THERMOELECTRIC POWER AND RESISTIVITY OF CrFe ALLOYS
3855 skn of vacancies. Considering the dilute alloys, thisbehavior S modified, and above about 14% Fe, the ç.rves are concave down. Mott and Jones5 have
sug-estcd that such behavior can be accounted for on the
.a5 of the filling of a d band. If this is the correct tplanation, it is the only evidence for s-dscattering found in an analysis of our data.
In Fig. 5 are plotted the resistivities of the alloys
hich clearly show the effects of ferromagnetic ordering.
In this case, where at a gwen temperature, the resis-ivitv of an alloy of higher Fe content is smaller than that of one of lower content, it is clearly seen that the reater degree of ordering for the larger Fe content decreases the resistivity below that of a more nearly
pure alloy. These curves too, show concavity toward the
T axis.
The peculiar peak in the curve of the 48.2% alloy is (1UitC interesting. At present, we have no explanation
for it. An examination of the TEP results (Fig. 3)
INTRODUCTION
The remarkable appearance of a splashing liquid Irop has been observed for centuries, but detailed
ludies of the process were not possible until the
de-:0pmeh1t of high-speed photographic techniques. The irst extensive description was given by Worthington.1
!ore recently the phenomenon has been investigated or a variety of specialized purposes; Edgerton and 'iillian2 used it to illustrate their photography tech-uiques; Gillespie and Rideal3 were concerned with he bouncing of drops from aerosol filters; Sahay4
xanuned the breaking of water drops falling into oil;
'.'. M Worthington, A Sludy of Splashes (The MacMillan o9any, New York, 1963).
'k. L. Edgerton and J. R. Killian, Jr. Flash (Charles .T. ranfod, Boston, 1954).
-'T. Gillespie and E. Rideal, J. Colloid Sci. 10, 281 (1955). 4B. K. Sahay, Indian J. Phys. 18, 306 (1944).shows that there is no effect on S, a fact which makes one suspect an artifact. However, the measurements were carefully repeated many times on three different samples2' and the peak in resistivity is quite
repro-ducible as is the absence of an effect in S. ACKNOWLEDGMENTS
The authors would like to thank Dr. R. A. Meussner
of the Physical Metallurgy Branch, U.S. Naval
Research Laboratory for his help in performing the
metallography on these samples; the Analytical
Chem-istry Branch, particularly O. Mylting, of the U.S.
Naval Research Laboratory for their work in the
quali-tative and quantiquali-tative analyses of these samples; and Professor Frank J. Blatt of Michigan State University for his helpful discussion regarding the data obtained
during this project.
21 Samples supplied by Dr. J. Goff, Naval Ordnance Laboratory,
White Oak, Silver Spring,
IAJSCHE
UVjjg
Labo!adwn vOor
Azchfef
Meketweg 2,2625 CD- De!ft
and Hobbs and Kezweeny5 studied the breakup of the rebounding splash jet.
These previous investigations are principally
experi-mental. Full theoretical studies have been precluded
by the great difficulty in solving the appropriate
equations. Our approach has therefore been numerical,
using the Marker-and-Cell technique3 to obtain
high-speed computer solutions of the full Navier-Stokes equa-tions for a viscous, incompressible fluid. The purpose is to determine the flow dynamics during the splash, and thereby to explain some of the experimental results that
'P. V. Hobbs and A. J. Kezweeny, Science 155, 1112 (1967). 'F. H. Harlow and J. E. Welch, Phys. Fluids, 8, 2182
(1965).
7J. E. Welch, F. H. Harlow, J. P. Shannon, and B. J. Daly, The MAC Method (Los Alamos Scientific Laboratory, Los
Alamos New Mexico, 1966) Vol. LA-3425.
'B. J. Daly, Phys. Fluids 10, 297 (1967).
JOURNAL OF APPLIED PHYSICS VOLUME 38, NUMBER 10 SEPTEMBER 1967
The Splash of a Liquid Drop
Fa.aNcIs H. HnLow AND Jo P. SEANNONUniversity of California, Las Alamos Scientifia Laboratory, Los Alamos, New Mexico (Received 17 May 1967)
The full Navier-Stokes equations are solved numerically in cylindrical coordinates in order to investigate
the splash of a liquid drop onto a flat plate, into a shallow pool, or into a deep pooL Solution is accomplished
with the Marker-and-Cell technique using a high-speed computer. Results include data on pressures,
velocities, oscillations, droplet rupture, and the effects of compressibility. They also show how the technique can be applied to a wide variety of other complicated fluid flow problems involving the transient behavior of
have been published with little or no correlative
inter-pretation.
