Interpolation of hydrodynamic velocity data with the continuity equation

Abstract

Meteorologists have developed a way of obtaining a velocity field V that satisfies the incompressible-flow continuity equation $\nabla ·\mathbf{V}=0$ in a bounded domain Ω , starting from a finite set of data ${{\mathbf{V}}_{\mathit{t}\mathit{r}\mathit{u}\mathit{e}}\mathrm{(}{\mathit{x}}_{\mathit{k}}\mathrm{)}}_{\mathit{k}=1}^{\mathit{N}}$ measured in Ω. In this work a mathematical framework of the scheme, is given. The approach can be extended to consider data provided by experimental methods such as the so-called particle image velocimetry technique. The viability of the scheme is tested with synthetic data from analytic solutions of the two-dimensional version of the equation $\nabla ·\mathbf{V}=0$ . Numerical examples show that the scheme can yield an accurate approximation V of the true velocity field ${\mathbf{V}}_{\mathit{t}\mathit{r}\mathit{u}\mathit{e}}$ .