Continuous Velocity Distribution Research Articles

Acoustic Monitoring of Tidal Flow and Salinity in a Tidal Channel

Fluvial Acoustic Tomography (FAT) is a powerful hydroacoustic system used in the investigations of estuarine dynamics. This research was designed to explore the effectiveness of using a novel and promising method of monitoring the continuous salinity intrusion and velocity distribution in a tidal channel using the FAT scheme. Four FAT units were installed near the riverbanks using a new zigzag system in a rectangular tomographic domain of 700 m × 170 m. The water velocities and salinities measured by FAT in this study were comparable to the data captured by traditional methods, including moving-boat Stream–Pro ADCP, CTD, and CT sensors. It was found that the delays in time between the maximum salinity and high water level along the channel ranged from 12 min to 1.5 h, with these time lags increasing seaward, primarily due to changes in freshwater flows upstream. In addition, the longitudinal salinity gradient was found to decrease toward the mouth of the river, with tide-driven mechanisms most likely being dominant in the dispersion process. The estuary is ebb-dominant, with an asymmetry in the ebb-tide and flood-tide velocities and the highest velocities occurring during the ebb tide. Furthermore, the residual current was found to be affected primarily by the freshwater discharge from upstream.

Linking mean pore velocity and dispersivity to pore velocity distribution by advection–dispersion and stream tube modeling

A large set of experimental solute tracer breakthrough data (corresponding to more than 350 individual tracer breakthrough curves) in eight granular filter materials, were used to investigate the links between solute dispersivity and the shape of the pore velocity—relative pore area distributions for the materials. Solute dispersivity values and the shape of the pore velocity—relative pore area distributions were determined by fitting the breakthrough data to the advection–dispersion equation and to the stream tube modeling (STM), respectively. For the STM calculations, stream tube diameter and average stream tube flow rate were allowed to vary between individual stream tubes. Shape parameters e.g., mean, coefficient of variation, skewness and excess kurtosis for the STM-based pore velocity—relative pore area distributions were subsequently calculated. Comparisons between dispersivity and shape parameter values showed strong correlation between dispersivity, coefficient of variation, skewness and excess kurtosis. These results also indicated a universal relationship between dispersivity, skewness and excess kurtosis across all materials and breakthrough curves. This relationship will assist in modelling of experimental data which cannot be characterized by a single or a limited number of pore velocities but require continuous pore velocity distributions. Results further indicated, that the shape of the observed pore velocity—relative pore area distributions could be well approximated by a log-normal distribution.

On the fundamental resonant mode of inhomogeneous soil deposits

The problem of estimating seismic ground deformation is central to state-of-practice procedures of designing and maintaining infrastructure in earthquake-prone areas. Particularly, the problem of estimating the displacement field in a soft shallow layer overlying rigid bedrock induced by simple shear wave excitation has been favored by engineers due to its simplicity combined with inherent relevance for practical scenarios. We here derive analytical estimates for both the fundamental frequency and the amplitude of the first resonant mode of such systems by applying an intuitive argument based on resonance of single-degree-of-freedom systems. Our estimates do not presuppose a continuous velocity distribution, and can be used for fast assessment of site response in seismic hazard assessment and engineering design. On the basis of the said estimates of fundamental frequency and amplitude, we next propose a novel definition of “equivalent homogeneous shear modulus” of the inhomogeneous deposit, and we show that the response of the fundamental mode is controlled primarily by the properties of the layers contiguous to the bedrock. We finally discuss the validity of our argument, and evaluate the accuracy of our results by comparison with analytical and numerical solutions.

