~/-

a> TECHNICAL REPORT ARBRL-TR-02493

(Supersedes IMR No. 766)

VISCOUS EFFECTS IN THE WEDEMEYER MODEL

OF SPIN-UP FROM REST

Raymond SedneyNathan Gerber

J u ne 1983

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND%BALLISTIC RESEARCH LABORATORY i,

ABERDEEN PROVING GROUND, MARYLAND

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TECHNICAL REPORT ARBRL-TR-02493 2 , / 2•.4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

VISCOUS EFFECTS IN THE WEDEMEYER MODEL FinalOF SPIN-UP FROM REST 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(*) 8. CONTRACT OR GRANT NUMBER(O)

Raymond SedneyNathan Gerber

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I8. SUPPLEMENTARY NOTES

This report supersedes IMR 766, dated January 1983.

I1, KEY WORDS (Continue on reverse si•e It nece•saro and idene._'y by block number)

Azimuthal Veloc 4 ty Profile Spin-Up FlowEkman Compatibility Condition Spin-Up TimeEkman Layer Unsteady Axisymmetric FlowRotating Fluid Wedemeyer Model

120. A93TIR ACT (Ceaua setm vwoe td~v if neseemy eaad fdmuiral by block n~umber) (bj a)

The problem of impulsive spin-up from rest of a liquid filling a right-circular cylinder is treated. A numerical solution is obtained to the non-linear partial differential equation that governs the spin-up flow by theWedemeyer model. An implicit, two-time level, second order, iterative finitedifference technique is used. Approximations to an impulsive start are dis-cussed. An analytical procedure is applied at the firsL time step of thecalculation to avoid errors caused by a discontinuity in the boundary conditions.

DO. A .• .. ,1473 EDITIONwOF 'OVSSISO4SOLETX UNCLASSIFIED jSECUPQTY CLASSIFICATION OF THIS PA(Ci (WImn Data IEntered)

UNCLASSIFIEDSECUIRITY CI.ASSI PIC ATION Of' TMgS PAQGE(Wd" DOeM XMacmd)

Conditions for the validity of the model are discussed and a detailed discus-sion Is presented of the Ekman compatibility condition relating radial toazimuthal velocity. Representative results are shown demonstrating the effectsof the problem parameters atid different Ekman compatibility conditions on spin-up profiles, i.e., curves of azimuthal velocity vs radial coordinate. Forseveral cases the spin-up results are validdted by comparison with those of theNavier-Stokes equations.

FAccosqian Far-NTI CFRA&IDTI' XA

Divilribution/

- Avwi I. •, t y O(ude s

_0 , I and/or

UNCLASSIFIEDL~~. SECUMITY CLASSIFICATION OF TMIS P9AGE('Wlin Date Ent.?d)j

IABLE OF CONTENTS fU eLIST OF ILLUSTRATIONS ............... o ....... *. ........ 5

1. INTRODUCT ION . . . o * . .o. .. 7

II. THE SPIN-UP MODEL..,........................... ....... ... 9

A. Time Scales ................................................. S

B. Approximation to Impulsive Start .... 10

Table 1. Time Scales for Impulsive Start, MS............... 11

C. The Physics of Spin-Up ...................................... 12

Do Spin-Up Theory . . . . .. . . . .. 12

E. Discuss~on of Ekman Compatibility Condition ................ 15

11 . METHOD OF SOLUTION .............................................. 18

A. Finite Difference Method .......... 18

B. Accuracy of Results ......................................... 19

Table 2. Effect of N on Spin-Up Velocity Profile,- Casve 1, k zRe 28.3 .... 20

Table 3. Effect of N on Spin-Up Velocity Profile,Case 2, kt Re - 736 ....................................... 20

C. Discontinuity Due to Impulsive Start ........................ 21

IV. RESULTS ........................................................ 22

V. APPROXIMATIONS FOR DIFFUSION EFFECTS .......... 23

V I . C O N C L U S I O N S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4

ACKNOWLEDGMENT .................................................. 25

LIST OF SYMBOLS ..................... ........................... 41

DISTRIBUTION LIST ......................................... ... 45

3

LISTOF SMBOL 41

LIST OF ILLUSTRATIONS

F i gU re Pg

1 Notation and Coordinate System for the Spinning Cylinder ..... 26

2 Normalized In-Bore Spin History for an M687 Shell LaunchedFrom the M109 Howitzer at Charge 3, i = 92.3 Rev/s ........... 26

3 Normalized Radial Mass Flux in the Boundary Layer vs Ratioof Outer Flow Spin to Disk Spin, Reference 13. Also U vsV/r When the Rl Curve is Used to Define an EC ................ 27

4 Grid Scheme in (r,t) Plane ................................... 27

5 V/r vs r for Re = 4974, c/a = 3.30 at t = 20 and 50Using Local Solution or Assuming V(1,O) = I ....... 28

6 Vorticity 4 vs r for Re = 4974, c/a = 3.30 at t = 100and 600 Using Local Solution or Assuming V(1,O) = 1 ....... ,.. 28

7 V vs r for Two Re and t With c/a 1 1. The Curves areFrom Navier-Stokes Solutions; Circled Points are Solutionsof (2.2) and (289.. ....... 29

8 V vs r for Case 1 With c/a n 3.120 Re = 4 x 104 at ThreeValues of t; t5 - 1245, Laminar Ekman Layers ................. 29

9 V vs r for Case 2 With c/a = 5.200, Re = 2 x 106 at Three 30Values of t; tt 2700, Turbulent Ekman Layers ............. 3

10. V vs r for Re = 3 x 105 at Two Values of c/a, t WithLaminar or Turbulent Ekman Layers ........................... 3

11. V vs r With Diffusion, (2.2), and Without Diffusion, (2.4),

for Re = 6.08 X 105, c/d = 2.679 at Two Values of t,Laminar Ekman Layers ......................................... 3

12, V vs r With Diffusion, (2.2), and Without Diffision, (2.4),

for Re = 2 x 10E, c/a = 5.200 at Two Values of ktt;

kt = 1/t st 1/2700, Turbulent Ekman Layers .................. 31

13. V vs r Using the EC From the RL Curve, Figure 3, and the ECFrom (2.8) for k Re = 5,000, c/a = 0.0485, t = 60, andAr = 11200 ................................................ 32

14. H (S) vs S, the Function in (5.2) .................... 32

1b. V vs r for c = 1/50 at t' = 0.80 From (2.2) and (2.4).The Venezian Approximation is the First Term of (5.2).The Points Are From Both Terms of (5.2) ..................... 33

5 .

i

I. INTRODUCTION

For many years it has been observed that liquid-filled projectiles ha\ve aproclivity for unusual flight behavior, often being unstable even though thesame projectile with a solid payload is stable. The spin of the projectileimparts rotation to the liquid which affects the dynamic behavior of theshell. Typically, the fluid is contained in a right-circular cylinder. Inthe problem considered here, the fluid fills the cylinder. The coordinatesystem and notation are shoin in Figure 1.

