Boundary layer scaling in Rayleigh-Benard convection

Accepted versio

Energy dispersion in turbulent jets. Part 2. A robust model for unsteady jets

In this paper we develop an integral model for an unsteady turbulent jet that incorporates longitudinal dispersion of two distinct types. The model accounts for the difference in the rate at which momentum and energy are advected (type I dispersion) and for the local deformation of velocity profiles that occurs in the vicinity of a sudden change in the momentum flux (type II dispersion). We adapt the description of dispersion in pipe flow by Taylor (Proc. R. Soc. Lond. A, vol. 219, 1953, pp. 186–203) to develop a dispersion closure for the longitudinal transportation of energy in unsteady jets. We compare our model’s predictions to results from direct numerical simulation and find a good agreement. The model described in this paper is robust and can be solved numerically using a simple central differencing scheme. Using the assumption that the longitudinal velocity profile in a jet has an approximately Gaussian form, we show that unsteady jets remain approximately straight-sided when their source area is fixed. Straight-sidedness provides an algebraic means of reducing the order of the governing equations and leads to a simple advection–dispersion relation. The physical process responsible for straight-sidedness is type I dispersion, which, in addition to determining the local response of the area of the jet, determines the growth rate of source perturbations. In this regard the Gaussian profile has the special feature of ensuring straight-sidedness and being insensitive to source perturbations. Profiles that are more peaked than the Gaussian profile attenuate perturbations and, following an increase (decrease) in the source momentum flux, lead to a local decrease (increase) in the area of the jet. Conversely, profiles that are flatter than the Gaussian amplify perturbations and lead to a local increase (decrease) in the area of the jet

Mixing and entrainment are suppressed in inclined gravity currents

We explore the dynamics of inclined temporal gravity currents using direct numerical simulation, and find that the current creates an environment in which the flux Richardson number $\mathit{Ri}_{f}$, gradient Richardson number $\mathit{Ri}_{g}$ and turbulent flux coefficient \unicode[STIX]{x1D6E4} are constant across a large portion of the depth. Changing the slope angle \unicode[STIX]{x1D6FC} modifies these mixing parameters, and the flow approaches a maximum Richardson number $\mathit{Ri}_{max}\approx 0.15$ as \unicode[STIX]{x1D6FC}\rightarrow 0 at which the entrainment coefficient $E\rightarrow 0$. The turbulent Prandtl number remains $O(1)$ for all slope angles, demonstrating that $E\rightarrow 0$ is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (J. Fluid Mech., vol. 2, 1957, pp. 456–466). Instead, $E\rightarrow 0$ occurs as the result of the turbulence intensity going to zero as \unicode[STIX]{x1D6FC}\rightarrow 0, due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small \unicode[STIX]{x1D6FC} which is able to predict accurately $\mathit{Ri}_{f}$, $\mathit{Ri}_{g}$ and \unicode[STIX]{x1D6E4} as a function of \unicode[STIX]{x1D6FC} and their maximum attainable values. The model predicts an entrainment law of the form $E=0.31(\mathit{Ri}_{max}-\mathit{Ri})$, which is in good agreement with the simulation data. The simulations and model presented here contribute to a growing body of evidence that an approach to a marginally or critically stable, relatively weakly stratified equilibrium for stratified shear flows may well be a generic property of turbulent stratified flows.</jats:p

A Lagrangian study of interfaces at the edges of cumulus clouds

Interfaces at the edge of an idealised, non-precipitating, warm cloud are studied using Direct Numerical Simulation (DNS) complemented with a Lagrangian particle tracking routine. Once a shell has formed, four zones can be distinguished: the cloud core, visible shell, invisible shell and the environment. The union of the visible and invisible regions is the shell commonly referred to in literature. The boundary between the invisible shell and the environment is the Turbulent-NonTurbulent Interface (TNTI) which is typically not considered in cloud studies. Three million particles were seeded homogeneously across the domain and properties were recorded along individual trajectories. The results demonstrate that the traditional cloud boundary (separating cloudy and non-cloudy regions using thresholds applied on liquid condensate or updraft velocity) are some distance away from the TNTI. Furthermore, there is no dynamic difference between the traditional liquid-condensate boundary and the region extending to the TNTI. However, particles crossing the TNTI exhibit a sharp jump in enstrophy and a smooth increase in buoyancy. The traditional cloud boundary coincides with the location of minimum buoyancy in the shell. The shell pre-mixes the entraining and detraining air and analysis reveals a highly skewed picture of entrainment and detrainment at the traditional cloud boundary. A preferential entrainment of particles with velocity and specific humidity higher than the mean values in the shell is observed. Large-eddy simulation of a more realistic setup detects an interface with similar properties using the same thresholds as in the DNS, indicating that the DNS results extrapolate beyond their idealised conditions