ICLASS-2006 Aug.27-Sept.1, 2006, Kyoto, Japan Paper ID ICLASS06-282 MODELING THE EFFECT OF THE INJECTION VELOCITY ON THE DISINTEGRATION OF ROUND TURBULENT LIQUID JETS USING LES/VOF TECHNIQUES Vedanth Srinivasan 1, Abraham J. Salazar 2 and Kozo Saito 3 1 Department of Mechanical Engineering, University of Kentucky, Lexington, USA 40506, vedanth@uky.edu 2 Department of Mechanical Engineering, University of Kentucky, Lexington, USA 40506, ajsala00@pop.uky.edu 3 Department of Mechanical Engineering, University of Kentucky, Lexington, USA 40506, saito@engr.uky.edu ABSTRACT The influence of liquid injection velocity on the disintegration behavior of round turbulent liquid jets is assessed using numerical techniques. The liquid-gas interface is captured using Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM). The effect of turbulence is computed using Large Eddy Simulation strategies, wherein large scale motions in the flow are resolved while modeling the small scale statistics. For this purpose, a one equation eddy viscosity model has been used to resolve the sub-grid scale stresses. The model predicts backscatter and hence provides accurate estimation of the multi-scales motions embedded in the multiphase regimes. The computation predicts growth surface waves leading to ligament drop formation. Several features associated with ligament dynamics viz., stretching, thinning, destabilization mechanisms are well accommodated. Interaction of ligament tip with shear vortices in the gas phase results in multiple drop sizes. The ligament alignment with respect to the liquid bulk jet is observed to vary appreciably by changing the liquid injection velocity. Disintegration of low-velocity liquid jets lead to bigger drops than those obtained from injection at higher velocities into stagnant atmosphere. The disintegration characteristics agree qualitatively well with experimental results. Velocity profile relaxation mechanism and near liquid-gas interface features have been discussed in detail. Keywords: Liquid Atomization, Droplet, LES/VOF, Primary breakup, CICSAM 1. INTRODUCTION The mechanism of primary breakup of turbulent liquid jets is of fundamental importance in various industrial processes such as spray coating, combustors, metal powder formation, etc. Breakup process of a liquid jet emanating from a nozzle is governed by a large number of parameters such as liquid jet velocity, interior nozzle design (influence of turbulence, cavitation, swirl components), liquid and ambient gas properties [1]. Distinct regimes arising during the jet breakup due to the action of dominant forces have been identified [2]. Of particular interest in the current study is the atomization phenomenon, where the liquid jet breaks up into droplets several orders of magnitude smaller than the jet diameter [2, 3]. The atomization process of liquid jets is thought to consist of two consecutive steps: primary and secondary breakup. During the primary breakup, the liquid jet exhibits large scale coherent structures that interact with the gas-phase and break into both large and small scale drops. Proceeding downstream, the liquid fragments formed due to primary breakup may split further into much smaller drops under the action of surrounding gas forces leading to secondary breakup [2, 4]. The process of atomization occurs in the turbulent flow environment which results in the presence of wide range of length and time scales of motion [2]. The foresaid variation in length and time scales differentiates the treatment of primary and secondary flow structure during atomization. The characteristics of primary breakup has significant influence on the properties of the dispersed phase affecting its mixing rate with the surrounding gas, the mechanisms of secondary breakup and droplet collisions, among others [1]. The mechanism of primary breakup was earlier observed by De Juhasz et al. [5]. Further studies taken up by McCarthy and Malloy [6] identified liquid turbulence properties behind the jet instability and subsequent breakup. Later, Hoyt and Taylor [7] concluded that the drop formation due to turbulent primary breakup was associated with the formation of discs and ligaments along the liquid surface and that the aerodynamic effects were generally of secondary importance. However, by increasing the liquid-gas density ratio, the aerodynamic forces become predominant and lead to early breakup. The interfacial instability is enhanced due to the force contribution of shear stresses arising from the denser gas [8, 9]. The breakup behavior of turbulent jets has several