numerical simulation of evaporating diesel sprays jasper

The Clean Thesis StyleNumerical simulation of evaporating diesel sprays
Academic year 2016-2017 Faculty of Engineering and Architecture Chair: Prof. dr. ir. Jan Vierendeels Department of Flow, Heat and Combustion Mechanics
Master of Science in Electromechanical Engineering Master's dissertation submitted in order to obtain the academic degree of
Counsellor: Gilles Decan Supervisors: Prof. dr. ir. Jan Vierendeels, Prof. dr. ir. Sebastian Verhelst
Jasper Van de Vyver
Numerical simulation of evaporating diesel sprays
Academic year 2016-2017 Faculty of Engineering and Architecture Chair: Prof. dr. ir. Jan Vierendeels Department of Flow, Heat and Combustion Mechanics
Master of Science in Electromechanical Engineering Master's dissertation submitted in order to obtain the academic degree of
Counsellor: Gilles Decan Supervisors: Prof. dr. ir. Jan Vierendeels, Prof. dr. ir. Sebastian Verhelst
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.
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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
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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 2016-2017
Abstract and extended abstract
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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 rec- ommended 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 pub- licly 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 temperature- dependent. 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. How- ever, 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
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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 63 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 Cmax.
C = ut
x ≤ Cmax (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] 0.25 500 ±575 ±1.15 Pei [9] 1 4000 595.3 2.381
Decan [7] 1 400 460 0.184 Lucchini [5] 0.25 500 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
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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 m(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 m(t)/(ρlA). A is the cross-sectional area of the nozzle (assumed to be constant) and ρl 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 Cd. The value can be calculated by estimating the friction losses or sometimes it is given for a particular nozzle.
Cd = m
ρlAutheor , utheor =
√ 2p
ρl (2)
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 high- pressure fuel sprays. Indeed, the high pressures lead to high injection velocities u and consequently also a high Weber number We = ρlu
2d/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
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n-dodecane at 363 K and 150 MPa is injected through a nozzle with a diameter of 0.090 mm into a constant volume combustion chamber. The ambient gas in the combustion chamber has a density ρa of 22.8 kg/m³, a temperature Ta of 900 K and a pressure pa 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 pa, injection velocity u and mass flow rate m 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 ρl, a value between 701 and 706 kg/m³ was used to account for the density change due to the ambient pressure variation.
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 Cd = 0.89.
Ta[K] ρa[kg/m³] pinj [MPa] pa[MPa] u[m/s] m[g/s] 900 22.8 150 6.09 569 2.55 900 22.8 100 6.09 460 2.06 900 22.8 50 6.09 314 1.41 900 7.6 150 2.03 578 2.58 900 15.2 150 4.06 574 2.56 700 22.8 150 4.74 572 2.56 1000 22.8 150 6.77 567 2.54 1100 22.8 150 7.45 566 2.54 700 22.8 50 4.74 319 1.43 440 22.8 150 2.98 576 2.57 303 22.8 150 2.05 578 2.58
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 0.675 mm in the axial direction, 0.135 mm in the radial direction and 6in the circumferential direction. This results in a total cell count of 138 754 cells. Mesh dependency was checked by refining the structured part of the mesh to a total cell count of 432 194. 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
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 quasi- identical results. This corresponds to a maximum Courant number of 0.337.
Figure 2: Quarter cubic mesh as created by Gilles Decan[7]
BREAKUP MODEL As a breakup model, the KH-RT model was used. It has two competing breakup mechanisms. B1 can be lowered to increase the breakup rate due to the KH instability and CRT 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 B1 of 1.73 and a CRT 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 B1 of 0.5 and a CRT 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-ε 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-ε models behaved similarly except for the initial transient which was better captured by the realisable k-ε model. Despite the fact that the k-ω SST model should behave similarly as a k-ε 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
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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-ε model, the vapour penetration could be brought within the limits of the experimental error by slightly decreasing C2ε from 1.9 to 1.77. C2ε determines the rate at which the turbulent dissipation ratio ε decays. So for lower values of C2ε, ε is higher, resulting in a lower turbulent kinetic energy and increased vapour penetration.
