THERMAL RIWATKIN~ AND MAGNETIC HELD. EFFECTS an THE now VARIABLES NEAR A STA-GNATION POINT ._.. 3.1523... .1 .1. Thesis for the Degree of Ph. D 2.1.1:? MICHIGAN STATE UNIVERSITY WENDELIN‘ SCHMIDT 1963- ., ....... . . f: : r . . V ‘ , . 233.... H . . .L» 5...”, 2.53123: .‘ ing% 3% {:an up.» if 1.... £3.51“. «I. ‘3 ”QSES IJBRARY Michigan State Lhnvmmfiy' This is to certify that the thesis entitled ‘mERMAL RADIATION AND MAGNETIC FIELD EFFECTS ON HIE FLOW VARIABLES NEAR A STAGNATION POINT presented by Hendelin Sch-1dt has been accepted towards fulfillment of the requirements for M— degree in We 0 /L/1Z 1}, /%th/Lt>zy flé’f/{J Major profgor Dme July 81 1968. 0-169 ABSTRACT THERMAL RADIATION AND MAGNETIC.FIELD EFFECTS ON THE FLOW VARIABLES NEAR A STAGNATION POINT by Wendelin Schmidt In this investigation the equations connecting the flow variables with the geonetric paraneters of the streanlines in three dinensional, inviscid and viscous radiation negnetohydrodynanic gas flow were derived. A sinplified nathcnatical nodel governing the flow variables distribution near a stagnation point in radiation nagneto- hydrodynanio flow was deve10ped and used to eetinate the eonbined effects of various physical phenonena on the flow field variables. Specifically we consider the combined effects of thermal radiation, nagnctic field, viscosity, heat conductivity and conpreesibility on the tenperature, pressure, electron density, and electric conductivity distribution near a stagnation point. The first order results obtained frcn the nunerical solutions of the governing equations indicated that the effects of thernal radiation, nagnetic field, and Vendelin Schnidt cclpressibility on the flow field variables are considerable, whereas the effects of viscosity and heat conductivity were found to be very enall in the case under considera- tion here. THERMAL RADIATION AND MAGNETIC FIELD EFFECTS ON THE FLOW VARIABLES NEAR A STAGNATION POINT By wondelin Schmidt tA THESIS Submitted to - Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Depart-cut of Mechanical Engineering 1968 ACKNOWLEDGMENTS The author expresses his appreciation to Professor M. Z.von Krsywoblocki for his assistance in the fornulation and analysis of the problen under consideration here; and to Professor J.E. Lay, G.E. Mass, and N.L. Hills for serving on the Guidance Ccnmittee. 11 TABLE OF CONTENTS Page AMOWLEDWNTSOOO00.00.000.000...000000000000000000 11 LIST OF FIGURES..................................... '1 NOMENCLATURE........................................ V111 1. INTRODUCTION............................ 1 PART I: INVISCID RADIATION MAGNETOHYDRODYNAMICS 2. INVISCID GOVERNING SYSTEM OF EQUATIONS 2.1. Fundamental Equations.............. 5 2.20 Roronul‘tionOOOOOOOOO0.00.00.00.00 6 2.3. Tensor Fern........................ 7 3. DYNAMIC AND KINEMATIC RELATIONS 3.1. Basic Deconposition................ 9 3.2. variation of Pressure along and Perpendicular to the Streanlinee... 12 3.3. Vorticity Conponents............... 17 3.&. variation of Energy along the Streamlines........................ 22 a. GOVERNING EQUATIONS IN CYLINDRICAL COORDINATES 4.1. General Cylindrical Coordinates.... 23 h.2. Axially Symnetric Case............. 25 e.3. Inconpressible case................ 27 e.a. Alternate Axially Symmetric Case... 28 111 5. 6. THERMAL RADIATION AND IONIZATION 5.1. The Equation of Transfer........... 5.2. Radiation Flux and Pressure........ 5.3. Ionization and Electric Con- ductivity.......................... SPECIAL CASE or v(r), v(z) ONLY 6.1. Equation of the Streanlines........ 6.2. Tenperaturc and Pressure Die- tribution along streamlines........ PART II: VISCOUS RADIATION MAGNETOHYDRODYNAMICS 7. GOVERNING EQUATIONS 0P VISCOUS FLOW 7.1. Fundamental Equations.............. 7.2. Transformation to Stroanlinee Coerdinatee........................ 7.3. Streamline Pressure variation, Curvature and Toreion.............. 7.e. vorticity.......................... 7.5. General Cylindrical Coordinates.... 7.6. Axially Symnetric Caee............. GEOMETRIC PARAMETERS OF STREAMLINES IN AXIAL SYMMETRY 8.1. Streamlines of the Porn 2 - f(r)... 8.2. Streamline Curvature k for the Streamfunction 1V. f(e,r).......... PART III: NUMERICAL SOLUTIONS AND RESULTS 9. INVISCID, INCOMPRESSIBLE FLOW RESULTS 9.1. Physical Streanlinee and Parameters iv Page 33 35 37 #2 #6 50 52 53 56 58 59 65 67 68 Page 9.2. Pressure Distribution along and Normal to Streanlines.............. 69 9.3. Temperature, Electron Density and Electric Conductivity Distribution. 71 9.