Investigation of deformation and breakup of a falling droplet using a multiple-relaxation-time lattice Boltzmann method

In this paper, a multi-relaxation-time lattice Boltzmann method for multiphase flows is employed to simulate different modes of deformation and fragmentation of an axisymmetric falling droplet under buoyancy force. To show the accuracy of the model, the Laplace law for stationary drops is conducted first. Then, drop deformation and breakup in a free fall is studied in an axially symmetric pipe. Surface tension effects as well as impacts of gas and drop viscosities are investigated for a wide range of Eötvös, Morton, and Archimedes numbers. The drag coefficient of the drop, as it falls, is measured and compared to the empirical correlations, and reasonable agreement is shown. The findings are further verified by comparing a typical bag breakup mechanism with experimental observations. It is seen that at low Eötvös numbers the drop deforms slightly and reaches a steady state. Increase of Eötvös number enhances the rate of deformation, and at a high enough Eötvös value breakup of the drop happens. While the gas viscosity is shown to have a trivial effect on the breakup of the droplet, drop viscosity is the overriding factor in the mechanism of disintegration. Consequently, various breakup modes of the falling droplet are observed just by varying the drop-based Archimedes number. By capturing different breakup mechanisms of a falling droplet such as bag breakup, shear breakup, and, particularly, multimode breakup, the present lattice Boltzmann method exhibits an excellent superiority over the sharp interface tracking schemes that fail to capture dissociation of the interface.     

Aggregate growth and breakup in particulate suspension flow through a micro-nozzle

A computational study is reported on the growth of aggregates in flow of a particulate suspension through a micro-nozzle. The study employs a soft-sphere discrete element method (DEM) with van der Waals adhesion force between the particles in two-dimensional, incompressible channel flow. A new computational approach for particle transport in complex domains is developed which uses a background Cartesian grid for efficient flow field interpolation at the particle locations, together with a level-set method to represent the nozzle boundaries in the particle computation. Three mechanisms for the growth or breakup of particulate aggregates in the micro-nozzle are examined: (1) enhanced particle collision due to lateral compression as fluid elements pass through the nozzle, (2) stretching of aggregates due to axial stretching of fluid elements, and (3) collision and intermittent adhesion of particles to the nozzle wall. The first of these mechanisms leads to aggregate growth, and the second to aggregate breakup. The wall collision and adhesion mechanism can enhance either aggregate growth or breakup, but it is found in most cases to be a primary agent in the breakup of incident aggregates as part of the aggregate attaches to the nozzle wall and is torn from the remainder of the aggregate due to the high shear near the walls. Simplified models for these processes are developed and used to interpret the trends observed in the DEM simulations. The effects of particle adhesion parameter, particle size and density, particle concentration, and nozzle geometry are examined. It is found that passage of a particulate suspension through a nozzle can lead to either a substantial decrease in aggregate size or a modest increase under different conditions, depending in part on the size of the incident aggregates.     

Nonlocal large deformation and localization behavior of metals

The present paper deals with a nonlocal continuum plasticity model which includes the dependence of the yield function on a nonlocal equivalent plastic strain measure. Particular attention is focused on the formulation of a generalized I₁–J₂ yield criterion to describe the effect of hydrostatic stress on the plastic flow properties of metals, and the nonlocal equivalent plastic strain is defined as a weighted average of the corresponding local measure taken over the neighboring material points of the body. The nonlocal yield condition leads to a partial differential equation which is solved using the finite difference method at each iteration of a loading step. Since this requires no additional boundary conditions, the displacement-based finite element procedure is governed by the standard principle of virtual work, and the associated linearized variational equations are obtained in the usual manner from a consistent linearization algorithm. Numerical simulations of the elastic–plastic deformation behavior of ductile metal specimens show the influence of the various model parameters on the deformation and localization prediction. The proposed nonlocal theory preserves well-posedness of the governing equations in the post-localization regime and prevents pathological mesh sensitivity of the numerical results. The internal length scale incorporated in the model determines the size of the localized shear bands.     

Smoothed Particle Hydrodynamics (SPH) simulation of a high-pressure homogenization process

High-pressure homogenization is a widely used process in the food, pharmaceutical, and cosmetic industry for producing emulsions. Because of small dimensions and high velocities, the experimental and numerical investigation of such a process is challenging. Hence, the development of products is mostly based on trial and error. In this paper, simulations of a generic high-pressure homogenization process using the Lagrangian, mesh-free smoothed particle hydrodynamics (SPH) method are presented and compared to experimental findings using Micro-Particle Image Velocimetry (μ-PIV). The SPH code has been developed and validated with the scope of simulating technical relevant multi-phase problems (Höfler et al. 2012). The present simulations cover the investigation of two different dynamic viscosities of the dispersed phase as well as different droplet trajectories. The comparison between the simulations and the experiments focusses on the velocity distribution of the continuous phase and the droplet deformation and breakup. In both cases a qualitatively good agreement is observed, demonstrating the ability of our SPH implementation for simulating technical relevant two-phase flows.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

