THE UNIVERSITY OF MICHIGAN College of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. UMICH 03'371-16-T LIQUID DROPLET IMPINGEMENT STUDIES AT UNIVERSITY OF MICHIGAN by F. Go HammWtt YC. Huang (Heat and Fluid Flow in Steam and Gas Turbine Plant Symposium) (University- of Warwick;: April 1973) Financial Support Provided by: National Science Foundation Grant No. GK-730 May 1972 * Professor-in-Charge, Cavitation and Multiphase Flow Laboratory, Mech. Engr. Dept., Univerisdy of Michigan, Ann Arbor, Michigan Engineer, Gibbs and Hill, N,-Y., N. Y., (formerly doctoral candidate, Mech. Engro Dept., University of Michigan, Ann Arbor, Michigan)

LIST OF FIGURES Figure 1 Shape-time-history of an initially cylindrical droplet with L/D = 1 at impact Mach number of 0.2 and under free-slip boundary condition 2 Shape-time history of an initially spherical droplet at Mach number =0.2, for free-slip boundary condition 3 Shape-time history of an initially cylindrical-spherical composite droplet with R/R= 0. 25 and L/D=1, at Mach number= 0. 2, for free-slip boundary condition 4 Pressure-time history at liquid-solid interface (z=fl) of an initially cylindrical droplet with L/D=1 for impact Mach number of 0. 2 and under free-slip boundary condition 5 Pressure-time history at liquid-solid interface (z=0) of an initially spherical droplet for impact Mach number of 0. 2 and for free-slip boundary condition 6 Pressure-time history at liquid-solid interface (z=0) of an initially cylindrical-spherical composite droplet with R/R=0. 25 and L/D:-l for impact Mach number of 0. 2 and for free-slip boundary condition 7 Isobar distribution in an initially cylindrical-spherical composite droplet with R/R =0. 25 and L/D=1, at time (Ct/D) = 1. 5 for impact Mach number of 0. 2 and for free-slip boundary condition 8 Photographs of the cavitation for a miter droplet following an impact on a solid plane (Brunton and Camus TABLES I Summary of Principal Numerical Results

F. G. Hammitt, PhD and Y. C. Hluang, PhD Cavitation and Multiphase Flow Laboratory, Dept. of Mech Engr., University of Michigan, Ann Arbor, (Y. C. Huang presently with Gibbs and Hill, N. Y., N. Y.) Michigan Liquid DropletImpingement Studies at University of Michigan SYNOPSIS Work at the University of Michigan relating to the turbine droplet impingement program is described. While this includes the development of a low-pressure wet-steam tunnel for studies of droplet formation and also experiments upon material resistance and droplet impingement, it primarily concerns numerical studies of droplet impingement upon target surfaces. Results include pressures and velocities within the drop and upon target surfaces. Pressures are compared with a "slab' collision model ("corrected water hammer pressure"' and with "simple water hammer", not involving correction for actual shock wave velocity in the compressed liquid. Maximum pressures for three-dimensional drops are well less than the "slab" model, but are essentially independent of impact velocity if normalized to this pressure. They exceed "simple water hammer pressure" only for velocities of the order of 750 m/s and larger. INTRCDUCTION particularly related to problems of moisture film 1. Liquid droplet impingement erosion is a build-up and droplet separation. serious and limiting