Equivalence of Two Different Integral Representations of Droplet Distribution Moments in Condensing Flow

It is proved that two different and independently derived integral representations of droplet size distribution moments encountered in the literature are equivalent and, moreover, consistent with the general dynamic equation that governs the droplet size distribution function. One of these representations consists of an integral over the droplet radius while the other representation consists of an integral over time. The proof is based on analytical solution of the general dynamic equation in the absence of coagulation but in the presence of both growth and nucleation. The solution derived is explicit in the droplet radius, which is in contrast with the literature where solutions are presented along characteristics. This difference is essential for the equivalence proof. Both the case of nonconvected vapor as well as the case of convected vapor are presented

On the invertibility of mappings arising in 2D grid generation problems

In adapting a grid for a Computational Fluid Dynamics problem one uses a mapping from the unit square onto itself that is the solution of an elliptic partial differential equation with rapidly varying coefficients. For a regular discretization this mapping has to be invertible. We will show that such result holds for general elliptic operators (in two dimensions). The Carleman-Hartman-Wintner Theorem will be fundamental in our proof. We will also explain why such a general result cannot be expected to hold for the (three-dimensional) cube

On a Dirichlet problem related to the invertibility of mappings arising in 2D grid generation problems

this paper depends strongly on a theorem of Carleman-HartmanWintner. This theorem is only true in two dimensional domains. In fact a straightforward generalization to more than two dimensional domains cannot be true. A counterexample to the proof of [15]forthe three dimensional case can be found by using a special harmonic function due to Kellogg [12]. This function is shown in [2]. A direct counterexample can be found in [13]. 2 Main result on smooth domain

Solution of the general dynamic equation along approximate fluid trajectories generated by the method of moments

We consider condensing flow with droplets that nucleate and grow, but do not slip with respect to the surrounding gas phase. To compute the local droplet size distribution, one could solve the general dynamic equation and the fluid dynamics equations simultaneously. To reduce the overall computational effort of this procedure by roughly an order of magnitude, we propose an alternative procedure, in which the general dynamic equation is initially replaced by moment equations complemented with a closure assumption. The key notion is that the flow field obtained from this so-called method of moments, i.e., solving the moment equations and the fluid dynamics equations simultaneously, approximately accommodates the thermodynamic effects of condensation. Instead of estimating the droplet size distribution from the obtained moments by making assumptions about its shape, we subsequently solve the exact general dynamic equation along a number of selected fluid trajectories, keeping the flow field fixed. This alternative procedure leads to fairly accurate size distribution estimates at low cost, and it eliminates the need for assumptions on the distribution shape. Furthermore, it leads to the exact size distribution whenever the closure of the moment equations is exact

Influence of water layer thickness on crater volume for nanosecond pulsed laser ablation of stainless steel

Under water laser ablation is a surface texturization method used to form micrometer-sized surface structures. Plasma confinement and cavitation bubble evolution play a critical role during the ablation process and their influence on material removal is strongly tied to liquid layer thickness. To influence the effects of these processes, such that material removal is at its maximum, an optimal layer thickness was found for various laser parameters. Specifically, for nanosecond pulsed laser ablation of stainless steel, however, the relation between layer thickness and volume removal is still unknown. Here, we show the relation between water layer thickness and removed material volume for a nanosecond pulsed laser. Results reveal that volume removal is at its maximum for a 1 mm water layer and drops by a factor of 2 when the layer thickness is increased to 2 mm. A further increase of layer thickness to 3 up to 10 mm shows a negligible effect on volume removal and removed volume amounts are shown to be similar to those obtained in ambient air in this water layer thickness range. This trend echo’s results obtained for nanosecond pulsed silicon ablation. The obtained results identify processing conditions which allow for faster and therefore more cost efficient texturization of stainless steel surfaces in the future.</p