Instantaneous Line Source Research Articles

Numerical and Experimental Validation of the Applicability of Active‐DTS Experiments to Estimate Thermal Conductivity and Groundwater Flux in Porous Media

AbstractGroundwater flow depends on the heterogeneity of hydraulic properties whose field characterization is challenging. Recently developed active‐Distributed Temperature Sensing (DTS) experiments offer the possibility to directly measure groundwater fluxes resulting from heterogeneous flow fields. Here, based on fundamental principles and numerical simulations, two interpretation methods of active‐DTS experiments are proposed to estimate both the porous media thermal conductivities and the groundwater fluxes in sediments. These methods rely on the interpretation of the temperature increase measured along a single heated fiber‐optic (FO) cable and consider heat transfer processes occurring both through the FO cable itself and through the porous media. The first method relies on the Moving Instantaneous Line Source model that reproduces the temperature increase and provides estimates of thermal conductivity and groundwater flux as well as an evaluation of the temperature rise due to the FO cable. The second method, based on the graphical identification of three characteristic times, provides complementary estimates of flux, fully independent of the effect of the FO cable. Sandbox experiments provide an experimental validation of the interpretation methods, demonstrate the excellent accuracy of groundwater flux estimates (<5%), and highlight the complementarity of both methods. Active‐DTS experiments allow investigating groundwater fluxes over a large range spanning 1 × 10−6−5 × 10−2 m/s, depending on the duration of the experiment. Considering the applicability of active‐DTS experiments in different contexts, we propose a general experimental framework for the application of both interpretation methods in the field, making active‐DTS field experiments especially promising for many subsurface applications.

Quality Measures of Mixing in Turbulent Flow and Effects of Molecular Diffusivity

Results from numerical simulations of the mixing of two puffs of scalars released in a turbulent flow channel are used to introduce a measure of mixing quality, and to investigate the effectiveness of turbulent mixing as a function of the location of the puff release and the molecular diffusivity of the puffs. The puffs are released from instantaneous line sources in the flow field with Schmidt numbers that range from 0.7 to 2400. The line sources are located at different distances from the channel wall, starting from the wall itself, the viscous wall layer, the logarithmic layer, and the channel center. The mixing effectiveness is quantified by following the trajectories of individual particles with a Lagrangian approach and carefully counting the number of particles from both puffs that arrive at different locations in the flow field as a function of time. A new measure, the mixing quality index Ø, is defined as the product of the normalized fraction of particles from the two puffs at a flow location. The mixing quality index can take values from 0, corresponding to no mixing, to 0.25, corresponding to full mixing. The mixing quality in the flow is found to depend on the Schmidt number of the puffs when the two puffs are released in the viscous wall region, while the Schmidt number is not important for the mixing of puffs released outside the logarithmic region.

Inverse Modeling for Aquatic Source and Transport Parameters Identification and its Application to Fukushima Nuclear Accident

Inverse modeling technique based on nonlinear least square regression method (LSRM) is developed for the identification of aquatic source and transport parameters. Instantaneous line source release model in two-dimensional domain and continuous point source release model in three-dimensional domain are used for the purpose. Case studies have been carried out for both types of releases to illustrate their application. Error analysis has been carried out to identify the maximum error that can be tolerated in the input concentration data used in the inverse model and to specify the minimum number of sampling points to generate such input data. The LSRM is compared with the well-established correlation coefficient optimization method for instantaneous line source release model, and good comparison is observed between them. The LSRM is used to quantitatively estimate the releases of different radionuclides into the Pacific Ocean which has resulted due to the discharge of highly radioactive liquid effluent from the affected Daiichi Nuclear Power Station at Fukushima in Japan. The measured concentrations of these radionuclides in seawater samples collected from two sampling points near Fukushima are used for the estimation. The average release works out to be 1.09 × 1016 for 131I, 3.4 × 1015 Bq for 134Cs, and 3.57 × 1015 Bq for 137Cs. Very good agreement is observed between the releases estimated in this study and those estimated by other different agencies.

