THE UNIVERSITY OF MICHIGAN College of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. UMICH 03371-12-T COMPUTER SIMULATION OF HIGH SPEED COLLISION WITH RAIN DROP (Combined Spherical- Cylindrical Shape) by Yen C. Huan.g,, F. G. Hammitt, Wen-Jei Yang (Submitted to ASME for Publication) Financial Support Provided by: National Science Foundation Grant No. GK-730 September 1971,. Research Associate (Formerly Doctoral Candidate) ^ Professor-In- Charge, Cavitation and Multiphase Flow Labortory Professor

NOMENC LATURE Symbol Des c ription A Exponent in Tait's equation of state B Constant in Tait's equation of state A, A2, B1, B2 Dimensionless coefficients C Shock wave velocity C Sonic velocity o D Diameter H1, H2 Dimensions of Computation Domain in z- and rdirection, respectively K Constant L Length M Mach number P Pressure p p/P C V 3R Radius of sphere R1 Radius of cylinder R Location of marker m in r- coordinate m r Radial coordinate 0r~ ~r/R t Time ~t0~ ~ ~Non-dimensional time, Ct/D U Marker velocity component in z-direction u Velocity component in z-direction u Velocity component in normal direction n V Marker velocity component in r-direction v Velocity component in r-direction vt Velocity component in tangential direction V Impact velocity o x Coordinate in normal direction n xt Coordinate in tangential direction z Vertical coordinate o z/L Z Location of marker m in z-coordinate m

ct Stability factor p Density A Inc rement Subscripts c Characteristic parameter m Marker index n Normal direction o Initial value t Tangential direction Superscripts o Non-dimensional variable

LIST OF FIGURES Figure Page 1. 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..... 14 2a. Isobar Distribution in an Initially Cylindrical-Spherical Composite Droplet with R /R = 0. 25 and L/D = 1, at Time (Ct/D) = 0. 125, for impact Mach Number of 0. 2 and for Free-Slip Boundary Condition................... 15 2b. Isobar Distribution in an Initially Cylindrical-Spherical Composite Droplet with R /R = 0. 25 and L/D = 1, at Time (Ct/D) = 0. 25, for Impact Mach Number of 0. 2 and for Free-Slip Boundary Condition........................ 16 2c. Isobar Distribution in an Initially Cylindrical-Spherical Composite Droplet with R,/R = 0. 25 and L/D - 1, at Time (Ct/D) = 0. 5, for Impact Mach Number of 0. 2 and for Free-Slip Boundary Condition.................... 17 Zd. 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.....................i8 2e Isobar Distribution in an Initially Cylindrical-Spherical Composite Droplet with R /R = 0. 25 and L/D = 1, at Time (Ct/D) = 2. 5, for Impact Mach Number of 0. 2 and for Free-Slip Boundary Condition.................... 19 3. Pressure-Time History at Liquid-Solid Interfact (z = 0) of an Initially Cylindrical-Spherical Composite Droplet with R /R = 0. 25 and L/D = 1, for Impact Mach Number of 0. 2 and for Free-Slip Boundary Condition.............. 20 4. Pressure-Time History along the Symmetrical Axis (r = 0) of an Initially Cylindrical-Spherical Composite Droplet with R /R = 0. 25 and L/D = 1, for Impact Mach Number of 0. 2 and for Free-Slip Boundary Condition..... 21 5. Local pressure-Time History at a (r = 0, z = 0. 5L), b (r = 0, z= 0), and c (r = 0. 75R, z = 0), in an Initially Cylindrical-Spherical Composite Droplet with R /R = 0. 25 and L/D = 1, for Impact Mach Number of 0. 2 and for FreeSlip Boundary Condition.......22

6. Radial Velocity-Time History at Liquid-Solid Interface (r = 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.......... 23 7. Maximum Pressure Gradient-Time and -Location Relation and Contact Edge Time History of an Initially CylindricalSpherical Composite Droplet with R /R = 0. 25 and L/D = 1, for Impact Mach Number of 0. 2 and for Free-Slip Boundary Condition................................................. 24 8. Photographs of the Cavitation for a Water Droplet (38 Following an Impact on a Solid Plane (Brunton and Camus... 25

