Paper ID ILASS8--7 ILASS 28 Sep. 8-, 28, Como Lake, Italy A44 NUMERICAL SIMULATION OF ATOMIZATION WITH ADAPTIVE JET REFINEMENT Anne Bagué, Daniel Fuster, Stéphane Popinet + & Stéphane Zaleski Université Pierre et Marie Curie, UMR-CNRS 79, Institut Jean Le Rond D Alembert 4 Plae Jussieu ase 62, Paris edex, 722, Frane + National Institute of Water and Atmospheri Researh P.O. Box 4-9, Kilbirnie, Wellington, New Zealand bague@lmm.jussieu.fr, fuster@lmm.jussieu.fr ABSTRACT Atomization proesses our in different systems like engine injetors. When a liquid is injeted with a surrounding gas, small instabilities appear at the interfae. When their amplitude is small enough, a linear instability appears and the problem an be solved in two dimensions with the Orr-Sommerfield equation. The temporal shear instability for two-phase laminar flow is studied for two different ases. The validity of the results obtained from the temporal theory applied to the spatial-temporal problem is also investigated. Atomization proesses our in numerous industrials fields. For example, in engine injetors, fuel is atomized into droplets when injeted inside the ombustion hamber in a omplex proess. Using simulation tehniques it is possible to estimate the ombustion quality, the engine effiieny and pollution rate. When a liquid jet is atomized, a strong interation is produed between the jet and the surrounding gas and, depending on gas and liquid properties and the injetor nozzle geometry, an instability appears. The flutuations introdued in the mean flow are responsible for the appearane of small disturbanes whih, initially, display a linear behavior. When these perturbations grow downstream they originate thin ligaments whih eventually break into droplets. This paper fouses on the simulation of the linear stability of shear for two-phase mixing layer flows in two dimensions and the analysis of the instability produed in the primary atomization. To perform the simulations, a numerial ode based on Volume of Fluid (VOF) advetion sheme for interfaial flows is used (GERRIS, http://gfs.sf.net). GERRIS [] is an adaptative mesh refinement ode for the time-dependent inompressible Navier-Stokes equations with a multigrid solver. The method is based on quad/otree spatial disretization with automati and dynami loal refinement. The pressure is obtained by means of a multilevel Poisson solver. Continuous surfae fore (CSF) is used with the height funtion method [3] to model surfae tension [4]. LINEAR STABILITY When a liquid is atomized with a surrounding gas, analysis of the evolution of the small instabilities introdued in the flow an be done with the aid of linear theory applied to two-phase mixing layer. In partiular, the theoretial solutions of this linear problem an be obtained by solving the Orr-Sommerfield equation.. Theory Let us onsider a domain defined by L l <y< for the liquid phase and <y<l g for the gas phase. The following properties ratios are defined: the density r = ρ g /ρ l, the visosity m = µ g /µ l and boundary layer thikness in the two fluids profiles n = δ g /δ l with the initial veloities profiles given by (Figure ): U l (y) = U int Ul erf (y/δ l ), () U g (y) = U int + Ug erf (y/δ g). (2) where U int is the interfae veloity and U is: { U l = U int U l, U g = U g U int. For this ase, it has been deided to use a system of referene moving with the interfae and therefore, U int equals to zero. (3)
At the edge of the domain, the following boundary onditions are imposed: φ l = φ l = at y = L l, () φ g = φ g = at y = L g. () At the interfae, the normal and tangential ontinuity of veloity and of stress onditions are applied: Figure : Parameters and profiles used for the liquid and gas phases. φ l + U l φ l = φ g + U g φ g, (2) A relation between these parameters an be obtained by applying the stress ontinuity ondition: U g U l = n m. (4) φ l + α 2 φ l + U l φ l φ l = φ g, (3) ( = m φ g + α 2 φ g + U g φ g ), (4) In order to interpret the results, it is useful to define the following Reynolds and Weber numbers. These dimensionless numbers will have an important effet on the type of observed instabilities []: Re l = ρ l U g δg µ l, Re g = ρg U g δg µ g, We l = ρ l (U g )2 δ g σ. We g = ρg (U g )2 δ g σ. The Orr-Sommerfield equation takes into aount the perturbation streamfuntions ψ g and ψ l, defined in relation to streamwise