The main focus of the PhD research has been to understand the dynamics of double-diffusive finger convection and investigate the role of various parameters that govern the finger structures, velocity, fluxes and their interdependence. Double-diffusive convection occurs in the presence of two components with different molecular diffusivity. When the faster diffusing component (T) stabilizes the system (dρ/dz < 0) and the slower diffusing component (S) makes the system unstable (dρ/dz > 0), with overall stratification remaining gravitationally stable (dρ/dz < 0), the convection takes the form of alternatively rising and sinking cells called ‘salt fingers’. ρ is the total density of the fluid. In this thesis, results from the numerical simulations, analytical and laboratory investigations of the dynamics of finger convection has been presented for a wide ranges of governing parameters.
In this work, 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 close to one (= 1.001) for Rayleigh numbers ranging from 7 × 10^3 to 7 × 10^8. We have identified many interesting features which has not been reported previously: (a) at high thermal Rayleigh number (RaT ), where thin fingers evolve, the salinity (S) and temperature (T ) fields are not distinguishable from each other, (b) at low RaT , wide fingers evolve, and the effect of molecular diffusion is observed in the T fields only. As the system moves from being advection dominated at high RaT to diffusion dominated at low RaT, velocity in the fingers decreases substantially.
We have demonstrated that Rayleigh number plays a significant role in determining the initial instability that develops at the interface and controls the dynamics of finger convection. Simulations were carried out for Rρ ranging from 1.001 to 10 and for the ranges of RaT from 10^4 to 10^9 . The density stability ratio is shown to be less important in controlling the instability compared to Rayleigh number. This result is in contrast to the previous investigations that has given significant weight to the only non-dimensional parameter, namely Rρ. We find that the convection onset time, t0 varies as t0 ~ RaS^(-3/5) . Before and after the onset of instability, we observed the two important features: (a) dS /dt < dT /dt , before the onset and, (b) dS /dt > dT /dt ,after the onset. Here S and T are the time averaged temperature and salinity in each layer. These conditions are satisfied in each layer for all values of Rρ and RaT in the fingering regime covered in the present studies.
Effect of Prandtl number on the finger system has been studied both numerically and with the help of laboratory experiments. Increase in Pr of the system was achieved by increasing the viscosity of the fluid (ν). Evolution of the salt finger structures were studied numerically for three cases: Pr = 7, Pr = 7 × 50 and Pr = 7 × 400. Simulating higher Pr system was computationally expensive. We observed interesting transition of finger system from low Pr to high Pr. At fixed Rρ , thin fingers evolve initially and later, strong convecting regions developed above and below the finger zone in the low Pr system. This limits the growth of the fingers. As Pr increases, wide fingers evolve and the sandwich structures observed at low Pr slowly disappears. This allows the fingers to penetrate deep into the reservoirs. Salt fingers in high Pr system never reach equilibrium. In laboratory experiments, Pr of the system was increased by adding carboxy-methyl cellulose (CMC) to distilled water. CMC increases the viscosity of the fluid exponentially without affecting the density of the fluid. Experiments were performed in salt-sugar and heat-sugar systems. We observed features that were similar to the numerical simulations at high and low Pr. Observations from laboratory and numerical simulations were extrapolated to suggest a new model for the formation of columnar basalt structures. (abstract in pdf)
The Project
FORTRAN 77 and Matlab code developed
FORTRAN 77 code for simulating non-dimensional continuity, conservation momentum, energy and concentration equation (Navier-Stokes equation)
Transient simulation, finite volume method (FVM) along with SIMPLER algorithm was used to solve these equations
Developed an experimental set-up for flow visualization and Particle Image Velocimetry (PIV)
Code validation with published literature and experiments
Developed analytical and theoretical model
Key results
Developed a new theory on relationship with convective finger width, velocity, heat and concentration fluxes variation with Rayleigh numbers and density stability ratio.
Double-diffusive finger convection is a two-parameter system and it cannot be described by a single non-dimensional parameter density stability ratio R_rho as reported by previous investigators.
The two parameters are thermal (or salinity) Rayleigh number and density stability ratio
Finger wave-number, heat and concentration fluxes, flux ratio, finger velocity, onset time to instability etc. are a strong function of Rayleigh numbers and weak function of density stability ratio.
