Moderate Stokes Numbers Research Articles

Granular pressure and particle velocity fluctuations prediction in liquid fluidized beds

A numerical modelling approach for the dynamic simulation of solid–liquid fluidized beds is evaluated. This approach is based on an Eulerian two-fluid formulation of the transport equations for mass, momentum and fluctuating kinetic energy. The solid-phase fluctuating motion model is derived in the frame of granular medium kinetic theory accounting for the viscous drag influence of the interstitial liquid phase. Solid–liquid fluidized bed two-dimensional simulations were performed for flow configurations taken from the experimental work of Zenit et al. [1997. Collisional particle pressure measurements in solid–liquid flows. Journal of Fluid Mechanics 353, 261–283], for three types of solid particles of contrasted inertia in water at high particle Reynolds number (nylon, glass and steel beads). Experimental and numerical granular pressures exhibit a satisfactory agreement with both low and high inertia particles, the best level of prediction being observed with the most inertial particles. Sensitivity of the predictions to the phenomenological laws used in the model is also presented and it appears that, due to non-linear correlations, the average granular pressure in the bed is a less sensitive variable than the fluctuating kinetic energy (or granular temperature). The transport mechanisms of the mean granular temperature in the bed are therefore analyzed as a function of the solid fraction and the particle inertia. At low and moderate Stokes number (nylon and glass beads) and in all range of solid-phase fraction, the dominant production mechanism of fluctuating kinetic energy is due to the mean velocity gradient, whereas the main dissipation term is that induced by the viscous drag. At higher Stokes number (steel beads) and concentration, the production of the granular temperature is controlled by the compressibility effects via the granular pressure. In this case, the dissipation is mainly provided by inter-particle collisions.

Direct numerical simulation of gas-solid two-phase mixing layer

In this paper, the spatially evolving of the higher Reynolds numbers gas-solid mixing layer under compressible conditions was investigated by a new direct numerical simulation technology. A high-resolution solver was performed for the gas-phase flow-field, particles with different Stokes numbers were traced by the Lagrangian approach based on one-way coupling. The processes of the vortex rolling up and pairing in the two-dimensional mixing layer were captured precisely. The large-scale structures developed from the initial inflow are characterized by the counter-rotating vortices. The mean velocity and the fluctuation intensities profiles agree well with the experimental data. Particles with smaller Stokes numbers accumulate at the vortex centers due to the smaller aerodynamic response time; particles with moderate Stokes numbers tend to orbit around individual streamwise vortices and in the periphery of paring vortices; particles with larger Stokes numbers disperse less evenly, showing a concentration distribution in the flow field.

Effects of nonreacting solid particle and liquid droplet loading on an exothermic reacting mixing layer

Numerical simulations are conducted of two-dimensional (2D) exothermic reacting mixing layers laden with either solid particles or evaporating liquid droplets. An irreversible reaction of the form fuel+r Oxidizer→(1+r) Products with exothermic Arrhenius kinetics is considered. The temporally developing mixing layers are formed by the merging of parallel flowing oxidizer and fuel streams, each uniformly laden with nonreacting particles or droplets. The gaseous phase is governed by the compressible form of the Navier–Stokes equations together with transport equations for the fuel, oxidizer, product, and evaporated vapor species concentrations. Particles and droplets are assumed smaller than the gas-phase length scales and are tracked individually in the Lagrangian reference frame. Complete “two-way” couplings of mass, momentum, and energy between phases are included in the formulation. The simulation parameters are chosen to study the effects of the mass loading ratio, particle Stokes number, vaporization, flow forcing, and reaction Zeldovich number on the flame evolutions. Quasi-one-dimensional simulations reveal that the asymptotic state of the laminar flames is independent of the particle or droplet loading. For forced 2D simulations, both particles and droplets are preferentially concentrated into the high-strain braid regions of the mixing layer. Cold solid particles entrained into the mixing zone cool the flame in the braid regions due to their finite thermal inertia. This results in flame suppression and, under certain conditions, local flame extinction in the braids. The potential for flame extinction is substantially enhanced by evaporating droplets through the latent heat, and also by the addition of nonreacting evaporated vapor which locally dilutes the reactant concentrations. In contrast, combustion proceeds robustly within vortex cores which have relatively dilute droplet distributions due to preferential concentration; particularly at moderate Stokes numbers. The extent of flame suppression and local extinction are increased with increasing reaction activation energy, dispersed phase mass loading, and also by decreasing particle or droplet Stokes number.

