Hydrodynamic forces and moments of streamlined bodies of revolution at large incidence

ARCHEF

_{Tecuclie HOSCh3o!}

De!fL

Hydrodynamic Forces and Moments of Streamlined Bodies

of Revolution at Large Incidence

1. Introduction

Deep submergence rescue-and-seardi vehicles, while operat-ing at low forward speeds against side currents and performoperat-ing mating and pickup maneuvers, encounter flow conditions with angles of attack varying from zero to 180 degrees. Unfortunate-ly, hydrodvnamic force and moment data for a systematic series of streamlined bodies of revolution are, for the time being, not publicly available for such a wide range of opera-tional conditions. In the meantime, a sort of theoretical make-shift approach, described in the following paragraphs, has been developed to make possible early systems feasibility and preliminary design and motion simulation studies. In this way, it will be possible to discover trends and tendencies and per-haps unusual, discontinuous, or anomalous behavior even if

the numbers involved are not too accurate.

2. Prediction Method

Forces and moments will be initially described in the follow-ing for the motion of yaw. Figure 1 gives the relation between the inertial axis X0 und Y0 and the body axis X and Y.

All forces and moments will he expressed in the body axis

system X, Y.

-- Figure 1

Orientation of Body Axes and Forces Relative to Fixed Axes

Since vehicles of the type considered are nearly rotationally symmetric configurations, all force and moment terms can also be applied in the pitch plane XZ by replacing Y, 3 p, y with

Z, ti, B w.

1) Senior Staff Scientist, North American Aviation, Inc., Los Angeles Division.

Dr.-Ing. Hans F. Mueller1)

An analytical method for predicting hydrodynamic forces and moments on a "closed" body of revolution at large incidence has been developed. The flow model utilized is primarily based on the independence of the cross-flow and the axial cross-flow and on the fact that the potential cross-flow over a closed body of revolution produces only a moment but no resultant force. Wind tunnel tests at the Los Angeles Division of North American Aviation, Inc. with the model of a Deep Submerged Rescue Vehicle DSRV) which was developed by the Navy Department's Special Project Office generally bear out the belief that this prediction method can be considered a useful tool for a qualitative evaluation of the hydrodynamic forces and moments at large incidence,

L.

y.

2.1 Closed Body of Revolution

2.1.1 Poteníial Flow Term

Max Munk [Reference 1] considered the potential flow about bodies of revolution and showed that at any station along a hull at angle of attack a local force per unit length of the magnitude

dYHp = (K9 - K1) _{/2 Ue2} dA

dx

should be experienced. From later work of G. N. Ward [Refe-rence 2], it may be shown that this force is directed midway between the normal to the axis of revolution and the normal to the flow direction. Hence, the right side of Equation 1 must be multiplied by cos 3/2. K9 and K1 are the transverse and longitudinal apparent mass coefficients, and A is the local cross-sectional area. The integration of the elementary forces and of their moments furnishes

= (K2 - K1) A1) /2 UO2 sin 21 cos 3/2 (2)

NHP (K9 - K1) l2 UO2 (Vol Ab xb) sin 213cos 13/2 (3)

where A1) is the base area, and xb its distance from the moment center. From Equations 2 and 3, it follows that the potential flow over an inclined "closed" body of revolution with zero base area generates only a moment

N11 = (K0 - K1) (Vol) (Sin 2 13 cos 3/2) l2 (u + v2) (4)

but no lift.

2.1.2 Viscous Fiole Terms

In applying the Navier-Stokes equations, together with the given boundary conditions, R. T. Jones showed in Reference 3 that various features of the viscous flow such as boundary-layer thickness and separation point, if observed in planes at right angels to the axis of an obliquely moving cylinder, will be determined solely by the component of the velocity of the cylinder in these planes, and the axial motion of the cylinder produces no effect on the flow. This postulated nondependence of the cross flow and the axial flow is, of course, the result of the law of shearing stress adopted in the Navier-Stokes equa-tions. Reif and Powell [Reference 4] carried out wind tunnel tests on circular wires at different angles of yaw. In these tests, the Reynolds number based on the crosswise velocity

compo-nent varied from 102 to 10, a range in which the drag

coeffi-Schcp.ouv.:!::

- 99 -

Schiffstechriïk Bd. 15 - 1968 - Heft 78

dent is nearly constant. Over this range, the crosswise com-ponent of the drag force on the wire should, therefore, be pro-portional to (U sin 13)2 and the tests do indicate this variation.