and ross-stream veloity omponents u and v in both phases [6]: () u = y ψ l,g, v = x ψ l,g. (6) The solutions are periodi in time and spae: ψ l,g = φ l,g (y)e iα(x t). (7) The omplex wave speed depends on the real wave number α. The perturbation equations for the liquid and the gas phase are [2]: (U l ) ( D 2 α 2) φ l U l φ l = iαre l ( D 2 α 2) 2 φl, (8) (U g ) ( D 2 α 2) φ g U g φ g = m r where D is the differential operator for y. ( D 2 α 2) 2 φg. iαre l (9) ( φ l 3α 2 φ ) l + irαre l r (φ l + U l φ l) () m ( φ g irαre 3α2 φ ( g) φ g + U g φ ) α 2 g = φ l. l rwe l.2 Results The examples used in this work for the ode validation are summarized in Table. referene m r δ l /L δ g /L A..2.2 B..2.2 C.99..2.2 D.99..2.2 referene Re l Re g We l We g A 2 2 B 2 2 C 98 2 D 98 2 Table : Charateristis of the simulations For ases A and B (m =. and r = ), the results are depited in Figure 2. The numerial results give a good approximation of the linear theory. Without surfae tension, the mean error between numerial and theoretial results is 2.22 %. When surfae tension is inluded, the simulations give a better estimation of linear theory with a mean error of.8 %. 2 2
α i δ g / U g.2.. theory We g = gerris We g = theory We g =4 gerris We g =4 referene 32 64 28 26 A 2.33.74 3.. B 7.3.28.48.4 C.7.24.4.9 D.39.76.7.4 Table 2: Perentage of the error for the different ases from Table and for mesh sizes of 32, 64, 28 and 26.... 2 2. 3 3. Figure 2: Growth rate of unstable modes and the orresponding results with GERRIS ode for ases A and B from Table. Figure 3 shows the numerial and theoretial urves when m =.99 and r =. (ases C and D). An improved approximation is obtained ompared to the previous results. The simulation results orretly predit the non-dimensionned growth rate α i provided by the linear theory. The mean error between numerial and theoretial results are.44 %, for the ase without surfae tension and.62 % with surfae tension. α i δ g / U g.2.2... theory We g = gerris We g = theory We g =4 gerris We g =4 2 3 4 6 Figure 3: Growth rate of unstable modes and the orresponding results with GERRIS ode for ases C and D the Table. Results obtained with numerial simulations ompare favorably with the theorerial solution obtained with the Orr-Sommerfield equation. The auray of the results is lower for ase A. For a small viosity ratio, slight differenes are observed whih beome more appreiable for small density ratios. Table 2 exhibits the perentage of the error between the theorial and numerial results are alulated for different mesh sizes. Generally, a better approah is obtained when the mesh is refined exept when surfae tension is inluded for the two most refine meshes. However, the error always remains lose to %. The solutions obtained by the numerial ode an therefore be onsidered as good approximations of the real solutions, and they will be used for the rest of this work to analyze the behavior of jets in more omplex situations. 2 INSTABILITIES PRODUCED IN THE PRIMARY ATOMIZATION In this setion, the jet behavior in the primary atomization zone is investigated. High resolution simulations are used. The mesh is adapted in the zones where the solution is more omplex, namely, at interfaes and vorties. The smallest mesh size is about.8µm. Output boundary onditions are applied in all boundaries of the domain exept at the entrane, where the veloity profile is presribed aording to Eq. (2) with an interfae veloity defined by the stress ondition at the interfae (Eq. 4). Moreover, a perturbation v is introdued in the vertial veloity, whih is a random funtion applied either in the liquid or in the gas giving values inside the range [. U :. U]. The rest of onditions required for the simulation are inluded in Table 3, and the relevant dimensionless parameters of these simulations an be found in Table 4. U (m/s) ρ (kg/m 3 ) µ (Pa s) Liquid 2 4 Gas 2.7 σ (N/m) Domain size (µm) δ (µm) Liquid.3 4.8 Gas 6 2. Table 3: Simulation onditions used for the analysis of the instabilities in the primary atomization zone. m r δ l /L δ g /L.34.2.8.2 Re l Re g We l We g 4 233 667 33 Table 4: Representative dimensionless numbers for the onditions inluded in Table 3. 3 3
