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The use of variable body forces to control the depth of towed submersibles
Ocean Engng. Vol. 4, pp. 57-74. Pergamon Press 1977. Printed in Great Britain
THE USE OF VARIABLE BODY FORCES TO C O N T R O L THE DEPTH OF TOWED SUBMERSIBLES P. J. WINGHAM School of Engineering, The University of Bath, England Abstract--The relationship between the depth and trail attained by a towed submerged vehicle and the magnitude and direction of the forces on it has been examined theoretically, for cases involving faired cables. It is shown that when the body force is varied the body in equilibrium moves on an almost circular path as if the cable were straight, and that depth changes depend primarily upon the change in the component of the body force that acts at right angles to the line joining the top and bottom of the cable. Examples are given which demonstrate that for bodies towed at large depression angles the use of water brakes is more effective in changing depth than the use of lifting wings. The data is presented so that a rapid preliminary assessment can be made of any proposed system to obtain the best combination of speed and scope for achieving a given depth with a body having known lift and drag coefficients.
D L S T W
NOTATION thickness of faired cable, m chord length of cable (Fig. 3), m depth of towed body, m drag force per unit length of faired cable, N/m length measured along cable, m scope (overall length) of cable, m tension coefficient T/r s sec (~) towing speed, m/s weight in water per unit length of cable, N/m coordinates of cable catenary, see Fig. 3, m trail of towed body, m drag (horizontal) force acting on body, N lift (vertical) force generated by body, N maximum cross section area of towed body, m ~ tension in cable, N weight in water of towed body, N
c~
body lift coefficient =
CL,
body drag coefficient -
Cw
body weight coefficient =
c~
faired cable drag coefficient per unit length of cable, based upon fairing thickness body force angle to horizontal, + ve downwards, deg depression angle of cable chord below surface, deg critical angle for cable, deg relative depression angle, deg relative body force angle, deg parametric angle of catenary, deg density of water, kg/m a 57
b ¢
d r
1 S t W X, y 2
12
Y xg y-~g ct -~1/ 0
L
½pv2S D ½pv2S W ½pv'~S
58 Subscripts 0 1
2 C
P.J. WINGriAM condition at (virtual) origin of cable condition at bottom point of cable condition at top point of cable (at zero depth) critical condition of cable
1. INTRODUCTION THERE ARE numerous applications of towed underwater sampling and sonar vehicles where control of their depth would offer considerable advantages, either for selecting given depth bands or maintaining a fixed distance above the sea bed. The depth can be varied either by driving the winch holding the top of the cable or by varying the hydrodynamic force generated by the towed body. The first method requires a sophisticated winching system that is unlikely to be available to many operators, and which becomes increasingly difficult to design as the towing speed is increased. The second method is then the only realistic alternative. The aim of this report is to examine the equilibrium conditions resulting from towed bodies able to generate hydrodynamic forces in a vertical plane by means of moving wings or fins. Attention is confined to cases where faired cables should be used, and these are assumed to have the properties of Eames's "heavy-fine" cable (Ref. 1). By defining the cable configuration relative to its critical direction instead of to the vertical, and by converting cable tensions to speed-dependent non-dimensional coefficients, a rapid means is achieved for assessing the requirements of any proposed variable depth system, particularly in the preliminary design stage. The mathematics of the cable in steady motion is presented first. This is followed by an analysis of the effect on the depth of an arbitrary body force, Finally, the use of variable lift or drag forces is investigated as alternative methods for changing the depth. 2. THE HEAVY-FINE CABLE This particular cable is one of several models presented and discussed by Eames (Ref. 1). It is recommended for applications where low cable drag is essential and where cable weight may not be ignored• It is characterised by the assumption that the hydrodynamic force on an element of faired cable in motion is proportional to the length of the element and acts in the direction of relative motion, but is independent of the inclination of the element to this direction. The drag force per unit length of faired cable is proportional to the square of the speed, the factor depending upon the shape and size of the fairing. This assumption, which ignores form drag, is reasonable for such low drag coefficients (of the order of 0.1) but is hydrodynamically unrealistic where the angle between an element and the direction of motion becomes less than about 40 degrees. The effect of such reduced angles is not, however, to change the direction of the resulting drag force but merely to reduce its magnitude, and for the applications we are to consider, the effects will be small. • 1 The catenary configuration
Figure 1 shows a loop of heavy-fine cable being moved by end tensions T at constant speed, v, through water. Under these assumptions the drag force acting on each equal increment of length ds will be the same and will act horizontally, and the weight force on equal elements will, of course, also be the same and will act vertically, The figure shows
The use of variable body forces to control the depth of towed submersibles
59
T ~V
.~_~Jwds
~
/I
/~,
Critical
ooot*
T V
/ FIG. 1.
Catenary form adopted by loop of 'heavy-fine' cable towed at speed v.
these forces as rds and wds, respectively, where r is the drag force per unit length at speed v and w is the cable weight in water per unit length. The cable will adopt the form of a common catenary, but with its axis of symmetry inclined to be parallel to the unique line of action of the resultant force on each element. This inclination, shown as angle ~b to the horizontal will be the critical angle for the given cable at the given speed. It is the angle adopted by the cable if towed by its top point, at the same speed, with no constraints applied to any other part of its length. Figure 2 shows (as the continuous straight line) a cable in this critical condition, supported only by tension Tc at the top.
7" ///t/
j/
//
r~
//
Z_-, i/
//
~"I
// /
Fla. 2. Cable in and near critical configuration. Since the cable is straight we have r d s . sin (~) ---- w d s . cos (qb)
(1)
so that tan (+) = w/r
(2)
60
P.J. WINGHAM
and the resultant force on each element dr is given by: (w~ + r")~ = r see (+~.
from (2) The magnitude of the top tension Tc in this critical case is therefore: T, .... s r sec (+)
(3)
where s is the overall length or scope of the cable.
2.2 The tension coefficient We shall find it convenient to define a tension coefficient t given by: T
t =~
s r sec
(+)
(4)
that is, the ratio of the tension at a point divided by the top tension of the total length of submerged cable in the critical condition at speed v.
2.3 Near-critical conditions We note that in the simple case of an unconstrained cable the value of the tension coefficient at the top is given by: to ~
1
(5)
and t will vary linearly from 0 at the bottom to unity at the top. If, now, an additional force/'1 is applied at the lower end, with its line of action along the line of the cable, the tension at all points will increase by T1 and the tension coefficient will vary linearly from tl to (1 q- tl) along the cable from the lower end. If the angle of the bottom force 7"1is changed the cable will turn upwards or downwards depending upon the direction of the turning moment about the top point created by the transverse component of 7"1. This is also illustrated in Fig. 2.
2.4 Cable equations Figure 3 defines the axes and notation governing the cable configuration in the general case. Axes 0x, 0y have their origin at the central point of the basic catenary, so that 0x is at the critical angle + to the horizontal. The cable moves from left to righL The equation for the catenary will be: x/[q- 1 =cosh(y/O
(6)
The use of variable body forces to control the depth of towed submersibles
61
/
x
7- ,~\ 7-
/~/"
~-Retotive depression
~'Y~:~
'Lwater
/ Y\
D~t, /
~
.x
rroil z
z/
OTo~__~ jCritical ____angle
Cotenary Fto. 3. Axis system and notation for heavy-fine cable with attached body.
where l i s a characteristic length defined as:
"l---- To/r sec (+)
(7)
where T o is the tension at the origin and r sec,~ is the resultant force per unit length at this speed (equation 2). If Xx, Yl is a point on the catenary, at a distance along the curve ll from the origin, and where the curve makes angle 01 with axis 0y we can show from equation (6) that:
Xx/i-= sec(00 yl/7
=
Ix~7=
-- 1,
log e (tan(01) q- sec (01)),
tan(01),
T1 = Tosec (01).
