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TECHNICAL REPORT ARBRL-TR-02489

(Supersedes IMR No. 762)

OSCILLATIONS OF A LIQUID IN A ROTATING

CYLINDER: PART II. SPIN-UP

Raymond SedneyNathan Gerber

May 1983

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

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OSCILLATIONS OF A LIQUID IN A ROTATING CYLINDER:' •' ~PART 11. SPIN-UPFiaPART S6. PERFORMING ORG. REPORT NUMBER

V 7. AUTHOR(*) S. CONTRACT OR GRANT NUMBER(*)

Ravmond SedneyNathan Gerber

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It. KEY woRDS (Cantltmg oun .veesie it awcoay and Woarf(r 6y Wach ,ýbet)

V Rotating Fluid Spin-UpInertial Oscillations Spin-Up EigenfrequenciesEigenfrequencies Spin-Up Damping RatesLinear Elgenvalue Theory Modal Analysis

2(kAldTRAE X~~t W 940fh VW&A& E i?*Wc (bj a)The unsteady motion of a fluid which fills a cylindrical container in a

spinning Projectile is considered. The spin is imparted inmulsively to thecylinder and spin-up of the fluid is the basic flow which is perturbed to studythe waves in the rotating fluid. The core flow is perturbed, not the Ekmanlayer flow. This is called the spin-up eigenvalue problem which is solved withmodal solutions. Viscous perturbations are needed because of the boundary la)yrand critical layer. A numerical method of solution is given and several results

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shown. There are two times at which prujectile instability might occur. Theresults of the theory and method are validated by comparison with experimentalresults.

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TABLE OF CONTENTS

LIST OF ILLUSTRATIONS..... ... 5

1. INTRODUCTION ............................ 7

I1. SPIN-UP BASIC FLOW ............................ .............. 9

Vi 11 i1. THE PERTURBED FLOW ....................................... ... 15

IV. THE MODAL SOLUTION AND EIGENVALUE PROBELM .................... 18

V. SOLUTION OF THE EIGENVALUE PROBLEM........................... 22

Z• •.VI. DISCUSSION .................. ........... . . . .. . . . . .... 24

Table 1. Experimental and Calculated Eigenvalues............ 26

Table 2. Experimental and Calculated Eigenfrequencies ....... 27

ACKNOWLEDGMENTS ............................................ 27

REFERENCES .. ................ ..... . ... ... . . . ...... t.*.... 36

APPENDIX A .. .... ...................... .... .......... . . ... . 39

APPENDIX B ........... ....... .... ............... % .. ... ... 45

L.IST OF SYMBOLS ......................... ..... ......... 49

O I S T R IB U T IO N L IS T . . . . . . ..5 1

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LIST OF ILLUSTRATIONS

Fiure1 V vs r for Case I with c/a = 3.120, Re = 4 x 104 at

Three Values of t; ts = 1245. Laminar Ekman Layer ......... 28

2 V vs r for Case 2 with c/a = 5.200, Re = 2 x 106 atThree Values of t; tt= 2700. Turbulent Ekman Layer ...... 29

3 V* vs z + c/a = z' for c/a = 1, Re = 104, ts = 200 at

(a) t = 62.5, (b) t = 250. The Solution is ObtainedFrom the Proqram in Reference 11 ........................... 30

.4 The Eigenfunction Real (w) vs r for Re = 5 x I05, c/a =2.679, Mode (5,1,1). (a)t = 7000, (b) t = 1000, (c)t = 400 .................................................... 31

5: CR vs t for Case I with c/a= 3.120, Re =4 x 104,

Mode 1, and Case 2 with = 5.200, Re = 2 3 106Mode (5,11 ................. ..................... . .... 32

6 Time Histories of CR and CI for c/a = 3.297, Re = 4,974,

Modes (3,1,1,) and (3,2,1) ................... .. *...... 33 p7 Trajectory of C2 With t as Parameter; c/a = 3.297,

Re = 4974, Mode (3,2,1). Approximate Regions in Which IFirst Guess for C2 Must Lie to Have Convergence to

n 2 Mode Are Shown for Four Times...... ............... 34.

8 The Surface IZI (CR, CI) for c/a = 3.297 and Re - 4,974 at

t - 180, in the Neighborhood of the Root CR2 = .101673,i: " ~ ~ ~C I ' 0 3 4 8 6 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 3 5 .

121

I .!

5 - a-

egg P"

: I. INTRODUCTION

For many years it has been observed that liquid-filled projectiles have aproclivity for unusual flight behavior, often being unstable even though thesamp projectile with a solid payload is stable. The basic states of motion ofthe liquid payload in a cylinder can be rationally separated into solid-bodyrotation and spin-up, the transient state that ends with the fluid in solid-body rotation. The determination, according to linear theory, of the fre-"quencies and decay rates of the waves in the rotating fluid is called the"eigenvalue problem." It was solved for the basic state of solid-body rota-

Vtion in Reference 1. The purpose of this report is to present the theory forthe basic state of spin-up and some representative results.

Some attempts to solve the eigenvalue, abbreviated e.v., problem havebeen made in the past. Reddi 2 attempted to solve the inviscid perturbationequations but had difficulty with a singularity in these equations, related tothe critical layer to be discussed in Chapter IV. A critique of this work isgiven by Lynn. An ad hoc method to solve the e.v. problem was consideredin Reference 4. The actual spin-up velocity profile was replaced by distri-butions in which the angular velocity was constant in an annular layer next tothe cylinder and zero elsewhere. These distributions were required to haveeither the same angular momentum or the same volume of rotating fluid as thecorrect spin-up profile. In this way the spin-up problem could be fitted intothe formalism of the Stewartson theory.5 This ad hoc approach was notsuccessful, and the e.v,'s disagreed with results from the present theory. Arational approach to the e.v. problem was given by Lynn 3 for a basic flowdesignated by Vw in Section II. The effect of the critical layer singularity

was discussed and the real part of the e.v. calculated for axisynvietricinviscid disturbances, for which no critical layer exists.

1. C. W. Kitohana, Jr., N. Gorber, and R. Sedney, "Os•ill.at.iono of a Liquildin a Rotating Cytinder: Part I. Soli4 Body Rotation," ARRR&M-TR-02081,June 1978 (A[) AO7759).

8. M. M, Rckidi, "t0, the KRieriaTluc of Couatte Flow in a Futly-FittedCyl indrioal Con tai ier," Fi-anki t Ine8titut Reaearoh Laboivtoiy,Phitadelphia, PA, Raport No. F-B2294, 196?.

3. Y. M. Lynn, "Free6 Ocillationa of a Liquid During Spin-Up.," BRL Report No.1663, August 1973 (AD 769710).

4. EApineer-Ing Deli•iLandbook. Liquid-Fitlled PooJ.ctile Poi.ign, AWC PaimphtotNo. 706. 165, Unitad States AzvV M4aterial Comwivnd, Waahington, D.C., AprilI1969 (AD 8.53719).

5. K. Stebt'arieon, 'On the Stability of a Spinning Top Containing Liquid,"Journal of PlZtid Machaniaa, Vol. 5, Part 4, 1959.

7

DI this report the linear e.v. problem is solved for viscous pertur-bations, axisymnetric or not, on a basic flow which is the spin-up core flowincluding diffusion. The modal analysis used allows slip on the cylinderendwalls. The perturbation equations have a singularity at the axis of thecylinder that is handled analytically. Because the Reynolds number is large,an integration scheme with orthonormalization is used that maintains linearlyindependent solutions. The determination of the eigenfunctions through thecritical layer is accomplished numerically.

The significance of spin-up effects on a projectile flight can be esti-

mated in an order of magnitude sense. Let a and c be the radius and half-height of the cylinder, R the spin in rad/sec of the projectile (say at themuzzle), v the kinematic viscosity of the liquid, and

2 .2Re = Q a2/v, E ý Vi 4 c,

the Reynolds number and Ekman number. Note that E is sometimes defined usingthe height, 2c, rather than c, and that for historical reasons we shall use Rein this report rather than the more conventional E. If the Ekman layers,i.e., endwall boundary layers, are laminar, the characteristic time for spin-upis

(2c/a)Re1/ 2 /il

= 2/E/11 2 (sec).

