Laminar diffusion flamelet models in non-premixed turbulent combustion

The laminar flamelet concept covers a regime in turbulent combustion where chemistry (as compared to transport processes) is fast such that it occurs in asymptotically thin layers—called flamelets—embedded within the turbulent flow field. This situation occurs in most practical combustion systems including reciprocating engines and gas turbine combustors. The inner structure of the flamelets is one-dimensional and time dependent. This is shown by an asymptotic expansion for the Damkohler number of the rate determining reaction which is assumed to be large. Other non-dimensional chemical parameters such as the nondimensional activation energy or Zeldovich number may also be large and may be related to the Damkohler number by a distinguished asymptoiic limit. Examples of the flamelet structure are presented using onestep model kinetics or a reduced four-step quasi-global mechanism for methane flames. For non-premixed combustion a formal coordinate transformation using the mixture fraction Z as independent variable leads to a universal description. The instantaneous scalar dissipation rate χ of the conserved scalar Z is identified to represent the diffusion time scale that is compared with the chemical time scale in the definition of the Damkohler number. Flame stretch increases the scalar dissipation rate in a turbulent flow field. If it exceeds a critical value χ q the diffusion flamelet will extinguish. Considering the probability density distribution of χ , it is shown how local extinction reduces the number of burnable flamelets and thereby the mean reaction rate. Furthermore, local extinction events may interrupt the connection to burnable flamelets which are not yet reached by an ignition source and will therefore not be ignited. This phenomenon, described by percolation theory, is used to derive criteria for the stability of lifted flames. It is shown how values of ∋ q obtained from laminar experiments scale with turbulent residence times to describe lift-off of turbulent jet diffusion flames. For non-premixed combustion it is concluded that the outer mixing field—by imposing the scalar dissipation rate—dominates the flamelet behaviour because the flamelet is attached to the surface of stoichiometric mixture. The flamelet response may be two-fold: burning or non-burning quasi-stationary states. This is the reason why classical turbulence models readily can be used in the flamelet regime of non-premixed combustion. The extent to which burnable yet non-burning flamelets and unsteady transition events contribute to the overall statistics in turbulent non-premixed flames needs still to be explored further. For premixed combustion the interaction between flamelets and the outer flow is much stronger because the flame front can propagate normal to itself. The chemical time scale and the thermal diffusivity determine the flame thickness and the flame velocity. The flamelet concept is valid if the flame thickness is smaller than the smallest length scale in the turbulent flow, the Kolmogorov scale. Also, if the turbulence intensity v′ is larger than the laminar flame velocity, there is a local interaction between the flame front and the turbulent flow which corrugates the front. A new length scale L G =v F 3 /∈ , the Gibson scale, is introduced which describes the smaller size of the burnt gas pockets of the front. Here v F is the laminar flame velocity and ∈ the dissipation of turbulent kinetic energy in the oncoming flow. Eddies smaller than L G cannot corrugate the flame front due to their smaller circumferential velocity while larger eddies up to the macro length scale will only convect the front within the flow field. Flame stretch effects are the most efficient at the smallest scale L G . If stretch combined with differential diffusion of temperature and the deficient reactant, represented by a Lewis number different from unity, is imposed on the flamelet, its inner structure will respond leading to a change in flame velocity and in some cases to extinction. Transient effects of this response are much more important than for diffusion flamelets. A new mechanism of premixed flamelet extinction, based on the diffusion of radicals out of the reaction zone, is described by Rogg. Recent progress in the Bray-Moss-Libby formulation and the pdf-transport equation approach by Pope are presented. Finally, different approaches to predict the turbulent flame velocity including an argument based on the fractal dimension of the flame front are discussed.

Laminar Flamelet Concepts in Turbulent Combustion

We focus attention on an idealized granular material comprised of identical, smooth, imperfectly elastic, spherical particles which is flowing at such a density and is being deformed at such a rate that particles interact only through binary collisions with their neighbours. Using general forms of the probability distribution functions for the velocity of a single particle and for the likelihood of binary collisions, we derive local expressions for the balance of mass, linear momentum and fluctuation kinetic energy, and integral expressions for the stress, energy flux and energy dissipation that appear in them. We next introduce simple, physically plausible, forms for the probability densities which contain as parameters the mean density, the mean velocity and the mean specific kinetic energy of the velocity fluctuations. This allows us to carry out the integrations for the stress, energy flux and energy dissipation and to express these in terms of the mean fields. Finally, we determine the behaviour of these fields as solutions to the balance laws. As an illustration of this we consider the shear flow maintained between two parallel horizontal plates in relative motion.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112083001044

