Numerical simulation of evaporating diesel sprays. Jasper Van de Vyver

Transcription

1 Numerical simulation of evaporating diesel sprays Jasper Van de Vyver Supervisors: Prof. dr. ir. Jan Vierendeels, Prof. dr. ir. Sebastian Verhelst Counsellor: Gilles Decan Master's dissertation submitted in order to obtain the academic degree of Master of Science in Electromechanical Engineering Department of Flow, Heat and Combustion Mechanics Chair: Prof. dr. ir. Jan Vierendeels Faculty of Engineering and Architecture Academic year

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5 Numerical simulation of evaporating diesel sprays Jasper Van de Vyver Supervisors: Prof. dr. ir. Jan Vierendeels, Prof. dr. ir. Sebastian Verhelst Counsellor: Gilles Decan Master's dissertation submitted in order to obtain the academic degree of Master of Science in Electromechanical Engineering Department of Flow, Heat and Combustion Mechanics Chair: Prof. dr. ir. Jan Vierendeels Faculty of Engineering and Architecture Academic year

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7 Acknowledgement I would like to express my sincere gratitude towards everyone who made the realisation of this master s thesis possible. First off, I would like to thank professor Vierendeels and professor Verhelst for their assistance during fortnightly meetings. Without their suggestions, constructive feedback and insights in the matter, this master s thesis would not be what it has become. Furthermore, a special mention goes to my counsellor Gilles Decan whose work provided the basis for this master s thesis. He was always available to answer any questions, even during his research stay at Politecnico di Milano. His extensive feedback on my writing is greatly valued as well. I would also like to thank Dr. Tarek Beji and others who were sporadically present at my thesis meetings. The department of flow, heat and combustion mechanics deserves to be mentioned for providing me with the necessary software and computing power for my spray simulations. Yves Maenhout was always there to fix any occasional IT-related problem. The people who contributed to the ECN should also not be forgotten. Without their experimental data, this thesis would have looked very different. I am also very grateful to my parents, grandparents and sister. They have always supported me both mentally and financially during my studies and during the writing of this thesis. Last but not least, my thanks go out to my friends and fellow engineering students whom I could turn to for serious and less serious talks. iii

8 Permission for usage The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the copyright terms have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. Jasper Van de Vyver, 02/06/2017 iv

9 Numerical simulation of evaporating diesel sprays Jasper Van de Vyver Supervisors: Prof. dr. ir. Jan Vierendeels, Prof. dr. ir. Sebastian Verhelst Counsellor: Gilles Decan Master s dissertation submitted in order to obtain the academic degree of Master of Science in Electromechanical Engineering Department of Flow, Heat and Combustion Mechanics Chair: Prof. dr. ir. Jan Vierendeels Faculty of Engineering and Architecture Academic year Abstract and extended abstract v

10 Numerical simulation of evaporating diesel sprays Jasper Van de Vyver Ghent University, Belgium Supervisor: Prof. dr. ir. Jan Vierendeels, Prof. dr. ir. Sebastian Verhelst Counsellor: ir. Gilles Decan ABSTRACT Within the context of more stringent emission legislations, researchers are searching for ways to better understand pollutant formations in internal combustion engines. One of the primary factors in pollutant formation in compression ignition engines is the spray formation, since it determines the degree to which the fuel mixes with the air. Spray formation, however, is still a complex and not fully understood multiphase phenomenon. The increase in accuracy of measurement techniques and the increase in computing power go hand in hand to develop the knowledge about this topic. The main objective of this research is to provide a recommended approach to simulate diesel sprays in a computational fluid dynamics (CFD) software package within a reasonable calculation time. Therefore, the commercial CFD package ANSYS Fluent was used to simulate ECN spray A, a widely tested n-dodecane spray with publicly available experimental data. A Eulerian-Lagrangian phase model was used together with various spray submodels governing the evaporation, breakup, collision and drag forces of the fuel droplets. As breakup model, the KH-RT model was found to be the most suitable. The sensitivity to fluid properties as well as to parameters of the various spray submodels and turbulence models was identified. Based on this sensitivity analysis, the values of the surface tension, liquid density and the thermal conductivity of the gas mixture were made temperaturedependent. Additionally, the model s capabilities to predict parametric variations of ECN spray A were explored. All trends were reproduced with a fairly good agreement, but the model is unable to take into account different conditions in the nozzle. Future research will still have to investigate if this approach can be successfully extended to reacting conditions. It also remains to be seen that this approach scales up well to the dimensions of marine diesel engines, as well as dual fuel engines with premixed natural gas. INTRODUCTION Recently, the International Maritime Organisation (IMO) introduced the IMO NOx Tier III requirement [1], which severely limits the allowable NOx emissions for marine diesel engines in global shipping. Compared to the last iteration, the limit has gone down by about 75%. This means that engine tuning alone is not sufficient any more to meet these new standards. A possible short-term solution is to install dual fuel engines. This technology can be retrofitted to existing diesel engines by adding a natural gas tank and a natural gas port fuel injection system. The natural gas-air mixture is sucked into the cylinder before being compressed and ignited by a small pilot injection of diesel. In this way, the engine mainly runs on natural gas and the NOx emissions are significantly improved. Dual fuel operation has a complex combustion behaviour. This leads to quite a few unsolved issues primarily due to a lack of optimisation of dual fuel engines at low loads. A lot of experimental research is being done on dual fuel engines but also more fundamental research has to be done to fully optimise and understand this kind of combustion. To this end, Computational Fluid Dynamics (CFD) can be a useful tool. The in-cylinder flow field, spray formation, combustion and heat transfer can all be modelled to have a better understanding of what happens inside the cylinder for different operating conditions. However, a CFD simulation of an entire dual fuel engine is a very daunting task in terms of modelling and calculation time. The flow in an engine is highly reactive and turbulent and moves in a geometrically complex, moving mesh. Furthermore, the different processes in an engine have 1

11 a large variety of characteristic time constants. In an effort to reach this goal, various processes are studied separately. In this research, there will be focussed on diesel spray breakup at engine-like ambient conditions, in particular automotive engine-like ambient conditions rather than marine diesel engines. The choice for a spray representative of an automotive engine is linked to the extensive experimental database of ECN [2] and the amount of research done on ECN spray A. A diesel spray could be simulated in a cylinder, but the cyclic variations and the uncertainty on the boundary conditions ask for a different approach. Most of the time, research on spray breakup is done in constant volume combustion chambers. They have the added benefit of being optically accessible for measuring the spray characteristics. LIQUID PHASE REPRESENTATION The Eulerian- Lagrangian approach is frequently used by many authors[3, 4, 5, 6, 7]. It is also called a discrete phase model or DPM. In this approach, droplets of similar properties are grouped in discrete particles and are tracked along the mesh. Additional submodels govern the evaporation, breakup, drag forces, coalescence and shedding of those particles. In this way, they interact with the continuous gas phase. Each time step, the continuous phase flow field is calculated. This flow field determines the trajectory of the discrete phase particles and these, in turn, determine the new continuous phase source terms. This process is then repeated every time step. One of the main assumptions of this approach is that the volume fraction of liquid droplets in a cell stays low. This creates a possible source for errors in the near-nozzle region and imposes a lower limit on the used cell sizes. Lucchini et al. [5] remark that the grid size generally adopted is much larger than the nozzle diameter (about 2-5 times). The need for a relatively coarser mesh is at the same time an advantage. It provides an efficient way of representing the small droplets (in the order of 0.1 µm) naturally present in high-pressure fuel sprays. This makes it a frequently used approach. If a high accuracy in the near-nozzle region is desirable, there is a tendency to use a Eulerian-Eulerian approach. Here, both the gas and liquid phase have a continuous representation. The interface between the two is reconstructed based on an extra scalar quantity that is transported over the mesh. In the Level-Set method, this scalar quantity is the distance to the interface. The interface can then be reconstructed by connecting the cells for which this quantity is 0. Herrmann [8] used this method to study the primary atomisation of sprays. He determined that at least 6 grid points are needed to resolve a droplet to get a grid independent result. In other words, at least 6 3 cells are needed to properly represent a droplet. When comparing the size of droplets in diesel sprays to the dimensions of a combustion chamber, it becomes clear that meshes of millions of cells are needed for this approach. This does not allow to obtain a solution within a reasonable calculation time on a cluster with a dozen cores. DISCRETISATION The design of the mesh depends on the choice of a Lagrangian or a Eulerian approach for the liquid phase. In a Lagrangian approach, the smallest grid size will seldom go below the nozzle diameter, while for a Eulerian approach the smallest grid size is generally a fraction of the nozzle diameter. It is clear that for optimal use of the mesh and for the most accurate results, the region near the nozzle and around the spray axis should be the most refined, as it is in these regions that biggest velocity gradients occur. The degree of refinement of the mesh is also related to the time step. One can imagine that if the flow travels more than one cell per time step the accuracy can suffer. This corresponds to a Courant number C of 1. Explicit time stepping schemes even become unstable above a certain C max. C = u t x C max (1) Table 1: Courant numbers used by different authors for their fuel spray simulations Author x [mm] t [ns] u [m/s] C Som [4] ±575 ±1.15 Pei [9] Decan [7] Lucchini [5] N/A 0.15 Som et al.[4] and Lucchini et al.[5] used an adaptive mesh refinement (AMR) technique, so the smallest cells do not necessarily coincide with the cells with the fastest flow. The x given in the table is the smallest possible grid size. u is an estimate of the injection velocity of the spray. The velocity of the entrained gas will be a bit lower. The Courant number is thus the largest possible one. It can be concluded that common Courant numbers are in the order of 10 1 to 1. INJECTION One of the challenges of using a discrete phase model is how to link the nozzle flow to an appropriate diameter distribution at the injection. The Lagrangian way of tracking particles does not allow to represent a fully liquid flow, such as in the nozzle. Furthermore, the levels of turbulence and cavitation in the nozzle greatly influence the speed and diameters of the droplets. Increased levels of turbulence in the nozzle flow destabilise the jet and increase its breakup rate. The same goes for the cavitation. One of the most basic ways of modelling the injection is imposing a distribution function of droplet sizes at the nozzle exit. This model assumes that the primary breakup has already occurred at the nozzle exit. Considering the quick primary atomisation of high-pressure fuel sprays, this is not a very hefty assumption. For example, Pei et al. [9] used a uniform distribution and tuned its initial droplet diameter to obtain approximately correct liquid 2

12 lengths. The Rosin-Rammler distribution function is also sometimes used. Probably, injecting droplets of the same diameter as the nozzle is the most prevalent method. This method is referred to as the Blob method. The breakup of these big droplets is handled by the secondary breakup models, dispersing the spray in much finer droplets as can be seen in figure 1. Its simplicity while retaining fairly accurate results is its main advantage and it is therefore used by many authors [4, 10, 5, 6] in their fuel spray simulation studies. Figure 1: Breakup of blobs by secondary breakup models using the concept of liquid core length, figure from ANSYS Fluent Theory Guide [11] The speed of the ejected droplets can be determined by the conservation of mass if the mass flow rate ṁ(t) through the nozzle is known. If there is no cavitation in the nozzle, the mean velocity u(t) of the ejected droplets is given by ṁ(t)/(ρ l A). A is the cross-sectional area of the nozzle (assumed to be constant) and ρ l the liquid density. If the mass flow rate is not known, Bernoulli s equation can be used to determine the maximum theoretical speed from the pressure difference p over the nozzle. The real speed will be lower, as friction losses are not taken into account here. The real speed is often compared to the theoretical one and their ratio is called the discharge coefficient C d. The value can be calculated by estimating the friction losses or sometimes it is given for a particular nozzle. C d = ṁ ṁ theor = ρ lau 2 p, u theor = (2) ρ l Au theor ρ l There still exist plenty of other models, some of which are commonly used but not available in ANSYS Fluent. For example, there is the Huh-Gosman model as used by various researchers [4, 5, 10] of which the first two used a modified version of the Huh-Gosman model called the Bianchi model. They both take into account the aerodynamically induced breakup by the Kelvin-Helmholtz instability and the turbulence induced breakup by introducing surface perturbations linked to the turbulent length scales of the flow in the nozzle. Other models also include the effect of cavitation, such as the KH-ACT model, developed by Som et al. [12]. SECONDARY BREAKUP MODELS The Wave, Kelvin- Helmholtz or KH model is a popular secondary breakup model for high-weber-number sprays, such as highpressure fuel sprays. Indeed, the high pressures lead to high injection velocities u and consequently also a high Weber number W e = ρ l u 2 d/σ. This model has been developed by Reitz and Diwakar [13] in 1987 and it has since been used by many people. For example, Som et al. [4] and Montanaro et al. [10] used it in their works. The Kelvin-Helmholtz instability describes the unstable behaviour of two fluids when there exists a relative speed difference between them. If this speed difference exceeds a critical value, the wave becomes unstable. It is an important breakup mechanism at the surface of fuel droplets. A common extension to the Wave model is to include the effect of the Rayleigh-Taylor instability. The model combining these two effects is called the KH-RT model. The RT instability is an unstable flow of two fluids, one of which is denser than the other. The less dense fluid pushes the heavier one and creates instabilities while doing so. This happens when a denser fluid sits atop a less dense fluid or when a less dense fluid is accelerated into a denser fluid. The latter is an important effect in the dilute region of the spray and the KH-RT model is therefore used by many researchers [4, 10, 14, 6]. COLLISION MODEL In a Lagrangian Particle Tracking approach, there is a possibility that particles collide. The outcome of such a collision can be that the particles coalesce and continue their way as one or that they bounce and continue their journey on a different path. Although high-pressure fuel sprays can have high collision rates, collision models are not used very often. Several authors [4, 9, 5, 7] declare not having used a collision algorithm, mainly because of its limited influence on the evaporation rate of a spray. TURBULENT DISPERSION Normally, the trajectory of Lagrangian particles in a flow field is calculated based on the mean flow velocities (u, v, w). However, in a real flow, the turbulent fluctuations create an additional dispersion of the particles. It would be better to calculate the trajectories based on the instantaneous flow velocities (u + u, v + v, w + w ), but these are not available in RANS models. In an effort to include this dispersion effect in spray simulations, some researchers [4, 9, 10] used turbulent dispersion models in their spray simulations. Lucchini et al. [5] chose not to include this effect in order to reduce the sensitivity of the results to the turbulence model. All those who did use a turbulent dispersion model chose for the stochastic tracking approach. SIMULATION SETUP ECN SPRAY A As already mentioned in the introduction, the choice for the studied case fell on ECN spray A. It is a spray representative for an automotive diesel engine with moderate exhaust gas recirculation. Liquid 3

13 n-dodecane at 363 K and 150 MPa is injected through a nozzle with a diameter of mm into a constant volume combustion chamber. The ambient gas in the combustion chamber has a density ρ a of 22.8 kg/m³, a temperature T a of 900 K and a pressure p a near 6.0 MPa. More specific boundary conditions can be found on the website of ECN [2]. In the 3rd ECN workshop [15], a fixed set of parametric variations of ECN spray A were defined. These will be used to validate the model. The ambient pressure p a, injection velocity u and mass flow rate ṁ were calculated with equation 2 and the ideal gas law with the gas constant of nitrogen R = 297 J/(kg.K). For the liquid density ρ l, a value between 701 and 706 kg/m³ was used to account for the density change due to the ambient pressure variation. way, but for a liquid fuel volume fraction of 0.1%.[16] The liquid penetration increased by about 10%, but this can be called a good result considering the difficulty in obtaining a grid-independent solution in spray simulations.[5] When a droplet exchanges momentum with a very small cell, the velocity of the continuous gas phase shoots up quickly. This leads to a lower relative velocity between the gas and liquid phase, and the spray penetrates further. Time steps smaller or equal to 0.4 µs produced quasiidentical results. This corresponds to a maximum Courant number of Table 2: Ambient temperature, density and injection pressure of the parametric variations of ECN spray A. The ambient pressure, injection speed and mass flow rate are calculated based on C d = T a [K] ρ a [kg/m³] p inj [MPa] p a [MPa] u[m/s] ṁ[g/s] MESH AND TIME STEP These experiments are carried out in constant volume combustion chambers. The one at Sandia National Laboratories is responsible for a sizeable part of the experimental data available on the website of ECN. Therefore, its geometry will serve as an inspiration for the calculation domain. It has a cubical-shaped combustion chamber with a characteristic length of 108 mm. Last year, Gilles Decan [7] already made a mesh for simulations of sprays in the GUCCI (Ghent University Combustion Chamber I). A rescaled version of his mesh was used in this work to simulate ECN spray A. Because of symmetry, the mesh represents a quarter of the combustion chamber. The zone around the spray axis is a structured mesh with a conical shape. The dimensions of the cells near the nozzle are mm in the axial direction, mm in the radial direction and 6 in the circumferential direction. This results in a total cell count of cells. Mesh dependency was checked by refining the structured part of the mesh to a total cell count of It barely altered the vapour penetration results. This is the maximum distance from the nozzle outlet to where the fuel vapour mass fraction is 0.1%. The liquid penetration can be defined in a similar Figure 2: Decan[7] Quarter cubic mesh as created by Gilles BREAKUP MODEL As a breakup model, the KH-RT model was used. It has two competing breakup mechanisms. B 1 can be lowered to increase the breakup rate due to the KH instability and C RT can be lowered to increase the breakup rate due to the RT instability. In both cases, the liquid penetration decreases due to the increased total surface area of the smaller droplets. After comparing with the experiments, a B 1 of 1.73 and a C RT of 0.3 gave a good agreement with the liquid penetration of the baseline condition of ECN spray A. These values were used to calculate all the parametric variations. Some cases were also calculated using a B 1 of 0.5 and a C RT of 0.5. For the baseline condition, this resulted in almost the same liquid length and thus increasing the relative importance of KH breakup. The results for the parametric variations look promising as explained in the results. TURBULENCE MODEL The realisable k-ɛ model was chosen by comparing the results of a wide range of turbulence models at standard settings. The choice has a profound effect on the entire spray formation, as can be seen in figure 3. The k-ɛ models behaved similarly except for the initial transient which was better captured by the realisable k-ɛ model. Despite the fact that the k-ω SST model should behave similarly as a k-ɛ model away from walls, the k-ω SST model severely underestimated the turbulent kinetic energy. This means that the spray penetrates further because it entrains less hot, stagnant air. The reduced mixing also leads to higher velocities, higher vapour mass fractions and lower temperatures on the spray axis, resulting in a lower evaporation rate and 4

14 an increased liquid penetration. On the other hand, the Spalart-Allmaras model heavily overestimated the turbulent viscosity ratio. It entrains so much air that it almost spreads out more quickly in the radial direction as in the axial direction. The Spalart-Allmaras model was never designed to simulate jet-like flows, so these inaccurate results were to be expected. For the realisable k-ɛ model, the vapour penetration could be brought within the limits of the experimental error by slightly decreasing C 2ɛ from 1.9 to C 2ɛ determines the rate at which the turbulent dissipation ratio ɛ decays. So for lower values of C 2ɛ, ɛ is higher, resulting in a lower turbulent kinetic energy and increased vapour penetration. VP [mm] Vapour penetration k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST experiment time ASOI [ms] Figure 3: Evolution of the vapour penetration in time COLLISION MODEL As expected, switching off the collision model did not alter the simulations results a lot, nor did it speed up the calculation time. Due to its positive influence on the convergence of the continuity equation, the O Rourke collision model was left enabled. TURBULENT DISPERSION Turbulent dispersion makes it easier for droplets to disperse away from the spray axis. This enhances the evaporation because more droplets end up in hotter neighbouring cells with a lower vapour mass fraction. When turbulent dispersion was disabled while keeping all other settings the same, the liquid length was almost doubled. It also slightly affected the vapour penetration, especially in the early stages of the spray (0.1 ms ms) when the droplets are still close to the vapour boundary. The results with turbulent dispersion (discrete random walk model) were closer to the experiments, so it was used in all further calculations. SENSITIVITY ANALYSIS A sensitivity analysis was performed by changing various fluid properties, initial conditions and other parameters to see which ones influenced the result the most so that extra care could be taken to provide accurate input. By looking at the equations of the predicted wavelengths of the KH-RT model, the wavelengths only scale approximately to the square root of the surface tension. However, at 60 bar, the surface tension drops quickly when it approaches its critical temperature of 659 K. [17] For this reason, surface tension values should be made temperature dependent. The viscosity only affects the KH breakup, so it had a rather small effect. The density of liquid n-dodecane varies between 704 kg/m³ and 436 kg/m³ in the temperature range 363 K K at a pressure of 60 bar. Drag and breakup will thus be more accurately predicted when a temperature-dependent density is used. The thermal conductivity of the air-fuel vapour mixture also influenced the liquid penetration quite a lot. A decrease of the thermal conductivity by a factor 4.5 increased the liquid penetration by 50%. DISCUSSION OF RESULTS The model as described in the last section was used to calculate the parametric variations of ECN spray A. The model constants and fluid properties remained unchanged, only the appropriate boundary conditions from table 2 were applied. The vapour penetration was well reproduced. The liquid penetration showed all the right trends but was sometimes quantitatively a bit off. Additionally, quite a few cases were lacking experimental data. Figure 4 shows the evolution of the vapour penetration for the first 3 cases in the table, i.e. a variation in injection pressure. The fuel vapour penetrates further into the domain for increasing injection pressures due to the increased momentum of the fuel leaving the nozzle. The simulation of 150 MPa was tuned to fit an experiment at MPa. The simulation at 50 MPa corresponds to an experiment at 53.9 MPa and the vapour penetration was mostly within the experimental error. The simulation at 100 MPa, however, corresponds to an experiment at 94.5 MPa which explains to a large extent why the vapour penetration is overestimated for that case. VP [mm] Vapour penetration 150 MPa 100 MPa 50 MPa experiments time ASOI [ms] Figure 4: Vapour penetration for different injection pressures Figure 5 shows the liquid penetration for the same range 5

