Aligned Fields Magneto-Fluid Dynamic Flow Past Bodies Citation Yonas, Gerold (1966) Aligned Fields Magneto-Fluid Dynamic Flow Past Bodies. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/XGWS-3X78. https://resolver.caltech.edu/CaltechETD:etd-09122002-094628 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The drag of spheres and disks has been measured in a flow of liquid sodium with an aligned magnetic field. The experiments were carried out for 10[superscript 4] < Re < 25 x 10 [superscript 4] and N , the interaction parameter, satisfying 0.1 < N < 80 . The sphere C[subscript D] was not a function of N for N < or = 0.3, began to increase appreciably for N ~ 1.0 , and reached an asymptotic dependence proportional to [square root]N for N > 10 . The disk gave a C[subscript D] which was relatively unchanged for N< 10 , began to increase for N~10 , and had approximately the same value as for spheres for N > 20 . We conclude, that for high N , flows are characterized by C[subscript D] insensitive to body shape and emphasize this range in our discussion. A physical model is presented which involves stagnant regions which grow in length as N increases, and are separated from the outer flow by thin dissipation layers. A singular perturbation technique is suggested for the theoretical treatment of such layers. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Engineering Science and Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Engineering Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Liepmann, Hans Wolfgang Thesis Committee: Unknown, Unknown Defense Date: 9 May 1966 Record Number: CaltechETD:etd-09122002-094628 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09122002-094628 DOI: 10.7907/XGWS-3X78 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3484 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 19 Sep 2002 Last Modified: 27 Feb 2024 19:08 Thesis Files Preview PDF (Yonas_g_1966.pdf) - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

ALIGNED FIELDS, MAGNSTO-FLUID DYNAMIC FLOW PAST BODIES Thesis by Gerold Yonas In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Californie Institute of Technology Pasadena, California 1966 (Submitted May 9, 1966) -ii- ACKNOWLEDGMENTS I would like to gratefully acknowledge the assistance of the . following people: Dr. H. W. Liepmann, my thesis advisor, who together with Dr. T. Maxworthy guided my research, Dr. J. D. Cole for his interest in the theoretical problem, and all of the mombers of the Fluid Physics Section at the Jet Propulsion Laboratory, who provided me with many invaluable discussions. I would like to thank the following people who assisted with the experiment: D. Griffith who operated and maintained the JPL Sodium Flow Facility, E. Coury who assisted with model fabrication and assembly, G Hotz and G, Bastien for their help with design of the test apparatus, and K. MacDavid and C. Dunlap who assisted with strain gauge aspects. = would also like to thank Mrs. B. Paul who typed the final manuscript and its many revisions. I am grateful for the financial assistance of ' the Daniel and Florence Guggenheim Fellowship (1962-63), and the support provided by the Jet Propulsion Laboratory (1963-66) for this research as well as for myself. finally, my heartfelt thanks must go to my wife Jane, who has provided the encouragement and patience which was so necessary to the completion of this work. -lii- Abstract The drag of spheres and disks has been measured in a flow of liquid sodium with an aligned magnetic field. The experiments were carried out for 10'< Re<2> x 10" and WN , the interaction parameter, satisfying O.l< N<80O. ‘The sphere Cy was not a function of N for NS&O.3, began to increase appreciably for N+-1.0 , and reached an asymptotic dependence proportional to VN for N>10. The disk gave a Cy which was relatively unchanged for N<10 , began to increase for Nwel0O , and had approximately the same value as for spheres for N>20. We conclude, that for high N , flows are characterized by Cp insen- sitive to body shape and emphasize this range in our discussion. A physical model is presented which involves stagnant regions which grow in length as N increases, and are separated from the outer flow by thin dissipation layers. A singular perturbation technique is suggested for the theoretical treatment of such layers. -ive Table of Contents Chapter a Introduction TT Description of Experiment III Analysis of Data IV Experimental Results Vv Discussion of Results vi Theoretical Discussion VII Conclusion References Figures Page 15 22 26 31 37 50 52 aT a D n Ag A K veV/V, B= 5'/3, P= (P' - P)/ AVS T= TF fe 4/8, H= P+ V°/o Rn = O 4 Va Re = V,a/Y Prn = O LV -V= List of Symbols velocity at infinity magnetia Pield at infinity pressure at infinity conductivity density permeability viscosity kinematic viscosity body diameter tunnel diameter frequency of drag oscillations tunnel area based on diameter D tunnel area minus projected area of body solid body blockage correetion factor dimensionless velocity dimensionless magnetic field dimensionless pressure dimensionless current density dimensionless Bernoulli function magnetic Reynolds number Reynolds number magnetic Prandtl number M= By avo/PY = B/V VPM L= XR =Bodyvu/P N =o? Rn = BOO a/P Vy Sen a/V, Q@ .2 Cy = 8 drag//P Vo 4°77 Cy, Cp, OQ x,¥ Vie 9 5. »P £(N) g(N) & (Rm , N) Riya Uj ,2 € -vi- Hartmann number Alfvén number Lundquist number interaction parameter Strouhal number drag coefficient drag coefficient without field drag coefficient corrected for blockage angular velocity dissipation layer coordinates dissipation layer variables order of magnitude of layer thickness order of magnitude of layer pressure ' order of magnitude of field perturbation generalized Reynolds number used in linear theory (Gourdine) perturbation velocity used in linear theory slenderness ratio used in thin airfoil theory =l- CHAPTER I introduction Magneto-fluid dynamics (MFD) has been widely heralded in recent years as a subject of promise and worthy of great interest. Its mar- riage of tne fundamental principles of electromagnetism and fluid mechanics has provided numerous authors with challenging new problems. Unfortunately, unless the work was motivated by actual engineering applications, its direction has generally been away from problems whieh are physically realizable. This is understandable because of the com- plexity of the exact equations which have forced various authors into using certain simplifying assumptions. Many of these assumptions are clearly non-physical, whereas others are meant to retain some of the characteristic features of certain flow regimes. One restricted problem, MFD flow over bodies, has been particularly popular as evidenced by an admittedly incomplete bibliography given by Sears and Resler (1) or some 71 articles. it is clear that in spite of their effort, few authors would have been willing to predict the outcome of any actual MFD measurement. This alone is sufficient reason for attempting such an experiment. The experiment discussed here is concerned with the drag of bodies in a Slow of liquid sodium with an applied magnetic field parallel to the free stream velocity. We will first discuss only that portion of the theoretical effort which deals with incompressible fluids and aligned fields. In practice, laboratory experiments would have to be done in liguid metals and we will emphasize the material properties of such fluids in our discussion. We must first define the important param- eters which appear in the steady dimensionless equations of motion: V.VV+VP =N(VAB) * Bt2, V7 V (a) VxB = Bm (VxB) () (2) V.B = V.V=O (ce) where _ 7 wee p Poe, cet y a- & the primed quantities being the physical variables. One can show from the TT theorem (2) that in addition to the force coefficient, we can choose three non-dimensional parameters which characterize the problem. The three found most convenient for our work are xa N= By od » interaction parameter PY, Re = Mad , Reynolds number (2) Rm = ouv,d » magnetic Reynolds number N and 1/Re are, respectively, the ratio of magnetic and viscous forces to the inertial force. The physical interpretation of Rm is not.as clear due to the coupling between the momentum equation (la) and the induction equation (1b). Childress (3) has pointed out that by making WN " arbitrarily large with Rm fixed, the relative importance of the induced field can be decreased. This is in contrast to the prevalent view of Rm alone defining the perturbation of the magnetic field. Let us, at this point, further limit our discussion by consid- ering only the case of aligned fields, i.e., problems in which the applied magnetic field is taken to be parallel to the velocity at infinity. Considerable interest has been shown in the more general problem of oblique fields (4, 5) or the crossed field case (6). Stewartson (7) considers the inviscid finite conductivity problem for oblique fields and shows that the aligned field case is nota continuous limit of the nonaligned problem due to a singular nature of the wake structure. We must therefore be careful in extending any conclusions from this special case to the more general one. We are unable to discuss the relationship of the three parameters already noted unless we consider the coupling between (la) and (1b). This is most easily done by linearizing the velocity and magnetic fields. Gourdine (8) has done this for the general steady problem of flow over a body, finding two rotational modes described by the equations (W*- Baga) U2 =O (a) (3) where ~ +t + a _N } A, ,= 2 Re [C1+ Pen) 201+ Pr, > 4 Prin CIB) (0) The ratio Rm/Re = OKAY has been replaced by the symbol Prm termed the magnetic Prandtl number. This material property has the following values: Sodium at 135°C. Prm = 107? Nak at 20°C. Prm = 107° Hg at 20°C, Prm = 107° om 2 The ratio N/Rm more commonly appears as “= 2 the PENG? Alfvén number, and we can consider it as further classifying the types of flows. For xX >? | (sub-Alfvénic), and Prm<¢ 1 =£Re [12/44 & |] ne Again, we consider typical values which could be realized in an a 1,2 experiment: B= 10,000 gauss, v = 50 cm/sec fe) ~2 Sodium at 135°C. N/Re = 10 -2 Nak at 20°C. N/Re = 10 Hg at 20°C. N/Re = 107" We therefore write: La a .