H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 86
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These experiments are difficult, and it appears in bothreferences 10.5 and 10.6 that unexplained datascatter can range to20 percent. It wouldseem possible, then, that a t most a 20-percentchange in force from one end to the other of theviscosity range covered in references 10.5 and10.6 might be present, If so, straight-lineextrapolation on a logarithmic plot of viscosityparamet'er (fig. 10.13) would indicate that pressures estimated a t a viscosity parameter of10" and extrapolated to the end of the prototype range using the pressure coefficient approach, equation (10.33) could be as much as40 percent in error. Exactly the same argumentand result applies t.o possible surface tensionscale effects.T h e scale effect work in reference 10.5covered a cavitation index range ofThe lower end of this range is iniportaxit fromthe cavitation standpoint.
Cryogenic fuels ina 9-meter tank with a pressurarit of 2 atmospheres may have a cavitation index corresponding t o the above of unity, and it niay be thatthe experiments of reference 10.5 have coveredmost of the pract~cairange ot cavitation indexwithout enormous scale effect.Impact Pressure DistributionsT h e data of reference 10.3 imply a relativelyconstant pressure distribuiion over the upperbulkhead, while those of reference 10.5 indicatea distribution with maximum a t the center.This is an apparent disagreement only. When369the pressure data for vertical firings with hemispherical head of reference 10.3 are examinedalone, i t appears that pressures near the edgesof the dome average about half of those near thetop.
This is the only comparable case with thedata of reference 10.5. The data for verticalfirings from reference 10.3 are subject to considerable scatter, and while all of the pieces ofdata from Cell No. I (fig. 10.5, hemisphericalhead) follow this trend, 3 out of 10 of the datumfrom Cell No. 4 do not. The pressure data inreference 10.3 for vertical firings for the conicalbulkhead display no consistent trend. Indicated maximum pressures from the cells a t thesides (Cells Sos.
1 and 4, fig. 10.5) range frorlihalf to twice those nieasured near the (*enter(Cell So. 2 ) , with no preponderance of eithertrend. The conclusio~iin reference 10.3 of a(Inot great" pressure variation over the head isapparently based on tlic*preponderance of datawhich are from the inclined firings. I n nll theinclined cases, variation iq maximutn pressurestippetws generally to be i25 percn_nt.Duration of ExwrimentsSince equation (10.24) implies that significantaxial deceleration resulting from rocket drag willpersist for Inany times the mininium timerequired for a free particle to arrive a t the upperbulkhead, t,, an assessment of the cited experimental work in this light is in order.
The tanksizes used in reference 10.3 indicate esperimeritduration times of two to three times 1,. Thework in reference 10.6 indicates that actualfirst-arrival times roughly correspond to t, forvertical firings. Since the vertical firing caseof references 10.3 and 10.6 is analogous to thesituation in the apparatus of reference 10.5, itwould seem that the experiments of reference10.5 also had durations of 3t, a t most.
Ifimpact forces reach a definlte maximum in thisinterval and seem unaffected by the prematureend of the experimental deceleration, then thisexperimental compromise with what may happen in reality is not important.I n t h e ' e ~ ~ e r i r n e noft s reference 10.3, a maximum, unaffected by the shortness of the experiment, was apparently experienced for inclined firings.
The experiments of reference370TRE DYNAMIC BEH AVIOR OF LIQUIDS10.6 (vertical firings) resulted in force maximaso obviously dictated by the end of the deceleration pulse that the maxima of the data were notconsidered of general use. I t is not clear fromreference 10.3 that a definite maximum is reachedfor vertical firings before the end of the deceleration pulse. Review of the motion picturesproduced at the time indicated that the fluidbreaks away from the initial quiescent freesurface in a large number of streamers slightlyafter a relatively thin annular ring of fluid movesup the walls of the tank.
This annular ring offluid is believed to be an effect produced bysurface tension. Thus, the first fluid reachingthe dome is thought to be composed of a verythin sheet of fluid moving up the sides, plus a"cloud" of droplets. The breaking away of thefluid from the main body a t the bottom of thetank apparently proceeds a t a rate dependingon the properties of the fluid and the acceleration. The motion pictures show most of thefluid in transit or remaining a t the upper bulkhead a t the end of the stroke, for the >;-full case.For the %-full case, however, about half theoriginal amount of fluid remains a t the bottomof the tank, while the amount in transit and themode of transit appears similar to the K-full case.These observations lead to the hypothesisthat when the deceleration is directed perpendicular to the free surface, the initial fluidimpact on the upper bulkhead is similar to thatof a hard rain on an empty bowl; high pressurescan be generated, but average pressures overreasonable areas are small.
