The unraveling dynamics of long, isolated, molecules of DNA subjected to an extensional flow in a crossed-slot device [T. T. Perkins, D. E. Smith, and S. Chu, "Single polymer dynamics in an elongational flow," Science 276, 2016-2021 (1997); D. E. Smith and S. Chu, "Response of Flexible Polymers to a Sudden Elongational Flow," Science 281, 1335-1340 (1998)] are predicted by Brownian dynamics simulations using measured elastic and viscous properties of the DNA as the only inputs. Quantitative agreement is obtained both in the percentages of various unraveling states, such as "folds," "kinks," "dumbbells," half-dumbbells," and "coils," and in the ensemble-averaged stretch and rate of stretch. Under fast flows (De greater than or similar to 10), unraveling is initially nearly affine, but for fractional stretch greater than approximate to 1/3, stretching is delayed to an extent that varies widely from molecule to molecule by flow-induced folded states, which are far-from-equilibrium kinetic hindrances not predicted by dumbbell models. From the computer simulations, the source of the high molecule-to-molecule heterogeneity in the experiments is traced to variability in the initial polymer configuration, which sets the unraveling path the molecule must take at De greater than or similar to 10. Formation of folds and kinks during unraveling can be predicted fairly reliably just by examining the initial state. The high-De unraveling behavior is consistent with the predictions of one-dimensional "kink dynamics" simulations.