Rayleigh-Benard Convection

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A new, more complete view of the mechanical energy budget for Rayleigh-Benard convection is developed and examined using three-dimensional numerical simulations at large Rayleigh numbers and Prandtl number of one. The driving role of available potential energy is highlighted.

The change in flow structure over the range of Ra (Fig. 1) is remarkable. At Ra ∼ 10⁷ the convection is more-or-less homogeneous and forms cell-like structures. Smaller scales appear with increasing Ra (Fig. 1b), and become increasingly organized at still larger Ra (Fig. 1c) as much larger scales of coherent motion emerge. Enhanced boundary shear from the ‘wind’ at the larger Ra is reflected in the flow structures (see Fig. 1). At Ra ∼ 10⁷ the cell structures are coherent and bounded by lines of flow convergence, which represent the base of ascending sheet-like plumes. The structure changes significantly at larger Ra; the sheet plumes tend to lie perpendicular to the ‘wind’ direction, and migrate and merge into a ‘megaplume’. Although there is already a tendency to form ‘megaplumes’ at Ra ∼ 10⁹, the signature of the large-scale structures and well organized convergence lines is clear at Ra ∼ 10¹³ (Fig. 1c). At these large Ra there is no longer a homogeneous distribution of small convective plumes inside the boundary layer.

The proposed energetics framework for RBC in a Boussinesq fluid is shown schematically in Fig. 2. This framework is broadly similar to that proposed for the global ocean circulation and recently implemented for horizontal convection by Gayen et al. 2013, but is modified here for the RBC case. Mechanical energy includes both kinetic energy E_k and gravitational potential energy E_p and energy transfer between these forms is facilitated by buoyancy fluxes, .Φ_z. The volume integrals are taken over the full depth of the fluid and horizontal distances suitably large compared to the significant scales of motion. The potential energy is further decomposed into background potential energy (BPE) E_b and available potential energy, E_a=E_p−E_b. The BPE of a volume of fluid corresponds to Ep for a state of no motion, or gravitational equilibrium, in which fluid parcels have been adiabatically rearranged to new vertical positions. Detailed derivations of the energy conversion rates in Fig. 3 show that exchange between APE and BPE is via irreversible mixing Φ_d, a lowering of the center of mass by molecular diffusion down the background gradient .Φ_i, and by buoyancy input through the boundaries .Φ_b2. For RBC Φ_b2. represents the rate of energy input required to maintain a density field away from an adiabatically relaxed state. Kinetic energy is dissipated by viscosity at the rate ε. Mechanical Eenergy is converted between internal and potential energy by both Φ_i and the net buoyancy input (i.e. the rate of change of available potential energy by net heating or cooling) at the boundaries: Φ_b1.

Progress is made here by decomposing the kinetic energy of the fluid into mean, E_k, and turbulent, E_k′ contributions. Small- scale turbulent motions must be sustained by extracting kinetic energy from the mean flow (the shear production Φ_T or from APE (the turbulent buoyancy flux Φ′_z; Fig. 2). The large- scale motions, on the other hand, are maintained only by the mean buoyancy flux Φ_z.

At the smallest Rayleigh number considered (6 × 10⁶) energy is transferred to both mean and fluctuating kinetic energy via buoyancy flux (the left and right pathways in Fig. 2) at comparable rates, although the fluctuating contribution is the smaller. Further generation of small-scale flows by shear production Φ_T is small. Thus small-scale convection is the primary source of turbulent motions. Viscous dissipation of the mean (large-scale) motions, ε, and of the small-scale motions, ε′, occur at similar rates. At larger Rayleigh numbers the large-scale circulation (or “wind”) develops (compare Fig. 1b and c). Accompanying this change in the flow is a substantial change in the energy pathways. More energy is pumped into kinetic energy through the mean buoyancy flux (left hand pathway in Fig. 2), as a result of small-scale convective plumes being swept along the boundary layer and merging into larger scales. The changes in Φz and Φ′z are gradual with Ra, but the ratio Φz /Φ′z increases rapidly at Ra > 10¹¹ . Viscous dissipation of the large-scale structures is inefficient and its relative role decreases with increasing Ra. The results also show that the wind velocity associated with the large-scale is larger at the larger Ra and the velocity boundary layer thickness is significantly smaller, resulting in enhancement of the boundary shear and production of small scale motions (the bottom arrow in Fig. 2). At Ra ∼ 10¹³ almost 80% of the turbulence is produced by shear; only 20% comes directly from APE via the turbulent buoyancy flux (the right hand pathway) associated with small-scale convective plumes. The associated turbulent kinetic energy is then dissipated (ε′ in Fig. 2). Thus convection at the higher Rayleigh numbers predominantly involves energy conversion from APE to mean KE, from mean KE to turbulence KE via shear instability, and dissipation of KE.

Ref:

B. Gayen, G. O. Hughes and R. W. Griffiths, (2013) Completing the mechanical energy pathways in turbulent Rayleigh-Benard convection
Phys. Rev. Lett. 111, 124301

G. O. Hughes, B. Gayen and R. W. Griffiths (2013) Available Potential Energy in Rayleigh-Benard Convection, J. Fluid Mech., 729, R2, 1-10

Bishakhdatta Gayen

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