Velocity Equation Research Articles

Summary of Experiments and Influencing Factors of Sediment Settling Velocity in Still Water

Sediment deposition significantly impacts soil erosion processes, consequently influencing the geographical morphology and surrounding environments of reservoirs and estuaries. Given the intricate nature of sediment deposition, it is imperative to consolidate and analyze existing research findings. Presently, studies on sediment settling velocity primarily employ theoretical, laboratory, and field experimentation methods. Theoretical approaches, rooted in mechanics, examine the various forces acting on sediment particles in water to derive settling velocity equations. However, they often overlook external factors like temperature, salinity, organic matter, and pH. Although laboratory experiments scrutinize the influence of these external factors on sedimentation velocity, sediment settling is not solely influenced by individual factors but rather by their collective interplay. Field observations offer the most accurate depiction of sediment deposition rates. However, the equipment used in such experiments may disrupt the natural sedimentation process and damage flocs. Moreover, measurements of sediment particle size from different instruments yield varied results. Additionally, this paper synthesizes the impact of suspended sediment concentration, particle size, shape, temperature, salinity, and organic matter on sediment settling velocity. Future research should focus on innovating new laboratory observation methods for sediment settling velocity and utilizing advanced scientific and technological tools for on-site measurements to provide valuable insights for further investigation into sediment settling velocity.

Stability for a system of the 2D incompressible magneto-micropolar fluid equations with partial mixed dissipation

This paper focuses on the 2D incompressible anisotropic magneto-micropolar fluid equations with vertical dissipation, horizontal magnetic diffusion, and horizontal vortex viscosity. The goal is to investigate the stability of perturbations near a background magnetic field in the 2D magneto-micropolar fluid equations. Two main results are obtained. The first result is based on the linear system. Global existence for any large initial data and asymptotic linear stability are established. The second result explores stability for the nonlinear system. It is proven that if the initial data are sufficiently small, then the solution for some perturbations near a background magnetic field remains small. Additionally, the long-time behaviour of the solution is presented. The most challenging terms in the proof are the linear terms in the velocity equation and the micro-rotation equation that will grow with respect to time t. We are able to find some background fields to control the growth of the linear terms. Our results reveal that some background fields can stabilise electrically conducting fluids.

A Kalman Filtering Algorithm for Measurement Interruption Based on Polynomial Interpolation and Taylor Expansion.

Combined SINS/GPS navigation systems have been widely used. However, when the traditional combined SINS/GPS navigation system travels between tall buildings, in the shade of trees, or through tunnels, the GPS encounters frequent signal blocking, which leads to the interruption of GPS signals, and as a result, the combined SINS/GPS-based navigation method degenerates into a pure inertial guidance system, which will lead to the accumulation of navigation errors. In this paper, an adaptive Kalman filtering algorithm based on polynomial fitting and a Taylor expansion is proposed. Through the navigation information output from the inertial guidance system, the polynomial interpolation method is used to construct the velocity equation and position equation of the carrier, and then the Taylor expansion is used to construct the virtual measurement at the moment of the GPS signal interruption, which can make up for the impact of the lack of measurement information on the combined SINS/GPS navigation system when the GPS signal is interrupted. The results of computer simulation experiments and road measurement tests based on the loosely combined SINS/GPS navigation system show that when the carrier faces a GPS signal interruption situation, compared with a combined SINS/GPS navigation algorithm that does not take any rescue measures, our proposed combined SINS/GPS navigation algorithm possesses a higher accuracy in the attitude angle estimation, a higher accuracy in the velocity estimation, and a higher accuracy in the positional localization, and the system possesses higher stability.

Soret Effect on Mixed Convection of Casson Fluid Flow in Presence of Porous Media in a Porous Channel

The purpose is to investigate the impact of Soret on free and mixed convective Casson fluid flow in presence of amplification in a porous channel with magnetic field is carried out analytically. The obtained are coupled PDEs are highly nonlinear and are converted to an ODEs by similarity transformation and further solved by using perturbation technique to obtain velocity, concentration and temperature equations of the physical system. The influence of non-dimensional parameters such as Prandtl number Pr, Schmidt number Sc, thermal buoyancy parameter λ, Reynolds number R, chemical reaction rate γ, concentration buoyancy parameter N, Darcy number Da, Casson fluid parameter β, Hartmann number M2 , Soret effect Sr on velocity, thickness of fluid and heat have been discussed with graphs. The few important computational outcomes reveal that the Soret effect Sr enriches the concentration, fluid flow and temperature of fluid whereas Casson fluid parameter β diminishes the profiles. The earlier work and present work have been compared in the absence of Soret effect and is establishes to be good contract.

