DISCUSSION OF RESULTS

In this Part, the experimental results are used to provide a reasonably complete description of this MHD flow, In so doing, the approximations and equations of motion

presented in Part I are assumed to apply. The measurements, the implications which can be drawn from them, and some re- sults available from previous investigation~ of such flows, are discussed in the process, The flow model presented is rather qualitative and is based on physical arguments more than mathematical analysis. As mentioned in Part 1,there are no theoretical solutions which apply in the limits appropriate to the experimental flow,

The experiments show ithat, compared to the zero field potential flow, the MHD flow for N 2 O(1) has a large up- stream disturbance extending far ahead of the body, The length of the disturbance, which grows as N 35

,

is 10 to 20 body diameters for the highest values of N used here.* On the other hand, the centerline velocity profiles show that only very much nearer the body is the fluid motion relative to the body actually reduced to the extent that it can be considered stagnated, Such flow did exist, however, just ahead of the bodyo as was dramatically demonstrated by the shaft-mounted sensor measurementse Although the sensor was sensitive to velocities as small as a few per cent of the

*

The corresponding length for flow at N = 0 is 0.85 body diameters.

d r i v e s h a f t v e l o c i t y , i t produced a s t e a d y , zero-velocity s i g n a l when mounted 0.5" ahead of t h e body a t N 2 17, and when mounted 1 " ahead of t h e body a t N 2 35. The measure- ments do n o t provide s u f f i c i e n t d a t a t o d e s c r i b e t h e depend- ence of t h e l e n g t h and shape of t h i s s t a g n a t e d p o r t i o n of t h e flow on N. However, t h e y do show t h a t t h e flow ahead of t h e body c o n s i s t s of a r e l a t i v e l y s h o r t region of s t a g n a n t f l u i d preceded by a much longer region i n which t h e f l u i d v e l o c i t y r a p i d l y approaches t h a t of t h e freestream flow.

Only i f t h e former grows a s N

3 ,

w i l l t h e c o r r e l a t i o n of u and %/N4 i n f i g u r e 1 2 remain v a l i d f o r a l l N. The s t a g - nated r e g i o n , however, could be i n c r e a s i n g i n l e n g t h a t a f a s t e r r a t e , such a s d i r e c t l y with N - I n such a c a s e , f o r i n c r e a s e d N , t h e t r a n s i t i o n from zero t o f r e e s t r e a m veloc- i t y would have t o t a k e p l a c e w i t h i n a region no longer

s i m i l a r t o t h e one measured h e r e , o r w i t h i n one i n which t h e d i s t a n c e , xb, i s r e f e r r e d t o t h e f r o n t of t h e s t a g n a n t

region and n o t t o t h e body. Such d e p a r t u r e s of t h e flow p a t t e r n from t h a t measured i n t h e s e experiments, could only occur f o r values of N much g r e a t e r than t h o s e obtained h e r e .

$hey would imply t h a t t h e l i m i t i n g flow f o r N ^-p i s ap- proached only very slowly

-

even a s N i s increased by o r d e r s of magnitude

.

The f u l l a x i a l v e l o c i t y r e s u l t s show t h a t , with i n - creased N, t h e magnitude of t h e v e l o c i t y d e f e c t a t a fixed p o i n t ahead of t h e body i n c r e a s e s on and near t h e flow

c e n t e r l i n e . I t i n c r e a s e s more slowly, o r may even d e c r e a s e , a t l a r g e r r a d i a l p o s i t i o n s . I n o t h e r words, a t t h e same time t h a t t h e t o t a l v e l o c i t y d e f e c t i n c r e a s e s , it a l s o becomes more concentrated about t h e a x i s of symmetry. I f t h e r a d i a l p o s i t i o n a t which t h e d e f e c t i s one-half i t s maximum i s used a s a measure of t h e width of each v e l o c i t y d e f e c t p r o f i l e , t h e width of t h e d i s t u r b a n c e slowly d e c r e a s e s a t a f i x e d a x i a l p o s i t i o n a s N i s increased. For a f i x e d N, it grows very slowly with increased d i s t a n c e from t h e body.