I. INTRODUCTION In recent years, there is growing concern with erosion from the severe local pressures and pressure changes due to liquid impingement on high speed machinery and aircraft components. How(1, 2) ever, the earliest attention' for liquid impact erosion arose in hydraulic (Pelton) and wet steam turbines. When the steam turbine stages operate in the region of vapro-liquid mixture, after the steam has expanded from the high pressure state, the liquid droplets are impacted by the rotating blades at about the operating speed of the turbine (, ). With the higher velocities in the very large new turbine designs, it is likely that the erosion problem can no longer be met through improved materials. Likewise, in the late 1940's when the speeds of aircraft exceeded Mach 0. 5, rapid erosion was experienced by aircraft flying through rain particularly on propellor blades. The problem has become increasingly severe with the present much higher speed aircraft, and also with helicopter blades. Research has been sponsored in this country) and also in many other countries, ) with attention focusing now on radome, window, and structural materials of fixed-wing aircraft and helicopters, as well as missiles to avoid degraded performance and possible failure. The phenomena are similar to the liquid droplet impact erosion of steam trubines and hydraulic Pelton turbines. (9, 10, etc. ) It has been suggested' that the damaging mechanism of cavitation may actually be a liquid jet impingement resulting from the non-symmetrical collapse of a bubble. Cavitation erosion is important for high speed ship propellers and other components as well as many other power plant components such as pumps, etc. is o a roble in urine(1213) and ums (1415)of sace It is also a problem in turbines and pumps of space

2 power plants using liquid metals as the working fluid, as well as in present sodium-cooled fast breeder reactor power plants and in many other cases involving many different fluids. ( Soil erosion caused by raindrop impact, a rather analogous (16) phenomenon, also has received attention in the present tide of interest in ecology. Liquid droplet impact on a solid surface has been observed for centuries. The first extensive description of this phenomena was (17) that given by Worthington around 1894. Considerable pioneering work since the early 1950's was conducted by Dr. Olive Engel 8) (19) Her work includes a chemical mapping of the radial water flow on the impact plane, the use of high-speed motion-picture photography to record the motion of the impacting water drop, and the use of Schlieren photography to study details in the radial flow. Bowden (20) and Brunton reported on fundamental studies of liquid jet impact (21) at supersonic speed. Recently, Fyall used a turbine-driven Barr and Stroud CP5 rotating mirror camera (2x0l - 8x10 pps) to record photographically the collision process of a moving target with a single stationary water drop in order to study the basic mechanisms of rain erosion, A rather similar camera is being used in the author's laboratory to study the impact of moving water droplets with stationary surfaces. The above are merely typical citations from a very numerous and growing literature in this field. Photographic studies19' 20 21) have shown that the maximum lateral velocity typically considerably exceeds the impact velocity. The resulting sheet of liquid spreads radially around the periphery of the contact zone, while the remaining portion of the drop is relatively undeformed. This suggests that compressibility effects are important in the liquid-solid impact phenomena. In the initial stages of the impact, the sudden deceleration of the liquid will establish a large pressure gradient. The spatial

