Classical Fluid Mechanics Research Articles

Turbulent particle transport in streams: Can exponential settling be reconciled with fluid mechanics?

Most ecological studies of particle transport in streams that focus on fine particulate organic matter or benthic invertebrates use the Exponential Settling Model (ESM) to characterize the longitudinal pattern of particle settling on the bed. The ESM predicts that if particles are released into a stream, the proportion that have not yet settled will decline exponentially with transport time or distance and will be independent of the release elevation above the bed. To date, no credible basis in fluid mechanics has been established for this model, nor has it been rigorously tested against more-mechanistic alternative models. One alternative is the Local Exchange Model (LEM), which is a stochastic advection–diffusion model that includes both longitudinal and vertical spatial dimensions and is based on classical fluid mechanics. The LEM predicts that particle settling will be non-exponential in the near field but will become exponential in the far field, providing a new theoretical justification for far-field exponential settling that is based on plausible fluid mechanics. We review properties of the ESM and LEM and compare these with available empirical evidence. Most evidence supports the prediction of both models that settling will be exponential in the far field but contradicts the ESM's prediction that a single exponential distribution will hold for all transport times and distances.

Freeze/Thaw-induced embolism: probability of critical bubble formation depends on speed of ice formation.

Bubble formation in the conduits of woody plants sets a challenge for uninterrupted water transportation from the soil up to the canopy. Freezing and thawing of stems has been shown to increase the number of air-filled (embolized) conduits, especially in trees with large conduit diameters. Despite numerous experimental studies, the mechanisms leading to bubble formation during freezing have not been addressed theoretically. We used classical nucleation theory and fluid mechanics to show which mechanisms are most likely to be responsible for bubble formation during freezing and what parameters determine the likelihood of the process. Our results confirm the common assumption that bubble formation during freezing is most likely due to gas segregation by ice. If xylem conduit walls are not permeable to the salts expelled by ice during the freezing process, osmotic pressures high enough for air seeding could be created. The build-up rate of segregated solutes in front of the ice-water interface depends equally on conduit diameter and freezing velocity. Therefore, bubble formation probability depends on these variables. The dependence of bubble formation probability on freezing velocity means that the experimental results obtained for cavitation threshold conduit diameters during freeze/thaw cycles depend on the experimental setup; namely sample size and cooling rate. The velocity dependence also suggests that to avoid bubble formation during freezing trees should have narrow conduits where freezing is likely to be fast (e.g., branches or outermost layer of the xylem). Avoidance of bubble formation during freezing could thus be one piece of the explanation why xylem conduit size of temperate and boreal zone trees varies quite systematically.

Electron effective mass in the strongly correlated two-dimensional uniform electron fluid from finite-temperature calculations

The very-low temperature thermal effective mass m* of paramagnetic and ferromagnetic electrons in a uniform electron fluid in two dimensions is studied. Analytical and numerical evaluations are used to meaningfully define an m*, even in the the Hartree-Fock approximation. The Hartree-Fock m* decreases linearly with the electron-disk radius r_s. Correlation effects lead to strong cancellations between exchange and correlation. Thus the effective mass is enhanced with increasing r_s for the unpolarized fluid, while m* decreases with the r_s of the polarized fluid. The effective mass is calculated from the coefficient of the quadratic temperature dependence of exchange-correlation free energy F_{xc}. This is calculated in a physically transparent manner using a new formula for the effective mass. This uses the T=0 pair-distribution functions of Gori-Giorgi et al., and the temperature derivative of a quantum analogue of the potential of mean-force well known in the statistical mechanics of classical fluids. The results are compared with recent quantum Monte-Carlo simulations at T=0, as well as with other available experimental and theoretical data for the effective mass.

On the validity of the Navier-Stokes equations for nanoscale liquid flows: The role of channel size

In this work, we investigate the validity of the Navier-Stokes (NS) equations for nanoscale liquid flows through molecular dynamics simulations. We focus on the role of channel size by considering the fluid-wall interaction. Liquid flows between two planar parallel walls driven by an external force with channel size ranging from 2 to 80 nm are studied. The volumetric flux is computed and the dependence of the volumetric flux on the channel size is explained both qualitatively and quantitatively. It is found that the flow is sensitive to the fluid-wall binding energy and the classical fluid mechanics falls apart in small nanochannels. However, the wall effects become insignificant and the NS equations are valid when the channel size is larger than about 150 molecular diameters (∼ 50 nm).

