Surface-averaged quantities in turbulent reacting flows and relevant evolution equations

Abstract

While quantities conditioned to an isosurface of reaction progress variable c, which characterizes fluid state in a turbulent reacting flow, have been attracting rapidly growing interest in the recent literature, a mathematical and physical framework required for research into such quantities has not yet been elaborated properly. This paper aims at filling two fundamental gaps in this area, i.e., (i) ambiguities associated with a definition of a surface-averaged quantity and (ii) the lack of rigorous equations that describe evolutions of such quantities. In the first (theoretical) part of the paper, (a) analytical relations between differently defined (area-weighted and unweighted) surface-averaged quantities are obtained and differences between them (quantities) are discussed, (b) a unified method for deriving an evolution equation for bulk area-weighted surface-averaged value of a local characteristic φ of a turbulent reacting flow is developed, and (c) the method is applied for deriving evolution equations for the bulk area-weighted surface-averaged reaction-surface density |∇c|, local reactionwave thickness 1/|∇c|, and local displacement speed S d , i.e., the speed of an isosurface of the c(x, t) field with respect to the local flow. In the second (numerical) part of the paper, direct numerical simulation data obtained recently from a highly turbulent reaction wave are analyzed in order to (1) highlight substantial differences between area-weighted and unweighted surface-averaged quantities and (2) show that various terms in the derived evolution equations are amenable to accurate numerical evaluation in spite of appearance of the so-called zero-gradient points [C. H. Gibson, Phys. Fluids 11, 2305 (1968)] in a highly turbulent medium. Finally, the obtained analytical and numerical results are used to shed light on the paradox of local flame thinning and broadening which is widely discussed in the turbulent combustion literature.

Assessment of an Evolution Equation for the Displacement Speed of a Constant-Density Reactive Scalar Field

“…A brief description will first be given for the derivation of the displacement speed evolution equation, both in an "averaged" as well as a "local" sense. More details on the derivation can be found in Yu and Lipatnikov (2019a). The explanations that follow will be directed toward physical interpretation of equation terms.…”

“…(6) can now be expanded using the product rule to yield [see Ref. Yu and Lipatnikov (2019a) for a more formal derivation] Equation (9) can be rewritten to give a general evolution equation for surface-averaged value of the quantity , which holds on all isosurfaces ĉ ∈ (0, 1) and at time instants and for an arbitrary variable . If is the displacement speed, S d , an equation for surface-averaged S d can then be obtained as long as * * t S d is known.…”

“…In the appendixes B and C of Ref. Yu and Lipatnikov (2019a) it is shown that the two terms in T 1 and T 2 can be recast into a form containing only spatial derivatives which facilitates their extraction from numerical simulations:…”

“…Here we should point out that zero gradient points have been excluded from the definitions of surface average in Eq. (5), see Yu and Lipatnikov (2019a), and Yu and Lipatnikov (2019c), however certain quantities of interest, e.g. S d and K , may locally grow unboundedly large in the neighborhood of a zero-gradient point, which poses a challenge for numerical calculation of surface-averaged quantities.…”

“…In the two frozen-velocity cases A and B, simulations of multiple transient waves c t are performed largely similarly to case C (further details on the difference are discussed in Yu and Lipatnikov (2019a)) but the duration of transient sampling is changed from 2 0 t to 2 F . It is worth noting that the transient data not only is of interest in itself because the vast majority of premixed turbulent flames are developing flames, as discussed in detail elsewhere (Lipatnikov and Chomiak 2002;Lipatnikov 2012), but also offer an opportunity to control the following numerical issue.…”