3 gradients of velocities are negligible compared with the large magnitude of local acceleration. For the one-dimensional case, during the early phases of impact, the integration of the appropriate equation of motion 6v bp - -t (1-1) yields the well-known "water hammer pressure" p =- PCV (1-2) where p is the density of the ambient liquid, V is the impact velocity, and C is the shock wave velocity with respect to the undisturbed liquid. For low impact velocity, C can be reasonably approximated by the acoustic velocity C. However, for high impact velocity, the compressibility effect must be taken into account. This requires the equation of state. Tait in 1888 proposed the following equation of state for water: p + B P ( o )A - o+B'o," (1-3) %+B Po where B and A are two empirical functions of temperature. A = 7.15 and B = 3047 bars at 20 C were given by Cole ). Li 24)concluded Tait's equation represents relationship of water properties very well by exhaustive examination of published experimental data. The combination of Equations (1-2) and (1-3) with the equation of continuity yields (25) an implicit function for the shock wave velocity as 2 P + B 1- (1- C (1-4) P V V 0 c~ (1 ~ A (26) Heymann proposed a linear relationship for the shock wave velocity as an explicit function of impact velocity only C = C (1+ 2V ) (1-5) o o where C is the sonic velocity in the undisturbed liquid. The above o expression is in good agreement with measured shock wave velocity (27, 28, 29, 30, 31) data for water from various sources It is recommended for use only up to V /C = 1. 2. A better approximation in quardratic form was derived in ref. 31.

4 V V The equation (1 -6) is applicable up to o 3. Within this range, C 0 o predictions for C from Equ (1-4) and (1-6) agree to within about 2%. (32) Savic and Boult presented a mathematical analysis of the low speed impact and spreading of a spherical liquid drop on a rigid surface but did not take compressibility into account. (19) Engel performed an approximate analysis of spherical droplet liquid-solid impact and modified the water hammer equation as follows, for the plane rigid surface, P - - C V (1-7) o o00 o0 where V is a coefficient giving the fraction of impact velocity. It was reported that' approaches unity for high impact velocities. (33) Heymann presented new quantitative results for the maximum impact pressure generated which is in some cases the order of 3 times the simple water hammer pressure. His analysis assumes that at the first instant of contact, the pressures have already (34) built up to p CV. The analysis shows that this assumption may not be valid. The bulk of prior research on high speed collision with rain drop has been limited to photographic studies, experimental tests and relatively simple analyses. No solution of a two dimensional pressure distribution has yet been made, to the author's knowledge, prior to the present study. The usage of high speed computer technique has made it feasible to solve this complex problem. The main objective of this study is to simulate high speed collision with rain drops by computer and solve for the flow patterns, impact pressure and velocity distribution developed in the rain drop as a function of time.

II. Analysis To solve the high speed collision of a rain drop on a target, it is desirable to simulate as close by as possible the physical phenomena of the impact process so that the simulation result will be close to the actual response. Depending upon the material, the solid target material may respond to the impact like a rigid body with negligible deformation, an elastic plate, a plastic plate or even as a fluid. The problem treated in its most general aspects requires a detailed investigation of the manner in which the material would behave rheologically under an impact. Some compromise must be made between the complexity of the physical problem and the practical difficulties involved in simulating the model selected. For the present study, the target surface has been assumed to be rigid. The model is significant in representing a limiting case of great theoretical importance, and is a close representation of many actual practical cases. The air surrounding the rain drop may or may not be influential, and the pressure variation within the air in the vicinity of the liquid-air interface may or may not significantly affect the deformation of the liquid-air interface. However, for the present study, constant atmosphere pressure will be assigned to the surrounding air. Actually, the appearance of shock-waves in the air induced by the target velocity may be of extreme importance in some cases by causing the drop to disintegrate before impact. Since we are dealing with high speed impact, the effect of compressibility of the liquid must be taken into account in any valid and realistic analysis. However, it is reasonable to neglect the effects of bodyforce and viscosity as compared with pressure and inertia. Similarly, the surfact tension, which is involved in the boundary condition, can also be neglected.

