Results have been presented from MFD drag measurements on t h r e e semi-infinite bodies over the range, 0 <
-
N C-
24. Two of the bodies, a Rankine halfbody and a 2-caliber ogive, were streamlinedshapes, and the third was a blunt halfbody. F o r N <
-
0(1), the drag coefficients of the streamlined halfbodies were found to increase linearly with N; but the drag coefficient of the blunt halfbody was relatively unaffected, F o r N > > 1, the drag coefficients of a l l t h r e e halfbodies were of 0(1) and appeared to be asymptotically converging to some common limiting value.A simple theoretical calculation of the drag coefficient of the Rankine halfboqy was possible for
N <
C 1 which agreed quite well with the experimental results. However, there was apparent agreementeven f o r N = 0(1j for which the theoretical calculation i s no longer valid. This was explained by a strictly inviscid theory which showed that a nondiffusive, vortical wake must exist a t downstream infinity.
By extending the results of this inviscid theory to the c a s e N> > 1 , a physical model of the flow was constructed which led to the conclusion that a s N-rm an infinitely-long stagnant slug must form in front of a halfbody and the drag coefficient of the body must approach a maximum value of unity. Although all the features of this model a r e consistent with the trend of the experimental results, m o r e complete experimental verification is needed.
]It i s particularly important to establish the extent to which the inviscid flow approximation i s valid. This might be studied by
measuring the wake velocities a t various distances downstream. Also i n question i s the existence of the thin c u r r e n t l a y e r s and long stag- nant slugs i n front of the halfbodies. The answer t o this question could be provided by detailed measurements of the velocity and magnetic field.
There is perhaps even a g r e a t e r need for further theoretical work on the problem. Although the MFD drag can be calculated f o r
small N and an upper bound on i t s value has been established for
N-.
a>, i t cannot a s yet be calculated for intermediate values of N.Hopefully, the physical framework that has been provided h e r e will grove beneficial in this endeavor.
Finally, results have also been p r e s e n t d from measurements made of the transient drag of flat disks which were started impulsively from rest. The drag of a disk was found to overshoot i t s steady-
state value by 30 to 50 p e r cent and required a distance of about 25 disk diameters to reach this steady+state value. This behavior was attributed t o the vortex formation process occurring in the wake of the disk. Additional experiments, p a r ~ c d a r l y visual flow studies, would probably lead to a better understanding of this interesting process.
REFERENCES
1. Liepmann, H. W.
,
Hoult, D. P.,
and Ahlstrom, H. G . , "Concept, Construction, and P r e l i m i n a r y Use of a Facility f o r Experi- mental Studies i n Magneta-Fluid Dynamics,"
Miszellen d e r Angewandten Mechanick, 175-189, 1960.2. Stewartson, K.
,
"Motion of a Sphere Through a Conducting Fluid i n t h e P r e s e n c e of a Strong Magnetic Field, P r o c . Camb.Phil. SOC.
-
52, 301-316 (1956).3. Chester, W.
,
"The Effect of a Magnetic Field on Stokes Flow i n a Conducting Fluid,$'
J. Fluid Mech,-
3, 304-308 (1957).4. S e a r s , W. R. and R e s l e r , E.
L.
Jr,,
"Theory of Thin Airfoils i n Fluids of High E l e c t r i c a l Conductivity, "J. Fluid Mech.-
5, 257-273 (1959).5. Stewartson,
K.,
!'On the Motion of a Non-Conducting Body Through a Perfectly Conducting Fluid, " J. Fluid Mech.-
8, 82-96 (1960).6, Gourdine, M, C.
,
''On Magnetohydrodynamic Flow over Solids,''
Ph. D. T h e s i s , California Institute of Technology, 1960. Also see: J. Fluid Mech.
-
10, 459-465 (1961).7. Maxworthy, T.
,
"Measurements of Drag and Wake Structure i n Magneto-Fluid Dynamic Flow about a Sphere,"
Heat T r a n s f e r and Fluid Mech. Inst.,
197-205, 1962.8. Ahlstrom, H. G . , "Experiments on the Upstream Wake i n Magneto- Fluid Dynamics, s' J. Fluid Mech,
-
15, 205-221 (1963).9. Motz,
R.