The fluid motions can be separated into two classes: solid-body rotationand spin-up. The meaning of the former is obvious. The latter referb to thetransient state produiced by changing the fluid motion from solid rotation atone spin (which may i)e zero) to that at a greater spin. The physics of spin-up from rest, the problem to be considered here, is different in some impor-tant respects from that of spin-up from a state of solid-body rotation. 1' 2

In particular, spin-up from rest is inherently a nonlinear problem, whereasspin-up between two finite states of solid-body rotation can be treated as alinear problem if the change in rotation is small. Spin-down is, in general,an unstable flow and often becomes turbulent. Relatively little work has beendone on spin-down; see Reference 3 for a discussion of this problem and somepertinent references on experimental work.

The basic reference on spin-up from rest is Wedemeyer's paper, our workis an extension of his. Wedemeyer's model was based on an order-of-magnitudeanalysis rather than on formal matched asymptotic expansions. A more formaltreatment was given by Greenspanl for a two-term expansion. In both cases aphenomenological approach was required to complete the theory.

Wedemeyer's model yielded a nonlinear partial differential equation ofthe diffusion type for the azimuthal velocity, V, as a function of time andradial coordinate but not of axial coordinate. Without the diffusion termsthe solution for V is elementary; Wedemeyer omitted them in his approximateanalysis, The main objective of this report is to present the numerical

1. H.P. Grcen.pan, The Thieory of Rotating Fluids, Cambridge University Press,London and New York, 1968.

2. E.R. Renton and A. Clark, Jr., "Spin-Up," article in Annual Review ofFluid Mechanics, Vol. 6, Annual Reviews, Inc., Palo Alto, California,f 97 4_.

3. G.P. Neitzel and Stephen H. Davis, 'ergy Stability Theory of Decel-erating Swirl Flows," The Physics of Fluids, Vol. 23, No. 3, March 1980,pp. 432-437.

4. E.H. Wedemeyer, "The Unsteady Flow Within a Spinning Cylinder," BRL ReportNo. 1225, October 1963 (AD 431846). Also Journal of Fluid &lechanics, Vol.20, Part 3, 1964, pp. 383-399.

7

PAGEyn%.Irl JkA

solution of Wedemeyer's equation. We shall state the necessary equations andboundary conditions and shall describe the numerical procedure. Also,we shallpresent results together with some discussion of the accuracy of the solu.,tion. Applications to projectile problems will be emphasized.

One of the main applications of spin-up theory is in the study of theeigenvalue problem, i.e., the determination of frequencies and decay rates ofthe waves in a rotating fluid. 5 Spin-up theory is also required to solve theforced oscillation, or, moment, problem. Previously thi, theory was used tostudy the spin decay of a liquid-filled projectile 6 after ejection from thegun.

Some &ppreciation of spin-up effects on a projectile flight can be gainedby considering orders of magnitude. Let a and c be the radius and half-heightof the cylinder, .Q the spin of the projectile in rad/sec (say at the muzzle),v the kinematic viscosity of the liquid, and

Re a a 2/v, E = v/8 c2 , (1.1)

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 Pein 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

S "(2c/a) Re &I/

/2 (1.2)= 21 (sec).

This expression is derivable from linear spii-up theory 1' 2 and from theWedemeyer solution withoutl diffusion. We also use a nondimensional spin-uptime ts = ot ts For Re > 10i, approxii;.f"Ty, the kmanlayer may be turbulent,

in which case the characteristic spin-up time can be estimated by

fst E (28.6 c/a) Re1 / 5 /S. (1.3)

This result can be obtained from the Wedemeyer solution without diffusion forturbulent Ekinan ldyers. Basically, these are time scales for the spin-upprocess and do not necessarily give a measure of the closeness of the processto solid-body rotation which can be measured in several ways, e.g.,

5. R. Sedney and N. Gerber, "Oscillations of a Liquid in a Rotating Cylinder:Part II. Spin-Up," BRL Technical Report ARBRL-TR-02489, May 1983.

6. C.W. Kitchens, Jr., N. Gerber, and R. Sedney, "Spin-Decay of Liquid-FilledProjectiLes," Journal of Spacecraft and Rockets Vol. 15, No. 6, November-December 1978, pp. 348-354.

the time for the angular momenturn, flow rate across a meridian p-larm, or otherproperty to attain a value within 10%., say, of the solid-body value. Thesemeasures require detailed calculations and so arQ not very convenient. A ruleof thumb is that solid-body rotation is rcached at about 4 C after an impul-sive start of the cylinder.

For projectile flight these characteristic spin-up time.; should becompared with tfl, the time of flight of the projectile, If t or t

<< t fl, spin.up effects can he disregarded and solid-body rotation can be

assumed. If f or E s f /10, or larger, spin-up effects probably have to

be considered. To put these estimates in perspective, consider the parametersfor two 155mm projectiles. For, one of these) Case 1, c/a = 3.120, Re

4 x 10", a = 754 rad/sec giving ts = 1.65 sec. For Case 2, c/a = 5.200, Re =

2 x 106, 1 = 754 rad/sec giving t st= 3.52 sec. For both, tfl = 40 sec, and the

above criteria indicate that spin-up effects may be important, especially forCase ?. The extent to which the projectile iotion is, in fact, affected bythe internal liquid motion requires a solution to the forced oscillation, ormoment, problem. Reference 7 treats the moment problem for solid-bodyrotation flow. The present report is restricted to the unperturbed time-dependent axisymmetric fluid motion.

11. THE SPIN-UP MODEL

A. Time Scales.

Consider the motion of a fluid which fills a cylinder, initially at rest,F which is "rapidly" brought to a constant angular velocity. The fluid motionis axisymmetric and time depzndent. The conditions for which the Wedemeyer

spin-up model 4 is valid can be presented in terms of the time scalesinvolved. It was derived on the basis of an impulsively started cylindricalcontainer, a theoretical concept which will he discussed below; for now thistime scale is taken to be zero. Only the laminar Ekman layer case will bedisco ,ed. There are three time scales that govern the spin-up process. Innondimensional form these are

fs 2 E -I112 Q_-1

-12 -1 1

S= (a/c) 2 E- s- 1 .

7. N. Gerber and R. Sedney, "Moment on a Liquid-Filted Spinning7 and NutatingProjectile: Solid Body Rotation," BRL Technical Report ARBRL-TR-02470,February 1983 (AD A125332).

9!4

| m .1

The first of these is the time for the cýmntainer to rotate through onerevolution. It can be shown that ( is the time scale for the formation of

quasi-steady Ekman layers on the endwalls, which layers are of thickness0 (v/ I) , Actually,the Ekman layers are quasi-steady after about 2 radians

of rotation, i.e., d - 2. Wedemeyer used this result; a more complete study

of Ekman layer formation is given by Benton.6 The thickness of the Lkman

layers is assumed to be small compared to either c or a so that E1/2 or Re-11 2

is required to be small. The second time scale is (1.2), anticipated to be

the characteristic spin-up time; spin-up occurs for f/fs N 0(1). The time

scale f is the time for velocity gradients or vorticity to diffuse viscously

across the radius of the cylinder. For c/a - 0(1), t is an order ofV

magnitude greater than fs so that viscous diffusion plays a small role inspin-up. For an infinite cylinder it is the only mechanism for spin-up; for

large c/a, f can become comparable to or less than t5 .V

For an impulsive start, the conditions for the validity of the spin-upmodel are

Q << fs<< fV

E >> I and c/a 0(1).

