  In turbulent suspensions, collision rates determine how rapidly particles
coalesce or react with each other. To determine the collision rate, many
numerical studies rely on the 'Ghost Collision Approximation' (GCA), which
simply records how often pairs of point particles come within a threshold
distance. In many applications, the suspended particles stick (or in the case
of liquid droplets, coalesce) upon collision, and it is the frequency of first
contact which is of interest. If a pair of 'ghost' particles undergoes multiple
collisions, the GCA may overestimate the true collision rate. Here, using fully
resolved Direct Numerical Simulations of turbulent flows at moderate Reynolds
number (R_\lambda = 130), we investigate the prevalence and properties of
multiple collisions. We demonstrate that the GCA leads to a systematic
overestimate of the collision rate, which is of the order of 15% when the
particle inertia is small, and slowly decreases when inertia increases. We
investigate the probability P(N) for a given pair of ghost particles colliding
N times. We find P(N)=\beta \alpha^N for N_>1, where \alpha and \beta are
coefficients which depend upon the particle inertia. This result is used to
explain the discrepancy between the GCA and the true collision rates. We also
investigate the statistics of the times that ghost particles remain in contact.
We show that the probability density function of the contact time is different
for the first collision. The difference is explained by the effect of caustics
in the phase space of the suspended particles.
