A correlation model of energy and impulse losses for vortex ring–porous wall interactions

An experimental study was conducted to investigate the impingement of a vortex ring onto a porous wall by laser-induced fluorescence and particle image velocimetry. The effects of different Reynolds numbers ( ${{Re}}_{\it\Gamma } = 700$ and $1800$ ) and hole diameters ( $d_{h}^{*} = 0.067$ , $0.10$ , $0.133$ and $0.20$ ) on the flow characteristics were examined at a constant porosity ( $\phi = 0.75$ ). To characterise fluid transport through a porous wall, we recall the model proposed by Naaktgeboren, Krueger & Lage (2012, J. Fluid Mech. , vol. 707, 260–286), which shows rough agreement with the experimental results due to the absence of vortex ring characteristics. This highlights the need for a more accurate model to correlate the losses in kinetic energy ( $\Delta E^{*}$ ) and impulse ( $\Delta I^{*}$ ) resulting from the vortex ring–porous wall interaction. Starting from Lamb’s vortex ring model and considering the flow transition from the upstream laminar state to the downstream turbulent state caused by the porous wall disturbance, a new model is derived theoretically: $\Delta E^{*} = 1 - k(1 - \Delta I^{*})^2$ , where $k$ is a parameter dependent on the dimensionless core radius $\varepsilon$ , with $k = 1$ when no flow state change occurs. This new model effectively correlates $\Delta E^{*}$ and $\Delta I^{*}$ across more than 70 cases from current and previous experiments, capturing the dominant flow physics of the vortex ring–porous wall interaction.