Abstract
Rigorous upper bounds are derived for large-scale turbulent flame speeds in a prototypical model problem. This model problem consists of a reaction-diffusion equation with KPP chemistry with random advection consisting of a turbulent unidirectional shear flow. When this velocity field is fractal with a Hurst exponentH with 0<H<1, the almost sure upper bounds suggest that there is an accelerating large-scale turbulent flame front with the enhanced anomalous propagation lawy=C H t 1+H for large renormalized times. In contrast, a similar rigorous almost sure upper bound for velocity fields with finite energy yields the turbulent flame propagation law\(y = \mathop C\limits^ \sim _H t\) within logarithmic corrections. Furthermore, rigorous theorems are developed here which show that upper bounds for turbulent flame speeds with fractal velocity fields are not self-averaging, i.e., bounds for the ensemble-averaged turbulent flame speed can be extremely pessimistic and misleading when compared with the bounds for every realization.
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Majda, A.J., Souganidis, P.E. Bounds on enhanced turbulent flame speeds for combustion with fractal velocity fields. J Stat Phys 83, 933–954 (1996). https://doi.org/10.1007/BF02179550
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DOI: https://doi.org/10.1007/BF02179550