the aerodynamics of shallow paraboloid antennas

THE AERODYNAMICS OF SHALLOW PARABOLOID ANTENNAS

T. A. Wyatt Department of Civil Engineering, Imperial College, London, England

Introduction

Wind forces are of great importance in many design aspects of steerable radio antennas, but the necessity to consider all possible attitudes of the antenna combined with consideration of the forces and moments in six components makes it difficult to grasp the problem and to appreciate the information and results obtained. The results of wind tunnel testing of a single specific design, in the simple case of a steady uniform wind, will extend to several pages of tables of values. It is therefore advisable to give some preliminary consideration to the probable design factors.

The discussion that follows is based largely on experience gained in design of paraboloid radio telescope reflectors on alt-azimuth mountings while the author was employed by Freeman, Fox and Partners of London.

The basic arrangement of such an antenna system is shown in FIGURE 1, the altitude (elevation) axis being carried in a trunnion or turret structure that rotates in plan, i.e., about the azimuth axis. The actual reflector is stiffened and supported by a complex truss system which can be generally referred to as the backup structure, and within the reflector is the feed and feed support structure. Even if a more complicated optical system is used, the structural arrangement is likely to be similar.

The influence of the wind must be considered in relation to: ( 1) accuracy of reflector as a paraboloid ( 2 ) pointing accuracy (3) loads on drive system (4) safety in extreme winds (non-operational) (5) freedom from oscillations. Factor ( 1) above strictly requires an investigation of the distribution of pres-

sure over the whole dish structure, but in fact this is normally not necessary. The wind load under which full accuracy is required is usually sufficiently small compared with the weight of the structure that an adequate approximation can be deduced from the overall forces and moments assuming, say, sinusoidal variation of pressure. This might not apply, however, for antennas designed for duties requiring accurate operation in very high winds.

If the antenna has a plated reflector surface, local distortions affecting individual panels near the rim may be important even in more normal wind speeds, as high local pressures may occur. There is little information on this aspect, but very high local suctions have been found in recent tests of hypar roof shapes, and possibly some special edge shape having a “spoiler” action would be beneficial. The stiffness required in the individual panels of the open mesh type is also dependent on wind loading, but this can be estimated as shown later.

The pointing error due to wind loading, factor (2) , is due to displacement of the effective radio axis of the dish and of the feed, due to deformation of the dish and any other parts not included within the control system, plus the control system errors. These latter include both steady and fluctuating components, the fluctuating components of the altitude drive moment when pointing towards the

222

Wyatt: Paraboloid Aerodynamics 223

t

FIGURE 1. Layout of altazimuth radio telescope.

wind being most significant. A complete spectral analysis is of direct application in design of the control system, and it is hoped that the methods being developed by Davenport will provide this information more quickly and more accurately than has been possible hitherto.

The design of the drive syslem clearly requires knowledge of the total forces on the moving structure, with the moments taken about the steering axes of the reflector. Drive power ratings ,and parts designed by considerations of wear, etc., depend on the steady values O F these moments, but transients may be important, particularly if backlash is to Ibe eliminated from the drive by ensuring a sufficiently large steady “bias” moment in still air.

Steerable antennas not otherwise protected are normally “stowed” in some specific attitude chosen to minimize the loading in extreme wind conditions. The loadings are obviously required for this attitude, overturning moments being required about lower sections as well as about the altitude axis.

An important consideration in determining the effect of transient loading is the natural frequency of the structure.’ To satisfy the functional requirements of the antenna for paraboloid accuracy and control, the natural frequencies are high, typically one to four cps. for current designs, and mechanical magnification effects are unlikely to be significant. Oscillations due to aerodynamic instability are a special field that will not be discussed here.

The predominant element ,as regards wind forces is invariably the reflector surface itself, and it is beneficilal to consider this separately.

224 Annals New York Academy of Sciences

The Plateid Paraboloid

The aerodynamic action of the reflector differs radically according to whether a solid plate is used or wire mesh or other permeable surface. The pressure distribution over a plated dish is fundamentally determined by its overall shape and must be investigated experimentally.

When pointing towards the wind the force is large, being greater than on a flat disc of the same diameter, but in a steady wind it is roughly in balance in respect of the moment about the vertex of the paraboloid. This moment is clearly very susceptible to gust action causing an increased speed over one-half relative to the other.