Density stability ratio (R_rho) decrease first and then increase for all values of Rayleigh numbers contrary to the popular notion that R_rho monotonically increase from the beginning.
Mixed layer or step like profile formation due strong horizontal convection in the layers is due to high Rayleigh numbers and not R_rho alone as reported by previous investigators.
Above equations are were solved numerically to simulation double-diffusive salt finger convection.
(a) Schematic diagram showing the conﬁguration of heat and salt in a two layer system that leads to the formation of salt ﬁngers. Excess ∆T and ∆S overlies relatively cold and fresh water. Due to the faster diﬀusion of temperature at the interface, potential energy stored in the unstable salinity ﬁeld is released via ﬁngers. (b) Heat diﬀuses rapidly down the temperature gradient (shown by gray arrows) separating ﬁngers, generating local density anomalies which drive vertical motion with velocity, W, and horizontal wavelength f = 2*pi/k, where k is the horizontal wavenumber.
Variation of flux ratio vs R_rho reported by various investigators. If flux ratio is a function of R_rho alone (as advocated by previous researchers) then why there is so much scatter in the plot? My research demonstrated that another important parameter called Rayleigh number has a strong influence on the flux ratio, which previous investigators overlooked.
Transition from non-linear connection to linear salt finger convection as Rayleigh number decreases (Source: Effect of Rayleigh numbers on the evolution of double‐diffusive salt fingers, 2014, O. P. Singh, J. Srinivasan, Phys. Fluids, 26(6), pp. 1-18)
Velocity field in fingers at high and low Rayleigh numbers at fixed buoyancy ratio (source: Effect of Rayleigh numbers on the evolution of double‐diffusive salt fingers, 2014, O. P. Singh, J. Srinivasan, Phys. Fluids, 26(6), pp. 1-18)
Double diffusive finger convection overturn at low Rayleigh numbers (Source: Effect of Rayleigh numbers on the evolution of double‐diffusive salt fingers, 2014, O. P. Singh, J. Srinivasan, Phys. Fluids, 26(6), pp. 1-18)
Variation of density stability ratio R_ρ (t) as function of time at different Rayleigh numbers for initial density stability ratio R_ρ0=2 and R_ρ0=10. Note that R_ρ (t) decreases along the same curve up to the onset time after which it commences to increase. (Source: Effect of Rayleigh numbers on the evolution of double‐diffusive salt fingers, 2014, O. P. Singh, J. Srinivasan, Phys. Fluids, 26(6), pp. 1-18)
Transient variation of dR_ρ/dt at various Rayleigh numbers at R_ρ0=2 (see legend). The slope changes from negative to positive for all 〖Ra〗_T (Source: Effect of Rayleigh numbers on the evolution of double‐diffusive salt fingers, 2014, O. P. Singh, J. Srinivasan, Phys. Fluids, 26(6), pp. 1-18)
CFD simulation of double diffusive salt fingers: evolution of concentration (left panel) and the corresponding temperature fields (right panel) at a fixed buoyancy ratio. Thermal Rayleigh number, Ra = 7e+6 (Source: On the relationship between ﬁnger width, velocity, and ﬂuxes in thermohaline convection, 2009, K. R. Sreenivas, O. P. Singh & J. Srinivasan, Phys. Fluids, 21(2), pp. 026601)
CFD simulation of salt fingers: Evolution of salinity fields indicating the formation of fingers after the onset of instability at the initial interface for various Rayleigh numbers at a fixed buoyancy ratio (Source: On the relationship between ﬁnger width, velocity, and ﬂuxes in thermohaline convection, 2009, K. R. Sreenivas, O. P. Singh & J. Srinivasan, Phys. Fluids, 21(2), pp. 026601)
Fingal cave: columnar basalt fingers; is the formation due to double-diffusive instability?
Experimental observation: Wide fingers evolve with high viscosity similar to the basaltic fingers shown above.
Velocity field comparison with simulation and PIV experiment
Double-diffusive finger system selects optimum scale: numerical and PIV experiment comparison
"Mistakes are a part of being human. Appreciate your mistakes for what they are: precious life lessons that can only be learned the hard way. Unless it's a fatal mistake, which, at least, others can learn from." -- Al Franken.