=-N,Re (5) 2 On the other hand, for K*Zé | (super-Alfvéniec): a, FRe[ 1 +Pr,* C1-B,,] a2" Re, Rm we can then deduce the general character of the far field from Eq. (3a) for these two cases. This is shown in Fig. 1 (9). The super-Alfvénic flow has two wakes extending downstream, whereas the sub-Alfvénic case has a diffuse forward wake and an ordinary viscous wake downstream. | By considering physical parameters, we have eliminated the -eases N/ReS>1 and Prm}>1. Few authors have treated Prn >» 1 problems, put tae case N/Re>? 1 has attracted a great deal of attention. Chester (10) and Chang (11) have solved the modified Stokes problem under the above restriction. Childress (12) emphasizes that the value of N/Re delineates the important cases. Ee considers the case N/Re >? 1 in detail using an inner and outer expansion technique. He also includes a brief discussion of the more zeneral case suggesting an outer expansion which is seen to depend on N only for N/Re<<l. Unfortunately, he does not discuss solutions of the outer problem for that case in this work, but only shows that this general limit is consistent with his conclusions for N/Re>> 1. Chang's work contains essentially the same conclusions but utilizes several expansions in various regions of the flow. Although the last two papers discussed do not solve the physical problem, they do point the way to techniques which seem very promising. Much emphasis has also been placed on flows of inviscid, perfect conductors. This rather singular case has proven to be the source of a great deal of controversy. Here the material property FPrm is indeterminate and Stewartson (13) argues that Prm-—?0O be taken as the proper limit in accordance with physical problems. If we consider sub-Alfvénie flows, then this describes the limit NP °° , Re-—»oo in the linear problem already discussed. Both wakes shown in Fig. 1 collapse into the x-axis and the problem appears to have the inherent ‘ uniqueness difficulties as does the infinite Re case in ordinary viscous flows. For this case, Stewartson (13, 14) finds that flows over bodies result in stagnant regions extending to infinity, whereas Sears and Resler (15) obtain a solution which involves irrotational fields outside of an infinitesimally thin "magnetic boundary layer" attached to the body. Stewartson (13), using a linear theory, and Ludford (16), by considering the manner in which Alfvén waves alter the flow field, treat an initial value problem and state that the Sears, Resler so- lution can not be set up from a state of rest. In addition, Stewartson (17) considers the magnetic boundary layer concept in detail showing that for sub-Alfvénice flows the boundary Layer would have to include reversed flow at the leading edge. He suggests that a more likely consequence would he separation of the layer and formation of a forward wake. It would indeed be exciting to be able to resolve this controversy through experiments. We are, in fact, tempted to compare the sub- Alfvénic, large N results with the predictions for large Rm. The linear theory seems to point to this since Rm is no longer a parameter for very sub-Alfvénic flows, Eq. (5). Unfortunately, this is solely a consequence of the linearization, and such a comparison would be mis- leading. in order to show this, we must consider the non-linear problem. This is best done by utilizing an equation derived by Tamada (18%). He considers the inviscid momentum equation rewritten in terms of the . dimensionless Bernoulli function H =P + v/2 VH+t VK (Vx V)=*5xB Dotting both sides with V > we get oa - 2m _ _ VeVHE-O 7. CVXBD (7) If Fm = © and we require V parallel to B at infinity, then one can show from (1b) that VxXB=O Hence - V-VH=O (8) On the other hand, we can have N--> S@with Rm finite by letting Ot» SO | omen — x . -- a 72 (9) VeVH Bm on and H can decrease along streamlines in contrast to the result for Rn = © . If we consider the physical problem again, we get for d=lcm, By = 10,000 gauss, Vo - 50 cm/sec Oo Sodium at 135 C. N = 100 Rm = 0.1 = 1000 Nak at 20°C. WN = 100 Rm = 0.1 ™ = 1000 (@) Hg at 20°C, N = 1.0 Rn = 0.01 = 100 We must conclude that for physical problems, large N flows are compatible only witn Rm—>O , and experiments are incapable of shedding light on the sub-Alfvénic infinite conductivity problem. Other authors have considered problems of greater physical interest, and also of much less mathematical complexity. Reitz and Foldy (19) consider the first-order perturbation to the pressure distribution by using the unperturhed potential velocity field and magnetic fields to calculate the current and the body force. ‘They also show that drag found from integrating the pressure gives the same result as integrating the total joule dissipation. Leonard (20) considers the same approximation, but in addition treats the case of small interaction but arbitrary Rm by using a relaxation method. Under similar conditions, Seebass and Tamada (21) find an exact solution for the magnetic field by using an integral equation derived from a Green's function technique. Finally, we must note that Tamada (18), by considering a perturbation expansion for small N , predicted that a nondiffusive velocity disturbance would exist downstream of the body, and Leonard confirmed this in his work. These small interaction approximations could describe flows in which boundary layer separation plays no role; however, one certainly could not expect that such solutions would be of any relevance to high Re flows. over bluff bodies. On the other hand, one could use such a technique to describe flows about semi-infinite bodies. Ahlstrom (22)has investigated the magnetic field distribution upstream of a -10- Rankine body moving through a tank of mereury. He utilizes such small interaction assumptions to predict the decay rate of the magnetic field in an unbounded fluid. He finds, however, that the snielding effect of the field coil on the moving field dis- turbance is dominant but is able to take this into account by considering an effective boundary value problem. One other experi- ment has been carried out that falls into this class. Motz (23) has measured the drag of an oscillating sphere immersed in mercury in an aligned field. There too, the zero order problem (no field) is describable by an inviscid flow field. He uses a theoretical explanation identical to that employed by Reitz and Foldy and finds excellent experimental correlation with those predictions. in the above discussions, the operator V.V in the equations of motion is replaced by Vy.V » where V5 is given by the poten- tial velocity field. In this way a simpler linear problem is obtained. Another approximation is that of Oseen, V,V and B,V replaced by & - This is done for arbitrary values of the parameters in dis- cussing the wake behavior as Gourdine (8) does. Another view is to consider a thin airfoil theory which permits one to obtain a uniformly valid approximation throughout the entire flow field. This technique is used by Lary (24) for inviscid flows, who finds limits on the 2 validity of such an approximation. He finds for O° >7] , Hm finite, two-dimensional problems: -ll- Cc, 64n for N<1 Cy-EVT tr E< xv ede where € is the slenderness ratio of the body. He attempts, however, to extend his results to the case N >> EZ by using the magnetic boundary layer concept developed by Sears and Resler (15). The resulting no~slip condition on the magnetic field may be a consistent model for steady infinite Fm flows, but it certainly can play no role where diffusion of the magnetic field is present. As we saw earlier, the analogy between infinite Rm and infinite WN flows was a mis-~ leading consequence of tne linearization, and that the infinite conductivity approximation can nave no physical relevance. in an attempt to resolve the infinite conductivity problen, Stewartson (7) extends the inviseid thin airfoil approximation for fluids of finite conductivity by considering three separate problems. He first considers a linear problem similar to Lary's, but unsteady. He then takes the limit of the non-aligned problem as the angle between the velocity and magnetic fields tend to zero. There is general agree- ment on the character of the flow field and the predicted drag if N and the elenderness ratio are limited as Lary deceribed. Stcwartson emphasizes the restrictions on this linear theory and points out that a different point of view is required for very sub-Alfvenic flows with finite Im, i.e, N becoming arbitrarily large. The third problem -12- he considers, the "strong field case," is not completely resolved though. He suggests that stagnant regions exist in this case, but he does not answer how one is to match such a near field to a far field approximation which is uniformly valid at infinity. We have seen that consideration of the non-linear aspects of MFD problems have led in many cases to conclusions entirely different from those of the linear versions of the same