(8)
(9)
(10) (11)
It is most useful to calculate all lengths as proportions of the cable scope s. This follows from the adoption of the tension coefficient t already defined (equation 4). because
t o --
To -- ~" s r sec(d?) s
(12)
62
P.J. WINGHAM
from equation (7), so that equations (8), (9), (11) become: x~/s = to(Sec(00 -- 1),
i13)
y~/s = t o logs (tan(01) + sec (00),
(14)
tl/t o = sec(01).
(I 5)
N o w letting xl, Yx define the lower end of the cable and xz, y~ the upper end at the water surface we can, by working from the origin, define the whole cable non-dimensionally, and derive the chord length c joining the cable ends and its angle 7 below the horizontal. We have (x2 - - x l ) / s =- AX/S = to(Sec(O~) -- sec(01)),
[tan(02) + sec(O,)~
(Y2 - - y l ) / s = A y / s = t01ogo \ t a n ( 0 0 + ~ ] '
t2/t x = sec(0~)/sec(00,
(16)
(17)
(18)
thus we m a y derive:
(19)
and
3' = ~ + t a n - !
or
7-- ~ =tan-l
<)
(20)
giving the trail as z / s = (c/s) cos(7)
(21)
and the depth as dis = (c/s) sin('/).
(22)
W e m a y also note that (from equations 10 and 11) (!~ - - l l ) / i = t a n ( O , ) -- tan(O0 = s / i
(23)
t o = t/(tan(O~) - - tan(O0)
(24)
so that tl = sec(OO/(tan(Od -- tan(01)).
(25)
The use of variable body forces to control the depth of towed submersibles
63
2.5 Application o f cable equations Consider a practical case where a known faired cable of scope s is towed at speed v. If the cable has maximum thickness b, and a drag coefficient Ca per unit length based upon this thickness, we can derive (27)
r = Cab½pv"
and
,) = tan-'(w/r)
from equation (2) where w is the cable weight in water per unit length. If the body force comprises lift, drag and weight forces, the former two defined as acting normal to, and rearwards along the direction of motion, respectively, we can derive T 1 and oc from T12 :
(W-
-ta
(28)
L) 2 + D"
X
Cable
~"/
Lift L
r~
¢--___W-L
Wei( ht W
FIG. 4.
Derivation of body force and body force angle.
as shown in Fig. 4, and determine t1
and
T1/s r sec(+)
01 = 9 0 - -
(oc - - t~)
as seen from Fig. 3. 02 is then given by equation (25) as 03
tan-l(sec(00/t~ + tan(01)).
(30) (31)
64
P.J. WIN(}HAM
Thus 7, c/s, z/s and dis are readily calculated from t~ and 0% using equations (15)-(22). In addition we may calculated T~, the tension at the top of the cable using T2 T1 see 0dsec 01 from equation (18). Summarizing, we see that, at given speed, the depth and trail depend on the body force coefficient and its angle to the horizontal. 2.6 Application to upward curving cables If oc < ~ the cable will be above the critical line, as in the upper line of Fig, 2. Because axis 0x is symmetrical with respect to the catenary (Fig. 3) solutions calculated at (oc -- ~) X (say) are applicable to (oc -- ~) . . . . X, the only effect being to change the sign of the resulting value of (7 -- ~) given by equation (20). We can show that if oc ;> ,~ 01 :=~
90 -- (oc
,~)
but ifoc <- + 01 = 90 .... ('# - oc). Thus a set of solutions calculated with
0,
+1,
over a range of values of q, will produce certain magnitudes for the resulting (7 .... +) which will be positive or negative according to whether (oc - +) is positive or negative. For this reason we treat the angle (7 -- ~) as a variable called the relative depression angle of the cable (see Fig. 3) and the angle (oc -- ~) as a variable called the relative body force angle. The two curved cables illustrated in Fig. 2 are examples of this, being determined from one calculation.
3. THE EFFECT ON DEPTH OF D I R E C T I O N AND SIZE OF BODY FORCE
3.1 Qualitative consideration Figure 5 shows a cable of fixed scope s in three positions of equilibrium under the action of an increasing body force at a fixed direction oc. The speed is constant so that ~ the critical angle is constant. The body force is expressed non-dimensionally as t~ (equation (4)). With finite tx the cable lower end will be at some intermediate position which may be determined from the chord length c and its angle 7 below the horizontal. As tx tends to zero the cable will be straight, along AF, at the critical angle, so that 7 -- ~ and c -- s. As tx tends to infinity the cable will be pulled straight along the line AC, giving 7 -- oc and, again, c -- s. Thus for 0 < q < ~ the relative depression angle (7 -- ~) will vary from 0 to ( ~ ~) and the cable will be within angle FAC.
The use of variable body forces to control the depth of towed submersibles
65
/ / /
Trail
/
A ~ ":,, 7"
\" ),,B tl >
0 if\t :/ """ " . . \
.
/
Depth
FIG. 5. Effect on cable of body force coefficient varying in magnilude but not direction, As t~ increases from zero the force making the cable swing down must be the component of tl acting normal to the chord line. This normal component is illustrated at B. Its magnitude is tlsin (oc -- y). If we assume as an approximation that the resultant overall weight and drag forces on the curved cable AB act at the mid-point of the chord, and put these forces in a coefficient form compatible with q, we have weight-force coefficient = ws/r sec (+) s -- sin (+) drag-force coefficient = rs/r sec ('~) s = cos (+) as shown in the figure. Taking moments about the top point A gives ta sin (~: -- y) = ~ sin (y -- +).
(32)
This simple equation suggests that there may be a close correlation between the relative cable depression angle and the component of t~ acting normal to the chord. Exact calculations prove this to be true. 3.2
Exact calculations
Figure 6 illustrates a set of cable configurations derived for a constant scope cable with increasing b o d y force coefficient tl at a constant relative body force angle of 45L Values o f tl and the ratio c/s are tabulated for each of the six points together with other relevant data. It is seen that c/s is never less than 98.8 ~ and that most o f the available range of relative depression angle is achieved with tl varying from 0 to about 2. The sector A O F and the curves contained in it could have been drawn at any value of without invalidating the solution. In addition the sector could be rotated about its angle bisector to display upward-curving cables resulting from have (oc -- ,~) of the same
P.
66
WINGHAM
J,
Trail
~
F Position
B_
{e-
I
~
Depth
~)
45" .
.
.
.
' .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
F .
--T-I--45*d--6~75;TS7~I £~5'T F
FIG. 6.
]
45 °
oo
i
45~ I 45°
I
. . .l. . . .[
45 °
J
.
.
.
.
.
.
0.2£~ 0 :~53
....
