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Register a new userTheory of weakly damped free-surface flows: a new formulation based on potential flow solutions
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Several theories for weakly damped free-surface flows have been formulated. In this paper we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoullis equation), but also to the kinematic boundary condition. The nonlinear Schrodinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.
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For ideal fluid flow with zero surface tension and gravity, it remains unknown whether local singularities on the free surface can develop in well-posed initial value problems with smooth initial data. This is so despite great advances over the last 25 years in the mathematical analysis of the Euler equations for water waves. Here we expand our earlier work (Chin. Ann. Math. Ser. B 40 (2019) 925) and review the mathematical literature and some of the history concerning Dirichlets ellipsoids and related hyperboloids associated with jet formation and flip-through, splash singularities, and recent constructions of singular free surfaces that however violate the Taylor sign condition for linear well-posedness. We illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation (whose derivation is detailed and discussed in an appendix). Additional numerical evidence strongly suggests that corner singularities may form in an unstable self-similar way from specially prepared initial data.
As wind energy continues to expand, increased interaction between wind farms and their surroundings can be expected. Using natural snowfall to visualize the air flow in the wake of a utility-scale wind turbine at unprecedented spatio-temporal resolution, we observe intermittent periods of strong interaction between the wake and the ground surface and quantify the momentum flux during these periods. Significantly, we identify two turbine operational-dependent pathways that lead to these periods of increased wake-ground interaction. Data from a nearby meteorological tower provides further insights into the strength and persistence of the enhanced flux for each pathway under different atmospheric conditions. These pathways allow us to resolve discrepancies between previous conflicting studies on the impact of wind turbines on surface fluxes. Furthermore, we use our results to generate a map of the potential impact of wind farms on surface momentum flux throughout the Continental United States, providing a valuable resource for wind farm siting decisions. These findings have implications for agriculture in particular, as crop growth is significantly affected by surface fluxes.
In this study, the stability dependence of turbulent Prandtl number ($Pr_t$) is quantified via a novel and simple analytical approach. Based on the variance and flux budget equations, a hybrid length scale formulation is first proposed and its functional relationships to well-known length scales are established. Next, the ratios of these length scales are utilized to derive an explicit relationship between $Pr_t$ and gradient Richardson number. In addition, theoretical predictions are made for several key turbulence variables (e.g., dissipation rates, normalized fluxes). The results from our proposed approach are compared against other competing formulations as well as published datasets. Overall, the agreement between the different approaches is rather good despite their different theoretical foundations and assumptions.
This paper has been withdrawn by the author due to a crucial error in the formulation.
Granular fronts are a common yet unexplained phenomenon emerging during the gravity driven free-surface flow of concentrated suspensions. They are usually believed to be the result of fluid convection in combination with particle size segregation. However, suspensions composed of uniformly sized particles also develop a granular front. Within a large rotating drum, a stationary recirculating avalanche is generated. The flowing material is a mixture of a visco-plastic fluid obtained from a kaolin-water dispersion, with spherical ceramic particles denser than the fluid. The goal is to mimic the composition of many common granular-fluid materials, like fresh concrete or debris flow. In these materials, granular and fluid phases have the natural tendency to segregate due to particle settling. However, through the shearing caused by the rotation of the drum, a reorganization of the phases is induced, leading to the formation of a granular front. By tuning the material properties and the drum velocity, it is possible to control this phenomenon. The setting is reproduced in a numerical environment, where the fluid is solved by a Lattice-Boltzmann Method, and the particles are explicitly represented using the Discrete Element Method. The simulations confirm the findings of the experiments, and provide insight into the internal mechanisms. Comparing the time-scale of particle settling with the one of particle recirculation, a non-dimensional number is defined, and is found to be effective in predicting the formation of a granular front.
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