Cavitation as Rapid Flash Boiling

ILASS-Americas 23rd Annual Conference on Liquid Atomization and Spray Systems, Ventura, CA, May 2011 Cavitation as Rapid Flash Boiling Bradley Shields, Kshitij Neroorkar, and David P. Schmidt Department of Mechanical and Industrial Engineering University of Massachusetts Amherst, Amherst, MA, 01003 Abstract Diesel injector nozzles often experience cavitation due to regions of extremely low pressure. There are computational models that deal only with high temperature flash-boiling flow [1, 2, 3], as well as those that focus on the lower-temperature process of cavitation[4, 5]. The ideal code would have the ability to represent both high-temperature flash-boiling flows and lower temperature cavitating flows. The current work uses the hypothesis that cavitation can be modeled as flash-boiling with rapid heat transfer between the liquid and vapor phases. The following paper examines a multi-dimensional computational fluid dynamics approach based on using an established flash-boiling model [6] to simulate cavitation in a fluid near room temperature. Coefficient of discharge is plotted against cavitation number, and the results are compared to the results of published cavitation code as well as experimental data. The flash-boiling model shows good agreement with accepted values for discharge coefficient. The flash-boiling model is proposed as a tool to simulate both flash-boiling and cavitation, and its accuracy is examined in non-cavitating cases. Nomenclature A 2 Nozzle outlet area C c Contraction coefficient C d Coefficient of discharge C p Specific heat h fg Latent heat of vaporization Ja Jakob number K Non-dimensional pressure ratio and cavitation parameter L/D Length/diameter ratio ṁ Mass flow rate P c critical pressure P v Vapor pressure P 1 Upstream pressure P 2 Downstream pressure P sat Saturation pressure Re Reynolds number r i /D Inlet rounding T Superheat U Flow velocity x Instantaneous quality x Equilibrium quality α Void fraction of vapor θ Quality relaxation time scale ρ Density ρ 1 Inlet Density ρ l Liquid density ρ v Vapor density τ Shear stress tensor φ Mass flux ψ Non-dimensional pressure ratio Corresponding Author: schmidt@ecs.umass.edu

Introduction The fuel injector is an extremely important component of the diesel engine, making the study of internal flow effects essential to lowering emissions and to improving the understanding of the atomization that occurs with a given injector. Most importantly, the spray community needs to know how the injector nozzle impacts the downstream spray. The injector characteristics represent the most significant parameters for adjusting spray behavior. Cavitation is the process by which liquid is converted to vapor by the low pressures within the nozzle. With a sharp-edged orifice, as fluid rushes into the nozzle, the flow often separates and contracts within an annulus of vapor, known as the vena contracta [7]. By conservation of momentum, as the flow contracts and speeds up to enter the nozzle, pressure falls. Should this pressure drop below the vapor pressure of the fluid, liquid will convert to vapor in the flow. This vapor can be manifested as individual bubbles in the flow or a foamy mixture of gas and liquid. Early modeling included the zero-dimensional model developed by Nurick [8] in 1976. Nurick developed a relation for a nozzle s coefficient of discharge, ( ) 1/2 P1 P V C D = C C (1) P 1 P 2 where P 1 is the upstream pressure, P 2 is the downstream or back pressure, and P v is the liquid s vapor pressure. This expression relates the pressure ratio and C C to the nozzle output. The variable C C represents the fraction of the nozzle cross-sectional area that liquid passes through, the fraction of the nozzle that is not taken up by vapor. The current study addresses slightly rounded nozzles, with the assumption that the area occupied by the vena contracta is constant in a given nozzle geometry. By varying the pressure ratio and nozzle geometry, Nurick observed hydraulic flip and cavitation. Nurick s results showed that by varying the pressure