0 0.5 1 1.5 time ASOI [ms]
0
20
40
60
80
100
120
k- Standard k- RNG k- Realisable Spalart-Allmaras k- SST experiment
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 - 0.3 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 per- formed 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 - 663 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 un- changed, 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. Addi- tionally, 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 152.7 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.
0 0.5 1 1.5 time ASOI [ms]
0
10
20
30
40
50
60
Figure 4: Vapour penetration for different injection pressures
Figure 5 shows the liquid penetration for the same range
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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 them- selves 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.
50 100 150 Injection Pressure [MPa]
0
2
4
6
8
10
12
7.6 15.2 22.6 Ambient density [kg/m³]
0
5
10
15
20
25
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.
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.
700 900 1100 Ambient temperature [K]
0
2
4
6
8
10
12
14
16
18
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 La- grangian 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 liquid- related 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.
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Contents
1 Introduction 1 1.1 IMO Tier 3 NOx limits . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Dual fuel internal combustion engines . . . . . . . . . . . . . . . . . 1
1.2.1 Working principle . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Advantages, disadvantages and current problems . . . . . . . 3
2 Computational Fluid Dynamics 5 2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Mass equation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.4 Species equations . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 RANS turbulence models . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Spalart-Allmaras model . . . . . . . . . . . . . . . . . . . . . 8 2.2.2 k-ε models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3 k-ω SST model . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.4 Reynolds Stress model . . . . . . . . . . . . . . . . . . . . . . 10
3 Literature review 11 3.1 CFD strategies in spray formation . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Eulerian-Lagrangian approach . . . . . . . . . . . . . . . . . . 11 3.1.2 Eulerian-Eulerian approach . . . . . . . . . . . . . . . . . . . 12 3.1.3 Hybrid approach . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Common practice in the simulation of fuel sprays . . . . . . . . . . . 13 3.2.1 Discretisation, Mesh and Time step . . . . . . . . . . . . . . . 13 3.2.2 Turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.3 Injection and primary breakup model . . . . . . . . . . . . . . 17 3.2.4 Secondary Breakup model . . . . . . . . . . . . . . . . . . . . 21 3.2.5 Collision model . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.6 Droplet drag model . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.7 Evaporation model . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.8 Turbulent dispersion . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Goal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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4 Simulation setup and model validation 34 4.1 ECN spray A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 Dimensions and structure of the mesh . . . . . . . . . . . . . 37 4.2.2 Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.3 Pressure-Velocity Coupling . . . . . . . . . . . . . . . . . . . . 48 4.2.4 Mesh dependence . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.5 Time step dependence . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Choice of turbulence model . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 Influence of fluid properties and initial conditions . . . . . . . . . . . 57
4.4.1 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4.2 Temperature-dependent fluid properties . . . . . . . . . . . . 59
4.5 Proposed methodology for fuel spray simulations . . . . . . . . . . . 61
5 Discussion of results 64 5.1 Spray penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.1 Evolution of the vapour boundary . . . . . . . . . . . . . . . . 64 5.1.2 Vapour mass fraction distribution . . . . . . . . . . . . . . . . 66 5.1.3 Streamlines and velocity field . . . . . . . . . . . . . . . . . . 67 5.1.4 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.5 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Near-nozzle region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2.1 Volume fraction of liquid n-dodecane . . . . . . . . . . . . . . 70 5.2.2 Droplet size distribution . . . . . . . . . . . . . . . . . . . . . 71 5.2.3 Breakup mechanisms . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Parametric variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3.1 Injection pressure . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3.2 Ambient density . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3.3 Ambient temperature . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.4 Combination of ambient density and temperature . . . . . . . 80