Near-wall velocity structures that drive turbulent transport from a line source at the wall

This work involves numerical experiments that were conducted in a turbulent channel flow using direct numerical simulation in conjunction with the tracking of thermal markers released from a single instantaneous line source at the wall of the channel. The simulations were conducted at Reτ = 300 and for different Prandtl number fluids (Pr = 0.1, 0.7, 6, 20, and 50), in order to investigate the physical mechanism of turbulent transport and the flow structures that promote turbulent transport. The behavior of these heat markers was observed in 12 locations ranging from 1 to 12 half-channel heights downstream from the source. In each of these locations the flow structures that carried thermal markers towards the channel center (i.e., those that transfer heat away from the wall) and flow structures that carried thermal markers towards the channel wall (i.e., those that transfer heat towards the wall) were identified and characterized as a function of the Prandtl number. The characteristic time and length scales were obtained based on the calculation of correlation coefficients for the markers that were captured in each of the downstream locations. The average width of the heat-transferring plumes that have positive vertical velocity fluctuations is larger than the plumes with negative vertical velocity. Even though areas where heat markers are present extend for a long distance downstream, consistent with a sheet-like behavior of thermal plumes, it is found that the eddies associated with these thermal structures are considerably shorter and a synergism among these short eddies, both close and farther away from the wall, is needed in order to transport heat.

New temperature response functions (G functions) for pile and borehole ground heat exchangers based on composite-medium line-source theory

This paper presents a new approach to modeling of heat transfer by ground heat exchangers, involving unsteady heat conduction in composite media together with complex geometry. Analytical solutions for continuous line and cylindrical-surface sources are developed based on Jaeger’s instantaneous line-source solution for composite media. New temperature response functions (G functions) are also presented for pile ground heat exchangers with spiral coils and for borehole ground heat exchangers with single or double U-shaped tubes. These temperature response functions can be used to analyze the impact of difference between properties of materials inside and outside boreholes or piles on the performance of ground heat exchangers. Theoretical results show that the difference in properties is an important factor affecting the temperature response of ground heat exchangers. Since the heat capacity of grout, inside boreholes, is fully considered by this line-source theory, the new G functions should be particularly suitable for predicting small-time dynamic behaviors of ground heat exchangers. Therefore, these G functions may be significant for energy analysis and in-situ thermal response tests of ground-coupled heat pump systems.

Unsteady state mathematical model of chemically reactive pollutants from an instantaneous line source into a stable atmospheric boundary layer

A time dependent atmospheric model represented for chemically reactive primary pollutants emitted from an elevated line source into a stable atmospheric boundary layer over a surface terrain. The model obtained from an analytical solution of the atmospheric diffusion equation with the quadratic diffusion coefficient (exchange coefficient) and the variable wind velocity taken to be of three different types’ viz. constant, constant shear and parabolic functions of vertical height. The pollutants considered to be of chemically reactive primary pollutants emitted from a time-dependent line source of Instantaneous type. In order to facilitate the application of the model the results for the general situation that includes chemical reaction rate & time dependent source incorporated in the model.

Time-domain Green functions for diffuse light in two adjoining turbid half-spaces

Propagation of light emitted by an instantaneous source located above a plane interface between two semi-infinite turbid media is considered using the diffusion approximation. Green functions are derived for an instantaneous line source and an instantaneous point source by the method of Bellman et al. [Philos. Mag. 40, 297 (1949)], which is based on integral transforms. Both two-dimensional and three-dimensional Green functions for diffuse light have been obtained in the form of single integrals that allow for fast calculation of the specific intensity in the whole space. The influence of the optical parameters of the two media (diffusion coefficients, absorptions, and refractive indices) on the shapes of the contour lines of the specific intensity is analyzed.