We also use a nondimensional spin-up time t 0 " For Re > 105,

approxirnately, the Ekman layer may be turbulent, in which case the charac-

teristic spin-up time can be estimated by Est=(28.6 c/a) Re

Basically, these are titme scales for the spin-up process and do not neces-sarily give a mea;sure of how close the spin-up process is to solid-bodyzotat ion. This can be measured in several ways, e.g., time for the angularmoment,,n, e.v., or other property to attain a value within 10%, say, of thesolid body value. These measures require detailed calculations and so are not

very convenient. A rule of thumb is that solid-body rotation is reached at

about 4 t after an impulsive start of the cylinder.5

The characteristic spin-up times can be com-pared with tf l I the time of

flight of the projectile. If ( or s < tf< ' spin-up effects can be

disregarded and solid-body rotation can be assumed. If tI or tst t/0'

or larger, spin-up effects probably have to be considered. To put theseestimates in perspective, consider the parameters for two ,f several recentfirings of 155tini projectiles conducted by Dr W. P. D'Amico, BRL. For one ofthese c/a = 3.120, Re t 4 x 104, fl - 754 rad/sec, giving ts a 1.65 sec; for

another case c/a = 5.200, Re = 2 x 101, ii = 754 rad/sec giving, s 3.58sec. These are referred to as Cases I and 2, respectively. For both of these

tfl 40 sec. and, from the e.v. histories, spin-up effects are significant.

8

A question of more practical concern is: To what extent does the spin-upprocess affect the moment exerted by the liquid on the projectile and its yawgrowth rate? The answer requires solution of the forced oscillation problemfor the fluid. The e.v. problem is the free oscillation problem, and itsrole, vis-a--vis the forced oscillation problem, is the same here as in anyvibrating system. For example, the e.v. and eigenfunctions can be used toexpand the solution of the forced oscillation problem. The e.v. can also beused to estimate when the projectile might be unstable. For inviscidperturbations on solid-body rotation, Stewartson5 showed that if the projec-tile nutational frequency is within a certain band of the eigenfrequencies,the moment has a resonant response and the projectile will be unstable. Ifthis idea is extended to viscous perturbations on spin-up, the real part ofthe e.v. would be compared with the nutational frequency. The comparison is,at best, a necessary condition for projectile instability. Confidence inapplication of this theory must be obtained by other means. This has beenprovided by experimentsb and numerical simulations. 7

II. SPIN-UP BASIC FLOW

Before the spin-up e.v. problem is considered,the basic flow requiressone discussion. Only those aspects of it germane to the e.v. problem will bediscussed. Our interest is in spin-up from rest which is inherently a non-linear problem. Spin-up from one state of solid-body rotation to another can

be treated as a linear problem if the change in rotation rate is small. Bothtypes are discussed by Greenspan8 and Renton and Clark. 9

Consider the motion of a fluid which fills a cylinder, initially at rest,which is "rapidly" brought to a constant angular velocity. The fluid motionis axisynrietric and tinme dependent. fhere are many variations on this problemwhich are of practical interest, e.g., the fluid only partially filling thecylinder, but they will not he considered here. For our application theangular acceleration of the cylinder is large during the time when the pro-jectile accelerates in the gun tube; the small deceleration in flight isneglected. For artillery projectiles this acceleration time is typically I0.020 sec. which is small compared to fl comparison of angular acceleration

6. S5. 8targiopoutos, "Apt eki~~iental .5tudy. of bwertiat Wwaea int a 1rluidContain, in a Rotating Cylindrical Cavity 1)uring Spin-ip From Rot," MD.The*sio, York Univareity, Tor'onto, Ontario, Febir-uary 19M?2

7. q. Sedney, N. cerba,,, a•d J.M. Barton, "Oe.itl.ation of a itquidI in aRotatinq Cylinder," ATAA Preprint 82-0296, 20tht Aeroopjace Scionoao Mecrtnj,OrlZando, F11 JanUary 1982.

8. H. P. Greenatpan, The Theory of Rotatiing Fluida, Cambridge lniver•iti P-,ass,London and New Yorw, 1968.

9. K:. R. Benton and A. Clark, Jr., "Spin-Up," artice in Annual Peview ofFluid M4eohanice Vol. 6, Annual Reoiomat, Inc., Palo Alto," CA, •974.

%L !I

time with spin-up time is also required. The assumption of an impulsive startfor the cylinder is a good approximation for the projectile problem; it maynot be in a laboratory apparatus. For an impulsive start our model requiresseparate treatment of the initial conditions.

The notation here will be the same as in Reference 1. Lengths, veloci-

ties, pressure, and time are made nondimensional by a, an, pn2 a2 , and-1

Srespectively, where p is the liquid density and a is the constant spinrate of the cylinder. In the inertial frame cylindrical coordinates r, 0, zare used, with the origin of z at the center of the cylinder, and velocitiesU, V, W, respectively. Dimensionless time is t. Note that in Reference I theorigin of z is at the bottom endwall. Derivatives are indicated by sub-

scripts.

The basic work on spin-up from rest was done by Wedemeyer. 10 In thisremarkable paper he shows that the flow must he divided into two regions: theEkman layers at the endwalls, which can be treated in a quasi-steady fdshion,and the rest of the flow, called the core flow. Wedemeyer did not point outthat an additional boundary layer, a Stewartson laj/r, is tequired at thecylinder wall. Starting with the Navier-Stokes equations for axisymmetricflow:

2 2Ut + U* U* + W* U* -V* 2 /r -P* + Rel (V2U* "U*/r ) (2.1a)t r z r 2

-1 2V * + +J V V + + U* V*/r Re (V2 V* V*/r ) (2.1b)t r z2

WJ* +U* W* W* * -ut r z z Re V2 W* (2.1c)L(rU*) r +rW*- 0 (2.1Id)

"r :

V2() Orr + (1/lr) r+ ()zz (2.10

where the asterisk indicates the exact solution, he used physical order-of-magnitude arguments to simplify them in the core flow. The approximationsreduce the three momentum equations. (2.1, a, b, c), to

V t + U (V r + V/r) Re"! [Vrr + (V/r) (2.2)

10. -'. H. Wale.dw,, r, mThe Una•twd Ftow Within a Spinning Cylinder,' USA BRLRopot No,, 1225, Ootober 1.963 (AD 431846). Also, Jaurw of Fluid-Machcnico, Vol. 20, Part 3, 1964, pp. 383-399.

10• ,• ,, ,

and

Uz Vz PZ 0, (2.3)

where the superscript is dropped for this approximate solution. For Re +Wedemeyer proposed neglecting the diffusion terms in (2.2) and arrived at

Vw + Uw (Vwr + Vw/r) =0 (2.4)

t rtwhere the sub w indicates this approximation. Wedemeyer used (2.4) ratherthan (2.2) when he applied his model.

Although Wedemeyer showed the crucial importance of the Ekman layers tothe spin-up process, his model did not require a solution for the flow inthese layers. He did discuss certain properties of the solution which wererequired in his model for the core flow; exclusion of this solution hasimportant consequences for the e.v. problem.

A more formal approach to this problem was given by Greenspan.8 In histreatment, lengths are made dimensionless by 2c rather than a, which is more

natural for the spin-up process; time is made dimensionless by f so that the

new time is t' t/ts t/(2c/a) Rel' 2 ; an expansion in the small parameter

(lts) (a/20) Re1"/ 2 (2.5)

is assumed. The forn of this expansion•: ~ U" ils UI + .. !I

V0 V (lit) V1 .+ . (2.6)

WO(lt) W 1

P * (lt ) Po s

follows from the facts that in the core flow Ui and W* are ()(1/ts) and V* is

0(1), if /t1s is small. See Reference 8 for details. The independent

variables are r, z, t'. Substituting the expansion into (2.1) after trans-forming tO t yields

11 •

iA

V + U (V + V/r) 0r

Vo2/r P (2.7)

0 0 1•<,P 0 Vo =UI 0.

z z z

These are the same as (2.3) and (2.4) since V0 Vw, Uw (lits) U1 , and

t' = t/ts. Although the formalism of matched asymptotic expansio.vs was not,

used, the first two terms of the outer expansion are apparently given by(2.6). The Ekman layer solution would be obtained from the inner expansionwhich was not considered in Reference 8.

