Numerical study of transition in the Rayleigh-B&#039 / enard problem of thermal convection between rigid plates heated from below under the influence of gravity with and without rotation is presented. The first numerical approach uses spectral element method with Fourier expansion for horizontal extent and Legendre polynomal for vertical extent for the purpose of generating a database for the subsequent analysis by using Karhunen-Lo&#039 / eve (KL) decomposition. KL decompositions is a statistical tool to decompose the dynamics underlying a database representing a physical phenomena to its basic components in the form of an orthogonal KL basis. The KL basis satisfies all the spatial constraints such as the boundary conditions and the solenoidal (divergence-free) character of the underlying flow field as much as carried by the flow database. The optimally representative character of the orthogonal basis is used to investigate the convective flow for different parameters, such as Rayleigh and Prandtl numbers. The second numerical approach uses divergence free basis functions that by construction satisfy the continuity equation and the boundary conditions in an expansion of the velocity flow field. The expansion bases for the thermal field are constructed to satisfy the boundary conditions. Both bases are based on the Legendre polynomials in the vertical direction in order to simplify the Galerkin projection procedure, while Fourier representation is used in the horizontal directions due to the horizontal extent of the computational domain taken as periodic. Dual bases are employed to reduce the governing Boussinesq equations to a dynamical system for the time dependent expansion coefficients. The dual bases are selected so that the pressure term is eliminated in the projection procedure. The resulting dynamical system is used to study the transitional regimes numerically. The main difference between the two approaches is the accuracy with which the solenoidal character of the flow is satisfied. The first approach needs a numerically or experimentally generated database for the generation of the divergence-free KL basis. The degree of the accuracy for the KL basis in satisfying the solenoidal character of the flow is limited to that of the database and in turn to the numerical technique used. This is a major challenge in most numerical simulation techniques for incompressible flow in literature. It is also dependent on the parameter values at which the underlying flow field is generated. However the second approach is parameter independent and it is based on analytically solenoidal basis that produces an almost exactly divergence-free flow field. This level of accuracy is especially important for the transition studies that explores the regions sensitive to parameter and flow perturbations.

High Rayleigh number(Ra) turbulent free convection has many unresolved issues related to the phenomenology behind the flux scaling, the presence of a mean wind and its effects, exponential probability distribution functions, the Prandtl number dependence and the nature of near wall structures. Few studies have been conducted in the high Prandtl number regime and the understanding of near wall coherent structures is inadequate for $Ra > 10^9$. The present thesis deals with the results of investigations conducted on high Rayleigh number turbulent free convection in the high Schmidt number(Sc) regime, focusing on the role of near wall coherent structures. We use a new method of driving the convection using concentration difference of NaCl across a horizontal membrane between two tanks to achieve high Ra utilising the low molecular diffusivity of NaCl. The near wall structures are visualised by planar laser induced fluorescence. Flux is estimated from transient measurement of concentration in the top tank by a conductivity probe. Experiments are conducted in tanks of $15\times15\times 23$cm (aspect ratio,AR = 0.65) and $10\times10\times 23$cm (AR = 0.435). Two membranes of 0.45$\mu$ and 35$\mu$ mean pore size were used. For the fine membrane (and for the coarse membrane at low driving potentials), the transport across the partition becomes diffusion dominated, while the transport above and below the partition becomes similar to unsteady non penetrative turbulent free convection above flat horizontal surfaces (Figure~\ref{fig:schem}(A)). In this type of convection, the flux scaled as $q\sim \Delta C_w ^{4/3}$,where $\Delta C_w$ is the near wall concentration difference, similar to that in Rayleigh - B\'nard convection . Hence, we are able to study turbulent free convection over horizontal surfaces in the Rayleigh Number range of $\sim 10^- 10 ^$ at Schmidt number of 602, focusing on the nature and role of near wall coherent structures. To our knowledge, this is the first study showing clear images of near wall structures in high Rayleigh Number - high Schmidt number turbulent free convection. We observe a weak flow across the membrane in the case of the coarser membrane at higher driving potentials (Figure \ref(B)). The effect of this through flow on the flux and the near wall structures is also investigated. In both the types of convection the near wall structure shows patterns formed by sheet plumes, the common properties of these patterns are also investigated. The major outcomes in the above three areas of the thesis can be summarised as follows \subsection* \label \subsubsection* \label The non-dimensional flux was similar to that reported by Goldstein\cite at Sc of 2750. Visualisations show that the near wall coherent structures are line plumes. Depending on the Rayleigh number and the Aspect ratio, different types of large scale flow cells which are driven by plume columns are observed. Multiple large scale flow cells are observed for AR = 0.65 and a single large scale flow for AR= 0.435. The large scale flow create a near wall mean shear, which is seen to vary across the cross section. The orientation of the large scale flow is seen to change at a time scale much larger than the time scale of one large scale circulation The near wall structures show interaction of the large scale flow with the line plumes. The plumes are initiated as points and then gets elongated along the mean shear direction in areas of larger mean shear. In areas of low mean shear, the plumes are initiated as points but