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.

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.

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 2. Simulations with gravitational effects

2016 ◽  
Vol 796 ◽  
pp. 659-711 ◽  
Author(s):  
Peter J. Ireland ◽  
Andrew D. Bragg ◽  
Lance R. Collins
Keyword(s):  
Reynolds Number ◽  
Single Particle ◽  
Isotropic Turbulence ◽  
Distribution Functions ◽  
Particle Collision ◽  
Collision Kernel ◽  
Finite Domain ◽  
Inertial Particles ◽  
Inertial Particle ◽  
Wide Range

In Part 1 of this study (Ireland et al., J. Fluid Mech., vol. 796, 2016, pp. 617–658), we analysed the motion of inertial particles in isotropic turbulence in the absence of gravity using direct numerical simulation (DNS). Here, in Part 2, we introduce gravity and study its effect on single-particle and particle-pair dynamics over a wide range of flow Reynolds numbers, Froude numbers and particle Stokes numbers. The overall goal of this study is to explore the mechanisms affecting particle collisions, and to thereby improve our understanding of droplet interactions in atmospheric clouds. We find that the dynamics of heavy particles falling under gravity can be artificially influenced by the finite domain size and the periodic boundary conditions, and we therefore perform our simulations on larger domains to reduce these effects. We first study single-particle statistics that influence the relative positions and velocities of inertial particles. We see that gravity causes particles to sample the flow more uniformly and reduces the time particles can spend interacting with the underlying turbulence. We also find that gravity tends to increase inertial particle accelerations, and we introduce a model to explain that effect. We then analyse the particle relative velocities and radial distribution functions (RDFs), which are generally seen to be independent of Reynolds number for low and moderate Kolmogorov-scale Stokes numbers $St$. We see that gravity causes particle relative velocities to decrease by reducing the degree of preferential sampling and the importance of path-history interactions, and that the relative velocities have higher scaling exponents with gravity. We observe that gravity has a non-trivial effect on clustering, acting to decrease clustering at low $St$ and to increase clustering at high $St$. By considering the effect of gravity on the clustering mechanisms described in the theory of Zaichik & Alipchenkov (New J. Phys., vol. 11, 2009, 103018), we provide an explanation for this non-trivial effect of gravity. We also show that when the effects of gravity are accounted for in the theory of Zaichik & Alipchenkov (2009), the results compare favourably with DNS. The relative velocities and RDFs exhibit considerable anisotropy at small separations, and this anisotropy is quantified using spherical harmonic functions. We use the relative velocities and the RDFs to compute the particle collision kernels, and find that the collision kernel remains as it was for the case without gravity, namely nearly independent of Reynolds number for low and moderate $St$. We conclude by discussing practical implications of the results for the cloud physics and turbulence communities and by suggesting possible avenues for future research.

scholarly journals Inertial-particle accelerations in turbulence: a Lagrangian closure

2016 ◽  
Vol 798 ◽  
pp. 187-200 ◽  
Author(s):  
S. Vajedi ◽  
K. Gustavsson ◽  
B. Mehlig ◽  
L. Biferale
Keyword(s):  
Numerical Simulations ◽  
Numerical Data ◽  
Direct Numerical Simulations ◽  
Inertial Particles ◽  
Inertial Particle ◽  
Order Unity ◽  
Heavy Particles ◽  
Preferential Sampling ◽  
Closure Scheme ◽  
Light Particles

The distribution of particle accelerations in turbulence is intermittent, with non-Gaussian tails that are quite different for light and heavy particles. In this article we analyse a closure scheme for the acceleration fluctuations of light and heavy inertial particles in turbulence, formulated in terms of Lagrangian correlation functions of fluid tracers. We compute the variance and the flatness of inertial-particle accelerations and we discuss their dependency on the Stokes number. The closure incorporates effects induced by the Lagrangian correlations along the trajectories of fluid tracers, and its predictions agree well with results of direct numerical simulations of inertial particles in turbulence, provided that the effects induced by inertial preferential sampling of heavy/light particles outside/inside vortices are negligible. In particular, the scheme predicts the correct functional behaviour of the acceleration variance, as a function of $St$, as well as the presence of a minimum/maximum for the flatness of the acceleration of heavy/light particles, in good qualitative agreement with numerical data. We also show that the closure works well when applied to the Lagrangian evolution of particles using a stochastic surrogate for the underlying Eulerian velocity field. Our results support the conclusion that there exist important contributions to the statistics of the acceleration of inertial particles independent of the preferential sampling. For heavy particles we observe deviations between the predictions of the closure scheme and direct numerical simulations, at Stokes numbers of order unity. For light particles the deviation occurs for larger Stokes numbers.

Fully developed MHD turbulence near critical magnetic Reynolds number

1981 ◽  
Vol 104 ◽  
pp. 419-443 ◽  
Author(s):  
J. Léorat ◽  
A. Pouquet ◽  
U. Frisch
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Mhd Equations ◽  
Liquid Sodium ◽  
Turbulent Heat ◽  
Mhd Turbulence