Dynamic crushing of cellular materials: a particle velocity-based analytical method and its application

Cellular material under high-velocity impact exhibits a typical feature of layerwise collapse. A cell-based finite element model is employed herein to simulate the direct impact of a closed-cell foam, and one-dimensional velocity field distributions are obtained to characterize the crushing band propagating through a cellular material. An explicit expression for the continuous velocity distribution is derived based on the features of the velocity gradient distribution. The velocity distribution function is adopted to determine the dynamic stress–strain states of cellular materials under dynamic loading. The local stress–strain history distribution reveals that sectional cells experience a process from the precursor elastic behavior to the shock stress state, passing through the dynamic initial crushing state. A power-law relation between the dynamic initial crushing stress and the strain rate is established, which confirms the strain rate effect of cellular materials. By extracting the critical points immediately before the unloading stage in the local dynamic stress–strain history curves, the dynamic stress–strain states of cellular materials are determined. They exhibit loading rate dependence but are independent of the initial impact velocity. Furthermore, with increase of the relative density, the dynamic hardening behavior of the cellular specimen is enhanced and the crushing process event is advanced. The particle velocity-based analytical method is applied to analyze the dynamic responses of cellular materials. This method is better than continuum-based shock models, since it does not require a preassumed constitutive relation. Therefore, the particle velocity-based analytical method proposed herein may provide new ideas to carry out dynamic experimental measurements, which is especially applicable to inhomogeneous materials.

Reactive contaminant transport simulation using the analytic element method, random walk particle tracking and kernel density estimator

In this study, we propose a simulation model to solve the two-dimensional groundwater flow and advection-dispersion-reaction equation (ADRE) by coupling analytic element method (AEM), random walk particle tracking (RWPT) and kernel density estimator (KDE). In the AEM-RWPT-KDE model, AEM solves the groundwater flow equation, RWPT solves the advection-dispersion-reaction equation and KDE enhances the accuracy and computational efficiency of RWPT. AEM generates a continuous velocity distribution which is suitable for RWPT. An analytic expression is modified to simulate radioactive decay for discrete transport time steps and is embedded in the RWPT model. Linear adsorption is also incorporated in the RWPT model by tracking particles with a retarded velocity. Unlike Eulerian transport models such as finite element or finite difference methods, RWPT is completely free from numerical dispersion. The comparison of results illustrates the superiority of the RWPT model over accurate Eulerian-Lagrangian models with reference to the analytical solution. Further, the AEM-RWPT-KDE model is used to simulate the transport processes of Radium-228 and Trichloroethene (TCE) in two hypothetical aquifer cases. Both case studies reflect the practical applicability of the proposed methodology.

Direct coupling between molecular dynamics and lattice Boltzmann method based on velocity distribution functions for steady-state isothermal flow

The direct coupling scheme between the molecular dynamics (MD) and lattice Boltzmann method (LBM) is established in this paper. Different from the existing coupling schemes which are based on the exchange of the density and velocities, the proposed coupling scheme is based on the velocity distribution functions. Firstly, the relations between the discrete velocity distribution functions of LBM and the continuous velocity distribution functions of MD are derived based on the Hermite expansions. Then, the coupling schemes between MD and LBM are proposed. The inconsistency between the equation of states of MD and LBM is specially treated and the deviatoric stresses are exchanged. The coupling simulations of the Poiseuille flow and Couette flow demonstrate that both the velocity and stress can be well exchanged by the coupling scheme. The coupling simulation of the flow past a nanotube shows that the proposed method can be further used in the study of microscopic fluid flow problems.

Specific features of the flow structure in a reactive type turbine stage

The results of experimental studies of the gas dynamics for a reactive type turbine stage are presented. The objective of the studies is the measurement of the 3D flow fields in reference cross sections, experimental determination of the stage characteristics, and analysis of the flow structure for detecting the sources of kinetic energy losses. The integral characteristics of the studied stage are obtained by averaging the results of traversing the 3D flow over the area of the reference cross sections before and behind the stage. The averaging is performed using the conservation equations for mass, total energy flux, angular momentum with respect to the axis z of the turbine, entropy flow, and the radial projection of the momentum flux equation. The flow parameter distributions along the channel height behind the stage are obtained in the same way. More thorough analysis of the flow structure is performed after interpolation of the experimentally measured point parameter values and 3D flow velocities behind the stage. The obtained continuous velocity distributions in the absolute and relative coordinate systems are presented in the form of vector fields. The coordinates of the centers and the vectors of secondary vortices are determined using the results of point measurements of velocity vectors in the cross section behind the turbine stage and their subsequent interpolation. The approach to analysis of experimental data on aerodynamics of the turbine stage applied in this study allows one to find the detailed space structure of the working medium flow, including secondary coherent vortices at the root and peripheral regions of the air-gas part of the stage. The measured 3D flow parameter fields and their interpolation, on the one hand, point to possible sources of increased power losses, and, on the other hand, may serve as the basis for detailed testing of CFD models of the flow using both integral and local characteristics. The comparison of the numerical and experimental results, as regards local characteristics, using statistical methods yields the quantitative estimate of their agreement.

