Efficient Numerical Calculation of Evaporating Sprays in Combustion Chamber Flows

Transcription

1 51-1 Efficient Numerical Calculation of Evaporating Sprays in Combustion Chamber Flows R. Schmehl, G. Klose, G. Maier and S. Wittig Lehrstuhl und Institut für Thermische Strömungsmaschinen Universität Karlsruhe (T.H.) Karlsruhe, Germany Summary Representing two different conceptual approaches, either Eulerian continuum models or Lagrangian particle models are commonly applied for the numerical description of dispersed two phase flows. Taking advantage of the positive features inherent to each model, a combination approach is presented in this study for the efficient computation of liquid fuel sprays in combustor flows. In the preconditioning stage, Eulerian transport equations for gas phase and droplet phase are solved simultaneously in a block-iterative scheme based on a coarse discretization of spray boundary conditions at the nozzle. Due to the close coupling of both phases, the time expense of this approximate flow field computation is not much higher as for single phase flows. In the refinement stage, Lagrangian droplet tracking is applied with a detailed discretization of initial conditions. To account for complete interaction between gas phase and droplets, gas flow solution and droplet tracking are concatenated by an iterative procedure. In this stage, the numerical description of the spray is enhanced by additional modeling of droplet breakup. Results of numerical simulations are compared with measurements of the two phase flow in a premix duct of a LPP research combustor. 1 Notation c p specific heat capacity U velocity component D droplet diameter We Weber number D.5 mass median diameter Y mass fraction Sauter mean diameter D.632 characteristic diameter Greek Symbols f body force α heat transfer coefficient h enthalpy α k liquid volume fraction Ḣ enthalpy flux β off axis angle H energy transfer rate ε dissipation rate of k I momentum transfer rate Γ diffusion coefficient k turbulent kinetic energy µ dynamic viscosity ṁ mass flux ν kinematic viscosity M mass transfer rate ρ density On Ohnesorge number τ shear stress P pressure Pr Prandtl number Subscripts Q conductive heat flux initial state Re Reynolds number g, d gas, droplet S source term int interface Sc Schmidt number t turbulent T temperature vap vapor Tu degree of turbulence 2 Introduction Improving modern gas turbine efficiencies by increasing pressure and temperature levels of the combustion process, essentially requires sophisticated combustion concepts in order to meet todays strict limitations on pollutant emissions. Fundamental to these low emissions concepts is a characteristic strategy to inject and mix the liquid fuel with the compressed air flow, avoiding local stochiometric combustion conditions as far as possible. Two promising approaches in this context are the concepts of Lean-Premix-Prevaporize (LPP) and Rich-Quench- Lean (RQL) combustion. In order to develop advanced combustor designs with the required flow characteristics, a better understanding of the two phase flow physics is necessary. Two phase flow effects typical for premix ducts of LPP combustors or prefilming air blast atomizers are summarized in Fig. 1. Atomization Droplet Dispersion + Evaporation Spray-wall Interaction Wall Film Flow Figure 1: Two phase flow effects in a LPP premix duct Due to the enormous increase in computing performance, Computational Fluid Dynamics (CFD) offers a promising potential for efficient combustor design and optimization. In particular when compared to experimental studies at elevated pressures, CFD analysis may be employed to reduce turn-around times and costs of combustor design significantly. On the other hand, complex flow phenomena such as turbulence, atomization or chemical reaction still represent some of the most challenging topics for CFD tools. Basically, two different conceptual approaches may be employed for the numerical description of dispersed two phase flows [3]. In analogy to single phase gas flow, the Eulerian approach is based on a continuum model of the spray, resulting in transport equations describing the propagation and evaporation of this droplet phase [28], [6]. In the Lagrangian approach, the spray is modeled by superposition of trajectories calculated for large numbers of representative droplets. Each of the two basic approaches is characterized by specific advantages and restrictions. In the Eulerian method, the transport equations of the droplet phase are appended to the gas phase transport equations, resulting in a compact description of the interacting two phase flow system. The essential advantage is a simultaneous solution of the interacting flow fields of gas phase and spray by a single numerical method. Applying a standard block-iterative solver for systems of linearized equations, the information exchange Paper presented at the RTO AVT Symposium on Gas Turbine Engine Combustion, Emissions and Alternative Fuels, held in Lisbon, Portugal, October 1998, and published in RTO MP-14.