Liquid-sodium-cooled breeder reactors may soon be operating at magnetic Reynolds numbers RM where magnetic fields can be self-excited by a dynamo mechanism (as first suggested by Bevir 1973). Such flows have kinetic Reynolds numbers RV of the order of 107 and are therefore highly turbulent.This leads us to investigate the behaviour of MHD turbulence with high RV and low magnetic Prandtl numbers. We use the eddy-damped quasi-normal Markovian closure applied to the MHD equations. For simplicity we restrict ourselves to homogeneous and isotropic turbulence, but we do include helicity.We obtain a critical magnetic Reynolds number RMc of the order of a few tens (non-helical case) above which magnetic energy is present. RMc is practically independent of RV (in the range 40 to 106). RMc can be considerably decreased by the presence of helicity: when the overall size of the flow L is much larger than the integral scale l0, RMc can drop below unity as suggested by an α-effect argument. When L ≈ l0 the drop can still be substantial (factor of 6) when helicity is a maximum. We examine how the turbulence is modified when RM crosses RMc: presence of magnetic energy, decreased kinetic energy, steepening of kinetic-energy spectrum, etc.We make no attempt to obtain quantitative estimates for a breeder reactor, but discuss some of the possible consequences of exceeding RMc, such as decreased turbulent heat transport. More precise information may be obtained from numerical simulations and experiments (including some in the subcritical regime).

scholarly journals Nonlinear amplification in hydrodynamic turbulence

2021 ◽  
Vol 930 ◽  
Author(s):  
Kartik P. Iyer ◽  
Katepalli R. Sreenivasan ◽  
P.K. Yeung
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Developed Turbulence ◽  
Advection Term ◽  
Nonlinear Amplification

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

scholarly journals Direct numerical simulation of turbulent scalar transport across a flat surface

2014 ◽  
Vol 744 ◽  
pp. 217-249 ◽  
Author(s):  
H. Herlina ◽  
J. G. Wissink
Keyword(s):  
Mass Transfer ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Gas Transfer ◽  
Computational Domain ◽  
Atmospheric Gases ◽  
Transfer Velocity ◽  
Physical Mechanisms ◽  
Schmidt Numbers ◽  
Large Eddies

AbstractTo elucidate the physical mechanisms that play a role in the interfacial transfer of atmospheric gases into water, a series of direct numerical simulations of mass transfer across the air–water interface driven by isotropic turbulence diffusing from below has been carried out for various turbulent Reynolds numbers ($R_T=84,195,507$). To allow a direct (unbiased) comparison of the instantaneous effects of scalar diffusivity, in each of the DNS up to six scalar advection–diffusion equations with different Schmidt numbers were solved simultaneously. As far as the authors are aware this is the first simulation that is capable to accurately resolve the realistic Schmidt number, $\mathit{Sc}=500$, that is typical for the transport of atmospheric gases such as oxygen in water. For the range of turbulent Reynolds numbers and Schmidt numbers considered, the normalized transfer velocity $K_L$ was found to scale with $R_T^{-{1/2}}$ and $\mathit{Sc}^{-{1/2}}$, which indicates that the largest eddies present in the isotropic turbulent flow introduced at the bottom of the computational domain tend to determine the mass transfer. The $K_L$ results were also found to be in good agreement with the surface divergence model of McCready, Vassiliadou & Hanratty (AIChE J., vol. 32, 1986, pp. 1108–1115) when using a constant of proportionality of 0.525. Although close to the surface large eddies are responsible for the bulk of the gas transfer, it was also observed that for higher $R_T$ the influence of smaller eddies becomes more important.

scholarly journals Analysis of the forward and backward in time pair-separation probability density functions for inertial particles in isotropic turbulence

2017 ◽  
Vol 830 ◽  
pp. 63-92 ◽  
Author(s):  
Andrew D. Bragg
Keyword(s):  
Probability Density ◽  
Isotropic Turbulence ◽  
Probability Density Functions ◽  
Density Functions ◽  
Inertial Particles ◽  
Mean Square ◽  
Inertial Particle ◽  
Particle Pair ◽  
Pair Separation ◽  
Dissipation Range

In this paper we investigate, using theory and direct numerical simulations (DNS), the forward in time (FIT) and backward in time (BIT) probability density functions (PDFs) of the separation of inertial particle pairs in isotropic turbulence. In agreement with our earlier study (Bragg et al., Phys. Fluids, vol. 28, 2016, 013305), where we compared the FIT and BIT mean-square separations, we find that inertial particles separate much faster BIT than FIT, with the strength of the irreversibility depending upon the final/initial separation of the particle pair and their Stokes number $St$. However, we also find that the irreversibility shows up in subtle ways in the behaviour of the full PDF that it does not in the mean-square separation. In the theory, we derive new predictions, including a prediction for the BIT/FIT PDF for $St\geqslant O(1)$, and for final/initial separations in the dissipation regime. The prediction shows how caustics in the particle relative velocities in the dissipation range affect the scaling of the pair-separation PDF, leading to a PDF with an algebraically decaying tail. The predicted functional behaviour of the PDFs is universal, in that it does not depend upon the level of intermittency in the underlying turbulence. We also analyse the pair-separation PDFs for fluid particles at short times, and construct theoretical predictions using the multifractal formalism to describe the fluid relative velocity distributions. The theoretical and numerical results both suggest that the extreme events in the inertial particle-pair dispersion at the small scales are dominated by their non-local interaction with the turbulent velocity field, rather than due to the strong dissipation range intermittency of the turbulence itself. In fact, our theoretical results predict that for final/initial separations in the dissipation range, when $St\gtrsim 1$, the tails of the pair-separation PDFs decay faster as the Taylor Reynolds number $Re_{\unicode[STIX]{x1D706}}$ is increased, the opposite of what would be expected for fluid particles.

scholarly journals Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence

1978 ◽  
Vol 88 (4) ◽  
pp. 685-709 ◽  
Author(s):  
U. Schumann ◽  
G. S. Patterson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Numerical Study ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Companion Paper ◽  
Real Space ◽  
Pressure Gradients ◽  
Velocity Statistics ◽  
Decaying Turbulence

The spectral method of Orszag & Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [lsim ] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

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