6 Under these assumptions, the high speed collision process of a rain drop on a flat rigid surface can be described as in the following. The equations of continutiy and momentum for the liquid drop become: bp b (pu) i ) (rpv) (2 + + 0 (2-1) bt bz r br a (pu) b (pu ) 1 a (rpvu) bp (2-2) bt 5z r 6r -6z (pv) + (pvu) 1 o(rpv ) P (2-3) bt bz r br br where u and v are the axial and radial velocity components for the cylindrical coordinates z and r respectively. t is the time variable, and p and p are the liquid density and pressure. Since there are three equations but four urknowns, an additional equation is required to solve the problem, i.e., the equation of state for water in which the density and pressure are coupled. p B ( p)A (2-4) po+B Since the liquid boundary is free to deform, it is required to keep track of its location. The marker particles imbedded in the boundary follow the equation of motion for a free body and the kinematic relations used in the Lagrangian calculation. d(mU) = F (2-5) dt z d(mV) = F (2-6) dt r dZm dt (2-7) dRm dt (2-8)

7 where mU and mV are the mommentums of a marker particle subject to external forces F and F in the z- and r- direction z r respectively. Zmand Rmdescribe the location of a marker particle. The above governing equations, after divided respectively by the characteristic parameters of the problemp, u, v, p, t, z, r, can be expressed in dimensionless forms: c c 6P + A1 b(u) + A- (rv) 0 (2- 9) c+B (p)A (2-19) pt 1 Pz 2 r 6r d(pu) b(pu_ ) 1 (rpvu) + A +A- = -B (2-10) bt 1 bz 2 r 5r 1 bz _____ - r (<2-11) p+B p0 +i~+B ( a)A (2-12) Po* B Po d(pU) dp (2-13) dt -'1 dz d(pV) _ dp -- B (2-14) dt 2 dr Zm = AjUdt (2-15) R = A2V dt (2-16) where u t u t ~ r 2 r c c ((2-17) p t p t p c c c B = B 1 p u z 2 pu r c c c c c c One may select the characteristic parameters Pc, u, p, t, z, r in such a way that all the coefficients A1, A2, B1, and B2 c c 2

8 considering the nature of the particular problem, the characteristic parameters will be chosen as follows. Characteristic density c will be the undisturbed density at 1 atm p. Characteristic velocity u will be the impact velocity V. Characteristic pressure p will be the simple water hammer pressure based on poCV. z and r are the cell size dimensions in the z and r directions, respectively. c It is convenient in many cases that they be chosen to be equal. There will be a definite relationship between z, r and the characteristic c c lengths of the problem, e.g., the length and the diameter of a cylindrical liquid droplet. Characteristic time t will be z /C, where C is the shock wave velocity defined in the Eqs. (1-4) and (1-6). After such selection of characteristic parameters, the four parameters in the governing equations: A,, A, B1, B can be reduced to A1 A = 2 1 1 2 1 2 M/(1+2M-0.lM ), and Bl=B2=1/(1+2M-0.lM ), where M=V /C. These coefficients, which are functions of impact Mach number, characterize the flow problem. The appropriate initial conditions over the domain of calculation are p = p u = u v v 0 0 0 where p is the environmental pressure, u and v are the initial n ~ ~~~~o o impact velocities in the z- and r- direction respectively. In the case of a normal impact (i. e., perpendicular), v -0 of course and V =u The appropriate boundary conditions are: i) along the axis of symmetry (z), r = 0, and symmetry requires v = 0, = O - br &r bv ii) along the impacted rigid surfaces, z = 0, - = 0 u=, bP = 0, for full-slip wall condition. 6z iii) along the free surface, the incompressible continuity yields bu bv P Po bx = x = n t

9 where u amd v are the moving velocity components of the liquid-air interface in the normal x and tangential n xt directions of the surface respectively. iv) along the sides of the finite computational domain, permeable boundary conditions will be imposed, in such a way that the normal spece derivative of the variable vanishes at the boundary, 5u 6v bp U=-0, = 0, 0 at z = H bz bz bz 1 =0,, = 0 at r = H2 br' r br 2 where H and H are sizes of computational domain in the z-and r- direction respectively. All the above equations are then approximated by finite difference expressions. The problem is solved by advancing the configuration through a set of finite time steps or computational cycles. Each numerical computational cycle consists of the following steps. 1). Marker particles on the fluid boundary are moved to appropriate new positions. 2). The continuity and momentum equations are used to advance the densities and velocities through the time change of one cycle by an explicit technique. 3). The pressures are calculated as a function of densities according to the equation of state, assuming quasi-steady process. 4). Boundary condition values and time counters are adjusted to prepare the next computational cycle. A detailed discussion of the numerical method, the stability, and the accuracy is given elsewhere (35).

