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WEAR ELSEVIER Wear186-187(1995)28-34
Thirty years of liquid impact research: a tutorial review
Martin Lesser Department of Mechanics, Royal Institute of Technology, S 100 44 Stockholm, Sweden
Abstract
Thirty years of research into the phenomena of liquid impact has led to a reasonable understanding of its basic mechanics. This understanding is summarized by the theory of guided acoustic waves. The basic equations and notions behind guided acoustic shock are derived and applied to several simple examples, most notably the so-called Brunton device. The theory is then applied to explain the main observations made in liquid impact experiments. A brief discussion is given of what major problems remain and what techniques may be used for their solution.
Keywords: Liquid impact; Guided acoustic waves; Brunton device
1. Introduction
The basic theory needed to understand the major features of high speed liquid impact has existed for over one hundred years. High speed in the context of the present article refers to impact speeds above 50 m s- ‘. Cook [ 11, writing early in this century, discusses liquid impact in terms of compression waves sweeping through the involved mass. He clearly rec- ognized the importance of the water hammer pressure, first analyzed by Joukowski in 1898 (see Ref. [ 21). While it did lead to some insight, this understanding did not lead to sig- nificant quantitative calculation. During the 196Os, the air- craft erosion community began to interest itself in the problem [ 31. It is interesting that the previous work appears to have had little influence aside from the simplest application of the water hammer idea. The turning point towards a renewed and extended application of the theory came in the mid 1960s with the work of Bowden and Field [4 1, and Brunton and coworkers [ 5-81. By the use of the developing techniques of high speed photography and by clever experimental design they were able to make visible a number of the features associated with the initial stages of impact. Field showed the importance of the contact edge speed being faster than the liquids acoustic speed. Brunton developed several new tech- niques for creating impact events and for viewing them. Camus and Rochester, working with Brunton, developed the technique of producing two dimensional impacts, which gave a clear visible picture of the drop dynamics. Rochester’s measurements clearly showed that the impact pressure reached its peak at the expanding contact edge. In addition Rochester applied some of the theoretical ideas developed by Skalak and Feit for the problem of water entry to the impact
0043.1648/95/$09.50 0 1995 Elsevier Science S.A. All rights reserved SSDIOO43-1648(95)07190-3
problem. Heymann [ 91 showed that near edge shock detach- ment and jetting it was necessary to calculate the edge pres- sure from nonlinear shock relations. Lesser and Field [ 101 treated a series of two dimensional impact problems involv- ing liquid wedges. This connected the impact problem to some earlier work on high speed transient liquid jets [ 111. Dear and Field [ 121, in an extensive thesis work using gels to produce shaped liquid masses, demonstrated the general validity of the acoustic shock concept for understanding liq- uid impact. Lesser [ 131 showed the applicability of geomet- ric wave theory, for calculating both wavefront geometry and pressure distributions. Adler [ 141 developed a careful series of experiments to produce highly accurate spherical drop shapes. Most recently Korobkin [ 151 has made a number of applications of the acoustic theory. He has also applied the ideas of transonic flow and modern perturbation methods to understand the jet formation process that is associated with edge shock detachment.
This short summary concentrates on what I feel are the high points of the work in this area over the last thirty years. It will probably be seen as incomplete by some, however the main purpose of this work is to provide the reader with a basic set of principles with which to understand liquid impact phenomena, and not to give an accurate historical summary of past efforts. It is probably notable that there is no mention of numerical investigations. This is deliberate as it is my opinion that such investigations, while occasionally useful, have given no real insight in the phenomena. Many have been poorly done, and those that have been well done have only confirmed their own validity by producing already well understood results. I do believe that numerics has an impor- tant role to play in the future and some mention of this will
Martin Lesser/ Wear 186-187 (1995) 28-34 29
Fig. 1. Two-dimensional liquid impact from Camus.
be made below. Reviews of past work can be found in the conference proceedings edited by Field [ 161 Korobkin and Pukhnachov [ 171, Lesser and Field [ lo] and Rein [ 181.