tightly coupled parameters that need to be taken into account during experiments. Circumventing these difficulties, several numerical techniques such as Volume of Fluid (VOF) [11], Cubic interpolated pseudo-particle (CIP) [12], Marker and Cell (MAC) [13], level set methods [14] have been utilized to simulate liquid jet breakup. Although several factors influence the disintegration of turbulent jets, the current study uses a VOF based interface tracking approach to assess the influence of liquid injection velocity on the breakup of a round turbulent liquid jet. Our calculations concern the initial stages of breakup, the primary breakup, restricted to few diameters downstream of nozzle exit [2]. In the primary breakup zone, the injected turbulent liquid interacts with the surrounding continuous gas phase on scales larger than the grid resolution, resulting in highly complex interface dynamics. Despite development of several models to accurately track the liquid-gas interface, inclusion of a very good turbulence model to account for the true influence of wide turbulence scales has been missing. In order to accurately quantify the interface features, it is essential that the large scale motions resulting in the multiphase dynamics be accurately resolved [15]. Large Eddy Simulation techniques offer very accurate representation of the geometry
dependent large scale motions behavior while modeling the geometry independent small scale motions [16]. In this paper, we apply LES formalism with a VOF based interface tracking methodology to evaluate disintegration of a round turbulent liquid jet emanating from a nozzle into still gases. Details of LES models and VOF methods are discussed in section 2. Relevant computational framework is briefed in section 3 while section 4 and 5 present the boundary conditions for the simulations and the results of our calculations, respectively. 2. NUMERICAL MODELING The present model treats the liquid-gas two-phase fields as a single incompressible continuum with an effective variable density ρ and an effective viscosity μ, which can be discontinuous across the phase interfaces. The effect of surface tension forces acting on the interfaces preserves the curvature. The governing mass and momentum conservation equations in the basic form can be represented as: ui = 0 (1) xi ρu ρuu P u i i i i + = + ν + F (2) s t t x x x i j j where F s is the force term arising due to surface tension. It is expressed as, F = ' n' ( x x' s σκ δ ) ds (3) S( t) In the above formulation u is the velocity, P the pressure field. The surface tension force is computed based on the curvature of the interface denoted by κ and the unit normal vector n on the interface S(t). The location of the transient interface S is determined by using a Volume of Fluid (VOF) surface capturing methodology employing the volume fraction indicator function equation. The volume fraction γ used in the VOF formulation is defined as: 1 for a point inside the liquid γ = 0 < γ < 1 for a point in the transition region 0 for a point completely outside the liquid The liquid-gas interface resides on the transition region defined in the above definition. The indicator function, a Lagrangian invariant, obeys the transport equation of the form, γ + t x i ( γ u ) = 0 i From the definitions of the phase indicator function γ as described above, the effective local density and viscosity of the fluid can be estimated as: ρ = γρ + 1 γ ρ (6) l l ( ) ( 1 ) g μ = γμ + γ μ (7) g (4) (5) where the subscripts l and g represents the liquid and gas phase, respectively. Since the interface is within the transitional zone, its exact shape and location are not known explicitly, and the surface integral in equation (3) cannot be evaluated directly. This problem is solved by using the Continuum Surface Force (CSF) approach [17], which represents the surface tension forces as continuous volumetric force acting within the transition region. The interfacial surface force is replaced by a smoothly varying volumetric force derived by integration of the surface tension force over the transition region. The surface tension force is transformed into a volumetric force by using a delta Dirac function centered at the interface. Expanding the surface integral forces, we have ( ') σκ' n' δ x x ds σκ γ (8) S where the curvature κ is given by: γ κ =. γ (9) The curvature is the surface divergence of the normal to an interface surface element. Finally, the normal vectors are determined based on the non-dimensional gradient of the volume fraction γ. 2.1Turbulence Modeling LES strategy The LES