Fractal structures of hydrodynamically unstable and diffusive-thermally unstable flames

This paper discusses the fractal structure of a hydrodynamically unstable flame with the background of the risk assessment of an explosion hazard. An accidental gas explosion usually occurs in a large-scale quiescent combustible mixture. A spherical flame outwardly propagates from the ignition point, and the flame accelerates owing to hydrodynamic instability. From the viewpoint of risk assessment, it is essential to consider such an increase in flame speed because the damage of an explosion is significantly influenced by the flame speed. Because hydrodynamically unstable flames have fractal structures and the flame area (and hence the flame speed) can be estimated using the fractal dimension, it is important to know the fractal dimension of the flame under the condition of a potential accidental explosion. Three methods (a box-counting method, a Fourier analysis, and a method based on the scale dependence of the flame speed) are tested to calculate the fractal dimension of a purely hydrodynamically unstable flame that is neutral in terms of diffusive-thermal instability. These methods are applied to the numerical solution of the Sivashinsky equation, but they can be also used to the result of an ordinary CFD calculation. The fractal structure of a purely diffusive-thermally unstable flame, which is neutral in terms of hydrodynamic instability, is also studied for comparison. The results show that all the three methods yield consistent fractal dimensions for the hydrodynamically unstable flame, whereas the diffusive-thermally unstable flame does not exhibit fractal characters. This is because the former flame has a hierarchical structure, whereas wrinkles of a specific wavelength mainly grow in the latter flame. The dependence of the fractal dimension on the thermal expansion ratio is also discussed.