Navier-Stokes Order Research Articles

Hydrodynamics of polydisperse gas-solid flows: Kinetic theory and multifluid simulation

Polydisperse gas-solid flows, which is notoriously difficult to model due to the complex gas-particle and particle-particle interactions, are widely encountered in industry. In this article, a refined kinetic theory for polydisperse flow is developed, which features single-parameter Chapman-Enskog expansion (the Knudsen number) and exact calculation of the integrations related to pair distribution function of particle velocity without any mathematical approximations. The Navier-Stokes order constitutive relations for multifluid modeling of polydisperse gas-solid flow are then obtained analytically, including the solid stress tensor, the solid-solid drag force, the granular heat flux and the energy dissipation rate. Finally, the model is preliminarily validated by comparing to the discrete element simulation data of one-dimensional granular shear flow and by showing that the hydrodynamic characteristics of gas-solid flows in a bubbling fluidized bed containing bidisperse particles can be successfully predicted.

Kinetic theory of polydisperse gas–solid flow: Navier–Stokes transport coefficients

The particulate phase stress and solid–solid drag force in the multifluid modeling of polydisperse gas–solid flows are usually closed using kinetic theory. This research aims to establish the hydrodynamic equations and constitutive relations of the multifluid model for polydisperse systems via species kinetic theory, in which the non-equipartition of energy and interphase slip velocity between different species are considered. Whereas previous studies have used approximations, such as Taylor series expansions, to simplify the calculation of collision integrals, the present study, for the first time, solves the collision integrals analytically without any approximations to obtain accurate constitutive relations. Explicit expressions for the constitutive laws are obtained, including the particle stress tensor, solid–solid drag force, heat flux, and energy dissipation rate up to the Navier–Stokes order. The present study offers more complete and mathematically rigorous constitutive laws for the multifluid modeling of polydisperse gas–solid flows.

An entropy criterion for the validity of Navier-Stokes order continuum theory for gas-solid flow: Kinetic theory analysis

Navier-Stokes order continuum theory (NS theory) established on the basis of local thermodynamic equilibrium (LTE) postulate has been widely used to study gas-solid flows. Qualitatively, the postulate is valid for systems sufficiently close to equilibrium, however, continuum theory provides no information on how small these deviations from equilibrium should be in order to guarantee the validity of LTE. To this end, kinetic theory analysis was carried out to formulate an entropy criterion, it was shown that (i) the entropy density up to the NS order is exactly the same as the one at the LTE condition, meaning that the LTE postulate is only valid for linear non-equilibrium regime, although NS theory and the LTE postulate are logically self-consistent; (ii) an entropy criterion characterizing the relative importance of nonlinear effects is derived, which should be small in order to make sure that the LTE postulate is valid and also sets the boundary of validity of NS theory; and (iii) the proposed entropy criterion, Is≪1, is not only a function of granular temperature gradient and velocity gradient, but also a function of 1-e2 characterizing the inelasticity of particle-particle collisions. It is more complex and systematic than the traditional criterion based on the Knudsen number, Kn≪1.

Plane shock waves and Haff’s law in a granular gas

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler- and Navier–Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff’s law (${\it\theta}(t)\sim t^{-2}$), but the downstream temperature decays faster than its upstream counterpart. Haff’s law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff’s law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be ‘regularized’ to arrest the maximum density.

Hydrodynamics for a model of a confined quasi-two-dimensional granular gas.

The hydrodynamic equations for a model of a confined quasi-two-dimensional gas of smooth inelastic hard spheres are derived from the Boltzmann equation for the model, using a generalization of the Chapman-Enskog method. The heat and momentum fluxes are calculated to Navier-Stokes order, and the associated transport coefficients are explicitly determined as functions of the coefficient of normal restitution and the velocity parameter involved in the definition of the model. Also an Euler transport term contributing to the energy transport equation is considered. This term arises from the gradient expansion of the rate of change of the temperature due to the inelasticity of collisions, and it vanishes for elastic systems. The hydrodynamic equations are particularized for the relevant case of a system in the homogeneous steady state. The relationship with previous works is analyzed.

Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method

AbstractThe non-Newtonian stress tensor, collisional dissipation rate and heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. For steady uniform shear flow, the balance equation for the second moment of velocity fluctuations is solved semi-analytically, yielding closed-form expressions for the shear viscosity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $, pressure $p$, first normal stress difference ${\mathcal{N}}_1$ and dissipation rate ${\mathcal{D}}$ as functions of (i) density or area fraction $\nu $, (ii) restitution coefficient $e$, (iii) dimensionless shear rate $R$, (iv) temperature anisotropy $\eta $ (the difference between the principal eigenvalues of the second-moment tensor) and (v) angle $\phi $ between the principal directions of the shear tensor and the second-moment tensor. The last two parameters are zero at the Navier–Stokes order, recovering the known exact transport coefficients from the present analysis in the limit $\eta ,\phi \to 0$, and are therefore measures of the non-Newtonian rheology of the medium. An exact analytical solution for leading-order moment equations is given, which helped to determine the scaling relations of $R$, $\eta $ and $\phi $ with inelasticity. We show that the terms at super-Burnett order must be retained for a quantitative prediction of transport coefficients, especially at moderate to large densities for small values of the restitution coefficient ($e \ll 1$). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for $p$, $\mu $ and ${\mathcal{N}}_1$ over a range of densities spanning from the dilute to close to the freezing point. In contrast, the predictions from a constitutive model at Navier–Stokes order are found to deviate significantly from both the simulation and the moment theory even at moderate values of the restitution coefficient ($e\sim 0.9$). Lastly, a generalized Fourier law for the granular heat flux, which vanishes identically in the uniform shear state, is derived for a dilute granular gas by analysing the non-uniform shear flow via an expansion around the anisotropic Gaussian state. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic second-rank tensor, for which explicit analytical expressions are given.

Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime

AbstractIn this work we quantitatively assess, via instabilities, a Navier–Stokes-order (small-Knudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers ($K{n}_{u} $) at the time of cluster detection reveals $K{n}_{u} \gg 1$ due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a small-$Kn$ (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier–Stokes-order (small-$Kn$) theories are surprisingly accurate outside their expected range of validity.

Time dependence and density inversion in simulations of vertically oscillated granular layers

We study a layer of grains atop a plate which oscillates sinusoidally in the direction of gravity, using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order as well as hard-sphere molecular dynamics simulations. For high accelerational amplitudes of the plate, the layer exhibits a steady-state "density inversion" in which a high-density portion of the layer is supported by a lower-density portion. At low accelerational amplitudes, the layer exhibits oscillatory time dependence that is strongly correlated to the motion of the plate. We show that continuum simulations yield results consistent with molecular dynamics results in both regimes.

Continuum representation of a continuous size distribution of particles engaged in rapid granular flow

Natural and industrial granular flows often consist of several particle sizes, approximately forming a continuous particle size distribution (PSD). Continuous PSDs are ubiquitous, though existing kinetic-theory-based, hydrodynamic models for rapid granular flows are limited to a discrete number of species. The objective of this work is twofold: (i) to determine the number of discrete species required to accurately approximate a continuous PSD and (ii) to validate these results via a comparison with molecular dynamics (MD) simulations of continuous PSDs. With regard to the former, several analytic (Gaussian and lognormal) and experimental (coal and lunar soil simulants) distributions are investigated. Transport coefficients (pressure, shear viscosity, etc.) of the granular mixture given by the polydisperse theory of Garzó et al. [“Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport,” Phys. Rev. E 76, 031303 (2007)10.1103/PhysRevE.76.031303;Garzó et al. “Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport,” Phys. Rev. E 76, 031304 (2007)10.1103/PhysRevE.76.031304] are compared using an increasing number of species s to approximate the given PSD. These discrete approximations are determined by matching the first 2s moments of the approximation and the given continuous distribution. Relatively few species are required to approximate moderately wide distributions (Gaussian, lognormal), whereas even wider distributions (coal and lunar soil simulants) require a larger number of species. Regarding the second objective, a comparison between MD simulations and kinetic-theory predictions for a simple shear flow of both Gaussian and lognormal PSDs reveal essentially no loss of accuracy stemming from the polydisperse theory itself (as compared to theories for monodisperse systems) or from the discrete approximations of continuous PSDs used in the polydisperse theory.

Convection in three-dimensional vibrofluidized granular beds

We study convective motion in vertically vibrated three-dimensional granular beds by comparing the predictions of a model based on a hydrodynamic description to Navier–Stokes order with experimental results obtained using positron emission particle tracking (PEPT). The three-dimensional conservation equations relating mass, momentum and energy are solved using the finite element (FE) method for a viscous vibrofluidized bed by using only observable system parameters such as particle number, size, mass and coefficients of restitution. The mean velocity profiles from the viscous model show reasonable agreement with the experimental results at relatively low altitudes for the range of experimental values studied, though the velocity fields at higher altitudes were systematically underestimated by the model. We confirm that the convection rolls are influenced by the sidewall coefficient of restitution and demonstrate the scaling relationships that operate, where increasing amplitude of vibration leads to a reduction in the angular velocity of the rolls.

Fluctuating Navier-Stokes equations for inelastic hard spheres or disks

Starting from the fluctuating Boltzmann equation for smooth inelastic hard spheres or disks, closed equations for the fluctuating hydrodynamic fields to Navier-Stokes order are derived. This requires deriving constitutive relations for both the fluctuating fluxes and the correlations of the random forces. The former are identified as having the same form as the macroscopic average fluxes and involving the same transport coefficients. On the other hand, the random force terms exhibit two peculiarities as compared with their elastic limit for molecular systems. First, they are not white but have some finite relaxation time. Second, their amplitude is not determined by the macroscopic transport coefficients but involves new coefficients.