15 of injection pressures. Experimentally, the liquid penetration stays almost constant due to two balancing effects. For lower injection pressures, the injection speed is lower, but the breakup and evaporation are also less intense. The latter effect seems more important in the simulation as in the experiments. A possible explanation is that in the simulation at lower pressures, the droplets themselves cannot create additional turbulence that improves the evaporation. In the simulation, the droplets act as momentum sources and turbulence is created by shear in the flow of the continuous gas phase, whereas in reality droplets can directly produce turbulent eddies. In figure 6, liquid penetration values for varying ambient density are presented. For decreasing ambient densities, the liquid penetration increases because there is less air hindering the movement of the spray. Drag forces and KH breakup decrease, but RT breakup increases. This could explain why the simulation is less sensitive to ambient density changes simulation experiment Introducing more temperature-dependent fluid properties such as the viscosity should increase the sensitivity of the model to changes in ambient temperature. Also, KH breakup and RT breakup do not respond in the same way to variations in ambient temperature and density. When the relative importance of KH breakup in the KH- RT model was increased, the liquid penetration changes due to variations in ambient density and temperature were more accurately followed. For the case with the lower ambient density of 7.6 kg/m³ the liquid penetration increased from 13.7 mm to 18.4 mm, and for the case with the lower ambient temperature of 700 K the liquid penetration increased from 11.8 mm to 13.2 mm. LP [mm] simulation experiment LP [mm] Ambient temperature [K] Figure 7: Liquid penetration for varying ambient temperatures Injection Pressure [MPa] Figure 5: Liquid penetration for different injection pressures LP [mm] simulation experiment Ambient density [kg/m³] Figure 6: Liquid penetration for varying ambient densities In figure 7, liquid penetration values for different ambient temperatures are shown. Here, the simulation also follows the right trend, while being less sensitive to changes in ambient temperature compared to the experiments. CONCLUSION The Eulerian-Lagrangian approach to spray modelling can accurately predict global quantities such as the vapour penetration and spray angle. These are quite insensitive to the breakup and are mainly determined by the total amount of momentum transferred to the gas phase. If local quantities near the nozzle are important, Eulerian phase models are better suited. The accuracy of Lagrangian phase models is determined by a rather complex interaction between various spray submodels. Some know-how is needed to tune various model constants in order to account for varying conditions in the nozzle. Although models exist which take into account the cavitation and turbulence in the nozzle [12], they are not yet available in a lot of CFD software packages. Additionally, it is hard to obtain a fully grid-independent solution for liquidrelated quantities. Experimental data for the parametric variations of ECN spray A is far from complete and additional experiments would benefit the modelling community. It also remains to be seen that the best approach developed for the ECN sprays scales up well to the dimensions of marine diesel engines. Later, this knowledge needs to be translated from marine diesel engines to dual fuel engines where the most notable difference is the presence of premixed natural gas. 6

16 Contents 1 Introduction IMO Tier 3 NOx limits Dual fuel internal combustion engines Working principle Advantages, disadvantages and current problems Computational Fluid Dynamics Governing equations Mass equation Navier-Stokes equations Energy equation Species equations RANS turbulence models Spalart-Allmaras model k-ɛ models k-ω SST model Reynolds Stress model Literature review CFD strategies in spray formation Eulerian-Lagrangian approach Eulerian-Eulerian approach Hybrid approach Common practice in the simulation of fuel sprays Discretisation, Mesh and Time step Turbulence model Injection and primary breakup model Secondary Breakup model Collision model Droplet drag model Evaporation model Turbulent dispersion Goal statement xii

17 4 Simulation setup and model validation ECN spray A Discretisation Dimensions and structure of the mesh Injection Pressure-Velocity Coupling Mesh dependence Time step dependence Choice of turbulence model Influence of fluid properties and initial conditions Sensitivity analysis Temperature-dependent fluid properties Proposed methodology for fuel spray simulations Discussion of results Spray penetration Evolution of the vapour boundary Vapour mass fraction distribution Streamlines and velocity field Momentum Temperature Near-nozzle region Volume fraction of liquid n-dodecane Droplet size distribution Breakup mechanisms Parametric variations Injection pressure Ambient density Ambient temperature Combination of ambient density and temperature Transition to reacting conditions 83 7 Conclusion 85 Bibliography 87 xiii

18 List of Figures 1.1 Regulations regarding NOx emissions issued by the IMO (figure (a) by dieselnet.org and figure (b) from IMO s Annex VI Working principle of a dual fuel engine (figure from targettrainingcentre.nl) Left: Mesh is fine enough to represent the droplet Right: Droplets smaller than the cell size cannot be represented [29] Adaptive mesh of Argonne National Laboratory at start of injection, used in the first ECN Workshop [31] Breakup of blobs by secondary breakup models using the concept of liquid core length, figure from ANSYS Fluent Theory Guide [11] Kelvin-Helmholtz instability: related to the existence of a difference in speed between two fluids [29] Comparison of the creation of child droplets due to the KH instability and RT instability [34] The constant volume combustion chamber at Sandia National Laboratories Quarter cubic mesh as created by Gilles Decan [7] Comparison of the liquid penetration in a 3D quarter cubic mesh and a 2D axisymmetric mesh Injection rate as given by the tool on the website of UPV compared to the mass flow rate obtained by applying the principle of Bernoulli and a C d of Influence on the liquid penetration of the B 1 breakup constant in the KH-RT model (C RT = 1) Influence on the liquid penetration of the C RT breakup constant in the KH-RT model (B 1 = 1.73) Effect of node based averaging on the liquid penetration Comparison between the Blob method (90 µm at 1 particle per parcel) and a uniform injection of 1 µm droplets at 500 particles per parcel Effect of Turbulent dispersion on the vapour boundary at 0.2 ms Liquid penetration for different levels of refinement of the mesh Vapour boundary at 1.44 ms for different levels of refinement of the mesh Liquid penetration for different time steps Vapour penetration for different time steps xiv

19 4.14 Effect of the turbulence model on the liquid penetration Effect of the turbulence model on various properties Liquid penetration and vapour boundary for different values of C 2ɛ Influence of various fluid properties and initial conditions on the liquid penetration Vapour and liquid boundary at 0.02 ms, 0.04 ms, 0.1 ms, 0.24 ms, 0.5 ms, 0.8 ms, 1.08 ms and 1.5 ms: blue line = vapour boundary of the simulation, black dotted line = experimental vapour boundary, edge of blue surface = liquid boundary of the simulation Vapour mass fraction distribution at 1.44 ms ASOI Pathlines coloured by velocity magnitude at 1.44 ms ASOI Momentum flow rate through planes orthogonal to the spray axis placed at different distances from the nozzle at 1.44 ms ASOI Temperature of the gas phase at 1.44 ms ASOI Temperatures of the discrete phase droplets at 1.44 ms ASOI Volume fraction of n-dodecane droplets in the cells near the nozzle at 1.44 ms ASOI Droplets coloured by particle diameter with a scaled diameter representation (top) and a constant diameter representation (bottom) Comparison of the WAVE breakup times (top) and KH-RT breakup times (bottom) Vapour penetration for different injection pressures Vapour boundary for different injection pressures at 1.44 ms ASOI Turbulent viscosity ratio and number of discrete phase particles along the spray axis at 1.44 ms ASOI Comparison of the liquid penetration at different injection pressures for simulation and experiment Comparison of the liquid penetration at different ambient densities for simulation and experiment Vapour penetration evolution and the vapour boundary at 1.44 ms ASOI for different ambient densities Velocity of the continuous gas phase and the discrete phase droplets along the spray axis at 1.44 ms ASOI for different ambient densities Turbulent viscosity ratio and density of the gas phase along the spray axis at 1.44 ms ASOI Comparison of the liquid penetration at different ambient temperatures for simulation and experiment Temperature of the gas phase and the fuel droplets along the spray axis at 1.44 ms ASOI Evolution of the vapour penetration for different ambient temperatures 80 xv

20 5.21 Evolution of the static pressure change in the combustion chamber for different ambient temperatures Comparison of the liquid penetration at different ambient densities for simulation and experiment Vapour penetration evolution and the vapour boundary at 1.44 ms ASOI for different ambient densities and temperatures Contours of constant temperature for ECN spray A in reacting conditions Contours of constant OH mass fraction for ECN spray A in reacting conditions List of Tables 1.1 Pros and Cons of Dual fuel engines as listed by Decan et al. [7] Courant numbers used by different authors for fuel spray simulations Most important specifications of the ECN Spray A baseline operating condition [2] Parametric variations of ECN spray A with the first one being the baseline condition [15] Cell count and dimensions of cells in the different zones of the mesh Ambient temperature, density and pressure, liquid density, injection velocity and mass flow rate calculated based on C d = 0.89 for the parametric variations of ECN spray A Standard breakup constants of the KH-RT model compared to the tuned breakup constants Change in values of various fluid properties and initial conditions for the sensitivity analysis Isobaric data of the density for dodecane at 60 bar [17] Temperature-dependent surface tension data for saturated dodecane [17] Other properties of liquid dodecane Droplet size statistics at 1.44 ms ASOI Ambient temperature, density and pressure, liquid density, injection velocity and mass flow rate calculated based on C d = 0.89 for some extra variations of ECN spray A xvi

21 Abbreviations and symbols Symbol Unit Description CFD Computational Fluid Dynamics x or x 1 m (distance to the nozzle along the) axial direction y or x 2 m (distance to the nozzle along the) radial direction z or x 3 m (distance to the nozzle along the) direction perpendicular to x and y u or u 1 m/s velocity along the x-direction v or u 2 m/s velocity along the y-direction w or u 3 m/s velocity along the z-direction ρ kg/m³ mass density ρ l kg/m³ mass density of the liquid phase ρ g kg/m³ mass density of the gas phase k m²/s² turbulent kinetic energy ɛ m²/s³ turbulent dissipation rate µ Pa.s (molecular) dynamic viscosity ν m²/s (molecular) kinematic viscosity µ t Pa.s turbulent dynamic viscosity ν t m²/s turbulent kinematic viscosity σ N/m surface tension c p J/(kg.K) heat capacity at constant pressure p inj Pa injection pressure p a Pa ambient pressure T a K ambient temperature ρ a kg/m³ ambient density ṁ kg/s mass flow rate RANS Reynolds-Averaged Navier-Stokes LES Large Eddy Simulation ASOI after start of injection KH Kelvin-Helmholtz RT Rayleigh-Taylor B 1 C RT model constant of KH breakup model (or WAVE) model constant of RT breakup model LP m liquid penetration or liquid length VP m vapour penetration ECN Engine Combustion Network xvii

22 Introduction IMO Tier 3 NOx limits The International Maritime Organisation (IMO) is a specialised agency of the United Nations. Their main goal is to create a set of global regulations to improve the safety, security and economical performance of international shipping. As of January 2016, they introduced the IMO NOx Tier III requirement.[1] This new iteration of their NOx limit for large marine diesel engines ( 130 kw) severely limits the allowable NOx emissions in certain emission control areas (ECAs) which are mostly found along the coasts of North America and Europe, as shown in figure 1.1b. Compared to Tier II, the limit has gone down by about 75% to 2.0 g/kwh for an engine with a rated speed above 2000 rpm and gradually going up to 3.4 g/kwh for rated engine speeds below 130 rpm as can be seen in figure 1.1a. This reflects the observation that at lower engine speeds generally more NOx is produced due to the higher available time for formation. Engine tuning alone is not sufficient any more to meet these new standards. (a)imo NOx limits with respect to rated engine speed (b)existing and possible future ECAs Figure 1.1: Regulations regarding NOx emissions issued by the IMO (figure (a) by dieselnet.org and figure (b) from IMO s Annex VI 1.2 Dual fuel internal combustion engines 1

23 1.2.1 Working principle To address the increasingly tougher emissions regulations, different solutions have to be considered. Besides long-term solutions, there is also a more immediate need to reduce emissions and improve the performance of the modern-day engines. One such solution is the use of dual fuel systems. It consists in mixing the air in the intake manifold with a high octane number fuel, such as alcohols or natural gas. This gaseous air-fuel mixture then gets compressed to finally be ignited with a small amount of pilot fuel. The pilot fuel has a low octane number, so it ignites spontaneously near the end of the compression stroke and further increases the heat and pressure in the cylinder to ignite the premixed air-fuel mixture. Most commonly diesel is used for this purpose. The entire process is schematically shown in figure 1.2. Figure 1.2: Working principle of a dual fuel engine (figure from targettrainingcentre.nl) The gaseous fuel is often natural gas because of its abundant availability and clean burning properties. Moreover, Yang et al. [18] mention that biogas, syngas and hydrogen are also sometimes used in dual fuel engines, but the high octane number of natural gas makes it a more suitable fuel for use in compression ignition engines due to their high compression ratio. The lower combustion rate and flame temperature also has a positive influence on the choice of natural gas. The goal is to mainly run on natural gas. When less diesel fuel is injected, NOx and PM emissions generally decrease but at a certain point combustion might become unstable. This amount is often fixed at around 10% of the mass flow rate at full load and stays constant at lower loads such as in the study of Wagemakers et al.[19]. This means that at lower loads a bigger relative amount of diesel is injected and less favourable emissions are obtained. By adding the natural gas circuit, existing diesel engines can be converted to a dual fuel engine at a relatively low cost and thus lowering the NOx emissions considerably to meet the demand of future emission standards. The engine also retains its capability to run in full diesel operation. Depending on the price of diesel 1.2 Dual fuel internal combustion engines 2

24 and heavy fuel oil, it can be economically preferable to operate solely on diesel outside of the ECAs and switch to dual fuel operation when entering the ECAs Advantages, disadvantages and current problems The explanation of the working principle already mentioned some advantages such as the flexible choice of fuel, the possibility of a retrofit on an existing diesel engine and reduced emissions. Dual fuel systems are most often used in marine diesel engines or other heavy-duty diesel engines where power-to-weight ratios are less important. Lately, the attention of the automotive sector has also increased, considering it could be an alternative to exhaust aftertreatment systems because it drastically reduces engine-out emissions. In this context, more advantages and disadvantages can be found. Decan et al. [7] listed the following pros and cons. Table 1.1: Pros and Cons of Dual fuel engines as listed by Decan et al. [7] Pros Cons - Allows retrofitting of current engines - Complexity of combustion - Low-cost retrofit - Lack of optimisation - Possibility of reduced PM and NOx emissions - Possibility of irregular / incomplete combustion and increased HC and CO emissions - Substitute diesel with plentiful natural gas - Issues present in gasoline engines appear - Similar performance to regular diesel - Higher specific fuel consumption engine - No need for additional maintenance It can be noticed that most of the cons are due to a lack of optimisation of dual fuel engines, especially at low load. Wagemakers et al.[19] explained that the higher emissions of hydrocarbons and CO are mainly caused by the over-lean fuel mixture at low loads. The lower temperatures and the lower amount of fuel can lead to unburnt fuel that gets left behind in crevice volumes in the cylinder and which later appears in the exhaust. Additionally, problems like knock and short circuiting of fuel from the inlet valve to the exhaust valve could appear when operating on dual fuel. The higher specific fuel consumption is again mainly at low loads. Srinivasan et al.[20] already found out early on that increasing the intake charge temperature is an effective strategy to improve combustion stability and emissions. However, methods to achieve this almost always need extra equipment such as throttle bodies, turbochargers or exhaust gas recirculation (EGR) systems. Other researchers, such as Yang et al.[18] looked at the ideal injection pressures and timing and natural gas injection timing. They concluded that delaying the natural 1.2 Dual fuel internal combustion engines 3

25 gas injection at low load creates a more stratified charge in the cylinder and thus improves emissions in most cases. A lot of experimental research is being done on dual fuel engines but also more fundamental research has to be done to fully optimise and understand this kind of combustion. To this end, Computational Fluid Dynamics (CFD) can be a useful tool. The in-cylinder flow field, spray formation, combustion and heat transfer can all be modelled to have a better understanding of what happens inside the cylinder for different operating conditions. However, a lot of factors influence the accuracy of a CFD calculation and everywhere models are used, modelling errors can occur. So apart from knowing the sensitivity of the CFD simulation to certain parameters, experimental validation of results is still needed. On the other hand, the benefits of CFD include the access to quantities which are otherwise unmeasurable and the ease at which initial conditions and boundary conditions can be changed. 1.2 Dual fuel internal combustion engines 4

26 Computational Fluid Dynamics 2 This chapter will give a brief introduction to Computational Fluid Dynamics or CFD. It will explain what its principles are while covering the governing equations and some turbulence models. It should provide a sufficient amount of background knowledge for those who are not so familiar with CFD to be able to follow the subsequent chapters. If the end of this chapter leaves you wanting to read more about CFD, the book of Versteeg et al. [21] is a good starting point. In their book, they define CFD as the analysis of systems involving fluid flow, heat transfer and associated phenomena such as chemical reactions by means of computer-based simulation. In other words, it tries to predict flow patterns and other related quantities such as the heat transfer, pressure differences and their associated forces on bodies in the flow,... A traditional example is trying to calculate the drag and lift of an aerofoil. Early attempts used the potential flow theory which made some heavy assumptions to be able to obtain an algebraic solution of the velocity field. It assumed the flow to be irrotational and inviscid (i.e. the effects of viscosity are neglected). Therefore, this theory is unable to represent vortices and boundary layers. The rise of the available computing power in the middle of the 20th century made it possible to numerically solve for the velocity field instead of assuming an inviscid, irrotational flow. As the computing power and algorithms progressed, more accurate solutions and increasingly complex geometries came within reach. Nowadays, CFD is used in a wide range of industries, ranging from aerospace and automotive to construction and the chemical process industry. It is even used in the healthcare industry to calculate blood flows in veins. 2.1 Governing equations The differential equations that define fluid flow are often called the transport equations because they express the conservation of several quantities in infinitesimal domains. These quantities are mass, the different momentum components (x and y in 2D and x, y and z in 3D) and energy. All these equations have the same basic form. The net change of the quantity in the volume is equal to the amount of this quantity that enters the volume minus what leaves the volume plus the net amount 5

27 of this quantity that gets produced in the volume. What leaves one volume enters other volumes and in this way the quantities get transported along the domain. The user of a CFD program needs to divide or discretise the domain into cells in a thoughtful manner, which is also called meshing. This allows the transport equations to be discretised into a system of algebraic equations. Most CFD programs use the finite volume method for this purpose. The resulting system of algebraic equations is then solved iteratively until a solution is obtained Mass equation The mass equation or the continuity equation is one of the basic conservation equations used in any CFD software. If the conservation of mass in an infinitesimal cube around a given point is expressed, the following equation is obtained. It is valid at any given point in an unsteady, compressible flow. or shorter, using tensor notation ρ t + (ρu) x + (ρv) + (ρw) = 0 (2.1) y z ρ t + (ρu i) = 0 (2.2) x i In these equations u or u 1, v or u 2 and w or u 3 denote the velocities along the x, y and z directions, respectively. When the density ρ in a point is constant, equation 2.2 simplifies to the divergence of the velocity vector being zero. u i x i = 0 (2.3) All flows which obey equation 2.3 are said to be incompressible Navier-Stokes equations The Navier-Stokes equations or the momentum equations use Newton s second law to state that the rate of momentum increase of a fluid particle is equal to the sum of forces acting on it. This includes forces due to a pressure difference, viscosity, gravity and other external body forces acting on the volume. The Navier-Stokes equations 2.1 Governing equations 6

28 given below are for a Newtonian fluid with a constant density ρ 0 and a constant kinematic viscosity ν. u i t }{{} variation + x j (u i u j ) }{{} convection = 1 ρ 0 p x i }{{} pressure + ν 2 u i + f i, x j x j }{{}}{{} body forces i = 1, 2, 3 (2.4) viscosity Energy equation Whereas the mass equation and the momentum equations are always included in CFD calculations, the energy equation can be optional if the flow is incompressible. In compressible flows, the energy equation is needed to determine the density in each point with the aid of an equation of state. The unsteady energy equation for a compressible fluid is given by t (ρe 0) + x j (ρu j e 0 ) = x j ( u j p q j + u i τ ij ) (2.5) In this equation e 0 is the total energy e + u ku k 2, q j is the heat flux calculated by the law of Fourier and τ ij is the viscous stress. For a Newtonian fluid τ ij is defined as follows. τ ij = µ ( ui + u j 2 x j x i 3 ) u k δ ij x k (2.6) Species equations In flows with different fluids, each fluid is tracked separately, the so-called species equations. The conservation of each species k results in the following equation t (ρy k) + (ρu i Y k ) = x i x i ( ρd k Y k x i ) + ω k (2.7) In which D k is the diffusion coefficient and ω k the source term for the species k. 2.2 RANS turbulence models A lot of flows in engineering applications are turbulent in nature. These chaotic 3D fluctuations make it hard to compare experiments that are seemingly carried out under the same circumstances. Therefore, one is often only interested in the mean values of a flow. This is the statistical approach to turbulence. In order to obtain these mean values, the Reynolds-Averaged Navier-Stokes (RANS) equations are used. They are derived by splitting up the velocity in its mean value u i and 2.2 RANS turbulence models 7