2.1.2.1 Viscous Crossflow Flull Forces and Moments In view of the facts described in the preceding paragraph, J. H. Allen and E. W. Perkins postulated in References 5 and 6 that a better evaluation of the cross-force distribution on a body of revolution of finite length moving at the velocity Ue with i = O could be obtained by adding to the potential cross-force distribution an additional cross cross-force calculated on the assumption that each circular element along the hull ex-periences a force equal to the drag force of an element of a circular cylinder of the same diameter in a stream moving at the crossflow component y. In the case of the DSRV which

is a "closed" body of revolution, the sum of the potential

cross forces is zero and their moments are already accounted for in the potential moment term described in paragraph 2.1.1. Hence, only the viscous cross flow terms need be considered. These transverse hydrodynamic forces depend on the local cross flow velocity due to the hull rotation 4i as well as due to

the transverse motion y of the entire craft. During

micro-maneuvering, these components can assume like magnitudes which rules out the simplification of assuming that the forces depend on the individual velocity components y and xii,. Thus, the reprensentation of the transverse hull forces will remain

complicated in that a cylindrical element of the hull

ex-periences a dynamic pressure which is formed by the square of the vector sum of velocities acting at its location.

(V + X

12 ¿

FIgure 2 Resultant Crossflow Velocity Acting on a Body Element

With the notation of Figure 2, the viscous cross force and moment can be described in their general forms.

£9

Hv = J' CJ) (Re) r (y + x

sign (y ± xi) dx (5)

- FI

£7

NHV =JCn (Re) rx (y + x)2sign (y + xi) dx .()

- Fi

2.1.2.2 Viscous Longitudinal Hull Forces

The principle of independence of the crossflow and the axial flow which served so conveniently to define the viscous cross-flow terms is, unfortunately, not fully reversible; i.e., the

crossflow may introduce a considerable influence on the axial flow. This arises because the longitudinal shear flow at the leeward side of the hull deteriorates with rising obliqueness and is gradually replaced by the dead-flow of the circular cylinder flow behind its separation lines. It can be expected that the resulting friction force coefficient based oli the total wetted area will decrease with increasing angle of attack. In the case of a longitudinally asymmetric hull, the resultant longitudinal force will ultimately reverse its sign at an

inter-mediaote angle of attack. For the time being, the viscous longi-tudinal hull force can therefore, be only reliably predicted for moderate angles of attack. For this restricted range it can be written in the form

X11 = - (CJ)) S /2 u2

. (8)

Until data for reversed motions of streamlined bodies of re-volution become available, the foregoing formula can only be applied to positive velocity components. The term (C15)

com-prises all drag components from surface friction, surface rough-ness, flow separation, and appendages. Data of the DTMB systematic 58 series of streamlined bodies of revolution [Refe-rence 7] may be utilized to predict the longitudinal resistance of the bare hull. A close inspection of the resistance curves of Reference 7 discloses the following facts:

The stimulation of turbulence by means of sandstrips at the bow is so effective that it reduces the transition effect within the measured speed range to a negligible magnitude. It can be reasonably assumed that a similar degree of turbulence will be introduced by the flow dis-turbances at the exits of the transverse thruster tubes and other openings.

The measured total resistance curves in the Reynolds number presentation are very nearly equidistant to the smooth flat plate friction coefficient. This indicates that the slenderness effect can be introduced as an additional constant quantity like the effect of surface roughness and of the appendages. This is contrary to Hoerner's approach, who introduced the slenderness effect by a fac-tor [Reference 8].

