Three component measurements on rectangular wings with cavitation

ARCHi45-Fl

Max -Planck -Institut fur Strom ungsfortEkAg

Gottingen

NAL REPORT

THREE -COMPONENT MEASUREMENTS

ON RECTANGULAR WINGS WITH 'CAVITATION

By

H. Reichardi and

W,..Satt.lier

Gottingen

March 1967..

The research reported In this document has been sponsored by the Office of Nava! Research, under Contract No. N 62558,3720

f

Yinal, Report

Three-Component Measurements on Rectangular Wings with Cavitation

By

H. Reichardt and W. Settler

Gottingen March 1967

The research reported in this document has been sponsored by the Office of Naval Research, under Contract No., N 62558-3720

Contents 1. Introduction 2. Cavitation Tunnel 3. Experimental Apparatus Wing Models Strain-Gage Balance Dynamic Pressure Reading

Determination of d

Performance of Measurements

4. Experimental Results

5. Discussion of Results and Comparison with Theory

Non-Cavitating Flow Past a Wing Cavitating Flow Past a Wing

6. Summary List of Symbols Figures References

a)

a.) b)

Abstract

Investigations of three-dimensional effects in non-cavitating and non-cavitating flow past rectangular flat plates are reported. Five rectangular wing models with aspect ratios AR 1.01; 1.5; 2.25; 2.49; and 3.12 were tested. Lift, drag, and pitching moment acting on these models were measured, and the characteristic development of cavities for several flow conditions were photographically recorded. To the extent that measurements of other authors [1] on cavitating hydrofoils were available, these results were used for comparison and supplementation of this work. Values according to the theory of T.Y. Wu [2] about two-dimensional fully cavitating flows were converted into values for a wing with finite aspect ratio

and were compared with experimental data.

The investigation was performed in a new cavitation tunnel with a free-jet directed vertically upward. Design and performance of the new tunnel are described.

1. Introduction

It is the objective of this work, to investigate the influence of aspect ratio and angle of attack on the forces on rectangular wings and to investigate how far the theories of two-dimensional flow about hydrofoils are applicable to wings with small aspect ratio - i.e., whether values of the

two-dimensional theories can be used after conversion to a wing of finite span.

Measurements were carried out fora range of aspect ratios (3.12) AR) 1.01) in which neither the conditions for wings with very small aspect ratio (AR 4 0.5, R.T. Jones [3]) nor the assumptions for the classical airfoil theory (AR 6, L. Prandtl [4]) are valid. In addition to the force

measurements, the development of the cavities on the different wings were photographically recorded. Several photos are added to the report for a better understanding of the measurements.

For small aspect ratios the equalization of pressure around the wing tips and flow separation determine the

configuration of flow and thus the configuration and size of the cavities on the wing. The shape of the wing tips and the wing section have a decisive effect. To avoid these additional

tip effects, the wing models were designed as flat plates with sharpened leading edges. Moreover, this model design conforms better with the assumptions of the theory.

The present investigation is restricted to very small cavitation numbers to avoid, on principle, a change of the cavity shape within the range of measurement. At small cavitation numbers a steady-state cavitation pocket develops between leading and trailing edges of the wing. Separated from this cavitation pocket two cavities originate within the two tip vortices of the wing. This cavity configuration (one

cavitation pocket originating from the leading edge and two cavities within the tip vortices) was found to persist over the entire range of angle of attack investigated. Within the

investigated range of cavitation numbers, 0.025 5 0.4,

a change of the tip-vortex cavity is caused by the vortex breakdown at larger angles of attack; this change occurs far enough behind the trailing edge that the forces on the wing are not affected by this process.

2. Cavitation Tunnel

The three-component measurements have been carried out

in the new cavitation tunnel of the Max-Planck-Institut fiir Stromungsforschung. Deviating from the former equipment, this

tunnel has a free jet directed vertically upward. The

advantages of a free jet tunnel compared with a conventional water tunnel with a closed test section for measurements at very low cavitation numbers have been described explicitly in

several reports [5],[6]. Therefore, only the most important characteristics will be quoted.

In a free jet, small cavitation numbers can be

obtained at a low dynamic pressure by reducting the pressure around the free jet. Because of the constancy of the pressure around the jet, the cavitation number in a cross-section of the jet is constant. The pressure of the air surrounding the jet can be measured exactly without disturbing the jet itself. Thus, very small cavitation numbers can be determined with a

good accuracy.

In the cavitation tunnel formerly used, it was not possible to recover the kinetic energy of the jet; this is the essential disadvantage of a horizontal free jet. The result is the generation of an air-water mixture and a rapid increase of water temperature. The air-water mixture is produced as the free jet pierces the water surface in a tank. In the former cavitation tunnel, air bubbles could only be separated at a jet velocity lower than 6 m/sec. At velocities larger than 6 m/sec, air bubbles get into the test section and cause undefinite test conditions. The air bubbles change the

cavitation pockets at the model in an uncontrollable way and prevent the view of the model. Therefore, exact measurements were possible at velocities below 6 m/sec only. However, low

velocity and, thus, low dynamic pressure, is disadvantageous for force measurements, especially if small models are used. This resulted in the desire to measure with larger models and at higher dynamic pressure and finally led to the design of a larger and more efficient cavitation tunnel. As a special

feature, it was intended to apply the principal, advanced in 1945 [5], to transform into potential energy most of the kinetic energy of a jet directed vertically upward.

Fig.1 shows a side view of the new tunnel. The jet enters the test section (b) through the nozzle (a). A large, pot-shaped deflector directs the flow into a reservoir (c). Within the deflector the now ring-shaped jet pierces the water surface of the reservoir. The water then flows through the vertical vessel (d) into the centrifugal pump (e) and from there back to the nozzle.

The jet piercing the water surface in the deflector introduces air bubbles into the water and produces a turbulent air-water mixture. A honeycomb flow straightener eliminates the large vortices and the bubbles are separated in the reservoir, which has a large cross section so that the water is flowing very slowly (approximately 0.4 m/sec).

