ST’dDY 0F TURBULENT AN ANALYTTCAL NAMlC CHANNEL FLOW MAGNETOHYDRODY Thesis for the Degree of Ph. D. MICHTGAN STATE UNNERSTTY ALBERT P. SCAGUONE 1969 “at; 0-169 This is to certify that the thesis entitled AN ANALYTICAL STUDY OF TURBULENT MAGNETOHYDRODYNAMIC CHANNEL FLOW presented by Albert P. Scaglione has been accepted towards fulfillment of the requirements for ph 12 degree in Mechanical Englneering J4! FAQ/{WV Major professor DateFX/QuZZ/y7fl LIBRARY ""5 Michigas' "rate .., U- v r: I "" L " BINDING av HMS & SONS' ‘. 300K HINDI" lllc. LIBRARY IlND‘ RS .— _— _‘ __——‘ A ,— — In 59"91013'1: channel 1 boundary and the : CEVelope Ba ABSTRACT AN ANALYTICAL STUDY OF TURBULENT MAGNETOHYDRODYNAMIC CHANNEL FLOW BY Albert P. Scaglione In this study an analytical investigation of both developing and fully developed magnetohydrodynamic (MHD) channel flow was conducted. A uniform flow field with zero boundary layer thickness at the leading edge was assumed, and the solutions were then carried out until fully developed flow conditions were asymptotically reached. Based on the theoretical arguments of Napolitano and Chandrasekhar a two-layer model was utilized to describe the turbulent MHD boundary layer. In the inner layer the eddy diffusion of momentum was based on a modified mixing length theory according to van Driest and Lykoudis. In the outer layer the eddy viscosity was based on a modifi— cation of the ”intermittent eddy viscosity" based on the studies of Klebanoff. Solutions to the developing turbulent MHD boundary layer equations were arrived at directly by a linearization technique and then utilization of an implicit finite __ .———_—.——_——_—_——§_————____ RiC$er9rr~, u b‘ five-H results ai q n+ ‘ de;.~. to AC Albert P. Scaglione difference method. The results of the solutions are pre— sented for Reynolds numbers ranging from 104 to 4(10)5 and for Hartmann numbers ranging from 0 to 1200. The results indicate that in the turbulent flow case the important physical parameters such as displacement thickness, mo— mentum thickness and skin friction coefficient show both a strong Hartmann and Reynolds number dependence. The results also exhibit the effect of detransition from turbu- lent to laminar flow when the ratio of Hartmann number over Reynolds number exceeds 1/225. The asymptotic solutions for fully developed flow agree quite closely with the existing experimental data and with the theoretical analy- sis of Lykoudis and Brouilette. A separate approximate analysis for the skin friction coefficient based on a laminar sub-layer, turbulent core model was conducted. The analysis yields the expression M . 60M 4 D4 2 C = 2(——)[Coth{-—77§eXp(6(10) (—-) )l 2'35 xp(2.72(10)4(3—4—)2)—1}/Sinh{-§%%§GXP(6(1°)4(£)2)H RE + { RE RE which is within 10% of the experimental data and agrees well with both the more exact asymptotic solution deve10ped herein and with the solution of Lykoudis and Brouillette. AN ANALYTICAL STUDY OF TURBULENT MAGNETOHYDRODYNAMIC CHANNEL FLOW BY Albert PL Scaglione A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1969 géz 835' 7—, /-7 0 ACKNOWLEDGMENTS The author wishes to express sincere thanks and appreciation to his major professor Dr. Mahlon C. Smith (Mechanical Engineering) and to Dr. Merle C. Potter (Mechanical Engineering), who served on the guidance com- mittee, for the generous way in which they helped and devoted much of their time during the completion of this work. Special thanks are extended to Dr. J. Sutherland Frame (Mathematics), who served on the guidance committee, and without whose help and understanding much of this thesis might not have been possible. Thanks are also extended to Dr. Kun MuChen (Electri— cal Engineering) for serving on the guidance committee. Finally the author wishes to thank his wife Diana and daughter Lisa for their help and understanding during the completion of this work, and his parents Joseph and .Agnes for their encouragement and faith in him. ii ! PYV'n-‘w .. . than! villuL . V. M , C p “‘54; " ‘.’,".‘T\.'f“ ‘ .- n.\/.'o~A\L“‘J . 1 T\V~». ‘ i.“ h z. V, ‘ DISC J) TABLE OF CONTENTS Page ACKNOWLEDGMENTS O O O O O O O O O O 0 i i LIST OF FIGURES . . . . . . . . . . . v NOMENCLATURE . . . . . . . . . . . Vii 1. INTRODUCTION . . . . . . . . l 2. LITERATURE SURVEY. . . . . . . . 5 2.1 Experimental Work in MHD channel flow . . 5 2.2 Analytical Work in Turbulent Fully Developed MHD Channel Flow . . 9 2.3 Analytical Work in Turbulent Develo- ping MHD Boundary Layers . . 12 3. ANALYTICAL STUDIES . . . . . . 17 3.1 Flow Configuration . . . . . 17 3.2 Basic Equations. . . . . . 19 3.3 Transformation of Momentum Equation 23 3.4 Method of Solution. . . . . . 25 3.5 Eddy Diffusion of Momentum . . 38 3.6 Approximate Analysis for Fully Developed Flow . . . . . . . . . . 51 3.7 Closure . . . . . . . . . 60 4. DISCUSSION OF RESULTS . . . . 65 4.1 Solutions for Laminar flow . . . 65 4.2 Displacement and Momentum Thickness for the Developing Boundary Layer. 67 4.3 Skin Friction Coefficient for the Developing Boundary Layer . . 70 4.4 Asymptotic Solution for Fully Developed Flow . . . 73 4.5 Approximate Solution for Fully Developed Flow . . . . . . . 76 iii Page 50 CONCLUSIONS 0 o o o o o o o o o o 99 REFERENCES. . . . . . . . . . . . . . 101 APPENDIX . . . . . . . . . . . . . . 106 iv p9s.~¢ .5 c c; 'C ’1‘- AAA. E me 2. .hu Fug n a .5. C. 3L an“ an 4 r; at c . E C I e huL mum n4. 1 LIST OF FIGURES Figure l. MHD Channel Flow Coordinate System and Flow Geometry . . . . . . . . . . . . 2. Finite Difference Molecule . . . . . . 3. Effect of the Magnetic Field on Displacement Thickness in Laminar Flow . . . . . . 4. Effect of the Magnetic Field on Momentum Thick- ness in Laminar Flow . . . . . . . . . 5. Effect of the Magnetic Field on the Skin Friction Coefficient in Laminar Flow. . . . 6. Effect of the Magnetic Field on the Displace- ment Thickness for a Reynolds Number of 10000. 7. Effect of the Magnetic Field on the Momentum Thickness for a Reynolds Number of 10000 . 8. Effect of the Magnetic Field on the Displace- ment thickness for a Reynolds Number of 25000. 9. Effect of the Magnetic Field on the Momentum Thickness for a Reynolds Number of 25000 . lC). Effect of the Magnetic Field on the Displace- ment Thickness for a Reynolds Number of 50000. 11.. Effect of the Magnetic Field on the Momentum Thickness for a Reynolds Number of 50000. . . 3L2.. Effect of the Magnetic Field on the Displace- ment Thickness for a Reynolds Number of 100000 1&3. Effect of the Magnetic Field on the Momentum Thickness for a Reynolds Number of 100000. 14, Effect of the Magnetic Field on the Displace- ment Thickness for a Reynolds Number of 200000 V Page 18 32 77 78 79 80 81 82 83 84 85 86 87 88 16. 19. ll l.(.l || “J C S . 1 \Au hr 22. 23. rk. PM “U. uni If. at 24. Figure 15. Effect of the Magnetic Field on the Momentum Thickness for a Reynolds Number of 200000. 16. Effect of the Magnetic Field on the Skin Friction Coefficient for a Reynolds Number of 10000 17. Effect of the Magnetic Field on the Skin Friction Coefficient for a Reynolds Number of 25000 .18. Effect of the Magnetic Field on the Skin Friction Coefficient for a Reynolds Number of 50000. . . L19. Effect of the Magnetic Field on the Skin Friction Coefficient for a Reynolds Number of 100000 . :20. Effect of the Magnetic Field on the Skin Friction Coefficient for a Reynolds Number of 200000 . . 21a Asymptotic Solution for Cf in Fully Developed Flow. . . . . . . . . . . . . . . :22. Comparison of Asymptotic Solution for Cf with Reference 21 . . . . . . . . . . 