A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles

Molecular Dynamics

The kinetics and microscopic mechanisms of laser melting and disintegration of thin Ni and Au films irradiated by a short, from 200 fs to 150 ps, laser pulse are investigated in a coupled atomistic-continuum computational model. The model provides a detailed atomic-level description of fast nonequilibrium processes of laser melting and film disintegration and, at the same time, ensures an adequate description of the laser light absorption by the conduction band electrons, the energy transfer to the lattice due to the electron-phonon coupling, and the fast electron heat conduction in metals. The interplay of two competing processes, the propagation of the liquid-crystal interfaces (melting fronts) from the external surfaces of the film and homogeneous nucleation and growth of liquid regions inside the crystal, is found to be responsible for melting of metal films irradiated by laser pulses at fluences close to the melting threshold. The relative contributions of the homogeneous and heterogeneous melting mechanisms are defined by the laser fluence, pulse duration, and the strength of the electron-phonon coupling. At high laser fluences, significantly exceeding the threshold for the melting onset, a collapse of the crystal structure overheated above the limit of crystal stability takes place simultaneously in the whole overheated region within \ensuremath{\sim}2 ps, skipping the intermediate liquid-crystal coexistence stage. Under conditions of the inertial stress confinement, realized in the case of short $\ensuremath{\tau}l~10\mathrm{ps}$ laser pulses and strong electron-phonon coupling (Ni films), the dynamics of the relaxation of the laser-induced pressure has a profound effect on the temperature distribution in the irradiated films as well as on both homogeneous and heterogeneous melting processes. Anisotropic lattice distortions and stress gradients associated with the relaxation of the laser-induced pressure destabilize the crystal lattice, reduce the overheating required for the initiation of homogeneous melting down to $T\ensuremath{\approx}{1.05T}_{m},$ and expand the range of pulse durations for which homogeneous melting is observed in 50 nm Ni films up to \ensuremath{\sim}150 ps. High tensile stresses generated in the middle of an irradiated film can also lead to the mechanical disintegration of the film.

Combined atomistic-continuum modeling of short-pulse laser melting and disintegration of metal films

A novel fluid-transport calculation by computer simulation, via nonequilibrium molecular dynamics, of laboratory methods of transport measurement is described. Shear viscosity of soft-sphere (${r}^{\ensuremath{-}12}$ potential) and Lennard-Jones particles (${r}^{\ensuremath{-}12}\ensuremath{-}{r}^{\ensuremath{-}6}$ potential) has been obtained from molecular dynamic modeling of Couette flow. Soft-sphere deviations from Enskog theory are similar to those found for hard spheres by Alder, Gass, and Wainwright, using time-correlations of equilibrium molecular dynamic system fluctuations. For the Lennard-Jones shear viscosity near the triple-point region, there is agreement between the equilibrium calculation of Levesque, Verlet, and Kurkijarvi and the nonequilibrium results using 108 atoms in a cube. However, systems two and three cubes wide give lower results, which, when extrapolated with inverse width, yield close agreement with the experimental argon shear viscosity. Comparison of the Lennard-Jones shear viscosity with experimental argon data along the saturated vapor-pressure line of argon confirms our successful simulation of macroscopic viscous flow with few-particle nonequilibrium molecular dynamic systems. A new result of the nonequilibrium molecular dynamics is the characterization of nonequilibrium distribution functions, which might provide the basis for a perturbation theory of transport. Since momentum transport is primarily accomplished by the repulsive potential core for high temperatures, the Lennard-Jones shear viscosity must behave like the soft-sphere system for high temperatures [viscosity divided by ${(\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e})}^{\frac{2}{3}}$ is a function of density divided by ${(\mathrm{temperature})}^{\mathrm{\textonequarter{}}}$]. In fact, the calculated excess shear viscosity (that part above the zero-density temperature dependence) has been successfully correlated in terms of the 12th-power scaling variables for temperatures as low as the critical value (along the freezing line). The utilization of soft-sphere scaling variables yields relatively simple functions for describing both the excess shear viscosity and the thermal-conductivity behavior throughout the fluid phase. The introduction of these scaling variables also clearly reveals two features: (i) weak temperature dependence, and (ii) the sign of the temperature derivative at constant density (negative for shear viscosity and positive for thermal conductivity). While both of these features have been experimentally observed in simple fluid experimental data, their cause has not been previously traced to the dominance of the core potential. Thus, the soft-sphere scaling variables should be useful for correlating experimental data.

