Model Of Velocity Distribution Research Articles

Approach to the Laboratory Modeling of the Flow Velocity Distribution behind a Hydro Turbine Runner. 1. Design of Swirlers Vanes

Approach to the Laboratory Modeling of the Flow Velocity Distribution behind a Hydro Turbine Runner. 1. Design of Swirlers Vanes

A probabilistic model on streamwise velocity profile in open channels using Tsallis relative entropy theory

The present study uses the concept of information entropy to derive a velocity distribution model in open channel flow. The Tsallis relative entropy theory is explored for that purpose, where the prior probability density function (PDF) is chosen from the maximum Tsallis entropy distribution. Both the cases of the fixed and varying entropy index are considered from the literature. The velocity equations are derived analytically using the homotopy analysis method (HAM) together with the Padé approximation technique in a general framework. The approximations and the HAM-based series solution are verified against laboratory and field data sets. Also, the prediction accuracies of the proposed models are assessed through the measures of statistical errors. It is seen that the model corresponding to the varying index is superior to the other model. The entropy index of Tsallis relative entropy is considered a variable, and the non-unity values of this index justify the applicability of the entropy function. Moreover, it is observed that one of the Lagrange multipliers corresponding to the velocity model with a varying index becomes negligible subject to the data sets considered, and hence, it simplifies the model development mathematically. The proposed approach can be further extended to develop models for estimating the suspended sediment concentration and shear stress distribution.

Modelling of Stratified River Bank Erosion Due to Undercutting

Theories of soil mechanics and basic hydraulic relations and principles of loose boundary channel hydraulics were combined and integrated to analyze stratified river bank failure due to undercutting (undermining) of a bottom loose layer. A near bank velocity distribution model was developed from which the boundary shear stress acting on the bank surface was determined. The model allows computation of the eroded soil volume from the bottom loose layer and the lateral undercut distance at any time in the flow hydrograph as long as the induced flow shear stress is greater than the particle critical shear stress resistance to hydraulic entrainment. At failure the size of the failure block and the ultimate critical value of the lateral undercut distance are determined. Therefore the annual rate of bank retreat and bank sediment load contribution can be identified for the reach under consideration. The application of the model to the River Nile hydrograph in Northern Sudan State showed a good agreement with field observations.

Enhanced ultrasonic total focusing imaging of CFRP corner with ray theory-based homogenization technique

Ultrasonic testing is effective in defect characterization and quality assurance of Carbon Fiber Reinforced Plastic (CFRP) components in the aerospace industry. Due to the coupling between complex shape and elastic anisotropy, the Phased Array Ultrasonic Testing (PAUT) and time-based Total Focusing Method (TFM) face significant challenges in the calculation of wave propagation. A wave velocity distribution model is established for a multidirectional convex corner of CFRP based on a homogenization theory and the above coupling effects are also incorporated. A ray-tracing method is proposed based on Dijkstra’s shortest path search algorithm. The predicted time of flight ensures that this technique, the homogenized TFM, could synthesize a high-quality focused image by post-processing on the full matrix capture data. Experiments on a laminate with three ϕ1.5 mm Side-Drilled Holes (SDHs) in different circumferential directions confirm a successful homogenized TFM imaging that all SDHs can be effectively detected. As compared to the isotropic scenario, the maximum positioning error is reduced to 0.12, 0.08, and 0.38 mm, and the Signal-to-Noise Ratios (SNRs) are increased by 2.1, 1.1, and 11.8 dB, respectively. It is suggested that the ray-tracing assisted TFM technique can effectively improve the imaging of corners in CFRP components.