Vectorized simulation of groundwater flow and contaminant transport using analytic element method and random walk particle tracking

AbstractGroundwater contaminant transport processes are usually simulated by the finite difference (FDM) or finite element methods (FEM). However, they are susceptible to numerical dispersion for advection‐dominated transport. In this study, a numerical dispersion‐free coupled flow and transport model is developed by combining the analytic element method (AEM) with random walk particle tracking (RWPT). As AEM produces continuous velocity distribution over the entire aquifer domain, it is more suitable for RWPT than FDM/finite element methods. Using the AEM solutions, RWPT tracks all the particles in a vectorized manner, thereby improving the computational efficiency. The present model performs a convolution integral of the response of an impulse contaminant injection to generate concentration distributions due to a permanent contaminant source. The RWPT model is validated with an available analytical solution and compared to an FDM solution, the RWPT model more accurately replicates the analytical solution. Further, the coupled AEM‐RWPT model has been applied to simulate the flow and transport in hypothetical and field aquifer problems. The results are compared with the FDM solutions and found to be satisfactory. The results demonstrate the efficacy of the proposed method.

Determination of two-dimensional velocity structure of Benioff zone based on inversion of body wave travel time (Case study of central Kamchatka)

A 2D variant of the inversion method for determining velocities within the Benioff zone of Kamchatka is developed with respect to the time of seismic wave travel from the foci group to Shipunskii station located in the region where the zone outcrops at the ocean bottom. The method is based on the idea of seismic tomography on the relationship between travel time discrepancies along the focus–station path and the value of seismic slowness, which is inverse to the velocity and corresponds to the gradient of the time field or the derivative of a hodograph with respect to the distance dt/dl. From this viewpoint, the field of discrepancies observed is the difference between the experimental and theoretical values of slowness. Its averaging with respect to depth and epicentral distance in 50 × 50 km rectangular windows and subsequent inversion make it possible to obtain a discrete velocity field using the GoldenSoftware Surfer program. Resmoothing with the same software leads to a variant of continuous velocity distribution = in the axial plane of the Benioff zone. The described procedure was used to calculate the velocities in this zone of the southern Kuril Islands and southern and central Kamchatka. The principal result in the latter case is identification of a sharp jump in the velocities of body waves in the upper mantle (up to 1.3 km/s for P-waves and up to 0.8 km/s for S-waves) beneath the Kronotskii Peninsula in the 7 years before the catastrophic Kronotskii earthquake that occurred in 1997 (M = 7.9) with an upthrow focal mechanism. This jump reflects the concentration of stresses in the epicentral zone of the earthquake. This result is important for medium-term forecasting of strong earthquakes.

Evidential Grid-Based Tracking and Mapping

Tracking and mapping the local environment form the basis of an autonomous vehicle system. They are often realized separately using occupancy grids, which do not require object or shape assumptions, and model-based object tracking algorithms. Many approaches require a binary classification of the sensor measurements into coming from a static or from a dynamic object, as otherwise inconsistencies between the different representations are likely to occur. This paper presents grid-based tracking and mapping (GTAM), a low-level grid-based approach that simultaneously estimates the static and the dynamic environment, their uncertainties, velocities, as well as information about free space. GTAM works on the level of grid cells, rather than creating object hypotheses. A particle filter is used to obtain continuous cell velocity distributions for all obstacles. Continuous evidences in a Dempster–Shafer model are derived without requiring a binary pre-classification of the sensor measurements. Results and evaluations using a vehicle moving in real dynamic street environments demonstrate the performance of the presented approach.