The main theme of this article is to provide what might be called a metaphor for understanding the important early stages of high speed liquid impact. For emphasis this is given a name and is called the guided acoustic shock or GAS theory. GAS provides a context with which to make basic calcula- tions, design experiments and interpret results. It gives us a falsifiable theory in the sense developed by Popper [ 191. Popper emphasized that we can never prove a theory, we can only find instances when it is incorrect, that is falsify it. This leads to new theories, the old theory still being useful in a domain restricted by the falsification. The point is that the original theory does explain many observations, hence we extend our understanding by trying to find cases where it fails. The regions of failure provide some of the most inter- esting extensions and new ideas. A major attraction of GAS is simplicity. GAS is not hard to learn or to use. While detailed calculations can involve considerable mathematics it is pos- sible to obtain many important and useful results with a min- imal amount of effort. With GAS one may becomes an effective consumer or producer of liquid impact research. Once mastered it effectively summarizes most of the known features of impact. Naturally it is unrealistic to expect to achieve mastery in the few pages of the present article. It is hoped that it will provide a starting point in the journey to understanding.
1. I. Explain the evidence
Fig. 1 from the work of Camus, shows the impact of a slider against an effectively two dimensional liquid drop sandwiched between two plates. The goal of this tutorial is to show how GAS can explain most of the phenomena seen in this picture. This includes the speed of the lateral motion or jets which are ten times faster than the impact speed, the formation of the dark zones where the fluid cavitates,the spray of material off the back face of the drop and the shape of the observed wavefront. In addition the theory explains the shape of the damage zone and the impact pressure distribution. The failures of the theory, that is its falsification, can also be used as a guide to achieve deeper understanding. This will be illustrated in one particular case.
Fig. 2. Jet generator.
2. Some basic GAS principles
The object of this section is to provide the reader with some simple explanations of basic phenomena such as the water hammer pressure equation and the jetting of part of the surface of an impacted fluid mass. For interest this will be done in the context of the so-called Brunton device for the study of liquid impact damage. The idea of this device is shown in Fig. 2.
A projectile strikes a shaped cavity containing liquid. The impact generates a shock wave which traverses the cavity and reflects off a free surface, creating a jet of fluid which is used for the liquid impact test. Such a device can achieve impact speeds in access of 300 m s-i at minimal cost and effort. Questions of optimizing the device’s performance can be assisted by GAS theory.
2.1. What is GAS?
GAS theory, as applied to liquid impact, rests on the fact that in typical cases the speed of the liquid mass is consid- erably lower than the liquids sound speed. This fact is expressed by the statement that the impact Mach number 44i or Vi/c the ratio of the impact speed to the liquids acoustic speed is small. The impact creates a sharp pulse, or shock, which moves through the liquid at the sound speed. Because this speed is so high the liquid mass is essentially undistorted during the first few passes of the shock and associated waves. The early distortion can be calculated by using the velocity created by this disturbance to calculate the surface displace- ments. Therefore the name guided acoustic shock theory was coined. The liquid’s original shape provides a guide for the compression and expansion waves created by the impact process. The resulting velocity field provides a means of calculating the liquid’s initial shape change. Critical points at which the frozen shape assumption is inaccurate can be understood by looking for effects either neglected or involv- ing new physics. For example when the local speed is com- parable to the acoustic speed or where the pressures are extremely high we meet the phenomena of the wave being
30 Martin Lesser/ Wear 186-187 (1995) 28-34
& ~c
m Fig. 3. Force-momentum balance on a liquid slug.
Fig. 4. x-t diagram for a simple jet generator.
convected or carried by the flow or the local state effecting the acoustic speed, An example of introducing new physics would be the problems involved with the state behavior of the liquid during violent expansions.
To introduce GAS it is useful to consider some simple situations. The simplest of what might be called a Brunton device is a straight tube of liquid being impacted by an “infi- nite” mass projectile moving at speed Ui. The infinite mass is simply a picturesque way of stating that the velocity of the projectile will not be altered by the impact. We assume that the disturbance created by the impact moves at the acoustic speed c. The basic quantitative relation needed to calculate pressure comes from Newton’s laws. The calculation is shown in Fig. 3.