Finite element modeling of the rotational dynamics of a single magnetic particle in a strong magnetic field and liquid medium

We have performed a comparative computational study of magnetic particle alignment in a strong magnetic field and ambient viscous fluid using the first-principles Navier–Stokes equation as well as numerical modeling using the finite element method (FEM). FEM solution has been compared with the solution of the unsteady rotations of a magnetic particle in a viscous fluid during the alignment process described with nonlocal integro-differential equations (IDEs) for torque and angular velocity. The assumption of nonlocality comes from the history term of acceleration torque with a non-Basset kernel function, which has its origin in the presence of vortices in the flow of a particle fluid environment due to its unsteady rotation and ambient fluid inertia and friction. The flow vortices of the ambient fluid are explicitly shown in the solution of the FEM model. Moreover, the solution of the time evolution of the alignment angle and angular velocity for the FEM model are in good agreement with an IDE model which we have recently developed. The obtained results are justification for interchangeability of the FEM and IDE models, which may have important consequences for large-scale simulations of magnetic microparticles.

Kinetic closures for unmagnetized and magnetized plasmas

Parallel and perpendicular closures with cyclotron resonance effects retained for the five-moment (density, temperature, and flow velocity) fluid equations are derived by solving the kinetic equation with the Bhatnagar–Gross–Krook operator in Fourier space. For parallel propagation, the parallel closures are reduced to those of Ji et al. [Phys. Plasmas 20, 082121 (2013)]. The closures when combined to the fluid equations reproduce the fully kinetic dispersion relation that can be directly derived from the kinetic equation. The closures for the five-moment fluid system can be utilized to derive closures for the extended fluid system, which is demonstrated by deriving closures for the ten-moment system consisting of density, flow velocity, temperature, and viscosity tensor equations.

A Strange Result Regarding Some MHD Motions of Generalized Burgers’ Fluids with a Differential Expression of Shear Stress on the Boundary

In this work, we investigate isothermal MHD motions of a large class of rate type fluids through a porous medium between two infinite horizontal parallel plates when a differential expression of the non-trivial shear stress is prescribed on the boundary. Exact expressions are provided for the dimensionless steady state velocities, shear stresses and Darcy’s resistances. Obtained solutions can be used to find the necessary time to touch the steady state or to bring to light certain characteristics of the fluid motion. Graphical representations showed the fluid moves slower in presence of a magnetic field or porous medium. In addition, contrary to our expectations, the volume flux across a plane orthogonal to the velocity vector per unit width of this plane is zero. Finally, based on a simple remark regarding the governing equations of velocity and shear stress for MHD motions of incompressible generalized Burgers’ fluids between infinite parallel plates, provided were the first exact solutions for MHD motions of these fluids when the two plates apply oscillatory or constant shear stresses to the fluid. This important remark offers the possibility to solve any isothermal MHD motion of these fluids between infinite parallel plates or over an infinite plate when the non-trivial shear stress is prescribed on the boundary. As an application, steady state solutions for MHD motions of same fluids have been developed when a differential expression of the fluid velocity is prescribed on the boundary.

A homogenized two-phase computational framework for meso- and macroscale blood flow simulations

Background and ObjectiveOwing to the complexity of physics linked with blood flow and its associated phenomena, appropriate modeling of the multi-constituent rheology of blood is of primary importance. To this effect, various kinds of computational fluid dynamic models have been developed, each with merits and limitations. However, when additional physics like thrombosis and embolization is included within the framework of these models, computationally efficient scalable translation becomes very difficult. Therefore, this paper presents a homogenized two-phase blood flow framework with similar characteristics to a single fluid model but retains the flow resolution of a classical two-fluid model. The presented framework is validated against four different sets of experiments. MethodsThe two-phase model of blood presented here is based on the classical diffusion-flux framework. Diffusion flux models are known to be less computationally expensive than two-fluid multiphase models since the numerical implementation resembles single-phase flow models. Diffusion flux models typically use empirical slip velocity correlations to resolve the motion between phases. However, such correlations do not exist for blood. Therefore, a modified slip velocity equation is proposed, derived rigorously from the two-fluid governing equations. An additional drag law for red blood cells (RBCs) as a function of volume fraction is evaluated using a previously published cell-resolved solver. A new hematocrit-dependent expression for lift force on RBCs is proposed. The final governing equations are discretized and solved using the open-source software OpenFOAM. ResultsThe framework is validated against four sets of experiments: (i) flow through a rectangular microchannel to validate RBC velocity profiles against experimental measurements and compare computed hematocrit distributions against previously reported simulation results (ii) flow through a sudden expansion microchannel for comparing experimentally obtained contours of hematocrit distributions and normalized cell-free region length obtained at different flowrates and inlet hematocrits, (iii) flow through two hyperbolic channels to evaluate model predictions of cell-free layer thickness, and (iv) flow through a microchannel that mimics crevices of a left ventricular assist device to predict hematocrit distributions observed experimentally. The simulation results exhibit good agreement with the results of all four experiments. ConclusionThe computational framework presented in this paper has the advantage of resolving the multiscale physics of blood flow while still leveraging numerical techniques used for solving single-phase flows. Therefore, it becomes an excellent candidate for addressing more complicated problems related to blood flow, such as modeling mechanical entrapment of RBCs within blood clots, predicting thrombus composition, and visualizing clot embolization.