Throughput analysis of one-dimensional vehicular ad hoc networks

We consider content distribution in a one-dimensional vehicular ad hoc network. We assume that a file is encoded using fountain code, and the encoded message is cached at infostations. Vehicles are allowed to download data packets from infostations, which are placed along a highway. In addition, two vehicles can exchange packets with each other when they are in proximity. As long as a vehicle has received enough packets from infostations or from other vehicles, the original file can be recovered. In this work, we derive closed-form expressions for the average per-node throughput, under both discrete and continuous velocity distributions for the vehicles. Our result shows that the average per-node throughput can be expressed as a linear function of the densities of different classes of nodes in the highway. This result implies that the average per-node throughput scales linearly with vehicle arrival rate, and the average system throughput scales quadratically with vehicle arrival rate. Besides, system throughput reduces when overall mobility increases.

Large-Scale Vortex Generation Modeling

Methods of modeling vortex generation in computational fluid dynamics calculations without meshing the vortex generating device are investigated. In this way, the effect of adding vortices to existing flows can be assessed without the need to modify the computational grid; this can represent a significant saving. Previous work in this area has focused on boundary layer control. This study looks at larger scale applications, such as using vortices for force augmentation or directing flow. Two different approaches are used: modeling the vortex generator and modeling just the vortex alone. For the former, an existing method, which acts to align the flow with the vortex generator by adding a forcing term to the governing equations, is tested, but found to be unsuitable for use on this scale. The other approach is to add specified vortex velocity profiles, allowing the introduction of arbitrary vortices. A new version is developed to add continuous 3D velocity distributions in regions where desired vortices are to be created. It is implemented using several different forms of forcing. After basic testing, all methods are applied in a practical engineering case using a commercial solver.

Creating realistic models based on combined forward modeling and tomographic inversion of seismic profiling data

Amplitudes and shapes of seismic patterns derived from tomographic images often are strongly biased with respect to real structures in the earth. In particular, tomography usually provides continuous velocity distributions, whereas major velocity changes in the earth often occur on first-order interfaces. We propose an approach that constructs a realistic structure of the earth that combines forward modeling and tomographic inversion (FM&TI). Using available a priori information, we first construct a synthetic model with realistic patterns. Then we compute synthetic times and invert them using the same tomographic code and the same parameters as in the case of observed data processing. We compare the reconstruction result with the tomographicimage of observed data inversion. If a discrepancy is observed, we correct the synthetic model and repeat the FM&TI process. After several trials, we obtain similar results of synthetic and observed data inversion. In this case, the derived synthetic model adequately represents the real structure of the earth. In a working scheme of this approach, we three authors used two different synthetic models with a realistic setup. One of us created models, but the other two performed the reconstruction with no knowledge of the models. We discovered that the synthetic models derived by FM&TI were closer to the true model than the tomographic inversion result. Our reconstruction results from modeling marine data acquired in the Musicians Seamount Province in the Pacific Ocean indicate the capacity and limitations of FM&TI.

Discovery of Molecular Loop 3 in the Galactic Center: Evidence for a Positive-Velocity Magnetically Floated Loop towards L = 355$^\circ$–359$^\circ$

