Christine M. Hrenya

Comparison of low Reynolds number k−ε turbulence models in predicting fully developed pipe flow

A comparative study is presented of ten different versions of the low Reynolds number k−e turbulence model. The individual models are briefly outlined and evaluated by application to fully developed pipe flow. Numerical simulations were performed at Reynolds numbers of 7000, 21,800, 50,000, and 500,000. Predictions of the mean axial velocity, turbulent kinetic energy, dissipation rate of turbulent kinetic energy, Reynolds shear stress, eddy viscosity, and skin friction coefficients are compared to both experimental and direct numerical simulation data. The relative performance of the models is assessed. It has been found that the predictions between the models can vary considerably, particularly for the turbulent kinetic energy, its dissipation rate, and the eddy viscosity. The results indicate that the model proposed by Myong and Kasagi (1990, JSME Int. J.33, 63–72) has the best overall performance in predicting fully developed, turbulent pipe flow.

Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport.

A hydrodynamic description for an s -component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation

The linear integral equations defining the Navier-Stokes (NS) transport coefficients for polydisperse granular mixtures of smooth inelastic hard disks or spheres are solved by using the leading terms in a Sonine polynomial expansion. Explicit expressions for all the NS transport coefficients are given in terms of the sizes, masses, compositions, density, and restitution coefficients. In addition, the cooling rate is also evaluated to first order in the gradients. The results hold for arbitrary degree of inelasticity and are not limited to specific values of the parameters of the mixture. Finally, a detailed comparison between the derivation of the current theory and previous theories for mixtures is made, with attention paid to the implication of the various treatments employed to date.

On the role of non-equipartition in the dynamics of rapidly flowing granular mixtures

For rapidly flowing granular mixtures, existing kinetic-theory descriptions based on an assumed form of the velocity distribution function typically contain one of two simplifying assumptions: a Maxwellian velocity distribution or an equipartition of energy. In the current work, the influence of non-equipartition effects is explored in the context of two flow types: flow in which species segregation does not occur (namely, simple shear flow) and a segregating flow. For the former case, a comparison between existing kinetic theories and molecular-dynamics simulations of a binary system indicates that the incorporation of a non-Maxwellian velocity distribution is critical for reliable stress predictions, as is consistent with previous findings. However, the predictions are fairly insensitive to the equipartition versus non-equipartition treatment, despite the presence of a significant non-equipartition of energy. Nevertheless, an analysis of the diffusion equation for a segregating flow indicates that the presence of a non-equipartition of energy gives rise to additional components of the driving forces associated with size segregation. These additional components involve gradients of the species temperature, whereas theories based on an equipartition assumption only involve gradients in the mixture temperature. Molecular-dynamics simulations of the segregating flow, in conjunction with kinetic theory of binary systems, show that the non-equipartition effects are non-negligible for systems characterized by moderate values of mass differences and restitution coefficients. These simulations also reveal that the more massive particle may exhibit a lower species temperature than its lighter counterpart, contrary to previous observations in non-segregating systems. A physical explanation for this behaviour is provided.

Kinetic temperatures for a granular mixture.

An isolated mixture of smooth, inelastic hard spheres supports a homogeneous cooling state with different kinetic temperatures for each species. This phenomenon is explored here by molecular dynamics simulation of a two component fluid, with comparison to predictions of the Enskog kinetic theory. The ratio of kinetic temperatures is studied for two values of the restitution coefficient alpha=0.95 and 0.80, as a function of mass ratio, size ratio, composition, and density. Good agreement between theory and simulation is found for the lower densities and higher restitution coefficient; significant disagreement is observed otherwise. The phenomenon of different temperatures is also discussed for driven systems, as occurs in recent experiments. Differences between the freely cooling state and driven steady states are illustrated.

Size segregation in rapid, granular flows with continuous size distributions

Two-dimensional (dissipative) molecular-dynamics simulations of particulate mixtures with Gaussian and lognormal particle size distributions are employed to gain insight on the segregation behavior of these mixtures when exposed to a granular temperature gradient. Simulations are performed for a collection of smooth, inelastic, hard disks (with constant material density and a constant coefficient of restitution) confined between two walls set to constant, though unequal, granular temperatures. As a result, a gradient in granular temperature develops across the domain. In general, particles of all sizes are found to move toward regions of low granular temperature (overall segregation). Species segregation is also observed. Specifically, large particles demonstrate a higher affinity for the low-temperature regions, and thus accumulate in these cool regions to a greater extent than their smaller counterparts. Furthermore, the local particle size distribution remains of the same form (Gaussian or lognormal) as the overall (including all particles) size distribution. In addition, the behaviors of Gaussian size distributions and narrow lognormal distributions are found to be quite similar.