To solve (2.2), (2.4), or (2.7) a relationship between U and V isnecessary. Wedemeyer used the facts that the Ekman layers are steady afterone revolution and that the radial mass flux in the core flow must be balancedby that in the Ekman layers to obtain some conditions on the U, V relation-ship. At this point he was forced to take a phenomenological approach. Therelationship is known at t + 0 and t and he proposed a linear inter-polation between them to obtain an approxitiate ,elationship for any t. Hetested this idea in some other problems where the solution was known anddecided it was satisfactory. Some confusion has appeared in the literaturebecause this step was misinterpreted; this is discussed in a separatereport. The result is

U k, (V -r)

f-11k at (a/c) Re" 1 2 (2.8)

for laminar Ekman layers. Wedemeyer proposed - 0.443 but Greenspan sug-gested K 0.5; the latter often gives results in better agreement withnumerical solutions to the Navier-Stokes equations. Other relationships havebeen proposed, but they will not be discussed here. For turbulent Ekmanlayers

12

U -kt (r-V) 8 / 5

kt= 0.035 (a/c) Re-I/5 (2.9)

l/tst,

where tst is the nondimensional , turbulent spin-up time., The core flow is

assumed to be laminar so that turbulent stresses are not introduced in theright-hand side of (2.2).

Using (2.8), (2.4) can be solved explicitly:

2ket 2kt -ktV :(re -i/r)/(e - 1) for r > e

(2.10)=0 for r < e- k,

-k t

so that r : e separates rotating and nonrotating fluid. Although Vw

is continuous there, a discontinuity in shear, i.e., V , exists there. UsingWr

(2.9) a numerical integration is necessary to obtain Vw; but the character of

the solution is the same as when (2.8) is used. The radial velocity isobtained fron. (2.8) or (2.9) and W is obtained from the continuity equation

W -(z/r) (rU)r

At r 1, W 0 0; thus, the Stewartson layer should be included at r = 1, butthis has yet to be done. Also W # (J at the endwalls z ± c/a.

Wedemeyer pointed out that the corner in the solution to (2.4) would besmoothed out if the diffusion terms were retained, as in (2.2). But this is an'inlinear second order equation and must be integrated by finite differencemethods. This can be done with some standard techniques for diffusion equa-tions, using (2.3) or (2.9). Special treatment is needed near the pointr 1, t ) because, for an impulsive start, a discontinuity in the boundaryconditions xists at that point. In our work a local, analytic solution wasderived to resolve the discontinuity. In most of our e.v. calculations weused the V from the numerical solution of (2.2) with either (2.8) or (2.9),but other options are also used. For laminar Ekman layers, the spin-upvelocity prnfile

V f (r, k t, k• Re);

334

for the turbulent case k is replaced by k These fincticoal forms follow

directly frmn (2.2) and (2.8) or (2.9) and rpresent the most efficient way todisplay the results.

Some examples of the solution of (2.2) will oe giver, simply to illustratethe form of V which must be perturbed in the c.v. pr-oblem. The parameters ofthe two cases of 155rmi projectiles, given in the intr-oduction, will be used.Case 1, with Re = 4 x i04, should have laminar Ekman layers. The solution to(2.2) with (2.8) gives the V vs r shown in Figure I for 3 values of t. One ofthem is the nondimensional spin-up time ts = 1245. For t = 4800, approxi-mately 4ts, solid-body rotation is achieved, essentially. Case 2, with Re

2 x 106, should have turbulent Ekman layers. The solution to (2.2) with

(2.9) gives the V vs ý shown in Figure 2 for 3 values of t includingt = 2700. For c = 10,000, approximately 4 tst, solid-body rotation has notbeen achieved to the same d(horee as in Figure 1. The profile for t = 1300 isrelatively steep; early time profiles such as this require more care in thee.v. calculatiofr.

Some results from the Wedemeyer &-pin-up model have been compared withexperiments; these tend to validate tite model. Another way to validate themodel is to compare its results with solutions to the Navier-Stokes equa-tions. The fir'te difference sohlu+tn presented by Kitchens,! was used forthis purpose.

The V's from both solutions were compared over a significant range ofparameters; in general, the differences were small for K = 0.5 and slightlylarger for c 0.443. Comparisons are given in a separate report. Somelimited comparisons are shown in the following discussion which actually has adifferent purpose, viz., to show the flow in the Ekman layer and its transi-tion to the core flow.

To the authors' knowledge no solution for the flow in the Ekman layersduring spin-up has been given other than finite difference solutions. InReferences 8 and 10 certain properties of these were used to determine thecore flow. The finite difference solutions to the Navier-Stokes equations byKitchens 11 provide the complete flow. In Figure 3 results from this methodare shown for c/a 1 1, Re = 104, t = 200 at t - 62.5 and 250. For threevalues of r, V* vs z + c/a is shown. According to Wedemeyer's spin-up model Vis independent of z; this is verified by the calculation for z + c/a > .05 att = 62.5 and for z + c/a Ž .025 at t - 250. These values of z + c/a c-an hetaken as one measure of Ekman layer thickness. The values of V from theWedemeyer model are also indicated; the differences from the results of theNavier-Stokes equation are relatively large at t 62.5 and essentially zeroat t 250. Significant undershoots exist In the V profile at t 62.5 but

il. C. W. Kitchens, Jr., "Nauier-Stokeo Solutions for Spin-lip in a PilledCy•tinder'," AIAA Journal, Vol. 18, No. 8, August 1980, pp. 929-934. AlsoTechnical Repor ARBRL-TR-02194, September 1979 (AD A07711S).

14 j

HIMPonly mild ones at t = 250. An examrpl2 of U vs z + c/a is given in Reference11. The azimuthal component of the basic flow which must be perturbed toobtain the e.v. is represented in Figure 3.

If only the core flow were perturbed, it would not be possible to satisfythe boundary conditions on the endwalls. Figure 3 also shows that a largeerror in azimuthal velocity is made in the Ekman layers if the core flow isextended to the endwalls. This is, in fact, what is done in the e.v, problemformulated in this report. Nevertheless, the wave frequency calculated in thisway agrees fairly well with some experimental results; however, the calculatedwave damping has a substantial error. This is expected since the dissipationin the Ekman layers should affect the wave damping.

III. THE PERTURBED FLOW

The procedure for obtaining the equations governing the perturbed flow isa standard one. It starts froa the Navier-Stokes equations for 3-D flow:

U . r + (V/r) U + W U V2/r = -P + Re-1(V2 U - U/r2 - 2V /r2U UUr Z r"

Vt + U +r ! (V/r) V + W V + U V/r: -P /r + Re" (v2 d V/r2 + 2Ui/r2)

W t + 11 W r + (V/r) W 0 + W Wz -P "z + Re'-1 2 W (3.1)

(r U)r + V + r Wz 0

v( ' % () + (1Ir)()oo

The velocity components and pressure are then expressed as the sum of thebasic and the perturbation:

U U* (rz,t) + u' (r,o,z,t) (3.2)

Iiand similarly for V, W, and P. For our application nonaxisynmetricd-;sturbances must be considered, hence the dependence of u' on 0.

The details of the derivation will be given only for the radial momentumequation, the first of (3.1). The ropresentat'.ton of the velocity cowponentsand pressure as a basic flow plus a perturbation, i.e., (3.2) anO the similar

expressions for V, W, and P, are subst'.tuted into (3.1). The terms areseparated into zeroth order terms which are independent of the u', v', w', p',

15

f:rs4 order terms which are linear in u', v', w, p', etc. The zeroth orderterms ýive exactly (2.1), by construction. The first order terms for theradial momentum equation, the first of (3.1), give

u't + U*u'r+ U' U* + (V*/r) u' + W* u' + w' - (2V*v')Ir (3.3)U rr 0 z Uz (Vv'/ 33

1ii o/ p2r 22]

-p'r + Re" [v2 u' - u'/r 2 - 2v' r.

One term does not appear because U* is independent of o.