§. velocity Distribution.............. 72 10. VISCOUS, COMPRESSIRLE FLOW RESULTS 10.1. Tenperature, Pressure, and Density. Distribution....................... 74 10.2. Electron Density and Electric Con- ductivity Distribution............. 80 10.3. Summery of Results and Conclusions. 80 APPENDIX Typical Fortran Progran and Results........ 98 BIBLIOGRAPHY.0....OOOOOOOOOOOOOOOOOOOOO...0.0.0.0.... 105 Figure 111. 1a 111. 2c 111. 3c III- 4. III- 50 111. 6c 111. 7a III- 8a 111’ 9c III-10. III-11. III-12. III-13. LIST OF FIGURES Streanline Pattern for Flow Against a Diskeeeeeeeeeeeeeeeccseeeeeeeeceeeeeeeeeeee Streanline Curvature k along Streamlinee‘TC Pressure Gradient variation Nornal to Stroanl1noaly.............................. Effect of Magnetic Field on Pressure Distribution along Streamlines (InconprOIIIDIO)........................... Temperature Distribution along Streamline"; (Inca-Pr0‘81b10)eeceeeceeceeeeeeeeceeeeases Electron Density Distribution along Strean- line'ya’(Incompreesible)................... Electron Conductivity Distribution along Streamline v; (Inconpressible)............. Conparieon of Tenperature Distribution along V1 and Y2 (Inconpreeeiblo). . . . . . . . . . . Comparison of Electron Density Distribution along‘V; and‘y; (Inconpreeeiblc)........... Comparison of Electron Conductivity Distribution along‘yi andV;(Inconpreee1ble) Velocity Distribution along Streamlines I’lee Density Distribution along Streamlines V. . . Pressure Distribution along Streamlines y} (cowro..1bl°)0000COO...00.000000000000000. vi Page 82 33 84 85 86 87 88 39 9O 91 92 93 9A Figure Page III-1e. Effect of Thernal Radiation Cooling on TC-porature D18tr1but10n.........o......... 95 III-15. Effect of Thernal Radiation Cooling on Electron Density Distribution.............. 96 III-16. Effect of Thermal Radiation Cooling on Electric Conductivity Distribution......... 97 vii ’6‘ GO ‘;3 as ns-re :n .o a ‘2 n: c. -43 ‘" be (h '6 I D I 'U 6' 0‘ NOMENCLATURE flow velocity, n/sec fluid density, kg/e3 fluid static pressure, n/n2 radiation pressure, n/m2 electric current density, amp./n2 magnetic field intensity, amp./n nagnetic perneability, websre/anp.-n internal energy of fluid, n-m/kg energy input, n-m/kg universal gas constant, n-q/kg-ox fluid tenperature, K a p + pB electric field intensity, n/coul. excess electric charge density, coul./'n3~ electric permittivity of free space, coul.2/n-m2 electric conductivity, ache/m tine, see. a (5 p)"1 nagnetic diffusivity, na/eec . frag nagnetic pressure, n/n2 e (p + pa + p-), total scalar pressure, n/m2 arc length along streamline, n arc length along nornal to etreanline, n arc length along binormal to streamlines, n viii streamline curvature, i/m torsion of streamline, 1/n rectangular coordinates, (i-i,2,3), x,y,z. cylindrical coordinate cylindrical coordinate cylindrical coordinate component of unit vector, Eq.(3.1.2) . (u . P/f . H2) netric tensor Mach nunber vorticity vector alternating unit tensor enthalpy of fluid, n-m/kg “PH nagnetic field ‘vebcrs/nz components of fluid velocity, n/ecc specific heat at constant volume, n-m/kg°K specific heat at constant pressure, n-m/kgox radiative heat flux vector, n-n/mz-sec velocity of light, n/eec Stefan-Boltznann constant, n-n/eec.-m2-°K Rosseland neon absorption coefficient, 1/m Planck mean absorption coefficient, 1/m streamfunction ix .’?Qb, b a constant, Eq.(6.1.2) AM,AR constants given by Eq.(6.1.21) 7“2 F K t it viscous stress tensor, Eq.(7.1.2), n/hz fluid viscosity, kg/n-sec thermal conductivity, n-n/mpsec-°X viscous dissipation function, Eq.(7.1.3), n-n/sec-n3 electron nunber density, #/cn3 1. INTRODUCTION The great interest in hypersonic flow around blunt vehicles has been stimulated in the last decade by the intercontinental ballistic missile, the satellite and the deep space programs. Among the phenonena that can be observed during hypersonic atmospheric entry of a vehicle are the thernal radiation emitted by the hot gas flowing around it and the reflection of nicrowaves by the ionized gas envelope surrounding the vehicle. It is well known that this ionized gas or plasma envelope is the cause of radio, and other communications black- out during atmospheric re-entry. (l, 2, 3, h) The influence of the electrons in the ionized gas around the vehicle is felt not only on electronagnetic signal attenuation through radio blackout, but also on aerodynamic quantities such as drag and heat transfer, and on physical quantities such as transport, radiative enission, and absorption properties. From the communica- tions problen