The kinetic energy of the jet is recovered except for a pressure drop of about 0.5 m, resulting from the deflection of the jet into the upper tank. Hence, the power requirement of the centrifugal pump is considerably reduced, and the pump of the former tunnel could be utilized with an effective

increase in efficiency (increase of the velocity from 6 m/sec to 10 m/sec and increase of the jet cross section from

15 x 20 cm2 to 20 x 26 cm2). The surface of the jet gradually disintegrates into droplets with increasing distance from the nozzle. Only a small part of these droplets reaches the upper tank, most of them falling down as spray (the quantity of the

spray water amounts to 6% of the total flow volume). The spray accumulates in the test section (b) and in the duct (f) and flows off into the small vessel (g). An additional pump (h) drains off this vessel into the upper water reservoir. Bubbles carried along during this process are separated in the upper tank (c).

The operation of the tunnel is as follows: With the slide valve (i) closed, the rotational speed of the centrifugal pump (e) is so regulated, that the pressure in the settling chamber before the nozzle corresponds to the dynamic pressure needed during a measuring run. After that, the slide valve is opened. At first, a narrow jet rises into the upper water reservoir. As the slide valve is opened further, the rotational speed of the pump is increased in such a way that the dynamic pressure is held constant. The pressure in the settling chamber corresponds to a velocity of 10 m/sec. This velocity is some-what higher than the minimum velocity at which the water circulation is just maintained.

At operation with minimum velocity, the spray water can just be drained off. For excessive amounts of spray (for instance caused by increasing the angle of attack of the model) it is possible, that the test section is submerged. In this situation, the free jet breaks down. Danger of jet submersion exists at the beginning of a run and at operations at small cavitation numbers (i.e., if the jet is splitted up into two divergent jets). A splitting-up of the jet at the start can be caused by ventilation of the trailing edge of the model support, especially, when the support crosses the whole jet. Only with a very slender support, as described below, the splitting-up can be avoided.

At large angles of attack, the cavities behind the models grow so large that the model is also ventilated at

small perturbations and the jet bursts up. In this state of flow, guide vanes in the test section direct the main quantity of the water into the upper tank.

3. Experimental Apparatus

Wing Models

In consideration of the relatively small cross-section of the jet (0.26 x 0.2 m2) only small wing models could be used. Model 1 has the largest wing chord of 29.8 mm and Model 5 the largest span of 56 mm (see Fig.2). The Reynolds number based on wing chord and jet velocity (10 m/sec) for Model 1 is

Re1 0.3-10 and for Model

5,

Re5 . 0.18-106. All the wings have an area of about 9 cm2. The models are made in the shape of flat plates. The leading edge is sharpened under a definite angle from both sides. This shape is best suited for an

accurate machining. The models were made out of heat-treated stainless steel Remanit 1790 V (Deutsche Edelstahlwerke, Material No,4112).

The dimensions of the spring elements of the balance are determined according to the forces on the models, which were estimated from the model dimensions and the dynamic pressure.

Strain-Gage Balance

As during the former investigations, the forces on the models were measured with three-component balances which work with strain gages: the strain gages, bonded to the spring blades, experience changes in length by bending or stretching of the springs, which induce alterations in resistance. These alterations can precisely be measured with the aid of electric

bridge circuits.

Two balances were built for the vertical-jet cavitation tunnel. The first design is shown in Fig.3. Concerning the arrangement of the measuring elements for the normal force and the pitching moment, this balance corresponds to those formerly used. The measuring unit for the tangential force was

reconstructed, using four instead of two strain gages. Here-with, a double sensitivity and a greater stiffness are achieved.

The support of the balance has a maximum thickness of 11 mm and a blunt trailing edge. Since the flow separates along this edge, the cavitating region within the wake of the support always develops at the same place. The influence of this cavitating region is hence only a function of the cavitation number and can be eliminated by a correction. For some angles of attack, however, the blunt trailing edge ventilates the models and splits up the jet.

To avoid this interference, a new balance with a much more slender support was built. The thickness of the support is only 5 mm, that of the measuring beam

3.5

mm. To attain a sufficient stiffness in spite of the slenderness of the balance and to avoid flutter, the distance between the model and the point of support was considerably shortend (Figs.4a and 5). The measuring unit for the tangential force is totally embedded in the support and protected against the outer flow by two thin plates. Most parts of the measuring units for the normal force and the pitching moment are also protected against the outer flow by the support. By this construction, the forces exerted on the balance itself remain small. The measuring springs for the normal force and the moment are made out of one piece, just as the spring and the spring support for the tangential force. The two parts are screwed together (Fig.5). It was found that the stainless steel Remanit 1740, manufactured by "Deutsche Edelstahlwerke" (Material No.4122) was best suited. The balance material was tempered to yield a

strength of 140 kp/mm2.

The arrangement of the measuring plates can be seen in Fig.5. Strain gages of the types FAB-12-12-S9 and FAB-06-12-39, manufactured by the Baldwin Lima Hamilton Corporation, were used; for bonding, EPY 150 was applied.

The pitching moment M is leasured by a differential bridge, consisting of the strain gages at the positions xl

and 13. The reference point for the moment is depicted in Fig.5. The pair of strain gages at position x2 is connected

to a half-bridge, which measures the moment with respect to this point. From both moments one can calculate the normal force N. The gages with the strains te4 and

-c5 are connected to an additive bridge; they yield the tangential force T. The interference between the single quantities is very small, in any case they are linear.

A detailed description of this sort of balance can be found in the report [8] and in our former paper [7].

Considerable difficulties were caused by the water-proof protection of the balance. On the one side, this protection must withstand the permanent pressure change between vacuum and atmospheric pressure and, on the other side, it must be so

elastic that no hysteresis is caused (thickness of the tangential-force spring is only 0.4 mutt). Both these requirements were satisfied by a coat of several layers of varnish on the basis of polyurethane, manufactured by Chemische und Lackfabrik Fritz Brandt, Porta-Neesen. In addition, this varnish prevents corrosion of the

balance1)-Fig.8 shows a diagram of the electric measuring

apparatus for gathering and reduction of data delivered by the balance. The unbalances of the bridges for normal force, tangential force, and moment are measured with a strain-gage amplifier. The setting and the fluctuation of the signals are observed with an oscilloscope.

