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Register a new userSpreading height and critical conditions for the collapse of turbulent fountains in stratified media
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Axisymmetric fountains in stratified environments rise until reaching a maximum height, where the vertical momentum vanishes, and then falls and spread radially as an annular plume following a well-known top-hat profile. Here, firstly, we generalize the model of Morton et al. (Proc. R. Soc. Lond. A textbf{234}, 1, 1956), in order to correctly determine the dependence of the maximum height and the spreading height with the parameters involved. We obtain the critical conditions for the collapse of the fountain, textit i.e. when the jet falls up to the source level, and show that the spreading height must be expressed as a function of at least two parameters. To improve the quantitative agreement with the experiments we modify the criterion to take the mixing process in the down flow into account. Numerical simulations were implemented to estimate the parameter values that characterizes this merging. We show that our generalized model agrees very well with the experimental measurements.
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A simple theory, based on observations of snowflake distribution in a turbulent flow, is proposed to model the growth of inertial particles as a result of dynamic clustering at scales larger than the Kolmogorov length scale. Particles able to stick or coalesce are expected to grow in size in flow regions where preferential concentration is predicted by a critical Stokes number $St=tau_p/tau_f simeq 1 $. We postulate that, during growth, $St$ remains critical, with the particle response time $tau_p$ evolving according to the specific flow time scale $tau_f$ defined by the vortices around which progressively larger particles end up orbiting. This mechanism leads to the prediction of the limiting size of aggregating particles in a turbulent flow. Such limit is determined by the extent of the turbulent inertial range, which can be formulated as a function of accessible integral-scale quantities. The proposed dynamically critical Stokes growth provides a framework to interpret particle aggregation, size growth and particle cluster growth in various geophysical multi-phase flows.
Imbibition plays a central role in diverse energy, environmental, and industrial processes. In many cases, the medium has multiple parallel strata of different permeabilities; however, how this stratification impacts imbibition is poorly understood. We address this gap in knowledge by directly visualizing forced imbibition in three-dimensional (3D) porous media with two parallel strata. We find that imbibition is spatially heterogeneous: for small capillary number Ca, the wetting fluid preferentially invades the fine stratum, while for Ca above a threshold value, the fluid instead preferentially invades the coarse stratum. This threshold value depends on the medium geometry, the fluid properties, and the presence of residual wetting films in the pore space. These findings are well described by a linear stability analysis that incorporates crossflow between the strata. Thus, our work provides quantitative guidelines for predicting and controlling flow in stratified porous media.
The Lagrangian (LA) and Eulerian Acceleration (EA) properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear flow. The probability density functions (pdfs) of both LA and EA have a stretched-exponential shape and they show a strong and similar influence on the Richardson number. The extreme values of the EA are stronger than those observed for the LA. Geometrical statistics explain that the magnitude of the EA is larger than its Lagrangian counterpart due to the mutual cancellation of the Eulerian and convective acceleration, as both vectors statistically show an anti-parallel preference. A wavelet-based scale-dependent decomposition of the LA and EA is performed. The tails of the acceleration pdfs grow heavier for smaller scales of turbulent motion. Hence the flatness increases with decreasing scale, indicating stronger intermittency at smaller scales. The joint pdfs of the LA and EA indicate a trend to stronger correlations with increasing Richardson number and at larger scales of the turbulent motion. A consideration of the terms in the Navier--Stokes equation shows that the LA is mainly determined by the pressure-gradient term, while the EA is dominated by the nonlinear convection term.
Boundary-layer transition triggered by a roughness element generates a turbulent wedge that spreads laterally as the flow proceeds downstream. The spreading half angle is about $6^{circ}$ in zero-pressure-gradient flows regardless of Reynolds number and roughness shape. Recent simulations and experiments have sought to explain the lateral-spreading mechanism and have observed high- and low-speed streaks along the flanks of the wedge that appear central to the spreading process. To better elucidate the role of streaks, a naphthalene flow-visualization survey and hotwire measurements are conducted over a wider range of Reynolds numbers and a longer streamwise domain than previous experiments. The results reconfirm the spreading half angle is insensitive to Reynolds numbers based on roughness location, $Re_{x,k}$, and roughness height, $Re_{kk}$. When made nondimensional by the unit Reynolds number, the distance from the roughness to the effective origin of the turbulent wedge and to the first high-speed flanking streaks depends on $Re_{kk}$ but not $Re_{x,k}$. The distance between the first and second high-speed streaks is also observed to depend on $Re_{kk}$. In spite of a long measurement domain, third streaks are not observed and it remains unknown whether subsequent streak-to-streak distances collapse to a universal value. The reason downstream streaks are not observed may be low-frequency meandering of streak structures. Hotwire measurements confirm breakdown to turbulence first occurs via a shear-layer instability above low-speed streaks. Farther downstream, high-intensity broadband fluctuations are observed in equivalent positions on secondary low-speed streaks.
Stratified turbulence shows scale- and direction-dependent anisotropy and the coexistence of weak turbulence of internal gravity waves and strong turbulence of eddies. Straightforward application of standard analyses developed in isotropic turbulence sometimes masks important aspects of the anisotropic turbulence. To capture detailed structures of the energy distribution in the wave-number space, it is indispensable to examine the energy distribution with non-integrated spectra by fixing the codimensional wave-number component or in the two-dimensional domain spanned by both the horizontal and vertical wave numbers. Indices which separate the range of the anisotropic weak-wave turbulence in the wave-number space are proposed based on the decomposed energies. In addition, the dominance of the waves in the range is also verified by the small frequency deviation from the linear dispersion relation. In the wave-dominant range, the linear wave periods given by the linear dispersion relation are smaller than approximately one third of the eddy-turnover time. The linear wave periods reflect the anisotropy of the system, while the isotropic Brunt-Vaisala period is used to evaluate the Ozmidov wave number, which is necessarily isotropic. It is found that the time scales in consideration of the anisotropy of the flow field must be appropriately selected to obtain the critical wave number separating the weak-wave turbulence.
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