B. Approximation to Iulsive Start.

The degree to which an impulsive start can be approximated in an experi-"ment depends on the system parameters. :or our application the angularacceleration is large during the time when the projectile is in the gun; thesmall deceleration in flight, the spin decay,6 is neglected. For artilleryprojectiles this acceleration time is typically 0.020 sec. A time history ofprojectile spin, b (f), is shown in Figure 2. To quantify the departure froman ideal impulse several approaches are possible; here time scales are used.

For a finite angular acceleration of the cylinder let the spin beS(), as in Figure 2, e.g. The time f is used as a reference. A chdr-

acteristic time fc is introduced, the time when the acceleration becomes zero,

The acciltration or impulse time scale is taken to be

8. E.R. Benton, "On the Flow Due to a Rotating Disk," Journal of Vluit4.eohanic, Vol. 24, Part 4, 1966, pp. 781-800.

10

dld

which was used by Weidman 9 '°10 f- 0 for a true impulse, For comparative

purposes Ti (t) is taken to be linear for 0 < , < and s (t 2 = for

t > " In most spin-up experiments enough information to define S (t)C

more precisely is not available; a linear approximation to 1 (J) in Figure 2is reasonable.

In Table 1 the above time scales are given for three experimentalarrangements that have been used in spin-up studies. (In some other experi-ments insufficient information was available to define the time scales.) Thefirst is the gun tube, using the parameters of Figure 2; the second is thespin generator from which Wedemeyer4 obtained observations of secondary flowduring spin-up; the third is the most nearly impulsive case in Weidman'sexperiments, Figure 2c of Reference 10.

TABLE 1. TIME SCALES FOR IMPULSIVE START, MS

SI •c

Gun tube 5.6 10.8 18

Spin generator ?6.6 22.2 200

Weidman 218 57.6 5,200

To approximate an impulsive start for the spin-up problem,two criteria

must be satisfied: fI << atc & a condition on the mechanical system and

f << to insure that the impulse time scale is small compared to the time

9. P.D. Weidman, "On the Spin-Up and Spin-Down of a Rotatin,? Fluid: Part, 1.Extendingj the wedemeyer Model," Jgurnal. of Fluid M•,echanicE, Vol. 77, Part4, 1976, pp. 685-708.

10. P.D. Weidman, "On the Spin-Up and Spin-Down of a Rotating Fluid: Part 2.Menaurements and Stability," JournaZ of Fluid Mfechanics, Vol. 77, Part 4,1976, pp. 709-735.

of formation of the quasi-steady Ekman layers. Table 1 shows that the guntube gives the best approximation to an impulsive start, but only marginallyaccording to the criteria. The spin generator does not give a impulsive startand Weidman's case Is far from it.

Since the Wedemeyer model assumes an impulsive start, deviations from itas shown in Table 1 would require nodification of the model. The necessaryanalysis has not been done for either small or large departures from animpulsive start. In References 9 and 11 the same type of modification of theWedemeyer model was made to account for a nonimpulsive start, The valie ofsl was replaced by the variable 5 (f). This should be a reaso,',ableapproximation for an almost impulsive start, In Reference 11 its validity waschecked by comparison with solutions to the Navier-Stokes equations.

C. The Physics of Spin-Up.

In time of O(f9) quasi-steady Ekman layers are formed on the endwalls.

For small time and near the center of the endwalls these layers are essen-tially the same as the boundary layer on a steady rotating disk, the vonKarman disk problem. The suction exerted by these layers draws external fluidinto the Ekman layers and imparts rotation to this fluid. With no pressuregradient acting, the fluid spirals out to larger radii, where the Ekman layerseject fluid. This fluid is now rotating. This is the hasic mechanism forspin-up.

0. Spin-Up Tý-_ory

The notation here will be the same as in Reference 5. Lengths, veloci-

ties, pressure, and time are made nondimensional by a, as, p) 2 a 2 , and-1s ,respectively, where p is the liquid density and 0 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. Derivatives are indicated bysubscripts. Wedemeyer4 showed that the flow can be divided into two regions:the quasi-steady Ekman layers at the endwalls and the rest of the flow, calledthe core flow. Wedemeyer did not point out that a boundary layer., i.e.,Stewartson layer, must be inserted at the cylinder wall. Starting with theNavier-Stokes equations for axisymmetric flow,

U* + U* U* + W* U* - V*2 /r -P* + Re 1 (v 2t1* - U*/r 2 ) (2.1a)t r z r

1h. C.W. Kitchens, Jr., and N. Gerber, "Prediction of Spin-Decay of Liquid-,Fi.lted Projct•s "B," BRL Report 1996, July 1977 (AD A043275)

12

~- --

I

V* + U* V* + W* V* + U* V*/r Re- (v 2V* - V*/r 2 ) (2.1b)t r Vz

W* + U* W* + W* W* -P* + Re- Vz W* (2.1c)t r z z

(rU*)r + rWz = 0 (2.1d)

v2() = ()rr + (1/r) ) + )zz' (2.1e)

where the asterisk indicates the exact solution, he used order-of-magnitudearguments as indicated in Sections NIA and 1IC to simplify them in the coreflow. The approximations reduce the three momentum equations, (2.1, a,b,c),to

-1V + U (V + V/r) Re [V + (V/r) (2.2)Vt+ ( Vr ) rrr

and

Uz = Vz = Pz = 0, (2.3)

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

Vw + Uw (V r + Vw/r) = 0, (2.4)

where the sub w indicates this approximation. In applying his model Wedemeyerused (2.4) rather than (2.2). Although he showed the crucial importance ofthe Ekman layers to the spin-up process, his model did not require a solutionfor the flow in these layers. He did discuss certain properties of thesolution which were required in his model for the core flow.

A mire formal approach to this problem was given by Greenspan, 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 E so that thes

new time is t' = t/ts = t/[(2c/a) Rel/]. An expansion in the small parameter

(I/t) = (a/2c) Re" 1 / 2 (2.5)

is assumed. The form of this expansion

U* (i/t) U1 +

V* V 0 + (i/t) V + (2.6)(2.6)W* (l/ts) W + .

P* P + (/t) P +"

13

follows from the knowledge that In the core flow U* and W* are 0(1/ts) and V*

is 0(1) if 1/ts is small. See Reference I for details. The independent

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

V ot + U1 (Vor + V0 /r) = 0

V 2/r = P (2.7)

Poz Voz = Ulz

These are the same as (2.3) and (2.4) if we set Vo = Vw, Uw = (i/ts) U, andt' = t/t . Although the formalism of matched asymptotic expansions 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 expansion,which was not considered in Reference 1.

To solve (2.2), (2.4), or (2.7) a relationship between U and V is neces-sary. This is called the Ekman compatibility condition, abbreviated EC,because the Ekman layer suction must be made compatible with the core flow.Wedemeyer used the facts that the Ekman layers are quasi-steady after onerevolution and that the radial mass flux in the core flow must be balanced bythat in the Ekman layers to obtain some conditions on the U, V relationship.At this point he was forced to take a phenomenological approach. The rela-tionship is known at t + 0 and t + - and he proposed a linear interpolationbetween them to obtain an approximate relationship for any t. He tested thisidea in some other problems where the solution was known and decided it wassatisfactory. Some confusion has appeared in the literature because this stepwas misinterpreted; further discussion of this is given below. His result is

U k (V - r)ký K (a/c) Re"-1/2 (2.8)

K E1/2= 2K/ts

for laminar Ekman layers. Wedemeyer proposed K = 0.443 but Greenspan sug-gested K = 0.5; the latter often gives results in better agreement withnumerical solutions to the Navier-Stokes equations. For turbulent Ekmanlayers

U = -kt (r-V) 8 / 5 (2. ~(2,9)•

k=t .035 (a/c) Re 5 st,

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).

