We consider the time-dependence of a hierarchy of scaled $L^{2m}$-norms $D_{m,ω}$ and $D_{m,θ}$ of the vorticity $\boldsymbol ω = \boldsymbol{\nabla} \times {\mathbf u}$ and the density gradient $\boldsymbol{\nabla} θ$, where $θ=\log (ρ^*/ρ^*_0)$, in a buoyancy-driven turbulent flow as simulated by \cite{LR2007}. $ρ^*({\mathbf x},\,t) $ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $ρ^*_2 > ρ^*_1$ and $ρ^*_{0}$ is a reference normalisation density. Using data from the publicly available Johns Hopkins Turbulence Database we present evidence that the $L^{2}$-spatial average of the density gradient $\boldsymbol{\nabla} θ$ can reach extremely large values, even in flows with low Atwood number $At = (ρ^*_{2} - ρ^*_{1})/(ρ^*_{2} + ρ^*_{1}) = 0.05$, implying that very strong mixing of the density field at small scales can arise in buoyancy-driven turbulence. This large growth raises the possibility that the density gradient $\boldsymbol{\nabla} θ$ might blow up in a finite time.
In this paper we present a unified shell model for stably stratified and convective turbulence. Numerical simulation of this model for stably stratified flow shows Bolgiano-Obukhbov scaling in which the kinetic energy spectrum varies as $k^{-11/5}$. We also observe a dual scaling ($k^{-11/5}$ and $k^{-5/3}$) for a limited range of parameters. The shell model of convective turbulence yields Kolmogorov's spectrum. These results are consistent with the energy flux and energy feed due to buoyancy, and are in good agreement with direct numerical simulations of Kumar {\em et al.} [Phys. Rev. E {\bf …
In this paper, we review the recent developments in the field of buoyancy-driven turbulence. Scaling and numerical arguments show that the stably-stratified turbulence with moderate stratification has kinetic energy spectrum $E_u(k) \sim k^{-11/5}$ and the kinetic energy flux $Π_u(k) \sim k^{-4/5}$, which is called Bolgiano-Obukhov scaling. The energy flux for the Rayleigh-Bénard convection (RBC) however is approximately constant in the inertial range that results in Kolmorogorv's spectrum ($E_u(k) \sim k^{-5/3}$) for the kinetic energy. The phenomenology of RBC should apply to other flows whe…
Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux $Π_u$, we demonstrate that for stably stratified flows, the kinetic energy spectrum $E_u(k) \sim k^{-11/5}$, the entropy spectrum $E_θ(k) \sim k^{-7/5}$, and $Π_u(k) \sim k^{-4/5}$, consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence $E_u(k)$ follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh Bénard convection, we show that the energy su…
We present a direct numerical simulation (DNS) study of buoyancy driven bubbly flows in two-dimensions. We employ volume of fluid (VOF) method to track the bubble interface. To investigate spectral properties of the flow, we derive the scale-by-scale energy budget equation. We show that the Galilei number (Ga) controls different scaling regimes in the energy spectrum. For high Galilei numbers, we find the presence of an inverse energy cascade. Our study indicates that the density ratio of the bubble with the ambient fluid or the presence of coalescence between the bubbles does not alter the sc…
Turbulent mixing layers in nature are often characterized by the presence of a mean shear and an unstable buoyancy gradient between two streams of different velocity. Depending on the relative strength of shear versus buoyancy, either the former or the latter may dominate the turbulence and mixing between the two streams. In this paper, we present a phenomenological theory that leads to the identification of two distinct turbulent regimes: an early regime, dominated by the mean shear, and a later regime dominated by the buoyancy. The main theoretical result consists of the identification of a …
A discussion on Li et al. (2019) [Numerical computations of resonant sloshing using the modified isoAdvector method and the buoyancy-modified turbulence closure model. Appl. Ocean Res. 93, article no. 101829, DOI:10.1016.j.apor.2019.05.014] is provided. Some mis-characterizations regarding the work of Larsen and Fuhrman (2018) [On the over-production of turbulence beneath surface waves using Reynolds-averaged Navier-Stokes models. J. Fluid Mech. 853, 419-460, DOI:10.1017/jfm.2018.577], on stabilizing two-equation turbulence closures beneath surface waves, are clarified.