UP to this point the basic flow is an exact solution of the Navier-Stokesequations, (2.1), which would have to be a finite difference solution. It isimpractical to use such a solution for the coefficients in (3.3). Therefore-,an approximation to it, the Wedemeyer model for the core flow, is used. This.leaves open the matter of treating the Ekman layer flow and the attendantdifficulties liscussed k-. the end of the last section. A correction is'ecuirad 'o ?' :ou,., for the Ekman layers; this will not be done here.

The next step is to estimate the order of magnitude of the terms in(3.3). They are not ali the same order of magnitude, and this step leads to aesignifica-t s.mplification. The expansion (2.6) is convenient for this pur-pose dIthough thR approach tKen in derlving (2.2) was actually used. It isassumed that all pertubations, the primed quantities, and their derivativesare 0(l). in particular u' W 0(1) which means that the dimensional time

scale for the perturbations s O0f(d) and is therefore small compared to t-s;

the Pond;mensional time scale for the perturbations is 0(1) which is smallcompared to ts. This estimation is used below to make another importantsimplification. It follows fru,.i (2.6), or the equivalent using Wedemeyer'sapproach, that all coefficients in (3.3) -ontaining U* and W* and theirderivatives with respect to r and z are 0(1/t s); all those containing V* andV*r are 3(l). Neglecting the 0(1/t s) termTs oti the left-hand side of (^.3)

deletes all terms that contain U* and W*. Only the tenr,, cont&ining V* andthose not involving the basic fiow reraiti. For V* we must su'stittite V of(2.2). Applying this same process to *ý'e r•maining equptions of (3.1) yieldsthe following set of perturbation equations:

ut + (V/r) u' a- 2Vv'/I - -pr + (v2 u - u'/r 2 2v' W. 2 (3.4d)

v't + [V + (V/r)] u' + (V/r) vi 0 -p' W/r + Re I (V 2v6 (3.4b)

Sv'I/r 2 + 2u' /r?)

16

w't + (V/r) w' : -p + Re 1 v2 w (3.4c)

(ru') r + v + r wz' = 0. (3.4d)

These are the equations that govern viscous perturbations of the core flow butnot the Ekman layers.

The presence of the viscous terms on the right-hand side of (3.4a, b, c)requires discussion. It is helpful to refer back to the spin-up basic flowmodel. A formal, rational expansion such as (2.6) yields (2.7) in which noviscous diffusion terms appear, but Wedemeyer's approach did have these terms,as in (2.2). Without them the predictions of the model would not agree nearlyas well with experimental or other numerical results for a range of r ald t(See Reference 12 for a discussion of this.) As mentioned before, at thediscontinuity in Vw the diffusion terms are required. They could be

rincluded in a "local" sense as Venezian 13 did or in a "global" sense asWedemeyer did. The situation with (3.4) is analogous. A formal, rationalexpansion would yield perturbation equations of the same form as (3.4) but

with Re-I = 0, i.e., the inviscid equations. These are adequate except fortwo regions where viscous effects are important. These are the boundary layerat r I and the critical layer, to be discussed below. Here also the viscousterms could be included in a local sense in both of these regions or in a

global sense as we have done by retaining the Re terms in (3.4).

There is another restriction on these equations arising fran the factthat Wedemeyer's model for the core flow is not correct at r = 1, as discussedin Section II. Specifically,W ; 0 at r = 1. One can either say that themodel is valid only outside a Stewartson layer at r = I or such a layer couldbe incorporated. In the second alternative an additional term would appear in

(3.4c) arising from the term 0 i in the z-momentum equation of (3.1). Therfirst alternative will be adopted here, but the presence of the Stewartsonlayer is neglected just as the presence of the Ekman layer is. The factthat W * 0 at z c t c/a is part of the overall problem of neglecting the Ekmanlayer in the basic flow.

The final step is to consider the applicability of the quasi-steadyassumption to (3.4). If V(r,t) varies significantly with t over the timescale 0(l) appropriate to (3.4), then several coefficients depend on both r

12. W. B. Watkins and R. G. Hlussey, 'tSpin-lVp Fr'om Rest: Limitations of theWedemeyep, Model," The Physics of, Fluids, Vol. 16, No. 9, September 1973,pp. 1530-1531.

13. C. Veneaian, Toptic in Ocean Engineering, Vol. 2, pp. 87-96, GulfPubliahing Co., R ouston, -7eao, ifO.

17

and t and separation of variables could not be used to solve (3.4). Asoointed out above the time scale for V, the spin-up time, is 0(t ) which is

large compared to 0(1). Therefore, over the time scale for the perturbationsV does not change appreciably. Thus,t can be regarded as a parameter in thesolution of (3.4); i.e., we can use the quasi-steady approximation. Apossible exception to this might exist for t + 0 and an impulsive start.

In Lynn's work 3 the same equations as (3.4) appeared except that hisequations have V in place of V. The basic flow used by Lynn was (0, Vw, 0,WP ) which is an exact solution to the steady Navier-Stokes equations; this

basic flow would not apply in the two wall layers. The philosophy behind hisapproach is different from ours even though the equations are the same if

V = Vw. Lynn solved only the inviscid equations, Re"1 0, for axisymmetric

perturbati ons.

From the discussion in the introduction, it is clear that wave-typesolutions to (3.4) are required. Although there may be several approaches toobtaining these, only one besides that actually employed will be mentioned.If the sinusoidal variation in e and t of (u',v',w',p') is factored out, a setof linear partial differential equations in r and z is obtained. These mighthave to be solved numerically. This approach would be especially useful ifthe Ekman layer effects were included. No serious attempt has been made touse this approach. Here a separation of variables is enployed to obtain modaltype solutions.

IV. THE MODAL SOLUTION AND EIGENVALUE PROBELM

In this section it is convenient to change the origin of the z-coordinateto the bottom endwall which was the convention in Reference 1. Let thatcoordinate be z' so that z' - z + (c/a); z' will be used instead of z.

It is assumed that the perturbation can be expressed as a superpositionof modes or a triple Fourier expansion in 0, z', and t with coefficientsfunctions of r. It is convenient to use complex notation and express the

•. perturbation as

Real {u (r) cos K z' exp Ci (Ct - me))] (4.la)

Real v (r) cos Kz exp i (Ct -mo)] (4.1b)

)Realw (r) sin Kz' exp Ci (Ct mo)) k4ac)

18

p'-= Real p (r) cos Kz' exp [i (Ct me)] (4.1d)

K = kna/2c.

The complex quantities u, v, w, and p are solutions of the ordinary differ-ential equations obtained by substituting these forms into (3.4). Theintegers m =, ±1,... and k = 1, 2,... are the azimuthal and axial wavenumbers, respectively. For axisymmetric perturbations m 0; m = 1 is thevalue relevant to the projectile problem, but we shall not specify m at thispoint. The nondimensional complex constant C = CR + i CI is the eigenvalue

of the system. The dimensional wave frequency is CR Q and the decay rate or

damping is CI S1.

For the free oscillation problem the boundary conditions at the endwalls,z' = 0, 2c/a, are u' = v' = w' = 0 if the complete flow is being perturbed.Since we are perturbing only the core flow,the boundary conditions are not the.same. From (4.1) w' = 0 at the endwalls but not u' and v'. Evidently,theform (4.1) represents the first term of an outer expansion of theperturbation. The inner expansion must be introduced to satisfy the no-slipboundary conditions, but for this the inner expansion of the basic flow, theEkman layers, must be perturbed. In lieu of this, the form (4.1) is usedwithout endwall correction; i.e., there is slip at the endwalls.

The no-slip boundary conditions at r = 1 are satisfied by requiring

u = v = w - 0 at r =1. (4.2)

Three more boundary conditions are necessary. For a filled cylinder theboundary conditions at r = 0 depend on m; they are presented in Reference 1

i and are simply restated here:

111: 0 u = v = dw/dr - 0u-iv w p 0 (4.3)

in> I u v w 0

Applying the two differential operators common to (3.4a,b,c) to thegeneric form

F f (r) sin Kzj exp [i (Ct - me)]cos Kz

yields

[a/at + (V/r) 3/3O] F i M (r) F

and

19

VK F = f cos KzK exp [t (Ct- me)]

where

M (r) C - m V/ r (4.4)

Alf frr + f/r- [(m2/ri) + K)2 f.

