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Bulk cavitation in a vertical water filled shock tube
Journal of Sound and Vibration (1982) 84(4), 563-572
B U L K C A V I T A T I O N IN A V E R T I C A L W A T E R FILLED SHOCK TUBE M. R. DRIELS
Department of Mechanical Engineering, University of Edinburgh, Edinburgh EH9 3JL, Scotland (Received 28 October 1981, and hz revised form 26 Februat3, 1982) A one dimensional lumped parameter theoretical model is developed to predict the occurrence of recompaction waves associated with bulk cavitation following the reflection of a pressure pulse from a free surface. Experimental data is obtained from a simple shock tube for comparison with the theoretical model and with that of Cushing whose work covers the classic explosion pulse. It is concluded that the model presented here predicts the occurrence of recompaction waves reasonably well and can have significant advantages over previously published work.
1. INTRODUCTION When studying the effects of an underwater shock wave either on the free surface or a submerged structure it is convenient to consider a simplified, one dimensional approach to the propagation of the wave. Recent work.J1] in which this assumption was used has shown that large scale rupture of the water (bulk cavitation) near an underwater structure can influence the interaction between a shock wave and the structure. This can lead to a greater degree of d a m a g e being inflicted on the structure than that in the case where the water does not cavitate. Several reasons were suggested in reference [1] why the water need not cavitate at the prevailing vapour pressure but might be able to sustain a considerable dynamic tension. It was then shown how the increased d a m a g e is influenced by a variable cavitation pressure or m o r e appropriately, breaking tension. Since the overall objective of the study is the prediction of damage, it became necessary to investigate the magnitude of breaking tensions and the conditions under which these values were non-zero. Much work has been done to this end by using a water filled shock tube, some of the m o r e comprehensive studies being those of Davies and T r e v e n a [2] and of Sedgewick and T r e v e n a [3]. In these experiments a short vertical steel tube with a freely sliding piston at the b o t t o m is filled with water. When a bullet is fired at the piston a shock wave resembling that of an underwater explosion is p r o p a g a t e d into the liquid column. This shock wave travels upwards to the free surface, is reflected, and begins to travel downwards. By considering the superposition of incident and reflected waves it can be shown that the water will cavitate at a certain depth, a void then occurring across the whole of the tube. This void is the region of bulk cavitation. T h e p h e n o m e n o n was developed one stage further by Cushing [4] who looked at the dynamics of the spall of water lying above the cavitation zone together with the effect of spray which is projected upward and accretes to the spall. Cushing, like ourselves, was concerned with underwater explosions which produce a characteristic pressure pulse consisting of a sharp rise to some p e a k value po followed by an exponential decay back to atmospheric. Unfortunately some initial experimental work intended to verify Cushing's theory showed that the pressure pulse generated by our experimental apparatus 563 0022-460X/82/200563 + 10 $03.00/0 9 1982 Academic Press Inc. (London) Limited
564
M . R . DRIELS
was not the classic shock but one with a significant rise time. Preliminary theoretical work also suggested that subsequent spall motion predicted by the model could be sensitive to the accuracy in specifying the incident pressure pulse. Further experimental work [5] in which a long horizontal tube was employed to investigate shockwave/target interactions demonstrated that although the pressure pulse initially has the form of the explosion wave, it becomes modified in shape as it progresses through the water. By the time the wave reaches the other end of the tube and interacts with a second sliding piston representing the target, it is not recognizable as a classic explosion shock. Because of these arguments it was decided that in order to study the shock wave/target interactions properly with the longer horizontal tube, a theoretical model would have to be developed capable of accommodating any shock wave input rather than being restricted to the exponential wave shape. This paper describes the formulation of such a model, together with some comparative experimental results obtained with the shorter vertical shock tube. 2. ANALYSIS Although the model is to be used for any wave shape the analysis will be done for the exponential wave. This will allow direct comparison with Cushing's results which have been summarized in the Appendix. The model is one in which one assumes that the input pressure pulse can be divided into a finite number of elements. Figure 1 shows
Pl
--'~Plle
P,,
<--z
[
I.
-11
.I Q
Figure 1. Definition of incident pressure pulse.
the input pulse travelling towards the free surface, the pulse itself being characterized by the number of data points, the peak pressure and the wavelength. The time datum t = 0 is when the wavefront arrives at the free surface and reflection begins. Subsequently the pressure field in the water is given by the sum of the incident wave, reflected wave, hydrostatic pressure and atmospheric pressure as shown in Figure 2. Basically, the wave is moved onwards one increment at a time and the new total pressure field is redetermined. If it is assumed that cavitation is prevented by the water sustaining any magnitude of tension then wave reflection is completed as shown in Figure 3(a). The water also acquires a velocity distribution due to the passage of the wave and this is shown in Figure 3(b). As would be expected, far from the surface, the reflected wave will be the negative of the incident wave although the increasing hydrostatic pressure will ensure that the tension finally disappears.