The splashing properties of a liquid drop of density
p are determined by several dimensionless parameters
that are functions of the drop radius R, the collision
speed u0, the depth of the splash pool D, the acceleration of gravity g, the kinematic viscosity coefficient z', and the surface tension coefficient T. (We assume that the
drop is spherical and that the fluid in the drop is the
same as that in the pool.) From these we form the four dimensionless parameters, R/D, (gR) "/uo, uoR/v, and
T/pRuo2, which characterize the flow.
If the drop had fallen with negligible air resistance from a height h, then the second dimensionless
pa-rameter would equal (R/2h) 1/2 Actually, air resistance
retards the drop, ultimately to a terminal speed that
depends upon drop size, but does not exceed about 10
rn/sec. The air tends also to deform the drop, and it
F. H. HARLOW AND J. P. SHANNON
FIG. 1. Splash of a drop of
radius 10.0 and impact speed
1.0 onto a fiat plate at times
¿=0,2, 6, 10, 14.
TIME
FIG. 2. Movenients of the droplet top, splash tip, and splash shoulder for the calculation shown in Fig. 1. Datum points are
Fic. 3. Photograph by Edgerton
)f the shallow-pool splash
of a
nilk drop.
nay even oscillate from prolate to oblate during fall,
)ut this effect is negligible for drop radii less than about
p.15 cm.9 Surface tension, which serves to maintain
phericity, also enters strongly into some aspects of the
nilision dynamics, but our calculations neglect these ifects. If T/pRuo2 is sufficiently small (a condition asily achieved with water drops) then the principal
ifects of surface tension occur at the edges of the thin
ateral sheet jets, which are of little concern to the
cincipal purposes of our calculations. The inclusion of urface tension in Marker-and-Cell calculations, how-ver, has been described by Daly and Pracht'° for some imilar investigations.
METHOD OF SOLUTION
The Marker-and-Cd! technique for high-speed corn-uter has been described in several publications,9-8 so iat only a brief description is given here.
The full (nonlinear) Navier-Stokes equations in lindrical coordinates (with axial symmetry) are
ap-roximated by finite-difference expressions correspond-ig to a mesh of computational cellsover the domain.
sing the finite difference incompressibility condition,
e obtain a Poisson equation for the pressure. The Ltial-value problems are solved by advancing the )flflguration through
a set of finite time steps, or
(cies. Each cycle consists of four steps:
(1) The finite-difference Poisson's equation for
D. M. A. Jones, J. Meteorol. 16, 504 (1959).
'°B. J. Daly and W. E. Pracht, Phys. Fluids, submitted for
tbhcat,on
pressure is solved over the mesh of calculational cells
by an iterative technique.
The momentum equations are used to advance the velocities through the time change of one cycle. Marker particles denoting center-of-m2q
posi-tions for fluid elements are moved through the cells to appropriate new locations.
Boundary condition values and time counters are, adjusted in such a way that the next cycle can begin. The results, like those of an experiment, are recorded periodically. These include the configuration of marker
particles, the velocities, pressures, and any desired functionals of those quantities. The illustrations of particle configurations and pressure surfaces in this paper were processed directly from the computer by the Stromberg-Carlson SC-4020 microfilm recorder and have not been retouched or otherwise altered.
Other functions have been graphed from printed data. The calculations themselves were performed by a code in Fortran IV on the CDC-6600 computer.
The numerical stability of these calculations has
been discussed by Daly and Pracht,'° extending the concepts described in the previous Marker-and-Cell method references, and illustrating an example of the powerful non-linear stability theory of Hirt."
The accuracy has been demonstrated by the usual tests of varying cell size and computational cycle
interval, and insisting that these be small enough for
negligible difference in the results.
