Topological groundwater hydrodynamics

Topological groundwater hydrodynamics is an emerging subdiscipline in the mechanics of fluids in porous media whose objective is to investigate the invariant geometric properties of subsurface flow and transport processes. In this paper, the topological characteristics of groundwater flows governed by the Darcy law are studied. It is demonstrated that: (i) the topological constraint of zero helicity density during flow is equivalent to the Darcy law; (ii) both steady and unsteady groundwater flows through aquifers whose hydraulic conductivity is an arbitrary scalar function of position and time are confined to surfaces on which the streamlines of the flow are geodesic curves; (iii) the surfaces to which the flows are confined either are flat or are tori; and (iv) chaotic streamlines are not possible for these flows, implying that they are inherently poorly mixing in advective solute transport.     

Transport properties in small-scale coastal flows: relative dispersion from VHF radar measurements in the Gulf of La Spezia

Lagrangian transport characteristics in the Gulf of La Spezia, a 5 × 10-km area along the western coast of Italy, are investigated using data collected from a very high frequency (VHF) radar system with 250 m and 30-min resolution and two clusters of Coastal Dynamics Experiment surface drifters during 2 weeks in the summer of 2007. The surface drifters are found to follow the temporal and spatial evolution of the finite-scale Lyapunov exponents (FSLEs) computed by the VHF radar, indicating the precision of both the radar measurements and the diagnostic FSLE in mapping accurately the transport pathways. In light of this agreement, an analysis of the relative dispersion is conducted. It is found that the average FSLE value varies within a narrow range of 4   day^-1 ￡ l ￡ 7   day^-14 \;\mbox{day}^{-1} \leq \lambda \leq 7 \;\mbox{day}^{-1} and displays an exponential regime over the entire extent of the measurements. The dynamical implication is that relative dispersion is controlled nonlocally, namely by slow, persistent, energetic mesoscale structures as opposed to the rapidly evolving high-gradient small-scale turbulent features. The value of the exponent is about an order of magnitude larger than those found in previous modeling studies and analysis of SCULP data in the Gulf of Mexico but somewhat smaller than that estimated from CLIMODE drifters in the Gulf Stream region. Scaling of the FSLE using a metric of resolved gradients of the Eulerian fields in the form of a positive Okubo–Weiss criterion is useful, but not as precise as in modeling studies. The horizontal flow convergence is found to have a small yet tangible effect on relative dispersion.     

Numerical investigation of chaotic advection past a groyne based on laboratory flow experiment

A laboratory flow past a groyne with complex hydrodynamics was investigated using surface Particle Tracking Velocimetry (PTV) technique for detecting chaotic features in fluvial mixing processes. In the reconstructed velocity field particles were deployed and tracked numerically in a Lagrangian way. Calculating some appropriate parameters (e.g. flushing times, finite-size Lyapunov exponent) originating from chaos theory, we are able to give a more detailed picture on surface mixing driven by aperiodic flows than traditional approaches, including the separation of sub-regions characterized by sharply different mixing efficiency.     

Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves

Abstract

We consider the mixing of passive tracers and vorticity by temporally fluctuating large scale flows in two dimensions. In analyzing this problem, we employ modern developments stemming from properties of Hamiltonian chaos in the particle trajectories; these developments generally come under the heading “chaotic advection” or “Lagrangian turbulence.” A review of the salient properties of this kind of mixing, and the mathematics used to analyze it, is presented in the context of passive tracer mixing by a vacillating barotropic Rossby wave. We then take up the characterization of subtler aspects of the mixing. It is shown the chaotic advection produces very nonlocal mixing which cannot be represented by eddy diffusivity. Also, the power spectrum of the tracer field is found to be k ^{? l} at shortwaves—precisely as for mixing by homogeneous, isotropic two dimensional turbulence,—even though the physics of the present case is very different. We have produced two independent arguments accounting for this behavior.

We then examine integrations of the unforced barotropic vorticity equation with initial conditions chosen to give a large scale streamline geometry similar to that analyzed in the passive case. It is found that vorticity mixing proceeds along lines similar to passive tracer mixing. Broad regions of homogenized vorticity ultimately surround the separatrices of the large scale streamline pattern, with vorticity gradients limited to nonchaotic regions (regions of tori) in the corresponding passive problem.