phenomenon today in various important technological fields such as large steam 4. The present paper discussed numerical (or other vapor) turbines, high-speed fixed-wing studies of pressures, velocities, stresses, and aircraft and helicopter rotors, to name three of the deformations occurring during droplet impact. most proninent. Also, cavitation applications, This work has been primarily based on the doctoral such as high-speed ships, should be included be- dissertation of one of the authors (6, 7). It has incause of the well-recognized similarities between cluded water Mach numbers up to 0. 5 (.^v750 m/s), these different forms of erosive attack. spherical and cylindrical droplets, as well as a combined shape vhich is perhaps more realistic of 2. In all cases the overall problem involves real impacts. For given fluid physical parameters several phases, e. g., and for a completely rigid surface (which was asa' Dynamics'of liquid phase (and droplets) during sumed in our initial study) we have shown that only their nucleation and traverse to the eroded region. the parameters of liquid Mach number and droplet Droplet size and shape, and impact velocity and shape are important. Cur study is continuing to angle, are thus determined, include the effects of surface deformation (8) and b) Interaction between droplet and eroded material finite material elasticity. These latter are particduring single impact. ularly important for elastomeric coating materials, c) Material failure after one of more impacts, thus not normally involved for steam turbine blading, creating erosion. but important in numerous other applications. Our laboratory is actively researching all of these three phases. However, of the three, we have in 5. The paper will present a summary of the the past concerned ourselves primarily with b), in- results from this continuing study obtained up to the teraction between droplet and material during single time of writing. impact, and c) material failure after one ormore impacts. The present paper will discuss the re- TURBINE-RELATED DROPLET IMPACT WOR A TURBINE-RELATED DROPLET IMPACT WORE, A2' suits, so far obtained and in progress, particularly UNIVERSITY OF MICHIGAN the analytical work which has been done under subject b'. A.. Numerical Studies of Droplet Impact 1. Solution Method 3. Our total program has included experiment-1. Since our work has been predonntly conal studies of material resistance and impact using numerical studies o cerned with numerical studies of impact, these will a "water gun" device capable of projecting liquid decried and ummaried be described and summarized first. To our knowdroplets (i.e., elongated slugs) at velocities up to rereses first relat y co600 isnaeetnasf /ledge,.his work represents the first relatively com600 m/s and at repetition rates of'30/min., as well as numerous cavitation damage studies. Theseve treatment of the Impact probl velocity range pertinent to the steam turbine prob.. are covered elsewhere (1-5\. In addition, we have lem and also to aircraft raindrop erosion, i.e., recently commenced the development of a low pres- a to ma. In this velocity range it sure steam tunnel for the study of wet steam flows, h possible to neglect ether fluid compressibilit' is not possible to neglect either fluid compressibility

