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General drag correlations for subsonic to supersonic flow past ordered particle arrays at low and moderate Reynolds numbers
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In the study of gas-particulate multiphase systems, the flow of high-speed gas through a distribution of solid
particulates is of utmost importance. While these aerodynamically interacting systems have been extensively
studied for low-speed gas flows in the gas continuum regime, less attention has been given to high-speed
systems where non-continuum effects are significant due to the high flow gradients. To address this, the flow
of rarefied gas through an aerodynamically interacting monodisperse spherical particle system is studied using
the Direct Simulation Monte Carlo (DSMC) gas-kinetic approach. Since the method provides the best resolution
of shocks at supersonic Mach numbers it is used to classify the weak separated shocks and strong collective
shocks in these systems based on particle spacing in a two-particulate system at different orientation angles.
The study used the two-particle system to help analyze more complex particle distributions of volume fractions,
1%, 5%, and 15%, exposed to gas flows in the slip and transitional gas regime for a free-stream Mach number
range of 0.2 < 𝑀𝑎∞ < 2.0. We observe that the weak separated shocks in the 1% distribution allow a higher
degree of gas penetration and shock-particle interactions or ‘‘hypersonic-surfing ’’, exposing a major fraction
of the particulates to higher force magnitudes. In contrast, the strong collective shock in the 5% and 15%
distributions only generates high particulate forces on the flow-facing particles. Finally, a simple stochastic
model is proposed for use in large-scale Eulerian–Lagrangian simulations that captures the non-monotonic
behavior of average drag and force variability generated by the complicated gas particulate interactions in the
compressible gas regime.
The continuum theory-based models, which include solid stress models and gas-solid drag models, are required for the modeling of gas-solid flows in the framework of the Eulerian–Eulerian method. The interactions among particles are characterized by their diverse behaviors at different flow regimes, including kinetic motion, particle–particle collision and enduring friction. It is difficult to describe the particle behaviors at various regimes by mathematical methods accurately. Therefore, it is very important to develop proper solid stress models that can capture the inherent characteristics of the flow behaviors. In addition, the gas-solid fluidization system is a typical heterogeneous system, which exhibits locally inhomogeneous structures such as bubbles or particle clusters with different shapes and sizes. Due to these inhomogeneous characteristics, the gas-solid drag model has become one of the key challenges in the simulation of gas-solid flows. Various forms of constitutive relations for solid stress models and gas-solid drag models have been reported in the literature. In this paper, we reviewed the solid stress models crossing various flow regimes and drag models in both micro- and mesoscales, which provide a useful reference for model selection in simulating gas-solid flows.
It is well accepted that the drag coefficients of incompressible flows past a sphere only depend on the Reynolds number Re. However, the influence of the Mach number Ma on the drag coefficient becomes increasingly obvious for flows with higher Mach numbers. Unfortunately, this influence has not yet been well understood. In this work, a numerical method coupling the hard-sphere and the pseudo-particle models (HS-PPM) is used to simulate flows past a sphere at 1≤Re≤20 and 0.1≤Ma≤3. The influences of the simulating domain and accommodation coefficients of the interactions between pseudo-particles and the solid sphere on drag coefficients are studied firstly. Then, approporiate simulation domains and reasonable accommodation coefficients are obtained to simulate flows past a sphere. Finally, based on the drag coefficients obtained from the HS-PPM simulations, a preliminary Mach-number correction model of drag coefficients is proposed.
Models for prediction of drag forces within a particle cloud following shock-acceleration are evaluated with the aid of results from particle-resolved simulations in order to quantify how much the disturbances introduced by the proximity of nearby particles affect the drag forces. The drag models evaluated here consist of quasi-steady forces, undisturbed flow forces, inviscid unsteady forces, and viscous unsteady forces. Two dense particle curtain correction schemes to these forces, based on volume fraction and input velocity, are also evaluated. The models are tested in two ways; first they are evaluated based on volume-averaged flow fields from particle-resolved simulations; secondly, they are applied in Eulerian-Lagrangian simulations, and the results are compared to the particle-resolved simulations.
The results show that both correction schemes significantly improve the particle force predictions, but the average total impulse on the particles is still underpredicted by both correction schemes in both tests. With the volume averaged flow fields as input, the volume fraction correction gives the best results. However, in the Eulerian-Lagrangian simulations it is demonstrated that the velocity fluctuation model, associated with the velocity correction scheme, is crucial for obtaining accurate predictions of the mean flow fields.
