THE UNIVERSITY OF MICHIGAN College of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. 02643-1-T 08466-6-T ON SHOCK-WAVE VELOCITY IN DROPLET IMPACT PHENOMENA by Y. C. Huang T. M. Mitchell F. G. Hammitt Financial Support Provided by: U. S. Naval Air Development Center Contract No. N62269-69-C-0285 and Army Research Office Contract No. DAHC04 67 0007 This document is subject to specialexport controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Commanding Officer, Naval Air Development Center. October 1969

i ABSTRACT A relationship between the shock wave and impact velocities in high-speed liquid-solid impact is determined from the fundamental equations (continuity, momentum, and state). Very good agreement is found with available experimental data.

ii TABLE OF CONTENTS I. INTRODUCTION.............. 1 II. FUNDAMENTAL RELATIONS........... 2 III. RESULTS................. 4 BIBLIOGRAPHY................. 5 TABLE.......... 6 FIGURE..................... 7

ON SHOCK-WAVE VELOCITY IN DROPLET IMPACT PHENOMENA INTRODUCTION The pressure developed in the high speed impact between a liquid and a solid surface is of interest and concern in the problems of turbine blades operating in moist vapors and of supersonic aircraft and missiles flying in rain. The pressure which acts on the material surface and causes the damage is related to the "water-hammer pressure", p = cv, O O where v is the impact velocity and e c is the mass flux, assuming o that a shock front can be regarded as a discontinuity and that a shock front has negligible volume. The velocity of propagation of the pressure or shock wave, c, can be approximated by the acoustic velocity, c, in the undisturbed liquid if the impact is just a small disturbance. However, for high velocity impact, the deviation from c becomes significant, and c must then be taken as an approximate shock wave velocity satisfying the continuity and momentum equations with an appropriate equation of state. It would be desirable to have an expression for c as a function of v, which could then be used to calculate p directly. Most of the fundamental studies ( 3 4 ) have tabulated shock front velocities, c, as a function of pressure, p. Heymann ) has proposed, as an approximate relationship between c and v, c = c + kv. However, no direct analytical formulation has been given, as yet, to justify this approximation. Such a relationship is developed in the present paper. 1

2 II. FUNDAMENTAL RELATIONS If a shock front is discontinuous and its thickness is negligible, the governing equations applying to such a shock front impinging upon a rigid boundary, derived from continuity and momentum considerations, are: oPc -Ccv) (1) P-P o o PcV (2) p p = e cv (2) where c is the shock wave velocity, v is the impact velocity, e is the density of liquid in the compressed state, E is the density of the undisturbed liquid, and p-p is the impulse of the net force per unit area across the shock front. The equation of state of water should provide the additional relation necessary for solving the problem, namely p+B ( n (3) 0o where B is a function of the thermal state (B has the value of 3. 047 kilobars at 20 C and 1 bar. Though somewhat sensitive to temperature, it is relatively insensitive to pressure.), and n is approximated as a constant equal to 7.15 for pressures up to 25 kilobars,

3 Solving equations (1) and (3) for e and equating, we obtain te ~o~ r n B 1 l/n c-v o +B (4) C-V 0 P +( _ "0 Raising both sides to the power n, and replacing p from eq. (2), eq. (4) becomes eoc n O cv (c0v) = n 1+ (5) o+B c-v p 4B Rearranging eq. (5) yields 2 v n c 1 - (1 — ) ------- -- c (6) p +B - ) ( ) ( 1-) p c c 0 where --: ( 1 always. Equation (6) now gives the shock wave vel ocity, c, directly in terms of -. The value of c is then used to determine impact c velocity v. It is more desirable to have explicit relations for c directly in terms of v. One such relation, we propose here, is = +a ( ) + b ( ) (7) c c c 0 o o In the present case, constants a - 1. 925 and b = -0. 083 v are determined by a least square fit computer program for up to 3 from the curve resulting from eq. (6). Within this range predictions 3 for crv r from eq. (6) and (7) agree to within this ran predictions for c from eq. (6) and (7) agree to within + 1%.