Using these the following ordinary differential equations are obtained:

[Re 1 (Al-r 2 ) - i M] u + (2/r)(V+i m r" Re- 1) v -Pr 0 (4.5a)

[Re-I (Al-r) - i M] v - (aV/ar+V/r+2i m Re- /r 2)u + i m p/r :0 (4.5b)

[Re" 1 Al-i M] w + K p = 0 (4.5c)

(r u)r - i m v + K r w 0. (4.5d)

Since V is obtained from the finite difference solution of (2.2), numericalintegration of these equations is required. Equations (4.5) are a homo-

eneous, sixth order system with homogeneous, two-point boundary conditions4.2) at r = I and (4.3) at r= 0. The system is not self-adjoint. Although

there is no proof, we assume the eigenvalues form a denumerable, discretespectrum. With n used to order the spectrum, the e.v. are C.; for brevity the

index n will sometimes be omitted. Eqs. (4.5) must be solved for Cn and the

eigenfunctions un, Vn, Wn, P'n given V, c/a, Re, m and k; t enters only through

V.

If the system were self-adjoint, the index n would indicate the radialmode number, where the mode can be defined in terms of the number of zeros ofthe eigenfunctions. For a system which is not self-adjoint, it is unclear howto define a mode in a general and unambiguous manner. However, the concept ofmode is a very useful one and will be employed here. The mode can be identi-fied by calculating the e.v. at large t, where the radial mode number is knownfrom the solid-body rotation solution, and then tracked as t decreases.

Consider the inviscid limit of Eqs. (4.5), Re 0. The order of thesystem is reduced to two, and, in fact, a single 2nd order equation isobtained (Reference 3); it is analogous to the Rayleigh equation inhydrodynamic stability theory. The coefficient of the highest order deriva-tive contains M - C - m V/r and the C is then real. If M- 0 has a root in0 < r < 1, the equation has a singular point and the theory breaks down in itsneighborhood. Let rc be the root of CR - m V/r. In analogy with the Orr-

20

A,,

4'

Sommerfeld equation and its inviscid limit, rc is called the critical level

and its neighborhood the critical layer. A review of critical layer dynamicsfor shear flows was given'by Stewartson.14 The physical interpretation of

CR ý m V/r

at r rc is that the wave frequency is an integral multiple of the angular

frequency of the basic flow, indicating a resonance. If m = 0 there is nocritical layer. If m = 1 there is a critical layer up to a time for whichLi• rc 0; after that a real rc does not exist.

For the viscous case, (4.5) do not have a singularity when M = 0.However, the critical layer still plays an important role just as it does inthe Orr-Sommerfeld equation. The r for which M = 0 is called a turning

point. Lynn 3 showed that the thickness of the critical layer is O(Re indirect analogy with the result for the Orr-Sommerfeld equation. The practicalconsequence of the existence of a critical layer is that the eigenfunctionsdevelop high frequency, large relative amplitude oscillations there. If theintegration scheme is not capable of resolving these, a solution to (4.5) willnot be obtained.

There are two other properties of (4.5) which make their numericalintegration not straightforward: (1) They have a singularity at r = 0; (2)

The coefficient of the highest order derivative, Re", is small,which classi-fies them as stiff equations. Both of these properties exist when the basicflow is solid body rotation. They were discussed in detail in Reference 1,soonly a summnary of these properties and the methods of treating them will begiven.

According to the theory of differential equations (Reference 1b),r = 0 is a singularity of the second kind. A singular transformation wasfound that transforms the system into one with a singularity of the firstkind. This is easier to treats but the transformed system has characteristicvalues that are not all distinct and differ by integers- in that case thereis no simple, systematic method of generating all solutions. (See page 136 ofReference 15.) For the sixth order system (4.5) there should be 6 independentsolutions near r = 0 and because of the houndary conditions, 3 of them shouldbe singular and the other 3 regular. It was more efficient to assume powerseries expansions near r = 0 and determine the 3 regular, linearly independentsolutions. The only significant difference, compared to the solid-body rota-tion case in Reference 1, is that in the present case V must be expressed as a

14. K. Stewaiton, "Marginally Stable Inviscid Fl.ows with Cý,itical Layers,"Journzal of Appli.od Matheviatica, Vol. 27, 1981, pp. 133-1?6.

"15. E. A. Coddington and N. Levinson, !'heo~I of Oidinary Di fer entialýzquationB MaGr•,-Hill Bock Co., New York, 1955.

21

power series in r. Numerical values of these power series solutions atr = E are used as initial conditions for the numerical integration of (4.5)over the interval c < r < 1. The number of terms in the series and c areparameters of the integration. A convergence test is applied for eachintegration. Formulae and recurrence relations for the power seriescoefficients are given in Appendix A.

The stiff nature of (4.5) was discussed in Reference 1,and ortho-normalization was applied to insure that the initially linearly independentsolutions remain so during the integration. No additional discussion isneeded for the spin-up case. General rules cannot be given for the number ofsubintervals, N, and the integration step size which are required for a givenaccuracy in the solution for C; they depend on a large number of parameters,in an interactive manner. What can be said is that spin-up and larger Rerequire larger N and smaller step size than solid-body rotation and smallerRe.

V. SOLUTION OF THE EIGENVALUE PROBLEM

The procedure for solving (4.5) for C, and the eigenfunctions u, v, w, pis the same as that given in Reference 1 for solid-body rotation. The majorsteps in the procedure are merely listed here except for the "first guess"strategy; this will be described in detail since it has no analog in the solid-body rotation case.

For numerical integration, (4.5) are put into canonical form, i.e., six- first order equations in the manner of Reference 1. A value of C must be

specified, the first guess. Next, the power series solutions are evaluated atr =. With these as initial conditions, (4.5) is integrated for a specifiedN and integration step size over : < r < 1. Orthonormalization insures that 3linearly independent solutions are "enerated. These are tested to see if theboundary conditiuns are satisfied at r 1 1; this test requires that thecharacteristic determinant, Z = 0; Z is defined in Appendix B. If Z * 0 towithin a certain tolerance, a new value of C is obtained using, essentially,Newton's method. The iteration then proceeds until Z = 0 and two successive Cvalues differ to within specified tolerances. The maximum number ofiterations allowed is typically 15. For a convergent iteration processthefirst guess must be reasonably close to the converged C.

Obtaining a first guess for a spin-up e.v. at an arbitrary t would bequite difficult. Usual ly,a spin-up history is needed, i.e., C vs t. Thecomputation is then started at large t, where a first guess can be obtainedfrom the C for solid-body rotation or an approximation to it. (See Reference 1,page 18.) For smaller t, the previously coiputed C, or the two previous ones,can be used to obtain the first guess. For some cases, at early times, therequired time interval between successive calculations of C becomes small, andadditional searching for a first guess may be necessary for the iterationprocess to converge.

The time to calculate one C depends mainly on the first guess for C, t,N, integration step size, tolerances, and severity of convergence test.Typical CPU times for one C calculation are one minute on the COC 7600 and 15minutes on the VAX. The output of the standard program is only the e.v., C.

22

j4

A separate program is used to print out the eigenfunctions with an option toplot real and imaginary parts of w and p vs r.

Some results for eigenfunctions and e.v. will now be shown. In present-ing these, it is convenient to designate the 3 wave numbers by the triplet(k, it, m) corresponding to the wave numbers in the (z, r, o) directions.

