Bolgiano Scale Research Articles

Evidence of strong wave turbulence and of Bolgiano temperature spectra in katabatic winds on steep slopes

The katabatic winds on steep slopes investigated in the present study reveal a novel spectral behavior, observed in the outer part of the jet. At low wavenumbers, the one-dimensional (1D) velocity spectra show evidence of a kx−1 range for the three components of the velocity vector: Eu(kx),Ev(kx),Ew(kx)∝kx−1 [as well as for the 1D temperature spectrum Eθ(kx)∝kx−1]. This suggests the existence of strong wave turbulence. A necessary condition for strong wave turbulence to be manifest is that the flow direction wavenumber, kx, extends to much lower values than the slope normal one, kz. This is satisfied in the present field experiment where wave energy is injected at wavenumber kx=kN=(Na sin α)/uj¯, while kz∼1/Δz, with Na the ambient stratification, α the slope angle, uj¯ the maximum wind velocity, and Δz the shear layer thickness of the jet. In the inertial range, the velocity spectra exhibit a power law kx−5/3 over two decades, whereas the temperature-buoyancy spectra show evidence of a −7/5 slope in the buoyancy sub-range, followed by a −5/3 slope. The change in spectral slopes occurs at the Bolgiano scale LB that is close to the Dougherty–Ozmidov scale LOZ. The high Reynolds number based on the Taylor micro-scale, Reλ∼103, allows clear identification of the spectral laws.

Feasibility of the Obukhov-Bolgiano scaling in buoyancy affected turbulence

The paper focuses on the effect of buoyancy on the spectral characteristics of turbulent flows. We obtain an analytical solution in the nondissipative limit for the spectrum containing both ranges with Obukhov-Bolgiano and Kolmogorov scalings. This leads to the derivation of an expression for the Bolgiano scale through the governing parameters of the problem.

Nonconservative Cascades in a Shell Model of Turbulence

Developed turbulent flows in which the intervention of external forces is fundamentally important at scales where the inertial range should exist are quite common. Then the cascade processes are not conservative any more and, therefore, it is necessary to adequately describe the external forces acting in the whole range of scales. If the work of these forces has a power law scaling, then one can assume that the integral of motion changes and the preserving value becomes a quadratic quantity, which includes the dependence on the scale. We develop this idea within the framework of shell models of turbulence. We show that, in terms of nonconservative cascades, one can describe various situations, including (as a particular case) the Obukhov – Bolgiano scaling proposed for turbulence in a stratified medium and for helical turbulence with a helicity injection distributed along the spectrum.

Shell models for confined Rayleigh–Taylor turbulent convection

For the confined Rayleigh–Taylor turbulent convection with large aspect ratio, different scaling behaviors such as Bolgiano-Obukhov (BO) scaling and Kolmogorov–Obukhov (KO) scaling are coexistent in the system. Traditional shell models cannot reproduce dual scaling properties with scale independent parameters. Boffetta et al. [Phys Rev E 2011; 83, 066302] firstly introduced geometrical constraints and two sets of model parameters into the original SabraT model to investigate the dual scaling properties of such complex convective turbulence. However, the present numerical results reveal that the enstrophy cascade scaling instead of BO scaling is setup at scales larger than the confinement scale for the modified 2D-3D SabraT model in the studied range of model parameter. Moreover, a 2D-3D Brandenburg model is constructed with the same idea and numerically investigated. Results show that two coexistence subranges of BO and KO scalings are respectively setup above and below the Bolgiano scale, which exactly relates to the confinement scale.

Numerical simulations of thermal convection on a hemisphere

In this paper we present numerical simulations of two-dimensional turbulent convection on a hemisphere. Recent experiments on a half soap bubble located on a heated plate have shown that such a configuration is ideal for studying thermal convection on a curved surface. Thermal convection and fluid flows on curved surfaces are relevant to a variety of situations, notably for simulating atmospheric and geophysical flows. As in experiments, our simulations show that the gradient of temperature between the base and the top of the hemisphere generates thermal plumes at the base that move up from near the equator to the pole. The movement of these plumes gives rise to a two-dimensional turbulent thermal convective flow. Our simulations turn out to be in qualitative and quantitative agreement with experiments and show strong similarities with Rayleigh-Benard convection in classical cells where a fluid is heated from below and cooled from above. To compare to results obtained in classical Rayleigh-Benard convection in standard three-dimensional cells (rectangular or cylindrical), a Nusselt number adapted to our geometry and a Reynolds number are calculated as a function of the Rayleigh number. We find that the Nusselt and Reynolds numbers verify scaling laws consistent with turbulent Rayleigh-Benard convection: Nu ∝ Ra 0.31 and Re ∝ Ra 1/2. Further, a Bolgiano regime is found with the Bolgiano scale scaling as Ra −1/4. All these elements show that despite the significant differences in geometry between our simulations and classical 3D cells, the scaling laws of thermal convection are robust.