For a parametric study, an explicit relation between the coefficient of total friction of a smooth flat plate, cf, and the Reynolds number is desirable. A formula which had been developed by Hama [Reference 9], bei Lap and Troost [Ref e-rence 101, and by Hughes [Refee-rence 11] is utilized here:

Cf 0.0816

(log Re1 1.703)2

The effects of slenderness, roughness, and of the appendages can be lumped together in a common constant C and the longi-tudinal force formula assumes the following final form:

xliv = -

0.0816

(log Re-- 1.703)2

+ C

(9)S /2 u2.

2.1.3 Mass Accession Forces and Moments

Mass accession forces are formally given by the theory of ideal fluid flow [Reference 1]. The terms in the yaw plane are:

X = K1 (Vol) û

(10)

= - K0

(Vol) ' (11)

N = - K'

r'ij3J'r x2 dx. (12)

2.2 Control and Stabilizing Surfaces

The following relationships are essentially identical for any configuration of control and stabilizing suifaces. They arc, however, specifically written for the case of annular shrouds selected for the DSRV. Data of lift, drag, and moment

coeffi-cients CL (13Se), C1) (13) and C21 (13s.) of annular airfoils are available from Reference 12. Their lift-curve slopes are about twice the lift-curve slopes of radial stabilizing fins of equal

-aspect ratios and the induced drag coefficients are one-half the induced drag coefficients of elliptic airfoils.

X = /2

A [u2 + (y + Ls 4t)2]

(13)

Ys = /2 Cy As [u2 + (y + Ls ,)2]

(14)

N5 = 9/2 A5 (Cy5 L + C11 C) [u! + (---v + L ,,)2] (15)

where A5 is the projected area of the shroud, C its chord

length, and L the distancof its quarter-chord from the center

of gravity.

Since CJ), C1, and C of the shroud in Reference 12 are conventionally referred to in the wind axis, they must be taken at the effective angle of attack

= ös +

tan'(

V + L)

The force coefficients C5 and Cy can then be obtained by the

following conversion

Cs = CL (Se) sin

_{[ tan1}

CYS = CL (I3Se) CO5

± CD (13s0)

Equations 13, 14, and 15 are also applicable for the calcu-lation of the forces and moments of the struts which hold the shroud.

Cy, As and L5.will be replaced by C, CYT,

AST und LST.

For the purpose of simulation, the movements of the vehic. les are assumed so slow that time rate of change effects on all force and moment coefficients can be neglected. All force and moment coefficients used in this report are the familar steady-state values, corresponding to instantanuous angle

of attack.

3. Wind Tunnel Experiments

3.1 Preliminary Force Esti mates

The previously described analysis was first employed to evaluate the hydrodynamic forces of a Deep Submergence Rescue Vehicle based on the dimensions given in the paper of S. Feldman and L. Cathers [Reference 13].

u

(_v± Ls1)]

CD (s) cos [

tant_V + £s1

u

_j]

[

tani(" ±

tan'( v +

u _,Ij

Figure 3 Model of DSRV in the NAA-LAD Low-speed Wind Tunnel

Since the power which the four ducted thrusters absorb to hold the vehicle against a one-knot cross current exceeded several times the power required to give the vehicle a 5-knot forward speed, particular attention was given to a reliable prediction of the crossfiow resistance.

The analysis predicted a sizable hump of the crossflow resistance curve between 0.3 to 0.6 knot crossflow velocity. The hump develops from the substantial reduction of the cylinder drag coefficient in the process of transition from laminar to turbulent cylinder flow. The energizing turbulent flow moves at higher Reynolds numbers the flow separation from the front half of the cylinder to its rear half which results in a much smaller wake and correspondingly lower resistartce. It was ex-pected that the shroud, the duct openings and especially the protuberent mating bell and the canopy inflict local separa-tions of the flow which wipe out, in part, the gains obtained from turbulent flow conditions on the remaining undisturbed sections of the hull.

3.2 Wind Tunnel Experiments - Part i

In order to gain more detailed knowledge of these inter-ference effects, a 1/8-scale mode of the SPO vehicle, as shown in figure 3, was investigated in crossflow condition in the Wind tunnel of North American Aviation's Los Angeles Division. The model tests were performed by measuring first the bare hull and then adding consecutively the various appendages.

