Optimization of the protrusion shape for a couette-type flow

A study is made of the longitudinal 2D viscous steady flow and heat flux between two plates. Optimal shape design problems are solved in explicit form and shown to have unique global extrema. Conformal mappings are used to bring the problems into a fixed domain and solve them as Dirichlet boundary value problems in the form of Cauchy integrals and series expansions. For the simplest problem statement the optimum is shown to coincide with the well-known concrete dam outline of constant hydraulic gradient

Estimation and optimization of transient seepage with free surface

Single sink depths providing maximum ground-water table decrease during a fixed time interval within a selected area are found for the 2-D and 3-D cases. The curve of the maximal phreatic surface position (underflooding curve) in the aquifer from flood induced variation in water level of the ground-water reservoir is calculated. Well-known analytical solutions based on nonlinear and linear potential theories and the Dupuit-Forchheimer approximation are applied to calculate the objective function, decision variables, and boundary of the fully saturated zone. In the linear case, an explicit analytic solution gives the unique maximum of the water table decrease at the compliance point for a given pumping duration. For small values of sink depth, the linear approach is invalid. In the nonlinear case, complex analysis and series expansions are used. For small values of drain depth, the series technique becomes untenable. For the reservoir-aquifer problem the spreading phreatic surface is a rotating straight line and the underflooding curve is a parabola. © ASCE

Dynamics of groundwater mounds: Analytical solutions and integral characteristics

Transient water table systems under the influence of decaying groundwater mounds are studied in terms of the Dupuit-Forcheimer approximation, its linearization and linear potential theory. Parabolic mounds spreading and shrinking due to gravity and evapotranspiration are derived from the general class defined by Polubarinova-Kochina (1945). The penetration curves are calculated as characteristics of the water table response at prescribed observation wells. The Polubarinova-Kochina solutions for rectangular mounds are used to derive isochrones, a "resting lens" into which a mound transforms, and a distortion picture of reference volumes. These characteristics are obtained from numerical solution of a system of ordinary differential equation. The plume illustrating the advective spread of a contaminant from a reference area is computed as a composition of all path lines passing through this area

Seepage optimization for trapezoidal channel

A complex-variable method and series expansions are applied to optimal- shape design problems for a channel bed. A dimensionless depth of a trapezoidal and rectangular channel is determined by minimizing the cost function constrained by specified hydraulic characteristics. The cost function includes seepage losses and the cost of lining. The hydraulic constraints are cross-sectional area, the hydraulic radius, and the discharge. The problem of steady-state two-dimensional seepage involves determination of a phreatic surface with geometrical parameters as control functions. The extremes found are stable for minor perturbations of channel shape. The optimal criterion value for trapezoidal channels is close to Preissmann's for an arbitrary bed outline. The including of supplementary factors (the cost of evaporating water, of the channel land area, and so on) as criteria are discussed. © ASCE

Steady seepage near an impermeable obstacle

The problem of an obstacle with maximum cross-sectional area has been analytically solved in terms of a model for 2-D seepage flow with a capillary fringe. The boundary of the obstacle appears to show a 'blunt configuration', that is, the pressure reaches its maximum value at the vertex and decreases monotonically downstream. With a sufficiently large size of obstacle, a positive pressure domain in the form of a 'bubble' is formed in the vicinity of its vertex. This result has been verified by computations for saturated-unsaturated flows in terms of the finite element method for the Richards equation. It has also been shown that an obstacle can transform the initial fully saturated flow into unsaturated flow inside the 'dry shadow' domain. The shape of an obstacle in a confined aquifer that provides a minimum water head drop has been determined within the scope of the Dupuit model. A 'critical cavity shape' has been found, i.e. a cavity for which the boundary is simultaneously an isobar and a stream line. © 1992

Explicit solutions for seepage infiltrating into a porous earth dam due to precipitation

