Particle-laden Mixing Layers Research Articles

Numerical study of the effects of unmatched pressure on the supersonic particle-laden mixing layer

The dispersion of monodisperse, inertial particles in a supersonic mixing layer consisting of two sheared flows with differing pressures (P1 for the particle-laden jet flow and P2 for the airflow) is numerically investigated using large Eddy simulation and Euler–Lagrange methods. The calculations reveal the following insights: The pressure disparity between the two flows induces a transverse gas flow effect, which swiftly deflects the mixing layer from the high-pressure side to the low-pressure side. The growth rate of mixing layer increases with the ratio of P2/P1 and while the deflected displacement correlates with the pressure difference |P2-P1|. However, the particles exhibit delayed tracking characteristics to the deflected mixing layer because of their relative relaxation to the transverse gas velocity, particularly in the upstream region of the mixing layer (also known as the Kelvin–Helmholtz instability developing zone or KH zone). Notably, when the P2 exceeds that of the P1, particles can more easily penetrate into the vortices of KH zone, significantly enhancing the downstream gas–particle mixing. This mixing enhancement is particularly pronounced for larger particles due to their increased inertia, which allows them to advance into the vortices of KH zone more effectively than smaller ones.

Numerical analysis of vortex formation and particle dispersion in a supersonic compressible particle-laden mixing layer

Numerical analysis of vortex formation and particle dispersion in a supersonic compressible particle-laden mixing layer

Particle dispersion in turbulent mixing layer at supercritical pressure

The heterogeneous structures of particles are generated due to the significant temperature changes when cold particles-dilute jet injects into a supercritical water fluidized bed, which is unfavorable for the reaction of coal particles. We expect to promote the particle dispersion by entrainment mechanism of large-scale vortical structures. The Eulerian-Lagrangian point-source method is used to numerically investigate the dynamics of dilute particles in mixing layer sustained by supercritical water with different temperatures. The effects of density ratio and velocity ratio between two streams are analyzed. In addition to the smaller particle numbers at high-density side, the particle clustering in the vortex core regions is found due to the inhibition of vortical structures by increasing density ratio. With increasing velocity ratio, however, the enhanced vortex stretching promotes the entrainment of particles. As a result, the increase in particle number density at high-density side is up to 26% compared with the constant-property flow. • Study on variable-density effect on dynamics of particle-laden mixing layer. • Enhancement of vortical structure by increasing velocity ratio between two streams. • Particle acceleration decrease with increasing density ratio between two streams. • Improvement of particle dispersion due to the entrainment of vortical structures.

Pressure-correlated dispersion of inertial particles in free shear flows

We investigate the relationship between dispersion of inertial particles and the pressure field in free shear flows. A three-dimensional temporally developing particle-laden mixing layer and a three-dimensional spatially developing particle-laden plane jet are studied by means of direct numerical simulation. The incompressible Navier-Stokes equations are accurately solved without any turbulence model. The dispersed inertial particles are traced in the Lagrangian framework. It is found that the instantaneous spatial distribution of inertial particles correlates well with the Laplacian of pressure inverted delta2p , and from the statistical viewpoint, the effect of particle size on the fraction of particle number distributed within different flow zones characterized by inverted delta2p for different particles is negligible when the flow is well developed. The potential explanations and applications for this feature are explored.

Particle transport and flow modification in planar temporally evolving laminar mixing layers. I. Particle transport under one-way coupling

Simulations of two-dimensional, particle-laden mixing layers were performed for particles with Stokes numbers of 0.3, 0.6, 1, and 2 under the assumption of one-way coupling using the Eulerian-Lagrangian method; two-way coupling is addressed in Part II. Analysis of interphase momentum transfer was performed in the Eulerian frame of reference by looking at the balance of fluid-phase mean momentum, mean kinetic energy, modal kinetic energy, and particle-phase mean momentum. The differences in the dominant mechanisms of vertical transport of streamwise momentum between the fluid and particle phases is clearly brought out. In the fluid phase, growth of the mixing layer is due to energy transfer from the mean flow to the unstable Kelvin-Helmholtz modes, and transport of mean momentum by these modes. In contrast, in the particle phase, the primary mechanism of vertical transport of streamwise momentum is convection due to the mean vertical velocity induced by the centrifuging of particles by the spanwise Kelvin-Helmholtz vortices. Although the drag force and the particle-phase modal stress play an important role in the early stages of the evolution of the mixing layer, their role is shown to decrease during the pairing process. After pairing, the particle-phase mean streamwise momentum balance is accounted for by the convection and drag force term. The particle-phase modal stress term is shown to be strongly connected to the fluid phase modal stress with a Stokes-number-dependent time lag in its evolution.

