Hydrodynamic Instability

This thesis deals with the dynamics of irreversible processes within the context of the general theory of relativity. In particular, we address the problem of the 'infinite' speed of propagation of thermal disturbances in a dissipative fluid. The present work builds on the multi-fluid variational approach to relativistic dissipation, pioneered by Carter, and provides a dynamical theory of heat conduction. The novel property of such approach is the thermodynamic interpretation associated with a two-fluid system whose constituents are matter and entropy. The dynamics of this model leads to a relativistic generalisation of the Cattaneo equation; the constitutive relation for causal heat transport. A comparison with the Israel and Stewart model is presented and its equivalence is shown. This discussion provides new insights into the not-well understood definition of a non-equilibrium temperature. The variational approach to heat conduction presented in this thesis constitutes a mathematically promising formalism to explore the relativistic evolution towards equilibrium of dissipative fluids in a dynamical manner and to get a deeper conceptual understanding of non-equilibrium thermodynamic quantities. Moreover, it might also be useful to explore the more fundamental issues of the irreversible dynamics of relativity and its connections with the time asymmetry of nature.

The dynamics of a three-dimensional axisymmetric bluff-body wake are examined at low Reynolds regimes where transitions take place through spatio-temporal symmetry breaking. A linear stability analysis is employed to identify the critical Reynolds number associated with symmetry breaking, and the associated eigenmodes, known as global modes. The analysis shows that the axisymmetric stable base flow breaks the rotational symmetry through a pitchfork m = 1 bifurcation, in agreement with previously reported results for axisymmetric wakes. Above this threshold, the stable base flow is steady and three-dimensional with planar symmetry. A three-dimensional global stability analysis around the steady reflectionally symmetric base flow, assuming no homogeneous directions , predicts accurately the Hopf bifurcation threshold, which leads to asymmetric vortex shedding. DNS simulations validate the stability results and characterize the flow topology during the early chaotic regime.

Turbulence is one of the most ubiquitous phenomena in all of fluid mechanics. In the ocean, no less than in the atmosphere, is its influence widespread. For example, the interplay between the ocean stratification and the diffusive turbulent motions is often crucial in determining the structure of each. Again, the response of the ocean to large-scale wind and thermal disturbances and the development of ocean currents is dependent on the transfer of matter, momentum and energy by irregular smaller scale motions of one kind or another.
Not all of the random motions found in the ocean, however, can be described properly as turbulence. The characteristic properties of turbulent motions are that they possess a random distribution of vorticity in which there is no unique relation between the frequency and wave number of the Fourier modes; that they are diffusive and dissipative. A distinction is drawn between turbulence in a stably stratified fluid on the one hand and random field of internal gravity waves on the other. This differentiation is useful, not only conceptually but also observationally since the mechanisms of energy transfer (in both physical and Fourier space) are essentially different.
When the wind blows across the surface of the water, a tangential surface stress is developed both directly from the interfacial stress, and indirectly by the rate of momentum loss from the surface waves by such processes as wave breaking. Below the surface, a turbulent mixed layer develops. If the underlying region is statically stable or neutral, the interface between turbulent and non-turbulent fluid is very sharp, and remains so as the turbulence erodes the lower fluid by entrainment. The temperature and salinity in the mixed layer are both virtually uniform as a result of turbulent diffusion, and the continued erosion results in an increasing contrast between the properties of the water in the mixed layer and that immediately below. In this way, a thermocline develops. If, on the other hand, there is substantial surface cooling, there may be a region below the mixed layer in which the density decreases with depth and which is statically unstable. Convective motions can develop and, particularly in polar waters, may extend to considerable depths.
Turbulence in the oceans can also be generated in more familiar ways, such as in shear zones produced by the confluence of different water messes, particularly in estuaries. Turbulence does not invariably occur, however; the stratification may be sufficiently great that the Richardson number is large enough everywhere to prevent the onset of dynamical instability.
Turbulent mixing, and especially turbulent mixing in a density stratified fluid, is a difficult problem in geophysical fluid mechanics, as well as in environmental and industrial studies. Of particular interest is the mixing in the neighbourhood of relatively sharp density interfaces in the oceans, lakes, reservoirs, and the atmosphere. In environmental fluid mechanics, an understanding of these mixing processes is essential in determining water quality in water bodies containing a sharp density interface. The presence of such an interface, by suppressing fluid turbulence, limits the downward transfer of pollutants and gases such as CO2 and O2 from the upper mixed layer. This has important consequences in the design of wastewater and thermal reservoirs. In the atmosphere serve air pollution problems may result from the presence of thermal inversions that inhibit the vertical transport of pollutants. Her again, a knowledge of the exchange processes across the inversion layer is important in predicting and controlling air quality in the atmosphere.
The information that can be obtained on mixing in a stratified fluid is essential for the development of computer models of geophysical phenomena, because, if better predictions are to be made, the distributions of potential and kinetic energy have to be correctly assessed for each process under study.

