Sediment-laden fresh water above salt water: nonlinear simulations

Abstract

When a layer of particle-laden fresh water is placed above clear, saline water, both double-diffusive and Rayleigh–Taylor instabilities may arise. The present investigation extends the linear stability analysis of Burns & Meiburg (J. Fluid Mech., vol. 691, 2012, pp. 279–314) into the nonlinear regime, by means of two- and three-dimensional direct numerical simulations (DNS). The initial instability growth in the DNS is seen to be consistent with the dominant modes predicted by the linear stability analysis. The subsequent vigorous growth of individual fingers gives rise to a secondary instability, and eventually to the formation of intense plumes that become detached from the interfacial region. The simulations show that the presence of particles with a Stokes settling velocity modifies the traditional double-diffusive fingering by creating an unstable ‘nose region’ in the horizontally averaged profiles, located between the upward-moving salinity and the downward-moving sediment interface. The effective thickness $l_{s}$ ($l_{c}$) of the salinity (sediment) interface grows diffusively, as does the height $H$ of the nose region. The ratio $H/l_{s}$ initially grows and then plateaus, at a value that is determined by the balance between the flux of sediment into the rose region from above, the double-diffusive/Rayleigh–Taylor flux out of the nose region below, and the rate of sediment accumulation within the nose region. For small values of $H/l_{s}\leqslant O(0.1)$, double-diffusive fingering dominates, while for larger values $H/l_{s}\geqslant O(0.1)$ the sediment and salinity interfaces become increasingly separated in space and the dominant instability mode becomes Rayleigh–Taylor like. A scaling analysis based on the results of a parametric study indicates that $H/l_{s}$ is a linear function of a single dimensionless grouping that can be interpreted as the ratio of inflow and outflow of sediment into the nose region. The simulation results furthermore indicate that double-diffusive and Rayleigh–Taylor instability mechanisms cause the effective settling velocity of the sediment to scale with the overall buoyancy velocity of the system, which can be orders of magnitude larger than the Stokes settling velocity. While the power spectra of double-diffusive and Rayleigh–Taylor-dominated flows are qualitatively similar, the difference between flows dominated by fingering and leaking is clearly seen when analysing the spectral phase shift. For leaking-dominated flows a phase-locking mechanism is observed, which intensifies with time. Hence, the leaking mode can be interpreted as a fingering mode which has become phase-locked due to large-scale overturning events in the nose region, as a result of a Rayleigh–Taylor instability.

Sedimentation from flocculated suspensions in the presence of settling‐driven gravitational interface instabilities

“…This produces an unstable band of sediment-laden saltwater, i.e., the so-called interface layer, sitting between the muddy freshwater and the underlying clear saltwater. This phenomenon has been experimentally observed by Hoyal et al [1999b], Parsons et al [2001], and Blanchette and Bush [2005] and examined through linear stability analysis by Burns and Meiburg [2012] and Yu et al [2013] and Direct Numerical Simulation (DNS) by Yu et al [2014] and Burns and Meiburg [2015]. A few defining characteristics of the settlingdriven mechanism are the general creation of the interface layer below the initial stratification contact [Hoyal et al, 1999b], the onset of Rayleigh-Taylor generated plume like structures, asymmetry in the downward and upward interface motions, and the overall larger length scales of the instabilities and instabilities spacing (centimeter scale) compared to the double-diffusive generated fingering (millimeter scale) [Hoyal et al, 1999b;Blanchette and Bush, 2005;Burns and Meiburg, 2012;Yu et al, 2013;Burns and Meiburg, 2015;Yu et al, 2014].…”

“…This phenomenon has been experimentally observed by Hoyal et al [1999b], Parsons et al [2001], and Blanchette and Bush [2005] and examined through linear stability analysis by Burns and Meiburg [2012] and Yu et al [2013] and Direct Numerical Simulation (DNS) by Yu et al [2014] and Burns and Meiburg [2015]. A few defining characteristics of the settlingdriven mechanism are the general creation of the interface layer below the initial stratification contact [Hoyal et al, 1999b], the onset of Rayleigh-Taylor generated plume like structures, asymmetry in the downward and upward interface motions, and the overall larger length scales of the instabilities and instabilities spacing (centimeter scale) compared to the double-diffusive generated fingering (millimeter scale) [Hoyal et al, 1999b;Blanchette and Bush, 2005;Burns and Meiburg, 2012;Yu et al, 2013;Burns and Meiburg, 2015;Yu et al, 2014]. Further key observations regarding settling-driven instabilities have been that (1) the thickness of the interface layer at the commencement of downward convection tends to thicken as concentration goes down [Hoyal et al, 1999b], (2) that the spacing between points of downward convection tends to also be inversely related to concentration [Hoyal et al, 1999b;Yu et al, 2013], and (3) that the diameters of the instabilities may increase with concentration or sediment settling velocity [Yu et al, 2013].…”

“…Double-diffusive fingering can also occur in thermally stratified systems with warm suspensions of sediment overlying cooler clear water [Houk and Green, 1973;Green, 1987;Hoyal et al, 1999a;Parsons et al, 2001]. Key in the development of double-diffusive fingering for suspensions of sediment is that the sediment in the buoyant layer settles very slowly so that substance diffusion, and not particle settling, is primarily responsible for changes in the vertical density profiles [Burns and Meiburg, 2012;Yu et al, 2013;Burns and Meiburg, 2015;Yu et al, 2014]. Within the band of conditions conducive to double diffusion, transport of sediment within the doublediffusive fingers can be a significant contribution to the total downward sediment flux [Hoyal et al, 1999a].…”

“…In their simulations, the difference in mode and instability morphology is clearly visible. Rather than using raw sediment size, Burns and Meiburg [2015] showed that the dominate instability mode could be related to a key interface thickness ratio, H=l s , where H is the thickness of the downward expanding sediment interface region and l s is the diffusive interface thickness of the salinity profile. For small values, H=l s < Oð0:1Þ, the interface develops double-diffusive fingering, while for larger values, H=l s > Oð0:1Þ, Rayleigh-Taylor processes dominate.…”

“…However, for buoyant mud plumes lofting over saltwater (hypopycnal river discharges), two additional mechanisms can potentially influence the vertical flux of sediment from the plume to the lower ocean waters. One of these processes is flocculation, which can complicate the estimation of Burns and Meiburg, 2015]. A distinction between these three mechanisms is whether or not there is turbulent mixing at the interface between the hypopycnal river discharge and the receiving water body.…”