A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media

AbstractThis paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.

Extended landslide velocity and analytical drag

The landslide velocity plays a dominant role in estimating the impact force and devastated area. Here, based on Pudasaini and Krautblatter (Earth Surf Dyn 10:165–189, 2022. https://doi.org/10.5194/esurf-10-165-2022), I develop a novel extended landslide velocity model that includes the force induced by the hydraulic pressure gradient, which was neglected by all the existing analytical landslide velocity models. By a rigorous conversion between this force and inertia, which facilitates constructing exact analytical solutions for velocity, I develop two peer systems expecting to produce the same result. However, this contradicts with our conventional wisdom. This raises a legitimate question of whether we should develop some new balance equations such that these phenomena can be better explained naturally. I compare the two velocity models that neglect and include the force induced by the hydraulic pressure gradient. Analytical solutions produced by the two systems are fundamentally different. The new model is comprehensive, elegant, and yet an extraordinary development as it reveals serendipitous circumstance resulting in a pressure–inertia paradox. Surprisingly, the mass first moves upstream for quite a while; then, it winds back and continues accelerating down slope. The difference between the extended and simple solution is significant, and widens strongly as the force associated with the hydraulic pressure gradient increases, demonstrating the importance of this force in the landslide velocity. The viscous drag is an essential dissipative force mechanism and plays an important role in controlling the landslide dynamics. However, no explicit mechanical and analytical model exists to date for this. The careful sagacity of the graceful form of new velocity equation results in a plain, yet mechanically extensive, analytical model for viscous drag, the first of this kind. It contains several physical and geometrical parameters, and evolves dynamically as it varies inversely with the flow depth. A dimensionless drag number is constructed characterizing the drag dynamics. Importance of the drag model is explained. In contrast with the prevailing practices, I have proved that drags are essentially different for the expanding and contracting motions. This is an entirely novel revelation. Drag coefficients are close to the empirical or numerical values often used in practice. But, now, I offer an innovative, physically founded analytical model for the drag that can be instantly applied in mass flow simulations.

On the Governing Equations for Velocity and Shear Stress of some Magnetohydrodynamic Motions of Rate-type Fluids and their Applications

The governing equations for the shear stress corresponding to some magnetohydrodynamic (MHD) motions of a large class of rate-type fluids are brought to light. In rectangular domains, the governing equations of velocity and shear stress are identical as form. The provided governing equations can be used to solve motion problems of such fluids when shear stress is prescribed on the boundary. For illustration, the motion in an infinite circular cylinder with shear stress on the boundary is discussed.

Theoretical derivation and a priori validation of a new scalar variance-based sub-grid drag force model for simulation of gas–solid fluidized beds

A novel sub-grid drag force model is proposed for coarse-grid Euler–Euler simulation of gas–solid fluidized beds. Starting from a transport equation for the drift velocity, an equilibrium condition is used as a basis to derive a new algebraic drift velocity model. The sub-grid correlations that show up are closed by a large-eddy PDF approach inspired from LES of turbulent reacting flows. The new analytical model only depends on the resolved slip velocity and on a few sub-grid moments of the solid volume fraction. Then, a conditional averaging procedure shows that the new model can be properly captured by a simple functional expression that only requires a closure for the sub-grid variance of the solid volume fraction. A priori validation studies show that the drift velocity is predicted with high accuracy (R2>0.90) for a large range of filter widths and for both Geldart A and Geldart B particles.