equations are obtained by applying a filtering procedure to the instantaneous conservation equations for mass and momentum represented by equations (1) and (2). This yields a sub-grid scale tensor that cannot be resolved in the flow and requires to be modeled. The large scale filtered components take the form, ρu ρuu P u i i i i + = + ν τ ij (10) t t x x x i j j where the sub-grid stress tensor is expressed as, τ = uu uu (11) ij i i i i In the current study, we use a one equation eddy viscosity model of Yoshizawa et al. [18] which utilizes an additional transport equation for the sub-grid scale turbulent kinetic energy to obtain correct statistics of the energy containing scales of motion. Solving an extra sub-grid scale (SGS) kinetic energy equation allows a better estimation of the velocity scale of the SGS fluctuations and the prediction of backscatter [12]. The SGS kinetic energy equation can be expressed as k k sgs sgs ksgs 2 i ( ) u + u = 2 S i ν + ν k t + ν + δ t ij sgs ij ε sgs t x x x 3 x i i i j (12) Equation (12) is based on the exact k sgs equation which represents the influence of turbulent diffusion, production term, dissipation, and viscous diffusion on the process. As with the case of other sub-grid scale models, the eddy viscosity approaches zero as the grid is refined such that all the momentum scales are captured. The filtering technique
is based on Δ, the SGS length scale which is taken proportional to the nominal cell dimension. In the current formulation, Δ is given by { x y z} Δ= min Δ, Δ, Δ (13) 2.2 Interface Capturing Methodology In general, the volume fraction indicator equation should be able to preserve a sharp interface with negligible diffusion. Due to numerical inaccuracy and grid refinement, discontinuity prevails over the reconstructed interface. Volume of fluid is an interface compression method which has been enhanced by Normalized Variable Diagram (NVD) based bounded compression schemes [19]. In the present study, interface capturing has been performed by using CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes)[20], a fully conservative technique based on the finite volume method. While capturing the liquid-gas interface, this method, based on Normalized Variable Diagram (NVD), switches between different high resolution differencing schemes (ULTIMATE QUICKEST and Hyper-C) depending on the orientation of the interface to the flow direction, to yield a bounded scalar field and preserve both the smoothness and sharpness of the interface. The derivation of the scheme is based on the recognition that no diffusion of the interface can occur. No explicit reconstruction of the interface is needed. This scheme is particularly well applicable to sharp fluid interfaces. Further, the scheme does not need any operator splitting and is implicit with second-order temporal accuracy. The capability of the scheme in handling interface rupture and coalescence has been well tested [20]. equation using Finite Volume method were solved using Incomplete Cholesky Conjugate Gradient (ICCG) based iterative solvers while the asymmetric matrices were solved using Incomplete-Cholesky preconditioned biconjugate gradient methods [22]. 4. BOUNDARY CONDITIONS The combined LES/VOF methodology presented in Section 2 is used to investigate the disintegration of a turbulent liquid jet emanating from a co-axial atomizer. The setup, experimented by Mayer et al. [26] consists of a two-fluid coaxial atomizer injecting liquid ethanol into gaseous nitrogen with gaseous nitrogen as the co-flowing gas. They reported results with both co-flow and no-flowing nitrogen gas stream. For our current study, we choose the no co-flowing scenario for discussion. A schematic of the physical and computational domain is shown in Fig (1). (a) Physical domain 3. COMPUTATIONAL FRAMEWORK The simulations were performed within the framework of OpenFoam C++ libraries available for continuum mechanics [21]. A finite volume method for arbitrary cell-shapes in combination with a segregated approach is used to discretize the equations. Our modeling with LES requires high-order numerical methods to avoid any masking of the sub-grid stress tensor by the leading order truncation error. The filter size Δ is proportional to the grid size, in our case the minimum of the grid size along three dimensions, makes the modeled sub-grid stress tensor an second order term. Finite volume based computations are performed with non-overlapping cells in the domain [22]. The cell average over a face of a given cell is used to derive semi-discretized LES equations. These