Surface Density Equation Research Articles

Modeling and LES of high-pressure liquid injection under evaporating and non-evaporating conditions by a real fluid model and surface density approach

Numerical modeling of high-pressure liquid fuel injection remains a challenge in various applications. Indeed, experimental observations have shown that injected liquid fuel jet undergoes a continuous change of state from classical two-phase atomization and spray droplets evaporation to a dense-fluid mixing phenomenon depending on the ambient pressure, temperature, and fuel properties. Accordingly, a predictive and efficient computational fluid dynamics (CFD) model that can represent the possible coexistence of subcritical and supercritical regimes during the fuel injection event is required. The widely used Lagrangian Discrete Droplet Method (DDM) requires parameter tuning of model constants and cannot model the dense near-nozzle region. Meanwhile, the high computational cost of Interface Capturing Methods (ICM) has prohibited their application to industrial cases. Thus, another alternative is an Eulerian Diffuse Interface Model (DIM), where the unresolved interface features are modeled instead of being tracked. Accordingly, the current work proposes a fully compressible multi-component two-phase real-fluid model (RFM) with a diffused interface and closed by a thermodynamic equilibrium tabulation method based on a real-fluid equation of state. The RFM model is complemented with a postulated surface density equation for fuel atomization modeling within the Large Eddy Simulation (LES) framework. The Engine Combustion Network (ECN) Spray A injector non-evaporating and nominal evaporating conditions are used as a reference for the proposed model validation. Simulations are performed using the proposed RFM model that has been implemented in the CONVERGE CFD solver. Under the non-evaporating condition, the RFM model can capture well the fuel mass distribution in the near-nozzle field, but also the interfacial surface area. Besides, the predicted drop size from simulations falls within the experimental data range. On the other hand, under the evaporating condition, spray liquid and vapor penetrations and fuel mixture fraction distribution are also accurately predicted. The vaporization effect on the surface area density is revealed to enhance surface generation in the dense spray region while reducing the surface density in the dilute spray region. The mean droplet size is also relatively reduced under the evaporating condition in the diluted spray region. Overall, the accuracy and computationally efficiency of the proposed RFM model coupled with the surface density equation for high-pressure fuel injection modeling are confirmed, allowing its use for high pressure industrial configurations in future studies.

Effects of Karlovitz Number on Flame Surface Wrinkling in Turbulent Lean Premixed Methane-Air Flames

ABSTRACTDirect numerical simulations are carried out to study the effects of the Karlovitz number (Ka) on the surface wrinkling of lean methane-air flames of equivalence ratio 0.5 and non-dimensional turbulence intensities (urms/SL) of 2–25. The mixture pressure is 20 bar and temperature is 810 K to simulate approximate conditions in lean-burn natural gas engines. The Karlovitz number varies from 1.1–49.4, and the Damköhler (Da) number from 0.26–3.2, corresponding to turbulent premixed combustion in the thin reaction zone regime. With increasing Ka, there is increased probability of forming highly curved surfaces locally. The distribution of the shape factor of the local surface does not vary with Ka. The surface-averaged tangential strain rate is proportional to the Ka for flames in the thin reaction zone regime. By examining the flame displacement speed on the flame surface, it is found that on a highly curved surface, irrespective of whether it is convex or concave, flame stretch becomes negative due to the negative propagation term. The functional dependence of the conditional mean flame displacement speed on curvature is only weakly dependent on Ka and the shape factor. The physical insights derived from the analysis are applied to the modeling of the flame surface density equation.

0D modeling aspects of flame stretch in spark ignition engines and comparison with experimental results

With the different kinds of fuel now available, modern spark ignition engines have to be adapted, owing not only to the difference between the characteristic heat of combustion of the different fuels, but also to the response of the flame to stretch. As the burning rate is a function of the laminar burning speed which is a function of the flame stretch, this parameter has to be taken into account by combustion models.In this study, a zero-dimensional combustion model, based on the Flame Surface Density equation, is enhanced with a model for the stretched laminar burning speed. Numerical results are compared with the experimental results of three lean air-fuel mixtures with isooctane, propane and methane as fuels, that have similar unstretched laminar burning speeds but different Lewis numbers. Laminar burning speed, flame radius, burnt mass fraction, and turbulent flame wrinkling are compared with experimental results. The simulation trends are similar to those of experimental results. Methane and propane (Lewis number 0.99 and 1.80 respectively) show similar wrinkling rates, while isooctane (Lewis number 2.90) has a lower wrinkling rate. The observed difference between computed burnt mass fraction with stretched and unstretched flame speed models reveals that the impact of stretch and Lewis number needs to be taken into account and that a stretched laminar burning speed must be used for modeling, especially for mixtures with a Lewis number much greater than unity.

Classical Solvability of a Free Boundary Problem for an Incompressible Viscous Fluid with a Surface Density Equation

We investigate a mathematical model introduced by Shikhmurzaev to remove singularities that arise when classical hydrodynamic models are applied to certain physical phenomena. The model is described as a free boundary problem consisting of the Navier-Stokes equations and a surface mass balance equation. We prove the local-in-time solvability in Hölder spaces.

