We describe a computational modeling study of a cold atmospheric pressure plasma jet interacting with a dielectric surface placed normal to the jet axis. The plasma jet is generated by the application of a nanosecond pulse voltage applied to a dielectric tube through which the jet issues into ambient air. A base fluid flow field is pre-computed using a Navier-Stokes model for the helium jet impinging on the dielectric target surface with a two-species description for laminar diffusional mixing of the helium and ambient air streams. A self-consistent, multiple species, two-temperature model is used to describe the non-equilibrium plasma discharge dynamics in the presence of the base jet flow field. A single nanosecond pulse discharge event starting from initial breakdown in the dielectric tube, to propagation into the open gap, and finally the interaction with the dielectric surface is simulated. Initially, the plasma forms within the dielectric tube and propagates along the tube surface as a surface discharge driven by large induced electric fields produced by trapped charge on the dielectric surface. When the discharge reaches the end of the dielectric tube, the discharge transitions to a constricted fast ionization wave that propagates along the helium-air interface. The fast ionization wave eventually reaches the dielectric target surface where charged species are deposited as the discharge propagates parallel to the wall as a surface driven discharge. The surface driven discharge ceases to propagate once the quantity of air to helium is sufficient enough to quench the hot electrons and prevent further ionization. Due to the low speed of the flow discharge and the short life times of the radical species such as O, most of the radical species delivered to the surface are a result of the surface discharge that forms after the plasma bullet impinges against the surface. It is found that factors such as the thickness of the target dielectric and the profile of the stagnation helium-air jet significantly impact the net quantity of reactive particles delivered to the surface.