point of view the electons are undesirable and should be elininated. However, the flight nagneto- hydrodynanics point of view considers the ionized gas as a phenonenon to be capitalized on by applying a strong magnetic field in such a nay so as to provide a re-entry vehicle with braking and other naneuvering capability. (5 to 17). A solution to the hypersonic blunt-body problen that 2 conbines the advantages of nininun conputational dif- ficulty with naxinun accuracy of results has been sought for nore than a decade. The problen under consideration is that of deternining the flow field properties (i.e. pressure, temperature, density etc.) around a blunt- nosed configuration traveling through a nniforn gas at a flight Mach nunber greater than unity. In general, the flow field about the body nay be divided into two regions based on the nagnitude of the local flow Mach nunber with respect to unity. In the region near the stagnation point of the body the flow Mach nunber is less than unity and the flow field is therefore subsonic. Rush of the effort expended on the blunt-body problem has been confined to the subsonic region since the solution to this region provides the starting data for the well known characteristic nethod of supersonic flow calcula- tions downstream of the stagnation point. The deter- nination of the fluid properties in the subsonic flow field over the blunt body also provides the necessary data for the subsequent evaluation of the radiant heat transfer as well as for initiating boundary layer cal- culations to determine wall shear and convective heat transfer to the nose of the body. Pron the aerodynamics point of view the nain problem associated with the re-entry of a space vehicle is that of convective and radiative heating, and aerodynanic drag. 3 Since the gas around the vehicle is in a plasma state and, therefore, electrically conducting, the possibility of utilizing an applied nagnetic field to reduce surface shear stress and heat transfer has been proposed by a nunber of authors. (5 to 8). The general approach to the problen consists of dividing the flow field into a viscous boundary layer and an outer inviscid flow. The solution to the boundary layer part of the problen requires a knowledge of the edge of the boundary layer flow condi- tions which are obtained from the solution to the outer inviscid part of the flow field. Part I of this thesis is directed towards the deterninatien of the cenbined thermal radiation and nagnetic field effects on the in- viscid flow field variables near a stagnation point. The division of the flow field into an inviscid flow region and a boundary layer is only possible for flight altitudes below which the ratio of the vehicle radius to nean free path of the gas nolecules is greater than about 75. i.e., lib/A > 75. For altitudes such that the ratio of vehicle radius to nean free path is be— tween about 75 and i, such a division of the flow field is not possible and either the full or a einplified fern ef the Nevier-Stokes equations nust be used as a flow nodel. (54). In Part II of this thesis we in- vestigate the combined effects of thernal radiation, nagnetic field, viscosity and heat conductivity on the flow field variables. k The central problem under consideration consists of the derivation of a nathenatical model which allows us to predict the effects of various physical phenonena on the flow field paraneters under given conditions. An exact nathenatical description of the flow field in- eluding thermal radiation, nagnetic field, viscosity, and heat conductivity effects, requires a ccnplicated set of non-linear partial differential equations which are very difficult to solve for a given vehicle con- figuration. Because cf this difficulty, nany simplified, approxinate nathenatical nodels have been proposed which held under various conditions. One such sinplified approach to the calculation of the flow field parameters ‘was proposed by several authors independently and consists of calculating the parameters along flow streamlines under an assuned flow field pressure distribution (49 to 52). The nethod was used for dissociating, inviscid, non- beatcenducting, non-radiating flow without the nagnetic field effect, and was found to be quite anenable to para- metric study of very complex flow fields. He shall use the etreanline approach in the present investigation with a nodification which consists of using an approxinate velocity distribution. PART I: INVISCID RADIATION MAGNETOHYDRODYNAMICS 2. INVISCID GOVERNING SYSTEM OF EQUATIONS 2.1. Fundanental Equations We consider an inviscid, non-heat-conducting steady flow of an ionized perfect gas in an electroqnagnetic field including thernal radiation. The governing hydro- dynanical systen of equations consists of the nathenatical fornulation of the physical laws of conservation of nass, nonentun, energy, and the equation of state of the gas; V‘U’?) . 