123. Approximate Solutions for Cf in Fully Developed Flow from Other Studies . . . . . . . . :24. Approximate Solution for Cf in Fully Developed Flow from this Study . . . . . . . . . vi Page 89 90 91 92 93 94 95 96 97 98 pr Al, A2, A NOMENCLATURE Channel half width Coefficients of matrix in equation (3.48) Matrix defined by equation (3.43) Function of the turbulence defined in equation (3.52) Coefficients for finite difference formulae defined in equation (3.35) Channel thickness Uniform magnetic field in y—direction Magnetic field intensity vector Functions defined in the Appendix Constant appearing in equation (3.57) vii "1 (“'1 P‘ {1.1L C-‘L Wt fr(n) f (gin) = f Skin friction coefficient Uniform electric field repre- sentation in z-direction Remainder terms in finite difference differentiation formulae (3.41) Electric field intensity vector Differentiation formulae de— fined in equation (3.41) Dimensionless stream function Ratio of free stream velocity to entrance velocity Initial Step size in n- direction Hartmann number based on Channel half width Current density in z-direction Current density vector Thermal conductivity Constant appearing in eddy viscosity expression Constant appearing in equation viii £i(n) pr -03: M-(E)Bod Constant appearing in equation (3.59) Constant appearing in equation (3.73) Variable step size constant Weighting functions defined in equation (3.41) Coefficients of matrix given by equation (3.45) Matrix defined in equation (3.44) Hartmann number based on hydraulic diameter Coefficients of matrix given by equation (3.44) Matrix given by equation (3.43) Pressure Magnetic field parameter given by equation (3.55) Radius of spherical turbulent fluid packet Reynolds number based on hydraulic diameter ix X,y n. II Q I: 3 C. H I-l 5| -2—J§—; and that when this condition was satisfied, RE 'the skin friction coefficient Cf followed the laminar flow line Cf = 2M/RE. Second, that there is a magnetic entrance .1ength where the flow cannot be considered fully developed aand as a result a restriction to the central l/2 of the (:hannel must be made if fully developed flow conditions ears to be satisfied. Third, it appeared that a minimum 1J1 the curve of Cf vs. M/RE with RE held constant would cxzcur at all values of the Reynolds number. The fact that tflie minima became shallower at larger Reynolds numbers sfliould have been a clue that the minima did not occur at Reynolds numbers beyond RE 2 1.5(10)5 which would be un- covered in later experiments. In experiments by Brouillette and Lykoudis [49] in 1J962 the range of skin friction coefficient data for rectangular cross section MHD channels was extended up to Reynolds numbers of 4(10)5. The main contribution of their 1.5(10)5. Branover and Lielausis [25] measured both Cf and mean velocity profiles for a range of Reynolds numbers of 5000—20000 in Open channel MHD flow. Their velocity pro- file data was quite qualitative in that they could get no Closer then y/a : 0.1 to the wall. In addition the open Channel flow only simulated the MHD channel flow problem. Cx MM. 339:; t5 Their data did show, however, the extreme flattening of the mean velocity profile with increasing Hartmann number. The skin friction coefficient data was in essential agree— ment with the aforementioned earlier findings. Finally, Brouillette and Lykoudis [20, 21] made skin friction and mean velocity measurements in a 1.4 inch wide rectangular duct of 5:1 aspect ratio. The Reynolds number range for the data was 58000-180000 and the Hartmann number range was high enough to bring about transition to laminar flow at all Reynolds numbers. The skin friction data found in these experiments was essentially in agreement with that of the earlier studies and confirmed the disappearance in the minimum in Cf with increasing M/RE at Reynolds numbers greater than 150000. Because of the instrumentation used, measurements for the mean.velocity profile could only be made for y/a distances greater than .05 from the wall. As a result, measurements deeptin the boundary layer were not possible. The mean velocity profile data did show the flattening of the velocity profiles with increasing Hartmann number. Some data was taken (although not with great accuracy [20]) between y/G = .05 and .1 which was not possible in the data of Branover and Lielausis. The conclusions which can be made concerning experi— mental MHD channel flow are the following: 1. Only data for the fully developed flow case is available. 2. Skin friction coefficient data have now been taken by several investigators and the corre— lation of the results is good. This data may now be considered reliable and conclusive. 3. Although some data exists for mean velocity profiles, correlation of this data in larger channels with more precise probes (such as hot film anemometers) is necessary before the structure of the mean velocity profile can be said to be well accepted experimentally. 3;} Analytical Work in Turbulent Fully Developed MHD Channel Flow As in the case for the experimental work, an excel— lent review of what has been done analytically in turbulent fully developed channel flow up until 1966 can be found in Brouillette [19]. Below is a brief summary of the work done until then and also the work done to this date. Harris [16] was the first to attempt to provide a model for the fully developed MHD channel flow. He ex— tended the dimensional analysis of Milliken [50] to include MHD effects utilizing the data of Murgatroyd. Expressions for both velocity profiles and skin friction coefficients were develOped. The main shortcoming of his ‘work.was that a minimum in Cf when plotted versus increas— ing M/RE was predicted for all Reynolds numbers up to 10 infinity because this was the trend which he deduced from the data of Murgatroyd. In the later experiments of Brouillette and Lykoudis [50] this assumption was proved invalid. In fact it was experimentally found that at Reynolds numbers beyond 1.5(10)5, Cf increased mono— tonically with M/RE' In addition the study of Harris was not based on an understanding of the eddy diffusion, or mechanism of turbulence in MHD flow and as a result no light was shed on the processes taking place in turbulent MHD channel flow. The damping of turbulence and detran- sition from turbulent to laminar flow were not predictable from his analysis. In the last several years several studies of the problem were made which were all based on a common theme. .As pointed out by Lykoudis and Brouillette [21] the anae lytical work of Maciulaitis and Loeffler [10], Tennekes [18], Khozhainov [23], Ryabinin and Khozhainov [24], and Branover and Lielausis [25] assume a functional relation- ship for either the velocity profiles or the skin friction coefficient directly. Free parameters occurring in these relationships are then adjusted by the application of an integral momentum technique and dimensional analysis. The results generated show varying degrees of agreement with time data. None however show complete agreement throughout alJ.:ranges of Reynolds and Hartmann number. In addition none are consistent in even qualitatively predicting all the trends exhibited by the experimental data. ll Nihoul [l7] attempted to utilize the Malkus postulates on the spectrum of the mean vorticity gradient and then develop the mean velocity profile and skin friction co- efficient by application of these postulates into the Navier Stokes equation along with the boundary conditions. How- ever in his derivation the assumption is made that the Hartmann number be sufficiently large that the flow is considered to be in the so-called inhibition region. That is a region between the fully turbulent flow and laminar flow at €:.: §%—, where the skin friction coefficient is little effected by Reynolds number. The results for skin friction coefficient agree quite well with the data for Reynolds numbers between 50,000 and 180,000 in the range 3 and 4(10)—3. This range appears to M/RE between 2.5(10)- define the inhibition region discussed in his theory. How— ever the existence of the inhibition region for all Reynolds numbers is incorrect. At low Reynolds numbers it is clear 3 from.the data that even at M/RE as high as 3(10)- the values of Cf are substantially higher than those on Nihoul's inhibition line (Figure 21). At high Reynolds numbers and as RE approaches infinity the values of Cf should approach the laminar line at all Hartmann numbers, based on the criteria for laminar flow §%-:.