  The field synergy principle has three criteria to qualitatively describe the
essence of convective heat transfer enhancement. However, in practice these
criteria are difficult to be applied to convective heat transfer analysis,
because there are no corresponding indicators available to quantitatively
describe them. Based on these three criteria, a unified formula for the field
synergy principle was developed in this study using probabilistic techniques,
which is applicable to incompressible flows with constant properties in both
laminar and turbulent flow regimes. The formula contains three categories of
non-dimensional indicators corresponding to the three criteria of the field
synergy principle respectively, including domain-averaged cosine of synergy
angle, the Pearson linear correlation coefficients between the scalar functions
contained in the energy governing equation of convective heat transfer, and the
variation coefficients of these functions. The physical meanings of these
indicators for the field synergy principle and their connections with the known
heat transfer enhancing mechanisms were then discussed. Based on the unified
formula, an improved analytical system for the field synergy principle was
proposed, which allows an efficient and quantitative analysis of all
single-phase constant-property convective heat transfer phenomena. This new
system overcomes the limitation of the conventional field synergy analytical
system that mainly analyzes the convective heat transfer mechanism from the
perspective of synergy angle.