Vorticity dynamics of dilute two-way-coupled particle-laden mixing layers

The two-way coupling mechanisms in particle-laden mixing layers are investigated, with and without particle settling, and with an emphasis on the resulting modifications to the fluid vorticity field. The governing equations are interpreted with respect to the production and cancellation of vorticity. These mechanisms are shown to be related to the misalignment of the concentration gradient and the slip velocity, as well as to the difference in fluid and particle vorticities. Preliminary insight into the physics is obtained from an analysis of the unidirectional base flow. For this model problem, the conditions are established under which the particle velocity remains a single-valued function of space for all times. The resulting simplified set of two-way-coupled equations governing the vorticity of the fluid and particulate phases, respectively, is solved numerically. The formation of a decaying travelling wave solution is demonstrated over a wide range of parameters. Interestingly, the downward propagation of the fluid vorticity field is not accomplished through convection, but rather by the production and loss of vorticity on opposite sides of the mixing layer. For moderate settling velocities, the simulation results reveal an optimal coupling mechanism between the fluid and particle vorticities at intermediate values of the mass loading parameter. For large settling velocities and intermediate mass loadings, more than one local maximum is seen to evolve in the vorticity field. A scaling law for the downward propagation rates of the vorticity fronts is derived.Two-dimensional particle-laden mixing layers are investigated by means of a mixed Lagrangian–Eulerian approach which is based on the vorticity variable. For uniformly seeded mixing layers, the simulations confirm some of the features observed by Druzhinin (1995b) for the model problem of a two-way-coupled particle-laden Stuart vortex, as well as by Dimas & Kiger (1998) in a linear stability analysis. For small values of the Stokes number, a mild destabilization of the mixing layer is observed. At moderate and large Stokes numbers, on the other hand, the transport of vorticity from the braids into the core of the evolving Kelvin–Helmholtz vortices is seen to be slowed by the two-way coupling effects. As a result, the particle ejection from the vortex cores is weakened. For constant mass loadings, the two-way coupling effects are strongest at intermediate Stokes number values. For moderately large Stokes numbers, the formation of two bands of high particle concentration is observed in the braids, which reflects the multi-valued nature of the particle velocity field. For mixing layers in which only one stream is seeded, the particle concentration gradient across the mixing layer leads to strong vorticity production and loss, which results in an effective net motion of the vortex in the flow direction of the seeded stream. Under particle settling, the vortex propagates downward as well. For the parameter range explored here, its settling velocity agrees well with the scaling law derived from the unidirectional flow analysis.

Study of the imposition of bulk flow pulsations on plane channel flow at moderate stokes numbers

Study of the imposition of bulk flow pulsations on plane channel flow at moderate stokes numbers

Study of the imposition of bulk flow pulsations on plane channel flow at moderate stokes numbers

A channel with rectangular cross section, 40 : 1 aspect ratio (height is 0.0127 m), and 4.27 m test section length (336 channel heights) was used to study the effects of imposed pulsations on laminar, transitional, and turbulent flow phenomena. Periodic velocity variations are produced in the test section by using a single rotating vane located in the flow downstream of the test section. Details of the imposed pulsation and phase-averaging techniques are given, along with data that show how the pulsating velocity varies in the test section, including changes to the shapes, amplitudes, and phase shifts of velocity waveforms at different channel locations. Flows with Reynolds numbers (Re) ranging from 1100 to 3700 are studied at Stokes numbers (St) of 4.08 and 5.79 and Strouhal numbers of 0.033 to 0.121 by imposing pulsations at 1 Hz and at 2 Hz. Pulsations at St = 4.08 have peak-to-peak amplitudes 1.4–16% of the local mean velocity, and at 5.79, peak-to-peak amplitudes are 4.5–25% of the local mean velocity. With these amplitudes and frequencies of imposed pulsation, time-averaged velocity profiles both within and outside of the Stokes layer are unaffected by the imposed pulsations for the entire range of Reynolds numbers studied. Phase-averaged traces show sinusoidal behavior across the channel height at Stokes numbers of 4.08 and 5.79 when flow is laminar (Re = 1100), transitional (Re = 1580), and turbulent (Re = 2550 and 3660). The sinusoidal nature of the waveforms is evident from spectra of voltage fluctuations from hot-wire sensors, which show principal pulsation amplitudes that are from one to three orders of magnitude greater than the amplitudes of harmonic peaks. Phase-averaged velocities are significantly nonsinusoidal at higher transitional Reynolds numbers near 2010, particularly for St = 4.08. These deviations coincide with significant changes in the amplitudes and phase lags of phase-averaged velocities compared to results at other Reynolds numbers where measurements were obtained.