—%§. Hence it can be said thatLNihoul's analysis predicts the existence of an inhi— lxition region and gives good results in that region for a range of Reynolds numbers where the region exists. 12 Finally, in a recent study by Lykoudis and Brouil- lette [21], an analysis was based on modeling the MHD turbu- lent shear, and introducing this model directly into the fully develOped flow momentum equation. Velocity profiles and skin friction coefficients then come about directly from the solutions of the differential equation. The important physical consideration in their study was the damping of the 5757 correlation due to the suppression of turbulence by the transverse magnetic field. This was coupled with a single layer model for the turbulent shear based on the work of Van Driest [35]. Their approach will be discussed in greater detail in section 3.5. Their analysis, in addition to providing an acceptable physical interpretation of the problem, yields results which are in good agreement with the data throughout all ranges of Hartmann and Reynolds number. As a result this work appears to be the only acceptable analytical study of the fully develOped MHD channel flow problem. 2.3 Analytical Work in Turbulent Developing MHD BoundaryiLayers Concerning the develOping turbulent MHD boundary layer there have only been four studies to date. Three of these, Napolitano [27], Kruger and Sonju [28], and Moffatt [51, 52], dealt with flow over a flat plate with zero pressure gradient. Only the study of Maciulaitis and Loeffler [10] dealt with channel flow. 13 Napolitano attempted to develop a model of the turbu— lence and then utilize this model in the fundamental equations governing the problem. To develop a model of the turbulence he utilized the theories of hydromagnetic turbulence developed by Chandresekhar. According to this the turbulent boundary layer is seen to have two regions with different "characteristic times of adjustment," the inner region and the outer region. In the outer region energy is extracted from the mean flow and converted into turbulent energy by the working of the mean flow against the Reynolds stresses. In the inner layer most of the turbulent energy input is dissipated by viscosity effects near the wall with a very small amount fed back into the outer layer. In the MHD boundary layer the above process is modi- fied in two ways: first the energy input into the outer layer is modified by the interaction between the mean velocity field and the magnetic field. Second the energy input into the inner layer is now dissipated by magnetic as well as viscous effects. Based on arguments of the kind described above, Napolitano concluded that in linearized* MHD a two—layer model of the turbulent boundary layer would follow. *Napolitano showed that when the magnetic Reynolds number is small there would be no transfer of energy from the turbulent velocity field to the turbulent magnetic field and the equations could be linearized. 14 The shortcoming of Napolitano's work lay in his scalution technique. To arrive at a solution he chose a searies expansion technique limiting his solutions to the case where his perturbation parameter REE was small since 116: only took one term involving it. Since RH€<<1 must be satisfied, the analysis was restricted to small values of Hartmann number in most cases, and in fact the analysis would have to break down at some point as 5 increased. The main contribution of the study was the extension of “trme theory of hydromagnetic turbulence to the turbulent MHD boundary layer and the prediction of the two-layer model for the turbulent MHD boundary layer. Kruger and Sonju developed numerical solutions of time momentum integral form of the MHD flat plate boundary liryer equations. The velocity profiles and skin friction Coefficient utilized were inferred from the work of Harris. Tfiieir results indicate that, the velocity profile is no 1&3nger similar along the plate, the boundary layer thick— rless approaches an asymptotic value, and the skin friction c=C>efficient is significantly changed from that in non MHD flddw. Since the fundamental assumptions of this study are based on the work of Harris all the shortcomings of Harris' Study as discussed in section 2.2 will also occur here. Moffatt also solved the momentum integral form of tflne MHD boundary layer equations. However, in his study I“3 assumed the Blasius form of the skin friction law and a 1/7 power velocity distribution law. The results of his 15 st:udy agreed within 5% of those of Kruger and Sonju for buoth displacement thickness and wall shear stress* [53]. Since the forms of the velocity profiles and skin friction chefficient differ so greatly in the two studies it is aipparent that the choice of using the momentum integral nmethod itself dominates the solution to the problem. The silortcomings of the integral methods in MHD boundary layer aJialyses are discussed by Heywood and Moffatt [53]. Their rtesult is that a great deal of care must be exercised be— ftJre the results of that kind of analysis may be con— sinered reliable. Based on the criteria set forth in their stzudy it is not yet possible to determine that the results of a turbulent MHD boundary layer analysis, using the momentum integral method, are reliable regardless of the fcxrms chosen for the velocity distribution and skin fric- tion coefficient. Maciulaitis and Loeffler conducted the only analysis t£> date for the develOping turbulent boundary layer in MHD Cfliannel flow. They utilized the momentum integral tech— nique and as in the study of Moffatt for flat plates afisSumed the Blasius form for the skin friction law and a 1/’7 power velocity distribution. Their results show only Ifljrnor success in predicting the skin friction coefficient ~ *The boundary layer thickness, 6 however differed 19y as much as a facter of two in these studies. This is insignificant, however, since the boundary layer thickness is ill-defined and as a result physically unimportant. 