Lian-Ping Wang

Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, has been shown to differ from that in still fluid owing to a bias from the particle inertia (Maxey 1987). Previous numerical results for particles in a random flow field, where the flow dynamics were not considered, showed an increase in the average settling velocity. Direct numerical simulations of the motion of heavy particles in isotropic homogeneous turbulence have been performed where the flow dynamics are included. These show that a significant increase in the average settling velocity can occur for particles with inertial response time and still-fluid terminal velocity comparable to the Kolmogorov scales of the turbulence. This increase may be as much as 50% of the terminal velocity, which is much larger than was previously found. The concentration field of the heavy particles, obtained from direct numerical simulations, shows the importance of the inertial bias with particles tending to collect in elongated sheets on the peripheries of local vortical structures. This is coupled then to a preferential sweeping of the particles in downward moving fluid. Again the importance of Kolmogorov scaling to these processes is demonstrated. Finally, some consideration is given to larger particles that are subject to a nonlinear drag force where it is found that the nonlinearity reduces the net increase in settling velocity.

Statistical mechanical description and modelling of turbulent collision of inertial particles

The collision rate of monodisperse solid particles in a turbulent gas is governed by a wide range of scales of motion in the flow. Recent studies have shown that large-scale energetic eddies are the dominant factor contributing to the relative velocity between two colliding particles (the turbulent transport effect), whereas small-scale dissipative eddies can enhance the collision rate significantly by inducing local nonuniform particle distribution (the accumulation effect). The turbulent transport effect is most noticeable when the particle inertial response time τ p is of the order of the flow integral timescale and the accumulation effect is most pronounced when τ p is comparable to the flow Kolmogorov time. We study these two contributions separately through direct numerical simulations. The two effects are quantified carefully with a numerical procedure that is independent of the computation of average collision rate. This facilitates the study of not only the statistical description of the collision kernel, but also the relative contributions and modelling of the two physical effects. Simulations at several flow Reynolds numbers were performed to suggest a model for the accumulation effect

Dispersion of heavy particles by turbulent motion

Abstract Accurate prediction of heavy particle dispersion in turbulent flows requires a simultaneous consideration of particles inertia and particles drift velocity. A mathematically simple and physically comprehensive analysis was developed to solve the dispersion statistics of heavy particles in a homogeneous and isotropic turbulent flow. Normalized particle diffusivity, rms fluctuating velocity, and Lagrangian integral time were related by algebraic equations to three dimensionless parameters: the inertia parameter, the drift parameter, and the turbulence structure parameter. When the drift parameter is large, dispersion scales are very sensitive to the inertia parameter. Heavy particles were found to disperse faster than fluid elements if the inertia parameter controls the dispersion and slower than fluid elements if the drift parameter governs the dispersion. This finding explains previous “contradictory” dispersion data observed in experimental measurements. Not only the particle time respective b...

Modelling turbulent collision of bidisperse inertial particles

We study finite-inertia effects on the collision rate of bidisperse heavy particles in a turbulent gas, using direct numerical simulations and kinematic descriptions. As shown previously for a monodisperse system (Sundaram & Collins 1997; Wang, Wexler & Zhou 2000), a statistical mechanical description of the average collision kernel consists of two parts, namely a description of the relative velocity between two colliding particles (the turbulent transport effect) and of the non-uniform particle distribution due to dynamic interaction of particles with coherent vortex structures (the accumulation effect). We first show that this description remains valid and accurate for a bidisperse system involving two groups of particles of inertial response time τ p 1 and τ p 2 , respectively. Numerical results for the turbulent transport effect and the accumulation effect have been obtained as a function of τ p 1 and τ p 2 . Interestingly, the accumulation effect in a bidisperse system is bounded above by that of a monodisperse system. An explanation for this observation is given, in terms of the correlation between concentration fields of the two size groups. Simulations show that particles from two size groups were found in different regions of a vortex, thus reducing the net accumulation effect in a bidisperse system. The turbulent transport effect, on the other hand, is bounded below by the level in a monodisperse system, due to a differential inertia effect. The above observations imply that the size polydispersity enhances the turbulent transport effect but weakens the accumulation effect, relative to a monodisperse system. A simple eddy–particle interaction (EPI) model was developed and shown to give a reasonable prediction of the collision kernel, except for a small parametric region where both τ p 1 and τ p 2 are on the order of the ow Kolmogorov time τ k and thus the accumulation effect must be included. A more accurate model incorporating both the turbulent transport effect and the accumulation effect has also been developed. The model would provide an upper bound on the collision rates for a non-dilute bidisperse system, since turbulence modulation and particle-particle interactions are not considered in this model. Finally, some consideration is given to the effect of nonlinear drag on the collision kernel. The results show that the drag nonlinearity can increase the collision kernel slightly (less than 10%) at large particle inertia.