An i n d i c a t i o n of t h e c u r r e n t d i s t r i b u t i o n can be ob- t a i n e d using t h e f a c t t h a t t h e c u r r e n t d e n s i t y a t a p o i n t i n t h e flow i s j u s t p r o p o r t i o n a l t o t h e r a d i a l v e l o c i t y a t t h a t p o i n t ( s e e P a r t I ) . The average r a d i a l v e l o c i t y be- tween

%

^and

+

1 a t r was c a l c u l a t e d by g r a p h i c a l l y i n t e g r a t i n g t h e c o n t i n u i t y equation v =

- ^-

r l e . 3 .

using t h e p r o f i l e s of a x i a l v e l o c i t y t o provide

-

a U The ax

r e s u l t s , shown i n f i g u r e 19, provide an e s t i m a t e of t h e magnitude and d i s t r i b u t i o n of v and j. These q u a n t i t i e s

i n c r e a s e from zero a t r = 0 almost l i n e a r l y with r , r e f l e c t - ing t h e weak r-dependence of

=

f o r small r. A s r + 0 ( l ) ,

ax

t h e c u r r e n t d e n s i t y and r a d i a l v e l o c i t y reach maximums and then decrease with f u r t h e r i n c r e a s e s i n r , Typical r a d i a l v e l o c i t i e s i n t h e flow a r e an o r d e r of magnitude s m a l l e r than t h e c h a r a c t e r i s t i c v e l o c i t y d e f e c t a t t h e same a x i a l p o s i t i o n .

These experimental r e s u l t s provide t h e magnitude and

d i s t r i b u t i o n of u, v , and j i n t h e flow ahead of t h e body f o r N > 10. The c u r r e n t l a y e r model mentioned i n P a r t I d e s c r i b e s t h e s e same q u a n t i t i e s i n t h e same flow region under t h e assumption t h a t N >> 1. I n t h e model flow, t h e r e e x i s t well-defined c u r r e n t l a y e r s , centered about r =

5

1

,

which maintain t h e r a d i a l pressure g r a d i e n t necessary t o s e p a r a t e t h e o u t e r freestream flow from a slug of n e a r l y stagnated f l u i d ahead of t h e body. The merging o f t h e l a y e r s , which terminates t h e stagnant region, t a k e s place a d i s t a n c e x = O ( N ) ahead of t h e body, I n t h e l i m i t N + co

,

f o r N / R ~ -+ 0 , t h e model flow becomes undisturbed freestream flow p a s t an i n f i n i t e l y long s l u g of f l u i d bounded by i n f i n - i t e s i m a l l y t h i n c u r r e n t s h e e t s . Therefore, while t h e mea- sured flow does n o t preclude t h e p o s s i b i l i t y of such a l i m i t i n g form, i t does n o t f i t t h e c u r r e n t l a y e r model.

I n s t e a d of well-defined c u r r e n t l a y e r s , t h e r e a r e broad r e - gions of maximum c u r r e n t d e n s i t y . The length of t h e upstream disturbance grows a s N

4 .

The small p o r t i o n of i t which con- t a i n s stagnated flow grows w i t h N a t an undetermined r a t e , b u t i s d e f i n i t e l y not of O ( N ) i n length, Although t h e en- t i r e disturbance becomes more "slug-like" i n t h a t it becomes more concentrated about r ₌ 0 a s N i n c r e a s e s o it i s not a

s l u g of very slowly moving f l u i d , The c u r r e n t l a y e r model d e s c r i b e s a region containing a s l u g of stagnant f l u i d bound- ed by c u r r e n t l a y e r s , b u t provides no d e s c r i p t i o n of t h e

intermediate region which must e x i s t between t h e s l u g and t h e

upstream f l u i d . I n t h e measured flow it i s found t h a t t h e stagnant region, which could not be measured i n d e t a i l , i s s h o r t , and t h a t most of t h e disturbance c o n s i s t s of a region of t r a n s i t i o n from zero t o freestream v e l o c i t y ,

The experimental flow can b e described f u r t h e r by r e - f e r r i n g t o t h e generalized Bernoulli law f o r i n v i s c i d , MHD flows derived by Tamada ( ~ e f , '91,

2 . V H = q e

l 2

The Bernoulli function H =

5

^(u2 + v + p, i s constant along streamlines on which

i

= 8 , and must decrease along a l l streamlines on which - j # 8 . This means t h a t s i n c e v

-

^{j ,} closed s t r e a m l i n e s a r e not p o s s i b l e i n steady flow,

and a l s o t h a t t h e maximum pressure i n t h e flow i n normal s t a g n a t i o n p r e s s u r e (Po =