Inertia effect on deformation of viscoelastic capsules in microscale flows

The deformation of capsules (i.e., cells, bacterial) in microscale flows plays an important role in biofluid flows such as blood flow in capillaries and cell manipulation in microfluidics. In previous studies on capsule deformation in microscale flows, the inertia effect was often assumed to be negligible and thus omitted. However, this assumption may not reflect real situations, as indicated by recent studies of inertial microfluidics. As such, we aimed to study the inertia effect on capsule deformation in microscale flows and to determine under which conditions this effect may be omitted. Using a collocated grid projection scheme, we developed a finite difference-front tracking method, and investigated the deformation of viscoelastic capsules in microscale flows for Reynolds number (Re) ranging from 0.01 to 10 as seen in vitro and in vivo. The results showed that the transient and steady-state deformation of capsules was significantly affected by inertia, and the flow structure varied considerably when Re was varied from 0.1 to 10. No significant changes were found for Re ranging from 0.01 to 0.1, and hence the inertia effect on capsule deformation in the microscale flows can be omitted when Re is less than 0.1. These findings improve the current understanding of the mechanism underlying cell movement in capillaries and can be applied to optimize the conditions for cell manipulation and separation in microfluidic devices.     

Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.     

A decomposition of Moran's I for clustering detection

The test statistics I_h, I_c, and I_n are derived by decomposing the numerator of the Moran's I test for high-value clustering, low-value clustering, and negative autocorrelation, respectively. Formulae to compute the means and variances of these test statistics are derived under a random permutation test scheme, and the p-values of the test statistics are computed by asymptotic normality. A set of simulations shows that test statistic I_h is likely to be significant only for high-value clustering, test statistic I_c is likely to be significant only for low-value clustering, and test statistic I_n is likely to be significant only for negatively correlated spatial structures. These test statistics were used to reexamine spatial distributions of sudden infant death syndrome in North Carolina and the pH values of streams in the Great Smoky Mountains. In both analyses, low-value clustering and high-value clustering were shown to exit simultaneously.     

A network rate management protocol with TCP congestion control and fairness for all

Our study is motivated by the need to enable quality of service (QoS), congestion control and fair rate allocation for all end applications. We propose a new approach to address these needs which is different from the current practice whereby end applications pursue their own rate control using TCP. Our approach comprises a network rate management protocol (RMP) that controls the rate of all flows (at an aggregate level based on routes) subject to QoS requirements. The RMP control also facilitates a new TCP sliding-window congestion control based on the fair target rates computed by the RMP. Each non-TCP aggregate flow is policed by its respective edge router and each TCP flow adapts its window size as to achieve the RMP suggested fair target rate. The stability analysis of the new TCP congestion control is performed in a linearly scalable framework, which is less restrictive than a fluid model. We show that our proposed control is linearly scalable and establish its global asymptotic stability under arbitrary and variable information time lags, aka totally asynchronous conditions. The stability and the vitality of our control is verified by two means. One is a simulation of a network comprising 74 core links and up to 768 flows, each using its own access link. The simulation is also used to compare our control with the congestion control algorithms used in Fast, Vegas and Reno TCPs. The second verification means is an actual implementation of the control in the Linux kernel and its experimentation in a WAN testbed network comprising six routers and long haul links running UDP flows as well as CUBIC, N-RENO and C-TCP flows. Our experiments demonstrate that our approach can guarantee fair rates for all flows and QoS to premium flows.     

A volume-of-fluid interfacial flow solver with advected normals

We presented a new approach to calculating normal vectors to fluid interfaces in [JCP, 2007;226:774-97], by advecting unit normals along with an interface. In this paper, we introduce an implementation of the method in an interfacial flow solver. The advected normals are used to compute the interface curvature for calculating the surface tension force, and for reconstructing the interface in a volume-conserving volume-of-fluid (VOF) method. To improve the performance of the method in under-resolved regions of the flow, where normals vary sharply, a curvature-based criterion is used to detect and correct poorly defined normals. We present two-dimensional results of advection as well as actual flow problems and demonstrate that the new method is well suited for problems that involve large interface deformation and breakup (i.e. problems that involve substantial interface movement).     

