Settling and asymptotic motion of aerosol particles in a cellular flow field

Summary This paper presents a proof that given a dilute concentration of aerosol particles in an infinite, periodic, cellular flow field, arbitrarily small inertial effects are sufficient to induce almost all particles to settle. It is shown that when inertia is taken as a small parameter, the equations of particle motion admit a slow manifold that is globally attracting. The proof proceeds by analyzing the motion on this slow manifold, wherein the flow is a small perturbation of the equation governing the motion of fluid particles. The perturbation is supplied by the inertia, which here occurs as a regular parameter. Further, it is shown that settling particles approach a finite number of attracting periodic paths. The structure of the set of attracting paths, including the nature of possible bifurcations of these paths and the resulting stability changes, is examined via a symmetric one-dimensional map derived from the flow.     

The effect of gravity on flow past a semi-circular cylinder with a constant pressure wake

The flow past a semi-cylinder with a trailing wake region is considered. In the absence of gravity the only known high Reynolds number solutions have tangential separation from the body and a cusped shape at the back of the wake. This flow can be a simple model for several situations, including the classical approximation of a constant pressure wake and the flow past an object with a region of trapped fluid of different density (or an air cavity) attached on the downstream side. Here we relax the assumption of high flow speeds to examine the effects of gravity. It is shown that there are situations in which a stagnation point can form either on the body or at the tail of the wake and that there is a minimum velocity beneath which a cavity will not form. Non-uniqueness in the parameter space is found in certain cases.     

A numerical study on statistical temporal scales in inertia particle dispersion

The statistical temporal scales involved in inertia particle dispersion are analyzed numerically. The numerical method of large eddy simulation, solving a filtered Navier-Stokes equation, is utilized to calculate fully developed turbulent channel flows with Reynolds numbers of 180 and 640, and the particle Lagrangian trajectory method is employed to track inertia particles released into the flow fields. The Lagrangian and Eulerian temporal scales are obtained statistically for fluid tracer particles and three different inertia particles with Stokes numbers of 1, 10 and 100. The Eulerian temporal scales, decreasing with the velocity of advection from the wall to the channel central plane, are smaller than the Lagrangian ones. The Lagrangian temporal scales of inertia particles increase with the particle Stokes number. The Lagrangian temporal scales of the fluid phase ‘seen’ by inertia particles are separate from those of the fluid phase, where inertia particles travel in turbulent vortices, due to the particle inertia and particle trajectory crossing effects. The effects of the Reynolds number on the integral temporal scales are also discussed. The results are worthy of use in examining and developing engineering prediction models of particle dispersion.     

Numerical Investigation of Particle Focusing Patterns in Laminar Pipe Flow With High Reynolds Numbers北大核心CSCD

The inertial focusing characteristics of particles in laminar flow pipes with high Re numbers were studied based on the “relative motion model”. In order to solve the problem of long pipes with high Re number flow, periodic boundary conditions were imposed on the inlet and outlet of the pipe. The research results show that the use of periodic boundary conditions can effectively reduce the computational, and the mechanical properties of particles in high Re flow can be calculated by using L=4D pipe. The difference from the low Re number is that as the Re number continues to increase,the lift force of the particles in the radial direction is no longer distributed as a parabola. The lift curve has a concave area between r+ =0.5 ~ 0.7, and there is a tendency for a new inertial focus point to appear in this section. By means of particles of a+ =1/17 for Re > 1 000, this new focus point position is solvable. In addition, in the analysis of the flow field, a secondary flow occurs around the particle, and its intensity gradually increases with the Re number and the closeness of the particle to the wall. The generation of the secondary flow affects the spatial distribution of the particle lift. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Dusty Supersonic Flow Past a Blunt Body: Viscous and Nonstationary Effects

High‐speed space‐ or aircrafts travelling through a dusty atmosphere may meet dust clouds in which the particles are often distributed very nonuniformly. Such nonuniformities may result in the onset of unsteady effects in the shock and boundary layer and (that is of prime interest) unsteady heat fluxes at the stagnation region of the vehicle. In the nearwall region of high‐speed dusty‐gas flow, there may take place regimes with and without particle inertial deposition, which require essentially different mathematical models for describing the heat transfer [1]. The present paper deals with two problems, considered within the framework of the two‐fluid model of dusty gas [2]: (i) determination of the limits of the particle inertial deposition regime and the distribution of dispersed‐phase parameters near the frontal surface of a sphere immersed in dusty supersonic flow (Mach number M = 6) at moderate flow Reynolds numbers (10² ≤ Re ≤ ∞); (ii) effect of free‐stream nonuniformities in the concentration of low inertial (non‐depositing) particles on the friction and heat transfer at the stagnation point of the body at high Re and M.     