Vorticity in the chaotic zone takes the form of an arrangement of strands which become progressively finer in scale and progressively more densely packed; this process transfers enstrophy to small scales. Although the enstrophy cascade is entirely controlled by the large scale wave, the shortwave enstrophy spectrum ultimately takes on the classical k ^{? l} form. If one accepts that the enstrophy cascade is indeed mediated by chaotic advection, this is the expected behavior. The extreme form of nonlocality (in wavenumber space) manifest in this example casts some doubt on the traditional picture of enstrophy cascade in the Atmosphere, which is based on homogeneous two dimensional turbulence theory. We advance the conjecture that these transfers are in large measure attributable to large scale, low frequency, planetary waves.

Upscale energy transfers amplifying the large scale wave do indeed occur in the course of the above-described process. However, the energy transfer is complete long before vorticity mixing has gotten very far, and therefore has little to do with chaotic advection. In this sense, the vorticity involved in the enstrophy cascade is “fossil vorticity,” which has already given up its energy to the large scale.

We conclude with some speculations concerning statistical mechanics of two dimensional flow, prompted by our finding that flows with identical initial energy and enstrophy can culminate in very different final states. We also outline prospects for further applications of chaotic mixing in atmospheric problems.     

Calculating the macrodispersion coefficient of the ensemble averaged solute transport equation in the discrete domain

Recognizing the spatial heterogeneity of hydraulic parameters, many researchers have studied the solute transport by both groundwater and channel flow in a stochastic framework. One of the methodologies used to up‐scale the stochastic solute transport equation, from a point‐location scale to a grid scale, is the cumulant expansion method combined with the calculus for the time‐ordered exponential and the calculus for the Lie operator. When the point‐location scale transport equation is scaled up to the grid scale, using the cumulant expansion method, a new dispersion coefficient emerges in the dispersive term of the solute transport equation in addition to the molecular dispersion coefficient. This velocity driven dispersion is called ‘macrodispersion’. The macrodispersion coefficient is the integral function of the time‐ordered covariance of the random velocity field. The integral is calculated over a Lagrangian trajectory of the flow. The Lagrangian trajectory depends on the following: (i) the spatial origin of the particle; (ii) the time when the macrodispersion is calculated; and (iii) the mean velocity field along the trajectory itself. The Lagrangian trajectory is a recursive function of time because the location of the particle along the trajectory at a particular time depends on the location of the particle at the previous time. This recursive functional form of the Lagrangian trajectory makes the calculation of the macrodispersion coefficient difficult. Especially for the unsteady, spatially non‐stationary, non‐uniform flow field, the macrodispersion coefficient is a highly complex expression and, so far, calculated using numerical methods in the discrete domains. Here, an analytical method was introduced to calculate the macrodispersion coefficient in the discrete domain for the unsteady and steady, spatially non‐stationary flow cases accurately and efficiently. This study can fill the gap between the theory of the ensemble averaged solute transport model and its numerical implementations. Copyright © 2011 John Wiley & Sons, Ltd.     

Experimental investigation of bedload transport processes under unsteady flow conditions

Hydraulic engineering is usually based on theoretical analysis and/or numerical modelling simulation. As the dynamic behaviour of sediment movement under unsteady flow is still unclear, and field measurement is comparatively difficult during a large flood, prior investigations through flume experiments are required. A series of flume experiments, conducted using different inflow hydrographs without sediment supply from upstream, was carried out to investigate the sediment transport process under unsteady flow conditions. A series of triangular hydrographs were performed in the experiments. The results indicate that a temporal lag was found between the flow hydrograph peak and the sediment hydrograph peak because large size sand dunes lasted for a short period in the falling limb of the flow hydrograph. The temporal lag was found to be about equal to 6–15% of the flow hydrograph duration. Owing to the temporal lag, the total bedload yield in the rising period was less than that in the falling period. Furthermore, the measured total bedload yield in the unsteady flow experiments was larger than the predicted value, which was estimated by using the results obtained from the equivalent steady flow experiment. The peak bedload transport rate for unsteady flow conditions was also larger than the predicted value. The ratios of the measured to the predicted quantities mentioned above were found to be constant values for different shapes of hydrographs. It is, therefore, expected that the analytical results of sediment transport from equivalent steady flow can be a good reference for sediment transport under unsteady flow conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

Magma mixing in a squeezed conduit

The mixing of mafic and silicic magmas in a squeezed conduit is simulated by fluid dynamics experiments using glue and dilute glycerin in a squeezed vinyl tube. Lighter and more viscous glue initially stably overlies dilute glycerin in a tube. As the tube is squeezed downward by a roller, the glue and dilute glycerin axisymmetrically circulate in the tube. The stable density stratification overturns due to the circulative motion. Because of viscosity contrast between the two liquids, the convective motion becomes unsteady and chaotic, which leads to efficient mixing of two liquids, even when the tube is squeezed very slowly. It is suggested that magma mixing in a squeezed conduit may explain some occurrences of natural mixed lavas with nearly homogeneous groundmass.     

