Session U19: Convection and Buoyancy-Driven Flows: Stratified Flow

An analytical model for a density current from a supercritical to a subcritical state

In the seminal paper by Ellison and Turner [1], the authors develop a theoretical model to calculate the longitudinal evolution of the characteristic quantities (velocity, thickness and density) of a continuous steady density current. These equations introduce the Richardson number Ri, which is a ratio between the buoyancy and the inertia forces of the current (Δρgh/ρ_aU²) and characterizes the stability of the flow. They also introduce two flow regimes related to the Richardson number: the subcritical regime when buoyancy forces dominate (Ri>1) and the supercritical regime when inertia forces dominate (Ri<1). For an initially supercritical flow, the set of equations developed by Ellison and Turner [1] presents a singularity when the flow transitions from the supercritical to the subcritical state through the critical state (Ri=1). In such a case, a mathematical discontinuity, similar to a jump, allows the super- and sub-critical solutions to be matched, enabling a complete computational resolution of the problem (Haddad et al. [2]).

To go further mathematically with this theoretical model, we propose in this study an algebraic development allowing a direct calculation of the evolution of the Richardson number as a function of the length of the studied domain. This method provides direct results, in contrast to those obtained by a computational resolution of coupled ordinary differential equations. This method also gives the variations of the primary variables from the Richardson number. It is even possible, for the limiting cases, to find strict analytical solutions by means of asymptotic developments. We perform numerical simulations using the large-eddy simulation code CALIF³S-Isis to obtain reference results and compare them with the theoretical solutions obtained. The extension proposed is interesting and allows, in a qualitatively acceptable way, the non-monotonic evolution of the density current primary variables to be reproduced.   

Irreversible mixing in stably-stratified wall-bounded turbulence

Evaluating irreversible vertical turbulent mixing of momentum and density in stably-stratified turbulent flows is complex and challenging. Precise quantification of irreversible mixing is of great importance in many industrial and geophysical flows. In this work, we focus on quantifying irreversible mixing in stably-stratified turbulent channel flow. To this aim, we conduct a campaign of pseudo-spectral direct numerical simulations (DNS) of the governing equations, written under the Boussinesq approximation, for shear Reynolds number Re_τ=1000 and Re_τ=180, and shear Richardson number in the range 50 ≤ Ri_τ ≤ 300 and 25 ≤ Ri_τ ≤ 75, respectively. We quantify the irreversible flux Richardson number Ri^*_f, which is a measure for irreversible mixing, and we propose a new parameterization for Ri^*_f as a function of the gradient Richardson number, Ri_g.   

Not Participating

We describe and model experimental results on the dynamics of a ”ludion” - a neutrally buoyant body - immersed

in a layer of stably stratified salt water. By varying periodically the hydrostatic pressure, the ludion

initially positioned at its equilibrium height oscillates vertically. Depending on the ratio of the forcing frequency to

the buyancy frequency N of the stratified fluid, the ludion can emit its own internal gravity waves that we

measure by Particle Image Velocimetry. Our experimental results describe first the resonance of the vertical

motions of the ludion when excited at different frequencies taking

into account added mass and added friction coefficients (Voisin, 2007). Then, for the larger oscillation amplitudes,

we observe and describe a bifurcation towardsfree horizontal swimming with chaotic trajectories.

 The statistical analysis suggests the existence of an underlying non equilibrium thermodynamics with even possible

”condensations” of ludions in pairs or in triplets when several

ludions are introduced in the container. Finally, we also observed that the ludion can interact with its own internal

gravity wave field and possibly become trapped in the container gravity eigenmodes. However, it seems that,

contrary to the surface waves associated with Couder walkers (Couder et al., 2005) the internal waves are

not the principal cause of the horizontal swimming (Deng et Caulfield, 2018). This does not however, exclude

possible hydrodynamic quantum analogies to be explored in the future (Couder et al., 2005, Bush, 2015).

