Abstract: A fifth-order upwind/sixth-order symmetric compact hybrid difference scheme coupled with a third-order explicit Runge-Kutta time-marching method is used as a direct numerical simulation algorithm to investigate a three-dimensional temporally-developing compressible plane free shear mixing layer with H$_{2}$/O$_{2}$ non-premixed combustion. The reacting mixing layer with product formation and energy release is perturbed by a pair of conjugate oblique waves, and hence experiences an instable transitional evolution. At the beginning of transition, some well-known large scale coherent structures, such as ${\it\Lambda}$ vortex and horseshoe vortex, are found, and a three-dimensional paring phenomenon of two horseshoe vortices following flow instabilities is also revealed. Furthermore, these large scale structures are approximately symmetric. At the last stage of the development of this flow, large scale structures break down continuously, and small scale structures gradually get dominant. The reacting mixing layer finally behaves very similar to turbulence and shows clear asymmetry. On the analogy of the stability theory for incompressible boundary layer, the large scale structures obtained here are due to a subharmonic secondary instability mechanism, which is well known as Herbert type secondary instability.

Key words: compressible mixing layer, combustion, transition, coherent structure, direct numerical simulation, compact scheme