Influence of bottom frictional effects in sill regions upon lee wave generation and implications for internal mixing

A cross-sectional nonhydrostatic model using idealized sill topography is used to examine the influence of bottom friction upon unsteady lee wave generation and flow in the region of a sill. The implications of changes in shear and lee wave intensity in terms of local mixing are also considered. Motion is induced by a barotropic tidal flow which produces a hydraulic transition, associated with which are convective overturning cells, wave breaking, and unsteady lee waves that give rise to mixing on the lee side of the sill. Calculations show that, as bottom friction is increased, current profiles on the shallow sill crest develop a highly sheared bottom boundary layer. This enhanced current shear changes the downwelling of isotherms downstream of the sill with an associated increase in the hydraulic transition, wave breaking, and convective mixing in the upper part of the water column. Both short and longer time calculations with wide and narrow sills for a number of sill depths and buoyancy frequencies confirm that increasing bottom friction modifies the flow and unsteady lee wave distribution on the downstream side of a sill. Associated with this increase in bottom friction coefficient, there is increased mixing in the upper part of the water column with an associated decrease in the vertical temperature gradient. However, this increase in mixing and decrease in temperature gradient in the upper part of the water column is very different from the conventional change in near-bed temperature gradient produced by increased bottom mixing that occurs in shallow sea regions as the bottom drag coefficient is increased.     

Deriving formulas for an unsteady virtual velocity of bedload tracers

In a flume experiment with steady flow conditions, H. A. Einstein recognised the transport of bedload particles as consisting of steps of rolling, sliding, or saltation with intermittent rest periods, and introduced the concept of an average, ‘virtual’ transport velocity. This virtual velocity then has also been derived from tracer studies in the field by dividing the travelled distance of a tracer by the duration of competent flow. As a consequence, the virtual velocity in the field is represented by one single value only, despite the unsteady flow variables. Tracer measurements in a river have not been yet used to express transport velocity as a direct function of these actual variables, and insights from tracer measurements into the processes of sediment transport remain limited. In particular, the unsteady conditions for bedload in the field have impeded the derivation of sediment transport characteristics as determined from laboratory experiments, as well as the transfer of laboratory insights to a field setting. We introduce a method of data regression for the derivation of an ‘unsteady’ virtual velocity from repeated surveys of tracer positions. The regression program called graVel (provided as supplementary material) relates the integral of an excess flow variable term to measured travel distances, yielding the most probable threshold value for entrainment and the coefficient of linear and non‐linear formulas. An extended regression allows additional fitting of the exponent in non‐linear formulas. Application to published tracer data from the Mameyes River, Puerto Rico, shows that the unsteady virtual velocity is more likely governed by non‐linear relations to excess Shields stress, similar to bedload transport, than by relations linking the particle velocity linearly to excess shear velocity. Partial agreements with non‐dimensional results derived from the larger, non‐wadeable Mur River encourage the establishment of a generalised formula for the unsteady virtual velocity. Copyright © 2017 John Wiley & Sons, Ltd.     

Field Trials of Chaotic Advection to Enhance Reagent Delivery

Chaotic advection is a novel approach that has the potential to enhance contact between an injected reagent and target contaminants, and thereby improve the effectiveness of in situ treatment technologies. One configuration that is capable of generating chaotic advection is termed the rotated potential mixing (RPM) flow. A conventional RPM flow system involves periodically reoriented dipole flow driven by transient switching of pressures at a series of radial wells. To determine whether chaotic advection can be engineered using such an RPM flow system, and to assess the consequent impact on the spatial distribution of a conservative tracer, a series of field-scale experiments were conducted. These experiments involved the injection of a tracer in the center of a circular array of wells followed by either mixing using an engineered RPM flow system to invoke chaotic advection, or by natural processes (advection and diffusion) as the control. Pressure fluctuations from the mixing tests using the RPM flow system showed consistent peak amplitudes during injection and extraction at a frequency corresponding to the switching time, suggesting that the target hydraulic behavior was achieved with the time-dependent flow field. The tracer breakthrough responses showed oscillatory behavior at all monitoring locations during the mixing tests which indicated that the desired RPM flow was generated. The presence of chaotic advection was supported by comparisons to observations from a previous laboratory experiment using RPM flow, and the Fourier spectrum of the temporal tracer data. Results from several quantitative metrics adopted to demonstrate field-scale evidence of chaotic advection showed that mixing led to improved lateral tracer spreading and approximately uniform concentrations across the monitoring network. The multiple lines of evidence assembled in this proof-of-concept study conclusively demonstrated that chaotic advection can be engineered at the field scale. This investigation is a critical step in the development of chaotic advection as a viable and efficient approach to enhance reagent delivery.     