We provide a numerical validation of a recently proposed phenomenological theory to characterize the space-time statistical properties of a turbulent puff, both in terms of bulk properties, such as the mean velocity, temperature and size, and scaling laws for velocity and temperature differences both in the viscous and in the inertial range of scales. In particular, apart from the more classical shear-dominated puff turbulence, our main focus is on the recently discovered new regime where turbulent fluctuations are dominated by buoyancy. The theory is based on an adiabaticity hypothesis which …
The evolution of buoyancy-driven homogeneous variable-density turbulence (HVDT) at Atwood numbers up to 0.75 and large Reynolds numbers is studied by using high-resolution Direct Numerical Simulations. To help understand the highly non-equilibrium nature of buoyancy-driven HVDT, the flow evolution is divided into four different regimes based on the behavior of turbulent kinetic energy derivatives. The results show that each regime has a unique type of dependency on both Atwood and Reynolds numbers. It is found that the local statistics of the flow based on the flow composition are more sensiti…
Quasigeostrophic flows are induced by spatial variations in interior potential vorticity and boundary buoyancy. We begin by developing the geostrophic turbulence theory of boundary buoyancy anomalies in a fluid with vanishing potential vorticity. We find that the vertical stratification controls both the interaction range of boundary buoyancy anomalies and the dispersion of boundary-trapped Rossby waves. Buoyancy anomalies generate longer range velocity fields and more dispersive Rossby waves over decreasing stratification [$\mathrm{d}N(z)/\mathrm{d}{z} \leq 0$, where $N(z)$ is the buoyancy fr…
It is shown, using results of recent direct numerical simulations, laboratory experiments and atmospheric measurements, that buoyancy driven turbulence exhibits a broad diversity of the types of distributed chaos with its stretched exponential spectrum $\exp(-k/k_β)^β$. The distributed chaos with $β= 1/3$ (determined by the helicity correlation integral) is the most common feature of the stably stratified turbulence (due to the strong helical waves presence). These waves mostly dominate spectral properties of the vertical component of velocity field, while the horizontal component is dominated…
Turbulence plays a major role in shaping marine community structure as it affects organism dispersal and guides fundamental ecological interactions. Below oceanographic mesoscale dynamics, turbulence also impinges on subtle physical-biological coupling at the single cell level, setting a sea of chemical gradients and determining microbial interactions with profound effects on scales much larger than the organisms themselves. It has been only recently that we have started to disentangle details of this coupling for swimming microorganisms. However, for non-motile species, which comprise some of…
Using direct numerical simulations of rotating Rayleigh-Bénard convection, we explore the transitions between turbulent states from a 3D flow state towards a quasi-2D condensate known as the large-scale vortex (LSV). We vary the Rayleigh number $Ra$ as control parameter and study the system response (strength of the LSV) in terms of order parameters assessing the energetic content in the flow and the upscale energy flux. By sensitively probing the boundaries of the region of existence of the LSV in parameter space, we find discontinuous transitions and we identify the presence of a hysteresis …
We present a method of disentangling the anisotropies produced by the cliff structures in turbulent velocity field and test it in the system of turbulent Rayleigh-Bénard (RB) convection. It is found that in the RB system the cliff structures in the velocity field are generated by thermal plumes. These cliff structures induce asymmetry in the velocity increments, which leads us to consider the plus and minus velocity structure functions (VSF). The plus velocity increments exclude cliff structures, while the minus ones include them. Our results show that the scaling exponents of the plus VSFs ar…
The effects of different initial density distributions on the evolution of buoyancy-driven homogeneous variable-density turbulence (HVDT) at low (0.05) and high (0.75) Atwood numbers are studied by using high-resolution direct numerical simulations. HVDT aims to mimic the acceleration-driven Rayleigh-Taylor and shock-driven Richtmyer-Meshkov instabilities and reveals new physics that arise from variable-density effects on the turbulent mixing. Here, the initial amounts of pure light and pure heavy flows are altered primarily to mimic the variable-density turbulence at the different locations o…
Rayleigh--Taylor fluid turbulence through a bed of rigid, finite-size, spheres is investigated by means of high-resolution Direct Numerical Simulations (DNS), fully coupling the fluid and the solid phase via a state-of-the art Immersed Boundary Method (IBM). The porous character of the medium reveals a totally different physics for the mixing process when compared to the well-known phenomenology of classical RT mixing. For sufficiently small porosity, the growth-rate of the mixing layer is linear in time (instead of quadratical) and the velocity fluctuations tend to saturate to a constant valu…