Simulations of dispersed turbulent multiphase flow

Abstract Direct numerical simulations of homogeneous isotropic turbulence are used to investigate the effects of turbulence on the transport of particles in gas flows or bubbles in liquid flows. The inertia associated with the bubbles or the particles leads to locally strong concentrations of these in regions of instantaneously strong vorticity for bubbles or strain-rate for particles. This alters the average settling rates and other processes. If the mass-loading of the dispersed phase is significant a random “turbulent” flow is generated by the particle settling. A simple demonstration of this is given, showing the statistically axisymmetric character of this flow and how it can modify an ambient turbulent flow.

Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 1. Results from direct numerical simulation

There have been relatively few studies of turbulent collision rate of sedimenting droplets in the context of cloud physics, for which both the gravitational settling and inertial effects must be simultaneously considered. In this study, direct numerical simulations (DNS) were used to study the geometric collision rates of cloud droplets. Both Stokes drag law and a nonlinear drag law were considered, but the droplet–droplet local aerodynamic interactions were not included. Typical droplet and turbulence parameters of convective clouds were used to determine the flow dissipation rate , characteristic Stokes numbers, and the nondimensional terminal velocities. DNS results from a large number of runs covering the range from 10 to 400 cm2 s− 3 and droplet sizes from 10 to 60 μm in radius are presented. These results show that air turbulence can increase the geometric collision kernel by up to 47%, relative to geometric collision by differential sedimentation. This is due to both a moderate enhancement of the radial relative velocity between droplets and a moderate level of pair nonuniform concentration due to local droplet clustering. The turbulence enhancements increase with the flow dissipation rate and flow Reynolds number. Comparisons with related DNS studies show that our results confirm and extend the previous findings. The mean settling velocity of droplets in a turbulent flow was also obtained, showing that a maximum increase relative to the terminal velocity occurs for 20 μm cloud droplets. This agrees with a previous theory based on simple vortex flows and confirms the importance of a new nondimensional parameter τp3g2/ν for sedimenting droplets, where τp is the droplet inertial response time, g is the gravitational acceleration and ν is the air kinematic viscosity. Limitations of DNS and future directions are also noted.

Theoretical Formulation of Collision Rate and Collision Efficiency of Hydrodynamically Interacting Cloud Droplets in Turbulent Atmosphere

A methodology for conducting direct numerical simulations (DNSs) of hydrodynamically interacting droplets in the context of cloud microphysics has been developed and used to validate a new kinematic formulation capable of describing the collision rate and collision efficiency of cloud droplets in turbulent air. The theoretical formulation is formally the same as the formulation recently developed for geometrical collision rate of finite-inertia, nonsettling particles. It is shown that its application to hydrodynamically interacting droplets requires corrections because of a nonoverlap requirement. An approximate method for correcting the kinematic properties has been developed and validated against DNS data. The formulation presented here is more general and accurate than previously published formulations that, in most cases, are some extension to the description of hydrodynamic–gravitational collision. General dynamic and kinematic representations of the properly defined collision efficiency in a turbulent flow have been discussed. In addition to augmenting the geometric collision rate, air turbulence has been found to enhance the collision efficiency because, in a turbulent flow, hydrodynamic interactions become less effective in reducing the average relative radial velocity. The level of increase in the collision efficiency depends on the flow dissipation rate. For example, the collision efficiency between droplet so f 20 and 25m in radii is increased by 59% and 10% by air turbulence at dissipation rates of 400 and 100 cm 2 s 3 , respectively. It is also shown that hydrodynamic interactions lead to higher droplet concentration fluctuations. The formulation presented here separates the effect of turbulence on collision efficiency from the previously observed effect of turbulence on the geometric collision rate.

Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization

The effect of air turbulence on the geometric collision kernel of cloud droplets can be predicted if the effects of air turbulence on two kinematic pair statistics can be modeled. The first is the average radial relative velocity and the second is the radial distribution function (RDF). A survey of the literature shows that no theory is available for predicting the radial relative velocity of finite-inertia sedimenting droplets in a turbulent flow. In this paper, a theory for the radial relative velocity is developed, using a statistical approach assuming that gravitational sedimentation dominates the relative motion of droplets before collision. In the weak-inertia limit, the theory reveals a new term making a positive contribution to the radial relative velocity resulting from a coupling between sedimentation and air turbulence on the motion of finite-inertia droplets. The theory is compared to the direct numerical simulations (DNS) results in part 1, showing a reasonable agreement with the DNS data for bidisperse cloud droplets. For droplets larger than 30??m in radius, a nonlinear drag (NLD) can also be included in the theory in terms of an effective inertial response time and an effective terminal velocity. In addition, an empirical model is developed to quantify the RDF. This, together with the theory for radial relative velocity, provides a parameterization for the turbulent geometric collision kernel. Using this integrated model, we find that turbulence could triple the geometric collision kernel, relative to the stagnant air case, for a droplet pair of 10 and 20??m sedimenting through a cumulus cloud at R?=2?104 and =600?cm2?s?3. For the self-collisions of 20??m droplets, the collision kernel depends sensitively on the flow dissipation rate.

Statistical mechanical descriptions of turbulent coagulation

A fundamental tenet of statistical mechanics is that the rate of collision of two objects is related to the expectation value of their relative velocities. In pioneering work by Saffman and Turner [J. Fluid Mech. 1, 16 (1956)], two different formulations of this tenet are used to calculate the collision kernel Γ between two arbitrary particle size groups in a turbulent flow. The first or spherical formulation is based on the radial component wr of the relative velocity w between two particles: Γsph=2πR2〈|wr|〉, where wr=w⋅R/R, R is the separation vector, and R=|R|. The second or cylindrical formulation is based on the vector velocity itself: Γcyl=2πR2〈|w|〉, which is supported by molecular collision statistical mechanics. Saffman and Turner obtained different results from the two formulations and attributed the difference to the form of the probability function of w used in their work. A more careful examination reveals that there is a fundamental difference between the two formulations. An underlying assum...

Effects of subgrid-scale modeling on time correlations in large eddy simulation

The effects of the unresolved subgrid-scale (SGS) motions on the energy balance of the resolved scales in large eddy simulation (LES) have been investigated actively because modeling the energy transfer between the resolved and unresolved scales is crucial to constructing accurate SGS models. But the subgrid scales not only modify the energy balance, they also contribute to temporal decorrelation of the resolved scales. The importance of this effect in applications including the predictability problem and the evaluation of sound radiation by turbulent flows motivates the present study of the effect of SGS modeling on turbulent time correlations. This paper compares the two-point, two-time Eulerian velocity correlation in isotropic homogeneous turbulence evaluated by direct numerical simulation (DNS) with the correlations evaluated by LES using a standard spectral eddy viscosity. It proves convenient to express the two-point correlations in terms of spatial Fourier decomposition of the velocity field. The LES fields are more coherent than the DNS fields: their time correlations decay more slowly at all resolved scales of motion and both their integral scales and microscales are larger than those of the DNS field. Filtering alone is not responsible for this effect: in the Fourier representation, the time correlations of the filtered DNS field are identical to those of the DNS field itself. The possibility of modeling the decorrelating effects of the unresolved scales of motion by including a random force in the model is briefly discussed. The results could have applications to the problem of computing sound sources in isotropic homogeneous turbulence by LES