equations are iterated over time using a multi-step method. Convective fluxes are discretized using a blending scheme combining upwinding and central differencing scheme to improve accuracy and preserve boundedness [23]. The pressure velocity coupling is handled with a Pressure Implicit Splitting of Operators (PISO) [24] procedure based on a modified Rhie-Chow interpolation for cell centered data storage [25]. The equations are solved in a segregated fashion with iteration over the coupling terms with time marching using a second order explicit scheme. The time step is limited by the CFL number, which in the present case is restricted to 0.2 to ensure accuracy and to resolve the dynamics present in the large scale motion. Symmetric matrices arising from the discretization of the governing (b) Computational domain Figure 1: Physical and computational domains The inner diameter of the liquid (ethanol) injector is D l = 2.2 mm. The experiments had co-flowing Nitrogen gas inlet which has been replaced with a wall for no co-flow test cases. Wall and pressure outlet (atmosphere) conditions are being imposed surrounding the injection domain, as shown. The computational model is axi-symmetric with a one cell extension in the azimuthal direction supplemented with rotational periodic boundary conditions. Typically, we can associate this model with a 2D flow with added flow and diffusion in the azimuthal direction. With one cell extension, volumetric fields required in LES computations can be computed. The effect of gravity is neglected in the current simulations. The domain extends 10 mm in the radial direction and 20 mm along the axial flow direction which can help capture the primary breakup regime [2]. Gaseous nitrogen, used in the computation as the gas phase, is assumed to
behave as an ideal gas and hence its density change is correlated to the change in chamber pressure using ideal gas law. Based on the experiments [26], all fluid properties in the simulation domain were taken at 296 o K. For our simulation purposes, the surface tension of the multiphase system was kept constant at σ EN =0.02 N/m. Structured hexahedral meshes were used for simulation purposes in order to increase the accuracy of the simulation. A total of 250,000 cells were used in the simulation domain. The lowest mesh size of 10 μm was made available in the simulation domain in the vicinity of nozzle exit and 5 diameters downstream. Due to very fine mesh size, the time stepping requirement was reduced < 10-8 sec. Accuracy of the simulation was monitored using a Courant number = 0.2. 5. RESULTS AND DISCUSSIONS For the current study, two different testing conditions experimented by Mayer et al. [26] have been used. In the first case, liquid velocity of U l = 20 m/s injected into a chamber with a pressure of 6 MPa while in the second case the liquid velocity is reduced to 5 m/s exiting to same chamber conditions. 5.1 Relevant dimensionless numbers The main factors affecting the liquid breakup with/without co-flowing gas stream are the surface tension, aerodynamic and liquid forces. The first two forces are bonded together as dimensionless Weber number given by 2 ρiul Dl Wei =, i = l, g (14) σ Exit injector diameter D l is used as the characteristic length for deriving the Weber number. The Ohnesorge number (Oh) gauges the effect of the liquid viscosity in opposing atomization and is defined as Wel μl ρlud l l Oh = =, Rel = (15) Re ρσd μ l l l Where, Re l is the liquid jet Reynolds number. Classification of liquid jet breakup modes based on Oh vs Re l map is shown in Fig. (2). In our study, Oh has a constant value = 6.2х10-3. Cases (1) and (2) are shown in the map. l 1 2 corresponding Weber number, We g = 2640 and liquid Reynolds number Re l = 30000 clearly lie in the atomization domain as shown in Fig. (2). The growth and disintegration characteristics of the turbulent liquid jet in still medium are presented as a contour plot of the liquid volume fraction in Fig. (3). (a) (b) (c) (d) Figure 3: Transient evolution of round turbulent ethanol jet into stagnant nitrogen gas: U l = 20 m/s, Re l = 30000, We g = 2640. (a)-(c) simulations, (d) experiment [26]. The scale of surface distortions increased progressively with increase in distance from the jet exit. The liquid-gas interface exhibits fine-grained structure near the nozzle exit (2D-4D where D is the liquid inlet diameter) while, larger-scale irregularities are seen as the distance from the nozzle exit increased (6D-8D) [27]. This behavior can be attributed to the fact that small scale disturbances complete their growth more rapidly than large disturbances and hence they appear first. Ligament-eddy interactions during the evolution of liquid jet are shown in Fig. (4). Figure 2: Disintegration modes using Oh Vs Re l map. 5.2 U l = 20 m/s In the first case reported, liquid ethanol is injected with a velocity U l = 20 m/s into still nitrogen gas medium. The Figure 4: Interaction of shear vortices with ligament tip leading to fragmentation Figure (5) shows the temporal growth of the surface