The eigenfunctions, of course, are determined only to within a multi-plicative constant. The eigenfunctions have not been normalized in thisprogram. In Figure 4 the eigenfunction Real (w) = wR vs r is shown at t

7000, 1000, and 400 for Re = 5 x 105, c/a = 2.679 (so that ts 3788) and

mode (5,1,1). In Figure 4a t/ts = 1.85, CR 8.354 x 10-2, C1 8.363

x 10 and there is no critical layer- the variation of wR through the

boundary layer can be barely discerned on this scale. In Figure 4b t/ts =

r.26, CR = 7.060 X 102, C1 2.269 x 102; r = 0.44 as indicated by the arrow

in the figure. The boundary layer is less discernible here than in Figure-2 -2

4a. In Figure 4c t/ts = .11, CR = 3.015 x 10, CI = 8.066 x 10-, and

rc 0.66; away from the critical layer wR is not equal to zero although it

appears to he on this scale. For the conditions of Figure 4c, if ý1 = 628rad/sec (100 Hz), t 0.64 sec. For both Figures 4b and 4c, the criticallayer oscillations are centered at r = r

Somie results for CR are presented next for the two sample cases discussed

in the introduction. In Figure 5 CRI vs t is shown for Case I with c/a

3.120, Re = 4 x 104, mode (3,1,1) and Case 2 with c/a = 5.200, Re2 6 , mode (5,1,1). Cases I and 2 are calculated with the laminar andturbulent Ekman layer compatibility conditions, (2.8) and (2.9), respec-tively. For Case 2 there is a maximum in C at t = 1500, a phenomenon first

determined by this investigation. If the calculation were continued forsmaller times, the curve would go through the origin because CR ' 0 for t

0. For Case I a maxinuium in CRI is not found for t > 180, the last point

computed. The maximum occurs for t < 180 since CR - 0 for t = 0. For both

cases, of course, CRi approaches the result for solid-body rotation as t .

The significance of a maximum in the CRI vs t curve is related to the

necessary condition for projectile instability discussed at the end of theintroduction. If the projectile nutational frequency is less than the maximumof CR, there are two times at which instability might develop. Whether or not

it does, at either tirme, is not within the scope of this report.

Finally, some results for another set of parameters are presented to showthe spin-up e.v. histories for n 1 1 and 2 and to Illustrate the sensitivityof the iteration process convergence to the first guess of C that sometimes

23

occurs at small t. These parameters were chosen by Dr. William P. D'Amico forI; a firing program, proposed a few years ago, to be conducted in the aero-

dynamics range for the purpose of obtaining data on spin-up instability of•I projectiles. Unfortunatelythe program was not fired,so that no such data

exist to compare with the theory. However, the e.v. calculations made forthis case are of interest.

In Figure 6, C1 and C2 are presented as time histories of CR and CI with

c/a = 3.297, Re = 4974, modes (3,1,1) and (3,2,1). For this case the Ekman"layers would be laminar, so that (2.8) is used in the spin-up basic flowcalculation. The CR1 curve has an unusual feature. For 180 < t < 220 it has

a shallow minimum and maximum; on the scale of this figure CRI appears to be

constant over this range. The variation of CI1 is rather typical. Obtaining

a first guess for C, for which the iteration process would converge, was mademore difficult because CR1 was not monotonic. However, the calculation

of C2 was, relative to C1 , considerably more difficult starting at about t

250. The range of values of CR2 , CI2 in which a first guess for C2 had to

lie, In order for the iteration process to converge to the n = 2 root of Z0, steadily decreased. This is shown in the CI vs CR plot in Figure 7. The

curve is the e.v. trajectory for 180 4 t < 250; the areas in which a firstguess must lie are shown for four values of t. As indicated for t = 200 and250, the areas are off-scale. For t - 180 the rectangle has dimensions(.0008, .0007). The search for a first guess must be done with differencesless than .001. The difficulty of obtaining a first guess is compounded bythe fact that C!2 is nonmonotonic near t - 200. When the iteration process

did not converge to the n - 2 root of Z - O, it almost always converged to then - I root; sometimes it converged to a root for lArger n,and a few times Itdid not converge. There was a very predominant preference for the n 1 1 root.

To visualize Z (C), IZI (CR CI) is plotted in Figure 8. Only the

shaded portion is the IZI surface; the planar regions from the surface to IZI0 are artifacts of the plotting program. The surface is plotted in the

neighborhood of the n - 2 reot for t 180, viz., CR? a .101673, C12

.034867. Only some of the values on the coordinate lines are labeled. The"root lies at the bottom of the steep valley (not visible in the plot). To theright of it is a steep peak, and this helps to explain why most guesses did notconverge to the n n 2 result. In a sense the valley is shaded by the peak.This case illustrates the fact that for early times the solution to the e.v.problem may become more laborious than for larger times.

VI. DISCUSSION

In this report the theory and a method for the solution of the spin-upe.v. problem are given. The theory for the perturbation flo,- is a linear onebut not for the basic flow. Because viscous effects are important in the e.v.problem at the cylinder wall and critical layer, viscous perturbations are

24

used. Several assumptions are made, and some limitations are contained in thetheory. The theory and methods are successful in the sense that they provideresults, as shown in the previous section, that are physically meaningful and

W do not violate intuition or the "physics of the problem." Other investigatorshave worked on this problem without success in that sense. The more importantgauge of success is validation by comparison with either numerical simulationor experimental results.

For the solid-body rotation case and m = 0, a detailed validation wasgiven in Reference 7 using finite difference solutions to Navier-Stokesequations. This provides confidence in the treatment of (4.5), and the solid-body rotation endwall correction, for that case. Spin-up and ni = 0 could bevalidated in the same way; it would require, relatively, more analysis toreduce the numerical data. At the present time there is no numericalsimulation to validate the m I case.

Although the importance of the spin-up problem for liquid-filled pro-jectiles has been recognized since the early 1960's, very few experimentsrelevant to spin-up have been conducted at BRL, and none of these weredirected to the modes of oscillation. The only published results seem to bethose quoted in References 4 and 10. Experiments in a spin generatordetermined particle paths during spin-up and range tests provided spin decayhistories; these were used by Wedemeyer 1 0 to validate his theory for the spin-up basic flow, described in section I1. Further validation of that flow wasobtained with measurements of velocity using the LDV, e.g., Reference 16.

The only attempts to measure the e.v. were those of Aldridge1 7 and thefurther development of that work reported in Reference 6. In Reference 17,measurements of CR (not C1) are given for modes (2,1,0), (Z,2,O), and (4,2,0)

over the range 1.4 < t/t < -and compared with Lynn's calculation forc

inviscid perturbations and m 0. For larger times the difference was small;but the trend, as t decreases, was for the differences to grow. Later,Aldridge found that a systematic error existed in the analysis of the data.Some of these experiments were redone using a different data analysistechnique and reported in Reference 6; they are presented below. For m - 0there is no critical layer and t/t M 1.4 is larger than one would want for a

spin-up test. In Reference 6 measurements of CR and CI are given for modes

(2,1,0), (1,2,1) and (1,1,1). The first of these replaces those of Reference11. The latter two were excited usin9 a rotating cylinder with a precessingtop. In Table 1, these experimental results are presented together withcalculated results using the method described here. The smallest t/ts

'6. W. H. Vat~kin nnd o. R. Muoaay, Mspioi~4J fnaw Root in a Cylinder," ThePhy•ica. o2f Fluitd, Vol. 2O, No, 10, Part 1, 1977, pp. 1596-1604.

17. X. DI. Aldridge, 't~i-ie,'ntat Voaification of thie Intertial Oscillations

of a Fluici in a Cylindler Duringi Spvii4'p " BRI, t'ontmct Report No .373,Albrdeamn Pi.-?ovi Gýiund, Ma?•*untn, Saptgepihr 1975 (AD A018797). Also

eoaophye. Aotrophys. Fluid Dynamics, Vol. 8, 1977, pp. 279-301.

25

1.51. The experimental CR has an estimated error of less than 1%; the error

in CI can be as much as 10%. The differences between the experimental and

calculated frequencies are less than 2%. For the damping the difference is afactor of 3 for t/ts = 1.51; this could be expected because we have neglected

the Ekman layers, as discussed in Sections III and IV. Solid-body rotation isachieved, essentially, for t/ts = 4.75; thus, the method of Reference 1 can be

used, including the solid-body rotation endwall correction (not Ekman layercorrection). This gives the CI in the last column of Table 1, and thedifference is now about 10%.