    

Wake structures and mixing of falling spheres and rising bubble in density-stratified fluids

Density stratification is common in nature. Oceans, lakes, and the atmosphere have density-stratified environments. Vertical movement of particles, droplets, and bubbles can induce mixing and subsequently enhance the transport of heat, carbon dioxide, and nutrients from the upper layer to the lower layers of the ocean. Studying the wake structure of moving bodies in the stratified fluid would help us to understand the underlying process that enhance mixing. Previous studies confirmed the presence of a reverse jet in the wake of descending spheres in a stratified fluid. As the sphere moves, it drags down the lighter fluid to deeper layers with higher density. Due to buoyancy forces, the lighter fluid moves back to upper layers and forms the inverse jet, making the mixing process less effective. The same phenomenon has been observed for bubbles rising in the stratified fluids. Even though the wake structure of different bodies has been studied extensively, there hasn’t been a study comparing the wake structure between falling spheres or rising bubbles in a stratified fluid. In this study, we investigate the wake structure of a falling sphere and a rising bubble with similar non-dimensional numbers (Re and Fr). The flow field and the reverse jet is quantified using 3-D Particle Tracking Velocimetry (PTV) method. The comparison between the two cases of rising bubbles and falling spheres will help us identify the more efficient way of induced mixing by moving bodies in stratified fluids.

    

Eulerian large-eddy simulations of deep-sea hydrocarbon plumes with hydrate formation and decomposition

In the case of a deep-sea oil spill incident, the gas bubbles, oil droplets, and entrained seawater form a multiphase hydrocarbon plume and rise together towards the sea surface through stratified seawater. Below the thermodynamic equilibrium depth, the natural gas molecules inside the bubbles can react with the surrounding seawater to form solid hydrate shells outside the bubbles. With sufficient reaction time, gas bubbles can even become solid hydrate particles completely. As the hydrate-covered gas bubbles and hydrate particles reach the thermodynamic equilibrium depth, hydrate decomposition occurs and gas molecules are dissolved into the surrounding seawater. The hydrate formation and decomposition can change the buoyancy force that drives the plume, resulting in considerable effects on the plume dynamics. In this study, an Eulerian large-eddy simulation model is developed to simulate the hydrocarbon plumes with the effects of hydrate formation and decomposition. Two different scenarios with the gas releasing depths of 630m and 1000m are modeled and the corresponding plume characteristics are analyzed.   

Modeling salt-finger convection in the oceanic parameter regimes

Re-scaled incompressible Navier-Stokes Equations (RiNSE) are formulated to simulate salt-finger convection in the parameter regime of τ → 0, relevant in the oceanic regime, with τ being the ratio of the salinity diffusion to the temperature diffusion. The equations are re-scaled with anisotropy in the z direction ∂_z ∼ τ▽_⊥, so τ serves as a control parameter for spatial anisotropy. These preconditioned equations are exact and alleviate the stiffness issues at a small τ. The two-dimensional RiNSE setup is used to simulate the salt finger convection for τ =0.01, which shows the patchiness of fingers at the large scales, confirming the presence of multiscale phenomena. Spontaneous emergence of mean horizontal shear is also observed, causing the tilt of salt fingers. Taking a further lead from RiNSE, we propose an asymptotically reduced model permitting an eddy flux feedback on the temperature flux. This reduced model is an extension of the previously proposed model: Inertia-Free Salt Finger Convection (IFSC) (Xie et al., 2017). Both RiNSE and the reduced model simulations are compared for the salt finger convection in the limit τ → 0.   

Direct Numerical Simulation of the Stable Stratification Transient in the HiRJET Facility

Stable stratification inhibits the natural circulation in the coolant loop and poses safety challenges in the sodium fast reactors. The poor performance of the present engineering turbulence models in buoyant mixing flows urges the high-fidelity data to support the improvement of turbulence modeling. In this work, direct numerical simulation is carried out to simulate the density stratification experiment in the High-Resolution Jet facility. Current simulation has covered the early stage of the stratification transient. It has been seen that the stratified front is highly turbulent initially due to the jet impinging and turbulent mixing effect. As time goes, the stratified front gradually stabilizes and separate from the mixing zone. Position of mixing zone, stratified fronts and the value of maximum density derivative are quantified by curve fitting and the agreement with the experimental results is observed.

    

Interaction between thermal stratification and turbulence in channel flow

In this work we investigate the behavior of stably-stratified channel turbulence by running a series of  Direct Numerical Simulations (DNS) under the Oberbeck-Boussinesq (OB) approximation, at shear Reynolds number Re_τ=1000 and shear Richardson number in the range  0≤Ri_τ≤300. By increasing stratification, active turbulence is sustained only in the near-wall region, whereas intermittent turbulence, modulated by the presence of non-turbulent wavy structures (Internal Gravity Waves, IGW), is observed at the channel core.  In  such  conditions,  the wall-normal transport  of  momentum  and  heat  is  considerably   reduced  compared  to the  case  of  non-stratified  turbulence. By performing a cross power spectral density analysis of temperature and the wall-normal velocity fluctuation signals we show the presence of a ∼π/2 phase delay between these two signals.This constitutes a blockage effect to the wall-normal exchange of energy. In addition, we also show the scaling laws for friction factor and Nusselt number. These scaling laws, which seem robust over the explored range of parameters, are consistent with previous experimental and numerical data, and are expected to help the development of improved models and parametrizations of stratified flows at large Re_τ.   