wave amplitude resulting in elongation of the protruding ligaments. Thinning of the cross wise extending ligament occurs due to drag forces which, pulls the ligament away from the jet surface to which it is attached, while the surface tension force act to stabilize this surface instability [28]. The thinning process continues as long as the ligament is connected to the liquid surface. When the aerodynamic forces overcome their surface tension counterpart, droplet formation occurs [29]. The stretching of the ejected ligament and surface deformation rate is a strong function of the local fluctuating velocity components. observed. Figure (7) shows different eddy locations, E1, E2, & E3, and their transport phenomena. Eddy E3 is captured very close to the interfacial region. The protrusions from the liquid surface extending into the gas domain introduces vorticity components and are shed off due to pull back mechanism of the surface tension. The recirculation regions existing in the close vicinity of disintegrated filaments contribute to stretching and later coalescence. The strength of these eddies is influential in determining the breakup modes and stretching. The continued influence of the vortex propagation can be seen in E1, E2 type eddies. Figure 5: Temporal characteristics: stretching and destabilization of a ligament from the bulk flow. From our numerical simulations, we conclude two major types of breakup mechanisms similar to the observation of Sallam et al. [30]: ligament tip breakup (related to formation of ligament type structure and their breakup) and ligament base breakup. Conforming to the current simulation parameters, the occurrence of ligament tip breakup was profound and can be viewed to be the most dominant disintegration mechanism during turbulent primary liquid jet breakup. In this mode, the breakup occurred at tip of the ligament, which was pulled away from the liquid surface due to the interfacial instability enhanced by radial fluctuating velocity components in the liquid [29]. The second mechanism of drop formation (individual liquid fragments), the ligament base breakup mode, was observed less frequently and involved breakup of the thick ligaments (discs at the liquid base) due to turbulent cross stream liquid velocity fluctuations that change the flow direction at the base of the ligament before Rayleigh type breakup can occur at the ligament tip [29]. The orientation of the ligaments with respect to the liquid bulk flow gives an indication of the nature of the breakup process. Figure (6) shows visualization of different breakup modes. Figure 7: Capturing eddies in the vicinity of liquid-gas interface reveals highly complicated interaction phenomena between the disintegrated ligaments and the local flow field. The perturbations in the form of growing surface waves lead to realignment of the liquid-gas interface in a non-linear fashion. The change in curvature of the interface demands matching conditions for stress distribution between the liquid and gas phases [2, 27]. This stress continuity condition shows up in the modified flow field near the interface. The intensity of flow field variation is subjected to the magnitude of perturbation that the interface suffers. As the jet surface expands and contracts, the local flow field is distracted from its quiescent mode. Figure (8) shows velocity relaxation phenomena 5D downstream. Disc base breakup Disc tip (ligament) breakup Very Figure 6: Disc (ligament) tip breakup and disc base breakup Low inclination from the liquid surface resulted in ligament tip breakup while, those with higher angles of inclination with respect to jet surface were associated with base breakup mode. Different types of vortices interaction with liquid fragments in the vicinity of the liquid-gas interface were Figure 8: Plot of velocity vectors at different sections (1 8) along the liquid jet axis. Shown in the background is the contour plot of liquid volume fraction. Note the velocity profile relaxation mechanism and localized recirculating flows prevalent in the vicinity of the liquid-gas interface.
