Is Stokes number an appropriate indicator for turbulence modulation by particles of Taylor-length-scale size?

It has been established both experimentally and numerically (e.g. Ferrante and Elghobashi (Phys. Fluids 2003)), that the Stokes number, $\tau_p/\tau_k$, can be used as an indicator to determine the extent to which small particles, $d_p < \eta$, modify the turbulence structure, for fixed values of their volume fraction, and mass fraction. Here, $\tau_p$, $d_p$, $\eta$ and $\tau_k$ are respectively the particle's response time and diameter, the Kolmogorov length- and time-scales. The objective of the present study is to investigate whether $\tau_p/\tau_k$ can also be used as an indicator for the modulation of turbulence by particles of the Taylor-lengthscale size, i.e. $ d_p \sim \lambda >> \eta$. We employ DNS with an immersed boundary method to fully resolve the flow around thousands of freely moving particles of Taylor-lengthscale size ($d_p \sim \lambda $) in decaying isotropic turbulence with initial $Re_\lambda = 110$ . Our results show that although the particles in different test cases have identical Stokes number and volume fraction, they have different effects on the turbulence kinetic energy, $E(t)$ and its dissipation rate $\varepsilon(t)$. For example, particles with smaller diameter and larger density ratio, $\rho_p/\rho_f$, augment $\varepsilon(t)$, resulting in a faster decay of $E(t)$. Our conclusion is that $\tau_p/\tau_k$ is not an appropriate indicator for determining the extent of turbulence modulation by particles with $d_p \sim \lambda $.