Staircase solutions and stability in vertically confined salt-finger convection

Bifurcation analysis of confined salt-finger convection using single-mode equations obtained from a severely truncated Fourier expansion in the horizontal is performed. Strongly nonlinear staircase-like solutions having, respectively, one (S1), two (S2) and three (S3) regions of well-mixed salinity in the vertical direction are computed using numerical continuation, and their stability properties are determined. Near onset, the one-layer S1 solution is stable and corresponds to maximum salinity transport among the three solutions. The S2 and S3 solutions are unstable but exert an influence on the statistics observed in direct numerical simulations (DNS) in larger two-dimensional (2D) domains. Secondary bifurcations of S1 lead either to tilted-finger (TF1) or to traveling wave (TW1) solutions, both accompanied by the spontaneous generation of large-scale shear, a process favored for lower density ratios. States breaking reflection symmetry in the midplane are also computed. The single-mode solutions close to the high wavenumber onset are in an excellent agreement with 2D DNS in small horizontal domains and compare well with 3D DNS.   

The effects of phase change on thermal and thermocapillary instabilities

Fluid flows over phase-changing boundaries find relevance in the study of many problems in astrophysical, geophysical, and engineering settings. A detailed understanding of the nonlinear coupling between fluid flow and the moving boundary is often critical to the evolution of the system. Growth of sea ice in the Earth's polar regions, the dynamics of ice sheets and the Earth's interior, and production of crystals for fabrication of semiconductors are a few examples that highlight their ubiquity across scales.

Here, we consider the stability of a layer of liquid that is bounded by its solid phase at the bottom and a free surface at the top. The liquid has a temperature of maximum density, T_m, and convective motions in the liquid layer ensue when the temperature of the free surface is greater than T_m. We use a combination of linear and asymptotic analyses to explore the effects of the phase-changing boundary, density anomaly, and the free surface on the thermal and thermocapillary instabilities in this system.   

Thermal convection under spatially varying magnetic fields

We study the properties of turbulent large-scale structures in an extended fluid layer of aspect ratio 16:32:1 under the influence of non-uniform magnetic field. The layer is heated from below and cooled from above (Rayleigh-Bénard configuration). The magnetic field is generated by two semi-infinite planar magnetic poles, with the convection layer located near the edge of the gap between the poles. We employ direct numerical simulations of the above setup for fixed Rayleigh and Prandtl numbers but vary the magnetic field profiles determined by the gap between the convection cell and the magnets. The simulations are conducted using an efficient parallel finite-difference solver. We investigate how the magnetic field profile impacts the evolution of superstructures (coherent large-scale flow patterns that extend over scales in the lateral direction that are significantly larger than the domain height) and their characteristic time and length scales. We also conduct a detailed study on the implications of these structures in turbulent heat transfer. Further, we study the properties of convective movements in the vicinity of sidewalls (wall modes) in the high magnetic flux region, and how their behavior changes with the magnetic field profile.   

Heat transport scaling theory for rotating Rayleigh-Benard convection

Thermally-induced buoyancy and rotation govern convection in many geophysical and astrophysical systems. The paradigm system with which to study such phenomena is rotating Rayleigh-Benard convection, where the strength of buoyancy is reflected in the Rayleigh number Ra and that of the Coriolis force in the Ekman number Ek. How the heat transport, measured by the Nusselt number Nu, depends on Ra and Ek, and how this dependence changes in the buoyancy- or rotation-dominated regimes, remain important questions and have been the subject of many years of debate. In Annu. Rev. Fluid Mech. 55 (2023) [1] we suggest a unifying heat transport scaling model, which relates the scaling exponents in these two regimes. The larger exponent in the buoyancy-dominated regime is related to a larger exponent in the rotation-dominated regime, and we find the limiting values of these exponents. The theoretical results are well supported by measurements and direct numerical simulations.