Polymer-induced change in scaling behavior in two-dimensional homogeneous turbulent thermal convection.

We study the effects of polymers in two-dimensional turbulent thermal convection using a shell model. In the absence of polymers, the inverse energy cascade in two dimensions leads to the observed Bolgiano-Obukhov scaling. When polymers are added, energy is extracted from the flow by the polymers, and as a result, the thermal balance between buoyancy and inertia in Bolgiano-Obukhov scaling is destroyed around the scales at which polymers interact strongly with the flow. This results in an increase in the Bolgiano scale and leads to a change in the scaling behavior of the velocity and temperature fluctuations for scales below the modified Bolgiano scale. We make theoretical estimates of the dependence of the mean rate of energy extracted by the polymers and the mean energy dissipation rate on the polymer relaxation time. Our theoretical analysis further leads to the prediction that the heat transport is not altered much by the polymers in two dimensions. We show that our theoretical estimates and prediction are in good agreement with the numerical results.

Attractive fixed-point solution study of a shell model for turbulent convection

Different scaling behaviors, such as Kolmogorov (K41) scaling and Bolgiano and Obukhov (BO) scaling, have been reported in various shell models proposed for turbulent thermal convection. However, two coexistent subranges with K41 and BO scaling are not set up with Bolgiano scale interlaying between the largest scale and the dissipation scale. In this paper, we summarize fixedpoint solution study of the Brandenburg model with small perturbation theory by introducing a small disturbance term as the impact of buoyancy. Three groups of fixed-point solutions with different locations of the so-called buoyancy scale, above/below which buoyancy is significant/insignifant. Both theoretical and numerical results show that a modified K41 scaling, instead of K41 and BO coexistent scaling, is set up even though buoyancy may be significant over the scaling range, which suggests that the buoyancy scale is not related exactly to the Bolgiano scale. Thus, a K41 and BO coexistent scaling behavior is not setup for the Brandenburg model.

Bolgiano scale in confined Rayleigh–Taylor turbulence

AbstractWe investigate the statistical properties of Rayleigh–Taylor turbulence in a three-dimensional convective cell of high aspect ratio, in which one transverse side is much smaller that the others. By means of high-resolution numerical simulation we study the development of the turbulent mixing layer and the scaling properties of the velocity and temperature fields. We show that the system undergoes a transition from a three- to two-dimensional turbulent regime when the width of the turbulent mixing layer becomes larger than the scale of confinement. In the late stage of the evolution the convective flow is characterized by the coexistence of Kolmogorov–Obukhov and Bolgiano–Obukhov scaling at small and large scales, respectively. These regimes are separated by the Bolgiano scale, which is determined by the scale of confinement of the flow. Our results show that the emergence of the Bolgiano–Obukhov scaling in Rayleigh–Taylor turbulence is connected to the onset of an upscale energy transfer induced by the geometrical constraint of the flow.

The mixing evolution and geometric properties of a passive scalar field in turbulent Rayleigh–Bénard convection

We report on measurements of a two-dimensional (2D) dye concentration field in turbulent Rayleigh–Bénard (RB) convection using the planar laser-induced fluorescence technique. The measurements were made in a vertical plane near the sidewall of a rectangular convection cell filled with water and with the Rayleigh number Ra varying from 109 to 1010, all at a fixed Prandtl number Pr=5.3 and Schmidt number Sc=2100. The mixing evolution and geometric properties of the measured passive scalar field are studied. It is shown that the mixing evolution of the passive scalar field in buoyancy-driven turbulence possesses certain features that are similar to those found in other types of turbulent flows, such as turbulent jets and grid-generated turbulence. These features include a power-law decrease in the mean concentration and exponential tails of single-point probability density function along the evolution path. Our results also show that the log-Poisson distribution is universal for the geometric measures of the passive scalar iso-contours, such as the perimeter, area, shape complexity and absolute value of local curvature; and the log-Poisson statistics may be used to model them. Regarding the influence of buoyancy on the turbulent scalar mixing, it is found that the buoyant scale, i.e. the Bolgiano scale ℓB, cannot be identified from the measured geometric properties.

