A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an ap- plication to a three-dimensional convection problem is presented.

/pdf/numerical-solution-of-the-navier-stokes-equations-1d4r986mvf.pdf

Numerical solution of the Navier-Stokes equations

The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large scale convection roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.

/pdf/heat-transfer-and-large-scale-dynamics-in-turbulent-rayleigh-39oz2czpiy.pdf

Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection

A systematic theory for the scaling of the Nusselt number Nu and of the A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [less, similar] 1011) the leading terms are Nu [similar] Ra1/4Pr1/8, Re [similar] Ra1/2Pr[minus sign]3/4 for Pr [less, similar] 1 and Nu [similar] Ra1/4Pr[minus sign]1/12, Re [similar] Ra1/2Pr[minus sign]5/6 for Pr [greater, similar] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu [similar] Ra1/2Pr1/2, Re [similar] Ra1/2Pr[minus sign]1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu [similar] Ra1/5Pr1/5, Re [similar] Ra2/5Pr[minus sign]3/5 for small Pr, a regime with Nu [similar] Ra1/3, Re [similar] Ra4/9Pr[minus sign]2/3 for larger Pr, and a regime with scaling Nu [similar] Ra3/7Pr[minus sign]1/7, Re [similar] Ra4/7Pr[minus sign]6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges.

Scaling in thermal convection: a unifying theory

Thermal convection in a layer heated form below is an exemplary case for the study of non-linear fluid dynamics and the transition to turbulence. An outline is given of the present knowledge of the simplest realisation of convection in a layer of fluid satisfying the Oberbeck-Boussinesq approximation. Non-linear properties such as the dependence of the heat transport on Rayleigh and Prandtl numbers and the stability properties of convection rolls are emphasised in the discussion. Whenever possible, theoretical results are compared with experimental observations. A section on convection in rotating systems has been included, but the influence of other additional physical effects such as magnetic fields, side wall geometry, etc., has not been considered.

https://iopscience.iop.org/article/10.1088/0034-4885/41/12/003/pdf

Non-linear properties of thermal convection

A finite-difference method for solving the time-dependent Navier-Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an application to a three-dimensional convection problem is presented.

Numerical Solution of the Navier-Stokes Equations*

An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112069001674

H. T. Rossby

Papers

A study of Bénard convection with and without rotation