Tahir Yavuz* Karadeniz Technical,

University, Trabzon,Turkey

Abstract

The total resistance of bluff bodies, such as hemispherical shell and cruciform canopies, moving unsteadily in fluid are experimentally studies. It is evaluated by force and moment measurements dur- ing relative motion of the model submerged in wat- er in a ship tank. Experiments performed to deter- mine the hydrodynamic resistance in unsteady mo- tion has been the lack of suitable correlation modulus. From this study, it can be said that for bodies whose drag coefficient remains approxima- tely constant over a wide range of Reynolds num- bers, the acceleration modulus can be considered to be a suitable correlation modulus in rectilin- ear or rectilinear+periodic unsteady motion.

Nomenclature

External forces External moments Velocity Acceletration Kinetic energy and period of the motion Potential energy Angular velocity Hidrodynamic force Density of the fluid Cross sectional area of the body Drag coefficient Total force coefficient Periodic velocity Periodic + linear velocity Acceleration number Apparent mass and angle of attadk. Apparent mas roefficient (=a/pv) Reference volume

1. Introduction

Attemts to determine the fluid resistance which is caused by the interaction between the fluid and the body moving in it, have been made by many auth- ors. The body acting on bodies submerged and un- steadily moving in a real fluid can be divided into three catogories; Static forces which are gravita- tional and buoyancy forces, Force that are depend- ent on the relative velocity between the body and surrounding fluid, which may be either pressure forces, skin friction or both and they are func- tion of Reynolds number, and Acceleration depend- ent forces which are function of the absulute ac- celeration of the body and the surrounding fluid. Static forces will not be considered here.

* : Associate Professor ** : Research Assistant

Copyright @ American Institute of Aeronautics and Astronautics. Inc.. 1989. All rights reserved.

Serpil Uzk~l~s** Erciyes University, Kayseri,Turkey

Velocity dependent forces are very familiar in that there are vast number amount of data pub- lished over a very wide range of Reynolds numbers for commonly used body shapes. But, almost all of these data are awailable for steady flow condi- tions. Very little is known and published for time dependent flow, even for simple body geometries such as sphere and cylinder. Continiously changing flow fields under time-dependent flows make it very difficult task to either calculate or measure the pressure and the frictional forces at any instant in time. These complications have full scale effect on bluff bodies where the variation of the separ- ation point and the changing width of the wake which carry the problem beyond the grasp of present knowledge of engineers.

Many flows of practical interest are unsteady, i.e., the characteristics of the ambient flow are time dependent. In the past 25 years, a large num- ber of theoretical and experimental studies have been carried out. These dealt primarily with un- separated laminar flows1. Very little has been attemted either theoratically or experimentally to analyse the wake-boundary layer interaction in time-dependent flows. Most of the numerical studies based on the use of the Navier-Stakes Equationsand some suitable spatial and temporal differencing schemes are limited, out of necessity, to low Reynolds numbers flows about cylinders and air- foils.

The numerical solution of unsteady incompress- ible Navier-Stokes Equation in their vorticity- stream function formulation have been investigated by numerous researchers through the use of various finite difference techniques. These studies concern mostly the separated flow about circular cylinder and prisms at relatively low Reynolds numbers2. In an attempt to obtain solutions at higher Reynolds Numbers and to provide a more natural and efficient description of the vortices, various versions of the discrete vortex model have been used. These models differ from each other in their treatment of the representation of the body, boundary layer, viscous diffusion, separation points, etc,( see e. g., chorin3, Sarpkaya - ~hoaf f 4 , Sarpkaya - lhrig% Using vortex methods, strickland6, McCoy - ~erme7 and Uzkilif8 analysed the aerodynamic character- istics of hemispherical cup which can be used as a model of parachute canopy.

It appairs that the existing numerical methods can not yet treat the high Reynolds number flows with sufficient accuracy for a number of reasonsg. It is clear that there is little hope of devising a saticfactory theoretical model before something is understood of the unsteady processes associated with the formation and reversal of the wake, the

sensitive dependence of the motion of vortices on small changes in the previous conditions, etc.