Figure 6 indicates that the appendages do not eliminate the step-wise transition effect. Hence, the anticipated hump in the crossflow resistance curve must be reckoned with even with all appendages attached. Only a simulation study of micrornaneu-vers will indicate if the effect of the transition of the cylinder flow from its laminar to the turbulent structure has undesirable consequences. If so, the development of effective trip devices with minimum drag penalties must be given further attention.

33 Wind Tunnel Experiments - Part 2

Part 1 of the wind tunnel test program was primarily con-cerned with obtaining reliable crossflow resistance data and to gain knowledge of interference effects. These tests were,

there-Figure 4 Model Support In Wind Tunnel for Crossflow Tests

fore, limited to flow conditions at an angle of attack i = 90 de-hull and also the complete vehicle configuration could be in-vestigated within the entire available range of Reynolds num-grees. It was likewise important for the analytical parametric treatment of deep submergence systems, in general, to prove or disprove the concept of independence of the crossflow and the axial flow which served as a basis for the analytical predic-tion method as described in paragraph 2. Since the model was sting-mounted on a turnable, as shown in figure 4, the bare - 101 - Schifistechnik Bd. 15 - 1968 - Heft 78

y CDC P1212 A sin2ß OP 0.91-0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 1 10° U0

cy

-,2 C D PI2DA sin2ß 1,3 _520° 1.2 1. 1 1.0 0.9 0.8 0.7 C ) 3 4 5 010 R 40_{sIn p}

Figure 5 Crossflow Drag Coefftcient of Bare Hull at Varying Crossflow Velocities and Angle of Attack

bers and at angles of attack varying from zero to 90 degrees. The shortcomings of this test setup which had no provision for a dummy strut arrangement (compare figure 4 with figure 9), must be fully born in mind. Since emphasis was originally solely on pure crossflow forces at 13 = 90 degrees, the

inter-ference of the sting at the leeward side of the model was con-sidered negligible which, of course, does not hold true at small angles of attack.

3.3.1 Transverse Forces Versus (3

If the transverse forces are solely determined by the cross-flow pattern, then the transverse force coefficients defined by

Y CONSTANT p - 90° VARIABLE CONSTANT U VARIABLU p' NASA TN D'540 /2Ue2A1 Sin 213

and plotted against the crossflow Reynolds number Ucd

Rey

sin(3 V RARE HULL MANIPULATOR OPERINO SHROUD MATI NC BELL * OPEN DICOS CONSTANT B 90° VARIABLE U, 000NSTANT U VARIABLE p' 50° 800 cao 2 3 4 5 6 smp

Figure 6 Crossfiow Drag Coefficient of tise Complete Configuration at Varying Crossilow Velocities and Angle of Attack

N N P1252 Ad 0. 4 0. 3 0. 2 O. 1 0,02 0,01 -0,01 -0,02 -0. 1 '0.2 -0. 3 -D. 4

Figure 7 Comparison of Predicted and Measured Yaw Moments

With and Without the Shroud

BARE HILL

'y,

/ B 300 6O°' 0 900 HULL SHROUD D _{EXPERIMENTS IN} O) NAAWIND7jpp., -- PREDICTED

must fall on the same curve whether they were obtained by varying the tunnel velocity 1-se at constant angle of attack 3 or by varying the angle of attack while maintaining constant tunnel velocity. This trend is strikingly demonstrated in figure5 for the case of the bare hull, where points obtained in both ways are shown. Surprisingly enough, it holds also true within an accuracy of 20 percent required for preliminary studies with all appendages attached as shown on figure 6.

3.3.2 Moments Versus1

It was expected that Monk's potential flow concept will only be applicable for small angles of attack, which area is most important for the evaluation of the longitudinal stability in yaw and pitch. It could also be concluded from the Part I tests that yawing and pitching moments will be reliably predict-able in the area of dominating crossflow - viz, at angles of

attack around 90 degrees. Figure 7 seems to bear out this

reasoning fairly well. It appears to give also the correct quali-tative trend with the shroud attached.

