Implementation of an Eulerian Atomization Model To Characterize Primary Spray Formation

Transcription

1 University of Massachusetts Amherst Amherst Masters Theses February Implementation of an Eulerian Atomization Model To Characterize Primary Spray Formation Nathaniel A. Trask University of Massachusetts - Amherst Follow this and additional works at: Part of the Aerodynamics and Fluid Mechanics Commons, Applied Mechanics Commons, Energy Systems Commons, and the Propulsion and Power Commons Trask, Nathaniel A., "Implementation of an Eulerian Atomization Model To Characterize Primary Spray Formation" (2010). Masters Theses February Retrieved from This thesis is brought to you for free and open access by ScholarWorks@UMass Amherst. It has been accepted for inclusion in Masters Theses February 2014 by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact scholarworks@library.umass.edu.

2 IMPLEMENTATION OF AN EULERIAN ATOMIZATION MODEL TO CHARACTERIZE PRIMARY SPRAY FORMATION A Thesis Presented by NATHANIEL TRASK Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING May 2010 Mechanical and Industrial Engineering

3 Copyright by Nathaniel Trask 2010 All Rights Reserved

4 IMPLEMENTATION OF AN EULERIAN ATOMIZATION MODEL TO CHARACTERIZE PRIMARY SPRAY FORMATION A Thesis Presented by NATHANIEL TRASK Approved as to style and content by: David P. Schmidt, Chair J. Blair Perot, Member Hans Johnston, Member Donald Fisher, Department Head Mechanical and Industrial Engineering

5 ACKNOWLEDGMENTS I d like to thank Professor Schmidt for providing me with a number of opportunities during my time here at UMass and for providing valuable insight and support throughout my entire project. Thanks to Professor Perot for providing me with advice and humoring my questions, from my days as an undergraduate here through the day I finished my graduate work. Thanks to my coworkers (Ali, Chris, Dnyanesh, Tim, Tom, Kshitij, Mihir, Martell, Shiva, Sandeep, Kyle, and Colarossi) for making the lab a fun place to come in to every day. Thanks to all the friends and classmates who have made my long stay at UMass enjoyable. Most importantly, thank you to my lovely wife for supporting me in my endeavors and putting up with my shenanigans. The author would also like to thank the Air Force Research Lab for financial support under grant number FA C Written in L A TEX2εusing BibT E X iv

6 ABSTRACT IMPLEMENTATION OF AN EULERIAN ATOMIZATION MODEL TO CHARACTERIZE PRIMARY SPRAY FORMATION MAY 2010 NATHANIEL TRASK B.S., UNIVERSITY OF MASSACHUSETTS Directed by: Professor David P. Schmidt Fuel injection elements in modern rocket propulsion and gas turbine power generation systems operate at extremely high Reynolds and Weber number, rendering traditional direct numerical simulation approaches computationally infeasible in resolving the small scales of the atomization process. Experimental techniques have also proven incapable of resolving the small scale features of the spray formation due to the primary atomization zone s topological complexity and optically dense nature. Gas-centered swirl-coaxial injectors (GCSC) in particular are very challenging to characterize due to the fact that atomization occurs within a small cup, preventing the use of traditional approaches such as laser diagnostics to measure spray density. To predict a GCSC injection element s ability to perform over a wide range of operating conditions, it is desirable to find a predictive method capable of characterizing atomization quality using a feasible amount of computational resources. v

7 A solver implementing an Eulerian model for the atomization process has been written using the OpenFOAM C++ library. The model assumes that at high Reynolds and Weber numbers, the atomization process occurs at small scales and does not effect the large scale motion of the flow. Using the Reynolds decomposition formalism typically employed in turbulence modeling, the mass averaged governing equations for the mean motion of the liquid/gas mixture can be calculated exactly while the small scale turbulent features of the flow are resolved through closure models. A review of previous applications of the model in the literature is conducted and details of the current implementation are presented. A series of validation cases are presented to demonstrate the accuracy of the model. The model is used to predict the internal flow of a GCSC injector and the spray formed by a subsonic jet in crossflow (JIC) and compared to experimental results. vi

8 TABLE OF CONTENTS Page ACKNOWLEDGMENTS iv ABSTRACT v LIST OF TABLES x LIST OF FIGURES xi CHAPTER 1. BACKGROUND Background Numerical atomization modeling LES Lagrangian Modeling Wall Film Modeling Eulerian Modeling Model Selection Σ Y MODEL Model Foundations Governing equations Surface area transport closure Background Original model Equation used by Lebas et al Turbulent liquid flux closure Reynolds stresses closure Evaluation of Model vii

9 3. IMPLEMENTATION Overview Pressure equation Foundation Multiphase mixing Compressibility effects Thermal expansion effects Final equation Mass transport equation Turbulence model Internal energy equation Lagrangian droplet tracking VALIDATION Compressibility: Converging-diverging nozzle case Introduction Analytical solution Validation case Liquid flux closure: Coaxial rocket injector Introduction Numerical results Surface area density closure: Diesel jet injection Introduction Numerical results APPLICATIONS - GAS-CENTERED SWIRL-COAXIAL INJECTOR (GCSC) Background Experimental Study Incompressible results APPLICATIONS - JET IN CROSS FLOW Background Case Setup Conclusions viii

10 7. CONCLUSIONS BIBLIOGRAPHY ix

11 LIST OF TABLES Table Page 4.1 CDN Validation run conditions Coaxial injector validation run conditions Diesel injector validation run conditions Simulation conditions Compressible conditions Simulation conditions x

12 LIST OF FIGURES Figure Page 1.1 Breakup regimes for a coaxial injector Fraser s idealized breakup of a wavy sheet [17] Direct numerical simulation of liquid jet using Menard et al. s level set/vof/ghost fluid method [57] Liquid structures downstream of injector [57] Ligaments that pinch off from the liquid core (a) are converted into Lagrangian droplets (b) [36] Comparison of actual and equilibrium SMD along center axis [4] Indicator function used to transition between source terms in dense and dilute part of spray [42] Experimental data for total dimensionless breakup time scale T as a function of Weber number [64] Spray SMD along symmetry axis for different gas jet velocities and a fixed liquid velocity of 0.5 m/s [4] Spray SMD along symmetry axis for different liquid jet velocities and a fixed gas velocity of 140 m/s [4] Liquid core profiles for fixed gas velocity of 140 m/s and liquid velocities of 0.55, 0.31, 0.17, and 0.13 m/s, respectively [4] Comparison of predicted liquid volume fraction profile to DNS results [42] Comparison of predicted interfacial surface area density profile to DNS results [42] xi

13 3.1 Process by which dilute liquid cells are identified, deleted, and replaced with Lagrangian parcels Preliminary results showing stripping of droplets by high-velocity co-flowing gas (bottom) from a liquid film (top) D axisymmetric mesh used for converging-diverging nozzle validation case Steady state profiles of Mach number (top) and static pressure (bottom) for each of the three test cases Steady state profiles of Mach number (top) and static pressure (bottom) for each of the three test cases Steady state profiles of Mach number (top) and static pressure (bottom) for each of the three test cases A schematic denoting the configuration and relevant length scales in coaxial fuel injection [40] Mesh used for 2D coaxial validation case Mass fraction profile for coaxial validation case using Rayleigh-Taylor instability production term Comparison of liquid volume fraction along center axis compared to experimental data [81] and results by Demoulin et al. [16] Schematic demonstrating fuel injection system for a diesel engine (Adapted from [73]) Mesh used for 2D validation case Comparison of 2D results for volume fraction profile to published ELSA results by Lebas et al. and to DNS results by Menard et al. (From top to bottom: Simulation results, Lebas et al. s results, Menard et al. s results, Simulation results) Comparison of 2D results for surface area density profile to published ELSA results by Lebas et al. and to DNS results by Menard et al. (From top to bottom: Simulation results, Lebas et al. s results, Menard et al. s results, Simulation results) xii

14 4.13 Plot of volume fraction profile along central axis compared to published ELSA results by Lebas et al. (top) and to DNS results by Menard et al. (bottom) Top left: GCSC geometric parameters. Top Right: Dimensions of tested inserts. Bottom Left: Film length over initial film height as a function of mixture ratio. [45] A typical image from the in-cup video is shown here. The edges of the injector body are highlighted including the sheltering lip Representative profiles of a long, medium and short film. These profiles are from the geometry shown in Fig. 5.2 at momentum flux ratios of 110, 484 and 823 respectively Transition from optimal diffuser performance (top) to stall conditions (bottom) (Figure adapted from [90]) Asymmetry in resulting spray at high momentum ratio. (Image courtesy of AFRL) Static pressure and volume fraction profile for ONPNTN-b1 case Experimental photograph of liquid profile for ONPNTN-b1 case Iso-contours of Ȳ = 0.5 compared to experimental profiles for standard k ǫ closure Iso-contours of Ȳ = 0.5 compared to experimental profiles for realizable k ǫ closure Experimental photograph of liquid profile for ONPNTN-a1 case Iso-contours of Ȳ = 0.5 compared to experimental profiles for standard k ǫ closure Iso-contours of Ȳ = 0.5 compared to experimental profiles for realizable k ǫ closure Instantaneous film profile and streamlines show the presence of a recirculation zone for the ONPNTN-a1 case Boundary conditions and mesh setup for compressible ONPNTN-a1 case xiii

15 5.15 Resulting profiles of pressure, density, velocity and temperature for ONPNTN-a1 case Isentropic compression followed by sudden expansion Comparison of CFD simulation to analytic results Schematic representing the two mechanisms of primary atomization and subsequent transition to secondary breakup. [92] Droplets from surface breakup mechanism visually obscure the primary liquid core. [83] Effect of liquid column on surrounding turbulent structures. [44] a) Experimental setup used by Tambe et al. b) Typical nozzle geometry [83] Shadowgraph of spray structure for case 5 by Tambe et al. [83] Experimental measurements of SMD (left) and liquid volume flux (right) [83] Liquid volume fraction measured 30 diameters downstream of liquid inlet Interfacial surface area density measured 30 diameters downstream of liquid inlet Predicted liquid mass fraction with superimposed correlations of spray penetration Predicted interfacial surface area density with superimposed correlations of spray penetration xiv

16 CHAPTER 1 BACKGROUND 1.1 Background Primary atomization and subsequent spray formation govern the efficiency and effectiveness of many industrial applications. In propulsion and power related applications, the extent of atomization directly effects the rate at which combustion will occur. The amount of fuel surface area present within a spray determines the evaporation rate which in turn plays an important role in the speed of the combustion reaction. Developing a means to predict the effectiveness and reliability of injector designs would be an invaluable tool. The primary challenge of analyzing the atomization process is the large range of length and time scales involved. For example, a typical diesel engine will have a spray penetration length of 10 2 m, a spray nozzle diameter on the order of 10 4 m, and typical droplet sizes of about 10 5 m [75]. Maintaining such a small resolution over the course of such a relatively large domain requires computation times well beyond the limit of current technology. Although different engines operate on many different principles, the atomization process is fairly universal. A liquid body, usually in the form of a sheet or jet, is discharged under pressure into either a co-flowing or quiescent fluid. A variety of mechanisms then transfer momentum and enhance interfacial instabilities between the two fluids. These mechanisms are complex and highly dependent upon the flow regime within which the injector is operating. For example, when a jet is issued into quiescent gas at low to moderate Reynolds number, breakup will initiate as a result of surface tension driven hydrodynamic instability. At large Reynolds number, atomization will 1

17 be driven by short-wavelength shear instability at a scale much shorter than the jet diameter. On the other hand, a jet exposed to a high-speed coaxial gas jet will undergo a fundamentally different process driven by the turbulent transfer of kinetic energy from one phase to the other [40]. Atomization can be broadly classified into either jet, sheet or film type atomization. In jet atomization liquid enters contact with the air as a cylindrical stream. Sheet atomization is characterized by a stream of liquid with two free surfaces. Film atomization is similar to sheet atomization but has only one free surface. Lightfoot offers a review of all three types of atomization, noting that very little literature presently exists with regard to film type atomization, most likely due to the lack of atomizers operating within this configuration [47]. Although the atomization process is dependent upon a variety of processes, such as internal flow characteristics, cavitation mechanisms, the jet velocity profile, turbulent mechanisms, and the thermodynamic states of both phases, linear stability analysis offers a qualitative approach to separating breakup phenomenon into separate regimes. The four main regimes of breakup within a coaxial injector are the Rayleigh regime, the first wind-induced regime, the second wind induced regime, and the atomization regime (Figure 1.1). Liquid sheets are commonly formed in a variety of injection schemes, such as the impingement of two liquid streams in a doublet injector, the deflection of a pipe flow through an annular orifice, or pressure-swirl and prefilming air blast nozzles [43]. The primary mechanism by which sheets disintegrate is via the perturbation and growth of interfacial instability between the two phases. Aerodynamic lift forces act to enhance instabilities, while surface tension forces act to bring the instability back to its initial equilibrium state [82]. A typical approach to model this mechanism is to perform a stability analysis for a small, sinusoidal perturbation of the interface and identify the wavelength of the fastest growing instability mode. Figure 1.2 illustrates Fraser et al. s 2

18 Figure 1.1. Breakup regimes for a coaxial injector model for the process, which assumes that the most rapidly growing wave detaches to form a cylindrical ligament with diameter equal to half the disturbance wavelength. These ligaments in turn disintegrate into spherical droplets of equal diameter via a Rayleigh instability like process [17]. Senecal et al. provide an assessment of the assumptions typically made within similar models [76]. Specifically, they outline a transitional Weber number of 27/16 where the sheet switches from the long wave mode of breakup, similar to the first wind-induced regime of cylindrical jets to a short-wave mode sensitive to viscosity, similar to the second wind-induced regime. Figure 1.2. Fraser s idealized breakup of a wavy sheet [17] 3

