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

Abstract

SUMMARYThis paper presents various ÿnite di erence schemes and compare their ability to simulate instability waves in a given ow ÿeld. The governing equations for two-dimensional, incompressible ows were solved in vorticity-velocity formulation. Four di erent space discretization schemes were tested, namely, a second-order central di erences, a fourth-order central di erences, a fourth-order compact scheme and a sixth-order compact scheme. A classic fourth-order Runge-Kutta scheme was used in time. The inuence of grid reÿnement 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 ow. Both the ampliÿcation rate and the wavenumber were considered as veriÿcation parameters, showing the degree of dissipation and dispersion introduced by the di erent numerical schemes. The results conÿrmed that high-order schemes are necessary for studying hydrodynamic instability problems by direct numerical simulation.

A compact finite differences exact projection method for the Navier–Stokes equations on a staggered grid with fourth-order spatial precision

“…Adam [3] and Hirsh [4] discussed some advantages of the fourth-order compact methods compared to traditional methods. Souza et al in [5] demonstrated that the sixth-order compact scheme has good agreement with linear stability theory.…”

“…the derivative, is included implicitly in the equation which approximates the derivative. Several papers have been published showing the advantages of this class of methods [2,5,26].…”

“…Detailed Fourier analysis for the interior central schemes can be found in [2,5,26,19]. In the following sections, only the boundary non-central schemes will be analyzed.…”

An Eulerian Immersed Boundary Method for flow simulations over stationary and moving rigid bodies

“…In the normal direction the same idea is adopted in the buffer domain regions. The spatial derivatives are calculated using high order compact finite difference schemes given by Souza et al (2005). The vPoisson is solved using a Full Approximation Scheme (FAS) multigrid with a V-cycle working with 4 grids (Stuben and Trottenberg, 1981).…”

Verification and validation of a Direct Numerical Simulation code

“…The equations presented above are discretized by high-order finite-difference schemes [17,13,18,19] and spectral approximations for the spatial derivatives. A fourth-order four-step Runge-Kutta method is used for the temporal integration.…”