scholarly journals Relative velocity of inertial particles in turbulent flows

2010 ◽  
Vol 661 ◽  
pp. 73-107 ◽  
Author(s):  
LIUBIN PAN ◽  
PAOLO PADOAN
Keyword(s):  
Turbulent Flows ◽  
Relative Velocity ◽  
Reynolds Numbers ◽  
Inertial Range ◽  
Inertial Particles ◽  
Inertial Particle ◽  
Two Phase ◽  
Time Dispersion ◽  
Particle Pairs ◽  
Acceleration Term

We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.

scholarly journals The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 1. Simulations without gravitational effects

2016 ◽  
Vol 796 ◽  
pp. 617-658 ◽  
Author(s):  
Peter J. Ireland ◽  
Andrew D. Bragg ◽  
Lance R. Collins
Keyword(s):  
Reynolds Number ◽  
Relative Velocity ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Inertial Particles ◽  
Inertial Particle ◽  
Physical Mechanisms ◽  
Preferential Sampling ◽  
Velocity Statistics ◽  
Particle Statistics

In this study, we analyse the statistics of both individual inertial particles and inertial particle pairs in direct numerical simulations of homogeneous isotropic turbulence in the absence of gravity. The effect of the Taylor microscale Reynolds number, $R_{{\it\lambda}}$, on the particle statistics is examined over the largest range to date (from $R_{{\it\lambda}}=88$ to 597), at small, intermediate and large Kolmogorov-scale Stokes numbers $St$. We first explore the effect of preferential sampling on the single-particle statistics and find that low-$St$ inertial particles are ejected from both vortex tubes and vortex sheets (the latter becoming increasingly prevalent at higher Reynolds numbers) and preferentially accumulate in regions of irrotational dissipation. We use this understanding of preferential sampling to provide a physical explanation for many of the trends in the particle velocity gradients, kinetic energies and accelerations at low $St$, which are well represented by the model of Chun et al. (J. Fluid Mech., vol. 536, 2005, pp. 219–251). As $St$ increases, inertial filtering effects become more important, causing the particle kinetic energies and accelerations to decrease. The effect of inertial filtering on the particle kinetic energies and accelerations diminishes with increasing Reynolds number and is well captured by the models of Abrahamson (Chem. Engng Sci., vol. 30, 1975, pp. 1371–1379) and Zaichik & Alipchenkov (Intl J. Multiphase Flow, vol. 34 (9), 2008, pp. 865–868), respectively. We then consider particle-pair statistics, and focus our attention on the relative velocities and radial distribution functions (RDFs) of the particles, with the aim of understanding the underlying physical mechanisms contributing to particle collisions. The relative velocity statistics indicate that preferential sampling effects are important for $St\lesssim 0.1$ and that path-history/non-local effects become increasingly important for $St\gtrsim 0.2$. While higher-order relative velocity statistics are influenced by the increased intermittency of the turbulence at high Reynolds numbers, the lower-order relative velocity statistics are only weakly sensitive to changes in Reynolds number at low $St$. The Reynolds-number trends in these quantities at intermediate and large $St$ are explained based on the influence of the available flow scales on the path-history and inertial filtering effects. We find that the RDFs peak near $St$ of order unity, that they exhibit power-law scaling for low and intermediate $St$ and that they are largely independent of Reynolds number for low and intermediate $St$. We use the model of Zaichik & Alipchenkov (New J. Phys., vol. 11, 2009, 103018) to explain the physical mechanisms responsible for these trends, and find that this model is able to capture the quantitative behaviour of the RDFs extremely well when direct numerical simulation data for the structure functions are specified, in agreement with Bragg & Collins (New J. Phys., vol. 16, 2014a, 055013). We also observe that at large $St$, changes in the RDF are related to changes in the scaling exponents of the relative velocity variances. The particle collision kernel closely matches that computed by Rosa et al. (New J. Phys., vol. 15, 2013, 045032) and is found to be largely insensitive to the flow Reynolds number. This suggests that relatively low-Reynolds-number simulations may be able to capture much of the relevant physics of droplet collisions and growth in the adiabatic cores of atmospheric clouds.

