  The use of simplified models of turbulent flows provides an appealing
possibility to study the collision rate of turbulent suspensions, especially in
conditions relevant to astrophysics, which require large time scale
separations. To check the validity of such approaches, we used a direct
numerical simulation (DNS) velocity field, which satisfies the Navier-Stokes
equations (although it neglects the effect of the suspended particles on the
flow field), and a kinematic simulation (KS) velocity field, which is a random
field designed so that its statistics are in accord with the Kolmogorov theory
for fully-developed turbulence. In the limit where the effects of particle
inertia (characterised by the Stokes number) are negligible, the collision
rates from the two approaches agree. As the Stokes number St increases,
however, we show that the DNS collision rate exceeds the KS collision rate by
orders of magnitude. We propose an explanation for this phenomenon and explore
its consequences. We discuss the collision rate $R$ for particles in high
Reynolds number flows at large Stokes number, and present evidence that
$R\propto \sqrt{{\rm St}}$.