Constant scope cables with varying body force coefficient at constant 45° angle to critical direction. (Drawn with ~ ~:: 30°, a = 75°).
magnitude but negative. The corresponding values of (y ..... ,~) at each t~ would then differ only in sign. Figure 7 shows a similar set but with {oc ~) .... 90% A l t h o u g h unrealistic in the sense that this b o d y force has a forward (propulsive) c o m p o n e n t these curves represent the extreme condition. Even here we see that the chord length is never less than -o4 o/~oo f the scope. Taken together Figs. 6 and 7 show that, if tl is in the range 0-2 (say), the most practical part o f the relative depression angle range can be achieved whatever the value of the body force angle. 3.3 Correlation of relative depression angle with body Jbrce normal component Figures 6 and 7 provide the data for Fig. 8 which correlates the relative depression angle with the b o d y force coefficient normal component. The upper continuous line is for (oc -- qb) = 90 ° from Fig. 7 and the lower for (cc -- +) = 45 ° from Fig. 6. The lower dashed line is equation (32) which becomes exact when y :, ~ . The b a n a n a shaped envelope, between the t o p m o s t line and the b o t t o m line, contains the whole family of curves generated by varying tl at constant values of (oc -~ ,~) between 0 and 90 °. Each curve of the family starts at the origin and terminates on the line of equation (32) at the condition & = ~3. The fact that the normal c o m p o n e n t has a finite value when & = oo and ~ -- y is due to a singularity at these end points.
The use of variable body forces to control the depth of towed submersibles / /
/~"
4" ve trail
--
Vetroi f
i
F Depth
Position
(e-~/)
fl
Oi I 02
90 °
0
O° O*
A
FIG. 7.
cls
()"-~):
I
0°
0
13.0 °
0.061
90 °
8 6 . 4 * 0.964
fl sin ( a - y )
/7
90 °
1/16
C
90 °
i/4
0°
76.0 ° 0.940
33.8 a
0.208
D
90 °
I
0 °
4 5 . 0 ~ 0.974
6 4 . 8 ~!
0.425
E
90 °
4
0 ° ~L 1 4 ° °
82.9 °
0.494
F
90 °
oo
0°
~
o°
0.997
I
,
90°
o.soo
Constant scope cables with varying body force coefficient at constant 90 ° angle to critical direction. (Drawn with t¢ = 30 °, c~ = 120°.)
/
90 ° 80*
- ~z = 90 ° from Fig. 7 ~
i
;x, 70 °
__
/I
/~////'
60 ° c o
50°
--
,~ from
~J 40 ° L
# ~ o
30° 20 °
o2
_ " i
/ 7~7
/
\
E.quot,on
(32)
//11,1,1,1,1,1,1,1,1~ ///
-~I~$.
/~,i'// //...,~'/; /
/ / j /
~./x//
Minimum c/s =0.94 here ~
I
i
Fig. 6 \
\
,~ Working opprox,mohon
I0 o
/ 0
i
L
OI
02 f[
FIG.
8.
__ I
0.5
_ _ . J j~~
04
0.5
sin(el- T )
"Banana curves" relating relative depression angle to normal component of body force coefficient.
67
68
P.J. WINGHAM
Since it is unlikely that any real problems will involve (oc --. +) ::: 9 0 or "z. - 7, we m a y adopt a median straight line approximation that is very useful for preliminary design purposes. Such a line is shown on Fig. 8. It corresponds to (y - '5) .... 140qsin(oc
-¥).
(33)
This approximation is valid for [~: ..... Y I > 90:~ and applies to both upward and downward-curving cables. Figure 9 is based u p o n it, and gives the relative depression angle as a function o f tx at constant values o f relative b o d y force angle. G~)"
i
..~Q
q~\"
c i?o ,
m
'
b,f)~
4C, ~
/
/~"~'~"
a~
40 ~ "{3
50 ~
0
r
20"
i
F3° i
If~
i
......
................. !~
i.
02
•
v
oG
04
Body
~1~
force coefficient,
ic,
i;2
',4
!,,
t
FiG. 9. Approximate variation of relative depression angle with magnitude and directnon of body force Coefficient. 4. S T R A I G H T - C A B L E M O D E L These results show that we can, with only small error, treat the cable as if it were a straight line o f length equal to the cable scope and pivoting about its top point with its relative depression angle given by equation (33), and involving only the value of tl and the relative b o d y force angle ( ~ -- ,~). This is evident because we can write this equation as: (?
-
-
9) --= 140 tl sin [(oc ..... ,.)) -- (V .... ,~)1
so that, for any known value of (oc -- ,5) we may obtain (y iteration.
(34)
+) by graphical means or by
4.1 Application to practical design problems Equation 33 enables us to determine the depth that can be achieved at any speed with any scope, once the body force and its direction have been determined (Fig; 4) and converted to tl and (oc -- +). It is most convenient to work at constant speed.
The use of variable body forces to control the depth of towed submersibles
69
If a graphical approach is desired one can use Fig. 9 to obtain (7 -- +) and determine the depth and trail from d -- s sin (7)
(35)
z = s cos (7)
(36)
Alternative values of scope at the given speed merely give alternative values of t~ (equation (4)), and for the same body the value of (~: -- +) will be unchanged. Thus the effects of scope changes are readily assessed. Another advantage of the concept of the tension coefficient is that, for given t~ and (oc -- +) the tension coefficient at the top of the cable, &, is also specified. Figure 10 which is derived from equations (25) and (18) shows the exact relation between ta, t2 and (~: -- '+). Thus the value of the maximum cable tension can also be found from tl and (~: -- +). Conversely, if the top tension is limited by cable strength, giving a known t2, Fig. 10 may be used to give the allowable value of tl that may be applied by the body at the given (cc -- ,~). The maximum depth that is achievable with a given top tension may thus be found by trial, using values of t2 and t~ related to successive changes in scope.
3I
//,
O°(C r i t i c a l
30 ° 40 ° 5 0 ° I a - d/I 60 ° deg, 70 ° 80 ° 90 °
u
~ o
o
i Bottom tension coefficient,
F[c. 10.
fl
Exact relation between tension coefficient at top of cable and size and direction of tension coefficient at the bottom.
The "straight-cable" model is intended primarily as a rapid means for assessing the preliminary requirements of a system, Derived values of depth, for example, will be within 5 or 6 % of exact values, except where one is operating at points on Fig. 8 that are demonstrably close to the boundaries of the envelope, when the errors can double. Bearing in mind the many other uncertainties that would exist in the initial phases of a design, such tolerances are not excessive. Calculated changes of depth, due to small variations of body force, would be acceptably accurate.
70
P. J, WINGHAM 5. C H O I C E
OF SYSTEM
FOR CONTROLLING
DEPTH
This model of the body-cable system illustrates a very important point. Because the depression angle of the 'straight' cable depends mainly on the component of body force normal to the cable the most efficient means of changing depth is by providing an additional body force in that direction. Thus, for bodies towed at large depression angles depth changes are most efficiently produced by changing the drag force on the body e.g. by some form of water brake, but for bodies towed at shallow depression angles a change in vertical body force is most efficient, as achieved by using a variable incidence lifting wing. Figure 11 illustrates this situation. ..... /1. . . . . /'
\ FiG. 11.
\
Showing desired direction of additional body force needed to change depth.
The choice of lifting versus braking bodies may also be influenced by the requirement governing the attitude of the body. The use of wings implies a change of body attitude when the lift is varied and although it is possible (see Ref. 2) to gear the pitch control to the moving wings and ensure that in equilibrium the body attitude is constant, there will always be a perturbation in attitude when the wings are being deflected. The use of brakes, which is favoured for changing depth at large depression angles, eliminates this problem but produces a system where the depth may be more sensitive to towing speed variations. Theoretically it would be possible to mount the brakes on a spring system so that speed increases will tend to reduce their deflection and thus correct for the drag increase that would otherwise occur. A further point is that drag brakes are hydrodynamically noisier than moving wings, and may generate eddies at frequencies near to the acoustic reception frequencies of any sonar equipment employed. 5.1 Available body forces The water forces acting on any body are normally specified by the aerodynamic lift and drag coefficients defined as cL -
I4½pv~S
Co ~ D/~pvZS
(37) (38)
where L and D are the lift and drag forces normal to, and (back) along the direction of motion, and S is the body cross-sectional area on which the coefficients are based.