F. G. Hammitt and Y. C. Huang (as might be possible in a very low velocity case) ord() (11) target material strength (as is sometimes done for dt 2 dz hypersonic impact analyses). As is shown by our analysis, for a fully rigid flat surface and forgiven Z ) dt (12) liquid properties, the only important parameters R dt (13) of the collision in the droplet size (perhaps 1-3mm m 2 v dia.) and velocity range considered are liquid im9. Characteristic density c is that of the unpact Mach number, droplet shape, and impact angle. n 2t t of the unpact disturbed liquid at 1 atm. and Hence the investigation has included the liquid Mach isti veli ty i the impact velit. Characternumbers of 0. 2 and 0. 5 (^300 and — 750 m/s for istic velocity is the impact velocity. Charactercold water). Three droplet shapes have been in- ti prsure is the "simple" water hammer prescluded, i.e., spherical, cylindrical, and a com- z and are the mathematical cell dimensions in the axial and radial directions, posite spherical-cylindrical droplet (cylindrical esetie o oneniene these droplet with rounded corners), which it is felt may o. aracterisic time these are chose be the most realistic of the three shapes for most C is the (corrected) shock wave veocitydefined by C is the (corrected) shock wave velocity defined by ~~~~impact cases. ~Eq. (10) below. We developed this semi-empirical 7. The analysis is described in detail else- second-power relation to give shock wave velocities 7. The analysis is described in detail elsecorrected for local density change for the purpose where (6, 7), but for convenience a brief summary corft ted for local density change for the purpose will be made here. For most of the numerical of this aalis 10) providing improved accuracy over relations already available (11, e. g. ). studies so far made, a completely rigid flat target surface was assumed. While compressibility of 10. The initial conditions over the domain of the liquid must be included to obtain realistic re- calculation are p = p u = u and v = v. u and sults in the velocity range of interest, surface v are the initial impact velocities in theaxiaCl and tension and viscosity can be safely ignored compar- radial directions respectively, so that v = 0 and ed to pressure and inertia effects for the droplet u = V for perpendicular impact, the onAly case size range of interest. Under these conditions the wAich we have so far studied. The appropriate equations of continuity and momentum for the liquid boundary conditions are then: drop become; i) along the axis of symmetry (z), r = 0 and 2 b (qu 1 6(rov).symmetry requires b4 + - = 0 (1) bt bz r r v=0, =0, ~ =0 or br b (u b (ou - -1 b (rpvu\ bo bv 4t + - r -- (21 ii) along the impacted rigid surfaces, z =0, - =0 bt bz r'Br bz', ab (oQv3 (~oVU 1 ab (rv b u = 0 =0, for the full-slip wall condition bt + rz r - += -br (3) here use. We have reported already similar results for a non-slip wall boundary condiand a suitable equation of state for water is that of tion. Tait (9): iii) along the free surface, the first order terms of the continuity equation yields p+ B (v)A (4) b u -v p0 + B - n _ 0 Po + B { o p = Po = — x bxt Values of the constants B and A were chosen to be I n t appropriate for 20 C water, i.e., B = 3047 bars where u and v are the moving velocity comand A.= 7.15. ponents of the iiquid-air interface in the normal 8. Marker particles are imbedded in the de- x and tangential x directions of the surface forming liquid boundary to give its location at all respectively times. The marker particles follow the equation iv) along the sides of the finite computational doof motion for a free body as described by kinematic main, permeable boundary conditions are inrelations in a Lagrangian formulation (6, 7). The posed, in such a way that the normal space derequations are then normalized by the characteristic ivative of the variable vanishes at the boundary, parameters of the problem to give the following u equation set. = 0 0, = atz=H set li. A -L z0 (5Z' z - z bt 1 z+ 2 r - r bu= 0, = 0 atr = H b 2 fib~u r Br e ar cota2 )+ ( + A 1 ( r vu) 1 B (6) t 1 z 2 r lwhere H and H areiz of computational pv) v) 2 domain n the z- and r- direction respectively. bt 1 bz 2 r br =-2 br 11. All the above equations are then approximated A by finite difference expressions. The problem is p +B = ( - ) (8) solved by advancing the configuration through a set po B of finite time steps, or computational cycles. Each numerical computational cycle consists of the folB(QU) = -B (9) lowing steps. dt 1 dz C/C = 1 + 2V /C - 0.1 (V/C )2 (10) 1) Marker particles on the fluid boundary o'o o * o0 o are moved to appropriate new positions.

