Cavitation inception mechanisms during vortex pair interactions and development of a compressible hybrid model

Abstract

The first part of this dissertation focuses on the interaction between a pair of counter-rotating vortices and the subsequent changes in the core pressure in (a) a canonical setting (axially periodic configuration) and (b) an experimental configuration (trailing vortices in the wake of a hydrofoil pair). In the second part, we present a compressible multi-scale model that captures cavitating structures over a wide range of length scales (both resolved and sub-grid) and also accounts for their mutual interactions. First, we examine the interaction between a pair of unequal strength counter-rotating vortices in an axially periodic configuration using Direct Numerical Simulation (DNS) at Reynolds number (Re𝞒,𝚪 - circulation strength) of 20000. Long-wavelength perturbations, representative of the Crow instability, are initially prescribed on the vortex filaments. In the linear regime, the two-dimensional core deformation due to mutual strain causes the numerical perturbation growth rates to differ from the analytical estimates. The non-linear regime is categorized into three phases. In the initial phase (Phase A), the axial stretching due to the growth of Crow instability causes the secondary core's pressure to reduce significantly and bring it closer to the primary core. The strong mutual strain due to the proximity of both cores results in the formation of a vortex sheet pair. In Phase B, these sheets undergo Kelvin--Helmholtz (KH) instability, resulting in vortex roll-up and the formation of a dipole with much smaller cores. Note that the transition to vortex sheets followed by the KH instability is a cyclic process. During the initial roll-up process, a sudden drop in the secondary core's pressure occurs, resulting in lower core pressure than that in Phase A. However, the strong viscous effects due to the close proximity of the cores result in a substantial loss of circulation strength of both cores over time and hence the secondary core pressure increases with time. Finally, the weaker secondary vortex transitions into a vortex ring that advects away from the stronger vortex due to self-induction in Phase C. During this phase, the core pressure in both cores remains relatively higher due to the lower circulation strength. Next, we use Large--Eddy Simulation (LES) to study the inception mechanism during the interaction between a pair of counter-rotating vortices in the wake of hydrofoils at Reynolds number, Re = 1.7x10⁶. The near wake measurements (up to x/c = 1.0) reveal decaying mean vorticity and velocity in both the cores due to the shear layer present between them. This causes the core pressure to rise, counter to the desired impact for inception. Beyond x/c = 1.0, the three--dimensional (3D) Crow instability develops on the weaker core periodically, causing it to stretch and wrap around the stronger core. Intermittent events of Cp < Cpv (Cp – pressure, CPV– saturated vapor pressure) primarily occur in the weaker vortex within 1.1 - 1.5 chord lengths downstream of the trailing edge. The temporal evolution of the weaker vortex over one Crow cycle shows that the peak axial stretching occurs in the initial stages, prior to the instance of lowest core pressure. The mean axial stretching (spatially averaged over one Crow wavelength) displays an oscillatory behavior with its amplitude decreasing over time. Probability density functions (pdfs) of the core pressure revealed a small portion of the weaker core having pressure lower than the saturated vapor pressure. The largest drop in weaker core pressure predominantly occurs in the regions where both cores are very close to each other. The impact of axial stretching is initially local, causing intense pressure reduction in a few regions along the vortex axis. However, its impact spreads along the axis in the later stages of the Crow cycle, causing more regions to have relatively lower pressure. Finally, we propose a compressible hybrid model that (i) captures the dynamics of both large vapor cavities (resolved vapor) and micro-bubbles (unresolved vapor), and (ii) accounts for medium compressibility. The vapor mass, momentum and energy in the compressible homogeneous mixture equations are explicitly decomposed into constituent resolved and unresolved components enabling their independent treatment. The homogeneous mixture of liquid and resolved vapor is tracked as a continuum in an Eulerian sense. The unresolved vapor terms are expressed in terms of subgrid bubble velocities and radii that are tracked in a Lagrangian sense using a novel `kR-RP equation’. The kR-RP equation is formally derived in terms of the pressure at a finite distance kR from the bubble while accounting for the effects of neighboring bubbles. p(kR) may, therefore, be either a near-field or far-field pressure. The equation exactly recovers the classical RP ad Keller--Miksis equations in the limits that k and c become very large. Also, the results are independent of k for a single bubble for all k, and for multiple bubbles when kR < d (where d denotes separation distance). Numerical results show this robustness of the model to the choice of k, which can be different for each bubble. The hybrid model is validated for the collapse of a single resolved/unresolved bubble. Its ability to capture inter-bubble interactions is demonstrated for multiple bubbles exposed to an acoustic pulse. The model is then applied to a problem where resolved and unresolved bubbles co-exist. Finally, it is validated using a cluster of 1200 bubbles exposed to a strong acoustic pulse. The results show the impact of the bubble cluster on the transmitted and reflected waves and the shielding effect, where bubbles at the edge of the cluster shield the interior bubbles by dampening the incident acoustic wave.