19 Within the atomization regime, the exact mechanism by which atomization occurs is unclear, due to the complex nonlinear interactions occurring. Lightfoot notes that the process can be characterized by the formation and growth of disturbances at the interface, followed by the subsequent breakdown and detachment of these structures. The process of disturbance formation can be characterized by mechanisms driven by liquid turbulence, hydrodynamic instabilities, gas turbulence, pressure fluctuations, and particle interactions. The following description of these mechanisms operating within the atomization regime is taken from Lightfoot s review of wall-bounded atomization [47]. The liquid turbulence mechanism is primarily driven by liquid eddies interacting with the free surface, causing the formation of ligaments. This has been observed experimentally by Dai et al. [14] and Sarpkaya and Merrill [74]. Gas turbulent structures may affect disturbance growth either through impact with the interface or by enhancing the growth of aerodynamic instabilities [52]. A large body of work exists to describe hydrodynamic instability mechanisms. Surface tension driven instabilities tend not to affect primary atomization within the atomization regime, although they do effect the secondary breakup atomization regime in which stripped-off ligaments are converted to droplets [47]. In addition to the linear stability analyses already mentioned [76] [82] [43], investigations specific to the film configuration have been performed. Boomkamp and Miesden [9] performed stability analysis of a film of infinite depth. Analogous to the analysis of hydrodynamic instability for a liquid sheet, both theoretical and experimental evidence suggest that droplet diameters produced in such a configuration are again proportional to the wavelength corresponding to the fastest growing instability mode [35, 51, 26, 82]. Once an initial disturbance has been created, waves will either shrink due to viscous dissipation or will grow by aerodynamic enhancement. Upon reaching a critical 4

20 height, the disturbance may be stripped from the wave crest, waves may break, or entire ligaments may detach. Whatever the dominant process may be, a successful model must be able to accurately capture the relevant physical mechanisms present. In the case of fully turbulent flows acting within the atomization regime, the model must resolve both the complex topological change of the interface and its coupled, small-scale interaction with the turbulent vortices[47] Numerical atomization modeling The only means to exactly describe the atomization process numerically is through the use of Direct Numerical Simulation (DNS), where the full Navier-Stokes equations are solved exactly for all relevant length and time scales. For a fully turbulent, multiphase flow, this involves discretizing the domain at a resolution on the scale of both the smallest turbulent velocity fluctuation (the Kolmogorov length scale) and the smallest liquid structure within the flow. Gorokhovski and Herrmann [20] summarize the challenges involved in applying this strategy to study primary atomization as Spatial and temporal scales span many orders of magnitude. In addition to resolving the Kolmogorov length scale η, it is also necessary to resolve the length scale of the smallest liquid structures. As topological change occurs, this length scale approaches zero as the liquid structure undergoes pinchoff. This necessitates a model to track topological changes. Discontinuities in material properties at the interface must be resolved. While some methods are able to approach a discontinuity in fluid properties well (for example, a level set method), some approaches, such as a finite difference method, will introduce numerical smearing along the interface. Surface-tension presents a singular force at the interface. 5

21 Surface tension effects are crucial to accurate resolution of breakup in the case of small local Weber number. A typical approach to this problem is the continuum surface force method, in which a local point force is applied proportional to the interfacial curvature. Both turbulence and turbulent atomization are inherently 3D, rendering 2D simulation inaccurate. The interface undergoes rapid topological distortion. Most current topological models to handle pinch-off exhibit grid dependence. A typical approach instantiates topological change when two separate interfacial segments enter the same cell. Oftentimes these methods will also introduce an error in liquid volume on the order of the grid size. Current attempts to break this dependency by using marker particles to track the interface are effective, but costly. The typical approaches to track the interface include front tracking methods [86], Volume of Fluid methods (VOF) [22] and level set methods [54]. Although DNS remains the only true way to resolve all atomization processes exactly, its prohibitive cost restricts its use to primarily academic applications. Menard et al produced a fully 3D DNS simulation of a diesel jet by using an artificially high surface tension to limit the minimum length scale of liquid structures. In Figure 1.3 and Figure 1.4 the topological complexity of the atomization process is apparent. Even these results are under-resolved, as the grid spacing of 2.36 µm is just below the minimum liquid length scale of 2.4 µm.[57] [20] Nonetheless, DNS results provide an invaluable tool for verifying models of primary atomization phenomenon for which no experimental data is available. 6

22 Figure 1.3. Direct numerical simulation of liquid jet using Menard et al. s level set/vof/ghost fluid method [57] Figure 1.4. Liquid structures downstream of injector [57] 1.2 LES Large Eddy Simulation (LES) is an approach in which the larger three-dimensional unsteady turbulence is explicitly resolved, while the small scale eddies are modeled. The smaller eddies, which are responsible for the majority of the energy dissipation but contain only a small amount of energy, are assumed to have a more universal, isotropic character, making them easier to model than the large eddies which are directly affected by the flow geometry and may have an unsteady character. This eases the turbulent resolution restriction significantly, and has been used successfully to simulate single phase flow. 7

23 In multiphase flow however, it is still necessary to resolve the length scales of the smallest liquid structures. Current LES approaches to resolving primary atomization either [20] Incorporate a coupled LES/DNS technique, performing a DNS calculation in regions containing multiple phases and LES in single-phase regions. This technique reduces the computational demand of DNS, but is still expensive. Current work done by de Villiers [15] and Bianchi [7] has attempted this approach at coarse resolutions. de Villier s results proved to be grid-dependent, with significantly different droplet distributions at increased grid resolution Develop a sub-grid scale atomization model. Several attempts have been made to couple a near DNS scale Eulerian simulation to a Lagrangian droplet formulation, removing stripped-off ligaments from the domain and replacing them with Lagrangian droplets with appropriate secondary atomization models. [36] Figure 1.5 displays Kim et. al s results using this approach to simulate a coaxial jet. Neglect sub-grid atomization terms. Figure 1.5. Ligaments that pinch off from the liquid core (a) are converted into Lagrangian droplets (b) [36] 8

24 While coupled LES/DNS is computationally feasible, successful resolution at the necessary scales remains at the limit of current technology, again suggesting that such an approach is useful in primarily academic applications. 1.3 Lagrangian Modeling Lagrangian spray methods derived from stability analysis of the Kelvin-Helmholtz and Rayleigh instabilities have been in use for the past twenty years. These methods have achieved this popularity due to their computational efficiency and ability to match experimental spray angles and penetrations [61]. The assumption is typically made that the liquid phase has negligible volume fraction in comparison to the gas phase, rendering such simulations to be accurate only in predicting secondary breakup in the far-field of the flow. As such, many applications require tuning of model constants based on empiricism. There have been significant attempts to generalize the Lagrangian formulation to accommodate accurate resolution of primary atomization. Gorokhovski et al. [21] generalized the stochastic approach traditionally used to model secondary atomization to predict primary air-blast atomization. 1.4 Wall Film Modeling In port-injected diesel engine simulations, several researchers have proposed models in which the dynamics of a thin wall film are simulated via a 2D formulation analogous to the shallow water equations. A variety of implementations have been proposed in both an Eulerian, a Lagrangian, and a coupled Eulerian film/lagrangian spray framework; see for example [85, 63, 3, 80]. 1.5 Eulerian Modeling Borghi et al. proposed an Eulerian model for resolving primary atomization in a liquid jet discharging into quiescent air. Their so-called Σ Y model is based on an 9

25 assumption that it is more general to track the growth of interfacial surface area rather than explicitly calculating droplet sizes. Their transport equation for surface area growth takes a form similar to the transport equation for a flame front in combustion. Since their model was originally proposed, it has been adapted to simulate the primary atomization of a diesel spray and air-blast atomizers. Several reports have evaluated the accuracy and range of applicability of the model, both experimentally in the sparse far-field of the spray and using direct numerical simulation in the dense primary atomization zone. Demoulin et al. modified the original dispersion and turbulence models to better account for the large density fluctuations. An extension of the model, termed the Eulerian Lagrangian Spray Atomization (ELSA) model, includes a mechanism for switching to a more accurate and computationally efficient Lagrangian formulation in the dilute region of the spray [18]. 1.6 Model Selection The objective of this project is to develop a means to predict primary atomization within a rocket fuel injector element. Although a DNS/LES approach would be ideal for accurately resolving this process, the cost of performing such a simulation with modern technology is prohibitive. It would be far more desirable to implement a model capable of quickly obtaining results so that either a parametric study could be performed, or simulations could be incorporated into a design optimization process. The only simulation techniques capable of feasible simulation time were the Eulerian Σ Y related models and the stochastic Lagrangian models. The Σ Y model has been selected for use within this project for the following reasons Even in its original form, the model is capable of predicting droplet size and distributions with order-of-magnitude accuracy. 10

26 In the past few years, there have been significantly more publications relating to this model, suggesting that the model shows promise to many researchers and that as the model matures, results will continue to increase in accuracy. The model has been applied to many configurations at comparable Weber and Reynolds number regimes to the rocket injector under consideration (for example, air-blast and coaxial injectors). Data to verify the model is therefore readily available, simplifying the model verification process and allowing more time to be spent producing results. The fully Eulerian formulation is naturally implemented and parallelized in OpenFOAM. 11

27 CHAPTER 2 Σ Y MODEL 2.1 Model Foundations Vallet et al. developed an Eulerian model to determine the sub-grid atomization of a liquid jet acting at high Reynolds and Weber number [87] [88]. The model treats the two-phase flow as a single multi-species flow with variable density. In analogy to the Kolmogorov hypothesis of turbulence, it is assumed that in the limit of infinite Reynolds and Weber numbers the large-scale fluid motion is independent of the surface tension and viscous effects that dominate the small-scale atomization mechanisms, and that the process is self-similar in the intermediate length scales. The model attempts to capture the features of primary atomization that are of practical interest to the design engineer while simplifying the complexities of the flow via simple closures within an Eulerian framework. By recasting the governing equations of motion in a Reynolds averaged formalism, the large scale bulk liquid motion is predicted. Rather than attempting to resolve individual atomization events, a transport equation for the development of surface tension density (Σ) is used to characterize the rate at which surface tension energy is created. A second transport equation tracking the transport of liquid mass fraction (Ỹ ) models the turbulent mixing of liquid. With knowledge of a local interfacial surface area and liquid volume, a characteristic mean diameter can then be determined. Predictions for the mean droplet size and spatial distribution can then be used as an input for Lagrangian secondary breakup models which can be used to predict fuel vaporization, which in turn provide the necessary inputs for combustion calculations. 12

28 The Σ Y model has been successfully applied and validated for a variety of coaxial and diesel jet type injections [58, 32, 5, 62, 56] and recently used for injection schemes with more complex geometries [84]. The original model proposed by Borghi is founded on four assumptions. Surface tension and viscosity act only at small length scales, which corresponds to infinite Reynolds and Weber numbers. This implies that the large scale features of the flow will be dependent only upon density variations. The surface tension and viscosity will only have an effect upon the small scale flow features, such as the droplet size. Although the small scale velocity fluctuations of the flow are unpredictable at the desired resolutions, the mean velocity field can be predicted using standard turbulence closures, such as the classical two equation k-epsilon model. The dispersion of the liquid phase into the gaseous phase can be modeled via a turbulent diffusion flux. The mean geometry of the liquid structures can be characterized by tracking the mean surface area of the liquid-gas interface per unit volume The model has seen successful application in cryogenic flame simulations. Meyers et al. implemented the model to simulate combustion within a LOx/GH2 shearcoaxial jet under cold flow operating conditions [59], obtaining promising initial results. Jay et al. used a modified form of the transport equations to simulate flames formed by cryogenic coaxial injectors under operating conditions comparable to those found in rocket engine applications [33]. They were able to produce good qualitative agreement between the predicted flame structure and available experimental data. They also noted that the model predicted mean droplet sizes with the correct order of magnitude, although direct quantitative analysis was not possible due to a lack 13

29 of experimental data. Both Jay s and Meyers works investigated atomization from coaxial injector configurations. Beheshti et al. wrote an initial assessment of the suitability of the model s ability to predict the effects of liquid properties and injection regimes on the atomization quality, using readily available experimental data of the downstream spray quality under air-assisted atomization conditions [4]. Lebas et al. used DNS results of a diesel spray performed by Menard [57] to assess the model s suitability in the primary atomization zone in close proximity to the nozzle exit [42]. Both assessments showed promising results, and will be reviewed in detail later. 2.2 Governing equations In the following derivation, an overbar denotes a volumetric time average and a denotes an unsteady turbulent fluctuation. To track the dispersion of the liquid phase an indicator function Y is introduced with value 1 in the liquid phase and 0 in the gas phase. The mean liquid volume fraction of the fluid is then given as Ȳ, and the mean mass averaged fraction of the fluid is defined as Ỹ = ρȳ ρ, where is used to denote a mass average. The governing equations are determined by performing the Reynolds decomposition u = ū + u (2.1) Y = Ȳ + Y (2.2) The balance equation for Ỹ is obtained by taking the mass average of the species transport equation and takes the form: ρỹ t + ρũ jỹ x j = ρũ i Y x i (2.3) Similarly, this allows the derivation of the time averaged momentum equation: 14