"C. W. Hirt, j. Computational Phys., submittedfor
pub-lication.
i
3858
F. H. HARLOW AND J. P. SHANNON
i
No novel methodological features have been intro-duced beyond those previously described. The use of cylindrical coordinates is a straightforward extension
of the technique as developed for Cartesian coordinates. (We now believe that even in Cartesian coordinates it is useful to base the difference equations on the viscous
term vVXVXu rather than i'V2ii,
as previously described. The result is considerable simplification,rather than increased accuracy.) The Marker-and-Cell
technique has now been generalized to incorporate the correct free surface stress conditions, but these are not
well demonstrated in the present studies, in which the
effects of viscosity are negligible. Recent calculations
by J. P. Shannon on the Coanda effect and by Hirt and Shannon on hydraulic jumps demonstrate the
FIG. 4. Shallow-pool splash of a drop of radius 10.0 and impact speed 1.0; R1D5.0. The frames are times ¡=0, 10, 20, 50.
tremendous importance of the viscous contributions to the free surface stress for flow problems with appreciable viscosity.
It should be emphasized that the Marker-and-Cdl
technique is equally applicable to confined-flow
prob-lems, and to circumstances in which there are two or more fluids of different densities.8 Even in those examples, the technique appears to have advantages over others that use the vorticity and streamfunction as primary variables.'
-SPLASH ON A FLAT PLATE
In all calculations described in this paper, the effects
of surface tension and viscosity have been neglected.
FIG. 5. Shallow-pool splash of a drop of radius 10.0 and impact speed 1.0; R/D= 1.0. The frames are at times £=0, 10, 30, 50. ccordingly, the results are completely determined by
he parameters R/D and (gR) 112/u0. The splash ontoa [at plate is characterized by R/D= .
Figure 1 shows a typical sequence of results. The iagnitude of (gR)"2/uo is zero. As observed experi ientally, such a drop does not splash upwards, but
stead forms a lateral sheet jet.
The calculation shows that the jet
moves with. constant speed,1.6 times greater than the initial
impact speed of the droplet. Actually the results of Worthington' show that the lateral jet speed decreases with time, presumably because of the surface tension. Measurements from Worthington's figures (Ref. 1,p. 137) show, nevertheless, rather close agreement with
...___s.s_
3860
F. H. HARLOW AND J. P. SHANNON
DROP BOTTOM HEIGHT
IO 20
- TIME
DROP TOP HEIGHT
FIG. 6. Movements of the drop top and bottom and of the splash
tip radius and height for the calculation shown ¡n Fig. 5.
the computer results until the time 1= 12, as shown in.
Fig. 2. Thereafter the experimental jet tip slows down and eventually is pulled back inwards by the surface
tension
The calculation shows a second outward moving wave 'iith speed u; this is the shoulder that showswell
I
in the latest time frame of Fig I Theflow behind th j
shoulder is like that which occurs in an impinging jet,. for which the Bernoulli theorem indicates a later velocity equal to uo in steady state. The novel featur here, therefore, is the thin sheet of much faster fluì that is initiated during the first stages of impact, when steady-state theories are not applicable. With its rela-tively great energy density, the fluid in this sheet jet
is capable of considerably enhanced erosive or cutting
power, an important consideration in agricultural engineering.'
The calculation also represents the collision of two drops that impact along the line of centers. The lateral sheet jet speed would then be 0.8 times the initial
relative collision speed.
'When the pool depth is comparable to the drop radius, then the splash properties vary rapidly with RID. To emphasize these we first examined anumber
of cases for which (gR) '12/u0 is negligible small. Forall
calculations in this section, R= 10.0 and uo= 1.0, thereby establishing the units of time, pressure-to-density ratio, etc.
Fia. 7. Pressure surfaces corresponding to the last two frames of Fig. 5. Pressure is vertical, z coordinate is to the right, rcoordinate is to the left. These are unretouched computer-generated plots.
THE SPLASH OF
The striking feature is that for RID large but finite,
the splash is distinctly different from that for R/D=oo.
Even a very thin film of fluid in the pool is enoughto
interact appreciably with the lateral sheet jet to produce an upward motion. This can be seen in the photograph
by Edgerton reproduced in Fig. 3, and in the shallow-pool calculation shown in Fig. 4, for which R/D=5.0.
The contrast with Fig. i is considerable.
One feature of disagreement between Figs. 3 and 4 is seen in the experimental tendency to form droplets
at the edge of the splash sheet. This is precluded in the calculations by the ommision of surface tension and by
the restriction of axial symmetry. (Three-dimensional
calculations cannot be resolved sufficiently by
present-day computers to make the full calculation practical. The Marker-and-Cell method, however, is completely
applicable to three-dimensional investigations without requiring any basic modifications.)
Figure 5 shows another calculation; in this case
¡2
L.
Io 20
TIME
FIG. 8. The pressure-to-density ratio at the plate, below the impact point, as a function of time for the calculation illustrated
in Fig. 5.