3.3.3 Longitudinal Force Versus(3

The longitudinal force X11 of the bare hull as plotted against the angle of attack (3 in figure-8 shows, as was predicted in paragraph 2.1.2.2, the typical gradual reversal from a rear-ward to a forrear-ward force with increasing angle of attack.

CXH pI2U A1 60° 90° DTMB MOO 4155 o -0.03

Figure $ Longitudinal Force Coefficient of the Bare Hull at Varying Angle of Attack

-For comparison, the resistañce coefficient of the DTMB Model No. 4155 is also shown in the diart. lt indicates that the measured resistance at zero angle of attack which also includes the ba'ance sting interference is 54 percent higher than that of a streamlined body of revolution of similar slenderness ratio. in an effort to evaluate the testing technique shown in Figure 4, the same model was again tested in the GALCIT 10-Foot wind tunnel supported on an internal strain gage balance by a tapered sting which passed through an opening in the base of the model (Reference 15). This rear-sting arrangement per-mitted at zero angle of attack the attachment of a dummy strut support of the shape used in the previous tests and its image in any desired peripheral position. Data from configuratuions which include a) no strut, b) the basic strut support, e) the basic strut support plus its image oriented at 180° to the basic support and d) the basic support plus an image oriented 90° lead to the following conclusions:

The sting interference at zero incidence is indeed of the magnitude shown in Figure 8 and nearly equal for the strut image at 90° and 180°. This is a very important and welcome result for many towing tank facilities where the 90° image technique is the only one available to measure support strut effects in a reasonable amount of time.

4. Concluding Remarks and Recommendations

The wind tunnel tests generally bear out the belief that the developed analytical method of predicting steady-state hydro-dynamic forces and moments can be considered a useful tool for early systems feasibility and preliminary design and mo-tion simulamo-tion studies. The qualitatively good agreement be-tween predicted values and test data despite the rather crude flow model adopted may be due to be deliberate neglect of secondary flow effects which seem to cancel each other. In this respect, it is particularly referred to recent experimental work by Sarpkaya [Reference 16] which indicates that, in the range

of Reynolds numbers convered in this report, the distance from the bow within which the crossflow developes to its steady-state condition, is negligibly small.

Major shortcomings of the prediction method are encoun-tered in the evaluation of the longitudinal drag. It does not predict a forward body force at large angles of attack. This deficiency is fortunately not too serious in that the longitudinal forward force at 3 90 degrees does amount to only 5 percent of the transverse force. Nevertheless, it is recommended to utilize the three-dimensional lifting body theory to obtain a first order magnitude solution with the Kutta condition fixed by assuming flow separation at the trailing edge.

As a follow-on research, - the previously described preli-minary wind tunnel tests will be repeated in an improved test rig and expanded to 1 = 180 degrees. As shown in figure 9, the model will be mounted on a vertical rotatable sting which carries at its upper end a six-component strain gage balance. A retractable dummy sting will be provided to determine for every tested angle of attack the force changes due to inter-ference with the flow about the model by the presence of the supporting balance sting.

In computing the viscous crossflow forces and moments according to Equations 5 and 6 for a predominantly trans-verse motion with small rotation, it appears permissible to evaluate the local cylinder drag coefficient from tests of cylin-ders with comparable slenderness ratio.

In the case of pure rotational motion, the local Reynolds number varies widely, being zero at the center of rotation. Part of the body will, therefore, be subjected to subcritical flow

Figure 9 Model Support In Wind Tunnel for the Investigation of the Effect of Large Angles of Attack

while others work at supercritical conditions. Most likely, there will be mutual interference and triggering of either the one or the other flow form in adjacent sections which makes the prediction of the forces uncertain.

A program is in preparation to find the integrated forces and moments experimentally through rotation tests of a cylin-drical body in water.

(Concluded in June 1967) References

Munk, Max M., 'The Aerodynamic Forces on Airship Hulls," NACA Report No. 184.