Statistics of the locally averaged thermal dissipation rate in turbulent Rayleigh–Bénard convection

From the measured thermal dissipation rate in turbulent Rayleigh–Bénard convection in a cylindrical cell, we construct a locally averaged thermal dissipation rate χ fτ by averaging over a time interval τ. We study how the statistical moments ⟨(χ fτ) p ⟩ depend on τ at various locations along the vertical axis of the convection cell. We find that ⟨(χ fτ) p ⟩ exhibits good scaling in τ, of about a decade long, with scaling exponents μ(p) for p = 1–6. For Rayleigh number (Ra) around 8×109, the scaling range is 1.4–21 s at the cell center and 4–21 s at the bottom plate. The dissipative and turnover times are about 0.8 s and 35 s respectively, while the timescale corresponding to the local Bolgiano scale is estimated to be about 31 s at the cell center and 3.5 s at the bottom plate. On the basis of several assumptions, we derive theoretical predictions for μ(p) at the different locations. The measured values of μ(p) are presented and shown to be in good agreement with our theoretical predictions.

Studying anomalous scaling and heat transport of turbulent thermal convection using a dynamical model

Turbulent thermal convection is a well-studied problem with various issues of interest. In this paper, we review our work which shows the nature and origin of anomalous scaling and heat transport in the limit of very strong thermal forcing, can be gained by studying a dynamical model, known as shell model, of homogeneous turbulent thermal convection in which buoyancy acts directly at most scales. Specifically, we have obtained two results. The first result is that when buoyancy acts directly at most scales such that the dynamics are governed by a cascade of entropy, the scaling behavior is described by Bolgiano and Obukhov scaling plus corrections that are due to the variations of the local entropy transfer rate. This result indicates the validity of the extension of refined similarity hypothesis to turbulent thermal convection. The second result is that when buoyancy is acting directly at most scales, a damping term acting on the largest scale, which has to be added for the system to achieve stationarity, plays a crucial role in heat transport, and that the heat transport depends on the strength of thermal forcing in the same manner as that predicted for the ultimate state of very strong thermal forcing. With our interpretation of the damping term representing the effect of the boundaries, this result indicates that in the ultimate state of turbulent thermal convection, when buoyancy is acting at most scales, boundaries would play a significant role in heat transport.

Temporal statistics in high Rayleigh number convective turbulence

Temporal statistics of temperature and velocity fluctuations are studied in a highly turbulent convective flow developed within a cylindrical cell of aspect ratio Γ=1/2. Numerical data are analyzed for Ra up to 2×10 11 at fixed Prandtl number ( Pr=0.7). Temperature and velocity time series are collected from numerical probes placed within the fluid volume in the bulk and close to the boundaries. It is shown that the effects of the boundaries and of the large scale recirculation are reflected in the temporal statistics of the analyzed quantities. Spectra and structure functions show that the temperature statistics follow the Bolgiano scaling while the velocity fluctuations exhibit scaling laws which are surprisingly very close to those of the temperature. These results are in contrast with the Bolgiano or Kolmogorov behaviors expected for the velocity statistics and a possible theoretical explanation is presented.

Turbulent Time Correlations and Generation of Acoustic Waves by Stellar or Solar Turbulent Convection

This paper considers the question of whether Eulerian or Lagrangian time correlations should be used to evaluate sound radiation by turbulent convection. Whereas previous analyses have either explicitly or implicitly used Lagrangian correlations, this paper argues in favor of Eulerian correlations. The physics of turbulent time correlations is briefly reviewed, and the implications of using Eulerian instead of Lagrangian correlations in sound-radiation calculations are discussed. The assumption made about time correlations alters the frequency distribution of the radiated sound; assuming Eulerian time correlations results in a more shallow acoustic power spectrum and consequently greater relative weighting of higher frequencies. The consequences of assuming Bolgiano scaling of buoyancy-generated turbulence are also discussed.

Scaling of the velocity power spectra in turbulent thermal convection.

We report measurements of the local velocity in a convection cell filled with water. In the buoyancy subrange (the scales between the Bolgiano scale f(B) and the integral scale f(o)), two scaling regions of the velocity frequency power spectra, separated by the peak frequency f(p) of the dissipation spectra, are found. For f(p)<f<f(B), we observe a power law with the Bolgiano-Obukhov scaling exponent -11/5. For f(o)<f<f(p), an unexpected scaling with an exponent -1.35 is observed. We also found that the velocity power spectra are universal functions with respect to the characteristic scales f(p). and f(B).

