skip to main content

Title: Generalization of Hop Distance‐Time Scaling and Particle Velocity Distributions via a Two‐Regime Formalism of Bedload Particle Motions

To date, there is no consensus on the probability distribution of particle velocities during bedload transport, with some studies suggesting an exponential‐like distribution while others a Gaussian‐like distribution. Yet, the form of this distribution is key for the determination of sediment flux and the dispersion characteristics of tracers in rivers. Combining theoretical analysis of the Fokker‐Planck equation for particle motions, numerical simulations of the corresponding Langevin equation, and measurements of motion in high‐speed imagery from particle‐tracking experiments, we examine the statistics of bedload particle trajectories, revealing a two‐regime distance‐time (LTp) scaling for the particle hops (measured from start to stop). We show that particles of short hop distances scale asL~giving rise to the Weibull‐like front of the hop distance distribution, while particles of long hop distances transition to a different scaling regime ofL~Tpleading to the exponential‐like tail of the hop distance distribution. By demonstrating that the predominance of mostly long hop particles results in a Gaussian‐like velocity distribution, while a mixture of both short and long hop distance particles leads to an exponential‐like velocity distribution, we argue that the form of the probability distribution of particle velocities can depend on the physical environment within which particle transport occurs, explaining and unifying disparate views on particle velocity statistics reported in the literature.

more » « less
Award ID(s):
1811909 1209402
Author(s) / Creator(s):
 ;  ;  
Publisher / Repository:
DOI PREFIX: 10.1029
Date Published:
Journal Name:
Water Resources Research
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    Bedload particle hops are defined as successive motions of a particle from start to stop, characterizing one of the most fundamental processes of bedload sediment transport in rivers. Although two transport regimes have been recently identified for short and long hops, respectively, there is still the lack of a theory explaining the mean hop distance–travel time scaling for particles performing short hops, which dominate the transport and may cover over 80 % of the total hop events. In this paper, we propose a velocity-variation-based formulation, the governing equation of which is intrinsically identical to that of Taylor dispersion for solute transport within shear flows. The key parameter, namely the diffusion coefficient, can be determined by hop distances and travel times, which are easier to measure and more accurate than particle accelerations. For the first time, we obtain an analytical solution for the mean hop distance–travel time relation valid for the entire range of travel times, which agrees well with the measured data. Regarding travel times, we identify three distinct regimes in terms of different scaling exponents: respectively, $\sim$ 1.5 for the initial regime and $\sim$ 5/3 for the transition regime, which define the short hops, and 1 for the Taylor dispersion regime defining long hops. The corresponding distribution of the hop distance is analytically obtained and experimentally verified. We also show that the conventionally used exponential distribution, as proposed by Einstein, is solely for long hops. Further validation of the present formulation is provided by comparing the simulated accelerations with measurements. 
    more » « less
  2. Abstract. Despite a rich history of studies investigating fluid dynamics over bedforms and dunes in rivers, the spatiotemporal patterns of sub-bedform bedload transport remain poorly understood. Previous experiments assessing the effects of flow separation on downstream fluid turbulent structures and bedload transport suggest that localized, intermittent, high-magnitude transport events (i.e., permeable splat events) play an important role in both downstream and cross-stream bedload transport near flow reattachment. Here, we report results from flume experiments that assess the combined effects of flow separation–reattachment and flow re-acceleration over fixed two-dimensional bedforms (1.7 cm high; 30 cm long). A high-speed camera observed bedload transport along the entirety of the bedform at 250 frames per second. Grain trajectories, grain velocities, and grain transport directions were acquired from bedload images using semiautomated particle-tracking techniques. Downstream and vertical fluid velocities were measured 3 mm above the bed using laser Doppler velocimetry (LDV) at 15 distances along the bedform profile. Mean downstream fluid velocity increases nonlinearly with increasing distance along the bedform. However, observed bedload transport increases linearly with increasing distance along the bedform, except at the crest of the bedform, where both mean downstream fluid velocity and bedload transport decrease substantially. Bedload transport time series and manual particle-tracking data show a zone of high-magnitude, cross-stream transport near flow reattachment, suggesting that permeable splat events play an essential role in the region downstream of flow reattachment.

    more » « less
  3. Abstract

    Relativistic magnetically dominated turbulence is an efficient engine for particle acceleration in a collisionless plasma. Ultrarelativistic particles accelerated by interactions with turbulent fluctuations form nonthermal power-law distribution functions in the momentum (or energy) space,f(γ)dγγαdγ, whereγis the Lorenz factor. We argue that in addition to exhibiting non-Gaussian distributions over energies, particles energized by relativistic turbulence also become highly intermittent in space. Based on particle-in-cell numerical simulations and phenomenological modeling, we propose that the bulk plasma density has lognormal statistics, while the density of the accelerated particles,n, has a power-law distribution function,P(n)dnnβdn. We argue that the scaling exponents are related asβα+ 1, which is broadly consistent with numerical simulations. Non-space-filling, intermittent distributions of plasma density and energy fluctuations may have implications for plasma heating and for radiation produced by relativistic turbulence.