A Semicoupled Shallow-Water Model for Vertical Velocity Distribution in an Open Channel with Submerged Flexible Vegetation

A Semicoupled Shallow-Water Model for Vertical Velocity Distribution in an Open Channel with Submerged Flexible Vegetation

Cotton canopy airflow simulation and velocity attenuation model based upon 3D phenotype and stratified sub-regional porous medium

Airflow assisted with sprayers can change not only the canopy’s porosity, but also allow the droplets to enter its interior. It is important to depict the airflow velocity distribution in a dense crop canopy quantitatively to optimize the goal of uniform deposition of droplets and full coverage within the canopy. However, there is yet no effective way to do so. The contribution of this paper is to propose a digital method to predict airflow distribution in the canopy using computational fluid dynamics (CFD) and machine-learning (ML) methods. Firstly, a virtual, simplified 3D cotton plant is generated based upon the phenotypic traits and the Logistic growth function to describe such canopy architecture parameters as leaf area density (LAD) and porosity quantitatively at any position in the canopy. Secondly, to solve the problem of uneven distribution of airflow in the canopy, in the CFD model, the target area surrounded by the main stems of 4 adjacent 3D cotton plants is selected as a stratified sub-regional porous medium (SSPM), and the accuracy of the CFD simulations is compared to the indoor measurements. Thereafter, a range of ML algorithms are trained with the velocity dataset obtained from the CFD results under different spray operating parameters (initial air velocity, canopy depth, LAD, and porosity). The comparison of CFD simulations and actual airflow measurements shows that the mean normalized mean absolute errors (NMAEs) of the lower, middle, and upper layers are 17.38 %, 21.35 %, and 9.75 % respectively. The Random Forest method has greater prediction accuracy, with a Coefficient of Determination (R2) of 0.9882 and Root Mean Squared Error (RMSE) of 0.5071, which indicates excellent agreement with the CFD simulation results. Therefore, such a data-driven model of airflow velocity distribution in a dense crop canopy can be used to optimize the operation parameters and eliminate complex CFD simulations or tedious physical experiments.

A dip-corrected discharge estimation method for a river system

This work proposes a novel dip corrected velocity distribution model in combination with the entropy theory for discharge estimation in a braided river. A modified form of the dip correction factor is derived by considering the topographical complexities and applied for assessing velocity profiles in river cross-sections. The velocity profiles at different verticals are computed by employing Shanon's entropy theory. The depth-averaged velocities at different cross-sections are estimated from the computed vertical velocity profiles and substituted in the area-velocity method for the discharge calculation. The model is applied to two study areas, Majuli and Umananda of the Brahmaputra River, having both simple and braided sections. The validation of the model is performed using the observed discharge data available at the nearby gauge site for low flow condition. Results indicate that the integration of bed rugosity factor in dip corrected velocity distributions improves the accuracy of discharge estimations.

A new calculation model of critical liquid-carrying velocity in inclined wells based on the principle of liquid film delamination slippage

Liquid loading has always been a production problem in the middle and late production of gas wells. Severe loading will even lead to the shutdown of gas wells. In actual production, critical liquid-carrying velocity is often used to judge the loading of gas wells. Most conventional calculation models are modified vertical pipe models for the critical liquid-carrying gas velocity of an inclined pipe. They are established based on liquid film inversion without considering the characteristics of liquid film delamination slippage. In this study, the elliptical distribution model of the liquid film of the inclined pipe is first established. Then the cross-sectional velocity distribution model is coupled to solve the cross-sectional flow rate integrally. When the cross-sectional liquid flow rate is 0, the gas velocity is the critical liquid carrying velocity. Model calculations reveal that the critical liquid-carrying velocity is the largest when the inclination angle is 50°, consistent with experimental data. Using the data of production wells in the Changning block, Sichuan, China, to analyze well loading, the accuracy between the prediction results of loading in this study and field production judgment reached 93.75%, which is 25% higher than that of the Belfroid model. The model in this study accounts for the delamination slippage of the liquid film for the first time, and the calculated critical liquid carrying velocity can better reflect the flow state of gas-liquid two-phase flow in inclined pipe, which is beneficial for researchers to accurately evaluate and understand the production status of gas wells.