30 ρũ j t + ρũ iũ j x i = p x j ρũ iu j x i (2.4) The turbulent diffusion liquid flux ũ i Y captures the effect of the relative velocity between the two phases. While the approach used here assumes that the resolved momentum of the liquid/gas mixtures can be characterized by a single bulk velocity, the relative velocity between the two phases can be defined as u l u g = 1 Ỹ (1 Ỹ ) ρũ Y (2.5) the derivation for which can be found in [16]. Models for both the turbulent diffusion liquid flux in the transport equation and the turbulent diffusion flux in the momentum equation must be formed to obtain closure. To close the system, an equation of state must be defined. A variety of approaches have been used in the literature. For subsonic flows, the assumption that both phases are incompressible is appropriate. Alternatively, for moderate Mach number flows, the density of the gas phase can be defined through the perfect gas law p = ρ gas RT (2.6) For liquid atomization occurring at high speed, a linearized equation of state for the liquid phase can be used ρ liq = ρ ref + ψ(p p ref ) (2.7) where ψ denotes the isothermal compressibility of the liquid and ref denotes a reference thermodynamic state about which the compressibility is linearized (generally atmospheric conditions). Regardless of the equation of state used to describe each phase, the total density of the fluid is defined as 15

31 1 ρ = Ỹ ρ liq + 1 Ỹ ρ gas (2.8) or equivalently ρ = Ȳ ρ liq + (1 Ȳ )ρ gas (2.9) It is assumed that the pressure acting upon both phases is equal. For the case where a compressible equation of state is used, it is necessary to define the evolution of the gas temperature through either an assumption that the fluid is isothermal or through the solution of an energy equation. The transport equation for the internal energy of a non-miscible two-phase mixture and its relation to the fluid temperature is given by the equations [37] ρẽ t + ρũ iẽ x i = p ũ i x i (2.10) ẽ = Cv T = (Ỹ Cv,l + (1 v,g) Ỹ )C T (2.11) where e denotes the total internal energy of the mixture, C v denotes a bulk equivalent specific heat, and C v,l and C v,g denote the specific heats of the liquid and gas respectively. With the specification of a transport equation for the surface area density Σ, a number of parameters can be defined to characterize the spray and atomization quality. The Sauter mean diameter gives the ratio of liquid volume to liquid surface area with the assumption that the spray is composed of spherical droplets and can be defined locally through 16

32 d 32 = 6 ρỹ ρ l Σ (2.12) Similarly, the droplet number density gives the number of droplets per a given volume with the assumption of spherical droplets, and is given by n = ρ2 l Σ 3 36π ρ 2Ỹ 2 (2.13) Knowledge of the droplet number density allows the characterization of the mean droplet spacing as L spacing = n 1 3 (2.14) 2.3 Surface area transport closure Background A number of different approaches have been taken to define the equation for the transport of interfacial surface area density. The original form of the model assumes that the transport equation takes a similar form to the transport equation used in combustion applications to track the development of flame surface area density [88]. Since the introduction of the model, a number of different formulations have been presented. The general strategy for all of the models is to determine an equilibrium droplet diameter and a characteristic time scale of the atomization process. This assumption gives a transport equation of the form Σ t + u j Σ x j = (D Σ ) + Σ ( 1 Σ ) x j τ char Σ eq (2.15) For the purposes of this work, the original formulation proposed by Borghi and Vallet [88] will be presented along with the most recent published work by Lebas et al. [42]. 17

33 2.3.2 Original model The original form of the transport equation proposed by Borghi and Vallet is Σ t + u j Σ x j = ( ) Σ D Σ + (A + a) Σ V s Σ2 x j x j (2.16) where the term D is a suitable diffusion coefficient usually taken as the turbulent eddy viscosity ν t over a Schmidt number, the terms A and a are inverse time scales that define the rate at which surface area is produced, and V s is a characteristic velocity scale that defines the rate at which surface area is destroyed through collision and coalescence. The A term models the creation of surface area via the stretching of the interface by mean velocity gradients. Vallet s original model takes this term to be proportional to the same time scale as that used in the production of kinetic energy in the traditional k-epsilon model. ũ A = α 0 i u j ũ i (2.17) k x j Similarly, the term a gives the inverse time scale at which interface is generated by turbulence. The simplest approach for this is to use the inverse integral time scale a = α 1 ǫ k (2.18) Alternatively, if it is assumed that the dominant mechanism is related to the collision and breakup of droplets, the inverse of the droplet collision time scale may also be used. a coll = L spacing v c haracteristic = α 2 (36π) 2/9(l t Σ) 2/3ρ l 4/9 k 1 (2.19) ρ ǫ Ỹ 4/9 18

34 The final term V s capturing the effects of interface destruction is determined by solving for the value of V s that will provide an equilibrium value of Σ eq set by a predicted equilibrium droplet radius. Σ eq = τ prod τ destr = 3 ρỹ ρ l r eq (2.20) Upon selection of a suitable r eq, the V s term is therefore given by V s = aρ lr eq 3 ρỹ (2.21) To fix this equilibrium radius, Vallet s original model assumes that, analogous to the viscous length scale η in Kolmogorov s theory of turbulence, there is also an internal equilibrium length scale r eq. While the Kolmogorov length scale η is defined such that Re cr = ηu η ν (2.22) has a value of 1, the critical droplet radius is similarly defined such that We cr = ρ g v 2 r r eq σ (2.23) also has a value of order one, for a suitably selected velocity scale v r. A detailed analysis of the range of values We cr takes for different flow conditions is discussed in depth by Pilch and Erdman [64]. Proper estimation of this velocity scale will depend upon whether the droplet radius is within the inertial or viscous range of the turbulent length scale spectrum. Assuming r eq > η and that the velocity scale is gives the equilibrium radius v r = k 1/2 (r eq /l t ) 1/3 (2.24) 19

35 ( ) 3/5 σ l 2/5 t r eq = ρ g k 3/5We3/5 cr (2.25) Alternatively, for a critical radius within the viscous scale, a velocity scale of v r = r eq /τ η gives ( ) 1/3 σlt ν r eq = We 1/3 ρ g k 3/2 cr (2.26) Vallet also proposes an alternate mechanism of breakup in which, assuming a two droplet collision, all kinetic energy is converted directly to surface tension energy, giving the following energy balance 4πρ L r 3 (δv) 2 = 4πσr 2 (2 1/3 1) (2.27) 3 Taking the difference in velocity between the droplets from the Kolmogorov scale δv = (ǫl) 1/3 where l is the mean droplet spacing, the equilibrium radius is then given by r eq = C σ3/5 l 2/5 t k 3/5 ρỹ 2/15 ρ 11/15 L (2.28) Vallet et al. then noted that, in the case of a water-air jet operating at a velocity scale k 1/2 on the order of 10 m/s and l t of 1 cm, the two models produce droplets of radii 1 mm and 15 µ m respectively. They took this to suggest that the second mechanism was the dominant one. Several authors have used the same equilibrium radius over a range of operating conditions and fuel injection schemes. It is important to note that this approach is based upon equilibrium processes only, and as such the results obtained may not be valid within short length scales of the atomization region. It is also important to note that the equilibrium radius defined here differs from the actual radius predicted by the model. Figure 2.1 demonstrates Beheshti s comparison of equilibrium diameter with predicted d 32 for a coaxial injection scheme [4]. 20

36 Figure 2.1. Comparison of actual and equilibrium SMD along center axis [4] Equation used by Lebas et al. Lebas et al. sought to validate the ELSA model s accuracy in the primary atomization region by comparing results to Ménard et al. s DNS of diesel jet atomization [42] [57]. Their implementation uses the same dispersion relation proposed by Demoulin et al. for the species transport [16]. They also implement a unique transport equation for the interfacial surface density, using an indicator function Ψ based upon the liquid volume fraction ranging from 0 to 1 to switch between two separate sources and sinks for Σ in the near and far field of the spray. The indicator function used in their paper is shown in Figure 2.2. They used values of φ dense = 0.5 and φ dilute = 0.2 in their work. Their transport equation takes the form ρ Ω t + u j ρ Ω x j = ( µt x j Sc ) Σ + Ψ (S init + S turb ) + (1 Ψ) (S coll + S 2ndBU ) + S vapo x j (2.29) where they have defined Ω = Σ. The S terms appearing on the right hand side ρ denote sources of surface area, each corresponding to a different mechanism. In the dense part of the spray S init gives the initial source of surface area at the injector 21

37 inlet, while S turb gives the source due to the interaction of gas phase turbulence with the interface. In the sparse part of the spray S coll gives a sink term due to droplet collision and coalescence, and S 2ndBU is a source corresponding to secondary breakup mechanisms. The contribution due to vaporization is given by S vapo. Figure 2.2. Indicator function used to transition between source terms in dense and dilute part of spray [42] The form assumed for the term S init is selected so that the liquid will tend toward a surface area density of where l t is the integral turbulent length scale k 3 2 ǫ Σ min = φ l(1 φ l ) l t (2.30). This is inspired by the assumption that the initial wrinkles of the liquid interface will have the same length scales as the gas-phase turbulent structures impacting the interface. The term is defined as S init = 12 ρµ t ρ l ρ g Scl t Ỹ x i Ỹ x i (2.31) 22

38 While this term has not been rigorously investigated, it has been shown to have negligible effect on the downstream solution behavior and is only necessary as a boundary condition to mark the location of the onset of surface area generation. The S turb term takes a form similar to the original derivation by Vallet. An identical argument is used to suggest that primary atomization occurs at a length scale satisfying a critical Weber number, giving the following relation We dense,crit = where σ l is the liquid surface tension coefficient. Ȳ k σ l Ω eq,dense (2.32) This critical Weber number is taken to be 1, the same value used in the original model formulation. The source term is defined as S turb = ρω τ t ( 1 Ω Ω eq,dense where τ t is the integral turbulent time scale k ǫ. ) (2.33) To determine the effects of collision in the transition to the dilute part of the spray, the source term again takes the form S coll = ρω ( 1 Ω ) τ coll Ω eq,coll (2.34) The characteristic time scale of droplet collision is taken to be τ coll = L 3 coll S eff V coll (2.35) where L coll is the mean droplet spacing, S eff is the cross-section of collision, and V is the characteristic collision velocity. These give a value for the droplet collision time scale of 23

39 1 τ coll = 2 ρ Ω 3 k (2.36) To determine the equilibrium droplet radius, the Weber number of the spray is calculated We coll = σ l 4Ỹ k (2.37) Ω eq,coll With the assumption that kinetic and surface energy is conserved throughout the breakup process, and that the critical Weber number occurs at We N = 12, the equilibrium droplet diameter is D 1 + We N 32 = D We coll 6 The equilibrium value for the surface area density is finally given by (2.38) D 32 = 6Ỹ ρ l Ω eq,coll (2.39) The final term describing breakup is given by [ S 2ndBU = Max 0, ρω τ 2ndBU ( 1 Ω Ω eq,2ndbu )] (2.40) and reflects the creation of surface area via hydrodynamic instabilities caused by the relative velocity between the liquid and the gas. Because these processes are purely unstable, this term can only act as a source of surface area. The critical Weber number at which the liquid achieves its equilibrium surface area is We 2ndBU = 6ρ g v 2 Ỹ ρ l σ l Ω (2.41) where the relative velocity between the two phases is taken from the model for the liquid diffusion flux 24

40 v 2 = The Weber number is taken from experiments [64] to be 1 Ỹ (1 Ỹ ) ρũ Y (2.42) We 2ndBU = 12( Oh 1.6 ) (2.43) where Oh is the Ohnesorge number characterizing the ratio of viscous forces to inertial and surface tension forces. The time scale is given by Oh = µ l (ρ l d 32 σ l ) 0.5 (2.44) τ 2ndBU = T d 32 ρl (2.45) v ρ g where T is a dimensionless time scale taken from the experimental work by Pilch et al. [64]. While Lebas did not note the value of T used in their work, the following correlations give T as a function of the Weber number provided in the work by Pilch et al. The experimental data from which these correlations were derived are given in Figure 2.3. T = 6(We 12) 0.25, 12 We 18 (2.46) T = 2.45(We 12) 0.25, 18 We 45 (2.47) T = 14.1(We 12) 0.25, 45 We 351 (2.48) T = 5.5, We 2670 (2.49) 25

41 Figure 2.3. Experimental data for total dimensionless breakup time scale T as a function of Weber number [64] 2.4 Turbulent liquid flux closure In the modeling of the diffusion of mass in laminar flows, kinetic theory shows that the molecular diffusion flux can, in its simplest form, be modeled via Fick s law of mass diffusion. To a higher order approximation, the general diffusion processes are also dependent upon the pressure gradient due to momentum flux caused by the random motion of molecules, and upon the Soret effect in which increased internal energy causes hot molecules to move faster than cold molecules, causing a net flow along negative temperature gradients [38]. In turbulent flows of comparable density, it is often assumed that the molecular viscosity used in Fick s law can be replaced with an effective turbulent viscosity with reasonable accuracy [8]. Accurate simulation of mixing between species of large density ratios remains an open research area today. While discussion of the modeling of this closure will be limited to the pseudo-fluid formulation employed by the Σ Y model, a general discussion of the closure of this relationship for a variety of modeling frameworks such as the two-fluid model and the drift-flux model can be found in Ishii and Habiki [29]. 26

42 In the original derivation of the model, Vallet proposes the following unclosed exact form for the transport equation of the correlation between velocity and mass fraction fluctuations. ρũ i Y t + ρũ jũ i Y x j = x j ( ( 1 γ Y ) ρũ j Y ũ i x j ) ρũ j u i Y Ỹ + p Y δ ij ρũ j u i x j ( 1 γ Y ) Ỹ p x i ρ a 1 τ t ũ i Y (2.50) The first term on the right hand side corresponds to the turbulent diffusion of mass fluctuations, the following three terms to production, and the final term to destruction. Equation 2.50 assumes that p Y x i can be expressed as p Y x i = ρ a 1 τ t ũ iy + γ Y ρũ jy ũ i x j + γ Y Ȳ p x i (2.51) where τ t is the integral turbulence time scale and a 1, γ Y and γ Y are constants. Assuming dominance by the production and destruction terms yields the following algebraic closure. ρũ iy = τ t a 1 [ Ỹ ρũ j u i + ( 1 γ ) Y ρũ x jy u ( ) ] i + 1 γ Y Ȳ p j x j x i (2.52) where Ȳ is defined exactly as Ȳ = ρỹ ( ) ( 1 Ỹ 1 1 ) ρ l ρ g (2.53) Under the further assumption that diffusion is dominated by the first term on the right hand side of Equation 2.52, this results in Fick s law of diffusion 27