0, (2.1.1) V'V‘t? a JV? + 1"(3 x p‘fi), (2.1.2) du + Piaf") - dQ .- o, > (2.1.3) p as far. (2.1.1.) The energy equation (2.1.3) nay be used in an integrated fern along a etreanline to be denoted as the generalized Bernoulli equation. The equations governing the electronagnetic field are Naxwell's equations and Ohn's lav; (55.58) V'E mg/E ; V x E m 0, (2.1.5) Vs-fi m 0 ; V! E m .3, (2.1.6) 3 -J(E + pV 2B). (2.1.7) 5 2.2. Refornulation The above systen of equations nay be reformulated so as to be nore suitable for the present analysis. Substituting for 3 fron Eq. (2.1.6) into the equation of notion (2.1.2) we get; G M7? . -r’v(p . pR . m2) . ”pd-v3. (2.2.1) The energy equation suitable for the present analysis may be obtained by starting with Eq.(2.i.3.) as follows; dQ . du . rd(’/p) . d(u . P/f ) -f"’dr, or 3-2 .- V-V (u + P/f ) -f’V-V P. (2.2.2) By expanding the left hand side of the equation of notion (2.1.2) and taking the scalar product with V we get, tvuv’) - NV 2 (V2 V» - J‘V-w . mass). Using this last result to elininate .p‘wp in Eq.(2.2.2) we get, v-wu . P/f . we) . 3.3 .r“'t.(3 x pH). (2.2.3) M; is the heat input fron all sources per unit nase, which in our case consists of the Joule heat and the radiation heat flux. Thus we have, 5-2 .f'uz/a) +F‘U'3- 7 Using this last result to elininate §§ in (2.2.3) and substituting (2.1.6) for 3 and expanding 3 x pH, we get; two. . P/f . 1v”) .F’w x H)*(Vx 3% .r’hqd-vfi) - FWVGH”) +f"V*‘5. (2.2.2) which is the energy equation in the required fern. Pron.Maxwell's equations and Ohm's law we develop the following ; solving for'E fron Eq.(2.1.7) and substituting for 3 er R we get, Eml/aVXE-PVXE; taking the curl of this last result and accounting for Eq.(2.1.5) we have, (1/P3)VI (v: s) -Vx (V x E) - 0. Expanding this last result and using (2.1.6) and re- placing 7]. 1/p3 we get, 7W3 . Vevfi - ewv + 3(7 43). (2.2.5) 2-3- We. The governing systen of equations will be recast in Cartesian tensor forn. In this forn the equation of continuity becones, 491(va . 0. (2.3.1) ¢ix The nonentun equation (2.2.1) becones, vav‘ 11Pt_ 331., .. 1° 33+s & P53: 0 (232) 8 2 ‘where Pt - (p + pn + 1P3 )- The equation of energy (2.2.t) gives, (“3'3 - 5"(sjr-317—33-s‘fig—gg) pvdfli-g—E} - “Lg—:3.- e #3::- , (2.3.3) I where I a (u + P/f 4 five); p. e {1132. The electromagnetic equations (2.2.3) and (2.1.6) becone, 31 _ J v1 + i v5 c k 31 V315: H ‘3: H -3': ”('3346 “3-3), (2030‘) .32; . 0, (2.3.5) 8 Th complete the systen of equations we add the equation of state of the gas p efRT. (2.3.6) The unknown quantities consist of three scalars and two vectors; i.e., p,f,T, V, and R. Il'he three scalar equations required are qu. (2.3.1), (2.3.3), and (2.3.6). The two vector equations are Eqs. (2.3.2) and (2.3.h). 3. DYNAMIC AND KINEMATIC RELATIONS 3.1. Basic Deccnposition The equations (2.3.1) to (2.3.5) will be transfcrned into a coordinate systen s1, hi, hi, where the synbole used denote the conponents of the unit tangent vector (e1), principal nornal (hi), and binormal (b1) vectors with respect to a etreanline at any point in the flow field. Denoting the magnitudes of the velocity and nagnetic field vectors as v, B, respectively we get, $1. . .1, (3e1e1) where s denotes arc length neasured along the stream- lines in the direction of the flow. Per the nagnetic field we have, E1- . hi, (3.1.2) where h1 is a constant unit vector. A set of relations involving the three unit vectors s1, n1, b1, is given by the well known Prenet-Serret fornulas of differential geonetry i 1 1 ds 1 d: . -Tn1, dn 1 31"“. as" 'rb‘ - ks , (3.1.3) where k is the curvature and 7* the torsion of the streamlines. 10 Substituting Eqs. (3.1.1), (3.1.2) into the appro- priate Eqs. (2.3.1) to (2.3.5) the systen of equations becones, “L'f(fv.1) ' 09 I (3.1.5) ax. ,1 its1 1.; Pt _ J an1 . fVI #1 4- g ‘3: FEB '35—! 09 (3.1.5) .1 I .. mik an _ mi" a”: + ... 2’:- “$44324 ”-3171 124 1 Rh p. i flVIJHh II3-iI-u - VlJ‘gj 4’ 'fi' s (3s1.6) N 12 a C 0, (3e1e7) a X Vets-31$ .. 311.1% + uni-$.21 4-3-3413 9 m1 )1 (3.1.8) Expanding Eq. (3.1.e) and using .1.-3: .- .37 we get, -3% e ~370an :- 0. (3.1.9) Expanding Eq. (3.1.7) we get, fig . 