14

Using (2.8), (2.4) can be solved explicitly for V(r,t) with V(r,O) k !and V(I,t) z 1:

2k 9t 2k t -k tV (re - I/r)/(e -1)t for r k (

, : 0 for r < e

-k t

so that r = e separates rotating and nonrotating fluid; it is called thefront. Although Vw is continuous there, a discontinuity in shear, i.e., Vw

exists because diffusion is neglected. Using (2.9) a numerical integration isnecessary, but the character of the solution is the same as when (2.8) isused. The radial velocity is obtained from (2.8) or (2.9) and W is obtainedfrom the continuity equation

W -(z/r) (rU)r

At r = 1, W • 0 so that a Stewartson layer must be inserted there, as men-tioned before; this has yet to be done. Also W • 0 at the endwalls z t ± c/aand 1 is discontinuous at the front.

Wedemeyer pointed out that the shear discontinuity in the solution to(2.4), see (2.10), would be smoothed out if the diffusion terms were retained,as in (2.2). But this is a nonlinear second order equation and must beintegrated by finite difference methods. This is discussed in Section I1l.For laminiar Ekman layers, the spin-up velocity profile is

V = f (r, k t, k Re),

where kYt = t' if K : 0.5; for the turbulent case k is replaced by kt. These

functional forms follow directly from (2.2) and (2.8) or (2.9) and representthe most efficient way to display the results.

E. Discussion of Ekman Compatibility Condition.

The original forms of the EC given by Wedemeyer, for laminar andturbulent Ekman layers, were discussed in Section liD. It is important torealize that an exact EC does not exist; it is necessarily approximate. Thiscan be deduced theoretically and is shown from finite difference solutions ofthe spin-up problem using the Navier-Stokes equations. 12 Three attempts toobtain EC which give improved approximations to V will be described.

12. C.W. Kitchens, Jr., "Ekman Compatibility Condition8 in Wedemeyer Spin-UpModel," The Physics of Fluids, Vol. 23, No.5, May 1980, pp. 10,62-1064.

155I-L ... 4

From his finite difference solutions Kitchens 1 2 obtained the radial massflow in the Ekman layer as a function of V/r for several r and t, The massflow can be used to form the EC. A unique relation was not found, rather aband of points. Based on these numerical data, Kitchens deduced an approxi-mate, monotonic, nonlinear EC. For 0 : V/r ! 0.75, U is a cubic in V/r andfor 0.75 • V/r 5 1, it is a linear function. Limited tests of this EC showonly small improvement in V compared to using (2.8).

The following EC is proposed as an extension of (2.8):

11 = -[r/(Re 11 2 c/a)] [.500 (1-V/r) - .057 (1-V/r) 3 ] (2.11)

This relation satisfies (9) in Reference 4 which holds for V/r = 0 at t = 0and the condition U (V/r = 1) = 0, for t + -, the two conditions imposed byWedemeyer in brriving at his linear expression. However, it also satisfi'es(14) in Reference 4, in slope and curvature, when the core flow is almostsolid-body rotation. Only one test of this EC has been made; it showed that Vwas significantly closer to the results from Navier-Stokes calculations thanthose using other EC's. This EC is the only one which is a rational extensionof Wedemeyer's linear expression.

The last EC to be discussed is of a somewhat different nature, and itsorigin is not clear. It utilizes the solutions of Rogers and Lance 1 3 to theboundary layer flow over an infinite disk rotating with angular velocity 'I andouter flow in solid-body rotation with angular velocity w . The main resultused from this paper is the computed radial mass flux in the boundary layer/

r Re m vs . It is plotted in Figure 3, and we shall call this the RLcurve. Using arguments somewhat different from Wedemeyer's, Greenspanl de-rived the EC (2.8) using a linear approximation to the RL curve. Possiblybecause of a suggestion of Greenspan,' page 169, several investigators haveattempted to improve (2.8) by using an analytical fit to the RL curve; theseinclude Goller and Ranov,1 4 Veneziar 1 5 Watkins and Hussey,' 6 and Weidman0This EC is determined from Figure 3 sing the alternate labels on the scales,

1/2 9IU (c/a) Re /r vs V/r. Weidman 9 uses a 7th order polynominal to approximatethe RL curve.

13. M, H. Rogers and G.N. Lance, "The Rotationally Symmetric Flow of a ViscousFluid in the Presence of an Infinite Rotating Disk," Journal of FluidMechanics, Vol. 7, Part 2, 1960, pp. 617-631.

14. H. Goller and J. Ranov, "Unsteady Rotating Flow in a Cylinder with a FreeSurface," Journal of Basic Engineering, TRANS. ASME, Vol. 90, Series ),December 1968, pp. 445-454.

15. G. Venezian, Topics in Ocean Engineering, Vol. 2, pp. 87-96, GulfPublishing Co., Houston, Texas, 7970.

16. 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.

16

.1

Note that Figure 3 appears in Wedemeyer's paper'" hut Pot for 1.hi pm1,!1)sq)f determinirng an E(.. After presenting his arguments fur (2. i) , h(v te e(!d it.against two other similar problems with known solutiors. Ono (, thof was thesolutions of Rogers and Lance. 1 3 There has heen some confusion on thismatter because it was assumed that Wedemeyer hbtained (2.2) as d 14n,>1,"

approximation to the RL curve.

As pointed out by t[vidman, one of tile assumptions needed to .ppl•y theRogers and Lance results to derive an EC is that the fluid in the core is

" "locally in solid-body rotation." It appears tnat this assumption is ratherunrealistic. Some arguments can be constructed to show that this EC isimplausible; but it cannot be proved that it is incorrect, becausc an exa-t FCdoes not exist. Some quelitative airguments were given by Benton;17 later,heprovided some arguments of a more quantitative nature18 which were based onthe computations described in this report. One can discuss the consequencesof using the EC obtained from the RL curve for either (2.2), the spin-upequation with diffusion, or (2.4), without diffusion. Some exawmpiles ofresults from the numerical solution of (2.2), using this EC, will he givenlater. The solution of (2.4) with the EC using the RL curve has some untsua!properties which actually provide the most compelling reason for rejectingthis EC. Since (2.4) is a first order, quasi-linear partial differentialequation, it can be solved by quadrature; this is true if (2.9) is used,e.g.. For this purpose the method of characteristics is used, there being aone parameter family of characteristics. Using the EC from the RL curve,Weidman 9 showed that the characteristics intersect nedr the front, implyingthat the solution to (2.4) is douhle-valued: for a fixed t, V is double-valued over a range of r, in tdie neiyhborhood of the r for which V = 0.

curve is not monotonic. Clearly this is an unacceptable result. Weidmncn saysthat there frnst be a discontinuity in V to resolve this dilenola. This remainsa conjecture since he did not determine the position and strength of thediscontinuity.