2 51-2 between phases is realized on the level of the non-linear iterations. As a consequence, computation times are generally small compared to the Lagrangian approach. However, each droplet initial condition to be simulated requires the solution of an individual set of 5 transport equations. This conceptual feature is a severe limitation for the discretization of complex sprays with wide ranges of initial droplet size, injection angles and velocities. Furthermore, Eulerian methods are generally not suited for the modeling of complex two phase phenomena like secondary atomization or spray-wall interaction. In the Lagrangian method, a large number of droplet trajectories has to be tracked to achieve a continuous distribution of the liquid phase. Due to the stochastic simulation of turbulent spray dispersion by random sampling of gas velocity fluctuations, identical droplet initial conditions lead to different trajectories. As a consequence, the liquid phase flow field is a statistical quantity. To limit the maximum field deviations, the number of simulated droplet trajectories has to be increased to values up to for typical combustor flows. Effects of the spray back on the gas flow such as aerodynamic dragging or evaporation cooling are recorded in droplet source terms, describing the local interfacial transfer of mass, momentum and energy. Including these source terms in the gas flow transport equations, complete phase interaction is taken into account by an iterative concatenation of gas flow computation and Lagrangian droplet tracking. Besides increased computation times, this iterative procedure entails an artificial decoupling of gas flow and spray. In particular for flows with intense phase interaction, this may cause severe convergence problems, requiring strong relaxation of source term fields. Nevertheless, Lagrangian methods are commonly preferred for practical CFD analysis due to significant advantages regarding complex spray discretization and modeling of flow phenomena such as secondary droplet breakup or droplet-wall interaction. In this study, a new hybrid approach is presented combining Eulerian and Lagrangian methods to take advantage of the capabilities inherent to both methods. In the first stage of this procedure, the Eulerian method is used for an efficient computation of an approximate two phase flow field. A coarse discretization of spray boundary conditions at the nozzle limits the size of the system of transport equations to a practical dimension. Good convergence rates are achieved due to the close coupling between gas flow and spray. In the refinement stage, iterative cycles of single phase gas flow computation and subsequent droplet tracking are employed to improve the quality of the preconditioned two phase flow field. Taking advantage of the stochastical nature of the tracking approach, a fine discretization of polydisperse sprays is achieved by random sampling of droplet initial conditions at the nozzle. The numerical description of the spray is enhanced by optional modeling of secondary droplet breakup and spray-wall interaction. Since modeling of spray-wall interaction has been described in detail in Ref. [24], only secondary atomization of droplets is considered in this study. To demonstrate the performance and accuracy of the numerical methods discussed in this paper, the evaporating spray in the premix duct of a LPP research combustor is simulated and assessed by measured droplet data. The experimental investigation of this premix section has been a focus of various studies [15], [16], [13] and represents a valuable source of experimental data. 3 Eulerian approach The Eulerian approach for the numerical description of dispersed two phase flows is based on the assumption that the liquid phase represents an additional continuum penetrating the gas phase. In analogy to the continuum approach of single phase flows, each phase is described by a set of transport equations for mass, momentum and energy extended by interfacial exchange terms. This set of transport equations can be recasted into a universial formulation which is discretized by a conservative Finite Volume method and solved by a block-iterative scheme fore systems of linearized equations. 3.1 Transport equations of the two phase flow Basic equations Except for the near region of the atomizer, the volume fraction of fuel in the flow field is low. In this dilute two phase flow regime, interactions between fuel fragments can be neglected. Starting from the basic Navier-Stokes equations, the instantaneous transport equations for gas and droplet phase are derived either by spatial, temporal [9] or ensemble phase averaging Gas phase: t αgρg + α gρ gu g,j x j = M int,g (1) t αgρgug,i + α gρ gu g,ju g,i x j = eu α g P g + τi,j +α gρ gf i x i x j + I int,g,i (2) t αgρghg + α gρ gu g,jh g x j = Droplet phase: qj x j +S h,g + H int,g (3) t α dρ d + α d ρ d U d,j x j = M int,d (4) t α dρ d U d,i + α d ρ d U d,j U d,i x j = α d P g +α d ρ d f i x i + I int,d,i (5) t α dρ d h d + α d ρ d U d,j h d x j = H int,d (6) The weighting factors α g and α d are a result of the averaging process and represent the local volume fractions of gas and liquid phases related by the following equation α g +α d = 1 (7) Dilute two phase flows are characterized by the conditions α d 1 ; α g 1 (8) In this flow regime, the transport equations of the gas phase approach the standard single phase transport equations extended by additional interfacial exchange terms M int,g,i int,g and H int,g Interfacial exchange terms The interfacial exchange terms describe the local rates of mass, momentum and energy transfer across the liquid-gas interface. Assuming a spherical shape of the droplets and a uniform internal temperature distribution, the transfer rates may be estimated from Lagrangian single droplet physics. The following model expressions are derived from Eq. (23) and the heat and mass

3 51-3 fluxes (28), (29) and (3) of an isolated droplet M int,d = M int,g = 6α d πd 3ṁ vap (9) I int,d,i = I int,g,i = 6α d πd 3 ( π 8 D2 ρ gc D U g U d (U g,i U d,i )+ṁ vapu d,i ) H int,d = H int,g = 6α d πd 3 (Ḣ vap,s + Q cond,s ) (1) (11) Transport equation of the droplet diameter According to Ref. [6], a transport equation of the mean droplet diameter can be established by combining transport equations of droplet number and droplet mass (4), giving t α dρ d D + α d ρ d U d,j D = 8α d x j πd 2 ṁvap (12) For non evaporating sprays this equation is identical to the continuity equation Turbulence modeling The transport equations derived so far are suited for the numerical description of sprays in laminar gas flows. Since combustors generally operate in the turbulent flow regime, the system of transport equations (1) - (6), (12) is extended by introducing turbulent fluctuations of the transport quantities followed by Reynolds averaging of the equations. With respect to the gas phase, the standard k-ε model is employed to model the transport terms resulting from correlations of fluctuating quantities. This procedure has been described in detail by several authors [12], [21]. The turbulence terms in the droplet phase transport equations are approximated by an algebraic model which is based on