III. RESULTS In examining the following results, one must keep in mind those boundary conditions at the interfaces of the axisymmetrical liquid droplet. Constant atmospheric pressure at the water-air free surface is imposed. The impact plane, which is perfectly rigid and smooth, has no movement. The properties of water are inviscid, without surface tension, but compressible as well as elastic, The tensile strength of water is taken to be -270 atm., an experiemental result of Briggs (. Of course, water will rupture in many circumstances at much smaller tensions, but perhaps not for the very short duration of tension involved. The two constants in the Tait's equation of state for water are A = 7.15 and B = 3008 atm. The shape of an actual rain drop is of course never precisely regular. However, a cylindrical-spherical composite shape (Fig. 1) may resemble an actual rain drop better than other simple shapes, because of the deformation of the rain drop by air resistance before impact. The numerical results for such a cylindrical-spherical i34) composite drop lie in between those for a cylindrical drop and (37) a spherical drop Fig. 1 shows the deformation history of the simulated rain drop following an impact on a rigid surface at Mach number of 0. 2 for free slip boundary condition. The liquid surface on the top side of the rain drop remains relatively undeformed until t0=1. It is also seen from the figure that the liquid on the top surface decelerates, while the liquid on the shoulder still rush downward with less effect received from the stopping action by the rigid surface. As time progresses, the drop shrinks in height and expands radially near the contact surface. Fig. 2a through 2e show the isobar distribution within the simulated rain drop at five different instants, t = 0.125, 0. 25, 0. 5,

11 1. 5, and 2. 5 respectively. The figures illustrate how the pressure waves propagate with time from the contact surface toward the free surface. The isobaric surfaces all end at or near the contact ring (the intersection of contact surface and free surface). Because of the condition of constant atmospheric pressure imposed on the free surface, the compression wave cannot proceed beyond the free surface. Each isobaric surface grows in size, radiating outward from the impact surface. Then, as time elapses, there is more outgoing lateral flow than incoming flow at the center area of the impact surface so that the isobaric surface near the contact area changes from a dome shape to a peanut shell shape (Fig. 2c). Another interesting phenomenon appears on Fig. 2d. A region of negative pressure (where actually a cavity might be expected) was generated by the reflection of a pressure wave from the top surface. At the moment as shown on Fig. 2d, the cavity at the center is shrinking, while the cavity in a form of annular ring is growing. Fig. 2e shows the isobar distribution as the impact parameters begin to approach roughly steady-state conditions. The nondimensional steady-state stagnation pressure in this case is 0.1, i. e., 1/2 poV divided by the simple water hammer pressure pC V, where impact Mach Number M = V /C - 0. 2. The peak pressure by this point has shifted on o o the impact surface from the center of impact to the approximate point where a vertical line, tangent to the initial shape of the rain drop, intersects the rigid surface (Fig. 2e). This shifting of peak pressure on the impact surface is caused by the interaction of radial flow and impacting liquid. The numerical computation was terminated at this instant because the steady state has been reasonably approached and thus most of the important features of the phenomenon have been disclosed. Better illustration of the pressure distributionn on the impact surface and along the axis of symmetry are given in Fig. 3 and 4