0.,
~ ' M a g n e t o h y d ~ o d p a m i c Drag on a Oscillating Sphereo" J.
Fluid Mech.-
24, 705-720 (1966).10, Yonas, G.
,
"Aligned Fields, Magneto-Fluid Dynamic Flow P a s t Bodies, " Ph. D. Thesis, California Institute of Technology,1966.
Dorman, L. I. and Mikhailov, Y. M . , "Investigation of Electro- Magnetic Phenomena Involved in the Motion of Bodies in a Conducting Fluid in a Magnetic Field," Sov. Phys.
JETP -
16, 531 (1963).Shercliff, J . A.
,
A Textbook of Magnetohydrodynamics, Permagon, 1965.Chester, W.
,
sfThe Effect of a Magnetic Field on the Flow of a Conducting Fluid P a s t a Body of Revolution," J. Fluid Mech.Hoerner, Fluid-Dynamic Drag; Practical Hnformation on A e r o
-
dynamic Drag and Hydrodynamic Resistance, Midland P a r k , 1965.
Childres s , S.
,
"Hnviscid Magnetohydrodynamic Flow in the Presence of a Strong Magnetic Field, JPL SPS No. 37-22, 254-255 (1963).Coldstein, S.
,
Modern Developments in Fluid Dynamics, Vol. 11, Clarendon, 1938.Rouse, H. and Howe, J. W., Basic Mechanics of Fluids, Wiley and Sons, 1953.
Schmidt, F. S.
,
f g Z ~ r bescPlHeunaigten Bewegung kugelf8rrniger IKSrper in ~derstehemederm Mitteln,'@
Diss,,
keipzig, 191 9.19. Schwabe, M.
,
" P r e s s u r e Distribution in Nonuniform Two- Dimensional Flow,"
NACA TM No. 1039, Jan. 1943.Translated from: Ingenieur-A rchiv, Vol. VI (Feb. 1939).
20. Sarpkaya, T,
,
"Separated Flow about Lifting Bodies andImpulsive Flow about Cylinders, 'I AIAA Jsurn.
-
4, 41 4-420, (1966).21. Kendall, J . M., J r .
,
"The Periodic Wake of a Sphere, l s J P L SPS No. 37-25 (1964).22. F r o m m , J. E, and Harlow, F. H.
,
s'Numerical Solution of the Problem of Vortex Street Development,"
Phys. Fluids-
6,23, , Jnt,ernational Critical Tables of Numerical Data, Physics, Chemistry and T echnology, McGraw-Hill, l 929, 24. Maxworthy, T.
,
private communication.d
25. Lake, B. M.
,
private communication.26. Liepmann, H. W.
,
"Hydromagnetic Effects in Couette and Stokes Flow,"
The Plasma in a Magnetic Field; A Symposium on Magnetohydrodynamics, (ed., Landshoff,R. K.
M.),117-130, 1958.
27, Chopra, K. P. and Singer, %.
F. ,
"Drag of a Sphere Moving in a Conducting Fluid i n the Presence of a Magnetic Field, 'I Heat Transfer and Fluid Me&. h s t . , 166-175, 1958.28. Reitz, J. R, and Fsldy,
L.
E.,
"The F o r c e on a Sphere Moving Through a Conducting Fluid i n the P r e s e n c e s f a Magnetic Field," 3, Fluid Mech.-
11, 133-142 (1961).29. Ludford, G. S. S.
,
"Inviscid Flow P a s t a Body a t Low Magnetic Reynolds Number," Rev. Mod. Phys.-
32, 1000-1003 (1960).30, Ludford, G. S. S. and Murray, J . D.
,
"On the Flow of a Conducting Fluid P a s t a Magnetized Sphere, " J . of Fluid Mech.-
7, 516-528 (1960).31. Milne-Thompson, L. M.
,
Theoretical Hydrodynamics, Macmillan, 1950.32. Tamada,
K. ,
ssFlow of a Slightly Conducting Fluid P a s t a C i r c u l a r Cylinder with Strong, Aligned Field,"
Phys. ofFluids
-
5, 817-823 (1962).33. Leonard, B.