a Bousinesq approach. t α dρ d + α d ρ d U d,j = x j ( ) µt,d α d + M int,d (13) x j Sc t,d x j t α dρ d U d,i + α d ρ d U d,j U d,i = x j [ ( Ud,i α d µ t,d + U )] d,j + x j x j x i α d ρ d f i α d P g + I int,d,i (14) x i t α dρ d h d + α d ρ d U d,j h d = x j ( ) µ t,d h d α d + H int,d (15) x j Pr t,d x j t α dρ d D + x j α d ρ d U d,j D = 8α d x j πd 2 ṁvap + ( ) µ t,d D α d (16) Pr t,d x j Double and triple correlations involving fluctuations of α d are neglected on the right hand side of Eq. (14). A value of.9 for the turbulent Schmidt and Prandtl numbers, Sc t,d, Pr t,d is chosen for the present calculations. Using this value, Eq. (13) effectively is a transport equation of a scalar in the asymptotical case of a vanishing droplet diameter. A fundamental assumption of this approach is the dependence of the turbulent viscosity of the droplet phase µ t,d on local mean flow properties [17], [1] ν t,d ν t,g = µ t,d µ t,g ρ g ρ d = ( τd τ g ) 2n (17) Thus, the ratio of the kinematic viscosities of droplet and gas phase is postulated to be a function of the characteristic time scales of both phases. In Ref. [1], a value of.25 for the empirical parametern is suggested. The time scalet d which is denoted as droplet relaxation time characterizes the ability of a droplet to follow turbulent gas velocity fluctuations: t d = 4 3 ρ d D 2. (18) ρ g C D Re d ν g Originally, the characteristic time of the gas flow turbulence t g is taken to be the dissipation time scale given by Eq. 25. In this formulation, the droplet phase turbulence model fails to describe the crossing trajectory effect which has a significant influence on turbulent droplet dispersion [27]. According to the turbulence modeling of the Lagrangian approach, the extended version of the model considers a second characteristic time scale. This crossing time t c is the time required by a droplet to cross the current coherent turbulence structure which is estimated from Eq. 26. Combining both time scales, the gas phase time scale is now defined as t g = min[t e,t c] (19) The validation of this enhanced turbulence model is based on the basic experiment described in Ref. [27]. 3.2 Discretization of polydisperse sprays To complete the numerical description of the spray, boundary conditions of the droplet phase have to be specified at the atomizer nozzle. However, most sprays of technical importance are characterized by a broad variety of initial droplet diameters and velocities. Since each individual droplet phase boundary condition theoretically requires the numerical solution of a separate set of transport equations (13) - (16), a polydisperse spray has to be discretized by a limited number of representative droplet classes. In practice, the computational effort increases at least linearly with the number of droplet classes employed. As a consequence, the maximum number of classes is restricted by the CPU time and memory capacity available. For dilute sprays, direct interaction between droplet classes can be neglected although each class is coupled to the gas phase by the closure equation M int,g I int,g H int,g nc = M int,d,k (2) k=1 nc = I int,d,k (21) k=1 nc = H int,d,k (22) k=1 4 Lagrangian approach 4.1 Spray dispersion The Lagrangian simulation of dispersed two phase flow is based on the tracking of statistically significant droplet parcels in the gas flow. Each parcel is represented by one droplet and is determined by discretization of the continuous spectra of droplet initial conditions in the near field of the atomizer. The tracking is based on the integration of the droplets equation of motion combined with an empirical correlation for the aerodynamic drag coefficient C D, d u d dt = 3 4 ρ g ρ d C D D u d u g ( u d u g) (23) C D = Re.573 d + 24 Re d (24)

4 51-4 In order to simulate the effect of turbulent spray dispersion, the turbulence structure of the gas flow field is modeled by a random process along the droplet trajectories [5], [18]. In this concept, the local turbulence structure is characterized by the length scalel e and dissipation time scalet e of eddies representing the coherent flow structures l e = C 1 2 µ k 3 2 ε, t e = le u g. (25) In addition to the life time scale t e, a crossing time scale t c is calculated from tc ( u g u d )dt = le, (26) t taking into account the droplet dynamics. Each time the smaller one of both time scales is elapsed, the droplet enters a new eddy. Consequently, the random process generates a new velocity fluctuation u g from a Gaussian distribution which is determined by 2 µ =, σ = k. (27) 3 This velocity fluctuation remains constant for the period of droplet-eddy interaction and is added to the local value of the gas flow velocity. 4.2 Spray evaporation In this study, droplet evaporation is simulated by means of the Uniform Temperature model [4], [26], [2]. This computationally effective droplet model is based on the assumption of a homogeneous internal temperature distribution in the droplet and phase equilibrium conditions at the surface. The analytical derivation of this model does not consider contributions to heat and mass transport by forced convection by the gas flow around the droplet. Since diffusive time scales in the surrounding gas phase are much smaller than in the droplet fluid, a quasi stationary description of the gas phase is applied. Using reference values for variable fluid properties (1/3-rule), an integration of the radially symmetric differential equations yields analytical expressions for the transport fluxes ṁ vap, Q cond,s and Ḣ vap,s. At this point, convective transport is taken into account by two empirical factors [1] resulting in the corrected fluxes ṁ vap, Q cond,s and Ḣ vap,s [23] ṁ vap = cfm ṁ vap, (28) Q cond,s = πd 2 α (T d T g), (29) Ḣ vap,s = ṁ vap c p,vap,ref (T d T g). (3) Vapor mass flux and heat transfer coefficient are calculated as follows ṁ vap = 2πD ρ g,ref Γ im,ref ln 1 Yvap,g 1 Y vap,s, (31) α = cfh exp ṁ vap c p,vap,ref πd 2 ) (ṁvap c p,vap,ref 2πD λ g,ref, (32) 1 cfm = Re 1 2 Sc 1 3, (33) cfh = Re 1 2 Pr 1 3. (34) The balance equations of the droplet reduce to ordinary differential equations, d dt m d = ṁ vap, (35) d dt (m d h d ) = Q cond,s Ḣ vap,s, (36) which can be appended to the differential equations describing the droplet motion, Eq Secondary droplet breakup At low relative velocities, the spherical shape of the droplets is preserved by the dominating effects of surface tension and viscous forces in the liquid. With increasing velocities, the destabilizing aerodynamic forces on the droplet surface are intensifying, resulting in deformation, oscillations and disintegration of the droplets Classification of breakup mechanisms A common practice to classify secondary droplet atomization processes is based on two characteristic groups of parameters, We = ρg u2 rel D µ d, On =. (37) σ d ρd D σ d The Weber number is a measure of the strength of aerodynamic forces relative to surface tension forces, whereas the Ohnesorge number assesses the damping effect of viscous friction in the droplet against surface tension. In the Weber number range from We = 1 up to a critical value We = We c, non-destructive droplet deformation and oscillation is observed. As illustrated in Fig. 