The use of variable body forces to control the depth of towed submersibles
71
Winged bodies. Reference (2) provi0es an example of a streamlined, stable, movingwing configuration designed to achieve a useful range of lift force with a realistic wing span (about 4 body diameters). It shows that the Cc may be varied from 0 to ± 2.2 before wing stall occurs, and that Co varies linearly with C [ ' from 0.2 at CL = 0 to 0.5 at Ct ~ = 5.0. Braked bodies. Alternatively, if some form of water brake were substituted for the wings one may expect to achieve a drag coefficient varying from 0.2 to about 4, depending upon the deflection of the brakes. CL would remain zero. 6.2. Determination oft1 These CL and Co values provide a basis for preliminary design work but it is necessary also to consider the additional contribution of body weight to the resultant body force. If the weight of the vehicle is converted into an 'aerodynamic' coefficient Cw, defined as Cw = W/½ov~S
(39)
which, unlike the values of CL and Co, will vary with speed, we can derive t I as a function of CL, CD and Cw and the drag coefficient of the cable fairing. If the cable fairing has a drag coefficient per unit length of Ca based upon its thickness b we have r = Cab½pv~
(27)
giving a revised set of coefficients campatible with tt, viz.
so that t I
CLS, Ct' = ~-7/s sec (+), c:ao
(40)
CoS Co' -- -~-7,/s sec (+), cdo
(41)
CwS. Cw' = -~--;-/ s sec (+), cdo
(42)
[(C w' -- CL') °" @ CD'2]½
and oc = tan
-1
( C w ' - - CL')
Co'
(43)
(44)
We note that, if Cc =: Co -- 0, tl = Cw' and oc = 90 °. This treatment has the advantage that, at given values of CL and Co (the basic aerodynamic properties of the body configuration) the values of tl and oc required for all further calculations depend only upon Cw' and the values of d? and the scope. Thus one can readily predict the effect that speed or weight change will have upon q.
72
P.J. WINGHAM
6. EXAMPLES OF LIFT AND DRAG FORCES USED TO VARY DEPTH The 'straight cable' model has been used to calculate the effect on the equilibrium depth of applying either lift or drag forces to a given size of body (S -----0.014 m 2) towed by a given faired cable (b = 2 cm, Ca = 0.1) at speeds of 3, 6 and 9 knots. The cable scope is constant at 300 m and the body weight in water is taken in turn as ½, 1 and 2 kN. These values are chosen to cover a useful range of the relevant parameters. They have no other significance. The results are shown in Fig. 12. Figure 12(a) and the left hand column of (b) (c) and (d) relate to the situation with the streamlined body having neither wings nor brakes, i.e. CL CD -: O. (The effect of putting Co = 0.2, the correct value, is negligible.) Figure 12(a) shows position of the cable chords at each speed and weight. The critical cable angles are 65.8, 29.1 and 13.9 deg at speeds 3, 6, 9 knots, respectively, but are not marked on the figure to avoid confusion. In one case (6 knots, W = 1) the actual cable configuration is shown as a broken line. The figures show the effects of applying (1) a realistic maximum hydrodynamic depressive force assuming wings that give Ct~ 2.25 with C1) 0.5, (2) a maximum upward force given by CL -: +2.25 with Co :~ 0.5, and (3) a maximum rearward force given by CD :-:: 4 with CL = 0. Figure 12(b) shows the resulting equilibrium depth plotted against speed, at each weight, for these alternative systems. Figure 12(c) shows these depth changes expressed as percentages of the depth of the basic body. We note first that at the lowest speed there is negligible effect of either wings or brakes, due to the low dynamic presstire, but at 9 knots it is possible to change the depth by about 20% using positive and negative lift. As far as the lifting system is concerned we see how small is the depth response to downward lift compared to that for upward. This is due to the fact that a negative deflection of the wings produces a downward lift force plus a backward drag force, the resultant of which tends to act more along the cable chord than normal to it, and is thus less effective in swinging the cable round. The relative effects of upward lift and rearward drag forces are explained by noting the cable depression angle associated with each case. For example, at 6 knots at W : 1 a drag force is a better depth changer than lift because the depression angle is large, but as the speed increases to 9 knots the depression angle decreases and the vertical lift force becomes more effective than the horizontal drag force. The falling effectiveness with speed increase, of both the positive lift and the drag systems when W : 0.5, is due to the cable depression angle approaching the critical angle, which nmrks the limit of depth reduction in this case (see Section 4.1). Figure 12(d) illustrates the rate of change of depth with towing speed, a parameter of interest where a close control of depth is required. The depth-speed gradient varies from 0 to 45 m/knot, over the speed and weight range examined. The adverse effect o f drag upon the speed sensitivity, at large depression angle, is evident from the result at 3 knots when W ~ 0.5. These examples provide a background against which to judge other depth-control systems. 7. CONCLUSION It is concluded that, for heavy faired cables, the use of axes aligned with the critical angle leads to cable equations of a relatively simple form. Furthermore, by adopting a tension coefficient embodying towing speed and cable scope the depth achieved by any combination
The use of variable body forces to control the depth of towed submersibles
73
(o) Depth ond troil
m
Trail,
0 300
300
9" Knots
/
•/
0 300
/ I Knots/ / / / ' ~" 91¢~ / /
/
/~ ~
IJ oo 5 VY=O.5kN
/I / I
0
Kn~fU /
I/
/fl
"
IJ oo 5
/~
5
W = I kN
CL
:o
°:°
W=2kN
(b) Depth vs speed
CL :
I00 m
CD
=0
L
CD = j /
I00
i,,,o¢/
200
300
6
(:3
9
Knols
9 5
G
9 3
6 Knots
9
6 Knots
9
6
9
c) Percentage depth change
-2o_,oF.......
_!
.
.
.
.
.
.
.
.
.
.
.
I
% 0
. . . . . . .
6
Knots
9
3
6
9
3
9
3
(d) Depth-speed gradient
~ 2o E
0 3
F r o . 12.
6
Knots
9
3
C
9
3
G
Knots
'Straight cable' models with various body forces at differing weights and speeds.
of cable and body force can be readily found, at any speed, by means of a single equation which is a close approximation of results given by exact treatment. Maximum tensions are also rapidly determined. A consequence of this approximation is that the cable may be regarded as a straight line pivoting about its top point so that the body effectively swings on a radius when the body force is varied. This leads to the important result that when the body force is varied it is only the component of the force normal to the cable chord that determines the angular swing of the cable and hence the change in depth.
74
P.J. WINGHAM
Calculations of the depth changes produced by using either moving wings or water brakes show that, for realistic body configurations, the depth may be varied by up to about 2 0 ~ . In the case of moving-wing systems the effect of induced depressive forces is small compared to the effect of induced upward forces. When large cable depression angles arc employed water brakes are more effective depth changers than moving wings Acknowledgement--The author is grateful for the help and advice of his colleagues, particularly W ( Chesterman and J. F. Henderson. REFERENCES EAMES,M. C. 1967. Steady-state theory of towing cables. Royal Institution of Naval Architecls. I~ND~gSON, J. F. 1974. Wind tunnel tests on a l/3rd scale model of a sea-bed-following research vehicle. University of Bath, School of Engineering, Report No. 292.
