Fractal dimension of premixed flames in multifractal turbulence

In turbulent premixed flames, the fractal dimension of flame iso-surface is argued to be $\mathbb{D}=7/3$ for Damk\"ohler's large-scale limit $(Da>>1)$ and $\mathbb{D}=8/3$ for Damk\"ohler's small-scale limit $(Da\sim\mathcal{O}(1))$ based on heuristic scaling arguments. However, such scaling arguments do not consider the effect of the multifractal nature of turbulent kinetic energy dissipation on the flame surface. In this paper, we account for the effects of multifractal dissipation on the fractal dimension of low $Da$ turbulent premixed flames. We derive two corrections to the upper-limit of fractal dimension -- $\mathbb{D}=8/3+3/4(1-D_{1/4})$ and $\mathbb{D}=8/3+2/3(3-D_{1/3})$ -- which correspond to the change in the scalar flux and the total area of flame interface due to fluctuations in the inner cut-off scale arising from the intermittent nature of turbulent dissipation, respectively. We further show that the second correction leads to an explicit dependence of the fractal dimension $(\mathbb{D})$ on the scaling exponent $(\xi)$ of the velocity structure function through the relation: $\mathbb{D}=11/3+\xi$. Thus, we explicitly quantify the effect of the multifractal nature of turbulence upon low $Da$ premixed combustion.