29 its fluctuating part u i, substituting this in the Navier-Stokes equations (equation 2.4) and time-averaging the entire equation. The result are equations that are very similar to the Navier-Stokes but with an added term. u i t + (u i u j ) = 1 p + ν x j ρ 0 x i 2 u i x j x j And similarly for the continuity equation (equation 2.3) x j (u i u j ) + f i, i = 1, 2, 3 (2.8) u i x i = 0 (2.9) ρ 0 u i u j is defined as the Reynolds stress tensor R ij. It represents the influence of the fluctuating velocities on the mean flow field. To close the system of equations, R ij has to be determined as a function of the mean velocities, mean pressure and other available quantities. In what follows, different RANS turbulence models will be introduced. They each close the system of equations in a different way. The models discussed here, except for the Reynolds Stress model, all rely on the Boussinesq hypothesis. He proposed that the Reynolds stress tensor is proportional to mean rates of deformation. In this way, turbulence can be modelled as an additional turbulent viscosity µ t (or eddy viscosity) that adds to the effect of the molecular viscosity. R ij = ρ 0 u i u j = µ t ( ui + u ) j 1 x j x i 3 ρ 0u i u i (2.10) RANS turbulence models are not the only way of modelling turbulence. Large eddy simulation and direct numerical simulation are briefly mentioned in section Spalart-Allmaras model The Spalart-Allmaras model [22] is one of the simplest and most robust turbulence models because it uses only one additional transport equation (equation 2.11). The variable ν is proportional to the turbulent viscosity, so it can be used to directly determine µ t in equation The model was developed with aerospace applications in mind, so it is commonly used to predict the flow over aerofoils. It produces quite accurate results in these applications, but in other applications the results can be quite inaccurate. The ANSYS Fluent User guide [23] even warns for relatively large errors in free shear flows, such as round and plane jets. ν t + u ν j = c b1 (1 f t2 ) x S ν j + 1 [ ( σ x j [ c w1 f w c b1 (ν + ν) ν x j ) κ 2 f t2] ( ν d ) 2 ] (2.11) ν ν + c b2 x i x i 2.2 RANS turbulence models 8

30 2.2.2 k-ɛ models The k-ɛ model is the most used turbulence model for general engineering applications in the industry. This rather old model still is a good compromise between its ease of use on the one hand and its performance on the other hand.[24] It uses two additional transport equations. One for the turbulent kinetic energy k = 1 2 u i u i and one for the turbulent dissipation rate ɛ. The turbulent kinematic viscosity is then based on the instantaneous values of these two quantities. ν t = C µ k 2 ɛ (2.12) where C µ is a model constant. The two transport equations are given here in vector notation for a fluid with a constant density and viscosity. Once again, it can be noticed that the transported quantities change by convection, diffusion, local production and destruction. k t [( k 2 ) ] + u. k =. ν + C K k + P ɛ (2.13) ɛ [ ɛ k 2 ] + u. ɛ =. C ɛ t ɛ ɛ + ɛ k (C 1ɛP C 2ɛ ɛ) (2.14) In these equations u is the mean flow velocity vector, P the local production term and the 5 C s are model constants. P = 1 2 C k 2 ( ui µ + u ) ( j ui + u ) j ɛ x j x i x j x i (2.15) The model constants are determined by experimentally comparing the results to basic flows. Commonly used values are: C µ = 0.09, C K = 1, C ɛ = 0.069, C 1ɛ = 1.44, C 2ɛ = 1.92 (2.16) Although the k-ɛ model performs well in a lot of applications, it still has a few shortcomings. Métais [24] states that the k-ɛ model is known to produce poor results in flows where normal stresses dominate the shear stresses, flows with strong streamline curvatures or flows with high levels of anisotropy. Examples of such flows include flows with a large adverse pressure gradient or flows in a rotating frame. The k-ɛ model is also known for the round jet / plane jet anomaly. Using the constants given above it can predict the flow field of a plane jet quite accurately, but when applied to an axisymmetric jet the spreading rate is severely overestimated. Because of its age and popularity, a lot of people have tried to improve the standard model. For example, the round jet / plane jet anomaly was solved in the realisable k-ɛ model by imposing some physical constraints on the solution. In the standard model u RANS turbulence models 9

31 could become negative, so one of the adjustments was to make C µ locally dependent on the mean flow field to guarantee that u 2 0 ( = condition for realisability). Another commonly used k-ɛ model is the RNG model, but as its principles do not help to further understand this thesis, the interested reader is referred to the literature k-ω SST model Not all two equation turbulence models use k and ɛ as the transported variables. The specific dissipation rate ω = ɛ/k can also be used instead of the dissipation rate ɛ. The result is a model that performs better near walls, but that is much more sensitive to the free stream boundary conditions. The idea of Menter [25] was to develop a hybrid model that switched between the k-ɛ model in free stream conditions and the k-ω model near the wall. In this manner, he got rid of the strong dependency of the free stream boundary conditions while retaining the superior performance of the k-ω model near the wall. In his SST (Shear Stress Transport) model, he also accounted for the transport of turbulent shear stress in the boundary layer by assuming the principal turbulent shear stress to be proportional to the turbulent kinetic energy. It is worth noting that when Menter applied his model on some free shear flows, the results were almost identical to the k-ɛ model.[25] Due to its versatile behaviour, the k-ω SST model has been successfully used in applications with wall-bounded flows, e.g., turbomachinery Reynolds Stress model The Reynolds Stress model is a 7 equation turbulence model. It has 6 transport equations for each of the components of the Reynolds stress tensor R ij = ρ 0 u i u j and a transport equation for ɛ similar to the one in the k-ɛ model (equation 2.14) but with a modified expression for P. The Reynolds Stress model does not use the eddy viscosity hypothesis (equation 2.10), so it is very well suited to handle anisotropic flows. However, this comes at a rather large additional computational cost. It has also proven itself to be less numerically stable than the aforementioned two equation turbulence models. 2.2 RANS turbulence models 10

32 Literature review CFD strategies in spray formation A CFD simulation of an entire dual fuel engine is a very daunting task in terms of modelling and calculation time. The flow in an engine is highly reactive and turbulent and moves in a geometrically complex, moving mesh. Furthermore, the different processes in an engine have a large variety of characteristic time constants. In an effort to reach this goal, various processes are studied separately. This thesis will focus on the diesel spray breakup at engine-like ambient conditions, in particular automotive engine-like ambient conditions rather than marine diesel engine-like ambient conditions. The choice for a spray representative of an automotive engine is linked to the extensive experimental database of ECN [2] (more info in section 4.1) and the amount of research done on ECN spray A. A diesel spray could be simulated in a cylinder, but the cyclic variations and the uncertainty on the boundary conditions ask for a different approach. Most of the time, research on spray breakup is done in constant volume combustion chambers. They have the added benefit of being optical accessible for measuring spray characteristics. In the next sections, different simulation approaches are discussed Eulerian-Lagrangian approach Most authors [3, 4, 5, 6, 7] use a Eulerian-Lagrangian approach. It is also called a discrete phase model or DPM. In this approach, droplets of similar properties are grouped in discrete particles and are tracked along the mesh. Additional models govern the evaporation, coalescence and shedding of those particles. In this way, they interact with the continuous gas phase. Each time step, the continuous phase flow field is calculated. This flow field determines the trajectory of the discrete phase particles and these in turn determine the new continuous phase source terms. This process is then repeated every time step. One of the main assumptions of this approach is that the volume fraction of liquid droplets in a cell stays low, generally less than 10 %. This creates a possible source for errors in the near-nozzle region and imposes a lower limit on the used cell sizes. Lucchini et al. [5] remark that the grid size generally adopted is much larger than the nozzle diameter (about 2-5 times). The need for a relatively coarser mesh is at the same time an advantage. 11

33 It provides an efficient way of representing the small droplets (in the order of 0.1 µm) naturally present in high-pressure fuel sprays. This makes it a frequently used approach Eulerian-Eulerian approach Some authors [26, 27, 28] use a Eulerian-Eulerian approach. Most of the time a Volume of Fluid (VOF) method is used. The transport equation 3.1 of the volume fraction α 1 of phase 1 governs the transport of the two phases, along with the conservation of momentum. The transport equation expresses the conservation of mass of phase 1. α 1 t + u α 1 x + v α 1 y + w α 1 z = 0 (3.1) with u,v and w the velocities according to the x, y and z-direction. The solution is a scalar field of volume fractions. The interface of the two phases is generally taken in the cells where the volume fraction is 0.5 (red line in figure 3.1). A general observation is the need of much smaller cell sizes. For being able to represent a droplet in a VOF model, the cell size must be smaller than the droplet. Figure 3.1: Left: Mesh is fine enough to represent the droplet Right: Droplets smaller than the cell size cannot be represented [29] As droplets in high-pressure diesel sprays can reach diameters smaller than 1 µm, one can easily imagine the large increase in cell count. Another model is the Level-Set method as used by Herrmann [8], but this method also experiences the same issues as VOF. The Level-Set method also expresses the conservation of a scalar quantity, but instead of the volume fraction of phase 1 a function that expresses the distance to the interface is used. The interface can then be reconstructed by the level surface for which the distance function is equal to 0. An advantage of the Level-Set method is that it is easier to reconstruct a smooth interface, but for very distorted surfaces the error on the conservation of mass increases.[29] Herrmann [8] determined that at least 6 grid points are needed to resolve a droplet to get a grid independent result. In other words, at least 6 3 cells are needed to properly represent a droplet. Ghiji [26] used a mesh of 20 million cells, a time step in the order of nanoseconds and a computational cluster with 384 cores. Even then simulations could take up to one month. These kinds of computational loads are out of the scope of this thesis, so 3.1 CFD strategies in spray formation 12

34 Eulerian methods will not be the main focus. With the presently available computing power, the main use of Eulerian methods is the validation of breakup models or for simulations including the needle and sac volume. The Eulerian approach solves the problem of being dependent on a breakup model, but it does not guarantee a mesh independent solution [26] Hybrid approach Sometimes, a Eulerian approach is used in the near-nozzle region and a Lagrangian approach outside of the liquid core region, as Bravo et al. [27] did in their highfidelity diesel spray simulation and Moriyoshi [30] in their study of a swirl-type injector. The coupling between the two is then done by an algorithm that detects small underresolved continuous liquid phase droplets and convert them to discrete phase droplets. The strategy that Bravo et al.[27] used, is to replace the continuous liquid phase droplets (contained within fewer than 5 3 contiguous cells) by discrete phase droplets of equal volume. The goal of this approach is to increase the precision in the near-nozzle region while retaining the advantages and disadvantages of the Lagrangian approach. The computational load lies between both approaches but it is still too big for the purpose of this thesis. Bravo et al. [27] reached a cell count of 60 million cells. 3.2 Common practice in the simulation of fuel sprays The result of a simulation in a CFD software package greatly depends on the used models and the temporal and spatial discretisation of the problem. This section will look at which models are commonly used in the literature to model fuel sprays using the Eulerian-Lagrangian approach. Special interest will be given to the models that are available in ANSYS Fluent, the CFD software package used in this thesis. One of the advantages of this software package is the availability of a comprehensive user manual and theory guide. So even while it is a commercial software package, the equations on which the different models are based are elaborately explained Discretisation, Mesh and Time step As already mentioned in the previous section, the design of the mesh depends on the choice of a Lagrangian or an Eulerian approach for the liquid phase. In a Lagrangian approach, the smallest grid size will seldom go below the nozzle diameter, while for a Eulerian approach the smallest grid size is generally a fraction of the nozzle diameter. It is clear that for optimal use of the mesh and for the most 3.2 Common practice in the simulation of fuel sprays 13

35 accurate results, the region near the nozzle and around the spray axis should be the most refined, as it is in these regions that biggest velocity gradients occur. Another peculiarity of meshes for fuel spray simulations is that a big part of the mesh sees no notable flow, especially in the first moments after injection. Therefore, adaptive mesh refinement (AMR) or adaptive local mesh refinement (ALMR) is sometimes used.[4, 10, 31] During simulation, cells for which a certain criterion is met, are divided into multiple cells (4 in 2D, 8 in 3D). The criterion compares a quantity in the cell to a predefined value. If the quantity becomes smaller or bigger than the predefined value, the cell is split. Possible quantities for fuel spray simulations are the velocity gradient, temperature gradient [31] or the fuel mass fraction [4]. The user can also specify a predetermined maximum number of levels of refinement or a maximum number of cells after which cells are no longer split. To ensure an accurate solution of the flow near the nozzle, a common choice is to permanently refine this region, regardless of the criterion. An example of this can be seen in figure 3.2. It shows the mesh that Argonne National Laboratory used for its contribution to the first ECN Workshop [31] at the start of injection. As the spray penetrates further into the domain, the refined region will grow, following the spray. In such a way, a pronounced improvement in calculation time can be noticed. Especially in the moments after the start of injection where the total number of cells is considerably lower as for the equivalent fixed grid. Lucchini et al. [5] report in their study an improvement in calculation time by a factor of 5 to 12. Figure 3.2: Adaptive mesh of Argonne National Laboratory at start of injection, used in the first ECN Workshop [31] Time steps in the order of 0.5 µs are most common for grids with the smallest cell size around 1 mm [31]. Decreasing grid sizes bring forth smaller time steps. This can be explained by looking at the Courant Friedrichs Lewy (CFL) condition. This 3.2 Common practice in the simulation of fuel sprays 14

36 condition needs to be satisfied in order for a partial differential equation problem to converge. In 1D, it takes the following form. C = u t x C max (3.2) In which C is the Courant number, u is the speed of the flow, x the length of the cell and t the time step. It only poses a strict limit when the time steps are solved explicitly. In such cases, C max is around 1 and depends on the spatial discretisation scheme. Implicit algorithms are less prone to numerical instabilities, so higher values of C max can be used.[21] Still, when decreasing the cell size, the time step is usually reduced as well, irrespective of the algorithm. In 3D, C is equal to the sum of the 1D cases along the different dimensions, but in the context of fuel sprays, the velocities perpendicular to the spray axis can be neglected with respect to the velocity along the spray axis. So for fuel sprays, equation 3.2 can still be used to determine the Courant number. Some articles in the literature provide enough information to calculate this number and in table 3.1 an overview is given of some Courant numbers used by different authors for fuel spray simulations with Lagrangian particle tracking methods. Table 3.1: Courant numbers used by different authors for fuel spray simulations Author x [mm] t [ns] u [m/s] C Mesh P-V coupling Som [4] ±575 ±1.15 3D PISO-SIMPLE Pei [9] D SIMPLE Decan [7] D Coupled Lucchini [5] N/A D PISO Som et al.[4] and Lucchini et al.[5] used an adaptive mesh refinement (AMR) technique, so the smallest cells do not necessarily coincide with the cells with the fastest flow. The x given in the table is the smallest possible grid size. u is an estimate of the injection velocity of the spray. The velocity of the entrained gas will be a bit lower. The Courant number is thus the largest possible one. Lucchini [5] explicitly stated that the AMR ensured a maximum Courant number of Common Courant numbers are in the order of 10 1 to 1. When using higher Courant numbers there is a tendency to use the generally more stable pressure-velocity coupling schemes, such as PISO and SIMPLE or even the combination of both for superior stability, but at the cost of added calculation time. When a good temporal discretisation is needed, one could opt for lower Courant numbers in combination with a coupled solver which is generally faster to converge but somewhat less stable. Still, most authors use PISO [4, 26, 5], SIMPLE [3, 9] or a combination of both. 3.2 Common practice in the simulation of fuel sprays 15

37 3.2.2 Turbulence model A lot of fluid mechanics problems of engineering interest have to handle turbulent flows which inherently have an unstable character. This is especially true for the flows inside the cylinder of an ICE. Due to the high velocities, the viscosity of the fluid is not able to dampen fluctuations and large turbulent eddies are formed. These turbulent eddies are unstable and split up in smaller eddies which in turn split up in even smaller eddies, carrying over their kinetic energy to smaller and smaller length scales until they get dissipated as heat by the viscous forces. To be able to represent the smallest eddies in a mesh, the mesh needs to be made very fine which is computationally very expensive. This strategy is called Direct Numerical Simulation (DNS). Another strategy is to explicitly resolve the large eddies, but model the turbulent eddies that are smaller than the cell sizes (also called the subgrid scales). This strategy called is Large Eddy Simulation (LES). The computational load is still relatively high, but LES calculations come within reach for a lot of applications and is becoming more and more popular. For complex geometries, the computational load of LES can be too high and that is why Reynolds-averaged Navier-Stokes (RANS) models are still used for modelling turbulence. RANS models only solve for the mean flow field and the effect of turbulence on this flow field is entirely modelled. This gives a significant reduction in computational load. Direct Numerical Simulation One option could be doing a Direct Numerical Simulation (DNS) of a fuel spray, but the use of DNS in spray breakup has been hindered by the available computing power. The high Reynolds number flow implies a mesh so fine it would be impossible to calculate a solution within a reasonable time frame, even with modern-day supercomputers. Just for resolving all the turbulent length scales of a low Reynolds number liquid jet, meshes of hundreds of millions of cells are needed. So, simulating the breakup of a high-pressure diesel spray is not feasible using DNS, although the result would be largely independent of the various models currently used in spray breakup. Reynolds-averaged Navier-Stokes The generally high computational load of DNS has led to the widespread use of turbulence models. Modelling the turbulence allows for much coarser meshes. The RANS equations were developed for this purpose. The turbulent flow field is split up into a mean part and a part due to the turbulent fluctuations. This is more 3.2 Common practice in the simulation of fuel sprays 16

38 elaborately explained in section 2.2. There are different models to determine the strength of the fluctuating part and its influence on the mean flow field. One of the most popular models is the k-ɛ model as explained in section Th is due to its relative simplicity and fairly accurate results in a lot of situations. RANS equations in general also allow for the use of symmetry, reducing the computational load even further. As only the mean flow field is solved, there is no need for a full 3D simulation to resolve the effect of the turbulence, which inherently creates a 3D flow. So when using RANS for fuel spray simulations, a quarter cube 3D domain or even a 2D axisymmetric domain could be used. It is generally agreed upon that k-ɛ is also the most widely used turbulence model for spray simulations. It is used by a lot of authors in all its different varieties (standard [4, 9, 6], realisable [3, 7] or RNG [4]). There is no clear preference to use one over the other and usually, more than one turbulence model is tried to see which one gives the best results. For the simulation of turbulent gas jets, it is common practice to increase constant C 1 of the standard k-ɛ model. [15] Sutradhar also used this strategy in his master s thesis [6]. Large Eddy Simulation If a more detailed solution is desirable, a Large Eddy Simulation (LES) is a possible solution. When using LES, the large turbulent flow structures are directly resolved and only the eddies with turbulent length scales smaller than the filter size (proportional to the grid size) are modelled. In this way, the modelling error decreases with the grid size. The simulation now also provides information about the turbulent fluctuations and gives a more accurate distribution of the evaporated fuel. However, it comes at an added computational cost. In general, LES requires a finer grid. Additionally, to obtain the mean flow field several solutions need to be averaged, whereas a RANS method directly provides the mean flow field. LES is gaining momentum in the modelling community at the expense of RANS modelling. High-fidelity simulations often use LES in combination with an Eulerian method for the liquid phase, as Bravo et al. [27] and Ghiji et al. [26] have done in their works. This is not to say that LES is not useful within the context of this thesis. Section 4.3 will investigate if LES can present an added value for the simulation of the spray in this thesis Injection and primary breakup model One of the challenges of using a discrete phase model is how to link the nozzle flow to an appropriate diameter distribution at the injection. The Lagrangian way 3.2 Common practice in the simulation of fuel sprays 17