Explosively Formed Projectile from Stepped Casing Shaped Charge: Formation Mechanism of Canted Fins

ABSTRACT Canted fins can significantly improve the flight stability and the on-target accuracy of Explosively Formed Projectiles (EFP). The current work uses the effective charge theory to derive the formation of periodic tails from shaped charge with stepped casing by considering the detonation impulse. According to the established model, the distribution of velocity along the circumference of the liner increases linearly with the square root of the casing thickness. There are two stages in the formation of canted fins influenced by the stepped casing: (1) the distribution of velocity along the circumference of the liner upon initiation, and (2) the torsion and folding of the liner edge during deformation. Numerical simulation was carried out with LS-DYNA for shaped charge (56 mm in diameter) with regular and stepped casing. The expansion of the casing and the deformation of the liner with velocity distribution at initial times were compared between the regular and the stepped casing. The distribution regularities and the final velocity of the canted fin EFP from the simulation agree well the established velocity distribution model. Soft recovery experiments were also carried out to prove the feasibility of using shaped charge with stepped casing.

Perturbed distribution functions with accurate action estimates for the Galactic disc

In the Gaia era, understanding the effects of the perturbations of the Galactic disc is of major importance in the context of dynamical modelling. In this theoretical paper we extend previous work in which, making use of the epicyclic approximation, the linearized Boltzmann equation had been used to explicitly compute, away from resonances, the perturbed distribution function of a Galactic thin-disc population in the presence of a non-axisymmetric perturbation of constant amplitude. Here we improve this theoretical framework in two distinct ways in the new code that we present. First, we use better estimates for the action-angle variables away from quasi-circular orbits, computed from the AGAMA software, and we present an efficient routine to numerically re-express any perturbing potential in these coordinates with a typical accuracy at the per cent level. The use of more accurate action estimates allows us to identify resonances such as the outer 1:1 bar resonance at higher azimuthal velocities than the outer Lindblad resonance, and to extend our previous theoretical results well above the Galactic plane, where we explicitly show how they differ from the epicyclic approximation. In particular, the displacement of resonances in velocity space as a function of height can in principle constrain the 3D structure of the Galactic potential. Second, we allow the perturbation to be time dependent, thereby allowing us to model the effect of transient spiral arms or a growing bar. The theoretical framework and tools presented here will be useful for a thorough analytical dynamical modelling of the complex velocity distribution of disc stars as measured by past and upcoming Gaia data releases.

Hydraulic flow mixing model

Based on the theory of a flooded jet moving in a concomitant flow, a hydraulic model of the downhole jet pump working process in the form of a central potential and coaxial annular flows is proposed. In the process of building a hydraulic model, the geometric dimensions of the elements of the flow part of the jet pump and the ratio of the costs of the working and ejected flow of the downhole ejection system are taken into account. The central potential flow is characterized by a constant velocity profile. Peripheral shear annular flow is characterized by an uneven distribution of kinematic parameters and can be approximated by elementary functions. The proposed velocity profile in the annular boundary layer maintains the approximate self-similarity throughout the initial section of theworking flow of the jet pump. To characterize the non-uniform distribution of kinematic parameters in the mixing chamber of the jet pump, the ratio of the average and maximum speed of the mixed flows is used. In the process of integration of the proposed velocity profiles, an analytical dependence is obtained, which connects the generalized coefficient of non-uniformity of distribution of kinematic parameters in the mixing chamber of the jet pump with the characteristics of the downhole ejection system. According to the obtained analytical dependence, the value of the coefficient of non-uniformity of the distribution of kinematic parameters is directly proportional to the distance between the working nozzle and the mixing chamber of the jet pump. The proposed model of velocity distribution in the inlet section of the mixing chamber allows to reduce the error of analytical determination of the jet pump pressure and increase the efficiency of designing the operating parameters of the well ejection system.