THE USE OF VARIABLE BODY FORCES TO C O N T R O L THE DEPTH OF TOWED SUBMERSIBLES P. J. WINGHAM School of Engineering, The University of Bath, England Abstract--The relationship between the depth and trail attained by a towed submerged vehicle and the magnitude and direction of the forces on it has been examined theoretically, for cases involving faired cables. It is shown that when the body force is varied the body in equilibrium moves on an almost circular path as if the cable were straight, and that depth changes depend primarily upon the change in the component of the body force that acts at right angles to the line joining the top and bottom of the cable. Examples are given which demonstrate that for bodies towed at large depression angles the use of water brakes is more effective in changing depth than the use of lifting wings. The data is presented so that a rapid preliminary assessment can be made of any proposed system to obtain the best combination of speed and scope for achieving a given depth with a body having known lift and drag coefficients.
D L S T W
NOTATION thickness of faired cable, m chord length of cable (Fig. 3), m depth of towed body, m drag force per unit length of faired cable, N/m length measured along cable, m scope (overall length) of cable, m tension coefficient T/r s sec (~) towing speed, m/s weight in water per unit length of cable, N/m coordinates of cable catenary, see Fig. 3, m trail of towed body, m drag (horizontal) force acting on body, N lift (vertical) force generated by body, N maximum cross section area of towed body, m ~ tension in cable, N weight in water of towed body, N
c~
body lift coefficient =
CL,
body drag coefficient -
Cw
body weight coefficient =
c~
faired cable drag coefficient per unit length of cable, based upon fairing thickness body force angle to horizontal, + ve downwards, deg depression angle of cable chord below surface, deg critical angle for cable, deg relative depression angle, deg relative body force angle, deg parametric angle of catenary, deg density of water, kg/m a 57
b ¢
d r
1 S t W X, y 2
12
Y xg y-~g ct -~1/ 0
L
½pv2S D ½pv2S W ½pv'~S
58 Subscripts 0 1
2 C
P.J. WINGriAM condition at (virtual) origin of cable condition at bottom point of cable condition at top point of cable (at zero depth) critical condition of cable
1. INTRODUCTION THERE ARE numerous applications of towed underwater sampling and sonar vehicles where control of their depth would offer considerable advantages, either for selecting given depth bands or maintaining a fixed distance above the sea bed. The depth can be varied either by driving the winch holding the top of the cable or by varying the hydrodynamic force generated by the towed body. The first method requires a sophisticated winching system that is unlikely to be available to many operators, and which becomes increasingly difficult to design as the towing speed is increased. The second method is then the only realistic alternative. The aim of this report is to examine the equilibrium conditions resulting from towed bodies able to generate hydrodynamic forces in a vertical plane by means of moving wings or fins. Attention is confined to cases where faired cables should be used, and these are assumed to have the properties of Eames's "heavy-fine" cable (Ref. 1). By defining the cable configuration relative to its critical direction instead of to the vertical, and by converting cable tensions to speed-dependent non-dimensional coefficients, a rapid means is achieved for assessing the requirements of any proposed variable depth system, particularly in the preliminary design stage. The mathematics of the cable in steady motion is presented first. This is followed by an analysis of the effect on the depth of an arbitrary body force, Finally, the use of variable lift or drag forces is investigated as alternative methods for changing the depth. 2. THE HEAVY-FINE CABLE This particular cable is one of several models presented and discussed by Eames (Ref. 1). It is recommended for applications where low cable drag is essential and where cable weight may not be ignored• It is characterised by the assumption that the hydrodynamic force on an element of faired cable in motion is proportional to the length of the element and acts in the direction of relative motion, but is independent of the inclination of the element to this direction. The drag force per unit length of faired cable is proportional to the square of the speed, the factor depending upon the shape and size of the fairing. This assumption, which ignores form drag, is reasonable for such low drag coefficients (of the order of 0.1) but is hydrodynamically unrealistic where the angle between an element and the direction of motion becomes less than about 40 degrees. The effect of such reduced angles is not, however, to change the direction of the resulting drag force but merely to reduce its magnitude, and for the applications we are to consider, the effects will be small. • 1 The catenary configuration
Figure 1 shows a loop of heavy-fine cable being moved by end tensions T at constant speed, v, through water. Under these assumptions the drag force acting on each equal increment of length ds will be the same and will act horizontally, and the weight force on equal elements will, of course, also be the same and will act vertically, The figure shows
The use of variable body forces to control the depth of towed submersibles
59
T ~V
.~_~Jwds
~
/I
/~,
Critical
ooot*
T V
/ FIG. 1.
Catenary form adopted by loop of 'heavy-fine' cable towed at speed v.
these forces as rds and wds, respectively, where r is the drag force per unit length at speed v and w is the cable weight in water per unit length. The cable will adopt the form of a common catenary, but with its axis of symmetry inclined to be parallel to the unique line of action of the resultant force on each element. This inclination, shown as angle ~b to the horizontal will be the critical angle for the given cable at the given speed. It is the angle adopted by the cable if towed by its top point, at the same speed, with no constraints applied to any other part of its length. Figure 2 shows (as the continuous straight line) a cable in this critical condition, supported only by tension Tc at the top.
7" ///t/
j/
//
r~
//
Z_-, i/
//
~"I
// /
Fla. 2. Cable in and near critical configuration. Since the cable is straight we have r d s . sin (~) ---- w d s . cos (qb)
(1)
so that tan (+) = w/r
(2)
60
P.J. WINGHAM
and the resultant force on each element dr is given by: (w~ + r")~ = r see (+~.
from (2) The magnitude of the top tension Tc in this critical case is therefore: T, .... s r sec (+)
(3)
where s is the overall length or scope of the cable.
2.2 The tension coefficient We shall find it convenient to define a tension coefficient t given by: T
t =~
s r sec
(+)
(4)
that is, the ratio of the tension at a point divided by the top tension of the total length of submerged cable in the critical condition at speed v.
2.3 Near-critical conditions We note that in the simple case of an unconstrained cable the value of the tension coefficient at the top is given by: to ~
1
(5)
and t will vary linearly from 0 at the bottom to unity at the top. If, now, an additional force/'1 is applied at the lower end, with its line of action along the line of the cable, the tension at all points will increase by T1 and the tension coefficient will vary linearly from tl to (1 q- tl) along the cable from the lower end. If the angle of the bottom force 7"1is changed the cable will turn upwards or downwards depending upon the direction of the turning moment about the top point created by the transverse component of 7"1. This is also illustrated in Fig. 2.
2.4 Cable equations Figure 3 defines the axes and notation governing the cable configuration in the general case. Axes 0x, 0y have their origin at the central point of the basic catenary, so that 0x is at the critical angle + to the horizontal. The cable moves from left to righL The equation for the catenary will be: x/[q- 1 =cosh(y/O
(6)
The use of variable body forces to control the depth of towed submersibles
61
/
x
7- ,~\ 7-
/~/"
~-Retotive depression
~'Y~:~
'Lwater
/ Y\
D~t, /
~
.x
rroil z
z/
OTo~__~ jCritical ____angle
Cotenary Fto. 3. Axis system and notation for heavy-fine cable with attached body.
where l i s a characteristic length defined as:
"l---- To/r sec (+)
(7)
where T o is the tension at the origin and r sec,~ is the resultant force per unit length at this speed (equation 2). If Xx, Yl is a point on the catenary, at a distance along the curve ll from the origin, and where the curve makes angle 01 with axis 0y we can show from equation (6) that:
Xx/i-= sec(00 yl/7
=
Ix~7=
-- 1,
log e (tan(01) q- sec (01)),
tan(01),
T1 = Tosec (01).