39 of tracking particles does not allow to represent a fully liquid flow, such as in the nozzle. Furthermore, the levels of turbulence and cavitation in the nozzle greatly influence the speed and diameters of the droplets. Increased levels of turbulence in the nozzle flow destabilise the jet and increase its breakup rate. The same goes for the cavitation. The high local speeds in the nozzle can cause pressures below the vapour pressure of the fuel. When the cavitation bubbles implode again at the nozzle exit, the jet breakup is accelerated. Another effect of the cavitation is the reduction of the effective cross-sectional flow area in the nozzle and thus an increase in speed of the droplets leaving the nozzle. The most accurate way to model the injection would be to use a Eulerian (section 3.1.2) or a hybrid (section 3.1.3) model to simulate the nozzle flow and the liquid core of the jet. However, the computational cost of this method has hindered its use. Usually, the domain of the simulation only extends to the nozzle orifice. The nozzle and sac volume are then excluded from the simulation. It s hard to account for all of the aforementioned influences in the boundary condition at the nozzle orifice. Therefore, different approximations can be made which are explained in the following sections. Droplet diameter distribution functions One of the most basic ways of modelling the injection is imposing a distribution function of droplet sizes at the nozzle exit. This model assumes that the primary breakup has already occurred at the nozzle exit. Considering the quick primary atomisation of high-pressure fuel sprays, this is not a very hefty assumption. ANSYS Fluent supports the use of a uniform distribution and a Rosin-Rammler distribution. For both the user has to specify a standard diameter. Also for the Rosin-Rammler distribution the spreading parameter has to be specified. According to the ANSYS Fluent Theory guide [11], the Rosin-Rammler distribution function is defined as follows. [ ( ) D n ] 1 Y = exp d (3.3) Y is the mass fraction of droplets for which the diameter is smaller than D, d is the Rosin-Rammler diameter and n is Rosin-Rammler exponent which determines the spread. The main difficulty is determining these parameters, as the current experimental measurement techniques do not allow to measure droplet sizes in the dense spray near the nozzle. The parameters need to be found iteratively by comparing the results of a simulation to experimental measurements in the region 3.2 Common practice in the simulation of fuel sprays 18

40 away from the nozzle. Alternatively, there also exist correlations to estimate d and n. The ANSYS Fluent Theory guide proposes the following value for d. d = (133.0 d ) ( 8 W e n) 1 n (3.4) with the Weber number W e = ρu2 d/8 σ (3.5) and d the nozzle diameter, u the injection velocity, ρ the liquid density and σ the surface tension of n-dodecane. There still are other ways to find an appropriate distribution of droplets. For example, Pei et al. [9] used a uniform distribution and tuned its initial droplet diameter to obtain approximately correct liquid lengths. Blob model Another more frequently used model is the Blob Method. It is based on the assumption that spherical droplets with a diameter equal to the nozzle diameter are introduced into the domain. The breakup of these big droplets is from thereon handled by the secondary breakup models, dispersing the spray in much finer droplets as can be seen in figure 3.3. Its simplicity while retaining fairly accurate results is its main advantage. It is therefore used by many authors [4, 10, 5, 6] in their fuel spray simulation studies. Figure 3.3: Breakup of blobs by secondary breakup models using the concept of liquid core length, figure from ANSYS Fluent Theory Guide [11] The speed of the ejected droplets can be determined by the conservation of mass if the mass flow rate ṁ(t) through the nozzle is known. If there is no cavitation in the nozzle, the mean velocity u(t) of the ejected droplets is given by equation 3.6. u(t) = ṁ(t) ρ l A (3.6) 3.2 Common practice in the simulation of fuel sprays 19

41 A is the cross-sectional area of the nozzle (assumed to be constant) and ρ l the liquid density. If the mass flow rate is not known, Bernoulli s equation can be used to determine the maximum theoretical speed from the pressure difference p over the nozzle. The real speed will be lower, as friction losses are not taken into account here. u theoretical = 2 p ρ l (3.7) The real speed is often compared to the theoretical one and their ratio is called the discharge coefficient C d. The value can be determined by calculating the friction losses. Sometimes, it is also given for a particular nozzle. C d = ṁ ṁ theoretical = ρ l Au ρ l Au theoretical (3.8) If there is cavitation, the effective cross-sectional area of the flow will decrease and the speed of the droplets leaving the nozzle will increase. Von Kuensberg Sarre et al. [32] have developed a method to quantify this effect. It is based on the existence of a vena contracta in the nozzle surrounded by evaporated fuel. They calculate the cross-sectional area of the vena contracta A vena with Nurick s expression. A vena = C c A, C c = [ ( ) r ] 0.5 (3.9) C c0 D C c0 is a constant equal to 0.61, r is the radius of the edge at the inlet of the nozzle and D is the diameter of the nozzle. Assuming a steady flow, the conservation of mass gives the speed at the vena contracta (u vena ). u vena = ṁ ρ l C c A = u C c (3.10) If the conservation of momentum between the vena contracta and the nozzle exit is expressed, equation 3.11 is obtained. It assumes that the cavitation keeps the pressure at the vapour pressure at the vena contracta. u eff = Ȧ m (p vap p exit ) + u vena (3.11) u eff is the effective speed of the droplets at the nozzle exit due to cavitation, p vap the vapour pressure for the fuel at a given temperature and p exit the static pressure at the nozzle exit. Using the mass flow rate, a decreased effective cross-sectional area at the nozzle exit can be calculated with u eff. The increased breakup rate due to implosions of cavitation bubbles is not accounted for in this model. 3.2 Common practice in the simulation of fuel sprays 20

42 Other models There still exist plenty of other models, some of which are commonly used but not available in ANSYS Fluent. For example, there is the Huh-Gosman model as used by various researchers [4, 5, 10] of which the first two used a modified version of the Huh-Gosman model called the Bianchi model. They both take into account the aerodynamically induced breakup by the Kelvin-Helmholtz instability (section 3.2.4) and the turbulence induced breakup by introducing surface perturbations linked to the turbulent length scales of the flow in the nozzle. Other models also include the effect of cavitation, such as the KH-ACT model, developed by Som et al. [12]. The cavitation induced breakup is based on the bubble collapse and burst times. The k-ɛ model forms the basis of the turbulence induced breakup. For the aerodynamically induced breakup, both the Kelvin-Helmholtz and the Rayleigh-Taylor instability are taken into account Secondary Breakup model Wave The Wave, Kelvin-Helmholtz or KH model is a popular secondary breakup model for high-weber-number sprays, such as high-pressure fuel sprays. Indeed, the high pressures lead to high injection velocities u and consequently also a high Weber number W e = ρ l u 2 d/σ. This model has been developed by Reitz and Diwakar [13] in 1987 and it has since been used by many people. For example, Som et al. [4] and Montanaro et al. [10] used it in their works. The Kelvin-Helmholtz instability describes the unstable behaviour of two fluids when there exists a relative speed difference between them. If this speed difference exceeds a critical value, the wave becomes unstable (figure 3.4). It is an important breakup mechanism at the surface of fuel droplets. Figure 3.4: Kelvin-Helmholtz instability: related to the existence of a difference in speed between two fluids [29] The Wave model as implemented in ANSYS Fluent and as described in its Theory Guide [11] will be further explained in this section. The model is derived from 3.2 Common practice in the simulation of fuel sprays 21

43 the stability analysis of a cylindrical, viscous, incompressible, liquid jet of radius a, velocity v and density ρ l. It is assumed that the jet penetrates in a stagnant, incompressible and inviscid gas of density ρ g. The details of the jet stability analysis will not be given here, as it does not help in understanding the use of this model. The reader is referred to the works of Reitz [13]. The analysis yields a dispersion relation linking the growth rate of a surface perturbation to its wave number. The Wave model assumes that the wavelength of the fastest-growing surface wave (and therefore also the most unstable one) determines the size of the droplets that are sheared off. Curve fits of the numerical solution for the maximum growth rate Ω in the dispersion relation give one corresponding value of the wave number K and thus also one corresponding wave length Λ = 2π/K. ρ l a Ω 3 σ = W e 1.5 g (1 + Oh)( T a 0.6 (3.12) ) Λ a = 9.02( Oh0.5 )( T a 0.7 ) ( W eg 1.67 ) 0.6 (3.13) In these equations, σ is the surface tension, U is the relative velocity between the liquid and the gas, W e g = ρgu 2 a σ the gas Weber number, W e l = ρ lu 2 a σ the liquid Weber number, ν l the kinematic viscosity of the liquid, Re l = Ua the Reynolds number, Oh = number. W el Re l the Ohnesorge number and T a = Oh W e g the Taylor ν l The radius of the newly-formed droplet is supposed to be proportional to the wavelength of the fastest-growing surface wave. Reitz proposed a factor B 0 of 0.61 and it is valid in almost any case. r = B 0 Λ (3.14) The rate of change of the radius a of the parent droplet is determined as follows. da dt r) = (a, r a (3.15) τ τ = 3.726B 1a ΛΩ (3.16) Equation 3.16 defines the breakup time of the parent droplet. The second model constant B 1 is generally taken to be 1.73, but can vary between 1 and 60. For example, if the high levels of turbulence and cavitation in the nozzle cause an increased breakup rate, B 1 can be lowered so that the parent droplet breaks up more quickly. In ANSYS Fluent, every time when equation 3.15 predicts a 5% mass loss of the 3.2 Common practice in the simulation of fuel sprays 22

44 initial mass of the parent parcel (a group of droplets with same properties), a parcel is shed in a random direction orthogonal to the direction the parent parcel was travelling. [11] The child parcel keeps the same properties as the parent parcel except for the direction of the speed and the droplet radius (which is equal to r as defined by equation 3.14). KH-RT A common extension to the Wave model is to include the effect of the Rayleigh-Taylor instability. The model combining these two effects is called the KH-RT model. The RT instability is an unstable flow of two fluids, one of which is denser than the other. The less dense fluid pushes the heavier one and creates instabilities while doing so. This happens when a denser fluid sits atop a less dense fluid or when a less dense fluid is accelerated into a denser fluid. The latter is an important effect in the dilute region of the spray and the KH-RT model is therefore used by many researchers [4, 10, 14, 6]. Hossainpour et al. [33] also explain one advantage of combining KH and RT breakup. WAVE breakup tends to create a bimodal droplet distribution because smaller droplets get separated from a big parent droplet. In RT breakup a parent droplet divides itself in a number of equally sized child droplets. These breakup mechanisms are visually illustrated in figure 3.5. In this way, RT breakup counteracts the formation of a bimodal droplet distribution by creating a number of equally sized droplets with intermediate size. (a)kh breakup mechanism (b)rt breakup mechanism Figure 3.5: Comparison of the creation of child droplets due to the KH instability and RT instability [34] The KH-RT model is also available in ANSYS Fluent. In what follows its implementation will be explained, based on the ANSYS Fluent Theory Guide [11]. Similarly as for the KH model, the RT model tracks the fastest growing surface 3.2 Common practice in the simulation of fuel sprays 23

45 waves to determine the breakup. In the KH-RT model, breakup will occur according to which of both instabilities is faster. The frequency of the fastest growing wave due to the RT instability is given by equation Ω RT = 2( g t(ρ l ρ g )) 3/2 3 3σ(ρ l + ρ g ) Where g t is the droplet acceleration in the direction the droplet travels. determined by the drag force acting on the droplet. (3.17) It is Multiple models exist to calculate this drag force and the most important ones are given in section The wavelength that corresponds to Ω RT is 3σ Λ RT = 2π g t (ρ l ρ g ) (3.18) Breakup can occur after the breakup time τ RT, to allow the RT waves to grow to adequate proportions. τ RT = C τ Ω RT (3.19) C τ is the Rayleigh-Taylor breakup time constant which has a default value of 0.5. The droplet is allowed to breakup when the predicted wavelength C RT Λ RT is smaller than the droplet diameter. Here C RT is the breakup radius constant which has a default value of 0.1. The radius of the child droplet r c is calculated similarly as for the KH model (proportional to the wavelength corresponding to the fastest growing wave). r c = 1 2 C RT Λ RT (3.20) This time, C RT can be adjusted to account for the effects of turbulence and cavitation in the nozzle. By decreasing C RT the predicted wavelength decreases and the condition for breakup becomes less severe, resulting in an increased breakup rate. The KH-RT model as implemented in ANSYS Fluent uses the concept of a liquid core length L. Inside the liquid core (see figure 3.3), only breakup due to the Kelvin-Helmholtz instability is considered. This strategy is often referred to in the literature as KHRT with breakup length. It is used to avoid an overestimation of the breakup rate in the liquid core. The high relative velocities between the gas and the liquid phase result in strong drag forces and thus high decelerations. This gives a 3.2 Common practice in the simulation of fuel sprays 24

46 high value for Ω RT and a small value for Λ RT, both resulting in a fast breakup. The length of the liquid core in ANSYS Fluent is computed by L = C L d 0 ρl ρ g (3.21) C L is the Levich constant with a default value of 5.7 and d 0 is a reference nozzle diameter. TAB Another model which is available in ANSYS Fluent is the Taylor Analogy Breakup or TAB model. It is based on the analogy between an oscillating droplet and a spring-mass system. It is hardly ever used for high-pressure fuel spray breakup, so no further attention will be given to this model. The ANSYS Fluent Theory guide [11] only recommends it for low-weber-number sprays. They explain that the shattering of droplets in high-weber-number sprays is not described well by the spring-mass analogy Collision model In a Lagrangian Particle Tracking approach, there is a possibility that particles collide. The outcome of such a collision can be that the particles coalesce and continue their way as one or that they bounce and continue their journey on a different path. Although high-pressure fuel sprays can have high collision rates, collision models are not used very often. Several authors [4, 9, 5, 7] declare not having used a collision algorithm. Lucchini et al. [5] state Collision models were also not used, because of their limited effects on the Sauter mean radius (SMR) of an evaporating spray, as it is illustrated in Baumgarten (2006). So collision algorithms can have a high computational load, without having a big influence on the result. This is especially true for sprays with a high number of particles. The total number of possible collisions scales quadratically with the number of particles. Hence, collision algorithms need to efficiently identify possible collision partners. Collision models can still be relevant for fuel spray breakup. Their importance will be studied in section The O Rourke collision algorithm is frequently used in ECN workshops [31, 16] (more info in section 4.1) and it is also the algorithm that is implemented in ANSYS Fluent. Two essential things make the O Rourke algorithm a viable option for practical use in spray breakup. First of all, it tries not to calculate the collisions geometrically, but it uses a stochastic approach. The actual number of 3.2 Common practice in the simulation of fuel sprays 25

47 collisions is taken from a Poisson distribution based on the mean expected number of collisions. This expected number is computed by taking the ratio of a collision volume (a volume in which collision is possible) and the cell volume. Secondly, it only allows particles in the same continuous-phase cell to collide. This greatly reduces the computational load of the algorithm, but this strategy cannot detect two particles in different cells which are still on their way to a collision. This is compensated by allowing particles in the same cell, but which are further apart, to collide. A further reduction in computation time is realised by the fact that ANSYS Fluent uses the concept of a parcel of particles. One parcel can group several thousand to millions of particles with the same properties. Once it has been determined that two particles will collide, the O Rourke algorithm also decides the outcome. There are three important factors that determine the outcome : the obliqueness of the trajectory of the collision, the ratio of the droplet radii r 1 /r 2 and the collisional Weber number W e c = ρ lurel 2 D σ with r 1 r 2, U rel the relative velocity between the two parcels and D the mean diameter of the two parcels. To determine if two droplets will coalesce, a critical offset b crit is calculated based on the radii and the collisional Weber number. If the actual offset b is smaller than the critical one, the droplets will coalesce. b crit = (r 1 + r 2 ) f( r 1 r 2 ) = ( r1 r 2 ) min(1.0, 2.4f W e c ) (3.22) ( r1 r 2 ) ( ) r1 r 2 (3.23) The impact parameter B = b/(r 1 + r 2 ) ranges between 0 and 1. B = 0 corresponds to a head-on collision and B = 1 is a grazing collision where the droplets barely touch each other. In the algorithm, it is just randomly generated as B = Y with Y a random number between 0 and 1. This defines b and after comparing with b crit, the outcome of the collision is known. If it is coalescence, the number of droplet collisions between two parcels are taken from a Poisson distribution. In case of a grazing collision, the new velocity components are calculated with the basic conservation laws, assuming some energy loss due to viscous dissipation and angular momentum generation Droplet drag model In CFD simulations of fuel sprays, droplet drag models play an important role. They have a large influence on the liquid penetration because they determine how quickly droplets leaving the nozzle are slowed done. The more they are slowed down, the less the liquid penetrates in the chamber before being evaporated. This, in turn, has 3.2 Common practice in the simulation of fuel sprays 26

48 an effect on the dispersion of the fuel. In reacting conditions, it further influences the equivalence ratios and thus the flame formation. Therefore, drag models are almost always used. In its simplest form, the drag of a droplet is modelled as the one of a sphere. The drag coefficient of a sphere is Re > 1000 C d,sphere = (3.24) 24 Re ( Re2/3 ) Re 1000 As one might imagine, droplets tend to deform so this approach is not very accurate. This led to the creation of lots of improved drag models. One of the most prevalent drag models is the dynamic drag model which takes the droplet deformation into account. Som et al. [4] and Montanaro et al. [10] used this model in their works and it is also repeatedly used in ECN Workshops [31, 16]. On the other hand, Pei et al. [9] preferred to use the High Mach number model, probably for its lower computational load. It is a variation of the spherical drag model including corrections for high particle Mach numbers. The spherical drag model assumes that the drag coefficient of a droplet is equal to that of a smooth sphere. There are still other drag models (such as the Stokes-Cunningham drag law) but they are used much less frequently for fuel spray simulations. The Stokes-Cunningham drag law performs better if used with fine particles ( 1µm). Of all the droplet drag models mentioned here, the dynamic drag model is the only one who takes the droplet deformation into account. For this reason, the dynamic drag model will be used in this thesis. For large Weber numbers, the surface tension cannot keep droplets spherical. In the extreme case, the droplet can take a disc-like form. The drag coefficient C d of a disc-shaped droplet is times higher than if it were a spherical droplet. [11] For that reason, the dynamic drag model calculates the droplet distortion y. When y = 0, the droplet is perfectly spherical and when y = 1, the droplet is a disc. The drag coefficient of the droplet can then be calculated as a linear variation between the drag coefficient of a sphere and that of a disc. C d = C d,sphere ( y) (3.25) In the dynamic drag model as implemented in ANSYS Fluent, the droplet distortion y is determined by the solution of a second order ODE. [11] d 2 y dt 2 = C F ρ g u 2 C b ρ l r 2 C kσ ρ l r 3 y C dµ l dy ρ l r 2 dt (3.26) Here C F, C k and C d are dimensionless constants with the values 1/3, 8 and 5, respectively. They have been judiciously chosen to match experiments. Once again, ρ l and ρ g are the densities of the liquid and the gas. u is the relative velocity of the 3.2 Common practice in the simulation of fuel sprays 27

49 droplet with respect to the continuous gas phase, r is the radius of the undistorted droplet, σ is the surface tension and µ l is the dynamic viscosity of the liquid. This equation represents a dimensionless version of a damped, forced oscillator based on Taylor s analogy. It is the same analogy as used in the TAB breakup model (section 3.2.4). The spring force corresponds to the surface tension forces, the external force to the drag force and the damping force to the viscous forces in the droplet Evaporation model The aim of a fuel spray in the context of internal combustion engines is providing the necessary fuel for combustion so that a net work can be obtained by expanding the combustion products. In order for the fuel to combust, it must first evaporate. Hence, modelling the evaporation is an inevitable part of fuel spray simulations. Evaporation is a combined heat and mass transfer problem. The surrounding gas needs to provide the necessary heat for the phase change and the evaporated fuel needs to be transported away from the surface in order for the evaporation to continue. Keeping the high relative velocities of fuel droplets in mind, one can easily derive that convection will be the dominant mechanism of transport. The convection coefficient h is often obtained by using the correlation of Ranz and Marshall (1952) [35]. It relates the Nusselt number to the Reynolds number and the Prandtl number for a sphere in a steady flow. Nu = hd k = Re 1/2 d P r 1/3 (3.27) in which d is the droplet diameter, k the thermal conductivity of the continuous phase, Re d the Reynolds number based on the particle diameter and the relative velocity and P r the Prandtl number of the continuous phase. 14 years earlier, Frössling found a similar correlation for the Sherwood number Sh AB of a sphere in a steady flow. In this way, the mass transfer coefficient k c can be calculated. Sh AB = k cd D i,m = Re 1/2 d Sc 1/3 (3.28) Where d is the droplet diameter, D i,m is the diffusion coefficient of the vapour in the bulk gas and Sc is the Schmidt number. Both correlations are used by Som et al. [4], Pei et al. [9] and Montanaro et al. [10] in their papers. These are also the correlations used in ANSYS Fluent to predict droplet evaporation. The rest of this section will be dedicated to explaining how droplet evaporation is handled by ANSYS Fluent as described in the ANSYS Fluent Theory Guide [11]. 3.2 Common practice in the simulation of fuel sprays 28