-

I ₂ i n t h i s n o t a t i o n ) , a t t h e s t a g - n a t i o n p o i n t , I n both t h e non-magnetic and MEID flows, t h e s t a t i c pressure along r = 0 r i s e s from i t s freestream value

( z e r o ) t o t h i s s t a g n a t i o n pressure a t t h e body. I n both eases t h e a x i a l pressure g r a d i e n t i s balanced only by t h e a x i a l f l u i d d e c e l e r a t i o n . The d i f f e r e n c e between the two flows appears i n t h e r a d i a l equation of motion, Radial

pressure g r a d i e n t s i n t h e ordinary flow can only be support- ed by t h e i n e r t i a % e m s r b u t i n t h e MHD flow Tor N >> 1 t h e s e may be neglected, s o t h a t the equation becomes

-

^~v.The ability of the radial flow to support a radial ar

pressure gradient is enhanced, while the net radial flux of fluid remains unchanged (for a given freestream velocity)

,

and the maximum possible radial pressure drop from r = 0 to r >> 1 is constant. Qualitatively, then,the magnetic force which acts on the fluid when it crosses field lines, tends to decrease the radial flow near the body and increase it away from the body. Thus, it straightens the streamlines and increases the axial distance over which the pressure rises from p = 0 upstream to p a 1 at the body.

Along the axis of the flow v = j = 0, and theBernoulli so that the local pressure is function is constant, H =

-

₂

_"

directly related to the known axial velocity, p(x, 0 ) =

-

₂ 1

-

^{@(X, 0 )} i

2

.

Therefore, the pressure change along r = 0 occurs over an axial distance proportional to N %

.

^An

example of this pressure profile for N = 29 is shown in

figure 20, along with the corresponding profile.for potential flow.

The pressure difference which exists between r ₌ 0 and the flow at large r is maintained by the net magnetic force,

L ,-

-

^N

4

v (x, r) ar. at x. Lacking an analytical expression for the distribution of radial velocity, the area under the

*

The results of the experiments are presented in terms of the normalized velocity defect in the fzeestream flow, so that the velocity ref erred to above is U = 1

-

^u

and is the normalized fluid velocity relative tod6~%dy.

v e l o c i t y p r o f i l e s of f i g u r e 19 can be used t o provide an e s t i m a t e of t h e magnitude of t h i s f o r c e . For example, a t N = 29, t h e values estimated i n t h i s way a r e -36 a t

%

⁼ 5 ,

.26 a t

%

= 6 , and .I45 a t

%

⁼ 8. The corresponding changes i n p r e s s u r e from f i g u r e 20, assuming p = 0 i n t h e o u t e r flow, a r e approximately -24, -18 and .13.* These admittedly crude e s t i m a t e s a r e c i t e d only t o demonstrate t h a t t h e r a d i a l

v e l o c i t i e s which correspond t o t h e measured a x i a l v e l o c i t y p r o f i l e s a r e of t h e c o r r e c t o r d e r , and s u f f i c i e n t , t o sup- p o r t t h e r a d i a l p r e s s u r e g r a d i e n t s which must e x i s t i n t h e flow.

I t may be noted h e r e t h a t had t h e r a d i a l v e l o c i t i e s been confined t o r e l a t i v e l y t h i n l a y e r s , t h e i r maximum values (which were,017@ ,012 and .006 a t t h e p o s i t i o n s r e -

f k r r e d t o above f o r N = 291, would have had t o have been considerably h i g h e r a t t h e same a x i a l p o s i t i o n s and same N.

And, f i n a l l y , t h a t i f Payers of some k i n d a r e assumed t o e x i s t very near t h e body where t h e r e i s a region s f s t a g - nated f l u i d , they must be a b l e t o support a r a d i a l p r e s s u r e change of o r d e r 0.5, Given an e s t i m a t e of t h e t h i c k n e s s of t h e l a y e r s i n t h i s r e g i o n p an approximate mean v e l o c i t y

*

These v a l u e s should be s l i g h t l y low because i n t h e o u t e r flow p ⁴ -.03 downstream of t h e body due t o t h e blockage e f f e c t of t h e body i n t h e t o w tank.

through them would then be known.