In the past, the three bridges were balanced one after the other and the temporal mean values of the data were

estimated and written down by hand. Since one must expect alterations of the flow and of the conditions of cavitation during this bothersome procedure, the old measuring apparatus was supplemented by electronic equipment in such a manner that the temporal mean values of the three components could be

1) Since the balance, the support, and the side plates are made of different materials (stainless steel, brass, tombac),

one must expect corrosion in the case of insufficient insulation.

measured and recorded in a single procedure and, thus, in an essentially shorter time.

The measuring procedure is now as follows. After settling of the desired flow parameters (pressure and

cavitation number), the recorder is switched on. The temporal mean values of the data are obtained and stored, the stored values are successively determined with a digital voltmeter and printed out. The whole procedure, beginning with the switching-on of the recorder up to the printing of the measured values, runs automatically. The time spent for averaging the data is marked on the record. By this method,

the data can be checked later by averaging the record graphically.

The control unit was developed at the Institute; for all other functions, commercial equipment was used.

Dynamic Pressure Reading

The dynamic pressure is measured through the pressure in the chamber approaching the nozzle. A pressure manometer with vertical tube after Prandtl, filled with mercury, is

used as recording instrument. The accuracy of reading is 0.1 mm of mercury. The dynamic pressure is regulated by adjusting the rotational speed of the pump.

Determination of 6

During the former investigations, it was possible to measure directly and without difficulty the pressure

difference po-pk dq , i.e.,the cavity pressure within the cavitation bubble which is generated by the tube slightly penetrating the guide wall. During the measurements within the vertical jet it turned out that spray water enters the tube in the case of higher cavitation numbers (i.e., shorter cavitation bubbles) and so falsifies the determination of

pk-po . Therefore merely the air pressure po in the test section is measured (with respect to vacuum of 10-3 Torr), and the cavity pressure pk for the momentary water

Since the temperature of the water rises slowly, the pressure difference p0-pt needed for a special 6 must be

continuously adjusted.

Comparing both methods of the 6-determination - on one hand by measuring pc) and the vapor pressure, on the

other hand by measuring pd-pk within the cavitation bubble -one cannot find measurable differences, provided the water is well degassed.

For the measurements of the air-pressure differences, the same pressure manometers after Prandtl were employed, filled with mercury for high pressure levels and with silicon oil for small pressure differences.

e) Performance of Measurements

With the slide valve closed, the test section is

filled with water to such a level, that both model and balance are completely submerged in stagnant water. In this way,

eventual temperature differences between the water and the balance are eliminated and zero drift of the strain gages due to changes in temperature is prevented. The bridges are then roughly balanced and the zero reading is recorded.

In order to avoid overstrain of the balance, which may occur with increased dynamic pressure during start, all runs begin with zero angle of attack. After draining the water out of the test section, the main pump is turned on, and the tunnel is set into operation in the afore mentioned way.

After each measurement, the inclination of the model is increased by 30 up to a maximum angle of attack of 210. After shutting down the tunnel, the zero reading of the

bridges is again recorded for the maximum angle of attack. Again, model and balance are in stagnant water. For the inter-mediate angles of attack the zero reading is assumed to vary

and the end of the run.

The forces measured in these runs are caused not only by the model but also by the model support. In order to determine the forces caused by the latter, the model is

disconnected from the balance and fastened to the support with the aid of two struts (see Fig.4b). Naturally, these two struts cause an additional disturbance of the flow, so that the flows around balance and model, on the one hand, and around the balance with the two struts attached, on the other hand, can only be called approximately similar. Compared with the corresponding forces of the wing measurements, the normal forces and the moments of the blind test are small. Therefore, the disturbance of the flow has no essential influence on the determination of these forces and moments. In contrast to that, the interference factor for the relative weak tangential force is not insignificant. Therefore, a correction of the blind test is required for the determination of the tangential force.

4. Experimental Results

Figs.9 to 14 are samples of the photographs taken at this investigation. They are to give a summary of the

development of cavity shapes for the entire range of

measurement and are to demonstrate several effects of aspect ratio, cavitation number, and angle of attack.

The photographs in Figs.9a and 9b show the development of cavity shapes for wing model 1 (AR 1.01) as a function of the cavitation number at constant angle of attack.

In comparison, Fig.10 shows the extent of the cavities

for wing model 4.

Figs.11, 12, and 13 illustrate the development of

cavities for wing model 1 as a function of the angle of attack at constant cavitation number.

Fig.14 gives information about the effect of aspect ratio on the length of the cavity originating from the leading edge.

Results of the three-component measurements are plotted in Figs.15 to 39.

Figs.15 to 29 show the lift, drag, and moment

coefficients as a function of angle of attack for a range of cavitation numbers 0.0254 0.4 for the five investigated wing models.

Figs.30 to 34 summarize the results of the measurements in the found polar diagrams (CL f(CD)).

For the purpose of discussing the experimental

results, the following diagrams were plotted (the discussion

shall be extended essentially to the behaviour of the lift coefficient).

In Fig.35 the lift coefficients for the non-cavitating flow are compared with the theory of N. Scholz [14].

Figs.36 and 37 give a comparison between CL-values obtained for a cavitating flow with wing models 1 and 5 and data according to the theory of T.Y. Wu [2].

Fig.38 shows CL-values for wing model 5 (AR 3.12) and model 1, which are converted for an aspect ratio 3.12 according to the theory of L. Prandtl [4].

Fig.39, finally, shows a comparison of lift coefficients for wing model 1 (AR = 1.01) with data of a 30° delta wing (AR = 1.07).

5.