43 ρũ iy = µ t Ỹ (2.54) Sc x i Demoulin showed that the use of Fick s law, while appropriate in the sparse area of the flow, is unable to predict the enhanced mixing within the primary atomization caused by Rayleigh-Taylor instability. MILES simulations conducted by Silvani et al. [78] demonstrated that the effective Schmidt number for primary atomization is characterized by ( 1 ρ 1 + ρ ) Sc eff ρ g ρ g (2.55) For buoyant, variable density flows, the force of gravity can act as either a stabilizing or destabilizing force through the Rayleigh-Benard instability. Lumley [53] and ( ) Launder [41] proposed the contribution to the correlation for p Y x i p Y = ρ a 1 ũ x i τ iy + γ Y ρũ jy ũ i + γ Y Ȳ p ( 1 + C y3 ρ 1 ) ρy t x j x i ρ l ρ Y g (2.56) g where g denotes the acceleration due to gravity. Motivated by these results, Demoulin suggested that in the case where Rayleigh-Taylor type instabilities are dominant, the g term should be replaced by a characteristic acceleration. If the turbulent acceleration is characterized by the turbulent kinetic energy divided by a characteristic length scale, and the characteristic length scale is taken as the inverse of the mass fraction gradient, the correlation then becomes p Y = ρ a 1 ũ x i τ iy +γ Y ρũ jy ũ i +γ Y Ȳ p ( 1 +C y3 ρ 1 ) Ỹ ρy t x j x i ρ l ρ Y k (2.57) g x i Through the same assumptions as those used to derive Equation 2.54, this reduces to the following algebraic closure for the turbulent liquid flux 28

44 [ ( ρu i Y νt k 2 1 = ρ + C P P t ǫ ρ 1 ) ] Ỹ (1 ρ g ρ Ỹ ) Ỹ (2.58) l x i The inclusion of this term has proven very successful in resolution of the primary liquid core length for both diesel [42] and coaxial [16] injections. While all currently published work with the Σ Y model implements this equation, the choice of k Ỹ x for a characteristic acceleration in Demoulin s modified transport equation for ũ Y makes the implicit assumption that the turbulent mixing is isentropic. In reality, if the predominant instability mechanism is Rayleigh-Taylor type breakup, the linear instability analysis suggests a form for a modeled production term as P source Ỹ a (2.59) where a denotes the acceleration acting on the fluid element. It is expected that instability should only be created if the acceleration is aligned with the density gradient of the fluid. If the acceleration is oriented parallel and opposite to the density gradient, the term should have a stabilizing effect. Therefore, the selection of k Ỹ x will not be able to capture the phenomenon where the turbulent acceleration actually stabilizes mixing at the interface, or where anisentropic turbulence may create mixing in one direction more favorably then another. For some of the applications demonstrated in the following section, this arrangement is typical of flows with swirl and recirculation. For such conditions, a more general formulation would preserve the directional character of the turbulence by defining the characteristic acceleration scale as R ij Ỹ x j. The algebraic closure in Equation 2.58 would then have the form: [ ( 1 1 ρu i Y = ρ + C P ρ 1 ) ] Ỹ (1 P t ρ g ρ Ỹ ) Rij Ỹ (2.60) l τ t x j where the Reynolds stresses R ij would be obtained from the turbulence model. 29

45 2.5 Reynolds stresses closure The topic of turbulence modeling even for single phase flows is an active area of research and the modeling of variable density flows, particularly flows with large density ratio, is certainly an unsolved problem. In the early published work using the Σ Y model, researchers applied the classical two equation k ǫ model by Jones and Launder [34] due to its robustness and accuracy in modeling shear driven flows with minimally curved streamlines. A general form of the k ǫ model typically employed for modeling variable density flows is given in Chassaing et al. [11]. This formulation of the model is appropriate for general variable density flows, regardless of whether density variation is caused by compressibility or species concentration mixing effects. The exact unclosed equation for the Farve averaged kinetic energy fluctuation is derived directly from the conservative form of the momentum equation after a Reynolds decomposition of velocity, and is given by ρ k t + ρũ j k x j = ρũ i u j ũ i x j x j [ ] 1 2 ρu i u i u j + p u j u i p p x u i + τ iju i u τ i ij i x i x j x j (2.61) The first term on the left hand defines the usual material derivative denoting the change in turbulent kinetic energy along a streamline. The first term on the right hand side denotes the production of kinetic energy by shear in the mean flow, and is identical to that found in the classic incompressible k ǫ formulation by Jones and Launder. The second term represents the turbulent advection or diffusion of kinetic energy. The third term is referred to as the mean pressure work, and is unique to Favre averaged variable density flows. It corresponds to the energy transfer caused by the coupling between the turbulent liquid flux and the mean pressure field. The fourth term is the pressure-dilation correlation and is specific to flows with 30

46 non-zero divergence velocity fluctuations. It is therefore expected that this term is only important in high Mach number flows. The last two terms are effects caused by the molecular velocity, the first being the work done by external forces on the velocity fluctuations, and the second being the Favre-averaged dissipation rate. For the purposes of high Weber and Reynolds number fuel injection, the first of these two effects is neglected. Similarly, a possible form for the Favre averaged turbulent dissipation rate is given by the equation ρ ǫ t + ρũ j ǫ x j = C ǫ1 ǫ k P +D ǫ +C ǫ2 ρ ǫ2 k +C ǫ3 ǫ i k p u x i C ǫ4 ǫ k u i p C ũ i ǫ5 ρ ǫ (2.62) x i x i The left hand side of the equation again gives the change of turbulent dissipation rate along a streamline. The first three terms on the right hand side correspond to the effects of shear production, diffusion and dissipation that turn up in the usual incompressible formulation. The last three terms are unique to Favre averaged variable density flows. The first two are take the same form as the third and fourth terms on the right hand side of the kinetic energy equation divided by the integral turbulent time scale k. The last term is used to account for the dependence of the integral ǫ length scale of the turbulence as a shock wave passes through it. [11]. In his work to determine a model for the enhanced liquid flux correlation occurring during primary atomization, Demoulin et al. also set out to define suitable modifications to the turbulence model used in Vallet s original closures [16, 88]. They distinguish that traditional variable density turbulence models are developed based on assumptions of compressibility mechanisms appropriate for high-velocity flow, such as the flow over an airfoil. These models are not necessarily applicable to variable density flows caused by the mixing of two immiscible fluids, such as the case of atomization between two fluids of different densities. Reinaud et al [71] have identified 31

47 unique secondary instability mechanisms in which a baroclinic effect caused by a flow acceleration coupled with a density gradient leads to an increase in mixing and destruction of the largest eddies [16]. A successful model must capture this coupling between pressure and density gradients, effects from density ratios between the two fluids on the order of a thousand, and the fact that, unlike in high-velocity flows, the density gradient is not dissipated by molecular diffusion, as capillary effects will become dominant first. Following the results proposed by Vallet, Demoulin suggests the following form for the Reynolds stress tensor transport equation. ρr ij t + ρũ kr ij x k = D ij + P ij + Φ ij + G ij ρ ǫ ij (2.63) where D ij denotes a diffusion term, P ij a production term, Φ ij a redistribution term, and G ij a term derived from the exact equations for the Reynolds stress proportional to the mean of the Favre velocity fluctuations. In traditional models, this term would be defined as G ij = ( ū p j ū p ) i x i x j (2.64) Vallet proposed that the redistribution term be modified from the traditional form to include a contribution from the G ij term Φ ij = φ ij + γ (G ij 3ū 2 ) p k δ ij x k (2.65) This leads to the following equations for kinetic energy and turbulent dissipation ρ k t + ρũ i k x i ( ) = µ t k x i Pr k x i ũ j p ρr ij x ū i ρ ǫ (2.66) i x i ρ ǫ t + ρũ i ǫ = ( ) ( ) µt ǫ ǫ ũ j p + C ǫ1 ρr ij x i x i Pr ǫ x i k x ū i C ǫ2 j x i ρ ǫ2 k (2.67) 32

48 The velocity fluctuation can be expressed exactly via where ρ u g i Y x i ū i = ( 1 ρ l 1 ρ g ) ρ ũ i Y x i (2.68) has been previously defined in Equation 2.3 as the turbulent flux. Together with the turbulent eddy viscosity hypothesis, the momentum equations are now closed upon a successful treatment of the turbulent flux. Interestingly enough, the results given by Demoulin are identical to those given by Chassaing et al. under the assumption that source terms in the k and ǫ equations due to dilatational effects and shock waves are negligible. The modified equations proposed by Demoulin have been used in recently published work [42] with excellent agreement to experimental and DNS results. 33

49 2.6 Evaluation of Model Beheshti set out to critique Vallet s original model by comparing the results it generated to experimental results of downstream measurements of Sauter Mean Diameter, using the correlations provided from Hopfinger and Lasheras [24] and Lasheras et al. s [39] work with coaxial air-water jets [4]. They noted that, although the model had proven accurate when applied to a single configuration, it could only prove useful if it continued to produce accurate results over a range of parameters typical of those found in practical scenarios. Similar to preceding work, a standard variable density k ǫ turbulence model was used in their work, justified under the assumption that air-assisted atomization tends to a self-similar state similar to a simple gaseous jet. This assumption has been supported to some extent by Prevost et al. s [66] observation of linear growth of the jet half-radius in air-assisted jet atomization within a pressurized chamber in accordance with traditional similarity analyses of a constant density turbulent jet (For example, [65]). These assumptions are only valid downstream of the primary atomization where density variations become less extreme. Beheshti acknowledged that this is inaccurate, but notes that the adoption of standard modifications to the k ǫ model for the presence of spherical droplets or particles is inappropriate for the ligament-type liquid structures occurring in the dense region of the spray. They noted that the model is limited in the fact that it only attempts to resolve the SMD and as such is unable to resolve effects caused by a wide distribution of droplet size in polydispersed sprays, such as ballistic drop spreading. They concluded that this is acceptable in the current application because existing experimental data for gaseous and aerosol jets show a lower spreading rate for an increasingly heavy central jet, suggesting that variable density effects are more dominant than ballistic spreading [19, 23]. 34

50 Figure 2.4. Spray SMD along symmetry axis for different gas jet velocities and a fixed liquid velocity of 0.5 m/s [4] Figure 2.5. Spray SMD along symmetry axis for different liquid jet velocities and a fixed gas velocity of 140 m/s [4] Beheshti s implementation provided excellent agreement with a variety of experimental data and correlations over a range of operating conditions. Figures 2.4 and 2.5 show the predicted SMD along the symmetry axis of the injector for a range of gas velocities. The only discrepancy that they noted is that, below a critical momentum ratio, a recirculation zone forms (See Figure 2.6). Within this range the 35

51 model under-predicts SMDs by as much as 200%. Beheshti attributes this discrepancy to the known inability of k ǫ models to accurately resolve recirculation zones [89]. While this is certainly true, recirculation zones are generally accompanied by an adverse pressure gradient which could have a significant effect on the Y p x i term in Demoulin s modification of the k ǫ model. Figure 2.6. Liquid core profiles for fixed gas velocity of 140 m/s and liquid velocities of 0.55, 0.31, 0.17, and 0.13 m/s, respectively [4] 36

52 Lebas et al. sought to validate the ELSA model s accuracy in the near field by comparing results to Menard et al. s DNS of diesel jet atomization [42, 57]. Although Menard s simulation only achieves a level of discretization on the order of half the atomized droplet size, their results provide the only currently available DNS data through which to make comparisons to the experimentally inaccessible primary atomization zone. They performed a simulation of Ménard s geometry using a 70,000 cell grid and a 140,000 cell grid. A comparison of the ELSA model results to the averaged DNS for Y and Σ are provided in Figures 2.7 and 2.8. Particularly noteworthy are the results shown in Figures 2.7 and 2.8 which demonstrate that, up to moderate lengths downstream, both the liquid volume fraction and surface area density profiles match almost perfectly with the DNS results. While the results given by the Σ Y model in the literature using consistent choice of modeling parameters have proven accurate over a range of operating conditions, all published work so far has been focused on shear-driven flows with simple geometry. Further work must be conducted to assess the models generality in predicting more complex injection schemes. Figure 2.7. Comparison of predicted liquid volume fraction profile to DNS results [42] 37

53 Figure 2.8. Comparison of predicted interfacial surface area density profile to DNS results [42] 38

54 CHAPTER 3 IMPLEMENTATION 3.1 Overview A finite volume solver has been written using the OpenFOAM C++ library implementing the Σ Y model. The implementation is based on the assumption that the fluid is composed of an ideal gas and a liquid of constant compressibility, and can be used to simulate flows at high Mach number. The solver has been validated against analytic solutions for compressible flow in a converging-diverging nozzle and against published results of the Σ Y model. A number of features have been implemented but have not yet been validated: the code has the capability to model interphase heat transfer, and a secondary formulation has been implemented allowing the transition to a Lagrangian-Eulerian formulation in the dilute part of the spray. The OpenFOAM library is an open source C++ library developed for the rapid implementation of computational continuum mechanics finite solvers. The library makes use of the templating features of C++ to allow an intuitive high-level approach to solver design. For example, implementation of the transport equation for mass fraction: ρỹ t requires only two lines of code: + ρũ iỹ x i = ρũ i Y x i (3.1) 39