The GDS model—a rapid computational technique for the calculation of aircraft spray drift buffer distances

A model for predicting the required spray drift buffer distance for a specified off target deposition level is described. The GDS model is based upon Gaussian diffusion and sedimentation of particles originating from an elevated instantaneous line source. Aircraft-induced near wake effects are ignored. Agreement between aircraft wake models FSCBG, AgDRIFT and the GDS model is reasonable for downwind distances greater than 50 m. The model has the advantage over Lagrangian models in that it is faster computationally and can readily provide real-time prediction in the cockpit over large distances (3 km). A sensitivity analysis has been performed on the model to elucidate the effects of the primary parameters on spray drift. The model has proved useful in the determination of spray drift buffer distances for regulatory purposes and in the development of appropriate spray drift management systems for aerial spraying. Further research work is required to refine the model to better account for air stability effects, collector type, evaporation and crop canopy.

Green functions for diffuse photon-density waves generated by a line source in two nonabsorbing turbid media in contact

Diffuse photon-density waves generated by an instantaneous line source that is parallel to the interface between two semi-infinite turbid media are studied by use of the diffusion approximation. For two nonabsorbing media the Green functions for diffuse light are obtained based on the Green functions for temperature fields that were derived with the Cagniard-de Hoop method. The boundary conditions for diffuse light take into account the discontinuity in the specific intensity at the interface between two media with different refractive indices. The results of the calculations of the specific intensities and the gradient lines for different sets of parameters are presented.

Transport properties for turbulent dispersion from wall sources

AbstractThe dispersion of a passive contaminant by turbulence is not only intrinsically interesting, but also is fundamental to practical heat‐ and mass‐transfer problems. The mechanism of turbulent dispersion from the wall and effects of the molecular Prandtl number are examined. Transport properties for scalar dispersion from an instantaneous line source at the wall of a parallel‐plate channel in which fully developed turbulent flow occurs, as well as from a continuous line source, are studied. Numerical experiments using a direct numerical simulation (DNS) of the flow, combined with Lagrangian scalar tracking (LST) of trajectories of thermal markers in the flow field, are conducted. The method is applied for a wide range of molecular Prandtl (Pr) or Schmidt (Sc) number fluids (0.1⩽Pr⩽50,000, which corresponds to liquid metals, gases, refrigerants, lubricants and electrochemical fluids). Results from the DNS/LST method are compared with experimental data, which are available for low Pr or Sc fluids. The turbulent diffusivity and turbulent Prandtl number for the plume that results from a line source are calculated, and an appropriate scaling is proposed. Finally, a model is developed for the prediction of the mean temperature or mean concentration profile as a function of Pr or Sc.

Thermal-diffusive ignition and flame initiation by a local energy source

The ignition and flame initiation in a gaseous reacting mixture subject to a local source of thermal energy is analysed by means of large activation energy asymptotics. The ignition transient is assumed to be long enough for heat conduction to be the dominant cooling mechanism. We show the existence of a critical value of the Damköhler number, defined as the ratio of appropriate characteristic times of conduction and chemical reaction, such that ignition only occurs for supercritical values. Additional conditions are required to ensure self-propagation of a flame after ignition. These are obtained, with the thermal-diffusive model, for a source of energy represented by an instantaneous point, line or planar source. The analysis, involving an unsteady free-boundary problem, shows that the initial flame kernel evolves to a self-propagating flame only if the energy released by the source is greater than a critical value.

Scalar dispersion from an instantaneous line source at the wall of a turbulent channel for medium and high Prandtl number fluids

Lagrangian methods have been used recently to reconstruct temperature profiles for relatively high Prandtl number, Pr, fluids (up to Pr=2400) in direct numerical simulations (DNS) of turbulent channel flow. The basic concept is that a heated surface is formed by an infinite number of continuous sources of heat. For example, the behavior of a heated plane can be synthesized by the behavior of an infinite number of continuous sources of heat that cover the plane. The building block for such a reconstruction is the behavior of a single instantaneous heat source located at the wall. The present work studies the behavior of such sources in turbulent channel flow. The trajectories of heat markers are monitored in space and time as they move in a hydrodynamic field created by a DNS. The fluids span several orders of magnitude of Pr (or Sc), Pr=0.1, 1, 10, 100, 200, 500, 2400, 7000, 15 000, 50 000, (liquid metals, gases, liquids, lubricants and electrochemical fluids). The effects of Pr in the evolution of the marker cloud are examined and quantified. The marker cloud is found to evolve in three stages, two of which are Pr dependent.