Discussion of Results and Comparison with Theory a) Non-Cavitating Flow Past a Wing

The first experimental investigations on rectangular wings with small aspect ratio already show discrepancies with

the airfoil theory of L. Prandtl [9]. The formula for the conversion of values from an aspect ratio to another one is

not exact for small aspect ratios. The measurements of

H. Winter [10] yield a larger than linear increase of lift with angle of attack.

A. Betz recognized that with decreasing aspect ratio, a nearly two-dimensional cross flow is established. For the lift coefficient of this cross flow he postulated

CLc CDc

= sin2a .

CD the drag coefficient of the cross flow. Betz proposed

the approximation CD a; 2a2 (more details can be found in

A.H. Flax and H.R. Lawrence [11]).

Other theories make the assumption, that the free vortices do not lie in one plane, but leave the wing tips at an angle with respect to the wing plane (W. Bollay [12], K. Gersten [13]).

The measurements of the present investigation show also the non-linear increase of the lift coefficient at increasing angle of attack. The experimental results are in good agreement with the theory of N. Scholz [14]. According to this theory, the lift coefficient CL becomes

dCL

CL

)

a + C/2 .

The factor C in this equation is independent of the aspect ratio for 0.54:AR4:3 and amounts to C = 3.6 .

b) Cavitating Flow Past a Wing

The model shapes are chosen in such a way that the flow conditions depend practically on the cavitation number, the angle of attack, and the aspect ratio alone. With the help of the photographs, it will first be illustrated in what manner these parameters control the formation of the cavities on the model. Figs.9a and 9b show the dependence upon the

cavitation number. The cavitation number varies

between-(

and 6= 0.025. Wing 1 with an aspect ratio of AR = 1.01 is

used as model. The angle of attack is cc = 70

As evident from the photographs, three separate cavities are formed in the entire 6-range. One cavity forms at the leading edge in the middle part of the flow past the wing, the two others are produced by the tip vortices. The cavitation phenomenon at the trailing edge of the wing will not be examined explicitly. For larger 6, the middle cavity ends on the upper side of the wing, whereas the tip vortices extend far in the downstream direction. With diminishing cavitation number (Fig.9: d<0.2), the middle cavity expands beyond the trailing edge of the wing, and the flow no longer reattaches to the suction side; the diameters of the tip vortices increase likewise. Finally, the three cavities cover the whole upper surface of the wing. Further reduction of the cavitation number yields practically no change of the flow configuration on the wing itself, but the cavities grow substantially in length.

Figa.11, 12, and 13 show the influence of the angle of attack. In general, one can say that all three cavities grow with increasing angle of attack. While the middle cavity and the tip-vortex cavities show a nearly uniform increase in

length, the tip-vortex cavities spread much stronger in the spanwise direction. This is very clearly discernible in Fig.12. The growth of the middle cavity takes place practically only

in flow direction. The photographs of Fig.12 were taken at a cavitation number of d= 0.075. If the cavitation number

exceeds a certain value, the tip-vortex cavities and the middle cavity join together as the angle of attack is increased

(Fig.13, above). In Fig.11, the tip vortices are seen to form long hoses behind the model up to an angle of attack of 16°. These cavities break down at an angle of attack of 19°. Fig.13 below once again shows both shapes of the tip-vortex cavities: the hose-formed type, reaching far downstream, and the bursted tip vortex. The upper photographs in Fig.13 were taken at a

cavitation number of- 0.4. No separated cavities can be seen at an angle of attack of a = 210.

At low cavitation numbers the aspect ratio has an essential influence on the length of the cavities (Figs.10 and 14). Fig.10 shows the growing of the cavities for a rectangular wing with aspect ratio AR . 2.49 at an angle of attack of 7°. Comparing these lengths with the lengths of the cavities at wing 1 (Fig.9) for corresponding cavitation numbers one recognizes the strong growth of the middle cavity. Once again this behavior is demonstrated in Fig.14 for a larger cavitation number. Whereas the middle cavity on the left wing (AR = 1.01) almost reaches the middle part of the wing, i.e., the flow reattaches, the middle cavity on the right wing (AR = 2.49) has already spread beyond the trailing edge of the wing. The area of influence of the tip flow remains constant for the various aspect ratios.

Converting the results of the investigation of H. Winter [10] on air flow around plates with small aspect ratios to a flow with cavitation, one finds the above communicated observation confirmed. Certain changes in the lift diagram can now be easily explained. For small angles of attack, the cavitation area in the region of the central wing flow causes a steeper gradient of the lift coefficient than at the wing without cavitation (Figs.15 to 19). It is known that the flow separation at the leading edge leads to the generation of the middle cavity. The flat plate gets, in effect, a positive camber, thus, the lift of the wing is increased. If the cavity extends roughly over the suction side, the pressure above the whole wing corresponds to the vapor pressure. Therefore, the upper side of the wing no larger contributes to the lift, as the angle of attack is increased further. The increase in lift is now caused by the pressure side only; thus, the slope of the lift curve is

In the two ranges, that of large lift-curve slope corresponding to small angles of attack and that of small lift-curve slope, the cavities on the model are steady-state cavities.

Fig.14 shows that the length of the cavities is largely dependent on aspect ratio. For a wing with a small aspect ratio, the cavity on the upper side reaches the

trailing edge at a larger angle of attack. Thus, the reduction of the lift-curve slope occurs at larger angles of attack. Fig.39 gives a comparison with a delta wing of equal aspect ratio. For this wing shape, the reduction of the lift-curve slope occurs at essentially larger angles of attack.

The area influenced by the tip flow (Fig.14) remains the same for different aspect ratios; thus, the tip-flow effect on the total wing is reduced with increasing aspect ratio. Correspondingly, the lift-curve slope is increased with increasing aspect ratio. With rising cavitation number,

the lift-curve slope shows a decrease.

An effect of the breakdown of the tip-vortex cavities on the coefficients could not be expected in the investigated range of angle of attack and cavitation number, because the breakdown takes place way off the trailing edge. Nevertheless, the reduction of the lift-curve slope for model 5 (1= 0.1; 0.2; 0.4; a>21°) might have been caused by a tip-vortex breakdown.