Identifying deficiencies of standard accretion disc theory: lessons from a mean-field approach

Turbulent viscosity is frequently used in accretion disk theory to replace the microphysical viscosity in order to accomodate the observational need for in- stabilities in disks that lead to enhanced transport. However, simply replacing the microphysical transport coefficient by a single turbulent transport coeffi- cient hides the fact that the procedure should formally arise as part of a closure in which the hydrodynamic or magnetohydrodynamic equations are averaged, and correlations of turbulent fluctuations are replaced by transport coefficients. Here we show how a mean field approach leads quite naturally two transport coefficients, not one, that govern mass and angular momentum transport. In particular, we highlight that the conventional approach suffers from a seemingly inconsistent neglect of turbulent diffusion in the surface density equation. We constrain these new transport coefficients for specific cases of inward, outward, and zero net mass transport. In addition, we find that one of the new transport terms can lead to oscillations in the mean surface density which then requires a constant or small inverse Rossby number for disks to maintain a monotonic power-law surface density.

The 3-Zones Extended Coherent Flame Model (Ecfm3z) for Computing Premixed/Diffusion Combustion

The Extended Coherent Flame Model of Colin et al. (2003) developed to model combustion in perfectly or partially mixed mixtures is adapted to also account for unmixed combustion. The ECFM model is based on a flame surface density equation which takes into account the wrinkling of the flame front surface by turbulent eddies and a conditioning averaging technique which allows precise reconstruction of local properties in fresh and burned gases even in the case of high levels of local fuel stratification. This model has been used with success in gasoline engines (Duclos et al., 1996; Duclos and Zolver, 1998; Lafossas et al., 2002; Henriot et al., 2003; Kleemann et al., 2003). In order to adapt the model to unmixed combustion for Diesel application, a description of the mixing state has been added. It is represented by three mixing zones: a pure fuel zone, a pure air plus possible residual gases zone and a mixed zone in which the ECFM combustion model is applied. A mixing model is presented which allows progressive mixing of the initially unmixed fuel and air. This new combustion model, called ECFM3Z (3-Zones Extended Coherent Flame Model), can therefore be seen as a simplified CMC (Conditional Moment Closure) type model where the mixture fraction space would be discretized by only three points. The conditioning technique is extended to the three mixing zones and allows to reconstruct, like in the ECFM model, the gas properties in the unburned and burned gases of the mixed zone. Application of the model to internal combustion engine calculations implies the necessity of auto-ignition modelling coupled to premixed and diffusion flames description. Auto-ignition is modelled following (Colin et al., 2004), while the premixed turbulent flame description is given by the ECFM. The diffusion flame is now accounted for thanks to the three zones mixing structure which represents phenomenologically the diffusion of fuel and air towards the reactive layer, that is the mixed zone. The ECFM3Z combustion model has already been presented (Beard et al., 2003) in a comparative work between Diesel experiments and corresponding calculations covering different engine operating points. Here, the model is presented in all its details and its behavior is analysed when the relative duration of injection and auto-ignition delay are varied in a direct injection Diesel engine. It is shown that the model is able to reproduce the relative importance of auto-ignition and diffusion flame on the total heat release, depending on the engine operating point considered.

Pocket formation and the flame surface density equation

The occurrence and properties of singularities in the equation for the surface density function σ≡|∇| are analyzed analytically and numerically using data from two-dimensional direct numerical simulation (DNS) of pocket formation in a premixed methane-air flame. The various stages and the relevant time scales associated with pocket formation were determined in a previous study. It was found that isolated pockets form if and only if a nondegenerate critical point of a saddle point type appears. The appearance of a singularity in the isoline representing the flame front may have implications to modeling of the terms in the surface density function (SDF) approach during such transient events as pocket formation. The sink and source terms in SDF are evaluated in the neighborhood of a critical point using DNS data during pocket formation and an analytic representation of a scalar in the vicinity of the critical point that allows for the computation of all kinematic properties. The analytic and computational results show that the normal restoration and dissipation terms in the SDF become singular at the critical point when the pocket emerges. Furthermore, the analytic results show that the singularities exactly cancel, and therefore, the main conclusion is that it is unnecessary to model the singular behavior of these terms at critical points. However, closure of their sum is recommended.