Using the EC from the RL curve in the spin-up equation with diffusion,(2.2), the numerical solution is noc double-valued or discontinuous. One suchresult will be shown later. It will be seen that when the limit with nodiffusion is approached, k Re - o, there is a tendency for the V vs r curve

to develop a vertical tangent. This might he an indication of an approach toa discontinuity. Both a vertical tangent in the theory with diffusion or adiscontinuity without diffusion are physically unacceptable and violate theassumptions of the model. Since the double-valued/discontinuous nature of thesolution is caused by the nonmonotonic RL curve, it seems evident that the ECbased on it should be discarded. Any suggestion that these anomalies indicatea breakdown of the Wedemeyer model is unjustified, of course. The numericalsolutions of (2.2), rather than (2.4), using this EC does give reasonable-

17. E.R. Benton, Memo to COTR, Taek Order 74-461, BKL, January 1975.

18. R.R. Bento'n, "Vorticity Dynamice in Spin-Up from Rest," The Phy.ei c ofFLuida, Vol. 22, No. 7, July 1979, pp. 1250-1251.

17

looking V(r,t) as long as the limit with no diffusion is not approached, i.e.,if k A Re is not large. For reasons given above this EC is not used except for

showing some comparative results.

I11. METHOD OF SOLUTION

A. Finite Difference Method.

The initial and boundary conr'itions for (2.2) are:

V = 0 for t < 0, 0 < r < 1 (3.1)

V = 1 for r = 1, t * 0 (3.2)

V = 0 for r - 0, t > 0 (3.3)

Since (2,2) is a parabolic, quasi-linear equation, it can be solved by march-ing in time; i.e., V(r,t) is obtained at t = kAt, where k = 1, 2,and 4t is the time increment used in approximating Vt. The grid used in the

finite difference scheme is shown in Figure 4. The scheme is an implicit oneand the discretization error is second order in both r and t, since centraldifferences are used in approximating all derivatives.

The interval 0 < r < 1 is divided into N segments of equal lengthAr = I/N. V is calculated at the N-i nodal points r. = j Ar, j 1,2, . .. , N-i. Given V at t - kAt, fo. all j, the solution at (k+l)At isubLained. At that time level, consider a typical point Q in Figure 4. The

finite difference approximation to (2.2) is centered at P. Thus,

VQ - VR = (1/2) (FR + FQ) At + 0 (At) 3 (3.4)

where

F -Re"I EV + (V/r)rI - U (V + V/r) (3.5)rr r rThe following difference approximations are made at the point R ( jar, kat):

Vr = (VB - VA)/( 2 Ar) (3.6a)

Vrr = (VB - 2VR + VA)/(Ar) 2 (3.6b)

(V/r)r = [(V/r)B - (V/r)A]/(2 Ar) (j > 1) (3.61-)

(V/r)r = (1/2) (VB - 2VR)/(Ar) 2 (j = 1). (3.6d)

The special form of (3.6d) is required because (V/r)A is indeterminate at

r = Ar. Identical approximations are made at point Q, with points A and Breplaced by points S and T, respectively, in (3.6).

i8

In (3.4) arid (3.5) IJQ, related to VQ by tho EC, requires an iterative

process. An initial fss ror" V) is used to calcuilate 0o. The c,1ci"u 1a J VQ

from (3.4) is then used to ]h'Cain a new '1, etc., until th ssired degree of

c.onvergence is obtained at all points for t n (kt1) •t. iitial gues

for VQ is obtained by extrapolating from VM and VR, IRecause of t.he implicit

nature of the finite difference scheme the values of V at all nodal points att = (k+1)At must be obtained simultaneoaisly. However, the matrix of t0• sotof N linear algebraic equations is tridiaIonal so that a straigihtfor'wardalgorithm can he used to solve the equations, which have the form

aU j-1, k+1 + b vj, k+1 +j vj+1, k+1 2 d. (3,7)

The expressions for aj, bit cj, and ,dj are given in the Appendilx.

3. Accuracy of Results.

Convergence criteria were set to insure that the numerical solution wasaccurate •o at least three decimal places. For, a number of cases, covering alarge range of k Re, or kt Re, calculations were made for different com-binations of v% and At to determine maximuri, permissible interval sizes. The

At was kept small encugh so that no more than four iterations were requiredfor convergence at any time step. The Ar = 1/N required to compute V to anaccuracy of, say, 1% or less depends on r and t and on the parameter k Re, or-

kt Re. Some representative results showing the accuracy of V are presented in

Tables 2 and 3 for the parameters of Cases I and ?, respectively. The per-centage error is largest for small r; V at r = 0.2 is chosen for Tables 2 and3. For k Re = 23 in Table 2, N = 100 is large enough for an accuracy of It

or less when t > 600. For k.. Re n 736 in Table 2, N • 200 is required for

t = 6,000. When this solution is used as the basic flow in our eigenvalue

calculations, the vorticity r = (rV)r/ 2 r is also required. The error in it is

usually largest near r = 1. A convenient test on the accuracy of ý is avail-able and discussed in Section III C. The CPU time for solving (2.2) varieswith the parameter but is usually less than one minute on the VAX.

Ultimately, the adequacy of the numerical solution, and of the modelitself, must be judged by comparisons with solutions to the Navier-Stokesequations. Some examples of these comparisons wil! be given in the nextsection.

iI

TABLE 2. EFFECT OF N 'N SPIN-UP VELOCITY PROFILE, CASE 1, kk Re 28.3

t - 1,200, r a 0.20N V AV.v [V(N) - V(100)]IV(100)

20 .07558 3.49%40 .07355 0.71%80 .07309 0.08%100 .07303 0.00%

t = 600, r 0.20

N V AV/V [V(N) - V(100)]/V(100)

20 .00240 72.6%40 .00160 5.2%80 .00141 1.4%100 .00139 0.0%

TABLE 3. EFFECT OF N ON SPIN-UP VELOCITY PROFILE, CASE 2, kt Re 736

t = 9,000, r = 0.20

N V AV. •. V(N) V(400)]/V(400)

50 .08626 1.90%100 .08517 0.63%200 .08476 0.14%400 .08464 0.00%

t 6,000, r = 0.20

N V AV/V EV(N) - V(1000)]/V(1000)

50 .01748 21.2%100 .01534 6.4%200 .01461 1.3%400 .01440 -0.14%800 .01443 0.07%1000 .01442 o.00%

2o

' 'I

C. flIscontinruit.. Due to Impulsive Start.

For an ifmulsive stdrt, conditions (3.1) and (3.2) require a discon-tinuity in V at r = 1, t = 0. If the finite Oifference scheme just describedis used without modification, the value of V(1,0) would be required but is notavailable. if values such as 0, 1, or 1/2 are used, a significarl- error ismade in a swdll neighborhood of r = 1. becaus2 of the diffusive nature of(2.2), this error will decrease as t increases. The error may be significantat the earliest time for which a solution is required. To reduce this errora local solution is needed to resolve the discontinuity.

