Conditioned moments in turbulent premixed flames: A DNS study (2013)

Behavior of conditioned moments of velocity field in turbulent premixed flames is investigated using a direct numerical simulation (DNS) approach. Assuming constant density flow, the simulation is setup as planar flame propagation in an inlet/outlet configuration, with the movement of an infinity thin flame being tracked by the level-set G equation. A parametric study is performed by varying laminar flame speed and Reynolds number. Differences between the conditioned and unconditioned statistics of the fluctuating velocity are shown for the first, second and third moments. It is found that an increase in the laminar flame speed results in thinning mean flame brush and increasing difference between unconditioned and conditioned first moments. For the second moments, both the burned and unburned conditioned values are smaller than the unconditioned one in most cases. It is also found that at certain laminar flame speed the unburned conditioned second moment behaves similar to the unconditioned one. The burned conditioned third moment is closer to the unconditioned value than the unburned conditioned value. Under a small flame speed, the mean flame brush is found to be thicken at lower Reynolds number, which, however, affects less on the conditioned statistics. Introduction To numerically study a premixed turbulent flame within Reynolds-Averaged Navier-Stokes (RANS) or Large Eddy Simulation (LES) framework, both the influence of turbulence on combustion and the influence of combustion on turbulence should be modeled. When addressing the former issue, it is a common practice (i) to consider the rms velocity u' to be the primary characteristic of turbulence and (ii) to invoke a closure relation that expresses either mean (or filtered) mass rate of product creation W or turbulent burning velocity Ut (or subfilter flame speed st) as a function of u', the laminar flame speed SL and thickness δL, an integral turbulence length , etc., with the dependence of W (or Ut, or st) on u' being particularly pronounced. Various such models developed for RANS and LES applications are reviewed in Refs. [1,2] and [3,4], respectively. Mainstream approaches to investigating the latter issue, reviewed elsewhere [5], place the focus of consideration on variations in u' within a flame, with the same definition of the rms turbulent velocity being applied both to constant-density non-reacting flows and flames. However, u' defined in a unified manner can have totally different physical meanings in the two cases. For example, let us consider the flamelet regime of premixed turbulent combustion, i.e. assume that fresh reactants and burned products are separated by a thin interface (flamelet), with the probability γ of finding intermediate states of the reacting mixture being much less than unity everywhere within a turbulent flame brush formed due to random motion of the interface. In such a case, the well-known BML equations [6,7] hold and, in particularly,                 O ~ ~ 1 ~ 1 ~ , , , ,                 b j i u j i u i b i u i b i j i u u c u u c u u u u c c u u . (1) Here, ρ is the density, c is the combustion progress variable (c=0 and 1 in the reactants and products, respectively), over-lines and overbars designate the Reynolds average with q q q    , while  q q  ~ is the Favre-averaged value of a quantity q with q q q ~     , subscripts u and b designate quantities conditionally averaged in the reactants (the unburned state) and products (the burned state), respectively, e.g. see Eq. (5) below, which is valid in the asymptotic case of infinitesimally thin (δL→0 and γ→0) flamelet,     u j u i u j i u j i u u u u u u , ,     and     b j b i b j i b j i u u u u u u , ,     . Equation (1) clearly shows that application of the standard definition of turbulent Reynolds stress tensor j iu u   to turbulent flow within a flame brush yields quantities that are controlled not only by turbulence, but also by the slip velocity vector u b u u u    associated with reactant-product intermittency. Accordingly, the use of the standard definition of the local       x x x   3 k ku u u       (here, the summation convention applies for the repeated index k) in order to model the local      , x u W  does not seem to be consistent with the underlying physics, because turbulence and the generation of the slip velocity by a flame front are totally different phenomena from the physical viewpoint. A widely accepted way to resolving the problem consists of considering the conditioned Reynolds stresses   u j i u u   and   b j i u u   to characterize the true turbulence in a flame and the conditioned Reynolds stresses were the main subject of numerous experimental research into premixed turbulent combustion, as reviewed elsewhere [5]. However, such a solution was recently put into question [8] by discussing a few very simple model examples. Subsequently, balance equations for conditioned Reynolds stresses were numerically studied in a hypothetical case of a statistically planar, 1D, constant-density premixed turbulent “flame” and substantial differences between the true u' and conditioned rms velocities were reported [9]. Because a number of closure relations that had not yet been validated were invoked to model conditioned balance equations in Ref. [9], the problem needs further study using a method that is based on substantially less number of assumptions. The present work aims at filling this gap by applying Direct Numerical Simulation (DNS) to investigating the difference between Reynolds-averaged and conditioned moments of turbulent velocity field within a premixed flame brush. Statement of the Problem 2D DNS study of the issue was performed by numerically integrating the constant-density continuity and Navier-Stokes equations (and p are the dynamic viscosity and pressure respectively),