Nonlinear Flow Solutions Research Articles

Determination of groove shape with strong destabilization and low hydraulic drag

Flow through a channel equipped with plane, longitudinal grooves is investigated. We focus on determining changes to the flow dynamics due to applied wall manipulations, especially the possibility of drag-reduction, potential for hydrodynamic destabilization and onset of secondary, nonlinear flow solutions. Considered patterns of geometrical manipulation consist of plane walled grooves of triangular, variations of trapezoidal and up to rectangular shapes and are compared with existing results obtained for channels with sinusoidal corrugations. We show that there exists a strong connection in the response of the flow system to the applied pattern of grooves with the leading Fourier component of the wall pattern. The analysis starts with undisturbed, two-dimensional flows investigated with the focus on hydraulic resistance for wide range of geometric parameters. Secondly, critical conditions for the onset of travelling wave instability are determined and compared with existing results. Thirdly, nonlinear flow solutions, obtained for flows through the geometry with lowest destabilization threshold are analyzed at supercritical conditions. Finally quantification of the diffusive transport intensification due to the kinematics of the nonlinear flow solution is attempted. It is shown that, irrespective of the groove shape, longitudinal wall patterns result in flow destabilization due to traveling wave mode already at very low values of the Reynolds number (<102). At the same time such configurations can be energy efficient since overall drag is marginally increased (or in fact reduced) and at the same time diffusive processes are intensified leading to improved mixing. Implications of this study might help in development of small-scale flow devices operating at low and moderate ranges of Reynolds number with the purpose of intensifying mixing, heat transfer or reaction of chemical or biological compounds.

Secondary flows in a longitudinally grooved channel and enhancement of diffusive transport

Flow in a longitudinally grooved channel is analysed with the primary objective of quantifying intensification of transport mechanism due to the onset of secondary flows resulting from hydrodynamic instability and amplification of unstable modes into the nonlinear regime. Considered geometry consists of a channel whose walls are fitted with sinusoidal corrugations, forming a system of longitudinal grooves parallel to the streamwise direction. Such configuration is energy efficient, because it reduces drag when compared to the smooth reference configuration and at the same time results in flow destabilization, due to travelling wave mode already at very low values of the Reynolds number ( < 102). The analysis is performed for a range of over-the-critical values of the Reynolds numbers and focuses on nonlinear flow solutions corresponding to the limit cycle to which flow transitions from a fixed-point laminar state through a supercritical Hopf bifurcation beyond which nonlinear solution is both three-dimensional and time-periodic allowing for development of complex advection patterns and consequently improved mixing. The main results of the current work are characterization of the nonlinear solutions, attained by the flow past bifurcation and for the first time illustration of changes in the diffusive transport mechanism due to nonstationary and three-dimensional form of the flow. Our analysis indicates that significant enhancement in advective transport may be attained and that it is not monotonically related to the Reynolds number, but rather, it is related to the ability of saturated flow to intensify spanwise motions. Implications of this study might help in development of small scale flow devices operating at low and moderate ranges of Reynolds number with the purpose of intensifying mixing, heat transfer or reaction of chemical or biological compounds.

Natural convection, Taylor dispersion, and diagenesis in a tilted porous layer.

As a model for natural convection in the Earth's upper crust, we discuss the flow in a porous tilted layer of infinite extent in an impermeable surrounding of different thermal conductivity across which a vertical temperature gradient acts. We calculate the basic nonlinear flow solution; it is a large-scale along-slope flow with zero mean flow. Two important geophysical consequences are discussed: (a) the transport of passive tracers by applying Taylor's theory of dispersion to our system, and (b) diagenetic changes of the porous matrix. For realistic geophysical parameters we estimate the order of magnitude of the effects and discuss the ranges of validity of the theory. The enhancement of diffusion of passive tracers due to the flow is quite significant and might have important implications for the migration of nuclear waste buried in the Earth.

Nonlinear Flow Toward Wells

A comprehensive review of the validity of Darcy’s law is presented with the emphasis on the necessity for a nonlinear flow law for Reynolds numbers especially greater than one. Subsequently, the nonequilibrium formula is developed for nonlinear flow and is useful in determining either the aquifer parameters from the time drawdown observations or in predicting the drawdown variations in a confined aquifer tapped by a fully penetrating well. The derivation of the necessary formula is based on the continuity equation in terms of the specific yield and a nonlinear flow law which is adopted in this paper as the exponential law. The general solution of nonlinear flow toward wells reduces to the Theis nonequilibrium equation when the turbulence exponent is taken as equal to one. Initial and moderate portions of nonlinear flow type curves prove to be very important for the estimation of aquifer parameters, especially for the turbulence exponent and for the nonlinear flow transmissivity. It is observed that even in the same aquifer with a constand pump discharge, differences in the observation well distances to the main well give rise to different type curves. The late time drawdown data can be used only for determining the conventional linear flow related parameters such as the storativity and the transmissivity. Finally, the application of the methodology has been performed for some field data available in the literature.