problems. We can see from Eg. (9) that any linear theory will satisfy Eq. (8) since J is a perturbation quantity, and the change of the Bernoulli function is totally lost. The principal consequence of this, as pointed out by Tamada (18), is that the Oseen approximation misses the existence of the downstream vortical disturbance for inviscid flows. By con- sidering the non-linear inviscid problem, Childress (3) concludes that general solutions are not necessarily continuous at infinity. He too admits the possibility of a “stationary wake" or a non-diffusive vortical region allowing finite velocity disturbances to reach to infinity downstream. The action of a small viscosity will be to diffuse this velocity perturbation leading to continuous behavior at infinity. The problem then appears to involve a consideration of a non- linear near field and the matching to the proper far Pield which “ineludes both joule and viscous dissipation. This theoretical problem is seen to coincide with the experimental restrictions which have already been delineated: -13- Re > 1 , N/Re<¢l, Pm<c 1 , and (10) Rm, WN arbitrary. If in addition N>>l , Rm<<l, then we could expect that magnetic forces are dominant near the body and an inviscid analysis could be attempted for the near field. The difficulty of the problem is apparent and one would like to have some experimental knowledge that at least the description of the problem is correct. It was hoped that a measurement of the drag would tell us which parameters were important and what ranges of those parameters were critical. Liepmann pointed out the importance of such an experiment, and _ attempts to measure MFD drag on bodies by timing the rise of buoyant spheres in the GALCIT mercury tow tank were begun in 1960 (2). These experiments proved to be unsuccessful and a technique using a sting mounted drag balance was started simultaneously with our work (25). Maxworthy, utilizing the terminal velocities of spheres falling through a column of sodium in an aligned field, noted a marked increase in drag for strong fields (26). The parametric range agreed with condi- tions (10) , but was limited to N< 3.7. He states that the drag coefficient scales with N/Re only. Private communications with him ‘since then indicated that those results should be viewed with skep- ticism due to difficulties involved with the experiment. Upon his suggestion, measurements of the drag of bodies were carried out using the JPL Sodiwn Flow Facility (27). There were definite advantages over ~- Lye the previous techniques: a) The field boundary conditions encountered in an unsteady experiment would not exist. b) Conventional drag balance techniques could be used. c) High values of N and Re could be obtained. In the following chapters we will discuss tne experiment and its results. We will present, in Chapter VT, a theoretical model motivated by the experiment, but a complete analytical treatment will not be attempted. We will, however, suggest the proper de- scription of the physical probiem and one possible way of solving it. -15- CHAPTER II Description of Experinent “The liquid sodium flow facility at the Jet Propulsion Laboratory, shown schematically in Fig. 2 (27), was designed to function as a basic research tool in magneto-fluid dynamics, its role being similar to that of the wind tunnel in ordinary fluid dynamics. Although much engineering interest has been stimulated in cross-field flows, we have seen the con- siderable amount of theoretical efforts contributed to the aligned field case. In particular, it was desirable to create flows in which the interaction parameter could be made much larger than one. Because of its high conductivity and low density, sodium seemed ideal for this purpose, but as Cowling (28) points out, "no liquid can be regarded as such when it forces an experimenter to wear asbestos clothing when approaching his apparatus." Not withstanding the difficulties involved with handling the fluid, the next most important concern is to really achieve a situation simu- lating aligned fields. Due to the physical constraints on the length of the test section and magnet, end effects are of concern. At the ends of the magnet, the fringing field interacts with the flow, giving currents which can be convected to the central portion of the test section. -In an attempt to minimize the currents generated at the entrance to the test section, an entrance nozzle which approximates the shape of the field was employed. Although this proved to be quite successful, ~16- no such attempt was made to minimize the disturbances created at the exit. As a result large disturbances were created there and under the most severe conditions did reach the central portion of the test section. It was therefore necessary to measure the velocity at the same iocation as that of the drag measurements. In order to do this, the stagnation pressure was measured in the entrance nozzle, and the static pressure was measured using a conventional static pressure probe. The results of these measurements at several field strengths -are shown in Fig. 3. An extensive experimental investigation of this "exit effect" has been carried out by Maxworthy (29). He discusses there the suitability of the Pitot tube in such flows and limitations on the usefulness of this flow facility. He coneludes that the velocity distribution is sufficiently uniform over the central half of the test section diameter to study flows over bodies under the most severe conditions that we can presently achieve. It is clear though that measurements at stronger fields than now presently employed would necessitate the use of either a longer test section or a shaped nozzle to minimize the interaction at the exit. We can see from Fig. 3 that the centerline velocity without field is not e linear function of the flow meter output voltage. An inte- gration of the velocity profiles shows that the flow meter voltage is indeed a non-linear function of the flow rate below 6 mv. The flow meter is of the standard circular, transverse-field type manufactured -17- by the MSA Research Corporation, and such a meter is generally con- sidered to be linear in its behavior. This is true only if the velocity profile is axially symmetric, and the field uniform. Shercliff (30) discusses this matter and emphasizes that difficulties can occur in liquid metals for low flow rates. We have already dis- cussed the importance of the interaction parameter and expect that for Nel the velocity profile can be altered by the field with a resulting loss of the axial symmetry. In fact, N , based on the flow meter diameter, has a value of 5 for an output voltage of about 6 mv and it is not surprising that it has non-linear behavior for lower flow rates. The magnet was dcoignea so as to maximize the length over which the interior field was uniform. The actual distribution was measured using a RFL Hall effect gaussmeter. The field was found to be uniform over the central 70% of its length to within 3. A calibration of the magnetic field at the body location, as a function of tne power supply shunt voltage, was carried out using a Bell model 240 incremental geussmeter. The Tall clement used with this insbrumeul was callbraled using a standard reference magnet giving an accuracy of z 1%. This was not done with flow and one might consider the possibility of the field being perturbed duc to currenuls propagallug upsbream Crom the exit. From the pressure distribution measurements, Maxworthy (29) has estimated the field perturbation due to the exit effect and states that 1b would be less than 1%. -18- Two magnet power supplies were used: A. 0. Smith Company rectifier power supply, 1100-2500 Amps, <= 8% ripple peak-to-peak; Cristie Electric Corporation rectifier power supply, O-1000 Amps, <= 1% ripple peak-to-peak. Fortunately, the ripple in both cases occurred at 360 cps which results in a skin depth in aluminum of —.3 cm. Since the test section wall is —.5 em thick, we can safely assume that the magnetic field was steady at the body location. A strain gauge balance was chosen for the drag measurement due to its simplicity, small size, and high temperature capability. A full strain gauge bridge was attached to an aluminim heam located in a chamber adjacent to the test section wall (see Figs. 4 and 5). A wire suspension system was chosen so as to minimize support drag and interference effects. The hodies were suspended from three tungsten-rhenium wires attached to three identical beams and pre- set at an angle that maximized the sensitivity. The wires pass from the interior of the test section into the gauge chamber through 1/16" holes. The chamber was packed with a silicone grease, Dow Corning il, to damp beam oscillations and to prevent sodium from reaching the gauges. No other protection was provided for the gauges, but all solder joints in the chamber were encapsulated in silicone rubber. The strain gauge bridge input voltage was provided by a Dynamics Corporation, Model 63:3, bridge supply, and the steady component of the drag was measured on 4 Keithly model #149 milli-microvoltmeter. -49- The drag balance also served as an indication of the unsteadiness of the flow, although it was by no means intended to provide quanti- tative information of thal kind. The unsteady component was amplified using a Dynamics Model 6496 differential D. C. amplifier and recorded on a Honneywell Visicorder oscillograph. The drag balance test section was fabricated in two halves to allow for easy installation of various bodies and to permit cali- bration before and after each run. The upper half of the