55 fvscalarmatrix YEqn ( fvm::ddt(rho,y) + fvm::div(rhophi,y) == - fvc::div(upyp) ); YEqn.solve(); which will cast the PDE into a matrix system of the form AY = b. The fvm:: namespace (finite volume method) returns the matrix coefficients of the analogous discrete calculus operators to each differential operator so that they can be solved for implicitly. Discretization schemes are run-time selectable allowing the use of higher order numerical schemes where desired. The fvc:: namespace (finite volume calculus) returns the explicit source terms for each cell. After building the appropriate matrix system, a run-time selectable matrix solver provides a solution to the system. Further details of the structure of the OpenFOAM code and its capabilities are available in [30, 31]. 3.2 Pressure equation Foundation In order to obtain a transport equation for pressure, the momentum equation must be forced to satisfy the continuity equation. The single-phase, incompressible algorithm outlined in Hrvoje Jasak s PhD thesis [30] has been extended to account for 40

56 compressible, multi-phase, variable temperature flow. This is achieved by recasting the momentum equation in the semi-discretized form: a p U p = H(U) p (3.2) where the momentum matrix has been split into a diagonal part a p and an offdiagonal part H(U). The term H(U) represents the fluxes acting at the cell faces and consists of two parts: the advection terms which includes the matrix coefficients of neighboring cells multiplied by the corresponding velocities and the source term, accounting for all point sources in the momentum equations (such as gravity, body forces, etc.). H(U) = N a N U N + U0 t (3.3) Equation 3.2 is rearranged and interpolated to faces. Taking the divergence of this gives (U p ) f = φ ( ) 1 p a p f (3.4) Where φ is the value of the velocity fluxes obtained from the predictor step in the PISO loop. In Jasak s case of a divergence-free velocity field, this yields the following Poisson equation for pressure ( ) ( ) 1 1 p f = φ a p a p (3.5) For the case of multiphase, compressible flow the velocity divergence is nonzero and can be split between the effects of turbulent mixing, thermal expansion and Mach effects by applying the chain rule to the continuity equation. 41

57 (U p ) f = Dρ Dt = 1 ρ DỸ ρ Ỹ Dt 1 ρ D p ρ p Dt 1 ρ ρ T D T Dt (3.6) In order to obtain a fully closed transport equation for pressure, each one of these terms must be treated in a numerically stable manner Multiphase mixing Along a streamline in the flow, the turbulent mixing of high density liquid with low density gas can cause a significant divergence in the velocity field. In the term ρ Ỹ is given by 1 ρ ρ Ỹ DỸ Dt (3.7) ( ρ 1 Ỹ = ρ2 1 ) ρ l ρ g and DỸ Dt is related to the transport equation for mass fraction via the equations (3.8) and [ DỸ Dt = 1 DρỸ ρ Dt ] Ỹ Dρ Dt (3.9) ρỹ t + ρũ iỹ x i = ρũ i Y x i (3.10) Combining these terms provides the contribution of the multiphase mixing to the velocity divergence 1 ρ ρ Ỹ [ DỸ 1 Dt = 1 ] ρ l ρ ρũ i Y (3.11) g 42

58 3.2.3 Compressibility effects term The divergence caused by compressibility at high Mach numbers is caused by the The derivative ρ p 1 ρ ρ D p p Dt (3.12) is the effective isothermal compressibility of the liquid/air mixture and is defined as ψ. The isothermal compressibility of the liquid phase is assumed constant and defined as ψ l. The compressibility of the gas phase is obtained by taking the derivative of density in the ideal gas equation of state with respect to pressure, giving ψ g = 1 R g T (3.13) The density is related to the mass fraction via the relation 1 ρ = Ỹ ρ l + 1 Ỹ ρ g (3.14) Taking the derivative of this with respect to pressure gives Simplifying with the relation Ỹ = ρȳ ρ l equation 3.13 gives 1 ρ 2ψ = Ỹ ρ l 2 ψ l 1 Ỹ ρ g 2 ψ g (3.15) and substituting in the ideal gas law and (Ȳ ψl ψ = ρ + 1 Ȳ ) ρ l p (3.16) The term defining the contribution of compressibility effect is therefore given by 1 ρ (Ȳ ρ D p p Dt = ψl + 1 Ȳ ) D p ρ l p Dt (3.17) 43

59 3.2.4 Thermal expansion effects To determine the effect of temperature change upon the density, a similar approach is used. The derivative of is taken with respect to temperature, to get 1 ρ = Ỹ + 1 Ỹ (3.18) ρ l ρ g 1 ρ ρ 2 T = Ỹ ρ l ρ 2 l T 1 Ỹ ρ 2 g ρ g T (3.19) By assuming that all thermal expansion effects are experienced purely by the gas phase, substituting in the ideal gas equation of state, and using the relation Ỹ = ρȳ ρ l, this equation simplifies to 1 ρ ρ T DT Dt = 1 Ȳ T The total effect of thermal expansion is then given by the term (3.20) 1 ρ ρ T = 1 Ȳ T DT Dt (3.21) Final equation Substituting each of these three effects back into Equations 3.6 and 3.4 gives the following final equation for pressure φ ( ) 1 p a p f (Ȳ ψl = ρ l + 1 Ȳ p ) D p Dt + 1 Ȳ T This equation is implemented in OpenFOAM as [ DT 1 Dt 1 ] ρ l ρ ũ i Y g (3.22) fvscalarmatrix peqn 44

60 ( fvm::laplacian(ruaf, p) - fvc::div(phi) == (Ybar*psil/rholiq + max(1.0-ybar,0.0)/p) * (fvm::ddt(p) + fvm::div(phi, p) - fvm::sp(fvc::div(phi), p)) - (max(1.0-ybar,0.0)*cvg/e) * (fvc::ddt(t) + fvc::div(phi, T) - fvc::sp(fvc::div(phi), T)) + Pr*fvc::laplacian(Dcoeff,Y) ); In order to maintain numerical stability, it is crucial to treat the total derivative of pressure implicitly as shown above. When incorporated into the PISO algorithm, solution of this equation will provide a pressure that will project the nonconservative predictor velocity flux field onto a mass conservative solution. The algorithm for each time step can be summarized as follows: while ( t < t_final) Solve density equation using fluxes from previous time step Solve secondary transport equations (mass fraction, internal energy) Solve momentum transport equations Do PISO loop (2-4 iterations) Iteratively solve pressure equation Update fluxes Update turbulence models Solve surface area density transport equation end 45

61 Using this algorithm, the solver will give a velocity field at each time step that satisfies the continuity equation, provided that it starts from an initial condition that also satisfies the continuity equation. 3.3 Mass transport equation Due to the large density ratios typical of atomization processes, precautions must be taken when handling the mass transport equation as seemingly small errors will be multiplied by a factor of In addition to predicting the evolution of liquid mass within the flow solution, the mass fraction variable is also used as an indicating function to specify transport properties of the fluid mixture. ρỹ t + ρũ iỹ x i = ρũ i Y x i (3.23) Boundary conditions for the mass fraction must be set consistently with those used for the density. Solution of the mass fraction equation requires special care in treating non-linearities that can arise in the source term caused by the turbulent liquid flux ũ i Y. In the simplest case where the turbulent liquid flux is closed via Fick s law of diffusion, the equation reduces to a well-behaved linear advection-diffusion equation. ρỹ t + ρũ iỹ x j ( = µ t x i Sc ) Ỹ x i (3.24) In the case where Demoulin s algebraic closure for the enhancement of mixing by Rayleigh-Taylor mechanisms is used a non-linearity arises in the equation ρỹ t + ρũ iỹ x j ( = [ ( νt k 2 1 ρ + C P x i P t ǫ ρ 1 ) ] ) Ỹ (1 ρ g ρ Ỹ ) Ỹ l x i (3.25) 46

62 which requires an iterative solution of the problem. Currently this is done using simple fixed point iteration, however if increased speed of convergence is required this could easily be done using a Newton-Rahpson method. The most significant numerical challenge arises when a second transport equation for the turbulent liquid flux is solved, such as the following from the original model proposal by Vallet [88]. ρũ i Y t + ρũ jũ i Y x j = x j ( ( 1 γ Y ) ρũ j Y ũ i x j ) ρũ j u i Y Ỹ + p Y δ ij ρũ j u i x j ( 1 γ Y ) Ỹ p x i ρ a 1 τ t ũ i Y (3.26) This equation is closely coupled with the mass fraction equation, the pressure equation, and the turbulence model. Ideally, a coupled block matrix solver would be used to treat both the mass fraction and the liquid flux term implicitly. Due to the constraints of the segregated solver paradigm implemented in OpenFOAM, the current implementation of the code uses the following simple iterative algorithm to couple the two systems of equations. while (residuals < TOLERANCE) solve Y equation solve upyp equation Y = Y_lastiteration + relax*(y_thisiteration - Y_lastiteration) upyp = upyp_lastiteration + relax*(upyp_thisiteration - upyp_lastiteration) Y=Y_lastiteration upyp=upyp_lastiteration 47

63 where relax is a constant ranging from 0 to 1 that can be lowered to improve stability. Although the above method requires a large number of costly matrix inversions to converge, the time required to solve the two equations is insignificant in comparison to the pressure correction loop of the algorithm, suggesting that the above strategy imposes an acceptably insignificant drop in solver performance in practical applications to incorporate into the solver. 3.4 Turbulence model The modified turbulence model proposed by Demoulin et al. is implemented as follows: ρ k t + ρũ i k x i ( ) = µ t k x i Pr k x i ũ j p ρr ij x ū i ρ ǫ (3.27) i x i ρ ǫ t + ρũ i ǫ = ( ) ( ) µt ǫ ǫ ũ j p + C ǫ1 ρr ij x i x i Pr ǫ x i k x ū i C ǫ2 j x i ρ ǫ2 k (3.28) While the standard k ǫ model used in OpenFoam has proven robust in most applications, the introduction of the ū i p x i term introduces a strong coupling into the equations, particularly in flows with strong pressure gradients typical of those encountered in high pressure diesel fuel injection. Again, to achieve convergence between the two strongly coupled equations, it is necessary to employ a relaxation technique. Unfortunately, unlike the approach used in the liquid flux and mass fraction transport equation, the k ǫ equations are non-linear and strong relaxation (relax = 0.5) is sometimes required in actual practice, with convergence sometimes requiring up to 10 repeated solutions of the two equations. For situations where this poses a computational burden, the following algorithm is implemented: 48

64 If the two transport equations are posed in the form of the block matrix system. k k ǫ k k ǫ ǫ ǫ k ǫ = E F where k k denotes the block of coefficients in the k transport equation to be multiplied by k, k ǫ denotes the block of coefficients in the k transport equation to be multiplied by ǫ, and so forth. The iteration of the process previously described amounts to one iteration of a block Gauss-Seidel inversion of the system, i.e. k n+1 = k 1 k (k RHS k ǫ ǫ n ) (3.29) ǫ n+1 = ǫ ( 1 ǫ ǫ RHS ǫ k k n+1) (3.30) Within the OpenFOAM code, the diagonal of the top-left block and bottom-right block can be accessed by calling the functions keqn.a() and epseqn.a(). The off diagonal contribution of both blocks is accessed through keqn.h() and epseqn.h(). In the segregated framework employed by OpenFOAM, the contribution of epsilon to the k equation and vice versa has to be treated as an explicit source term (accessible via keqn.source(), etc.) that gets lumped in with the other source terms Pulling all those together allows the use of a less expensive Jacobi inversion within each block, via the algorithm: while (residuals < TOL){ k = (1.0/kEqn.A()) * (keqn.source() - keqn.h()); epsilon = (1.0/epsEqn.A()) * (epseqn.source() - epseqn.h()); } Which will converge to the same result as that obtained when solving the fully coupled block matrix system. Limited testing performed with this technique showed 49

65 promise, but required in some cases excessive relaxation to obtain convergence, due to the restrictive convergence criteria of Jacobi matrix inversion. For the time being, full inversion of the system is implemented in the code with relaxation, but developers of the OpenFOAM library have suggested that the fully coupled block matrix solver is currently under development and should be available soon. 3.5 Internal energy equation To determine the evolution of temperature, the following equations are solved ρẽ t + ρũ iẽ x j = p ũ i x i (3.31) ẽ = Cv T = (Ỹ Cv,l + (1 Ỹ )C v,g) T (3.32) T = ẽ Cv (3.33) where e denotes the total internal energy of the mixture, C v denotes a bulk equivalent specific heat, and C v,l and C v,g denote the specific heats of the liquid and gas respectively. The solver has only been validated for non-isothermal flows and for cases with a single gas phase. All other calculations presented in this work make use of the assumption that the liquid/gas mixture is isothermal. 3.6 Lagrangian droplet tracking One of the most attractive features of the Σ Y model is that, as the spray becomes more sparse far from the primary atomization zone, it becomes numerically advantageous to transition to a Lagrangian formulation. Published work with this so called Eulerian-Lagrangian Spray Atomization (ELSA) model has been very encouraging, and produced accurate predictions for diesel injection flows [18]. While 50