16 cisita.in fully developed flow. They are inaccurate not only in magnitude but also in qualitatively even predicting the trends of the data (Figure 23) . In addition their analysis <3c>nnpletely fails to predict the law C = 2M/RE when f ha/IIEE > §%§ and transition back to laminar flow occurs. :F::c>m.the previous discussion concerning the application of tiles momentum integral method to the flat plate turbulent 13c>t1ndary layer it becomes apparent that without a much Inc>zre precise understanding of the turbulent velocity pro— file, the turbulent shear stress and the relationship between them the momentum integral method is presently uxissuitable for approaching the developing turbulent MHD channel flow problem. The conclusions which can be drawn concerning the state of knowledge of the developing turbulent MHD boundary layer are the following: 1. For the case of small magnetic Reynolds number the turbulent MHD boundary layer is describable by means of a two—layer model. 2. Some work has been done for the flat plate case but due to limitations discussed above is yet far from conclusive. 3. In the case of channel flow the only study is that of Maciulaitis and Loeffler and the results are not in agreement with the data nor do they provide any physically reasonable model for the flow. 3. ANALYTICAL STUDIES 3.1 Flow Configuration Numerous studies have been made for both laminar and turbulent MHD channel flow. However, only in the laminar flow case has the problem of the develOping boundary layer in the entrance section been extensively investigated. In the turbulent flow case the only analysis is that of Maciulaitis and Loeffler [10]. As discussed in Chapter 2, this study falls short of accurately describing the flow phenomena for several reasons; and even in the fully developed flow case the agreement with the experimental data and trends is not good. The important feature which must be exhibited by any flow model is an accurate eXpres— sion for eddy diffusion which goes to zero as the magnetic field is increased, and detransition back to laminar flow is experienced. The detransition effect has been well proven both experimentally and semi-analytically. In this study the flow model consists of an electri- cally'conducting fluid flowing between parallel plates with a uniform magnetic field transverse to the flow direction (Figure l) . Viscous boundary layers starting at the leading edge of the plates are symmetrically growing on both plates. 17 18 FDARALLEL TO DIRECTION OF INFORM ELECTRIC Z FIELD REPRESENTATION-E. I la TI‘ < I . X b. / DIRECTION OF omen”, OF MEAN VELOCITY-U WIFORM APPLIED MAGNETIC FIELD (b) FIGURE l (a) MHD CHANNEL FLOW COORDINATE SYSTEM (b) FLOW GEOMET RY 19 3.2 Basic Equations The basic equations governing MHD flow can be written as the Maxwell equations, Ohms law, the equation of con- tinuity, the equation of motion with J X B body force, the energy equation with Joule heating, and the equation of state. In this study the fluid will be considered to be a a homogenous isotrOpic continuum and high frequency phe— 1: xuxmena will not be considered so that the displacement currents are negligible compared to the conduction current ‘. J. I Maxwell Equations: Vxfi=3 (3.1) v-§=o (3.2) —»_31'3’ VXE—B—E (3.3) v.3=o (3.4) Ohms Law: 3=o<fi+Vx§> (3.5) Conservation of Momentum: _\ p(%¥+(V-V)v)=-Vp+V-T+3xfi (3.6) 20 Conservation of Energy: p——E = pv - V’+ kV2 T + o + E - 3 (3.7) 30 . -*_ 8? + V p V — 0 (3.8) Equation of State: In our case we will consider the fludxi to be incompressible so the equation of state reduces to the following simple form. p = Constant (3-9) Equations (3.1) through (3.8) can be simplified by Utilqizing the Prandtl order of magnitude analysis and mak- ing the following assumptions. 1. The channel walls are assumed to be electrically non-conducting. 2. Electrical conductivity of the fluid is suf— ficiently low so that magnetic Reynolds number is small (cumuoa< p) (3.50) pr pr s

1 (3.60) which for the sake of convenience may reasonably be approxi- mated by [30] Y = [1 + s.s<§>61‘1 (3.60a) Therefore, in the outer layer the expression for the eddy viscosity in the non MHD case may be written as 1 so = k2ue6*[l + 5.5(§)6]’ (3.61) The model chosen for the MHD outer layer eddy viscosity in the present study will be a modification of (3.61) based on the 3rd damping factor developed earlier in this section. The assumption then is that the basic intermittent character of the turbulence is unchanged by the magnetic field interaction, but that the level of the turbulence is suppressed by the MHD ponderomotive body force. The choice of the 3rd damping factor as the turbu- lence suppressing mechanism in the outer layer is a rather Obvious one based on the success it yielded in predicting eXperimental results when applied by Lykoudis and Brouil- lette in the fully developed MHD channel flow problem. '. 3 - inundy-txum' ' N I ._ ‘J .. _, _ P 49 However, it makes sense in the present study to extend to the two-layer model in the developing boundary layer prob— lem under investigation. This presumption is based not only on the amazing agreement it yielded with the experi- mental data in Cebecei and Smith [33] for non MHD flows, but also on the basis of the theoretical arguments pre— sented by Napolitano and Chandresekhar for MHD flows as mentioned earlier. Thus, the expression for the MHD eddy diffusivity in the outer layer is given by 1 so = k2u86*[l + 5.5(§)6]‘ exp(—CA2) (3.62) At this point a note should be made concerning the validity of the above expression for the developing turbu- lent boundary layer in MHD channel flow. For zero mag- netic field it is clear that at some point the effect of the boundary layer growing on the opposing wall will begin to interfere with the boundary layer development. At this point (which occurs when the boundary layer is approxi- mately 85 per cent of the channel half width [39]) the intermittently turbulent region in the central core of the channel is interferred with by the boundary layer. Therefore when 6‘3‘.85a the concept of the intermittently turbulent outer layer begins to break down for the channel flow problem with zero magnetic field. From the results, however, it appears as though this deviation from the proposed model, for the zero magnetic field case, is not 50 significant. This can be seen in the agreement with the computed results for the asymptotic (fully deve10ped flow) case with eXperimental data [this is shown in (4.3)]. For the case of finite magnetic field the prOposed model does not deviate from the physical problem since the fully developed flow boundary layer thickness is almost always much less than .85a. For the Hartmann numbers _‘._§-y ’ _ h p which will be presented in the curves in Chapter 4 this will always be the case.* The validity of the above approximation can be Er directly checked only by experimental observation of the intermittency factor in the MHD boundary layer as was done by Klebanoff and Corrsin and Kistler in the non-MHD bound- ary layer. However, the state of the art in experimental MHD appears far from this point at the present time since only recently have mean velocity profiles been measured [21, 25]. However, well validated data for Cf exists in the fully developed flow case since the recent experimental data of Brouillette and Lykoudis, and Branover and Lielau-l sis, fill in the gaps** and confirm some of the earlier *For Hartmann number M > 2 it appears as though the boundary layer thickness is always less than .85a and an intermittently turbulent core will exist. **The shortcomings of the data of Murgatroyd and of Hartmann and Lazarus are pointed out by Shercoiff [47] in his review of the work of Harris. One of the key features which did not appear in that data or in the theoretical study of Harris was the disappearance in the decrease in the skin friction coefficient at high Reynolds number when plotted versus increasing Hartmann number divided by 51 findings of Murgatroyd, and of Hartmann and Lazarus as discussed in Chapter 2. The key then to the validity of the above physical model would seem to be the ability of the present analysis to asymptotically approach the now well accepted fully developed channel flow experimental results and show agreement with the theoretical analysis P1 of Lykoudis and Brouillette. The results of the present ‘ analysis do indeed show extremely good agreement with those of Lykoudis and Brouillette. In addition, unlike any of the other theoretical studies done for turbulent MHD #3, channel flow, except for the approximate fully deve10ped flow analysis developed herein, all the trends and fea- tures exhibited by the eXperimental data and the theory of Lykoudis and Brouillette are exhibited as a special limiting case (namely the asymptotic solution for fully developed flow) of this analysis. 