Except for theoretical investigations on cavitating slender delta wings, theoretical considerations about cavitating wings in general concern two-dimensional flow (T.Y. Wu [2], M.P. Tulin). To render possible a comparison with the present experimental data, the theoretical results must be converted to a finite aspect ratio.

This problem is known from the airfoil theory. The lifting line theory of L. Prandt1 gives a formula, which permits converting experimental data for a wing with aspect

ratio AR1 to one with aspect ratio AR2. Whereas a certain lift coefficient CL for wing 1 with the aspect ratio AR.'

corresponds to the angle of attack al, for wing 2 with AR2 it ia obtained at I2 According to L. Prandtl,

CL

= a

+-2 +-2 1 x 1

)

- AT-2 1

This conversion formula is exactly valid for elliptic loading only. For rectangular wings, wrong values are obviously obtained.

Several methods take this fact into consideration (H. Glauert [15]) by means of a correction of the induced angle of downflow or of the lift coefficient, respectively. But the application of the Glauert-formula

1+t2 1+T )1

AR2 AR1

= correction considering the deviation from the elliptic loading) gives differences between theoretical and experimental

data as well.

The classical airfoil theory of L. Prandtl is valid for large aspect ratios (AR >6) . For small aspect ratios, the theoretical lift coefficients are too large. If values of the classical airfoil theory are corrected according to R.T. Jones

[16], or if the values are calculated by the Weissinger theory [17], the theoretical lift agrees better with

experiments.

As noted in the chapter on non-cavitating flow, the flow past a wing with very small aspect ratio is not

completely determined by the cited theories [4],[17]. The cross flow which occurs in that case causes additional lift.

For rectangular wings at the cavitation numbers= 0,

according to V.E. Johnson Jr. [18] one can write

1

C = 0.88 sin2a cosa .

-The factor 0.88 = 211-CDc

corresponds to the drag 4+n

coefficient of a plate placed normal to the flow.

For

eS>o,

it follows

27t 1

C (, +. + d2)

Dc 4+n 6(4+n)

For the present investigation, the values according to the theory of T.Y. Wu were converted for wings of aspect ratios 3.12; 2.49; 2.25; 1.5; and 1.01 by the more accurate correction of R.T. Jones. To the resulting values, the increase due to cross flow was added.

Figs.35 and 36 show a comparison of the converted theoretical values with experimental results. Whereas the converted data for the aspect ratio 3.12 show a good agreement, an increasing difference between theoretical and experimental

data is obtained with decreasing aspect ratio.

Fig.37 compares experimental data for wing model 1

converted for the aspect ratio of wing model 5 with

experimental data of wing model 5. The comparison turns out quite satisfactory.

6. Summary

An investigation of the flow pattern and the forces on rectangular wings with aspect ratios 1.01,<AR4:3.12 at angles of attack up to 22° and cavitation numbers 0.025<d40.4 is described.

The flow pattern shows the development of three

separated steady-state cavities on the upper side of the wing. Two cavities are found within the tip vortices and the third originates at the leading edge in the middle section of the wing.

The dimensions of the cavities depend greatly on cavitation number and aspect ratio.

In non-cavitating flow, the C()-diagrams are in very good agreement with diagrams obtained from nonlinear theories.

For small angles of attack, the lift-curve slope in cavitating flow is larger than in non-cavitating flow.

If the cavity in the middle-flow region of the wing extends up to the trailing edge, and the upper side of the wing is thus covered by a cavitation pocket, the lift-curve slope is substantially reduced.

Values according to the theory of the cavitating two-dimensional, flat plate were converted for an aspect ratio of 3.12. The converted lift coefficients show good agreement with measured lift coefficients. The difference between theoretical and experimental data increases with decreasing aspect ratio.

When transformed with the conversion formula of the airfoil theory, the measured lift coefficients for the different wing models correspond well to one another.

The investigations were carried out in a new

cavitation tunnel with a free jet directed vertically upward. Design and operation of the tunnel are described in detail.

List of Symbols Lift Lift Coefficient

-ag

CD w Dr _{Drag Coefficient} CL'

C _Moment Moment Coefficient (about

70

M m PI--;

d =

PbPk

Cavitation Number q V lc,, Re

wv

Reynolds Number AR mg Aspect Ratio, ciu * v2 Dynamic Pressure Wing Area Wing Span Wing Chord Static Pressure Cavity Pressure

Velocity of Undisturbed Flow Near the Model

cc Angla of Attack

Kinematic Viscosity of Water

Density of Water

Glauert Correction Factor [15]

Po Pk b c /-1 v2A F

dzeor

# ZAW All

II

II r 472:46W49% 41 II P

3=EE

et

Fig.1 Cavitation Tunnel of the Max-Planck-Institut fUr StrOmungsforschung with a Jet Directed Vertically Upward.

Vacuum

iNh

_Nil

oil

Moment Reference Axis

_L

Fig.2

WingModels.

56

i

r*--

30 .1

n

44,75 Model 1 AR =1,01 Model 2 AR =1,5 Model 3 AR= 225

t

ol

r,

Model 5

i_

0

AR= 3,12 Leng h in mm 50 Model 4 AR = 249 37,5 VI L__

L-L I r1 -. I Q t. It

ill

I-1 i M 1 .A

Fig.4a Final Design of Strain-Gage-Balance (Fairing

Eliminated)-=61

Fig.4b Above: Model Attached to Balance Below: Model Attached

to Two Struts. , 7 I

I°

1\

Half - Bridge 70-z x. Strain- Gages 1032

.=1=1.

Differencing- Bridge

Fig.5 Schematic Drowings of Balance.

Fig.6 Gage-Bridge Circuits.

40

ow- _{MMEMMEDDRAMMEWAMMEMEM}

Summing - Bridge -o

Fig.7 Water-Tunnel Test Section with Balance and Balance Support.