When the dish is “feathered,” or pointing normal to the wind, the force is very much smaller, but as the pressures are basically normal to the surface with little skin friction effect, their resultant tends to act far above the vertex, causing a large moment. This moment should not be very susceptible to gust action, the condition for the maximum being roughly the greatest possible average speed over the whole dish.

The normal to the paraboloid surface at any point intersects the generating axis at 2f measured along the axis from that point, where f is the focal length. For a shallow paraboloid it is therefore expected that the pressure resultant resolved transverse to the axis will act at an eccentricity from the vertex of 2f plus, say, half the depth of the dish (FIGURE 2) . In terms of the focal ratio:

The behavior described is shown in detail in FIGURE 2, the force in the plane containing the generator axis of the dish and the wind direction being resolved into axial and transverse components, PA and P,.

No other components of force or moment are possible, due to the symmetry of the dish.

The results shown are based on detailed tests carried out by the author on behalf of Freeman, Fox and Partners. Two dishes were investigated, focal lengths f = 0.29D and f = 0.46D. The variation of CA with focal ratio is small, and other values could be interpolated. The transverse force is defined in terms

f of CT - : this is analogous to expressing the coefficient in terms of the feathered D projected area. This was found to be virtually independent of the focal ratio, except at about 45’ incidence, which is discussed later.

The moments measured correspond very closely to the predicted eccentricity, and it follows that the moment about the vertex of the paraboloid is only dependent on the focal ratio to the small extent that the eccentricity e is affected. This is an interesting result.

Due to the size limitations of the wind tunnel used, the tests were carried out at a very low Reynolds Number, of the order 2 X lo4. The results have been corrected for tunnel blockage according to Maskell’s proposals.’

The results have been compared with the results of investigations at the Na-


tional Physical Laboratory, England, on two actual designs: the Jodrell Bank radio telescope3j4 and a radar The contribution to the vertex moment in the feathered attitude made by wind force on the other structural elements was less than 10 per cent of the total, and so could be estimated with sufficient accuracy to show the net moment due to the paraboloid in this position. The values obtained were:

Jodrell Bank, f / D = 0.25. eC,/D = 0.093 (0.110) Radar antenna, f / D = 0.285. eC,/D = 0.108 (0.105)

0.5 - Incidence 8 degrees

0 I " ! ' -0 30 GO

-0.5 -

-1.0 -I

0

FIGURE 2. Force coefficients for paraboloid dish.


The values predicted from FIGURE 2 are shown in brackets. Some further comparison with these results is given below, by analysis of tests for the N.R.C. 150' radio telescope.

Large moments occur over a range of wind incidence to the rear quarter of the dish, and design factors requiring high coexistant moments and forces may be expected to be critical at about 135' incidence. It should be noted, however, that a large moment about the altitude axis coexistant with a large force can arise where the dish is supported by a central tower, due to shielding of the lower part of the dish when pointing downwind ( 180° incidence).

For angles of incidence less than about 60' the moment is of negative sign and can be quite large. This is somewhat surprising, but it is not likely to be critical for design purposes as the additional effects of the back-up structure, etc., will act in the opposite sense.

Over a range of incidence of about 8", up to an angle roughly equal to the slope of the surface at the rim so that the leading edge is tangent to the wind, two values of the coefficients are possible. As the angle of incidence increases the force and moment increase rapidly at the end of this range: if the incidence is then reduced the coefficients reach even higher values and then collapse sud- denly at the lower limit of the range. As each limit is passed there is thus a sud- den change of torte on the dish, but fortunately the width of the loop is sufficiently great that it is unlikely that changes of incidence due to gusts or to motion of the dish could cause worse than isolated, infrequent passage through the change.

The moments due to wind loading on the feed and back-up structures are relatively small, so that sufficient accuracy for practical purposes may be achieved by relatively simple estimates. Two stages may be distinguished; estimation of

F..d structure

lncidcncs X

3(b), Yoriahon of wmd i w d with pointing direction Moa.nt M i n o n * n t r h w poinhng r o r n o l h vmd x R,

FIGURE 3. Shielding effects.


wind action on these parts irrespective of the reflector surface, and then allowance for interaction with the latter.

