (a) Mean velocity profiles and (b) Mean temperature profiles for friction Richardson numbers $Ri_{\tau} = \Delta \rho g h/\rho_0 u_{\tau}^2 = 0, \ 60,\ 240,\ 720$, represented with increasing darkness. Instantaneous contours at the mid-spanwise plane illustrate flow structures for neutral ($Ri_{\tau} = 0$) and stratified ($Ri_{\tau} = 720$) cases. All profiles and contours correspond to a friction Reynolds number $Re_{\tau} = \rho_0 u_{\tau} h/\mu_0 = 550$. In these governing parameters, $u_{\tau}$ is the friction velocity, $\rho_0$ and $\mu_0$ denote the density and dynamic viscosity of the fluid, $h$ is the half-channel height, $g$ is the gravitational acceleration (pointing in the negative $z$ direction), and $\Delta \rho$ is the imposed density difference across walls.
• Figure 2: Summary of the parameter space sampled in the DNS database, with data at $Re_{\tau} = 1000$ from zonta2022interaction. Data for each case will be presented consistently with these marker colors.
• Figure 3: The dimensionless velocity gradients, $(\kappa z\ \mathrm{d} u^+/\mathrm{d}z)$, obtained from DNS across a range of stratification levels as a function of $z/L$ in panel (a.) and $z/\Lambda$ in panel (b.) Increasing shades of blue correspond to $Ri_{\tau} = 60,\ 240,\ 720$ at $Re_{\tau} = 550$, the green line corresponds to $Ri_{\tau} = 720$ at $Re_{\tau} = 395$, and the red one to $Ri_{\tau} = 600$ at $Re_{\tau} = 1000$. These gradients are compared against the Businger relation, $1+4.7\ \zeta$, where $\zeta$ is the stability parameter.
• Figure 4: (a) Schematic illustrating the distinct regions in a stably stratified turbulent channel flow, classified based on the influence of stratification (quantified by $z/\Lambda$) and the relative contributions of viscous stress ($\tau_{\mathrm{v}}$), turbulent stress ($\tau_{\mathrm{t}}$), to total stress ($\tau$). The identified regions are: viscous sublayer (I), shear-dominated sublayer (II), stratified outer region (III), turbulent-viscous transition layer (IV), and viscous core (V). The illustration corresponds to the case with $Ri_{\tau} = 720$ at $Re_{\tau} = 550$. (b) Stress profiles for various cases. Blue curves: $Re_{\tau} = 550$, $Ri_{\tau} = 0,\ 60,\ 240,\ 720$ (increasing darkness). Green: $Re_{\tau} = 395$, $Ri_{\tau} = 720$. Red: $Re_{\tau} = 1000$, $Ri_{\tau} = 600$zonta2022interaction. (c) Close-up of the profiles close in the near wall region to illustrate the slow rise of turbulent stresses with increasing stratification. (d) Close-up near the center of the channel, highlighting the drop-off of turbulent stresses to zero.
• Figure 5: Comparison of the mixing length obtained from DNS ($\ell_m^{\mathrm{DNS}} = (-\overline{u'w'})^{1/2}/(\mathrm{d}\overline{u}/\mathrm{d}z)$, solid lines) with the mixing lengths used in this study. The short dashed lines correspond to the mixing length expression for the shear-dominated sublayer, $\ell_m^{\mathrm{II}}$ (Eq. \ref{['eq:ml_iner_sl']}), while the long-dashed lines correspond to the mixing length formulation in the stratified outer region, $\ell_m^{\mathrm{III}}$ (Eq. \ref{['eq:mixinglength_sol']}). Line colors indicate different cases: green for $Re_{\tau} = 395$, $Ri_{\tau} = 720$; blue for $Re_{\tau} = 550$, $Ri_{\tau} = 720$; and red for $Re_{\tau} = 1000$, $Ri_{\tau} = 600$.