BULK CAVITATION IN A WATER SHOCK TUBE
565
Pi ,...1 1
/ k Figure 2. Pressure distribution following reflection.
{ol Figure 3. Spatial pressure and velocity distributions for non-cavitating reflection.
As the wave is incremented along, the pressure at section m from the free surface is given by pm =p~ = p , +Pa + p g ( m x s ) ,
(1)
where s is the length of each incremental section. If however the water is unable to sustain any tension then cavitation will occur at the section where the pressure falls to zero first. Up to this instant the fluid particle velocity is given by u = (1~pc)[p, +p,].
(2)
Following the onset of cavitation at depth Zo (defined by Cushing as the onset depth) the program continues to move the incident wave towards the surface and the reflected wave away from it. This results in the cavitation of more water and the spread of a cavitation front downwards from the onset depth. As each incremental layer is traversed by the cavitation front its velocity of projection is calculated from u = u~ + u , ,
us = pdpc,
u, = [pa + p g ( m x s) +pi]/pc,
(3)
(4)
since the reflected wave can only reduce the pressure to absolute zero. It might be expected that cavitation ceases once the hydrostatic pressure makes the total pressure positive; however Cushing has argued that the lower limit of cavitation occurs when
du (z)/dz > 0.
(5)
566
M.R.
DRIELS
It is assumed that below this limiting or termination depth zr, the water is essentially quiescent. Cushing's "instant cavitation" approximation is also adopted for the model whereby it is assumed that the cavitation front spreads so quickly through the cavitated zone that the water particles do not move appreciably during that interval. Representative results from this first phase of the analysis are shown in Figure 4. H e r e an exponential wave of 500 data points, peak pressure 20 atm and wavelength 0.5 m is reflected from the surface. Each spatial increment corresponds to 0.010 m. Onset of cavitation occurs after two incremental movements of the pulse at Zo = 0.020 m, cavitation terminating at zT: 1.86 m. Figure 4 also shows the velocity distribution between the surface and zr. -3
187
186
du dz -2
Zf
-3
Zo
Figure 4. Spatial velocitydistribution for Po= 20 bar, h = 0.5 m. In the second phase each element shown in Figure 4 is considered to be a rigid body with the appropriate velocity obtained from the associated distribution. This distribution is shown in Figure 3 and indicates that the spall velocity is upwards. Near the surface is the spall consisting of two elements. Its velocity is calculated by averaging the component Velocities. Although its velocity is greater than that of the underlying water it will have a higher downward acceleration due to the air pressure a b o v e it. This acceleration is given by
:
as = g + pa/ps',
(6)
where s' is the spall thickness. Each layer of spray, of thickness S, below the original spall has zero pressure acting on both upper and lower faces. The layer deceleration is thus due to gravity alone. Hence the first layer of spray will catch up with the spall and collide with it. If one assumes m o m e n t u m to be conserved at impact the new velocity and mass of the spall+ 1st layer can be calculated. The general collision case is shown in Figure 5 where the ith layer of spray collides with the spall after a time t. This time is the solution of
t2[pd2ps'] + tc[ui -tts] + [zi + s - zs] = 0,
(7)
where ui and tts are the spray layer and spall velocities at the beginning of the ballistic flight interval. Hence the momenta of spall and spray just prior to impact are
Ms = p s ' [ u s - ( g +p~/ps')tc],
M1 = p s ( u i - g t ~ )
(8, 9)
and the new spall velocity is
u's = (Ms + M1)/(s' + s).
(10)
BULK C A V I T A T I O N IN A W A T E R S H O C K T U B E
567
Impact Iqallislic
flighl
I Dolurn Figure 5. General case of spall/layer collision.
This process has been defined by Cushing as c!osure from the top. Closure of the cavitation void can also occur from the bottom since the acceleration due to gravity may result in a layer of water failing to catch up the spall. If this happens then the layer follows a ballistic trajectory returning to its starting position without colliding with the spall. Closure from the bottom occurs at the layer in question if tc > 2 u o / g ,
(11)
where llo is the initial velocity of the layer. Final closure of the cavitation zone occurs when closure from the bottom spreading upwards meets closure from the top spreading down. At this instant the spall together with the reconstituted layer of water from projected spray is moving downwards at an appreciable velocity causing a significant recompaction pressure wave to be propagated downwards. The study of this recompaction wave is important since its magnitude suggests that it could also contribute to the damage sustained by an underwater structure. Assuming that at recompaction as much energy is propagated into the underlying water as is propagated back up the spall, one has that the amplitude of the recompaction pulse is given by ~pc u,,
(12)
where tic is the spall velocity relative to the underlying liquid at closure. The time of occurrence of the pulse is determined by the spall dynamics outlined above while the pulse duration, if closure occurs above the upper transducer, is r = 2s'/c.