test section is shown in Fig. 6 attached to the balance used for calibration. The accuracy of the balance was Zl gram and the full scale of 1600 grams (force) was utilized. Although the strain gauges were mounted on a thin strip of aluminum within close proximity of each other providing excellent temperature compensation, the drag balance itself was still quite temperature sensitive. This was due to the lengthening of the wire and slight changes in the gauge factors of the strain gauges. it was therefore necessary to carry out the calibration at the operating temperature of 135°C. The temperature of the gauges was measured using a chromel-alumel thermocouple embedded in the beam. During the cali- bration, the test section was filled with Dow Corning 200 fluid, wrapped with an electrical-neating tape and insulated. It was necessary to heat, then cool, the test section many times prior to calibration to remove air bubbles trapped in the grease. On cooling, the volume which had been occupied by the air was filled with the silicone fluid. If this -20- were not done, the gas bubbles adjacent to the beams would expand or contract with changes in test section static pressure. Some difficulties that arose for small loads, less than 10 grams, were attributable to this effect. The two types of models used in this experiment, disks and spheres, were made from either non-magnetica stainlescs steel or brass. Tungsten-rhenium wire was employed for its high tensile strength and improved ductility over ordinary tungsten wire. Dif- ficulties in supporting bodies arose due to fracture under fatigue, ‘particularly at any edges over which the wire passed. In the final configurations, shown in Fig. 7, we see that all sharp bends are contained within the bodies and that the wires are constrained between two surfaces as they pass from the body. Data was taken by pre-setting the magnet current and varying the pump speed. With the field on, all zeros were set and the flow rate was increased in regular increments. The following quantities were measured for each data point: 1. bridge output voltage; 2. strain-gauge beam temperature; 3. magnet power supply current; 4, flow-meter output voltage. A selector switch was used to choose between these inputs which were all measured with the milli-microvoltmeter. A separate apparatus was employed to study possible interference of the bodies on the tunnel boundary layer. The Reynolds number based ~2]- on the length from the start of the test section, Re, » ranged 3 and 3,500 x 10°, The laminar wall boundary layer between 140 x 10 which would exist at least at the lowest velocities employed might have separated due to a small region of adverse pressure gradient created by the presence of the sphere (31). In order to investigate this possibility, an air Plow visualization experiment was set up using an entrance nozzle previously employed in another experiment (32). An appropriately scaled test section made of plexiglass was used, and the velocity of flow was measured with a Pitot tube and a Betz manometer. Two visualization techniques were employed, oil injected into the boundary layer and cotton tufts attached to the suspension wires, neither of which gave any evidence of separation. We concluded that the tunnel boundary layer remained attached for Rez, > 100,000. -22— CHAPTER IIL Analysis of Data The following configurations were utilized: Run #1 -250-in. sphere - .005-in. support wires Run #2 -500-in. sphere = .01Q-in. support wires Run #3 -750-in. sphere ~ .O01O-in. support wires Run #4 .500-in. sphere - .CO5-in. support wires Run #5 -500-in. disk ~ .Q010-in. support wires. In each case the drag balance was calibrated before and after each run to ensure reliable results. A typical calibration curve is shown in Fic. (8). For this particular mim the balance was linear for loads less than 300 grams. The extent of the linear range varied greatly for different runs, being extremely sensitive to the initial angle of the suspension wires. The linear behavior existed generally only over a small portion of the total range of values measured, and most of the measurements had to be converted to drag in grams using such a curve. The calinration and interpolation could be carried out with an accuracy of * 1%, Raw data for the .750-in. sphere is shown in Fig. (9). Smooth curves were drawn through the data points, and the values used in the final computation were taken Prom these curves at flow rates denoted by the dashed lines. This facilitated computation and reduced the scatter for a particular field strength. There was some latitude in drawing these curves, and we can attribute an accuracy of z 2% at best to this technique. The effect of strong fields on the cenlerlinue velucily has already been noted. The results of the velocity measurement were shown in Fig. (3). We see there that the largest magnet current, 2500 amps or 7050 gauss, has a measurable effect over the entire velocity range, whereas currents less than 1500 amps, 4260 gauss, do not affect the velocity for flow meter voltages greater than i millivolt. For each combination of magnet current and flow meter voltage, we find a free~stream velocity either from these curves or by interpolating for other fields. Inaccuracy in this separate measurement, curve ritting, and interpolation contribute an inac- curacy which can be estimated as * 24, The material properties for sodium at the operating temperature of 135°C were found by interpolation from data appearing in the Liquid Metals Handbook (33). ‘They are Pf = .918 x 10° kg/m? o A Although the temperature of the entire facility is regulated, records .911 x 10! mho/meter 575 X 10° ke/meter second. fo) of the test section temperature show a variation between 132°C and © + 138°C. This results in an additional inaccuracy in LP of - .1% ; in © of 51%, and in 4 of = oh. ~2h. In calculating the sphere drag we must subtract the wire drag from the total force measured. We assumed that the wire drag coef-~ ficient was not affected hy the field and used values for this quantity given in Schlichting (31). This assumption will be discussed later. The entire length of the suspension wires was used in this calculation with no allowance made for the test seetion wall boundary layer. On the other hand, the free stream velocity was used in this calculation which does not take into effect the accelerated flow near the body or the blackage effect. Fortunately, this uncertainty is relatively unimportant in most of the measurements because the wire drag itself is only a small per- eentage of the total drag. This is true because the ratio of the projected area of the wires to that of the sphere for Runs #-4 are, respectively, .25, .10, .05, and .05. We see that the greatest diffi- culty arises for Run #i, ond we cxpeet that inaccuracics in cotimating the wire drag will appear most strongly there. If we estimate an inaccuracy of t 5% in the wire drag calculation, the resulting in- accuracy in the calculated drag cocfficient will be, respectively, 2.5%, 1%, .5%, and .5% for Runs #1-4 for the no-field case. As the field strength is increased, the wire drag becomes a smaller percentage or the toval, and this inaccuracy becomes unimportant. The calibration of the magnet as a function of the power supply shunt voltage was carried out with an accuracy of z 1%. However, due to the changing temperature of the magnet and drift of the power supply 25 itself during a run at a given current setting, we must attribute an inaccuracy of = 2% to the value of magnet field used. in the calculation. Without field the resulting maximum inaccuracy in the calculated drag coefficient is, respectively, t 9.5%, : 8%, = 7.5%, and z 7.5% for Runs 1-4. For strong fields the Lnaccurucy in wire drag becomes negligible and this estimate becomes ~ 7% for all runs. We will see later that the only other important nondimensional parameter is N, . . . . t aud we eeblmate lbs maximum iuaccuracy as = 3.5%. -25- CHAPTER IV Experimental Results The drag coefficient, Cp , is plotted in Figs. 10, 11, 12 as a function of Re for Runs 1-3. The ranges of parameters can be summarized: 10,000 < Re < 250,000 O< M< 1660 The range of Re is that which is characterized, for no field, by a laminar sphere boundary layer, separation upstream of the center of the sphere, and Cy relatively insensitive to Re. For a fixed Re the Cp) increases monotonically with field, and for a fixed M it decreases monotonically with Re. Any exceptions to this are attributed to experimental scatter. In Figs. 13, 14, 15 we see Cp plotted as a function of the interaction parameter N for various values of the Hartmann number M. The range here is o<N< 80 B2cd / Neo completely checked except by carrying out experiments in different The correlation with the paraueler, WN = » Cau uol be fluids. In each run we only vary B, and V, and we see that Cp behaves like a function of By /V, . The results of these runs are summarized in Fig. (16) where we show smooth curves fitted to the data shown in Figs. 13, 14, 15. We see that d/D , where D is the test -27- section diameter, remains as an independent parameter. For N210 the dependence of Cp is seen to be proportional to N@ and for NS .3 it is independent of N. In both of these ranges we can estimate the blockage correction using a straightforward technique and compare it with the results shown in Fig. (17). ‘There we see Cp plotted as a function of a/D for N=O and N20. The experimental scatter, shown by the horizontal bars, is approxi- mately that predicted in Chapter III. Smooth curves are drawn and an extrapolation is made to d/D=0. We assume that the blockage constraint has the simple effect of increasing the velocity that shouid be used in calculating the appro- priate nondimensional parameters. Maskell (34) and others have discussed a blockage correction for bluff bodies, but this author was unable to find any experimental or theoretical results for spheres. For flows about bodies with a sharp edge, the separation point is fixed; however, it is possible that the blockage might have the additional effect of shifting the point of separation on a sphere. We attempt, nevertheless, to use a blockage correction based simply on the solid body blockage constraint: v,/V = A/A, = (1 - a“/D°) =K For the no-field case the extrapolated value is found to be Cy = 40, and for N>10 it is given by Cp/VN = .33 . This then yields -28- . and Cc, = .33 VT «3/2. S40 The following table gives the results of applying such a correction: ao «2 G ge xe Cyto x3/8o7.33 vi N=0O N>10 N=0O N>10 0 1.00 1.00 40 33 1.00 1.00 -125 .963 .977 .42 34 1.02 * 1.01 -250 .878 .906 45 38 .988 1.04 -375 .738 «795i 24.3 - 90). 