    more » « less
  4. Abstract

    We report the Earth's rate of radiogenic heat production and (anti)neutrino luminosity from geologically relevant short‐lived radionuclides (SLR) and long‐lived radionuclides (LLR) using decay constants from the geological community, updated nuclear physics parameters, and calculations of theβspectra. We track the time evolution of the radiogenic power and luminosity of the Earth over the last 4.57 billion years, assuming an absolute abundance for the refractory elements in the silicate Earth and key volatile/refractory element ratios (e.g., Fe/Al, K/U, and Rb/Sr) to set the abundance levels for the moderately volatile elements. The relevant decays for the present‐day heat production in the Earth (19.9 ± 3.0 TW) are from40K,87Rb,147Sm,232Th,235U, and238U. Given element concentrations in kg‐element/kg‐rock and densityρin kg/m3, a simplified equation to calculate the present‐day heat production in a rock isurn:x-wiley:ggge:media:ggge22244:ggge22244-math-0001

    The radiogenic heating rate of Earth‐like material at solar system formation was some 103to 104times greater than present‐day values, largely due to decay of26Al in the silicate fraction, which was the dominant radiogenic heat source for the first10 Ma. Assuming instantaneous Earth formation, the upper bound on radiogenic energy supplied by the most powerful short‐lived radionuclide26Al (t1/2= 0.7 Ma) is 5.5×1031 J, which is comparable (within a factor of a few) to the planet's gravitational binding energy.

    more » « less
  5. The relative velocities and positions of monodisperse high-inertia particle pairs in isotropic turbulence are studied using direct numerical simulations (DNS), as well as Langevin simulations (LS) based on a probability density function (PDF) kinetic model for pair relative motion. In a prior study (Rani et al. , J. Fluid Mech. , vol. 756, 2014, pp. 870–902), the authors developed a stochastic theory that involved deriving closures in the limit of high Stokes number for the diffusivity tensor in the PDF equation for monodisperse particle pairs. The diffusivity contained the time integral of the Eulerian two-time correlation of fluid relative velocities seen by pairs that are nearly stationary. The two-time correlation was analytically resolved through the approximation that the temporal change in the fluid relative velocities seen by a pair occurs principally due to the advection of smaller eddies past the pair by large-scale eddies. Accordingly, two diffusivity expressions were obtained based on whether the pair centre of mass remained fixed during flow time scales, or moved in response to integral-scale eddies. In the current study, a quantitative analysis of the (Rani et al. 2014) stochastic theory is performed through a comparison of the pair statistics obtained using LS with those from DNS. LS consist of evolving the Langevin equations for pair separation and relative velocity, which is statistically equivalent to solving the classical Fokker–Planck form of the pair PDF equation. Langevin simulations of particle-pair dispersion were performed using three closure forms of the diffusivity – i.e. the one containing the time integral of the Eulerian two-time correlation of the seen fluid relative velocities and the two analytical diffusivity expressions. In the first closure form, the two-time correlation was computed using DNS of forced isotropic turbulence laden with stationary particles. The two analytical closure forms have the advantage that they can be evaluated using a model for the turbulence energy spectrum that closely matched the DNS spectrum. The three diffusivities are analysed to quantify the effects of the approximations made in deriving them. Pair relative-motion statistics obtained from the three sets of Langevin simulations are compared with the results from the DNS of (moving) particle-laden forced isotropic turbulence for $St_{\unicode[STIX]{x1D702}}=10,20,40,80$ and $Re_{\unicode[STIX]{x1D706}}=76,131$ . Here, $St_{\unicode[STIX]{x1D702}}$ is the particle Stokes number based on the Kolmogorov time scale and $Re_{\unicode[STIX]{x1D706}}$  is the Taylor micro-scale Reynolds number. Statistics such as the radial distribution function (RDF), the variance and kurtosis of particle-pair relative velocities and the particle collision kernel were computed using both Langevin and DNS runs, and compared. The RDFs from the stochastic runs were in good agreement with those from the DNS. Also computed were the PDFs $\unicode[STIX]{x1D6FA}(U|r)$ and $\unicode[STIX]{x1D6FA}(U_{r}|r)$ of relative velocity $U$ and of the radial component of relative velocity $U_{r}$ respectively, both PDFs conditioned on separation $r$ . The first closure form, involving the Eulerian two-time correlation of fluid relative velocities, showed the best agreement with the DNS results for the PDFs. 
    more » « less