Why chaotic mixing of particles is inevitable in the deep lung

Fine/ultrafine particles can easily reach the pulmonary acinus, where gas is exchanged, but they need to mix with alveolar residual air to land on the septal surface. Classical fluid mechanics theory excludes flow-induced mixing mechanisms because of the low Reynolds number nature of the acinar flow. For more than a decade, we have been challenging this classical view, proposing the idea that chaotic mixing is a potent mechanism in determining the transport of inhaled particles in the pulmonary acinus. We have demonstrated this in numerical simulations, experimental studies in both physical models and in animals, and mathematical modeling. However, the mathematical theory that describes chaotic mixing in small airways and alveoli is highly complex; it not readily accessible by non-mathematicians. The purpose of this paper is to make the basic mechanisms that operate in acinar chaotic mixing more accessible, by translating the key mathematical ideas into physics-oriented language. The key to understanding chaotic mixing is to identify two types of frequency in the system, each of which is induced by a different mechanism. The way in which their interplay creates chaos is explained with instructive illustrations but without any equations. We also explain why self-similarity occurs in the alveolar system and was indeed observed as a fractal pattern deep in rat lungs (Proc. Natl. Acad. Sci. USA. 99:10173-10178, 2002).

Recollections of Chandra

Recollections of Chandra

Statistical derivation of modified hydrodynamic equations for nanotube flows

The problem of flow through a nanotube is considered. Classical fluid mechanics does not give an adequate description of the process. Modified hydrodynamic equations are derived from a microscopic model using the dynamical Bogolyubov approach and the Zwanzig projection operator method.

3D Engineering Model of Downburst Evolution in Thunderstorm

Downburst in thunderstorm mainly refers to the outflow on frontier and the strong wind shear caused by the downburst. In this paper, the thunderstorm evolution is divided into three stages as growing, mature and dissipating stages according to its development. Based on meteorological data, trends of relevant parameters are presented. Then with the 3D ring vortices method, the classical fluid mechanics theory is applied through adding linear source model to simulate the distribution of velocity in the wind shear field and its evolution over time. This modeling method could effectively simulate the evolution of downburst, and is suitable for engineering application.

Synthetic turbulence using artificial boundary layers

Turbulent vortices in fluid flows are crucial for a visually interesting appearance. Although there has been a significant amount of work on turbulence in graphics recently, these algorithms rely on the underlying simulation to resolve the flow around objects. We build upon work from classical fluid mechanics to design an algorithm that allows us to accurately precompute the turbulence being generated around an object immersed in a flow. This is made possible by modeling turbulence formation based on an averaged flow field, and relying on universal laws describing the flow near a wall. We precompute the confined vorticity in the boundary layer around an object, and simulate the boundary layer separation during a fluid simulation. Then, a turbulence model is used to identify areas where this separated layer will transition into actual turbulence. We sample these regions with vortex particles, and simulate the further dynamics of the vortices based on these particles. We will show how our method complements previous work on synthetic turbulence, and yields physically plausible results. In addition, we demonstrate that our method can efficiently compute turbulent flows around a variety of objects including cars, whisks, as well as boulders in a river flow. We can even apply our model to precomputed static flow fields, yielding turbulent dynamics without a costly simulation.

A continuum model for the flow of thin liquid films over intermittently chemically patterned surfaces

It is known from both experiments and molecular dynamics simulations that chemically patterning a solid surface has an effect on the flow of an adjacent liquid. This fact is in stark contrast with predictions of classical fluid mechanics where the no-slip boundary condition is insensitive to the chemistry of the solid substrate. It has been shown that the influence on the flow caused by a steep change in the wettability of the solid substrate can be described in the framework of continuum mechanics using the interface formation theory. The present work extends this study to the case of intermittent patterning. Results show that variations in wettability of the substrate can significantly affect the flow, especially of thin films, which may have applications to the design of microfluidic devices.

PESTICIDE ATOMIZATION MODELING FOR HOLLOW-CONE NOZZLE

This paper presents a new approach to model the pesticide atomization in order to get the droplet size and velocity very close to the nozzle exit. The two-phase flow was calculated inside and outside the nozzle. The model was based on classical fluid mechanics transport equations for the liquid dispersion, velocity and turbulence. Moreover, a novel transport equation was developed for the mean liquid/gas surface area, coming from studies in automotive and aeronautics fields. Coupling the transport equations for the liquid mass fraction and the surface area led to an estimation of a Sauter Mean Diameter. These equations have been implemented in the commercial CFD code Fluent. A swirling flow was found inside the nozzle. Moreover, a hollow cone liquid sheet expanded outside the nozzle. Calculations have been conducted with various injection pressure values, leading to a mass flow rate in good accordance with manufacturer data. Surfactant influence has been studied by varying the surface tension coefficient in the surface area transport equation: as expected, droplets obtained are smaller than when water is considered.