1.04 The correction is seen to be well within the scatter for N>10 but seems to overestimate the blockage correction for N =0 > L/D = 375 . For N< 10 neither Cp nor Ch ~ 1 obeys any simple power law with N. It was found that although no simple blockage correction could be applied, Cy/Cp, ) where CD, is the drag coefficient without field, is independent of d/D . This is shown in Fig. (18). As we have seen, the sphere drag coefficient is relatively unchanged for N<1. In reducing the data we assume that the wire drag has similar behavior, and since N € 1.0 , based on the wire diameter of -OlO-in., we have strong support that this is a good approximation. In addition, Run + was specifically carried out to check this. The drag coefficient for the .500-in. sphere was remeasured using .005~in. support wires. The results of this measurement are shown in Fig. (19), super- imposed on those for Run #2. In order to investigate the effect of body shape, the drag of 4 -500-in. disk with .010-in. suspension wires was measured. Cy vs Re for Run #5 is shown in Fig. (20). We see that the simple monotonic behavior which was characteristic of sphere drag does nob appear here. At nigh Re , Cp is in fact lower with a field applied than without. In order to clarify this, we must again consider the effect of the blockage constraint. Fortunately, considerable experimental and theoretical work has been carried out concerning the problem of blockage effect on disks. Maskell (34) points out that solid body blockage alone is insufficient to explain measured drag coefficients. With no blockage the disk wake is roughly twice as wide as the disk itself, and wake blockage must be taken into account. He establishes a model which is partly based on empirical evi- dence since no complete theory for bluff-body wakes exists. He presents a formula which is supported by the experiments of Fail, Lawford, and Eyre (35). 2/2 Cp = Cy/(1 + 2.5 Cy a/D°) where Cy is the measured value and Cy, the corrected value. The effect of this correction applied to the M=0 result is shown in Fig. (21) along with the behavior of Cp as a function of N. We see that -30- although Cp drops below the no-field value in the range 1<N<10 , it rises again for N< 1. Within the accuracy of the experiment, it never drops below the corrected value. | Figure (21) also shows a comparison with the results for the .500-in. sphere. We see that for N> 20 there is good agreement between the two, and the asymptotic behavior of VN is again reached. The drag balance is sensitive to both unsteady lift and drag. Its response to an unsteady force is dependent on not only the frequency range, but also the steady force. Quantitative results would necessitate a difficult calibration procedure and at least one additional gauged beam. Hopefully, by measuring unsteady force at locations 120° apart, one could separate the iift and drag. This was not attempted, and oscillograph traces of the unsteady force were used only to give a rough idea of the dominant frequencies. The traces demonstrated the existence of unsteadiness under certain conditions, and one could define the approximate range of the Strouhal number, 5S = nd/V, ,» that appeared. Without field S was approximately .20 and it was clear that marked changes could occur even for weak fields. Fields of about 0.30 web/m@ eappearec to give steady drag over the full velocity range. Unsteadiness was apparent again for fields of approxi- mately 0.50 web/m* and the frequencies decreased as the field was further increased. The Strouhal number of these strong field oscil- lations was quite small and dropped from a value of approximately .09 at 0.50 web/m@ to .03 at 0.70 web/m= . -31- CHAPTER V Discussion of Results The results of these measurements as illustrated in Figs. (17), (18), and (21) can be summarized here: 1 f (4) Cy = 40 (1 - a&/D*) “ N= 0 sphere -3/2 drag (44) Cp? .33VN (2 - a°/D*) N > 10 (iii) Cy # Cy F(N) O0<N~<10 where F(N) is relatively unchanged from F(N=0) for O<N<l , and approaches the behavior WN as N approaches the value 10. ~1 (iv) Cy # .11 (1 - 2.8 a“/D°) N=0 disk | 2, 2-3/2 drag (v) C+ .33 VN (1-4 /D) WN > 20 (vi) Cy G({N) Oo<N<20 where G(N) is relatively unchanged from G(N=0) for O<N<l1, decreases to the value ~ 1.10 for 1<N<10 , and approaches the _ behavior VN as WN approaches 20. We must first compare the measured sphere Cp Yor N= 0 with results quoted by others. Although we have loosely described the drag coefficient as being constant in the range 10,000 < Re < 250,000, ~32- previous empirical evidence does in fact show a systematic variation. Schlichting (31) exhibits a compilation of the work of various experi- menters clearly showing this. One can see there that Cp increases from .39 to .48, then decreases to .40 as Re increases from 10,000 to 250,000. Due to the scatter in our zero field data, no such system- atie variation ean be obscrvcd. The extrapolated value of Cp = 40 Por a/D =O should be compared with the range of Cy that exists for the same Re range. The accuracy attributed to our value is z 8% so the range of C, found here is -43 - .37, and overlaps the previous experi- mental evidence. We have already pointed out that the solid body blockage correction seems to overestimate tne increase in drag for a/D = .375 with no field. It is possible that movement of the separation point downstream is respon- sible for this effect. If this exists, it is also possible that the extrapolation to d/D=0 does not properly take it into account. It is beyond the scope of this experiment, or for that matter, its intention to investigate this effect more fully. We are primarily interested in the strong field case, N>10 , and we noted that the solid body correction factor adequately described the behavior of Cp as a function of a/D . We can therefore surmise that the field acts both to fix the point of separation and to constrain the wake. We conclude from this that the boundary only increases the effective free stream velocity which must be used in calculating the proper nondimensional parameters. For N~<10 we saw that Cy/ Cp, was only a function of N, d/D being eliminated by this normalization. Additionally, the critical value -33- of N at which c,/Cy began to increase appreciably was 1, and the fe) value of N at which the final behavior of VN was reached is given approximately by 10. N is thus seen to delineate various flow regimes, which we might describe as follows: N/Re << 1 Rn“ 1 N<l Separation of laminar viscous boundary layer dominates. Field has little effect. Ll< N <10 { Intermediate range where separation of viscous boundary Rn Sl layer and MFD forces both play a role. N>10 {MPD forces dominate. Rn << 1 Asymptotic high N behavior is reached. We have shown that N/Re was also an important parameter in our discus- sion of the theory. We must emphasize that N/Re < 107° throughout this experiment so that the asymptotic behavior N>10 must include the restriction N/Re<<1. We expect that N/Re would again enter if we continued to increase N , holding Re fixed, so that N/Re exceeded 1. As already mentioned, this is not possible using conventional field strengths, Bowl web/m= . In the laboratory these high values of N are achieved by reducing the free stream velocity, and as a result we are also restricted to Rm <<1. -34- We argued that the wire Cy had similar behavior with N as that of the sphere. Since Njs,.,< 1.0 we assumed that C. is given by its D zero field value and had no interference effect on sphere drag. This was also supported by experimental results shown in Fig. (19). Such wire supports have been investigated by others, and it is known that as long as the flow is subcritical (laminar boundary layer separation on upstream side of sphere), the wires have little effect (36). Such a support does, however, have a marked effect for the supercritical range which is never reached here. We conclude that the wire interference is negligible without field as long as the separation point is upstream of the point of attachment of the wires, and with field and N >1 the magnetic field dominates the sphere wake behavior and drag coerticient. The disk drag coefficient was shown to be adequately corrected for nero field fron Cp = 1.35.15 to 2.10 1.10. ‘This shows good agreement with the value given by Fail, Lawford, and Eyre (35) of 1.12. For N< 1 the disk drag coefficient is little affected, and as before, 1<N< 10 appears to be an intermediate range. Although the separation point is fixed, the blockage effect leads to an additional