gets elongated in directions decided by the flow induced by the adjacent plumes. The effect of near wall mean shear is to align the plumes and reduce their lateral movement and merging. The time scale for the merger of the near wall line plumes is an order smaller than the time scale of the one large scale circulation. With increase in Rayleigh number, plumes become more closely and regularly spaced. We propose that the near wall boundary layers in high Rayleigh number turbulent free convection are laminar natural convection boundary layers. The above proposition is verified by a near wall model, similar to the one proposed by \cite{tjfm}, based on the similarity solutions of laminar natural convection boundary layer equations as Pr$\rightarrow\infty$. The model prediction of the non dimensional mean plume spacing $Ra_\lambda^~=~\lambda /Z_w~=~91.7$ - where $Ra_\lambda$ is the Rayleigh number based on the plume spacing $\lambda$, and $Z_w$ is a near wall length scale for turbulent free convection - matches the experimental measurements. Therefore, higher driving potentials, resulting in higher flux, give rise to lower mean plume spacing so that $\lambda \Delta C_w^$ or $\lambda q^$ is a constant for a given fluid. We also show that the laminar boundary layer assumption is consistent with the flux scaling obtained from integral relations. Integral equations for the Nusselt number(Nu) from the scalar variance equations for unsteady non penetrative convection are derived. Estimating the boundary layer dissipation using laminar natural convection boundary layers and using the mean plume spacing relation, we obtain $Nu\sim Ra^$ when the boundary layer scalar dissipation is only considered. The contribution of bulk dissipation is found to be a small perturbation on the dominant 1/3 scaling, the effect of which is to reduce the effective scaling exponent. In the appendix to the thesis, continuing the above line of reasoning, we conduct an exploratory re-analysis (for $Pr\sim 1$) of the Grossman and Lohse's\cite scaling theory for turbulent Rayleigh - B\'enard convection. We replace the Blasius boundary layer assumption of the theory with a pair of externally forced laminar natural convection boundary layers per plume. Integral equations of the externally forced laminar natural convection boundary layer show that the mixed convection boundary layer thickness is decided by a $5^{th}$ order algebraic equation, which asymptotes to the laminar natural convection boundary layer for zero mean wind and to Blasius boundary layer at large mean winds. \subsubsection*{Effect of wall normal flow on flux and near wall structures} \label{sec:effect-wall-normal} For experiments with the coarser($35\mu$) membrane, we observe three regimes viz. the strong through flow regime (Figure~\ref{fig:schem}(b)), the diffusion regime (Figure \ref{fig:schem}(a)), and a transition regime between the above two regimes that we term as the weak through flow regime. At higher driving potentials, only half the area above the coarser membrane is covered by plumes, with the other half having plumes below the membrane. A wall normal through flow driven by impingement of the large scale flow is inferred to be the cause of this (Figure \ref{fig:schem}(b)). In this strong through flow regime, only a single large scale flow circulation cell oriented along the diagonal or parallel to the walls is detected. The plume structure is more dendritic than the no through flow case. The flux scales as $\Delta C_w^n$, with $7/3\leq n\leq 3$ and is about four times that observed with the fine membrane. The phenomenology of a flow across the membrane driven by the impingement of the large scale flow of strength $W_*$, the Deardorff velocity scale, explains the cubic scaling. We find the surprising result that the non-dimensional flux is smaller than that in the no through flow case for similar parameters. The mean plume spacings in the strong through flow regime are larger and show a different Rayleigh number dependence vis-a-vis the no through flow case. Using integral analysis, an expression for the boundary layer thickness is derived for high Schmidt number laminar natural convection boundary layer with a normal velocity at the wall. (Also, solutions to the integral equations are obtained for the $Sc\sim 1$ case, which are given as an Appendix.) Assuming the gravitational stability condition to hold true, we show that the plume spacing in the high Schmidt number strong through flow regime is proportional to $\sqrt{Z_w\,Z{_{v_i}}}$, where $Z{_{v_i}}$ is a length scale from the through flow velocity. This inference is fairly supported by the plume spacing measurements At lower driving potentials corresponding to the transition regime, the whole membrane surface is seen to be covered by plumes and the flux scaled as $\Delta C_w^{4/3}$. The non-dimensional flux is about the same as in turbulent free convection over flat surfaces if $\frac{1}{2}\Delta C $ is assumed to occur on one side of the membrane. This is expected to occur in the area averaged sense with different parts of the membrane having predominance of diffusion or through flow dominant transport. At very low driving potentials corresponding to the diffusion regime, the diffusion corrected non dimensional flux match the turbulent free convection values, implying a similar phenomena as in the fine membrane. \subsubsection*{Universal probability distribution of near wall structures} \label{sec:univ-prob-distr} We discover that the probability distribution function of the plume spacings show a standard log normal distribution, invariant of the presence or the absence of wall normal through flow and at all the Rayleigh numbers and aspect ratios investigated. These plume structures showed the same underlying multifractal spectrum of singularities in all these cases. As the multifractal curve indirectly represents the processes by which these structures are formed, we conclude that the plume structures are created by a common generating mechanism involving nucleation at points, growth along lines and then merging, influenced by the external mean shear. Inferring from the thermodynamic analogy of multifractal analysis, we hypothesise that the near wall plume structure in turbulent free convection might be formed so that the entropy of the structure is maximised within the given constraints.