Abstract We have discovered a molecular dome-like feature towards 355$^{\circ}$$\leq$$l$$\leq$ 359$^{\circ}$ and 0$^{\circ}$$\leq$$b$$\leq$ 2$^{\circ}$. The large velocity dispersions of 50–100 km s$^{-1}$ of this feature are much larger than those in the Galactic disk, and indicate that the feature is located in the Galactic center, probably within $\sim $1 kpc of Sgr A$^{\ast}$. The distribution has a projected length of $\sim $600 pc and a height of $\sim $300 pc from the Galactic disk, and shows a large-scale monotonic velocity gradient of $\sim $130 km s$^{-1}$ per $\sim $600 pc. The feature is also associated with HI gas having a more continuous spatial and velocity distribution than that of ${}^{12}$CO. We interpret the feature as being a magnetically floated loop similar to loops 1 and 2, and name it “loop 3”. Loop 3 is similar to loops 1 and 2 in its height and length, but is different from loops 1 and 2 in that the inner part of loop 3 is filled with molecular emission. We have identified two foot points at both ends of loop 3. HI, ${}^{12}$CO, and ${}^{13}$CO datasets were used to estimate the total mass and the kinetic energy of loop 3 to be $\sim $3.0 $\times$ 10$^{6} M_\odot$ and $\sim $1.7 $\times$ 10$^{52} $erg. The huge size, velocity dispersions, and energy are consistent with the magnetic origin of the Parker instability, as in the case of loops 1 and 2, but is difficult to be explained by multiple stellar explosions. We argue that loop 3 is in an earlier evolutionary phase than loops 1 and 2 based on the inner-filled morphology and the relative weakness of the foot points. This discovery indicates that the western part of the nuclear gas disk of $\sim $1 kpc radius is dominated by the three well-developed magnetically floated loops, and suggests that the dynamics of the nuclear gas disk is strongly affected by magnetic instabilities.

Estimation of Vertical Continuous Stochastic Parameters from Seismic Reflection Data

This paper presents a scheme to invert reflected seismic wave fields for vertical spatial statistics of the Earth’s continuous stochastic velocity distribution. The forward model is based on a single scattering theory, where a seismogram is modeled as a convolution of a source wavelet and stochastic reflectivity, which obeys a von Karman autocorrelation function. Tests with synthetic data indicate that the method presented here recovers vertical characteristic length from seismic reflection data that is similar to that of the original velocity model; however, the inversion does not reliably recover the Hurst exponent, ν. The input wavelet’s power spectrum is a free parameter in the inversion and is estimated as well, but is band-limited relative to the original source wavelet. Noise degrades the quality of the estimates of the stochastic parameters such that the recovered stochastic parameters are biased according to the power spectrum of the noise.

Analysis of Short-Time Transient Dynamics of a Diffusion-Controlled Reaction in a Hard-Sphere Fluid Based on Fokker–Planck–Kramers Equation

Abstract Molecular dynamics (MD) simulations are performed for a simple diffusion-controlled reaction in a hard-sphere fluid. The short-time transient rate constants are compared with the Markovian Langevin dynamics (LD), two theories based on the Fokker–Planck–Kramers equation (FPKE), and the Smoluchowski–Collins–Kimball (SCK) theory based on the diffusion equation. At n* = 0.7856, where n* is the total number density reduced by the diameter of the molecules, molecular motions are diffusive and Markovian, and the MD results agree well with the LD results. At n* = 0.9428, the MD results are much larger than the LD ones at short times, and the discrepancy can be explained by non-Markovian effects. Also at n* = 0.2000, where inertia effects are important, the time profile of the MD rate constant is satisfactorily explained by LD, while the MD rate constant shows a smaller decay than at higher densities. FPKE theory with a continuous velocity distribution reproduces the LD results under all conditions indicating that the theory is valid over a wide density range. FPKE theory with a discontinuous velocity distribution cannot explain the simulation results at the lowest density despite taking into account inertia effects. Although SCK theory is expected to be valid only at high densities, the predicted rate constant is close to that from the simulation even at the lowest density, if the intrinsic rate constant is chosen to reproduce the exact value of the initial rate constant.

Application of Fokker-Planck-Kramers equation treatment for short-time dynamics of diffusion-controlled reaction in supercritical Lennard-Jones fluids over a wide density range

