A fundamental dimensionless number appropriate for studying laminar and turbulent buoyancy-driven flames.

is developed by considering the balance of momentum, thermal energy and species flux.

Two different natural flame configurations are studied: pool fires and ceiling fires. For turbulent pool fires, the sublayer thickness, ηβ, is shown in terms of Π_β to be

The fuel consumption is a turbulent pool fire expressed in terms of ηβ,(Π_β) and correlated by the experimental data leads to

The model agrees well with a previous model based on stagnant film hypothesis.

For turbulent ceiling fires, the sublayer thickness, ηβ, is shown in terms of a modified Π_β to be

where Π_f is based on a flame Rayleigh number, Ra_f, which is defined as

The fundamental relation for turbulent ceiling fires is correlated to experimentally measured flame stand-off distances. Based on the limited available data, the agreement with experiments is very good. Additional experiments are needed to further verify the validity of the scaling laws in the turbulent flow regime.