To assist in the first stage for the case where the back-up structure is basically a set of trusses radiating from a central hub, tests have been made on two arrays of 16 Warren trusses (FIGURE 3 ) . The members were of circular section, and the shadow fraction of each truss was 0.27 and 0.40 respectively. The measured total force was expressed as the value predicted by summation of the individual parts multiplied by a reduction factor R,. The effect of the differing incidence of the various trusses is allowed for in the summation. The reduction factor allows for the shielding of each truss by the others, and depends on the effective solidity of the whole array: it is therefore related in FIGURE 3a to the predicted total drag expressed as a coefficient in terms of the outline projected area of the whole.

The interaction with the paraboloid surface has been investigated by tests of simple additions to the small model dish of focal ratio 0.46. The total moment was measured about an axis 0.2D behind the vertex. First, a simple tetrapod feed support was added, springing from the rim. The addition to the moment is shown in FIGURE 3b, expressed in terms of the reduction factor to be applied to the value for the feed structure alonie as measured when pointing normal to the wind.

Secondly, a very simple back-up structure was added, in the form of 16 straight members, each of diameter D/60 and tangent to the dish near the rim. The extra moment when pointing normal to the wind was found to be the same as measured for the back-up structure without the dish. The reduction factor by which the value for any pointing direction can be obtained from this basic value is also shown in FIGURE 3b.

ApplicO.rtion to the Steering Axes

In order to apply these results in terms of alt-azimuth components, the geo- metric relations required are shown in FIGURE 4. FIGURE 4a defines the pointing direction relative to the wind, whence the true angle of incidence can be found from

cosy = COS a Sin p (3) For any rotationally symmetric body the wind force may be resolved into two

components in the plane OABC as shown. Resolving and evaluating moments about 0, six components are obtained referred to the steering axes as shown in figure 4b.

It is instructive to analyze wind tunnel tests using these results. A complete investigation of loading in terms of the six components referred to the telescope axes was made during the design of the 150' radio telescope now under construc- tion for the National Research Council of Canada.6 The photographs (FIGURES 5 and 6) show the 1/60 scale model of this instrument in the 7' X 7' wind tunnel at the National Physical Laboratory. The dish is plated to a diameter of 120' and the remainder is panelled with wire mesh of about 32 per cent shadow area (viewed normal to the mesh). Mesh of actual full size spacing of wires was used on the model to minimize viscosity effects; reduced to 1/60 scale the mesh would have had extremely fine openings. The focal length is 60', and the altitude axis about 20' from the vertex.

The six components were measured by balance at seven azimuth angles for six different zenith angles. FIGURES 7, 8 and 9 show this large volume of results reduced to C,, C, and eC,/D. The results are quite consistent.

Fig . 4 (6).

Fig. 4 (a).

horizonfat X norma/ PO aIMhde axis

Y horironfa.o/

parollel to altitude axis

sin OL PT -sinll Y =

z = P* c o s p

'T s i n a cos Z-r P =

Q = - pT = c o s a cos p s in 8

sin 8 R = sin oc s i n p FIGURE 4. Forces related to steering axes.


FIGURE 10 shows the moments compared with the results for the plain dish plus calculated values for the mlesh portion and for the feed and back-up structures. It will be seen that the agreement is quite good and it may be concluded that the presence of the other elements, and the high roughness of the rear of the dish caused by the stiffeners shown in the photograph, does not seriously alter the flow pattern of the paraboloid. The only anomalous result is for the tail-on position, where the moment is largely due to the screening effect of the tower.

FIGURE 5 . Model of N.R.C. 150-foot radio telescope. (Reproduced by permission of the Director, National Physical Laboratory.)


FIGURE 6 . Model of N.R.C. 150-foot radio telescope. (Reproduced by permission of the Director, National Physical Laboratory.)

The Wire Mesh Reflector

When the dish surface is composed of panels of wire mesh permitting relatively free passage for the wind, the overall forces and moments can be evaluated assuming there is no “overall shape factor,” i.e., assuming that the pressure on each element of the surface is as measured on an isolated plane piece at the appropriate angle of incidence in a wind tunnel.

Wyatt: P'araboloid Aerodynamics 23 1

1.0

C A

0 5

0

-0.5

-1.0

. 8 ..

