César Dopazo

An approach to the autoignition of a turbulent mixture

Abstract This paper considers the turbulent homogeneous mixing of two reactants undergoing a one step, second order, irreversible, exothermic chemical reaction with a rate constant of the Arrhenius type. A statistically stationary turbulent velocity field is assumed given and unaffected by mass or heat production due to the chemical reaction. Relative density fluctuations are neglected. A Hopf-like functional formalism is presented, with application to both statistically inhomogeneous and statistically homogeneous flows. Single and double point probability density function differential equations are derived from those functional equations. The limit of very large activation energies is considered; a low degree of statistical correlation between temperature and concentration fields during the ignition period is hypothesized. After making use of the homogeneity assumption a closure problem is still present due to the nonlocalness of the molecular diffusion term. The problem is rendered closed by assuming a Gaussian conditional expected value for the temperature at a point given the temperature at a neighboring point. The closure is seen to preserve very important mathematical and physical properties. A linear first order hyperbolic differential equation with variable coefficients for the probability density function of the temperature field is obtained. A second Damkohler number based on Taylors microscale turns out to be an important controlling parameter. A numerical integration for different values of the second Damkohler number and the initial stochastic parameters is carried out. The mixture is seen to evolve towards an eventual thermal runaway, the detailed behavior however being different for different systems. Some peculiarities during the ignition period evolution are uncovered.

''Relaxation'' of initial probability density functions in the turbulent convection of scalar fields

The evolution of an initially binary (zero unity) scalar field undergoing turbulent and molecular mixing is studied in terms of conservation equations for the probability density function of the scalar property. Attention is focused on the relaxation of the dynamic system to a state independent of the intial conditions. A few existing methods are discussed and evaluated and a new mechanistic model is proposed. Classical iteration techniques are used to obtain an equation for the single point probability density and the unperturbed Green’s function. It is suggested that use of the true Green’s function or perturbed propagator of the system might be necessary in order to obtain the correct evolution of the probability density function.

A binomial Langevin model for turbulent mixing

A Langevin model with binomial random diffusion replacing the classical Wiener process is proposed to model the turbulent mixing of a scalar convected by a field of statistically homogeneous turbulence. A Monte Carlo simulation is performed. The results display an excellent agreement with existing data from the numerical experiment of Eswaran and Pope [Phys. Fluids 31, 506 (1988)].

Combustion characteristics of heavy oil-water emulsions

The combustion of heavy oil and its emulsions with water was investigated in experiments on a semi-industrial scale. Two comparisons between heavy oil and oil-water emulsion flames are presented that, due to the different initial conditions of the spray, provide complementary information. Reported results include spatial distributions in the flame of temperature and species concentrations (O2, CO, UHC, NOx) as well as gaseous and solid emissions in the flue gases. The measurements inside the emulsion flame display a remarkable improvement in the combustion process with respect to that of the neat oil with poor atomization; differences are much less important if a fine spray is achieved with the heavy oil. Solid emissions are significantly reduced in the emulsion tests and the morphology of the particle samples demonstrates the fragmentation of the drops and/or the coke particles initially formed. The flame temperatures are reduced by ∼65 K. The heat absorbed by the water injected in the emulsion and enhanced radiative heat transfer due to the higher particle number density could explain this difference. The spatial distribution of NOx indicates that a significant reduction is obtained in the final part of the flame; this may be attributed to a decrease in the rate of thermal-NO formation as a consequence of lower gas temperatures. No measurable difference in NOx concentration is found in the inner core of the flames.

Longitudinal instabilities in an air-blasted liquid sheet

An experimental and numerical study has been performed to improve the understanding of the air/liquid interaction in an air-blasted breaking water sheet. This research is focused in the near eld close to the exit slit, because it is in this region where instabilities develop and grow, leading to the sheet breakup. In the experiments, several relevant parameters were measured including the sheet oscillation frequency and wavelength, as well as the droplet size distribution and the amplication growth rate. The flow was also investigated using linear instability theory. In the context of existing papers on instability analysis, the numerical part of this work presents two unique features. First, the air boundary layer is taken into account, and the eects of air and liquid viscosity are revealed. Second, the equations are solved for the same parameter values as those in the experiments, enabling a direct comparison between calculations and measurements; although qualitatively the behaviour of the measured variables is properly described, quantitative agreement is not satisfactory. Limitations of the instability analysis in describing this problem are discussed. From all the collected data, it is conrmed that the oscillation frequency strongly depends on the air speed due to the near-nozzle air/water interaction. The wave propagates with accelerating interface velocity which in our study ranges between the velocity of the water and twice that value, depending on the air velocity. For a xed water velocity, the oscillation frequency varies linearly with the air velocity. This behaviour can only be explained if the air boundary layer is considered.