4 III. RESULTS Results calculated from eq. (6) agree extremely well with (7, 8, 9, 10) the whole spectrum of experimental data as shown in Fig. 1. The asymptotic value of acoustic velocity, c, calculated from eq. (6), is about 4850 ft/sec which is slightly lower than the 4900 ft/sec that we adopted from Heymann's paper for calculation of v/co. The relations c v - = 1 + k, k = 2. 0, (8) c c o 0 and - 1 + ) + b(-, a = 1.925 c c c' c (9) o0U~ 0 0 b = -0. 083 have been evaluated and compared with the numerical results of P B --- --- o( o c v c~Cc c o o c' c 0 0 V v deviation of - increases+ 1. It925 - is abouless than c c o0 0 vv v. On the other hand the whole rantage of - up to 3. ~~~~c c~ ~o O o deviation of c I + 1. 925 _ 0.083 (v 2 is less than 0 0 0~

5 REFERENCES 1. Heymann, F. J., "On the Shock Wave Velocity and Impact Pressure in High-Speed Liquid-Solid Impact". Trans. of ASME, Sept. 1968, pp. 400-02. 2. Cole, R. H., Underwater Explosions, Dover. 1965. 3. Brunton, J. H. "High Speed Liquid Impact", Phil. Trans of Roy. Soc., (London!, Series A, Vol. 260, Part 1110, 1966, pp. 79-85. 4. Jenkins, D. C., and Booker, J. D., "The Impingement of Water Drops on a Surface Moving at High Speed", Aerodynamic Capture of Particles, ed. Richardson, E.G., Pergamon Press, 1960, pp. 97-103. 5. Skalak, R., and Feit, D., "Impact on the Surface of a Compressible Liquid", Journal of Engineering for Industry, Trans. ASME, Series B, Vol. 88. No. 3, Aug. 1966, pp. 325-331. 6. Kirkwood, J. G. and Bethe, H. A.,"The Pressure Wave Produced by an Underwater Explosion; I —Basic Propagation Theory", OSRD Report 588, 1942. 7. Kirkwood, J. G., and Richardson, J. M.,'The Pressure Wave Produced by an Underwater Explosion; III —Properties of Salt Water at a Shock Front", OSRD Report 813, 1942. 8. Richardson, J. M. Arons, A. B., and UTalverson, R. R., "Hydrodynamic Properties of Sea Water at the Frontof a Shock Wave", Journal of Chemical Physics, Vol. 15, No. 11, Nov. 1947, pp. 785-794. 9. Rice, M. H., and Walsh, J. M., "Equation of State for Water to 250 Kilobars,", Journal of Chemical Physics, Vol. 26, No. 4, April,1957, pp. 824-830. 10. Cook, M. A., Keyes, R. T., and Ursenbach, W. O., "Measurements of Detonation Pressure", Journal of Applied Physics, Vol. 33, No. 12, Dec. 1962, pp. 3413-3421. 11. Jones, A. H., Isbell, WM., and Maiden, C. J. "Measurement of the Very-High-Pressure Properties of Materials Using a Light- Gas -Gun", Journal of Applied Physics, Vol. 37, No. 9, Aug. 1966, pp. 3493-3499.

TABLE I Comparison of Values for Shock Wave Velocity as Function of Impact Velocity Calculated from Three Different Equations Nondimensional Nondimensional Shock Wave Velocity Deviation % Impact Velocity Impact Velocity Computed Computed Computed c c c -c v/c - 3 1 o from eq. (1)* from eq. (2)*from eq. (3)* c1/c cc2/c c3/c c1 C1 0.1 1.1924 1.2 1.1917 0.627 -0.065 0. 5 1. 9643 2.0 1. 9418 1. 808 -1.145 1.0 2.8547 3.0 2..8422 5.073 -0.439 1.5 3.6947 4.0 3. 7012 8.263 0.177 2.0 4. 5020 5. 0 4.5186 11.06 0.370 2.5 5.2863 6.0 5.2945 13.50 0.159 3.0 6. 0528 7.0 6. 0290 15.65 -0.391 c = 4900 ft./sec. = speed of sound in the undisturbed liquid. o 2 4 2 v n = 1.935 lb. sec. /ft eq. (1) c 1 - ( 1 - ) o 7 _':- Bq= (1)1 B 0.6367 x 107 psf p+B (v v n Po_ (v- (1 —-) ( c1-) c(1 _ n = 7.15 *0 eq. (2) c2 v -- = 1 +2 c c o 0 eq. (3) c3 v v 2 eq. (3) = 1 + 1. 925 ( ) - 0.083 ( )2 c c c 0 0 0

5.0 _..._.i-_. EQUATION (8) /EQUATIONS (6) a (9) 4.0 0 I0 3.0 0. /~ _ n /2 DATA SOU)RCES n 0 RICE a WALSH (9) O| Pure water, 20 C _I zo 0] COOK et al (10) iuC Pure water, 20 C z z 0 A RICHARDSON et al (8) Salt water, 20 C V KIRKWOOD a RICHARDSON (6) Salt water, 25 C (also in refs. 7 8 8) 2938 0 1.0 2.0 3.0 PARTICLE VELOCITY MACH NUMBER Co Figure 1. Shock Wave Velocity versus Particle Velocity in Water