ratio, thereby including or excluding cavitation, the discharge was directly affected. Schmidt and Corradini [7] compiled the work of several experimentalists, as shown in Figure 1. The experimental data validated the accuracy of the model. However, some of the experimental data show a coefficient of discharge that increases linearly from K=1 to K=2 but then falls as K increases. The pressure ratio K is a form of cavitation parameter which appears in Eqn. 1, and is defined as: ( ) P1 P V K = (2) P 1 P 2 Figure 1. Nurick Theory vs. Experiments. Data are plotted on log-log axes from [8, 9, 10, 11, 12, 13, 14, 15] As the value of K exceeds a threshold of about 1.7 to 1.9, the cavitating flow transitions to noncavitating flow, and Schmidt notes that the variability of the data could indicate other effects, such as Reynolds number dependency. These other effects may include the scale of Hiroyasu s [10] nozzles (indicating scale-dependant factors) or manufacturing imperfections. Som [16] breaks modern cavitation modeling into two major groups: single fluid/continuum models, and two fluid models. Single fluid/continuum models utilize the vapor volume fraction to average vapor and liquid phase properties of a fluid. Schmidt et al [17] is an example of the pseudo-fluid approach, assuming liquid and vapor to be in a thermal equilibrium, evenly mixed in each cell, and with no-slip conditions between the phases. The separate phases and mixture were treated as compressible. The model s two-phase sound speed was approximated using Wallis HEM closure [18]. Two-phase approaches are those that handle each phase with its own set of conservation equations. Som [16] breaks these down further into two categories, Eulerian-Eulerian approaches and Eulerian-Lagrangian approaches. Eulerian-Eulerian models, such as that proposed by Singhal et al [4], are similar to single fluid models in that they have fluid density as a function of vapor mass fraction. The vapor mass fraction is found through a transport equation that includes mass and momentum conservation equations. Source terms define vapor generation/condensation rates, and stem from flow parameters and fluid properties. A generalized Rayleigh-Plesset equation is used to derive the bub- 2

ble dynamics equation. Eulerian-Lagrangian models instead view the liquid flow in a Eulerian sense, but follow the bubbles of the vapor phase as Lagrangian particles, as in Gavaises and Arcoumanis [19]. The bubbles are handled with the nonlinear Rayleigh-Plesset equation and offer predictions of bubble dynamics. Usually assumptions or empirical estimates of bubble number density are required. The vaporization process that occurs through cavitation is very similar to that of flash-boiling, with a few important caveats. Where cavitation represents the vapor formed through a constant temperature system experiencing a drop in pressure, flash boiling represents the same system with a lower pressure drop but elevated temperatures. In flash boiling, the availability of energy required for phase change is limited by the speed of interphase heat transfer while cavitation is often inertially dominated [17, 20]. This can be shown by the Jakob number, Ja = ρ lc p T ρ v h fg (3) where ρ represents density, and C p represents specific heat at constant pressure, T is the amount of superheat, and h fg is the latent heat of vaporization. As Ja 1, more energy per unit volume is required for vaporization than is available in the form of sensible heat. As temperature increases, so does the ρ v h fg term. Therefore, at elevated temperatures the energy required for vaporization increases, increasing the time required for heat transfer between phases, even approaching flow transit time through the nozzle. In contrast, in cavitating flows, the time required for vaporization is very small, ensuring that vaporization is essentially instantaneous. Another important consideration is that phase change is a continuous process in flash-boiling nozzles. Every fluid molecule will experience a local pressure that is less than the vapor pressure prior