2, three different mechanisms govern the breakup of droplets for increased Weber numbers typical for flows in gas turbine combustors. From these visualizations it is obvious that Bag We=1 Multimode We=2 Shear We=7 Figure 2: mechanisms of water droplets (Ref. [22]) a common feature of all three mechanisms is an initial deformation of the droplet into a disc shape. After this deformation period, various complex flow phenomena lead to the final droplet breakup depending on the intensity of the aerodynamic forces. Exceeding the critical Weber number, the first mechanism observed is bag breakup. This process is characterized by the formation of a thin hollow bag of droplet fluid stretching from a toroidal rim. The thin film of this bag is eventually bursting into a cloud of tiny droplets, followed by a disintegration of the toroidal rim into significantly larger fragments. With increasing aerodynamic forces, a transition to more complex bag structures is observed. In this multimode or stamen breakup regime the aerodynamic flow interaction is forming an additional fluid filament in the center of the bag structures which is aligned with the relative flow velocity. For even higher Weber numbers, shear breakup is observed. This mechanism is fundamentally different to the preceding mechanisms and is characterized by a rapidly disintegrating film of fluid continuously stripped off the rim of the disc shaped droplet by shear forces. Fig. 3 summarizes the results of various experimental studies [11], [19] in a breakup regime map, indicating the relevant mechanism corresponding to a specific combination of Ohnesorge and Weber number. For On >.1 a significant influence of viscosity is observed. The transitions between the different mechanisms are given by the following functions

5 51-5 We Shear Multimode Bag Deformation On Figure 3: regime map ( : Present flow simulation) Critical Weber number (Transition to bag breakup): We c = 12 ( On 1.6 ) (38) Transition to multimode breakup: Transition to shear breakup: We = 2 ( On 1.5 ) (39) We = 4 ( On 1.4 ) (4) Deformation and breakup times Basically, the breakup process can be subdivided in two stages: Initial deformation and further deformation with disintegration. It is convenient to express the relevant times of the breakup process in terms of the characteristic time of shear breakup t = D u rel ρd ρ g. (41) As stated in Ref. [8], the time of initial droplet deformation t def has a constant value independent of the specific breakup t mechanism def = 1.6. (42) t However, the breakup time t b measured from begin of deformation until final destruction strongly depends on the specific mechanism. A fit to a large number of experimental data is given in Ref. [19] 6(We 12).25 12< We < 18 t 2.45(We 12).25 18< We < 45 b t = 14.1(We 12).25 45< We < 351 (43).766(We 12) < We < < We. For On > 1, liquid viscosity is the dominating parameter of the breakup process resulting in the following correlation t b t = 4.5 ( On.74 ). (44) Droplet drag Deformation of the droplet prior to breakup leads to a significant increase of aerodynamic drag. Due to the resulting acceleration, the droplet generally experiences substantial displacement from initial deformation until final breakup. With respect to the deformation period, several authors [8] report a linear increase of the droplet size from D up to a maximum value given by (On <.1, We < 1) D max D = We, for (45) The effect of higher Ohnesorge numbers is taken into account by using a corrected Weber number in the above and following equations. We We corr = On 1.6. (46) As suggested in Ref. [8], a linear transition of the drag coefficient from the sphere shape value to the disc shape value is used in the present study to model the aerodynamic properties of the flattening droplet. In the following period of disintegration, droplet drag depends on the specific breakup mechanism. As illustrated in Fig. 3, the bag and filament structures observed prior to breakup are very complex. According to Refs. [11], [22] the toroidal rim evolving in bag breakup is expanding to seven times the initial droplet diameter, whereas in multimode breakup a maximum diameter of six times the initial diameter is reached (see Fig. 6). At this time however, a major part of the droplet cross section consists of a thin fluid film accelerated in direction of the relative velocity thus decreasing the aerodynamic drag. To bypass the difficulties of describing these opposing effects, the drag of the disintegrating droplet is calculated from the constant disc state reached at the end of the deformation period. In shear breakup, the size of the disc shaped droplet is continuously decreasing to its final value att b Secondary droplet sizes Based on an extended experimental study covering the complete range of breakup mechanisms, a single correlation for the Sauter mean diameter has been derived in Ref. [8] for all three mechanisms (On <.1, We < 1) D = 6.2 On.5 We.25 (47) The exponents in this correlation have been determined by the authors from an approximate analysis of the droplet internal flow during shear breakup, leaving only a constant factor as a parameter for the fitting to experimental data. Using the Weber number given by Eq. 46 to account for viscosity effects, additional fitting of the exponents leads to an improved correlation = 1.5 On.2 We.25 corr. (48) D This correlation which is used for the flow simulations in the present study and the experimental data is shown in Fig. 4. Although the above correlation is valid for the complete range /D Water Glycerol 42% Glycerol 63% n-heptane Ethyl Alcohol 1.5 On.2 We corr On We corr Figure 4: Improved correlation for (Data from Ref. [8]) of breakup conditions, the distribution function of the droplet diameter is substantially different for the various mechanisms of secondary breakup. Bag and multimode breakup Considering bag or multimode breakup, the volume distribution of the droplet fragments is approximated by a root normal distribution [25], given by the following density function f(d) = x 2 2π σ D exp { 1 2 [ x µ σ ] 2 }, (49)

6 51-6 and the parameters D x =, D.5 µ = 1, σ =.238 (5) For a volume distribution with this distribution function, the mass median diameter D.5 is related to the Sauter mean diameter by D.5 = 1.2 (51) Shear breakup According to Ref. [8], the volume distribution resulting from shear breakup is characterized by a bimodal density function with a maximum at small diameters and a second maximum at large diameters. From the experimental data illustrated in Fig. 5, it is concluded that the fine fraction of the droplet fragments corresponds to approximately 8% of the total cumulated volume. These droplets result from film fragments stripped off the Cumulative Volume % Experiment Root normal distribution 1 D/D.5 2 Figure 5: Cumulative droplet volume (Data from Ref. [8]) disc-shaped droplet by shear forces. The 2% in the large diameter range not specified in Fig. 5 represent the contribution of the core droplet fragment left by the stripping process. The diameter D c of this core droplet is estimated from the critical Weber number, evaluating Eq. (38) at flow conditions at the instant of breakup. As illustrated by the curve in Fig. 5, the