Particle transport and flow modification in planar temporally evolving mixing layers. II. Flow modification due to two-way coupling

Simulations of two-dimensional, two-way coupled particle-laden mixing layers were performed for particles with various Stokes numbers, at mass loadings between 0.1 and 0.5. This component complements Part I of the study, where particle and fluid transport was analyzed under one-way coupling. Under two-way coupling, the accumulation of particles in the periphery of the Kelvin-Helmholtz vortices results in the formation of intricate undulating patterns, and rupture and break-up of the vortices. At higher mass loadings the vortex structure is completely destroyed. The overall accumulation at the edges of the mixing layer is significantly reduced due to two-way coupling. The rate of evacuation of the vortex core was found to be much slower compared to one-way coupling. In a global sense, particles delay the development of the mixing layer in terms of saturation of the fundamental and the subharmonic modes. Significant generation of small-scale vorticity and higher energy in the small scales is observed at higher mass loadings. Particles are shown to increase the modal kinetic energy dissipation rate. The mean fluid kinetic energy balance shows that most of the kinetic energy exchange between the particle and fluid phases takes place at the edges of the mixing layer. As the mixing layer evolves, the kinetic energy exchange with the particle phase was shown to decrease in importance compared to the other terms in the mean kinetic energy balance; namely, the energy exchange between the mean and the modal fields and the modal transport term.

Direct numerical simulation of a particle-laden mixing layer with a chemical reaction

A direct numerical simulation (DNS) is applied to a particle-laden turbulent mixing layer with a chemical reaction, and the effects of particles on turbulence and chemical species’ diffusion and reaction in both zero and finite gravity cases are investigated. Unreactive particles, whose response time, τ P, is smaller than the Kolmogorov time scale, τ K [ τ P/ τ K = O(10 −1)], are uniformly injected into the high-speed side of the mixing layer. Two reactive chemical species are separately introduced through different sides. The results show that although laden particles generally depress turbulent intensities, they begin to enhance turbulent intensities downstream as the particle size decreases provided that the inlet particle volume fraction is fixed. This is because that the small particles with small particle response time easily accumulate at the circumference of coherent vortices and act to suppress the growth of primitive small-scale coherent vortices upstream but enhance that of relatively developing large-scale ones downstream. Also, since the small-scale turbulence, which promotes the chemical reaction, is suppressed by the laden particles in the entire region, chemical product decreases overall. Furthermore, the presence of finite gravity on the particles acts to depress the turbulent intensities, but its effects on chemical species’ diffusion and reaction are quite small.

Nonlinear Development of Particle-Laden Mixing Layers at Low Mach Number

The two-phase coupling effects on the particle-laden compressible mixing layer are studied for low convective Mach numbers. A full Eulerian approach is used to obtain a set of conservative partial differential equations. The initial condition is provided by the solution of the solution of the linear problem. The numerical solutions show that the presence of particles in the flow does not alter the typical development of the mixing layer observed for one-phase flows: the amplification of linear instability waves, the formation of two-dimensional large-scale vortices, and the pairing of vortices. The shape of the vortical structures is modified compared to single-phase calculations. The particle density distribution is greatly dependant on their inertia that can lead to a particle-free zone in the vortex. The compressibility effects are clearly seen on the modal kinetic energy, which decreases when the convective Mach number increases

Numerical analysis of the continuum formulation for the initial evolution of mixing layers with particles

Numerical analysis of the standard continuum description of a dilute dispersed phase as applied to a laminar, particle-laden, mixing layer during its initial evolution has been performed. The flow has been previously analyzed under the framework of linear stability analysis where both the continuous and the dispersed phases are considered as continua. Earlier studies had neglected the closure terms resulting from the averaging of the nonlinear transport term involved in the derivation of the dispersed-phase momentum equations. In this work, Lagrangian particle tracking was coupled to an incompressible Navier–Stokes solver to directly estimate the closure terms (referred to as the averaging-stress terms) and compare them to the other terms balancing the dispersed-phase continuum equations. Calculations were performed for particle Stokes numbers of 1, 10, and 50, and for a mass loading of one. Dispersed-phase flow quantities such as the number density and velocity were determined by averaging the data in the spanwise direction. A parametric study of the influence of the number of particles, for Stokes number of one, showed that an improved approximation to a continuum can be obtained by increasing the number of particles. Examining the momentum balance in detail revealed that the main contributors were the time-derivative, convective, and the interfacial force terms. The averaging stress was at least two orders of magnitude smaller for all the Stokes numbers studied. However, the averaging stress, though negligible in magnitude, showed a deterministic variation in the center of the mixing layer. The results lend support to the currently used continuum equations for analyzing the stability of laminar, particle-laden mixing layers, and possibly other free-shear flows such as jet and wake flows.