66 the implementation and validation of a secondary Lagrangian breakup model is beyond the scope of this project, a preliminary implementation is introduced here with preliminary results. The coupling operates by determining at the beginning of each time step which liquid-containing cells should be switched. An indicator function β is used to signal transition to the Lagrangian framework after the ratio of mean droplet spacing to droplet mean diameter reaches a critical value. The function β takes a value of 1 in cells where transition is occurring and 0 everywhere else. The value of the mass and volume fraction fields for these cells are set to zero, and a sink term is introduced into the continuity equation balancing the mass to be added to the Lagrangian phase: ρ t + ρũ i t = βȳ ρ l (3.34) A Lagrangian parcel is then injected into the flow with a diameter and number density determined by the Σ and Y fields. The velocity of these particles is selected randomly from the interval u parcel = ũ ± k (3.35) A schematic illustrating this process is given in Figure 3.1. The equations of motion for each parcel are then solved via the preexisting Lagrangian sub-models in OpenFOAM, and coupled to the Eulerian phase through point sources in the momentum equation for drag. Because the framework employed uses the native OpenFOAM Lagrangian libraries, the particle tracking is fully parallelized. While this approach has not been rigorously validated, preliminary calculations do show that the coupling is mass conservative. A sample calculation of stripping from a liquid film is presented in Figure 3.2. The framework implemented in the code provides all of the necessary data structures and coupling to the PISO loop to allow an interested party to fully develop a secondary Lagrangian model. 51

67 Figure 3.1. Process by which dilute liquid cells are identified, deleted, and replaced with Lagrangian parcels Figure 3.2. Preliminary results showing stripping of droplets by high-velocity coflowing gas (bottom) from a liquid film (top) 52

68 CHAPTER 4 VALIDATION 4.1 Compressibility: Converging-diverging nozzle case Introduction The converging-diverging nozzle (CDN) or Laval nozzle, as it is sometimes referred to after its inventor Gustaf de Laval, is used in energy and propulsion applications to accelerate a fluid past the speed of sound. Traditional subsonic nozzles become choked when the flow within them reaches the speed of sound, meaning that for further decreases in downstream pressure, the mass flow rate will remain fixed. The flow within a CDN is particularly interesting as a validation case because it can [1]: 1. Verify that global mass conservation is satisfied throughout the nozzle 2. Verify that total pressure and temperature is conserved throughout the nozzle, as is the case for all isentropic flows of ideal gases [91] 3. Verify the accuracy of the numerical solver against an analytic solution for the flow behavior 4. Verify the suitability of the solver for resolving discontinuities across shock waves Analytical solution The following derivation of the analytic flow through a CDN is taken from Wilcox [91]. It is assumed that the flow is one dimensional, inviscid, thermally non-conductive, 53

69 isentropic, and that boundary layer effects near the walls are negligible (slip velocity boundary condition). Implementing a control volume analysis, the integral form of the mass conservation equation gives ρua = ρ U A (4.1) where () denotes conditions at the throat of the nozzle. For an arbitrary point along the nozzle, A A = ρ U ρu = ρ a Ma ρ ama (4.2) where a denotes the speed of sound and Ma = U a the Mach number. Substituting the speed of sound for a perfect gas a pg = γrt gives the relation A A = ρ T Ma ρ T Ma (4.3) With the assumption that the flow becomes sonic at the throat of the nozzle (i.e. Ma = 1), the area ratio is given by A A = ρ T 1 ρ T Ma (4.4) While for a given inlet total pressure and outlet pressure the flow at the throat may not actually be sonic, the above relation still holds but A becomes instead a fictitious reference area with no physical meaning other than the area that would be necessary to isentropically accelerate the flow to Mach 1. For isentropic flow, total temperature and density are constant, giving and [ T 1 + γ 1 ] [ Ma 2 = T 1 + γ 1 ] Ma (4.5) 54

70 [ ρ 1 + γ 1 ] 1 Ma 2 γ 1 = ρ [1 + γ 1 ] 1 Ma 2 γ With the assumption again that Ma = 1 these reduce to (4.6) and T T = 2 [ 1 + γ 1 ] Ma 2 γ (4.7) ρ ρ = { 2 [ 1 + γ 1 ] Ma 2 } 1 γ 1 (4.8) γ Substitution of these back into the continuity equation gives the following relation showing that the area ratio can be expressed as a function of Mach number only. A A = 1 Ma { 2 [ 1 + γ 1 ] Ma 2 } γ+1 γ (γ 1) (4.9) Because the total pressure p t is also a function of Mach number only [ p t p = 1 + γ 1 ] γ Ma 2 2 γ 1 (4.10) it can also be shown that the area ratio can also be equivalently expressed as a function of total pressure only A A = ( ) [ 1 ( ) γ 1]1 2 p γ p t 1 p γ p t ( γ 1 ) ( ) γ+1 2 2(γ 1 γ+1 (4.11) Therefore, for a given nozzle geometry and inlet reservoir pressure, 1D isentropic flow through a CDN can be completely characterized by the implicit solution of Equations 4.9 and 4.11 for the Mach number and static pressure. 55

71 4.1.3 Validation case The validation case geometry and analytic solution for the pressure and Mach number distribution were taken from the NPARC Alliance Verification and Validation Archive [1]. The nozzle is 10in long and has an inlet cross sectional area of 2.5in 2, a throat cross-sectional area of 1in 2, and an outlet cross-sectional area of 1.5in 2 The profile of the nozzle is defined via the function if ( x.lt. 5.0 ) then area = * cos( ( 0.2 * x ) * pi ) else area = * cos( ( 0.2 * x ) * pi ) endif where x denotes the stream-wise distance from the nozzle inlet. A 2D-axisymmetric model of the geometry was meshed using 900 cells (Figure 4.1). Figure D axisymmetric mesh used for converging-diverging nozzle validation case A total pressure of kpa and a total temperature of 55.5K were imposed at the inlet. A non-reflective fixed static pressure boundary condition was imposed at the outlet for each of three cases (Table 4.1), each of which are indicative of the possible types of solutions that can arise in compressible flow (hyperbolic, shock discontinuities, and parabolic, respectively). 56

72 Table 4.1. CDN Validation run conditions Case p outlet (kpa) Description A Subsonic B Normal shock forming in diverging part of nozzle C Supersonic Steady state results of static pressure and Mach number are presented in Figure 4.2. The resulting profile is plotted along the central axis and compared to the analytic solution of Equations 4.9 and 4.11 in Figures 4.3 and 4.4. Agreement is excellent for the Cases A and C. The solver has difficulty resolving the shock discontinuity in Case B. Without explicit attention to the numerics in the vicinity of the discontinuity, this error is unavoidable. Figure 4.2. Steady state profiles of Mach number (top) and static pressure (bottom) for each of the three test cases 4.2 Liquid flux closure: Coaxial rocket injector Introduction A coaxial fuel injector operates by injecting high velocity annular oxidizing gas coaxially to a central low velocity liquid fuel. The shear layer that forms between the 57

73 Figure 4.3. Steady state profiles of Mach number (top) and static pressure (bottom) for each of the three test cases Figure 4.4. Steady state profiles of Mach number (top) and static pressure (bottom) for each of the three test cases 58

74 two phases enhances mixing. Oftentimes flow separation can occur at the separation plate between the two injectors, causing a small region of recirculation to form. A diagram of the process is shown in Figure 4.5. The coaxial injector is a valuable validation case because it can be used to: 1. Verify that global mass conservation is satisfied throughout the domain 2. Verify the ability of the solver to accurately predict liquid mass distribution 3. Verify the ability of the solver to resolve flows with recirculating regions. Experimental measurements of liquid volume fraction conducted by Stepowski et al. are used to validate results [81]. This case was used by Demoulin et al. to validate their implementation of the Σ Y model solving a parabolized form of the Navier-Stokes equations, and their results are presented for comparison. Figure 4.5. A schematic denoting the configuration and relevant length scales in coaxial fuel injection [40] Numerical results The case uses a 2D axisymmetric mesh with ten thousand cells (Figure 4.6). The pressure at the outlet was fixed to 1atm. Temperature was set to 300K throughout the domain and assumed to remain constant. Turbulent boundary conditions were 59

75 imposed at the inlets using correlations for fully developed pipe flow to give the conditions in Table 4.2. The incompressible formulation of the code was used, although preliminary calculations using the compressible formulation provided identical results. Figure 4.6. Mesh used for 2D coaxial validation case In Demoulin s original work, simulations were performed using a variety of closures. For this validation case, the liquid flux term is closed through either Fick s law (Equation 2.54) or Demoulin s proposed term for the enhancement due to Rayleigh- Taylor instabilities (Equation 2.58), given again below for convenience. ρũ iy = µ t Ỹ (4.12) Sc x i [ ( ρu i Y νt k 2 1 = ρ + C P P t ǫ ρ 1 ) ] Ỹ (1 ρ g ρ Ỹ ) Ỹ (4.13) l x i Turbulence is modeled using both the standard k ǫ formulation and the high density ratio formulation ρ k t + ρũ i k x i ( ) = µ t k x i Pr k x i ũ j p ρr ij x ū i ρ ǫ (4.14) i x i ρ ǫ t + ρũ i ǫ = ( ) ( ) µt ǫ ǫ ũ j p + C ǫ1 ρr ij x i x i Pr ǫ x i k x ū i C ǫ2 j x i ρ ǫ2 k (4.15) 60

76 Table 4.2. Coaxial injector validation run conditions Boundary Velocity m s k m2 s 2 ǫ m2 s 3 Gas inlet Liquid inlet Results of the validation case are given in Figures 4.7 and 4.8. While the results are in agreement with the experimental data in that the liquid core begins to diminish after 1 diameter downstream, the transition is significantly more abrupt in the experimental case. Demoulin s published work is able to match the experimental profile exactly. This difference is attributed to the fact that Demoulin s implementation uses a parabolized solution technique that is inappropriate for flows with recirculation zones. Private correspondence with the authors suggest that the discrepancy may be caused by artificial numerical diffusion. It was decided that an additional validation case would be necessary to fully develop confidence in the current implementation of the solver. Figure 4.7. Mass fraction profile for coaxial validation case using Rayleigh-Taylor instability production term 61

77 Figure 4.8. Comparison of liquid volume fraction along center axis compared to experimental data [81] and results by Demoulin et al. [16] 4.3 Surface area density closure: Diesel jet injection Introduction Diesel fuel atomization generally functions by injecting liquid diesel through a plain circular orifice under high pressure. A schematic outlining the fuel injection system for a diesel fuel injector is shown in Figure 4.9. Menard et al. performed high fidelity direct numerical simulation of this arrangement by using a level-set volume of fluid method and artificially increasing the surface tension of the liquid to ease the resolution requirement of droplet size. Because the primary atomization is typically visibly inaccessible to experimental measurement, measurements of turbulent correlations and small scale flow features are difficult to obtain. This DNS is unique in that it allows an exact validation of the Σ Y model in the primary atomization zone. Lebas et al. performed fully three dimensional simulations using their implementation of the model and obtained excellent agreement. The diesel injector case is an invaluable validation case because it can be used to: 62

78 1. Verify that global mass conservation is satisfied throughout the domain 2. Verify the ability of the solver to accurately predict liquid mass distribution 3. Verify the ability of the solver to predict Rayleigh-Taylor mixing in the primary atomization zone. 4. Verify the prediction of surface area density 5. Verify the implementation of turbulence model closures Figure 4.9. Schematic demonstrating fuel injection system for a diesel engine (Adapted from [73]) Numerical results Two dimensional simulations of the case were performed using the compressible formulation of the code. A regular hex mesh containing 50k cells was used (Figure 63

79 4.10). The Rayleigh-Taylor enhanced closure for the liquid flux proposed by Demoulin (Equation 2.58) was used along with the variable density k ǫ model with the production term coming from high density ratio effects (Equations 2.66 and 2.67). The transport equation for surface area density proposed by Menard (Equation 2.16) was used. A non-reflective pressure boundary condition was used to impose 1atm at the outlet of the domain. The flow was assumed to be isothermal at 300K. Inlet boundary conditions are defined in Table 4.3. Table 4.3. Diesel injector validation run conditions Boundary Velocity m s k m2 s 2 ǫ m2 s 3 Liquid inlet Results from the simulation are compared to both the Menard s DNS results and Lebas results using their implementation of the model in Figures 4.11, 4.12 and Agreement between the three cases is excellent and suggests that the implementation of the code is accurate. Figure Mesh used for 2D validation case 64

80 Figure Comparison of 2D results for volume fraction profile to published ELSA results by Lebas et al. and to DNS results by Menard et al. (From top to bottom: Simulation results, Lebas et al. s results, Menard et al. s results, Simulation results) 65

81 Figure Comparison of 2D results for surface area density profile to published ELSA results by Lebas et al. and to DNS results by Menard et al. (From top to bottom: Simulation results, Lebas et al. s results, Menard et al. s results, Simulation results) 66

82 Figure Plot of volume fraction profile along central axis compared to published ELSA results by Lebas et al. (top) and to DNS results by Menard et al. (bottom) 67

83 CHAPTER 5 APPLICATIONS - GAS-CENTERED SWIRL-COAXIAL INJECTOR (GCSC) 5.1 Background Fuel injection elements in modern rocket propulsion systems operate at extremely high Reynolds and Weber number, producing a wide range of relevant length scales. Rendering these disparate length scales with direct numerical simulation techniques is extremely expensive, since the mesh resolution would need to be fine enough to resolve the smallest droplets yet have an extent that spans the entire nozzle. Experimental techniques, though able to provide macroscopic insights, have difficulty quantifying the small scale features of the spray formation due to the primary atomization zone s topological complexity and optically dense nature. While traditional coaxial injectors have been prevalent in American rocket injector designs, there has been significant recent interest in evaluating the potential of the gas-centered swirl coaxial injector (GCSC) in rocket combustion applications. Until recently the GCSC had mostly seen application in Soviet flight engines. The injector functions by injecting high velocity propellant axially through the center of the element, while liquid propellant is injected tangentially along the element wall to produce a swirling liquid film. A spray is formed by a combination of instability growth along the surface of the film and shear induced by the high velocity co-flowing gas. Although the injector resembles other injection schemes such as pressure-swirl and coaxial injectors, the GCSC differs in that the majority of atomization occurs within the injector cup before the liquid enters the combustion chamber without forming a 68