High resolution finite volume scheme for the quantum hydrodynamic equations

The theory of quantum fluid dynamics (QFD) helps nanotechnology engineers to understand the physical effect of quantum forces. Although the governing equations of quantum fluid dynamics and classical fluid mechanics have the same form, there are two numerical simulation problems must be solved in QFD. The first is that the quantum potential term becomes singular and causes a divergence in the numerical simulation when the probability density is very small and close to zero. The second is that the unitarity in the time evolution of the quantum wave packet is significant. Accurate numerical evaluations are critical to the simulations of the flow fields that are generated by various quantum fluid systems. A finite volume scheme is developed herein to solve the quantum hydrodynamic equations of motion, which significantly improve the accuracy and stability of this method. The QFD equation is numerically implemented within the Eulerian method. A third-order modified Osher–Chakravarthy (MOC) upwind-centered finite volume scheme was constructed for conservation law to evaluate the convective terms, and a second-order central finite volume scheme was used to map the quantum potential field. An explicit Runge–Kutta method is used to perform the time integration to achieve fast convergence of the proposed scheme. In order to meet the numerical result can conform to the physical phenomenon and avoid numerical divergence happening due to extremely low probability density, the minimum value setting of probability density must exceed zero and smaller than certain value. The optimal value was found in the proposed numerical approach to maintain a converging numerical simulation when the minimum probability density is 10 −5 to 10 −12. The normalization of the wave packet remains close to unity through a long numerical simulation and the deviations from 1.0 is about 10 −4. To check the QFD finite difference numerical computations, one- and two-dimensional particle motions were solved for an Eckart barrier and a downhill ramp barrier, respectively. The results were compared to the solution of the Schrödinger equation, using the same potentials, which was obtained using by a finite difference method. Finally, the new approach was applied to simulate a quantum nanojet system and offer more intact theory in quantum computational fluid dynamics.

Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes

Despite the enormous scientific and technological importance of micro-channel gas flows, the understanding of these flows, by classical fluid mechanics, remains incomplete including the prediction of flow rates. In this paper, we revisit the problem of micro-channel compressible gas flows and show that the axial diffusion of mass engendered by the density (pressure) gradient becomes increasingly significant with increased Knudsen number compared to the pressure driven convection. The present theoretical treatment is based on a recently proposed modification (Durst et al. in Proceeding of the international symposium on turbulence, heat and mass transfer, Dubrovnik, 3-18 September, pp 25-29, 2006) of the Navier-Stokes equations that include the diffusion of mass caused by the density and temperature gradients. The theoretical predictions using the modified Navier-Stokes equations are found to be in good agreement with the available experimental data spanning the continuum, transition and free-molecular (Knudsen) flow regimes, without invoking the concept of Maxwellian wall-slip boundary condition. The simple theory also results in excellent agreement with the results of linearized Boltzmann equations and Direct Simulation Monte Carlo (DSMC) method. Finally, the theory explains the Knudsen minimum and suggests the design of future micro-channel flow experiments and their employment to complete the present days understanding of micro-channel flows.

Numerical simulations of viscous flows using a meshless method

AbstractThis paper uses the element‐free Galerkin (EFG) method to simulate 2D, viscous, incompressible flows. The control equations are discretized with the standard Galerkin method in space and a fractional step finite element scheme in time. Regular background cells are used for the quadrature. Several classical fluid mechanics problems were analyzed including flow in a pipe, flow past a step and flow in a driven cavity. The flow field computed with the EFG method compared well with those calculated using the finite element method (FEM) and finite difference method. The simulations show that although EFG is more expensive computationally than FEM, it is capable of dealing with cases where the nodes are poorly distributed or even overlap with each other; hence, it may be used to resolve remeshing problems in direct numerical simulations. Flows around a cylinder for different Reynolds numbers are also simulated to study the flow patterns for various conditions and the drag and lift forces exerted by the fluid on the cylinder. These forces are calculated by integrating the pressure and shear forces over the cylinder surface. The results show how the drag and lift forces oscillate for high Reynolds numbers. The calculated Strouhal number agrees well with previous results. Copyright © 2008 John Wiley & Sons, Ltd.

A Generalization of the Stillinger–Lovett Sum Rules for the Two-Dimensional Jellium

In the equilibrium statistical mechanics of classical Coulomb fluids, the long-range tail of the Coulomb potential gives rise to the Stillinger–Lovett sum rules for the charge correlation functions. For the jellium model of mobile particles of charge q immersed in a neutralizing background, the Stillinger–Lovett sum rules give the charge and second moment of the screening cloud around a particle of the jellium. In this paper, we generalize these sum rules to the screening cloud induced around a pointlike guest charge Zq immersed in the bulk interior of the 2D jellium with the coupling constant Γ=β q 2 (β is the inverse temperature), in the whole region of the thermodynamic stability of the guest charge amplitude Z>−2/Γ. The derivation is based on a mapping technique of the 2D jellium at the coupling Γ = (even positive integer) onto a discrete 1D anticommuting-field theory; we assume that the final results remain valid for all real values of Γ corresponding to the fluid regime. The generalized sum rules reproduce for arbitrary coupling Γ the standard Z=1 and the trivial Z=0 results. They are also checked in the Debye–Hückel limit Γ→0 and at the free-fermion point Γ=2. The generalized second-moment sum rule provides some exact information about possible sign oscillations of the induced charge density in space.