50 When a particle is below the vaporisation temperature or its volatile fraction has been entirely consumed, the particle obeys the inert heating and cooling law. The vaporisation temperature is a parameter with no physical significance and only indicates when vaporisation starts. The temperature change of the particle dtp dt determined by a heat balance. m p c p dt p dt = ha p(t T p ) + ɛ p A p σ(θ 4 R T 4 p ) (3.29) In this equation m p is the mass of the particle, c p the heat capacity of the particle, h the convection coefficient as determined by the Ranz-Marshall correlation (equation 3.27), A p the surface area of the particle and T the local temperature of the continuous phase. The last term accounts for the radiative heat transfer, but it is often neglected in the context of fuel spray simulations. is Once the vaporisation temperature has been reached, the droplet will start evaporating. For slow vaporisation rates, ANSYS Fluent assumes that it is governed by gradient diffusion. N i = k c (C i,s C i, ) (3.30) N i is the molar flux of vapour. k c is the mass transfer coefficient determined by the Frössling correlation (equation 3.28). C i,s is the vapour concentration at the droplet surface and it is evaluated by using the ideal gas law and by assuming that the partial pressure of vapour at the interface is equal to the saturated vapour pressure. The vapour concentration in the bulk gas C i, is derived from its mole fraction and the absolute pressure by applying Dalton s law of partial pressures. For high vaporisation rates, the evaporation is no longer diffusion controlled. Vapour gets convected away from the droplet surface and equation 3.30 is no longer valid. It is replaced by dm p dt = k c A p ρ ln(1 + B m ) (3.31) where ρ is the density of the bulk gas. k c is still determined by the Frössling correlation (equation 3.28). B m is the Spalding mass number. It is equal to the ratio of the difference of vapour mass fraction at the surface and in the bulk gas to the mass fraction at the surface of all non-vapour species. B m = Y i,s Y i, 1 Y i,s (3.32) Of course, when vaporisation has started the heat balance in equation 3.29 needs to adjusted to include the latent heat of vaporisation h fg. m p c p dt p dt = ha p(t T p ) dm p dt h fg + ɛ p A p σ(θ 4 R T 4 p ) (3.33) 3.2 Common practice in the simulation of fuel sprays 29

51 In the case of high vaporisation rates, the convection coefficient h is calculated with a modified version of the Ranz-Marshall correlation. B T Nu = hd k = ln(1 + B T ) B T ( Re 1/2 d P r 1/3 ) (3.34) is the Spalding heat transfer number and it is assumed to be equal to the Spalding mass transfer number (equation 3.32). When the heat transfer to the droplet increases its temperature to the boiling point, the vapour mass transfer is calculated by a boiling rate equation. ANSYS Fluent uses a boiling rate equation as proposed by Kuo (1986) in the book Principles of Combustion. d(d) dt = 4k [ ρ p c p, d ( Re d )ln 1 + c ] p, (T T p ) h fg (3.35) with c p, and k the heat capacity and the thermal conductivity of the gas, respectively. ρ p is the droplet density. There still exist states other than inert, vaporising or boiling in which droplets are categorised by ANSYS Fluent, but they are only relevant for combustible particles. For fuel sprays in non-reacting conditions, the ones mentioned in this section are the most relevant, in particular the convection/diffusion controlled regime with high vaporisation rates. The high ambient temperatures and high relative velocities suggest high vaporisation rates Turbulent dispersion Normally, the trajectory of Lagrangian particles in a flow field is calculated based on the mean flow velocities (u, v, w). However, in a real flow, the turbulent fluctuations create an additional dispersion of the particles. It would be better to calculate the trajectories based on the instantaneous flow velocities (u + u, v + v, w + w ), but these are not available in RANS models. In an effort to include this dispersion effect in spray simulations, some researchers [4, 9, 10] used turbulent dispersion models in their spray simulations. Lucchini et al. [5] chose not to include this effect in order to reduce the sensitivity of the results to the turbulence model. All those who did use a turbulent dispersion model chose for the stochastic tracking approach. This model is also implemented in ANSYS Fluent in its discrete version, the so-called Discrete Random Walk Model (DRW model). The next paragraph will explain this model, according to the ANSYS Fluent Theory Guide [11]. Other models still exist, such as the Particle Cloud Tracking model, but it is hardly ever mentioned in the literature concerning fuel sprays. Therefore, it will not be explained here and the 3.2 Common practice in the simulation of fuel sprays 30

52 reader is referred to the literature. When keeping the highly turbulent character of a fuel spray in mind, an influence of a turbulent dispersion model is to be expected and its influence will be studied in section The Discrete Random Walk model calculates the fluctuating fluid velocities (u, v, w ) based on the turbulent kinetic energy k and the turbulent dissipation rate ɛ provided by the turbulence model. One remark that can be made is that the turbulence model needs to be able to provide these quantities. 1-equation turbulence models such as Spalart-Allmaras cannot be used for this purpose. The DRW model is discrete because the fluctuating fluid velocities are kept constant during a characteristic lifetime of an eddy. That is to say, the fluctuating fluid velocities are a piecewise constant function of time. The random nature is explained by looking at the determination of these fluctuating fluid velocities. They are obtained by multiplying a standard normally distributed random number ζ with an RMS value of the fluctuating fluid velocities (see equation 3.38). Computing this random walk for a sufficient number of particles models the influence of the turbulence. The RMS value of the fluctuating fluid velocities in a node can be calculated from the turbulent kinetic energy. k is defined as k = 1/2(u 2 + v 2 + w 2 ) (3.36) Assuming isotropy (u 2 = v 2 = w 2 ) results in u 2 = v 2 = w 2 2k = 3 (3.37) As explained higher up, the fluctuating fluid velocities are found by multiplying with the random number ζ u = ζ u 2, v = ζ v 2, w = ζ w 2 (3.38) The particles interacts with the eddy during the lifetime of the eddy τ e or when the particle eddy crossing time has passed. After this, the particle passes to a new eddy with new values of ζ. In ANSYS Fluent the eddy lifetime τ e can be defined as a constant. In this case, it is approximated by: τ e = 0.30 k ɛ (3.39) Or the eddy lifetime can be calculated randomly as well τ e = 0.15 k ln(r) (3.40) ɛ 3.2 Common practice in the simulation of fuel sprays 31

53 with r a uniform random number between 0 and 1. The random τ e should do a better job at representing the inherent random character of turbulence. 3.3 Goal statement The first part of this thesis is dedicated to exploring the possibilities of the ANSYS Fluent software package for simulating non-reacting, evaporating diesel sprays at engine-like ambient conditions. The influence of different parameters of turbulence models, droplet breakup models, injection strategies and material properties will be studied. When a reliable CFD model is found that is able to predict the spray formation at different working conditions, the study will be extended to simulating a reacting spray. Simulation results of both cases will be validated against the extensive database of ECN [2] containing the experimental data of sprays in numerous conditions. Spray formation is still a complex and not completely understood multiphase phenomenon. The literature does not provide a single best way to simulate the atomisation of a liquid jet. The plethora of breakup models (Blob, Huh Gosman, TAB, ETAB, WAVE, KH-RT,...) further illustrates this point. The increase in accuracy of measurement techniques and the increase in computing power go hand in hand to develop the knowledge about this topic. Nevertheless, the goal here is not to do fundamental research on jet breakup models using meshes of millions of cells, but rather have a reliable model with reasonable calculation times. The possible grid dependent results and errors due to the use of simplified models could lead to need of tuning some model parameters. However, by doing so valuable experience will be gained to finally be able to implement the spray combustion in a complete in-cylinder simulation of a dual fuel engine. This is one of the bigger goals of the Transport Technology research group at Ghent University in which PhD student Gilles Decan is using CFD to understand and optimise the operation of marine dual fuel engines at varying engine conditions. Future research will still have to tackle the problem of simulating a diesel spray in an atmosphere of natural gas and air. At the moment, ECN does not offer experimental spray data in atmospheres containing fuel. The in-house constant volume combustion chamber of Ghent University (GUCCI) could be used to gain valuable experimental insights. It is already being used to conduct research on diesel sprays in marine engine conditions and it could play a central role for research on diesel sprays in marine dual fuel engines. The engine ambient conditions are quite different from the sprays of the ECN. Most notably, the use of bigger nozzle diameters and the general 3.3 Goal statement 32

54 dimensions of the spray. It remains to be seen that the best approach developed for the ECN sprays scales up well to the dimensions of marine diesel engines. Later, this knowledge needs to be translated from marine diesel engines to dual fuel engines where the most notable difference is the presence of premixed natural gas. 3.3 Goal statement 33

55 Simulation setup and model validation 4 This chapter will go through the different steps used to set up the simulation. First, the experiments on which the simulations are based will be introduced. The next section will mainly be about the meshing process and modelling the injection. Thereafter, the influence of using different turbulence models and the sensitivity to various fluid properties will be studied. The chapter will end by proposing a general methodology for simulating fuel sprays. 4.1 ECN spray A As already mentioned before, there exists a need to experimentally validate CFD simulations. But even validation is not as straightforward as it sounds. As Som et al. [4] put it, the term validation is used very loosely in the literature, since models predict a CFD expert s interpretation of the experimental parameter being measured. This is where the Engine Combustion Network (ECN)[2] comes in. They provide a database of high-fidelity experimental diesel spray data, in a wide range of reacting and non-reacting conditions. Additionally, they provide guidelines for comparing experimental results with simulation results to streamline the collaboration in engine combustion research. One of these efforts is to propose a fixed set of boundary condition for a diesel spray. The so-called ECN Spray A is one of the most widely used and its specifications can be found in table 4.1. Multiple institutions across the world have already contributed experimental data for this spray. As a fuel, n-dodecane is used. It is a diesel surrogate with clearly defined chemical properties. This is to eliminate the influences of the variable composition of diesel fuel around the world. They also propose a set of parametric variations of injection temperatures, ambient density and ambient temperature to further validate spray models. These parametric variations are given in table 4.2 as proposed during the 3rd ECN workshop [15]. The ECN workshops are yearly conferences that promote the exchange of knowledge regarding spray combustion and help to coordinate future research. They are attended by experimental researchers as well as researchers conducting numerical research. 34

56 Table 4.1: Most important specifications of the ECN Spray A baseline operating condition [2] Spray A operating condition (baseline) Ambient gas temperature 900 K Ambient gas pressure near 6.0 MPa Ambient gas density 22.8 kg/m³ Ambient gas oxygen (by volume) 15% O2 (reacting); 0% O2 (non-reacting) Ambient gas velocity Near-quiescent, less than 1 m/s Fuel injector nominal nozzle outlet diameter mm Nozzle K factor K = (dinlet doutlet)/10 [use µm] = 1.5 Discharge coefficient C d = 0.86, using 10 MPa pressure drop Number of holes 1 (single hole) Fuel injection pressure 150 MPa (1500 bar), prior to start of injection Fuel n-dodecane Fuel temperature at nozzle 363 K (90 C) Injection duration 1.5 ms Injection mass mg Table 4.2: Parametric variations of ECN spray A with the first one being the baseline condition [15] Priority level T [K] density [kg/m³] Injection pressure [MPa] These experiments are carried out in constant volume combustion chambers such as the one at Sandia National Laboratories (figure 4.1). This ensures a proper control over the operating conditions. The one at Sandia National Laboratories is responsible for a sizeable part of the experimental data available on the website of ECN. Therefore, its geometry will serve as an inspiration for the calculation domain. It has a cubical-shaped combustion chamber with a characteristic length of 108 mm. The injector is placed in the middle of one of the faces of the cube. Auxiliary equipment such as the thermocouple, spark plug and fan at the top face are supposed to be placed sufficiently far from the injector so that their influence on the 4.1 ECN spray A 35

57 flow field can be neglected. That is why they are not included in the calculation domain. If this reasoning is continued, a perfect cube with the injector in the middle of one of the faces seems as an adequate representation of the domain for simulation purposes. The parametric variations in table 4.2 represent different operating conditions of an automotive compression ignition engine. 1, 2 and 3 are variations in injection pressure. This is very important in ICEs using unit injectors or pump-line-nozzle injection systems because the fuel pump is directly driven by the engine. This means that the maximum injection pressure is engine speed dependent. In common rail systems, the injection pressure is independently regulated, so the injection pressure can be chosen to optimise the performance and emissions at a given speed and load. 1, 4 and 5 are ambient density variations. They correspond to engine load variations at the same speed. At a higher load, more air is sucked into the cylinder because of the turbocharger. 1, 6, 7 and 8 are ambient temperature variations. They could be explained by variable conditions at the start of injection, e.g., a cold engine. 9 and 10 are the mid- and low-temperature conditions, respectively. The importance of these conditions is explained by the bigger role spray breakup plays at reduced ambient temperatures and injection pressures.[15] The non-evaporating condition 11 is to test if the model still works in this extreme case, but it is questionable whether tuning at non-evaporating conditions makes sense for high-temperature fuel sprays. (a)photo of the inside of the combustion chamber (b)schematic cross-section Figure 4.1: The constant volume combustion chamber at Sandia National Laboratories When conducting experiments at non-reacting conditions, one is mainly interested in measuring two phenomena: the atomisation and subsequent evaporation of the liquid fuel jet and the extent to which the fuel vapour penetrates in the combustion chamber. Optical techniques with high speed cameras allow to follow the evolution 4.1 ECN spray A 36

58 of the spray and distinguish liquid fuel from fuel vapour and air. For instance, the evolution of the liquid phase can be tracked by using the Mie scattering technique. The penetration of the fuel vapour can be followed using the Schlieren technique. From these images a boundary of the liquid phase and the vapour phase can be extracted. The distance of the injector nozzle hole to the rightmost point on the spray axis that also lies on the liquid boundary can be called the liquid length or liquid penetration. In an analogous way, the vapour penetration can be defined. ECN has proposed some guidelines to compare these quantities to CFD simulation results.[16] Their preferred definitions are given here. Liquid penetration: Place a set of spheres (with radius 1 mm) along the injector axis every 0.1 mm or less. Then look for all the droplets being in each of such spheres and compute the void fraction in them. The distance of the farthest one having a void fraction higher than 0.1% represents the spray penetration Vapour penetration: Maximum distance from the nozzle outlet to where the fuel mass fraction is 0.1% These definitions were followed as closely as possible but only the void fraction of each cell was available to determine the liquid penetration. This represents mainly a loss in accuracy because the cells have a length of mm. This is fairly close to the size of a sphere with radius 1 mm, so no systematic error should be made. If cells were smaller, one small droplet could easily bring forth a cell with a volume fraction above the threshold of 0.1%. 4.2 Discretisation Dimensions and structure of the mesh RANS turbulence models (section 2.2) allow for the use of symmetry, so a quarter cubic domain was used to simulate the diesel spray. Two orthogonal planes intersecting each other on te spray axis divide the cubic domain in 4 parts. Only one of those parts is considered in the simulation and the cutting planes are defined as a symmetry boundary condition. In theory, also a 2D axisymmetric mesh can be used for this problem. Later in this section, a 2D axisymmetric mesh and a 3D quarter cubic are compared, but the results of the 3D mesh look more promising so this mesh will be used in the rest of the thesis. 4.2 Discretisation 37

59 3D quarter cubic mesh Last year, Gilles Decan [7] already started doing CFD simulations of diesel sprays for marine diesel engine applications. His mesh also seemed appropriate to simulate ECN spray A, so a rescaled version of his mesh was used in this thesis. His mesh is shown in figure 4.2. Three zones can clearly be distinguished. A central core part in green is the part of the mesh were the most refinement is needed. A refined structured grid offers the most details of the flow field and should be able to properly resolve the flow in the region with the largest velocity gradients. Around this, a slightly coarser structured grid is used. In the figure, this part is coloured in blue. The rest of the domain, in the figure shown in grey, is filled with an unstructured grid to limit the total cell count. Figure 4.2: Quarter cubic mesh as created by Gilles Decan [7] As explained in section 4.1, the length of the domain is 108 mm. In the green core part along the centreline, the domain is divided in 160 cells, so the length x of one cell is mm. The dimensions in the radial and circumferential direction as well as the dimensions of the other parts of the rescaled mesh are summarised in table 4.3. Due to the conical shape, cells in the structured part get bigger in the radial direction. The r given in table 4.3 are for the cells near the injector nozzle hole. Table 4.3: Cell count and dimensions of cells in the different zones of the mesh Zone Cell count x [mm] r [mm] θ [ ] core (green) fine (blue) chamber (grey) sum Discretisation 38

60 2D axisymmetric mesh In an early stage, a 2D axisymmetric mesh of similar dimensions to the 3D quarter cubic mesh was made. The 2D axisymmetric mesh is rectangular and structured. In the zone near the injector nozzle hole, special care was taken to obtain cells with similar dimensions as the 3D quarter cubic mesh. Furthermore, the simulation settings were chosen to be as similar as possible. In both cases, the realisable k-ɛ with the same model constants was used. The mass flow rate of the injection was adjusted so that the total mass injected in the combustion chamber was the same. In ANSYS Fluent, the mass flow rate of a discrete phase injection for a 2D axisymmetric domain is defined as the total mass flow rate divided by 2π radians. For the 3D quarter cubic domain, one fourth of the total mass flow rate was taken. Apart from this, the discrete phase and the injection were defined in the same way. The results for the evolution of the liquid penetration are shown in figure 4.3. Using the standard model constants, the liquid penetration of the 3D quarter cubic mesh better approximates the experimental liquid penetration than the one of the 2D axisymmetric mesh. By looking at the collision rates and mean droplet diameters, it could be seen that droplets in the 2D mesh collide and coalesce much less frequently. Due to their bigger surface-to-volume ratio, smaller droplets evaporate more quickly and get slowed down more by drag forces. A possible explanation is that droplets in the 2D mesh cannot collide in the circumferential direction because they are only able to move in a plane. Intuitively, one can understand that rings of droplets with the same properties, moving away from the nozzle, do not accurately represent a diesel spray. As Pei et al. [9] have shown, a 2D axisymmetric mesh can still be used effectively. In most cases though, model constants will need more heavy tweaking. 12 Liquid penetration 10 LP [mm] D quarter cubic 2D axisymmetric experiment time [ms] Figure 4.3: Comparison of the liquid penetration in a 3D quarter cubic mesh and a 2D axisymmetric mesh 4.2 Discretisation 39

61 There was only an improvement in calculation time for the continuous phase of the 2D mesh. Because parcels of discrete phase droplets were injected at the same rate, the calculations for the discrete phase took approximately the same time. The small decrease in calculation time was not convincing enough to prefer the 2D mesh over the 3D mesh. Additionally, the proceedings of the first ECN workshop [31] mention that full 3D simulations are probably more accurate than 2D simulations Injection Injection type, mass flow rate and velocity There are multiple injection types in ANSYS Fluent. The most important ones being releasing particles from a surface or releasing particles from a point. As the injector nozzle hole is smaller than the cell right next to it, there is no point in creating a surface to release particles from. Indeed, table 4.1 indicates a nozzle hole diameter of 90 µm, while table 4.3 gives a r of mm. This gives a ratio of 1.5, consistent with Lucchini s remark in section From table 4.1 with the operating conditions of ECN spray A, the mass flow rate can be derived by dividing the injection mass by the injection duration. ṁ = 3.5 mg = 2.3g/s (4.1) 1.5 ms The injection shape of ECN spray A is a square wave, so the actual instantaneous values will not vary much from this mean value. Due to the finite duration of the needle lift event, the actual mass flow rate is slightly smaller at the beginning and the end of injection and slightly bigger in the middle of the injection. ECN recommends the use of a virtual injection rate generator hosted on the website of Universitat Politècnica de València. ( Mass flow rate [g/s] UPV Bernoulli Time ASOI [ms] Figure 4.4: Injection rate as given by the tool on the website of UPV compared to the mass flow rate obtained by applying the principle of Bernoulli and a C d of Discretisation 40

62 In order to be independent of the availability of the virtual injection rate generator, the preference has been given to use the representative square wave as the injection rate shape in the simulations here. The injection velocity can then be obtained by the density of the liquid fuel and the area of the nozzle hole, as illustrated in equation 3.6. This strategy was only used for the calculations in this chapter. For the parametric variations of ECN spray A however, no information about the mass flow rate was available, so the injection velocity and mass flow rate were determined based on the definition of the discharge coefficient C d (equation 3.8). Despite table 4.1 specifying a C d of 0.86, the injectors used in the experiments had a C D of around The result of these calculations is given in table 4.4. The ambient pressure p a was also calculated to account for the small changes in density of n-dodecane. To calculate the liquid density of n-dodecane, equation 7 in the SAE paper of Desantes et al. [28] has been used. The results are very similar to the liquid densities given by the NIST [17]. p a = ρ a R N2 T a (4.2) with R N2 = 297 J/(kg.K) the gas constant of nitrogen, ρ a the ambient density and T a the ambient temperature as given in table 4.2. Table 4.4: Ambient temperature, density and pressure, liquid density, injection velocity and mass flow rate calculated based on C d = 0.89 for the parametric variations of ECN spray A Priority level T a [K] ρ a [kg/m³] p a [MPa] ρ l [kg/m³] u [m/s] ṁ [g/s] The simulations are not so sensitive to variations in injection velocity. The product of mass flow rate and injection speed (= flow of momentum) has a bigger influence on the spray penetration. If only the injection velocity is varied, e.g. an increase of 20%, the vapour penetration increased by 4.5% and the liquid penetration decreased by 12.9%. The last observation might sound surprising, but the increased velocity 4.2 Discretisation 41