12 respectively. Fig. 3 shows the pressure distribution on the contact surface as it develops with time. The pressures fall sharply at the contact edge because of the boundary condition of constant atmospheric pressure at this location. The figures also depict the peak pressures at various instants as they shift from the center to the contact edge. Fig. 4 shows how the pressure waves propagate from the impact surface along the axis of symmetry. They reach the top surface of the drop, and then relfect as tension waves creating a zone of negative pressure. Fig. 5 shows the local pressures as a function of time. The maximum pressure was recorded at the stagnation point "b". The pressure at "c" will not increase until contact between liquid and solid occurs at that point. The time rate of pressure increases at locations "b" and "c" are about the same for the initial period of impact. As the radial flow commences the pressure distribution is affected so that the pressure at point "b" starts to decrease while that at "c" continues to increase, reaches to its peak and then falls along with the pressures on the impact surface and finally oscillates about the steady-state stagnation pressure. The pressure at "a" also does not increase until the pressure wave approaches that point, and it will not reach as great a value as that at "b" because of the effect of the free surface. The fact that a "bursting out" of the top surface does not occur supports the fact that the pressure at all locations such as "a" will be lower than that at location "b", increasing so as the distance from the impact surface is increased. Fig. 6 shows the radial velocity distribution on the impact surface at various instants. The peak radial velocities appear slightly inside the contact edge. The dotted portion of the curves indicate regions where the jetting liquid does not remain in contact with the solid surface. The jetting velocity betins to exceed the impact velocity at t~ - 0. 25 and increases to - 3 x the impact velocity.

13 Fig. 7 shows the maximum radial pressure gradient, and its location at any given time on the impact surface. The peak maximum pressure gradient occurs at t 0.5 (or about 0.. 5 usec for a typical rain drop of 2 mm at 1000 ft/sec impact velocity). The locus of the maximum pressure gradient lies closely within the locus of contact edge. It is conjectured that the erosion may be caused by the shear forces associated with these maxiumu pressure gradients. (38) Fig. 8 illustrates photographs, by Brunton and Camus, of cavitation within a water drop following an impact on a solid plane, as predicted by these calculations (Fig. 2d).

14 03 0 02 - 0 oc'J ccd HO L 0 if cr~~~~~~~~~~~~~~~~~~~C a4Q) rd s 000 rd a H _ _ _ _ _ _ 0 ~~~~~~~~~~~-iH 0 O6 11.Hcd 4-) rd 00 f r i Pipi 0 0) aH 6PI ~ ~ ~ ~ W NIO E - id~~~~~~~PRQ ~PIP~L~L ~~ r. ~ s~ ~~~pBs ~~lB0W N~~'F rz ~

15 z D - p- 0.1 \ DP~ooCoVo~ / 1 0 1 r R Fig. 2a. Isobar Distribution in an Initially CylindricalSpherical Composite Droplet with R1/R = 0.25 and L/D = 1, at Time (Ct/D) = 0.125, for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition.

16 Z D 0 1 I r R Fig. 2b. Isobar Distribution in an Initially CylindricalSpherical Composite Droplet with R1/R = 0.25 and L/D = 1, at Time (Ct/D) = 0.25, for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition.

17 D 1 I: =0.5 I~~ j /Ppo~Vo 1 1 0 1 r R Fig. 2c. Isobar Distribution in an Initially CylindricalSpherical Composite Droplet with R1/R = 0.25 and L/D = 1, at Time (Ct/D) = 0.5, for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition.

18 z D 0I05 r1 0 R Fig. 2d. Isobar Distribution in an Initially CylindricalSpherical Composite Droplet with R1/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.

T~~-OJ ~4-) 0P4 o ~- Ii 0 ~k0 10 0 Cd / / _ ln n0, r.-44 ~ 4 0 ddr / I-H II 0 /H X I I l O 1o AJ t TOM/~~~~ $ / /~0 IJn' C /d (x, / /ri I r It - H P: 4 \>\ 0 0 0 III o (~."7~~- 1^ 1 ot i4 Co \TN N \-d 0 c \ \0 C\J~~~~( \ ~ ~~~~~~~~~~~~ \*Hf \~ ~ ~~~~~t \^pT \ \-H O~~~~~~~~~~~~~~~~ \ \ >^ ^~~~~~~~ \ ~ ~~~~~~~ \~ /^ C N. \ (^'~?~~~~~~~( ^< \ I^ r^~~~~~b ^S. \ \^ Pi ^

20 p p~oCoVo Z ~1 t~- I~! ~r J Mach =0.2 Time — )=0.25 Free-Slip 0.5 195 0~05 2.5 t5 0 0 1 2 r R Fig. 3. Pressure-Time History at Liquid-Solid Interface (z = 0) of an Initially Cylindrical-Spherical Composite Droplet with RI/R = 0.25 and L/D = 1, for Impact Mach Number of 0,2 and for Free-Slip Boundary Condition.