The wave created by the impact travels the length of the slug L in time L/c. The cross-sectional area of the packet A multiplied by the pressure created behind the shock provides the force. The packet momentum changes from zero to pdL during the wave traversal time. Therefore equating the change in momentum to the applied force multiplied by the time interval shows that
Ap= fpac Au
The minus sign applies if we reverse the direction of the impact while maintaining the direction to the right as positive. Most of the most important results concerning liquid impact arise from these relations. They will be called the acoustic shock jump conditions. Their use is extended to multiple dimensions by considering the shocks to propagate in ray tubes. Acoustic shocks can be either expansive or compres- sive. Particular problems are solved by applying the jump relations and suitable boundary conditions. A few examples illustrate the idea. Thus a straight open ended tube (the sim- plest Brunton device) is filled with liquid and one end is
impacted as described above. This is best pictured in an x-t diagram, as shown in Fig. 4, where we plot the shock fronts and the boundary positions. The slope of the line representing a boundary or wavefront indicates the objects speed. The projectile moves with its original velocity after the impact as illustrated. Thus the surface of the liquid is forced to move at the impact velocity. The impact creates a shock which moves into the still liquid and strikes the free surface on the right. The shock reflects from the surface and this free boundary starts moving. This is the jet. Label the regions in the tube as 0, 1 and 2. Zero denotes the undisturbed state of the liquid, 1 the state induced by the impacter and 3 the state created by the shock reflection. The pressure here is taken as the value over ambient. The boundary condition at the free surface is thatp, = 0 while at the impact end we have ui = Ui. The shock jump condition over the advancing shock gives
PI = PGJI = POCU,
while for the reflecting wave we have
Pz-Pl= -Poc(u,-h)
Using the condition that p2 = 0 it is seen that
lJ2 = 2u,
The reflecting wave is an expansion, which doubles the fluid velocity, producing in the ideal case a jet with twice the impact velocity.
In a similar manner we can consider the situation where the steady motion of a fluid is stopped by a valve closing. This sends a wave upstream, as shown in the Fig. 5. The boundary condition on the closed valve is the stoppage of the flow, i.e. u2 = 0. The acoustic over-pressure in the flowing liquid is zero. Application of the jump relations gives the immediate result that the valve causes a pressure rise of p,cu,. This is the classic water hammer case.
To analyze a more complex Brunton device where the cavity undergoes a sudden decrease in cross-sectional area from A to B requires a new boundary condition to relate the local flow in the two regions. This is illustrated on the x-t diagram of Fig. 6.
The additional condition is that the flux of volume flow Au, on the left equals the volume flow into the right Bu,.
Pp-0=-p&(0- v,)
Fig. 5. Water hammer x-t diagram.
Martin L.esser/ Wear 186-187 (1995) 28-34 31
p,-O=+p,c(v,- 0) Pa - P, = -pp,e @a- VI) 1 p3 - 0 = +pOc (v3 - 0)
0 - P, = -pot (v, - VJ
Fig. 6. x-t diagram for jetting device.
Using this condition and the jump conditions between the regions of the diagram it is found that if Ui is the impact velocity and u, the “jet” velocity:
4ui *r=1+BIA
Before leaving this device it is useful to consider another approach where it is assumed that the area change is made up of a gradual transition. Taking limits in the situation where B the area change is given by A + dA one obtains the equation
P A
Or after integration
Ap2 =Apocup = constant
If p and u are taken as the jump across the shock we see that the jump in power flux Apu is a constant for a shock travelling down a tube of slowly varying area. It can be seen that this gives superior performance for a jet generation device. Most important for our purpose is that we can apply this relation to ray tubes in a higher dimensional situation. A more exacting analysis shows that this relation arises from seeking shock like solutions to the linear wave equation.