Continuum simulations of shocks and patterns in vertically oscillated granular layers

We study interactions between shocks and standing-wave patterns in vertically oscillated layers of granular media using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order. We simulate a layer of grains atop a plate that oscillates sinusoidally in the direction of gravity. Standing waves form stripe patterns when the accelerational amplitude of the plate's oscillation exceeds a critical value. Shocks also form with each collision between the layer and the plate; we show that pressure gradients formed by these shocks cause the flow to reverse direction within the layer. This reversal leads to an oscillatory state of the pattern that is subharmonic with respect to the plate's oscillation. Finally, we study the relationship between shocks and patterns in layers oscillated at various frequencies and show that the pattern wavelength increases monotonically as the shock strength increases.

Effects of thermal noise on pattern onset in continuum simulations of shaken granular layers

The author investigates the onset of patterns in vertically oscillated layers of dissipative particles using numerical solutions of continuum equations to Navier-Stokes order. Above a critical accelerational amplitude of the cell, standing waves form stripe patterns which oscillate subharmonically with respect to the cell. Continuum simulations neglecting interparticle friction yield pattern wavelengths consistent with experiments using frictional particles. However, the critical acceleration for standing-wave formation is approximately 10% lower in continuum simulations without added noise than in molecular-dynamics simulations. This Brief Report incorporates fluctuating hydrodynamics theory into continuum simulations by adding noise terms with no fit parameters; this modification yields a critical acceleration in agreement with molecular-dynamics simulations.

Linear response and hydrodynamics for granular fluids

A formal derivation of linear hydrodynamics for a granular fluid is given. The linear response to small spatial perturbations of a homogeneous reference state is studied in detail, using methods of nonequilibrium statistical mechanics. A transport matrix for macroscopic excitations in the fluid is defined in terms of the response functions. An expansion in the wave vector to second order allows identification of all phenomenological susceptibilities and transport coefficients through Navier-Stokes order in terms of appropriate time correlation functions. The transport coefficients in this representation are the generalization to granular fluids of the familiar Helfand and Green-Kubo relations for normal fluids. The analysis applies to a variety of collision rules. Important differences in both the analysis and results from those for normal fluids are identified and discussed. A scaling limit is described corresponding to the conditions under which idealized inelastic hard sphere models can apply. Further details and interpretation are provided in the paper following this one, by specialization to the case of smooth, inelastic hard spheres with constant coefficient of restitution.

Transport coefficients for the hard-sphere granular fluid

In the preceding paper, linear response methods have been applied to obtain formally exact expressions for the parameters of Navier-Stokes order hydrodynamics. The analysis there is general, applying to both normal and granular fluids with a wide range of collision rules. Those results are specialized here to the case of smooth, inelastic, hard spheres with constant coefficient of normal restitution, for further elaboration. Explicit expressions for the cooling rate, pressure, and transport coefficients are given and compared with the corresponding expressions for a system of elastic hard spheres. The scope of the results for further analytical explorations and possible numerical evaluation is discussed.

Modified gas-kinetic scheme for shock structures in argon

The gas-kinetic scheme (GKS) is a finite-volume method in which the fluxes are constructed from the single particle velocity distribution function f. The distribution function f is obtained from the linearised Boltzmann equation and is retained only to the Navier-Stokes order in terms of the Chapman-Enskog expansion. Higher order non-equilibrium effects are included in a variable local particle collision time depending on local gradients of the hydrodynamic variables up to second order. The fluxes so constructed possess the non-equilibrium information beyond the linear constitutive relations and Fourier law. The modified gas-kinetic scheme with a local particle collision time λ* depending on the gradients of hydrodynamic variables is used to compute shock structures in argon for Mach numbers between 1.2 and 25.0. We validate our results with existing numerical and experimental ones. Our results show that the modified gas-kinetic scheme is effective and efficient for shock structure calculations.

Kinetic theory of response functions for the hard sphere granular fluid

The response functions for small spatial perturbations of a homogeneous granular fluidhave been described recently. They are represented here as stationary state time correlationfunctions. These functions are then expressed in terms of reduced single particlefunctions that are expected to obey a linear kinetic equation. The functionalassumption required for such a kinetic equation, and a Markov approximationfor its implementation are discussed. If, in addition, static velocity correlationsare neglected, a granular fluid version of the linearized Enskog kinetic theory isobtained. The derivation imposes no a priori limitation on the density, space andtimescale, or degree of inelasticity. As an illustration, recently derived Helfand andGreen–Kubo expressions for the Navier–Stokes order transport coefficients areevaluated with this kinetic theory. The results are in agreement with those obtainedfrom the Chapman–Enskog solution to the non-linear Enskog kinetic equation.

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.

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.

Mass and heat fluxes for a binary granular mixture at low density

The Navier–Stokes order hydrodynamic equations for a low-density granular mixture obtained previously from the Chapman–Enskog solution to the Boltzmann equation are considered further. The six transport coefficients associated with mass and heat flux in a binary mixture are given as functions of the mass ratio, size ratio, composition, and coefficients of restitution. Their quantitative variation across this parameter set is demonstrated using low-order Sonine polynomial approximations to solve the exact integral equations. The results are also used to quantify the violation of the Onsager reciprocal relations for a granular mixture. Finally, the stability of the homogeneous cooling state is discussed.