Tight end-to-end per-flow delay bounds in FIFO multiplexing sink-tree networks

Aggregate scheduling has been proposed as a solution for achieving scalability in large-size networks. However, in order to enable the provisioning of real-time services, such as video delivery or voice conversations, in aggregate scheduling networks, end-to-end delay bounds for single flows are required. In this paper, we derive per-flow end-to-end delay bounds in aggregate scheduling networks in which per-egress (or sink-tree) aggregation is in place, and flows traffic is aggregated according to a FIFO policy. The derivation process is based on Network Calculus, which is suitably extended to this purpose. We show that the bound is tight by deriving the scenario in which it is attained. A tight delay bound can be employed for a variety of purposes: for example, devising optimal aggregation criteria and rate provisioning policies based on pre-specified flow delay bounds.     

Numerical model of a two-dimensional,non-plane transient magnetohydrodynamic flow

The equations describing two-dimensional three-component magnetohydrodynamic (MHD) transient flows are formulated for a system of spherical coordinates. With the numerical code based on Implicit Continuous Fluid Eulerian (ICE) scheme, MHD flows resulting from a sudden energy release in a stratified medium are examined. Because of the inclusion of out-of-plane components of velocity and magnetic fields, MHD transverse waves are observed in addition to fast, slow and entropy waves. Numerical results for compressible MHD shocks are found in satisfactory agreement with the theoretical predictions.     

A zonal RANS-LES method to determine the flow over a high-lift configuration

High Reynolds number flows are particularly challenging problems for large-eddy simulations (LES) since small-scale structures in thin and often transitional boundary layers are to be resolved. The range of the turbulent scales is enormous, especially when high-lift configuration flows are considered. For this reason, the prediction of high Reynolds number flow over the entire airfoil using LES requires huge computer resources. To remedy this problem a zonal RANS-LES method for the flow over an airfoil in high-lift configuration at Re_c=1.0×10⁶ is presented. In a first step, a 2D RANS solution is sought, from which boundary conditions are formulated for an embedded LES domain, which comprises the flap and a sub-part of the main airfoil. The turbulent fluctuations in the boundary layers at the inflow region of the LES domain are generated by controlled forcing terms, which use the turbulent shear stress profiles obtained from the RANS solution. The comparison with an LES solution for the full domain and with experimental data shows likewise results for the velocity profiles and wall pressure distributions. The zonal RANS-LES method reduces the computational effort of a full domain LES by approx. 50%.     

A New Multiple-relaxation-time Lattice Boltzmann Method for Natural Convection

This article is devoted to the study of multiple-relaxation-time (MRT) lattice Boltzmann method with eight-by-eight collision matrix for natural convection flow. In the velocity space, eight speed directions are used and the corresponding incompressible multiple-relaxation-time model with force term is presented. D2Q4 model is for temperature field. The coupled double distribution functions (DDF) overcome artificial compressible effect corresponding to the standard MRT model. The simulations of natural convection flows with Pr=0.71 for air and Ra=10³–10⁹ are carried out and excellent agreements are obtained to demonstrate the numerical accuracy and stability of the proposed model.     

Computation of shape and drag of a deforming elastic body under fluid dynamic force

This paper describes numerical simulations for the shape and its drag of an elastic body deforming under the fluid dynamic force. The simulations were carried out by coupling the Navier-Stokes equations and the equations of motion of the elastic body. The equations of motion are formulated for an elastic shell model which is composed of material particles connected with elastic springs and dampers. The relation between deforming elastic body shape in response to the fluid dynamic force and its drag force was investigated under the constraint of constant volume and fixed center of gravity of the elastic body for incompressible and compressible supersonic flows. In these simulations, an initial shape of the elastic body is a circular cylinder and starts deformation under the fluid dynamic force.     

Routing Internet traffic in heterogeneous mesh networks: Analysis and algorithms

In practical wireless mesh networks (WMNs), gateways are subject to hard capacity limits on the aggregate number of flows (in terms of bit rate) that they can support. Thus, if traffic is routed in the mesh network without considering those constraints, as well as the traffic distribution, some gateways or intermediate mesh routers may rapidly get overloaded, and the network resources can be unevenly utilized. To address this problem, in this paper we firstly develop a multi-class queuing network model to analyze feasible throughput allocations, as well as average end-to-end delay, in heterogeneous WMNs. Guided by our analysis, we design a Capacity-Aware Route Selection algorithm (CARS), which allocates network paths to downstream and upstream Internet flows so as to ensure a more balanced utilization of wireless network resources and gateways’ fixed connections. Through simulations in a number of different network scenarios we show that the CARS scheme significantly outperforms conventional shortest path routing, as well as an alternative routing method that distributes the traffic load on the gateway nodes to minimize its variance.     

