The advantages of using high-order finite differences schemes in laminar-turbulent transition studies

This paper presents various finite difference schemes and compare their ability to simulate instability waves in a given flow field. The governing equations for two-dimensional, incompressible flows were solved in vorticity–velocity formulation. Four different space discretization schemes were tested, namely, a second-order central differences, a fourth-order central differences, a fourth-order compact scheme and a sixth-order compact scheme. A classic fourth-order Runge–Kutta scheme was used in time. The influence of grid refinement in the streamwise and wall normal directions were evaluated. The results were compared with linear stability theory for the evolution of small-amplitude Tollmien–Schlichting waves in a plane Poiseuille flow. Both the amplification rate and the wavenumber were considered as verification parameters, showing the degree of dissipation and dispersion introduced by the different numerical schemes. The results confirmed that high-order schemes are necessary for studying hydrodynamic instability problems by direct numerical simulation.