This paper describes analyses of the nonlinear dynamics of harmonically forced, turbulent premixed flames. A key objective of this work is to analyse the ensemble-averaged dynamics of the flame front position, $\langle \xi \rangle $, excited by harmonic forcing of amplitude $\varepsilon $, in the presence of stochastic flow fluctuations of amplitude $\mu $. Low-amplitude and/or near-field effects are quantified by a third-order perturbation analysis, while the more general case is analysed computationally by solving the three-dimensional level-set equation, extracting the instantaneous flame position, and ensemble averaging the results. We show that different mechanisms contribute to smoothing of flame wrinkles, manifested as progressive decay in the magnitude of $\langle \xi \rangle $. Near the flame holder, random phase jitter, associated with stochastic velocity fluctuations tangential to the flame, is dominant. Farther downstream, propagation of the ensemble-averaged front normal to itself at the time-averaged turbulent burning velocity, $ \overline{{S}_{T, eff} } $, leads to destruction of wrinkles, analogous to the laminar case, an effect that scales with $\mu $. A second, new result is the demonstration that the ensemble-averaged turbulent burning velocity, ${S}_{T, eff} (s, t)$, is modulated by the harmonic forcing, i.e. ${S}_{T, eff} (s, t)= \overline{{S}_{T, eff} (s)} + { S}_{T, eff}^{\prime } (s, t)$, where ${ S}_{T, eff}^{\prime } $ has an inverse dependence upon the instantaneous, ensemble averaged-flame curvature, an effect that scales with $\varepsilon $ and $\mu $. We show that this curvature dependence follows from basic application of Huygens propagation to flames with stochastic wrinkling superimposed upon base curvature. This effect also leads to smoothing of flame wrinkles and is analogous to stretch processes in positive-Markstein-length, laminar flames.
Fluid residence time is a key concept in the understanding and design of chemically reacting flows. In order to investigate how turbulent mixing affects the residence time distribution within a flow, this study examines statistics of fluid residence time from a direct numerical simulation (DNS) of a statistically stationary turbulent round jet with a jet Reynolds number of 7290. The residence time distribution in the flow is characterised by solving transport equations for the residence time of the jet fluid and for the jet fluid mass fraction. The product of the jet fluid residence time and the jet fluid mass fraction, referred to as the mass-weighted stream age, gives a quantity that has stationary statistics in the turbulent jet. Based on the observation that the statistics of the mass fraction and velocity are self-similar downstream of an initial development region, the transport equation for the jet fluid residence time is used to derive a model describing a self-similar profile for the mean of the mass-weighted stream age. The self-similar profile predicted is dependent on, but different from, the self-similar profiles for the mass fraction and the axial velocity. The DNS data confirm that the first four moments and the shape of the one-point probability density function of mass-weighted stream age are indeed self-similar, and that the model derived for the mean mass-weighted stream-age profile provides a useful approximation. Using the self-similar form of the moments and probability density functions presented it is therefore possible to estimate the local residence time distribution in a wide range of practical situations in which fluid is introduced by a high-Reynolds-number jet of fluid.
The flow in a decelerating turbulent round jet is investigated using direct numerical simulation. The simulations are initialised with a flow field from a statistically stationary turbulent jet. Upon stopping the inflow, a deceleration wave passes through the jet, behind which the velocity field evolves towards a new statistically unsteady self-similar state. Assumption of unsteady self-similar behaviour leads to analytical relations concerning the evolution of the centreline mean axial velocity and the shapes of the radial profiles of the velocity statistics. Consistency between these predictions and the simulation data supports the use of the assumption of self-similarity. The mean radial velocity is predicted to reverse in direction near to the jet centreline as the deceleration wave passes, contributing to an approximately threefold increase in the normalised mass entrainment rate. The shape of the mean axial velocity profile undergoes a relatively small change across the deceleration transient, and this observation provides direct evidence in support of previous models that have assumed that the mean axial velocity profile, and in some cases also the jet spreading angle, remain approximately constant within unsteady jets.
This paper describes an investigation of the response of bluff body stabilized flames to harmonic oscillations. This problem involves two key elements: the excitation of hydrodynamic flow instabilities by acoustic waves, and the response of the flame to these harmonic flow instabilities. In the present work, data were obtained with inlet temperatures from 297 to 870 K and flow velocities from 38 to 170 m=s. These data show that the flame-front response at the acoustic forcing frequency first increases linearly with downstream distance, then peaks and decays. The corresponding phase decreases linearly with axial distance, showing that wrinkles on the flame propagate with a nearly constant convection velocity. These results are compared with those obtained from a theoretical solution of the G-equation excited by a harmonically oscillating, convecting disturbance. This kinematic model shows that the key processes controlling the response are 1) the anchoring of the flame at the bluff body, 2) the excitation of flame-front wrinkles by the oscillating velocity, 3) interference of wrinkles on the flame front, and 4) flame propagation normal to itself at the local flame speed. The first two processes control the growth of the flame response and the last two processes control the axial decrease observed farther downstream. These predictions are shown to describe many of the key features of the measured flame response characteristics. Nomenclature A = flame area D = burner depth d = bluff body width f = frequency f o = forcing frequency f BvK = Bénard-von Kármán instability frequency f KH = Kelvin-Helmholtz frequency G = isoscalar contour variable describing flame position H = burner height I t = edge detection threshold k = wave number L = flame position L 0 = fluctuation of flame position jL 0 j = gain of Fourier transformed flame position Q = heat release R 2 = goodness-of-fit parameter Re = Reynolds number S = distance along time-averaged flame front S L = laminar flame speed S T = turbulent flame speed t = time u = flow velocity in the x-direction u 0 = unsteady axial flow velocity u 0 = mean flow velocity u c;eff = effective convection velocity of a flame disturbance u c;v = convective velocity of a flow disturbance u 0 n = fluctuating flow velocity normal to the flame u t;0 = mean flow velocity tangential to the flame V = excitation voltage v = flow velocity in the y-direction W = burner width x = position along mean flow direction y = position perpendicular to mean flow direction = parametric flame-front angle " = nondimensionalized disturbance amplitude = time-averaged flame-front angle c = convective wavelength, u o =f 0 f = phase of a flame wrinkle x = phase of velocity perturbations in the x-direction y = phase of velocity perturbations in the y-direction ! = disturbance radial frequency
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