3.6 Approximate Analysis for Fully Developed Flow In the present fully developed flow analysis a two- layer model will be utilized. That is, in the "inner" or "laminar sub-layer" the turbulent energy input is nearly all dissipated by viscous action with only a very small surplus being fed back into the outer layer. In the outer layer a wake or intermittently turbulent type region Reynolds number. This effect has been well confirmed both theoretically and experimentally by Lykoudis and Brouillette. 52 exists where there is an interaction between the mean flow and the magnetic field. This interaction leads to a re- duction of the mean kinetic energy in the outer layer and as a result the turbulent energy input into the inner layer. A theoretical justification for the two-layer model has been discussed by Napolitano [27] based on the work of F1 Chandresekhar [26]. ' Considering the inner or so-called "laminar sub— layef'to be describable by an MHD Couette flow with pres- 3 sure gradient the fundamental equations (3.1) to (3.9) can £3 be reduced to the following form: d2u + 0B 2(11 - u) = gE- (3 63) udyz o o x ' where as in section 3 the uniform electric field repre- sentation E2 is given by -uoBo since there is zero net current flow across the channel. The boundary conditions for (3.63) are the following: at y = 0 u = O (3.64) II o» :3 II x: at y b where 5b represents the thickness of the MHD sub-layer and 11b represents the velocity at the edge of the sub-layer The solution to (3.63) subjected to the boundary conditions (3.64) may be written as follows: 53 u u .. 9.1: O l cosh[(u) Boy] + Hub - l)/ ' h[(0);’B 61+ th[(°)1‘B 6 1} a: 8111 i O 1'.) CO {1- O b . sinh(%)kBOy (3.65) where gE-has been found from the MHD Bernoulli equation %2 = 0B 2u - OF B (3.66) x o e z o Fruuruumaz " . «1... ' ' I To evaluate the skin friction coefficient, (3.65) must be differentiated and evaluated at y = 0, ”(9.2) since C = By y=0 f k u 2 p O _ o 2 o b o . o k . . cf _ 2(fi— 5714 (—u———)/Sinh[(fi-) Boébl O O + coth[(g-);5B (S 1} (3 67) u o b ‘ If the Hartmann and Reynolds numbers based on hydraulic diameter are now introduced uodh v -032 _ M - (_. Bodh RE — then (3.67) becomes u M6 u M6 cf = :E—M [coth(—a-f) + (59 - l)/sinh(-ah—b—)] (3.68) O 54 Upon examination of (3.68) it is clear that the form of the equation is qualitatively correct. That is functions of this type may exhibit an initial decrease with increasing M/RE followed by an increase to the value 2M/RE as M/RE be— comes large. It is now necessary to introduce physically plausible arguments for evaluation of 6b and ub the laminar sub-layer thickness and the velocity at the edge of the sub-layer respectively. The first step in this process will be to consider the values of 6b and 11b from the non MHD case and see if insertion of these values into (3.68) will reduce (3.68) to the prOper closed form expression in the non MHD case. Namely, (3.68) should then agree with the Blasius expres- sion [54]. If we consider a two-layer model for the turbu- lence where the inner layer is entirely laminar and the outer layer is describable by means of a 1/7 power velocity profile, it is possible to develop expressions for 6b and uh. This has been done by Eckert [55] and his results (modified for channel flow) may be eXpressed as follows: 6 b 60 a— = —77§ (3.69) h RE u . b 2.35 ‘10 REI;§ 55 Utilizing (3.69) and (3.70) in (3.68) and applying L'Hospital's rule at M/RE = 0 the result for Cf for non MHD flow becomes C = .0783 f R217? which is in agreement with the Blasius expression, accord- ing to Schlicting [54], as it should be. As a result we see that this approximate eXpression for Cf is now accepta— ble on two counts. It reduces exactly to the value for non MHD flow when M/RE = 0 and it tends to the laminar line when M/RE becomes large. The expression used for the laminar sub-layer thick- ness and the velocity at the edge of the sub-layer thus far have been taken from the non MHD case. Clearly this cannot be correct since we know that the sub-layer thick— ness must increase finally to the laminar MHD boundary layer thickness as the Hartmann number is increased to the point of detransition back to laminar flow. Also the velocity at the edge of the sub-layer must then increase to the freestream velocity. To gain an insight into the way in which the sub— layer thickness and velocity at the edge of the sub—layer .are effected by the magnetic field, let us consider what happens to the turbulent core as the magnetic field is increased. Suppose we consider a small element of fluid in the turbulent core to be represented by a sphere. If we 56 choose our coordinate system to be attached to the average 110 velocity of the fluid at any location we would find that the spherical element of fluid would have a velocity u' due to turbulence in this direction. We would like to know how this velocity is affected by a transverse magnetic field Bo' In general if we were to consider the forces on this fluid particle we would say that its inertia must be balanced by viscous shear and magnetic forces. However, in the turbulent core we will neglect the viscous forces in comparison to the magnetic forces. As a result, a force balance on the fluid particle would yield du Dm Dv DE=T+T W“) where Duz Dm = -CDm—§— (3.718.) where Duz Dv = —CDv—§— (3.7lb) From the paper by Reitz and Foldy [56] the drag on the sphere moving through an inviscid conducting fluid in the presence of a uniform magnetic field has been found for the case of small magnetic Reynolds number. He deve10ps the expression _- i: --:;m :- 57 CD = ClBlRm (3.72) m vallere C1 = constant (3.72a) 01302 V] mm + Eirud Rm = noru (3.72c) . du OB 2ru k OB 2rt 11 exp( 3 and u 0 T) (3-73) IQcmy if we consider that the time t required to have the Sphere experience, a certain decrease in velocity from uO ‘tuo u can be represented by a characteristic time for the Sphere, namely t = r/uO and also since D <>6 V which is the assumption made here; then -OB 21:2 _ o u — uoexp(——E;r-—) Considering r = y1£* it is postulated that the radius of the sphere is related to the distance 9. moved ‘ *Since the radius r is related to the scale of the 'tltrbulence then the assumption is made that there is a idaear relationship between distance moved during the de- cay of the turbulence and the scale of the turbulence. 