This study focuses on the mathematical and numerical modelling of how different properties, such as channel length and hot source height, affect the flow of thermal plumes and their interaction with the surrounding environment. Studying the behaviour of a directed thermal plume flow can provide a good visualisation of the control parameters for various applications related to fires and pollutants. To determine the control parameters affecting this flow, it is imperative to initially analyse the flow through a natural convection consisting of a disc-shaped heat source at the inlet of a vertical cylinder using mathematical and numerical methods. The flow in this study is characterised by natural convection, turbulent, steady, two-dimensional and incompressible. Numerical results were obtained via computational fluid dynamics analysis, which is based on mathematical flow modelling using the system of Navier–Stokes equations and the finite volume method. The SIMPLE algorithm was used for the velocity–pressure coupling. After introducing the concept of Reynolds averaging and analysis, different RANS turbulence models are presented, particularly two-equation models, using the Ansys Fluent software. Comparison of the different profiles revealed the ideal flow control parameters. An improvement in thermal properties was observed when the height of the heat source was = 1.5 cm and the vertical length of the channel was D = 20cm.