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Motivated by wind blowing over water, we use asymptotic methods to study the evolution of short wavelength interfacial waves driven by the combined action of these flows. We solve the Rayleigh equation for the stability of the shear flow, and construct a uniformly valid approximation for the perturbed streamfunction, or eigenfunction. We then expand the real part of the eigenvalue, the phase speed, in a power series of the inverse wavenumber and show that the imaginary part is exponentially small. We give expressions for the growth rates of the Miles (J. Fluid Mech., vol. 3, 1957, pp. 185-204) and rippling (e.g. Young & Wolfe, J. Fluid Mech., vol. 739,2014, pp. 276-307) instabilities that are valid for an arbitrary shear flow. The accuracy of the results is demonstrated by a comparison with the exact solution of the eigenvalue problem in the case when both the wind and the current have an exponential profile. [ABSTRACT FROM AUTHOR]

As expected, we observe Kolmogorov scaling Graph $E(k)\sim k^{-5/3}$ for Graph $k and a pseudo-turbulence scaling Graph $E(k) \sim k^{-3}$ for Graph $k>k {d}$. Hence for both of these cases we obtain Graph $E(k) \sim k^{-3}$ for Graph $k > k {d}$ and Graph $E(k)\sim k^{-5/3}$ for Graph $k. In figure 13 we plot the p.d.f. of Graph ${\rm \pi} {K}(\boldsymbol {x})$ with Graph $K=k {d}$ for Graph $b=0$, Graph $b=0.13$ and Graph $b=\infty$. For all the runs, Graph $L=2{\rm \pi}$ and Graph $d=0.46$, and the dimensionless numbers Graph $Ga= 302$, Graph $Bo =1.8$, Graph $At = 0.04$ and Graph $\varPhi = 1.64\,\%$ are kept fixed. \end{equation} In figure 3( I a i , I b i ) we plot the velocity streamlines of the average velocity field Graph ${\boldsymbol {u}}^{CM}({\boldsymbol {\xi }})$ for Graph $b=\infty$ (Graph ${\texttt {R0}}$) and Graph $b=0.13$ (Graph ${\texttt {R3}}$). [Extracted from the article]

The effect of turbulence on the mass transfer between a fluid and embedded small heavy inertial particles that experience surface reactions is studied. For simplicity, the surface reaction, which takes place when a gas phase reactant is converted to a gas phase product at the external surface of the particles, is unimolar and isothermal. Two effects are identified. The first effect is due to the relative velocity between the fluid and the particles, and a model for the relative velocity is presented. The second effect is due to the clustering of particles, where the mass transfer rate is inhibited due to the rapid depletion of the consumed species inside the dense particle clusters. This last effect is relevant for large Damköhler numbers, where the Damköhler number is defined as the ratio of the turbulent and chemical time scales, and it may totally control the mass transfer rate for Damköhler numbers larger than unity. A model that describes how this effect should be incorporated into existing simulation tools that utilize the Reynolds averaged Navier-Stokes approach is presented. [ABSTRACT FROM AUTHOR]