I

.. * True incidence 8 , degrees

1 " 1 ' " . ' 1 . ' ~ ' ' ~ 3 30 60 90 120 I 50 180

1

-. .. .. .. .. . . .

FIGURE 7. Coefficient C, for N.R.C. radio telescope.

The results of several wind tunnel investigations are available for this purpose. Some results not previously published are shown in FIGURE 11 : tests on woven wire mesh were carried out at the N.P.L.7 and at Bristol University! In each case the mesh was 0.044" dia wires at 0.31" centers, giving a shadow area fraction of 0.27, and the wind speed was about 30 ft./sec. The two results are indis- tinguishable.

The maximum value of the drag coefficient, 0.42, corresponds to 1.55 based on the shadow area. This is higher than other sources suggest, but such sources for the most part refer either to wire meshes of lower shadow fraction or expanded metals of greater solidity: at both extremes, isolated wires or a solid plate (of "square" aspect ratio), a value of about 1.2 would be expected. T h e high value

0.34

0.2 CT

O-I 1 0

-0.1

* . a . * .*.*.: . .. .. - 0 .

* . . . .. . . . .

0' 9 . ' ' I ' ' I ' ' I ' ' I ' ' 0 30 60 90 i 20 I50 180

True incidence 'd , dqqmes

FIGURE 8. Coefficient for CT for N.R.C. radio telescope.


0.10-

e Tic, 005-

0

-0.05-

.

.

- . A : . A *: **.. . . = . .. 0 . .

.. I , ' , , , , , , , , , , , ,

- 0 , 30 . 60 90 120 150 180 Incidence 8 , degrees

Note: @ mtusured f- hterscctian o f rhcnirg a x e s .

FIGURE 9. Coefficient e C, for N.R.C. radio telescope. I

D


may also partly be due to twine1 blockage effects in the N.P.L. tests, while at Bristol, where an open jet tunnel was used, the samples were abnormally rough due to poor quality galvanising.

The possibility of overall shape effects was investigated. At the N.P.L. a panel of aspect ratio 1.5 was compared with an approximately square panel. At Bristol University the coefficient at normal incidence was also derived from the pressure drop through a mesh screen extended over the full cross-section of the tunnel. In both cases the previous results were duplicated exactly. Some variation with Reynolds number was noted, the coefficients at 100-150 ft./sec. being some 5 per cent lower than at 30 ft./r;ec.

The coefficients obtained at Bristol for a 36" "square mesh" expanded metal 0.06" thick, shadow fraction 0.33, are also shown. These results have been aver- aged for all orientations of the mesh. The drag coefficient of 0.64 at normal

0 20 40 60 80 90

1 - Wire mesh 0.31" x o.o++~ yalvanised

2 - ExpandeB meid (sgcrore me&) 0.37' x 0 . 0 ~ ~

Angle o f /ncicfencc )k degrees

FIGURE 11. Force coefficients for mesh.


incidence, o r 1.93 on the shadow area, is also high. When this mesh was galvan- ized the edges of the “wires” were significantly smoothed, so that the force coefficients increased only 5 per cent although the shadow fraction rose 12 per cent to 0.37. The coefficients obtained from the pressure drop tests were in this case about 5 per cent higher than the results shown.

The geometry required for application to the paraboloid form is somewhat tedious. FIGURE 12a shows an elemental ring of mesh at radius r from the generating axis. For simplicity of description this is drawn for zero azimuth angle, when the pointing direction and the wind lie in a vertical plane: as previously, however, results expressed in components P, and P,. in the wind/pointing plane, with eccentricity e , are general, varying with angle y only.

The angle of incidence +, defined as the angle between the wind and the normal to the mesh as in FIGURE 11, at any point A is given by:

C o s t , b = C o s ~ # ~ C o s y + S i n 4 S i n y C o s O (4) The drag ( d ) and cross-wind (L) forces on an element a t A, acting in plane

OAC, can then be determined. Consideration of the symmetrically opposite point A together with A leads to the forces shown in FIGURE 126, which lie in the plane OBC. The angle 7 must be evaluated from the expression shown. The forces and moment for the whole dish are then found by arithmetrical summation.

