  We prove exponential decay of correlations for a realistic model of piecewise
hyperbolic flows preserving a contact form, in dimension three. This is the
first time exponential decay of correlations is proved for continuous-time
dynamics with singularities on a manifold. Our proof combines the second
author's version of Dolgopyat's estimates for contact flows and the first
author's work with Gou\"ezel on piecewise hyperbolic discrete-time dynamics.
  (Presentation revised.)