TABLE 1. EXPERIMENTAL AND CALCULATED EIGENVALUES

(2,1,0) MODE * SPIN-UP

Re 43,000 c/a 0.995

Frequency Damping

t/ts Exp. Calc. Exp. Calc. Uncorr. Calc. CorrsI

4.75 1.269 1.268 .0061 .00297 .0073.04 1.266 1.265 .0062 .002952.02 1.236 1.242 .0068 .003111.68 1.202 1.219 .0101 .003341.51 1.184 1.202 .0105 .00342

The comparisons of CR for the m I modes are given in Table 2. For eacht, Re varies slightly; the values given are representative. Agair• thedifferences between the experimental CR and our calculated results are, at most,

2%. Further validation is needed for smaller t/ts, for both smaller and

larger Re and for larger c/a.

For projectile applications m 1 Is the pertinent azimuthal wave numberand, to apply the necessary condition for projectile instability, CR is

required. The comparison shown in Table 2 gives the significant result thatthe method presented here is validated within the limitations noted.

The lack of projectile tests in ballistic ranges has already been noted.Field tests wherein data is obtained from yawsondes can provide some informa-tion, but not as detailed as range tests. For the field tests on 155Wiprojectiles mentioned in the introduction, some of the parameters were chosenon the basis of our calculations to obtain information an instabilities duringspin-up. The results are in publication by O'Amico.IB

18. W. P. D'Amico., "Flight Data on Liquid-kPillad S•hol for' Sin-LpTNotabilifits, PRL IMOpovt in prei'altio'n, 2983.

26

TABLE 2. EXPERIMENTAL AND CALCULhAiED EIGENFREQUENCIES

c/a 0.60G

Mode (1,2,1) Re 50,000

5t/ts Cal culated Experimental Difference(!-CR) (I-CR)

1.I,473 .8B474 .8658 2.1i%

1.804 .8352 .8488 1.6%

2.800 .8276 .8305 .35%

4.947 .82,5,5 .8295 .49%

Mode (I,II) Re -- 30,000

1.357 1.3359 1.3487 .95%

1,634 1.3707 1.3736 .2%

3.180 1.4275 1.426 .1%

4110 1.4278 1.4290 .1%

Free oscillation of the fluid during spin-up was considered in thisreport. For projectile motion the forced oscillation problem must besol ,'ed. The modal solutions to (4.5) would be needed there also, but theboundary conditions are nonhomogeneous. As shown in Reference 19 for thesolid-body rotation case, the same techniques discussed in this report areneeded in the forced oscillations.

4 ACKNOWLEDGMENTS

The authors wish to thank Ms Joan M. fiartos 'or her work on a largenumber of tasks which made the programs used here operational. Herprogramming and editing were essential to the completion of this work, WP areindebted to Dr. C. W. Kitchens, Jr., senior author of Part I of this report,for his analysis, programming and insight. This work draws heavily on that inPart I. We thank Mr. William H. mermagen for acquiring the surface plotprogram and dnaiyzing it to produce Lhe 1ZI plot. We are also indebted toProfessor A. Davey who provided us with a computer program for his ortho-,nurmal ization technique.

19. N. Gerber., R. Srdney, J. A!. Bartos, "Pressure Mcnent on a riquid-Fi lednhoiecti, 7e: So?,d Body Rctation," BRI Technioal Report ARBRL-TR-02422,(Ontober 1982 (AP A129567).

27

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SV

WII

.6

.6 -t :48C00

1245

•;• .4 -600

.2 -

.0 .2 .4 .6 .8 1.0

Fiqure 1. V vs r for Case 1 with c/a 3.120, Re =4 x 104 atThree Values of t; ts 1245. Laminar Ekman Layer.

28

................................ -a

!;

1.0 - -••-•--•,

S°8

• v•i.6 - t = 10,000

! 2700 !

•,•

°4 •

!•.s.

• .0- "•

i: .0 .2 .4 .6 .8 1.0

i Figure 2, V vs r for Case 2 with cla •- 5.200, Re • 2 x 106 at•:)ii.:, Three Values of t; tst ",' 2700. Turbulent Ekman Layer.

•:,

z~c/ A t62.5

.05

.00 -

(a)0 .5 1.0

t 250.0 M

z+la- - r= .475a

r .728I.05 I

Sr~.917

V~ Vfrom (2.2)

.00

.0 .5 1.0

(b)

Fioure 3. V~ vs z +c/a= z' for c/a 1, Re 10 t~ 200

at (a) t -62. 5, (b) t -250. The Solution i s ObtainedFromn the Program in Reference 11.

itt J30

*1 102w R 0.0----

t7000

0.0 0.2 0.4 0.6 0.01.

(a) r

0.10- --

0.05-10 wR

t =1000

(b) r

Figure 4. The Eigenfunction Real (w) vs r for Re -5 x 10,c/a =2.679, Mode (5.1,1). (a) t 7000, (b) t1000$ (c) t 400.

31

t =400

0002040.6 0.68 1.0(c) r

Figure 4. The Eigenfunction Real (w) vs r for Re =5 x 10,c/a = 2.679, Mode (5,1,1). (a) t 7000, (b) t1000, (c) t =400 (continued).

.3

CR

.2

.0

0 t 5,000 IOSOOO

Figure 5. C R1 vs t for Case 1 With c/a 3.120, Re 4 x 10 Mode (3,1,1)and Case 2 With c/a =5.200, Re 2 2x 10b Mode (5,1,1).

32j

9-' * c.3 •

... .... ...... .... ...

-4

Cr)

C.-, 1.,~-4,,

r-

U,

t!

C,,

C11

LO~

44-

33

S(:., " ..

or,-

C--

0iS.... I ••• .... . i ....... . . -I-............. '" i ... . .4.-

• * U • " C ,.

S33-

S... . . . US........................ i . .. . ........ !... .. . . . ..

co

CJc~ C)2

4-)

~4J

CV\3 C%j

4- 4-

4. ) 4n

. (4.

6- U- a

Csj cci4.9

-C c

4- w

c~~.0 V 0

II344-i

VICS,Lf4) 00

c,,

CY)

wCIO-CDr

C'j

4-)

U 4-)00

u0

4-0

S-

C3 0)

L) U

%:T- Li

CD 0 C

35 -

REFERENCES

1. C. W. Kitchens, Jr., No Gerber, and R. Sedney, "Oscillations of aLiquid in a Rotating Cylinder: Part I. Solid Body Rotation," ARBRL-TR-02081, June 1978 (AD A057759).

2. M. M. Reddi, "On the Eigenvalues of Couette Flow in a Fully-FilledCylindrical Container," Franklin Institute Research Laboratory,Philadelphia, PA, Report No. F-B2294, 1967.

3. Y. M. Lynn, "Free Oscillations of a Liquid During Spin-Up," BRL ReportNo. 1663, August 1973, AD 769710).

4. Engineering Design Handbook, Liquid-Filled Projectile Design, AMCPamphlet No. 706-165, United States Army Materiel Command, Washington,D.C., April 1969 (AD 853719).

5. K. Stewartson, "On the Stability of a Spinning Top Containing Liquid,"Journal of Fluid Mechanics, Vol. 5, Part 4, 1959.

6. S. Stergiopoulos, "An Experimental Study of Inertial Waves in a FluidContained in a Rotating Cylindrical Cavity During Spin-Up from Rest,"Ph.D Thesis, York University, Toronto, Ontario, February 1982.

7. R. Sedney, N. Gerber, and J. M. Bartos, "Oscillations of a Liquid in aRotating Cylinder," AIAA Preprint 82-0296, 20th Aerospace SciencesMeeting, Orlando, Florida, January 1982.

8. H. P. Greenspan, The Theory of Rotating Fluids, Cambridge UniversityPress, London and New York, 1968.

9. E. R. Benton and A. Clark, Jr., "Spin-Up," article in Annual Review ofFluid Mechanics, Vol. 6, Annual Reviews, Inc., Palo Alto, California,1974.

10. E. H. Wedemeyer, "The Unsteady Flow Within a Spinning Cylinder," USA BRLReport No. 1225. October 1963 (AD 431846). Also Journal of FluidMechanics, Vol. 20, Part 3, 1964, pp. 383-399.

11. C. W. Kitchens, Jr., "Navier-Stokes Solutions for Spin.-Up in a FilledCylinder," AIAA Journal, Vol. 18, No. 8, August 198 0,pp. 929-934. AlsoTechnical Report ARI3•R-•R-02193, September 1979 (AD A077115).