(8)
(9)
(10) (11)
It is most useful to calculate all lengths as proportions of the cable scope s. This follows from the adoption of the tension coefficient t already defined (equation 4). because
t o --
To -- ~" s r sec(d?) s
(12)
62
P.J. WINGHAM
from equation (7), so that equations (8), (9), (11) become: x~/s = to(Sec(00 -- 1),
i13)
y~/s = t o logs (tan(01) + sec (00),
(14)
tl/t o = sec(01).
(I 5)
N o w letting xl, Yx define the lower end of the cable and xz, y~ the upper end at the water surface we can, by working from the origin, define the whole cable non-dimensionally, and derive the chord length c joining the cable ends and its angle 7 below the horizontal. We have (x2 - - x l ) / s =- AX/S = to(Sec(O~) -- sec(01)),
[tan(02) + sec(O,)~
(Y2 - - y l ) / s = A y / s = t01ogo \ t a n ( 0 0 + ~ ] '
t2/t x = sec(0~)/sec(00,
(16)
(17)
(18)
thus we m a y derive:
(19)
and
3' = ~ + t a n - !
or
7-- ~ =tan-l
<)
(20)
giving the trail as z / s = (c/s) cos(7)
(21)
and the depth as dis = (c/s) sin('/).
(22)
W e m a y also note that (from equations 10 and 11) (!~ - - l l ) / i = t a n ( O , ) -- tan(O0 = s / i
(23)
t o = t/(tan(O~) - - tan(O0)
(24)
so that tl = sec(OO/(tan(Od -- tan(01)).
(25)
The use of variable body forces to control the depth of towed submersibles
63
2.5 Application o f cable equations Consider a practical case where a known faired cable of scope s is towed at speed v. If the cable has maximum thickness b, and a drag coefficient Ca per unit length based upon this thickness, we can derive (27)
r = Cab½pv"
and
,) = tan-'(w/r)
from equation (2) where w is the cable weight in water per unit length. If the body force comprises lift, drag and weight forces, the former two defined as acting normal to, and rearwards along the direction of motion, respectively, we can derive T 1 and oc from T12 :
(W-
-ta
(28)
L) 2 + D"
X
Cable
~"/
Lift L
r~
¢--___W-L
Wei( ht W
FIG. 4.
Derivation of body force and body force angle.
as shown in Fig. 4, and determine t1
and
T1/s r sec(+)
01 = 9 0 - -
(oc - - t~)
as seen from Fig. 3. 02 is then given by equation (25) as 03
tan-l(sec(00/t~ + tan(01)).
(30) (31)
64
P.J. WIN(}HAM
Thus 7, c/s, z/s and dis are readily calculated from t~ and 0% using equations (15)-(22). In addition we may calculated T~, the tension at the top of the cable using T2 T1 see 0dsec 01 from equation (18). Summarizing, we see that, at given speed, the depth and trail depend on the body force coefficient and its angle to the horizontal. 2.6 Application to upward curving cables If oc < ~ the cable will be above the critical line, as in the upper line of Fig, 2. Because axis 0x is symmetrical with respect to the catenary (Fig. 3) solutions calculated at (oc -- ~) X (say) are applicable to (oc -- ~) . . . . X, the only effect being to change the sign of the resulting value of (7 -- ~) given by equation (20). We can show that if oc ;> ,~ 01 :=~
90 -- (oc
,~)
but ifoc <- + 01 = 90 .... ('# - oc). Thus a set of solutions calculated with
0,
+1,
over a range of values of q, will produce certain magnitudes for the resulting (7 .... +) which will be positive or negative according to whether (oc - +) is positive or negative. For this reason we treat the angle (7 -- ~) as a variable called the relative depression angle of the cable (see Fig. 3) and the angle (oc -- ~) as a variable called the relative body force angle. The two curved cables illustrated in Fig. 2 are examples of this, being determined from one calculation.
3. THE EFFECT ON DEPTH OF D I R E C T I O N AND SIZE OF BODY FORCE
3.1 Qualitative consideration Figure 5 shows a cable of fixed scope s in three positions of equilibrium under the action of an increasing body force at a fixed direction oc. The speed is constant so that ~ the critical angle is constant. The body force is expressed non-dimensionally as t~ (equation (4)). With finite tx the cable lower end will be at some intermediate position which may be determined from the chord length c and its angle 7 below the horizontal. As tx tends to zero the cable will be straight, along AF, at the critical angle, so that 7 -- ~ and c -- s. As tx tends to infinity the cable will be pulled straight along the line AC, giving 7 -- oc and, again, c -- s. Thus for 0 < q < ~ the relative depression angle (7 -- ~) will vary from 0 to ( ~ ~) and the cable will be within angle FAC.
The use of variable body forces to control the depth of towed submersibles
65
/ / /
Trail
/
A ~ ":,, 7"
\" ),,B tl >
0 if\t :/ """ " . . \
.
/
Depth
FIG. 5. Effect on cable of body force coefficient varying in magnilude but not direction, As t~ increases from zero the force making the cable swing down must be the component of tl acting normal to the chord line. This normal component is illustrated at B. Its magnitude is tlsin (oc -- y). If we assume as an approximation that the resultant overall weight and drag forces on the curved cable AB act at the mid-point of the chord, and put these forces in a coefficient form compatible with q, we have weight-force coefficient = ws/r sec (+) s -- sin (+) drag-force coefficient = rs/r sec ('~) s = cos (+) as shown in the figure. Taking moments about the top point A gives ta sin (~: -- y) = ~ sin (y -- +).
(32)
This simple equation suggests that there may be a close correlation between the relative cable depression angle and the component of t~ acting normal to the chord. Exact calculations prove this to be true. 3.2
Exact calculations
Figure 6 illustrates a set of cable configurations derived for a constant scope cable with increasing b o d y force coefficient tl at a constant relative body force angle of 45L Values o f tl and the ratio c/s are tabulated for each of the six points together with other relevant data. It is seen that c/s is never less than 98.8 ~ and that most o f the available range of relative depression angle is achieved with tl varying from 0 to about 2. The sector A O F and the curves contained in it could have been drawn at any value of without invalidating the solution. In addition the sector could be rotated about its angle bisector to display upward-curving cables resulting from have (oc -- ,~) of the same
P.
66
WINGHAM
J,
Trail
~
F Position
B_
{e-
I
~
Depth
~)
45" .
.
.
.
' .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
F .
--T-I--45*d--6~75;TS7~I £~5'T F
FIG. 6.
]
45 °
oo
i
45~ I 45°
I
. . .l. . . .[
45 °
J
.
.
.
.
.
.
0.2£~ 0 :~53
....