complication. For 1<N~<10 C) is actually lower than the value for N=0, but it does not fall below the corrected value within the accuracy of the experi- ment. The magnetic field in this intermediate range nas the apparent effect of eliminating the blockage constraint. We must emphasize, however, that we do not expect that the physical situation is a simple one. One can conclude that more detailed measurements of the flow and magnetic fields are required before we could hope to understand any such range ~35- where two or more different effects are important. For N>i1cC the disk Cy is very close to that of the sphere and appears to reach the sume asymptotic dependence for N> 20. This strongly implies that for N>10 the flow field is relatively insen- sitive to the shape of the body. We have already stated that insufficient evidence was obtained to draw quantitative conclusions concerning wake stability. There were indications though that weak fields were able to completely damp the dominant frequencies that existed without an applied field. This is consistent with the physical concept of the field providing a strong restoring force for oscillations normal to it. Similar results were obtained by Maxworthy (26) who noted non-oseillatory wake behavior for Nw .5. In addition, the suppression of turbulence due to an aligned field was noted by Globe (37) using a pipe flow of mercury in an aligned field. In this Light, the reappearance of unsteadiness at much lower frequencies for strong fields is somewhat surprising. These low fre- quency oscillations appeared when the steady drag and presumably the flow configuration were considerably different from that with no field. it is possible that the wake was able to selectively amplify long wavelength disturbances which existed in the flow. Wake measurements of the fluctuating Pield or velocity would be required in order to understand this result. The trend toward lower Strouhal numbers with increasing field seemed clear, although there seemed to be no dependence on N itself. On the contrary, Cyn was a function of N only, over the entire range of N. We concluded from -36= this that the oscillations were not an important factor in determining the steady drag. - The converse is not necessarily true and more quanti- tative wake measurements appear to be required if one is to more fully understand this interesting observation. We suggest a steady model, but realize that our support is based on incomplete evidence. Complete justification of the use of this model could follow only from further measurements of wake structure or a self- consistent anaiytic solution based on the model. We can not present either here, but suggest the former as being the most likely of success. We will, nevertheless, propose a steady model which we hope would contain the dominant features of the flow. The experiment suggests at least one | range of parameters where such a model might be simple. This range was discussed in the introduction as being one that has not been emphasized in the Literature: Re >>1, N/Re<< 1, Prm<<1, %Rox<<1, N>>i (12) With the additional information derived from this experiment, it is hoped that further theoretical progress can he made for this ease. The key results which would have to be an integral part of such a theory would be: a) Cp . 7 (12) vb) Cp is insensitive to shape of the body, -37- CHAPTER VI Theoretical Discussion Our objective is to provide a theoretical model which will include the results (12) under the conditions (11). Unfortunately, we have not succeeded in this although certain general concepts can be exposed. Our purpose then is to present a picture of the relevant problem and a pos- sible physical model. We suggest that for high N , finite disturbances in the velocity can propagate along the magnetic field to large distances by means of weakly damped Alfvén waves. Stewartson (7), (14) proposes that for >> 1 » N —® co , such waves can bring the flow to rest witnin the area projected by the body in the direction of the applied magnetic field. Consequently, the final steady flow is characterized by stagnant regions extending to infinity upstream and downstream of the body. We consider a flow configuration for finite but large N which involves stagnant regions of Pinite length and ask: How long will such regions be, and what will the pressures be in them? let us first consider the question of Alfvén wave propagation for finite N. In low N experiments such as that of Ahlstrom (22) or Motz (23), propagation of Alfvén waves can play no role. Motz has shown that one can safely use the local potential flow field in calculating Currents and body forces. On the other hand, i? Noeoeil , Alfvén wave propagation is almost ideal in an unbounded fluid and must be important in setting up any final steady state from a fluid initially at rest. Lunaquist (38) points oub thal the necessary coudlliou fur the existence -38- of undamped Alfvén waves in a medium of infinite extent is 2 w << Bo 7 (33) If we substitute = v,/4 as an effective frequency for the initial motion of a body, this condition becomes N>>1 (14) Lundquist and Lehnert (39) consider the problem of torsional waves set up in a tank by a rotating plate for both mercury and sodium. In their problem the currents flow radially and the condition J=0 at insulating walls results in an additional mechanism for dispersion. Lundquist shows that low frequencies are strongly damped since the corresponding wave lengths normal to the applied field are larger than the dimension of the apparatus and gives the additional condition: | D*Ouw Z| (15) where D is a length on the order of the dimension of the apparatus normal to the applied field. He combines conditions (12) and (14) to give the requirement: Lp Zp2 2 2 Bs O"DM 5 | LY (16) -39- OD ; The number L = Boo Dv can be interpreted as a magnetic Reynolds VP v4 number based on the Alfvén speed rather than the fluid free stream speed, and has been called the Lundquist number (40). It is not clear to this author that the Iundquist number is. an important parameter in high N flows over bodies. Thais thought is pased on the above model that involves disturbances strongly constrained _to the projected area of the body, in the direction of the field. If such a model is correct, the only importan. boundary condition is fluid dynamic, no flow normal to the walls. In such a case condition, (14) rather than (16) is appropriate. The experiment confirms this since WN alone appears to be the important parameter and a simple blockage cor- rection is adequate for N>10. We must emphasize though that for our experiment L~ 1 and one would have to carry out a similar experiment for L<<1, N>>1 #42to show that L does not enter. Such an experiment might be carried out in mercury since, for the same field strength and length scale, Looqiyn A 39 Lnercury » Since the fluid velocity does not enter into LL, the same value of NWN could be obtained but at a lower value of LL. | If we allow that slightly diffused Alfvén waves can exist, what support is there for a description of the flow field which involves such large stagnant regions? First, such a model would explain the experi- mental result that Cy is insensitive to the shape of the body for large N. Second, we think that work in rotating flows lends support to such an idea. Many authors have suggested that certain analogies can exist between the two problems. One case in particular is quite similar, -lLiO- that being for strong rotation, slow flow, so that: 20d? (Ekman number) = coriolis force/viscous force = a7 1 v (Rossby number) = coriolis force/inertial force = ag >> 1 in that case the results ror flow about a body are summarized in the “Taylor -Proudman theorem (41): "All slow steady motions in a rotating fluid are two-dimensional." Hide and Roberts (42) state that the analogy is not complete, but this seems to be a result of the order of which they take the strong field, inviscid Limits in the MFD case. Gourdine (8), on the other hand, utilizes field splitting prior to taking this limit and avoids the ambiguity of taking limits in one equation which involves two dissipation mechanisms. The result is given in equations (3)-(5) and for N—»og, Re —» oo is: which is identical to the conclusion in the rotating case. Recent experi- mental evidence due to T. Maxworthy (private communication) lends edditional support to this model for rotating flows. Te has found that spheres moving through strongly rotating flows do show such stagnant regions. It is noped that detailed measurements of the flow field as well as the drag in that case will assist in our understanding of the MFD problen. ~-1- Let us consider the two-dimensional problem of steady flow over a finite conductivity body which contains no sources or sinks of magnetic field. SGince the problem is steady, wx HE=0O , and it follows from conditions at infinity that E is a constant equal to zero throughout the flow field. The tangential electrical field is continuous at the surface of the body, so it must be zero inside. Hence,no current flows inside the body. and the Lorentz force does not contribute directly to the force on the body. The body force does affect the pressure distri- bution though, and thusly the drag of the body. We begin with the inviscid problem for N large, but finite, ana will show that in order to have a consistent picture including arbi- trarily large drag, non-zero viscosity will again be required. Jn the Limit Rn—> 0, N-—-»co , Re —» co we consider the thin dissipation layers which separate the stagnant regions from the undisturbed outer