A scalar subgrid model with flow structure for large-eddy simulations of scalar variances

A new model to simulate passive scalar fields in large-eddy simulations of turbulence is presented. The scalar field is described by clouds of tracer particles and the subgrid contribution of the tracer displacement is modelled by a kinematic model which obeys Kolmogorov's inertial-range scaling, is incompressible and incorporates turbulent-like flow structure of the turbulent small scales. This makes it possible to study the scalar variance field with inertial-range effects explicitly resolved by the kinematic subgrid field while the LES determines the value of the Lagrangian integral time scale TL. In this way, the modelling approach does not rely on unknown Lagrangian input parameters which determine the absolute value of the scalar variance.The mean separation of particle pairs displays a well-defined Richardson scaling in the inertial range, and we find that the Richardson constant GΔ ≈ 0.07 which is small compared to the value obtained from stochastic models with the same TL. The probability density function of the separation of particle pairs is found to be highly non-Gaussian in the inertial range of times and for long times becomes Gaussian. We compute the scalar variance field for an instantaneous line source and find good agreement with experimental data.

Time-resolved plasma diagnostics and mass removal during single-pulse laser ablation

Laser ablation processes occurring over several orders of magnitude in time were investigated by using time-resolved spectroscopy, shadowgraphs and interferograms. A picosecond ablation plasma was measured with an electron density on the order of 1020 cm-3 originating from the breakdown of air. The longitudinal expansion of this plasma was suppressed due to the development of a strong space-charge field. At post-pulse times, the lateral (radial) expansion of the plasma was found to follow the relation, r∼t1/2, consistent with the expansion from an instantaneous line source of energy. The electron number density and temperature were deduced by measuring spectroscopic emission-line broadening during the early phase (30–300 ns) of a mass (atomic/ionic) plasma. These properties were measured as a function of the delay time and irradiance. Possible mechanisms such as inverse bremsstrahlung and self-regulation were used to describe the data before an explosion threshold of 20 GW/cm2. The laser self-focusing and critical temperature are discussed to explain dramatic changes in these properties after the irradiance threshold. On the microsecond time scale, the surface explodes and large (>μm) particles are ejected. Mass removed from single-crystal silicon by high power (109–1011 W/cm2) single-pulse laser ablation is studied by measuring the crater morphology. Time-resolved shadowgraph images show that the rapid increase in the crater depth at the threshold corresponds to large-size droplets leaving the surface. This rapid growth of the crater volume is attributed to explosive boiling.

Dual Thermal Probes near Plane Interfaces

The dual probe estimates the thermal properties of soils. Because it is strongly dependent on water content, volumetric heat capacity is useful for moisture estimation. A previous study of probe errors due to soil heterogeneity is taken further with the aid of a new solution for instantaneous line sources near a plane interface between two regions. The solution allows for different conductivities and heat capacities in the regions, but requires uniform diffusivities. This condition holds reasonably well near wetting fronts in a range of soils. Previous error bounds are confirmed and sharpened. The conclusion in the previous study that errors are small when the heterogeneity is no closer to the probes than the probe separation (typically 0.006 m) is reinforced. The expectation that the dual probe gives good resolution of water content close to fronts and interfaces is strengthened.