*

However, despite the fact that near the body the distribution of radial velocity must become reasonably concentrated around r =

-

₂ ^{the flow}

in this region (where the stagnated flow, the curved body face, and the outer flow come together) is certainly too complex to be described in terms of a simple current layer.

The effects of the Joule dissipation on the pressure and velocity in the flow can be considered by referring again to the Bernoulli Paw (Eq. 4.1)- The stagnation pres- sure a Po = p

+(U2 )

is decreased on all streamlines which have passed through regions of non-zero current den-

sity, The decrease at any point in the flow is equal to the total Joule dissipation which has occurred upstream of the point along the streamline passing through it,

Since the stagnation pressure can only decrease, and the amount of Joule dissipation which occurs along different streamlines varies, radial as well as axial stagnation pressure gradients occur in the flow, At axial positions ahead of the body the stagnation pressure falls from

P

1 ^on r = 0 to lower values at r > 0,in the region where stream-

*

For example, a mean radial velocity sf

-

^{-09 would be}

required for N = 29 through layers of thickness ~r-1/N4

-

^.18

^-

the thickness at

x

-- O(1) which results from the order of magnitude arguments used in the current layer model.

lines pass which have undergone dissipation, and then rises again to

-

1 at large r where there are streamlines on which

2

the radial velocity has been continuously zero. As shown by Tamada (Ref. 7), far downstream of the body, where the flow again becomes uniform in the sense that v = 0, there can be no radial static pressure gradient and the axial velocity must be directly related to the stagnation pressure, Due to the loss in stagnation pressure suffered by the flow along streamlines which have experienced Joule dissipation, the axial velocities near the body are lower than those in the flow far from the body. This vortical wake does not diffuse or dissipate in inviscid flow. Looked at in terms of

vorticity and the equation for the change in vorticity along dsg

- -

^N

^-

a streamline,

-

_ds

-

^aV this means that the vorticity Bx

'

in the direction created by the positive

aV

in the flow ax

ahead of the body, is greater than the vorticity suppressed near the body where

-

av becomes negative, For Re large, but

ax

not infinite, this wake will ultimately diffuse to produce uniform flow far downstream.

Because of the stagnation pressure variations in the flow, the velocity measurements cannot be used directly to determine static pressures. This is particularly true over the front of the body near r =

where

I the stagnation pres- sure must be close to its minimum value, Measurements of static and stagnation pressures over the surface of a sphere in aligned-fields MHD flow, for 1.5 < N < 40, have been made

by Maxworthy (Ref. 9). These show that the stagnation pres- sure in the flow past the sphere, at the position corre- sponding to r =

T

1 on the halfbody used here, is from 40% to 90% less than its freestream value and decreases as N

%

, At the same position, static pressures are found to be of O(1) and negative, so that the corresponding velocities must be greater than freestream by as much as 50%. The resultant net pressure force on the front of the sphere rises only to about CD = 0.5 at N -- 12, and then falls toward zero as N is increased further, due to the effect of the increasingly large contribution of the negative pressure. In addition, large negative pressures near the body can only exist if maintained by a magnetic force, so that the flow past the body must contain negative radial velocities. These re- sults are cited to demonstrate the possible effects of the stagnation pressure losses due to Joule dissipation in such flows, and to introduce a discussion of whether such an effect could occur in the flow under consideration here.

Suzuki (Ref, 4) has measured the drag sf a Rankine halfbody in these flows, His measurements show that, as N

is increased from N << 1, the drag increases linearly with N to a value of CD -- 0.5 at N = 6. For N > 6, CD increases more slowly with increased N I and appears $0 be approaching

-

^0.8 asymptoticaPly for N > 20. Although at low N there are certainly regions of negative pressure on the front of the body (as there are for N = O), these measurements

indicate that they are probably not increasing in strength with increased N. The highest drag possible for a halfbody

in these flows is CD = 1.0, for which the entire frontal area must be at freestream stagnation pressure. The veloc- ity measurements indicate that it is unlikely that this con- dition exists in the flow even at N > 20, so CD -- 0.8 can occur only if there is very little negative pressure on the body. Another indication of the pressure near the body

comes from some velocity measurements made in the flow down- stream of the body, Due to physical limitations imposed by the experimental apparatus, these could only be made for 0.625 g r s 1-25 and for only 2 to 3 body diameters past the stagnation point.* Nevertheless, the results are sufficient to indicate that axial velocities in this region were not greater than freestream, but were, in fact, smaller, Typical results of such measurements are included in figures 14 and 20. On the basis of these drag and velocity measurements, it must be concluded that the static pressure on the body near r = - is not large and negative, but is probably close to

