H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 84
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In th2 s c r ~ d ~ s m i ~ ~ l ystable vehicle, this normal acceleration maydecay rapidly enough to zero that the effect ofnormal acceleration on the impact problem mayonly be to bias the motion of the fluid toward oneside of the tank or the other. I n the aerodynamically unstable case, the very presenceof an appreciably inclined initial fluid free surfaceimplies an appreciable angle of attack which, in362THE DYNAMIC BEHAVIOR OF LIQUIDSturn (depending on the degree of instability),practically guarantees a radical increase innormal relative acceleration during times of thesame magnitude required for the fluid to "fall"from one position in the tank to another.
Thissituation also indicates that the probable modeof motion of the fluid will be to run "up" oneside of the tank toward the upper bulkhead.It should also be noted that the fluid can haveinitial motion in the form of normal lateralsloshing and that this may appreciably affectthe behavior of the fluid after thrust termination.This has been verified experimentally.Since all practical tanks are elastic, the relative acceleration conditions before thrust termination may influence the impact indirectly bydeforming the lower bulkhead under the inertialloads of the liquid.
Upon thrust termination,the lower bulkhead could impart additionalrelative acceleration to the fluid, thus havingthe effect of shortening time to impact andprobably accentuating impact pressures. Thispoint has not, to date, been investigatedthoroughly.10.3 REVIEW OF EXPERIMENTAL WORKSimilitude requirements were found to bequite closely satisfied by a 14.3-centimeterdiameter tank containing carbon tetrachlorideas the liquid and 8-butylene a t 1%atmospheresas the pressurant. The associated model toprototype time ratio was 0.031, the model toprototype pressure ratio was 12.4, and the modelto prototype acceleration ratio mas 82.5. Implicit in the modeling assumption was that theacceleration field immediately prior to the acceleration change would have no effect on inipact forces; i.e., no lower bulkhead springback effect.The apparatus used (pictured in ch.
5,fig. 5.28) was a pneumatically driven devicewhich was capable of accelerating a smallmodel toward the Earth over a distance ofabout 60 centimeters a t up to 50 g's. Thisapparatus is shown schematically in figure 10.3.Since it was desired to explore the effects ofnormal relative acceleration, it was possible toincline the apparatus to the vertical by theangle 8, as shown. The tank is supported ona piston rod which is constrained to moveaxially. Just before firing, the acceleratingforce is balanced by a restraining force, F,,applied through a mechanical latch. Fixing aEarly Experiments a t Southwest Research Institute(refs.
10.3 and 10.4)The objective of this first study was todetermine the pressures resulting from the impact of fuel on the head of a fuel tank in aparticular suddenly decelerated guided missile.Though the initial prototype acceleration beforethrust termination was not specified, a changein acceleration of up to 0.6 g was specified for a1.8-meter-diameter fuel tank containing kerosene and a helium pressurant a t 3 atmospheres.A similarity analysis was performed, assumingthat the parameters of importance were:(1) Characteristic linear dimensions(2) Acceleration change(3) Time(4) Liquid impact pressures(5) Surface tension of liquid(6) Density of liquid(7) Density of pressurant(8) Viscosity of liquid(9) Viscosity of gas/Frt~ts";;RGURE10.3.-Schematic of liquid impact teat facility(ref.
10.4).LIQUID IMPACT O Ncoordinate system at the tank position beforerelease, the equations of motion for the tankassembly after release areAn isolated particle mi, in the absence of reaction with other part,icles or the tank, hasequations of motionmiyf= -mig cos 8miff=-mig sin 8Thus, the relative accelerations between a hypothetical free-falling particle and the tank areThe quantity g.n(t) is proportional to thechange in signal from an accelerometer mountedwith sensitive axis parallel to the tank axis.The time history of the acceleration pulse hada roughly trmpezoidal shape, as shown in figure10.4.
It is the character of the apparatusthat the duration of the acceleration stroke isquite sharply defined, thus making it possibleto compute an acceleration index, n,, which isthe uniform acceleration necessary for the tankto travel the constant stroke of the apparatus,S, in the observed time, T,. Thus45 Runs:3gs(ns- 1 1 ~ 1 9 9'7=I=".The quantity (Q n,) is inclusive of (g cos 8), and,therefore, the relative acceleration of the fluidin the tank can be normalized by the quantityb(n,-cos B)], as shown in figure 10.4 (for 8=0).The shaded band in figure 10.4 denotes therange of values of normalized relative acceleration pulse obtained in a series of vertical f i g s .I t can be noted from equation (10.28) thatthe relative acceleration normal to the tankaxis is always constant.