As the decelerationcontinues, "rain" is generated continuouslyfrom the main body of fluid, and builds upagainst the upper bulkhead, slowly increasingan essentially fluid-static pressure until all thefluid is in contact. If this be the case for thevertical firings of reference 10.3, the maximumpressures measured were not the maximumwhich would have been attained had thedeceleration continued for twice or three timesthe attained interval.Judging from the sample force and pressuretime histories in reference 10.5, the maximumforce or pressure was reached a t about threefourths of the total duration of the half-sinerelative acceleration pulse. This point in timeis about where the acceleration pulse may bethought to be beginning the most rapid half ofits decay from maximum to zero.
Whether theobserved forces or pressures in reference 10.5may be expected to hold for longer decelerationpulses of the same magnitude is perhaps anopen question.I n summary, the question of the duration ofthe experiments relative to possible prototypedurations has been consistently omitted fromthe cited references, but is a consideration inthe problem.Variation of Impact Pressures With AccelerationThe conclusions of reference 10.3 recommended, for design purposes, that a designimpact pressure could be taken as a constanttimes the deceleration.
This approach is consistent with the pressure coefficient approachnoted in equation (10.33). Reference 10.5, onthe other hand, noted that maximum pressuresand forces are dependent on acceleration, andthe force and pressure results of that referencedo not appear to br proportional to acceleration.The comparativr \it~l;ltion is summurized infigure 10.14. This figure was prepared from allthe pressure data contained in references 10.3and 10.5 (which together may very well containall existing pressure data) as follows:The pressure datrr of figure 10.11 were converted topressure coefficient form using the indicated accelerations of that figure.
(These accelerations are thosederived for the moment of maximum pressure.) Boththe "quiescent" and "modal" data are shown, and thepressure coefficients are plotted against indicated relative acceleration in the left-hand portion of figure 10.14.The pressure data of reference 10.3 are too voluminousto fit on a single plot with clarity if the results of allpressure cells are included.
Consequently, the highestmaximum pressure, P,,,, measured on any cell duringa single firing was arbitrarily chosen as indicative andthis pressure was nondimensionalized by the averageacceleration measured during that firing. The resultsare plotted against average acceleration on the righthand side of figure 10.14. Each series of connectedsolid or dashed line segments connect the test pointsobtained in a series of firings at different accelerationswith the same inclination, head geometry, baffling, andamount of fluid.I t appears that the results of the two references are not consistent. The pressure coefficients derived from the data of reference10.3 appear to be relatively independent ofLIQUID IMPACT O N TANK BULKHEADSFIGURE10.14.-MaximumI1pressule coefficient versus acceleration.acceleration within attainable experimentalaccuracy, which is as it shduld be if this extrapolation approach is valid.
The trend of thepressure coefficients from reference 10.5 indicate just the opposite; in fact, to the sameaccuracy that the data of reference 10.3 indicateproportionality of pressure and acceleration, thedata of reference 10.5 indicate virtual independence of pressure and acceleration.1i-.i.1--.--1.I.-IiEffect of Lateral Sloshing and Lateral AccelerationFigure 10.14 illustrates the differences ofpressures between the rotational and "modal"mode of fluid motion. The data of referencei0.6 quite piainiy indicate t h a ~about the samepressures are induced by the modal and thequiescent" mode. I n the results of reference10.3, even though the fluid behaves in a mannersimilar to the "modal".
of reference 10.5 forinclined firings, the pressures induced areseveral times higher than those from vertic:.lfirings. The obvious difference is that thefirst-mode lateral sloshing of reference 10.5dictates initial conditions, while the constant((J,,-I+ -:I*-ai. --11371lateral acceleration of reference 10.3 actscontinuously on the fluid.From the point of view of the estimates ofsection 10.2, the variation of impact pressureswith relative normal acceleration is of greaterinterest than the variation with the inclinationof apparatus in the laboratory.
Accordingly,the maximum pressure coefficients from reference 10.3 are shown in figure 10.15 as afunction of the ratio of relative normal toaxial acceleration (x/?i). The data points havebeen segregated according to initial angle ofinclination of the free surface and according tohead shape. Included, but not denoted byspcis! sj~.hn!s, sre bsth the C B S ~ S nf nnefourth and one-half full and the cases where ringframes were installed. Quantitative data forItn apparatus inclination of 12.5' were obtainedfrom the authors of refen nce 10.3 and areincluded.
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