Over the last few decades, numerous analytical and numerical correlations have been devised for the drag force of particle in packed arrays. Lots of them, however, may not be applicable to microspheres because of the invalid of the no-slip assumption at the solid wall. In this paper, slip flow through assemblages of spheres is investigated by the lattice Boltzmann method (LBM). The velocity slippage at the particle surface is captured by a curved kinetic boundary condition with well preserving the geometric feature of sphere. Three periodic arrays of static and mono-disperse particles, i.e., the simple cubic (SC), the body-centered cubic (BCC) and the face-centered cubic (FCC), each with a relatively wide range of solid volume fraction, 0.0631 ≤ ϕ ≤ 0.431, are considered. The Knudsen number range is 0.01 ≤ Kn ≤ 0.1. The LBM is validated in two cases, including the slip flow over a single unbounded sphere, and the continuum flow through spheres in the SC array. The LBM results agree well with the experimental and numerical data in the literature. Simulations of slip flow through the three ordered arrays of spheres are then performed. The effects of solid volume fraction and slip are both quantified within the developed drag laws. It is found that increasing ϕ contributes to a larger drag, while increasing Kn decreases the drag. The drag is nonlinearly correlated with the ϕ and the Kn, respectively. The difference of drag between the three arrays is obvious only for high ϕ. Moreover, the extent of drag reduction caused by the slip effect is more pronounced for dense arrays.
A study was conducted to demonstrate improved drag correlation for spheres and its application to shock-tube investigations. The improved correlation was based on a number of assumptions, such as limited attention paid to continuum flows and constant and equal particle temperature to the surrounding gas temperature. The resulting improved drag-coefficient correlation consisted of three separate correlations for subcritical, supersonic, and intermediate Mach number regimes. The drag coefficient was expressed as a nonlinear interpolation between the drag coefficients at M = 1 and M = 1:75 in the supersonic regime. The flow around a spherical particle was shock free for subcritical Mach numbers, resulting in the drag coefficient being weakly affected by compressibility effects. A shock wave of limited radial extent existed on the sphere for supercritical but subsonic Mach numbers and the drag coefficient became more strongly dependent on such numbers.
Mass flow rate measurements in a single silicon microchannel were carried out for various gases in isothermal steady flows. The results obtained from hydrodynamic to near free molecular regime by using a powerful experimental platform allowed us to deduce interesting information, notably about the reflection/accommodation process at the wall. In the 0–0.3 Knudsen range, a continuum analytic approach was derived from the NS equations, associated with first or second order slip boundary conditions. Identifying the experimental mass flow rate curves to the theoretical ones the tangential momentum accommodation coefficient TMAC of various gases was extracted. Over the full Knudsen range 0–30 the experimental results were compared with theoretical values calculated from the kinetic approaches: using variable accommodation coefficient values as fitting parameter, the theoretical curves were fitted to the experimental ones. Whatever the Knudsen range and whatever the theoretical approach, the TMAC values are found decreasing when the molecular weights of the gas increase as long as the different gases are compared using the same approach. Moreover, the values of the various accommodation coefficients are rather close to one another but sufficiently smaller than unity indicating that the full accommodation modeling is not satisfactory to describe the gas/wall interaction.
Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.
High-speed disperse multiphase flows are present in numerous environmental and engineering applications with complex interactions between turbulence, shock waves, and particles. Compared with its incompressible counterpart, compressible two-phase flows introduce new scales of motion that challenge simulations and experiments. This review focuses on gas–particle interactions spanning subsonic to supersonic flow conditions. An overview of existing Mach-number-dependent drag laws is presented, with origins from eighteenth-century cannon firings and new insights from particle-resolved numerical simulations. The equations of motion and phenomenology for a single particle are first reviewed. Multiparticle systems spanning dusty gases to dense suspensions are then discussed from numerical and experimental perspectives.
Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 56 is January 2024. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
We present results from experiments within Sandia National Labs’ multiphase shock tube on the shock-induced dispersal of dense particle curtains. The curtain spread rate was measured by tracking the position of the upstream and downstream fronts using high-speed schlieren images. The effect of particle density on the curtain spread rate was examined by comparing curtains comprised of soda lime, stainless steel, and tungsten particles at volume fractions φp=9% and φp≈20%. Various incident shock strengths were investigated, with shock Mach numbers ranging from 1.4 to 1.7. Non-dimensionalized time scales of the spreading process were generated using two scaling methods from literature; one related to the pressure ratio across a reflected shock and the other to the incompressible drag through a grid. Both scaling methods successfully collapse the spread rate of curtains with different particle densities, while only the drag-based scaling properly accounts for variation in volume fraction. A new scaling based on a simple force balance and the experimentally measured pressure differential across the curtain achieves the tightest collapse of all methods tested. Correlations are introduced for the pressure upstream and downstream of the curtain which incorporate the effect of volume fraction and successfully collapse the curtain spread rate when used with the simple force balance scaling. These results expand the parameter space over which the examined scaling methods are valid and introduce correlations that accurately estimate the pressure behind the transmitted and reflected shocks.
Shock–particle interaction is an important phenomenon in a wide range of technological applications and natural phenomena, and the development of accurate models for this interaction is therefore of interest. This study investigates the transient forces during shock–particle interaction at particle Reynolds numbers between 100 and 1000, and incident shock wave Mach numbers between 1.22 and 2.51. This is achieved with the aid of particle-resolved large-eddy simulations. The simulation results show that shock–particle interaction differs qualitatively for subcritical and supercritical incident flow conditions. By decomposing the total force, the inviscid and viscous unsteady forces are estimated. The inviscid unsteady component is significantly larger than the viscous contribution, but the magnitude of the viscous component is comparable to steady-state drag. The predictions of current state of the art force models are compared to the computed particle forces. For subcritical flows, the models are quite successful in predicting the drag. For these conditions, the magnitudes of both the inviscid and viscous unsteady force models agree well with the simulation results, but the transient nature of the viscous unsteady force history is not well captured. For supercritical flows, the inviscid unsteady force model is not able to capture the force dynamics. This highlights the need for the development of unsteady force models for supercritical flow conditions.
A generalized physics-based expression for the drag coefficient of spherical particles moving in a fluid is developed. The proposed correlation incorporates essential rarefied effects, low-speed hydrodynamics, and shock-wave physics to accurately model the particle-drag force for a wide range of Mach and Knudsen numbers (and therefore Reynolds number). Owing to the basis of the derivation in physics-based scaling laws, the proposed correlation embeds gas-specific properties and has explicit dependence on the ratio of specific heat capacities. The correlation is applicable for arbitrary particle relative velocity, particle diameter, gas pressure, gas temperature, and surface temperature. Compared with existing drag models, the correlation is shown to more accurately reproduce a wide range of experimental data. Finally, the new correlation is applied to simulate particle trajectories in high-speed dusty flows, relevant to a spacecraft entering the Martian atmosphere. The enhanced surface heat flux due to particle impact is found to be sensitive to the particle drag model.
A comprehensive review of all relevant experimental data was completed, including recent data for the drag coefficient for a sphere in supersonic and hypersonic flows. The primary characterization parameter included the relative Mach, Knudsen, and Reynolds numbers based on the relative velocity, the sphere diameter, and other parameters. This review of data showed that the previously proposed nexus at a Reynolds number below 45 was not strictly met, and it instead included a weak transonic bump, which was identified numerically for the first time with the present simulations. New continuum-gas and rarefied-gas simulations were conducted and were combined with the expanded experimental dataset to improve the quantitative description of the drag coefficient in this region. The results indicated that a quasi nexus bridges the rarefaction regime and the compressible flow regimes. The comprehensive dataset was then used to develop new empirical models for the drag coefficient that showed improved robustness and accuracy as compared to previous models. These models are limited by the critical Reynolds number associated with boundary-layer transition on the sphere, which was found to increase substantially with the sphere Mach number.
Particle-resolved simulation is performed to study the drag force on arrays of ellipsoidal particles with a temperature difference with respect to the surrounding fluid. The effect of particle shape, arrangement, solid volume fraction and particle temperature is examined. We found that the particle shape and arrangement strongly influence the drag coefficient CD when the particle volume fraction ϕ is small. The influence, however, is inhibited as ϕ increases. We also show that the drag coefficient decreases with the increase of the solid volume fraction, which is a reversal of the trend found in previous uniform suspension systems. Moreover, CD increases with the increase of particle temperature while slightly decreases with the fluid temperature. Through a novel decomposition analysis, we show that the increase of fluid viscosity causes the increase of CD when the particle temperature is increased, while a combined effect of the fluid density, inlet velocity and the integral of the velocity gradient around particle on CD was observed when the fluid temperature changes. Finally, new drag correlations, as power functions of Reynolds number, particle temperature and volume fraction, are proposed for clusters with different particle shapes and arrangements.