Finite-amplitude convective steady flows that do not bifurcate from the respective conduction state are discovered. They arise as the compensating horizontal gradients of two density-affecting components with equal diffusivities but different boundary conditions are applied to the Boussinesq fluid at rest with and without stable vertical stratification. These flows emanate from convection in a laterally heated stably stratified slot. Their relevance to convective states in a horizontal slot with two vertical gradients, emphasizing universality of the underlying type of convection, is discussed.

Hydro-geology (hydro- meaning water, and -geology meaning the study of the Earth) is the area of geology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust (commonly in aquifers). The term hydrology is often used interchangeably. Some make the minor distinction between a hydrologist or engineer applying themselves to geology (hydrology), and a geologist applying themselves to hydrology (hydro geology).Hydrogeology is an interdisciplinary subject; it can be difficult to account fully for the chemical, physical, biological and even legal interactions between soil, water, nature and society. The study of the interaction between groundwater movement and geology can be quite complex. Groundwater does not always flow in the subsurface down-hill following the surface topography; groundwater follows pressure gradients (flow from high pressure to low) often following fractures and conduits in circuitous paths. Taking into account the interplay of the different facets of a multi-component system often requires knowledge in several diverse fields at both the experimental and theoretical levels. The following is a more traditional introduction to the methods and nomenclature of saturated subsurface hydrology, or simply the study of ground water content.Hydrogeology, as stated above, is a branch of the earth sciences dealing with the flow of water through aquifers and other shallow porous media (typically less than 450 m or 1,500 ft below the land surface.) The very shallow flow of water in the subsurface (the upper 3 m or 10 ft) is pertinent to the fields of soil science, agriculture and civil engineering, as well as to hydrogeology. The general flow of fluids (water, hydrocarbons, geothermal fluids, etc.) in deeper formations is also a concern of geologists, geophysicists and petroleum geologists. Groundwater is a slow-moving, viscous fluid (with a Reynolds number less than unity); many of the empirically derived laws of groundwater flow can be alternately derived in fluid mechanics from the special case of Stokes flow (viscosity and pressure terms, but no inertial term). The mathematical relationships used to describe the flow of water through porous media are the diffusion and Laplace equations, which have applications in many diverse fields. Steady groundwater flow (Laplace equation) has been simulated using electrical, elastic and heat conduction analogies. Transient groundwater flow is analogous to the diffusion of heat in a solid, therefore some solutions to hydrological problems have been adapted from heat transfer literature.Traditionally, the movement of groundwater has been studied separately from surface water, climatology, and even the chemical and microbiological aspects of hydrogeology (the processes are uncoupled). As the field of hydrogeology matures, the strong interactions between groundwater, surface water, water chemistry, soil moisture and even climate are becoming more clear.For example: Aquifer drawdown or overdrafting and the pumping of fossil water may be a contributing factor to sea-level rise

Transient laminar opposing mixed convection in a gravity driven downward flow confined inside a vertical rectangular channel has been investigated, with both walls suddenly subjected to symmetrical isothermal heat sources over a finite portion of the channel walls. The unsteady two-dimensional Navier-Stokes and energy equations have been solved numerically for a wide parametric set. Studies are carried out for Reynolds numbers of 100 and 200 and several values of buoyancy strength or Richardson number. The effect of Reynolds number and opposing buoyancy on the temporal evolution of the overall flow structure, temperature field, and Nusselt number from the heated surfaces is investigated using fixed geometrical parameters and considering heat losses to the channel walls. In this parameter space, for a given Reynolds number and relatively small values of the buoyancy parameter, the transient process leads to a final symmetric or asymmetric steady-state. However, as the value of buoyancy strength increases, the flow and temperature fields become more complex and an oscillatory flow with a fundamental frequency sets in when a critical value of the Richardson number is reached. Numerical predictions show that the critical value of the Richardson number between the two regimes strongly depends on the value of the Reynolds number, and the time scales, natural frequencies, and phasespace portraits of flow oscillation are presented and discussed in detail. Stability of the symmetric response has been analyzed. The results include the effects of Prandtl number and heat losses to the channel walls on the evolution of the final flow and thermal responses.