Figure 4 displays a representative view of the behavior of the jet for the onditions indiated above. As it an be seen, the simulation domain is foused only in the primary atomization zone, before the jet is definitely broken into droplets. (y-y int, )/δ g - x/δ g =. 2 7 Figure 4: Interfae profile along a distane of 4µm downstream of the injetor at t =µs (top). Zoom to the transition between the linear and non-linear regime (bottom). Two main zones an be distinguished: In the first zone, the disturbanes at the entrane generate small instabilities whih are quikly amplified downstream. The intensity of the instabilities is measured using the kineti turbulent energy. Figure depits this variable at different positions from the entrane. In the very early stage of injetion the initial turbulene indued in the liquid is firstly transferred to the interfae (transition from x/δ g =to x/δ g =2). Then, even when the small disturbanes introdued at the entrane are quikly dissipated in the bulk liquid due to visous effets, they are strong enough to reate some instabilities at the interfae. As the instabilities grow downstream, the level of turbulene inside the gas and liquid inreases, mainly in zones near the interfae. It an be stated that the small perturbation originally indued in the liquid is firstly transferred to the interfae, whih finally generates a strong turbulene downstream not only in the liquid, but also in the gas. Remarkably, this phenomenon is also enountered when the instabilities are indued in the gas (Figure 6). In this ase, the gas momentum is powerful enough to generate appreiable perturbations at the interfae, whih are finally amplified produing a strong turbulene on both phases. In the seond zone, the instabilities beome nonlinear and ligaments appear. These ligaments are finally broken into droplets whih interat with the vortex appearing in the gas phase near the interfae. It has been found that for large Reynolds numbers, the flow strutures in the gas vortex generated under the ligaments strongly interat with the drops whih have been generated. Therefore, simulation results are extremely sensitive to the mesh size and DNS simulation of the jet is required, onsiderably restriting the Reynolds numbers whih ould be reahed with a high degree of auray. (y-y int, )/δ g - e- e-4... E k / E T 3 2 2 - - - -2 x/δ g = 2 7 2 7 e- e-4... E k / E T Figure : Profiles of the turbulent kineti energy (E k = /2(u i )2 ) at different distanes from the injetor. Perturbation is solely indued in the liquid phase. (y-y int, )/δ g - - x/δ g =. 2 7 e- e-4... E k / E T Figure 6: Profiles of the turbulent kineti energy (E k = /2(u i )2 ) at different distanes from the injetor. The perturbation is solely indued in the gas phase. Thus, it an be onluded that turbulene at the entrane either of both phases is going to promote the appearane of instabilities at the interfae, whose amplifiation is responsible for the transition from the linear to the nonlinear regime. This transition is aptured in Figure 7, where the time-averaged maximum of the interfae position at different loations is plotted as a funtion of the distane from the injetor. During the first zone, a lear exponential growth is observed and one the amplitude of the waves is muh larger than the thikness of the boundary layer, the growth rate is saturated. 4 4
4 (y - y int, ) / δ g (y - y int, ) / δ g 3 2 - -2 x/δ g = x/δ g = 2 x/δ g = x/δ g = 7-3. 2 4 6 8 2 4 6 8 x / δ g -4 - -... 2 (U(y) - U,LT ) / U,LT Figure 7: Evolution of the averaged amplitude of the waves along the jet. Transition to the linear regime with an exponential growth of the waves (x/δ g 8) to the non-linear regime. In the linear regime, this initial growth rate is a funtion of parameters like the Reynolds and Weber number or the gas and visosity ratios. Some attempts have been done in order to use the linear theory to predit the harateristi frequenies whih are going to be amplified downstream. However, the lak of a omplete developed theory for the temporal-spatial problem and the diffiulties related with the temporal hange of the base flow makes it is neessary to resort to numerial simulations in order to investigate these type of situations. Nevertheless, it is interesting to ompare some of the results from simulations with those obtained from the linear theory already presented in setion. Firstly, it is important to emphasize that numerial results are ompared with theoretial results based on the assumption that the Gaster relation is appliable, whih is