84 conical sheet under typical operating conditions. It is essential to effectively predict atomization quality, both to achieve a uniform spray and to accurately predict film length. For designs in which the injector cup is significantly longer than the film length, overheating of the injector faceplate can occur. For cup lengths shorter than the film, incomplete atomization may occur [45]. In an extension of their 2006 paper in which they outline general mechanisms by which atomization takes place, Lightfoot et al. published a report in which they determine the relevant processes at play within the GCSC. They suggest, based on a inviscid linear instability analysis[25], that liquid instability growth cannot be a dominant mechanism based on the prediction that few waves would reach a height capable of causing atomization, and that they would only be capable of atomizing approximately 2.5% of the total film volume. Under similar arguments, they also suggest that liquid-phase turbulence cannot be a dominating factor, and that the dominant mechanism taking place is the impact of gas-phase turbulent structures with the liquid interface. Based on this assumption, they propose a model by which cylindrical gas eddies scoop out a disturbance in the liquid. Although the majority of the assumptions made are only to give a rough characterization of entrainment rate, they propose a useful criterion for gas turbulence driven atomization to occur: the kinetic energy of the turbulent eddy must exceed the surface tension energy of the interface. ρ g v 2 eddy σ d eddy (5.1) They use this equation to develop a criteria to judge what fraction of eddies are large enough to cause stripping, and use turbulence data from fully developed pipe flow to estimate an eddy size distribution. This relation is also useful for verifying the assumption that gas-phase turbulence is dominant. Under the assumption that 69

85 the dominant eddies lie within the range of the larger, energy containing scale, the characteristic length and velocity scales are [65] L = k 3 2 ǫ (5.2) U = k 1 2 (5.3) The assumption that d eddy and v eddy are on the order of L and U respectively, gives the following relation k g 5 2 ǫ g C σ ρ g (5.4) Lightfoot et al. later performed experiments upon a variety of GCSC geometries to obtain measurements of film length as a function of relevant non-dimensional groups, such as momentum flux ratio and mixture ratio. Figure 5.1 demonstrates the range of geometry considered and a sample correlation. Lightfoot s work provides a quantitative characterization of the film within the injector cup. Downstream measurements of atomization quality have been performed over a wide variety of operating conditions over the past decade. For example, Rahman et al. performed hot and cold experiments of a GCSC to obtain visualizations of the spray structure and measurements of the pressure drop across the element [69], obtaining spray angle measurements but no direct calculations of droplet size. In a preliminary assessment of the potential of the GCSC versus traditional coaxial injectors, Cohn et al. used a laser based Doppler interferometer to obtain measurements of droplet size and velocity, obtaining a range of diameters from 3.8 µm to 440 µm during typical operating conditions [13]. Soltani et al. investigated sprays produced by a liquid-liquid GCSC in a non-combusting environment to obtain measurements of average droplet size and velocity, noting that both quantities achieve a self-similar structure [79]. 70

86 Figure 5.1. Top left: GCSC geometric parameters. Top Right: Dimensions of tested inserts. Bottom Left: Film length over initial film height as a function of mixture ratio. [45] The effort described below focuses on predictions of the internal flow of a GCSC injector using the Σ Y model, and attempts to predict the model s ability to perform over a wide range of operating conditions by characterizing the internal flow within the injector cup. The GCSC is a challenging case because it has a number of features that differ from previous applications of the model. Atomization within the cup is effected by swirl, by the strength of the recirculation zone formed in the wake of the injector lip, and potentially by compressibility effects for cases with high gas mass flow-rates. A summary of the experimental work carried out at the Air Force Research Laboratory (AFRL) at Edwards Air Force Base is presented, followed by a summary of the numerical simulations performed. Results of an analysis of both a high and low momentum ratio case carried out using an incompressible formulation are shown, 71

87 along with presentation of progress that was made toward achieving a compressible analysis. 5.2 Experimental Study Experimental data was collected using an injector body composed of acrylic. This acrylic body is modular, allowing differing injector geometries to be tested. Nominally, the outlet diameter of the injector is mm (0.75 inches). The initial film thickness is nominally 1.65 mm (0.065 inches) with the step separating the gas and liquid having a height of 1.52 mm (0.06 inches). Upstream of the modular acrylic section is a stainless steel section which consists of 180 mm of gas inlet with a fixed radius of 6.35 mm (0.25 inches). All tests were performed with atmospheric back pressure using working fluids (simulants) of water and nitrogen at room temperature. The gas flow rates were varied from kg/s. The liquid flow rates were varied from kg/s. The momentum flux ratio, defined using the mass flow rates along with flow areas based on the initial film thickness for the liquid and the average gas post height (r p + r o )/2 = r p + (s + τ)/2 varied from around 10 to around Film length was determined from video images. For lighting, a laser beam was split and expanded into two sheets which were oriented 180 from one another along the centerline of the injector or spray. A Vision Research Phantom v7.3 camera positioned 90 from the laser sheets captured the images at 6006 fps with an exposure time of 150 microseconds. A typical image, with flow from left to right, is shown in Fig An in-house Matlab code was used to determine the film profile in each image using the change in intensity to determine the film boundary. Once the film profile has been determined, the code uses input information about the wall location to determine film length. Because the film length is not steady, the profiles extracted from 5000 images are averaged together to determine an average film length (and 72

88 Figure 5.2. A typical image from the in-cup video is shown here. The edges of the injector body are highlighted including the sheltering lip. Figure 5.3. Representative profiles of a long, medium and short film. These profiles are from the geometry shown in Fig. 5.2 at momentum flux ratios of 110, 484 and 823 respectively standard deviation). Analysis shows that 5000 images are sufficient to give good statistical results. Sample profiles are shown in Fig At sufficiently high momentum ratios, an asymmetry appears between the top and bottom profiles. A number of hypothesizes have been proposed to explain this phenomenon: either it is caused by experimental error or a physical mechanism. Assuming that the experimental setup is perfect (which is reasonable due to the extensive literature and work that has been done with the setup over the past few years), two main physical mechanisms are assumed here. The first is that as the sheet of light hits the first film surface, the irregular topology of the interface caused the photons to stochastically diffuse outward. When these diffuse photons impact the film on the opposite side of the injector, they impact a broader region of interface, therefore causing the intensity of the light to decrease. At higher momentum ratios, the interface topology would become more irregular and increase the intensity of this diffusion process. In the image processing software, the mean interface location is determined by taking iso-contours of the light intensity. This could bias one side of the injector over the other. The second mechanism, which seems much more likely, is that at a critical momentum ratio, the pressure drop caused by the flow separation is sufficiently low to trigger an instability mechanism analogous to diffuser stall. In single-phase diffuser stall, the adverse pressure gradient facing the flow causes the 73

89 boundary layers to separate from the wall. For sufficiently large pressure gradients, the flow will lean to either one side or the other (Figure 5.4). It is proposed that a similar instability is occurring within the GCSC and that, while the liquid film does geometrically resemble a single phase diffuser, the film profile is tightly coupled with the instability in the gas flow. When separation occurs, a significantly strong pressure drop will cause the film to lift up off the cup, increasing the intensity of the recirculation zone on one side of the cup and increasing the mass entrainment rate on the side that is no longer sheltered by the separation zone. This hypothesis is supported by measurements of the resulting spray formed by the GCSC [46]. High momentum ratio experiments demonstrate an asymmetry in the spray with the large droplets on one side or the other. Figure 5.5 demonstrates this effect. Note the presence of darker spray on either side of the cup exit, suggesting the presence of a larger, more dense spray. With increasing swirl number this feature is diminished. Swirl number has a well documented stabilizing effect on diffuser stall. To definitively define the cause of this phenomenon, the actual gas-phase flow path must be defined to identify separation regions within the flow. With the current experimental setup this is not possible: the remainder of this work will therefore focus primarily on low momentum-ratio conditions and any flow asymmetries will be presented for reference but not discussed in detail. 5.3 Incompressible results The conditions and geometry used for the incompressible study are discussed in Lightfoot et al. [45]. Specifically, the ONPNTN geometry (Figure 5.1) was considered under the operating conditions listed in Table 5.1. The computational boundary conditions were set by matching liquid flow rate and gas flow rate, assuming a uniform incoming velocity profile, and a fixed total pressure condition at the injector outlet. It is important to note that for the higher momentum ratio case, the gas flow-rate yields 74

90 Figure 5.4. Transition from optimal diffuser performance (top) to stall conditions (bottom) (Figure adapted from [90]) 75

91 Figure 5.5. Asymmetry in resulting spray at high momentum ratio. (Image courtesy of AFRL) 76

92 flow velocities in the transonic regime, rendering the assumption of incompressibility inaccurate. Experimental measurements of the total pressure at the inlet suggest that choking is not occurring, but the difference in gas density could substantially alter the momentum ratio at the interface. Table 5.1. Simulation conditions Case name ρ gas ρ liq U gas,in U liq,in,z U liq,in,θ p out kg kg m m m m 3 m 3 s s s kpa ONPNTN-a ONPNTN-b In the computation, the domain is in two-dimensional polar coordinates (r, z), with r > 0, with a two-dimensional mesh containing eleven thousand cells. The assumption of axisymmetric flow was validated by performing a 3D simulation of a 90 sector of the full geometry. While the experimental photographs did exhibit asymmetry between the top and bottom film profiles, this has been attributed to experimental conditions and is discussed by Schumaker et al. [48] and will be addressed at length later. Turbulence was modeled using both a classical two equation k ǫ model and the realizable k ǫ model developed by Shih [77]. All previous implementations [88, 33, 4, 42] of the Σ Y model have used the k ǫ model because of its stability and ability to resolve mixing layers, which was well suited for the plain orifice and coaxial arrangements previously considered. The k ǫ model is well known to be unable to accurately predict the recirculation zone following a backward facing step. While any turbulence model based on the eddy viscosity assumption will have difficulty capturing the effect of swirl, the realizable k ǫ model should be capable of more accurately resolving the strength of the recirculation zone, which is assumed to be the dominant turbulent mechanism affecting the film profile. Simulations were run using the standard k ǫ model to predict the Reynolds stresses and Fick s law of diffusion to model the turbulent liquid flux for the ONPNTN- 77

93 b1 case using the default value for the Schmidt number of 0.9. Figure 5.6 shows the resulting pressure and volume fraction profiles. Figure 5.7 shows an instantaneous photograph of the injector under the same operating conditions. The experimental photograph suggests that the liquid should be pushed up against the backward facing step by the recirculation zone, and the lack of this feature suggests that the model is over-predicting the turbulent dispersion. Also visible in Figure 5.6 is a static pressure gradient caused by the centripetal acceleration present in the swirling liquid film. Figure 5.6. Static pressure and volume fraction profile for ONPNTN-b1 case Preliminary results using Demoulin et al s modified closure for the enhancement of mixing via Rayleigh-Taylor instabilities (Equation 2.58) further underestimated the film length. This is attributed to the fact that, for the swirling liquid film under consideration, the centripetal acceleration acting upon the interface has a stabilizing effect that suppresses the growth of the Rayleigh-Taylor instability. As a first-order approximation of the reduced turbulent diffusion caused by the swirl, the Schmidt number was increased to determine its effect upon the liquid film profile. Figure 5.8 demonstrates that increasing the Schmidt number from the stan- 78

94 Figure 5.7. Experimental photograph of liquid profile for ONPNTN-b1 case 2.5 x Experiment Top Experiment Bottom Sc=0.9 Sc=5 Sc=10 Sc=15 Sc=20 h (m) z (m) Figure 5.8. Iso-contours of Ȳ = 0.5 compared to experimental profiles for standard k ǫ closure 79

95 dard value of 0.9 up to a value of 20 allows the model to accurately reproduce the experimental data. 2.5 x Experiment Top Experiment Bottom Sc=0.9 Sc=5 Sc=10 Sc=15 h (m) z (m) Figure 5.9. Iso-contours of Ȳ = 0.5 compared to experimental profiles for realizable k ǫ closure Figure 5.9 gives a similar parametric study of the Schmidt number when the realizable k ǫ turbulence closure is used instead. For this case, a Schmidt number of only 15 provides a more accurate representation of the film profile. To characterize the generality of these results, a case with a significantly shorter film length was investigated. Figure 5.10 shows an experimental photograph of the film profile. For this case, the light scattering at the interface is substantially more diffuse, suggesting the presence of more violent instability growth at the liquid interface. Figures 5.11 and 5.12 compare the predicted liquid profiles to the time-averaged experimental data. Both simulations show good initial agreement, but are abruptly cut short as a recirculation zone forms at the end of the liquid film (Figure 5.13). As expected, the realizable k ǫ model comes closer to predicting the experimental 80

96 Figure Experimental photograph of liquid profile for ONPNTN-a1 case result but also fails to accurately resolve the correct behavior near the recirculation zone. In Beheshti et al. s assessment of the Σ Y model [4], they noted that in its current form the model is incapable of accurately resolving flows with areas of large recirculation. In this scenario, the fundamental assumption that the flow is acting at large Reynold s and Weber number is violated, as the velocity within the recirculation zone is substantially less than the free-stream velocity. To properly predict the flow behavior at high momentum ratios, the compressible formulation of the code was used. Additional measurements has to be made to fully define the boundary conditions, and are given in Table 5.2. The experimental rig consists of a pressurized tank followed by a sonic nozzle and many meters of both hard and soft line leading to the injector. By assuming adiabatic flow within the line, the total temperature is used to set the boundary condition for temperature at the inlet through the following relation. T = T tot u2 2c p (5.5) A non-reflecting total pressure condition was used to set the outlet pressure, and the mass flow-rate was fixed to match experimental measurement. Static pressure 81