0. (3.1.10) 11 Expanding Eq. (3.1.5) and using sing): a: .3; , hi—g—x-f a ‘3? , we get, fv-s—Eel + fvz-S-‘g- + 813-33 - pit-3%} :- 0. (3.1.11) Using Eqs. (3.1.3) and (3.1.10), Eq.(3.i.11) becones, fv-s-gei e fv’m‘ + 83-3-3» . 0. (3.1.12) Expanding Eq. (3.1.8) we get, wage:l - (93%} - HIV-Si: e 803% e vfih‘ 4‘33““ ()3 )b‘, (3.1.13) or (W + air-3%- - '713315‘“ ()3 ))h1 - H-fi-Ei- a '0. (3.1.14) Expanding Eq.(3.1.6) we get, "3% .34(‘Jrhk all? J r)H _ erhk 31:13] an ) + pVHsJ-QEGhfl - V-S-E-E e jg . (3.1.15) Using Eq.(3.1.10), equation (3.1.15) reduces to, p. 1 fv-gé . -5-'(g3’.§.:.r. .313) - “'37 4» %. (3.1.16) 12 3.2. Variation of Pressure Alon and Fe endicular to the Streamlines To deternine the variation of the total pressure Pt along the tangent, principal nornal, and binornal directions of the streamlines, we take the scalar product of Eq. k (3.1.12) with 31k. , giknk, gikbk, respectively and get, P iv :Hgiks e fvzknigiksk + gugiks sk-gfi m 0. (3.2.1) P j’v-3%s1g1rnk e j’vzknj’giknk + Quaint-3;; e 0, (3.2.2) r flag-15.1w. fvzkn‘gub .g‘Jgubk-gj» . 0. (3.2.3) Makingi use of the orthogonal properties of s1, n1, b1, and g 3311'51' fig:- -3-,nJ-d:-m .63.“... we get from Eqs. (3.2.1) to (3.2.3). ”-3-: + .3;1 n 0, (3.2.11) fvzk + 6:3 s 0, (3.2.3) .3? . 0, (3.2.6) where Pt e (p tpR e 1).). 13 From the result of Eq. (3.2.6) we note that the total pressure renains constant along the binornal direction of the streanlines. Pron the result of Eq. (3.2.5) we obtain an expression for the curvature k of the streanlines as a function of the fluid density P , the velocity v, and the nornal pressure gradient as, r . -(pv2)-1.3.;1, (3.2.7) Tb obtain a relation for the normal vector of the stream- lines 11: as a function of k, f, v, and the velocity and total pressure gradients along the streamlines we solve Eq. (3.8.12) for n1 n1 - -(PV2K)'1(PV-3-z41 + til-33'). (3.2.8) and get, Multiplying the last term of Eq.(3.2.8) by the scalar product of s1 we get, (lieu-521L33- - 01%;. (3.2.9) Substituting Eq.(3.2.9) for the last tern of (3.2.8) and using Eq.(3.1.1) we get from (3.2.8), n1 . -(fv3r)’1(fv-3% . -3-?-)v‘. (3.2.10) 15 we next obtain a relation for the binornal vector of the streamlines by starting fron the definition of hi, b1 n e1Jngpgkqspnq. , (3.2.11) Fron Eq.(3.1.12) we have, .9 . -(pv21)'1(pv-§§.9 . g" :1"). (5.2.12) xr Substituting Eq.(3.2.12) into (3.2.11) we get, 2 b1 - ~(PV2k)'1(fV-3-‘,Ie”ksjpskqopoq + ouksjpskqo‘qu-gfi). (3.2.13) °13k Since, nggkqspsq - 0. ‘kqsqr II 8: s CJPBP - VJV-i. we get the binornal vector as a function of the flow field parameters, b1 . ~(fv3k)-1(e1kaJ-g%). (3.2.14) Streanline Torsion To obtain a relation connecting the torsionflT; of the streamlines with the flow field parameters we nake use of the Prenet formula Eq.(3.1.3) which is, 1 ~121- §-}-. (3.2.15) 15 Differentiating Eq.(3.2.1h) along a etreanline we get, i . 2 (31'3" . ~013k-§;((PV33)’1VJ 433‘)- (3.2.16) Expanding Eq.(3.2.16) and substituting the result into Eq.(3.2.15) we get for the torsion of the streanlines, -'l"n1 . .135 (1’1"):de .3; 37(fv3k) .- (Modes); 5:1 + ”137%?” . (3.2.17) We next deternine the static fluid pressure-gradient along a etreanline as a function of the Mach nunber M and the other four variables and their gradients along a streamline; i.e.,F, V, T, R, and their gradients. Pron Eq. (3.1.9) we have, fi + v‘1-3-‘J— . r16; . 0. (3.2.18) Solving Eq.(3.2.§) for v and substituting it into 3s Eq.(3.2.18) we get, A 1 -vzf-g—il- - (72-3; 4- -35 e -3;-2 + -3-:5 e 0. (3.2.19) The velocity of sound is defined by a2 a -35-, or a'a-g-E .- -3-§ . (3.2.20) 16 Substitutine (3.2.20) into (3.2.19) we get, '3'; . (52-1)‘1(-3-:3 , .3? .pv“ a5). (3.2.21) 1 Substituting for “—33-!- by usins (3.2.18), Eq. (3.2.21) x becones, {3% . window-3% . '3'? e ”'3': . Ila-3;). (3.2.22) Equation (3.2.22) indicates the influence that each variable has upon the fluid pressure variation along a streamline. Expanding Eq.(3.2.5) and solving for the fluid pressure variation in the nornal direction to the streamlines we get, fit; . -(pfl-3-E . 3:3 . flak). (3.2.23) From equation (3.2.6) we obtain the following expression for the pressure variation in the binornal direction to the streamlines fig . -(pfl-3% . 5-153). (3.2.21) 17 3.3. vorticitngonponents The vorticity conponents are defined as, wk . .MJgJ’LA-Vi A (3.3.1) ax” ' Substituting v1 a Vs1 and expanding we get, 1 wk . v.kij‘Jr-g-;? . '1'113‘Jr'gir . (3.3.2) k k k Taking the scalar product of Eq.(3.3.2) with s , n , b , respectively and recalling the cross-product relations eujsisk .