1504 固気混相混合層中での乱流およびスカラ拡散に及ぼす粒子の効果

1504 固気混相混合層中での乱流およびスカラ拡散に及ぼす粒子の効果

Temporal instabilities of a mixing layer with uniform and nonuniform particle loadings

Temporal stability analysis of particle-laden mixing layers with uniform and nonuniform particle loadings is presented. New analytical results have been derived in the limit of small and large Stokes numbers using small-parameter expansions. A dichotomy in the behavior of the fluid–particle system, based on whether the particle diameter or density is increased to achieve large Stokes numbers, is clarified. Good agreement between the limiting analytical expressions and numerical results is obtained. Additional unstable modes are observed for nonuniform particle loadings for large Stokes numbers and high mass loadings conditions. The effect of the steepness of the nonuniformity is presented for the first time. The primary effect of nonuniformity, is an increase in the range of unstable wave numbers for small to intermediate Stokes numbers, and a change in the nature of the dominant Kelvin–Helmholtz instability from stationary to dispersive. The value of the most unstable wave number is shown to remain unaffected by the nonuniformity.

Parallel Algorithms for Particles–Turbulence Two-Way Interaction Direct Numerical Simulation

Understanding the demixing effect on the dispersion of particles by large-scale turbulence is very important in practical applications. Using pseudo-spectral and Lagrangian approaches, we have simulated a three-dimensional particle-laden mixing layer under one-way coupling effect. However, the computer resource required to simulate such a two-phase flow with high Reynolds number and two-way momentum coupling effect exceeds the limit of the current single processor. In this paper, the computation of particles and the two-way momentum coupling terms are partitioned in the span-wise direction because particles are distributed most evenly in this direction. The computation of the tree-dimensional flow field is first partitioned into three groups of processors because of the most independence of the computation among the three spatial dimensions. In each group, the domain is then partitioned using two different schemes based on the property of the fast Fourier transformation. The first one, the master–slave scheme, is employed for Algorithm MS due to its simplicity and overlapping of communication and computation. The second one, the transpose approach, is used for Algorithm TP order to partition all of the flow field computation. An analysis shows that compared to Algorithm MS, Algorithm TP can also reduce nearly a half of the amount of communication work. Experiments show that Algorithm MS has obtained a speedup of 4.3 using 9HP workstations and a speedup of 6.4 using 15 nodes of IBM SP-2 for a problem size on the order of 643, and Algorithm TP has achieved speedups 44% higher than Algorithm MS.

Vortex pairing in two-way coupled, particle laden mixing layers

The role of two-way interphase coupling effects in subharmonically forced, dilute, particle laden mixing layers is investigated. Temporally growing Lagrangian–Eulerian numerical simulations address both uniformly as well as differentially seeded flows. Emphasis is placed on a vorticity-based interpretation of the simulation results. Expanding the fluid vorticity source term due to two-way coupling demonstrates the importance of particle concentration gradients and `slip vorticity'. For uniformly laden mixing layers and intermediate St particles, the reduction in the growth rate of the Kelvin–Helmholtz instability by the two-way coupling effects also slows down vortex pairing. At large St, a layered vorticity structure in the outer regions of the vortex cores results from the thick or even double banded nature of the braids. By causing the vortices to rotate around each other, the subharmonic forcing also enhances the particle transport away from the center of the mixing layer. Intermediate St particles are seen to benefit the most from this mechanism, so that they can, at least temporarily, exceed the dispersion rate of the passive case. For differentially loaded mixing layers, the vorticity source term is seen to cause an asymmetry in the pairing process, by weakening the upstream and strengthening the downstream vortex, which subsequently dominates the pairing. For intermediate St particles, this asymmetry leads to a more rapid ejection of particles from the downstream vortex. For large St particles, the braid region in between vortex pairs is seen to thicken and to roll up, thereby causing the emergence of a satellite vortex. Somewhat surprisingly, the results indicate that in a streamwise averaged sense the two-way coupling effects do not significantly alter the integral particle dispersion measure for subharmonically forced, differentially loaded mixing layers. Throughout, frequent comparisons with experimental data are made.

Linear stability analysis of particle-laden mixing layers using Lagrangian particle tracking

This paper presents salient features of the temporal stability of particle-laden mixing layers under uniform particle loadings. Results show the strong stabilizing influence of particles with Stokes numbers of the order of unity. The particles also significantly change the most unstable wavenumber which dominates the initial evolution of the mixing layer. Results from the linear stability analysis were used to evaluate the Eulerian–Lagrangian (particle tracking) methodology and excellent agreement was observed for the instability growth rates. Two methods of implementing the fluid–particle coupling, both based on projecting the coupling force onto the fluid nodes, were compared and found to give almost identical results; indicating little practical difference between the two in the present case. Analyzing the accuracy requirements for the velocity interpolation found second-order interpolation to be inadequate. Converged results were obtained with fourth- and sixth-order accurate interpolation schemes.