6 Transition to reacting conditions 83
7 Conclusion 85
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 . . . . . . . . . . . . 1
1.2 Working principle of a dual fuel engine (figure from targettrainingcentre.nl) . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1 Left: Mesh is fine enough to represent the droplet Right: Droplets smaller than the cell size cannot be represented [29] . . . . . . . . . . 12
3.2 Adaptive mesh of Argonne National Laboratory at start of injection, used in the first ECN Workshop [31] . . . . . . . . . . . . . . . . . . . 14
3.3 Breakup of blobs by secondary breakup models using the concept of liquid core length, figure from ANSYS Fluent Theory Guide [11] . . . . 19
3.4 Kelvin-Helmholtz instability: related to the existence of a difference in speed between two fluids [29] . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Comparison of the creation of child droplets due to the KH instability and RT instability [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 The constant volume combustion chamber at Sandia National Laboratories 36 4.2 Quarter cubic mesh as created by Gilles Decan [7] . . . . . . . . . . . . 38 4.3 Comparison of the liquid penetration in a 3D quarter cubic mesh and a
2D axisymmetric mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 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 Cd of 0.89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Influence on the liquid penetration of the B1 breakup constant in the KH-RT model (CRT = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Influence on the liquid penetration of the CRT breakup constant in the KH-RT model (B1 = 1.73) . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.7 Effect of node based averaging on the liquid penetration . . . . . . . . 45 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 . . 46 4.9 Effect of Turbulent dispersion on the vapour boundary at 0.2 ms . . . . 48 4.10 Liquid penetration for different levels of refinement of the mesh . . . . 50 4.11 Vapour boundary at 1.44 ms for different levels of refinement of the mesh 50 4.12 Liquid penetration for different time steps . . . . . . . . . . . . . . . . 51 4.13 Vapour penetration for different time steps . . . . . . . . . . . . . . . . 52
xiv
4.14 Effect of the turbulence model on the liquid penetration . . . . . . . . 54 4.15 Effect of the turbulence model on various properties . . . . . . . . . . 56 4.16 Liquid penetration and vapour boundary for different values of C2ε . . 57 4.17 Influence of various fluid properties and initial conditions on the liquid
penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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 . . . . . . . . . . . . 65
5.2 Vapour mass fraction distribution at 1.44 ms ASOI . . . . . . . . . . . 66 5.3 Pathlines coloured by velocity magnitude at 1.44 ms ASOI . . . . . . . 67 5.4 Momentum flow rate through planes orthogonal to the spray axis placed
at different distances from the nozzle at 1.44 ms ASOI . . . . . . . . . 68 5.5 Temperature of the gas phase at 1.44 ms ASOI . . . . . . . . . . . . . . 69 5.6 Temperatures of the discrete phase droplets at 1.44 ms ASOI . . . . . . 70 5.7 Volume fraction of n-dodecane droplets in the cells near the nozzle at
1.44 ms ASOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.8 Droplets coloured by particle diameter with a scaled diameter
representation (top) and a constant diameter representation (bottom) 72 5.9 Comparison of the WAVE breakup times (top) and KH-RT breakup times
(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.10 Vapour penetration for different injection pressures . . . . . . . . . . . 73 5.11 Vapour boundary for different injection pressures at 1.44 ms ASOI . . . 74 5.13 Turbulent viscosity ratio and number of discrete phase particles along
the spray axis at 1.44 ms ASOI . . . . . . . . . . . . . . . . . . . . . . 75 5.12 Comparison of the liquid penetration at different injection pressures for
simulation and experiment . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.14 Comparison of the liquid penetration at different ambient densities for
simulation and experiment . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.15 Vapour penetration evolution and the vapour boundary at 1.44 ms ASOI
for different ambient densities . . . . . . . . . . . . . . . . . . . . . . . 77 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 . . 77 5.17 Turbulent viscosity ratio and density of the gas phase along the spray
axis at 1.44 ms ASOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.18 Comparison of the liquid penetration at different ambient temperatures
for simulation and experiment . . . . . . . . . . . . . . . . . . . . . . . 78 5.19 Temperature of the gas phase and the fuel droplets along the spray axis
