Decay Of Scalar Fluctuations Research Articles

Passive scalar mixing and decay at finite correlation times in the Batchelor regime

An elegant model for passive scalar mixing and decay was given by Kraichnan (Phys. Fluids, vol. 11, 1968, pp. 945–953) assuming the velocity to be delta correlated in time. For realistic random flows this assumption becomes invalid. We generalize the Kraichnan model to include the effects of a finite correlation time, $\unicode[STIX]{x1D70F}$, using renewing flows. The generalized evolution equation for the three-dimensional (3-D) passive scalar spectrum $\hat{M}(k,t)$ or its correlation function $M(r,t)$, gives the Kraichnan equation when $\unicode[STIX]{x1D70F}\rightarrow 0$, and extends it to the next order in $\unicode[STIX]{x1D70F}$. It involves third- and fourth-order derivatives of $M$ or $\hat{M}$ (in the high $k$ limit). For small-$\unicode[STIX]{x1D70F}$ (or small Kubo number), it can be recast using the Landau–Lifshitz approach to one with at most second derivatives of $\hat{M}$. We present both a scaling solution to this equation neglecting diffusion and a more exact solution including diffusive effects. To leading order in $\unicode[STIX]{x1D70F}$, we first show that the steady state 1-D passive scalar spectrum, preserves the Batchelor (J. Fluid Mech., vol. 5, 1959, pp. 113–133) form, $E_{\unicode[STIX]{x1D703}}(k)\propto k^{-1}$, in the viscous–convective limit, independent of $\unicode[STIX]{x1D70F}$. This result can also be obtained in a general manner using Lagrangian methods. Interestingly, in the absence of sources, when passive scalar fluctuations decay, we show that the spectrum in the Batchelor regime at late times is of the form $E_{\unicode[STIX]{x1D703}}(k)\propto k^{1/2}$ and also independent of $\unicode[STIX]{x1D70F}$. More generally, finite $\unicode[STIX]{x1D70F}$ does not qualitatively change the shape of the spectrum during decay. The decay rate is however reduced for finite $\unicode[STIX]{x1D70F}$. We also present results from high resolution ($1024^{3}$) direct numerical simulations of passive scalar mixing and decay. We find reasonable agreement with predictions of the Batchelor spectrum during steady state. The scalar spectrum during decay is however dependent on initial conditions. It agrees qualitatively with analytic predictions when power is dominantly in wavenumbers corresponding to the Batchelor regime, but is shallower when box-scale fluctuations dominate during decay.

Turbulent dispersion from line sources in grid turbulence

Probability density function (PDF) calculations are reported for the dispersion from line sources in decaying grid turbulence. The calculations are performed using a modified form of the interaction by exchange with the conditional mean (IECM) mixing model. These flows pose a significant challenge to statistical models because the scalar length scale (of the initial plume) is much smaller than the turbulence integral scale. Consequently, this necessitates incorporating the effects of molecular diffusion in order to model laboratory experiments. Previously, Sawford [Flow Turb. Combust. 72, 133 (2004)] performed PDF calculations in conjunction with the IECM mixing model, modeling the effects of molecular diffusion as a random walk in physical space and using a mixing time scale empirically fit to the experimental data of Warhaft [J. Fluid Mech. 144, 363 (1984)]. The resulting transport equation for the scalar variance contains a spurious production term. In the present work, the effects of molecular diffusion are instead modeled by adding a conditional mean scalar drift term, thus avoiding the spurious production of scalar variance. A laminar wake model is used to obtain an analytic expression for the mixing time scale at small times, and this is used as part of a general specification of the mixing time scale. Based on this modeling, PDF calculations are performed, and comparison is made primarily with the experimental data of Warhaft on single and multiple line sources and with the previous calculations of Sawford. A heated mandoline is also considered with comparison to the experimental data of Warhaft and Lumley [J. Fluid Mech. 88, 659 (1978)]. This establishes the validity of the proposed model and the significant effect of molecular diffusion on the decay of scalar fluctuations. The following are the significant predictions of the model. For the line source, the effect of the source size is limited to early times and can be completely accounted for by simple transformations. The peak centerline ratio of the rms to the mean of the scalar increases with the Reynolds number (approximately as Rλ1/3), whereas this ratio tends to a constant (approximately 0.4) at large times independent of Rλ. In addition, the model yields a universal long-time decay exponent for the temperature variance.

