<p>Large-eddy simulation based on a newly developed wall-adaptive local eddy-viscosity model is used to study the viscous oscillatory flow around a cylinder at high Keulegan-Carpenter (<i>KC</i>) and Reynolds (<i>Re</i>) numbers. The efficacy of the new model, which was implemented in OpenFOAM, was first checked via simulations of a standard benchmark flow, namely that around a single cylinder in oscillatory flow at <i>Re</i> = 600 and <i>KC</i> = 2.08. These simulations accurately captured the Honji vortices observed in the measurements of Honji. Thereafter, simulations were performed for values of <i>KC</i> in the range 1.67 ⩽ <i>KC</i> ⩽ 66.7, where it was found that the evolution of the Honji vortices and streaked flow is influenced by the interactions that occur between the outer oscillation flow and the inner separated boundary layer. Further, it was found that these interactions are strongly dependent on this parameter, as are all the dimensionless coefficients for both the mean and fluctuating components of the hydrodynamic forces. At low values of <i>KC</i> and <i>Re</i>, frequency doubling resonance between the vortex shedding (<i>f</i>_vo) and that of the oscillatory flow (<i>f</i>_os) was observed. A new empirical relationship between <i>f</i>_vo/<i>f</i>_os, <i>KC</i> and <i>Re</i> is proposed.</p>