(13)
The computer program which embodies this theoretical model keeps track of each element within the cavitation zone, calculating the collision time of the enlarging spall with the layer of spray beneath it. Then, by using equations (8)-(10), the new spall
568
M.R. DRIELS
parameters are calculated, the new spatial co-ordinates of the other layers calculated and the process repeated for the next layer down. The final spall thickness is identified when equation (11) is satisfied. 3. EXPERIMENTAL RESULTS Although a comparative assessment of the shock tube may be found in reference [5], a brief functional description will be given here, for completeness. The tube itself is 2.7 m long, with a 30 mm bore and a 35 mm outside diameter, and is rigidly mounted on a vertical steel girder (see Figure 6). A series of five 6 mm tapped holes are located
Wo~er level /,~
Pressure transducers
/
1"5'm
,I
Storage oscilloscope Sliding piston Guide tube
~
I I
Chort r
_,-..--Steel slug
I Poeomot,c tom
Figure 6. Schematicof experimentalapparatus. along the length of the tube to accommodate AVL 6QP 500 quartz crystal pressure transducers which are connected to charge amplifiers..The signals from the charge amplifiers are recorded on a Gould]Advance OS400011 digital storage oscilloscope which has the facility for producing hard copies on an X - Y plotter. Water is contained within the tube by a freely sliding piston at the lower end. This piston is suspended on springs and a water seal is effected by means of a pressurized greasing system. Mounted directly beneath the shock tube is another similar tube containing a small (2 kg) steel projectile. A high speed pneumatic ram accelerates the projectile in an upward direction but the ram itself reaches the end of its stroke when the projectile is about 1 cm from the piston. By ensuring that the projectile continues its motion under free flight conditions, bodily motion of the water in the shock tube is avoided. Usually two transducers are
569
B U L K C A V I T A T I O N IN A W A T E R S H O C K T U B E
I
0"5 ms
I f0 bar
Upper transducer
-Lower transducer Figure 7. Wave propagation in the shock tube.
f
0 bar
I . 5 ms
I
[] ' ~<~Predicted
I
"
Predicted
,:J
1
;
Upper transducer
Predicted R
i
Lower transducer (o)
(b)
Figure 8. Experimental result for (a) Po = 28 bar and (b) Po = 14 bar.
f
0 bar
I
O0 HS I _ _
(o)
Ib) Figure 9. Effect of filtering. (a) 50 kHz; (b) 10 kHz.
10
M, R. DRIELS
;ed in each test and the oscilloscope is triggered when the pressure wave passes the )wer one. Figure 7 shows representative transducer outputs, over a short time scale. Representative results can be seen in Figures 8(a) and (b) which show the wave ropagating from the bottom of the tube upwards past the two transducers, and reflecting 9om the free surface (point R, Figure 8(b)), together with the recompaction wave some me later. The precise form of the generated pulse is shown in Figure 9 for two different harge amplifier filter frequencies. It can be seen that this filter attenuates the high requency oscillation attributable to pipe vibration and has little effect on the measured ise time of the pulse. 4. DISCUSSION OF RESULTS The overall validity of the theoretical model is assessed by the prediction of the ecompaction pulse, specifically its time of occurrence, (relative to the passage of the ncident pulse past the upper transducer) magnitude and duration. These results are ;ummarized in Table 1 for the two sets of experimental data shown in Figures 8(a) ~nd (b). TABLE 1
Summary of results Time of pulse (ms)
28 14
Magnitude,/~ (bar)
Duration, r (ms)
Predicted
Measured
Predicted
Measured
Predicted
Measured
11.1 5.5
10.2 6.5
11.8 5.6
7,0 4.6
0.4 0.4
0.5 0.5
As the rise time of the measured pressure pulse is a function of the length of the impacting steel slug, and the decay a function of the mass of the sliding piston, it was assumed that the generated pulse would have the same form for all tests. This enabled the input to the theoretical model to be taken from Figure 9(b) in the form of 20 data points with linear interpolation between them. The magnitudes were then scaled so that the maximum pressure corresponded to the test in question. In terms of the comparative figures tabulated in Table 1, the model appears to predict the time of occurrence to within 10% and the duration to within 20%. The magnitude and duration of the recompaction pulse is calculated less successfully although previous work [6] on similar mechanisms has also failed to compute this quantity successfully. One possible explanation is that the recompacting spall is not all liquid but a mixture of air bubbles and liquid, resulting in a greater energy loss at impact and a lower wave speed extending the interval over which the momentum of the spall is dissipated. Confirmation that the theoretical results are sensitive to the specification of the input pulse was obtained by changing the total theoretical duration of the pulse data from 600 ixs to 500 ixs. This reduced the predicted time of occurrence of the recompaction wave for p o = 2 8 b a r from 11.1 ms to around 7.0ms. This would suggest that more accurate input data is desirable, preferably from the same test from which Figures 8(a) and (b) were derived. At this stage the model was provided with data for an exponentially decaying shock pulse for comparison with results predicted by Cushing. Although it may seem unfair to
BULK CAVITATION
IN A x,V A T E R
SHOCK
TUBE
571
