We report the results of recent numerical simulations of the head-on merger of a cooling flow cluster with an infalling subcluster of galaxies. The objective of these simulations was to examine the effects of different types of cluster mergers (with 16:1 and 4:1 mass ratios) on the evolution of cluster cooling flows (with mass accretion rates of 100 and 400 Msolar yr-1). The two-dimensional simulations were performed with a combined hydrodynamics/N-body code on a uniform grid with a resolution of 20 kpc (~12 zones per core radius). In our simulations, cooling flow disruption is indicated by a dramatic increase (by a factor of 10-40) in the central cooling time of the primary cluster. We find that the ram pressure of the infalling gas is crucial in determining the fate of the cooling flow, because disruption occurs when a substantial amount of subcluster gas reaches the primary's core. In such cases, the subcluster gas can increase the central cooling time by displacing the high-density cooling gas and by heating it via shocks and turbulent gas motions. However, the fate of a merging cooling flow is also dependent on its initial cooling time. In cases where the initial cooling time is very short (i.e., 10-40 times shorter than the Hubble time), then even if the flow is disrupted, the central cooling time will remain less than a Hubble time, and the flow will likely reestablish itself. This has an important observational consequence, because such clusters will be classified as cooling flows on the basis of their cooling times, even though they have undergone a significant merger. In addition, we find that there is a time delay between core crossing and the point at which the central cooling time of a disrupted flow becomes of order a Hubble time. Thus, even in the case of disruption, a cluster can be classified as a cooling flow and exhibit substructure (indicative of a merger) for 1-2 Gyr after merging with a subcluster. We argue that our results make it possible to reconcile the high cooling flow frequency inferred by some observations with both high merger rates and a high frequency of substructure.