Characteristics of double -diffusive finger evolution : numerical, analytical and experimental study (PhD)

Abstract

The main focus of the research work reported in this thesis is to study the characteristics of double diffusive finger evolution and effective parameters that govern them. The double diffusive instabilities, salt fingers, arises in oceans when hot and salty water lies over cold and fresh water of greater density. Convection is driven by the varying diffusivity of the two components that contributes in the density variation, that is heat (T) and salinity (S), where diffusivity of heat is hundred times faster than the salt. Faster diffusing component (T) stabilizes the system whereas slower diffusing components (S) destabilizes the system with overall density stratification remaining stable. The convection in this state takes the form of rising and sinking fingers known as salt fingers. Salt fingers are found to be highly interactive and have an extensive contribution in vertical mixing, nutrients up-welling and trans port of heat and salt fluxes in oceans. In some tropical oceans it is considered as the major source of transportation of heat and salt fluxes which is higher than turbulent mixing. For example, in tropical western Pacific Ocean, the estimated flux and associated vertical diffusivity due to double diffusion is approximately 1 order of magnitude higher for temperature and 2 orders of magnitude higher for salinity compared to values calculated from a turbulence model. Also, in several provinces of Atlantic, pacific and Indian oceans, nitrate diffusion mediated by salt fingers is responsible for 20% of the new nitrogen supply. Scope of double diffusion convection is not restricted to oceanography; it is widely spread in other field also, like stellar and planetary dynamics and evolution, crystal growth, magma chambers, geology etc. To explore the salt finger phenomenon, a non-dimensional density ratio (Rρ) is determined which is a measure of the degree of compensation between temperature and salinity gradients in terms of their effects on density stratification. For fingers to form in the heat-salt system, 1 < Rρ < τ−1. Here, τ is the diffusivity ratio of the slowest to the fastest diffusing components. It is suggested by literature that finger formation takes place at low Rρ in oceans and most distinctive fingers are formed when Rρ approaches unity. Many linear theories have been proposed to understand the finger behaviour as a function of density stability ratio (Rρ) . Theories of Stern (1960), Schmitt (1979, 2011) and Kunze (1987) are prominent and acceptable in scientific community. However, these theories are based on various assumptions which have some repercussions. One of the outcome of these assumptions is lack of accuracy when Rρ → 1. Unbounded results are obtained at neutral buoyancy ( Rρ = 1). Whereas many experimental studies in literature clearly indicates the formation of salt fingers at Rρ = 1. Now the question rises that why this kind of unusual behaviour is observed only at low Rρ, what is the physical interpretation of the assumptions that leads to the unboundedness at Rρ and how finger behaviour is different from other Rρ, let say Rρ = 2. We addressed this issue in this thesis and have shown that these assumptions in the previous theories are invalid in certain range of governing parameters. Moreover, it is believed in the scientific community that wide fingers transport large fluxes of heat and salinity compared to thin fingers. Theories proposed on this basis are also scrutinized. New insights into the physical understanding of the assumptions behind the previous linear theories are discussed with the help of numerical simulation. Numerical study is carried out to study the assumption taken by these linear theories that wide fingers transport large fluxes of heat and salt compared to thin finger. Simulations were run for the wide range of Rayleigh number 7×103 to 7×108 and it was observed that the flux ratio of the narrower fingers is more than that of the wide fingers, which is contrary to the assumptions taken by these linear theories. In the context of recent observations, which states that fingers growth rate, kinetic energy, evolution pattern, finger width etc. were demonstrated to be a strong function of Rayleigh numbers and weak dependence on Rρ, it would be difficult to predict finger characteristics from the previous linear theories, which are only the strong function of Rρ and the role of Rayleigh number is not considered. Therefore, a new theory has been developed for growth rate from the linearized governing equations with explicit dependence on Rayleigh numbers, density stability ratio, Schmitt number, Prandtl number and diffusivity ratio (of the components contributing to density of the system). Expressions for power-law equations of maximum