63 results in a more intense breakup due to the higher relative velocities, increased drag forces and thus higher droplet accelerations in equation Breakup model From section in the literature review, it can be concluded that the WAVE and KH-RT model are the best possible candidates for the breakup model. As KH-RT can be seen as an extension of the WAVE model, the KH-RT model was the model of choice in this thesis. It also seems to be a bit more frequently used than the WAVE model in recent research papers. The KH-RT model cannot account for the effects of turbulence and cavitation in the nozzle, so some tuning of the breakup constants of this model was needed to obtain correct liquid length results. The values of these constants that were used here are given in table 4.5. The default values in ANSYS Fluent differ from the standard values in the ANSYS Theory Guide [11], so they are also included in the table. The default values were used for the calculations in this chapter and the tuned values in chapter 5. In this chapter, also the effect of increasing the KH breakup relative to the RT breakup while keeping the liquid length for the baseline case the same will be investigated. The conclusion is that this improves the response to parametric variations of ambient density and temperature. Table 4.5: Standard breakup constants of the KH-RT model compared to the tuned breakup constants B 0 B 1 C τ C RT C L standard [11] default tuned increased KH The KH-RT model has two competing breakup mechanisms. B 1 can be lowered to increase the breakup rate due to the KH instability and C RT can be lowered to increase the breakup rate due to the RT instability. In both cases, the liquid penetration decreases due to the increased total surface area of the smaller droplets. During the the tuning of the breakup constants, different sets of values for B 1 and C RT were tried. The effect on the liquid penetration of changing B 1 from 10 to 1.73 is shown in figure 4.5. The liquid penetration drops from 12.7 mm to 11.3 mm. After the initial transient, the time-dependent liquid penetration data fluctuates around these values. The experimental data also shows this fluctuating behaviour due to clusters of droplets that are constantly separating themselves from the liquid core of the jet. These clusters of droplets travel further in front of the jet and when 4.2 Discretisation 42

64 they totally evaporate the liquid penetration value falls back abruptly to a lower value. The numerical simulations exhibit the same behaviour (see figure 5.1), but the amplitudes of these fluctuations have a strong dependency on the values of the breakup constants and the rate at which parcels are injected. In figure 4.6, the result of changing C RT from 0.5 to 0.3 is shown. The liquid length decreases from 11.2 mm to 8.9 mm. It can be noticed the relative change in C RT is smaller than the relative change in B 1, but still the liquid length decreases more than in the case of changing B 1. From this, it can be concluded that in this case the breakup due to the RT instability is more important than the breakup due to the KH instability. A second remark is that the fluctuations of the liquid length increase with decreasing C RT. This is even more clear when comparing the curves of C RT = 0.5 and C RT = 0.3 in figure 4.6 to the curves in figure 4.5 which used a C RT value of 1. A possible explanation is that decreasing C RT increases the relative importance of RT breakup and consequently more clusters of equally sized droplets are separated from the liquid core of the jet. In WAVE breakup smaller child droplets are separated from a big parent droplet, while in RT breakup a parent droplet is split in a number of equally sized child droplets as illustrated in figure 3.5. This leads to a more gradual evaporation in case of WAVE breakup and more fluctuations in case of RT breakup. However, simulations that keep the mean liquid penetration equal and vary the relative importance of KH and RT breakup do not support this explanation. A simulation with B 1 = 0.5 and C RT = 0.5 resulted in almost the same liquid penetration behaviour as the simulation with B 1 = 1.73 and C RT = 0.3. The mean liquid penetration barely changed from 8.9 mm to 8.8 mm. The standard deviation of the fluctuations even slightly increased from 1.4 mm to 1.6 mm. The more gradual evaporation of small KH child droplets also has as a result that there are more big parent droplets in the liquid core of the spray. For shorter liquid lengths, these big droplets can reach the end of the liquid core more easily, separate themselves from this core, travel further in front of the jet and cause bigger fluctuations. This also means that in the case the relative importance of KH breakup is increased (B 1 = 0.5 and C RT = 0.5), the liquid part of the spray is less dispersed as the parcels with smaller child droplets quickly evaporate. This leads to a decreased overall parcel count and a higher overall Sauter diameter of the spray. Comparing the two sprays at 1.44 ms ASOI, the spray with increased KH breakup has 118 parcels in the domain with an overall Sauter diameter of 1.14 µm as opposed to the case with B 1 = 1.73 and C RT = 0.3 which has 108 parcels with an overall Sauter diameter of µm. 4.2 Discretisation 43

65 16 Liquid penetration LP [mm] B1=10 B1=1.73 experiment time ASOI [ms] Figure 4.5: Influence on the liquid penetration of the B 1 breakup constant in the KH-RT model (C RT = 1) 16 Liquid penetration LP [mm] C RT =0.5 C RT =0.3 experiment time ASOI [ms] Figure 4.6: Influence on the liquid penetration of the C RT breakup constant in the KH-RT model (B 1 = 1.73) Node based averaging Node based averaging is a technique to lower the grid dependence of simulations using a Lagrangian particle tracking approach. The subject of grid dependence in spray simulations will be handled in section 4.2.4, but here it is tried to see if it can lower the fluctuations in liquid length. In standard simulations without node based averaging, the particles act as a source term for the continuous phase cell in which they are in. This can produce locally very high values. By dividing the source term of the particle over the neighbouring cells depending on their distance, this can be 4.2 Discretisation 44

66 avoided. In this way, grid dependence is reduced and the overall solution is smoother. By looking at figure 4.7, it can be seen that it also has a clear effect on the liquid penetration. It has succeeded in its objective to lower the fluctuations, but the mean value has also risen. The smoother near-nozzle velocity distribution results in less turbulence being created. This reduces the evaporation rate and consequently also the liquid penetration, as will be explained in section 4.3. Although the fluctuations of the liquid penetration were closer to the experiment, node based averaging was not used in all the simulations further on, because it seemed less physically correct. Velocity differences in the near-nozzle region are an important source of turbulence. 14 Liquid penetration LP [mm] Standard Node Based Avg experiment time ASOI [ms] Figure 4.7: Effect of node based averaging on the liquid penetration Blob method As introduced in section 3.2.3, the Blob method consists in choosing the diameter of the injected droplets equal to the nozzle diameter. Injecting very big droplets is the closest way a discrete phase model can represent the liquid core of the jet. The liquid core concept of the KH-RT model even prevents excessive breakup of these very big droplets near the nozzle. This was illustrated in figure 3.3 with the breakup length L defined by equation Also, by fixing the initial droplet diameter, there is one constant less to be tuned in order to find agreement between experiments and simulations. Injecting droplets of 90 µm according to the Blob method instead of droplets of 1 µm (a typical size for droplets downstream of the nozzle) has an influence on a lot of parameters. First of all, the liquid core length is proportional to the reference diameter. The reference diameter is supposed to be equal to the nozzle diameter, so 4.2 Discretisation 45

67 only in case of the Blob method the liquid core concept is used in the correct way. Secondly, the Sauter mean diameter 1 is significantly higher in the first 3 mm of the spray. This leads to a lower heat transfer rate to the droplets. As a result, the droplets evaporate less quickly further downstream. Obviously, this means that the liquid penetration increases, as can be seen in figure 4.8. LP [mm] Liquid penetration Blob (90 µm) uniform 1 µm experiment time ASOI [ms] (a)liquid penetration data after start of injection evaporation rate [mg/s] Blob (90 µm) uniform 1 µm x [mm] (b)evaporation rate along the spray axis at 1.44 ms Figure 4.8: Comparison between the Blob method (90 µm at 1 particle per parcel) and a uniform injection of 1 µm droplets at 500 particles per parcel It also has a minor effect on the vapour penetration. The vapour penetration at 1.5 ms increased from 46 mm for the 1 µm injection to 48 mm for the 90 µm injection. This can be explained by the difference in density between liquid n-dodecane and n-dodecane vapour. In the case of the 1 µm injection, the droplets evaporate more quickly. The extra fuel vapour entrains a bit more air and in this way the vapour penetration is slowed down a bit more. Collision model Switching off the collision models did not alter the simulations results a lot, nor did it speed up the calculation time. The liquid penetration decreased by less than 1 mm, while the vapour penetration stayed virtually the same. Another trend that could be noticed was that vapour mass fractions near the nozzle (at less than half the liquid length) were generally higher when no collision model was used. The most noticeable influence is with regard to the residual errors when solving the 1 The Sauter mean diameter, SMD or d 32 of a group of droplets is equal to the diameter of a droplet with the same surface-to-volume ratio as the entire group of droplets. It is a measure for the fineness of a spray. 4.2 Discretisation 46

68 flow equations. For a simulation without collision model and using B 1 = 1.73 and C RT = 1 as breakup constants, the residual error of the continuity equation had more troubles converging. It was difficult to go below 1e-1. Nevertheless, the residuals still dropped by 6 orders of magnitude, but when compared to a similar simulation with a collision model, the residuals easily dropped below 2e-4. For this reason, the O Rourke collision algorithm was used in all subsequent simulations. Turbulent dispersion Section explained that for modelling the influence of the turbulent fluctuations on a discrete phase droplet, a random number proportional with the square root of the turbulent kinetic energy can be added to the mean flow velocities. This makes it easier for droplets to disperse away from the spray axis, hence the name turbulent dispersion. This enhances the evaporation because more droplets end up in hotter neighbouring cells with a lower vapour mass fraction. When turbulent dispersion was disabled while keeping all other settings the same, the liquid length was almost doubled. It went up from about 9 mm to almost 16 mm. The earlier mentioned fluctuations which are for example clearly visible in figure 4.6 decreased by disabling turbulent dispersion. Disabling turbulent dispersion did not only affect the liquid penetration but also the vapour penetration, especially in the early stages of the spray. Between 0.1 ms and 0.3 ms, the vapour penetrated noticeably further because the limited amount of evaporated diesel was not able to spread as much in the radial direction. This is illustrated in figure 4.9. Later the vapour penetration of the turbulent dispersion case catches up with the case without turbulent dispersion. After that their evolution is quasi identical. Also, the vapour mass fraction distribution in general became smoother with less local extrema along the spray axis. On a more general note, the same trend can be seen in temperature distributions, velocity fields and turbulent properties. 4.2 Discretisation 47

69 5 Vapour boundary at 0.2 ms y [mm] with turbulent dispersion without turbulent dispersion experiment x [mm] Figure 4.9: Effect of Turbulent dispersion on the vapour boundary at 0.2 ms Pressure-Velocity Coupling As can be expected, changing the pressure-velocity coupling scheme had no noticeable effect on the solution. Different schemes mainly influenced the rate at which the calculations converged and some had more stability issues than others. The coupled solver converged faster than the pressure-based segregated algorithms (SIMPLE, SIMPLEC, PISO,...) because the coupled algorithm solves the momentum and continuity equation at the same time. The pressure-based segregated algorithms solve the momentum and continuity equation separately, using a predictor-corrector approach. This results in slower convergence, but it generally has a superior stability. Introducing the blob method and temperature-dependent fluid properties made convergence of the coupled solver more difficult, so the SIMPLE algorithm was the pressure-velocity coupling scheme of choice. However, when looking back on this choice, the very small gain in accuracy did not outweigh the additional calculation time Mesh dependence Solutions of CFD calculations can be dependent on the mesh. Coarse meshes are often unable to accurately describe certain flow configurations. If these coarse meshes are refined, some physical quantities of interest may change in value. The solution is said to be mesh or grid dependent. In a lot of CFD calculations, there exists a point after which refining the mesh no longer changes the values of interest. 4.2 Discretisation 48

70 The solution is said to be mesh or grid independent. To accurately predict physical flows, users of CFD programs should strive to obtain a solution that is as grid independent as possible. Lucchini et al.[5] accurately explain the importance of grid dependence in spray simulation. Grid dependency plays also a very important role in spray simulation, since coarse meshes are not able to correctly describe the interaction between the liquid and gas phase causing an underestimation of the spray penetration, while excessively refined meshes lead to an unphysical fast diffusion of momentum from the liquid to the gas phase, resulting in high gas velocities and spray penetration. Indeed, when a droplet exchanges momentum with a very small cell, the velocity of the continuous gas phase shoots up quickly. This leads to a lower relative velocity between the gas and liquid phase, and the spray penetrates even further. Following this reasoning, one can understand why there exists a lower limit for the used cell size in a Eulerian-Lagrangian approach (section 3.1.1). To limit the grid dependence, Lucchini et al.[5] recommend the use of a spray-adapted grid, i.e. a grid where the spray penetrates each cell as perpendicularly as possible. In the mesh used in this thesis, the conical shape of the structured mesh helps to achieve this. When using a general mesh, the relative position of the spray axis and grid influences the results because it changes the number of cells with which the liquid phase exchanges momentum. To investigate the influence of the mesh on the solution, a finer mesh was created. The 3 dimensions of the cells in the zones core and fine as defined in table 4.3 were halved, leading to an 8 times higher cell count. The chamber zone using an unstructured mesh remained unchanged, as the spray always stays in the structured part. The total cell count of the fine mesh ended up being The main physical quantities of interest in the context of spray simulation are the liquid penetration and the vapour boundary. The latter is the isosurface of 0.1% vapour mass fraction and it defines the vapour penetration and the spray angle 2. The effect of refining the mesh on the liquid penetration is shown in figure The simulation in the refined mesh shows a slightly higher liquid penetration, as could be expected from the explanation in the last paragraph. Another thing worth mentioning is that as the volume fractions of the discrete phase increased by refining the mesh, the convergence of the continuity equation became more difficult. 2 The local spray angle in a point on the vapour boundary can be defined as the angle between the spray axis and a line connecting this point and the centre of the injector nozzle hole. It is an important indicator for the rate at which the spray mixes with air. 4.2 Discretisation 49

71 Figure 4.11 shows the effect of refining the mesh on the vapour boundary. The vapour boundary remains almost identical after refining the mesh. The solution cannot be called fully grid independent, but is sufficient for our purposes. This is particularly true considering the difficulty of obtaining a grid independent solution for spray simulations. 14 Liquid penetration LP [mm] normal fine time ASOI [ms] Figure 4.10: Liquid penetration for different levels of refinement of the mesh 15 Vapour boundary at 1.44 ms 10 5 y [mm] normal fine x [mm] Figure 4.11: Vapour boundary at 1.44 ms for different levels of refinement of the mesh Time step dependence The solution can also depend on the time step. For explicit time schemes, one can intuitively understand that if flow travels across more than one cell per time step, stability issues will occur. This condition corresponds to a Courant number of 1, see 4.2 Discretisation 50

72 equation 3.2. Here, implicit time schemes are used, so a strict upper limit does not exist. The accuracy of the calculation however will suffer if too large time steps are chosen. Therefore, the same simulation was run using different time steps and the liquid and vapour penetrations were compared. At first, the liquid penetration seemed to decrease for decreasing time steps. The cause was the standard setting of ANSYS Fluent to inject only 1 parcel per time step. When this setting is used, the smaller the time step is, the bigger the rate at which particles are injected in the domain and the lower the number of particles per parcel. This number needs to be sufficiently high for an accurate solution and it also influences the amplitude of the time fluctuations of the liquid penetration. This effect was studied in more detail by Johnson et al. [36]. The Blob method imposes a lower limit on this number because at injection, parcels of 1 big particle per parcel are injected and the overall particles-per-parcel count is governed by the breakup algorithm. The time step dependence study was repeated with a constant number of particles per parcel and this time the results of the simulations with time steps smaller or equal to 4e-7 s were virtually the same. The results for the liquid penetration are shown in figure 4.12 and for the vapour penetration in figure The simulation with a time step of 1e-7 s was stopped a bit earlier at around 1 ms, but the trend remains clear. 18 Liquid penetration LP [mm] e-7 4e-7 2e-7 1e time ASOI [ms] Figure 4.12: Liquid penetration for different time steps 4.2 Discretisation 51

73 50 Vapour penetration VP [mm] e-7 4e-7 2e-7 1e time ASOI [ms] Figure 4.13: Vapour penetration for different time steps 4.3 Choice of turbulence model The choice of turbulence model is arguably the most defining choice for spray simulation. Almost every submodel relies directly or indirectly on information provided by the turbulence model. Therefore, various turbulence models were tested using the standard parameters to see which one gives qualitatively the most accurate results. If the basic shape of the solution corresponds to the experimental curves, there is always the possibility of changing parameters of the turbulence model to account for the unknown effects of turbulence in the nozzle. The parameters of turbulence models are calibrated on multiple basic flow configurations, but these values do not always give the correct solution in more complex flows. A slight variation is often allowed. It comes as no surprise that the choice of turbulence model heavily influenced the vapour penetration. Figure 4.15a shows the time evolution of the vapour penetration for various turbulence models which were already introduced in section 2.2: the Spalart-Allmaras turbulence model, the k-ω SST model, the Reynolds Stress model and the standard, RNG and Realisable k-ɛ models. The calculations with the Reynolds Stress model became unstable after a while, so the results from this model are not presented here. A simulation with the dynamic Smagorinsky LES model in a full cubic mesh was also tried, but the results were too grid-dependent, indicating that the mesh was too coarse for LES calculations. The 3 k-ɛ models gave rather similar results except for the overestimated vapour penetration values right after the start of injection for the k-ɛ RNG model. The overestimation is linked to the initial overshoot of liquid n-dodecane in the initial 0.1 ms after the start of injection (ASOI) that 4.3 Choice of turbulence model 52

74 can be seen in figure The vapour penetration of the Spalart-Allmaras model is underestimated due to a severely overestimated turbulence viscosity ratio as shown in figure 4.15i. This leads to an overestimated momentum exchange with the surrounding stagnant air. The entrainment of more air leads to a quicker spreading rate in the radial direction (y) as can be seen on the vapour boundary plot 4.15b. Figure 4.15c shows that the n-dodecane vapour is transported away from the spray axis, resulting in lower vapour mass fraction along the spray axis. The spray is also slowed down more quickly in the axial direction, as illustrated in figure 4.15f. The Spalart-Allmaras model was never designed to simulate jet-like flows, so these inaccurate results were to be expected. The opposite was true for the k-ω SST model. The vapour penetration was severely overestimated. After 1 ms, it even reaches the other side of the combustion chamber. This was a surprising result because the k-ω SST model is designed to work as a k-ɛ model far away from wall boundary layers. The severe overestimation of the vapour penetration can be attributed to a heavily underestimated turbulent kinetic energy and consequently also a underestimated turbulent viscosity ratio. This can be seen in figure 4.15g and figure 4.15i. The diesel spray shoots out forward out of the nozzle while barely mixing with the surrounding stagnant air. The liquid penetration is also significantly higher as for the k-ɛ models. There are 4 important reasons why lower levels of turbulence lead to higher liquid lengths. 1. The gas phase along the spray axis gets slowed down less by the surrounding stagnant air. As a result droplets get transported away from the nozzle at higher speeds. 2. The core of the spray cools down because of the evaporation of fuel droplets. Less hot surrounding air is mixed in with the spray, resulting in less available energy for evaporation. 3. The fuel vapour stays concentrated along the spray axis, close to the fuel droplets. An atmosphere with a higher vapour mass fraction makes it more difficult for fuel droplets to evaporate. 4. Since the effect of turbulent dispersion is included in the simulation, higher turbulent kinetic energies lead to higher fluctuating fluid velocities, see equation This means that droplets can travel further away from the spray axis to hotter neighbouring cells and evaporate more quickly. In the end, the realisable k-ɛ model was chosen because it qualitatively matched the experimental data the best. The liquid penetration was predicted without a large overshoot, see figure After the initial transient, the liquid length quickly evolved to a steady state regime, just as experimental results show. The evolution of the vapour penetration shows the same trend as the experimental data and it 4.3 Choice of turbulence model 53

75 seems like the simulation can match the experiments, provided that the turbulent dissipation is slightly increased. 70 Liquid penetration LP [mm] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST experiment time ASOI [ms] Figure 4.14: Effect of the turbulence model on the liquid penetration VP [mm] Vapour penetration k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST experiment y [mm] Vapour boundary at 1.44 ms k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST experiment time ASOI [ms] (a)evolution of the vapour penetration in time x [mm] (b)vapour boundary at 1.44 ms 4.3 Choice of turbulence model 54

76 mass fraction [%] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST mass fraction [%] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST x [mm] (c)vapour mass fraction along the spray axis at 1.44 ms y [mm] (d)radial plot of vapour mass fraction at 80% of the vapour penetration at 1.44 ms velocity [m/s] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST x-velocity [m/s] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST x [mm] (e)velocity of the fuel droplets along the spray axis at 1.44 ms x [mm] (f)axial velocity of the gas phase along the spray axis at 1.44 ms k [m 2 /s 2 ] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST [10 6 m 2 /s 3 ] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST x [mm] (g)turbulent kinetic energy along the spray axis at 1.44 ms, No data available for Spalart-Allmaras x [mm] (h)turbulent dissipation rate along the spray axis at 1.44 ms, No data available for Spalart-Allmaras 4.3 Choice of turbulence model 55