The paper presents an experimental study of the capillary electrostatic instability occurring under effect of a constant electric field on a magnetic fluid individual peak. The peaks under study occur at disintegration of a magnetic fluid layer applied on a flat electrode surface under effect of a perpendicular magnetic field. The electrocapillary instability shows itself as an emission of charged drops jets from the peak point in direction of the opposing electrode. The charged drops emission repeats periodically and results in the peak shape pulsations. It is shown that a magnetic field affects the electrocapillary instability occurrence regularities and can stimulate its development. The critical electric and magnetic field strengths at which the instability occurs have been measured; their dependence on the peak size is shown. The hysteresis in the system has been studied; it consists in that the charged drops emission stops at a lesser electric (or magnetic) field strength than that of the initial occurrence. The peak pulsations frequency depending on the magnetic and electric field strengths and on the peak size has been measured.

This work is concerned with the investigation of fluid-mechanical behaviour and the performance of different subgrid-scale models for LES in the numerical prediction of a confined axisymmetrical bluff-body flow. Four subgrid-scale turbulence models comprising the Smagorinsky model, Dynamic Smagorinsky model, WALE model and subgrid turbulent kinetic energy model, are validated and compared directly against the experimental data. Two different mesh counts are used for the LES studies, one with a higher mesh resolution in the shear layer than the other. It is found that increasing the mesh resolution improves the time-averaged fluctuating velocity profiles, but has less effect on the time-averaged filtered velocity profiles. A comparison against experiment shows that the recirculation zone length is well predicted using LES. The accuracy of the four different subgrid scale models is then assessed by comparing the LES results using the dense mesh with the experiment. Comparisons with the time-averaged axial and radial velocity profiles demonstrate that LES displays good agreement with the experimental data, with the essential flow features captured both qualitative and quantitatively. The subgrid velocity also matches well with the experimental results, but a slight underprediction of the inner shear layer is observed for all subgrid models. In general, it is found that the Smagorinsky and WALE models are more dissi-pative than the Dynamic Smagorinsky model and subgrid TKE model. Comparison of the spectra against the experiment shows that LES can capture dominant features of the turbulent flow with reasonable accuracy, and weak spectral peaks related to the Kevin-Helmholtz instability and helical vortex shedding are present.

The Correlates of War project scrupulously collects information about disputes between the countries over a long historical period together with other data relevant to the character and reasons of international conflicts. Using methods of modern Data Science implemented in the R statistical software, we investigate the datasets from the project. We study political, economic, and religious factors with respect to the emergence of conflicts and wars between the countries. The results obtained lead to certain conclusions about variances and causalities between the factors considered. Some unpredictable features are presented.

[1] This is a computational study of gravity-driven fingering instabilities in unsaturated porous media. The governing equations and corresponding numerical scheme are based on the work of Nieber et al. (2003) in which nonmonotonic saturation profiles are obtained by supplementing the Richards equation with a nonequilibrium capillary pressure-saturation relationship, as well as including hysteretic effects. The first part of the study takes an extensive look at the sensitivity of the finger solutions to certain key parameters in the model such as capillary shape parameter, initial saturation, and capillary relaxation coefficient. The second part is a comparison to published experimental results that demonstrates the ability of the model to capture realistic fingering behavior.

In a number of studies concerning the lateral heating of a vertical rectangular enclosure containing a stably stratified brine solution, diverse flow regimes have been observed for different values of the buoyancy ratio. In this Brief Communication, certain intervals of the buoyancy ratio have been revealed to possess multiple steady flows. The presence of the multiple solutions was demonstrated to be underlain by complex bifurcation phenomena including numerous limit points and the symmetry-breaking bifurcation. The results appear to be insightful for the understanding of various findings of previous works.

A stable gradient of one density-affecting component having Dirichlet conditions at two horizontal boundaries of an infinite slot or a finite enclosure is considered in combination with an unstable gradient of another such a component having the Neumann boundary conditions. Even when the component diffusivities are equal, steady instability driven by the different component boundary conditions arises when the overall background stratification is statically stable. Its nature is demonstrated to be analogous to that of the finger instability in conventional double-diffusive convection. Potentially relevant to some large-scale oceanographic applications, unequal vertical and horizontal (eddy) diffusivities are also examined. Eventually, these give rise to a multilayer convection pattern.