Experiments were performed within Sandia National Labs’ Multiphase Shock Tube to measure and quantify the shock-induced dispersal of a shock/dense particle curtain interaction. Following interaction with a planar travelling shock wave, schlieren imaging at 75 kHz was used to track the upstream and downstream edges of the curtain. Data were obtained for two particle diameter ranges ( , 300{-}355~\unicode[STIX]{x03BC}\text{m} ) across Mach numbers ranging from 1.24 to 2.02. Using these data, along with data compiled from the literature, the dispersion of a dense curtain was studied for multiple Mach numbers (1.2–2.6), particle sizes ( 100{-}1000~\unicode[STIX]{x03BC}\text{m} ) and volume fractions (9–32 %). Data were non-dimensionalized according to two different scaling methods found within the literature, with time scales defined based on either particle propagation time or pressure ratio across a reflected shock. The data show that spreading of the particle curtain is a function of the volume fraction, with the effectiveness of each time scale based on the proximity of a given curtain’s volume fraction to the dilute mixture regime. It is seen that volume fraction corrections applied to a traditional particle propagation time scale result in the best collapse of the data between the two time scales tested here. In addition, a constant-thickness regime has been identified, which has not been noted within previous literature.
A novel approach to improve the performance of supersonic and hypersonic intake systems with nano-particle injection has been studied. A parametric study using Mach number (M ∞ ), Stokes number (Stk), particle Eckert number (Ec p ), and thermal transport number (α t ) was conducted across a quasi-1D converging-diverging (C-D) supersonic intake at idealized and single-shock compression cases. Gains in stagnation pressure recovery were achieved for both the cases. Gains were observed in the idealized compression case when: Ec p >0.25 and α t >0.5 for a Mach number of 2.5; and Ec p >0.5 and α t >0.7 for a Mach number of 5. The results also showed that a combination of cooling, momentum exchange, and particle size is required to enhance intake performance. A rectangular mixed-compression intake at Mach 3 was simulated using an unsteady compressible gas-particle CFD solver in OpenFOAM. CFD simulations with nano-particle injection predicted a 16% gain in the exit pressure recovery.
High-speed, time-resolved particle image velocimetry with a pulse-burst laser was used to measure the gas-phase velocity upstream and downstream of a shock wave–particle curtain interaction at three shock Mach numbers (1.22, 1.40, and 1.45) at a repetition rate of 37.5 kHz. The particle curtain was formed from free-falling soda-lime particles resulting in volume fractions of 9% or 23% at mid-height, depending on particle diameter (106–125 and 300–355 μm, respectively). Following impingement by a shock wave, a pressure difference was created between the upstream and downstream sides of the curtain, which accelerated flow through the curtain. Jetting of flow through the curtain was observed downstream once deformation of the curtain began, demonstrating a long-term unsteady effect. Using a control volume approach, the unsteady drag on the curtain was estimated from velocity and pressure data. The drag imposed on the curtain has a strong volume fraction dependence with a prolonged unsteadiness following initial shock impingement. In addition, the data suggest that the resulting pressure difference following the propagation of the reflected and transmitted shock waves is the primary component to curtain drag.
Continuum methods are not accurate enough for flows at high Knudsen numbers, whereas rigorous molecular dynamics (MD) methods are too costly for simulations at practical dimensions. Hard-sphere (HS) model is a simplified MD method efficient for dilute gaseous flow but is of poor parallelism due to its event-driven nature, which sets a strong limitation to its large-scale applications. In this work, pseudo-particle modelling, a time-driven modelling approach is coupled with HS model to construct a scalable parallel method capable of simulating flows and transport processes at high Knudsen numbers without losing necessary molecular details in describing their macro-scale behaviours. The method is validated in several classical simulation cases and its performance is evaluated to be favourable. To demonstrate the potential applications of this method, we also simulate the diffusion of small molecules in multi-scale porous media which is related to catalysis, material preparation and micro chemical engineering in the long term.
