  In order to improve the application maturity of high-order difference
schemes, the free-stream preservation property, whose importance has been
widely recognized in recent years, has been developed into a focus of study..
In past literatures, only central schemes are considered to be suitable for
free-stream preservation. In this study, the methodology for arbitrary linear
schemes to achieve the property is investigated. First, derivations of grid
metric by Thomas, Lombard and Neier are reviewed, through which linear schemes
for the metric and unsplit flux could attain the property by the proof of
Vinokur and Yee firstly. In practical applications, flux splittings are usually
at presence and therefore the direct use of upwind schemes seems difficult to
fulfill free-stream preservation. To overcome the difficulty, two attempts are
made: firstly, a central-scheme-decomposition is worked out, through which a
central difference scheme is derived to approximate the first-order partial
derivative in metric evaluations; secondly, treatments are proposed for flux
splitting and a concrete example is presented. Through these two attempts, the
linear upwind node schemes can achieve the property. For half-node or mixed
type schemes, interpolations should be used to derive variables at half nodes.
As a result, a directionally consistent interpolation is proposed, which is
shown to be necessary in order to avoid violation of the metric identity and
free-stream preservation. Two numerical problems are also tested, i.e., the
free-stream and vortex preservation on wavy, largely randomized and triangular
grids. Numerical results validate aforementioned theoretical outcomes;
especially, simulations on the triangular grids indicate the methods discussed
in this study, which are typical algorithms on structured grids, has the
application potential for problems on unstructured meshes.