Turbulence in the flow of fluid through a pipe can be suppressed by buoyancy forces. As the suppression of turbulence leads to severe heat transfer deterioration, this is an important and undesirable phenomenon in both heating and cooling applications. Vertical flow is often considered, as the axial buoyancy force can help drive the flow. With heating measured by the buoyancy parameter $C$, our DNS show that shear-driven turbulence may either be completely laminarised or transitions to a relatively quiescent convection-driven state. Buoyancy forces cause a flattening of the base flow profile, …
Numerical and observational evidence indicates that, in regions where mixed-layer instability is active, the surface geostrophic velocity is largely induced by surface buoyancy anomalies. Yet, in these regions, the observed surface kinetic energy spectrum is steeper than predicted by uniformly stratified surface quasigeostrophic theory. By generalizing surface quasigeostrophic theory to account for variable stratification, we show that surface buoyancy anomalies can generate a variety of dynamical regimes depending on the stratification's vertical structure. Buoyancy anomalies generate longer …
We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_B^2$ imposed in the third, vertical direction in a fluid with kinematic viscosity $ν$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this sy…
The velocity-vorticity correlation integral (Chkhetiani invariant) is an invariant of a viscous Karman-Howarth equation. In the tree-dimensional space this invariant can naturally determine a turbulent diffusivity (viscosity). It is shown, using results of direct numerical simulations, that distributed chaos dominated by this integral can provide an adequate description of the turbulent Rayleigh-Bénard convection, stably stratified turbulence and Rayleigh-Taylor mixing at Prandtl number $Pr \sim 1$.
We investigate the stability of stratified fluid layers undergoing homogeneous and periodic tidal deformation. We first introduce a local model which allows to study velocity and buoyancy fluctuations in a Lagrangian domain periodically stretched and sheared by the tidal base flow. While keeping the key physical ingredients only, such a model is efficient to simulate planetary regimes where tidal amplitudes and dissipation are small. With this model, we prove that tidal flows are able to drive parametric subharmonic resonances of internal waves, in a way reminiscent of the elliptical instabili…
In this paper, the properties of stably stratified turbulence (SST) and turbulent thermal convection are contrasted. A key difference between these flows is the sign of the kinetic energy feed by buoyancy, $\mathcal{F}_B$. For SST, $\mathcal{F}_B < 0$ due to its stable nature; consequently, the kinetic energy flux $Π_u(k)$ decreases with wavenumber $k$ that leads to a steep kinetic energy spectrum, $E_u(k) \sim k^{-11/5}$. Turbulent convection is unstable, hence $\mathcal{F}_B > 0$ that leads to an increase of $Π_u(k)$ with $k$; this increase however is marginal due to relatively weak buoyancy…
\citet{turner1957} proposed that dry thermals entrain because of buoyancy (via a constraint which requires an increase in the radius $a$). This however, runs counter to the scaling arguments commonly used to derive the entrainment rate, which rely on either the self-similarity of \citet{scorer1957} or the turbulent entrainment hypothesis of \citet{morton1956}. The assumption of turbulence-driven entrainment was investigated by \citet{lecoanet2018}, who found that the entrainment efficiency $e$ varies by less than $20\%$ between laminar (Re = 630) and turbulent (Re = 6300) thermals. This motiva…
It is commonly accepted that the breakup criteria of drops or bubbles in turbulence is governed by surface tension and inertia. However, also {\it{buoyancy}} can play an important role at breakup. In order to better understand this role, here we numerically study Rayleigh-Bénard convection for two immiscible fluid layers, in order to identify the effects of buoyancy on interface breakup. We explore the parameter space spanned by the Weber number $5\leq We \leq 5000$ (the ratio of inertia to surface tension) and the density ratio between the two fluids $0.001 \leq Λ\leq 1$, at fixed Rayleigh nu…
We use Direct Numerical Simulations to study the two-dimensional flow of a rotating, half soap bubble that is heated at its equator. The heating produces buoyancy and rotation generates a Coriolis forces in the fluid. However, due to the curved surface of the bubble, the buoyancy and Coriolis forces vary with latitude on the bubble, giving rise to rich flow behavior. We first explore the single-point properties of the flow, including the Reynolds and Nusselt numbers, mean fields, and Reynolds stresses, all as a function of latitude. For a given Rayleigh number, we observe a non-monotonic depen…
Budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales $\ell$ in sheared, stably stratified turbulence are analyzed using a filtering approach. Competing effects in the flow are considered, along with the physical mechanisms governing the energy fluxes between scales, and the budgets are used to analyze data from direct numerical simulation (DNS) at buoyancy Reynolds number $Re_b=O(100)$. The mean TKE exceeds TPE by an order of magnitude at the large scales, with the difference reducing as $\ell$ is decreased. At larger scales, buoyancy is never obse…
Spatial buoyancy-helical distributed chaos (turbulence) in the cumulus, stratiform, stratocumulus, cirrus and cirrus mammatus clouds have been studied using results of direct numerical simulations and measurements in cloudy atmosphere. It is shown that in the considered cases the moments of helicity distribution dominate the kinetic energy spectra both in the middle of the clouds and in the cloud-top regions.
Recent research has shed light on the role of coherent structures in forming layers when stably stratified turbulence is forced with horizontal shear (Lucas, Caulfield & Kerswell, J. Fluid Mech., vol. 832, 2017, pp. 409-437). Here we extend our previous work to investigate the effect of rigid boundaries on the dynamics by studying stably-stratified plane Couette flow with gravity oriented in the spanwise direction. We observe near-wall layering and associated new mean flows in the form of large scale spanwise-flattened streamwise rolls. The layers exhibit the expected buoyancy scaling $l_z\sim…
We study the turbulent flow of the density-stratified fluid around a small translating (either passively or self-propelled) particle. It was found recently [A. M. Ardekani and R. Stocker, Phys. Rev. Lett. vol. 105, 084502 (2010)] that without turbulence, the familiar Stokes flow is dramatically altered by the stratification. Stratification-induced inhomogeneity "turns on" the buoyancy introducing a new "cutoff" or "screening" length scale for the flow, yielding closed streamlines and a faster (exponential-like) decay of velocity. This result, however, did not account for the potential role of …
This paper is the second part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. In a first paper, we showed that the dynamics of the flow was dominated by coherent structures made of a superposition of nonlinear waves. (Mixed)-ribbons and (mixed)-cross-spirals are generated by interactions between a pair of linearly unstable helical modes of opposite `handedness', and appear to be responsible for the formation of well-mixed layers and sharp density interfaces. In this paper, we show that…
We present a direct numerical simulation (DNS) study of pseudo-turbulence in buoyancy driven bubbly flows for a range of Reynolds ($\Rey$) and Atwood ($\At$) numbers. We study the probability distribution function of the horizontal and vertical liquid velocity fluctuations and find them to be in quantitative agreement with the experiments. The energy spectrum shows the $k^{-3}$ scaling at high $\Rey$ and becomes steeper on reducing the $\Rey$. To investigate the spectral transfers in the flow, we derive the scale-by-scale energy budget equation. Our analysis shows that, for scales smaller than…
Lagrangian dispersion in experimental stratified turbulence
Laboratoire des Ecoulements Geophysiques et Industriels (LEGI), Universite Grenoble Alpes, CNRS, Grenoble-INP（地球物理和工业流动实验室）；Imperial College London（伦敦帝国理工学院）；Laboratoire de Mécanique et ses Interfaces, ENSTA Paris, Institut Polytechnique de Paris（力学及其界面实验室）
We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size $\varepsilon$. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the density behav…
We investigate numerically the dynamics and statistics of inertial particles transported by stratified turbulence, in the case of particle density intermediate in the average density profile of the fluid. In these conditions, particles tend to form a thin layer around the corresponding fluid isopycnal. The thickness of the resulting layer is determined by a balance between buoyancy (which attracts the particle to the isopycnal) and inertia (which prevents them from following it exactly). By means of extensive numerical simulations, we explore the parameter space of the system and we find that …
The vectoral wind structure of daytime atmospheric boundary layer (ABL) turbulence is strongly dependent on the balance between shear-driven turbulence production of horizontal fluctuations (driven by winds at the mesoscale), and buoyancy-driven turbulence production of vertical velocity fluctuations (driven by solar heating), characterized by the global instability state parameter $-z_i/L > 0$. In the fully shear-driven neutral limit $-z_i/L \rightarrow 0$, the surface layer is dominated by coherent streamwise-elongated concentrations of negative streamwise fluctuating velocity (low-speed str…
A robust energy transfer mechanism is found in nonlinear wave systems, which favours transfers towards modes interacting via non-resonant triads, applicable in meteorology, nonlinear optics and plasma wave turbulence. Transfer efficiency is maximal when the frequency mismatch of the non-resonant triad balances the system's nonlinear frequency: at intermediate levels of oscillation amplitudes an instability is triggered that explores unstable manifolds of periodic orbits, so turbulent cascades are most efficient at intermediate nonlinearity. Numerical simulations confirm analytical predictions.
