H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 83
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I n addition to the condition (a=O), which meansp=180° in this case (since the fluid must endup in the opposite end of the tank), an upperand lower bound can be put on fl as a functionof a by inserting the previously cited ranges ofkl, ka, k3, and Goa_, into equation (10.10).One "boundJ' is found for the "high lift," subsonic velocity case (Goa_,.=0.15) and anotherfor the "low lift," sonic velocity case. Bothare plotted in figure 10.2 along with the rangeon fl found previously for the case before thrusttermination, and some sketches indicating therange of quiescent fluid surface positions fora=5O.The results for before thrust terminationindicate a possible range of initial conditionsfor the fluid.
Normal sloshing can easily bevisualized superimposed on the initial fluidangle. Even in a vehicle under control towithin f3' angle of attack, the initial fluidangle may approach 10' from the tank axisnormal. The estimates for the final angle of flafter thrust termination presuppose that rocketvelocity, angle of attack, and so forth, do notchange for the time required for the fluid tocome to rest.
This assumption is, of course,incorrect and the meaning of these final anglesis subject to further considerations on thedynamics of the vehicle.When thrust is terminated at t=O, the secondof equations (10.6) for the hypo thetical particleis unchanged. The first of equations (10.6)becomes. R, FlJI=--Y=-ml .mD cos a-L+mmlR,sin a (10.11)Since (D cos a) predominates over (L sin a),the particle is accelerated upward.
If R, isalso assumed zero a t t=Oand the distance through which the particletravels toward the upper bulkhead, 5,in time,t, becomes-y=-v -F -=t2 Fvot2m 2 m 2assuming F, is constant during time t, (Fu=-F,,) and that the particle mas initially a t rest.If b denotes the distance of the free liquidsurface from the upper bulkhead, the minimumtime required for a particle on the free surfaceto reach the bulkhead will be approximatelyThrust-B 160O8 114"WithoutThrust-09FIGURE10.2.-Variation of liquid free surface orientationwith respect to tank&.The minimum time required for a particle onthe bottom of the tank to reach the positionof the hypothetical ultimate free surface(fig.
10.2) will be about the same. Since thefluid particles react with one another andthere will be a resistance to flow, the firstof tho fluid should reach the upper bulkheadslightly later than tiin, and the time requiredfor the fluid to move to an even approximatelyfinal position may be several times equation(10.13). As the tank empties, b+h thusraising the time to impact. Equation (10.13)sugg&ts a nondimensional representation fortime in the form.LIQUID IMPACT ON TANIC BULKHEADS359Substituting equation (10.18) . in the dragforce approximation then givesAs mentioned in a previous paragraph, thequestion of how long the relative accelerationof particle and tank persists is of considerableimportance. After t=O, and assuming no interaction between fluid particles, from equa t,ions(10.6) and (10.12)What is sought, consequently, is the variationwith time of the aerodynamic forces which areexplicit functions of rocket velocity and angleof attack and, thus, indirectly functions offlight path angle.
This amounts to a generalintegration of the nonlinear trajectory equations (10.1) and is not feasible, in general.Some rough magnitudes may be found fromspecial cases. If angle of attack, a, and flightpath angle, v, are zero, and thrust, T, is zeroa t t=O, and gravity is negligible, equations(10.1) becomev= -(10.16);=o.Assuming that axial aerodynamic force, F,, islargely drag, the first of equations (10.15)beconieswhere (y,)o=relative acceleration at t=O. Nondimensionalizing time with equations (10.14)giveswith respect to the quantity ( D i ) 2 / ~ oinequation (10.2 1)Di=O1.,If it is assumed that the drag can be linearizedasD=Do+Di (v-vO)wherethenSubstitution of the standard analytic forms forCD as functions of velocity into the bracketedportion of equation (10.22) indicates that thevalue of the expression in brackets does notvary greatly with velocity and has a maximumof unity.
ConsequentlyFrom equation (10.17)) the rocket velocitymight behave for short ranges of velocity asFor orders of magnitude (CD),,.= 1 and1360THE DYNAMIC BEHAVIOR OF LIQUIDSISubstituting in equation (10.21)Ifi-<e-[+?$1":(10.23)(YJOThe quantity(9)represents the mass ofdisplaced air over rocket mass and could easilyhave a maximum of only 0.001 for typicalbooster vehicles. Since b is by definition a tmost the length of the tank, a plausible upperis 2.0, assuming a t least two tanksin the vehicle.