As has recently been reported by Tsitverblit and Kit [Phys. Fluids A 5, 1062 (1993)], a vertical rectangular enclosure containing stably stratified brine and differentially heated from its side walls is characterized by complex steady bifurcation phenomena. In the present work, the structure of steady solutions in the enclosure has been studied in detail for several values of the salinity Rayleigh number, Ra_S , fixed near the commencement of the double-diffusive region. It was found that when the thermal Rayleigh number, Ra_T, is either very small or sufficiently large, the steady solution is unique while in an intermediate region of this parameter, there exists a great variety of the multiple steady flows, being the result of nondegenerate hysteresis points and isolas of asymmetric solutions forming as Ra_S, is increased. In particular, at the maximal value of Ra_S considered there have been observed symmetric and asymmetric one-, two-, three-, four-, and five-cell flows. Despite the multiplicity of the flow patterns, a critical interval of the buoyancy ratio has been distinguished, above and below which the generic characteristics of the steady solutions were found to resemble the respective features of the " successive " and " simultaneous " regimes of layer formation whose existence was established in previous studies. Although the set of the steady solutions has been found to contain no linearly stable multicell flows, the perturbation was so long retained in the close proximity of the unstable steady solutions that such flows could be easily observable in the experiment. In spite of the appreciably different range of the Rayleigh numbers, the physically meaningful parameters suggested in previous studies were found to be represented in the present results.

Double-component instabilities and the resulting multiple equilibrium states arising in laterally heated stably stratified fluid systems are briefly reviewed from the perspective of the roles of unequal diffusivities and different sidewall boundary conditions for heat and solute played in the nature of these phenomena.

Onset of small-amplitude oscillatory and both small-and finite-amplitude steady double-component convection arising due to component different boundary conditions in an infinite slot is studied for various slot orientations to the gravity. The main focus is on two compensating background gradients of the components. The physical mechanisms underlying steady and oscillatory convection are analyzed from the perspective of a universally consistent understanding of the effects of different boundary conditions. In a horizontal slot with inviscid fluid addressed by Welander [P. Welander, Tellus Ser. A 41 (1989) 66], oscillatory convection sets in with the most unstable wave number and oscillation frequency being zero. Exact expressions for the critical fixed-value background gradient and the respective group velocity at zero wave number are derived from the long-wavelength expansion both for the horizontal slot with independently varying background gradients and for the inclined slot with the compensating gradients. In the horizontal slot with viscous fluid, the dissipa-tion of along-slot perturbation-cell motion reduces efficiency of the oscillatory instability feedback and thus prevents the most unstable wavelength from being infinite. Based on this interpretation, the oscillatory instability of a three-dimensional (3D) nature is predicted for an interval of long two-dimensional (2D) wavelengths in an inclined slot, and such 3D instability is indeed shown to arise. Related general conditions for three-dimensionality of most unstable disturbances are also formulated. As the slot orientation changes from the horizontal by angle h (Pp/2), the oscillatory 2D marginal-stability boundaries in inviscid and viscous fluid are expected to eventually transform into respective steady ones. Oscillatory instability in the vertical slot with viscous fluid, first reported by Tsitverblit [N. Tsitverblit, Phys. Rev. E 62 (2000) R7591], is of a quasi-steady nature. Its (new) mechanism is identified. It is underlain by differential gradient diffusion. As the horizontal slot at θ = π, addressed by Tsitverblit [N. Tsitverblit, Phys. Fluids 9 (1997) 2458], changes its orientation to vertical, the wave number interval of linear steady instability shrinks to the vicinity of the most unstable zero wave number and vanishes. Consistently with the basic nature of finite-amplitude steady convection being the same in the horizontal and vertical slots, the respective convective flows are continuously transformed into each other. The dissimilarity between the nature of finite-amplitude steady convective flows in the horizontal slot with θ = 0, revealed by Tsitverblit [N. Tsitverblit, Phys. Lett. A 329 (2004) 445], and that in the vertical slot is shown to eventually give rise to a region of hysteresis in θ ∈ (0, π/2).