,
"Some Aspects of Magnetohydrodynamic Flow about a Blunt Body, ' I AFOSR 2714, 1962.34. Childress, S.
,
"On the Flow s f a Conducting Fluid of Small Viscosity," J P L Tech. Wept. No. 32-351, Jan. 1963.35. Ludford, G . S. S. and Singh, M. P. "On the Motion of a Sphere Through a Conducting Fluid i n the P r e s e n c e of a Magnetic Field, I f P r o c . Camb. Phil. Soc.
-
59, 625-635 ( 1963).36, L a r y , E. C.
,
"A Theory of Thin Airfoils and Slender Bodies i n Fluids of Finite Electrical Conductivity with Aligned Fields,"J . Fluid Mech.
-
112, 209-226 (1962).37. Ckang, I. D.
,
"On a Singular Perturbation P r o b l e m in Magneto- hydrodynamics," Zeits. %. angew, Math, und Phys,-
14,38, Cole, J. D.
,
Perturbation Methods i n Applied Mathematics, Blaisdell, (to b e p u b l i ~ h e d ) .39. Childress, S . , "The Effect of a Strong Magnetic F i e l d on Two- Dimensional Flows of a Conducting Fluid:'
J.
Fluid Mech.-
15, 429-441 (1963).40. Kovasznay, L. S. G. and Fung, F. C. W.
,
"AsymptoticSolutions f o r Sink Flow i n a Strong Magnetic Field,
''
Phys.of Fluids
-
5, 661 -664 (1962).41. S c h l i c h t i ~ g , H.
,
Boundary L a y e r Theory, McGraw-Hill, 1960.42. Greenspag, H. P. and C a r r i e r , G. F.
,
'#The Magnetohydro- d y n a q i c Flow P a s t a F l a t Plate, J, Fluid Mech.-
6 , 77-9643. Stewartson,
X.,
''On Magnetic Boundary J . Inst, Math.AppPic,
-
1, 29-41 (1965).-76-
APPENDICES
A . Calculation of Base Drag without Dissipation
The base drag will be calculated assuming that the flow i s inviscid and non-dissipative. The coordinate f r a m e i s fixed to the halfbody and a control volume i s selected which consists of the tow tank walls and two cross-sections, I and I I, of the tow tank a s shown in figure 10. The entry cross-section I of a r e a A 1 i s chosen f a r enough upstream such that the velocity ul and p r e s s u r e pl a r e uniform a c r o s s it. Likewise, the exit cross-section 1 1 of a r e a A 2 i s chosen f a r
enough downstream such that the velocity u2 and p r e s s u r e p2 a r e uniform a c r o s s it. These l a t t e r conditions a r e assumed t o exist by the time the flow reaches the lower bellows of the drag balance. Under the above assumptions, the conservation equations may be written a s follows :
ulAl
=
u2A2 ( Continuity)2 2
p1Al
*
WlAl=
pZAZ*
puZAZ 3- (Momentum)3 3
plulAI 4-
*
~ u 1 . A ~'
P 2 2 u A 2*
h 2 A Z {Energy, o r B ernoulli law) Elimination of u2 and p2 f r o m these equations leads toSince the drag is always measured with reference to p l , we can simply set p
P =
0 . The drag coefficient i s then given bySince A1/A2 = 1.035 for the tow tank, CD
=
0.0357.B.
