Methods for Measuring and Computing the Reference Temperature in Newton’s Law of Cooling for External Flows

Newton’s law of cooling requires a reference temperature ( T r e f ) to define the heat-transfer coefficient ( h ). For external flows with multiple temperatures in the freestream, obtaining T r e f is a challenge. One widely used method, referred to as the adiabatic-wall (AW) method, obtains T r e f by requiring the surface of the solid exposed to convective heat transfer to be adiabatic. Another widely used method, referred to as the linear-extrapolation (LE) method, obtains T r e f by measuring/computing the heat flux ( q s ′ ′ ) on the solid surface at two different surface temperatures ( T s ) and then linearly extrapolating to q s ′ ′ = 0 . A third recently developed method, referred to as the state-space (SS) method, obtains T r e f by probing the temperature space between the highest and lowest in the flow to account for the effects of T s or q s ′ ′ on T r e f . This study examines the foundation and accuracy of these methods via a test problem involving film cooling of a flat plate where q s ′ ′ switches signs on the plate’s surface. Results obtained show that only the SS method could guarantee a unique and physically meaningful T r e f where T s = T r e f on a nonadiabatic surface q s ′ ′ = 0 . The AW and LE methods both assume T r e f to be independent of T s , which the SS method shows to be incorrect. Though this study also showed the adiabatic-wall temperature, T A W , to be a good approximation of T r e f (<10% relative error), huge errors can occur in h about the solid surface where | T s − T A W | is near zero because where T s = T A W , q s ′ ′ ≠ 0 .