Constant scope cables with varying body force coefficient at constant 45° angle to critical direction. (Drawn with ~ ~:: 30°, a = 75°).
magnitude but negative. The corresponding values of (y ..... ,~) at each t~ would then differ only in sign. Figure 7 shows a similar set but with {oc ~) .... 90% A l t h o u g h unrealistic in the sense that this b o d y force has a forward (propulsive) c o m p o n e n t these curves represent the extreme condition. Even here we see that the chord length is never less than -o4 o/~oo f the scope. Taken together Figs. 6 and 7 show that, if tl is in the range 0-2 (say), the most practical part o f the relative depression angle range can be achieved whatever the value of the body force angle. 3.3 Correlation of relative depression angle with body Jbrce normal component Figures 6 and 7 provide the data for Fig. 8 which correlates the relative depression angle with the b o d y force coefficient normal component. The upper continuous line is for (oc -- qb) = 90 ° from Fig. 7 and the lower for (cc -- +) = 45 ° from Fig. 6. The lower dashed line is equation (32) which becomes exact when y :, ~ . The b a n a n a shaped envelope, between the t o p m o s t line and the b o t t o m line, contains the whole family of curves generated by varying tl at constant values of (oc -~ ,~) between 0 and 90 °. Each curve of the family starts at the origin and terminates on the line of equation (32) at the condition & = ~3. The fact that the normal c o m p o n e n t has a finite value when & = oo and ~ -- y is due to a singularity at these end points.
The use of variable body forces to control the depth of towed submersibles / /
/~"
4" ve trail
--
Vetroi f
i
F Depth
Position
(e-~/)
fl
Oi I 02
90 °
0
O° O*
A
FIG. 7.
cls
()"-~):
I
0°
0
13.0 °
0.061
90 °
8 6 . 4 * 0.964
fl sin ( a - y )
/7
90 °
1/16
C
90 °
i/4
0°
76.0 ° 0.940
33.8 a
0.208
D
90 °
I
0 °
4 5 . 0 ~ 0.974
6 4 . 8 ~!
0.425
E
90 °
4
0 ° ~L 1 4 ° °
82.9 °
0.494
F
90 °
oo
0°
~
o°
0.997
I
,
90°
o.soo
Constant scope cables with varying body force coefficient at constant 90 ° angle to critical direction. (Drawn with t¢ = 30 °, c~ = 120°.)
/
90 ° 80*
- ~z = 90 ° from Fig. 7 ~
i
;x, 70 °
__
/I
/~////'
60 ° c o
50°
--
,~ from
~J 40 ° L
# ~ o
30° 20 °
o2
_ " i
/ 7~7
/
\
E.quot,on
(32)
//11,1,1,1,1,1,1,1,1~ ///
-~I~$.
/~,i'// //...,~'/; /
/ / j /
~./x//
Minimum c/s =0.94 here ~
I
i
Fig. 6 \
\
,~ Working opprox,mohon
I0 o
/ 0
i
L
OI
02 f[
FIG.
8.
__ I
0.5
_ _ . J j~~
04
0.5
sin(el- T )
"Banana curves" relating relative depression angle to normal component of body force coefficient.
67
68
P.J. WINGHAM
Since it is unlikely that any real problems will involve (oc --. +) ::: 9 0 or "z. - 7, we m a y adopt a median straight line approximation that is very useful for preliminary design purposes. Such a line is shown on Fig. 8. It corresponds to (y - '5) .... 140qsin(oc
-¥).
(33)
This approximation is valid for [~: ..... Y I > 90:~ and applies to both upward and downward-curving cables. Figure 9 is based u p o n it, and gives the relative depression angle as a function o f tx at constant values o f relative b o d y force angle. G~)"
i
..~Q
q~\"
c i?o ,
m
'
b,f)~
4C, ~
/
/~"~'~"
a~
40 ~ "{3
50 ~
0
r
20"
i
F3° i
If~
i
......
................. !~
i.
02
•
v
oG
04
Body
~1~
force coefficient,
ic,
i;2
',4
!,,
t
FiG. 9. Approximate variation of relative depression angle with magnitude and directnon of body force Coefficient. 4. S T R A I G H T - C A B L E M O D E L These results show that we can, with only small error, treat the cable as if it were a straight line o f length equal to the cable scope and pivoting about its top point with its relative depression angle given by equation (33), and involving only the value of tl and the relative b o d y force angle ( ~ -- ,~). This is evident because we can write this equation as: (?
-
-
9) --= 140 tl sin [(oc ..... ,.)) -- (V .... ,~)1
so that, for any known value of (oc -- ,5) we may obtain (y iteration.
(34)
+) by graphical means or by
4.1 Application to practical design problems Equation 33 enables us to determine the depth that can be achieved at any speed with any scope, once the body force and its direction have been determined (Fig; 4) and converted to tl and (oc -- +). It is most convenient to work at constant speed.
The use of variable body forces to control the depth of towed submersibles
69
If a graphical approach is desired one can use Fig. 9 to obtain (7 -- +) and determine the depth and trail from d -- s sin (7)
(35)
z = s cos (7)
(36)
Alternative values of scope at the given speed merely give alternative values of t~ (equation (4)), and for the same body the value of (~: -- +) will be unchanged. Thus the effects of scope changes are readily assessed. Another advantage of the concept of the tension coefficient is that, for given t~ and (oc -- +) the tension coefficient at the top of the cable, &, is also specified. Figure 10 which is derived from equations (25) and (18) shows the exact relation between ta, t2 and (~: -- '+). Thus the value of the maximum cable tension can also be found from tl and (~: -- +). Conversely, if the top tension is limited by cable strength, giving a known t2, Fig. 10 may be used to give the allowable value of tl that may be applied by the body at the given (cc -- ,~). The maximum depth that is achievable with a given top tension may thus be found by trial, using values of t2 and t~ related to successive changes in scope.
3I
//,
O°(C r i t i c a l
30 ° 40 ° 5 0 ° I a - d/I 60 ° deg, 70 ° 80 ° 90 °
u
~ o
o
i Bottom tension coefficient,
F[c. 10.
fl
Exact relation between tension coefficient at top of cable and size and direction of tension coefficient at the bottom.
The "straight-cable" model is intended primarily as a rapid means for assessing the preliminary requirements of a system, Derived values of depth, for example, will be within 5 or 6 % of exact values, except where one is operating at points on Fig. 8 that are demonstrably close to the boundaries of the envelope, when the errors can double. Bearing in mind the many other uncertainties that would exist in the initial phases of a design, such tolerances are not excessive. Calculated changes of depth, due to small variations of body force, would be acceptably accurate.
70
P. J, WINGHAM 5. C H O I C E
OF SYSTEM
FOR CONTROLLING
DEPTH
This model of the body-cable system illustrates a very important point. Because the depression angle of the 'straight' cable depends mainly on the component of body force normal to the cable the most efficient means of changing depth is by providing an additional body force in that direction. Thus, for bodies towed at large depression angles depth changes are most efficiently produced by changing the drag force on the body e.g. by some form of water brake, but for bodies towed at shallow depression angles a change in vertical body force is most efficient, as achieved by using a variable incidence lifting wing. Figure 11 illustrates this situation. ..... /1. . . . . /'
\ FiG. 11.
\
Showing desired direction of additional body force needed to change depth.
The choice of lifting versus braking bodies may also be influenced by the requirement governing the attitude of the body. The use of wings implies a change of body attitude when the lift is varied and although it is possible (see Ref. 2) to gear the pitch control to the moving wings and ensure that in equilibrium the body attitude is constant, there will always be a perturbation in attitude when the wings are being deflected. The use of brakes, which is favoured for changing depth at large depression angles, eliminates this problem but produces a system where the depth may be more sensitive to towing speed variations. Theoretically it would be possible to mount the brakes on a spring system so that speed increases will tend to reduce their deflection and thus correct for the drag increase that would otherwise occur. A further point is that drag brakes are hydrodynamically noisier than moving wings, and may generate eddies at frequencies near to the acoustic reception frequencies of any sonar equipment employed. 5.1 Available body forces The water forces acting on any body are normally specified by the aerodynamic lift and drag coefficients defined as cL -
I4½pv~S
Co ~ D/~pvZS
(37) (38)
where L and D are the lift and drag forces normal to, and (back) along the direction of motion, and S is the body cross-sectional area on which the coefficients are based.
