Collisions of Small Drops in a Turbulent Flow. Part III: Relative Droplet Fluxes and Swept Volumes

M. B. Pinsky Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel

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A. P. Khain Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel

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B. Grits Department of Atmospheric Sciences, Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel

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M. Shapiro Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa, Israel

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Abstract

Swept volumes of cloud droplets with radii below 20 μm are calculated under conditions typical of atmospheric cloud turbulence characterized by enormous values of Reynolds numbers, high turbulent intermittency, and characteristic values of the dissipation rate. To perform the calculations, the motion equation for small droplets proposed by Maxey is generalized for Stokes numbers St > 0.1, which allows one to simulate relative droplet motion even for very high turbulence intensities typical of deep cumulus clouds. Analytical considerations show that droplet motion is fully determined by turbulent shears and the Lagrangian accelerations.

A new statistical representation of a turbulent flow has been proposed based on the results of the scale analysis of turbulence characteristics and those related to the droplet motion. According to the method proposed, statistical properties of turbulent flow are represented by a set of noncorrelated samples of turbulent shears and Lagrangian accelerations. Each sample can be assigned to a certain point of the turbulent flow. Each such point can be surrounded by a small “elementary” volume with linear length scales of the Kolmogorov length scale, in which the Lagrangian acceleration and turbulent shears can be considered as uniform in space and invariable in time.

This present study (Part III) investigates the droplet collisions in a turbulent flow when hydrodynamic droplet interaction (HDI) is disregarded. Using a statistical model, long series of turbulent shears and accelerations were generated, reproducing probability distribution functions (PDF) at high Reynolds numbers, as they were obtained in recent laboratory and theoretical studies. Swept volumes of droplets are calculated for each sample of an acceleration–shear pair, and the PDF of swept volumes is calculated for turbulent parameters typical of cloud turbulence. The effect of turbulent flow intermittency manifests itself in two aspects: 1) an increase of swept volume variance with increasing Reynolds number, and 2) formation of the swept volume PDF that has a sharp maximum and an elongated tail.

In spite of the fact that the magnitude of the mean swept volume increases significantly with Reynolds number and the dissipation rate, this increase does not exceed ∼60% of pure gravity values even under turbulent conditions typical of strong cumulus clouds. A comparison with the classical results of Saffman and Turner is presented and discussed.

Corresponding author address: Prof. Alexander Khain, The Institute of Earth Sciences, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel. Email: khain@vms.huji.ac.il

Abstract

Swept volumes of cloud droplets with radii below 20 μm are calculated under conditions typical of atmospheric cloud turbulence characterized by enormous values of Reynolds numbers, high turbulent intermittency, and characteristic values of the dissipation rate. To perform the calculations, the motion equation for small droplets proposed by Maxey is generalized for Stokes numbers St > 0.1, which allows one to simulate relative droplet motion even for very high turbulence intensities typical of deep cumulus clouds. Analytical considerations show that droplet motion is fully determined by turbulent shears and the Lagrangian accelerations.

A new statistical representation of a turbulent flow has been proposed based on the results of the scale analysis of turbulence characteristics and those related to the droplet motion. According to the method proposed, statistical properties of turbulent flow are represented by a set of noncorrelated samples of turbulent shears and Lagrangian accelerations. Each sample can be assigned to a certain point of the turbulent flow. Each such point can be surrounded by a small “elementary” volume with linear length scales of the Kolmogorov length scale, in which the Lagrangian acceleration and turbulent shears can be considered as uniform in space and invariable in time.

This present study (Part III) investigates the droplet collisions in a turbulent flow when hydrodynamic droplet interaction (HDI) is disregarded. Using a statistical model, long series of turbulent shears and accelerations were generated, reproducing probability distribution functions (PDF) at high Reynolds numbers, as they were obtained in recent laboratory and theoretical studies. Swept volumes of droplets are calculated for each sample of an acceleration–shear pair, and the PDF of swept volumes is calculated for turbulent parameters typical of cloud turbulence. The effect of turbulent flow intermittency manifests itself in two aspects: 1) an increase of swept volume variance with increasing Reynolds number, and 2) formation of the swept volume PDF that has a sharp maximum and an elongated tail.

In spite of the fact that the magnitude of the mean swept volume increases significantly with Reynolds number and the dissipation rate, this increase does not exceed ∼60% of pure gravity values even under turbulent conditions typical of strong cumulus clouds. A comparison with the classical results of Saffman and Turner is presented and discussed.

Corresponding author address: Prof. Alexander Khain, The Institute of Earth Sciences, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel. Email: khain@vms.huji.ac.il

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