A Near-Field Lagrangian Dispersion Model for Fuel Sprays
Given that conventional Lagrangian-Eulerian (LE) computations of sprays are typically performed with the near-field region almost completely unresolved, key physics associated with radial dispersion are not well captured. A new Near-field Lagrangian Dispersion Model (NFLDM) is presented to improve stochastic radial dispersion of the liquid phase. This Langevin-based model employs self-similar representations of mean velocity and Reynolds stress fields obtained from Volume-of-Fluid (VoF) simulations. A data-fitted self-similarity model is developed using VoF results from a range of spray simulations employing 2-4.5 MPa ambient pressure, multiple injection velocities, injections of n-dodecane and methanol, and the Engine Combustion Network Spray-A and Spray-D nozzle geometries. The NFLDM is first compared against the conventional LE model, where a given spray angle is typically imposed. In addition to correctly predicting the variation of liquid dispersion as a function of distance downstream of the injector, the NFLDM yields an approximate 50% reduction of error in comparison to the conventional LE treatment. It is shown that the radial transport of liquid in the near field is governed by stochastic fluid motions rather than by the mean radial flow. The NFLDM is further assessed over a range of conditions yielding satisfactory agreement with experimentally-validated VoF results.
Figure: Comparison of the near-field spray shape from the nozzle injection point to the end of the near-field ($x/D_{noz}=60$) for the case A1 conditions using VoF, the NFLDM, and conventional LE simulations.
A hybrid VoF Lagrangian-Eulerian Treatment for simulating sprays
The hybrid Volume-of-Fluid and Lagrangian–Eulerian (VoFLE) strategy is an attractive approach for reducing the computational cost of spray simulations while retaining a reasonable amount of fidelity. It is based on the concept of transitioning small liquid bodies or droplets to a Lagrangian–Eulerian (LE) representation, alleviating the burden of maintaining high resolution for small droplets. This hybrid VoFLEmethodology is extended in the present work by incorporating a hydrodynamic breakup model based on maximum entropy formalism (MEF). This approach is particularly suitable for realistic spray conditions, such as high-pressure fuel injectors, where adequate numerical resolution of the smallest droplets is extremely difficult. The first step in the present VoFLE treatment is the identification of unresolved liquid structures targeted for LE transition. This step is followed by the application of the MEF breakup model for those structures that are hydrodynamically unstable, resulting in the assignment of secondary drop sizes and velocities. The model is evaluated statistically and tested against experimental data from the Engine Combustion Network and the breakup of a water jet. Relatively favorable results are encountered in these tests.
Figure: External view of the evolution of the fuel injection process utilizing the ECN Spray D. The light blue iso-surface is the portion of the domain captured by the
VoF, and the dark blue particles are the Lagrangian elements.
A Maximum Entropy Formalism Model for the Breakup of a Droplet
A model for the prediction of the size and velocity distribution of daughter droplets created by the breakup of an unstable parent droplet is proposed. The basis of the model is the Maximum Entropy Formalism, which states that the most probable joint probability density function (JPDF) for the daughter droplet population is the one that maximizes the Bayesian entropy conditional on the enforcement of a set of constraints, which are the conservation laws for the problem. The result is a closed-form expression for the JPDF. Validation against experimental and Direct Numerical Simulations (DNS) data over the bag, multimode, and sheet-thinning breakup regimes is included. Predictions from the model show that the daughter droplet velocity distribution widens as the droplet size decreases. This result is due to a heightened sensitivity to drag force with lower droplet inertia and coincides with spray behavior. The velocity distribution is found to be near Gaussian. The model does not treat size and velocity as independently distributed, as generally assumed in the literature. In fact, marginal conditional densities derived from JPDF show that the distribution of size and velocity are interrelated.
Figure: Distributions of the daughter size and velocity distribution in (a) bag (W eG=20) (b) multimode (WeG=60) and (c) sheeting-thinning breakup regimes (WeG=100)
An Analysis of the Performance Enhancement with Adaptive Mesh Refinement for Spray Problems
Adaptive mesh refinement (AMR) provides an attractive means of significantly reducing computational costs while simultaneously maintaining a high degree of fidelity in regions of the domain requiring it. In the present work, an analysis of the performance of AMR supported by simulations is undertaken for liquid injection and spray formation problems. These problems are particularly challenging from a computational cost perspective since the associated interfacial area typically grows by orders of magnitude, leading to similar growth in the number of highly refined cells. While this increase in cell numbers directly contributes to a declining performance for AMR, a second less obvious factor is the decaying trend for the cell-based speedup. A theoretical analysis is presented, leading to a closed- form estimate for this cell-based speedup. It is shown that for spray formation problems, AMR performance is diminishing. Additional contributing sources are also examined, which include the role of load balancing and the choice of linear solvers for the Poisson system.
Figure: Evolution of the AMR grid corresponding to liquid injection at 300 m/s.