Reynolds number scaling of inertial particle statistics in turbulent channel flows

2014 ◽  
Vol 758 ◽  
Author(s):  
Matteo Bernardini
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
Large Scale ◽  
Spatial Organization ◽  
Reynolds Numbers ◽  
Stokes Number ◽  
Wall Region ◽  
Inertial Particles ◽  
Turbulent Channel Flows ◽  
Strong Segregation

AbstractThe effect of the Reynolds number on the behaviour of inertial particles in wall-bounded turbulent flows is investigated through large-scale direct numerical simulations (DNS) of particle-laden canonical channel flow spanning almost a decade in the friction Reynolds number, from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau } = 150$ to $\mathit{Re}_{\tau } = 1000$. Lagrangian particle tracking is used to study the motion of six different particle sets, described by a Stokes number in the range $\mathit{St} = 1\text {--}1000$. At all Reynolds numbers a strong segregation in the near-wall region is observed for particles characterized by intermediate Stokes number, in the range $\mathit{St} =10\text {--}100$. The wall-normal concentration profiles of such particles collapse in inner scaling, thus suggesting the independence of the turbophoretic drift from the large-scale outer motions. This observation is also supported by the spatial organization of the suspended phase in the inner layer, which is found to be universal with the Reynolds number. The deposition rate coefficient increases with $\mathit{Re}_{\tau }$ for a given $\mathit{St}$. Suitable inner and outer scalings are proposed to collapse the deposition curves across the available ranges of Reynolds and Stokes numbers for the different deposition regimes.

Annular Two-Phase Flow—Part 2: Additional Effects

1970 ◽  
Vol 92 (1) ◽  
pp. 73-81 ◽  
Author(s):  
G. B. Wallis
Keyword(s):  
Shear Stress ◽  
Stress Distribution ◽  
Liquid Film ◽  
Relative Velocity ◽  
Two Phase Flow ◽  
Reynolds Numbers ◽  
Shear Stress Distribution ◽  
Phase Flow ◽  
Two Phase ◽  
Flow Part

The theory of Part 1 is modified by taking additional phenomena into account. The liquid film, the gas core, and the interface are considered separately. The effects of entrainment, compressibility, liquid and gas Reynolds numbers, shear-stress distribution, relative velocity, and the various types of interfacial waves are discussed.

Inertial particle relative velocity statistics in homogeneous isotropic turbulence

2012 ◽  
Vol 696 ◽  
pp. 45-66 ◽  
Author(s):  
Juan P. L. C. Salazar ◽  
Lance R. Collins
Keyword(s):  
Structure Function ◽  
Relative Velocity ◽  
Velocity Structure ◽  
Structure Functions ◽  
Inertial Subrange ◽  
Inertial Particles ◽  
Inertial Particle ◽  
Scaling Exponents ◽  
Velocity Statistics ◽  
Dissipation Range

AbstractIn the present study, we investigate the scaling of relative velocity structure functions, of order two and higher, for inertial particles, both in the dissipation range and the inertial subrange using direct numerical simulations (DNS). Within the inertial subrange our findings show that contrary to the well-known attenuation in the tails of the one-point acceleration probability density function (p.d.f.) with increasing inertia (Bec et al., J. Fluid Mech., vol. 550, 2006, pp. 349–358), the opposite occurs with the velocity structure function at sufficiently large Stokes numbers. We observe reduced scaling exponents for the structure function when compared to those of the fluid, and correspondingly broader p.d.f.s, similar to what occurs with a passive scalar. DNS allows us to isolate the two effects of inertia, namely biased sampling of the velocity field, a result of preferential concentration, and filtering, i.e. the tendency for the inertial particle velocity to attenuate the velocity fluctuations in the fluid. By isolating these effects, we show that sampling is playing the dominant role for low-order moments of the structure function, whereas filtering accounts for most of the scaling behaviour observed with the higher-order structure functions in the inertial subrange. In the dissipation range, we see evidence of so-called ‘crossing trajectories’, the ‘sling effect’ or ‘caustics’, and find good agreement with the theory put forth by Wilkinson et al. (Phys. Rev. Lett., vol. 97, 2006, 048501) and Falkovich & Pumir (J. Atmos. Sci., vol. 64, 2007, 4497) for Stokes numbers greater than 0.5. We also look at the scaling exponents within the context of the model proposed by Bec et al. (J. Fluid Mech., vol. 646, 2010, pp. 527–536). Another interesting finding is that inertial particles at low Stokes numbers sample regions of higher kinetic energy than the fluid particle field, the converse occurring at high Stokes numbers. The trend at low Stokes numbers is predicted by the theory of Chun et al. (J. Fluid Mech., vol. 536, 2005, 219–251). This work is relevant to modelling the particle collision rate (Sundaram & Collins, J. Fluid Mech., vol. 335, 1997, pp. 75–109), and highlights the interesting array of phenomena induced by inertia.