Introducing new coordinates

R (] - r) t'-I 2 , T = t',

a solution for small t' was obtained in the form

112 vV =V 0 (R) + T I (R) +

The first term is given by

Vo=erfc (R/W /2c erfc ([l-r] [Re/t]I//2)

1/(k Ze).

For r near unity, this is the same as the soluticn for the impulsivelySstarted, infinite plate (the Rayleigh problem) as expected. This solutiont satisfies (3.1) and (3.2) but not (3.3). It is used to provide the initial

condition which is applied at the first time step t = At. Since (3.3) is notsatisfied, an error is made at r = 0. A limit is placed on this error: V

(0, At) < I0-6, which is satisfied if (Re/At) 1 / 2 > 7. The latter inequalityS~must be satisfied for the calculation to proceed. In practice it has never

been violated. The second term. I has not been needed in our calculations.

Using this initial condition we obtain V/r vs r shown in Figure 5 for

Re =4974, c/a =3.30 and t =20 and 50. Also shown are the results obtained

if V(1,0) = 1 is assumed; there is a relatively large error for t = 20. The"intersection of the dashed lines is a consequence of the error. In Figure 6the nondimensional vorticity c (rV)r/2r vs r is shown for t =100 and

600. The two sets of curves are again obtained using this initial conditionand V(1,0) = 1. For r > 0.95 and t = 100 the errors resulting from use ofV(1,0) = 1 are quite large. The vorticity plot has the advantage that a

check is readily available. It can be shown from (2.2) and (2.8) that[Ir (r = 1) = 0. The slope of the solid curve at r = 1 for t = 100 - slightlynegative; for t =600 it is zero. The slopes of the dashed curves at r = 1

are quite different from zero.

21

_ _

-. 4

IV. RESULTS

The main features of the thec, y will be Illustrated by some examples.Usually the parameters are chosen to be of interest in liquid-filled p-ojec-tile applications, but sometimes they are chosen to make a particular point.

The Wedemeyer model for spin-up simplifies the Navier-Stokes equations sothat a more tractable mathematical problem is obtained while retaining theessential physics of the problem, Since it is now feasible to solve theNavler-Stokes equations by finite difference techniques, it may be that themodel is not needed; it is premature to drew that conclusion. However, theresults of the model can b6 compared with accurate finite difference solu-tions.19 One such comparison is shown in Figure 7. Using the program ofReference 19, V vs r was computed for Re 103, t = 62.5 and for Re - 104,t - 62.5 and 125, both for c/a 1 1. The results at z - 0 are plotted. Thecircled points are solutions of (2.2) and (2.8). For these parameters thedifferences between the spin-up model and the Navier-Stokes solutions are ofthe order of a few percent, except possibly for very small V. The differencesare not always so small; - have no general rules for what values of theparameters give such small differences.

Consider the solution to the spin-up equation for the parameters of thetwo cases of 155mm projectiles, given in the Introduction. For Case 1, withRe = 4 x i0k, we should expect laminar Ekman layers. The solution is shown in

Figure 8 for three values of t. One of them is ts = 1245. For t 4800,

approximately 4 ts, solid-body rotation is achieved, essentially. For Case 2,

,vith Re = 2 x 106, we should expect turbulent Ekman layers. The solution to(2.2) with (2.9) gives the velocity profile shown in Figure 9 for threevalues of t including tst 2 2700. For t = 10,000, approximately 4 tst, solid-

body rotation has not been achieved to the same degree as in Figure 8.

The Ekman compatibility conditions (2.8) and (2.9) were proposed byWedemeyer for either laminar or turbulent Ekman layers, there being no pro-vision for transition from one to the other. Quite different results can beobtained using (2.8) or (2.9). This is illustrated in Figure 10 with Re3 x 105, a value at which either condition might apply. The results using(2.8) and (2.9) are shown for two values of c/a and t. The difference islarge for c/a = 1.0 but small for c/a = 0.05, an extremely small aspect ratio.

The need for including diffusion in the spin-up model was discussed inSection II. The differences in V, with and without diffusion, are illustratedin Figure 11 for Re = 6.08 x I05, c/a - 2.679 and t = 800 and 3200. Thedifferences can be smaller or larger, depending on the parameters; generallythey are smaller at larger values of k Re. For given t, the largest

lI f. C. W. Kitolo~s, ,r,. , "Naet.' ,-tBi,)k•L •So I i oU fo, ,7p ? n- Up f•,or' hr i n (19ntoacnr, c,-,c12hnimcn/ Rorpomt AhfRI.-7'K-O..1,, f.rmpl,-.',bci -, aO

(AD A077115).

22!

dit f(oren., occurs at the front and this ifncreases as t decr'as,•s. Even thouuhRe is large, lamionar Lkman layers are assumed in this exauplio.

A similar comparison is shown in Figure 1? for turbulent. Lkman layers forRe = 2 x 10(" and c/a = 5.2. The results including diffusion are the same asthose in Figure 9 for t = 1JO0 and 2700. Withowc diffusion, V depends only onr anid k t. The differences are smaller than for, the laminar case.

The offect on the velocity profile of using the [C from the R1. curve(Iigure 3) is shown in Figure 13. A rather abrupt change in curvature isobtained for small V; recall that the maximum in the RL curve occurs for smallV. To obtain this result it was necess.iry to use ,ýr = 1/200; a valueuf Ar = 1/50 did not show the abrupt change in curvature. The purpose of thiscalculation was to illustrate the approach to the no-diffusion limit in whichdouble-valuec', or possibly discontinuous, V is predicted by Weidman'stheory.9 Thus the rather large value k Re = 5000. It is not unreasonable to

conclude that the curve in Figure 13 is tending toward one with a verticaltangent, which might be related to the predicted douhle-valued or discon-tinuous V. As the slope of the tangent increases,the calculation would hecomceincreasingly more difficult. For comparison the V calculated from (2.2) with(2.8) is also shown. The difference between the two is about 0.1 at r r 0.4,

which is quite large; for smaller k, Re the differences are smaller. Theresults obtained by using (2.8) show no unusual behavior. This comparisongives further evidence for not Using the EC from the RL curve.

V. APPROXIMATIONS FOR DIFFUSION EFFECTS

Inclusion of diffusion effects in the solution to the spin-up equationwas amply demonstrated in the previous section, but the numerical results donot orovidc much understanding of the physical process. Without diffusion ashear discontinuity exists at the front which implies that viscous diffusion,neglected in arriving at (2.4), cannot be neglected in the neighborhood of thefront. A local solution, including diffusion, can be sought to examine thestructure of the shear layer. This was done by Venezian,1 who did a boundarylayer type analysis of (2.2) on the moving front. His result will be shownbelow as pait of a more general solution. He found that the moving shear

discontinuity is a layer of thickness 0(E 1 / 4). His interpretation is that the

E1/4 Stewartson layer that exists at the sidewall in linear problems breaksaway from the sidewall for this strongly nonlinear case and propagates into

1/3the interior. The E Stewartson layer apparently remains attached; it isthe one mentioned just before (?.1a).

