A new high-order method for the simulation of incompressible wall-bounded turbulent flows

A new high-order method for the accurate simulation of incompressible wall-bounded flows is presented. In the stream- and spanwise directions the discretisation is performed by standard Fourier series, while in the wall-normal direction the method combines high-order collocated compact finite differences with the influence matrix method to calculate the pressure boundary conditions that render the velocity field exactly divergence-free. The main advantage over Chebyshev collocation is that in wall-normal direction, the grid can be chosen freely and thus excessive clustering near the wall is avoided. This can be done while maintaining the high-order approximation as offered by compact finite differences. The discrete Poisson equation is solved in a novel way that avoids any full matrices and thus improves numerical efficiency. Both explicit and implicit discretisations of the viscous terms are described, with the implicit method being more complex, but also having a wider range of applications. The method is validated by simulating two-dimensional Tollmien–Schlichting waves, forced transition in turbulent channel flow, and fully turbulent channel flow at friction Reynolds number Reτ=395Reτ=395, and comparing our data with analytical and existing numerical results. In all cases, the results show excellent agreement showing that the method simulates all physical processes correctly.