Similarities between the structure functions of thermal convection and hydrodynamic turbulence

Abstract

In this paper, we analyze the scaling of velocity structure functions of turbulent thermal convection. Using high-resolution numerical simulations, we show that the structure functions scale similar to those of hydrodynamic turbulence, with the scaling exponents in agreement with She and Leveque's predictions [Phys. Rev. Lett. 72, 336-339 (1994)]. The probability distribution functions of velocity increments are non-Gaussian with wide tails in the dissipative scales and become close to Gaussian in the inertial range. The tails of the probability distribution follow a stretched exponential. We also show that in thermal convection, the energy flux in the inertial range is less than the viscous dissipation rate. This is unlike in hydrodynamic turbulence where the energy flux and the dissipation rate are equal.

Prandtl number dependence of the small-scale properties in turbulent Rayleigh-Bénard convection

“…In turbulent convection, buoyancy injects kinetic energy at all scales, including the dissipation range. Bhattacharya et al [50] showed that for Pr = 1 convection, although the modal kinetic energy injection by buoyancy is small at intermediate wavenumbers (as discussed earlier), it adds up to a significant fraction of the total kinetic energy injection when summed over the inertial and dissipative scales.The inertial-range kinetic energy flux is due to the fraction of energy injected only at large scales and hence less than the total kinetic-energy dissipation rate. We expect the kinetic-energy dissipation rate and the energy injection rates by buoyancy to exhibit a similar Pr dependence as the energy flux.…”

“…Consequently, Π u < u . Bhattacharya et al [50] showed that for Pr = 1, the inertial-range kinetic energy flux is approximately one-third of the total dissipation rate. In this subsection, we will describe these quantities for various Prandtl numbers.…”

“…On the other hand, Sun et al [46], and Kaczorowski and Xia [33] reported KO41 scaling for the lower order structure functions of RBC. Recently, Bhattacharya et al [50] showed that the third-order velocity structure function of RBC for Pr = 1 obeys Eq. ( 19), similar to hydrodynamic turbulence, except that u is replaced with Π u .…”

“…Although there have been many studies on structure functions of thermal convection for moderate Prandtl numbers [12,33,[41][42][43][44][45][46][47][48][49], their reported scalings are not very conclusive. Recently, using high resolution numerical simulations of convection for Pr = 1, Bhattacharya et al [50] showed that the structure functions of thermal convection scale similarly as that of 3D hydrodynamic turbulence. Note that these works are for Pr ≤ 7.…”

Predictions of Reynolds and Nusselt numbers in turbulent convection using machine-learning models

“…Recent studies, however, show that these rates get an additional correction of approximately Ra −0.2 for Pr ∼ 1 10,26,29,30,44,45 . The above correction is because of the inhibition of nonlinear interactions due to the presence of walls and buoyancy 6,29,30,57 . In addition, the above studies reveal that the thickness of the viscous boundary layer in RBC deviates from Re −1/2 , contrary to what is assumed in GL model 44,58,59 .…”