Flocking Dynamics of the Inertial Spin Model with a Multiplicative Communication Weight

In this paper, we study a flocking dynamics of the deterministic inertial spin (IS) model. The IS model was introduced for the collective dynamics of active particles with an internal angular momentum, or spin. When the generalized moment of inertia becomes negligible compared to spin dissipation (overdamped limit) and mutual communication weight is a function of a relative distance between interacting particles, the deterministic inertial spin model formally reduces to the Cucker–Smale (CS) model with constant speed constraint whose emergent dynamics has been extensively studied in the previous literature. We present several sufficient frameworks leading to the asymptotic mono-cluster flocking, in which spins and relative velocities tend to zero asymptotically. We also provide several numerical simulations for the decoupled and coupled inertial spin models to see the effect of the C–S velocity flocking and compare them with our analytical results.

     

Intrusive Gravity Currents

Intrusive gravity currents arise when a fluid of intermediate density intrudes into an ambient fluid. These intrusions may occur in both natural and human-made settings and may be the result of a sudden release of a fixed volume of fluid or the steady or time-dependent injection of such a fluid. In this article we analytically and numerically analyze intrusive gravity currents arising both from the sudden release of a fixed volume and the steady injection of fluid having a density that is intermediate between the densities of an upper layer bounded by a free surface and a heavier lower layer resting on a flat bottom. For the physical problems of interest we assume that the dynamics of the flow are dominated by a balance between inertial and buoyancy forces with viscous forces being negligible. The three-layer shallow-water equations used to model the two-dimensional flow regime include the effects of the surrounding fluid on the intrusive gravity current. These effects become more pronounced as the fraction of the total depth occupied by the intrusive current increases. To obtain some analytical information concerning the factors effecting bore formation we further reduce the complexity of our three-layer model by assuming small density differences among the different layers. This reduces the model equations from a 6×6 to a 4×4 system. The limit of applicability of this weakly stratified model for various ranges of density differences is examined numerically. Numerical results, in most instances, are obtained using MacCormack's method. It is found that the intrusive gravity current displays a wide range of flow behavior and that this behavior is a strong function of the fractional depth occupied by the release volume and any asymmetries in the density differences among the various layers. For example, in the initially symmetric sudden release problem it is found that an interior bore does not form when the fractional depth of the release volume is equal to or less than 50% of the total depth. The numerical simulations of fixed-volume releases of the intermediate layer for various density and initial depth ratios demonstrate that the intermediate layer quickly slumps from any isostatically uncompensated state to its Archimedean level thereby creating a wave of opposite sign ahead of the intrusion on the interface between the upper and lower layers. Similarity solutions are obtained for several cases that include both steady injection and sudden releases and these are in agreement with the numerical solutions of the shallow-water equations. The 4×4 weak stratification system is also subjected to a wavefront analysis to determine conditions for the initiation of leading-edge bores. These results also appear to be in agreement with numerical solutions of the shallow-water equations.     

Numerical Study on the Influence of Particle Inertia in Particle-Driven Gravity Currents

We present two-dimensional numerical simulations of particle-driven gravity currents in a lock-exchange configuration. The fluid is described in an Eulerian framework, whereas the particles are tracked in a Lagrangian manner. The study is restricted to dilute suspensions, allowing to neglect particle-particle interactions. The particle forces considered are buoyancy and the Stokes drag. We study the influence of particle inertia on the flow evolution by performing simulations with different Stokes numbers. We also consider the case where particle inertia is neglected. Generally, we observe significant changes in the form and structure of the gravity current with increasing particle Stokes numbers. Particularly, the formation of Kelvin-Helmholtz vortices is more and more suppressed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interaction of a finite-size particle with the moving lid of a cavity