to exiting the nozzle. In cavitating flow, it is possible for an annular vapor region to form, after which liquid need not further change phase. Consequently neglecting the temporal nature of the heat transfer process is erroneous when P v > P 2, and requires deliberate consideration. Schmidt [21] investigated the accuracy of using a cavitation model to simulate Reitz s [22] flash-boiling experiment using two sets of assumptions, thermal equilibrium and thermal non-equilibrium. As temperature increased, the equilibrium model s results became erroneous, while the non-equilibrium model remained accurate. The present work attempts to confirm the accuracy of the flash boiling code s ability to quantify cavitating flows. The flash boiling CFD code described by Schmidt et al. [6] is employed under cavitating conditions and evaluated for its accuracy in predicting coefficient of discharge. This cavitating regime is outside of the applicability for which the model was designed. To test the accuracy of the non-equilibrium model, simulation results were compared to accepted cavitating and non-cavitating coefficient of discharge data. Methodology A parametric study was conducted, using a 2D axisymmetric nozzle 3 mm in diameter with an L/D ratio of 4. The inlet rounding of the nozzle, r i /D, was 1/40. The nozzle was examined at upstream pressures ranging from 6 to 200 MPa, with a constant backpressure of 5 MPa. The working fluid was water at an average temperature of 18 C, to ensure a low enough temperature for instantaneous heat transfer between the fluid s liquid and gas phases. The properties of the working fluid were provided by the REFPROP database and code library. REF- PROP uses the Wagner and Pruss equation of state for water. The HRMFoam model was used to predict mass flow rate, and thus discharge coefficient, for the axisymmetric flow. Derived from Bernoulli s Equation evaluated at the nozzle inlet and outlet, the ideal nozzle mass flowrate is given by ṁ ideal = A 2 2ρ1 (P 1 P 2 ) (4) The symbol A 2 is the nozzle outlet area, ρ 1 is the inlet density, and P 1 and P 2 are upstream and downstream pressure, respectively. The ratio of the numerically computed flowrate to ideal output flow is the coefficient of discharge. The CFD code, HRMFoam, was created using a pseudo-fluid paradigm. The full description is given in Schmidt et al. [6] and is only summarized here. The governing equations considered in this case include the continuity and momentum equations denoted by Eqns. 5 and 6 respectively. ρ + φ = 0 (5) t ρu + (φu) = p + τ (6) t The term φ represents the mass flux and is given as φ = ρu (7) The variable τ represents the shear stress tensor. At present, no turbulence model has been incorporated and laminar flow is assumed. The reason for 3

this assumption is that our main focus here is understanding the effect of flash boiling/cavitation on the flow. The study of those phenomena coupled with turbulence is left for future work. The pressure equation is ρ ( H(U ) 1 ρ Dx ) ρ p + p,h =0 ap ap x Dt (8) where the subscript p represents the computational cell under consideration. The variable a represents the contributions from the specific cells. The operator H is defined as H(U ) = r X an U N Figure 2. Computational Grid (9) The most important consideration while using the HRM model is the formulation of the time scale. Downar-Zapolski [1] used the pressure profile and mass flux from the Moby Dick experiments of Reocreux [24] and combined their governing equations to derive an equation for the time scale. They found that in all cases, the time scale was a monotonically decreasing function of the void fraction and a non dimensional pressure. Based on their observations, they proposed the following equation for the time scale θ N where r is the contribution from the source terms to the linear system matrix, and N represents the neighboring cells. H is a convenient replacement for the off-diagonal and source term contributions. The flash boiling model is used to calculate the last term of Eqn. 8. The homogeneous relaxation model was used