volume distribution of the fine fraction of the droplet spectrum after shear breakup can be approximated by a root normal distribution based on a reduced Sauter mean diameter,red = 4D32 Dc 5D c. (52) In this equation, the Sauter mean diameter of the complete droplet spectrum is evaluated from Eq Secondary droplet velocities Due to the dominating influence of aerodynamic forces on small droplets, tiny breakup products are immediately dragged with the gas flow. So, an accurate modeling of initial velocities is not necessary in general. These considerations apply in particular to the tiny droplet fragments generated by bursting of film bags or by shear induced film stripping. The motion of large droplets in turn is dominated by inertia forces. As a consequence, the modeling of initial velocities of large droplet fragments has a significant influence on the dispersion behavior of the spray. As a first approximation, the fragments generated by droplet breakup inherit the velocity of the parent droplet due to momentum conservation. In bag or multimode breakup, a transverse velocity component of droplet fragments is observed induced by the transverse spreading motion of droplet fluid during the expansion of the toroidal rim. This transverse velocity component is responsible for increased dispersion of sprays with secondary atomization. For an approximate estimation, the growth velocity of the rim is determined from time-resolved visualizations of breakup processes. Fig. 6 indicates that the ring has D r,max /D We=1 We=15 We= t/t b Figure 6: Growth of the toroidal rim (Water droplet, [22]) D r,max D r,max an extension of about seven times the original droplet diameter in bag breakup against six times in multimode breakup. These observations agree with the values reported in Ref. [11]. Based on these results, the transverse velocity component is estimated as Dr,max D v t = 2(t b t def ). (53) With respect to multimode breakup, a fraction of the droplet fluid is concentrating on the axis of the disintegrating droplet (see Fig. 2, We = 2). Due to the alignment of this prolate filament with the flow, the fragments of this structure have no transverse velocity component. The volume fraction of the filament is estimated from a Weber number based interpolation between the limiting values of % for bag breakup and 2% (core droplet) for shear breakup Stochastical simulation of droplet breakup The period of droplet disintegration is specified by the characteristic times t def, and t b of the breakup process. In shear breakup mode,the secondary droplets are continuously generated in the time from t def to t b, whereas in bag or multimode breakup mode significant generation of fragments is observed during the second half of this time period [22]. Instead of focusing on a realistic simulation of each individual breakup event, the computational implementation makes use of the Lagrangian trajectory superposition approach involving large numbers of droplet parcels. During droplet deformation, the cross sectional area of the droplet and the drag coefficient are continuously increased up to their maximum values at t def. A certain time later, the parent droplet trajectory is terminated and a fixed number of child droplets is generated by random sampling of initial conditions. Each secondary trajectory is assigned an equal fraction of the volume flux. To limit the number of secondary droplets to be tracked, only 3 child droplets have been modeled per breakup event in the present flow simulation in which 1 parent droplets are injected per Lagrangian step. In analogy to the stochastical modeling of fuel atomization and gas flow turbulence effects, the superposition of large numbers of trajectories leads to continuous and thus realistic representation of secondary atomization. Modeling a single shear breakup event, the time of droplet disintegration is determined as a random number with uniform distribution between t def and t b. To model bag and multimode breakup events, the time of disintegration is sampled in the second half of this interval. The initial size of the child droplets is determined as a random number with a root normal distribution or from stability criteria (core droplet in shear

7 51-7 breakup). Droplets generated by rim fragmentation are provided with an additional transverse velocity component. In bag breakup mode, virtually all larger fragments possess a transverse velocity component whereas in multimode breakup a volume fraction of up to 2% of the largest fragments is starting without dispersive transverse momentum. The limiting value of 2% represents the transition to shear breakup and corresponds to the volume fraction of the core droplet. In this breakup regime all secondary droplets inherit the velocity of the parent droplet since no significant spreading motion is observed. Injection Beginning Deformation Formation of Bag Figure 7: of a 6µm droplet 1mm Fig. 7 illustrates the computation of a deforming and disintegrating droplet in the near field of the nozzle. Compared to the trajectory of a rigid spherical droplet (dashed line), a significant deflection of the deforming droplet due to increased aerodynamic forces is observed. This single droplet computation clearly demonstrates that advanced modeling of secondary breakup has to take into account the time scales of the breakup process since the droplets experience considerable displacements before their final disintegration. 4.4 Iterative solution procedure Tracking of a statistically significant number of droplet parcels and superposition of their trajectories yields a flow field approximation of the dispersed liquid phase. However, a simple Lagrangian two step calculation consisting of a gas flow computation and subsequent droplet tracking does not take account of spray effects on the gas flow such as aerodynamic dragging or evaporation cooling. In particular for evaporating sprays in combustor flows, these effects have a significant influence on the overall two phase flow field. To establish mutual information exchange between both phases, Lagrangian two step cycles are concatenated in an iterative procedure with droplet source terms being updated during each tracking step. Representing local transfer rates of mass, momentum and energy from spray to gas flow, droplet source terms are included in the gas flow computation of the following iteration cycle. This iterative approach is illustrated in the lower part of Fig. 8. The separated computation of gas and liquid phase flow fields and the iterative exchange of interfacial transfer data entails an artificial decoupling of both phases. In particular for two phase flows with intense phase interaction, this effect leads to a critical overestimation of droplet source terms in the first iteration cycles. To achieve convergence of the iterative procedure, a relaxation of the droplet source term fields is employed. Recursive damping of the source terms on the level of the iteration cycles is realized by the following equation S i+1 d,φ = α φ S i d,φ + (1 α φ )S i d,φ. (54) According to Eq. 54, only a fraction of the source terms recorded during the previous droplet tracking, Sd,φ, i is contributing to the source terms i+1 d,φ included in the current gas gas flow solution. The second contribution is calculated from the source term included in the previous gas flow computation. For gas flows which are substantially influenced by the fuel spray, strong relaxation of droplet source terms may be necessary to enforce convergence of the iterative procedure. In these cases an increased number of two stage iterations may be necessary for complete consideration of phase interaction [24]. In the present flow simulation, only weak relaxation (α φ >.5) is required to achieve a convergent solution for the two phase flow field within 1 two stage iterations. 