Inertial particle dynamics: Coherent structures in the presence of the Basset--Boussinesq memory term

We present an equivalent formulation of the Maxey--Riley equation in the presence of the Basset--Boussinesq memory term. A physical advantage of this formulation is that it reveals drag- and pressure-type forces within the memory term. The computational advantage of the new form is that it turns the Maxey--Riley equation from an implicit differential equation into an explicit one, enabling the use of classic numerical schemes in its solution. We further simplify the Maxey--Riley equation for small particles by deriving its reduction to its attractor. The reduced equation obtained in this fashion reveals that the memory term is asymptotically of the order of $\mbox{St}^{3/2}$, with $\mbox{St}$ being the Stokes number. This explains recent numerical findings on the relative importance of the Basset--Boussinesq term. Finally, we compute inertial Lagrangian coherent structures (ILCS) for vortex shedding behind a cylinder. The reduced ILCS closely capture the full inertial dynamics while providing significant savings in computational cost and complexity.