The effect of turbulence on mixing in prototype reaction-diffusion systems

Abstract

The effect of turbulence on mixing in prototype reaction-diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large-scale mean flow plus a small-scale spatiotemporal periodic flow. In the limit of fast reaction and slow diffusion, it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large-scale velocity field. This surprising result contrasts strongly with earlier work of the authors that always yields a large-scale propagation speed enhanced by small-scale turbulence for propagating fronts. The mathematical reasons for these differences are pointed out. This main theorem rigorously justifies the limit equilibrium approximations utilized in nonpremixed turbulent diffusion flames and condensation-evaporation modeling in cloud physics in the fast reaction limit. The subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero-diffusion limit with a nontrivial effect of turbulence on mixing in the limit. The situation with slow reaction and slow diffusion is also studied in the present work. Here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction-diffusion equations. Physical examples from non-premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results.

A Rigorous Asymptotic Perspective on the Large Scale Simulations of Turbulent Premixed Flames

“…For that choice of f (T ), T = 1 is a stable equilibrium point of the ODE T t = f (T ) while T = 0 is an unstable equilibrium point. The irreversible conversion from fresh gas (unstable T = 0) to burnt gas (stable T = 1) corresponds to the propagation of the flame front toward the fresh gas [27].…”

A Review of Some Monte Carlo Simulation Methods for Turbulent Systems

“…On the other hand, models which account for turbulence only through an enhanced diffusivity coefficient are often insufficient [43,54,77]. This is particularly the case in reaction and mixing processes where the fine-scale fluctuations in the immersed chemical species play an important role, and these are completely ignored in a crude eddy diffusion model and not well represented in large eddy simulations [33,45].Monte Carlo approaches are an attractive option for turbulence simulations due both to their capacity for investigating systems with many degrees of freedom and to their natural generation of a disordered velocity field structure and irregular particle trajectories. Indeed, it is difficult to conceive of a deterministic mechanism for generating a velocity field with disordered fluctuations over a wide range of scales, other than by an expensive direct numerical simulation which resolves all those scales!…”

“…The key difficulty is the inability for even large computers to resolve all of the active scales in systems with strong turbulence in a direct numerical simulation based on the fundamental Navier-Stokes equations [43,54]. A numerical representation of turbulence, however, is crucial in computer simulation studies in many applied fields, ranging from atmosphere-ocean dynamics (including weather and climate prediction) [11,43,47], combustion [7,33,45,62,80,85], turbulent diffusion [48], mixing processes and numerous other engineering situations [36,54]. These applications do not require a turbulence simulation with full fidelity -only certain key features of the turbulence need to be represented.…”

“…On the other hand, models which account for turbulence only through an enhanced diffusivity coefficient are often insufficient [43,54,77]. This is particularly the case in reaction and mixing processes where the fine-scale fluctuations in the immersed chemical species play an important role, and these are completely ignored in a crude eddy diffusion model and not well represented in large eddy simulations [33,45].…”

Statistical analysis of a semilinear hyperbolic system advected by a white in time random velocity field

“…Such a system arises in isothermal non-premixed turbulent combustion under single step reaction: Y +νO x → P , where P is product, ν stoichiometric constant, see [3] (chapters 2 and 6), [9] (chapters 3 and 5), and [17]. We shall be concerned with the inviscid regime where κ = 0.…”