Uniformly High Order Methods for Unsteady Incompressible Flows

The paper presents the development of uniformly high-order (UHO) schemes for unsteady incompressible flows. The schemes are designed via high-order (rth-order) polynomial reconstruction of the fluxes at the cell faces. The similarity of the present schemes with the essentially nonoscillatory (ENO) and weighted ENO (WENO) schemes, originally developed for flows with discontinuities, lies in the concept of high-order flux reconstruction. In the present case the combination of all neighbouring fluxes is obtained by an interpolation procedure based on constant weight coefficients. The latter are defined by posing numerical conditions for minimising the numerical dispersion and dissipation. The high-order flux reconstruction can be implemented in conjunction with any first- or second-order Godunov-type method. In the present work, the high-order reconstruction is combined with a characteristic-based (CB) scheme. The latter is a flux averaging procedure according to which the flow variables, and subsequently the fuxes, are calculated at the cell faces by an upwind Godunov-type scheme. For the time integration we employ a fourth-order TVD Runge-Kutta scheme and a non-linear multigrid method in conjunction with dual-time stepping. The primary goal of the paper is to present the development of the schemes. However, their capability to resolve complex unsteady flow features, is also demonstrated for the case of two- and three-dimensional direct numerical simulations of a jet in a doubly period geometry and laminar-to-turbulent transition in a mixing layer.

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