The practical needs, coupled with the comp- lexity of the problems, led to the proposal of an ampirical equation, known as the MOJS equation (Morison, O'brien, Johnson, schafflO) to predict the time-dependent in line-force acting on a sub- merged bodies in wake or other unsteady flows. Its justifications rest with experimental conforma- tions. A characteristics feature of experiments performed to determine the hydrodynamics resis- tance in unsteady motion has been the lackof suit- able correlation modulus. In liturature, various correlation modulus have been used by authors to express the fluid resistance and to justify MOJS equation in unsteady motion. However, for some bodies the scatter of their results is consider- able.

Earl experimental work conducted by Yavuz and CockrellT1 aimed to define and to determine the apparent mass of the prachute canopy which is hemispherical shell to use in the mathematical model of the parachute-store system. In this study, reanalsing the force-time output obtained from this experimental work, the variations of the to- tal force coefficient with various non-dimensional numbers are presented and a suitable correlation modulus which can be used to define the fluid re- sistance acting on bluff bodies in unsteady motion is given.

2. The Nature of the Total Force in Unsteady Motion of Submerged Bodies

In any study of the motion of immersed bodies through fluids the motion of the fluid in which the body is immersed is always an integral part of the total dynamic system whether it constitutes a significant part of the whole will depend on the displaced mass of the fluid compared with that of the immersed body. External forces and moments act- ing on an immersed body can be expressed in terms of the changes in the sum of the kinetic energies for the body and for the necessarily ideal fluid in which it is immersed. In a real fluid, the ex- ternal forces and moments which act on the system are partly conservative and partly non-conserva- tive. It can be shown12 that the form of Lagrange's equations, when written in terms of the Lagrangien function L, defined as L = T - U where T and U are respectively the kinetic energy and potantial en- ergy and potantial energy functions for the system is identical to that

and

in which V and w are linear and angularvelocities respectively. Since the form of these equations is appropriate for a non-conservative system, such as motion of parachute-store system, they can be used as the basis for experimental determinations of the

fluid resistance in which the instantaneous exter- nal forces or moments acting on moving system is measured simultaneously with the system's instanta- neous velocity and its acceleration.

However, all fluids are viscous, thus shear stresses develop in regions of velocity gradient. Where shear stresses are present the local flow is rotational and thus, in a real fluid vorticity is generated. As it is known, rather different state of affairs exists when a bluff body moves through a real fluid. The flow round this body will separ- ate from it and at the rear of it a wide wake will develop. Velocities at points in this wake do not readily scale on that of the blnff body and the energy of significant part of the surrounding flow field becomes much more difficult to calculate. In order to obtain credible values for total fluid re- sistance in real fluid flows it is necessary to turn the experimental techniques. As stated above, the total force acting on a body in unsteady motion can be expressed by MOJS equation in terms of the immersed body's instanteneous velocity V and its instantaneous accelerations $, as

in which p is density of fluid, A is the cross sec- tion area of the body, Cg is the total fluid resis- tance coefficient, Cn is the resistance coefficient in steady motion, a zs the apparent mass and m is the mass of the body.

During the past nine years the experimental towing-carriage f acilitiesl l (shown in Figure 1) has been used to investigate and to understand the unsteady behaviour of the parachute-store system. In 1981, at San Diego, in the paper written by Ya- vuz and cockrelll1, based on the emphasis was on which of the apparent mass components were of

Figure 1: Drive mechanism for water tunnel experiments.

greatest significant in predicting the dynamic sta- bility of parachute pitching motion. Following method adopted much earlier by Frazer and ~immonsl3, the 61 m long ship tank, having a cross section area of 3.66 m x 1.83 m at the Institute of Higher Education, Southampton, England was first used for this purpose. The experimental rig detailed in

Ref.11, basically consists of two circular ring, the outher one of which is rigitly attached to the towing curriage while the inner one carries a slider-krank mechanism driven by electricmotor. Strain-gauged sting which supports model is free to rotate in the slider. While being towed through the water by the motion of the carriage, the slider-krank mechanism can cause the sting mounted model to be harmonically oscillated at a low fre- quency along any line in a plane parallel to tank bed. The method which they adopted to evaluate the two unknown c6efficients CD and k, from test con- ducted on the parachute canopy with one degree of freedom motion was to assume that at an unsteady velocity, the coefficient of CD would be equal to its steady-state value. They found that the appar- ent masses were strongly dependent on the instan- taneous acceleration modulus b/v2, and the angle of attackcx. Using this results in the mathemat- ical model, the performance characteristics of parachutestore system are analvsedby yavuz14.