Quantitative Analysis of the Complex Time Evolution of a Camphor Boat

The motion of a camphor boat on the water’s surface is a long-studied example of the direct transformation of chemical energy into a mechanical one. Recent experimental papers have reported a complex character of boat motion depending on the location of the camphor source. If the source is close to the stern, the boat moves at a constant speed. When it is shifted towards the boat center, oscillations of speed are observed. When the source is close to the boat center, pulses of speed followed by oscillations appear. Here, we focus on numerical simulations of camphor boat motion. We discuss approximations that allow us to reduce the numerical complexity of the problem and formulate a model in which the equation for boat velocity is coupled with a one-dimensional reaction–diffusion equation for camphor surface concentration. We scanned the phase space of model parameters and found the values that give qualitative agreement with the experiments. The model predicts all types of boat motion (continuous, oscillating, and pulsating) observed in experiments. Moreover, the model with selected parameter values shows that for specific locations of the camphor source, a spike in speed is followed by transient oscillations, which are an inherent part of speed relaxation.

Partial Fourier Transform Method for Solution Formula of Stokes Equation with Robin Boundary Condition in Half-space

The area of applied science known as fluid dynamics studied how gases and liquids moved. The motion of the fluid in the liquid and vapour phases is described by a special system of partial differential equations. The research purpose of this article investigated the solution formula of incompressible Stokes equation with the Robin boundary condition in half-space case. The solution formula for Stokes equation was calculated using the partial Fourier transform. This calculation was carried out over the Weis’s multipliers theorem. Our calculation showed that the solution formula of Stokes equation with Robin boundary condition in half-space for velocity and pressure were contained multipliers as due to work Shibata & Shimada. Due to our consideration of the half-space situation, the partial Fourier transform approach is the most appropriate one to use to get the velocity and pressure for the Stokes equation with Robin boundary condition. Furthermore, research methods in this article, in the first stage, we use the resolvent problem of the model. Secondly, we apply the partial Fourier transform to the model problem and finally, we use inverse partial Fourier transform to get the solution formula of the incompressible type of Stokes equation for velocity and pressure. This result indicates that Weis' multiplier theorem also allows us to find the local well-posedness of the model problem in addition to the maximal Lp-Lq regularity class (Gerard-Varet et al., 2020).

Study on the Influence of Missile Cabin on Fragment Velocity under Explosive Detonation Impact

For the air-to-air missile warhead, there is a cabin with a certain thickness at a distance around the fragments. At present, the influence of missile cabin has not yet been taken into account in the study of fragment velocity. In this paper, based on the law of conservation of energy, the theoretical equation of fragment velocity considering the kinetic energy of cabin debris was deduced. Then, the rationality of the theoretical formula is validated through the static explosion experiments of two prototype warheads, one with a titanium alloy cabin and the other without any cabin. It was found that after the warhead is equipped with the cabin, part of the energy is consumed to drive the cabin debris, resulting in a decrease in fragment velocity, but the velocity of cabin debris was greater than that of fragment of warheads without any cabin. Besides, through numerical simulation, the driving process of fragments and cabin debris during explosive detonation loading of the warhead with the cabin was studied, which can be divided into six stages, and the error between numerical result and experimental value is not more than 4.8%. Finally, the variety regulation of fragment velocity and cabin debris velocity at different interval distances was further studied by numerical simulation. The results indicate that fragment velocity of warheads with cabin at different interval distances is basically the same, but cabin debris velocity decreases with the increase of interval distance. This conclusion can provide a reference for the structural design and fragment velocity evaluation of warheads with cabin.

Breakdown of the velocity and turbulence in the wake of a wind turbine – Part 2: Analytical modelling

Abstract. This work aims to develop an analytical model for the streamwise velocity and turbulence in the wake of a wind turbine where the expansion and the meandering are taken into account independently. The velocity and turbulence breakdown equations presented in the companion paper are simplified and resolved analytically, using shape functions chosen in the moving frame of reference. This methodology allows us to propose a physically based model for the added turbulence and thus to have a better interpretation of the physical phenomena at stake, in particular when it comes to wakes in a non-neutral atmosphere. Five input parameters are used: the widths (in vertical and horizontal directions) of the non-meandering wake, the standard deviation of wake meandering (in both directions) and a modified mixing length. Two calibrations for these parameters are proposed: one if the users have access to velocity time series and the other if they do not. The results are tested on a neutral and an unstable large-eddy simulation (LES) that were both computed with Meso-NH. The model shows good results for the streamwise velocity in both directions and can accurately predict modifications due to atmospheric instability. For the axial turbulence, the model misses the maximum turbulence at the top tip in the neutral case, and the proposed calibrations lead to an overestimation in the unstable case. However, the model shows encouraging behaviour as it can predict a modification of the shape function (from bimodal to unimodal) as instability and thus meandering increases.