Premixed flame–wall interaction in a turbulent channel flow: budget for the flame surface density evolution equation and modelling

Turbulent premixed flame propagation in the vicinity of a wall is studied using a three-dimensional constant-density simulation of flames propagating in a channel. The influence of the walls is investigated in terms of the flamelet approach, where flamelet speed and flame surface density transport are used to describe the flame. The walls have constant temperature and lead to flamelet quenching for sufficiently small wall–flame distances. Starting from the exact evolution equation for the surface density of propagating interfaces (Trouvé & Poinsot 1994; Candel & Poinsot 1990; Pope 1988), a budget for the flame surface density equation is presented before, during, and after the interaction with the wall. Before the flame interacts with the wall, flame propagation is controlled by a balance between surface production and annihilation. During the interaction, high flame surface density gradients near the wall are responsible for the predominance of the transport terms. Closures of all terms of the flame surface density equation are proposed. These models are based on flamelet ideas and take into account wall effects. Enthalpy loss through the wall affects flamelet speed, flamelet annihilation and flame propagation. Decrease of turbulent scales near the wall affects turbulent diffusion and flame strain. This model is compared to DNS results using two types of tests: (i) a priori tests, where individual terms of the modelled flame surface density equation are compared to the terms of the exact interface density propagation equation, calculated with the DNS; (ii) a posteriori tests, where the final model is used to obtain total reaction rate, mean fuel mass fraction, heat flux at the wall and fuel mass fraction at the wall in the configuration used in the DNS. For both types of tests the model compares well with the DNS results.

Direct numerical simulation analysis of flame surface density equation in non-premixed turbulent combustion

In flame surface density models used in turbulent cmbustion, the flame is described as a collection of laminar flame elements quantified in terms of available flame surface per unit volume Σ ¯ . A modeled transport equation for this central quantity was first derived from phenomenological considenrations [6] for non-poremixed turbulent combustion. More recently, an exact but unclosed transport equation for the density of a propagating surface has been established, leading to improvements in premixed turbulent combustion modeling. The same general framework, together with concepts borrowed from Vervisch et al. [16] may be used to describe non-premixed turbulent combustion. In the opresent work, two equivalent transport equations for the flame surface density Σ ¯ corresponding to the stoichiometric iso‐level value of the mixture fraction Σ ¯ are analyzed. The first equation is similar to a PDF transport equation whereas the second one introduces the concept of an effective propagation speed. Flame surface density Σ ¯ and terms in Σ ¯ equations are examined using direct simulations of a two-dimensional spatial turbulent mixing layer. In a first step, our study is limited to a constant density flow without shear. Simple models are also proposed for the source and snk terms in the Σ ¯ equations.

Experimental analysis of flamelet models for premixed turbulent combustion

Flamelet models are now widely used to predict turbulent premixed combustion because they allow theseparation of chemical features from the description of the turbulent flow field. Some of them introduce a flame surface density (flame surface per unit volume) modelled either by an algebraic closure (Bray-Moss-Libby model) or from a transport equation. This equation may be exactly written but needs closure assumptions. In the present work, a two-dimensional turbulent premixed propane-air V flame is studied. Velocity profiles are obtained from laser-Doppler velocimetry. Flame front visualizations are achieved by tomography with high-speed cinematography using a copper vapor laser. Images are then processed to extract flame front characteristics (flame surface density, vector normal to the flame front, curvature). The Bray-Moss-Libby model is found to have a good trend but needs a precise closure of the flame wrinkling length scale. Then some terms of the exact transport equation for flame surface density are examined. A simple model is proposed to close the strain rate term acting on the flame surface and due to the mean flow. Curvature and propagation terms are found to act as a source term on the fresh gases side and as a consumption term on the burnt one. This fact points out a lack in the generally used closures of the flame surface density equation where the propagation term is neglected and the curvature term modelled as a destruction one.

Strain rate modelling for a flame surface density equation with application to non-premixed turbulent combustion

Flamelet models of turbulent combustion have been the subject of continuous interest because theydecouple the description of the flamelet structure from that of the turbulent flow field. While these concepts are now widely used to calculate premixed and non-premixed flames, many issues remain unsettled. Formulations based on a transport equation for the flame surface density, like that used in this article, require a set of closure rules that were previously defined from physical considerations. The objective of this article is to show how these rules may be improved by making use of direct numerical simulations. A refined description of the strain rate that effectively produces flame area is proposed and included in the model. Unsteady flamelet effects are also taken into account. Experimental data on flame surface density are obtained by processing a series of instantaneous planar laser-induced fluorescence (PLIF) images of a 2D non-premixed turbulent burner. Results from the model are compared with these experimental measurements of flame surface density. Their excellent agreement shows the accuracy of the new formulation of the strain rate.

Direct simulation and modeling of flame-wall interaction for premixed turbulent combustion

The interaction between turbulent premixed flames and walls is studied using a two-dimensional full Navier-Stokes solver with simple chemistry. The effects of wall distance on the local and global flame structure are investigated. Quenching distances and maximum wall heat fluxes during quenching are computed in laminar cases and are found to be comparable to experimental and analytical results. For turbulent cases, it is shown that quenching distances and maximum heat fluxes remain of the same order as for laminar flames. Based on simulation results, a “law-of-the-wall” model is derived to describe the interaction between a turbulent premixed flame and a wall. This model is constructed to provide reasonable behavior of flame surface density near a wall under the assumption that flame-wall interaction takes place at scales smaller than the computational mesh. It can be implemented in conjunction with any of several recent flamelet models based on a modeled surface density equation, with no additional constraints on mesh size or time step. Preliminary tests of this model are presented for the case of a spark-ignited piston engine.