at 1.44 ms ASOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.20 Evolution of the vapour penetration for different ambient temperatures 80
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5.21 Evolution of the static pressure change in the combustion chamber for different ambient temperatures . . . . . . . . . . . . . . . . . . . . . . 80
5.22 Comparison of the liquid penetration at different ambient densities for simulation and experiment . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.23 Vapour penetration evolution and the vapour boundary at 1.44 ms ASOI for different ambient densities and temperatures . . . . . . . . . . . . 82
6.1 Contours of constant temperature for ECN spray A in reacting conditions 84 6.2 Contours of constant OH mass fraction for ECN spray A in reacting
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
List of Tables
1.1 Pros and Cons of Dual fuel engines as listed by Decan et al. [7] . . . . 3
3.1 Courant numbers used by different authors for fuel spray simulations . 15
4.1 Most important specifications of the ECN Spray A baseline operating condition [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Parametric variations of ECN spray A with the first one being the baseline condition [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Cell count and dimensions of cells in the different zones of the mesh . 38 4.4 Ambient temperature, density and pressure, liquid density, injection
velocity and mass flow rate calculated based on Cd = 0.89 for the parametric variations of ECN spray A . . . . . . . . . . . . . . . . . . . 41
4.5 Standard breakup constants of the KH-RT model compared to the tuned breakup constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.6 Change in values of various fluid properties and initial conditions for the sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Isobaric data of the density for dodecane at 60 bar [17] . . . . . . . . . 59 4.8 Temperature-dependent surface tension data for saturated dodecane [17] 60 4.9 Other properties of liquid dodecane . . . . . . . . . . . . . . . . . . . . 60
5.1 Droplet size statistics at 1.44 ms ASOI . . . . . . . . . . . . . . . . . . 71 5.2 Ambient temperature, density and pressure, liquid density, injection
velocity and mass flow rate calculated based on Cd = 0.89 for some extra variations of ECN spray A . . . . . . . . . . . . . . . . . . . . . . 81
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Abbreviations and symbols
Symbol Unit Description
CFD Computational Fluid Dynamics x or x1 m (distance to the nozzle along the) axial direction y or x2 m (distance to the nozzle along the) radial
direction z or x3 m (distance to the nozzle along the) direction
perpendicular to x and y u or u1 m/s velocity along the x-direction v or u2 m/s velocity along the y-direction w or u3 m/s velocity along the z-direction ρ kg/m³ mass density ρl kg/m³ mass density of the liquid phase ρg kg/m³ mass density of the gas phase k m²/s² turbulent kinetic energy ε m²/s³ turbulent dissipation rate µ Pa.s (molecular) dynamic viscosity ν m²/s (molecular) kinematic viscosity µt Pa.s turbulent dynamic viscosity νt m²/s turbulent kinematic viscosity σ N/m surface tension cp J/(kg.K) heat capacity at constant pressure pinj Pa injection pressure pa Pa ambient pressure Ta K ambient temperature ρa kg/m³ ambient density m 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 B1 model constant of KH breakup model (or WAVE) CRT model constant of RT breakup model LP m liquid penetration or liquid length VP m vapour penetration ECN Engine Combustion Network
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1Introduction
1.1 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
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
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.
1.2.2 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 engine
- Higher specific fuel consumption
- 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
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.
2Computational Fluid Dynamics
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
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.
2.1.1 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.
∂ρ
∂ρ
∂xi = 0 (2.2)
In these equations u or u1, v or u2 and w or u3 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.
∂ui ∂xi
= 0 (2.3)
All flows which obey equation 2.3 are said to be incompressible.
2.1.2 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
given below are for a Newtonian fluid with a constant density ρ0 and a constant kinematic viscosity ν.
∂ui ∂t
+ fi body forces
2.1.3 Energy equation
∂
∂xj (−ujp− qj + uiτij) (2.5)
In this equation e0 is the total energy e+ ukuk 2 , qj is the heat flux calculated by the
law of Fourier and τij is the viscous stress. For a Newtonian fluid τij is defined as follows.
τij = µ
( ∂ui ∂xj
2.1.4 Species equations
∂
∂Yk ∂xi
) + ωk (2.7)
In which Dk is the diffusion coefficient and ωk 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 ui and
2.2 RANS turbulence models 7
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.