3. Guided acoustic shocks in two and three dimensions
The most interesting problems involve more than one dimension. To deal with them we need a few additional notions. These are the generalization of the shock jump con- ditions and the idea of a region of influence for a disturbance. The generalization of the jump condition is immediate. At any point on a curved shock surface we take the tangent plane to the surface as a local representation of the shock. This reduces the problem to the one dimensional one treated in the last section. In most cases we apply the additional condition that there is no jump in the velocity components tangent to this plane. Such a jump can be shown to be another type of discontinuity, called a vortex sheet, which will not be included in this basic discussion.
point disturbance line disturbance
rbance Boundary at time cl
Disturbance Boundary at time t confined region
PA incident wave on wedge rsates comer wave 1 lsturbance while reflected
1 wave tangents comer wave
Fig. 7. Regions of influence.
The idea of region of influence is basic to understanding any type of transient wave propagation problem. Disturb a point in the wave bearing medium. The fact of this distur- bance must now travel to other parts of the medium. This occurs through the action of the moving waves, which takes a finite amount of time. Therefore at any given time there is a distinct region which has been influenced by the distur- bance. This is the region of influence of the disturbance! While very simple this provides a powerful tool for under- standing wave motion. The frontier, i.e. the outer boundary, of the region of influence is the wavefront. Thus find the outer boundary of the region of influence and you have found the wavefront for the disturbance. This idea is illustrated for some important cases in Fig. 7.
4. Wedge and cylinder impact
As an example of these ideas consider the two dimensional problem of a liquid wedge impacting a rigid surface, a situ- ation treated extensively in the experiments of Field et al. [ 201. The rectangular wedge moves towards the surface at velocity Ui, which is assumed to be well below the acoustic speed of approximately 1500 m s - i . The disturbance region is the front edge of the wedge, as illustrated in Fig. 8.
After a time t the entire rectangular region with height ct will be in the range of influence of the front edge. This is not the whole story as we must consider the conditions at the boundary. At the front edge the condition is that the liquid must come to rest, hence the normal velocity must vanish.
Cl’ SF
t dt
P=mv,
F3: PC t= pOcvl ct m = (1/4)p, R czF
& = (2/X) Vi
I
Fig. 8. Rectangular wedge impact.
32 Martin Lesser/ Wear IS-187 (1995) 28-34
But the side of the wedge is not constrained, hence the corner point separates two very distinct regions. Above the corner point the condition is that the pressure at the free surface must match the ambient pressure, hence in the GAS theory the over-pressure must vanish! This condition “travels” from the corner point into the liquid. Therefore the range of influ- ence of the comer point denotes the region of fluid that has knowledge of both the impact and the free edge condition. This region is seen to be the indicated quarter circle of radius ct. Thus we are concerned with two wavefronts, the plane “shock” produced by the impact and the corner wave, car- rying the information that the edges are free into the region of shocked liquid. The fluid in region S is at rest, so the pressure in this region is easily calculated to be the “water hammer” pressure pocui, as it is exactly the same problem we dealt with above for the closing valve. The region C, encompassed by the comer wave is somewhat more complex in structure, however we can apply simple Newtonian ideas to obtain the average velocity of this region. Again referring to Fig. 8, note that the mass of the involved material is purrs hence its momentum is V, times this. V, is the average lateral velocity of this region. Note that we are work- ing in the context of GAS hence linear approximations, so we do not take into account the mass disturbance in the region. The lateral force on the expanding quarter circle region is given by the normal pressure. The pressure is the water ham- mer pressure and the resultant lateral force term is the pressure times the projection of the circle onto its radius, i.e. PocVi ct. Applying Newton’s law that the rate of change of momentum equals the applied force we find that the lateral “jetting” velocity is:
VA J Tr.’
This shows some of the power of GAS in that some very simple ideas lead to a useful result which can be checked by photographic observation. It is not very hard to extend this calculation to the impact of a cylinder on a surface. The impact shock is a plane moving away from the surface. Instead of a comer wave we have a quarter toroid centred on the impact circle. The formula for the average velocity is a bit more involved, but the concepts are the same.