2

zero, The stagnation pressure in this region would then be

*

^The measurements in the downstream flow are notl therefore, complete enough to allow calculation of the drag using the downstream wake profile and a momentum balance,Such a cal- culation would have been difficult in any case, since the flow in this region may vary more or less continuously from the body to the tow tank walls. The velocity near the walls, for example, must be

5

> 1.03 due to the

blockage effect of the body in the tank and the presence of the wake near the body.

u ² 1

of o r d e r Po =

-

^{+ p <}

^- ,

^{where u} < 1 and p i s approxi-

2 2

mately zero o r a t most s l i g h t l y negative. I t should be noted h e r e t h a t because t h e upstream flow i s being forced away from t h e a x i s i n o r d e r t o pass t h e body, t h e flow o u t - s i d e t h e d i s t u r b a n c e must be a c c e l e r a t i n g . Since t h e cur- r e n t regions a r e broad and extend w e l l beyond r =

-

1

2

,

e s p e c i a l l y a t l a r g e xb, flow a t f a i r l y l a r g e r a d i a l d i s - tances i s turned and a c c e l e r a t e d , A s a r e s u l t , t h e r e g i o n s of a c c e l e r a t e d flow and s t a g n a t i o n p r e s s u r e l o s s p a s t t h e body a r e widespread. There i s no l a r g e r a d i a l g r a d i e n t i n a x i a l v e l o c i t y o r , except p o s s i b l y v e r y c l o s e t o t h e body, i n s t a g n a t i o n pressure.

V

.

CONCLUSION

V e l o c i t i e s i n a l i g n e d - f i e l d s MHD flow ahead of a semi- i n f i n i t e Rankine body have been measured over a wide range of N. C e n t e r l i n e flow v e l o c i t i e s have been measured t o with- i n one-half body diameter of t h e s t a g n a t i o n p o i n t , and veloc- i t y p r o f i l e s a c r o s s t h e flow t o w i t h i n about f i v e body diam- e t e r s . I t was found t h a t with increased N , t h e upstream d i s t u r b a n c e t e n d s t o become more confined r a d i a l l y w i t h i n t h e region d i r e c t l y ahead of t h e body, and t h a t i t s length i n c r e a s e s a s N

% .

The flow was found t o c o n t a i n a region of s t a g n a n t f l u i d ahead of t h e body, and a much longer region over which t h e t r a n s i t i o n i s made from freestream c o n d i t i o n s t o c o n d i t i o n s near t h e body. The r a t e a t which t h e length of t h e s t a g n a n t region i n c r e a s e s with N was n o t determined.

However, t h e region was found t o be much s h o r t e r i n l e n g t h than i s p r e d i c t e d by a t h i n c u r r e n t Payer model, The r a d i a l g r a d i e n t s of v e l o c i t y components, p r e s s u r e s , and c u r r e n t d e n s i t y were found t o be considerably smaller than suggested by such a model, The r e s u l t s a r e c o n s i s t e n t with a drag

c o e f f i c i e n t which i n c r e a s e s a s N i s increased and approaches O(1), In t h e corresponding flow p a s t t h e body, t h e r a d i a l v e l o c i t y and c u r r e n t d e n s i t y go t o z e r o , and t h e r e a r e small r a d i a l g r a d i e n t s i n t h e a x i a l v e l o c i t y and t h e s t a g n a t i o n p r e s s u r e .

Although a reasonably complete d e s c r i p t i o n of t h e flow

h a s been obtained using t h e measurements and t h e i n v i s c i d equations of motion f o r N >7 1 and Rm << 1, a d i f f e r e n t

flow may evolve a s N + so, The l i m i t i n g flow which develops, as N i s g r e a t l y increased over t h e v a l u e s used h e r e , w i l l depend on how t h e l e n g t h of t h e s t a g n a n t region i n c r e a s e s i n r e l a t i o n t o t h e t r a n s i t i o n r e g i o n ahead of it, The i n t e r - a c t i o n parameter may have t o be much l a r g e r than i n t h e s e experiments b e f o r e such a l i m i t i s approached.