Stratified Flow Approximation for Sloping Aquifers

The Dupuit and sharp interface approximations are used to derive a nonlinear partial differential equation that describes the movement of a lens of injected fluid through a sloping aquifer that contains a second fluid with a different mass density. This equation describes the movement of either a dense fluid along a lower sloping boundary or a light fluid along an upper sloping boundary when the aquifer has a vertical thickness that is large compared with the thickness of the lens of injected fluid. Similarity methods are used to obtain exact solutions of this equation for instantaneous line and point sources, and approximate solutions are obtained for steady flow from both point and finite sources. Comparisons with experiment show that the mathematical model is a close approximation to reality when transition zone thicknesses along the interface are small compared with the thickness of the lens of injected fluid.

Simulation of coastal oil spills using the random walk particle method with Gaussian kernel weighting

Oil spills, amongst the most common marine hazards, are a major community concern, particularly in coastal areas where there are competing requirements due to population centres, conservation of the natural environment, industrial development and transport. It has become common practice to include simulations and modelling exercises in applications for development, so that oil spill trajectory studies are regularly carried out for planning and assessment purposes as well as for making decisions in the event of actual incidents. In this paper we present a procedure for the simulation of coastal oil spills which uses the random walk particle method with Gaussian kernel weighting to simulate the movement of oil in the sea, and includes special beaching and refloating algorithms for shorelines. This procedure allows one to simulate coastal spills rapidly and accurately and to perform calculations at closed boundaries consisting of a variety of shoreline types such as exposed headlands, sand beaches, sheltered marshes, etc., and at open boundaries with an extended computational region. Moreover, the mass concentration of the spill can be calculated at any point of a finite diffusion domain and conveniently represented by contour and mesh plots. Simulations in a typical coastal region are presented to illustrate the procedure for spills from instantaneous point sources, instantaneous line sources, instantaneous area sources, continuous point sources and continuous moving sources.

The second-order moment structure of dispersing plumes and puffs

A description of the behaviour of the second-order moments of concentration for a variety of source types is derived within the context of the classical phenomenology of isotropic turbulence. The sources considered include instantaneous area, line and point sources and can also be interpreted as relating to plumes from continuous point sources and continuous crosswind line sources in a strong uniform mean flow. A large number of different regimes are identified corresponding to different relative sizes of the many length scales involved. Perhaps the most interesting result is the identification of an ‘Inertial–meander’ subrange when the inertial-subrange eddies contribute to the meandering of the plume.

Eulerian‐Lagrangian Analysis of Transport Conditioned on Hydraulic Data: 3. Spatial Moments, Travel Time Distribution, Mass Flow Rate, and Cumulative Release Across a Compliance Surface

In paper 1 of this series we described an analytical‐numerical approach to predict deterministically solute transport under uncertainty. The approach allows conditioning such predictions on hydraulic measurements and assessing the corresponding reduction in uncertainty. In paper 2 we examined the effects of log transmissivity and hydraulic head data on conditional predictions of concentration due to instantaneous point and nonpoint sources. In this paper we show how the same approach can be used directly to estimate mass flow rate across a “compliance surface,” cumulative mass release, and the probability distribution of travel times across this surface and the associated uncertainty. Contrary to some other methods in the literature, our approach requires neither a special theory for travel times nor a prior assumption about their probability distribution. We also show how one can compute explicitly the second spatial moment of the conditional mean plume about its center of mass, the conditional mean second spatial moment of the actual plume about its center of mass, and the conditional covariance of the plume center of mass. We illustrate these quantities and the effect of conditioning on some of them by considering instantaneous point, line, and area sources in a two‐dimensional, statistically homogeneous, isotropic, mildly varying log transmissivity field under uniform prior (unconditional) mean flow.

Dispersion of pollutants in an asymmetric flow through a channel

The longitudinal dispersion of a pollutant due to asymmetric flow through a two-dimensional (2-D) channel has been studied for two different inputs: (i) instantaneous uniform line source and (ii) oscillatory source. This study reveals the process whereby the dispersion reaches a stationary state after the release of pollutant as an instantaneous line source; and an estimate is obtained for the axial distance along which fluctuations in concentration decay in the case of low frequency release.