polpitiye15, using same experimental rig, de- scribed further experiments performed on the para- chute models. He argued that when the periodic motion undergone by a bluff which is very complex, then in place of two free parameters, CD and k in the equation (3), a three parameter model

in which is the time lag, is better to describe this force. Polpitiye take CD to be equal to its steady-state value and k to be the potential flow estimate, leaving one free parameter T , which he determined from experimental data. He found the time lag of 0.3 second for hemispherical canopy having five second of oscillation period. He ar- gued that this time lag is independent of the characteristics of the motion. In 1986, at Al- bequarque, in the paper16, it indicates that the performance characteristics of the parachute de- scending in a real fluid can not be described by three-parameters model.

Following the method suggested by Sarpkaya and 1saacson17, the Leicester University Bluff Body Research ~ r o u b l ~ conducted further experiments on hemispherical and cruciform models to determine average values of CD and k which are aimed to be used in the mathematical model of the parachute- store system. They used basically same experimen- tal facilities described above. Mulfiplying equa- tion (3) through either by V(t) or V(t) and taking mean value over the oscillation period, the aver- age values can be determined as

and

The results obtained are given in the table 1 for three canopy shapes with different porosities. The applicabilities of these average values in the mathematical model are always questionable. Same values can also be deduced by averaging values de- termined in Reference ll.

Table 1: Experimental characteristics of Parachute canopiesin steady and unsteady motions1?

3. Experimental Evaluations of Fluid Resistance and Correlation Modulus

As pointed out earlier, the important fact in defining the total fluid resistance in unsteady motion is to choose a suitable correlation modulus. In this study, the total fluid resistance acting on the hemispherical and cruciform models which have linear+periodic unsteady motion are defined as

in which VD/V is Rqynolds number, $D/v2=6 is accel- ation modulus, V = VD~/VV is a parameter, and u/Um is the velocity ratio.

Following Dalton, Hunt, and ~asanl9, the total fluid resistance coefficient CR is defined in terms of the instantaneous forces and maximum vel- ocity, then

Thus, instead of separating the total force into velocity dependent force (drag) and accelaration dependent force (force due to the apparent mass) the variation of the total fluid resistance coef- ficient, CR with the instantaneous Reynolds number, accelaration modulus, velocity ratio and A = 6D2/vv as a parameter is determined.

( 6 ) 4. Results and Discussions

Reanalysing the force-time output obtained from experimental workl1 (see in Fig.2) for hemi- spherical and cruciform models, with zero angle

of attacks the variations of CR with the non- dimensional numbers defined above are given in the Figures. As seen in Figures 3,4 and 5, the value of CR increases with increasing 6 and A . As 6 and A approache to the zero, the value of CR reduces to the steady-state value. The scatter of CR With 6 is less tha that of A.

Figure 2: Tangentional Force-Time Output for Hemispherical ~odelll.

In the case of the negative acceleration, the variation'of CR with 6,A and Re are given Figure 6. Because of the geometry changes and effective surface, CR shows different behaviour. Variation of CR with 6 and A determined by linear acceler- ation and linear deceleration of the carriagewith- out periodic motion are given in Figure 7. Also, CR vs u/Um in which u is the periodic velocity and UM is the linear + periodic velocity is presented in Figure 8.

Ithasbeenappared thevariousresearcherhaveused variouscorrelation modulus to define the total fluid resistance and appropriate apparent mass coefficient. The resistance coefficient has been plotted ;gainst acceleration modulus by Iversen and ~alent~O and by Yavuz and cockrellll, the period parameter, VT/D by Keulagen and carpenderZ1 and by ~ a r ~ k a ~ a ~ ~ , and different correlation modulus by other authors.

