Boundary layer scaling in Rayleigh-Benard convection

Accepted versio

Wind and boundary layers in Rayleigh-Benard convection. Part 2: boundary layer character and scaling

The effect of the wind of Rayleigh-Benard convection on the boundary layers is studied by direct numerical simulation of an L/H=4 aspect-ratio domain with periodic side boundary conditions for Ra={10^5, 10^6, 10^7} and Pr=1. It is shown that the kinetic boundary layers on the top- and bottom plate have some features of both laminar and turbulent boundary layers. A continuous spectrum, as well as significant forcing due to Reynolds stresses indicates undoubtedly a turbulent character, whereas the classical integral boundary layer parameters -- the shape factor and friction factor (the latter is shown to be dominated by the pressure gradient) -- scale with Reynolds number more akin to laminar boundary layers. This apparent dual behavior is caused by the large influence of plumes impinging onto and detaching from the boundary layer. The plume-generated Reynolds stresses have a negligible effect on the friction factor at the Rayleigh numbers we consider, which indicates that they are passive with respect to momentum transfer in the wall-parallel direction. However, the effect of Reynolds stresses cannot be neglected for the thickness of the kinetic boundary layer. Using a conceptual wind model, we find that the friction factor C_f should scale proportional to the thermal boundary layer thickness as C_f ~ lambda_Theta, while the kinetic boundary layer thickness lambda_u scales inversely proportional to the thermal boundary layer thickness and wind Reynolds number lambda_u ~ lambda_Theta^{-1} Re^{-1}. The predicted trends for C_f and \lambda_u are in agreement with DNS results

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

Asymptotic solutions for turbulent mass transfer at high Schmidt number

International audienceThis paper introduces a new methodology for the complexity analysis of higher-order functional programs, which is based on three ingredients: a powerful type system for size analysis and a sound type inference procedure for it, a ticking monadic transformation, and constraint solving. Noticeably, the presented methodology can be fully automated, and is able to analyse a series of examples which cannot be handled by most competitor methodologies. This is possible due to the choice of adopting an abstract index language and index polymorphism at higher ranks. A prototype implementation is available

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

Small-scale entrainment in inclined gravity currents

We investigate the effect of buoyancy on the small-scale aspe cts of turbulent entrainment by performing direct numerical simu lation of a gravity cur- rent and a wall jet. In both flows, we detect the turbulent/non turbulent interface separating turbulent from irrotational ambient flow region s using a range of en- strophy iso-levels spanning many orders of magnitude. Conf orm to expectation, the relative enstrophy isosurface velocity v n in the viscous superlayer scales with the Kolmogorov velocity for both flow cases. We connect the in tegral entrainment coefficient E to the small-scale entrainment and observe excellent agree ment be- tween the two estimates throughout the viscous superlayer. The contribution of baroclinic torque to v n is negligible, and we show that the primary reason for reduced entrainment in the gravity current as compared to th e wall-jet is the reduction in the surface area of the isosurfaces