97 2.5 x Experiment a1 top Experiment a1 bottom Experiment b1 top Experiment b1 bottom Simulation a1 Simulation b1 h (m) z (m) Figure Iso-contours of Ȳ = 0.5 compared to experimental profiles for standard k ǫ closure 2.5 x Experiment a1 top Experiment a1 bottom Experiment b1 top Experiment b1 bottom Simulation a1 Simulation b1 h (m) z (m) Figure Iso-contours of Ȳ = 0.5 compared to experimental profiles for realizable k ǫ closure 82

98 Figure Instantaneous film profile and streamlines show the presence of a recirculation zone for the ONPNTN-a1 case 1.63in upstream of the injector was measured and used to validate the predicted pressure drop across the injector for a single phase simulation. Thermocouples were used to measure the temperature within the cup (T ), but these measurements were deemed suspect due to heat transfer occurring along the thermocouple. The case setup and results for the ONPNTN-a1 case are provided in Figures 5.14 and 5.15, respectively. Table 5.2. Compressible conditions Case name p atm p T tot T ṁ kg kpa kpa K K s ONPNTN-a ONPNTN-b In order to match the predicted pressure drop from experiment, the total temperature at the gas inlet had to be raised to 310 K. It is expected that this temperature rise is due to some combination of viscous heating and heat transfer to the gas from the laboratory. Assuming 1D isenthalpic flow between the pressurized tank and the injector and uniform heat transfer, the following equation relates the discrepancy in total temperature between the experiment and simulation [2]. 83

99 Figure Boundary conditions and mesh setup for compressible ONPNTN-a1 case q = c p (T tot,out T tot,in ) (5.6) This gives a rough estimate of 1.2 kw of heat transfer occurring between the laboratory and the injector line. Unfortunately, shortly after taking these measurements, the insert used in the experiments was damaged, preventing additional validation. To verify the accuracy of the compressible results, a 1D analytic model of the setup was developed (Figure 5.16). The injector setup was modeled by considering 1D isentropic acceleration of a gas from a specified total temperature and pressure (A). The injector geometry was modeled as a sudden expansion (B), followed by discharge to ambient conditions (C). Flow between A and B is given by [2] T b = T a u b 2 2c p (5.7) ( p a = p b 1 + γ 1 2 Ma b 2 ) γ γ 1 (5.8) 84

100 Figure Resulting profiles of pressure, density, velocity and temperature for ONPNTN-a1 case Figure Isentropic compression followed by sudden expansion 85

101 where Ma denotes the Mach number of the gas and γ is the ratio of specific heats. The pressure drop across the expansion is defined by considering conservation of mass, momentum, and assuming a uniform pressure distribution directly following the expansion: [60] A b V b = A c V c (5.9) A c (p b p c ) = ρa c V c (V c V b ) (5.10) where A is the cross sectional area following the expansion. Solution of these equations allows an analytic relation between pressure drop and mass flow-rate. Figure 5.17 gives a comparison of single phase, axisymmetric simulation to predictions from the analytic model. Good agreement suggests that the simulations are in fact accurate. The presence of expansion waves and transonic velocities within the cup suggest that sonic effects are present in high momentum ratio cases. Future work will focus on developing experimental comparisons of film length under these transonic conditions, and conducting fully three-dimensional simulations of the injector geometry to investigate the cause of the asymmetries in the liquid profile. The results obtained with the model are encouraging: it has been shown that the Σ Y formalism can in fact be adapted to flows with complex geometries and flow features. A first order approach for accounting for swirl has been proposed by reducing the effective Schmidt number in the closure for the turbulent liquid flux. Additional research is currently underway to determine higher-order models to compensate for swirl effects. 86

102 Figure Comparison of CFD simulation to analytic results 87

103 CHAPTER 6 APPLICATIONS - JET IN CROSS FLOW 6.1 Background The quality of atomization and its effect upon the subsequent evaporation and mixing of fuel occurring within combustion systems is fundamental in determining both the combustion efficiency and the production of emissions for a given system. In particular, the transverse injection of a liquid jet into a high velocity cross-flowing gas is common in a variety of fuel injection schemes, such as the fuel injection process for lean premixed prevaporized (LPP) gas turbine engines, augmenters, and for ramjet and scram jet applications [92, 83, 67, 28, 27, 50, 68]. For these applications, cross flow injection has the desirable benefit of achieving rapid fuel atomization with control over the location of fuel placement. Of particular interest to design engineers is a predictive tool for determining the penetration of the resulting spray to prevent impingement of liquid fuel upon walls, and for determining the resulting spray atomization quality, both of which are prerequisite for modeling fuel vaporization and subsequent combustion processes. The process by which liquid disintegrates into droplets upon injection from an orifice under conditions typical of gas turbine injection is as follows [67]: directly after injection, the liquid jet forms a column subjected to a drag force which bends to align with the co-flowing gas. At the same time, the pressure distribution across the liquid interface causes the jet to adopt a flattened profile. Progressing down the length of the jet, the profile continually flattens until it becomes subject to wave-like instabilities and eventually pinches off to form ligaments oriented transversely to the 88

104 flow. This breakup mechanism is referred to as column breakup and qualitatively resembles the first wind-induced regime of jet atomization. While this mechanism is predominant at lower momentum ratios, for high pressure flows typical of those found in combustors a second mechanism referred to as surface breakup occurs. Surface breakup is characterized by the high Weber number stripping of ligaments by shear forces acting at the liquid surface and tends to form smaller droplets than the column breakup mechanism. The ligaments formed by both mechanisms then undergo secondary breakup typically driven by shear breakup mechanisms [70]. Figure 6.1. Schematic representing the two mechanisms of primary atomization and subsequent transition to secondary breakup. [92] Experimental work has been conducted to characterize maps of the two breakup regimes, and correlations for jet trajectory, penetration, and jet width. For much of the experimental work at high Weber number and momentum ratio, the cloud of small droplets form by the surface breakup process (Figure 6.2) rendering the primary atomization zone and point of jet breakup optically inaccessible. The majority of published experimental work utilizes either a shadowgraphy or Phase Doppler and Particle Analyzer (PDPA) technique to characterize the breakup zone. Shadowgraphy fails to resolve spray regions of small droplet density, and as a result tend to yield shorter jet penetrations than PDPA experiments [83] [49]. Promising work has 89

105 recently been conducted by Linne et al. [50] using ballistic imaging techniques to resolve both the small scale features occurring due to surface breakup and large scale, periodic features of the primary liquid core. Figure 6.2. Droplets from surface breakup mechanism visually obscure the primary liquid core. [83] Of all the numerical approaches typically used to model atomization processes, the Lagrangian blob formulation employed by Reitz [72] is probably the most ubiquitous and is readily implemented in a number of commercial CFD packages. While this framework has given excellent agreement in cases where a liquid jet is injected into either quiescent or co-flowing gas, the assumption that the gas and liquid phases are only coupled through momentum transfer is inadequate to resolve the jet in cross flow behavior. The wake formed by the presence of the liquid column alters the turbulent structures (Figure 6.3). Also to the leeward side of the jet, there is a low pressure zone similar to that found in the wake of a solid cylinder, and that low pressure contributes to the bending of the liquid column. A number of other modeling approaches have been carried out in the literature. Bellofiore et al. coupled an Eulerian analysis of the jet to a Boundary Layer Stripping model to predict the depletion of mass from the jet due to stripping, obtaining good predictions of penetration and trajectory at high jet velocities [6]. Mashayek and 90

106 Ashgritz developed an analytic-numerical model of the spreading of the jet via an analogy to the deformation of a liquid droplet in quasi-static liquid flow [55] and were able to reproduce photographs of the jet deformation at low momentum ratios. Figure 6.3. Effect of liquid column on surrounding turbulent structures. [44] The Σ Y model, originally proposed by Borghi and Vallet [10], attempts to capture the features of the flow of practical interest to the design engineer while simplifying the complexities of the flow via simple closures within an Eulerian framework. By recasting the governing equations of motion in a Reynolds averaged paradigm, the bulk liquid movement is predicted. Rather than attempting to resolve individual atomization events, a transport equation for the development of surface tension energy is used to characterize the rate at which surface area is created. With knowledge of a local interfacial surface area and liquid volume, the Sauter mean diameter can then be characterized. Predictions for the mean droplet size and spatial distribution can then be used as an input for Lagrangian secondary breakup models which can be used to predict fuel vaporization, which in turn provide the necessary inputs for combustion calculations. The Σ Y model has been successfully applied and validated for a variety of coaxial and diesel jet type injections [58, 32, 5, 62, 56] and recently used for injection schemes with more complex geometries [84]. 91

107 For this work, the accuracy of the Σ Y model in reproducing the liquid trajectory and liquid volume distribution given by the work of Tambe et al. [83] will be evaluated. Tambe s experimental work investigated the effects of altering momentum ratio, free stream velocity, nozzle diameter, and surface tension coefficient upon the structure of the resulting atomized spray. 6.2 Case Setup The experimental conditions investigated by Tambe et al. [83] were matched to determine the Σ Y model s ability to predict mass distribution for sprays in cross flow. Tambe investigated injections of water, Jet-A, and N-Heptane and the effect of varying injection diameter, momentum ratio, cross flow velocity, and surface tension coefficient in the downstream liquid mass flux and Sauter mean diameter profiles. In their work they noted that increasing the surface tension coefficient had little effect upon the penetration of the liquid jet, and only served to increase the resulting droplet size resulting from surface breakup. This suggests that the hypothesis of the Σ Y model that the bulk fluid motion remains separate from the breakup behavior should be satisfied for this case. Figure 6.4. a) Experimental setup used by Tambe et al. b) Typical nozzle geometry [83] 92

108 Figure 6.4 shows the experimental setup used by Tambe et al. [83]. For this numerical investigation, the test chamber following the reducer throat was simulated using a hexahedral mesh of 500k cells. The mesh was refined in the vicinity of the liquid jet and near the boundary layer at the lower wall such that the characteristic cell size in the vicinity of the injector and within the boundary layer was 70 microns. The mesh far from the domain occupied by the liquid jet was coarsened to increase computational efficiency. The operating conditions modeled are given in Table 6.1. Uniform turbulent boundary conditions were imposed to match fully developed turbulent pipe flow conditions of an equivalent hydraulic diameter. Experimental evidence suggests that it is necessary to resolve momentum boundary layer effects [12, 83]. Wall functions were applied to obtain turbulent boundary layer characteristics. The momentum boundary layer was modeled at the inlet using the following relations for a turbulent boundary layer over a flat plate. δ L = 0.37 Re 1 5 u ( y )1 7 = U inf δ (6.1) (6.2) where δ denotes the boundary layer thickness and Re is the Reynolds number based on the stream wise plate length. This length was taken to be the distance from the end of the reducer to the beginning of the simulated domain ( 10 cm), giving a boundary layer thickness at the beginning of the computational domain of 2.5 mm. As seen in Figure 6.6, the simulation captures a number of the relevant qualitative flow features. A relatively solid primary liquid core forms with a lesser amount of mass in its wake, corresponding to the expected high Weber number mixing caused by the shear-driven surface breakup. A comparison of the results to measurements of normalized liquid volume flux and Sauter mean diameter 30 inlet diameters downstream of the injection point (Figures ) show reasonable agreement in mass 93

109 Figure 6.5. Shadowgraph of spray structure for case 5 by Tambe et al. [83] Table 6.1. Simulation conditions Case number 5 Liquid Water D (mm) We 96.2 Momentum ratio 5.2 U ( m ) s U liq ( m) s 9.7 p kpa placement and location of interfacial surface area. While it is not possible to directly compare values of volume fraction due to the fact that Tambe et al. only published volume fluxes normalized by their maximum value, the results do indicate that the majority of mass is within 6 mm of the wall. Of particular relevance is the fact that this level of agreement has been achieved using the exact same values as those used to predict coaxial and diesel jet injections. Previous work [84] has shown that altering either the Schmidt number or the constant C p found in Equation 2.58 can be used to better fit experimental values for the position of the primary liquid core. 94

110 Figure 6.6. Experimental measurements of SMD (left) and liquid volume flux (right) [83] Figure 6.7. Liquid volume fraction measured 30 diameters downstream of liquid inlet. Figures 6.9 and 6.10 compare the results of the simulation to empirical correlations for spray penetration cited in [83]. The differences between the correlations have been attributed by Tambe et al. to both differences in experimental techniques between studies and differences in ambient operating conditions. While the penetration of the primary liquid core is significantly less than that predicted by the correlations, the outer sparse region of the spray containing the majority of the interfacial surface area shows excellent agreement. 95

111 Figure 6.8. Interfacial surface area density measured 30 diameters downstream of liquid inlet. Figure 6.9. Predicted liquid mass fraction with superimposed correlations of spray penetration. 6.3 Conclusions The Σ Y model for predicting primary atomization has shown to give promising preliminary results in predicting atomization in spray in cross flow injection schemes with no alteration of the model. This is particularly encouraging, as previous work with the model has focused primarily upon simple shear driven atomization arrangements. Additional work is required to determine the generality of the model and whether better accuracy can be obtained through the use of more suitable model- 96

112 Figure Predicted interfacial surface area density with superimposed correlations of spray penetration. ing parameters and by incorporating the high density ratio modification of the k ǫ turbulence model. The results are additionally encouraging for design engineers looking for practical tools for predicting atomization. Unlike most atomization models in use today, the Σ Y model has been shown here to provide reasonably accurate predictions with no empiricism or a priori knowledge of the flow. 97