REFERENCES

1. Liepmann, H. W., Hoult, D. P., and Ahlstrom, H. G.,

"Concept, Construction, and Preliminary Use of a Facility for Experimental Studies in ~agneto-~luid Dynamics," Mizellen der Ansewandten Mechanick,

175-189, 1960.

2. Yonas, G., "Aligned-Fields, Magneto-Fluid Dynamic Flow Past Bodies," Ph.D. Thesis, California Institute of Technology, 1966.

3. Shercliff,J. A., A Textbook of Magnetohydrodynamics, Pergamon, 1965.

4. Suzuki, B. H., "Magneto-~luid Dynamic Drag Measurements on Semi-Infinite Bodies in Aligned Fields," Ph.D.

Thesis, California Institute of Technology, 1967, 5. Childress, S., "On the Flow of a Conducting Fluid of

Small Viscosity," JPL Tech, Rept, No, 32-351, Jan, 1963.

6 Childress, S., "The Effect of a Strong Magnetic Field on TWO-Dimensional Flows sf a Conducting Fluid, '' J. Fluid. Mech.

z,

429-441 (1963),

7. Tamada, K., "Flow of a Slightly Conducting Fluid Past a Circular Cylinder with Strong, Aligned Field,"

Phys. of Fluids

5,

817-823 (1962).

8. Maxworthy, T., "Measurements of Drag and Wake Structure in Magneto-Fluid Dynamic Flow about a Sphere," Heat Transfer and Fluid Mech. Inst., 197-205, 1962,

9- Maxworthy, T o , "Experimental Studies in Magneto-Fluid Dynamics: Pressure Distribution Measurements Around a Sphere," J. Fluid Mech. 3J, 801-814 (1968),

10. Ahlstrom, H. G,, "Experiments on the Upstream Wake in Magneto-Fluid Dynamics," J. ~ l u i d Mech. 3.5, 205-221

-

(1963) ,

11, Sajben, M., "Hot Wire Anemometer in Liquid Mercury,"

Rev. Sci. Instr.

36,

7, 945-949 (1965).

12. Lumley, J. E., "The Constant Temperature Hot Thermistor Anemometer," ASME Symposium on Measurements in Un-

steady Flow, Worcester, Mass,, May, 1962.

REFERENCES (cont

'

d

.

⁾

13. Malcolm, D. G., "Some Aspects of Turbulence Measurement in Liquid Mercury Using Cylindrical Quartz-Insulated Hot-Film Sensors." To be published J. Fluid Mech.

14, Chapman, A. J., Heat Transfer, Macmillan, 1960.

15. Hill, J. C., "The Directional Sensitivity of a Hot-Film Anemometer in Mercury," P h , D . Thesis, University of Washington, Seattle, 1968.

APPENDIX

The average uncertainties in the directly measured quantities which apply to all of the data in general were:

magnetic field. Bo...+

-

^1%

drive shaft velocity, U...+

-

^3%

mercury temperature, Tfluidw

.. .... ^{+ -} .

^1°c

Magnetic field changes produced temperature changes of up to 2o0c due to the heating of the magnet. These would have introduced uncertainties of 1% to 2% in the values of param- eters such as N or PQ if the mercury physical properties had been considered constant, Because it was possible to cal- culate all such parameters using mercury physical properties evaluated at the appropriate temperatures, this effect was not present.

Estimates for

x

(P4)

The use of the parameter. X (Pk) a

(& ^-

requires that the two measurements necessary to determine each data point be made at the same value of AT, and for the same sensor coating conditions, The uncertainty in

x ( P ~ ) which could arise due to temperature or coating changes

during the course of a run can be estimated from the measure- ments of fluid temperature and zero velocity sensor output made before and after each run.

There was no measurable temperature change during the

course of any run, and u s u a l l y none during each s e t of runs a t a f i x e d magnetic f i e l d . Therefore, s i n c e AT -- 50°c, each p a i r of measurements was made a t constant AT t o w i t h i n

- ⁺

^-2%.