A kinetic-theory analysis of the scale-up of circulating fluidized beds

Abstract In response to the challenges associated with the scale-up of high-velocity, gas–solid systems such as circulating fluidized bed (CFB) reactors, numerous sets of scaling parameters have been proposed in the literature. Although the scaling sets are typically derived via a non-dimensionalization of the continuum equations for gas–solid flows, variations between sets arise due to differences in the assumed forms of the constitutive relations. In the current study, an existing kinetic-theory model is used to assess the ability of the various scaling laws. In particular, for unlike systems with identical values of the scaling parameters, the level of similarity between the radial profiles for solids concentration and solids velocity is examined. The model predictions indicate that detailed hydrodynamic similitude is not achieved for “reduced” scaling sets in which the ratio of the particle diameter to tube diameter is omitted. Furthermore, the model results also show that the properties characterizing particle collisions (e.g., coefficient of restitution) must also be matched to ensure similarity. Both of these provisions can be traced to the kinetic-theory description of solid-phase stresses, which is not accounted for in the derivation of existing scaling laws, but is included in the mathematical model used in this investigation.

On the role of the Knudsen layer in rapid granular flows

A combination of molecular dynamics simulations, theoretical predictions and previous experiments are used in a two-part study to determine the role of the Knudsen layer in rapid granular flows. First, a robust criterion for the identification of the thickness of the Knudsen layer is established: a rapid deterioration in Navier–Stokes order prediction of the heat flux is found to occur in the Knudsen layer. For (experimental) systems in which heat flux measurements are not easily obtained, a rule-of-thumb for estimating the Knudsen layer thickness follows, namely that such effects are evident within 2.5 (local) mean free paths of a given boundary. Secondly, comparisons of simulation and experimental data with Navier–Stokes order theory are used to provide a measure as to when Knudsen-layer effects become non-negligible. Specifically, predictions that do not account for the presence of a Knudsen layer appear reliable for Knudsen layers collectively composing up to 20% of the domain, whereas deterioration of such predictions becomes apparent when the domain is fully comprised of the Knudsen layer.

Evidence of Higher-Order Effects in Thermally-Driven, Rapid Granular Flows

Molecular dynamic (MD) simulations are used to probe the ability of Navier–Stokes-order theories to predict each of the constitutive quantities – heat flux, stress tensor and dissipation rate – associated with granular materials. The system under investigation is bounded by two opposite walls of set granular temperature and is characterized by zero mean flow. The comparisons between MD and theory provide evidence of higher-order effects in each of the constitutive quantities. Furthermore, the size of these effects is roughly one order of magnitude greater, on a percentage basis, for heat flux than it is for stress or dissipation rate. For the case of heat flux, these effects are attributed to super-Burnett-order contributions (third order in gradients) or greater, since Burnett-order contributions to the heat flux do not exist. Finally, for the system considered, these higher-order contributions to the heat flux outweigh the first-order contribution arising from a gradient in concentration (i.e. the Dufour effect).

The effects of continuous size distributions on the rapid flow of inelastic particles

Molecular-dynamics simulations are employed to investigate the stresses and granular energy in granular materials with Gaussian and lognormal size distributions. Specifically, smooth circular disks of uniform material density engaged in unbounded two-dimensional shear flow are simulated using an event-driven algorithm. Particle collisions are treated as hard-sphere collisions and all collisions have the same coefficient of restitution. The resulting stresses, when nondimensionalized with the root-mean-square diameter, are found to remain relatively constant as the widths of the particle size distributions are increased away from the monodisperse limit. As a consequence, the stresses predicted by monodisperse kinetic theory (using the root-mean-square diameter) are reasonably accurate in the Gaussian and lognormal systems studied herein. This width-independent nature of the total stresses is traced to an effective balancing of the stresses between the larger particles, which generate relatively high stresses, and smaller particles, which generate lower stresses. Moreover, similar to binary-sized systems, the granular energy in Gaussian and lognormal systems is found to be unequally distributed among the various sizes of particles, with large particles possessing more granular energy than their smaller counterparts (i.e., an equipartition of energy is not observed). This difference in granular energy between two particles increases with both inelasticity and the size difference.