Considering, Fraser and Simmon's experimental work13 conducted on a sphere and a streamlined body with a 4:l fineness ratio, both of which were towed through water, and for which these results were plotted against velocity and acceleration. From this work the total fluid resistance coef- ficient, CR is here determined and plotted against acceleration modulus for various Reynolds numbers. As seen in Fig.9 for sphere, the Reynolds number effect on the total fluid resistance is consider- able, and it is evident that correlation of the coefficient only with the acceleration modulus is not very convincing. However, as seen in Figure 10, for streamlined body which has a 4:l fineness ratio, scatter is quite small and the effect of Reynolds number on the coefficient is negligible. This is because with sphere the separation points vary considerably with Reynolds number, but little variation accurs with streamlined bodies. Vari- ation of CR with aceleration modulus for various Reynolds number determined by ~ e i m ~ ~ for circular cylinder is presended in Figure 11. Hence, for bodies on which the point of separation is effectively independent of Reynolds number the

total fluid resistance coefficient correlates well with the acceleration modulus.

5. Concluding Remarks

The fluid resistance acting on submerged and unsteadily moving bluff bodies is defined by one parameter, so called the total fluid resistance coefficient, CR . This total fluid resistance coefficient varies with non-dimensional charac- teristics of the motfon, such as Reynolds number, acceleration number VD/V', velocity ratio u/Um, and VD'/VV as a parameter for linear or linear + peri- odic unsteady motions, and Reynolds number, period parameter VT/D and D//VT as a parameter for peri- odic unsteady motion.

Conclusions drawn from this study is that for bodies whose steady drag coefficient remains con- stant over a wide range of the Reynolds number, such as hemispherical shell and streamlined bodies the acceleration modulus can be considered to be a suitable correlation modulus for linear or linear+ periodic unsteady motions.

References

Bradbury,L.J.S., Rurst,F.,Launder,B.E., Schmidt, F.W., and Whitelaw,J.H.(eds), "Turbulent Shear Flows-111," Springer-Verlag,New York,1982.

Davis,R.W. and Moore,E.F. "A Numerical Study of Vortex Shedding from Rectangles", T.Fluid Mechanics, vol. 116, 1982.

Chorin,A.J. "Vortex Sheet Approjimation of Boundary Layers", J.Computationa1 Phys., Vol. 27, 1978.

Sarpkaya,T. and Shoaff,R.L. "Inviscid Model of Two-dimensionalvortex Shedding by a Circular Cylinder," AIAA Journal, Vo1.17, 1979.

Sarpkaya,T. and Ihrig,C.J. "Impulsively-Started Steady Flow about Rectangular Prisms:Experi- ments and Discrete Vortex Analysis", Journal of Fluid Engineering, Vol. 108, 1986.

Strickland,J.H. "On the Utilization of Vortex Methods for Parachute Aerodynamics Predic- tions", Proceeding of 9th AIAA Conference, Albuquerque, Oct. 7th-9th, 1986.

McCoy,H.H. and Werme,T.D. "Axisymetric Vortex Lattice Method Applied to Parachute Shapes", Proceeding of 9th AIAA Conference, Oct. 7th- 9th, Albuquerque, 1986.

Uzkilic,S. "Kut Cisimlerin Etrafindaki Dalga Hareketinin Vorteks Metodu ile Analizi", M.Sc Thesis, Erciye University, 1987.

Sarpkaya,T. "Oscillating Flow over Bluff Bodies in A U-Shaped Water Tunnel", Aerodynamic and Related Hydrodynamic Studies Using Water Faci- lities, AGARD Conference No: 413, Monterey, 1986.

Morison,J.R., O1Brien,M.P., Johnson,J.W. and

Schaaf,S.A. "The Forces exerted by Surface Waves on Piles", Petroluem Trans., AIME, Vol. 189, 1950.

11. Yavuz,T. and Cockrel1,D.J. "Experimental De- termination of Parachute Apparent Mass and Its Significance in Predicting Dynamic Stabil- ity", Proceeding of 7th AIAAA Conference, San Diego, 1981.