- 0, ekuei’nk s bJ, eanib" . --nJ , we get, wk.k . vow-"rmggh (3.3.3) wknk s “chunky-3%) + {g . ' (3.3.5) wkbk . v(.ukag‘lr-g-;-:=) - a; . (3.3.5) .221 In order to obtain the tern. d r as a linear ccnbination x of grpsp, grpnp, grpbp, we make use of the following identities, Irfi; I —3¥:1 , nrfi; . dag-1n, brfi; In fig, (3.3.6) 18 Each identity of Eq. (3.3.6) is the scalar product of gel r r r 0 r with s , n , b , respectively and since the later x are orthogonal we have, 1 1 1 1 .1? .- -3—:--grpsp + -3%—grpnp + -3-E—grpbp . (3.3.7) Multiplying this last result by ng and using gJ rrp‘ .- 8-1 we get, s’SJ}-- 3 +~3—-nJ. 'girbj- (3.3.8) Substitutins (3.3.3) int. the Eqs. (3.3.3) to (3.3.3) we get frcn the later, 1 1 1 ‘ k 1. V(-3-:—ekusksJ + -3-;—ekusknJ e -3-£-0quka)1(3.3.9) k as v(-§%1ok:unks'1 + -3-:T1-ekunknj e -3-Eienjnkbj) + '3'; (3.3.10) 1 1 1 k .- V(-3-:—~ek“bksJ e -3-l-:-ok:l_3bkn"I .- ~3la-ekubkbj) - -3-§. (3.3.11) Taking account of the cross-product relations in equations (3.3.9) to (3.3.11) we get the following results, 19 1 1 wksk .- V(g1rbr-3€— - girnr-3%-), (3.3.12) 1 wknk a V(-g1rbrn1k e O + first-3%.) e -3% e -3% , (3.3.13) wkbk . v11 — «3% . (3.3.111) We note again that each equation (3.3.12) to (3.3.1h) is the scalar product of wk with ex, nk, bk, and thus we get the vorticity components as, 1 1 wt - v(g1rbr-fi-§- - glrnr-fi-g—kkpsp 4- (-3-E)gkpnp + (Vk "' '3%)8kpbp e (303e15) Multiplying this last result by at and using pubsk .- IvJ EJKSkp ' 8:) s '0 S“ w‘1 as V(g1rbr—37:1 - girnr-Swfiihi e njcsrg + (V): - -£%)b'1. (3.3.16) Using the identity developed in Eq.(3.3.8) on v we get, Sir-3i; I -3-§s1 + $111 + -3§b1. (3.3.17) Multiplying this last result by v and solving for v :s“ we get, 2O fV-g-Esin PVgir-glxr-r- - fV(-3iv-n1 e '3'¥°1L (3.3.18) Substituting Bq.(3.3.18) into Eq.(3.1.12) we get from the later, 1 v2 v1 v1 1 1 I’t ifs”-(‘};;-fV(-3;n +-3-5b).fv22n +gr-§;;.0. (3.3.19) Adding and subtracting “legit-345,- fron Eq.(3.3.19) and rearraging we get, -’-1‘1r-3-x?(p 4. DR 4- {W2 .,. p.) * *rlvisilzgff _ vm: - -3§)n1 — magi} . (3.3.20) Introducing a function U’defined by sir-3%; - -F121r-3-;;(p + Pa + p. + 1W2) + ”Ash-di- ax" ’ (3.3.21) With this last result equation (3.3.20) becones, 1r '5’ v 1 v i g -3—x? I V(Vk - -3;)n - V , (3.3.22) or 3%. . v(vu - -3%)g11n1 - v-ggnbi. (3.3.23) 21 Equation (3.3.23) is a vector nornal to the surfaces '5 .- constant, and if we let its magnitude be %’we get 11'0‘ qu(3e3e23)’ [3% . (may + vzm - £922 . (3.3.21) Taking the scalar product of equation (3.3.23) with Eq. (3.3.16) we get, 111-3% . v(vr - {gr-3% gnninJ - vm: - flag-Rubin1 m 0. (3.3.25) Also, taking the scalar product of equation (3.3.23) with sJ we get, J F. I . I . . 6 8—3: 0 (332) Thus, equations (3.3.25) and (3.3.26) imply that the surfaces 3’: constant contain both the streamlines and the vortex lines. 22 3.§. Variation of Energz Along the Streamlines d 31’ 3 1 Substituting the i entity .m . f( .(p.f'1) _p- . ) into equation (3.1.16) we get, _ ' 1 1 fV-S-I; +PV-3;(p_r‘) - -a war-35; 13%) + m3? «33 . (3.11.1) Dividins (3.10.1) by (W) and conbining the two terms on the left we get, I _ _ l1 - 1 ‘321- -a 1(3Jr‘3'EE‘SLBNPV) 1 + p.431; + (m 13.3,, (3......) where It s (u e pF'1 + ply-1 4- p-f’1 4- fva). Since ngsrsJ :- 1, we get for the first term on the right of Eq.(3.§.2) 34(83r'r038‘1r-3-E’: ‘38:.) . &-1('3%)2~(3°'“3) Usingthe itintity developed in Eq.(3.3.8) on Q1 and substituting the result together with Eq. (3.13.3) into Eq.(3.§.2) we get, {7‘- = -a"(‘3%)2(rvr‘ + p.-3é:1'+ <9v>‘1<-3§-‘-1 + $22.1 + 13234:”. (3.4...) Equation (3.4.h) shews that the change of total energy per unit mass, per unit distance along the streamlines depends on: (i) the Jeule heat generated, (2) the work of coapression done by the magnetic pressure, and (3) the variation of the heat flux vector along and perpendicular to the streamlines. 23 h. GOVERNING EQUATIONS IN CYLINDRICAL COORDINATES In this section we formulate the equations of section 2. in cylindrical coordinates for later application to a specific flow problen. 5.1. General Cylindrical Coordinates Introducing cylindrical coordinates (r,0,z), we get from Eq.(2.i.i), W,S¢l..§§:fl.1§gfl.o, (5.1.1) and from equation (2.1.2) we have v v v v2 v F(Vr'3'r£ I $237!.— " 3'2- I vz'37£) ' ' ‘33).21 I ”0324230) (§.1.2) V V V V V V f(Vr-g-i,3 4- 9%; + —:?-°- 4- V2632) - - 33-; +(JzBr—Jrnz) (s.1.3) v v v v P(V,.-3-;.3 + é—f}; + V5335). - - .3; + (arse-Jog). (5.1.5) The energy equation becones, mt}; + Hi + v.%f:—> - ”133% - 93% + V2439 * g?