58 during time t in which the velocity has changed from uO to 'ul. Where Y1 is a value determined by the characteristic ‘tzime or velocity and size of the sphere . M2 . . u = uoexp(-f(RE)——) RE (3.74) Hence it is seen that the velocity decrease or decay of iii? trurbulence will be affected by the damping factor I .3 I a ‘. a exp[(-f(RE) Mz/RE] where f(RE) is a function of the turbu- Juence to be partially determined from the data. One ‘vvomld expect that as the turbulence is decreased in the izurbulent core the velocity at the edge of the laminar saub-layer would increase and the laminar sub-layer thick— IleSS increase as discussed previously. In this approximate analysis it will be further loostulated that this process would progress in an inverse Inanner to the decay of turbulence velocity, i.e., 2 a = a /{exp[-f ( >fl—1} (3.75a) bmagn bnon-magn 1 RE RE /{ [ < P421} < > u = u exp —f _. 3.751: bmagn bnon-magn 2 RE RE SSeveral forms were tried for the parameters fl(RE) and i52(RE). The following expressions 4 fl(RE) = 6(10) /RE (3.76a) 59 f2(RE) = 2.72(lO)4/RE (3.76b) were found to give the best agreement with the data for Cf when used in (3.68) over all ranges of Hartmann number and Reynolds numbers. The physical interpretation for the selection of the inverse Reynolds number function may be described as follows. Consider that larger distances traveled during turbulence decay indicate smaller turbulence levels. This can be appreciated by comparing 2 identical fluid spheres with the same characteristic time of decay but traveling different distances z during that time. Clearly then the turbulent fluid packet traveling the larger R would experi- ence a smaller change in velocity. The turbulent fluid packet experiencing the smaller change in velocity over the same time may be considered to have a lower turbulence level. This is certainly true in the limit as the velocity change goes to zero since then we have laminar flow. The Reynolds number varies only with l since no, and t are constants in this case. Therefore, we see that the effect of Reynolds number in this case is to decrease the damping of the turbulence. This was first pointed out in Lykoudis and Brouillette [21] and may be seen by considering the cnxrves of Cf versus M/RE for different Reynolds numbers (Figure 21). The simplest function of Reynolds number which satisfies this criteria is C/RE where C is a con— stant. That is the function selected in this study. 'Wn.mm nun-my _J_f__ . rh—b 6O Utilizing equations (3.75) and (3.76) in (3.68) there results the following final form for Cf cf = 2(31) [Coth{—§%§exp(6(10)4(l)2)} RE RE RE 2.35 60M 4 b4 2 4 b4 2 . + { xp(2.72(10) (——) )-l}/Sinh{ xp(6(10) (—-) )}] F 178 RE F 778 RE (3.77) This expression for Cf is shown in Figure 24. The results exhibit all the necessary features of the fully developed turbulent MHD channel flow. In addition the results agree quite well with the data over its range of validity. This will be discussed in greater detail in section 4.5. 3.7 Closure In attempting to provide a thorough description of the analysis of the problem studied herein it is quite possible that the real contributions of the present study may have become obscured for some readers. In the pre- ceding develOpment a model for the deveIOping turbulent MHD boundary layer was devised and solutions for the problems based on this model were developed. In Chapter 4 it will be shown that these solutions are in excellent agreement with the only well accepted available data for turbulent MHD flow, namely the data of Cf vs. M/RE for 61 fully developed channel flow. Based on this theoretical prediction of the experimental data the present analysis is considered to provide an accurate and consistent model for the turbulent MHD boundary layer. As pointed out in section 2 there were only four previous studies for the turbulent MHD boundary layer and none of them were seen to provide an accurate model for the flow. In light of the above the contributions of the pre- sent analytical study may be listed as follows: 1. Since the present solution will (in Chapter 4) be shown to be accurate for both laminar and turbulent developing and fully deve10ped MHD channel flow, the present study poses the complete and consistent solution to this problem as its basic contribution. Only one prior study for the developing turbulent MHD channel flow has been made. That study neither agreed quantitatively with experimental data nor qualitatively with expected physical trends. Therefore, the present study presents the first acceptable theoretical model and analysis for the deve10p— ing turbulent MHD channel flow. 2. The author has prOposed a two—layer model for the eddy diffusion using a modified mixing length concept in the inner layer and a magnetically damped intermittently turbulent hypothesis in the outer layer. In light of the agreement with the experimental data the prOposed model for the MHD channel flow may be considered acceptable. 62 Therefore, a model for the magnetoturbulence in the developing turbulent MHD boundary layer has been deve10ped. 3. It has been shown that the single layer model proposed by Lykoudis and Brouillette for fully developed MHD channel flow is applicable in a modified form in the inner layer of the turbulent MHD boundary layer, but that in the develOping boundary layer an extension to the two— layer model and the magnetically damped intermittently turbulent concept should be made. 4. Based on the agreement of the MHD laminar (6+ = boundary layer (see section 4.1) results with those of other studies the author has extended the solution tech— nique of Cebeci and Smith [30] to include laminar MHD boundary layers. Most former studies for the laminar MHD boundary layer were based on either momentum integral techniques, series expansion techniques or some other specialized techniques where the technique was usually valid only for the particular problem studied. In addi- tion these techniques often had implicit error involved in them. For example, the arbitrary termination of a series expansion limiting the results to small values of the perturbation parameter. The author is presently compiling a series of such problems with the intent of writing a review paper on the area of laminar MHD boundary layer 0) flows so that a consistent set of results will be developed and erroneous results in the literature can be pointed out and corrected. In addition a variety of unsolved laminar -‘umn‘fl 63 MHD boundary layer problems are of interest such as the prediction of the point of separation of the laminar MHD boundary layer over bodies of revolution. A class of these problems is presently being formulated by the author to be solved using the present technique. Therefore, it is seen that the author has extended the technique of Cebeci and Smith [30] to include laminar MHD boundary layers and veri- fied its reliability through agreement with previous studies. 5. As stated previously the present model for the [MHD turbulent boundary layer has been verified by agreement saith the only available acceptable data. It may be said to provide an accurate model for the turbulent MHD boundary layer. The present analysis may then be applied to solving the turbulent MHD boundary layer in a variety of cases. In contrast to the situation with the laminar MHD boundary layer, there has been very little work done for the turbu— lent MHD boundary layer. In fact, of the work which has been done (three studies for flow over a flat plate, and One in the entrance region of a channel) it was shown in Section 2 that none of these offered an accurate solution to the problem. The present study therefore presents a model for the MHD turbulence and a method of solution for the turbulent MHD boundary layers so that a large class of Problems involving the MHD turbulent boundary layer may now be solved. Very little has previously been known about unconfined turbulent MHD boundary layer flows. It is hOped it: 64 that the present work will open the door for studies of this kind to begin. 