Inclusion of diffusion over the entire radius can be studied usingmatched asymptotic expansions. It is convenient to work with 'he circu-lation r - rV. Using K 0.5 and introducing the time tV, (2.2) and (2.R)yield

rt,- (r- r/r) 1 = • rrr - rr/r) (5.1)

23

I

where

S- (k Re)"_

Coordinates centered on the front are introduced(r2 e t '- 22t' _

x ( a 1)/(e I 1

2t'

After transforming (5.1) to the (x,y) coordinates, it is solved by expansionsin c. The first term in the outer solution is (2.i0); the first term in theinner solution is Venezian's solution. Using a 2-term inner expansion and a4-term outer expansion, the composite solution is

1 /12 2(21/) 1/2 Y_ 112 e-S2 [erfc (S/81/2)] I + e H (S) (5.2)

where

S -x(y/)�1/2.

The first t.rm is essentially the same as Venezian's solution. H (S) satis-fies a linear, 2nd order, ordinary differential equation. It was most con-venient to solve that equation numerically, although asymptotic forms werealso derived. The solution H (S) is shown in Figure 14.

in Figure 15 V vs r is shown for • = 1/50 at t' 0.80. The solution(2.10) with the shear discontinuity is shown and the solution to (2.2) with(2.8). Venezian's result, the first term of (5.2), is also presented; his

L solution is very successful in correctiny (2.10) in the neighborhood of thefront and for larger radii. As r decreases from the radius of the front, hissolution deviates increasingly from the solution to (2.2) and has a minimum.Of course it was intended to apply only near the front. Also shown are a fewpoints calculated from (5.2). To plotting accuracy they are the same as thesolution to (2.2) even though the value of e is not very small. Thusdiffusion can be included analytically, giving results essentially the same as(2.2). The front, or shear discontinuity, is at S - (I and from (5.2) itE

thickness is 0(c 1/2) or O(E1/4

VI. CONCLUSIONS

The Wedemeyer model for spin-up of a fluid contained in a cylinder whichis impulsively rotated about its axis has been implemented, and numericalsolutions to the spin-up equation have been obtained. The approximation to animpulsive start was discussed. A critique of the Ekman compatibility con-dition based on the Rogers and Lance boundary layer mass flow relation wasgiven; there are compelling reasons for not using it.

24

SiI

1he finite difference method of solviny the spin-up equat Ion is presente(iin detail and the results of error studies given. Error doe to a disronti-nuity in boundary conditions, caused ry the impulsive start, were avoided bydeveloping a local solution.

A number of results were presented to illustrate the main features of thetheory and calculations. Comparison of the results with those from tfinitedifference Solutions to the Navier-Stokes equations validale thL spin-uptheory and numerical method. The effects of including diffusion art. .se-quately shown by various examples but an analytical approach is also derived.

ACKNOWLEIMfME NT

The authors wish to thank Ms Joan Bartos for her work on a large nuherof tasks which made the program used here operational.

25

I

Z'2c I

I - r, U

Figure 1. Notation and Coordinate System forthe Spinnina Cylinder.

SHELL EXITS TUBE1 AFTER 0.94 REVOLTONb •-

IN-BORE 7

S~0.5

00 10 20t(ins)

Figure 2. Normalized In-Bore Spin History for an M687Shell Launched From the M109 Howitzer atCharge 3, s2 = 92.3 Rev/s.

?6

0.5-

0.4-

or 0.3

U c Re 0.2

a r

0.1

0 I0 0.2 0.4 0.6 0.8 1.0

w10 or V/r

Figure 3. Normalized Radial Mass Flux in the Boundary Layer vs Ratio ofOuter Flow Spin to Disk Spin, Reference 13. Also U vs V/r whenthe RL Curve is Used to Define an EC.

rj_1 rj .rj+

S Q T* * • • t = (k+1) At

S• t= k AtA R B

* M 0 t (k-I) AtM

r=0 r=Ar r=OAr r=1 NAr

Figure 4. Grid Scheme in (r,t) Plane.

27

At

10

V/r "0.6 -

0.4 ~ -~ _____Using local solution

----- Assuming V(1.0) =1

o 't. 20

0.2

0.0•

0 90 .92 .94 .96 .98 1oor 1

Figure 5. V/r vs r for Re = 4974, c/a = 3.30 at t = 20 and50 Using Local Solution or Assuming V(1,0)=i.

2.5 1

2.0 t=10 0 -

1.5 2

1.0 t=600 - - -

Using local solution

0.5 ...... Assuming V(1,0) = I

0.0 "

.95 .96 .97 .98 .99 1.00

r

Figure 6. Vorticity c vs r for Re = 4974, c/a = 3.30 at t = 100and 600 Using Local Solution or Assuming V(1,0) = 1.

28

i ,.. ' • .-

1.0 RRe=40 3, Navier.-Stokes<

. R =104, Navier- S,.0.8 Stokes

z z(2.2) with (2.8) . 'V'-

- /- , ,,

0.4 "I /%;/t•=62.5 ' t=62.5St=62Z.5 ,",

0.2N.,.

o~o wt=125.0

0.0 0.2 0.4 0.6 0.8 1.0r

Figure 7. V vs r for Two Re and t With c/a 1. The Curvesare From Navier-Stokes Solutions; Circled Pointsare Solutions of (2.2) and (2.8).

.3

V

600

,0 I I

.0 .2 .4 .6 .3 1.3

r

Figure 8. V vs r for Case 1 With c/a = 3.120 Re 4 x 10at Three Values of t; t = 1245, Laminar EkmanLayers.

29

.

.3

V

130a

.0 I I

r

Figure 9. V vs r for Case 2 With c/a 5.200, Re = 2 x 106 at

Three Values of t; tSC 2700, Turbulent Ekman Layers.

'1.

V 1.0~--------Laminar

"i t 360

0.01 7 7c/a=1 .0

0.0 0.5 1.0r

Figure 10. Vvs r for Re -3 *~10 5at Two Values of c/a, tWith Laminar or Turbulent Eknan Layers.

30

- t d

.- with diffusion8 - --.. without diffusion /

6 tz

t ~ a

/ I

t=3200/.2 1 / / t

/t 8o I

I#

a .2 .4 .6 .8 1.0

r

Figure II. V vs r With Diffusion, (2.2), and Without Diffusion,(2.4), for Re = 6.08 x 105, c/a = 2.679 at Two Valuesof t, Laminar Ekman Layers.

tt

.1.0 . 1.

r/

-with diffusion

.31

------------------------------------ without diffusion

.6

, t1. I

.0 .Z .4 .6 .3 1.0rFigure 12. V vs r With Diffusion, (2.2), and Without Diffusion,

I(2.4), for Re = 2 x 106, c/a = 5.200 at Two Values ofktt; kt = i/tst 1/2700, Turbulent Ekman Layers.

31

.6

.5

VEC from RL ve

.3

.2 EC from (2.8)

.1I

.2 .3 . 4 .5 .6 .7r

Figure 13. V vs r Using the EC From the RL Curve, Figure 3,and the EC From (2.8) for k Re = 5,000, c/a

0.0485, t = 60, and Ar = 1/200.

0.2

H(S) :--0.2

-0.4

-0.6

-0.8

-10 10 20 30 40S

Figure 14. H (S) vs S. the Function in (5.2).

32

1.0

(2.4) /

(2.2)

V Venezian approx

/1!7//

0.5 - ( 5 .2

0.5 -,,.. __i.- ,, /J., /

/

0.0 0.5 1.0r

Figure 15. V vs r for c = 1/50 at t' 0.80 From (2.2) and (2.4).The Venezian Approximation is the First Term of (5.2).The Points are From Both Terms of (5.2).