- + (‘3? + :5 ‘9ng2 *‘3'Elb (“01-5) 2h 2 2 2 where P-(p+pn), es(u+P/f), -J 'JrIJOIJz From Eq. (2.1.5) we get (705-; - fl?) . o, < (t. ("3':-- T) I 0. (h. . Egg-:9 - {(35-5) =0. (1. . and fron Eq. (2.1.6) we have, ££+§E+;£33+-3;£-0, (‘h- (1.05%."3':2)'-on (k. 2 1.6) 1.10) n n 933—" .. 8-5;) . Joy, A (4.1.11) @322- T) - J2)! . (4.1.12) and by Eq.(2.i.7) Jr .5013, + (v03; vzaon, (4.1.13) Jo .3010 + (vznr- vrnzn, (11.1.15) J2 . 3(3, + (v1.30. vast», (1.1.15) The above equations (§.i.1) to (5.1.15) are a set of relations for the following unknown quantities, p’f’ T’v r’ v0, vz’ Br, Bo, Dz, gr, 80’ Ba, Jr, J0, J2. 25 5.2. Axiallz Szggetric Case In this section we consider Eqs. (5.1.1) to (5.1.15) in axial synmetry for which we have the conditions, -3-5 s 0, VO 3 0, B0 c O, and we set v-Vr, w-V‘. Introducing the above conditions into the Eqs. (5.1.1) to (5.1.15) we get from (5.1.10) and (5.1.12) Jr I JI a 0, (§.2.1) from Eqs.(5.1.13) and (5.1.15) Jr uazr, a, .an', (5.2.2) thus, by (5.2.1) we find that Er'Ez' a, (5.2.3) and by Eqs. (5.1.6) to (5.1.8) we find 1go E0 «3;- . 0, '32- .0, so the Boa-constant. Fro. Eq.(5.1.i5) we find that JO :0, for vuw-O, so that Eg a 0, and therefore it is zero everywhere in the flow field, since it is a constant. The system of equations (5.1.1) to (5.1.15) now reduce to the following, .3411!)- . 93'.)- . fifll . 0, (4.2.1.) 26 (F311; + «~39 - - {3% + a(vB,-vn,)n,. (5.2.5) (5&3; + r39 = - -3§ -a(wBr-vn')nr, (5.2.6) (v-s-g + Ffi) :- “-3-; + w-S-E) 53(wBr-sz)2 + 03;! + :5 + -3-33), (5.2.7) where we have used J° from Eq. (5.1.15). Equations (5.1.9) and (5.1.11) now becoae by using (5.1.15), -3-:—11 «I» :5 4—3-31- . 0, “02-3) Br B2 (.3;— - ‘3'?) . pa (wBr-vB‘). (5.2.9) The equations (5.2.5) to (5.2.9) together with the equation of state p- RT, are seven equation for the seven unknown quantities; i.e., p,f, ‘1', v, w, Br’ 8‘. The above quantities are functions of a and r, only. 27 5.3. Incompressible Case If the fluid density can be considered as renaining essentially constant in some flow region, the systen of equations (5.2.5) to (5.2.9) reduces to the followins; 8.; . g, . £3; . 0, (5.3.1) fur-3% + "3%) 5 fig ’533('3r"9z) - 0, (5.3.2) fur-3; 5 *3!!!) 5 fig +5Br(wBr-vnz) .- 0, (5.3.3) p.'(v-3-§ . w 3) -3('Br-'B.)2 - (.335! . :5 . $.22) . 0, (5.3.5) B B - .3—:£ q. I“! 4- &3 I 0, (#0305) Dr B! (-I$- «- IleI) - Pa ('Br-VBz) 3 0e (Q0306) Equations (5.3.1) to (5.3.6) are six equations for the six unknowns; 1,e., p, 1', v, w, B 8', fa constant. r, 5.5. Alternate Axiallz Szggetric Case In this section we consider equations (5.1.1) to (5.1.15) in axial synnetry for which we have the following conditions; In steady flow the electric field 2' may be taken as constant or in the case of no applied electric field it any be taken as zero, (55). Introducing the above conditions into the Eqs.(5.1.i) to (5.1.15) we get from (5.1.1), Sifiv) + g + sip" . o , (5.5.1) from (5.1.2) f('._3_;_ + r332) . .. .3; .. .1230 , (5.5.2) from (5.1.5) f(v-fi 5- F339 a - -3§ + JrBO’ (5.5.3) from (5.1.5) r++++7+—+ fron (5.1.13) (5.5.5) Jr I «II-3'30 , (‘e‘e5) 29 from (5.1.15) Jz 83730 , (5.5.6) from (5.1.10) and (5.1.12) .3122 . .. er, —3—:£ . J2)“ ~ (5.5.7) Eliminating the current density J fron Eqs. (5.5.2) to (5.5.5) by using (5.5.5) to (5.5.7) we get, from (5.5.2) 90-3;- + ”3%) .- - $3, - lid-3:230 , (5.5.8) from (5.5.3) ?('-3-',-'. + '83,!) - - -3-E - Fifi-:98, . (5.5.9) from (5.5.5) fives-g e “3%) a (57-3; 5- w—S—E) 5 B:(v2 + '2) + @3135 + :33 + £31). (5.5.10) Adding the components of (5.5.7) and substituting (5.5.5) and (5.5.6) for the current density we get, 3;: + -3-:£ - P330“ 5- I). (’5-‘1-11) Equations (5.5.8) and (5.5.9) may be written as 2 (”h-3f.- + v-S—E) 5 fi; + 43-14;!!!) - 0, (5.5.12) “#31:. + "'33) + ‘3'; + -3;(~§) . 0, (5.5.13) 30 To complete the system of equations we add the equation of state of the gas and the equation of continuity, p IFRT, (11.5.15) .342). + g! + $22.). . 0. (5.5.15) Equations (5.5.10) to (5.5.15) are six equations for the six unknown quantities p,f, T, v, w, and Bo . We will now integrate the equations of motion and energy along a streamline. Thus, multiplying (5.5.12) by dr, (5.5.13) by da, and (5.5.10) by dz and using (5.3.13) for the streamlines we get, f(v-S—Islr . v E z) . $4.- . 33(3):)" 8 0, (5.5.16) I’(w-3-;dr . “3.542) + $5. + 33%;)“ - o, ‘ (5.5.17) (nu-37:4: . 0-355.) . (”3751: . 