Flow characteristics of dispersed two-phase flows in an 8 × 8 rod bundle

Two-phase flows in rod bundles or tube bundles appear in heat transfer systems. The bundle-average (or one-dimensional) void fraction is a critical parameter in overall system design, performance evaluation, and safety assessment. Accurate bundle-average void fraction prediction requires a precise interfacial momentum transfer term formulated using the distribution parameter and drift velocity, two essential drift-flux parameters in the drift-flux model. The currently utilized modeling approach is based on the distribution parameter and drift velocity determined together through drift-flux plots. This results in possible compensation errors between the distribution parameter and drift velocity in the model validation process. The independent validation of the constitutive equations for the distribution parameter and drift velocity has been challenging due to experimental difficulty in measuring detailed two-phase flow structures in rod bundles. The present study proposes an approximation methodology to obtain subchannel average two-phase flow parameters using local two-phase flow data measured at two local points: the subchannel center and minimum gap center in a rod bundle. The distributions of subchannel average two-phase flow parameters are invaluable in benchmarking subchannel analysis codes. Comprehensive subchannel average void fraction and gas velocity mappings are given to discuss the effects of gas and liquid velocities, pressure, spacer grid, and developing length on the two-phase flow characteristics in the rod bundle. The bundle-average distribution parameter values calculated by subchannel average two-phase flow parameters are used for independent validation of the constitutive equations of the distribution parameter. The bundle-average drift velocity values back-calculated using the distribution parameter values are also used for independent validation of the constitutive equation of the drift velocity.

Some controllability results for linearized compressible Navier-Stokes system with Maxwell's law

In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in density or stress equation is everywhere in the domain. This is the best possible null controllability result obtained for this system. Next, we show that the system is approximately controllable at large time using localized controls. Thus, our results give a complete understanding of the system in this direction.

N-Euler mathematical modelling and numerical simulation of liquid–gas–solid flows

Three phase flows are omnipresent in industrial as well as natural configurations. Understanding the behaviour of solid particles interacting with liquid–air flows can be of great interest for both topics. In this paper we present a new liquid–gas–solid modelling technique based on the hybrid method combining liquid–gas two-fluid phase-average approach and probability density function (PDF) methods for solid dispersed phases. The PDF approach is based on single particle dynamic equation taking into account its interaction with more than one fluid and separate stochastic Lagrangian equations for each fluid phase velocity seen by the particles. This method does not assume the carrier phases typology. The solid particles can interact with bubbles, droplets, pockets and stratified flow through drag only (other interaction forces are supposed to be negligible for our cases). In this work, it is assumed that the total drag applied to the particle is a linear combination of the drags computed if the particles were interacting separately with each fluid phases. Finding the linear combination coefficient represent the main modelling step. Particle collisions and particle turbulent dispersion are taken into account in the model. Dispersed bubbles can undergo coalescence and fragmentation. The effect of all these sub-models is studied in the comparison with experimental data.The fluid and particle equations are first derived before introducing the appropriate coupling terms. Then, the closure terms for turbulence and multiphase drag are introduced. The model is implemented in neptune_cfd, a finite volume solver developed by EDF, CEA, IRSN and Framatome which allows for the numerical resolution of separate Eulerian equations (mass, momentum and energy) for n phases coupled by interfacial transfer terms. The model is then compared against experimental data obtained in a bubble column charged with particles. The results obtained with it are satisfactory and enable us to study more in details its hypothesis and to confirm comments made in the first section. Eventually, the model is tested on an industrial case which shows its efficiency.

Implementing vorticity–velocity formulation in a finite difference lattice Boltzmann method for two-dimensional incompressible generalized Newtonian fluids

A finite difference lattice Boltzmann approach is introduced to address the two-dimensional macroscopic equations of velocity–vorticity for generalized Newtonian fluids (GNFs). The study involves equations governing macroscopic momentum, energy, and concentration, along with constitutive models applicable to Newtonian, power-law, and viscoplastic fluids. Subsequently, the lattice Boltzmann method, which recovers these macroscopic equations, is detailed, along with proof of its capability to reproduce the aforementioned equations. In order to evaluate the effectiveness and time efficiency of the method, it is validated against various benchmarks. The results demonstrate the efficacy of the proposed method in successfully solving isothermal, thermal, and solutal problems of GNFs, while significantly reducing computational time compared to our previously suggested approach in this domain.