F.G. Hammitt and Y.C. Huang 2) The continuity and momentum equations and impact angle) and impact liquid Mach number. are used to advance the densities and velocities Hence, the results are general in terms, of scale. through the time change of one cycle by an explicit To assist physical interpretation of the results, technique. the units of the non-dimensional time are micro3) The pressures are calculated as a func- seconds if droplet diameter is 2 mm, a typical tion of densities according to the equation of state, diameter for damaging droplets in turbine flows assuming a quasi-steady process for each time or aircraft rain erosion. For such collisions, increment. pressures exerted upon the target surface are of 4) Boundary condition values and time the order of the water hammer pressure (rather counters are adjusted to prepare the next computa- than of the normal stagnation pressure) only durtional cycle. ing the initial small fraction of a microsecond. 12. Our numerical scheme (ComCAM) was They approach quite closely the ordinary stagnation developed from a combination of the Particle-in- pressure after 2-3 (s. Cell (PIC) method (13), the Marker-and-Cell (MAC) 18. Computed quantities include pressure and method (14), and, the Lax-Wendroff two-step nine - velocity throughout the impacting droplet (including point Eulerian scheme (15). It overcomes some of the interface with target surface), as well as the the difficulties of each, while maintaining most of coordinates of the droplet boundary, as a function the advantages at least for the present problem, of normalized time. Both pressures and velocities such as the compressibility of a continuous fluid along the interface are of vital importance for the and the history of a free surface. prediction of damage, and pressures within the drop 13. The program was first used to solve the may also be of considerable interest in that they The program was first used to solve the two-dimensional axisymmetrical droplet impact become strongly negative at times during the improblem, which is that leading to the classical pact, thus giving rise to the possibility of cavitation one-dimensional water-hammer result when a during impact (16 which may then add to the rigid-tube non-permeable shell boundary is impos- damage. ed. Good agreement with the exact solution for b. Specific results this idealized problem was found (6, 7) and numeric- 19. Table 1 lists the most important numerical al oscillations with our ConCAM method were found results, i.e., maximum pressures and radial to be suitably small. velocities along the target surface for all the cases 2. Numerical Results of ConCAMstudied. The pressures are presented as multiples. General scope of results of both the "simple water-hammer pressure" and 14 a The numerical results of tr e studies s far the "corrected water-hammer pressure". For the 14. The numerical results of the studies so far performed are very numerous and cannot be fully frmer case, the water-hammer pressure is calreported here. Full details are available else- culated using the density and sonic velocity for the where (6, 7,8,10, e~.g. ). However, the general undisturbed fluid; for the latter, the sonic velocity where (6,of th e results cn be gver, thand sme o is corrected according to Eq. (10). The effect of scope of the results can be given, and some of the correction s substantial even at the lower those which are considered of primary importance. Mach number (about 1tantial even at the lower Mach number (about 1.4 for M = 0. 2), and much 15. In general, all but one preliminary numeric- more so for the higher ( about 2.0 at M = 0. 5). al study (8) have utilized a fully-rigid flat target Table I shows that the target surface pressure surface, upon which both full-slip and non-slip is nearly independent of Mach number if normalized boundary conditions were studied. Ref. 8 involves to the corrected water-hammer pressure (hence, a non-rigid surface, such as very soft elastomeric the results in this form can be considered valid over coating, for which material reaction exists only in a much larger range of Mach number), and is in all terms of inertia. However, only tentative prelim- cases substantially less than unity. This fact is imary results have been obtained to the time of ascribed to the additional degrees of freedom for this paper. the flow in these two-dimensional problems as com16. only perpendicular impacts have so far been pared to the impacting slab one-dimensional case, 16. Only perpendicular impacts have so far been where the "corrected" water-hammer pressure studied, and these have been limited to the impact where the "corrected" water-hammer pressure would indeed be obtained. Mach numbers (referred to the sonic velocity of the undisturbed liquid) of 0.2 and 0.5 (300 and 750 20. The target surface pressure exceeds the m/s). Cylindrical, spherical, and a composite "simple" (conventional) water-hammer pressure cylindrical-spherical droplet shape have been at the lower Mach number only for the cylindrical investigated. Their initial profiles and later pres shape (which is the most unrealistic of those studied sure contours are shown in Fig. 1, 2.3. The com- due to its sharp corners), and is substantially less posite shape is believed most typical of actual im- in the other cases. However, for 0. 5 Mach it expacting drops such as encountered in the turbine ceeds the simple water hammer pressure in all application, cases (but only by 7% for the spherical drop). 17. The independent variables of the problem 21. Fig. 4, 5, and 6 show the interface pressures are normalized axial and radial coordinates and at various times during the impact in more detail time, In terms of these, for the other restrict- than do the pressure profiles of Fig. 1, 2, and 3. ing conditions discussed above, the problem is fully described in terms of geometry (droplet shape 22 As is well known (17, e. g.), the radial _Compreessble-Cell-and-Marker method, --— B velocity during impact is usually several times the impact velocity. This result, previously based