The model tests for the NRC. 150’ radio telescope included investigation of the effect of the wire mesh, which made up 36 per cent of the total reflector area. Forces were measured for the critical attitudes with the mesh removed and with it plated over, for comparison with the basic form. F~GURE 13 shows the moment about the steering axes, the full line representing the values for the wire mesh alone calculated by the above method, the plotted points being derived from the difference of the measured values with and without the mesh. No allowance was

tan

.. 2d

. S u m of Forces points A‘& A

r 9 = n sin 8

sin 8 cot# - C O S ~ C O S ~ 12 (a). €/ementat ring of mesh tanr;, =

FIGURE 12. Forces on wire mesh paraboloid.


FIGURE 13. Effect of wire mesh and of plating for outer part of dish; N.R.C. radio telescope.

made for shielding by the mesh, which would appear to be important near the feathered position. Also shown (broken line) are the theoretical and experimental effects of adding plating from 12.0' to 150' diameter.

Non-uniform Wind Loading

The analysis of wind loading under gust action normally requires consideration of two effects, the strictly aerodynamic analysis of pressures in nonuniform, non- steady, flow and of the mechanical response of the structure. It has already been stated that the natural frequency (no) achieved in steerable antennas in fulfil- ment of the functional requirement for stiffness is sufficiently high that dynamic magnification of loading is unlikely to be significant. This conclusion is based on the value of the parameter V/n,D, which expresses the gust size corresponding to the natural frequency as a fraction of the diameter of the antenna. For the typical example D = 150', n = 2cps., V = 100 fps., V/n,D = 0.33, and for such small gust size the net effect integrated over the whole antenna would be slight.

There is very little information on the remaining, aerodynamic, aspect of the problem. Some conclusions can be drawn if distinction can be made between the effects of the variations of the f low with time and those of variation with position across the dish. The former can perhaps be neglected, although the evidence for some other bluff bodies is that tht: force coefficients for fluctuating flow are larger than for steady flow. The effect of the nonuniformity across the region of flow affecting the antenna can be estiimated on the assumption that the pressure on the structure at any point is proportional to the velocity at that point only. As previously discussed this would appear to be closely satisfied for mesh-type


antennas, up to a mesh having a maximum drag coefficient of about 0.5, but tests described later suggest this may give a n over-estimate in the important case of the moment about a diameter when face-on to the wind.

Prior to the publication of mathematical description of the wind structure to facilitate estimation of the integral force or moment by this assumption, the author had used a much simplified arithmetic analysis based on actual simultane- ous anenometer readings. Results were provided by Sherlock showing the speed at stations at 25‘ intervals on a vertical mast: the same speed was assumed to apply across the full width of the antenna at the corresponding level, and Simpsons rule used for evaluation of the force and moment for each time interval. The design value was obtained from a statistical analysis, as the mean plus a multiple of the standard deviation, the multiplying factor being defined so that the same method applied to the individual wind speed records would give the value for the gust factor at a point recommended by Sherlock. Since rapid response anemometers had been used, a high value of gust factor at a point was appropriate, and the value chosen was 1.8 on speed or 3.25 on pressure as for a half-second duration. The short response time of the structure was also taken into consideration. The multiple of the standard deviation required varied widely according to the records used, but consistent predictions were thus obtained for the maximum loading.

For a 150’ diameter antenna facing the wind the maximum predicted force was equivalent to 0.87 times the value caused by uniform application of the peak gust pressure. The maximum moment about a diameter was equivalent to the peak gust pressure acting on one-half. combined with the steady pressure plus 20 per cent of the gust pressure increase on the remaining half. Alternatively 55 per cent of the maximum wind pressure could be considered as a live load to be applied only where its action is unfavorable. These results exclude wind gradient. The method gives no information, however, on the actual force co- existent with the maximum moment about the diameter, and to find the moment about any other axis a new set of instantaneous values must be analyzed.