Steady two-dimensional gravity-driven seepage in homogeneous porous lumps is studied with the help of conformal mappings and boundary value problem technique. The Terzaghi flow pattern for a trapezoidal dam exposed to a heavy rainstorm is analysed. For a semi-circular massif, the influence of impervious bed inclination is studied. Recharge-discharge distributions, hinge points, gradients along the lump contour as well as the total flow rate exhibiting water-bearing capacity of the unit are found in explicit form. Generalizations for non-isobaric boundary conditions are discussed

Steady, two-dimensional flow of ground water to a trench

The problem under consideration is a steady ground water inflow to a single trench which drains a water-bearing layer of infinite extent. The equipotential corresponding to the trench outline is determined from the solution for extremum problems. The isoperimetric constraints selected for solution of these problems include cross-sectional area, seepage flow rate and size of a region with a guaranteed head loss. The equations for the required extremals and variable functions are written explicitly in terms of the solution for the Dirichlet problem. © 1991

Analytical estimation of ground-water flow around cutoff walls and into interceptor trenches

An impermeable boundary of the grout cutoff has been found that maximizes the seepage flow for a specified grouting volume. The shape of a trench has been defined that maximizes the quantity of fluid entering the ground-water flow from the trench. The seepage characteristics of the grout cutoff and the trench which have arbitrary shapes are estimated in terms of the derived exact analytical solutions of variational problems

Accumulation of a light non-aqueous phase liquid on a flat barrier baffling a descending groundwater flow

The pioneering solution of Zhukovskii for a steady two-dimensional flow of an ideal heavy fluid with a nonlinear free boundary condition is extended to a Darcian flow of groundwater encumbered by an impermeable barrier. The stoss or/and lee sides of the barrier are covered by a macrovolume of a liquid contaminant. Explicit parametric equations of the sharp interface are obtained by inversion of the hodograph domain. Zhukovskiis gas-finger shape is shown to be a particular case of our new class of free surfaces. For a cap of a light liquid, partially covering the roof, from the given crosssectional area of the cap, the affixes of the conformal mapping are found as a solution of a system of two nonlinear equations. The horizontal width and vertical height of the cap are determined. If the dimensionless incident velocity is higher than the density contrast, then the interface (cap boundary) cusps at its apex. For a relatively small velocity, the interface spreads to the vertices of the barrier, the apex zone remaining blunt shaped. We depict all the relevant domains and plot the flow nets using computer algebra routines. © 2012 The Royal Society

Conduction through a grooved surface and Sierpinsky fractals

Conduction in a semi-infinite wall with a grooved line of contact between the wall material and convective environment is studied using series expansions. A periodic composition of semicircles is shown to result in a uniform gradient distribution at specific values of the groove radius and the convection heat transfer coefficient. Two fractal parquets exposed to natural thermal gradients are studied by the methods of complex analysis. In double periodic patterns each elementary cell is fractal (Sierpinsky's carpet and Sierpinsky's gasket) in which 'dark' and 'light' phases have arbitrary conductivities. The Maxwell approximation is used to calculate effective characteristics of both fractal structures by 'homogenization' of the environment of an 'inclusion'. Solution of an exact two-dimensional refraction problem within an elementary cell including two components is used for upscaling, i.e. recalculation of effective conductivities and dissipations of subfractals of consequently increasing order. | Conduction in a semi-infinite wall with a grooved line of contact between the wall material and convective environment is studied using series expansions. A periodic composition of semicircles is shown to result in a uniform gradient distribution at specific values of the groove radius and the convection heat transfer coefficient. Two fractal parquets exposed to natural thermal gradients are studied by the methods of complex analysis. In double periodic patterns each elementary cell is fractal (Sierpinsky's carpet and Sierpinsky's gasket) in which 'dark' and 'light' phases have arbitrary conductivities. The Maxwell approximation is used to calculate effective characteristics of both fractal structures by 'homogenization' of the environment of an 'inclusion'. Solution of an exact two-dimensional refraction problem within an elementary cell including two components is used for upscaling, i.e. recalculation of effective conductivities and dissipations of subfractals of consequently increasing order