A generalization of Yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence

1999 ◽  
Vol 391 ◽  
pp. 359-372 ◽  
Author(s):  
L. DANAILA ◽  
F. ANSELMET ◽  
T. ZHOU ◽  
R. A. ANTONIA
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Inertial Range ◽  
Second Order ◽  
Grid Turbulence ◽  
Third Order ◽  
Production Term ◽  
Decaying Turbulence ◽  
Energy Dissipated

In most real or numerically simulated turbulent flows, the energy dissipated at small scales is equal to that injected at very large scales, which are anisotropic. Despite this injection-scale anisotropy, one generally expects the inertial-range scales to be locally isotropic. For moderate Reynolds numbers, the isotropic relations between second-order and third-order moments for temperature (Yaglom's equation) or velocity increments (Kolmogorov's equation) are not respected, reflecting a non-negligible correlation between the scales responsible for the injection, the transfer and the dissipation of energy. In order to shed some light on the influence of the large scales on inertial-range properties, a generalization of Yaglom's equation is deduced and tested, in heated grid turbulence (Rλ=66). In this case, the main phenomenon responsible for the non-universal inertial-range behaviour is the non-stationarity of the second-order moments, acting as a negative production term.

scholarly journals Dynamics of inertial particles in a turbulent von Kármán flow

2011 ◽  
Vol 668 ◽  
pp. 223-235 ◽  
Author(s):  
R. VOLK ◽  
E. CALZAVARINI ◽  
E. LÉVÊQUE ◽  
J.-F. PINTON
Keyword(s):  
Turbulent Flows ◽  
Laser Doppler Velocimetry ◽  
Reynolds Numbers ◽  
Probability Density Functions ◽  
Density Functions ◽  
Inertial Particles ◽  
Von Karman ◽  
Kolmogorov Scale ◽  
Von Kármán ◽  
Neutrally Buoyant

We study the dynamics of neutrally buoyant particles with diameters varying in the range [1, 45] in Kolmogorov scale units (η) and Reynolds numbers based on Taylor scale (Reλ) between 590 and 1050. One component of the particle velocity is measured using an extended laser Doppler velocimetry at the centre of a von Kármán flow, and acceleration is derived by differentiation. We find that the particle acceleration variance decreases with increasing diameter with scaling close to (D/η)−2/3, in agreement with previous observations, and with a hint for an intermittent correction as suggested by arguments based on scaling of pressure spatial increments. The characteristic time of acceleration autocorrelation increases more strongly than previously reported in other experiments, and possibly varying linearly with D/η. Further analysis shows that the probability density functions of the acceleration have smaller wings for larger particles; their flatness decreases as well, as expected from the behaviour of pressure increments in turbulence when intermittency corrections are taken into account. We contrast our measurements with previous observations in wind-tunnel turbulent flows and numerical simulations.

Effects of Turbulence and Containing Wall on Rise Velocities of Single Bubbles in a Vertical Circular Pipe

2002 ◽  
Author(s):  
Tomio Okawa ◽  
Kazuhiro Torimoto ◽  
Masanori Nishiura ◽  
Isao Kataoka ◽  
Michitsugu Mori
Keyword(s):  
Turbulent Flows ◽  
Relative Velocity ◽  
Phase Distribution ◽  
Liquid Velocity ◽  
Simulation Models ◽  
Continuous Phase ◽  
Rise Velocity ◽  
Two Phase ◽  
Flow Channels ◽  
Two Factors

Experiments were conducted to clarify the single bubble rise characteristics in turbulent flows in vertical flow channels. It was revealed that the rise velocity of a bubble relative to the time-averaged local liquid velocity could be much smaller in turbulent upflows than in stagnant liquids. The reduction of relative velocity was estimated to be caused by the two factors: turbulence in continuous phase and steep velocity gradient near wall; new correlations for describing these two effects were proposed. The relative velocity between the phases significantly affects the lateral phase distribution in multidimensional simulation of bubbly two-phase flow and the present correlations can give reasonable predictions for the relative velocity in turbulent flow. It is hence expected that the new correlations can contribute to the further improvement of the simulation models of bubbly two-phase flows.

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