∂ui ∂t
∂xj (u′iu′j) + fi, i = 1, 2, 3 (2.8)
And similarly for the continuity equation (equation 2.3)
∂ui ∂xi
= 0 (2.9)
−ρ0u′iu ′ j is defined as the Reynolds stress tensor Rij . It represents the influence of
the fluctuating velocities on the mean flow field. To close the system of equations, Rij 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 µt (or eddy viscosity) that adds to the effect of the molecular viscosity.
Rij = −ρ0u′iu ′ j = µt
( ∂ui ∂xj
+ ∂uj ∂xi
3ρ0u′iu ′ 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 3.2.2.
2.2.1 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 µt in equation 2.10. 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
2.2.2 k-ε models
The k-εmodel 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
2u ′ iu ′ i and
one for the turbulent dissipation rate ε. The turbulent kinematic viscosity is then based on the instantaneous values of these two quantities.
νt = Cµ k2
ε (2.12)
where Cµ 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
k (C1εP − C2εε) (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 2Cµ
) (2.15)
The model constants are determined by experimentally comparing the results to basic flows. Commonly used values are:
Cµ = 0.09, CK = 1, Cε = 0.069, C1ε = 1.44, C2ε = 1.92 (2.16)
Although the k-ε model performs well in a lot of applications, it still has a few shortcomings. Métais [24] states that the k-ε 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-ε 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-ε model by imposing some physical constraints on the solution. In the standard model u2
could become negative, so one of the adjustments was to make Cµ locally dependent on the mean flow field to guarantee that u2 ≥ 0 ( = condition for realisability). Another commonly used k-ε 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.
2.2.3 k-ω SST model
Not all two equation turbulence models use k and ε as the transported variables. The specific dissipation rate ω = ε/k can also be used instead of the dissipation rate ε. 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-ε 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-ε 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.
2.2.4 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 Rij = −ρ0u′iu
′ j
and a transport equation for ε similar to the one in the k-ε 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.
3Literature review
3.1 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.
3.1.1 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
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.
3.1.2 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 α1 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.
∂α1 ∂t
= 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 63 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
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].
3.1.3 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 high- fidelity 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 53 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.
3.2.1 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
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
condition needs to be satisfied in order for a partial differential equation problem to converge. In 1D, it takes the following form.
C = ut x ≤ Cmax (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, Cmax is around 1 and depends on the spatial discretisation scheme. Implicit algorithms are less prone to numerical instabilities, so higher values of Cmax 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] 0.25 500 ±575 ±1.15 3D PISO-SIMPLE Pei [9] 1 4000 595.3 2.381 2D SIMPLE
Decan [7] 1 400 460 0.184 3D Coupled Lucchini [5] 0.25 500 N/A ≤0.15 3D 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 0.15. 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.
3.2.2 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
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-ε model as explained in section 2.2.2. 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-ε 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 C1 of the standard k-ε 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.
3.2.3 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
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.
1− Y = exp
)n] (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
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 = 1.2726 (
133.0d8We−0.74 )(
(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 m(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) = m(t) ρlA
A is the cross-sectional area of the nozzle (assumed to be constant) and ρl 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.
utheoretical = √
(3.7)
The real speed is often compared to the theoretical one and their ratio is called the discharge coefficient Cd. The value can be determined by calculating the friction losses. Sometimes, it is also given for a particular nozzle.
Cd = m
mtheoretical = ρlAu
ρlAutheoretical (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 Avena with Nurick’s expression.
Avena = CcA, Cc = [( 1
D
]−0.5
(3.9)
Cc0 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 (uvena).
uvena = m
ρlCcA = u
Cc (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.
ueff = A
m (pvap − pexit) + uvena (3.11)
ueff is the effective speed of the droplets at the nozzle exit due to cavitation, pvap the vapour pressure for the fuel at a given temperature and pexit 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 ueff . The increased breakup rate due to implosions of cavitation bubbles is not accounted for in this model.
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

numerical simulation of evaporating diesel sprays jasper

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