Strain- Gage-Amplifier

Normal-Force _{Tangential F021} Moment Low- Pass _Filter

Relay-Unit

._---Oscillograph Recorder

Ill

Integrator Storage- Unit Control- Unit Fig.8

Block Diagram of Tunnel Instrumentation for Force

Measurements. ..-.1

Digital-Voltmeter

6= 0,4

Fig.9a

Wing-Model

1

(AR = 1,01); Angle of Attack a = 70. Development of Cavitation

G = 0,075

Fig.9b

Wing-Model

1

(AR = 1,01); Angle of Attack a = 70 Development of Cavitation Bubbles

6= 0,3

Fig.10

Wing-Model 4 (AR . 2,49); Angle of Attack a . 70 Development of Cavitation Bubbles on Upper Side of the Wing at Cavitation Numbers d. 0,3 and 0,05.

10o

16°

19°

22°

Fig.11 Wing-Model 1 (AR 1,01); Cavitation Number C.

0,2.

Development of Cavitation Bubbles on Upper Side of the Wing as a Function of Angle of Attack (Oblique Views).

Fig.12

Wing-Model

1

(AR = 1,01); Cavitation Number d= 0,075. Development of

Cavitation

Bubbles on Upper Side of the Wing as a Function of Angle of Attack.

= 0,4

a

= 70

14°

21°

6= 0,2 a =

Fig.13 Wing-Model 1 (AR = 1,01); Cavitation Number 6= 0,4

and 0,2. Development of Cavitation Bubbles on Upper Side of the Wing as a Function of Angle of Attack.

ari

Fig.. 14

Effect of Aspect Ratio on Length of Cavitation Pocket in

the Middle

Of the Wing. a ft const

7(); 6;7. const = 0,44

AR.ft 141

AR = 2 AR =

'erzo Q55 050 CL 0,45 0,40 0,35 0,30 0,25 0,20 035 030 0,05

r

..

I

ME

,w,

mg

_

V

ME'

1 ! I . ,t !!

.

a 1 op

Willy

a 1 0 G m + A 1.0 IVoncavitating .0,4 =0,2 =al Flow 10 1, Model. It . i .0975 : ';°, 5 i 3 6 9 - 12 ,F5 IS 21' 24 m'

Fig.15 Wing-Model 1 (AR = 1,101). Lift Coefficient C

as a Function of Angle of Attack.

0

0,65 0,60 0,55 CL A 050 0,45 040 Q35 Q30 0,25 Q20 0,15 Q10 0,05

IP

o_i_o_

IN

ME

EMMEN

li

SR

I

A

111.1.

ilEr171

BEM

Noncawtating Flow

. 6 .04

MI

Model 2 . =Q2 =01 + a r 075 .(105 A _o

..Q025

i I 0 3 6 9 12 15 18 21 24 cr.

Fig.16 Wing-Model 2 (AR . 1,5). Lift Coefficient CL as a Function of Angle of Attack.

0,05

Fig.17 Wing-Model 3 (AR 2,25). Lift Coefficient CL ae a Function of Angle of Attack.

.1

PI

MEM

. . . .

wawa

FA

A

IA

wig

wm.A

Arm-

Noncawtating

=0,4 =01 Flow

ri--

₇

-0 6 Model 3 +

.

r 0075 o =po5 ° =0025 0 6 9 12 15 18 21 24 0,65 0,6 0,55 050 CL 0,45 0,40 035 030 025 0,20 0,15 -=

7ig.18 Wing-Model 4 (AR

mg 2,49).

Lift Coefficient as a Function of Angle of Attack.

A

ME

IEM:

PAN

WM

ir

PP"

WA

ME

_rAimm.

AN

ME

ati:Q204:tatm now

V

FAA

m

3

6

72 15' 18 24 0.6 Q6 Q55 .Q5 CL 1 0,45 0.4 Q35 Q30 Q25 Q20 0,15 0,10 0,05 0 Model 4 2

Q65 0,60 c, 0,55 1 0,50 0,45 040 0,35 0,30 Q25 0,20 Q15 0,10 0,05

Fig.19 Wing-Model 5 (AR 3,12). Lift Coefficient CL as a Function of Angle of Attack.

. . .

EMMA

MN

El.

ir

A

BINIIIIIIMMI

En

11111MINEr

Ara

o 6

5 "a2

Nomawtging zQ4 . QI - Q075 =005 =0,025 I Flow Mode

_,

.

.

a 0 3 6 9 12 15 18 21 2,

a VD, A 12( ,ce S1g.20 Wing-Model 1

(AR = 1,01)4

Drag Coefficient CD

as

A Function of Angle of Attack. V.) 0,2 5 ) -II 1 1 1 . I I I

i

II

i

---''''m

zr;/:::%:;

Model 1 V G: Nondavitating F104

t 04

=02

=0/

= 0075 = 005 t. 0,025 , .

.---_-71

x ,x---____

,____:

,

t a (3 12 /lc 112 91 0

/

o 6

03 ₀₂₅ Co 02 _0.15 0.1 005 0 Fig.21 Wing-Model 2 (AR

1,5). Drag Coefiicient CD as a Function of Angle

of Attack. x i--.

g

Model 2 0 e' Noncavitatmg Flo, = 04

i

= 0.1 = 0075 = 005 = 0.025 .

-''''''

+ A

°

13 3 6 9 12 15 18 21

025 0.1 005 0 I 1 [ 1 el 1 4,./ a o 1 1 , Model 3 AP 0 + A 6 0 G Noncavitatingrflow . OA . OJ = 0075 = 005 = 0,025 : 12 /5 18 21 -1111-(10 Wing-Oiodel 3 (AR

= 2,25),

.Drag Coefficient C at a Function (if

Angle

of Attack, 02 ₀₁₅ 0.

E1g.22

3 6 9

0.3 005 0 12. /5 /8

ao

Pig.23

Wiug-Modei

4 (AR =. 2,49). Drug CoeffiCient CD ao a Punt:firm of Angle

pf Attack.

'9 21 I 1 . 1 1 1 x , 1 li , +." I 1 I + o -o V NoxowWing Floii

.40

Model 4_

°

6 4.4

. _

_..---I 1-= OJ 0075

o:-..=-.1-5-3;

-.

