Two-dimensional reactive Navier-Stokes equations for an acetylene-air mixture are solved numerically to simulate the interaction of a shock wave and an expanding flame front, the formation of a flame brush, and deflagration-to-detonation transition (DDT). The effects of viscosity, thermal conduction, molecular diffusion, and chemical reactions are included. A new method for adaptive mesh refinement was used to ensure that the structure of the flame front was resolved. The shock-flame interactions, through the Richtmyer-Meshkov instability, create and maintain a highly turbulent flame brush. The turbulence is not Kolmogorov turbulence, but it is driven at all scales by repeated shock-flame interactions. Pressure fluctuations generated in the region of the turbulent flame brush create, in turn, hot spots in unreacted material. These hot spots may then transition to DDT through the gradient mechanism. Repeated shock-flame interactions and merging shocks in unreacted material lead to the development of a high-speed shock that moves out in front of the turbulent flame. The region between this shock and the flame is subject to intense fluctuations generated in the flame. The simulations show that the interactions of shocks and flames create the conditions under which deflagration-to-detonation transition may occur.