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Vortex dynamics and fractal structures in reactive Richtmyer-Meshkov instability

In nuclear and process plant safety, the accurate prediction of explosion loads requires understanding of various mechanisms responsible for ﬂame acceleration (FA) and deﬂagration-to-detonation transition (DDT). In conﬁned geometries, shock-ﬂame interactions due to shock reﬂections at obstacles and walls are a key contributing factor to FA and DDT. In this context, the wrinkling of the ﬂame surface due to the production of baroclinic torque and in consequence the increase of the integral reaction rate is referred to as the reactive Richtmyer-Meshkov Instability (RMI). With the goal of developing a RMI sub-grid model, compressible 3D-DNS of a shock-ﬂame interaction in a lean and a stoichiometric homogeneous H2/Air mixture have been performed. The chemistry is modeled using a modiﬁed one-step Arrhenius approach, since the problem is dominated by ﬂuid dynamic phenomena rather than complex chemical kinetics. Temporal and spatial discretization are achieved using a 3rd order Runge-Kutta method and a high-order WENO scheme respectively. The set-up consists of a planar shock propagating towards a perturbed but statistically planar ﬂame. After interacting with the ﬂame, the shock is reﬂected at the back-end of the domain, causing a second shock-ﬂame interaction (re-shock). The development of the RMI, measured by the evolution of the ﬂame surface area and turbulent mixing width, is inﬂuenced by key physical parameters. One important parameter is the equivalence ratio, as it affects the Lewis number and laminar ﬂame speed. Using the DNS database, sub-grid modeling approaches for closure of the averaged/ﬁltered reaction rate can be quantiﬁed. For this purpose individual terms of the enstrophy transport equation are analyzed, identifying dominant mechanisms during the development of the instability. Furthermore, the fractal behavior of the RMI is analyzed in an approach of power-law based modeling of the ﬂame wrinkling factor. For the investigated cases it is found that the baroclinic torque is the most dominant enstrophy transport term, showing high peaks after every shock-ﬂame interaction. The baroclinic torque is heavily dependent on the ﬂame thickness, which varies with the chosen equivalence ratio and in time, as the ﬂame thickness is reduced by about 50% after interacting with the shock and re-shock. Each shock interaction also causes a strong increase of the ﬂame surface area and turbulent mixing width, with an maximum increase in area of about 400%. The ﬁnal area increase of the stoichiometric case is about 30% lower than the lean case due to funnel-closure and smoothing effects, caused by the increased reaction rate. The fractal dimension D f , as found in power law based modeling, is investigated using explicit ﬁltering of the present DNS data. There a maximum fractal dimension of 2.92 to 2.96 is determined for both cases, before the stoichiometric and lean cases settle at a value of 2.38 and 2.67 respectively. Since the maximum fractal dimension of the ﬂame surface is expected to be 8 / 3 for a premixed ﬂame, values of D f > 8 / 3 are a strong indicator for additional wrinkling caused by the RMI. For the lean case with an effective Lewis number Le eff < 1, thermo-diffusive instabilities can provide an additional contribution towards ﬂame wrinkling.