The motion of a solid particle in a lid-driven cavity is investigated. If the tangential velocity of the lid is large the streamlines are dense near the moving lid and the finite size of a particle can have a profound effect on its trajectory. To assess this effect different particle-motion models are examined: inertial point particles (Maxey–Riley equation) one-way coupled to the flow and finite-size particles the flow around which is fully resolved (two-way coupling). We compare the corresponding trajectories with those obtained using the particle–surface interaction model originally introduced by Hofmann and Kuhlmann [Phys. Fluids 23, 0721106 (2011)]. The finite-size effect on the particle's trajectory is quantified and discussed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Einordnung der Gleichungen der inkompressiblen,reibenden und schweren Flüssigkeit

Summary The four forcesK _p ,K _ϱ ,K _η , andK _g due to pressure, inertia, viscosity and gravity respectively, may be represented by the four corners of a tetrahedron whose 6 edges correspond to the dimensionless groups formed from the forces. The equations of hydrodynamics for the steady state which are shown to be represented by the volume, sides, or edges of the tetrahedron, can then be classified in eleven sets of 3, 2 or 1 dimensionless groups. For flow with two and more phases, many independent dimensionless groups occur which may or may not be a complete set to a given problem. A simple construction gives the answer.      

Hydraulic Theory and Particle-Driven Gravity Currents

Hydraulic theory, as it has been applied to compositionally driven gravity flows, involves the single simplifying assumption that the pressure in the fluid is hydrostatic [1]. This assumption provides, as a consequence, a depth independent horizontal velocity field. This approach has led to a greatly increased understanding of many of the phenomena associated with these complex flows, including issues surrounding internal hydraulic jumps and energy loss [2]. Recently, investigations into flow and deposition of particles from particle-driven gravity currents have been carried out using an approach that employs the hydraulic theory that had proved so successful in the case of homogeneous flows [3]–[7]. Unfortunately, as we show here, there is a fundamental contradiction in adopting this simplifying assumption when particles drive the flow. This contradiction is essentially that one cannot have a hydrostatic pressure that arises from the presence of particles while at the same time maintaining a depth-independent horizontal velocity field, as was assumed in references [3]–[7].     

A novel car-following inertia gray model and its application in forecasting short-term traffic flow

Real-time and accurate short-term traffic flow prediction results can provide real-time and effective information for traffic information systems. Based on classic car-following models, this paper establishes differential equations according to the traffic state and proposes a car-following inertial gray model based on the information difference of the differential and gray system, in combination with the mechanical characteristics of traffic flow data and the characteristics of an inertial model. Furthermore, analytical methods are used to study the parameter estimation and model solution of the new model, and the important properties, such as the original data, inertia coefficient and simulation accuracy, are studied. The effectiveness of the model is verified in two cases. The performance of the model is better than that of six other prediction models, and the structural design of the new model is more reasonable than that of the existing gray models. Moreover, the new model is applied to short-term traffic flow prediction for three urban roads. The results show that the simulation and prediction effects of the model are better than those of other gray models. In terms of the traffic flow state, an optimal match between short-term traffic flow prediction and the new model is achieved.     

On the stability and uniqueness of the flow of a fluid through a porous medium

In this short note, we study the stability of flows of a fluid through porous media that satisfies a generalization of Brinkman’s equation to include inertial effects. Such flows could have relevance to enhanced oil recovery and also to the flow of dense liquids through porous media. In any event, one cannot ignore the fact that flows through porous media are inherently unsteady, and thus, at least a part of the inertial term needs to be retained in many situations. We study the stability of the rest state and find it to be asymptotically stable. Next, we study the stability of a base flow and find that the flow is asymptotically stable, provided the base flow is sufficiently slow. Finally, we establish results concerning the uniqueness of the flow under appropriate conditions, and present some corresponding numerical results.     

Conservation laws of turbulence models

The conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow's structure over time. In turbulent flow, conservation laws remain important in the inertial range of wave numbers, where viscous effects are negligible. It is in the inertial range where energy, helicity (3d), and enstrophy (2d) must be accurately cascaded for a turbulence model to be qualitatively correct. A first and necessary step for an accurate cascade is conservation; however, many turbulent flow simulations are based on turbulence models whose conservation properties are little explored and might be very different from those of the Navier-Stokes equations.We explore conservation laws and approximate conservation laws satisfied by LES turbulence models. For the Leray, Leray deconvolution, Bardina, and Nth order deconvolution models, we give exact or approximate laws for a model mass, momentum, energy, enstrophy and helicity. The possibility of cascades for model quantities is also discussed.     