We develop an analytical model for the thermal boundary conductance between a solid and a gas. By considering the thermal fluxes in the solid and the gas, we describe the transmission of energy across the solid/gas interface with diffuse mismatch theory. From the predicted thermal boundary conductances across solid/gas interfaces, the equilibrium thermal accommodation coefficient is determined and compared to predictions from molecular dynamics simulations on the modelsolid-gas systems. We show that our model is applicable for modeling the thermal accommodation of gases on solid surfaces at non-cryogenic temperatures and relatively strong solid-gas interactions (εsf ≳ kBT).
The particle velocity in cold gas dynamic spraying (CGDS) is one of the most important factors that can determine the properties of the bonding to the substrate. The acceleration of gas to particles is strongly dependent on the densities of particles and the particle size. In this paper, the acceleration process of micro-scale and nano-scale copper (Cu) and platinum (Pt) particles in De-Laval-Type nozzle is investigated. A numerical simulation is performed for the gas-particle two phase flow with particle diameter ranging from 100nm to 50μm, which are accelerated by carrier gas Nitrogen in a supersonic De-Laval-type nozzle. The results show that cone-shape weak shocks (compression waves) occur at the exit of divergent section and the particle density has significant effect on the accele ration of micro-scale particles. At same inlet condition, the velocity of the smaller particles is larger than the larger particles at the exit of the divergent section of the nozzle.
Following a critical review of previous publications on the relative velocity between a cloud of spherical solid particles and the continuous fluid med
Pseudo-particle modeling (PPM), a molecular modeling method which combines time-driven algorithms and hard molecule modeling, was originally developed for simulating gas in complex multiphase systems (Ge & Li, 2003; Ge et al., 2005; Ge, 1998). In this work, the properties of two- and three-dimensional pseudo-particle systems, namely, mean free path, compressibility factor, self-diffusion coefficient and shear viscosity, are systematically measured by using PPM. It is found that in terms of an effective diameter, the results well conform to the Chapman–Enskog theory, thus suggesting that PPM can be employed to simulate the micro- and meso-scale behavior of ordinary gas and fluid flows.
The dynamics of unstart of a floor-mounted inlet/isolator model in a Mach 5 flow are investigated experimentally. The inlet section contains a 6-deg compression ramp, and the isolator is a rectangular straight duct that is 25.4-mm high by 50.8-mm wide by 242.3-mm long. Measurements made include 8-kHz schlieren imaging and simultaneous fast-response wall pressures along the length of the inlet/isolator. Unstart is initiated by deflecting a flap at the downstream end of the isolator. The shock system, induced by unstart, initially propagates upstream through the isolator at a velocity of about 35 m/s (in the laboratory frame of reference), then decelerates to about 20 m/s near the isolator entrance, and then accelerates to a velocity of about 74 m/s within the inlet. Throughout the isolator, unstart is seen to be strongly associated with boundary-layer separation. Once the inlet has unstarted, a highamplitude oscillatory (periodic) unstarted flow ensues, for which the oscillation frequency is about 124 Hz. However, under some conditions, an 84-Hz oscillatory unstarted flow mode, with lower pressure fluctuations, is observed. Under other conditions, a nonoscillatory unstarted flow, with much lower pressure fluctuations, is observed.
An extensive series of measurements has been made in a ballistic range to establish accurate values of sphere drag coefficients in a flight regime characterized by freestream Mach numbers from 0. 1 to 6. 0 and Reynolds numbers from 20 to 200. 000. These measurements, together with other published data, permit the derivation of sphere drag coefficients with an uncertainty of plus or minus 2% in this flight regime.
Separation of ultrafine particles by inertial effect in a supersonic flow field has been experimentally studied. The experiment was carried out in a Laval nozzle with different configurations. As a result, the cut size was found to decrease as Mach number increases, i.e., gas velocity becomes faster and the smallest cut size by this experiment has an aerodynamic diameter of about 0.04 μm at Mach number of about 2.8, which is about one order of magnitude smaller than that of conventional impactor. It is also much smaller than the performance of so-called low-pressure impactor and a fairly clear difference was observed among the configuration of nozzle and clearance between nozzle outlet to collection plate. Separation efficiency of the present impactor was evaluated by a Stokes number Stk and was found to increase very steeply against Stk, which is much steeper than the obtained separation curve for conventional impactor operating under atmospheric pressure.