just justified when the veloity of the waves is larger than the group veloity. This property is revealed by stopping the perturbation at the inlet. For the ase studied here, the waves are then propagated downstream and the equilibrium state given by the input boundary onditions is reovered. Another effet whih should be taken into aount is the evolution of the base flow downstream. The hanges in the veloity profiles as well as those in the boundary layer thikness an alter the behavior of the different modes. Figure 8 depits different veloity profiles at different positions downstream of the injetor. Just in a very short distane from the entrane, the veloity profile is similar to that introdued as an input boundary ondition. The boundary layer then grows due to visous effets, and when the flow beomes nonlinear (for distanes larger than x/δ g = 8) the veloity profile is signifiantly modified due to the turbulene generated for the waves. For a more aurate omparison between simulation and theoretial results, four spetra at different loations are plotted in Figure 9. The interpretation of the results has to be understood under the following assumptions: (y - y int, ) / δ g 2 - - - x/δ g = 2 x/δ g = 7 x/δ g = 2 x/δ g = 7-2 - -... 2 (U(y) - U,LT ) / U,LT Figure 8: Evolution of the veloity profile and interfae position in different setions. Top graph: Evolution of the veloity profile in the zone where waves are still linear. Bottom: Transition from the linear to the nonlinear zone. Due to the instability has been identified as onvetive, it is onsidered that some onvetive veloity relates the frequenies measured in the simulations, f, andthewavelenghts predited by theory α. As a first approximation this veloity is taken as a onstant and it is obtained from Eq.??. The Gaster relation is used to relate frequenies and wavelenghts: U,LT = f (6) α The initial perturbation amplitude is assumed to be equal for all the frequenies. Thus, the amplitude at some distane x from the injetor would be proportional to the growthrate i.
(Amp / δ g ) f (δ g / U,LT ).. x/δ g = 2 LT. e-4... f δ g / U,LT α i δ g / U,LT The results provided by the linear theory have been shown to also orretly predit the main behavior of the instabilities in the primary atomization zone when the spatial-temporal is studied. GERRIS ode has allowed us to observe the different instabilities reated in this zone. REFERENCES [] T. Boek, J. Li, E. López-Pagés, P. Yeko & S. Zaleski, Ligament formation in sheared liquid gas layers, Theoretial and Computational Fluid Dynamis, vol. 2, pp. 9-76, 27. [2] T. Boek & S. Zaleski, Visous versus invisid instability of two-phase mixing layers with ontinuous veloity profile, Phys. Fluids, vol. 7, 326, 2. (Amp / δ g ) f (δ g / U,LT ).. e-4.. f δ g / U,LT.. α i δ g / U,LT [3] M. Franois, S. Cummins, E. Dendy, D. Kothe, J. Siilian & M. Williams, A balaned-fore algorithm for ontinuous and sharp interfaial surfae tension models within a volume traking framework, J. Comput. Phys., vol. 23 (), pp. 4-73, 26. [4] S. Popinet, An aurate adaptive solver for surfae-tensiondriven interfaial flows, (In preparation). [] S. Popinet, Gerris: a tree-based adaptive solver for the inompressible Euler equations in omplex geometries, J. Comput. Phys., vol. 9,pp. 72-6, 23. x/δ g = 2 x/δ g = x/δ g = 7 x/δ g = LT [6] P. Yeko & S. Zaleski, Transient growth in two-phase mixing layers, J. Fluid Meh., vol. 28,pp. 43-2, 2. Figure 9: FFT of the temporal signal at different loations (lines) and theoretial growthrate (dots). Despite the simplifiations done to perform the omparison, a good fitting is displayed between the numerial and theoretial spetra. Results fit speially well for the first spetrum (x/δ g =2), where the small instabilities are still linear. As explained, the most amplified frequenies tend to derease downstream whih is a onsequene of the vortex growth in the two phase mixing boundary layer. When the instability is large enough, the vortex generated in the gas phase predominately exites low frequenies whih beome even smaller as the vortex grows downstream. CONCLUSION The temporal visous linear theory in two dimensions for twophase flow has been used to validate the numerial ode. The adaptative mesh refinement ode used in this work (GERRIS) orretly verifies the linear theory independently of the value of the surfae tension. 6 6