flow. We write the variables x=xX=0(i); (y-1) = y/f(N) = 0 (1/f(N)) Ve = ¥, = 0(2) 5 vy = W/e(n) = 0 (a/e(n)) B= Bp=0(1) 5 a, = B/e(a) = 0 (2/2(H)) p = g(N) P = 0 (g(N)) lou where all tilde quantities are 0O(1), and el (y) and g(N) give the order of the layer thickness and pressure. We assume that f(N)—~p co as N-»oco , and since Rm —»O , that the magnetic field is only slightly disturbed. We take a linear approximation for the magnetic field: Bo=l+ ¥ by By = ¥ by . (17) Viaci The induction equation (1b) yields ab, _%< ay -Yy (18) ¥ = Rn/e(n)* The momentum equation becomes: Yas Y ag] \ Te sa TN EQoy Was) and C+ eS + $60) gen) 3 = Fix) Vy (20) The inertial term in (20) can be neglected for a thin layer and the pressure gradient across the layer is balanced by the body force; as N-w oo: a oe = -%, 4 (21) 2 g(N) = N/f(N) By combining with (18), we get. ) “_ aa (P + bx) = 0 (22) which is analogous to the condition of constant pressure across an ordinary viscous boundary layer. Since Rm-—~—»0O , (19) becomes . mm A ~~ oo -™ 2: =a Ft — + =35 =O (23) Cv. On \y aq) V, gC) On In order to choose g({N) , we must consider the consequences of the non-linear equation for the Bernoulli function (9). VeV oH = - (C/E) I We cee that the Bernoulli function H is strictly dcercasing along streamlines that pass through any regions of current. Along the center streamline J =O , from symmetry, and the Bernoulli function is constant up to the face of the body. Consequently, the maximum pressure on the front of the body is stagnation pressure and within the stagnant region VP=0. olde For X*>>1 and any WN , Joule losses provide a dissipation mechanism within the forward wake and undisturbed conditions exist at infinity upstream. The non-linear dissipation mechanism which results in a change in the Bernoulli function can exist within the thin layers described. Any streamline that passes through such a layer will reach downstream infinity, resulting in a distribution of Bernoulli function there. The linearized equations apply far downstream but allow for only exponentially small currents outside of the parabolic wake which lies upstream for x > 1. Parallel flow is reached if we go far enough downstream for any finite N. If we require uniform pressure at infinity, a non-zero NL exists there, and this is termed a "non-dittusive vortex trail" by Tamada. Let us consider a model consistent with these requirements. We choose f(N) = VN (24) g(N) = 1 giving vn - - P 0, XE “Yew BE = 0 (25 ) and oP wok ag oe OE (26) “5 = Such layers are suggested by Childress (12) for N/Re << 1 but never used in his later work. Let us consider the maximum drag which such a model would provide. As upstream, the Bernoulli function is conserved along the center streamline downstream. If we require uniform pressure at infinity, the Bernoulli function can never be negative, and the base pressure is constant equal to zero (relative to Py) within a stagnant region. Dissipation layers spreading like i/vN extend only upstream, and the mathematical description of the flow field would have to be completed by describing the matching of such layers to the upstream wake which also spreads like VN . Since the pressure is everywhere uniform down- stream, no current flows and the linear approximation is valid everywhere except possibly at the edges of the body. The magnetic field is uniform downstream and equal to its undisturbed value. We can integrate (22) and determine the constant at YF-—-—»oo . We then have the field dis- tribution in the layer from the solution for the pressure. The determination of the magnetic field is completed by matching the layer solution to a potential distribution in the stagnant region and body. The consequence of this model is that the maximum drag coefficient equals one. Childress (43) utilizes a successive approximation technique and only considers terms of the first.order in solving equations (25)- (26). He finds Cy (first approximation) = .721. It seems clear that for finite N , inviscid, steady flow, we can not account for the large drag measured. The model we have described for the flow upstream seems to be on firm ground, but we need to propose another model for the 4 6~ downstream flow. In order to have drag which increases with VN » we must allow e(N) = VN on the body. Since the maximum pressure on the front face is stagnation pressure, we must have an increasingly large suction on the back as N increases. Making this assumption, we get g(N) = VN L (27) £(N) = NS and the equations aP 3K 2° 78) and >P - 9 by. ag 77 Yr" IG (29) We note that the only difference between the equations valid for layers of thickness 1/ VN and the equations valid for layers 1/n is the existence cof the inertial term in equation (25). The solution to equations (28), (99), along with the continuity equation, ineludes two undetermined functions 3(}) and B, (#) : ww AL P | Vy = a C6 a 2 «.d*P Vo = xX m7) + Py 4 qe LT whereas equations (25) and (26) can in principle provide complete solutions. This difficulty might be resolved by substituting a full expansion for the tilde variables in terms of a small parameter such as 1/ YN , each higher order providing sufficient information to determine the next lower order. The model we propose is identical on the upstream side to that deseribed by equations (2k). - (26) , but differs downstream due to the large but negative pressure inside the stagnant region. For N large but finite, the layers spread and merge at a distance x = o( YN ) Gownstream and x = O(N) upstream. The behavior of the far field for N large but finite is describable by the solutions of Gourdine (8). Upstream a wake spreading parabolically like 1/VvVN can be matched to the layer solution, but no diffusion mechanism exists downstream. Since the Bernoulli function is conserved along the center streamline, it must be large and negative at downstream infinity. This is inconsistent with the required far wake behavior. Such a near field can exist, though, if we relax the condition on Re and only require that Re be large but finite: Re >>1 ; N/Re << 1 We then have a mechanism to permit the pressure to return to P, and the drag. to be distributed as momentum defect in the viscous wake. For N finite we are still faced with the difficult problem of matching the layer solution with the viscous wake. However, a rather simple flow results for N—~» ce while still maintaining the condition N/Re<<1. -48- The layers become infinitesimally thin sheets extending to infinity, and the flow is superficially similar to the infinite conductivity, zero viscosity model proposed by Stewartson (13). In that case infinitely long slugs are formed, both sides of the body contribute equally. to the drag, and the drag coefficient is proportional to the Alfvén number. Here we find that for: Re —» co N/Re —» O that Com vir and the large drag is due to suction on the base. The pressure jump . across the layer tends to infinity with YN, and is balanced by current flowing in the thin layer which interacts with the applied Tield. The current is estimated for finite N to be: A Bx AY A 0 (Rn/NS) Ju and for Rn-0O, N-»sco The ultimate flow for N—» co is shown in figure 22. -4g. Initial measurements, carried out by T. Maxworthy (private communi- cation), of the pressure distribution on a sphere for high WN show excellent agreement with the drag measurement. They show that the drag is in fact due to low base pressures, but do not in themselves confirm or deny the model proposed here. The question of how viscosity enters for large but finite N seems to have features in common with the ordinary high Re problem of sepa~ rated steady flow,which is itself unresolved (44), The problem here is different in that the separation and the base pressure are governed by another body force. In both cases we must connect a boundary layer solution with the viscous wake through a rather poorly understood inter- mediate region. We expect that progress in either problem would be beneficial to the other, but are unable to make that contribution here. “50- CHAPTER VII Conclusion In our survey of the literature of this field, we noted that many authors have chosen to emphasize what are clearly non-physical problems. There seemed to be a tendency to treat MPD problems as of an exceptional nature and to divorce ones thinking not only from physical MFD flows, but from other problems in fluid mechanics as well. On the other hand, consideration of the physically realizable cases led to a definitive set of conditions and to similarities with two other areas of fluid mechanics currently under investigation. These were strongly rotating flows and steady high Re flows. One must be careful in carrying over exact results from one problem to another, but a successful approach in one might prove to be fruitful in the others. A measurement of the drag coeffieient of spheres and disks was carried out over a fairly wide range of N . We found that the sphere Cp was not a function of N for N&.3 , began to increase appreciably for Neal , and reached an asymptotic dependence proportional ta VY for Nw10. The disk Cp was relatively unchanged for N<10 , began to increase for N10 , and for N>20 the drag for spheres and disks was approximately the same. We eoneluded that the high N range was charac- terized by Cp insensitive to body shape. The intermediate range (15 N10) did not give any simple dependence on N , and we expect that the physical situation involves a complex combination of both viscous and Lorentz forces. We chose to emphasize the high WN behavior because of its simplicity. -51- We noted the analogy