Compaction–dissolution waves in porosity and melt pressure form spontaneously in numerical simulations of melt migration in an upwelling, viscously compacting, porous column in a solubility gradient. The melt fraction is assumed to be small and the solid comprises olivine and orthopyroxene. The solubility of orthopyroxene in the melt is assumed to increase linearly with height and induces a gradient reaction, assumed to be at local equilibrium. Approximations for the vertical, 1-D, steady-state solutions are derived assuming negligible resistance to compaction. The linear stability of the steady-state solutions is characterized by complex eigenvalues and an oscillatory instability with strong wavenumber selection. This instability leads to the formation of checkerboard compaction–dissolution waves observed in the non-linear numerical simulations. The phase velocity of these waves is larger than the solid velocity but smaller than the melt velocity. The oscillatory instability is realized over a range of parameters and the variation in wave properties is explored. A power-law bulk-viscosity formulation, ξ = η/φ m f , is shown to decrease growth rates linearly in the exponent, m. For small perturbations, the growth rates and phase velocities measured from high-resolution numerical simulations are predicted by the linear theory as well as the dominant wavenumber in the non-linear regime. We present a regime diagram for reaction infiltration instabilities in viscously compacting porous media and show that compaction–dissolution waves are favored by increasing solid upwelling and small solubility gradients relative to high-porosity channels. The regime diagram suggests that the formation of compaction–dissolution waves is a feasible new physical mechanism for melt transport beneath mid-ocean ridges.

This paper presents the research methodology followed at the von Karman Institute (VKI) to study jet wiping. In this process, a strip is withdrawn from a bath of liquid and then wiped by impinging gas jets to control the final coating thickness. The methodology developed at VKI aims to capture the physical mechanism responsible for surface wiggles limiting the product quality. At the scope, the jet process is analysed with a set of simplified scenarios, each of which providing an insight on the phenomenon from a different perspective.

We developed a numerical setup to simulate swirling jet flow undergoing vortex breakdown. Our simulation code CONCYL solves the compressible Navier-Stokes equations in cylindrical coordinates using high-order numerical schemes. A nozzle is included in the computational domain to account for more realistic inflow boundary conditions. Preliminary results of a Re = 5000 compressible swirling jet at Mach number M a = 0.6 with an azimuthal velocity as high as the maximum axial velocity (swirl number S = 1.0) capture the fundamental characteristics of this flow type: At a certain point in time the jet spreads and develops into a conical vortex breakdown. A stagnation point-flow in the vicinity of the jet axis is clearly visible with the stagnation point located close to the nozzle exit. The stagnation point precesses in time around the jet axis, moving up- and downstream. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The first subject treated in this thesis is bifurcation phenomena in laterally heated thermosolutal systems. They are considered in the context of explaining previous diverse-flow observations in such systems. The second subject is the absence of a relation between vortex breakdown in such confined flows as a cylindrical container
and an annulus with flow instabilities in these geometries, particularly with the axisymmetric oscillatory instability. A more detailed abstract is provided in the full text itself.

The steady axisymmetric Navier-Stokes equations for the flow in a cylindrical container and in an annulus have been solved by the continuation method, being a standard tool of numerical methods for bifurcation problems. By alteration of the boundary condition for vorticity at the axis, the transition of the flow structure, including the separation bubble, being present in a conventional container, to that in an annulus with an infinitesimal inner radius was proceeded. By further continuation in the ratio between inner and outer radii the flow pattern in the annulus was computed. It was shown that even for very high values of this ratio a recirculation region occurred at the inner wall of the annulus. The transition to a nonsteady motion through supercritical Hopf bifurcation was studied by following the time development of a small axisymmetric disturbance introduced to the full or linearized system of equations.

We study the effects of zero-mean stochastic gravity modulation on the mixing characteristics of two miscible fluids initially separated by a thin diffusion layer. Both correlated and uncorrelated gravity modulation are considered and the 2D time-dependent Boussinesq equations are solved numerically. The flow is characterized by a Grashof number, Gr=δρρgl^ 3ν&2 circ;, based on the viscous length scale, lν=√ν/φ, where φ is a frequency defined through the power spectrum; the Schmidt number, Sc; and geometric aspect ratios. We vary Gr, holding the other parameters constant, and study the evolution of the interface. For both correlated and uncorrelated jitter, the ensemble averages of realizations exhibit dispersive spreading of the interface at a rate which is amplified by Gr, and which is larger for correlated jitter. Individual realizations exhibit the formation of folds, which are more pronounced in the case of correlated jitter. Many of the phenomena occurring for deterministic jitter also occur in the stochastic case, but at a lower equivalent Gr. Accordingly, the rate of mixing for stochastic jitter is higher than that for deterministic harmonic jitter.

The article presents a liquid film instability model designed using results of the set of CFD simulations. The governing equations of the model are derived using a linear equation of motion. The stability analysis is carried out by imposing a liquid surface disturbance which growth rate is investigated in dependence on the geometrical and physical configuration. The gas effect parameters, which are decisive variables in the model, are derived using results of the set of CFD simulations of turbulent flow in channel with wavy surface. The agreement between predicted and measured critical gas velocities and wavelengths in dependence on the liquid film thickness is very good.