The use of variable body forces to control the depth of towed submersibles
71
Winged bodies. Reference (2) provi0es an example of a streamlined, stable, movingwing configuration designed to achieve a useful range of lift force with a realistic wing span (about 4 body diameters). It shows that the Cc may be varied from 0 to ± 2.2 before wing stall occurs, and that Co varies linearly with C [ ' from 0.2 at CL = 0 to 0.5 at Ct ~ = 5.0. Braked bodies. Alternatively, if some form of water brake were substituted for the wings one may expect to achieve a drag coefficient varying from 0.2 to about 4, depending upon the deflection of the brakes. CL would remain zero. 6.2. Determination oft1 These CL and Co values provide a basis for preliminary design work but it is necessary also to consider the additional contribution of body weight to the resultant body force. If the weight of the vehicle is converted into an 'aerodynamic' coefficient Cw, defined as Cw = W/½ov~S
(39)
which, unlike the values of CL and Co, will vary with speed, we can derive t I as a function of CL, CD and Cw and the drag coefficient of the cable fairing. If the cable fairing has a drag coefficient per unit length of Ca based upon its thickness b we have r = Cab½pv~
(27)
giving a revised set of coefficients campatible with tt, viz.
so that t I
CLS, Ct' = ~-7/s sec (+), c:ao
(40)
CoS Co' -- -~-7,/s sec (+), cdo
(41)
CwS. Cw' = -~--;-/ s sec (+), cdo
(42)
[(C w' -- CL') °" @ CD'2]½
and oc = tan
-1
( C w ' - - CL')
Co'
(43)
(44)
We note that, if Cc =: Co -- 0, tl = Cw' and oc = 90 °. This treatment has the advantage that, at given values of CL and Co (the basic aerodynamic properties of the body configuration) the values of tl and oc required for all further calculations depend only upon Cw' and the values of d? and the scope. Thus one can readily predict the effect that speed or weight change will have upon q.
72
P.J. WINGHAM
6. EXAMPLES OF LIFT AND DRAG FORCES USED TO VARY DEPTH The 'straight cable' model has been used to calculate the effect on the equilibrium depth of applying either lift or drag forces to a given size of body (S -----0.014 m 2) towed by a given faired cable (b = 2 cm, Ca = 0.1) at speeds of 3, 6 and 9 knots. The cable scope is constant at 300 m and the body weight in water is taken in turn as ½, 1 and 2 kN. These values are chosen to cover a useful range of the relevant parameters. They have no other significance. The results are shown in Fig. 12. Figure 12(a) and the left hand column of (b) (c) and (d) relate to the situation with the streamlined body having neither wings nor brakes, i.e. CL CD -: O. (The effect of putting Co = 0.2, the correct value, is negligible.) Figure 12(a) shows position of the cable chords at each speed and weight. The critical cable angles are 65.8, 29.1 and 13.9 deg at speeds 3, 6, 9 knots, respectively, but are not marked on the figure to avoid confusion. In one case (6 knots, W = 1) the actual cable configuration is shown as a broken line. The figures show the effects of applying (1) a realistic maximum hydrodynamic depressive force assuming wings that give Ct~ 2.25 with C1) 0.5, (2) a maximum upward force given by CL -: +2.25 with Co :~ 0.5, and (3) a maximum rearward force given by CD :-:: 4 with CL = 0. Figure 12(b) shows the resulting equilibrium depth plotted against speed, at each weight, for these alternative systems. Figure 12(c) shows these depth changes expressed as percentages of the depth of the basic body. We note first that at the lowest speed there is negligible effect of either wings or brakes, due to the low dynamic presstire, but at 9 knots it is possible to change the depth by about 20% using positive and negative lift. As far as the lifting system is concerned we see how small is the depth response to downward lift compared to that for upward. This is due to the fact that a negative deflection of the wings produces a downward lift force plus a backward drag force, the resultant of which tends to act more along the cable chord than normal to it, and is thus less effective in swinging the cable round. The relative effects of upward lift and rearward drag forces are explained by noting the cable depression angle associated with each case. For example, at 6 knots at W : 1 a drag force is a better depth changer than lift because the depression angle is large, but as the speed increases to 9 knots the depression angle decreases and the vertical lift force becomes more effective than the horizontal drag force. The falling effectiveness with speed increase, of both the positive lift and the drag systems when W : 0.5, is due to the cable depression angle approaching the critical angle, which nmrks the limit of depth reduction in this case (see Section 4.1). Figure 12(d) illustrates the rate of change of depth with towing speed, a parameter of interest where a close control of depth is required. The depth-speed gradient varies from 0 to 45 m/knot, over the speed and weight range examined. The adverse effect o f drag upon the speed sensitivity, at large depression angle, is evident from the result at 3 knots when W ~ 0.5. These examples provide a background against which to judge other depth-control systems. 7. CONCLUSION It is concluded that, for heavy faired cables, the use of axes aligned with the critical angle leads to cable equations of a relatively simple form. Furthermore, by adopting a tension coefficient embodying towing speed and cable scope the depth achieved by any combination
The use of variable body forces to control the depth of towed submersibles
73
(o) Depth ond troil
m
Trail,
0 300
300
9" Knots
/
•/
0 300
/ I Knots/ / / / ' ~" 91¢~ / /
/
/~ ~
IJ oo 5 VY=O.5kN
/I / I
0
Kn~fU /
I/
/fl
"
IJ oo 5
/~
5
W = I kN
CL
:o
°:°
W=2kN
(b) Depth vs speed
CL :
I00 m
CD
=0
L
CD = j /
I00
i,,,o¢/
200
300
6
(:3
9
Knols
9 5
G
9 3
6 Knots
9
6 Knots
9
6
9
c) Percentage depth change
-2o_,oF.......
_!
.
.
.
.
.
.
.
.
.
.
.
I
% 0
. . . . . . .
6
Knots
9
3
6
9
3
9
3
(d) Depth-speed gradient
~ 2o E
0 3
F r o . 12.
6
Knots
9
3
C
9
3
G
Knots
'Straight cable' models with various body forces at differing weights and speeds.
of cable and body force can be readily found, at any speed, by means of a single equation which is a close approximation of results given by exact treatment. Maximum tensions are also rapidly determined. A consequence of this approximation is that the cable may be regarded as a straight line pivoting about its top point so that the body effectively swings on a radius when the body force is varied. This leads to the important result that when the body force is varied it is only the component of the force normal to the cable chord that determines the angular swing of the cable and hence the change in depth.
74
P.J. WINGHAM
Calculations of the depth changes produced by using either moving wings or water brakes show that, for realistic body configurations, the depth may be varied by up to about 2 0 ~ . In the case of moving-wing systems the effect of induced depressive forces is small compared to the effect of induced upward forces. When large cable depression angles arc employed water brakes are more effective depth changers than moving wings Acknowledgement--The author is grateful for the help and advice of his colleagues, particularly W ( Chesterman and J. F. Henderson. REFERENCES EAMES,M. C. 1967. Steady-state theory of towing cables. Royal Institution of Naval Architecls. I~ND~gSON, J. F. 1974. Wind tunnel tests on a l/3rd scale model of a sea-bed-following research vehicle. University of Bath, School of Engineering, Report No. 292.