Natural turbulent convection studies encompass a wide range of flows occurring in nature, for example, atmospheric and oceanic flows, con-vection in the Earth’s mantle, convection in the stars and also in many engineering applications. Rayleigh-Benard convection (RBC), i.e. con-vection in a horizontal fluid layer confined between two plates with a temperature differential maintained across them, has been a proto-type problem in the studies of turbulent natural convection. Many small scale and global features of the flow in the turbulent regime of RBC are known, yet the flow dynamics is not fully understood, es-pecially at high Rayleigh numbers (Ra). Present work comprises of experimental investigations of a different type of flow, high Rayleigh number turbulent convection in a long vertical tube (abbreviated as tube convection or TC). The tube of aspect ratio (length to diameter) of about 10, open at both the ends interconnects two large tanks. The flow driven by an unstable density difference created between the two tanks, has some unique features, different from RBC. The net flow at any tube cross-section is zero and the time averages of the velocities, the Reynolds shear stress and the mean shear are also zero. Turbu-lent energy production is therefore solely due to buoyancy. The flow is axially homogeneous and axisymmetric. In the homogeneous region, the mean density gradient is linear. Rayleigh number in TC is conve-niently defined based on the mean (linear) density gradient (denoted by Rag). Two sets of experiments are carried out. In one set of experiments, the density difference is created using brine and fresh water and in another set, it is created using heat. The ranges of Rag achieved are 3 × 108 < Rag < 8 × 109 in the experiments using salt (Schmidt number, Sc ≈ 600) and 5 × 104 < Rag < 5 × 106 in the experiments using heat (Prandtl number, P r ≈ 6). From the measured salt and heat fluxes in both the sets of experiments, the non dimensional flux 1 1 scaling above a certain value of Rag is obtained as N ug ∼ Rag2 P r 2 and from the velocity measurements in the experiments using salt, the 1 Reynolds number scaling is obtained as Re ∼ Rag2 P r− 12 . Both these are as per the predicted scalings by the mixing length model proposed by Arakeri et al. (2000) for high Rag convection in the vertical tube. The flux scaling N u ∼ (RaP r)2 , also known as the ‘ultimate regime’ of convection, expected at very high Ra but not yet observed in the experiments in classical RBC, is easily achieved in TC at relatively lower values of Ra. The fluxes and Reynolds numbers in TC are orders of magnitude higher as compared to those obtained in RBC for similar values of Ra and P r. In the lower range of Rag values for P r ≈ 6, a transition to a new flux scaling, N u ∼ (RaP r)0.29 is found. Similar transitions are also found to be present in the results of Tovar (2002) for Sc ≈ 600 and in the DNS results of Schmidt et al. (2012) for P r = 1, at different values of Rag. Collecting all these data, it is shown that the transition occurs at a fixed Grashof number of 1.6 × 105, independent of P r. Velocity measurements are carried out using particle image velocime-try (PIV) in the salt experiments. Kinetic energy spectra computed from the velocity fields are presented for the locations from the tube axis to the wall, for the lowest and the highest values of Rag achieved in the experiments. The spatial energy spectrum of lateral velocity at the tube axis follows Kolmogorov-Obukhov (KO) scaling (−5/3 scaling exponent) while the spatial spectrum of longitudinal velocity shows a scaling slightly higher than −5/3 but lower than −11/5 (the Bolgiano-Obukhov (BO) scaling). The scalar spectra is computed from the concentration fields obtained from planar laser induced fluorescence (PLIF) in the experiments using salt, and also from the temperature measurements from the experiments using heat. Both the concentra-tion and temperature fluctuations spectra show some evidence of dual scaling - BO scaling (−7/5 scaling exponent) in the inertial subrange followed by Obukhov-Corrsin (OC) scaling (−5/3 scaling exponent) over a narrow range of scales. Light propagation through the buoyancy driven turbulent flow in TC has also been experimentally investigated. Light propagation through convective turbulence is encountered in many situations. In some cases e.g. in observational astronomy it is undesirable, while in some other cases it is useful, e.g. in remote sensing of meteorological parameters. In the present study, light intensity and angle of arrival fluctuations in a parallel beam of light are measured. Laser shadowgraphy is used in the intensity measurements while the angle of arrival is obtained by measuring deflections of narrow laser beams, created by passing collimated laser light through a mask having equispaced grid of holes. Background oriented schlieren (BOS) measurements have also been carried out to obtain the displacements, which are proportional to the angle of arrivals. The equations for frequency spectrum of intensity and angle of arrival from the literature, developed for isotropic, ho-mogeneous turbulent media, are modified for the flow in the present case and the asymptotic scalings for high and low frequency ranges are obtained. The scalings in the frequency spectra computed from the measurements of intensity and angle of arrival fluctuations are com-pared with the obtained asymptotic scalings. The results from the present work are also compared with results from studies in the atmo-sphere and lab experiments.