use Cushing's results as a comparison with the experimental work, a superficial inspection of Figure 7 might provide justification for considering the pulse to be of a classic explosion form, The pulse used in the model was obtained from Figure 9(b) by having thesame maximum pressure and the same energy content as the measured pulse. The result was that both the discrete model and Cushing's theory predicted an onset depth a few millimetres below the surface, and a termination depth of about 1.4 m (for Po = 28 bar). Cushing's theory, however, predicts a closure depth of 0.92 m from the bottom transducer. The recompaction wave would therefore propagate both up and down the tube from this location resulting in a time delay of 0.2 ms between passing the upper and lower transducers. However, it can be seen from both Figures 8(a) and (b) that closure occurs above the upper transducer and propagates downwards in both cases, since the interval between the recompaction pulses is the same as the interval between the incident pulses. Thecomputer model further revealed that with non-shock type pulses, e.g., half sine waves, triangular waves and the wave taken from Figure 9(b), the termination and closure depths were identical and equal to half of the total length of the specified pulse. Hence for the theoretical results predicted by the discrete model, closure occurs about 0.3 m from the free surface or 0.2 m above the upper transducer, a value confirmed by the experimental results. The accuracy of the theoretical model used in this analysis will obviously depend upon the magnitude of the spatial increment which, for the results quoted here, is equal to 10 mm. The computer model was subsequently operated with half this value being used and produced values of Zo, zT and zc differing by less than 0.1% of those produced by the standard increment. A further reduction of one half produced a change of about 0.001% in the same parameters. 5. CONCLUSIONS 1. The lumped parameter model described here predicts reasonably well the time of occurrence and duration of the recompaction wave though its magnitude is calculated less accurately. 2. Because the recompaction mechanism appears to be sensitive to the form of the reflected pulse, the lumped parameter model may have significant advantages over Cushing's model when the pulse is marginally non-shock. 3. Again due to the sensitivity of the process a real time pulse data acquisition system would seem desirable. ACKNOWLEDGMENT This work has been undertaken with support from the Ministry of Defence and AMTE to whom grateful acknowledgment is made. Particular thanks are also due to Dr A. Hicks for his assistance and helpful comments. REFERENCES 1. M. R. DRIELS 1980 Journal of Sound and Vibration 73, 533-545. The effect of a non-zero cavitation tension on the damage sustained by a target plate subject to an underwater explosion. 2. R. M. DAVIES and D. H. TREVENA 1955 Proceedings of the National Physical Laboratory Symposhtm on Cavitation in Hydrodynamics. The tensile strength of liquids under dynamic stressing. 3. S. A. SEDGEWICKand D. H. TREVENA 1976 Journal of Physics D9, 1983-1990. Limiting negative pressure of water under dynamic stressing.
72
M . R . DRIELS
9 V. J. CUSHING 1969 Final Report, Office of Naval Research Washington, U.S.A., Contract Nonr. 3709(00). On the theory of bulk cavitation9 9 M. R. DRIELS 1981 Journal of Sound and Vibration 77, 287-290. An improved experimental technique for the laboratory investigation of cavitation induced by underwater shock waves9 9 M. R. DRIEI~.S 1973 Transaction of the American Society of Mechanical Etlgineers, Journal of Fluids Engineering 95, 408--414. An investigation of pressure transients in a system containing a liquid capable of air absorbtion.
APPENDIX: BRIEF REVIEW OF CUSHING'S THEORY [4] Consider a pressure wave of the form p = P o e -(t§ where po is the peak value, t is le time measured from reflection from a free surface, z is the spatial co-ordinate, + v e own f r o m surface, c is the acoustic wave speed, and 3. is the pulse wave length. T h e )tal pressure field due to this wave and its free surface reflection is p = p~ +pgz - p o ( 1 - e-2Z/x),
(A1)
'here p~ is the atmospheric pressure, p is the water mass density, and g is the acceleration ue to gravity. Cavitation will begin at z = Zo when p = 0 in equation (A1). This leads to z o~3. = P d (2po-pg3. ).
(A2)
]ae particle velocity distribution just after the passage of the reflected rarefaction wave u (z) = - (p~ + pgz + 2po e -2z/x)/pc.
(A3)
'his velocity distribution indicates that since the velocity is in an upward direction, a avitation void will p r o p a g a t e down into the fluid as spray is projected from underlying later across the void on to the detached spall. The propagation will continue so long as accessive layers have a velocity greater than the layer below it enabling it to detach self and m o v e across the void. H e n c e cavitation will continue to depth Zr at which oint it will cease because du ( z ) / d z > 0.
(A4)
;ushing has shown that at this point equation (A3) gives ZT = (h/2) In [4po/pgh ].
(A5)
:losure of the cavitation void can begin f r o m the top as the spall thickness increases nd f r o m the b o t t o m as layers of spray m a n a g e to detach themselves from the layers elow, but do not have sufficient velocity to cross the cavitation void. As a result they all back on to the underlying water 9 Final closure occurs when the closing front spreading ownwards f r o m the top meets the closing front moving upwards. This can be shown to e at zc = (h/2) In [ 2 p J p g h ].