Linear instability of two-way coupled particle-laden jet

The two-way coupled particle-laden jet is studied numerically in order to understand how the addition of solid particles affects the stability of the jet. In contrast with the particle-laden mixing layer studied, the jet flow stability is complicated by a variable wave (or phase) velocity and an additional curvature parameter, i.e., the ratio of jet radius to shear layer thickness. Nevertheless, we have demonstrated that the following similar scenario holds for a particle-laden jet: the addition of particles can destabilize the flow at small particle Stokes number, while the stabilizing effect prevails for intermediate to large Stokes numbers. Both these competing effects scale approximately linearly with the particle mass loading. We also found that the addition of particles increases the wave velocity at high wave numbers but decreases the wave velocity at low wave numbers. The fact that the addition of particles can destabilize the gas flow in the absence of gravity has been shown to follow the previous speculation. Two general asymptotic relations proposed by Saffman have been confirmed numerically for the case of the particle-laden jet. Physically, the increase of effective inertia of the fluid–particle mixture causes a destabilization effect while the enhanced viscous dissipation around the particles gives a stabilization effect. Results for various mass loadings, Stokes numbers, and wave numbers show that for a given mass loading and wave number, there is an intermediate Stokes number that corresponds to a maximum flow stability. We have shown that this Stokes number is on the order of 1 and depends weakly on the wave number.

固気二相自由乱流の渦法解析

Vortex methods have been thus far applied to simulate gas-particle free turbulent flows, such as mixing layer, wake and jet, to investigate the particle diffusion due to the vortex structure of the gas-phase. The simulations are performed only under low mass loading ratio, because the vortex methods are based on a one-way coupling between the two phases. This article describes a two-way vortex method proposed by the author. The gas flow and the particle motion are simultaneously calculated by the Lagrangian method. The applicability of the method is also demonstrated by presenting the numerical simulations of a particle-laden mixing layer and a slit nozzle gas-particle two-phase jet.

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.

Two-way coupled particle-laden mixing layer. Part 1: Linear instability

The two-way coupled particle-laden mixing layer is studied numerically and theoretically in order to understand how the addition of solid particles affects the stability of the gas flow. Numerical simulations are undertaken to obtain results equivalent to solving the Orr–Sommerfeld equation of a dusty gas mixing layer first derived by Saffman. The growth rate of a viscous particle-laden mixing layer depends on the wavenumber, flow Reynolds number, Stokes number, and bulk particulate mass loading. Two asymptotic relations proposed by Saffman have been confirmed for the first time by numerical simulations. In addition to the stabilizing effect of particles on the gas flow at large Stokes number, a destabilizing influence at small Stokes number is also observed at finite flow Reynolds number. The fact that the addition of particles can destabilize the gas flow in the absence of gravity has been shown to follow the original speculation of Saffman. Physically, the increase of effective inertia of the fluid–particle mixture causes a destabilization effect, while the enhanced viscous dissipation around particles gives a stabilization effect. These qualitatively different effects have been shown to be directly related to the direction of interphase energy transfer. Results at arbitrary mass loading, Stokes number, and wavenumber show that for a given mass loading and wavenumber, there is an intermediate Stokes number which corresponds to a maximum flow stability. We have shown that this Stokes number is on the order of one, and depends on the wavenumber. An analytical model for predicting the growth rate in a viscous, particle-laden gas mixing layer is proposed and compared with the simulation results.

Linear instability of a particle-laden mixing layer with a dynamic dispersed phase

The linear, inviscid, spatial instability of a mixing layer uniformly laden with a dilute concentration of heavy particles is studied numerically. The effect of the particles is modeled using an ensemble averaged Eulerian description of the velocity field and Stokes’ drag formula to compute an averaged force, and the carrier fluid and the particle motions are assumed to be fully coupled. The behavior of the linear instability (for a given mean shear) depends on two dimensionless parameters: Cf, representing the product of the inverse Stokes number and mass loading, and Cp, representing the inverse Stokes number. For finite values of Cf and large values of Cp, the particles respond as fluid elements and the growth rate is equal to the one of the single-phase flow, while decreasing Cp results in a growth rate decrease. The growth rate also decreases with increasing Cf. Beyond certain critical values of increasing Cf and decreasing Cp, a second unstable low-frequency mode appears which is distinct from the fundamental mode. The fully coupled character of the instability reveals three important aspects of the particle effect on the flow structure: (1) the particle concentration field is organized into alternating bands of increased and decreased concentration corresponding to the braid and core regions of the vortices, respectively, with peak perturbations occurring at intermediate Cp values (0.01⩽Cp⩽0.1), (2) the streamwise particle velocity is higher than the streamwise fluid velocity for a substantial range of Cp values and every finite Cf, and (3) the modification of the fluid vorticity field structure with respect to the corresponding field in single-phase flow is driven by the divergence of the particle velocity field.