to provide closure to the above mentioned system of equations. This model describes the rate at which the instantaneous quality, the mass fraction of vapor in a two-phase mixture, will tend towards its equilibrium value. The simple linearized form proposed by Bilicki and Kestin [23] for this rate equation is shown in Eqn 10 x x Dx = Dt θ θ = θ0 α 0.54 ψ 1.76 The value of the coefficient is θ0 = 3.84 10 7 [s] and ψ= (10) αρv ρ (11) where α is the void fraction of vapor, and ρv represents saturated vapor density. The void fraction is calculated as follows α= ρl ρ ρl ρv Psat P Pc Psat (14) where Pc is the critical pressure. This empirical equation (Eqn. 13) is being used beyond the range and fluids for which it was formulated. Though there is no guarantee that the model will be accurate under such conditions, previous studies have produced very encouraging results [17, 25, 26, 27, 6]. Additionally, the same time scale correlation was used for all the fluids considered. The flow was represented by a coarse mesh of 12000 cells, with greater cell density near the inlet corner and along the wall above the inlet (Figure 2). The mesh is made up of mainly quadrilateral prisms. The cases were re-run with a 500% finer mesh as well, yielding a mean change of 4.51% in coefficient of discharge. The coarse mesh requires further refinement to be verified as convergent. In the above equation, x represents the instantaneous quality, x represents the equilibrium quality and θ represents the time scale over which x relaxes to x. Eqn. 10 is an approximation to the extremely complicated processes that are associated with the flash boiling process. It can be noted that the HRM equation is inserted into the last term of the pressure equation formulation, Eqn. 8. The value of x is obtained from a look-up table as a function of pressure and enthalpy. The instantaneous quality is calculated from the void fraction as shown below x= (13) Results The findings of this parametric study were compared to the results of several experimentalists as well as Singhal s Full Cavitation Model (Figure 3). (12) 4

Figure 3. HRMFoam vs. experimental and computational results. Data plotted on log-log axes from [14, 8, 15, 4] The accepted onset of cavitation is K = 1.7 [7]; larger values of K indicate non-cavitating flow, while smaller values indicate cavitating flow. HRMFoam models the flow accurately, with coefficients of discharge differing from the Nurick trend by an average of 0.022, and good agreement with the Full Cavitation model. Part of this difference from Nurick s theory is the challenge of discretizing the region near the sharp inlet corner. Whereas a more rounded corner allows for velocity and pressure to transition smoothly (if rapidly) to their downstream values, a sharp inlet does not. It represents a discontinuity where velocity changes instantaneously. In non-cavitating cases, the flow clearly exhibits non-trivial transient vapor formation due to periodic vortex shedding (Figure 4). As this region represents essentially single phase flow, HRMFoam should not indicate significant generation of vapor. As time goes on, vortices form just inside the nozzle inlet (Figure 4, inset), growing in number as time continues. Periodically, one of these vortices is pinched off by the flow and separates, moving down the nozzle. The shedded vortex is carried out through the nozzle exit. The velocity changes the vortices intro- Figure 4. Vortex streamlines highlighted against nozzle flow [m/s] Inset: vena contracta close-up, immediately inside nozzle inlet 5