1 1 11

IL

A

.

o = 005 0

. 0025

<a 02 CD 0. r0.15 A 0.1

7

02 6

03 025 02 015 0/ 005 0 o

.

V V

V

i

0-7'"----.-'

7

+

o-''''''''''

Noncawtating Mode! 5 0 c = 04

-02

K + =01 -,===

_

-a = 0075 0 r005 0 =Q025 6 3 9 21 18

of Attack.

12 /5

Fir.24

0.0 *0,02 Cm

t

= 0,04 1 0,06 _,0,08 -90;12 ,10,14 Fie.25 Wing-Model 1 (AR F 1,0 ). Zoment Coefficient C as a Function of Angle of Attack., . 12 .18 21 1 1 - . 1

.

a 'Noncavitating = 04 = 02 0.1 - 0,075, = Q05 = Q25 0 ,6 Mode( 1 lc Flow

--+ . a 0 ci 0

Q02 0 Q02 Cm -0,04 Q06 0,08 0,10 0,12

Fig.26

Wing-Model 2 (AR . 1,5). Moment Coefiicient Cm as a F.Inction of Anele

of Attack.

/8 0 0---____,.... ___

\.

g + g

go

x 9 + x o

00

Noncavrtating Flow o = 0,4 Model 2 x = Q2 + = 0,1 a _ Q075 =Q05 0 0

r0025

1

0,02 0 002 Cm A 0,04 -0,06 -0,08 -0,10 -0,12

Fig.27

WingModel 3 (AR = 2,25). Moment Coefficient Cm as a Function of Angle

of Attack.

9 12 15 18 21 + 0 e .11114 + ' e IL. o 9 a o +--o

.

1

Noncavitating How

°--.0 6 = 0,4 x = Q2 Model 3 +

.

0

A 0,075 = = Q05 0 0

r025

1

092 -Q02 CM 4, -Q04 Q06 Q08 0)0

-7

/5 18 -111.-CC Noncawtating now o 5 = 0,4 = Q2 = 0./ 0,075 0 = Q05 Q025 Fig.28

0,02 0 002 Cm

-04

- 006 -0,08 0,10 0,12 Fig.29 Winp-Mojel 5

(AR = 3,12). Moment Coefficient C,1

a Function of Angle of Attack. 9 12 1 1 ₁₅ 18 a° woo

----v .

h.

o_______ Model 0 5 x NoncavitatIng Flow G = 0.4 = Q2 z0,1 = 0,075 = 005

r0025

+ A 0 0

0,6 0,55 Q50 0,45 CL

I354°

0,30 0251 0,20, 0,75 0,10 0,05 0 0,04

---0"

cD

Fig.30 Polar Diagram for the Wing ae Affected by Cavitation. Model 1 (AR

im 1,01).

Noncavitating i Flow

:::uiiiiiii

1 1

El

iple-Q2z

I

0,05 -

-I c

go. li

0,025 =-,

MI

1

I.

1.

4

Model 1

cc-.

= 12 . 42 i

:

:71GI2 .142 =192 .1: 0,12 0,16 0,20 24 0.05

0,65 Q60 0,55 Q60 0,45 CL ; 0,35 0,30 q2 0,10 Q25 0)5 go 004 Q08 0,22

Q6

0,20 "

Fig.31 Polar Diagram for the Wing as Affected by Cavitation. Model 2 (AR

m 1,5).

Q24

11111

w

Mill

Al

Pi

Noncavitatmg Flow

em

=0,4 02

mi

..4111

tkomm_woo-d05%

VIilIUL

r1.UUU

l'

111

I '

FM

, , , , 0,40

CD

Fig. 32 Polar Diagram for the Wing as Affected

by Cavitation. Model 3 (AR = 2,25).

I 1 Noncavita 11 , ing Flow 1 '

AllA

I

rjr

I 1 I! 1 1

MIMI

=pr.._

Info,--

005,075

TAMP

_0,025

ir

. , , , Model 3

A .12

0 0 ,

II

H l 402₄₃₂

U

462

42

1 Q04 0 . 0,12 016 020 024 0,70 Q65 060 055 0,50 045 CL 0,40 0,35 1030 025 Q20 .0,15 0,10 0,05 0

.

a

Co

Fig.33 Polar Diagram for the Wing as Affected by Cavitation. Model 4 (AR 2,49).

i 1 , Noncavitating 1 , Row 1 , 1 P5 , 1 I 50

MP"

IFF M

... 45 , _

111111

1

(5 I140

wiAmin-35

ii

6'-e-jg75 0,1 02 1 30 ?5

rA

N

A

la_

...111P

_,r7r4/

0,025'

-?0 1 _ ,

UI

1,2 1 1

1

'

rag

FM

)5

11.111.11

-

11

1 1 Model 4

..

N 2 '

: 2-?2

c :02

0

ria?

0 =la2

0 :222 nnn

11

1 ri II PVICI ri f - --.,.-, -

-CL 0, a Q 0 = 015 Q24

0,7 0,65 0,55 0,50 0,45 CL Q40 0,35 030 ;0,25 0,20 0,15 0,70 0,05 8 0,16 020 7 Co

Fig.34 Polar Diagram for the Wing as Affected by Cavitation. Model 5 (AR = 3,12).

Q24

IIIIIIIIU

Ellsrioncavitating,

IIIII

Flow

III!

, , EC 0,4 --.'

MEMO

II!

1,1

11

111 *,,,,'0,5

:Q°5

1//10/%,

FAIN=

eQ02,

.

e' i

tillE

,

EVE

... .,

IFA

,

1111

-!uuuuI

iu,1I

, -A075 Q1 Mode/ 5 =72 .19,2

o=22

0.12

IQ

Q

a'

Fig.35 Comparison of Lift Coefficients in Noncavitating Flow with Dat# According to the Theory of

A.

