Pressure drop in engineered (TPMS-based) porous media

Triply periodic minimal surfaces (TPMS)-based media (a type of metamaterial) are defined by mathematical expressions, which are amenable to additive manufacturing, and are finding increasing practical applications owing to their porous nature. We present experimental pressure drop measurements for a range of velocities spanning laminar to turbulent regimes for three TPMS geometries – gyroid, primitive and body-centred cubic (BCC) – with different porosity, unit cell length and surface finish. Dimensional Darcy and Forchheimer permeabilities are estimated via quadratic fitting for the gyroid geometry, which closely resembles random packed porous media. Subsequently, the non-dimensional drag ( ${\kern-0.5pt}f$ ) is plotted against Reynolds number ( $Re$ ) yielding distinct curves for each case. The lack of collapse stems from varying definitions of pore diameter, complicating comparisons across porous media (not just TPMS). Therefore, a method is developed to estimate an equivalent hydraulic diameter $d_{{H\hbox{-}\textit{equ}}}$ from pressure drop data by matching the laminar drag $f$ of packed spheres via the Ergun equation, allowing the collapse of all porous media $f-Re$ curves in the laminar regime. The value of $d_{ {H\hbox{-}\textit{equ}}}$ is related to the ‘true’ Darcy permeability defined strictly in the linear regime (unlike permeability from quadratic fitting). We observe an approximate linear relationship between the $d_{ {H\hbox{-}\textit{equ}}}$ and the hydraulic diameter for self-similar TPMS configurations. The common basis of $d_{ {H\hbox{-}\textit{equ}}}$ allows intercomparison of TPMS geometries, and shows that BCC achieves significant drag reduction compared with packed spheres in the turbulent regime partially because of their open tube-like structure, whereas some configurations show drag increase. Although gyroid can be represented using the traditional quadratic drag law, primitive and BCC show an increase in $f$ with increasing $Re$ immediately before transitioning to fully turbulent regime – akin to rough-wall pipe flows, likely owing to their periodic streamwise elongated open structures.