(A6)
B U L K C A V I T A T I O N IN A V E R T I C A L W A T E R FILLED SHOCK TUBE M. R. DRIELS
Department of Mechanical Engineering, University of Edinburgh, Edinburgh EH9 3JL, Scotland (Received 28 October 1981, and hz revised form 26 Februat3, 1982) A one dimensional lumped parameter theoretical model is developed to predict the occurrence of recompaction waves associated with bulk cavitation following the reflection of a pressure pulse from a free surface. Experimental data is obtained from a simple shock tube for comparison with the theoretical model and with that of Cushing whose work covers the classic explosion pulse. It is concluded that the model presented here predicts the occurrence of recompaction waves reasonably well and can have significant advantages over previously published work.
1. INTRODUCTION When studying the effects of an underwater shock wave either on the free surface or a submerged structure it is convenient to consider a simplified, one dimensional approach to the propagation of the wave. Recent work.J1] in which this assumption was used has shown that large scale rupture of the water (bulk cavitation) near an underwater structure can influence the interaction between a shock wave and the structure. This can lead to a greater degree of d a m a g e being inflicted on the structure than that in the case where the water does not cavitate. Several reasons were suggested in reference [1] why the water need not cavitate at the prevailing vapour pressure but might be able to sustain a considerable dynamic tension. It was then shown how the increased d a m a g e is influenced by a variable cavitation pressure or m o r e appropriately, breaking tension. Since the overall objective of the study is the prediction of damage, it became necessary to investigate the magnitude of breaking tensions and the conditions under which these values were non-zero. Much work has been done to this end by using a water filled shock tube, some of the m o r e comprehensive studies being those of Davies and T r e v e n a [2] and of Sedgewick and T r e v e n a [3]. In these experiments a short vertical steel tube with a freely sliding piston at the b o t t o m is filled with water. When a bullet is fired at the piston a shock wave resembling that of an underwater explosion is p r o p a g a t e d into the liquid column. This shock wave travels upwards to the free surface, is reflected, and begins to travel downwards. By considering the superposition of incident and reflected waves it can be shown that the water will cavitate at a certain depth, a void then occurring across the whole of the tube. This void is the region of bulk cavitation. T h e p h e n o m e n o n was developed one stage further by Cushing [4] who looked at the dynamics of the spall of water lying above the cavitation zone together with the effect of spray which is projected upward and accretes to the spall. Cushing, like ourselves, was concerned with underwater explosions which produce a characteristic pressure pulse consisting of a sharp rise to some p e a k value po followed by an exponential decay back to atmospheric. Unfortunately some initial experimental work intended to verify Cushing's theory showed that the pressure pulse generated by our experimental apparatus 563 0022-460X/82/200563 + 10 $03.00/0 9 1982 Academic Press Inc. (London) Limited
564
M . R . DRIELS
was not the classic shock but one with a significant rise time. Preliminary theoretical work also suggested that subsequent spall motion predicted by the model could be sensitive to the accuracy in specifying the incident pressure pulse. Further experimental work [5] in which a long horizontal tube was employed to investigate shockwave/target interactions demonstrated that although the pressure pulse initially has the form of the explosion wave, it becomes modified in shape as it progresses through the water. By the time the wave reaches the other end of the tube and interacts with a second sliding piston representing the target, it is not recognizable as a classic explosion shock. Because of these arguments it was decided that in order to study the shock wave/target interactions properly with the longer horizontal tube, a theoretical model would have to be developed capable of accommodating any shock wave input rather than being restricted to the exponential wave shape. This paper describes the formulation of such a model, together with some comparative experimental results obtained with the shorter vertical shock tube. 2. ANALYSIS Although the model is to be used for any wave shape the analysis will be done for the exponential wave. This will allow direct comparison with Cushing's results which have been summarized in the Appendix. The model is one in which one assumes that the input pressure pulse can be divided into a finite number of elements. Figure 1 shows
Pl
--'~Plle
P,,
<--z
[
I.
-11
.I Q
Figure 1. Definition of incident pressure pulse.
the input pulse travelling towards the free surface, the pulse itself being characterized by the number of data points, the peak pressure and the wavelength. The time datum t = 0 is when the wavefront arrives at the free surface and reflection begins. Subsequently the pressure field in the water is given by the sum of the incident wave, reflected wave, hydrostatic pressure and atmospheric pressure as shown in Figure 2. Basically, the wave is moved onwards one increment at a time and the new total pressure field is redetermined. If it is assumed that cavitation is prevented by the water sustaining any magnitude of tension then wave reflection is completed as shown in Figure 3(a). The water also acquires a velocity distribution due to the passage of the wave and this is shown in Figure 3(b). As would be expected, far from the surface, the reflected wave will be the negative of the incident wave although the increasing hydrostatic pressure will ensure that the tension finally disappears.
BULK CAVITATION IN A WATER SHOCK TUBE
565
Pi ,...1 1
/ k Figure 2. Pressure distribution following reflection.