33

i il m i i "I il I "•i :: I{":

REFERENCES

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

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

3. G.P. Naitzel and Stephen H. Davis, "Energy Stability Theory of Decel-erating Swirl Flows," The Physics of Fluids, Vol. 23, No. 3, March1980, pp. 432-437.

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

5. R. Sedney and N. Gerber, "Oscillations of a Liquid in a RotatingCylinder: Part II. Spin-Up," BRL Technical Report ARBRL-TR-02489, May 1,83.

6. C.W. Kitchens, Jr., N. Gerber, and R. Sedney, "Spin-Decay of Liquid-Filled Projectiles," Journal of Spacecraft and Rockets, Vol. 15, No. 6,November-December 1978, pp. 348-354.

7. N. Gerber and R. Sedney, "Moment on a Liquid-Filled Spinning and NutatingProjectile: Solid Body Rotation," BRL Technical Report ARBRL-TR-02470,February 1983 (AD A125332).

R E.R. Benton, "On the Flow Due to a Rotating Disk," Journal of FluidMechanics, Vol. 24, Part 4, 1966, pp. 781-800.

9 .D. Weidman, "On the Spin-Up and Spin-Down of a Rotating Fluid: Part 1.Extending the Wedemeyer Model," Journal of Fluid Mechanics, Vol. 77, Part4, 1976, pp. 685-708.

10. P... Weidman, "On the Spin-Up and Spin-Down of a Rotating Fluid: Part 2.P'asurements and Stability," Journal of Fluid Mechanics, Vol. 77, Part 4,

'6, pp. 709-735.

11. C.W. Kitchens, Jr., and N. Gerber, "Prediction of Spin-Decay of Liquid-Filled Projectiles," BRL Report 1996, July 1977 (AD A043275).

12. C.W. Kitchens, Jr., "Ekman Compatibility Conditions in Wedemeyer Spin-UpModel," The Physics of Fluids, Vol. 23, No. 5, May 1980, pp. 1062-1064.

13. M.H. Rogers and G.N. Lance, "The Rotationally Symmetric Flow of a ViscousFluid in the Presence of an Infinite Rotating Disk," Journal of FluidMechanics, Vol. 7, Part 2, 1960, pp. 617 631.

14. H. Goller and J. Ranov, "Unsteady Rotating Flow in a Cylinder with a FreeSurface," Journal of Basic Engineering, TRANS. ASME, Vol. 90, Series D,December 16, pp. 445-454.

34

J

REFERENCES (Continued)

15. G. Venezian, Topics in Ocean Engineering, Vol. 2, pp. 87-96, GulfPublishing Co. ouston, Texas, 1970

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

17. E.R. Benton, Memo to COTR, Task Order 74-461, BRL, January 1975.

18. E.R. Benton, "Vorticity Dynamics in Spin-Up from Rest," ThePhysics ofFluids, Vol. 22, No. 7, July 1979, pp. 1250-1251.

19, C.W. Kitchens, Jr., "Navier-Stokes Solutions for Spin-Up from Rest in a

Cylindrical Container," Technical Report ARBRL-TR-02193, September 1979(AD A077115).

3

!35

- - - -- fli7L___ __

APPENDIX A

Coefficieots in Eq. (3.7)

37

kj~cLjw ac sw -on L

APPENDIX A: COEFFICIENTS IN EQ. (3.7)

Upj - A. [ r -(12) (vj,k + vj,k+)]-- laminar Ekman layer

u. E-k [j 6r - (1/2) (v + v j 1/5 -- turbulent Ekman layert j,k j,k+l

2Aj B (At/Ar)u , [At/(Ar) ]/Re

3~pi

al = 4N 0

a. = -(A./4) + [(4(j-l)}- - (1/2)]L for j 1 1,N

b I + (AI/2) + (3D/2)

bN I

bj = 1 + [Aj/(2j)] + D for j 1 I,N

c= (A,/4) - (3D/4)

* CN=O

cj. (Aj/4) - [{4(j-1)}- + (112)]F0 for j * 1,N

3 3ý

d Ip V1I,k - (6/4 (v2,k + 2Vl~k) + (D/4) (3v2,k -6vlk

dN -w

d j .vJ-l,k ((%j/4) + (0/4) [2-(J-1)'])} +4 j (I-D-(Aj/[2j])}

d -V+J+l k (-(j/ 4) + (D/4) [2+(J+1)4-1 for j I ,N

43i

LISI OF SYMBOLS

a cross-secti onal radi us of cylinder

c half-hcight of cylinder

-l c, Lkman number

EC abbreviation for "Ekman compatibility condition"

j index indicating r-coordinate of nodal point in finitedifference solution (r = j Ar)

k index indicating time in finite-difference calculation(t = k At)

k < (a/c) Re"/2 (see (?,8))

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

radial mass flux in boundary layer/r

N number of subintervals in r in finite-difference solution

p pressure/(pa 2 ý12 ) in Wedemeyer model (see (223)) iiPo zeroth order approximation, in (1/ts), to P* (see (2.6))

P coefficient of first order approxiimation to P* in (2.6)991

p* pressure/(pa2'-) in solution to Navier-Stokes equations(see (2.1))

r radial coordinate/a

R (i - r)/t'I/2

Re a 2 , /v, Reynolds number

RL abbreviation for "Rogers-Lance"

S =-x / )/

t time x

ts t '

t t/ts

time

C time required for spin of cylinder to reach Q2; i.e.,cc

41

tfl characteristic time of flight of projectile

1/2£I-- (d T•/dt)" /, acceleration or impulse time

s (2 c/a) Re1 / 2 /s, characteristic spin-up time for laminarEkman layer

st (28,6 c/a) ke/ 5 /I, characteristic spin-up time for turbulentEkman layer

t (a/c) 2/(EQ), time scale for viscous diffusion of vorticitv

21y/a, time for one revolution

U, V, W radial, azimuthal, axial velocity components x I/(as) ofWedemeyer model spin-up flow with diffusion (see (2.2) and(2 .3) )

Uw , V w radial and azimuthal velocity components x 1/(as) ofWedemeyer model spin-up flow without diffusion (see (2.4))

U1 , V1 9 W1 coefficients in first order approximations, in I/t , toU*, V*, W* (see (2.6))

U*, V*, W* radial, azimuthal, axial velocity components x 1/(ai) ofNavier-Stokes flow (see (2.1))

V zeroth approximation to V* (see (2.6))

0

V0 , VI functions of R in early time solution in Chapter IIIC

X (r2 e - 1)/(e 2t- 1)

2t'-y e -

z axial coordinate (z 0 at cylinder midplane)

r rV, circulation

Ar I/N, r-irterval size in finite-difference solution

At t-interval size in finite-difference solution

e1/(k 9 Re)

a (rV)r/( 2 r), nondimensional vorticity

e azimuthal angle

K constant in expression for radial velocity with laminarEkman layer (see (2.8))

42

V kinematic viscosity of fluid

P density of fluid

angular velocity of spinning cylinde-

•2 (t) time history of projectile spin

4I

43

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