Ward, G. N., Superson1c Flow Past Slender Pointed Bodies,"

Qu a r t e rl y Jo u r n a 1 o f Me ch a ni e s a n d Appi i ed

Mathematics, Vol 2, Part I, March 1949, pp 75-97,

Jones, Robert T,, "Effects of Sweepback and Boundary Layer and Separation," NACA Report No. 884.

(4] Reif, E. H., and Powell, C. H., "Tests on Smooth and Stranded Wires Inclined to the Wind Direction and a Comparison of the Results on Stranded Wires in Air and Water," British A.R.C., R. & M. No. 307, 1917,

Allen, H. Julian, "Estimation of the Forces and Moments

Acting on Inclined Bodies of Revolution," NACA HM A9.i26,

1949.

Allen, H. Julian, and Perkins, W. Edward, "Characteristics of Flow Over Inclined Bodies of Revolution," NACA HM A50L07.

Gertler, Morton, "Resistance Experiment on a Systematic

Series of Streamlined Bodies of Revolution for Application to the Design of High-Speed Submarines," DTMB Report - 297 NS 715-1)80, April 1950.

Hoerner, Sighard F., "Fluid Dynamic Drag" published by

author, 1965, pp 5-8.

Hams, F. R., "Boundary-Layer Characteristics for Smooth and Rough Surfaces," Trans SNAME, Vol. 62, 1954.

Lap, A, J. W., and Troost, L., "Frictional Drag of Ship Forms,"

Northern California Sec SNAME, February 1952. Also Full.

SNAME, June 1953.

Hughes, F., "Friction and Form Resistance in Turbulent Flow

and a Proposed Formulation for Use in Model-Ship

Cor-relation," Trans Institute of Naval Architects

I N A ) Vol 96 (1954).

Fletcher, Herman S., "Experimental Investigation of Lift, Drag and Pitching Moment of Five Annular Airfoils," NACA TN4117, October 1957.

Feldman S., and Cathers, L., "Design of the Deep Submergence

Rescue Vehicle," 0cc an Science and Ocean

Engi-neering, Transactions of

the Joint

Con-ference 14-17 June1965, Washington, D.C. Voll.

Linwood, W. McKinney, "Effects of Fineness Ratio and

Rey-nolds Number on the Low Speed Crosswind Drag

Characte-ristics of Circular and Modified Square Cylinders," NASA TN D-540, October 1960.

William H. Bettes, "Report on Wind Tunnel Tests of a

0.125-Scale Model of the North American (Autonetics) Deep

sub-mergence Rescue Vehicle." GALCIT Report 839, January 1967.

Turgut, Sarpkaya, "Separated Flow about Lifting Bodies and Impulsive Flow about Cylinders," Paper given at the AIAA

Second Annual Meeting, San Francisco, CalifOrnia, July 26-29,

1965.

Symbols

A local cross-sectional area, also projected area

Ab base area

Ap planform area of body of revolution

CL, CD, C lift, drag, and moment coefficients, based on pro-jected area AS and chord length CS of shroud

C1) cylinder drag coefficient, based on projected area

Cp longitudinal drag coefficient, based on wetted surface area

Cf friction coefficient of a smooth fiat plate

d maximum body diameter

I moment inertia

K1 K.), K' longitudinal, transverse, and rotational apparent mass coefficients

£ body length between perpendiculars

£ distance of the shroud's quarter chord from the center-of-gravity

M mass

N yaw moment about center of gravity r radius of local cross section

S wetted area

U5 velocity, relative to fluid, of origin of X and Y axis in yaw plane

u, y components of Ue in the direction of the X and Y axes

Vol displaced volume

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components of the hydrodynamics force In the direction of the x, y, z axes

a right-hand orthogonal system of moving axis in the yaw plane

distance between base area and moment center angle of attack

shroud deflection angle kinematic viscosity mass density of fluid yaw angle

time rate of change of yaw angle

Schiffstechnik Bd. 15 1968 Heft 78 ₁₀₄ -Subscripts H hull S shroud P potential flow V viscous flow c crossflow condition e effective quantity