Scholz [14j. / LA 1 ' . 65 , ' . E49. 55. 50 45 A' All' ,

Firmirm

I Aviv

iiirsA

1,1111

49 35

illIM

FIA111111

1 , 10

WAWA

?5 ?0

Tr

1 '

ir,,,,,,r

0

. z

)5 ,/'

/

A

Model 1

Noncavr atIng ;now

0 2

.,.

,7

x 3

..,

---aihtori_N Scholz IN

0 3 6 9 12 IS 11P 0 0 CL 0,15 + 4

0,05 -57-, . 1 I.., .. 1 1 .,

.

t

.

a ..._

IliMM-MILiffild

:

Effign

a

...is

Model 1 -. 6 =02 1 4' +, 12' Theory

roj

=0025 1 T Y Wu (2j 3 6 9 12 15 18 21 24 CC'

Fig.36 Comparison of Lift Coefficients for Wing-Model 1 (AR = 1,01) at Constant Cavitation Numbers wit4 Data According to the Theory of T.Y. Wu 12J. 0,45 0,40 Q35 CL 0,30 0.25 020 015 010 0

I.-0 0 o 0 o x .. x x + o Model 0 6 r0,4

r025

TV Theory . .Q2 4. :0,1 0 Wu123 0 3 6 9 12 15 16 21 24

Fig.37 Comparison of Lift Coefficients for Wing-Model 5 (AR 3,12) at Constant Cavitation Numbers with Data According to the Theory of T.Y. Wu 0,65 060 0,55 0,50 CL 0,45 0,40 Q35 0,30 Q25 Q20 0,15 0,10 Q05 = ?

Q5 CL.

I04

J0,4 Q3 Q3 Q2 Q2 0,15 , , 0,05 1 A Noncawtating

ri

1 '

ll

Illarr

111

1 Flow ,1 It 1 Iril1 1

opm

J

of

FA

,...,-.

pm.

_

mil

_Mall

i 1 Model 1₅ I i 0,08 Q12 0,16 . Q20 Co

Lift Coefficients, for Wing-Model 1 (AR = 1,01), Converted for an Aspect Ratio 3,12 and Compared with Data for Wing-Model 5 (AR = 3,12).

0,7 0,6 '06 0,10 0 dig.38 4

0,45 040 0,35 CL. 0,30 025 0,20 0,15 0,10 0,05

/

/+----/

, ----Model I d . 0,1 =0925 Rectangular Delta-Wing +

--.05

/

/

a Wing 30*

---0 6 9 2 /5 18 21 24

a'

Fig.39 Comparison of Lift Coefficients for Wing-Mogel 1 (AR = 1,01) with Data of a

30 -Delta-Wing (AR = 1,07).

[1] R.W. Kermeen

References

Experimental Investigations of Three-dimensional Effects on Cavitating Hydrofoils.

Cal.Inst. of Techn. Rep.No.47-14, Sept. 1960.

[2] T.Y. Wu A Wake Model for Free-Streamline Flow Theory Part 1: Fully and

Partially Developed Wake Flows and Cavity Flows Past an Oblique Flat Plate.

Journ. of Fluid Mech. Vol.13, Part 2, pp.161...181, 1962. [3] R.T. Jones Properties of Low-Aspect-Ratio

Pointed Wings at Speeds Below and Above the Speed of Sound.

NACA TR 835, 1946.

[4] L. Prandtl TragflUgeltheorie 1. und 2.Mittlg. Nachr.d.Kg1.Ges.Wiss.Gottingen, Math.-Phys.Klasse 1918,

S.451...477 and 1919, S.107...137.

[5]

H. Reichardt Uber Kavitationsanlagen fUr kleine Kavitationszahlen.

UM 6620, Kaiser Wilhelm-Institut fUr Strbmungsforschung 1945-[6] Ch.D. Christopherson Experimental Design Studies on

L.G. Straub

Free-Jet Water Tunnels.,

St.Anthony Falls Hydraulic Lab. Sept.1951.

[7] H. Reichardt Three Component-Measurements on W. Settler

Delta-Wings with Cavitation. Max-Planck-Institut fUr

Strbmungs-forschung, Gbttingen, July 1962.

-[8] E. Wedemeyer Windkanalwaagen mit Dehnungsme8-streifen fUr den AVA-Hoch-geschwindigkeitskanal. AVA FB 59/04/1959.

[9] L. Prandtl Ergebnisse der AVA Gottingen

A. Betz

I.Lieferung (1920), S.50...53.

[10] H. Winter Stromungsvorgange an Platten und profilierten KOrpern bei kleinen Spannweiten.

Forsch.Ing.-Wes. Bd.6 (1935), S.67...71.

[11] A.H. Flax The Aerodynamics of Low-Aspect-H.R. Lawrence

Ratio Wings and Wing-Body Combinations.

Third Anglo-American Aeronautical Conference Brighton 1951.

[12] W. Bollay A Non-Linear Wing Theory and its Application to Rectangular Wings of Small Aspect Ratio.

ZAMM Bd.19 (1939), S.21...35.

[13] K. Gersten Nichtlineare Tragflachentheorie fUr RechteckflUgel bei

inkompres-sibler Strbmung.

Z.Flugwiss. Bd.5 (1957), Heft 9, S.276...280.

[14] N. Scholz Kraft- und

Druckverteilungs-messungen an Tragflachen kleiner Streckung.

Forsch.Ing.-Wes. Teil B, Bd.16 (1949/50), S.85...91.

[15] H. Glauert Grundlagen der TragflUgel- und Luftschraubentheorie.

NACA RM L 57 I 16, 1958.

[16] R.T. Jones Correction of Lifting-Line Theory for the Effect of the Chord. NACA TN 817, 1941.

[17] J. Weissinger Uber eine Erweiterung der Prandtl-schen Theorie der tragenden Linie. Math.Nachr. Bd.2 (1949),

S.45...106.

[18] V.E. Johnson, Jr. Theoretical and Experimental Investigation of Arbitrary Aspect Ratio, Supercavitating Hydrofoils Near the Free Water Surface.