F. G. Hammitt and Y. C. Huang primarily upon photographic evidence, is confirm- and Brunton (21), using their single-shot momentum by the present calculations. Table 1 shows that exchange device (also used in our laboratory for its maxima range between 2 and 3 times impact more exacting studies of single-drop impact). velocity (depending upon droplet shape) for the However, the automated device is far more suitable lower Mach number, and between 1. 6 and 2.3 for for experiments involving material resistance to the higher Mach number. The effect of this ex- repeated impact. Results of some of this work tremely high liquid velocity upon the target surface will be published in the near future. Fig. 9 shows through shear and also direct impact upon surface a typical liquid slug from the automated device. asperities may be very instrumental in the damage process in some cases. CONCLUDINGREMARKS 23. A final result of the numerical studies of 27, Work from our laboratory related to the considerable importance is the discovery of regions turbine droplet impingement problem has been of stropgly negative pressure within the impacting described, while the development of a low-presdrops due both to the reflection of pressure waves sure wet-steam tunnel for the study of droplet foras tensile waves from the droplet free surfaces, mation and also experiments upon material resisand to the inertia of the outwardly accelerating tance to droplet impingement utilizing an automated radial flow along the surface. The pressure pro- water gun device are included. The program files of Fig. 7 are illustrative of this situation, concerns primarily numerical studies of the actual which is discussed in more detail elsewhere (6, 7, impingement upon a surface of liquid droplets 16). These results confirm recent photographic of various shapes. We believe these to be the first evidence showing vapor cavities within impacting relatively comprehensive numerical studies of this drops by Brunton and Camus (18)!'included here phenomenon. They have produced several relativefor convenience as Fig. 8. Cavitation within the ly important and basic results: impacting drop could of course contribute to dam- 28.1. The maximum pressures generated upon the age and could help explain observed rapid erosion target surface at least over an appreciable portion in pure impact tests at relatively low impact of a microsecond, are in general considerably less velocity (rapid erosion of stainless steels at 100 than the water-hammer pressure, if this is corm/s is common - (Q9, e. g.). rected for the increase of shock wave velocity in the compressed liquid. While this pressure would B. Additional University of Michigan Turbine be obtained exactly for a "slab" collision, the Droplet Research additional degrees of freedom inherent in actual 24. Additional portions of our research effort droplet shapes are responsible for the reduced upon the turbine droplet erosion problem include pressure. Cf course, the pressures will be further both a newly commenced study of droplet formation reduced if target surface elasticity (or plasticity) in wet steam flow, and actual impact erosion tests. is considered. For impact velocities of the order These efforts are described very briefly below. of 300 m/s, the pressures are also less than the conventional water hammer pressure (shock-wave 1, Wet-Steam Tunnel Facility velocity assumed that of ambient liquid), though 25. The construction of a low-pressure, wet- they are greater than this value for impact veloci. steam tunnel is underway in our laboratory at the ties of the order of 750 m/s. The increase over time of this writing. It is designed to model vel- conventional water hammer pressure increases ocity, moisture, and pressure conditions pertin- with impact velocity. ent to the low-pressure stages of large steam turbines. Moisture film build-up upon submerged 29 These results are roughly confirmed by the surfaces and the subsequent formation and break- nly applicable experimental measurements of off of liquid droplets will be studied photograph- which we are aware, those reported by Brunton ically. No results are as yet available. (24 and summarized by Hays (23) in 1961. He measured the maximum impact pressure for a jet 2. Liquid Impact Experiments of 720 m/s velocity(about equivalent to our Mach 26. We have developed and extensively utilized 0.5 condition), and found that it was 0.875 x "simple an automated water-gun device (patterned after water hammer pressure". Since the shape of the that of Kenyon (20) producing up to 30 elongated droplet is perhaps most similar to our composite liquid slugs per minute at velocities up to 600 shape, his pressure value is less than that which m/s. Liquid slug diameter is about 1 mm (although we predict by 1. 30/0. 875 = 1.50 (see Table 1). The different diameters can be easily obtained), and discrepancy may be partly due to inadequate respontheir length is the order of 50 dia. The striking se rate of his transducer. edge is approximately hemispherical. As is generally agreed (20, e.g.) and confirmed by our 30 His measurements confirm our estimates numerical studies already discussed in this paper, of the duration of the high pressure portion of the generally damagingly high surface pressures and impact, which he states to be "less than 3 ts, velocities occur only during the very initial portion for a drop of about 3 mm. diameter. Duration is of impact (fraction of 1 H~s), so that only the nose of course approximately proportional to diameter. of the liquid slug is important. Thus, while spher- 31. 2. If normalized to "corrected "water hammer" ical droplets would be preferable for damage stud- pressure" the maximum target surface pressures ies, the elongated slugs are adequate for the pur- are nearly independent of impact velocit pose. The shape of slug produced by the gun de- <i.e., the classical one-dimensional case. vice is very similar to that obtained by Bowden