4.1. Edge transition
We can come to grips with what occurs when a curved liquid surface impacts a target, by considering a family of wedge impact problems as indicated in Fig. 9.
What is important here is the speed of the edge contact point Q, which can be obtained from the geometry shown in figure. When this is faster than the wave speed the free surface is unchanged by the impact. At transition it moves at the wave speed and above this the free surface is effected and relief waves move into the shocked region. The shapes of the regions are found by applying the idea of region of influence. The distinctive feature of the pre-transition case is the
t M -vi
i- C
Mi = sin pc Fig. 9. Angular wedge impact.
attached sloping planar shock. The pressure in this region is easily calculated using the jump relations and the condition that the velocity component normal to the impact boundary must vanish. The calculation of pressure is simple in principle and gives the result that
PW Mi -= pocvi sin’/l + cos /3 (M,’ - sin’p) ’ ”
The geometry also shows that the critical angle for transition is given by
sin & = Mi
The critical pressure is
PC = pot*
The picture this gives us is that the maximum pressures occur at the edge of the impact. When release occurs they will drive the fluid at speeds in the order of the acoustic speed. While the picture is true qualitatively, it must be wrong in quantitative detail as the relative speeds are too rapid for the GAS assumption of wave propagation over an undistorted region. At other positions in the flow GAS is both qualita- tively and quantitatively correct. Experiments however show that the transition appears to occur at angles somewhat larger than PC. The possible reason for this is due to the deformation of the target. This is discussed together with the problem of spherical and circular drop impact below. Free Surface Reflection In the case of a collision with /l > PC the region of influence argument leads to the need to consider a reflected wave at the free surface. The reason for this is seen by con- sidering the envelope curve of circular zones of influence generated by the travelling edge point. This also shows that the angle of incidence of the reflected wave must equal the angle of incidence of the edge shock, due to the assumption of constant wave speed. A simple way to solve this problem is to consider two plane waves moving in space and crossing at a point. The requirement is that one is a compression the other an expansion and that the pressure vanishes in the region effected by both waves. This leads to the result that the veloc- ity imparted normal to the free surface is given by 2u, sin y where y is the angle between the shock normal and the free surface and v, is the velocity behind the advancing shock.
Martin Lesser/ Wear 186-187 (1995) 28-34 33
The reader can check that this gives the required result we found in the discussion of the Brunton device above. Using this velocity we can find the shape of the free surface in the zone between the reflected wave and the advancing corner wave.
5. Drop impact
Fig. 10 shows a typical drop impact sequence. The pic- tures come from the experiments of Camus, however they have been processed so as to remove “noise” and elements that are not essential to the present discussion. Thus they emphasize the wavefronts and surface jetting phenomena. Most of what we observe in the figure can be understood in terms of the GAS picture. The initial impact leads to a shock leaving the contact zone. On a length scale of the critical radius, i.e. the radius at which the side slope angle matches p, this front will have a non circular geometry which can be found using the region of influence principle. When the crit- ical angle is reached the high edge pressure is released gen- erating a downward jet of liquid, leading to a damage mark on the impacted surface. Release waves sweep into the drop and the shock detaches from the contact surface. On the scale of the drop size the initial high pressure impact zone is small and the wave appears as a circle originating at the impact vertex. As the shock sweeps up the drop the angle with the edge changes. At a point near the rear of the drop this leads to velocities of the edge that are high enough to generate an observable motion of liquid. This effect peaks at the rear of the drop. Meanwhile the relief wave generated along the drop edge converge on some point near the drop rear edge. This leads to the focusing of relief waves at some point near the rear edge.
We can obtain some better understanding of the observa- tions by introducing some new physics into the picture. Thus we can expect the relief waves to generate negative pressures or tensile stresses in the liquid. This occurs at the rear of the drop, the rear focus point and at the impact boundary where the edge decompression waves cross. This leads to cavitation
Fig. 10. Two-dimensional drop impact sequence
in the interior zones and a jetting of material at the drops rear. Experimental evidence also leads to damage zones and pho- tographic evidence that show a larger value of p, then obtained from GAS. The simplest explanation of this is that the target material can deform. Since acoustic shocks are also generated in the target material, and the wave speed is in general larger, it can be expected that the target will deform before the arrival of the contact edge. This will change the angle between the surface and the drop edge in such a manner as to delay the onset of shock detachment. A simplified dis- cussion of this is to be found in Lesser [ 131. Thus we see how falsification of GAS leads to improved understanding of the entire process.
