Abstract
We address the flow field-dependent variation (FDV) methods in which complex physical phenomena are taken into account in the final form of partial differential equations to be solved so that finite difference methods (FDM) or finite element methods (FEM) themselves will not dictate the physics, but rather they are no more than simply the options how to discretize between adjacent nodal points or within an element. The variation parameters introduced in the formulation are calculated from the current flow field based on changes of Mach numbers, Reynolds numbers, Peclet numbers and Damkohler numbers between adjacent nodal points, which play many significant roles, such as adjusting the governing equations (hyperbolic, parabolic and/or elliptic), resolving various physical phenomena and controlling the accuracy and stability of the numerical solution. The theory is verified by a number of example problems addressing the physical implications of the variation parameters, which resemble the flow field itself, shock capturing mechanism, transitions and interactions between inviscid/viscous, compressibility/incompressibility and laminar/turbulent flows
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