Statistics of local temperature dissipation in high rayleigh number convection

We study the statistics of the local temperature dissipation in high Rayleigh number convection. We find that its probability distribution deviates from a lognormal, although very low order moments can be approximated by a lognormal distribution. Instead, the moments satisfy a hierarchy similar to that proposed by She and Leveque [Phys. Rev. Lett. 72, 336 (1994)] for the local energy dissipation. Moreover, the moments scale with the separation time. No change in the scaling behavior is observed when the Bolgiano scale is crossed, indicating that the statistics have the same nature in the buoyancy-driven and the inertia-driven regimes.

Intermittency of temperature field in turbulent convection

The scaling behavior of the temperature structure functions in turbulent convection is found to be different for length scales below and above the Bolgiano scale. Both sets of the exponents are well described by log-Poisson statistics. The parameter beta(T) which measures the degree of intermittency is the same for the two regimes of scales and is consistent with the corresponding value for the passive scalar field. A balance between thermal forcing and nonlinear velocity advection, which is a key ingredient leading to Bolgiano scaling, is also checked.

On the Heat Transfer in Rayleigh–Bénard Systems

In this paper we discuss some theoretical aspects concerning the scaling laws of the Nusselt number versus the Rayleigh number in a Rayleigh–Benard cell. We present a new set of numerical simulations and compare our findings against the predictions of existing models. We then propose a new theory which relies on the hypothesis of Bolgiano scaling. Our approach generalizes the one proposed by Kadanoff, Libchaber, and coworkers and solves some of the inconsistencies raised in the recent literature.

Three applications of scaling to inhomogeneous, anisotropic turbulence

The energy spectrum in three examples of inhomogeneous, anisotropic turbulence, namely, purely mechanical wall turbulence, the Bolgiano-Obukhov cascade, and helical turbulence, is analyzed. As one could expect, simple dimensional reasoning leads to incorrect results and must be supplemented by information on the dynamics. In the case of wall turbulence, a hypothesis of Kolmogorov cascade, starting locally from the gradients in the mean flow, produces an energy spectrum that obeys the standard ${k}^{\ensuremath{-}5/3}$ law only for ${\mathrm{kx}}_{3}&gt;1$, with ${x}_{3}$ the distance from the wall, and an inverse power law for ${\mathrm{kx}}_{3}&lt;1$. An analysis of the energy budget for turbulence in stratified flows shows the unrealizability of an asymptotic Bolgiano scaling. Simulation with a Gledzer-Ohkitani-Yamada shell model leads instead to a ${k}^{\ensuremath{-}\ensuremath{\alpha}}$ spectrum for both temperature and velocity, with $\ensuremath{\alpha}\ensuremath{\simeq}2$, and a cross correlation between the two vanishing at large scales. In the case of non-reflection-invariant turbulence, closure analysis suggests that a purely helical cascade, associated with a ${k}^{\ensuremath{-}7/3}$ energy spectrum cannot take place, unless external forcing terms are present at all scales in the Navier-Stokes equation.

Analytical theory of the destruction terms in dissipation rate transport equations

Modeled dissipation rate transport equations are often derived by invoking various hypotheses to close correlations in the corresponding exact equations. D. C. Leslie [Modern Developments in the Theory of Turbulence (Oxford University, Oxford, 1972)] suggested that these models might be derived instead from Kraichnan’s [J. Fluid Mech. 47 (1971)] wavenumber space integrals for inertial range transport power. This suggestion is applied to the destruction terms in the dissipation rate equations for incompressible turbulence, buoyant turbulence, rotating incompressible turbulence, and rotating buoyant turbulence. Model constants like Cε2 are expressed as integrals; convergence of these integrals implies the absence of Reynolds number dependence in the corresponding destruction term. The dependence of Cε2 on rotation rate emerges naturally; sensitization of the modeled dissipation rate equation to rotation is not required. A buoyancy related effect which is absent in the exact transport equation for temperature variance dissipation, but which sometimes improves computational predictions, also arises naturally. The time scale in the modeled transport equation depends on whether Bolgiano or Kolmogorov inertial range scaling applies. A simple extension of these methods leads to a preliminary dissipation rate equation for rotating buoyant turbulence.