The validity of a Fokker-Planck-Kramers equation (FPKE) treatment of the rate of diffusion-controlled reaction at short times [K. Ibuki and M. Ueno, J. Chem. Phys. 119, 7054 (2003)] is tested in a supercritical Lennard-Jones fluid over a wide density range by comparing it with the Langevin dynamics and molecular dynamics simulations and other theories. The density n range studied is 0.323n(c)< or =n< or =2.58n(c) and the temperature 1.52T(c), where n(c) and T(c) are the critical density and temperature, respectively. For the rate of bimolecular reactions, the transition between the collision-limited and diffusion-limited regimes is expected to take place in this density range. The simulations show that the rate constant decays with time extensively at high densities, and that the magnitude of decay decreases gradually with decreasing density. The decay profiles of the rate constants obtained by the simulations are reproduced reasonably well by the FPKE treatment in the whole density range studied if a continuous velocity distribution is used in solving the FPKE approximately. If a discontinuous velocity distribution is used instead of the continuous one, the FPKE treatment leads to a rate constant much larger than the simulation results at medium and low densities. The rate constants calculated from the Smoluchowski-Collins-Kimball (SCK) theory based on the diffusion equation are somewhat smaller than the simulation results in medium and low densities when the intrinsic rate constant is chosen to adjust the steady state rate constant in the low density limit to that derived by the kinetic collision theory. The discrepancy is relatively small, so that the SCK theory provides a useful guideline for a qualitative discussion of the density effect on the rate constant.

Probabilistic ballistic annihilation with continuous velocity distributions.

We investigate the problem of ballistically controlled reactions where particles either annihilate upon collision with probability p, or undergo an elastic shock with probability 1-p. Restricting to homogeneous systems, we provide in the scaling regime that emerges in the long time limit, analytical expressions for the exponents describing the time decay of the density and the root-mean-square velocity, as continuous functions of the probability p and of a parameter related to the dissipation of energy. We work at the level of molecular chaos (nonlinear Boltzmann equation), and using a systematic Sonine polynomials expansion of the velocity distribution, we obtain in arbitrary dimension the first non-Gaussian correction and the corresponding expressions for the decay exponents. We implement Monte Carlo simulations in two dimensions, which are in excellent agreement with our analytical predictions. For p<1, numerical simulations lead to the conjecture that unlike for pure annihilation (p=1), the velocity distribution becomes universal, i.e., does not depend on the initial conditions.

Fokker–Planck–Kramers equation treatment of dynamics of diffusion-controlled reactions using continuous velocity distribution in three dimensions

A theory has been developed for the short-time dynamics of diffusion-controlled reactions based on the Fokker–Planck–Kramers equation (FPKE) in three dimensions. A continuous velocity distribution function has been proposed to solve the FPKE approximately. The present theory agrees better with the Langevin dynamics results than the earlier theory using a discontinuous velocity distribution. This indicates the validity of the present theory in three dimensions, because the Langevin dynamics results can be assumed to be the exact solutions to the FPKE. The theory is compared with molecular dynamics (MD) simulations in Lennard-Jones fluids to examine the applicability for realistic systems. The present theory predicts a somewhat smaller rate constant than the MD simulation in the time range of a few picoseconds. The discrepancies can be explained qualitatively in terms of the non-Markovian effect on the molecular motions.

Improved Velocity Distribution Applied to Fokker–Planck–Kramers Equation Treatment for Dynamics of Diffusion-Controlled Reactions in Two Dimensions

Abstract The validity of theoretical treatments of short-time dynamics of diffusion-controlled reaction based on the Fokker–Planck–Kramers’ equation (FPKE) has been examined using molecular dynamics (MD) simulation in two-dimensional Lennard–Jones fluids. First, we made Langevin dynamics (LD) simulation assuming that the relative diffusion coefficient is given by the sum of the self-diffusion coefficients and that the potential between the reactants is given by the potential of mean force. The LD and the MD simulation results for the time dependence of the survival probabilities of the reactants agree well. This indicates that the FPKE derived from the Langevin equation can be applied to the problem of diffusion-controlled reaction. The time-dependent Harris theory based on the FPKE and the approximation of the half-range Maxwellian velocity distribution showed disagreements with the MD simulation. Such a limitation of the theory has not been recognized in three-dimensional fluids. In the approximation used in the time-dependent Harris theory, the velocity distribution function is not continuous. We have shown that the agreement between the MD simulation and the theory can be improved when we use a new approximation based on a continuous velocity distribution function.