{ol Figure 3. Spatial pressure and velocity distributions for non-cavitating reflection.
As the wave is incremented along, the pressure at section m from the free surface is given by pm =p~ = p , +Pa + p g ( m x s ) ,
(1)
where s is the length of each incremental section. If however the water is unable to sustain any tension then cavitation will occur at the section where the pressure falls to zero first. Up to this instant the fluid particle velocity is given by u = (1~pc)[p, +p,].
(2)
Following the onset of cavitation at depth Zo (defined by Cushing as the onset depth) the program continues to move the incident wave towards the surface and the reflected wave away from it. This results in the cavitation of more water and the spread of a cavitation front downwards from the onset depth. As each incremental layer is traversed by the cavitation front its velocity of projection is calculated from u = u~ + u , ,
us = pdpc,
u, = [pa + p g ( m x s) +pi]/pc,
(3)
(4)
since the reflected wave can only reduce the pressure to absolute zero. It might be expected that cavitation ceases once the hydrostatic pressure makes the total pressure positive; however Cushing has argued that the lower limit of cavitation occurs when
du (z)/dz > 0.
(5)
566
M.R.
DRIELS
It is assumed that below this limiting or termination depth zr, the water is essentially quiescent. Cushing's "instant cavitation" approximation is also adopted for the model whereby it is assumed that the cavitation front spreads so quickly through the cavitated zone that the water particles do not move appreciably during that interval. Representative results from this first phase of the analysis are shown in Figure 4. H e r e an exponential wave of 500 data points, peak pressure 20 atm and wavelength 0.5 m is reflected from the surface. Each spatial increment corresponds to 0.010 m. Onset of cavitation occurs after two incremental movements of the pulse at Zo = 0.020 m, cavitation terminating at zT: 1.86 m. Figure 4 also shows the velocity distribution between the surface and zr. -3
187
186
du dz -2
Zf
-3
Zo
Figure 4. Spatial velocitydistribution for Po= 20 bar, h = 0.5 m. In the second phase each element shown in Figure 4 is considered to be a rigid body with the appropriate velocity obtained from the associated distribution. This distribution is shown in Figure 3 and indicates that the spall velocity is upwards. Near the surface is the spall consisting of two elements. Its velocity is calculated by averaging the component Velocities. Although its velocity is greater than that of the underlying water it will have a higher downward acceleration due to the air pressure a b o v e it. This acceleration is given by
:
as = g + pa/ps',
(6)
where s' is the spall thickness. Each layer of spray, of thickness S, below the original spall has zero pressure acting on both upper and lower faces. The layer deceleration is thus due to gravity alone. Hence the first layer of spray will catch up with the spall and collide with it. If one assumes m o m e n t u m to be conserved at impact the new velocity and mass of the spall+ 1st layer can be calculated. The general collision case is shown in Figure 5 where the ith layer of spray collides with the spall after a time t. This time is the solution of
t2[pd2ps'] + tc[ui -tts] + [zi + s - zs] = 0,
(7)
where ui and tts are the spray layer and spall velocities at the beginning of the ballistic flight interval. Hence the momenta of spall and spray just prior to impact are
Ms = p s ' [ u s - ( g +p~/ps')tc],
M1 = p s ( u i - g t ~ )
(8, 9)
and the new spall velocity is
u's = (Ms + M1)/(s' + s).
(10)
BULK C A V I T A T I O N IN A W A T E R S H O C K T U B E
567
Impact Iqallislic
flighl
I Dolurn Figure 5. General case of spall/layer collision.
This process has been defined by Cushing as c!osure from the top. Closure of the cavitation void can also occur from the bottom since the acceleration due to gravity may result in a layer of water failing to catch up the spall. If this happens then the layer follows a ballistic trajectory returning to its starting position without colliding with the spall. Closure from the bottom occurs at the layer in question if tc > 2 u o / g ,
(11)
where llo is the initial velocity of the layer. Final closure of the cavitation zone occurs when closure from the bottom spreading upwards meets closure from the top spreading down. At this instant the spall together with the reconstituted layer of water from projected spray is moving downwards at an appreciable velocity causing a significant recompaction pressure wave to be propagated downwards. The study of this recompaction wave is important since its magnitude suggests that it could also contribute to the damage sustained by an underwater structure. Assuming that at recompaction as much energy is propagated into the underlying water as is propagated back up the spall, one has that the amplitude of the recompaction pulse is given by ~pc u,,
(12)
where tic is the spall velocity relative to the underlying liquid at closure. The time of occurrence of the pulse is determined by the spall dynamics outlined above while the pulse duration, if closure occurs above the upper transducer, is r = 2s'/c.