growth rate are also derived as a function various non-dimensional parameters. The model predicts the finger characteristics reasonably well such as wavelength and fluxes reported by previous investigators. In the limit of parameter value approaching unity i.e. (RaT,Rρ → 1), the growth rate model collapse into a single curve and follow Schmitts theory (1979). The predicted results of the new theory corroborate well with the data reported from the field measurements, experiments and numerical simulations. Lately, a new finger regime has been observed in some experimental study, where finger convection occurs even for Rρ < 1. Moreover, linear theories also indicated the unusual behaviour of salt fingers at neutral buoyancy (Rρ = 1). But critical information such as convective structures and fluxes are still unknown as to what exactly happens at neutral buoyancy. There has been comprehensive study of salt finger convection at Rρ > 1 but scarce literature exist that explored finger behaviour at Rρ in large range of governing parameters. The intriguing behaviour of fingers at neutral buoyancy motivated us to further investigate it. In this thesis we present the unexplored finger convection numerically at various ranges of parameters where initial system is at Rρ = 1. We have solved numerically the partial differential equations governing the continuity of mass, momentum, energy and species in two dimension. We have considered a two-layer system similar to the laboratory setup. A series of simulations have been conducted in the heat-salt system at a fixed Rρ equal to one for Rayleigh numbers ranging from 7×103 to 7×108. It is noticed that salt fingers characteristics and evolution pattern vary drastically with the change in Rayleigh number. We have reported finger structures, mean profiles and kinetic energy at a wide range of Rayleigh number. To provide useful insight of the system, i.e. how fingers transport heat and salt fluxes vertically, the layer properties are investigated in details by analysing upper and lower layer of the system as the fingers run down. The ratio of heat to salt fluxes called flux ratio (Rf) has been studied at Rρ = 1 and the most common belief of scientific community that when Rρ → 1 =⇒ Rf → 1 has been investigated and it is observed that this condition is true only for high Rayleigh number. A new insight is developed on the finger behaviour at neutral buoyancy which is further investigated experimentally. Recent experimental study of Hage and Tilgner (2010) has revealed a surprising new finger regime where thin fingers form even for a small stabilizing temperature gradient, results in Rρ < 1, which is not considered as a usual domain for finger formation. Finger formation below neutral buoyancy has not been studied much in the past much. In this thesis, we have explored this new regime of finger convection numerically for the first time at a large range of governing parameters where initial system is at Rρ = 1/10. Finger formation below neutral buoyancy has not been studied much in the past. Finger evolution, flux ratio, kinetic energy, layer proper ties, density variation and Rρ variation with time is examined for this regime and some first-hand results are reported. At low Rayleigh number, initial layer proper ties of average concentration and temperature exceed the mean value. However, at high Rayleigh numbers, layer properties do not change as rapidly and remains near the initial layer properties. Moreover, thin fingers were seen only at high Rayleigh number however at low Rayleigh number, scarcely any fingers formed. Experimental study of double-diffusive salt finger convection is conducted in a Hele Shaw cell to investigate the effect of two most significant governing parameters, Rayleigh number and density stability ratio. Finger characteristics like structure, evolution time, growth rate and finger width at variable Rayleigh number and density stability ratio are investigated. The two layer system was formed using salt and sucrose solution(τ = 1/3). Rhoda mine-B was used as a dye for fluid visualization which was added to salt solution. Rayleigh number and density stability ratio was systematically changed one at a time by controlling the concentration of the solute. Experiments are conducted for a wide range of Rayleigh number, O(104)−O(106) , for Rρ = 1−1.5. Salt finger characteristics like finger structures, growth rate, finger width and evolution time is reported and it is observed that they strongly vary as a function of Rayleigh number. It is observed that salt finger behaviour observed at neutral buoyancy ration in experimental study matches with that of the numerical simulations. Bulbous tips are observed at neutral buoyancy ratio at high Rayleigh number in both experiments and numerical simulations. However, such behaviour is delayed at low Rayleigh number.