Despite its wide applications, fluidization is not understood enough to satisfy our technical or academic interests. Cascading simulation approaches on different scales, with small-scale approaches provide constitutional correlations to larger scale approaches, is considered a practical way toward this direction. However, by physically reproducing many macro-scale phenomena in fluid flow and fluidization on micro-scales even below the traditional continuum limit, pseudo-particle modeling (PPM, Ge and Li (Proceedings of the Fifth International Conference on Circulating Fluidized Bed, Beijing, China, Science Press, Beijing, 1996) has suggested the possibility of a more straightforward and penetrating way. In this paper, traditional approaches are reviewed first and then PPM is discussed in full length and validated further. We demonstrate that it has maintained all necessities on the molecular level for comprehensive flow description, and the reproduced phenomena, such as bubbling, clustering and radial heterogeneity, have reflected the fundamental mechanism of their macro-scale counterparts despite the vast scale difference. With this digital miniature, every detail of the flow can be traced non-intrusively until the lowest level in classic physics and experiment with flexible parameters, which provides a unique tool for theoretical study and engineering predictions. Therefore, PPM is at least a useful complement, if not substitute, to traditional approaches.
Accommodation coefficient (alpha) values which can be confidently applied to a solution of satellite-drag and heat-transfer problems must be obtained from experiments with gas molecules whose velocities match those of satellites. The problem of generating in the laboratory a nearly monoenergetic beam of molecules in the range 1 - 10 eV with adequate flux has not been solved. Experimental facilities such as shock and arc tunnels which may produce molecular flows of nearly adequate velocities are not readily adaptable to alpha measurements. For these measurements the transient flow obtained in a shock tunnel is extremely inconvenient, while the composition and energy distribution of the gas obtained in an arc tunnel is not known with sufficient accuracy. The composition of a satellite surface in orbit, and hence the accommodation coefficient, will most probably differ appreciably from that of an untreated surface of the same material exposed to a laboratory vacuum of, say only, 10 to the -7th power mm of Hg. Furthermore, in orbit, satellite surface composition may change with changing surface temperature and ambient gas pressure, resulting in corresponding changes in alpha.
The temperature dependence of the tangential momentum accommodation coefficient (TMAC) is investigated by examining gas flows in a submicron channel using molecular dynamics simulations. The results show that the TMAC decreases with the increasing temperature following an exponential decay law, and is more sensitive to lower temperatures than to higher ones. The molecular trapping-desorption behaviors near the channel surface are found to be responsible for this dependence. (C) 2005 American Institute of Physics.
Linearized viscous compressible Navier-Stokes equations are solved for the transient force on a spherical particle undergoing unsteady motion in an inhomogeneous unsteady ambient flow. The problem is formulated in a reference frame attached to the particle and the force contributions from the undisturbed ambient flow and the perturbation flow are separated. Using a density-weighted velocity transformation and reciprocal relation, the total force is first obtained in the Laplace domain and then transformed to the time domain. The total force is separated into the quasi-steady, inviscid unsteady, and viscous unsteady contributions. The above rigorously derived particle equation of motion can be considered as the compressible extension of the Maxey-Riley-Gatignol equation of motion and it incorporates interesting physics that arises from the combined effects of inhomogeneity and compressibility.
A review of compressibility and rarefaction effects on spherical particle drag was conducted based on existing experimental data, theoretical limits, and direct simulation Monte Carlo method results. The data indicated a nexus point with respect to effects of Mach number and Knudsen number. In particular, it was found that a single drag coefficient (of about 1.63) is obtained for all particle conditions when the particle Reynolds number is about 45, and is independent of compressibility or rarefaction effects. At lower Reynolds numbers, the drag is dominated by rarefaction, and at higher Reynolds numbers, it is dominated by compressibility. The nexus, therefore, allows construction of two separate models for these two regimes. The compression-dominated regime is obtained using a modification of the Clift-Gauvin model to specifically incorporate Mach number effects. The resulting model was based on a wide range of experimental data and showed superior prediction robustness compared with previous models. For the rarefaction-dominated regime, the present model was constructed to directly integrate the theoretical creeping flow limits, including the incompressible continuum How limit (Stokes drag), the incompressible weak rarefaction limit (Basset-Knudsen correction), and the incompressible free-molecular flow limit (Epstein theory). Empirical correlations are used to extend this model to finite particle Reynolds numbers within the rarefaction-dominated regime.
