Abstract: A static air mixing apparatus for mixing of air passing through a space or duct. The apparatus includes an inner enclosure at least partially traversing the duct, an outer enclosure surrounding the inner enclosure, a first plurality of radial vanes extending outwardly from a center of the inner enclosure to adjacent the inner enclosure, a second plurality of radial vanes extending from between the inner and outer enclosures, and a transition member forming a divergent retention region downstream of the enclosures for prolonging mixing of different temperature airstreams as they pass through the air mixing apparatus. Further, the apparatus are derived from an optimized depth ratio between the depth of the enclosures to the minimum diameter of the outer enclosure of between 0.37 and 0.40 and core ratio between the core area defined by the inner enclosure to the outer enclosure area including the core area of between about 0.60 to about 0.65.