Figure 5. Nozzle side view of nozzle throat, centerline to outer radius. Top: Void fraction Bottom: Pressure [Pa] duce to the flow and the vapor they generate are possible sources of error. This phenomenon has also been reported by Canino and Heister [28]. Canino and Heister noted a reduction in C D as the radius of a nozzle s inlet corner approached perfect sharpness. Figures 4, 5, and 6 show the (non-cavitating) K = 2.4 case. Figure 5 is an image of the nozzle mirrored over the horizontal axis, showing alpha and pressure on the top and bottom, respectively. The low pressure regions are vortex centers, which drop well below the vapor pressure of the working fluid. The regions of low pressure correspond to the regions of vapor formation, at the eye of each vortex. Figure 6 shows the high velocity gradient and magnitude at the entrance to the nozzle. The red region at the inlet corner is a product of the large gradient there. Conclusion A parametric study was conducted, testing the ability of the Homogenous Relaxation Model to accurately depict cavitating flow conditions. It remains to be seen if vortex shedding and the error it causes is a physical reality or error in numerical approximation. However, HRMFoam shows reasonable accuracy when dealing with either cavitating or flash boiling flows. The results presented here are slightly mesh dependent, and further mesh convergence study is left to future work. Acknowledgments We thank General Motors Research Center for supporting this research. References [1] P. Downar-Zapolski, Z. Bilicki, L. Bolle, and F. Franco. 3rd ASME/JSME Joint Fluids Engineering Conference, 208(616), 1999. [2] V. N. Blinkov, O. C. Jones, and B. I. Nigmatulin. International Journal of Multiphase Flow, 19:965 986, 1993. [3] H.J. Richter. International Journal of Multiphase Flow, 9(5):511 530, 1983. [4] A.K. Singhal, M.M. Athavale, H. Li, and Y. Jiang. Journal of Fluids Engineering, 124:617, 2002. [5] A Kubota, H. Kato, and H Yamaguchi. Journal of Fluid Mechanics, 240:59 96, 1992. [6] D.P. Schmidt, S. Gopalakrishnan, and H. Jasak. Intl. J. of Multiphase Flow, 36:284 292, 2010. [7] D P Schmidt and M L Corradini. Int J Engine Research, 2(1):1 2, January 2001. [8] WH Nurick. ASME Transactions Journal of Fluids Engineering, 98:681 687, 1976. [9] A.L. Knox-Kelecy and P.V. Farrell. International Fuels & Lubricants Meeting & Exposition, San Francisco, CA. SAE International, 1992. [10] H. Hiroyasu, M. Arai, and M. Shimizu. ICLASS-91 Gaithersburg, MD, pp. 275 282, 1991. [11] R.D. Reitz. PhD thesis, Princeton Univ., NJ, 1978. Figure 6. Flow velocity at inlet [m/s] [12] TR Ohrn, D.W. Senser, and A.H. Lefebvre. Atomization and Sprays, 1(3), 1991. 6

[13] W. Bergwerk. ARCHIVE: Proceedings of the Institution of Mechanical Engineers 1847-1982 (vols 1-196), 173(1959):655 660, 1959. [28] J. Canino and S.D. Heister. Atomization and Sprays, 19(1):91 102, 2009. [14] AG Gelalles and NACA. Langley Research Center. Coefficients of Discharge of Fuel Injection Nozzles for Compression-Ignition Engines. National Advisory Committee for Aeronautics, 1932. [15] C. Soteriou and R.J. Andrews. Direct injection diesel sprays and the effect of cavitation and hydraulic flip on atomization. Technical report, Society of Automotive Engineers, 400 Commonwealth Dr, Warrendale, PA, 15096, USA,, 1995. [16] S. Som, SK Aggarwal, EM El-Hannouny, and DE Longman. Journal of Engineering for Gas Turbines and Power, 132, 2010. [17] D.P. Schmidt, C.J. Rutland, ML Corradini, P. Roosen, and O. Genge. SAE transactions, 108(3):613 629, 1999. [18] G.B. Wallis. One-dimensional two-phase flow, volume 409. McGraw-Hill New York, 1969. [19] M. Gavaises and C. Arcoumanis. International Journal of Engine Research, 2(2):95 117, 2001. [20] D.P. Schmidt, S. Rakshit, and K. Neroorkar. 11th Triennial International Conference on Liquid Atomization and Spray Systems, 2009. [21] D. P. Schmidt. PhD thesis, The University of Wisconsin-Madison, 1997. [22] R.D. Reitz. Aerosol Science & Technology, 12(3):561 569, 1990. [23] Z. Bilicki and J. Kestin. Proceedings of the Royal Society of London: Series A, 428:379 397, 1990. [24] M. Reocreux. PhD thesis, Universite Scientifique et Medicale de Grenoble, France, 1974. [25] S. Gopalakrishnan and D.P. Schmidt. SAE Paper-2008-01-0141, 2008. [26] J. Lee, R. Madabhushi, C. Fotache, S. Gopalakrishnan, and D. Schmidt. Proceedings of the Combustion Institute, 32(2):3215 3222, 2009. [27] K. Neroorkar, S. Gopalakrishnan, D. Schmidt, and R. O. Grover Jr. 11th Triennial International Conference on Liquid Atomization and Spray Systems, 2009. Vail, Colorado USA, July 2009. 7