Study on Flow Velocity Distribution in Open Channel With Flexible Vegetation

Ecological management of river channels is a hot topic for current sustainable development and flow measurement of ecological river is an important part. In this article, a flow velocity distribution model of the channel containing flexible vegetation is constructed from the vegetation riverbed theory and the bursting phenomenon to reveal the microscopic mechanism of the flow velocity distribution in the upper layer of vegetation. In the vegetation riverbed law, the effect of flexible vegetation is evaluated by the mixed length formula. The bursting phenomenon law considers the influence of the channel sidewalls on the flow and a two-dimensional velocity model is established by introducing the concept of average turbulence structure. The mechanism of the downward shift of the maximum flow velocity point on the channel sidewall is explained. The verification of the calculated velocity profiles is carried out based on data obtained in laboratory experiments. The results show that the combination of the two models can well describe the velocity distribution of the whole channel. At the end, the phenomenon of flow velocity zoning in open channel is discussed, which provides a solution for flow measurement in ecological channel.

An analytical two-layer model for velocity distribution in open-channel flows with submerged flexible canopies considering multiply fluids mechanics

Submerged flexible vegetation such as reeds and sedge are quite common in rivers during flood seasons. Different from rigid aquatic vegetation like shurbs, flexible vegetation bends under the effect of streamwise flow, making the velocity profile even more complicated. This study aims to build an analytical model for predicting the vertical velocity profile in an open channel flow fully covered by submerged flexible vegetation whose projection area varies in vertical direction. A classical two-layer approach is applied which divides the flow into a vegetation layer and a free surface layer using the top of the vegetation patch as the boundary. For the vegetation layer, the vegetation-induced drag force is assumed to be proportion to the local streamwise velocity and projection area considering the influence of vegetation bending. In order to obtain the vertical velocity profile, width-averaged Navier-Stokes equation is solved with a first order closure scheme for the Reynolds stress. For the free surface layer, similar approach as the vegetation layer is applied while a wake function is added into the solution and the vegetation-induced drag force is considered to be zero in this layer. The average relative error between the results of current analytical model and experimental measurements is 4%–11%, indicating the applicability and accuracy of the model to predict the vertical velocity profile.

Deep Velocity Structure of Southeast Asia from Rayleigh Wave Group Velocities: 3D Isotropic Model of the S-Wave Velocity Distribution in the Upper Mantle

Deep Velocity Structure of Southeast Asia from Rayleigh Wave Group Velocities: 3D Isotropic Model of the S-Wave Velocity Distribution in the Upper Mantle

Study on the Horizontal Distribution Law of Flood Water and Sediment Factors under the Effect of Vegetation on a Curved Beach

The sediment-laden floodplain flood is affected by beach vegetation and the shape of curved compound channels. The laws of water and sediment exchange and deposition distribution in beach troughs are very complex and play a significant role in the formation and development of secondary suspended rivers, the adjustment of beach horizontal gradients, and even the evolution of flood control situations. This study used a combination of experimental simulations and theoretical research to carry out a generalized model test of floodplain flood evolution, analyzing the transverse distribution characteristics of sediment-laden flow and sediment factors in a curved compound channel under the conditions of beach vegetation, proposing a theoretical model of transverse distribution of velocity and sediment concentration that is based on the momentum equation considering the inertial force of the lateral secondary flow and river curvature. The results showed the following: (1) The model test results for floodplain flood in the compound channel with curved vegetation showed that the main stream was not only concentrated in the main channel but also appeared near the foot of the left and right bank levees and formed flood discharge along the embankment, as the beach siltation was mainly concentrated in the beach lip; (2) The arrangement of full vegetation on the beach had a uniform effect on the velocity distribution of the beach, which can reduce the phenomenon of excessive velocity at the foot of the beach and increase the velocity effect in the main channel; and (3) Through five numerical examples, the lateral velocity distribution model of a curved compound channel with beach vegetation was tested and, in general, the analysis model was consistent with the experimental results. The research results will provide a theoretical basis for river management and have great significance for enriching the basic theory of water and sediment movement and promoting the integration of hydraulics, river dynamics, and ecology.