The gust action when pointing vertically will depend approximately on the average speed over the structure and also on the vertical components of gustiness (V,, e.g.) . The greatest moment would be expected with upwards inclined wind on the leading edge and downwards over the trailing edge, which can be compared in aerodynamic effect with decreasing the focal length of the dish. The inverse relationship between the transverse force coefficient and the focal length implies that:

(5) P, D V analogous to the usual linearisation of gust action in the direction of the steady wind :

SPT = 4f v, -

The effect of vertical gustiness may thus be about 20 per cent of the horizontal, so that allowing for lack of correlation between the two, the maximum moment in the feathered altitude would be little different from the value under uniform peak gust speed. For a plated reflector this is about 160 per cent of the moment in the face-on position. Other gust effects can be considered similarly.

Another aspect of nonuniform flow is the variation of the steady wind speed with height. The mesh type antenna can be analyzed as before. The effect is important in increasing the moment in the face wind position.

To study the behavior of a plated dish, the N.R.C. radio telescope model was

Wyatt : Paraboloid Aerodynamics 237

tested with a velocity profile corresponding roughly to a power law index of 0.18. The forces and the total moment were found to depend principally on the speed at the highest point of the dish., and did not agree with the assumption of pressure proportional to the square of the speed at any level. In the nearest position to facing the wind (a = 0' B = 80") the addition to the moment coefficient caused by the wind gradient, after deducting a calculated allowance for the effect of the mesh, was equivalent to an eccentricity of the axial force of 0.012D, less than half the value that would be predicted. Pointing downwind (a = 180' /3 = 80') the wind gradient caused no change in the moment.

Tests with the wind incident obliquely onto the rear of the paraboloid showed, however, that although there was little change in the total force and moment coefficients referred to the speed at the top of the dish, the transverse force com- ponent was rotated towards the vertical out of the plane containing the pointing and wind directions. This increased the moment about the altitude axis and de- creased the azimuth moment: the variation of the moments with zenith angle at constant azimuth angle 135O is shown by FIGURE 14. The maximum displacement of the transverse force was about 20°.

- C 0.08-

0.04- -

0 20 40 60 00 90 Z e n i t h angle , degrees.

Azimuth angle 1350 Contkuous lines show values in u n i f r m flw. PIoHFed poink show vduss with grod&n/. hdex 048.

FIGURE 14. Effect of variation of wind speed with height; N.R.C. radio telescope.


Acknowledgments

The author is grateful to Gilbert Roberts, Esq., Freeman, Fox and Partners; and to the Director of the National Physical Laboratory, for permission to pub- lish this paper.

References 1. DAVENPORT, A. 0. 1961. The application of statistical concepts to the wind loading of

2. WHITBREAD, R. E. 1963. Model simulation of wind effects on structures. Conference on

3. HUSBAND, H. C. 4. National Physical Laboratory. Unpublished report. NPL. Aero. 275. 5. National Physical Laboratory. Unpublished report. NPL. Aero. 250. 6. WHITBREAD, R. E. N.P.L. Aero Report. 1023. (Unpublished) 7. National Physical Laboratory. Unpublished report. NPL. Aero. 354. 8. TINKLER, J. Lift and drag of round wire and expanded metal gauzes. (Unpublished

Discussion of the Paper

structures. Proc. I.C.E. 19: 449.

the wind effects on structures. N.P.L. London. 1958. The Jodrell Bank radio telescope. Proc. I.C.E. 9: 65.

report.)

E. KOSKO (National Research Council of Canada): On what assumptions were the corrections for wind-tunnel blockage calculated?

T. A. WYATT (Department of Civil Engineering, Imperial College, London, England) : The effects of blockage of the wind tunnel caused by the models used had been studied, and the method of correction proposed by Maskell had been shown to be generally satisfactory. The corrections made amounted to 20 per cent in some of the results quoted.

The formulae for correction can be found in the reference given in the paper. For application to a parabolic antenna the formula becomes:

measured force - 1 4- 5 R (Cdm - ci> -

true force where Cdln = drag coefficient based on forces as measured

Ci = - C,2 theoretical correction for induced drag

The induced drag term is normally fairly small, but becomes large at around 50' incidence. Tests on three geometrically similar dish models (depth equal to 0.13 diameter) of 13 per cent, 6% per cent, 3% per cent of the tunnel area respectively suggest that the induced drag term is too large: for the examples tested close agreement would have been obtained by taking Ci as 60 per cent of the theoretical value quoted above.

1 4

R = ratio of reference area of model to tunnel area

the aerodynamics of shallow paraboloid antennas

Documents