Turbulent Energy Spectrum via an Interaction Potential

For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed by the Bogoliubov–Born–Green–Kirkwood–Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas–Fermi interatomic potential (as well as its variations, such as the Ziegler–Biersack–Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas–Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas–Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.

     

On the development of rational approximations incorporating inertial effects in coating and rimming flows: a multiple-scales approach

Analysis of thin-film rimming and coating flows is extendedto create a more general model, which can be identified as anew distinguished limit that has the work of certain previousstudies as specific limiting cases. Specifically, three-dimensionalflows under the influence of gravity are considered in whichthe Reynolds number is large enough to necessitate considerationof both inertial and centrifugal effects, along with those ofviscous, gravitational and surface-tension forces. Reductionof dimensionality is shown to be attainable within a systematic(multiple-timescale) asymptotic framework that encompasses allthese effects, leading to a two-dimensional formulation thatnot only is more general than many of the existing ones butalso represents a rational approximation in the sense that,in particular, the sizes of the dimensionless physical parametersto which it corresponds, together with the orders of magnitudeof the errors resulting from the reduction, can be preciselycharacterized.     

Electrohydrodynamic flow in a retarding electric field with allowance for charged particle inertia : PMM vol. 43, no. 6, 1979, pp. 1029–1039

The one-dimensional electrohydrodynamic flow is considered in a retarding electric field when, owing to the inertia of charged particles, it is necessary to use the complete equation of momenta for the charged component. It is shown that in spite of the negligibly small relative volume occupies by particles in the initial cross section of the stream, there is a section in which particles move at low but finite velocity with a finite relative volume of the charged component, when the particles are retarded by an external homogeneous electric field (without taking into account the induced field). Interaction of the charged and neutral components, which may be absent in the initial cross section, is always substantial. The pressure drop required for such flow is determined. The flow is investigated with allowance for induced electric fields that diminish the effect of concentration increase of charged particles.     

Simple calculation of moments of inertia for polygons

Recently the problem of finding the mass, centre of gravity and moments of inertia for laminae bounded by polygons arose in a programming course for first year civil engineering degree students. It was suggested that it would be useful for them to construct a program that would take as input the vertices of the polygon, and print out area, centroid and moment of inertia. In order to make the program have a logically simple structure, the method presented in this note was devised. After consultation with many standard textbooks and engineering handbooks it was discovered that the method was either original or completely neglected. Since in many respects the method compares favourably with the subdivision techniques of the standard sources it would appear to be worth publicising the technique.     

Optimization methods for pipeline transportation of natural gas with variable specific gravity and compressibility

In this paper, the problem of flow maximization in pipeline systems for transmission of natural gas is addressed. We extend previously suggested models by incorporating the variation in pipeline flow capacities with gas specific gravity and compressibility. Flow capacities are modeled as functions of pressure, compressibility and specific gravity by the commonly-used Weymouth equation, and the California Natural Gas Association method is used to model compressibility as a function of specific gravity and pressure. The sources feeding the transmission network do not necessarily supply gas with equal specific gravity. In our model, it is assumed that when different flow streams enter a junction point, the specific gravity of the resulting flow is a weighted average of the specific gravities of entering flows. We also assume the temperature to be constant, and the system to be in steady state. Since the proposed model is non-convex, and global optimization hence can be time consuming, we also propose a heuristic method based on an iterative scheme in which a simpler NLP model is solved in each iteration. Computational experiments are conducted in order to assess the computability of the model by applying a global optimizer, and to evaluate the performance of the heuristic approach. When applied to a wide set of test instances, the heuristic method provides solutions with deviations less than 10% from optimality, and in many instances turns out to be exact. We also report several experiments demonstrating that letting the compressibility and the specific gravity be global constants can lead to significant errors in the estimates of the total network capacity.     

Chaotic mixing simulations

Numerical simulations of high Reynolds number flows in the unit driven cavity have been performed. The system is shown to become unsteady at Re=8125 and chaotic at Re=17,000. In between this range the system switches between periodic and quasi-periodic states with step-wise changes in period. A passive concentration field and passive tracer particles are introduced into the flow at its asymptotic state to show the effects of chaos on mixing.