Functional formulation of nonisothermal turbulent reactive flows

The two familiar functional formalisms in turbulence are applied to the simultaneous turbulent mixing and chemical reaction of scalar fields. One‐step, second‐order, irreversible, exothermic chemical reactions with an Arrhenius‐type rate constant are considered. The problem is formally posed in terms of initial and boundary conditions. For the special case of equal mass‐diffusivities and a Lewis number of one the functional equations are decoupled into a turbulent binary mixing study and a reactive problem. The Lewis‐Kraichnan formalism is used in order to obtain exact functional solutions of the binary mixing case in final period turbulence. Driving forces are included in the thermal energy equation. These solutions are used to obtain detailed information about the binary mixing problem and the behavior of very rapidly reacting species in the final period.

Dynamics of velocity gradient invariants in turbulence: Restricted Euler and linear diffusion models

A complete system of dynamical equations for the invariants of the velocity gradient, the strain rate, and the rate-of-rotation tensors is deduced for an incompressible flow. The equations for the velocity gradient invariants R and Q were first deduced by Cantwell [Phys. Fluids A 4, 782 (1992)] in terms of Hij, the tensor containing the anisotropic part of the pressure Hessian and the viscous diffusion term in the velocity gradient equation. These equations are extended here for the strain rate tensor invariants, RS and QS, and for the rate-of-rotation tensor invariant, QW, using HijS and HijW, the symmetric and the skew-symmetric parts of Hij, respectively. In order to obtain a complete system, an equation for the square of the vortex stretching vector, Vi≡Sijωj, is required. The resulting dynamical system of invariants is closed using a simple model for the velocity gradient evolution: an isotropic approximation for the pressure term and a linear model for the viscous diffusion term. The local topology ...

A binomial sampling model for scalar turbulent mixing

The closure problem generated by the molecular mixing term in the turbulent convection of scalars is studied. The statistical average of this term both in moment formulations and in the probability density function (pdf) approach implicitly encloses the turbulence straining action on scalar gradients leading to a significant enhancement of the molecular dissipative effects. Previous pdf model equations are examined in terms of cumulants evolution and reasons for their failure are diagnosed. A new noninteractive model is proposed, combining a linear mean square estimation (LMSE) deterministic subprocess affecting all the Monte Carlo particles, used to represent the pdf, and a binomial sampling acting on a fraction of them. The scalar lower and/or upper bounds are naturally considered in the formulation. For unbounded scalars, or when the scalar standard deviation is much smaller than the absolute value of the difference between the bounds and the scalar mean, the binomial sampling tends to a Gaussian one. ...

Local flow topologies and scalar structures in a turbulent premixed flame

A three-dimensional direct numerical simulation of a propagating turbulent premixed flame is performed using one-step Arrhenius model chemistry. The interaction of the flame thermochemical processes with the local geometries of the scalar field and flow topologies is studied. Four regions (“fresh reactants,” “preheating,” “burning,” and “hot products”), characterized by their reaction rate and mass fraction values, are examined. Thermochemical processes in the “preheating” and “burning” regions smooth out highly contorted iso-scalar surfaces, present in the “fresh reactants,” and annihilate large curvatures. Positive volumetric dilatation rates, −P = ∇ · u, display maxima for elliptic concave and minima for convex scalar micro-structures. Constant average tangential strain rates, aT, with large fluctuations, occur throughout the flow domain, whereas normal strain rates, aN, follow the trends of volumetric dilatation rates. Focal topologies, present in the “fresh reactants,” tend to disappear in favor of n...

STATISTICAL DESCRIPTION OF THE TURBULENT MIXING OF SCALAR FIELDS

A formulation in terms of probability density function (PDF) transport equations is presented for inert and reactive scalar fields undergoing turbulen mixing. The PDF methodology is related to the classical moment equations. The hierarchy of PDF transport equations resembles the BBGKY equations in statistical mechanics. Closure hypothesis, approximating the molecular mixing term, are described and their predictions for simple systems are compared with direct numerical simulations (DNS). Solution algorithms in terms of Monte Carlo particles are also discussed.