Critical conditions and breakup of non-squashed microconfined droplets: effects of fluid viscoelasticity

Droplet breakup in systems with either a viscoelastic matrix or a viscoelastic droplet is studied microscopically in bulk and confined shear flow, using a parallel plate counter rotating shear flow cell. The ratio of droplet diameter to gap spacing is systematically varied between 0.1 and 0.85. In bulk shear flow, the effects of matrix and droplet viscoelasticity on the critical capillary number for breakup are very moderate under the studied conditions. However, in confined conditions a profoundly different behaviour is observed: the critical capillary numbers of a viscoelastic droplet are similar to those of a Newtonian droplet, whereas matrix viscoelasticity causes breakup at a much lower capillary number. The critical capillary numbers are compared with the predictions of a phenomenological model by Minale et al. (Langmuir 26:126–132, 2010); the model results are in qualitative disagreement with the experimental data. It is also found that the critical dimensionless droplet length, the critical capillary number, and the dimensionless droplet length at breakup show a similar dependency on confinement ratio. As a result, confined droplets in a viscoelastic matrix have a smaller dimensionless length at breakup than droplets in a Newtonian matrix, which affects the breakup mode. Whereas confined droplets in a Newtonian matrix can break up into multiple parts, only two daughter droplets are obtained after breakup in a viscoelastic matrix, up to very large confinement ratios.     

Effects of topological changes in microchannel geometries on the asymmetric breakup of a droplet

Passive asymmetric breakups of a droplet could be done in many microchannels of various geometries. In order to study the effects of different geometries on the asymmetric breakup of a droplet, four types of asymmetric microchannels with the topological equivalence of geometry are designed, which are T-90, Y-120, Y-150, and I-180 microchannels. A three-dimensional volume of fluid multiphase model is employed to investigate the asymmetric rheological behaviors of a droplet numerically. Three regimes of rheological behaviors as a function of the capillary numbers Ca and the asymmetries As defined by As = (b1 ? b2)/(b1 + b2) (where b1 and b2 are the widths of two asymmetric sidearms) have been observed. A power law model based on three major factors (Ca, As and the initial volume ratio r ₀) is employed to describe the volume ratio of two daughter droplets. The analysis of pressure fields shows that the pressure gradient inside the droplet is one of the major factors causing the droplet translation during its asymmetric breakup. Besides the above similarities among various microchannels, the asymmetric breakup in them also have some slight differences as various geometries have different enhancement or constraint effects on the translation of the droplet and the cutting action of flows. It is disclosed that I-180 microchannel has the smallest critical capillary number, the shortest splitting time, and is hardest to generate satellite droplets.     

An experimental and numerical study of the structure and stability of laminar opposed-jet flows

Experiments, simulations, and numerical bifurcation analysis are used to study the incompressible flow between two opposed tubes with disks mounted at their exits. The experiments in this axisymmetric geometry show that for low and equal Reynolds numbers, Re, at both nozzles, the flow remains symmetric about the plane halfway through the nozzle exits and the stagnation plane is located halfway between the two jets. When Re is increased past a critical value, asymmetric flow fields are obtained even when the momentum fluxes of the two opposed streams are equal. For unequal Re at the jet exits, when the fixed velocity (and the corresponding Reynolds number, Re₁) of one stream is low, the stagnation plane location, SPL, changes smoothly with the Re₂. For high enough Re₁, a hysteretic jump of SPL is observed. Particle Image Velocimetry and flow visualization demonstrate that within the hysteretic range, the two stable flow fields are anti-symmetric. The experimental setup is also studied with transient incompressible flow simulations using a spectral element solver. It is found that to accurately model the flow, we either need to extend the domain into the nozzles, or impose experimental velocity profiles at the nozzle exits. As in the experiments asymmetric flows are obtained past a critical Re. Finally, bifurcation analysis using a Newton-Picard method shows that the transition from symmetric to asymmetric flows results from the loss of stability of the symmetric flows at a pitchfork bifurcation.     

Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles

The drag, lift and moment coefficients of differently shaped single particles have been determined as a function of the angle of incidence at particle Reynolds numbers between Re = 0.3 and 240 under different conditions. For this purpose simulations of the flow around these particles have been performed using the three-dimensional Lattice Boltzmann method. In the first case studied a particle is fixed in a uniform flow, in the second case the particle is rotating in a uniform flow to determine, among others, the Magnus lift force and in the third case the particle is fixed in a linear shear flow. In the first case six particle shapes are considered, i.e. a sphere, a spheroid, a cube, a cuboid and two cylinders with an axis ratio of 1 and 1.5, respectively. In the second and third case the sphere and the spheroid are considered. At the higher Re considered, the drag depends strongly on particle shape, the angle of incidence and particle rotation. The lift and the torque of both the sphere and the spheroid are strongly affected by particle rotation and fluid shear. For approximately Re ? 1, the shear induced lift for unbounded flow could not be simulated as the top and bottom wall have a significant influence in the current flow configuration. The shear induced lift of the sphere changes direction at approximately Re = 50 and the mean (over the orientation) shear induced lift of the spheroid changes direction at approximately Re = 90.