12. W. B. Watkins and R. G. Hussey, "Spin-Up from Rest: Limitations of theWedemeyer Model," The Physics of Fluids, Vol. 16, No. 9, September 1973,pp. 1530-1531.

13. G. Venezian, Topics in Ocean Engineering, Vol. 2, pp. 87-96, GulfPublishing Co. -, t-on, Texas,"197U.

14. K. Stewartson, "Marginally Stable Inviscid Flows with Critical Layers,"Journal of Applied Mathematics, Vol. 27, 1981, pp. 133-175.

36

15. E. A. Coddington and N. Levinson, Theory of Ordinary DifferentialEquations, McGraw-Hill Book Co., New York, 1955.

16. W. B. Watkins and G. R. Hussey, "Spin-Up from Rest in a Cylinder," The

Physics of Fluids, Vol. 20, No. 10, Part 1, 1977, pp. 1596-1604.

17. K. D. Aldridge, "Experimental Verification of the Inertial Oscillationsof a Fluid in a Cylinder During Spin-Up," BRL Contract Report No 273,Aberdeen Proving Ground, Maryland, September 1975 (AD A018797). AlsoGeop.nys. Astrophys. Fluid Dynamics, Vol. 8, 1977, pp. 279-301.

18. W. P. D'Amico, "Flight Data on Liquid-Filled Shell for Spin-UpInstabilities," BRL Report in preparation, 1983.

19. i1. Gerber, R. Sedney, J. M. Bartos, "Pressure Moment on a Liquid-FilledProjectile: Solid Body Rotation," BRL Technical Report ARBRL-TR-02422,October 1982 (AD A120567).

Si

'I

37 i

' _

APPENDIX A

Power Series Coefficients

A

39 00..-.4

•.•lm2.•39

-'G, I I

APPENDIX A. POWER SERIES COEFFICIENTS

The need for the power series expansions near r = 0 of u, v, w, and p isdescribed in Section IV. The coefficients of these series are recorded here;they are obtained by substituting the series into (4.5). V must also beexpressed in a power series

V = Vj rj •

which is permissible since V is an analytic function of r, near r :0, except I Vthat it may not be uniformly so for t + 0. The coefficients Vj could be

determined from (2.2) in terms of DV/ar, a (aV/at)/ar, etc., at r = 0. Thesewould have to be found numerically from the solution V (r,t). Instead weevaluate the V. directly, fitting a quartic to V (r,t).

Because the boundary conditions (4.3) are different for mn 0 and m 1,the coefficients must be given separately for these two cases.

Mn 0:V

The forms of the power series are

u uri vu V, vr1 j j j

j W j +

The coefficients are given by the following sequence of formulas, wherew., p0 , and vI are arbitrary complex constants:

H K2 R" + i C

U - (K/2)wo, w= 0

•Ttvi vit P1 0

u Ow2- (Re/4) (lwo -Kpo)

v2 0, P2 (1/2) Hul + (K/Re) w, - V1 v1

For j 1, 2, 3 ...

w+ Re (j+2)" 2 [Hw -p

41

I1U~ A

sj

S+2 -K(j+3) Wj+j

j+2 Re(j + 4j + 3)]-EHvj + + 1 V

Pj+ 2 = (j+2)' 1 [-K(j+2)' (Hw Kpj) - Huj+I1 + 2 V t vj+2.]

The solution for the azimuthal mode m ! is given by

j=0 j j

Summations are taken from 0 for convenience,

-The coefficients are given by the following sequence of formulas, whereb0, d,, and e1 are- arbitrary comprlex constants:

a ibo, bo =bo, d •0, e= 0

a1 •0, b =0, d Id1 , e1 =Qe

H Re- K2 + i C

a2 -(1/4)KdI + (1/8) Re (b0 (iH - VI) + e l

b2 = -i(1/4)Kd1 - (31/8) Re [b° (iH - VI) + e13

For j 0, 1, 2,

N I = Re [Ha - j VE+ (2b + iaJ+1-E)A

J+1

N2j = Re Hbj+I + V+ ((2)a, .- ib

2 42

-j i(j+2)N 2~ir +1 (+

dj+2 Re oH ei i 0v11 jp/(~) +)

2EN[K (j+4)N/( d 8j 15)

,j+3 jj+2( +

2

e =[2a +i +j + 7)b -N.]Rtj+2 j+3 + Uj+3 A ]/R

43

APPENDIX Bf Characteristic Determinant, z

45_

(APPENDIX B. CHARACTERISTIC DETERMINANT, Z

The characteristic determinant, Z (C), denoted by D (C) in (25) ofReference 1, is defined here for convenience. In the integration procedure wecalculate three linearly independent solutions that satisfy three boundaryconditions at r = 0. These solutions are denoted by (uj, uj - i vj, w.)9

j 1, 2, 3; pj is not needed here. The no-slip conditions at the sidewall

require the velocity components to vanish at r = 1. These components arelinear combinations of the three solutions above. Thus

Y1 u1(1) + Y2 [u,(1) " i v1(1)] + Y3 W(1) =0

Y1 u2 (1) + Y2 [u 2 (1) - i v2 (1)] + Y3 w2 (1) 0 (BI)

Y1 u3 (1) + Y2 [u 3 (1) " i v3 (1)] + Y3 w3 (1) 0,

where Y1 . Y2, and are constants. In order that a non-trivial solution for

Yi' Y2, and Y3 be found, the determinant of the coefficients in (B), which is

designated Z, must equal zero.

ii.i

47

AN"O A UAWWW-W, N

•_,:,• .

w•~

LIST OF SYMBOLS

a cross-sectional radius of cylinder

c half-height of cylinder

C CR + i CI, complex eigenvalue

CI disturbance decay rate/a

CR disturbance frequency/si

CR1, CR2 CR'S for n = 1 and n = 2 modes, respectively

E = v/a c 2 , Ekman number

k axial mode number (see (4.1))

k- K (a/c) Re (see (2.8))

kt 0.035 (a/c) Re"1 / 5 (see (2.9))

K = k na/(2c) (see (4.1))

m aziiiiuthal mode number (see (4.1))

M C - mV/r

n radial mode number

N number of subintervals used in orthonornalization

p function describing radial variation of p' (see (4.1))

, p'3-0 pressure perturbation/(pa2 j 2)

P pressure/(pa 2Q2) of approximate basic flow (see (2.3))

A P pressure/(pa 202 ) io 3-0 Navier-Stokes equations (see (3.1))

P zeroth order approximation, in 1/t., to P* (see (2.6))0

PI coefficient in first order approximation to P* in (2.6)

SP* pressure/(pa2 a2) of exact basic flow (see (2.1))

r radial coordinate / a

rc critical level radius / a

Re a2j/v, Reynolds number

t time a

49•

Xi ~ SU

tf characteristic flight time of projectile

t (2c/a) Re1/2' t t sAl spin-up time (laminar)

= (28.6c/a) Re1 /5, t5 : tst/2 = spin-up time (turbulent)

:÷, t i t/ts

u, v, w functions describing radial variation of u', v', w' (see (4.1))

U', v', w' radial, azimuthal, axial velocity components x 1I/(a i) of 3-Dperturbed flow (see (3.2))

U, V, W radial, azimuthal, axial velocity components x 1/(a ý-) ofapproximate basic flow (see (2.2))

U1, V1 , W1 coefficients in first order approximations, in 1/ts, to U*, V*,W* (see (2.6))

Uw, Vw radial and azimuthal velocity components x 1/(a s) of basic flowwithout diffusion given by (2.4)

U, V, W radial, azimuthal, axial velocity components x 1/(a a) of the3-D Navier-Stokes equations (see (3.1))

U*, V*, W* radial, azimuthal, axial velocity components x 1/(a a) ofexact basic flow (see (2.1))

Vo zeroth order approximation, in i/ts, to V* (see (2.6))

z axial coordinate/a (z 0 at cylinder midplane)

;Z: Z + c/a

- characteristic determinant whose roots are elgenvalues

value of r where numerical integration is initiated

- azimuthal angle

constant in expression for radial velocity in approximate basicflow with laminar Ekman layer (see (2.8))

v kinematic viscosity of fluid

.p density of fluid

spin rate of cylinder

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