R,/D= 1.0. Even for this great
a pool depth, the
tendency to empty the region of impact is quite marked.
Many of the properties of the shallow-pool splash are shown by the sequences of fluid configurations.
Figure 6, for example, illustrates several of the features
that can be measured. Notice that in the absence of gravity, the lateral splash sheet grows in height and
radius with uniform speed after an initial establishment period. This is true for all of the calculations with RID
in the interval from 1.0 to 3.0. The results can be expressed for the rate of radius increase, u,., and of
edge height rise, u6, as follows:
Ur/'UO=0.77O.2Ø(D/R)2
u.Iuo=0.36_0.Ô5(D/R)2
The results for a shallow pool are in agreement with
those that can be measured from the photographs of
Edgerton and Killian,2 from which we obtain u,./u0= 0.8±0.1 and uJuo=O.4±O.1.
A. LIQUID DROP
.35j
I-u, z 0.4 ¡Q u, u, w o-0.2 20 RADIUSFm. 9. The pressure-to-density ratio at the plate, as a function of radius, for various times, for the calculation with R/D=2.0.
In addition, there are properties of the flow, suchas
pressure, that cannot be measured directly from the fluid configurations. Figure 7 shows pressure surface plots for the same problem as illustrated in Fig. 3, and for the same times as the last two frames shown inthat figure. In Fig. 7, the vertical axis in the distance is the
ordinate of pressure, whose base rests at the inter-section of the cylindrical axis with the bottom of the pool. Radiating from that base are the cylindrical axis
to the right and the bottom of the pool to the left. Each plot is scaled relative to the maximum fluid pressure at that instant; this variation with time is
shown in Fig. 8.
Another plot of pressures is given in Fig. 9, which shows, pressure-to-density ratio along the bottom of
the pool at various times for a calculation in which.
Q6 0.4 0.2 ¡O I 20 -u IO 20 RADIUS
Fic. 10. Radial velocity component at the plate, as a function of radius, for various times, for the calculation with R/D=2.0.
.R/D= 2.0. From data of this sort impulse distribution on the bottom can be calculated in whatever forni one
desires.
For this same calculation, the radial velocity along
the bottom is plotted for several times in Fig. 10. These results can be used to estimate shearing stress and drag erosion, provided that the boundary layer is sufficiently
thin
SPLASH IN A DEEP POOL
For a deep pool, the crucial splash parameter is (gR)"2/u. With present computers, calculations can
FiG. 11. Deep-pool splash of a drop of radius 5.0 and impact speed 1.0, with (gR)1"/uo=O.707. The frames are at times £= 10, 20
25, 35, 50, 75.
be performed only for relatively large values of the parameter; for low values (high impact speed) the
early-stage crater size is too large to be resolved.
The three examples chosen for illustration have R=5.O and g=0.l, and initial impact speeds such that (gR)"2/uo=0.707 (Fig. 11), 0.354 (Fig. 12), and
.177 (Fig. 13). These correspond, respectively, to fall
distances of R, 4R, and 16R. At the extremes, they
demonstrate quite different types of droplet distortion.
In Fig. 11, the original droplet fluid maintains its
integrity, whereas in Fig. 13 the droplet is broken into
two parts, one remaining below the surface in a fairly
Pio. 13. Decp.poolsplash ola drop of radius 5.0 and impact speed 4.0, with (gR) "/uo=O.177. The frmes are at times i=10,20, 25, 35.
compact volume, the other spread out as a film on the
surface.
Sahay' has observed this distinction experimentally, and has determined the critical falling distance as a
function of drop size for water falling into viscous oils
of lower density than water. For shorter falls than the critical, the water drops retain their integrity during splash; otherwise they break into two. Sahay shows that the critical fall distance is inversely proportional to the droplet mass. In the absence of viscosity or surface tension, the scaling parameter shows that the critical height is proportional to the cube root of the
droplet mass, while if viscous forces are dominant it is
inversely proportional to the two-thirds power of droplet mass. Thus, the results of Sahay indicate a strong effect of viscosity, in contrast to the results
presented here. Nevertheless, we believe that the basic
D 20 Io -Io- . -20 40 60 TIME
Fzc. 14. Height above initial surface as a function of time of the bottom and top of the initial drop material for the calculations of Fig. 11 (solid) and Fig. 12 (dashed).
