Stratification and temporal evolution of mixing regimes in diurnally heated river flows

Abstract

AbstractDirect numerical simulations of stratified open channel flows subject to a varying surface heat flux are performed. The influence of the diurnal heating time on the spatial and temporal variation of mixing in the flow and the characteristics of the mean flow state are examined. The control parameters are the bulk stability parameter $$\lambda_{B}$$ λ B , defined through the ratio of the channel height $$\delta$$ δ and a bulk Obukhov length scale $$\mathscr{L}_{B}$$ L B , and the diurnal time scale $$\hat{t}$$ t ^ , defined as the ratio of the heating time to an eddy turnover time. The Prandtl number Pr and Reynolds number $$Re_{\tau}$$ R e τ have values of 1 and 400. Simulations are performed over $$\hat{t} = 1$$ t ^ = 1 to 24 and $$\lambda_{B} = 0.6$$ λ B = 0.6 to 26. Two key flow features are used to classify the flow regimes observed, namely the laminar layer depth (LLD) and stratified layer depth (SLD) where the LLD is defined as the depth from the free surface when the buoyancy Reynolds number $$Re_{B} \approx 7$$ R e B ≈ 7 and the SLD is the depth from the free surface when the turbulent Froude number $$Fr \approx 1$$ F r ≈ 1 . This study attempts to characterise how these length scales vary across the diel cycle. The LLD is a viscous length scale and a regime map of a viscous parameter, the bulk Obhukov Reynolds number $$Re_\mathscr{L}$$ R e L , and $$\hat{t}$$ t ^ is presented to classify the LLD behaviour. A regime map of $$\lambda _{B}$$ λ B and $$\hat{t}$$ t ^ is presented to classify the behaviour of the SLD. Three classifications for each layer depth behaviour within a diel cycle form the basis of the regime maps for this paper: a neutral flow where the LLD or SLD does not exist (denoted by NL and NS), a stratified flow where the LLD or SLD are diurnally varying (denoted as DL and DS) and a persistent layer of the LLD or SLD (denoted as PL and PS). The transition between the NL to DL is $$\hat{t} \propto Re_{\mathscr{L}}^{4.5}$$ t ^ ∝ R e L 4.5 , DL to PL is $$\hat{t} \propto Re_{\mathscr{L}}^{- 0.5}$$ t ^ ∝ R e L - 0.5 , NS to DS is $$\hat{t} \propto \lambda _{B}^{0}$$ t ^ ∝ λ B 0 and DS to PS is $$\hat{t} \propto \lambda _{B}^{1}$$ t ^ ∝ λ B 1 . The regime maps may be used as a predictive tool to determine when suppressed mixing regimes occur in rivers. At each flow depth, the flow sweeps though a range of mixing states across the diel cycle. The local mixing efficiency are briefly assessed and found to scale well with the instantaneous $$Fr$$ Fr  number according to the regimes proposed by Garanaik and Venayagamoorthy (J. Fluid Mech., vol. 867, 2019, pp. 323-333).