Reconciled bedload sediment transport rates in ephemeral and perennial rivers

It has been thought for some time that bedload sediment transport rates may differ markedly in ephemeral and perennial rivers and, supporting this thought, there has been observation of very high rates of bedload transport by flash floods in the ephemeral river Nahal Yatir. However, until now, there has been no quantitative model resolving the observation, nor a theory capable of explaining why bedload transport rates by unsteady flash floods can be reasonably well described by bedload transport capacity formulae initially derived for steady flows. Here a time scale analysis of bedload transport is presented as pertaining to Nahal Yatir, which demonstrates that bedload transport can adapt sufficiently rapidly to capacity determined exclusively by local flow regime, and accordingly the transport capacity formulations developed for steady flows can be applied even under unsteady flows such as flash floods. Complementing the time scale analysis, a series of computational exercises using a coupled shallow water hydrodynamic model are shown to adequately resolve the observation of the very high rates of bedload transport by flash floods in Nahal Yatir. While bedload transport rates in ephemeral and perennial rivers differ remarkably when evaluated against a pure flow parameter such as specific stream power, they are essentially reconciled if assessed with a physically sensible parameter incorporating not only the flow regime but also the sediment particle size. The present finding underpins the practice of fluvial geomorphologists relating measured bedload transport to local flow and sediment characteristics only, irrespective of whether the flow is unsteady or steady. Copyright © 2010 John Wiley & Sons, Ltd.     

An experimental study of Stewartson layer separation

Abstract

The separation of sidewall boundary layers in a rotating annulus of homogeneous fluid is studied experimentally. The flow is driven by a differentially rotating lid, and a very small uniform slope of the bottom causes a weak mountain vortex pair to form in the interior, away from the sidewalls. A necessary condition for aerodynamic separation of the sidewall boundary layers is derived and compared with the experimental results. The laboratory flow separates for parameters that are just slightly more inviscid than those required by the necessary condition for the existence of adverse pressure gradients at the wall. As the bottom friction is decreased further, the flow becomes unsteady and chaotic. The most interesting aspect of this problem is that chaotic interior behavior, associated with the separated boundary layer, is observed for parameter values for which the interior topographically forced flow is, by itself, essentially linear.     

Lagrangian simulations of unstable gravity-driven flow of fluids with variable density in randomly heterogeneous porous media

A new Lagrangian particle model based on smoothed particle hydrodynamics (SPH) is developed and used to simulate Darcy scale flow and transport in porous media. The method has excellent conservation properties and treats advection exactly. The Lagrangian method is used in stochastic analysis of miscible density-driven fluid flows. Results show that heterogeneity significantly increases dispersion and slows development of Rayleigh–Taylor instability. The presented numerical examples illustrate the advantages of Lagrangian methods for stochastic transport simulations.     

SEDIMENT TRANSPORT EXPERIMENTSIN UNSTEADY FLOWS

l INTRODUCTIONThe theories and formulas of sediment dynamics were established based on steady and uniform flows.These theories and formulas often fail to aPply in engineering projects because, in nthee, sediment istransported often by unsteady and non-uniform fiows (Wang et al., l997).TWo cases ofnon-stationarity can be distinguished f long-tertn and short-term (Plate, l994). Long-termnon-stationarity tfansport can be defined as a sediment process which can be treated by sequences ofstat…     

Simulation of sediment transport during flood events: laboratory work and field experiments

Abstract

This paper aims at initiating a fundamental understanding of the suspended load transport of river sediment in unsteady flow. Laboratory erosion tests as well as artificial flood experiments are used to evaluate the influence of the transient regime on the transport efficiency of the flow. The erosion experiments reveal that the transport capacity is augmented when the unsteadiness of the flow increases. However, the influence of the transient regime is counteracted by the cohesive properties of the river bed. Field experiments with artificial floods released from a reservoir into a small canal confirm these findings and show a relationship between the friction velocity and the suspended load transport. An appropriate parameter β is proposed to evaluate the impact of the transient regime on the transport of suspended sediment.     