5.3 U l = 5 m/s The Weber number We g = 165 and Re l = 7500 lie in the atomization regime as observed from Fig. (2). The simulation indicated presence of long-wavelength waves which correlates well with the increase in perturbation wavelength with decrease in liquid jet velocity [2]. The evolution of liquid jet ethanol in the domain is shown in Fig. (9). and larger blobs. This explains the primary reason behind the formation of bigger droplet sizes. The interaction of vortices with fragments is shown in Fig. (11). (a) (b) Figure 10: Spatial and temporal break up events leading to a variety of fragment sizes (c) Figure 11: Visualization of unsteady flow structures: Traveling eddies and their interaction with ligaments. Note that the strength of the wake behind a disintegrated liquid fragment is a strong function of the fragment size. (d) Figure 9: Transient evolution of a liquid ethanol issuing into stagnant nitrogen: U l = 5 m/s, Re l = 7500, We g = 165. (a) (c) simulations, (d) experimental results [26]. Following the overall disintegration structure, we analyze the spatial and temporal evolution of unstable waves and the resulting fragmentation process. The ligament length extending from the liquid bulk was reduced noticeably indicating the decreased turbulent fluctuating components inside liquid core. In addition, due to reduced relative velocity, U r = 5 m/s, between the two phases, thinning and stretching of the ligaments was less intensive. Ligaments ejected from the main stream formed large fragments which did not result in secondary breakup unlike our previous case. Figure (10) represents sequences of spatial and temporal fragment stretching, breakup events. Observe from Fig. (10), different size blobs emanating from the detached ligament coalesce thereby forming larger In order to characterize the velocity relaxation process occurring spatially along the jet, we plot the velocity profile along sections of liquid jet, 5D downstream. The relaxation of the velocity from the liquid stream to the free stream condition is similar to our previous case (section (5.2)). Figure 12: Plot of velocity vectors superimposed on contours of volume fraction along different sections of the liquid jet. Low velocity of the liquid jet core results in larger velocity relaxation periods.
However, the relaxation time is less rapid to the reduced velocity difference between the stagnant free stream and low velocity jet. Further, the long wavelength waves dominating the growth of unstable surface undulations in the low speed jet are associated with low surface relaxation time scales as seen from Fig. (12). The ligament dynamics arising from two different injection cases show unique features. A Schematic of variables used in the ligament analysis is shown in Fig. (13). Ligament orientation extended as far as 60 o for the former while it was 70 o for the low speed case. Figure 13: Schematic of Ligament variables Weber numbers, based on ligament length and cross stream velocity components, were constructed for both cases at different ligament locations and plotted against the Ligament length to thickness ratio in Fig. (14). Figure 14: Plot of Ligament slenderness ratio to Ligament Weber number Figure (14) shows increased L lig /D lig ratio of the ligaments with increased injection velocity. The plot agrees with the theories that increase in injection velocity leads to increased turbulent fluctuations which then enforce radial expulsion of liquid fragments [29]. Now, with increase relative velocity with respect to gas phase in the first case, higher thinning rates result in smaller ligament thickness thereby increasing the L lig /D lig ratio for a given ligament size. Similarly, the orientation of ligaments in the domain can be observed from Fig. (15). A sample size of 75 ligaments was used for constructing the probability density function. Note that for the first case with mean liquid velocity = 20 m/s into a still chamber, the most frequent distribution of ligaments were obtained at 25 o while the curve shifted to 32 o in the case of decreasing the liquid jet speed to 5 m/s. Figure 15: Probability density function plot of ligament orientation distribution in the domain for different cases studied in this article. 6. CONCLUSIONS Numerical simulations with a combined Large Eddy Simulation and Volume of Fluid (VOF) approach has been used in studying the disintegration of round turbulent liquid jets. Our technique has been able to effectively capture the non-linear interaction between the liquid and gas phases. Formation of surface waves leading to stripping of liquid fragments from the core flow has been captured well. The shear between the liquid and the gas interface due to the relative movement of the phases resulted in very small scale eddies with a length scale lower than the disc (ligament) thickness has been comfortably identified using the current model. Different breakup modes, ligament tip breakup and ligament base breakup, arising due to the non-linear interaction between the gas stream and the liquid bulk have been recognized. Interaction of the ligament with shear vortices leading to a wide range of droplet sizes gives clear indication of the stochastic nature of atomization. The ligament characteristics, thickness and protrusion length, are strictly modified when the relative velocity and liquid turbulence is altered. The current study demonstrates the capability of LES formalism in simulating multiphase flows arising in atomization phenomena. At this juncture, grid convergence studies have not been performed. LES formalism for multiphase flows has not been fully tested and may have unknown effects on the computations. Full 3-D simulations are underway to obtain more accurate flow statistics present in the turbulent multiphase regime. 7. NOMENCLATURE u velocity [m/s] U average velocity [m/s] P Pressure [N/m 2 ] t time [s] Δx,Δy,Δz cell thickness [m] F force [N] n normal vector
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