6. The Approximate Analysis for fully developed flow offers a simple yet physically tractable description of the flow mechanism in fully developed turbulent MHD channel flow. The model yields (with the selection of two new E3 empirical constants) an explicit expression for the skin friction coefficient in fully developed MHD channel flow which is in good agreement with the experimental data. I No prior study provided an explicit expression for Cf in y, agreement with the experimental data over any significant range of parameters. 4. DISCUSSION OF RESULTS 4.1 Solutions for laminar flow From equation (3.23) it can be seen that when 8+ — 0 v- the solution to the momentum equation depends on the param- eter Haz/RE and not Ha or RE independently. Also from o o (3.23) we see that alternately the solution is expressible EDI-mugs... ' n. in terms of the independent parameters Ha and (x/a)/RE . In Figures 3, 4, and 5 the results of the solutions f0: displacement thickness, momentum thickness, and skin friction coefficient are shown respectively. It is seen that the displacement and momentum thick- ness, starting from zero at the leading edge, show a significant decrease with Hartmann number at a fixed value c>f the parameter (x/a)/RE . For low values of the Hartmann IIumber the displacement aid momentum thickness do not Eisymptotically approach the fully developed so called "Hartmann values." Clearly this is the case for zero magnetic field ssince the boundary layer thickness, and as a result the (iisplacement and momentum thicknesses, increases monoto- Ilically until it reaches the half width of the channel. lit that point the boundary layer thickness would continue 12<> grow if it were not for the confinement of the channel 65 66 walls. Therefore, there is an abrupt termination in the increase in boundary layer thickness with distance accom- panied by a discontinuity in the first derivative. At low Hartmann number this is still the case; however, here the confinement is due to the presence of the inviscid core and the channel walls, and the mechanism is governed by conservation of mass. As the Hartmann number is increased to very large values the tendency is for the discontinuity in the derivative to become smaller and smaller until finally the fully developed flow "Hartmann values" are :reached asymptotically. This trend is shown in Figures 3 and 4 (the trends are also shown in Figures 6 through 15 Vnhich give results for turbulent flow.) The results for skin friction coefficient in Figure 5 show Cf, starting from infinity at the leading edge and (decreasing to its fully developed flow value asymptotically. (Phe effect of Hartmann number is to decrease Cf at a fixed inalue of (x/a)/RE . The results for‘Cf asymptotically o aliproach the fully developed "Hartmann values." In addi— tiJDn a significant decrease in entrance length is ex- Per ienced with increasing Hartmann number. As a check on the numerical method the results were C=<311upared with those of references S, 10, 12, 14 and 15. 67 No plotable differences in 6*, 6, or Cf could be found in those cases where comparisons could be made? 4.2 Displacement and Momentum Thickness for the Developing Turbulent Boundary Layer The growth of the displacement and momentum thick- nesses with increasing distance from the leading edge are shown for the turbulent MHD boundary layer in Figures 6 through 15. The results are given for five Reynolds :numbers ranging from 0 to 400. ‘v‘; 1' m mu ham-nun?!“ Both the displacement and momentum thickness show 61 strong Reynolds number as well as Hartmann number de- foendence. As expected, the values decrease with increasing IReynolds number at a fixed Hartmann number and distance. 188 in the laminar flow case the values decrease with in— <:reasing Hartmann number at a fixed Reynolds number and (iistance. In the turbulent flow case, however, the value <>f the Hartmann number, necessary to effect a change in tflie growth of the displacement or momentum thickness with distance, is seen to increase with increasing Reynolds number. Comparing Figures 6 and 14 at M = 80, for example, SGVeral interesting features of the flow can be brought out. \ . *In reference 53 it is shown that certain of the solu- tll<>ns (based on the momentum integral method) of the above InEtntioned references breakdown at a fixed value of (X/a) (Efiaz/RE ) due to the selection of the form of the velocity Fxr<>file. Agreement is found between the present work and t1'l<:>se studies in their range of validity given in ref. 53. 68 For RE = 10,000, 6*/a initially increases much more rapidly than that for RE = 200,000 with X/a due to the presence of turbulence. However, at a value of X/a between 16 and 18 the value of 6*/a for RE = 200,000 exceeds that for RE = 10,000 and continues to grow at a significant rate. At this distance the flow for RE = 10,000 is already r§ fully developed. Theoretically, this is exactly what is expected since the mechanism in the RE = 10,000 case is <:ompletely laminar (due to the fact that M/RE 1 1/225) and :for M = 80 the flow develops rapidly and the displacement w] 'thickness is significantly decreased. However, at RE - 200,000 and M = 80 the value of M/RE is still quite small eand the boundary layer behaves almost like that for fully ‘turbulent non MHD. Since the RE = 10,000 case has under- ggone transition from turbulent to laminar flow at M = 80 .it is expected that the value of 6*/a be equal to that (obtained from the "Hartmann solution." That is 6*/a = 4/M fkor large M. From Figure 6 it is seen that this is the CEise for M = 60 and 80 and 6*/a is approaching 4/M for M = 40. In general if the analysis is theoretically consis- t*3r1t 6*/a should approach 4/M whenever the value of (WI/RE 3 1/225. Examing Figures 6, 8, 10, 12, and 14 it is Seen that this is exactly the case. Similarly the value <3f' G/a should be given by the Hartmann solution whenever M/RE 3 1/225. That is g- = 1%- for large M. Examining 69 Figures 7, 9, ll, 13, and 15 it is seen that this also is exactly the case. At large M it is clear that the values of the dis- placement and momentum thickness calculated in this analysis for large M asymptotically approach their fully developed flow values. However, for small M as in the Jenninar flow case the fully developed flow values are not rtuached asymptotically as discussed in (4.1). In addition, tfluere is no well accepted solution for the fully developed flLIW'values as in the laminar flow cases*. The method for den:ermining the fully developed flow values in this anerlysis is based on the value of the skin friction co- efificient asymptotically reaching a constant for fully denreloped flow. The distance at which this occurred was tine distance at which the displacement and momentum thick- nesses were considered to stop increasing and arrive at thueir constant values for fully developed flow. They are Sfuown as dashed lines in the curves. In summary it is seen that, the increase in dis- Efilacement and momentum thickness with distance from the JJaading