Bulk Flow Parameterization of Temporally Evolving Turbulent Stratified Open Channel Flow Subject to Radiative Heating

The transient response of an initially neutral turbulent open channel subject to sudden stable stratification through radiative surface heating is investigated through direct numerical simulations. We find the convergence of the vertical buoyancy and momentum fluxes toward their respective analytical equilibrium values to be a global monotonic process such that locally, buoyancy and momentum flux equilibrium is obtained simultaneously at all locations within the channel. We present scaling arguments to show that the evolution of the flux convergence ratios toward equilibrium scale directly with $R{i}_{\tau }^{-1/2}(t/T_{\tau })$ and reach equilibrium at $R{i}_{\tau }^{-1/2}(t/T_{\tau })\approx 2$, consistent with results from the literature, where $R{i}_{\tau }$ is the friction Richardson number, $T_{\tau }$ is the friction time scale, and $t$ is the measured time from the initial isothermal state. We define the upper laminar layer thickness ${\delta }_i$ and present scaling arguments to show that the laminar layer thickness is well-parameterized by a hybrid bulk parameter such that ${\delta }_i/\delta \approx 0.45(C_R^{1/2}PrR{i}_{\tau }^{-1}R{e}_{\tau }{)}^{-1/2}$ across the full external parameter set. Here $\delta$ is the channel height, $C_R$ is the equilibrium convergence ratio, $Pr$ is the molecular Prandtl number, and $R{e}_{\tau }$ is the friction Reynolds number.