5 The Hybrid procedure As indicated in the preceding sections, the continuum description of the Eulerian method has the advantage of close coupling of gas and liquid phase in a single numerical scheme. Thus, computation times are rather short as long as the number of droplet classes used for the discretization of the spray is small. Consequently, Eulerian methods are particularly suited for an approximate but efficient computation of polydisperse two phase flows in combustors. Lagrangian methods in turn achieve a high resolution discretization of complex spray structures by tracking large numbers of droplet parcels of various initial sizes and velocities. However, the price to be payed for Eulerian Method Gas Phase Gas Phase Droplet Sources Flow Field, Droplet Sources Droplet Phase Lagrangian Method Flow Field Trajectories Figure 8: Structure of the Hybrid procedure a realistic spray representation is high. Due to the artificial decoupling of the two phase flow computation by separate solution schemes for each phase and iterative realization of phase interaction, total computation times are rather large [24]. For flow cases where strong relaxation of droplet source terms is required, time expenses can grow to practically unmanagable extents. For such two phase flows, a reduction of computational effort is achieved by preconditioning the two phase flow field by means of an Eulerian method based on a coarse discretization of the spray. This is in fact the basic idea of the Hybrid procedure: A

8 51-8 two stage combination of both methods in order to reduce total computation times by Eulerian preconditioning yet maintaining the detailed modeling of spray physics in a Lagrangian refinement stage. This approach is schematically illustrated in Fig. 8. Following an approximate computation of the two phase flow, the flow field and the droplet source terms are passed to the refinement stage. Here, Lagrangian iteration cycles are based on a fine discretization of droplet injection conditions and an advanced modeling of secondary droplet breakup. Since the two phase flow field calculated by the Eulerian method already accounts for spray effects on the gas flow, the droplet source terms recorded during subsequent tracking steps are rather close to the final flow result. Consequently, the number of iterations is significantly reduced compared to a standard Lagrangian simulation. 6 Simulation of a LPP premix duct flow The performance and accuracy of the numerical methods presented is demonstrated by a simulation of the two phase flow in the premix duct of a LPP combustor. The experimental investigation of this combustor has been part of an extended research project on low emission combustion concepts [15], [16], [13]. A detailed description of the test rig and the measurement techniques is presented in a parallel study [2]. The combustor section of interest for the present flow simulation is illustrated in Fig. 9. Compressed air is supplied to the cylindrical duct Figure 9: Premix zone of the LPP research combustor (l = 124mm,d i = 44.6mm) by two coaxial annular ducts. The fuel is injected into the gas flow by a pressure swirl atomizer aligned with the duct axis. The nozzle diameter is about 1 mm. In order to perform PDPA measurements, optical access to the flow is given by circumferencial slits in the duct liner at various axial positions. For spray visualizations, the metallic liner is substituted by a quartz glass cylinder. Premix and reaction zones are separated by an arrangement of swirler vanes acting as a flame stabilizer. The complete configuration illustrated in Fig. 9 is enclosed in a pressure casing with water cooled window ports. The outer coaxial annular air flow (bypass air) is shielding the duct liner from droplet impact and film formation. The highly accelerated inner flow (atomization air) is focused directly onto the conical fuel sheet generated by the pressure swirl atomizer. The fundamental idea of this atomization concept is a further reduction of droplet sizes by high velocity aerodynamic interaction between spray and gas flow. The operating point of the premix duct investigated in this study is specified by the flow parameters summarized in Table 1. The computational domain of the flow simulation is illustrated in Fig. 1 including (from left to right) intake section, coaxial annular ducts, premix zone, swirler vanes, reaction zone, dilution holes and burnout zone. The axial coordinate is measured from the atomizer nozzle. To model the evaporation behavior of the diesel spray, tetradecane is used as a single component diesel substitute in the present flow simulation. A detailed descrip- Gas Flow (Air) Fuel (Diesel) ṁ g 213 g/s ṁ fuel 6 g/s T g 753 K T fuel 35 K p g 4bar v sheet 3 m/s Tu g 15% β sheet 4 Table 1: Parameters at the inlet of the premix duct (z = mm) Premix zone Figure 1: Computational domain (3 -Segment) tion of the calculated non-reacting single phase gas flow in the intake and premix zone is given in Ref. [2]. 6.1 Discretization of the spray In order to derive droplet phase boundary conditions for the Eulerian method and droplet initial conditions for the Lagrangian method, the spray is visualized in the near field of the atomizer ( < z < 1 mm). A side view on the three dimensional spray cone is given by the flashlight shadowgraph in Fig. 11(a). The picture was taken under atmospheric, cold conditions without bypass air flow, using water as a fuel substitute. It is evident that the outer region of the spray is dominated by larger fuel fragments. To get an impression of the spray structure inside Figure 11: (a) Flashlight shadowgraphy, (b) Laser light sheet the cone, a laser light sheet photograph is shown in Fig. 11(b), which was taken under real operation conditions of the combustor. In contrast to the coarse structure of the outer spray region, this cross view reveals a very fine droplet distribution in the spray cone. The spray visualizations indicate two basic processes that govern the atomization of the liquid fuel [15]. Due to its high swirl, the fuel is leaving the nozzle as a conical sheet. According to Fig. 11(a), this sheet is completely disintegrating along a distance of 1 to 2 mm. In this immediate near zone of the nozzle, prompt atomization is the governing process. This mechanism is controlled mainly by the internal sheet dynamics [14]. Further disintegration of sheet fragments is induced by aerodynamic interaction with the high velocity gas flow which penetrates the spray. The resulting small secondary fragments are dragged by the gas flow into the core flow as indicated in Fig.