Calculation of P r e s s u r e Jump due to Flow in StandpipeThe p r e s s u r e jump in the tow tank due t o the acceleration of the m e r c u r y flowin@ into the standpipe may be easily calculated by consid-
k
ering the balance of m a s s and energy of the system shown in the sketch
Again, the flow i s assumed to be inviscid and the velocities and p r e s - s u r e s a t sections 1 I I and IV a r e assumed uniform. The velocity u3 i s just the velocity of the fluid displaced by the drive shaft, and since the ID of the standpipe equals the OD of the drive shaft, u4 i s just
equal t o t h e drive shaft velocity, u2. Conservation of m a s s and energy thein gives:
A little algebra then gives the p r e s s u r e jump a s
since A ~ / A ~
=
1/30.Now i f the difference in the effective a r e a s of the bellows i s AA and the cross-sectional a r e a of a model is A , then the drag coef-
b
ficient corresponmding to the apparent drag force produced by the p r e s -
The ratio A / A b , where A i s the average effective a r e a of the bellows, b
i s 5.5 and hAb/Ab i s about -025 (this was determined by increasing the mercury hydrostatic head by a known amount and measuring the change in the drag balance output). Therefore. CD
=
.025/5.5 6 .004.It should also be noted that there may be an additional p r e s s u r e i n c r e a s e in the tow t a d due t o viscous effects in the standpipe flow.
However, this i n c r e a s e was calculated assuming fully-developed
turbulent flow and was found t o be only about 10 percent of the p r e e - s u r e jump calculated above.
C .
Calculation of B a s e Drag with DissipationWe again choose the control volume shown in figure 10. A s in the non-dissipative c a s e (appendix A ) , the conditions a c r o s s section I a r e assumed to be uniform. However, we can no longer make this assumption about the conditions a c r o s s section I I. If kinetic energy is being dissipated in the region n e a r the nose, due either to viscous o r ohmic l o s s e s , then a vortical wake f o r m s downstream and the velocity a c r o s s section I I ' m a y appear a s shown by the dashed line denoted a s
5
i d figure i0. In any r e a l flow, this wake i s ultimately dissipated very f a r downstream by the action of viscosity and uniform flow conditions a r e regained.Although the velocity
5
is nonuniform a c r o s s section 11, if this section is taken sufficiently f a r downstrearm ( s a y at the location of the lower bellows), then the streamlines should become nearly parallel and the p r e s s u r e a c r o s s the section may be assumed constant.Under this assumption, the continuity and energy equations f o r the system 'may b e written a s follows:
where Q is t h e dissipation p e r unit t i m e within the control volume.
1
Using (C. 1) i n (C. 2), we get
It will b e shown below that the t e r m i n brackets may differ f r o m unity by only 1 p e r cent s o that i t will simply b e taken equal t o unity. We
estimate Ql by soting that t h e work done by the d r a g f o r c e , Du2, is not only dissipated into heat but a n appxeciable fraction m a y a l s o b e c a r r i e d out of the control volume a s kinetic energy by the vortical wake. This is shown eqlicitlly i n Sec. 4.1 f o r t h e c a s e i l l u s t r a t e d i n f i g u r e 18a. Thus, if we l e t Ql
=
$ Du2 (Oc f,<
I ) , then (C. 3) can b e written a swhere
F o r the tow tank, A 1 / ~ 2
=
1.035 soThis result i s e9sentially a statement of the simple fqct that the p r e s - s u r e in a pipe should drop i f there i s any dissipation. Without
dissipation the p r e s s u r e drop i s given by C =
-.
070, so that the P 2t e r m - . 0 3 5 $ CD i s a measure of the p r e s s u r e drop due to dissipation.
We shall now go back and estimate the t e r m in brackets in eqn.
(C. 3) which wa$ taken to be unity in arriving a t eqn. (6.4). This t e r m will be estimated by considering the fictitious process in which the vortical wake, u2, flowing out of section PI is ultimately dissipated a s N
heat, Q2, such that the velocity becomes uniform again a c r o s s some section P I 1 farther downstream I a The m a s s and energy balances for the fluid between sections I f ant3 I P P give
and
The process i s fictitious because in the real flow the effects of the halfbody boundary layer a r e bound to become important very f a r downstream and the flow will approach cylindrical Couette flow.
However, by considering euch a fictitious process, we can ignore these effects.
Since A j
=
A2, ug=
u2 and the energy equation may b e rewritten a sBut p2
-
pg>
0 , soBy the same argument used above f o r Q l , we can s e t
Qf
YDu, Q < y < l . In particular, we a s s u m eY
= * which is probably a s g o d a guess a sny, Then the above relation becomes
CD= 1,015 for %=1
which gives the required result.
