  A hybrid-parallel direct-numerical-simulation method with application to
turbulent Taylor-Couette flow is presented. The Navier-Stokes equations are
discretized in cylindrical coordinates with the spectral Fourier-Galerkin
method in the axial and azimuthal directions, and high-order finite differences
in the radial direction. Time is advanced by a second-order, semi-implicit
projection scheme, which requires the solution of five Helmholtz/Poisson
equations, avoids staggered grids and renders very small slip velocities.
Nonlinear terms are computed with the pseudospectral method. The code is
parallelized using a hybrid MPI-OpenMP strategy, which is simpler to implement,
reduces inter-node communications and is more efficient compared to a flat MPI
parallelization. A strong scaling study shows that the hybrid code maintains
very good scalability up to more than 20000 processor cores and thus allows to
perform simulations at higher resolutions than previously feasible, and opens
up the possibility to simulate turbulent Taylor-Couette flows at Reynolds
numbers up to $\mathcal{O}(10^5)$. This enables to probe hydrodynamic
turbulence in Keplerian flows in experimentally relevant regimes.
