H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 82
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As far ascurrent practice is concerned, whether or notthe upper bulkhead (or dome) of a boosterruptures will be 0: z o consequczcs t c theultimate disposition of the booster; however, itis necessary during an abort to separate theman-carrying upper stages and anything elserecoverable and to insure that this equipmentand the men aboard are a safe distance fromthe booster before it breaks up or a fireball isformed.
Hence, the possibility of rupturingtank bulkheads by fluid impact during an abort353'354THE DYNAMIC BEHAVIOR OF LIQUIDSor engine malfunction may be an importrantsafety consideration.Experimental investigations of dome-impactphenomena are limited in number. Chronologically, the first of three such studies wascarried out in 1957 a t Southwest ResearchInstitute (ref. 10.3), later published in condensed form (ref. 10.4).
The second and thirdstudies (refs. 10.5 and 10.6) were carried outalmost simultaneously 7 years later at NASALangley and Southwest Research Institute.Owing to the di£Eculty in documenting theimportance of this problem from operationalfailures, and to the great diversity of rocketattitudes, accelerations, etc., which are possiblewhen thrust is terminated, the overall problemis not presently well set from the fluid dynamicist's point of view, either experimentally ortheoretically.
As a consequence, much of theexperimental work to be reviewed is exploratoryin nature, and an adequate theoretical treatment has yet to be developed. Engineeringanalyses are of the "order of magnitude"variety, impact pressures estimated being verymuch a function of initial assumptions regardingfluid behavior (refs.
10.7 and 10.8).10.2 SOME ESTIMATES OF THECONDITIONSIMPOSED ON THE TANK FLUID BY THRUSTTERMINATIONAs mentioned previously, the diversity ofpossible rocket attitudes, accelerations, andoperational situations when thrust is unexpectedly terminated is so great that simple generalizations covering all possibilities are not feasible.I t must be assumed that the data wvill be a thand to estimate the drag deceleration and subsequent behavior of the vehicle for thrusttermination or for loss of attitude controlduring any portion of the flight.
With suchestimates, the possibilities of fluid impact onthe dome can be assessed and the boundargconditions for the fluid dynamic problem set.However, i t is not possible for interpretativemasons to sidestep this question altogether inthe present treatment, and some crude estimates of conditions which may be expected arenecessarp. For simplicity, i t will be assumedthat initially a rocket vehicle is proceeding onsome assigned trajectory under power and con-trol in the Earth's atmosphere when enginefailure or intentional shutdown occurs. I t isfurther assumed that staging does not alsooccur, and that any portions of the vehiclewhich are separated after thrust termination areof negligible mass relative to the origind vehicleassembly.
The wealth of data contained inreference 10.9 dews some estimates of thissituation to be made which may serve presentpurposes.For a gross treatment of the accelerationsimposed on the vehicle, it will be assumed thatthe vehicle is a material point located a t thevehicle center of gravity. Simplified twodimensional trajectory eclutitiolls arc derived inreference 10.0. sec.tion 6.32.
in tangential(direction of velocity vector) and normal coordinates, neglecting Earth rotation. Figure 10.1indicates the forces on the rocket as a pointmass (from ref. 10.9). In this figureW=mgm=rocket massi]=locul gri;-tit,yL=lift forceD=drag forceT= engine t,hrustu, v=nonnal and tangential velocitiesa=angle of attackv =&ght path anglero=Earth radiusThe corresponding differential equations fortrajectory analysis are given in reference 10.9 as. -- T c o s a D gcosv=mm. Tsina L g vv=m,+;;;u+(G---)(10.1)sin vIntegration of these coupled equations givesvelocity, v, and Bight ~ a t angle,hv, as functionsof time. For present purposes, i t is convenientto r e d t e the equations for an approldmation totangential and normal accelerations ( b , u )v.=---- T cos a Dmm,Tsina Lu=m +,+gcos(10.2)sin v355LIQUID IMPACT ON TANK BULKHEADSml, is assumed acted upon by forces, R, and R,,as shown in figure 10.1(b).
Conceptually,these forces are exerted on the particle by thet m k ; aerodynamic forces and thrust do not actdirectly on the particle, which is assumed to beof a much smaller order of magnitude than, andcontained, in the "material point" of figure( a ) . Rotation of the material point(rocket) is also neglected. From figure 10.1(b),the equations of motion arey1+icos a+7i sin a=--gRvm1R,11-i sin a+2i cos a=-+gm1at X,,Y,,of mass m,cos (,,+a)(10.3)sin (,,+a)(Magnitudes of y, and z1are grossly distorted infigure lO.l(b) relative to r; the particle weightvector (myg) is sensibly parallel to local vertical.)Substitution of the accelerations of the originof the moving coordinate system, equation(10.2) into equation (10.3) yieldsy,=-+ R, D cos a- L sin a- Tm1m..RDsina+Lcosax1=2mlmEarthCenter(10.4)In effect, this result illustrates the point thatboth the fluid and the tank are in free fall.Consequently, the relative acceleration betweenfluid and tank is independent of the flight pathangle, V , and of local gravity, g.The axial aerodynamic force on the rocket,F,, and the normal aerodynamic force, F,, are*Fv=L sina-D cosaF,=L cos a + D sin a(10.5)Substituting in equation (10.4) givesFIG^ 10.1.--Coordinate system.I n a similar way, crude equations of motionof a single small particle of fluid in one of therocket tanks can be written.