Quantitative results can be obtained by use of the region of influence principle and the acoustic jump conditions. This is not discussed here, but it should be appreciated that the calculations are not complex and can be easily extended to three dimensional geometries. In fact the jump conditions together with the acoustic shock propagation law give the exact solution of the full acoustic problem at the wavefront.
Space limitations have only allowed a discussion of a few of the interesting situations which can be understood. Others are shock induced cavitation, the generation of “micro-jets” at curved free surfaces, central pits inside impact craters in multiple impact and focusing due to complex impact geom- etries. One important lesson is that the impact geometry is very crucial in determining the impact events. For example an oblate drop will maintain the high impact pressures for a longer period due to the lower edge angle.
6. Future efforts
Much of the significant theory is in place, including useful calculational tools based on the acoustic jump conditions and the region of influence principle. In some cases complex geometry makes it necessary and reasonable to use simple numeric procedures. What remains is to make use of the theory as part of general computational schemes that treat the surface motion, both numeric and analytical. Therefore con- siderable future efforts may simply make more use of basic GAS theory. GAS also can be extended in several useful ways. For higher speed impacts the nonlinear effects of con- vection of wavefronts and change of wave speed with state can be included. One way to do this is to use the methods of geometric shock dynamics or GSD as developed by Whitham [ 211 and later by Prasad [ 221. Some of this is indicated in the paper of Apazidis et al. [ 231 presented at this meeting. Within the context of GAS there remains the problems of dealing with zones where the theory undergoes a local break- down. For example the shock detachment point in curved surface impact. Korobkin [ 151 has made some important advances in dealing with these problems. In many cases they also require some new physical ideas as the thermodynamic behavior of the medium appears to play an important role. Experimental work can also be expected to continue with the
34 Martin Lesser / Wear 186-187 (1995) 28-34
tremendous improvements we have seen already. The mul- tiple impact methods being developed by Field and his group [24] at Cavendish Laboratories is an example of one such direction. Naturally the new experimental data will need new explanations. In the final analysis GAS will still remain the basic tool for understanding. If we have no expectations we cannot be surprised, for all is possible. GAS provides us with expectations. The surprise we have when they are not what we expect is the grist for the mill of creativity in this subject. Anyone truly concerned with these problems must understand the basic picture.
7. Literature
The basic methods used in GAS and its nonlinear modifi- cations are to be found in the monographs of Bleistein [ 2.51, Friedlander [ 261, Prasad and Whitham. Whitham’s book is probably the easiest to start with. Friedlander’s book provides most of the basic techniques needed to obtain quantitative results using GAS. Bleistein gives some introduction to the treatment of diffraction of waves by geometric methods. Pra- sad introduces a technique for obtaining a rational series of improvements to the nonlinear shock propagation problem. The geometric theory of diffraction has become a standard tool in the area of electromagnetic wave propagation. Some introduction to this can be found in the book by James [ 271. A complete discussion of the physics of shocks in water can be found in Ridah [ 281.
Acknowledgements
It is a pleasure to thank John Field, John Dear, John Brun- ton and John Camus for the many insights, corrections and generally interesting conversations they have provided me with over the years I have been interested in this area. I would also like to thank Alexander Korobkin for the new insights and important calculations he has added to this subject.
References
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approach to liquid impact. ELSI VIII, 1994. 1241 C.S.J. Pickles, CR. Seward and J.E. Field, Important considerations
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Cambridge, 1958. 1271 G.L. James, Geometrical Theory of Diffraction for Electromagnetic
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[28] S. Ridah, Shock waves in water, J. Appl. Phys., 64 (1988) 152-158.