(13)
The computer program which embodies this theoretical model keeps track of each element within the cavitation zone, calculating the collision time of the enlarging spall with the layer of spray beneath it. Then, by using equations (8)-(10), the new spall
568
M.R. DRIELS
parameters are calculated, the new spatial co-ordinates of the other layers calculated and the process repeated for the next layer down. The final spall thickness is identified when equation (11) is satisfied. 3. EXPERIMENTAL RESULTS Although a comparative assessment of the shock tube may be found in reference [5], a brief functional description will be given here, for completeness. The tube itself is 2.7 m long, with a 30 mm bore and a 35 mm outside diameter, and is rigidly mounted on a vertical steel girder (see Figure 6). A series of five 6 mm tapped holes are located
Wo~er level /,~
Pressure transducers
/
1"5'm
,I
Storage oscilloscope Sliding piston Guide tube
~
I I
Chort r
_,-..--Steel slug
I Poeomot,c tom
Figure 6. Schematicof experimentalapparatus. along the length of the tube to accommodate AVL 6QP 500 quartz crystal pressure transducers which are connected to charge amplifiers..The signals from the charge amplifiers are recorded on a Gould]Advance OS400011 digital storage oscilloscope which has the facility for producing hard copies on an X - Y plotter. Water is contained within the tube by a freely sliding piston at the lower end. This piston is suspended on springs and a water seal is effected by means of a pressurized greasing system. Mounted directly beneath the shock tube is another similar tube containing a small (2 kg) steel projectile. A high speed pneumatic ram accelerates the projectile in an upward direction but the ram itself reaches the end of its stroke when the projectile is about 1 cm from the piston. By ensuring that the projectile continues its motion under free flight conditions, bodily motion of the water in the shock tube is avoided. Usually two transducers are
569
B U L K C A V I T A T I O N IN A W A T E R S H O C K T U B E
I
0"5 ms
I f0 bar
Upper transducer
-Lower transducer Figure 7. Wave propagation in the shock tube.
f
0 bar
I . 5 ms
I
[] ' ~<~Predicted
I
"
Predicted
,:J
1
;
Upper transducer
Predicted R
i
Lower transducer (o)
(b)
Figure 8. Experimental result for (a) Po = 28 bar and (b) Po = 14 bar.
f
0 bar
I
O0 HS I _ _
(o)
Ib) Figure 9. Effect of filtering. (a) 50 kHz; (b) 10 kHz.
10
M, R. DRIELS
;ed in each test and the oscilloscope is triggered when the pressure wave passes the )wer one. Figure 7 shows representative transducer outputs, over a short time scale. Representative results can be seen in Figures 8(a) and (b) which show the wave ropagating from the bottom of the tube upwards past the two transducers, and reflecting 9om the free surface (point R, Figure 8(b)), together with the recompaction wave some me later. The precise form of the generated pulse is shown in Figure 9 for two different harge amplifier filter frequencies. It can be seen that this filter attenuates the high requency oscillation attributable to pipe vibration and has little effect on the measured ise time of the pulse. 4. DISCUSSION OF RESULTS The overall validity of the theoretical model is assessed by the prediction of the ecompaction pulse, specifically its time of occurrence, (relative to the passage of the ncident pulse past the upper transducer) magnitude and duration. These results are ;ummarized in Table 1 for the two sets of experimental data shown in Figures 8(a) ~nd (b). TABLE 1
Summary of results Time of pulse (ms)
28 14
Magnitude,/~ (bar)
Duration, r (ms)
Predicted
Measured
Predicted
Measured
Predicted
Measured
11.1 5.5
10.2 6.5
11.8 5.6
7,0 4.6
0.4 0.4
0.5 0.5
As the rise time of the measured pressure pulse is a function of the length of the impacting steel slug, and the decay a function of the mass of the sliding piston, it was assumed that the generated pulse would have the same form for all tests. This enabled the input to the theoretical model to be taken from Figure 9(b) in the form of 20 data points with linear interpolation between them. The magnitudes were then scaled so that the maximum pressure corresponded to the test in question. In terms of the comparative figures tabulated in Table 1, the model appears to predict the time of occurrence to within 10% and the duration to within 20%. The magnitude and duration of the recompaction pulse is calculated less successfully although previous work [6] on similar mechanisms has also failed to compute this quantity successfully. One possible explanation is that the recompacting spall is not all liquid but a mixture of air bubbles and liquid, resulting in a greater energy loss at impact and a lower wave speed extending the interval over which the momentum of the spall is dissipated. Confirmation that the theoretical results are sensitive to the specification of the input pulse was obtained by changing the total theoretical duration of the pulse data from 600 ixs to 500 ixs. This reduced the predicted time of occurrence of the recompaction wave for p o = 2 8 b a r from 11.1 ms to around 7.0ms. This would suggest that more accurate input data is desirable, preferably from the same test from which Figures 8(a) and (b) were derived. At this stage the model was provided with data for an exponentially decaying shock pulse for comparison with results predicted by Cushing. Although it may seem unfair to
BULK CAVITATION
IN A x,V A T E R
SHOCK
TUBE
571