Currently there is a substantial lack of data for interactions of shock waves with particle fields having volume fractions residing between the dilute and granular regimes. To close this gap, a novel multiphase shock tube has been constructed to drive a planar shock wave into a dense gas–solid field of particles. A nearly spatially isotropic field of particles is generated in the test section by a gravity-fed method that results in a spanwise curtain of spherical 100-micron particles having a volume fraction of about 20%. Interactions with incident shock Mach numbers of 1.66, 1.92, and 2.02 are reported. High-speed schlieren imaging simultaneous with high-frequency wall pressure measurements are used to reveal the complex wave structure associated with the interaction. Following incident shock impingement, transmitted and reflected shocks are observed, which lead to differences in particle drag across the streamwise dimension of the curtain. Shortly thereafter, the particle field begins to propagate downstream and spread. For all three Mach numbers tested, the energy and momentum fluxes in the induced flow far downstream are reduced about 30–40% by the presence of the particle field.
The interaction of a planar shock wave with a dense particle curtain is investigated through modeling and experiments. The physics in the interaction between a shock wave with a dense gas-particle mixture is markedly differently from that with a dilute mixture. Following the passage of the shock wave, the dense particle curtain expands rapidly as it propagates downstream and pressures equilibrate throughout the flow field. In the simulations, the particles are viewed as point-particles and are traced in a Lagrangian framework. A physics-based model is then developed to account for interphase coupling. Compared to the standard drag law, four major improvements are made in the present interphase coupling model to take into account: (1) unsteady force contributions to particle force; (2) effect of compressibility on hydrodynamic forces; (3) effect of particle volume fraction on hydrodynamic forces; (4) effect of inter-particle collision. The complex behavior of the dense particle curtain is due to the interplay between two-way coupling, finite particle inertia, and unsteady forces. Incorporation of these effects through significant modeling improvements is essential for the simulation results to agree well with the experimental data. As a result of the large pressure gradient inside the particle curtain, the unsteady forces remain significant for a long time compared to the quasi-steady force and greatly influence the particle curtain motion.
The drag force on a particle in a fluid—multiparticle interaction system may be expressed as the product of the drag force on an unhindered particle, subject to the same volumetric flux of fluid, and a voidage function. It is demonstrated that for a wide varicty of both fixed-bed and.suspended-particle systems, file voidage function may be expressed as ϵ−β, where the exponent β is dependent on the particle Reynolds number but independent of other system variables.
The problem of a mathematical description of the relation between the distribution functions of impinging and emerging molecules at a solid wall is considered. Under suitable assumptions, of a rather general nature, certain properties of the acceptable models are set forth. These properties are then used to prove certain basic inequalities (including the one necessary to prove the H-theorem in presence of physical walls) as. well as constructing a specific model containing two disposable parameters.
We calculate the force on a periodic array of spheres in a viscous
flow
at small
Reynolds number and for small volume fraction. This generalizes the known
results
for the force on a periodic array due to Stokes flow (zero Reynolds number)
and the
Oseen correction to the Stokes formula for the force on a single sphere
(zero volume
fraction). We use a generalization of Hasimoto's approach that is
based on an
analysis of periodic Green's functions. We compare our results to
the
phenomenological ones of Kaneda for viscous flow past a random array
of spheres.
Theory and lattice-Boltzmann simulations are used to examine the effects of fluid
inertia, at small Reynolds numbers, on flows in simple cubic, face-centred cubic and
random arrays of spheres. The drag force on the spheres, and hence the permeability
of the arrays, is determined at small but finite Reynolds numbers, at solid volume
fractions up to the close-packed limits of the arrays. For small solid volume fraction,
the simulations are compared to theory, showing that the first inertial contribution
to the drag force, when scaled with the Stokes drag force on a single sphere in
an unbounded fluid, is proportional to the square of the Reynolds number. The
simulations show that this scaling persists at solid volume fractions up to the
close-packed limits of the arrays, and that the first inertial contribution to the drag
force relative to the Stokes-flow drag force decreases with increasing solid volume
fraction. The temporal evolution of the spatially averaged velocity and the drag
force is examined when the fluid is accelerated from rest by a constant average
pressure gradient toward a steady Stokes flow. Theory for the short- and long-time
behaviour is in good agreement with simulations, showing that the unsteady force
is dominated by quasi-steady drag and added-mass forces. The short- and long-time
added-mass coefficients are obtained from potential-flow and quasi-steady
viscous-flow approximations, respectively.
Lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at
moderate Reynolds numbers, on flows in simple cubic, face-centred cubic and random
arrays of spheres. The drag force on the spheres, and hence the permeability of the
arrays, is calculated as a function of the Reynolds number at solid volume fractions
up to the close-packed limits of the arrays. At Reynolds numbers up to O(102), the
non-dimensional drag force has a more complex dependence on the Reynolds number
and the solid volume fraction than suggested by the well-known Ergun correlation,
particularly at solid volume fractions smaller than those that can be achieved in
physical experiments. However, good agreement is found between the simulations
and Ergun's correlation at solid volume fractions approaching the close-packed limit.
For ordered arrays, the drag force is further complicated by its dependence on the
direction of the flow relative to the axes of the arrays, even though in the absence
of fluid inertia the permeability is isotropic. Visualizations of the flows are used to
help interpret the numerical results. For random arrays, the transition to unsteady
flow and the effect of moderate Reynolds numbers on hydrodynamic dispersion are
discussed.
Sphere drag has been measured in a low-density supersonic wind tunnel by a simple displacement technique. A conical nozzle of fixed geometry, operated at a constant supply temperature of about 300 °K was used. Test section Mach numbers ranged between 3.8 and 4.3 depending on supply pressure. The Reynolds number and Knudsen number range, based on free-stream conditions and sphere diameter, were 50 < Re[infty infinity] < 1000 and 0·106 < Kn[infty infinity] < 0·006, respectively. This range was achieved by varying sphere size and supply pressure. The drag coefficient was found to increase from a value of CD = 1·17 at Re[infty infinity] = 1000 to CD = 1·73 at Re[infty infinity] = 50.
A linear stability analysis is performed for the homogeneous state of a monodisperse
gas-fluidized bed of spherical particles undergoing hydrodynamic interactions and
solid-body collisions at small particle Reynolds number and finite Stokes number.
A prerequisite for the stability analysis is the determination of the particle velocity
variance which controls the particle-phase pressure. In the absence of an imposed
shear, this velocity variance arises solely due to the hydrodynamic interactions among
the particles. Since the uniform state of these suspensions is unstable over a wide range
of values of particle volume fraction [phi] and Stokes number St, full dynamic simulations
cannot be used in general to characterize the properties of the homogeneous state.
Instead, we use an asymptotic analysis for large Stokes numbers together with
numerical simulations of the hydrodynamic interactions among particles with specified
velocities to determine the hydrodynamic sources and sinks of particle-phase energy.
In this limit, the velocity distribution to leading order is Maxwellian and therefore
standard kinetic theories for granular/hard-sphere molecular systems can be used to
predict the particle-phase pressure and rheology of the bed once the velocity variance
of the particles is determined. The analysis is then extended to moderately large Stokes
numbers for which the anisotropy of the velocity distribution is considerable by using
a kinetic theory which combines the theoretical analysis of Koch (1990) for dilute
suspensions ([phi] [double less-than sign] 1) with numerical simulation results for non-dilute suspensions
at large Stokes numbers. A linear stability analysis of the resulting equations of
motion provides the first a
priori predictions of the marginal stability limits for
the homogeneous state of a gas-fluidized bed. Dynamical simulations following the
detailed motions of the particles in small periodic unit cells confirm the theoretical
predictions for the particle velocity variance. Simulations using larger unit cells
exhibit an inhomogeneous structure consistent with the predicted instability of the
homogeneous gas–solid suspension.
We treat the problem of slow flow through a periodic array of spheres. Our interest is in the drag force exerted on the array, and hence the permeability of such arrays. It is shown to be convenient to formulate the problem as a set of two-dimensional integral equations for the unknown surface stress vector, thus lowering the dimension of the problem. This set is solved numerically to obtain the drag as a function of particle concentration and packing characteristics. Results are given over the full concentration range for simple cubic, body-centred cubic and face-centred cubic arrays and these agree well with previous limited experimental, asymptotic and numerical results.