F r o m eqn. ((2.4) we can now obtain an estimate f o r C
.
F o rP, &#
the blunt halfbody we take CD
=
.8 and rather a r b i t r a r i l y assume p= B.
The p r e s s u r e drop is then given by
This admittedly crude estimate i s probably a s good ( o r bad) for N
=
0 a s i t i s for N>O since CD does not vary much with N (c. f . , Sec. 3 , l ) and the dissipation within the control volume probably stays about the s a m e even though the dominant di s sipative mechanism may change from viscous to ohmic. Hn any case, the estimate cannot be too f a r offsince for C
=
1 and $=
l , ( C , 4) givesD
In the c a s e of the Rankine and ogive halfbodies, the dissipation was almost zero at N = 0, but surely increased a s
N
increased 2,
c o r - r e s p o n d i n g t o i i n i n c r e a s e i n BC D i n e q n . (C.4). Thus, C P2 m u s t h a v e decreased accordingly a s N increased. However, i t i s shown in Sec.4. 2 that CD< 1, so that C cannot decrease below the value given by
3
5
( 6 . 5 )
,
which corresponds t o a maximum base drag coefficient ofC = t. lQ5. Therefore, the base drag coefficients s f the Rankine and Db
ogive halfbodies must vary from their zero-field value of .036 (c, f.
appendix A ) to a maximum value of
.
105 a s N becomes very large.D. Derivation of MFD Boundary Layer Equations
We consider the laminar flow of a conducting fluid past a n insulated flat plate for the following conditions:
In Sec. 3. 1 i t will be shown that CD increases monotonically with N.
3 Some stagnation p r e s s u r e measurements described in Sec. 4. l (pp. 54 -55) indicate that the p r e s s u r e at the lower bePBows of the drag balance does not in fact decrease below this value.
We shall assume that Rm-, 0 so that eqns. ( 1. l a ) and (1.1 b) can be used with
-
B=
-i. The variables a r e then defined a s follows:-
where V , P and Y a r e unknown scaling factors which depend on Re and N. Since we want to relate the results of this analysis to the boundary layer flow over a halfbody, i t was convenient to retain the halfbody diameter, d, a s the characteristic length i n defining the parameters in (D. 1) and the dimensionless variables, x and y, i n (D. 2). Note that this implies x*
=
xd=
O(d). In t e r m s of the variable@ defined in (D. 2 ) , the continuity and momentum equations take the following forms:(D. 3a)
(D. 3b)
1
2-
a';; v - a';; - P s
-NV';;+v v i & + 1 u s j -
--
I!
aY
Now eqn. (D. 3a) can only be satisfied i f V = Y. The viscous and inertia forces in eqn,
(D,
3b) can then be balanced by requiring thatThe only t e r m s that can now be balanced in eqn, (D. 3c) is the p r e s s u r e gradient against the magnetic force. This requires that
Thus, the following estimates have been obtained:
where 6 = W/d i'tq the dimensionless boundary layer thickness. In the limit, We-, ao, N + w , eqns, (D. 3) reduce t o
The f i r s t two equations a r e just the ordinary boundary layer equations f o r a flat plate. The third equation gives the p r e s s u r e a c r o s s the boundary layer once
7
i s obtained from the solution s f the f i r s t twoequations. However, the Blasius solution for
7
leads to 85/& > 0 s s i f this t e r m were included i n the second equation above, it would cause the flow t o decelerate and boundary layer to thicken. Therefore, we must examine this effect m o r e closely in o r d e r t o determine i f i t can cause the boundary layer t o separate, A crude estimate can b e made by evaluating the KBrmgn-PolhPhatosen parameter (Ref. 41):Now from (D.4)
Hence,
Since the criterion for separation i s
=
-12, this estimate indicates that separation i s very unlikely.Thus, we have shown that the ordinary boundary layer -
equations a r e very likely to be valid for the conditions given by
( D
1). 1\90 unusual effects, such a s upstream-growing boundary l a y e r s and reversed flow (indicating a breakdown i n the boundary layer equations) which have been predicted by some theoreticalsolutions (e. g.