The strength of the nonlinearity is measured in decaying two-dimensional turbulence, by comparing its value to that found in a Gaussian field. It is shown how the nonlinearity drops following a two-step process. First a fast relaxation is observed on a timescale comparable to the time of for-mation of vortical structures, then at long times the nonlinearity relaxes further during the phase when the eddies merge to form the final dynamic state of decay. Both processes seem roughly independent of the value of the Reynolds number.
A weakly nonlinear spectrum and a strongly nonlinear spectrum coexist in a statistically steady state of elastic wave turbulence. The analytical representation of the nonlinear frequency is obtained by evaluating the extended self-nonlinear interactions. The {\em critical\/} wavenumbers at which the nonlinear frequencies are comparable with the linear frequencies agree with the {\em separation\/} wavenumbers between the weak and strong turbulence spectra. We also confirm the validity of our analytical representation of the separation wavenumbers through comparison with the results of direct nu…
It is shown how the variable density model (VDM) that governs the Rayleigh-Taylor instability (RTI) for the miscible mixing of two incompressible fluids can be transformed into a diffusive version of the inhomogeneous, incompressible Navier-Stokes equations forced by gradients of the composition density $ρ$ of the mixing layer. This demonstrates how buoyancy-driven flows drive and enhance Navier-Stokes turbulence. The role of the potential vorticity $q = ω\cdot\nablaρ$ is also discussed.
We study the dispersion of light particles floating on a flat shear-free surface of an open channel in which the flow is turbulent. This configuration mimics the motion of buoyant matter (e.g. phytoplankton, pollutants or nutrients) in water bodies when surface waves and ripples are smooth or absent. We perform direct numerical simulation of turbulence coupled with Lagrangian particle tracking, considering different values of the shear Reynolds number (Reτ = 171 and 509) and of the Stokes number (0.06 < St < 1 in viscous units). Results show that particle buoyancy induces clusters that evolve …
Wall-bounded turbulent flows are widely observed in natural and engineering systems, such as air flows near the Earth's surface, water flows in rivers, and flows around a car or a plane. The universal logarithmic velocity profile in wall-bounded turbulent flows proposed by von Kármán in 1930 is one of the few exact physical descriptions of turbulence. However, the mean velocity and temperature profiles cannot be adequately described by this universal log law when buoyancy effects are present. Monin-Obukhov similarity theory (MOST), proposed in 1954, has been the cornerstone theory to account f…
This work extends the framework of the partially-averaged Navier-Stokes (PANS) equations to variable-density flow, \text{i.e.}, multi-material and/or compressible mixing problems with density variations and production of turbulence kinetic energy by both shear and buoyancy mechanisms. The proposed methodology is utilized to derive the PANS BHR-LEVM closure. This includes \textit{a-priori} testing to analyze and develop guidelines toward the efficient selection of the parameters controlling the physical resolution and, consequently, the range of resolved scales of PANS. Two archetypal test-case…
On dissipation time scales of the basic second-order moments: the effect on the Energy and Flux-Budget (EFB) turbulence closure for stably stratified turbulence
