University of Delaware
Newark, Delaware, U. S. A.
Turbulent coagulation is a key process in many chemical and energy industries such as the production of particulate commodities (pigments and carbon blacks) and spray combustion. An accurate predictive model of turbulent coagulation can aid design, optimization, and control of industrial processes involving particle formation. In open literature a number of theoretical models of turbulent coagulation exist for weak inertia and finite inertia particles. There are, however, inconsistencies among these models which were not resolved in the past due to the lack of experimental evidences. It is now possible to compare various models and formulations as direct numerical simulations can provide well-controlled data base of turbulent collision rates. This talk focuses on the geometric collision rates of particles in a turbulent flow.
We first address the collision rates of weak inertia particles. In pioneering work by Saffman and Turner (1956), two different formulations of collision kernel $\Gamma $ were given. The first or the spherical formulation is based on the radial component $w_r$ of the relative velocity ${\bf w}$ between two particles: $\Gamma^{(1)} = 2\pi R^2 <|w_r|>$, where $w_r = {\bf w}\cdot {\bf R}/R$, ${\bf R}$ is the separation vector, and $R=|{\bf R}|$. The second or the cylindrical formulation is based on the vector velocity itself: $\Gamma^{(2)} = \pi R^2 <|{\bf w}|>$, which is supported by molecular collision statistical mechanics ({\it e.g.}, McQuarrie, 1976). In the past both formulations were used in turbulent coagulation studies and the question of whether they are equivalent has never been clearly answered. We show that there is a fundamental difference between the two formulations (Wang {\it et al.} 1998a). An underlying assumption in the cylindrical formulation is that the relative velocity at any instant is locally uniform over a spatial scale on the order of the collision radius $R$, which is certainly not the case in turbulent flow. Therefore, the cylindrical formulation is not expected to be rigorously correct. We demonstrate analytically and numerically that the cylindrical formulation leads to a collision kernel about $25\% $ larger than the first formulation in isotropic turbulence. The spherical formulation always gives correct predictions, under the assumptions that particles can overlap in space and are retained in the system after collisions (Wang {\it et al.} 1998b).
We then discuss the collision rates of finite-inertia, monodisperse, solid particles in a turbulent gas (Zhou {\it et al.} 1998). Numerical results show that the collision kernel peaked at a particle response time between the Kolmogorov and the large-eddy turnover times, implying that both the large-scale and small-scale fluid motions contribute, although in very different manners, to the collision rate. Both numerical results and a stochastic theory based on the concept of particle-eddy interactions show that the collision kernel approaches the kinetic theory of Abrahamson (1975) only at very large $\tau_p/T_e$, where $\tau_p$ is the particle response time and $T_e$ is the flow integral time scale. For arbitrary particle response time, numerical results are used as a basis to compare previous theoretical results. A rigorous approach for evaluating the statistics of relative velocities between two-colliding particles will be suggested.
An additional complication for the case of finite-inertia particles is the effect of preferentical concentration, namely, that particles tend to accumulate in regions of low vorticity and high strain rate. This results in a rapid increase of the collision kernel with the particle response time for small $\tau_p/\tau_k$, where $\tau_k$ is the flow Kolmogorov time scale. A small inertia of $\tau_p/\tau_k =0.5$ can lead to an order of magnitude increase in the collision kernel relative to the zero-inertia particles. A scaling law for the collision kernel at small $\tau_p/\tau_k$ has been proposed and confirmed numerically by varying the particle size, inertial response time, and flow Reynolds number. A leading-order theory for small $\tau_p/\tau_k$ is developed to demonstrate the effect of nonuniform particle concentration on the collision kernel.
Finally, numerical results and theoretical models for collision rates in a suspension of polydisperse solid particles will be discussed, as well as directions for future studies on this topic.
References:
[1] J. Abrahamson, ``Collision rates of small particles in a vigorously turbulent fluid,'' Chemical Engineering Science {\bf 30}, 1371 (1975).
[2] D. A. McQuarrie, {\it Statistical Mechanics} (Happer and Row Publishers, New York, 1976), pp. 406-411.
[3] P. G. Saffman and J. S. Turner, ``On the collision of drops in turbulent clouds,'' J. Fluid Mech. {\bf 1}, 16 (1956). Also Corrigendum, J. Fluid Mech. {\bf 196}, 599 (1988).
[4] L.-P. Wang, A. S. Wexler, and Y. Zhou, ``On Statistical Mechanical Descriptions of Turbulent Coagulation.'' Submitted to Phys. Fluids (1998a).
[5] L.-P. Wang, A. S. Wexler, and Y. Zhou, ``On the collision rate of small particles in isotropic turbulence. Part 1. Zero-inertia case,'' Phys. Fluids {\bf 10}, 266, (1998b).
[6] Y. Zhou, L.-P. Wang, and A. S. Wexler, ``On the collision rate of small particles in isotropic turbulence. Part 2: Finite inertia case,'' Phys. Fluids, in press (1998).