F. G. Hammitt and Y. C. Huang 32. 3. Radial velocities arising from impact are 10. HUANG, Y. C., HAMMITT, F. G. the order of 2-3 times impact velocity. The mul- MITCHELL, T. M. "A Note on Shock Wave Veltiplying factor decreases with impact velocity ocity in High-Speed Liquid-Solid Impact, " to over the range studied. be published J. Appel. Phys.; a lso ORA Rept. 33. 4. Strongly negative pressures arise within UMICH 033710-11-T, Sept., 1071. the droplet during impact which may give rise to 11. HEYMANN, F. J., " On the Shock Wave Velcavitation as already observed photographically ocity and Impact Pressure in High-Speed Liquid(17). Thus the potential for damage may be in- Solid Impact, " Trans. ASME, J. of Basic Engr., creased by this phenomenon. 90, p. 400, July 1968. 34, 5. Damaging pressure occur only during the 12. HUANG, Y. C., HAMMITT, F. G., and initial fraction of a microsecond during a collision YANG, W-J, "Mathematical Modelling of Normal for a drop of 2mm dia. The pressure duration is Impact between a Finite Cylindrical Liquid Jet less for smaller drops. and Non- Slip Flat Ridgid Surface, " Submitted for publication to 1st Internation Symposium on Jet ACKNOWLEDGEMENTS Cutting Technology, BHRA; also available as ORA Financial support for this work was pro- Report No. UMICH - 03371-13-T, Univ. of Mich. vided primarily under National Science Foundation 13. AMSDEN, A. A., "The Particle-in-Cell Method Grant GK 730. for the Calculation of the Dynamics of Compressed Fluids," Los Alamos Scientific Lab., Report No. REFERENCES LA-3466 (1966). 1. HAMMITT, F. G. "Collapsing Bubble Damage 14. WELCH, J. E., HARLOW, F. W., SHANNON, to Solids, " ASME Cavitation State of Art Sympos- J. P., and DALY, B. J., "The MAC Methods, " ium, June 1969, pp. 87-102. Los Alamos Scientific Lab., Report No. LA-3425 (1960). 2. GARCIA, R andHAMMITT, F.G., "Cavitation (1960) Damage and Correlations with Material and Fluid 15. LAX, P. and WENDOROFF, B., "Syster.i of Properties", Trans. ASME, J. Basic Engr., Vol. Conservation Laws, " Ccrmnar.'nication on Pure 89, Dec. 1967, pp. 755-763. and Applied Mathematics, XIII, pp. 217-239 (1960). 3. HAMMITT, F. G., "Damage to Solids Caused 16. HUANG, Y. C., HAMMITT, F. G., "Cavitation by Cavitation," Phil. Trans. Roy. Soc. A. No. within an Impinging Liquid Droplet, " 1972 Poly1110, 260, 245-255, 1966. phase Flow Forum, ASME; also available as ORA UMICH 03371-15-I, 1972. 4. HAMMITT, F. G.,etal., "Laboratory Scale UMICH 03371-15-, 1972. Devices for Rain Erosion Simulation", 2nd Meers- 17. FYALL, A. A., "Single Impact Studies of burg Conference on Rain Erosion and Associated Rain Erosion, " Shell Aviation News, 374, (1969). Phenomena. August -5 1967, Meersburg,. enomena August 2-25, 1967, Meersburg, 18. BRIGGS, L. J., "Limiting Negative Pressure Federal German Republic. of Water," J. of Applied Physics, 21, July 1950, 5,.HAMMITT, F. G,,, "Impact and Cavitation pp. 721-722. Erosion and Material Properties," Proc. 3rd International Rain Erosion Congress,Farnborough, 19 CANAVELIS, R., "Comparison of the ResEngland, Aug. 1970. istance of Different Material with a JetImpact Test Rig, " HC/061-230-9, Electricite de France, 6. HUANG, Y. C., "Numerical Studies of Unstea- Chatou, France, Nov., 1967. dy, Two-Dimensional Liquid Impact Phenomena, " PhD Thesis, The University of Michigan 1971; 20 KENYON, H. F., personal communications also available as ORA Report No. UMICH 033710-8- to F. G. Hammitt, 1967-1970. T, 1971. 21. BOWDEN, F. P., and Bruton, J. H., "The 7. HUANG, Y. C., HAMMITT, F. G., YANG W-J, Deformation of Solids by Liquid Impact at Super"Hydrodynamic Phenomena During High Speed Col- onic Speeds, " Proc. of the Royal Society, A, lision between Liquid Droplet and Rigid Plane, " 263, pp. 433450 (1969). May 1972, submitted to ASME; also available CRA 22 BRUNTON, J. H., "Deformation of Solids by Report UMICH 03371-16-T. Impact of Liquids at High Speed, " ASTM STP 307, 8. HUANG, Y. C., HAMMITT, F. G., "Liquid Im- 1961, pp. 83-98. pact on Elastic Solid Boundary, " IProg. 1th Symp. 23. HAYS, L. G., "Turbine Erosion Research in on Electromagnetic Windows, Geo. Inst. Tech., Great Britain, " NASA Tech. Memo No. 33-271, July, 1972. Jet Prop. Lab., Pasadena, Cal., March 1966. 9. TAIT, P.G., "Report on Some of the Physical Properties of Fresh Water and Sea Water, " Report on Scientific Results of Voy, H. M. S., and Challenger,Phys. Chem., 2, 1-71 (1888 ). *The steam tunnel has been designed by Prof. J. Krzyzanowski of IMP PAN, Gdansk, Poland, who was resident in our laboratory during the spring of 1972.