Turbulence-particle interactions under surface gravity waves

The dispersion and transport of single inertial particles through an oscillatory turbulent aquatic environment are examined numerically by a Lagrangian particle tracking model using a series of idealised test cases. The turbulent mixing is incorporated into the Lagrangian model by the means of a stochastic scheme in which the inhomogeneous turbulent quantities are governed by a one-dimensional k- ε turbulence closure scheme. This vertical mixing model is further modified to include the effects of surface gravity waves including Coriolis-Stokes forcing, wave breaking, and Langmuir circulations. To simplify the complex interactions between the deterministic and the stochastic phases of flow, we assume a time-invariant turbulent flow field and exclude the hydrodynamic biases due to the effects of ambient mean current. The numerical results show that the inertial particles acquire perturbed oscillations traced out as time-varying sinking/rising orbits in the vicinity of the sea surface under linear and cnoidal waves and acquire a non-looping single arc superimposed with the high-frequency fluctuations beneath the nonlinear solitary waves. Furthermore, we briefly summarise some recipes through the course of this paper on the implementation of the stochastic particle tracking models to realistically describe the drift and suspension of inertial particles throughout the water column.     

Probabilistic collocation and lagrangian sampling for advective tracer transport in randomly heterogeneous porous media

The Karhunen-Loeve (KL) decomposition and the polynomial chaos (PC) expansion are elegant and efficient tools for uncertainty propagation in porous media. Over recent years, KL/PC-based frameworks have successfully been applied in several contributions for the flow problem in the subsurface context. It was also shown, however, that the accurate solution of the transport problem with KL/PC techniques is more challenging. We propose a framework that utilizes KL/PC in combination with sparse Smolyak quadrature for the flow problem only. In a subsequent step, a Lagrangian sampling technique is used for transport. The flow field samples are calculated based on a PC expansion derived from the solutions at relatively few quadrature points. To increase the computational efficiency of the PC-based flow field sampling, a new reduction method is applied. For advection dominated transport scenarios, where a Lagrangian approach is applicable, the proposed PC/Monte Carlo method (PCMCM) is very efficient and avoids accuracy problems that arise when applying KL/PC techniques to both flow and transport. The applicability of PCMCM is demonstrated for transport simulations in multivariate Gaussian log-conductivity fields that are unconditional and conditional on conductivity measurements.     

EXPERIMENTAL STUDY OF BEDLOAD TRANSPORT IN UNSTEADY OPEN-CHANNEL FLOW

I.INTRODUCTIONBedloadtransportinsteadyuniformopenchannelflowhasbeenextensiVelystudied.Manyoftheformulasdevelopedforthepredictionofbedloadtransportinuniformopen-channelflowcanbebroughtinthefollowingform(ChienandWan,1983);ac=f(O)(l)xvhereacisthedimensionlessparameterofbedloadtranSPortandOisthedimensionlessparameterofflowintensity.TheseparametersaredefinedasfwheregsisthebedloadtranspoftratePerunitwidthindryweight;disthesedimentdiameter,Sisthebedslopeofthechannel;Rbisthehydraulicradiusdue…     

DEPTH-AVERAGE ANALYSIS OF HYSTERESIS BETWEEN FLOW AND SEDIMENT TRANSPORT UNDER UNSTEADY CONDITIONS

1 INTRODUCTION Flow and sediment transport in natural rivers are generally unsteady, and exhibit temporal and spatial lags. Traditionally, in most hydraulic engineering problems the unsteady flow and sediment transport are approximately treated as steady …     

Advection of passive scalars induced by a bay-trapped nonstationary vortex

A simple model of fluid particle advection induced by the interaction of a point vortex and incident plane flow occurring near a curved boundary is analyzed. The use of the curved boundary in this case is aimed at mimicking the geometry of an isolated bay of a circular shape. An introduction of such a boundary to the model results in the appearance of retention zones, where the vortex can be permanently trapped being either stationary or periodically oscillating. When stationary, it induces a steady velocity field that in turn ensures regular advection of nearby fluid particles. When the vortex oscillates periodically, the induced velocity field turns unsteady leading to the manifestation of chaotic advection of fluid particles. We show that the size of the fluid region engaged into chaotic advection increases almost monotonically with the increased magnitude of the vortex oscillations provided the magnitude remains relatively small. The monotonicity is accounted for the fact that the frequency of the vortex oscillations incommensurable with the proper frequency of fluid particle rotations in the steady state. Another point of interest is that it is demonstrated that bounded regions, in which the vortex may be trapped, can appear even at a significant distance from the bay. Making use of a Lagrangian indicator, examples of fluid particle advection induced by the periodic motion of the vortex inside the bay are adduced.