77 turbulent viscosity ratio [-] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST x [mm] (i)turbulent viscosity ratio along the spray axis at 1.44 ms temperature [K] k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST x [mm] (j)temperature of the gas phase along the spray axis at 1.44 ms Figure 4.15: Effect of the turbulence model on various properties Adjusting parameter C 2ɛ of the Realisable k-ɛ model Figure 4.15b showed the effect of overestimating and underestimating the turbulence. The realisable k-ɛ model predicted a vapour penetration that was slightly lower as in the experiment. If the predicted turbulent viscosity ratio 3 was lower, the spray would penetrate further, matching the experiment better. A lower turbulent viscosity ratio corresponds to a lower turbulent kinetic energy. Turbulent kinetic energy is produced in regions with high shear rates (equation 2.15) 4. Changing C µ would directly influence the rate at which turbulent kinetic energy is produced from shear rates, but this would violate a condition of realisability of the realisable k-ɛ model. Also, the value of C µ is quite universal for a wide range of flows. On the other hand, C 2ɛ can be slightly modified. It determines the rate at which the turbulent dissipation ratio ɛ decays. So for lower values of C 2ɛ, the turbulent dissipation rate is higher, resulting in a lower turbulent kinetic energy and turbulent viscosity. Figure 4.16 shows the effect on the liquid and vapour penetration of changing C 2ɛ from the standard value of 1.9 to a value of ν t/ν with ν t as defined in equation It should be noted that the realisable k-ɛ model uses transport equations which are not entirely the same as the standard model. Still, the same qualitative results of changing model constants apply for the realisable k-ɛ model. 4.3 Choice of turbulence model 56

78 LP [mm] Liquid penetration C 2 =1.9 C 2 =1.8 experiment time ASOI [ms] (a)evolution of the liquid penetration in time y [mm] Vapour boundary at 1.44 ms C 2 =1.9 C 2 =1.8 experiment x [mm] (b)vapour boundary at 1.44 ms ASOI Figure 4.16: Liquid penetration and vapour boundary for different values of C 2ɛ The vapour boundary now nearly coincides with the experimental vapour boundary. Another consequence is that the liquid penetration has increased by a few mm due to the 4 reasons explained earlier in this section. This can be counteracted by tuning the breakup constants of the KH-RT model in table Influence of fluid properties and initial conditions Sensitivity analysis A sensitivity analysis was performed by changing various fluid properties, initial conditions and other parameters. The goal was to find which ones influenced the result the most, so that extra care could be taken to provide accurate input. For example, some constant fluid properties can be changed to be temperaturedependent. The fluid properties and initial conditions that were taken into consideration are the following. The viscosity of liquid dodecane was doubled. The thermal conductivity of the air-dodecane vapour mixture was lowered. The initial turbulent kinetic energy was changed to This corresponds to ambient velocities of around 0.05 m/s when a turbulent intensity u /u of 5% is used in equation The initial turbulent dissipation rate was changed to an arbitrarily high value. The surface tension of liquid dodecane was lowered because a large drop from the value at room temperature was expected. The turbulent Schmidt number and Prandtl number were also changed. The values of the reference case and the changed case are given in table 4.6. The fluid properties of the reference 4.4 Influence of fluid properties and initial conditions 57

79 case correspond to the values of kerosene (C 12 H 23 ) in the ANSYS Fluent Material database. Table 4.6: Change in values of various fluid properties and initial conditions for the sensitivity analysis Property reference changed to Viscosity [Pa.s] 1.72E E-05 Thermal conductivity [W/(m.K)] Initial TKE [m²/s²] Initial TDR [m²/s³] Surface tension [N/m] Turbulent Schmidt Prandtl (Energy and Wall) The results of these simulations are summarised in figure The change in properties mainly influenced the liquid penetration, while the vapour penetration was almost the same in all cases. The most noticeable changes were seen for the surface tension and the thermal conductivity. The latter directly influences the evaporation rate by determining the heat transfer rate with the surrounding hot air. The surface tension is the restoring force which keeps the droplets spherical. If there is less surface tension, the droplets break up more easily. This is also reflected in the equations. It influences both the KH breakup and the RT breakup. Considering equation 3.18 and equation 3.13, the predicted RT wavelength scales to σ and for the conditions in this problem, the KH wavelength scales approximately to σ 0.8. Some minor effects could also be seen. The decrease of the turbulent Schmidt number led to a small decrease of vapour mass fraction values on the spray axis together with an almost negligible increase in spray angle of 1 to 2. These changes can be attributed to the higher turbulent mass transfer. As given in equation 4.3, the turbulent Schmidt number defines the coupling between the turbulent hydrodynamics and the mass transfer, just like the turbulent Prandtl number does for the heat transfer. ν t is the turbulent viscosity and D t is the turbulent mass diffusion coefficient. Sc t = ν t D t (4.3) A second minor effect was that the increase in viscosity resulted in an increase in liquid length of less than 1 mm. The effect is relatively smaller than the surface tension because the change in viscosity only affects the KH breakup. The Ohnesorge and Taylor numbers in equation 3.13 are directly proportional to the viscosity. For the given conditions, the predicted KH wavelength scales approximately to the viscosity to the 0.2 power. 4.4 Influence of fluid properties and initial conditions 58

80 18 Liquid penetration LP [mm] reference viscosity thermal conductivity initial TKE initial TDR surface tension Schmidt number Prandtl number experiment time ASOI [ms] Figure 4.17: Influence of various fluid properties and initial conditions on the liquid penetration Temperature-dependent fluid properties Some fluid properties influence the liquid penetration more than others. If these properties vary a lot with temperature, introducing temperature-dependent properties will make the simulation more accurate. In fact, this is true for all properties, but the additional calculation cost has to be weighed up against the improved accuracy. Therefore, the temperature-dependence of various properties has been studied using the database of material properties of the National Institute of Standards and Technology (NIST)[17]. Liquid n-dodecane The density of liquid n-dodecane varies between 704 kg/m³ and 436 kg/m³ in the temperature range 363 K K at a pressure of 60 bar. This means that a droplet increases in size when it is heated up. Consequently, the drag forces on the droplet increase as well. To account for this effect, the density was made temperaturedependent. The 7 density-temperature couples from table 4.7 were used to construct a piecewise linear density-temperature function. Outside of this temperature range, the density was taken constant equal to the value at 363 K or 663 K. Table 4.7: Isobaric data of the density for dodecane at 60 bar [17] Temperature [K] Density [kg/m³] Influence of fluid properties and initial conditions 59

81 As demonstrated in the previous section, the surface tension plays an important role in droplet breakup. Additionally, the surface tension varies between 20.3 mn/m to 0 N/m in a temperature range of 350 K to 659 K for saturated dodecane. Indeed, dodecane becomes supercritical at temperatures above 659 K at a pressure of 60 bar. No interface between the liquid phase and the gas phase can be distinguished and the surface tension disappears. Simulations indicate that the droplets evaporate before reaching the critical temperature, but the surface tension still drops quickly at higher temperatures. Hence, the surface tension was also made temperature dependent by constructing a piecewise linear surface tension-temperature function based on the data in table 4.8. Table 4.8: Temperature-dependent surface tension data for saturated dodecane [17] T[K] σ[mn/m] Temperature-dependent saturated vapour pressure data for kerosene (C 12 H 23 ) was already available in the ANSYS Fluent material database and its values did not differ a lot from the values provided by the NIST. It is only used to determine C i,s in equation 3.30, so the values of ANSYS are sufficiently accurate. For the other properties of liquid n-dodecane, a constant value representative for the entire temperature range was chosen. Either this value did not vary a lot over the temperature range or the value did not considerably influence the results of the simulations. Droplets often have a temperature around 500 K, so 513 K and 6 MPa were chosen as a representative condition. Values that correspond as closely as possible to this condition were extracted from the database of the NIST and are presented in table 4.9. Table 4.9: Other properties of liquid dodecane latent heat viscosity c p J/(kg.K) Pa.s J/(kg.K) N-dodecane-air mixture The air mixture initially present in the combustion chamber is approximated by N 2 as an ideal gas. The n-dodecane vapour is approximated by kerosene vapour (C 12 H 23 ) which is present in the ANSYS Fluent material database. The thermal conductivity of the air-fuel mixture is calculated by a mass weighted mixing law with a constant thermal conductivity of W/(m.K) for N 2 at 6 MPa and 900 K 4.4 Influence of fluid properties and initial conditions 60

82 (supercritical) and a constant thermal conductivity of W/(m.K) for dodecane at 700 K and 6 MPa (supercritical). 4.5 Proposed methodology for fuel spray simulations This section proposes a methodology for simulating fuel sprays using RANS turbulence models and a Lagrangian particle tracking approach. If high-performance computing equipment is available and a more accurate solution is desirable, it might be a better idea to combine an LES turbulence model and a Eulerian phase model in a finer mesh. This methodology does not claim to be the most efficient way of setting up the simulation, but it provides a quick way to get fairly accurate results. 1. Create a structured mesh in such a way that the spray penetrates the cells as perpendicularly as possible. The minimum cell size should be bigger than the diameter of the injector nozzle hole. Finer meshes will have trouble converging. Coarser meshes provide a less accurate solution, but it takes less time to calculate this solution. For optimal use of the computational resources, cells should become bigger the further away from the nozzle hole and spray axis. Regions where the spray does not penetrate can be made even coarser. 3D meshes are often better at capturing the spray formation. Symmetry can be used to reduce the computational load. More information can be found in sections and Choose a turbulence model. The k-ɛ models probably give the most satisfactory results. Among these models, one can choose the realisable k-ɛ model for its ability to better predict the spreading rate of a round jet. It is recommendable to try more than one turbulence model since it is one of the most defining choices in spray simulations. 3. Define the injection according to the Blob method (section 3.2.3). Droplets with the size of the injector nozzle hole are injected each time step. Because the droplets are relatively large, the injection can be safely set to 1 particle per parcel without risking excessive calculation times 5. The velocity can be calculated based on the pressure difference using Bernoulli or if accurate mass flow rate data is available, the velocity can be calculated based on the liquid density and the nozzle area. Measuring the velocity experimentally is another 5 At least in ANSYS Fluent, where the amount of new parcels generated due to breakup is limited. Only when the shed mass accumulates to 5% of the mass of the initial droplet, a new parcel is created. 4.5 Proposed methodology for fuel spray simulations 61

83 option, e.g. by writing out a momentum balance for a plate held in front of the injector. 4. Decide which discrete phase models to use. As a breakup model, KH-RT is often used for high-pressure fuel sprays. WAVE is a good alternative. A drag model should also be enabled. The use of the dynamic drag model is probably the most widespread. Collision and turbulent dispersion models can be omitted without much loss of accuracy if the breakup constants are retuned. Their effect on the calculation time is limited, so there are also no strong reasons not to use them. 5. Enter the correct boundary conditions and initial conditions. No inlet boundary conditions have to be foreseen because the fuel is introduced using a discrete phase mass source. Pay special attention to the ambient pressure and temperature as these define the ambient density, an important factor in spray penetration. 6. Because the minimum cell dimensions are limited in a Lagrangian particle tracking approach, higher order spatial discretisation schemes should be used to obtain a more accurate result. These higher order schemes generally lead to a more difficult convergence, so second order schemes are a good compromise. 7. The time step should be chosen small enough to have a sufficient temporal resolution, but it should also be small enough to resolve transient effects. This can be done by ensuring a maximum Courant number as defined in equation is a rather conservative value and provides a good initial value for the time step, but higher values should be tried to see if calculation time can be saved without a noticeable loss in accuracy. 8. Check the mesh dependence by recalculating the solution in a finer mesh. Take care to lower the time step to keep the Courant number constant. The vapour boundaries should be almost identical. However, a small increase in liquid penetration should be expected. 9. Especially if the model needs to be used in a relatively wide temperature range, consider adding temperature-dependent fluid properties, starting with the surface tension, liquid density and thermal conductivity of the gases. The viscosity also has a minor effect. See section Several flow phenomena in the nozzle cannot be accounted for in the Blob method. Therefore, tuning model constants is needed to obtain agreement between simulations and experiments. First, some model constants of the 4.5 Proposed methodology for fuel spray simulations 62

84 turbulence model should be appropriately tuned. There are model constants that should not be changed since they are valid in almost every flow. Other model constants are determined by the other model constants to satisfy certain consistency conditions. For the realisable k-ɛ model, a slight change in C 2ɛ should be allowed. If the liquid penetration is overestimated, the vapour penetration might still decrease a few millimetre when the breakup constants are tuned and vice versa. The breakup constants should be tuned by changing B 1 and C RT. Higher values result in a reduced breakup rate and a higher liquid penetration. Typical ranges for these constants are 1-60 for B 1 and for C RT. The ratio of B 1 to C RT can be chosen in such a way that the simulation responds well to ambient density and temperature variations, as will be discussed in section Proposed methodology for fuel spray simulations 63

85 Discussion of results 5 Now that the simulation is properly set up, its results can be analysed, starting with the baseline case of ECN spray A (section 4.1). In the first section of this chapter, there will be focused on the vapour penetration, vapour mass fraction distribution, momentum, streamlines and other things related to the mixture of air and fuel vapour. In the next section, the focus will shift towards the near-nozzle region where the liquid penetration, the volume fraction, the breakup mechanism and so on will be studied in more detail. Section 5.3 will be dedicated to the parametric variations of ECN spray A. The model as described in the previous chapter was used to calculate these parametric variations. The model constants and fluid properties remained unchanged, only the appropriate boundary conditions from table 4.4 were applied. By comparing the results to the experimental parametric variations, one can see how accurate the model follows changes in injection pressure, ambient density and ambient temperature. In this way, the validity of the modelling approach can be checked. Experimental data was not always available for every parametric variation of ECN spray A, so two other well-documented cases on the website of ECN[2] were also simulated. In these cases, both the ambient density and temperature are varied at the same time compared to the baseline condition of ECN spray A. The results show that vapour penetration lengths are well reproduced. The liquid penetration showed all the right trends but was sometimes quantitatively a bit off. Increasing the relative importance of KH breakup improved the ability of the simulation to follow ambient density and temperature variations. 5.1 Spray penetration Evolution of the vapour boundary Quite a few graphs of the liquid or vapour penetration as a function of time were already shown in the previous chapter. As defined in section 4.1, the liquid and vapour penetration are defined based on isosurfaces of volume fraction and vapour mass fraction, so it might be interesting to show the development of these liquid and vapour boundaries in 2D. This is the purpose of figure 5.1. Additionally, the experimental vapour boundary is also shown to be able to compare it to the simulation. The vapour boundaries show a good agreement. The vapour 64

86 Vapour and liquid boundary at 0.02 ms Vapour and liquid boundary at 0.5 ms y [mm] 0 y [mm] x [mm] x [mm] Vapour and liquid boundary at 0.04 ms Vapour and liquid boundary at 0.8 ms y [mm] 0 y [mm] x [mm] x [mm] Vapour and liquid boundary at 0.1 ms 10 5 y [mm] x [mm] Vapour and liquid boundary at 0.24 ms Vapour and liquid boundary at 1.5 ms y [mm] 0 y [mm] x [mm] x [mm] Figure 5.1: Vapour and liquid boundary at 0.02 ms, 0.04 ms, 0.1 ms, 0.24 ms, 0.5 ms, 0.8 ms, 1.08 ms and 1.5 ms: blue line = vapour boundary of the simulation, black dotted line = experimental vapour boundary, edge of blue surface = liquid boundary of the simulation 5.1 Spray penetration 65

87 penetration, as well as the local spray angles, are reproduced accurately for the different time steps. The experimental vapour boundary corresponds to 1 experiment, thus the turbulent fluctuations explain why it has a more irregular shape. If an average of a sufficient number of experiments is taken, the turbulent fluctuations would be filtered out, and a smooth vapour boundary as in the RANS simulations would be found. During the first few µs, the liquid fuel shoots out of the nozzle. At the surface of fuel droplets, there is always some fuel vapour present, so a sudden rise can be seen in liquid and vapour penetration graphs. At 0.04 ms, the fuel the furthest away from the nozzle really starts to evaporate and the difference between the vapour and liquid penetration starts to grow. At 0.24 ms, the high-speed fuel spray starts to entrain more air and the fuel vapour spreads out axially and radially. At the same time, the liquid penetration reaches an equilibrium length because the fuel evaporates at nearly the same rate as it is injected through the nozzle. Some small fluctuations around this equilibrium value are possible, as already explained in section A cluster of droplets separating itself from the liquid core can be observed at 0.5 ms. Later, this cluster evaporates and the liquid penetration drops abruptly. At 1.08 ms, even an entirely separated cluster can be seen Vapour mass fraction distribution Figure 5.2 shows the mass fraction distribution of n-dodecane vapour at 1.44 ms after the start of injection. The mesh is shown for spatial reference. From table 4.3, it is known that a cell is mm long in the axial direction, so the maximum vapour Figure 5.2: Vapour mass fraction distribution at 1.44 ms ASOI 5.1 Spray penetration 66

88 mass fraction of 41% is situated at 6 mm from the nozzle. This is a bit upstream of the liquid boundary where the bulk of the evaporation takes place. Downstream of this location, the vapour mass fraction is diluted by the entrained air. The isoline of 0.1% vapour mass fraction defines the vapour boundary. It delineates a similar shape as in figure Streamlines and velocity field Streamlines are lines that are tangent to the velocity vectors at a given instant. These can be approximately visualised by following massless particles in the flow field at that given instant. Strictly speaking, pathlines are the result of this calculation, but they also give a good view of the flow field as can be seen in figure 5.3. Close to the nozzle, the entrained air enters the spray perpendicularly to the spray axis, just like a fully developed turbulent jet. Near the tip of the spray, a transient phenomenon is observed. A toroidal vortex ring has been developed, moving air away from the area in front of the spray and feeding it to the spray in a similar way as close to the nozzle. The maximum velocity of the continuous gas phase is 324 m/s and is located near the nozzle. This value is somewhat lower as the injection velocity of 569 m/s. The value of 324 m/s is not a very meaningful value because smaller cells will lead to a higher value due to increased importance of the momentum sources of the discrete phase droplets, as already mentioned in section In reality, the fuel vapour at the surface of the droplets moves at roughly the same speed as the droplets. The value of 324 m/s does, however, indicate that its better to take compressibility Figure 5.3: Pathlines coloured by velocity magnitude at 1.44 ms ASOI 5.1 Spray penetration 67

89 into account for the turbulence model, as Mach numbers are close to unity. It also shows that the choice of defining a Courant number based on the injection velocity was rather conservative (table 3.1). Downstream of the maximum, the velocity in sections at different distances from the nozzle decreases rapidly when moving away from the nozzle. At the same time, the mass flow rate through these section increases when moving away from the nozzle due to the entrainment of air. The combined effect is that the momentum flow rate stays almost constant Momentum In a fully developed turbulent jet, the momentum flow rate stays almost constant along the spray axis. This is because the pressure field is approximately uniform, and thus there is a conservation of momentum flow rate. Due to the resemblance of a turbulent round jet and a diesel spray, it can be interesting to see if this observation still holds true for diesel sprays. In order to find the momentum flow rate M(x) through a plane A orthogonal to the spray axis, the axial momentum flux ρu 2 is integrated over this plane. For this integration, the negative axial velocities u due to the tip vortex were neglected. M(x) = ρ g u 2 da (5.1) In this equation, the gas density ρ g and u are location-dependent. Figure 5.4 shows the result of this integral over planes at different distances x from the nozzle. The momentum enters the domain through the nozzle at a rate of ρ l u 2 inj πd2 /4 = 1.45 kg.m/s 2 with d the diameter of the nozzle, ρ l the liquid density and u inj the A along spray axis at inlet Momentum along spray axis Momentum [kg m/s²] x [mm] Figure 5.4: Momentum flow rate through planes orthogonal to the spray axis placed at different distances from the nozzle at 1.44 ms ASOI 5.1 Spray penetration 68

90 injection velocity from table 4.4. The value of 1.45 kg.m/s 2 is also drawn on the figure as a reference. Close to the nozzle where the fuel has not been totally evaporated yet, the momentum flow rate is low and rises steadily, as the liquid phase gradually transfer its momentum to the gas phase. From there on to around 35 mm where the influence of the vortex becomes noticeable, the momentum flow rate stays approximately constant. The recirculating air in the vortex creates additional momentum at the tip of the spray because the negative velocities are neglected. At even bigger distances, the momentum flow rate drops quickly because this zone contains nearly stagnant air that is pushed forward by the spray Temperature As shown in figure 5.5, the temperature field looks similar as the vapour mass fraction. The evaporation of droplets cools down the zone near the nozzle to about 446 K. Further downstream, the mixing with the surrounding hot air gradually heats up the air-fuel vapour mixture. Figure 5.5: Temperature of the gas phase at 1.44 ms ASOI In figure 5.6, temperatures of the discrete phase droplets are shown at 1.44 ms after the start of injection. It can be seen that the droplets gradually warm up from the temperature of the fuel at injection (363 K) to around 400 K K due to the heat exchange with the hotter surrounding air. The evaporation of the droplets limits the temperature rise. The simulation indicates that the droplets remain below the critical temperature, even at higher ambient temperatures. However, experiments show the existence of a supercritical mixing layer for ECN spray A [37]. The observation that the n-dodecane droplets stay subcritical in the simulation can be explained by the 5.1 Spray penetration 69