Dense-fluid shear viscosity via nonequilibrium molecular dynamics

The evolution of the two- and three-dimensional structures in a temporally growing plane shear layer is numerically simulated with the discrete vortex dynamics method. We include two signs of vorticity and thus account for the effect of the weaker boundary layer leaving the splitter plate which is used to create a spatially developing mixing layer. The interaction between the two layers changes the symmetry properties seen in a single vorticity-layer calculation and results in closer agreement with experimental observations of the interface between the two streams. Our calculations show the formation of concentrated streamwise vortices in the braid region between the spanwise rollers, whereas the spanwise core instability is observed to grow only initially. Comparison with flow visualization experiments is given, and we find that the processes dominating the early stages of the mixing-layer development can be understood in terms of essentially inviscid vortex dynamics.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112088000928

Three-dimensional shear layers via vortex dynamics

With a Biot-Savart model of vortex filaments to provide initial conditions, a finite difference scheme for the incompressible Navier-Stokes equation is used in the region of closest approach of two vortex rings. In the Navier-Stokes solution, we see that the low pressure which develops between the interacting vorticity regions causes the distortion of the initially circular vortex cross section and forces the rearrangement of vorticity on a convective time scale which is much faster than that estimated from viscous transport.

/pdf/numerical-study-of-vortex-reconnection-3j7j4d4zmp.pdf

Numerical study of vortex reconnection.

The mathematical analogy between the elastic stress due to particle displacements in Hooke's law solids and the viscous stress due to velocity gradients in incompressible fluids correlates two interesting phenomena. In a two‐dimensional crystal the elastic restoring force opposing particle displacements approaches zero with increasing crystal size, leading to a logarithmically diverging rms displacement in the large‐system limit. The vanishing of the solid‐phase force is mathematically analogous to the lack of viscous damping for a particle moving slowly through a two‐dimensional incompressible fluid. These two continuum results are compared with discrete‐particle computer simulations of two‐dimensional solids and fluids. The divergence predicted by macroscopic elasticity theory agrees quantitatively with computer results for two‐dimensional harmonic crystals. These same results can also be correlated with White's experimental study of the viscous resistance to a cylinder (a falling wire) moving slowly th...

/pdf/two-dimensional-computer-studies-of-crystal-stability-and-55qtsarmpp.pdf

Two‐dimensional computer studies of crystal stability and fluid viscosity

One-dimensional turbulence (ODT), a method for one-dimensional stochastic simulation of turbulent flow, is generalized to incorporate variable-density effects. This formulation is used to investigate variable-density effects in planar mixing layers. Computed results are compared to direct numerical simulations of temporally developing mixing layers and to measurements performed in spatially developing mixing layers. Dependencies of mean flow structure and fluctuation statistics on the free-stream density ratio s are examined, including s values beyond the range of previous experimental and computational studies. For temporally developing mixing layers, the previously noted decrease of the layer growth rate as s deviates from unity is reproduced. For spatially developing mixing layers, dependence on s is sensitive to whether the high-speed or the low-speed stream is denser; by convention, the latter case corresponds to s>1. Experimental results indicating that layer growth is an increasing function of s have previously been interpreted on the basis of models that imply the continuation of this monotonic trend for all s. ODT reproduces the observed trend within the experimentally accessible range, but predicts a reversal of the trend slightly beyond that range and a subsequent decrease of the growth rate as s increases. This and related results suggest a closer analogy between the behaviors of temporally and spatially developing mixing layers than has previously been recognized. An experimental test of the predicted trend reversal in spatially developing mixing layers is proposed.

Wm. T. Ashurst

Papers

Dense-fluid shear viscosity via nonequilibrium molecular dynamics

Three-dimensional shear layers via vortex dynamics

Numerical study of vortex reconnection.

Two‐dimensional computer studies of crystal stability and fluid viscosity

One-dimensional turbulence: Variable-density formulation and application to mixing layers