9 (b). With respect to the numerical simulation, it is obvious that a representation of the complex two phase flow in the nozzle near field by non-interacting droplets is a rather crude approximation of the physical reality. But despite of this simplified modeling of fuel atomization, the simulation of spray dispersion and evaporation in the premix duct agrees well with the experimental data. In the following sections, two strategies are presented to derive droplet injection conditions at the nozzle. The Eulerian calculation is essentially based on droplet data measured at a downstream position of the atomizer. It is important, that secondary atomization has ceased completely and droplets are spherical at this position. Basically, the procedure starts from an assumed discretization of the injection conditions of the droplet phase. In subsequent optimization iterations, the computed droplet data is fitted to the corresponding measured data. In contrast to this approach, the Lagrangian calculation is based on a rather crude approximation of the fuel sheet disintegration in the nozzle near field. Here, a far more physical description of the spray in the secondary atomization region is achieved by modeling droplet deformation and breakup. A similar strategy is described in Ref. [7] where pressure swirl fuel injection into a diesel engine combustion chamber is discussed Eulerian method, droplet phase boundary conditions Since secondary droplet breakup is observed in a spray region up to 5 mm downstream of the atomizer, measured droplet data atz = 55mm is used for an optimization of droplet phase boundary conditions. The droplet size distribution of the spray is discretized by means of 3 diameter classes. The volume flux fraction (or normalized volume flux) of each class is described by a Rosin-Rammler distribution evaluated at the representative class diameter D i [ ( ) n ] V i V = 1 exp Di,D.632 = D.5 D n 1. (55) Values of D.5 = 46µm for the mass median diameter and n = 4.8 for the spreading parameter are determined as a good fit to the reference droplet data atz = 55mm. The mean injection velocity of the droplet phase of3m/s is derived from the mean velocity of the liquid sheet. The remaining parameters with significant influence on the spray structure are the injection angles of the droplet phases. Introducing 3 angle classes D i [µm] V i/ V [1] β i [ ] / /3 7 1/ /15 3 1/3 1 1/ / / /3 2.5 Table 2: Droplet phase boundary conditions! I per diameter class leads to a final discretization of the spray by means of 9 droplet phases with different boundary conditions at the nozzle. Table 2 summarizes the correlations between droplet phase diameter, normalized volume flux and injection angle employed for the Eulerian two phase flow simulation in the present study. L B Lagrangian method, droplet initial conditions In the Eulerian calculation, the boundary conditions of the droplet phase at the nozzle have to account implicitly for secondary atomization in an extended downstream flow region. It is evident that small secondary droplets originating from breakup of larger sheet fragments in outer flow regions may not be reproduced by a spray representation as described in the previous section. The approximation by rigid spherical particles of different size injected into the contracting, high velocity gas flow leads to a separation of droplet sizes. As a consequence, tiny and small droplets are captured in the axis region of the core flow unless they are injected with unphysically large radial velocity components. In the Lagrangian calculation, secondary breakup of droplets is taken into account during trajectory integration. Thus, only prompt atomization of the conical sheet has to be considered for the formulation of droplet initial conditions. The basic idea is to inject most of the fuel in form of large droplets with sizes similar to the characteristic sheet thickness of about1 to2µm. Due to this coarse primary spray structure and the high relative velocities in the nozzle near zone, the critical conditions of droplet breakup are significantly exceeded as indicated by the data points mapped Fig. 3. Modeling of delayed droplet deformation, drag increase and breakup results in a fine secondary spray contribution to the core flow region in the premix duct. Very good agreement to the experimental droplet data is Droplet diameter: 1 size classes equally spaced from to2 µm Rosin-Rammler distribution of V i/ V D.5 = 88.5 µm, n = 3 11 Droplet velocities: 1 1 Sampled, Gaussian distribution 11 v = 3 m/s, σ v = 5m/s Injection angle: Sampled, Gaussian distribution 11 β = 4, σ β,i = 3,...,5 11 β max = (clipping value) 1 1 Table 3: Droplet initial conditions β sheet v sheet achieved by using the spray discretization summarized in Table 3. Droplet velocity and injection angle are sampled as random numbers with Gaussian distributions. From Fig. 11(b) it is obvious that the disintegration of the conical sheet is responsible for a fine primary contribution to the spray in the core flow. To model this effect approximately, the variance σ β,i of the injection angle β is correlated with droplet size resulting in values of 3 for the smallest droplets (D from to 2µm) up to 5 for the largest droplets (D from 18µm to2µm) 6.2 Results To illustrate the calculated two phase flow in the premix duct, contour plots of the Eulerian flow simulation are discussed first. Comparing the axial gas velocities of the single phase and the two phase flow calculations from Fig. 12 indicates that sprayinduced deceleration of the gas flow is limited to the core flow of the duct. In particular in the nozzle near zone, the axial gas