use Cushing's results as a comparison with the experimental work, a superficial inspection of Figure 7 might provide justification for considering the pulse to be of a classic explosion form, The pulse used in the model was obtained from Figure 9(b) by having thesame maximum pressure and the same energy content as the measured pulse. The result was that both the discrete model and Cushing's theory predicted an onset depth a few millimetres below the surface, and a termination depth of about 1.4 m (for Po = 28 bar). Cushing's theory, however, predicts a closure depth of 0.92 m from the bottom transducer. The recompaction wave would therefore propagate both up and down the tube from this location resulting in a time delay of 0.2 ms between passing the upper and lower transducers. However, it can be seen from both Figures 8(a) and (b) that closure occurs above the upper transducer and propagates downwards in both cases, since the interval between the recompaction pulses is the same as the interval between the incident pulses. Thecomputer model further revealed that with non-shock type pulses, e.g., half sine waves, triangular waves and the wave taken from Figure 9(b), the termination and closure depths were identical and equal to half of the total length of the specified pulse. Hence for the theoretical results predicted by the discrete model, closure occurs about 0.3 m from the free surface or 0.2 m above the upper transducer, a value confirmed by the experimental results. The accuracy of the theoretical model used in this analysis will obviously depend upon the magnitude of the spatial increment which, for the results quoted here, is equal to 10 mm. The computer model was subsequently operated with half this value being used and produced values of Zo, zT and zc differing by less than 0.1% of those produced by the standard increment. A further reduction of one half produced a change of about 0.001% in the same parameters. 5. CONCLUSIONS 1. The lumped parameter model described here predicts reasonably well the time of occurrence and duration of the recompaction wave though its magnitude is calculated less accurately. 2. Because the recompaction mechanism appears to be sensitive to the form of the reflected pulse, the lumped parameter model may have significant advantages over Cushing's model when the pulse is marginally non-shock. 3. Again due to the sensitivity of the process a real time pulse data acquisition system would seem desirable. ACKNOWLEDGMENT This work has been undertaken with support from the Ministry of Defence and AMTE to whom grateful acknowledgment is made. Particular thanks are also due to Dr A. Hicks for his assistance and helpful comments. REFERENCES 1. M. R. DRIELS 1980 Journal of Sound and Vibration 73, 533-545. The effect of a non-zero cavitation tension on the damage sustained by a target plate subject to an underwater explosion. 2. R. M. DAVIES and D. H. TREVENA 1955 Proceedings of the National Physical Laboratory Symposhtm on Cavitation in Hydrodynamics. The tensile strength of liquids under dynamic stressing. 3. S. A. SEDGEWICKand D. H. TREVENA 1976 Journal of Physics D9, 1983-1990. Limiting negative pressure of water under dynamic stressing.
72
M . R . DRIELS
9 V. J. CUSHING 1969 Final Report, Office of Naval Research Washington, U.S.A., Contract Nonr. 3709(00). On the theory of bulk cavitation9 9 M. R. DRIELS 1981 Journal of Sound and Vibration 77, 287-290. An improved experimental technique for the laboratory investigation of cavitation induced by underwater shock waves9 9 M. R. DRIEI~.S 1973 Transaction of the American Society of Mechanical Etlgineers, Journal of Fluids Engineering 95, 408--414. An investigation of pressure transients in a system containing a liquid capable of air absorbtion.
APPENDIX: BRIEF REVIEW OF CUSHING'S THEORY [4] Consider a pressure wave of the form p = P o e -(t§ where po is the peak value, t is le time measured from reflection from a free surface, z is the spatial co-ordinate, + v e own f r o m surface, c is the acoustic wave speed, and 3. is the pulse wave length. T h e )tal pressure field due to this wave and its free surface reflection is p = p~ +pgz - p o ( 1 - e-2Z/x),
(A1)
'here p~ is the atmospheric pressure, p is the water mass density, and g is the acceleration ue to gravity. Cavitation will begin at z = Zo when p = 0 in equation (A1). This leads to z o~3. = P d (2po-pg3. ).
(A2)
]ae particle velocity distribution just after the passage of the reflected rarefaction wave u (z) = - (p~ + pgz + 2po e -2z/x)/pc.
(A3)
'his velocity distribution indicates that since the velocity is in an upward direction, a avitation void will p r o p a g a t e down into the fluid as spray is projected from underlying later across the void on to the detached spall. The propagation will continue so long as accessive layers have a velocity greater than the layer below it enabling it to detach self and m o v e across the void. H e n c e cavitation will continue to depth Zr at which oint it will cease because du ( z ) / d z > 0.
(A4)
;ushing has shown that at this point equation (A3) gives ZT = (h/2) In [4po/pgh ].
(A5)
:losure of the cavitation void can begin f r o m the top as the spall thickness increases nd f r o m the b o t t o m as layers of spray m a n a g e to detach themselves from the layers elow, but do not have sufficient velocity to cross the cavitation void. As a result they all back on to the underlying water 9 Final closure occurs when the closing front spreading ownwards f r o m the top meets the closing front moving upwards. This can be shown to e at zc = (h/2) In [ 2 p J p g h ].
(A6)