These considerations lead toWriting a grossly simplified equation ofmotion for the vehicle, after thrust and controlterminationIii=Ma.a(10.25)(Moments are taken about the vehicle centerof gravity.)whereI=rnass nionient of inertia of the vehicleM= aerodynamic momentao=angle of attack at t=Ofrom which the angle of attack, a, would beexpected to diverge as (ref.
10.9)(At the end of a time interval 10 times as longas the time required for the first particle toimpact the upper bulkhead.)Most of the assumptions above are fairlyconservative, and it seems plausible to assumethat the relative axial acceleration of equation(10.15) changes quite slowly with respect tothe time required for initial portions of thefluid impact.
Equation (10.22) for [ ( D i ) 2 / ~ o ]and the subsequent orders of magnitude alsoimply that the velocity,, v, varies slowly. Thisindicates that the flight path angle, v, of therocket having zero angle of attack during andafter thrust termination will be very littledifferent that that of the free particle in thetank "falling" toward the upper bulkhead, ora t least the period of negligible change willnot be much shorter than that for relative axialacceleration.The normal relative acceleration of the fluid,x1 (eq.
(10.15)), is angle of attack, a, sensit,ive.Given thrust and control termination a t someangle of attack, a,, the lateral forces, F,, willtend to zero for the aerodynamically stablevehicle, and tend to increase for the unstablevehicle. In terms of large launch vehicles, theaerodynamically unstable case is the morepertinent.Substituting equat,ion (10.14)Assuming, as before, thatF,= qS#a (for small a)(Fz0=normal force at thrust termination)and then using equation (10.15)The order of magnitude of the radical ofequation (10.27) is of interestwhere2= distance of "center of pressure" forwardof the center of gravityLIQUID IMPACT ONUsing previously estimated expressions forF, and F,or (using previously estirna ted coefficients)where the low end roughly corresponds to verylow angle of attack, transonic velocity, and theupper end roughly corresponds to subsonicvelocity and a,== 1.5O.For the aerodynamically unstable vehicle, aminimum for 2 would be near zero and a maximum perhaps, 0.41.
Under these assumptionswhereN=for low a , transonic velocity{ 0.88.0 for subsonic velocity,= 1.5'a0The moment of inertia of a slender vehicle maybe approximatelyConsequentlyThe preceding approximations lead towhereO<J<5 for low a , transonic velocityO< J< 16 for a = 1.5', subsonic velocityIf (v/v,)~is constant, the time, 7 , for the relativenormal acceleration to double will beAssuming l/b=4, the relative normal acceleration could double in one-half the time requiredfor a particle a t the initial free surface to"fall" to the upper bulkhead under the influence of a constant initial relative axialacceleration of (y,),.
I t seems possible thatthe normal relative acceleration can appreciablyincrease during time intervals of the samemagnitude as those required for initial portionsof impact, indeed under the worst of circumstances a free particle starting to "fall" fromone side of the tank may tend to "fall" acrossthe tank rather than along the tank axis.I n general, then, the "final" fluid anglesshown in figure 10.2 could only occur for aneutrally stable vehicle.The treatment above excludes the possibilityof upper stage exhaust-gas impingeinent on aseparated lower stage during premature staging.This was considered for some special cases inreference 10.7 and, as f ~ asr the present treatment is concerned, can very appreciably increase the relative axial acceleration of thefluid for a time which is of the same magnitudeas the time required for initial portioos ofimpact.
Consequently, while the crude analysispreceding implies a suddenly applied M- to 2-grelative axial acceleration which decays veryslowly, exhaust-gas impingement may produce arelative acceleration pulse. Both types ofaxial acceleration time histories have beenused in simulation.From the simulation viewpoint, the relativenormal acceleration is the most diicult torationalize. If the fluid has an initial quiescentinclination when thrust is terminated, it isprobably the result of vehicle angle of attack,and in all cases (except the unlikely one ofneutral aerodynamic stability) this angle ofattack may change quite rapidly with a consequent rapid variation of normal relativenrce!crati~-, $th t:,-,e.
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