The blue whirl is a small, stable, spinning blue flame that evolved spontaneously in recent laboratory experiments while studying turbulent, sooty fire whirls. It burns a range of different liquid hydrocarbon fuels cleanly with no soot production, presenting a new potential way for low-emission combustion. This thesis uses numerical simulations to present, for the first time, the flame and flow structure of the blue whirl. These simulations show that the blue whirl is composed of three different flames - a diffusion flame and a premixed rich and lean flame - all of which meet in a fourth structure, a triple flame which appears as a whirling blue ring. The results also show that the flow structure emerges as the result of vortex breakdown, a fluid instability which occurs in swirling flows. This thesis also presents the development and testing of the numerical algorithms used in the simulation of the blue whirl. This work is a critical step forward in understanding how to use this new form of clean combustion.