edge is strongly effected by both Hartmann and Reynolds number; the effect of Hartmann number decreases \ *Brouillette and Lykoudis developed what appears tC> be the best analysis for turbulent fully developed MHD 9fuannel flow, however, their technique was numerical and 1iterative in nature. In the absence of conclusive data fCnr'velocity profiles it is not worthwhile to attempt to make a logical comparison at this time. 70 in almost direct proportion to the increase in Reynolds number; for M/RE :_l/225 the fully deve10ped flow values agree exactly with the “Hartmann values" for laminar flow showing the ability of the theory to correctly predict transition from turbulent to laminar flow in agreement with prior theory and experiment. _ T1 4.3 Skin Friction Coefficient for the Developing Q“ merbulent Boundary Layer ' The decrease of the skin friction coefficient from art asymptote of infinity at X/a = O to a fully developed 9’ asynmptotic value is shown for the turbulent MHD boundary laéger in Figures 16 through 20. As in the curves for the didsplacement and momentum thickness the results are pre- Senuted for five Reynolds numbers ranging from 10,000 to 20(),000 and for a variety of Hartmann numbers ranging from 0 to 400. By a comparison of Figures 16 through 20 it is seen tfllat for a fixed value of the Hartmann number M, the skin fItiction coefficient Cf decreases with increasing Reynolds number at a fixed distance X/a, while the entrance length JIncreases with increasing Reynolds number. The effect of Hartmann number on Cf is significantly (iifferent at different Reynolds numbers. At RE = 10,000 U?igure 16) it is seen that all the curves come from a CKNnmon branch at small values of X/a but that the curve for M = 20 actually lies below that for M = o. This is 71 due to the damping of turbulence. At values of M of 40 to 80 the curves all lie above that for M = O. In these cases the effect of the MHD ponderomotive forces outweighs the turbulence damping effect resulting in a net increase in Cf. At high values of M the fully developed flow value approaches the Hartmann value Cf = 2M/RE' That is,*when DL/RE approaches l/225 it is expected that the transition frcnn turbulent to laminar flow takes place and the fully denreloped flow solution should approach that for laminar flxyw. For M = 20 the entrance length is seen to increase slsightly. For M = 40 through 80 the entrance length is semen to then decrease with increasing M to a very small VEfilue of X/a when M = 80. The slight increase in entrance lenngth is attributable to the turbulence damping effect discussed previously. For RE = 25,000 the slope, of the curves near X/a = 0 irucreases with increasing M. However, the curves cross at SCnne value of X/a and the effect of turbulence damping can be: seen as the curves for M = 20 and 40 then lie below the Clurve for M = O. For M = 60 to 100 the MHD pondermotive ffxrces dominate and the curves for Cf lie successively ahxrve each other with increasing M. Again as in the case for RE = 10,000 the entrance lengths for the Hartmann rlumbers (M = 20, 40) where turbulence damping dominates arts seen to increase slightly. At values of M where the 72 MHD ponderomotive forces dominate the entrance lengths are again seen to decrease to a very small value at the highest M. The results for RE = 50,000 and 100,000 show the same trends as those for 10,000 and 25,000. The effect <3f domination by the turbulence damping is not as signi- Pi .ficant as it was at the lower Reynolds numbers. At a P Reynolds number of 100,000 the curves for M = 0, and M = 40 area identical indicating that the effect of turbulence thumping and the MHD ponderomotive forces are just balanced k3 resnalting in no increase or decrease in Cf. For R = 200,000 the curves for C versus X/a lie E f Sucncessively above each other as M increases. In addition the: entrance lengths are then seen to show a monotonic dexzrease with increasing M. As first pointed out by Lyd 400,000. Data from four separate studies are presented afui the results show excellent agreement with the data. It: is seen that the minimum in Cf with increasing M/RE djAsappears at RE between 105 and 2(10)S and this is also cOrrelated by the data. At this point it is worthwhile to discuss the nu- merical method for arriving at the asymptotic fully dfinreloped flow value. In the numerical results it was seen (KMTIW. .0 9' iv ‘3': W? '5. War OK: 7.9 Jud-r. 74 that the value of Cf would decrease monotonically with X/a until it reached a constant value about which it oscillated with very small amplitude. The amplitude of the oscillation was generally .000001 times the value of Cf at the point. This oscillation was due to the value of .000001 selected for Si the degree of convergence. ‘When Cf reached a value to which it remained constant :3 g \Mithin .000001, the asymptotic value was reached within ‘the limits of accuracy of the numerical method and the ciegree of convergence chosen. Several different values i; for 61 were selected. For 8i between .0001 and .000001 the asymptotic value for Cf, 6* and 6 remained relatively ‘unchanged. However for eii>.0001differences began to nly about 11%, with the average difference being about 6%. It is believed that the failure of the present alualysis to approach the laminar line more rapidly at 75 RE = 10,000 is due to the choice of taking the parameter C to be a constant rather then a weak function of Reynolds number which it really should be as explained in the work of Lykoudis and Brouillette. In the fully developed flow case this choice was adequate as can be seen from the results of Lykoudis and Brouillette for RE = 10,000. It . becomes apparent that in the developing boundary layer it :1“ is somewhat more important to include this functional relationship at low Reynolds number. Physically this makes sense since it is clear that L? in an attempt to arrive at asymptotic results by solutions of the developing boundary layer equations an accurate description of the turbulence damping and its effect on the inner or laminar sub layer is of great importance, and, at lower Reynolds numbers the effect of the turbulence damping is of greater significance as previously discussed. The asymptotic solution is therefore seen to be not only in 100,000 are not shown since the P; E (approximate solution is an extension of the Blasius ex- jpression for non MHD flow which is accurate only up to about RE = 100,000. It would be expected that deviations from the data and other solutions would start to become appreciable as the Reynolds number continued to increase .beyond 100,000. Figure 23 shows results of other studies for the :fully developed flow case. It is clear from Figure 23 fihat none of these studies provide a solution which is \Lalid over any significant range of parameters. Consider- :Lng then that no closed form expression for Cf was pre- \fiously available (3.77) provides a useful approximate Jrelationship which is reasonably accurate in the Reynolds nl-‘lmber range between 10,000 and 100,000. 77 m mewE Sm\s\xvooo_ *0 m6 m.¢ 0:? NM ¢.N 6.. m6 0 No.0 8.»: \ if ,¢0.0 \ \\ owns.P . \ mod IV \\ \ om»: \\ \ o\._m “Hulk \ mod 011/ \\M\ o... 6 28.528 $2.23 On: Li ~ ‘ O_.O 78 1. mass: o.m\s\xv ooo. .v.m m6 9v of Nd TN 0. _ 0.0 o 5.0 02"!“ i& . \\\ NO 0 “\ 84.7W “11% ,Iluhull no.0 \\ o\o 0.? u z A V0.0 \ oflw H zoEozoo $2.24.. _ _ _ \ AH é no.0 79 V0 o.m\s\xv 80. 0.0 06 06 N .m ¢.N 0.. m 0.0 mmDOE o . l _ 0n .0 “20:.5200 m<2_2