Transversely isotropic lower crust of Variscan central Europe imaged by ambient noise tomography of the Bohemian Massif

Abstract. The recent development of ambient noise tomography, in combination with the increasing number of permanent seismic stations and dense networks of temporary stations operated during passive seismic experiments, provides a unique opportunity to build the first high-resolution 3-D shear wave velocity (vS) model of the entire crust of the Bohemian Massif (BM). This paper provides a regional-scale model of velocity distribution in the BM crust. The velocity model with a cell size of 22 km is built using a conventional two-step inversion approach from Rayleigh wave group velocity dispersion curves measured at more than 400 stations. The shear velocities within the upper crust of the BM are ∼0.2 km s−1 higher than those in its surroundings. The highest crustal velocities appear in its southern part, the Moldanubian unit. The Cadomian part of the region has a thinner crust, whereas the crust assembled, or tectonically transformed in the Variscan period, is thicker. The sharp Moho discontinuity preserves traces of its dynamic development expressed in remnants of Variscan subductions imprinted in bands of crustal thickening. A significant feature of the presented model is the velocity-drop interface (VDI) modelled in the lower part of the crust. We explain this feature by the anisotropic fabric of the lower crust, which is characterised as vertical transverse isotropy with the low velocity being the symmetry axis. The VDI is often interrupted around the boundaries of the crustal units, usually above locally increased velocities in the lowermost crust. Due to the north-west–south-east shortening of the crust and the late-Variscan strike-slip movements along the north-east–south-west oriented sutures preserved in the BM lithosphere, the anisotropic fabric of the lower crust was partly or fully erased along the boundaries of original microplates. These weakened zones accompanied by a velocity increase above the Moho (which indicate an emplacement of mantle rocks into the lower crust) can represent channels through which portions of subducted and later molten rocks have percolated upwards providing magma to subsequently form granitoid plutons.

Study of micro-dimples fabricated on alumina-based ceramics using micro-abrasive jet machining

• Micro-texturing with desired morphology characteristics was firstly explored on ceramics using micro-air jet machining. • The model of abrasive velocity distribution was amended and found to be a good indicator to present the micro-texturing performance. • Introducing micro-patterns on ceramics using micro-AJM is verified to be an effective way to enhance wear resistance in a lubricant system. This paper presents a study of fabricating micro-dimples with desired morphology characteristics using micro-abrasive jet machining. Experiments are conducted under different processing conditions so that blasted dimples with various sizes and geometries could be obtained and analysed. The results indicated that among all the processing factors used in this study, changing processing time and air pressure is found to be the most efficient and controllable way to vary dimple dimension. On the other hand, nozzle stand-off distance and air pressure both play a pivotal role in determining dimple geometry. The results indicate that a variety of desired morphologies on the blasted area could be obtained by optimizing parameter combination. To further reveal the mechanism of shape variation, an amended model of cross-sectional particle velocity distribution is proposed. By correlating the working conditions and the final form of machined micro-structures, the model is confirmed to be an effective indicator to present the machining performance. In the end, friction tests and wear characterization are conducted to verify the positive effect of the introduced micro-texture on improving the lifespan of ceramic-based workpieces.

Large gyro-orbit model of ion velocity distribution in plasma near a wall in a grazing-angle magnetic field

