Effect of Turbulence on Collisional Growth of Cloud Droplets

We investigate the effect of turbulence on the collisional growth of micrometer-sized droplets through high-resolution numerical simulations with well-resolved Kolmogorov scales, assuming a collision and coalescence efficiency of unity. The droplet dynamics and collisions are approximated using a superparticle approach. In the absence of gravity, we show that the time evolution of the shape of the droplet-size distribution due to turbulence-induced collisions depends strongly on the turbulent energy-dissipation rate [Formula: see text], but only weakly on the Reynolds number. This can be explained through the [Formula: see text] dependence of the mean collision rate described by the Saffman–Turner collision model. Consistent with the Saffman–Turner collision model and its extensions, the collision rate increases as [Formula: see text] even when coalescence is invoked. The size distribution exhibits power-law behavior with a slope of −3.7 from a maximum at approximately 10 up to about 40 μm. When gravity is invoked, turbulence is found to dominate the time evolution of an initially monodisperse droplet distribution at early times. At later times, however, gravity takes over and dominates the collisional growth. We find that the formation of large droplets is very sensitive to the turbulent energy dissipation rate. This is because turbulence enhances the collisional growth between similar-sized droplets at the early stage of raindrop formation. The mean collision rate grows exponentially, which is consistent with the theoretical prediction of the continuous collisional growth even when turbulence-generated collisions are invoked. This consistency only reflects the mean effect of turbulence on collisional growth.

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Transport equation for the mean turbulent energy dissipation rate on the centreline of a fully developed channel flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.342 ◽

2015 ◽

Vol 777 ◽

pp. 151-177 ◽

Cited By ~ 17

Author(s):

S. L. Tang ◽

R. A. Antonia ◽

L. Djenidi ◽

H. Abe ◽

T. Zhou ◽

...

Keyword(s):

Energy Dissipation ◽

Transport Equation ◽

Channel Flow ◽

Dissipation Rate ◽

Turbulent Energy ◽

Energy Dissipation Rate ◽

Grid Turbulence ◽

Normal Velocity ◽

Turbulent Energy Dissipation Rate ◽

The Mean

The transport equation for the mean turbulent energy dissipation rate $\overline{{\it\epsilon}}$ along the centreline of a fully developed channel flow is derived by applying the limit at small separations to the two-point budget equation. Since the ratio of the isotropic energy dissipation rate to the mean turbulent energy dissipation rate $\overline{{\it\epsilon}}_{iso}/\overline{{\it\epsilon}}$ is sufficiently close to 1 on the centreline, our main focus is on the isotropic form of the transport equation. It is found that the imbalance between the production of $\overline{{\it\epsilon}}$ due to vortex stretching and the destruction of $\overline{{\it\epsilon}}$ caused by the action of viscosity is governed by the diffusion of $\overline{{\it\epsilon}}$ by the wall-normal velocity fluctuation. This imbalance is intrinsically different from the advection-driven imbalance in decaying-type flows, such as grid turbulence, jets and wakes. In effect, the different types of imbalance represent different constraints on the relation between the skewness of the longitudinal velocity derivative $S_{1,1}$ and the destruction coefficient $G$ of enstrophy in different flows, thus resulting in non-universal approaches of $S_{1,1}$ towards a constant value as the Taylor microscale Reynolds number, $R_{{\it\lambda}}$, increases. For example, the approach is slower for the measured values of $S_{1,1}$ along either the channel or pipe centreline than along the axis in the self-preserving region of a round jet. The data for $S_{1,1}$ collected in different flows strongly suggest that, in each flow, the magnitude of $S_{1,1}$ is bounded, the value being slightly larger than 0.5.

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Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.299 ◽

2016 ◽

Vol 798 ◽

pp. 140-164 ◽

Cited By ~ 19

Author(s):

Hiroyuki Abe ◽

Robert Anthony Antonia

Keyword(s):

Energy Dissipation ◽

Skin Friction ◽

Channel Flow ◽

Dissipation Rate ◽

Turbulent Channel Flow ◽

Turbulent Energy ◽

Energy Dissipation Rate ◽

Turbulent Energy Dissipation Rate ◽

The Mean ◽

Overlap Region

Integrals of the mean and turbulent energy dissipation rates are examined using direct numerical simulation (DNS) databases in a turbulent channel flow. Four values of the Kármán number ($h^{+}=180$, 395, 640 and 1020;$h$is the channel half-width) are used. Particular attention is given to the functional$h^{+}$dependence by comparing existing DNS and experimental data up to$h^{+}=10^{4}$. The logarithmic$h^{+}$dependence of the integrated turbulent energy dissipation rate is established for$300\leqslant h^{+}\leqslant 10^{4}$, and is intimately linked to the logarithmic skin friction law,viz.$U_{b}^{+}=2.54\ln (h^{+})+2.41$($U_{b}$ is the bulk mean velocity). This latter relationship is established on the basis of energy balances for both the mean and turbulent kinetic energy. When$h^{+}$is smaller than 300, viscosity affects the integrals of both the mean and turbulent energy dissipation rates significantly due to the lack of distinct separation between inner and outer regions. The logarithmic$h^{+}$dependence of$U_{b}^{+}$is clarified through the scaling behaviour of the turbulent energy dissipation rate$\overline{{\it\varepsilon}}$in different parts of the flow. The overlap between inner and outer regions is readily established in the region$30/h^{+}\leqslant y/h\leqslant 0.2$for$h^{+}\geqslant 300$. At large$h^{+}$(${\geqslant}$5000) when the finite Reynolds number effect disappears, the magnitude of$\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$approaches 2.54 near the lower bound of the overlap region. This value is identical between the channel, pipe and boundary layer as a result of similarity in the constant stress region. As$h^{+}$becomes large, the overlap region tends to contribute exclusively to the$2.54\ln (h^{+})$dependence of the integrated turbulent energy dissipation rate. The present logarithmic$h^{+}$dependence of$U_{b}^{+}$is essentially linked to the overlap region, even at small$h^{+}$.

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