21 z D 2 rt ~Mach=0.2 I \ O>^ ^^^ r Free- Slip Time ( —)=0.5.5 2.5 05 0 0.5 1 P q CoVo Fig. 4. Pressure-Time History along the Symmetrical Axis (r = 0) of an Initially CylindricalSpherical Composite Droplet with R1/R = 0.25 and L/D = 1, for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition.

22 z D p..< fI*)5 05 X " (RI+R),I I,, 0.5 - \ ~~~~~~~~~~~poCoVo~ 0.5 \~ f' \b 01 2 Time (-) Fig. 5. Local Pressure-Time History at a (r = 0, z = 0.5L), b (r = 0, z = 0), and c (r = 0.75R, z = 0), in 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.

23 3 I \ - L 0 ^ 05,,..., / 1 2 ~R~I Fig. 6. Radial Velocity-Time History at Liquid-Solid I I I I 0.5 Fig. 6. Radial Velocity-Time History at Liquid-Solid Interface (r = O) of an Initially CylindricalSpherical Composite Droplet with R1/R = 0.25 and L/D = 1, for Impact Mach Number of 0.2 and for Free-Slip Boundary Condition.

24 Locus of Contact Edge 11 / I 8 —! / KLocus of Maximum Pressure 1.0 d(C X) / Gradient on the Impacted Surface di ()/ r ~~I I;~~~~~~R, / / \ Maximum Pressure Gradient -'// \ ^\ on the Impacted Surface 40~~~~ ~~~~~~-I _05:f { j } Mach=0.2 2 1 r'''- ~ 0.5 1 1.5 2 2.5 Time ( —Ct) Fig. 7. Maximum Pressure Gradient-Time and -Location Relation and Contact Edge-Time History 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.

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

26 IV. Concluding Remarks The objective of this paper was to investigate the collision process between a rain drop and a rigid plane. Effort was directed to simulate this impact phenomenon. High-speed computation was employed to carry our the task. Results such as the deformation of the simulated rain drop, isobar distribution within the droplet at various instants, pressure distributions on the impact surface and everywhere within the computation domain were recorded by computer. Other variables such as density, velocity and momentum were also found as a function of time. Since the liquid surface is free to deform, the pressure buildup is affected, from the first instant of impact, by the immediate radial release flow. Although an earlier more approximate analy(39) sis postulates there is an initial stage during which no gross spreading or lateral outflow occurs, the present dynamic analysis finds that by the time incremented numerical approach used, the pressure gradient which exists at the contact edge will result in lateral flow immediately. For an impact Mach number of 0. 2, which is equivalent to an impact velocity of 980 ft/sec. the maximum radial velocity is about 2750 ft/sec. These calculation results are in good agreement with the photographic observation in which radial velocity of 3056 ft/sec. resulted from an impact speed of 990 ft/sec., according to Fyall 1). Negative pressure within the drop was computed, which could lead to actual cavitation, as a result of reflection of the pressure wave from the top surface of the droplet as a tension wave, as (19) suggested earlier by Engel. Very recent photographic evidence (38) by Brunton and Camus actually shows the fromation of such vapor pockets.

27 The maximum pressure in this simulated cylindrical-spherical rain drop is -0. 9 x the simple water hammer pressure p C V. The numerical results are thus between 0. 7 for spherical drop (34) and 1. 2 for cylindrical jet). ACKNOWLEDGEMENT The work was supported by National Science Foundation Grant No. GK 730.

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