I f AT had changed, would have been c a l c u l a t e d using a s opposed t o t h e c o r r e c t value

which would have been x ( P & )

)

By compar-

ing t h e s e two expressions, ~ a l c o l m (Ref. 1 3 ) has shown t h a t t h e e r r o r introduced i n t o x ( P & ) by t h e u n c e r t a i n t y i n AT can be up t o n e a r l y an o r d e r of magnitude g r e a t e r than t h a t of t h e l a t t e r , depending upon t h e value of P&, For t h e s e ex- periments t h i s would correspond t o a maximum u n c e r t a i n t y i n X ( P & ) of

+ -

^2%, a s a r e s u l t of t h e u n c e r t a i n t y i n t h e temper- a t u r e measurements,

The sensor output a t zero v e l o c i t y was s e n s i t i v e t o v a r i a t i o n s i n both f l u i d temperature and coating p r o p e r t i e s . Measurements made before and a f t e r each run showed t h a t it changed by l e s s than +

-

^1% f o r runs made a t high v e l o c i t i e s

(P& > . 5 ) , and by +

-

^-1% o r l e s s f o r runs made a t low veloc- i t i e s (Pd < . 5 ) . Equation (2.4), when w r i t t e n a s an expres- s i o n f o r t h e measured q u a n t i t y q ( O ) ,

shows t h a t q ( 0 ) may vary due t o changes i n AT, I

Nu ( 0 ) and t h e velocity-independent coating terms, Since AT was known

t o be constant t o within

+ -

^.2%, and s i n c e i t i s known t h a t f o r c y l i n d e r s i n f r e e convection with c o n s t a n t f l u i d prop- e r t i e s NU ( 0 )

=

( A T )

% ,

changes i n q ( 0 ) were due p r i m a r i l y t o t h e coating terms, The above equation can be re-written:

where C r e p r e s e n t s t h e coating terms. The use of X ( P Q ) = nkfLAT (-1 _{q ( 0 )} 1

- _PI)'

¹ = ( N u ( o ) ^l

^-

N U ( P & ) ) assumes t h a t t h e value of C i s t h e same f o r t h e measurements a t PC% = 0 and PB

#

0, Although l a r g e c o a t i n g changes occurred when t h e sensor was passed through t h e mercury f r e e surface, t h e output from a s t a t i o n a r y sensor a f t e r immersion i n d i c a t e d t h a t coating p r o p e r t i e s were then constant except, on

occasionp f o r a very slow d r i f t (see Sec. 2 . 4 ) . Comparison of q ( 0 ) values obtained b e f o r e and a f t e r each run provides an e s t i m a t e of t h e e f f e c t s of whatever coating changes might have occurred during t h e course of a runo For high v e l o c i t y runs t h e v a r i a t i o n i n q ( 0 ) was l e s s than 2%

,

TTkfEAT

(q ( 0 )

-(Fensoq

^output

^,

^{a a 2%} change i n ( 0 ) corresponds t o a 4% t o 6% change i n x ( P ~ ) , s i n c e X ( P & ) i s 30% t o 50% of

nkfLAT rrkfLA T

4 ( 0 ) A t low v e l o c i t i e s both t h e changes i n

q (0) and t h e values of

x

^(Pd) were smaller by an order of magnitude s o t h a t t h e percentage v a r i a t i o n i n X ( P & ) i s again 6% o r l e s s .

Estimated i n t h i s wayp t h e average u n c e r t a i n t y i n

x(P&) due to temperature and coating effects is something less than

+ -

^6%. However, the most meaningful indication of the average uncertainty in the parameter X(PQ) is that provided by the scatter in the calibration curve. This is of the order of +

-

^5%.

Estimates for P6 and u

In the low velocity range where x(P&) < 0.7, an un- certainty of

+ -

5% in the value of x(P&) calculated from the measurements, corresponds to an uncertainty of +

-

7% in the value of P& found using the calibration curve. At higher velocities where x(P&) becomes a progressively weaker func- tion of P4, the same uncertainty in X(P6) corresponds to variations in P& which increase to the order of

+ -

^{15 to}

+

20% as X(P&) + 1.4.

-

Values of the P& from the calibration curves were used to calculate the normalized velocities. Although the latter were determined using two different expressions, according to the way in which the data were obtained (see Sec. 2.4),

both expressions involved only the values of P& correspond- ing to the measured flow velocity, the drive shaft velocityo and the velocity of the displacement flow. The velocity profile of the displacement flow could have become peaked near the axis sf the tow tank if the mercury had been per- mitted to rise through the fringing magnetic field near the top of the tank. For this reason these experiments were