Taking a movingcoordinate system, origin a t the material pointof figure lO.l(a), with y-axis alined with therocket longitudinal axis, a fluid particle of mass,I t appears from reference 10.9 that angles ofattack, a, of 10' or less are generally desirablefrom str~icturalconsiderations. Consequently,the angle of attack might be assumed small for356THE DYNAMIC BEHAVIOR OF LIQUIDSpresent purposes, since the present exampleassumes a controlled vehicle up to the point ofthrust termination. (The perturbations indirection of thrust were also neglected as smalldeviations in the above analysis.) For smallangles of attack the aerodynamic lift, L, on anunfinned body of revolution might be expectedto be (from ref. 10.9, sec.
5.2)and the increment of drag due to liftD=q[(kz--kl)Sbar2+tlC~Awherekp varies from 0.6 to 0.9I=overall vehicle lengthd =base diameterThusFrom reference 10.9, section 5.2, both (kz-k,)and 7 are functions of (Ild). Again, verycrudely, between 1/d= 5 and 20whereq= dynamic pressure&=area of base of rocket(ka- k,) =inertia coefficientsqCm= adjusted crossflo~v drag coefficientA,=projected lateral areaThe above approximations should be representative for transonic velocity as well as subsonic,according to reference 10.9. Thus, the increment to axial force due to lift will be (linearized)The crossflow drag coefficient C, is a functionof Mach number and angle of attack, but itstotal variation is between 1.2 and 1.8, approximately.
Thus,where k1 might range from 2 to 20 for vehicleswith l/d from 5 to 20. Similarly from reference10.90.8<(k2-kl)<land the increment to normal force due to liftwill beThus, for orders of magnitudeA 3AF,=L+AD.aManipulating(The omitted powers of a introduce only a smallpercentage error for a _< lo0.)For orders of magnitude for most vehiclesfor 5<l/d<20F,_kza+klei!lSbwhereThe basic drag a t zero angle of attack for conecylinders without t i n s in the transonic rangeseems to be made up of friction, base and wavedrag, and it might be assumed that these components of drag will not be greatly changed forsmall deviations in anglo of attack.
Consequently, it will be assumed that the dragforce a t zero angle of attack is the remainingpart of F,LIQUID IMPACT O N TANK BULKHEADSIwhere, for orders of magnitude, CD,_, variesbetween 0.15 and 0.8. Combining all theabove results giveswhere the coefficients have the ranges previouslycited. Substituting the above approximationsinto equations (10.6) then leads towhere&If the fluid in t,he tanks is not moving priorto thrust termination and has no relative velocityor acceleration, f and y for each particle iszero and the forces exerted on each particle are357For angles of attack of lo0 or less, the denominator in the expression above is insensitive to a.Depending on rocket velocity and altitude, themagnitude of rocket thrust, T, can be verymuch larger than the drag [qSb(CDa_o-k3a!2)]with the result that the denominator in equation(10.9) can be very large.
During flight throughthe atmosphere, the dynamic pressure, q, isgenerally a maximum somewhere. If thisoccurs at the transonic peak in the drag coefficient (CD,,, ~ 0 . 8 the) ~ denominator in equation(10.9) is approximately minimized with respectto drag variation and is approximately equaltoFrom reference 10.9, the practical variation inthe quantity, T/Sb, for multistage rocketsbuilt or projected as of the date of that reference,appears to lie between 1.0 and 2.5 kg/cm2, forthe first st,ages of multistage vehicles havingthrusts varying from 13,000 to 4 million kg,with values of 2 to 2.5 kg/clllVor man-carryingvehicles.
If maximum q is assumed to be 1 or1.5 kg/cm2, then a plausible lower limit on thedenominator of equation (10.9) is approximatelyunity for the maximum q condition for mancarrying vehicles. These considerations implya plausible range for tan B of0 5 tan 81 ( h f20a2)The sketch given in figure 10.1(c) indicates thepositive directions of R,and R,. The quiescentfluid free surface will be approximately normalto the resultant, R, of R, and Rut as shown.Then, the angle the free surface makes with thenonnai the tank axis isThe lower limit corresponds to a = O or verylow q; the upper end of the range to m&xim~unq.Evaluation of this upper range indicates thatbetween angles of attack of 0 and lo0, theangle 8 varies between 0 and 2 to 4% times a! forlarge booster vehicles in normal flight (beforethrust termination).18 t"YCkthm-at T, --chn111t-l'I-- tnminete; and theangle of attack and velocity does not changewhile the fluid is reorienting itself, the finalangle, fi, is approximated byV L u i uuv,2tan 19sThe right-hand side of equation (10.10) isessentially the ratio between the normal and358THE DYNAMIC BEHAVIOR OF LIQUIDSaxial aerodynamic force coefficients.
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