10 51-1 velocity is decreased by up to 4 m/s due to the aerodynamic acceleration of the droplet phase. The influence of fuel evaporation is indicated by the gas flow temperature and fuel vapor concentration in Figs. 13 and 14. Although the gas phase experiences a substantial temperature drop across the whole core flow region, significant concentrations of fuel vapor are not calculated in the first half of the duct. This delay in vapor generation is a consequence of low evaporation rates in the transient heating phase of the droplets. This conclusion is confirmed by an analysis of Lagrangian single droplet computations, which indicate that heatup and propagation time scales of typical droplets are of the same order. In total, 28% of the injected fuel is evaporated in the Eulerian simulation, in contrast to a value of42% in the Lagrangian simulation. The difference is caused by the secondary atomization modeling in the tracking algorithm, resulting in considerable numbers of small, rapidly evaporating droplet fragments. At this point it should be noted that the total fraction of evaporated fuel substantially depends on the -correlation used for the secondary breakup modeling. Summarized over all calculated breakup events, Eq. 47, which is actually not used, leads to a fragment mass median diameter ofd.5 = 53µm, whereas Eq. 48 results in a value of D.5 = 38µm. To compare the calculated axial volume flux densityα d U d with PDPA measurements, it is weighted by the annular area and normalized by the total axial volume flux r+.5mm 2π α d U d r dr V r V = r.5mm. (56) 24mm 2π α d U d r dr This normalized volume flux is illustrated in Fig. 15 and represents the fraction of the total liquid volume flux which passes an annular fraction of the duct cross section. The contour plot includes mean trajectories determined by integration of the mean axial velocity of the droplet phase. The trajectories are evaluated for the two limiting size classes and clearly demonstrate the influence of initial droplet momentum on the propagation of the droplet phase. The second point of discussion is concerning the comparison of experimental droplet data with Eulerian and Lagrangian two phase flow simulations. In this context, the Hybrid procedure is used as a computational tool to accelerate the Lagrangian calculation. The overall reduction of computation time achieved is about 3% of the time required for a standard Lagrangian flow simulation without Eulerian preconditioning. Radial profiles of number averaged two phase flow variables are presented at axial positions z = 2,55 and 9 mm. With respect to the mean axial velocities of the droplets shown in Figs. 16 and 17, both numerical methods predict a maximum in the core flow (r < 6 mm) which is not observed in experiment. In particular at z = 55 and 9 mm the axial droplet velocities in the core flow region are overestimated by the Lagrangian calculation. Basically, this effect is a result of an insufficient sprayinduced flow deceleration. This conclusion is supported by the underestimated liquid volume flux in the core flow region at the corresponding axial positions as shown in Fig. 19. With respect to the Eulerian result shown in Fig. 18, it is evident that the unavailability of secondary breakup models requires small injection angles of the droplet phase to meet the radial volume flux profile in the second half of the duct flow. Consequently, substantial deviations are observed in the upstream flow region where secondary atomization occurs. As illustrated by Figs. 2 and 21, Eulerian and Lagrangian flow simulations predict rather similar radial profiles of the Sauter mean diameter. Although the calculated values are deviating from experimental data by up to 2 µm, both flow simulations reproduce the trend in the evolution of the droplet size spectrum. 7 Conclusions Evaporating fuel sprays in combustor flows are characterized by high rates of mass, momentum and enthalpy transfer between spray and gas flow. Fuel atomization, spray dispersion and evaporation are often complicated by additional physical effects such as droplet-wall interaction, shear driven evaporating wall films or secondary breakup of droplets. The primary objective in this study is the design of a computational tool for an efficient numerical simulation of combustor two phase flows including advanced modeling of secondary atomization physics. Compared to state of the art Lagrangian spray simulations, a significant reduction of computation time is achieved by the presented Hybrid procedure. Basically, this computational strategy is a two stage combination of an Eulerian and a Lagrangian method. The Eulerian method is used as an efficient preconditioner for the interacting two phase flow field, in order to reduce the number of subsequent Lagrangian gas flow computation - droplet tracking iterations. In this refinement stage, advanced modeling of secondary breakup of droplets including bag, multimode and shear mechanisms is used to improve the physical description of the spray. To assess the accuracy of the two fundamental numerical approaches, Eulerian and Lagrangian flow simulations are compared with droplet data measured in the two phase flow of a LPP combustor premix duct. The atomization concept employed in this premix duct achieves a substantial improvement of fuel atomization by aerodynamic breakup of droplets in an extended flow region downstream of the nozzle. With respect to the Eulerian flow simulation, extrapolation of measured droplet data to the point of injection with implicit consideration of secondary atomization effects results in a rather crude approximation of the spray structure. Since secondary breakup of droplets is modeled in detail by the tracking algorithm, the numerical description of the spray structure in the Lagrangian flow simulation is significantly improved. It is evident that the computational acceleration achieved by the Hybrid procedure significantly depends on the structure of the two phase flow. With respect to the rather uncritical premix duct flow presented in this study, it is certainly questionable whether the moderate time savings are worth the additional complexity of the CFD program. However, technical practice offers a sufficient number of critical two phase flow applications, where strong relaxation of droplet source terms requires an excessive number of Lagrangian coupling iterations [24]. In these flow cases, the Hybrid procedure has an increased potential for substantial speedup of flow simulations. Acknowledgement This work has been founded by the Department of Education, Science, Research and Technology of Germany under contract No The support is gratefully acknowledged. The authors would like to thank Mr. Michael Willmann, BMW Rolls-Royce, Germany, for his support concerning the development of droplet breakup models. References [1] B. Abramzon and W.A. Sirignano. Droplet Vaporisation Models for Spray Combustion Calculations. International Journal of Heat and Mass Transfer, 32: , 1989.