A model is presented for the ion distribution function in a plasma at a solid target with a magnetic field $\boldsymbol {B}$ inclined at a small angle, $\alpha \ll 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $\alpha \gg \sqrt {Zm_{e}/m_{i}}$ , where $m_{e}$ and $m_{i}$ are the electron and ion mass, respectively, and $Z$ is the charge state of the ion. An electric field $\boldsymbol {E}$ is present to repel electrons, and so the characteristic size of the electrostatic potential $\phi$ is set by the electron temperature $T_{e}$ , $e\phi \sim T_{e}$ , where $e$ is the proton charge. An asymptotic scale separation between the Debye length $\lambda _{D} = \sqrt {\epsilon _0 T_{{e}} / e^{2} n_{{e}} }$ , the ion sound gyro-radius $\rho _{s} = \sqrt { m_{i} ( ZT_{e} + T_{i} ) } / (ZeB)$ and the size of the collisional region $d_{c} = \alpha \lambda _{\textrm {mfp}}$ is assumed, $\lambda _{D} \ll \rho _{s} \ll d_{c}$ . Here $\epsilon _0$ is the permittivity of free space, $n_{e}$ is the electron density, $T_{i}$ is the ion temperature, $B= |\boldsymbol {B}|$ and $\lambda _{\textrm {mfp}}$ is the collisional mean free path of an ion. The form of the ion distribution function is assumed at distances $x$ from the wall such that $\rho _{s} \ll x \ll d_{c}$ , that is, collisions are not treated. A self-consistent solution of the electrostatic potential for $x \sim \rho _{s}$ is required to solve for the quasi-periodic ion trajectories and for the ion distribution function at the target. The large gyro-orbit model presented here allows to bypass the numerical solution of $\phi (x)$ and results in an analytical expression for the ion distribution function at the target. It assumes that $\tau =T_{i}/(ZT_{e})\gg 1$ , and ignores the electric force on the quasi-periodic ion trajectory until close to the target. For $\tau \gtrsim 1$ , the model provides an extremely fast approximation to energy–angle distributions of ions at the target. These can be used to make sputtering predictions.

A Semianalytic Model of the Pairwise Velocity Distribution between Dark Matter Halos

Abstract We study the probability distribution function (PDF) of relative velocity between two different dark matter halos (i.e., pairwise velocity) with high-resolution cosmological N-body simulations. We revisit a non-Gaussian framework to predict pairwise velocity statistics developed in Tinker. We investigate the pairwise velocity PDFs over a wide range of halo masses of 1012.5 ≲ M [h −1 M ⊙] ≲ 1015 and redshifts of 0 < z < 1. At a given set of masses, redshift, and separation length between two halos, our model requires three parameters to set the pairwise velocity PDF, whereas previous non-Gaussian models in the literature assumed four or more free parameters. At length scales of 5 < r [h −1 Mpc] < 40, our model predicts the mean and dispersion of the pairwise velocity for dark matter halos with masses of 1012.5 ≲ M [h −1 M ⊙] ≲ 1013.5 at 0.3 < z < 1 with a 5%-level precision. We demonstrate that our model of the pairwise velocity PDF provides an accurate mapping of the two-point clustering of massive-galaxy-sized halos at scales of O(10)h −1 Mpc between redshift and real space for a given real-space correlation function. For a mass-limited halo sample with masses greater than 1013.5 h −1 M ⊙ at z = 0.55, our model can explain the monopole and quadrupole moments of the redshift-space two-point correlations with a precision better than 5% at the scales of 5−40 and 10–30 h −1 Mpc, respectively. Our model of the pairwise velocity PDF will give a detailed explanation of the statistics of massive galaxies at intermediate scales in redshift surveys.

A quasi three dimensional hydrodynamic model for velocity distribution in open channel

ABSTRACT This work aims at developing a fully coupled numerical model by integrating the two-dimensional shallow water equations with entropy theory. TVD Mac-Cormack predictor-corrector finite difference scheme is employed to solve the modified form of shallow water equation in MATLAB. Computed two-dimensional hydraulic parameters are substituted in the entropy theory to calculate the vertical velocity profile in the domain. The coupled model applied at different locations in the Brahmaputra river and the computed outputs validated with the observed field data. The robustness and stability of the linked model are found satisfactory. The influence of the hydraulic structures on secondary current generation and the velocity dip phenomenon is investigated during the study.