91 fact that a Lagrangian particle tracking approach is unable to represent supercritical droplets. When the droplets approach the critical temperature, the surface tension drops quickly and the breakup is accelerated. Figure 5.6: Temperatures of the discrete phase droplets at 1.44 ms ASOI 5.2 Near-nozzle region Volume fraction of liquid n-dodecane The volume of the discrete phase droplets in a cell can be compared to the volume of the cell itself to find the volume fraction of liquid n-dodecane. Figure 5.7 shows the volume fraction in the cells near the nozzle. Once again, the mesh is shown for Figure 5.7: Volume fraction of n-dodecane droplets in the cells near the nozzle at 1.44 ms ASOI 5.2 Near-nozzle region 70

92 spatial reference. Also here, the tendency of the droplets to separate themselves from the liquid core in clusters is clearly visible. Only the cells bordering the spray axis are cool enough to sustain a considerable amount of liquid droplets. Droplets that do make it to the neighbouring cells will evaporate almost instantly. In the figure, some cells with a volume fraction of 12.5 % are found. This value is quite high for a Lagrangian particle tracking method (section 3.1.1) but still acceptable Droplet size distribution The volume fraction of droplets is not the only interesting characteristic of the liquid part of the spray. The surface-to-volume ratio of the droplets also plays an important role in the evaporation process. The heat transfer to the droplets is proportional to its surface and the heat capacity of the droplet is proportional to its volume. Hence, the temperature of the droplets changes according to their surface-to-volume ratio. This implies that smaller droplets heat up and evaporate more quickly. Smaller droplets also slow down more quickly because the drag forces scale approximately quadratically to the droplet diameter and the mass cubically. The acceleration is equal to the ratio of the drag force and the mass. This means that the acceleration of droplets is also proportional to the surface-to-volume ratio. Therefore, it is useful to study different statistics that quantify the diameters, surface and volume of the spray, as given in table 5.1. Table 5.1: Droplet size statistics at 1.44 ms ASOI Total number of parcels 108 Total number of particles 3.23e+08 Total mass µg Maximum RMS distance from injector mm Maximum particle diameter µm Minimum particle diameter µm Overall mean diameter (D 10 ) µm Overall mean surface area (D 20 ) µm Overall mean volume (D 30 ) µm Overall surface diameter (D 21 ) µm Overall volume diameter (D 31 ) µm Overall Sauter diameter (D 32 ) µm Overall De Brouckere diameter (D 43 ) µm 5.2 Near-nozzle region 71

93 The mean diameters D jk are calculated from the following expression in its discrete form. D jk = ( i=1 n i D j i i=1 n i D k i ) 1 j k (5.2) Here, n i stands for the number of droplets of diameter D i, and the sum goes over the entire range of droplet diameters. The resulting values are rather small compared to experimentally measured values. Then again, current experimental measurements do not always allow to measure the smallest droplets. Also, Lagrangian particles cannot fully represent the liquid core of a jet as can be seen in figure 5.8 by looking at the red, orange and yellow droplets. These are the big blobs with a diameter close to the nozzle diameter. The secondary breakup model breaks down these blobs rather quickly. One final remark is that at higher temperatures and pressures, these concepts become obsolete. As the fuel reaches its critical temperature, no liquid surface can be distinguished and the concept of a droplet fades away. Figure 5.8: Droplets coloured by particle diameter with a scaled diameter representation (top) and a constant diameter representation (bottom) Breakup mechanisms Figure 5.9 compares the KH breakup time (equation 3.16) and the KH-RT breakup time (minimum of equation 3.16 and 3.19). Due to the nature of the KH-RT model, the KH-RT model will always have a higher breakup rate as the WAVE model when the same value for B 1 is used. By comparing figure 5.8 and 5.9, it can be seen that this is especially true for the bigger droplets, but the blobs near the nozzle still break up according to the WAVE breakup because they are within the liquid core length of 2.85 mm (equation 3.21). Beyond the liquid length at about 1/3 of the distance between the nozzle and the furthest droplet, there is a big cluster of droplets where RT breakup is clearly dominant. In this zone, a lot of droplets of intermediary sizes are created whereas in the liquid core big parent droplets are surrounded by small child droplets. 5.2 Near-nozzle region 72

94 5.3 Parametric variations Injection pressure Figure 5.10 shows the evolution of the vapour penetration for the first 3 cases in table 4.4, i.e. a variation in injection pressure. The corresponding data of the 3 experiments is overlaid on the graph. The fuel vapour penetrates further into the domain for increasing injection pressures due to the increased momentum of the fuel leaving the nozzle. The general shape of the curves are very similar except for the slight kink in the curves of the simulation at around 0.1 ms. The kink is more pronounced at lower injection pressures, but it seems to gradually disappear when using smaller time steps. Quantitatively, the baseline case of 150 MPa and the case Figure 5.9: Comparison of the WAVE breakup times (top) and KH-RT breakup times (bottom) 60 Vapour penetration VP [mm] MPa 100 MPa 50 MPa experiments time ASOI [ms] Figure 5.10: Vapour penetration for different injection pressures 5.3 Parametric variations 73

95 Vapour boundary at 1.44 ms 10 5 y [mm] MPa 100 MPa 50 MPa experiments x [mm] Figure 5.11: Vapour boundary for different injection pressures at 1.44 ms ASOI of 50 MPa are very close, but the case of 100 MPa is somewhat off. This can be explained as follows. The simulation of 150 MPa was tuned to fit an experiment at MPa. The simulation at 50 MPa corresponds to an experiment at 53.9 MPa, and the vapour penetration was mostly within the experimental error. The simulation at 100 MPa, however, corresponds to an experiment at 94.5 MPa which explains to a large extent why the vapour penetration is overestimated for that case. The same can be seen in figure 5.11 where the vapour boundary at 1.44 ms ASOI is shown. Figure 5.12a shows the liquid penetration for the same range of injection pressures. Experimentally, the liquid penetration stays almost constant due to two balancing effects. For lower injection pressures, the injection speed is lower, but the breakup and evaporation are also less intense. The latter effect seems more important in the simulation than in the experiments. A possible explanation is that in the simulation at lower pressures, the droplets themselves cannot create additional turbulence that improves the evaporation. In the simulation, the droplets act as momentum sources and turbulence is created by shear in the flow of the continuous gas phase, whereas in reality droplets can directly produce turbulent eddies. This is illustrated in figure The lower injection velocities for lower injection pressures lead a reduced breakup rate due to the smaller relative velocity between the droplets and the air. The lower velocities also lead to lower turbulent kinetic energies and turbulent viscosity ratios. Both effects have a negative influence on the evaporation since a reduced breakup rate has a larger Sauter mean diameter as a result. The effect of turbulence on the liquid penetration was already elaborately explained in section Parametric variations 74

96 turbulent viscosity ratio [-] MPa 100 MPa 50 MPa Number MPa 100 MPa 50 MPa x [mm] (a)turbulent viscosity ratio x [mm] (b)number of discrete phase particles per cell Figure 5.13: Turbulent viscosity ratio and number of discrete phase particles along the spray axis at 1.44 ms ASOI Ambient density The ambient density in a constant volume combustion chamber can be lowered by decreasing the pressure in the chamber. The reason for choosing a density variation is based on the observation that a spray in different gases with the same ambient density behaves more similarly as a spray in different gases with same ambient pressure. The effect of changing the ambient density on the liquid penetration is shown in figure For decreasing ambient densities, the liquid penetration increases because there is less air hindering the movement of the spray. Drag forces and KH breakup decrease, but RT breakup increases (see equation 3.13 and 3.18) simulation experiment Liquid penetration 8 10 LP [mm] 6 LP [mm] Injection Pressure [MPa] (a)mean liquid penetration MPa 100 MPa 50 MPa experiment time ASOI [ms] (b)evolution in time (experiment at 150 MPa) Figure 5.12: Comparison of the liquid penetration at different injection pressures for simulation and experiment 5.3 Parametric variations 75

97 25 simulation experiment 25 Liquid penetration LP [mm] LP [mm] Ambient density [kg/m³] (a)mean liquid penetration kg/m³ 15.2 kg/m³ 7.6 kg/m³ experiment 7.6 kg/m³ time ASOI [ms] (b)evolution in time (experiment at 7.6 kg/m³) Figure 5.14: Comparison of the liquid penetration at different ambient densities for simulation and experiment This could explain why the simulation is less sensitive to ambient density changes. A simulation where the relative part of the KH breakup was increased, resulted in a higher liquid penetration, more closely following the experiment. To set up this simulation, the same breakup constants (B 1 = 0.5 and C RT = 0.5) as in section were used. For the baseline case, the liquid penetration remained unchanged, but for the case with the lower ambient density of 7.6 kg/m³ the liquid penetration increased from 13.7 mm to 18.4 mm, better following the experimental liquid penetration in terms of mean value as well as the amplitude of the fluctuations. The standard deviation of the liquid penetration over the period 0.15 ms to 1.5 ms dropped from 2.5 mm to 1.4 mm. A second reason why the liquid n-dodecane penetrates further into the combustion chamber for lower ambient densities is the increased velocities of both the gas phase and liquid phase for lower ambient densities, as can be seen in figure This is a consequence of the lower momentum needed to accelerate a low-density gas. Another minor effect is the increased pressure difference over the nozzle due to the lower ambient pressure, but when looking at the velocities of the droplets near the nozzle in figure 5.16b, the difference is very small. A third reason for the increasing liquid penetration at lower ambient densities is that the turbulent viscosity ratio decreases. Although the velocity and turbulent kinetic energy are higher, the turbulent viscosity ratio is lower. By comparing figure 5.16a and 5.17b, one can see that the density decreases more than the velocity of the gas phase increases, resulting in a flow with a lower Reynolds number. 5.3 Parametric variations 76

98 VP [mm] kg/m³ 15.2 kg/m³ 7.6 kg/m³ time ASOI [ms] (a)vapour penetration evolution y [mm] kg/m³ 15.2 kg/m³ 7.6 kg/m³ x [mm] (b)vapour boundary at 1.44 ms ASOI Figure 5.15: Vapour penetration evolution and the vapour boundary at 1.44 ms ASOI for different ambient densities x-velocity [m/s] kg/m³ 15.2 kg/m³ 7.6 kg/m³ velocity [m/s] kg/m³ 15.2 kg/m³ 7.6 kg/m³ x [mm] (a)velocity of the continuous gas phase x [mm] (b)velocity of the discrete phase particles Figure 5.16: Velocity of the continuous gas phase and the discrete phase droplets along the spray axis at 1.44 ms ASOI for different ambient densities turbulent viscosity ratio [-] kg/m³ 15.2 kg/m³ 7.6 kg/m³ density [kg/m 3 ] kg/m³ 15.2 kg/m³ 7.6 kg/m³ x [mm] (a)turbulent viscosity ratio x [mm] (b)density of the gas phase Figure 5.17: Turbulent viscosity ratio and density of the gas phase along the spray axis at 1.44 ms ASOI 5.3 Parametric variations 77

99 A final remark needs to be made about the experimental data. The liquid length measurement of the case with a lower ambient density of 7.6 kg/m³ used another injector (675 instead of 677) with the same nominal diameter. The variation of the ECN injectors is quantified in the SAE paper of Malbec et al.[38]. Variations in liquid length of up to 2 mm can be found. Also, the diffuse back-illumination (DBI) technique was used to measure the liquid length instead of the Mie-scattering technique. The same remark can be made about the experimental data in section about the variation of the ambient temperature. The vapour penetration in figure 5.15 increases for lower ambient densities. The same reason as for the increased liquid penetration also applies here. There is less air to hinder the movement of the fuel vapour cloud. The entrained air has a lower mass, and it slows down the spray to a lesser degree. The increased axial velocity also results in a slightly lower spray angle Ambient temperature In figure 5.18, liquid penetration values for different ambient temperatures are shown. Higher ambient temperatures lead to a higher heat transfer to the fuel droplets, and thus the droplets evaporate more quickly. Consequently, the liquid penetration decreases. For the parametric variation of the ambient temperature, the simulation also follows this trend, while being less sensitive to changes in ambient temperature compared to the experiments. The relative intensity of KH and RT breakup plays again an important role in explaining the difference between the experiment and the simulation. As shown in figure 5.19, the higher ambient temperatures lead to higher temperatures of the fuel droplets. The accompanying simulation experiment Liquid penetration LP [mm] 10 8 LP [mm] Ambient temperature [K] (a)mean liquid penetration K 900 K 1100 K experiment (900 K) time ASOI [ms] (b)evolution in time (experiment at 900 K) Figure 5.18: Comparison of the liquid penetration at different ambient temperatures for simulation and experiment 5.3 Parametric variations 78

100 temperature [K] K 900 K 1100 K x [mm] (a)continuous gas phase temperature [K] K 900 K 1100 K x [mm] (b)discrete phase droplets Figure 5.19: Temperature of the gas phase and the fuel droplets along the spray axis at 1.44 ms ASOI drop in surface tension influences the breakup considerably. The sensitivity of the breakup process to the surface tension was already touched upon in section 4.4. There, it was calculated that the predicted KH wavelength is more sensitive to changes in surface tension than the predicted RT wavelength. So when neglecting the small differences in densities, the liquid penetration should increase more for lower ambient temperatures when the relative part of KH breakup is increased. This is also confirmed by a simulation where the relative part of the KH breakup was increased. To set up this simulation, the breakup constants for increased KH breakup (B 1 = 0.5 and C RT = 0.5) from section were used. For the baseline case, the liquid penetration remained unchanged, but for the case with the lower ambient temperature of 700 K the liquid penetration increased from 11.8 mm to 13.2 mm. Introducing more temperature-dependent fluid properties such as the viscosity should also increase the sensitivity of the model to changes in ambient temperature. Additionally, the remark about the experimental data in the previous section also applies here. The vapour penetration is relatively insensitive to the ambient temperature, as illustrated in figure The reason for this is that the vapour penetration is primarily determined by the momentum of the fuel leaving the nozzle and the ambient density. Both of these quantities do not vary a lot when only the ambient temperature is changed. Therefore, the velocities and turbulent properties are also very similar. The pressure in the combustion chamber was also tracked during the simulation by calculating the volume-averaged pressure in the unstructured part of the mesh (see figure 4.2). The result is shown in figure In this figure, also the simulation 5.3 Parametric variations 79

101 at an ambient temperature of 303 K is included. By including the non-evaporating condition in the graph, two balancing effects can be seen. The pressure in the combustion chamber rises due to the fuel vapour that is added during an injection. At higher temperatures, the effect of the evaporation becomes visible. The evaporation withdraws heat from the combustion chamber, and thus it lowers the pressure in the combustion chamber. The latter effect dominates so that for fuel sprays in evaporating conditions a pressure drop can be expected. The experimental measurements of the pressure drop are quite noisy, but they show a similar pressure drop. 60 Vapour penetration VP [mm] K 900 K 1100 K experiment (900 K) time ASOI [ms] Figure 5.20: Evolution of the vapour penetration for different ambient temperatures 500 Relative ambient pressure in vessel P rel [Pa] K 900 K 1100 K 303 K time ASOI [ms] Figure 5.21: Evolution of the static pressure change in the combustion chamber for different ambient temperatures Combination of ambient density and temperature In this section, both the ambient density and temperature are changed. These experiments are not a part of the parametric variations of ECN spray A, but they 5.3 Parametric variations 80

102 were included to see the combined effect. In addition, these experiments are well documented on the site of ECN [2]. The ambient pressure, injection velocity and mass flow rate were calculated similarly as in table 4.4. The result is given in the table below. The three cases still use an injection pressure of 150 MPa. Table 5.2: Ambient temperature, density and pressure, liquid density, injection velocity and mass flow rate calculated based on C d = 0.89 for some extra variations of ECN spray A Case T a [K] ρ a [kg/m³] p a [MPa] ρ l [kg/m³] u [m/s] ṁ [g/s] Compared to the baseline case (1), the ambient temperatures have been increased and the ambient densities have been decreased. Higher ambient temperatures lead to a shorter liquid length, but lower ambient densities lead to a longer liquid length. So when combined, these changes counteract each other, as can be seen in figure In reality, the changes do not cancel out completely, but the liquid length increases slightly. In the simulation, however, the changes do cancel out almost completely, but this is not surprising, considering the reduced sensitivity to ambient density as explained in section Probably, the simulations would also follow the experiments better when the breakup constants for increased KH breakup are used (table 4.5). When switching from the tuned breakup constants to the increased KH breakup constants, the liquid length improved more for ambient density variations than for ambient temperature variations. However, the combined effect was not Liquid penetration 12 simulation experiment LP [mm] 8 6 LP [mm] case (a)mean liquid penetration K, 22.8 kg/m³ 1100 K, 15.2 kg/m³ 1400 K, 7.6 kg/m³ experiments time ASOI [ms] (b)evolution in time (experiment for case 2 and 3) Figure 5.22: Comparison of the liquid penetration at different ambient densities for simulation and experiment 5.3 Parametric variations 81

103 tested. Because of time constraints, not every case could be rerun using the breakup constants for increased KH breakup. The vapour penetration has increased compared to the baseline case mainly due to the lower ambient density. Figure 5.23 shows the evolution in time for the three cases mentioned in table 5.2. Even for these cases which are quite different from the baseline case, the vapour penetration is predicted reasonably well. At these higher temperatures, n-dodecane can become supercritical more easily. Fluid properties deviate from the ones provided at 900 K, and the solution is a bit less accurate. When keeping the high temperatures and pressures in mind, it is becoming hard to justify the ideal gas approximation for the air and fuel vapour. Also, the experimental injection pressure, ambient density and ambient temperature are not exactly equal to the nominal values used in the simulation. The difference can be up to a few percent. All this can explain why the vapour penetration and spray angle is slightly overestimated. 80 Vapour penetration 15 Vapour boundary at 1.44 ms VP [mm] 40 y [mm] K, 22.8 kg/m³ 1100 K, 15.2 kg/m³ 1400 K, 7.6 kg/m³ experiments K, 22.8 kg/m³ 1100 K, 15.2 kg/m³ 1400 K, 7.6 kg/m³ experiments time ASOI [ms] x [mm] (a)vapour penetration evolution (b)vapour boundary at 1.44 ms ASOI Figure 5.23: Vapour penetration evolution and the vapour boundary at 1.44 ms ASOI for different ambient densities and temperatures 5.3 Parametric variations 82

104 Transition to reacting conditions 6 This chapter briefly reports on a first attempt to simulate ECN spray A in reacting conditions. It serves as a stepping stone towards a possible future thesis. The simulation method is described and the preliminary results are presented. Instead of a combustion chamber without oxygen, the atmosphere now contains 15% oxygen which corresponds to a light use of EGR. The oxygen allows the spray to ignite and combust, raising the temperatures in the process. The interaction of the flow, turbulence and chemistry is very complex, and the different models will not be explained here. Different strategies include assuming infinitely fast reactions in a well-mixed model or tracking the transport of the mixture fraction of fuel and air over the mesh in a transported probability density function (TPDF) model. An important characteristic of reacting fuel sprays is the lift-off length (LOL). It is the distance from the nozzle to the point where the fuel starts to react. In this zone, the fuel evaporates and the mixture is too rich or too cold to ignite. Longer LOLs allows more premixing of air and fuel, so this significantly influences the combustion and emissions formation. As described on the site of ECN[2], the LOL is often experimentally determined by measuring the light emissions at 310 nm of OH in its excited state (OH*) with a high speed camera. Not all CFD simulations of reacting sprays provide information about OH*, so often OH in its ground state or the temperature are used to determine the LOL in simulations.[9] Pei et al.[9] used a transported probability density function model where OH* was included, and integrated the OH* concentration along a line of sight to be able to better compare his results with experiments. To setup the simulation, ANSYS Fluent s non-premixed combustion model with standard settings was used. It assumes that all species concentrations and other chemical and thermal properties can be related to the mixture fraction.[11] A PDF lookup table was created based on the properties of C 12 H 23 in the ANSYS Fluent material database to relate the mixture fraction and its variance to these thermal and chemical properties. C 12 H 23 represent kerosene and is the species that most resembles n-dodecane in the database. Each time step, the mixture fraction is transported along the mesh and the local equilibrium concentrations of the different species are calculated based on the Gibbs Free energy. For simplicity, no radiation 83

105 model was included. Node based averaging was enabled to aid convergence even though it was not really necessary. As for the rest, the turbulence and injection were handled in the same way as the cases in the previous chapter. As shown in figures 6.1 and 6.2, the result is an underestimated LOL compared to the experiment. In the simulation, the droplets evaporate very quickly and the fuel vapour ignites almost instantly at 2.7 mm from the nozzle whereas in experiments values around 16 mm are found. Apart from the inaccurate material properties, an appropriate combustion model with the right model constants will have to be chosen to get quantitatively accurate results. Probably, it was not justified to assume a local chemical equilibrium. Switching to a composition PDF model with finite-rate chemistry should help despite being computationally more expensive. Pei et al.[9] have already proven that it is possible to obtain good results using this model in a 2D mesh. Figure 6.1: Contours of constant temperature for ECN spray A in reacting conditions Figure 6.2: Contours of constant OH mass fraction for ECN spray A in reacting conditions 84