Symbol Notation A Exponent in Tait's equation of state B Constant in Tait's equation of state A1, A2, B1, B2 Dimensionless coefficients C Shock wave velocity C Soric velocity H1,2 Dimensions of Computa- - tion Domain in z- and rdirection, respectively P Pressure Rm Location of marker m in r- coordinate r Radial coordinate t Time U Marker velocity component in z-direction u Velocity component in zdirection V Marker velocity component in r- direction v Velocity component in r- direction vt Velocity component in tangential direction z Vertical coordinate Q Density Subsc ripts m Marker index n Normal direction o Initial value t Tangential; direction

TABLE I Summary of Principal'Numerical Results M =0.2 M 0.5 (V'= 300 m/a) (V = 750 m/s) 0 0 Pmax Pmax Vmax Pmax Pmax Vmax o C V o CV V C V CV V o 0 0 v0 0o o o 0 0o Free Slip Wall Boundary;^ 1.17 0.84 2.00 1.61 0.82 1.65,: o0.69 0.495 2.65 1.07 0.52 2.15 RP, 0.90 0.65 2.80 1.30 0.66 2.25 Non-Slip Wall Boundary, _g_ 1.20 0.87 2.10 1.675 0.85 1.70,,- 0.80 0.59 2.85 1.229 0.625 2.30 o C V = 2 /M = 4450 bars 11,100 bars vo 0 0 0 0 3428

Fig. 1 Shape-Time HTime(story of an Initially Cylindrical 0.5 R (t= 1 LS for 2mm drop) Fig. 1 Shape-Time History of an Initially Cylindrical Droplet with L/D = 1, at Mach Number = 0.2, for Free-Slip Boundary Condition. 3386

Time (c)0. ~1 0 10.5 2.5 (t = Ct 1 L.s for 2mm drop) Fig. 2 Shape-Time History of an Initially Spherical Droplet at Mach Number = 0.2, for Free-Slip Boundary Condition. 3393

Time =. D t: _ I 2 1 0 1 2 (to Ct r (to t = 1 s for 2mm drop) r D R Fig. 3 Shape-Time History of an Initially Cylindrical-Spherical Composite Droplet with R1/R = 0.25 and L/D = 1, at Mach Number = 0.2, for Free-Slip Boundary Condition. 3400

p PoCoVo O,25 0. 125 \\^<~~ jt~ Mach=0.2 Free-Slip 0.5 Time (-t) 0.5 2.5 0 Q5 1 15 2 r - Ct (t = 1 fls for 2mm drop) D Fig. A;. Pressure-Time History at Liquid-Solid Interface (z = 0) of an Initially Cylindrical Droplet with L/D = 1, for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition. 3395

_P poCoVo z 0.25 C.2 - t a t M ach = 0.2 0.5 Free -Slip 0.5 i~= Dt 1 l.s for 2mm drop) 1.5 r R ( C t. - D = 1 ~s for 2rmm drop) ig. 5 Pressure-T)me n istory at Liquid-Solid Interface (z = O) of an Initially Spherical Droplet for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition. 3395

70 P 1 * I ) Mach =0.2 TiTef) =0.25 Free-Slip 0. 0'R O 1 2: R Fig. -6. Pressure-Time History at Liquid-Solid Interface (z = 0) of an Initially Cylindrical-Spherical Composite Droplet with R1/R = 0.25 and L/D = 1, for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition. 3402

D 005 0.1 1 0 1 R (t = Ct 1 pIs for 2mm drop) Fig. 7 Isobar Distribution in an Initially CylindricalSpherical Composite Droplet with R1/R = 0.25 and L/D = 1, at Time (Ct/D) = 1.5for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition. 3401

Fig. 8 Photographs of the Cavitation for a Water Droplet Following an Impact on a Solid Plane (Brunton and Camus ) 3385