Heavy hydroelastic travelling waves

This is a mathematical study of steady two-dimensional waves of prescribed period which propagate without change of form on the surface of an infinitely deep expanse of fluid that is moving under gravity and bounded above by heavy, frictionless, thin elastic sheet. The flow, which is supposed irrotational, is at rest at infinite depth and its velocity is stationary relative to a frame moving with the wave. In that frame, the elastic sheet coincides with the zero streamline and its material points move according to the equations of hyperelasticity. From the mechanics of the surface material, the pressure on a point of the sheet in the moving frame is shown to be a function of its height, its slope and derivatives with respect to arc length of these quantities. Independently, according to classical fluid mechanics, the pressure in the fluid at the same point is determined by its height and the fluid velocity tangent to the zero streamline. Therefore, this is a free-boundary problem: to find a non-self-intersecting curve in the plane which is the zero contour of a harmonic function (the stream function) and at which the normal derivative of the same harmonic function is a prescribed function of the shape of the curve. With the wavelength fixed, the parameters are the density ρ of the undeformed sheet, the wave velocity c 0 , the mean velocity c of the surface sheet relative to the wave and the gravitational acceleration g . The case of a weightless sheet, in which ρ =0 and c does not appear, is a special case. Under quite general hypotheses on the elastic response of the sheet, the existence of rapidly propagating or very slow (depending on parameter values) steady waves is established by finding a critical point of the Lagrangian. This is a saddle-point problem in the calculus of variations.

Heat transfer in porous media

In this work, the effects of electrical double layer (EDL) near the solid/ liquid interface, on three dimensional heat transfer characteristic and pressure drop of water flow through a rectangular microchannel numerically are investigated. An additional body force originating from the existence of EDL is considered to modify the conventional Navier-stokes and energy equations. These modified equations are solved numerically for steady laminar flow on the basis of control volume approaches. Fluid velocity distribution and temperature with presence and absence of EDL effects are presented for various geometric cases and different boundary conditions. The results illustrate that, the liquid flow in rectangular microchannels is influenced significantly by the EDL, particularly in the high electric potentials, and hence deviates from flow characteristics described by classical fluid mechanics.

Particle Image Velocimetry Studies of Counterflow Heat Transport in Superfluid Helium II

Heat is transported in helium II by a mechanism known as thermal counterflow where the liquid behaves as if it consists of two fluid components: the viscous normal fluid that carries the entropy and the inviscid superfluid that flows in opposition to conserve mass and momentum. Although the two fluid model has been successful at interpreting a number of unique transport processes in helium II, only recently has there been a significant effort to actually observe the associated fluid component motion. We have previously shown that Particle Image Velocimetry (PIV) can be used to visualize the motion of micron scale particles in helium II in response to thermal counterflow. These studies have led to some exceptional observations. The present paper summarizes our recent PIV experimental results on counterflow in helium II including the observed flow patterns around a cylinder and through a rectangular channel with backward facing step change in cross section, configurations which have been studied extensively in classical fluid mechanics. In addition to contributing to the fundamental understanding of helium II turbulence, as is discussed in the last section of the article, we show how this work also has a potential application in micron-scale particle classification and separation.

Cosserat Modeling of Turbulent Plane-Couette and Pressure-Driven Channel Flows

The theory of micropolar fluids based on a Cosserat continuum model is utilized for analysis of two benchmarks, namely, plane-Couette and pressure-driven channel flows. In the obtained theoretical velocity distributions, some new terms have appeared in addition to linear and parabolic distributions of classical fluid mechanics based on the Navier-Stokes equations. Utilizing the principles of irreversible thermodynamics, a new dissipative boundary condition is developed for angular velocity at flat plates by taking the couple-stress vector into account. The obtained results for the velocity profiles have been compared to results of recent and classical experiments. This paper demonstrates that continuum mechanical theories of higher orders, for instance Cosserat model, are able to describe a complex phenomenon, such as hydrodynamic turbulence, more precisely.

Nano boundary layer equation with nonlinear Navier boundary condition

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n > 0 , which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n = 1 / 2 , is solved analytically. Another analytical but non-unique solution is found corresponding to the value n = 1 / 3 , while other values of n for n > 1 / 2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.