Experimental and numerical analysis of stratified turbulent V-shaped flames

The present paper is devoted to (i) the experimental study of partially premixed combustion with strong equivalence ratio gradients, i.e., stratification of the reactive mixture and (ii) the numerical modeling of turbulent reactive flows in such situations where reactants are far from being ideally premixed. From a practical point of view, at least two variables are necessary to describe the local thermochemistry in this case: the mixture fraction ξ and the fuel mass fraction Y f are considered to represent respectively the local composition of the fresh mixture and the progress of chemical reactions. From the experimental point of view, the use of simultaneous imaging techniques allows the evaluation of both variables in terms of fuel mole fraction and temperature. In the present study, a combined acetone PLIF measurement and Rayleigh scattering technique is used. The influence of temperature on the fluorescence signal is corrected thanks to the knowledge of the local temperature through Rayleigh scattering measurements. Conversely, the influence of the acetone Rayleigh cross section can be evaluated with the local value of acetone mole fraction. Using the iterative procedure already described by Degardin et al. [Exp. Fluids 40 (2006) 452–463], the corrected fuel mole fraction and temperature fields can be obtained. Here the particular flow configuration under study is a stratified turbulent V-shaped flame of methane and air. In a first step of the analysis, the optical diagnostics are used to perform a detailed investigation of the flame thickness with a special emphasis on the influence of partially premixed conditions. In a second step, experimental data are used to evaluate the LW-P model as defined by Robin et al. [Combust. Sci. Technol. 178 (10–11) (2006) 1843–1870] to calculate turbulent reactive flows with partially premixed conditions based on an earlier analysis by Libby and Williams [Combust. Sci. Technol. 161 (2000) 351–390]. The closure problem raised by the mean scalar dissipation terms is also discussed in the light of experimental results. Since the usual closures for nonreactive flows are expected to be unsuitable to describe reactive scalar fluctuations decay a new modeling proposal based on the recent developments of Mura et al. [Combust. Flame 149 (2007) 217–224] is used. After a preliminary validation step where numerical predictions of the flame mean quantities are compared successfully with the experimental database, numerical simulations are used to describe the mean structure of stratified flames and in particular the evolution of the mean chemical reaction rate for different partially premixed conditions.

Modeling of scalar dissipation in partially premixed turbulent flames

The question of the closure of scalar dissipation terms for both passive and reactive scalars is addressed. In a first step of the analysis, modeled transport equations for the mean reactive scalar dissipation are reviewed in the two limiting cases of (i) the flamelet regime and (ii) the thickened flame regime of turbulent premixed combustion. The corresponding asymptotic algebraic closures obtained for large values of the turbulent Reynolds Re T , i.e., eddy breakup and linear relaxation model for scalar fluctuations decay, are recovered. These two limiting closures are then used to build a general algebraic closure of the mean dissipation of reactive scalar fluctuations. The resulting expression is applicable to modeling turbulent premixed combustion whatever the combustion regime. In a second step, the same strategy leads to the development of an algebraic closure for partially premixed situations, a situation requiring the use of at least two variables. In this process, a relationship between reactive, passive, and cross-scalar dissipations is derived. This general result is then simplified to elaborate new closed expressions of mean dissipation terms. There are applicable to a wide range of conditions, from premixed to partially premixed situations and from the flamelet to the thickened flame regime.

Decay of scalar turbulence revisited.

We demonstrate that at long times the rate of passive scalar decay in a turbulent, or simply chaotic, flow is dominated by regions where mixing is less efficient. We examine two situations. The first is of a spatially homogeneous stationary turbulent flow with both viscous and inertial scales present. It is shown that at large times scalar fluctuations decay algebraically in time at all spatial scales. The second example explains chaotic stationary flow in a disk/pipe. The boundary region of the flow controls the long-time decay, which is algebraic at some transient times, but becomes exponential, with the decay rate dependent on the scalar diffusion coefficient, at longer times.

The approach to self-preservation of scalar fluctuations decay in isotropic turbulence

An analytical study of isotropic scalar fluctuations decay in isotropic turbulence is undertaken. From a fixed-point analysis, the existence of two complete self-preserving regimes is demonstrated. One of them relates to the final period of decay whereas the other one corresponds to the decay at large Reynolds and Péclet numbers. In both cases, the scalar-to-velocity timescale ratio is constant. In the final period, its asymptotic value is determined by the values of the coefficients of velocity and scalar enstrophy destruction. At large Reynolds and Péclet numbers, it depends on the latter as well as on the values of the mixed-derivative and velocity derivative skewness coefficients. A model for the approach to full self-preservation of scalar fluctuations decay is proposed. This model accounts for non-equilibrium resulting from initial unbalanced vortex stretching. In the conditions of experiments on temperature fluctuations decay in grid-turbulence, it displays a transient regime, extending over the region of measurements, during which the decay exponent is not constant but slowly varying.

Consistent modeling of scalars in turbulent flows

Conserved, passive scalars with equal diffusivities evolve independently according to the same linear transport equation. This observation leads to independence and linearity principles that impose constraints on the construction of consistent turbulence closure approximations. It is shown that the functional form of the scalar dissipation equation is more restrictive than has generally been assumed, and that the consistently modeled equation leads to qualitatively incorrect results for the decay of scalar fluctuations. It is also shown that the scalar-pressure-gradient correlation can be obtained from the velocity-pressure-gradient correlation, and thus an independent model for the scalar correlation is not required. (This last result depends upon an additional assumption that is certainly valid for nearly homogeneous flows.)