Analysis of Performance of Cavitation Models with Analytically Calculated Coefficients

Cavitation is often simulated using a mixture model, which considers the transport of an active scalar, namely the vapor fraction αv. Source and sink terms of the transport equation of αv, namely vaporization and condensation terms, rule the dynamics of the cavity and are described through different models. These models contain empirical coefficients generally calibrated through optimization processes. The purpose of this paper is to propose an analytical approach for the calculation of the coefficients, based on the time scales of vaporization and condensation processes. Four different models are compared considering as a test-case a two-dimensional flow around a cylinder. Some relevant quantities are analyzed both for standard value of coefficients, as found in the literature, and the coefficients calculated through the analytical approach. The study shows that the analytical computation of the coefficients of the model substantially improve the results, and the models considered give similar results, both in terms of cavitation regime and mean vapor fraction produced.

On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

<p style='text-indent:20px;'>This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> is of lower singularity, i.e., <inline-formula><tex-math id="M3">\begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a logarithmic smoothing operator and <inline-formula><tex-math id="M5">\begin{document}$ \beta \in [0, 1] $\end{document}</tex-math></inline-formula>. We complete this study by considering the more singular regime <inline-formula><tex-math id="M6">\begin{document}$ \beta\in(1, 2) $\end{document}</tex-math></inline-formula>. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.</p>