3864
F. H. BARLOW AND J. P. SHANNON
z
'J
THE SPLASH OF A LIQUID DROP
V
3866
F. H. HARLOW AND J. P SHANNON
mechanism responsible for the rupturing of splashingdrops is as we present it here, and that viscosity serves
only to modify the quantitative aspects by giving the
droplet an apparently smaller impact speed
In all three illustrated examples, the initial stages
are quite similar. The droplet material is deposited onto the bottom and sides of a crater, the bulk of the material lying near the bottom. As the cavity sides collapse, the
bottom simultaneously tends to rise back towards the original surface level. Thus, two processes compete. For a low-speed impact (Fig. 11) the crater is shallow and the droplet rebounds to the surface before the sides collapse. For a high-speed impact (Fig. 13), the crater is much deeper and the sides collapse earlier, trapping much of the original drop material well below
the surface. In this latter case, the materialdeposited
on the sides of the crater is carried up in the rebound
splash, isolating two different parts of the original drop fluid from each other.
If the pool fluid were the less dense and the two were immiscible, then each part of the droplet would coalesce and the two would drift slowly downwards. In our cal-culation, the larger of the two parts would be the lower,
so that they would not tend to reunite. In the absence of any commeuts in this regard by Sahay,4 we expect that this explanation pertains directly to his
observa-tions.
Worthington' implies that the drop material is all
contained in the rebounding central column. The
experi-mental evidence is based upon photographs of milk drops falling into water and of droplets coated with a thin layer of soot. Our calculations show that part of the original droplet fluid does indeed rise as a coating
to the rebound jet, thereby agreeing with Worthington's
observations. But much of the droplet fluid can be
trapped in a pocket below the surface, and we
suppose-that this was not visible in Worthington's experiments because of the refractive properties of the water.
Figure 14 illustrates the motion of the top and
bottom of the droplet for the calculations shown inFigs. 11 and 12. The oscillatory motions are related to
those of the Cauchy-Poisson problem discussed by Lamb'4 and Stoker,'5 but an analytical treatment of
low-impact-speed splash problems appears to be much
more difficult.
Estimation of the CauchyPoisson
"H. Lamb, Hydrodynamics (Dover Publications, New York, 1945) 6th Ed., pp. 384 430.
"J. J. Stoker, Waler Waves (Interscience Publishers, New York, 1957), p. 156.
period of oscillation for the examples of Fig. 14 gvr results that are approximately one-third of those inri.
cated at the latest times in the figure.
The heaving oscillations of a freely floating bod a: also related. John" has discussed this for some siinpliÍii
examples, obtaining results that qualitatively rcn'i,
those in Fig. 14. Again, a really pertinent anah-sis fr,e the splash problem appears intractable.
EFFECTS OF COMPRESSIBILITY The impact of certain plastic or metallic spheres rns
produce melting, in which case the splash resemb
the incompressible processes already discussed. As t'.r
impact-speed increases (approaching the order 4
5 mm/sec) the effects of compressibility b«oee
progressively more important; the material vaporierduring impact and subsequently behaves much like a gas. An example to show this extreme is given in )a
13. The material is a polytropic gas with y=. Tz
radius of 10.0 and impact speed of 1.0 are the saine &
in Figs. 4 and 5, but the contrast with those figwr
shows several important effects of compressibility. At first there is almost no splash. Shocks are gencrat'l in both the drop and the pool, but their lateral mot'..
in the pool is relatively slow so that very little expans.i
occurs in the surface. As soon as the shock reaches thr
top of the droplet, there is a strong reexpansion IbIs
violently expulses the material back out in the dirrtt
from which it entered. The outward moving
sxt'!
5the droplet top is very close to the initial impart s;w'
This set of calculations was performed by A. .1.
Ainsden, using the Particle-in-Cell technique forbr
speed flows.'7"8
ACKNOWLEDGMENTS
We are grateful to J. E. Welsh for theMarke, a
computer code, to A. A. Amsden for the compr''-flow calculations, and to H. E. Edgerton
lot 1d
splashing drop photograph. This work was
perfr.
under the auspices of the United States AtomicCommission.
"F. John, Comm. on Pure and App!. Math. 2, 13 17 F. H. Harlow, Proccedings of She Sy,no3iJ i
.fl' "
Mathemalics (American Math Society, Providence, 115, p. 269.
-"A. A. Arnsden, The Pariide-in-CdlMethod r th( Cii.
*4"
of She Dynamics of Compressible Fluids L*s M]