between the Taylor-Proudman theorem for strong rotation and similar results for high N flows. This suggested a physical model involving large stagnant regions both upstream and downstream of the body. Equations were presented to describe thin dissipation layers separating the inner stagnant region from the uniform outer flow. The similarity between these Joule dissipation layers for high N and viscous dissipation layers for ordinary high Re flows became apparent. Finally, the importance of matching the near field to a viscous wake downstream appeared, and we pointed out that this unan- swered question was crucial to both steady high Re flows and high N flows. We expect that such a physical model should be amenable to a theoretical analysis. We suggested a singular perturbation technique similar to that used by others, but were unable to provide a solution here. We feel that work should continue both on this theoretical approach and on further measurements. Detailed flow and field measure- ments in the vicinity of the body would be particularly useful and we "intend to pursue such an experimental program. This experiment is to our knowledge the first successful measurement of a gross physical quantity for MFD flows with strong interaction. In itself, it has solved few,if any, of the many unanswered questions which have appeared. Its principal value is to lend direction to the research which must continue if one is to understand these phenomena. By demon- strating a physical set of conditions and a consequence of them, we have gained an orientation in our view to this problem. It is hoped that because of this, further advance can be made. 2. 52- References Sears, W. R. and Resler, E. L., "Magneto-Aerodynamic Flow Past Bodies," Advances in Applied Mech., Vol. 8, 1-68, 1964. Liepmann, H. W., Hoult, D. P., and Ahlstrom, H. G., "Concept, Construction, and Preliminary Use of a Facility for Experi- mental Studies in Magneto-Fluid Dynamics," Miszellaneen Angewandten Mechanick, 175-189, 1960, Acad. Verlag, Berlin. Childress, S., "On the Flow of a Conducting Fluid of Small Viscosity," Proc. of a Symposium on Fundamental Topics in Relativistic Fluid Mechanics and Magnetohydrodynamies, Academic Press, 1963. Resler, E. L. and McCune, J. E., "Some Exact Solutions in Linearized Magnetoaerodynamics for Arbitrary Magnetic Reynolds Numbers," Rev. Mod. Phys. 32 (848-854) 1960. Stewartson, K., "Magneto-Fluid Dynamics of Thin Bodies in Oblique Pields - I," Zeits. £. angew. Math. und Phys. 12, 261-271 (1961). ludford, G. S. S., "The Effect of a Very Strong Magnetic Cross-Field on Steady Motion through a Slightly Conducting Fluid," J. Fluid Mech. 10, 141-155 (1961). | Stewartson, K,, "Magneto-Fluid Dynamics of Bodies in Aligned Fields," Proc. Roy. Soc. A 275, 70-86 (1963). Gourdine, M., "Magnetohydrodynamic Flow Constructions with Funda- mental Solutions," J. Fluid Mech. 10, 459-465 (1961). Gourdine, M., "On the Role of Viscosity and Conductivity in Magneto- hydrodynamics," JPL Tech. Rept. 32-3, 1960. 10, il. Le. 13. 14. 15. 16. 17. 18. 19. =53- Chester, W., "The Effect of a Magnetic Field on the Flow of a Conducting Fluid Past a Body of Revolution," J. Fluid Mech. 10, 459-465 (1961). Chang, I., "On a Singular Perturbation Problem in Magnetohydro- dynamics," Zeits. f. angew. Math. und Phys. 14, 134-147 (1963). Childress, S., "The Effect of a Strong Magnetic Field on Two- Dimensional Flows of a Conducting Fluid," J. Fluid Mech. 15, yog-hhi (1963). Stewartson, K., "On the Motion of a Non-Conducting Body through a Perfectly Conducting Fluid," J. Fluid Mech. 8, 82-96 (1960). Stewartson, K., "Motion of Bodies through Conducting Fluids," Rev. of Mod. Phys. 32, 855-859 (1960). Sears, W. R. and Resler, E. L. Jr., "Theory of Thin Airfoils in Fluids of High Electrical Conductivity,” J. Fluid Mech. 5, 257-273 (1959). Ludford, G S. S. and Leibovich, S., "The Transient Hydromagnetic Flow Past an Airfoil for an Aligned Magnetic Field," Journal de Mécanique 4, 21-50 (1965). Stewartson, K., "On Magnetic Boundary Layers," J. Inst. Maths. Applics. 1, 29-41 (1965). Tamada, K., "Flow of a Slightly Conducting Fluid Past a Circular Cylinder with Strong, Aligned Field," Phys. of Fluids 5, 817-823 (1962). Reitz, J. R. and Foldy, L. L., "The Force on a Sphere Moving through a Conducting Fluid in the Presence of a Magnetic Fiela," J. Fluid Mech. 11, 133-142 (1961). 20. al. 22. 23. 2h. 25. 26. 27. 28. 29. 30. 31. 5h Leonard, B., "Some Aspects of Magnetohydrodynamic Flow about a Blunt Body," AFOSR 2714 (1962). Seebass, R. and Tamada, K., "The Distortion of a Magnetic Field by the Flow of a Conducting Fluid Past a Circular Cylinder,” J. Fluid Mech. 22, 561-578 (1965). Ahlstrom, H., "Experiments on the Upstream Wake in Magneto-Fluid Dynamics," J. Fluid Mech. 15, 205-221 (1963). Motz, R. O., "Magnetohydrodynamics of an Oscillating Dielectric Sphere," Plasma Research Laboratory, Columbia University, Report No. 21 (1965). Lary, E. C., "A Theory of Thin Airfoils and Slender Bodies in Fluids of Finite Electrical Conductivity with Aligned Fields," J. Fluid Mech. 12, 209-226 (1962). Suzuki, B., (to be published). Maxworthy, T., "Measurcments of Drag and Wake Structure in Magneto- Fluid Dynamic Flow about a Sphere," Heat Transfer and Fluid Mech. Inst., 197-205 (1962). Maxworthy, T., “Liquid Sodium Flow Facility for Magneto-Fluid Dynamic Research," Rey. of Sci. Inst. 32, 1260-1261 (1961). Cowling, T. G., "Magnetohydrodynamics," Interscience (1957). Maxworthy, T., JPL Tech, Rept. (to be published). Sherecliff, J. A., "The Theory of Electromagnetic Flow Measurement," Cambridge (1962). Schlichting, H., “Boundary Layer Theory," Fourth Edivion, McGraw- Hill (1960). 32. 33: 3h. 336 36. 38. 39. 40. 42. he, -55- Reshotko, E., "Experimental Study of Stability of Pipe Flow," JPL - Progress Report No. 20-364 (1958). Liguid Mctals Handbook, Third Edition, Atomic Energy Commission (1955). Maskell, E. C., "A Theory of the Blockage Effects on Bluff Bodies and Stalled Wings in a Closed Wind Tunnel," R.A.E. Report 2685 (1963). | Fail, R. A., Lawford, J, A. and Eyre, R. C. W., "Low-Speed Experi- ments on the Wake Characteristics of Flat Plates Normal to an Air Stream," R. & M. Report 3120 (1959). Goldstein, S., “Modern Developments in Fluid Dynamics," Vol. II, Oxford (1957). Globe, S., "The Effect of a Longitudinal Magnetic Field on Pipe Flow of Mercury," Trans. Amer. Soc. Mech. Eng. 83, Series C, 445- 453 (1961). . Lundquist, S., "Experimental Investigation of Magneto~-Hydrodynamic Waves," Phys. Rev. 76, 1805-1809 (1949). Lehnert, B., "Magneto-Hydrodynamic Waves in Liquid Sodium," Phys. Rev. 9h, 815-82h (195h). . Cambel, A. B., "Plasma Physics and Magneto-Fluid Mechanics," McGraw-Hill (1963). Squire, H. B., "Rotating Fluids," Surveys in Mechanics, 139-161 (1956). Hide, R. and Roberts, P., "Some Elementary Problems in Magneto- Hydrodynamics," Advances in Applied Mech. 7, 215-316 (1962). ~56- 43. Childress, S., "Inviscid Magnetohydrodynamic Flow in the Presence of a Strong Magnetic Field," JPL SPS 37-22, 254-255 (1963). 4, Acrivos, A., Snowden, D. D., Grove, A. S., and Petersen, E E., "The Steady Separated Flow Past a Circular Cylinder at Large Reynolds Numbers," J. Fluid Mech. 21, 737-760 (1965). J SECOND MODE Rm Ve FIRST MODE.“ ox t (Re) “2m 0 | (a << [) SECOND MODE y > i or, Nile | Z FIRST MODE * se y > (Fe) Y2— 0 (a >>1) FIG. 1 Wake Behavior for Pry¢<l, W/Re<l EXPANSION TANK ELECTRO- WIDE- MAGNETIC ANGLE b DIFFUSER ~— HEATING FLOWMETER WITH BLANKET SCREENS MAGNET TEST SECTION fh YPASS CENTRIFUGAL PROBE——> PUMP ' oe 1000 g.p.m. FILTER | () PUMP 60-ft HEAD) VERSING UME ) OVERFLOW NARROW-ANGLE —+ | DIFFUSER J OTOR- Ht—SUMP TANK cata ~— MAGNETIC FIELD LINES ——-—— FLUID STREAMLINES SUPPLY FIG. 2 FLOW DIAGRAM FOR THE J.P.L. LIQUID SODIUM TEST FACILITY VELOCITY AT BODY LOCATION, cm/sec NZ. 1200 | Vi NOMINAL MAGNET Hooke CURRENT, amp / ane) v 1500 1000 — Vv 1850 y, a 2200 0 2500 900 7 800 a 700 ff YW 600 HA 500 7 Wi 300 400 d — 200 300 — i0o 200 _ 100 ok 1+ 6 8 Te) 12 14 is 18 FIG. 3 FLOWMETER OUTPUT, mv Fliow Meter Calibration ENTRANCE NOZZLE ~60- FLOW MAGNET | mee THERMOCOLPI FE —~ FULL. STRAIN GAUGE BRIDGE —™~_ THREE IDENTICAL BEAMS LOCATED 120 deg APART soon twee 1 RHENIUM WIRE — NONMAGNETIC STAINLESS STEEL SL 0.500-inD SPHERE FIG. 4 _ Drag Balance Test Section | BEAM CHAMBER PACKED WITH SILICONE GREASE ~6l1- ‘ul OSZO—>J weeg osney utez4s G ‘DTA q1/A/Aw OF'O :ALIALLISNSS WV38 IVNIWON JoOZI : LINIT SUNLVYAdWAL YaddN TWNIWON DoGEl : SUNLIVYAdGNAL ONILVYSdO “Ul Oe 31dNODOWNSHL V3WN1V-13WOYHO 7. 2 i) eee — oie <t—(b) Sa9nv9 0,9 NIVYLS 1104 a3LV 1NSdVON3Z at -62. onbruyoay, UOTYBIqET Teg 9 *DIa SONVIVE WYSE 31diyl HOTS (9SeN) +-OINI4 SNODIS [~3dV1L SNULV3H / NOILVINSNI co (g0u04) § 008! QvO1 WNWIXVA \ Sep 23 J uoTPeINSt Tue TSpow L *Sra G3013aM-LOdS SYIM \ BAEK bap Gp + oogo— | ASIC VaSLS SSSTINIVLS OILANSVNNON YOOH NOILVesl wd STE YAOW SY ‘ul OSLO a ‘ul QOS — ‘ul OSZ'0 S3Y3HdS SSVug YO ‘ul OIOO YO SOOO V33LS SSAINIVLS - S3tiIM WOINSHY-NALSONNL OILSNOVINNON STRAIN GAGE BRIDGE OUTPUT, mv ~6h- FIG. 8 200 400 600 600 i000 1Z00 1400 1600 18600 DRAG, g (FORCE) Drag Balance Calibration; 0.750-In. Sphere STRAIN GAGE BRIDGE OUTPUT, mv ler wo I NOMINAL MAGNET CURRENT, amp 0 600 850 1140 1500 1850 2200 2500 i HARTMANN NUMBER fe) 360 580 707 1020 1250 i470 i670 Onupawoop-x I I 0.750-in. SPHERE @ wl o a $ FIG. 9 10 FLOWMETER OUTPUT, my Raw Data for 0.750-In. Sphere DRAG COEFFICIENT, Cp A ow T T 0.250-in. SPHERE 0.005 -in. WIRE 10° ray Q ) <4 S oo O-0- io7 FIG. 10 Drag Coefficient Vs Reynolds Number for 0.250-In. 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