Strictly, then, therelative acceleration conditions imposed on thefluid in this apparatus are what might beexpected in the initial stages of thrust termination on a vehicle having neutral aerodynamicstability. In light of the possible radical variation of normal relative acceleration during thetime of interest, this inherent constancy of normal acceleration is an experimental deficiency.However, in light of what is practical in anexploratory investigation, it comes to rationalizing this deficiency or doing nothing aboutthe problem. The results from this apparatusremain the only available approximations towhat may happen when relative normal accelerations are appreciable.Some additional characteristics of the apparatus may be approximated. Figure 10.4 justifies an assumption of roughly constant acceleration over a length of time, T,,orIntegrating the first of equation (10.28) twice,under the assumptions thatn(t) =;=constant from t=O to t=to (average relative acceleration)&=minimum time for a particle initiallyon the fluid surface to impact theupper bulkhead@'=O a t t=Oyi=O a t t=Owe havewhereFIGURE10.4.-Normalized acceleration pulses obtained intest facility (ref.
10.6).b=shortest axial distance from initial freeliquid surface to the upper bulkhead.364THE DYNAMIC BEHAVIOR OF LIQUIDSCombining equations (10.29) through (10.31)Equation (10.32) relates the duration of theexperiment with the approximate minimumtime required for the first particles of fluid to"fall" t o the upper bulkhead in the absence ofany interaction with other particles or the tank,and it is important to note that (t,/T,) isrelatively insensitive to acceleration levels, Z,greater than 3 or 4 g's.The test program, proper, in references 10.3and 10.4, involved two 14.3-centimeter-diametertanks, one having a conical upper bulkhend andone having an upper bulkhead in the shape of nspherical segment, as shown in figure 10.5.Quantitative data were obtained by means ofpressure taps on the upper bulkheads.
Anextensive experimental program included testswith each head configuration, with and withoutring frames added per figure 10.5 for %- and Xfull tank conditions. In each of these cases,the acceleration level, 5 , was varied between 10and 50 g's for each of three angles of inclination(0°, 25O, and 50°), requiring approximately 90tests to cover the parameter range. Additionally, many of these tests were repeated in orderto obtain high-speed motion picture coverage.A typical oscillograph record of pressure andaccelerations obtained during a test of asmooth walled tank, one-quarter full withspherical head and inclined a t 50' to thevertical is shown in figure 10.6.
Worthy ofnote is that pressures reached a definite maximum during the acceleration stroke of theapparatus and that the pressure rise startsa t about the midpoint of the stroke in time.This last roughly checks with equation (10.32)since, in the case of figure 10.7, b was about20 centimeters and the stroke of the apparatusis about 71 centimeters [ta=0.53T, fromequation (10.32)]. Figure 10.7 is typical ofthe many results obtained under this program.Immediately notable is the great difference inhead pressures between vertical and inclinedfirings. The conclusions from references 10.3and 10.4 were as follows:(1) The pattern of motion of the fuel isrelatively independent of the magnitude ofthe acceleration, but is quite sensitive to angleof inclination and head shape.(2) I n the vertical configuration, the fuelbreaks away from the surface, first as anouter ring follon-ed by a progressive breakingaway of droplets toward the center, forming ahollow truncated cone shape.
This is in turnfollowed by a general formation of streamers orTop plateSpherical head 7Small tankPressure cellSmall tankt.0TankISpherical headConical headFIGURE10.5.-Model tank geornetriea (ref. 10.4).365LIQUID IMPACT ON TANK BULRHEADSFIGURE10.6.-Typical test record (ref. 10.4).Average acceleration i n g'sFIGURE 10.7.-Sphericalbulkhead pressure data in a112-full tank as a function of acceleration and for threeangles of inclination (ref.
10.4).colt~nlnsover the entire surface which move11p and impinge on the head. A large increasein viscosity tends to exaggerate the ringcone effect and results in the formation of asingle center column of fuel. An increase ingas pressure tends to delay the formation ofthe outer ring.(3) I n the inclined positions with a sphericaltank head. a relatively smooth circtdatorymotion is set up, with the fuel moving up thefar wall, around the head and down the nearwall.(4) In the inclined positions with a conicaltank head, the fuel splits, flowing around thetwo sides of the cone, converging in an arrowhead shape and rebounding toward the tankbottom.(5) There is little difference in the flow'pattern between the 25" configuration and the50" configuration.(6) Head pressures corresponding to the50" angle of inclination are from 5 to 15 timesgreater than the corresponding pressures withthe tank oriented vertically.
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