Evaluation and validation of LES sub-grid spray dispersion models
(In collaboration with C.-W. Tsang and C. Rutland) A sub-grid model accounting for the interaction of spray and sub-grid turbulence was developed and tested. The model predicts the sub-grid scale dispersion velocity used for calculating the slip velocity in Lagrangian–Eulerian Large-eddy simulation spray models. The dispersion velocity is assumed to be decomposed into a deterministic and a stochastic part, and it is updated in every turbulence correlation time for each computational parcel. The model was validated against two datasets: volume-of-fluid simulations and Engine Combustion Network experiments. The volume-of-fluid data showed that dispersion velocities at the centerline are anisotropic. This qualitative feature is well captured by the current model. For the Engine Combustion Network Spray A cases, it was found that sub-grid scale dispersion has profound impact on the prediction of the spatial distribution of liquid mass. Neglecting the sub-grid scale dispersion model results in underprediction of the width of the lateral projected liquid mass density profiles. Also, the prediction of the projected liquid mass density is sensitive to the two model constants determining the sub-grid scale dispersion velocity magnitude and turbulence time scale. However, the predictions of resolved gas-phase statistics are relatively insensitive to different sub-grid scale dispersion model setups. The primary reason for this was investigated. It was found that the motion of high-momentum liquid blobs in the near-nozzle region leading to air entrainment and subsequent gas jet development is minimally influenced by sub-grid scale dispersion. The importance of sub-grid scale dispersion inversely correlates with drag force magnitude: the larger the drag force, the less critical the sub-grid scale dispersion. Moving further downstream, quasi-equilibrium between the two phases is established, resulting in relatively small slip velocity and drag force.
Figure: Instantaneous PMD contours at 0.55?ms predicted by the different values of the model constants, Csig and Cturb, in the stochastic part of the SGS dispersion model.
Modeling Urea-Water Solution Droplet Evaporation
A computational solver has been developed for the calculation of UWS droplet vaporization. It is based on the solution of the mass density, chemical species transport, and energy within the droplet, and it is fully-coupled to the jump conditions for species and energy transport at the droplet interface, and the phase-equilibrium conditions. Pressure-Volume-Temperature relationships and fugacities are predicted using the Peng-Robinson Equation of State. The numerical code is validated by testing its ability to resolve the dynamics of internal species and temperature fields during phase change, predict phase equilibrium for UWS and hydrocarbon systems, and predict vaporization for C7H16, C7H16/C10H22, and UWS droplets. Results show that UWS droplet vaporization can be divided into three different phases consisting of: i) temperature rise at nearly constant composition, ii) overall urea enrichment at nearly constant temperature, and iii) simultaneous overall heating and urea enrichment of the entire droplet. The third phase is typically characterized by solidification of the gas-liquid interface, producing a urea shell, a state that can potentially lead to micro-explosions. Higher ambient temperatures are shown to promote urea solidification more readily than lower temperatures to the decreasing role of liquid species diffusion with increasing temperature.
Figure: Evolution of the distribution of thermodynamic state of the droplet at various ambient temperatures.
Large-eddy simulation of shear flows and high-speed vaporizing liquid fuel sprays
(In collaboration with C.-W. Tsang and C. Rutland) Large eddy simulation models are tested for use on high speed evaporating liquid sprays. Three models are tested: standard Smagorinsky, a one-equation viscosity based model, and the dynamic structure model. The models are tested using channel flow, planar gas jet, and a diesel spray. The motivation of the work is to evaluate the accuracy and suitability of LES models that can be used for internal combustion engine modeling with direct injection of fuel in which moderate mesh resolution is the norm. Results from the channel flow and the gas jet provide additional insight into model capabilities that impact performance in liquid spray simulations. Simulation results were analyzed using contour images to compare the general structure of the flows, and experimental data for comparison of ensemble averaged mean and rms velocity profiles, liquid and vapor penetration, and mass fraction profiles. Results show that all three models perform similarly in the channel flow with good matches on the mean velocity results and some under prediction of rms values. The dynamic structure model showed slightly steeper near wall rms values closer to the experimental data. In the jet flow, the one-equation model results showed unexpected flow structures in images of the velocity magnitude. Mean velocity profiles matched data well for all three models. But centerline velocity decay rates and rms velocity radial profiles matched data much better for the dynamic structure model results. In the liquid spray the one-equation model performed poorly when compared to experimental data. The Smagorinsky model gave reasonable results and the dynamic structure model gave very good results in these comparisons. The overall conclusion is that the dynamic structure model is the best of the three models for direct injection engine simulations.
Figure: Comparisons of the instantaneous temperature contours at 2.0 ms for the 0.5 mm mesh predicted by the (a) Smagorinsky model, (b) the one-equation model, and (c) the dynamic structure model, respectively. The ensemble-averaged temperature contour measured by the ECN spray experiment
Mass Loading Limits in Spray Desuperheating Applications
Desuperheating is defined as the cooling of superheated vapor, usually steam, and can be performed by mixing the vapor with saturated or subcooled liquid or by convecting the steam through a cooled wall environment. Desuperheating is essential for systems which need to regulate the temperature of superheated steam and is often used to protect downstream piping and equipment. It has widespread applications in desalination plants, power generation, food processing, and petrochemical fields.In the present work, an analytical expression is developed for the mass loading limit, defined as the limit beyond which liquid is unable to be vaporized in a general desuperheating system. This limit is subsequently compared to predictions originating from 3D numerical simulations based on a Lagrangian-Eulerian framework in combination with a RANS treatment for the vapor phase. The computations show that even for cases having much smaller mass loadings than the theoretical limit yield significant accumulation of liquid along the walls. Furthermore, the numerical findings presented in terms of streamwise profiles of mean droplet diameter, average vapor temperature, vapor-droplet slip velocity, and liquid mass show that the desuperheating process can be described as taking place in two distinct zones. In the first zone, located in the near-field, the flow process is characterized by vigorous liquid atomization and significant exchanges of mass, momentum, and energy between the liquid and vapor phases. In the second zone, which resides beyond the near-field, the desuperheating process displays a significantly reduced degree of vaporization, a near-equilibration of phasic velocities, and a milder change in the vapor temperature along the streamwise direction.
Figure: Results corresponding to 50% mass loading case showing averaged temperature field in (a) and instantaneous spray droplet colored by slip velocity in (b). Both images show a close up view of the thermal sleeve region and the main pipe section and clearly illustrate the reduction in local vapor temperature coincident with the spray plume.