12. Wells,D.A. "Theory acd Problems of Legrangien Dynamics", Schaum Outline Series, McGraw-Hill Book Company, 1967.

13. Frazer,R.A. and Simmons,L.F.G. "The Dependence of the Resistance of Bodies Upon Acceleration as Determined by Choronograph Analysis", Advisory Committee for Aeronautics, Report and Memoranda, No.590, 1919.

14. Yavuz,T. "Effect of Changes in Physical Para- meters on the Stability Characteristics of Parachute Descending Through Real Fulid", The Aerounautical Journal, Vo1.89,No.889,1985.

15. Polpitiye,S.J. "Ph.D.Thesis, Leicester Univer- sity, 1986.

16. Yavuz,T. "Performance Prediction for Fully- Deployed Parachute Canopies", Proceeding of 9th AIAA Conference, Oct.7th-9th, Albuquerque, 1986.

17. Sarpkaya,T. and Isaacson,M. "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold, New York,N.Y., 1981.

18. Cockrell,D.J., Shen,C.Q., Harwood,R.J. and Baxter,A.C. "Aerodynamic Forces Acting on Parachutes in Unsteady Motion and the Conse- quential Dynamic Stability Characteristics", Proceeding of 9th AIAA Conference, Oct.7th- 9th, Albuquerque, 1986.

19. Dalton,C., Hunt,J.P. and Hassan,A.K.M.F. "The Forces on a Cylinder Oscillating Sinusoidally in Water, Firther Experiments", J. of Pressure Vessel Tech., Vol. 1001297, 1978.

20. Iversen,H.W. and Balent,R., "A Correlating Modulus for Fluid Resistance in Accelerated Motion", J.appy.Phys. Vo1.22,No.3, 1951.

21. Keulagen,G.H. and Carpender,L.H. "Forces on Cylinden and Plates in a Ocsillating Fluid", J. of Research of the National Bureau of Standarts, Vo1.60, No.5, 1958.

22. Sarpkaya,T. "Forces on Cylinders and Sphere in a Sinusoidally Oscillating Fluid", J. of App. Mech., Vo1.32, 1975.

Figure 3: Variation of CR with 6 and A for Hemispherical Model.

23. Keim,S.R. "Fluid Resistance to Cylinder in Accelerated Motion", J.Hid.Div.,ASCE. 72,HV6, Paper No.1113,Dec.1956.

o :Increasing accde ration CR 1

x :Decreasing acceleration

1.0 0.9 0.8 - x 0 . 7 " " " ' " " "

0 0.02 004 0.06 OD8 OJO 032

Figure 4 : Variation of CR with 6 andA for Cruciform Model ( 4 : 1)

Figure 5: Variation of CR with 6 and A for Cruciform Model ( 3 : l )

0.41 , , , - 200 250 300 350

Rex 1 6 ~

Figure 6: Variation of CR with 6 ,A and Re numbers for Hemispherical Model (negative acceleration)

Hernispherica I

I . . . . . . . I,

aio 00s 0.02 o cxi 0.3 o. .0.7

Figure 7: Variation of CR with 6 and D for Hemispherical Model in Linear Acceleration.

Hemisp h e r d 4

Figure 8: Variation of CR with u/Um in Periodic Unsteady Motion.

J OD 01 Q2 0.3 0.4 0.5 0.6 0.7 0.E

Acceleration modulus a \12

Figure 9: Effect of Reynolds Number on the Total Fluid Resistance Coefficient CR for Sphere

~ . d . . . * . . . - ~ 0.02 0.04 0.06 010 0.14 V. D Acceleration modulus - - v2

Figure 10: Effect of Reynolds Number on the To- tal Fluid Resistance Coefficient CR for Streamlined Body with a Fineness Ratio of 4 : l .

Reynolds number

1 : 1L70 to L750 2 : 1320 3 : 1180 L : 1030 5 : 880 6 7LO 7 : 590 8 : LLO 9 : 3 0 0

I a .

0' Q02 0.04 0.1 0.2 04.6 10 2 3 4 6 10 v. D Acceleration modulus - v2

Figure 11: Total Fluid Resistance Coefficient of Cylinder CR, vs. Acceleration Modulus with Various Reynolds Numbers.