231;...) +33%? . .2)“ + (3.3; . gs . gim, (a...) Factoring v, and w, and noting that dv a fidr + «aids etc. we get, fvdv + -3-:dr 5 -37(%2)dr s: 0, (5.5.19) fwdw 5- %: + aibgéks - 0, (5.5.20) 31 fwde - de 533202 5- w2)dz 5- “3'13: 5- :3:- + -3-.Q-z-Ms,(5.5.21) Adding equations (5.5.19) and (5.5.20) we get, d(‘}v2 5- §w2) 5- GP 5- (102;) a 0. (5.5.22) Dividing Eq. (5.5.22) by the fluid density and integrating ‘we get, 2 B five 5- fwz 5 f9; 5 [91%)- . constant. (5.5.23) For constant fluid density we get from (5.5.23) by integration from some reference point, 2 22 (m2 + 50-") - (59-: + 55:) + (24,) + (g; - ,3) . 0. (5.5.25) Solving equation (5.5.25) for the fluid static pressure p, we get I 2 p 8 Po +‘zfi-5 ivafi + wg) - iva2 + w2) - pR -1%F .(5.5.25) We note that equation (5.5.25) reduces to the classical Bernoulli equation for the non-radiating, non-magnetic case. From equation (5.5.21) we get with e a cpT, 2 Q Q Q a; a (Ford 3% 5 fi-Efivz 5- w2) 5- (fopw)"'1(-$-i,£ 5- 5,-!- 5- 5-333). y-censt . (5.5.26) 32 For the incompressible case f a constant we get fro- equation (5.5.21) fwd(ch 5- P/f) a fwd(ch) 5- ml? as de 5382(v2 5- w2)dz 54-3-33- 5- :3: 5- -3—zQ-'-)dz. (5.5.27) Upon cancelling de and dividing by fwov we get from equation (5.5.27) ‘31; 8 ghz 5 '2) 5- (Fc'w)’1(_3;£ 5 :5 5- %). (5.5.28) fgoonst. He will return to the above equations in section 6., after we establish the general fluid flow conditions and the fluid properties. 33 5. THERMAL RADIATION AND IONIZATION In this section we give a brief outline of the governing equations of radiative transfer, and develop the equations for calculating ionisation and electric conductivity of the gas. 5.1. The Equation of Transfer A high temperature gas emits radiation energy as a result of rotational, vibrational, and electronic transitions from exited energy levels to lower energy levels. The emitted radiant energy corresponding to these transitions is distributed over a wide wave length region. The total radiant intensity emitted from a volume of gas is obtained by summing the radiant intensities from the individual energy transitions. For gas dynamic calculations the simplest approach to the determination of the radiative intensity of gases is to determine overall emissivities as a function of pressure and temperature of the gas. The fundamental quantity sought in radiative transfer of energy through an absorbing, emitting, and scattering medium is the specific intensity 1,,defined by, as, no. ' I Iv, (5e1e1) ‘where 18,1. the amount of energy transmitted in the frequency interval (1!, 7542/), through dA in time dt, in a direction making an angle 0 ‘with the normal to dA, and lying within the solid angle dw. 35 The distribution of the intensity' I,)in the radiation field is governed by a conservation equation called the radiative transfer equation. This equation, as given by Chandrasekhar and Kcurgancff, is (36,37) d - 3%! . p151, - my . (5.1.2) where, P - fluid density k,,s absorption coefficient 3,,- emission coefficient The emission coefficient 1,,for the case in which both scattering and absorption and emission are present, is given by Kourganoff as (36), J,,- kiigI' + (1 -‘§§)kpRV(T). (5.1.3) where i; represents the fraction of energy loss due to scattering and is called the albedo for single scattering, 53d 37(T) is the Planck function given by, 321'”) - zhflso'ghxpéq) --1)~"'1 (5.1.5) where k and h are the Boltzmann and Planck constants respectively. The two special cases of local thermodynamic equilibrium and perfect isotropic scattering are obtained from Eq. (5.1.3) by letting 57°. 0, and 7°. 1, respectively. 35 Substituting Eq. (5.1.3) into (5.1.2) and dividing by Pky we get, 01,, .. .. W a 17’ - (“OI-7+ (1 - V°)BV(T))9 (5.1.5) 5: where L, - 5£17dyo , (poa- cos 0). For local thermodynamic equilibrium ‘fio- 0, and Eq. (5.1.5) becomes, My - ma;- 3 IV - 32/(1'). (5.1.6) For isotropic scattering ‘F‘. 1, and Eq. (5.1.5) gives, d1), - m . I7, - I, (5.1.7) The optical thickness of the medium between the points s' and s is defined by, 71.5.) mffky ds, (5.1.8) 5! so that 07: . kyfds. (5.1.9) 5.2. Radiation Flux and Pressure In the general case equation (5.1.5) must be solved for the specific intensity 1,). The heat flux vector ‘fih is then obtained by integration as, 192” '63 a [flysineccsededddy , (5.2.1) o o 0 36 and the radiation pressure is given by, +1 2%"1 I 2 d -) . . Since the fluid dynamic equations of motion and energy in which the above two terms appear are a set of differential equations, it is desireable to obtain the expressions for ‘5h and pR as a function of the fluid properties or their derivatives. this is possible if local thermodynaaic equilibrium nay be assumed such that a local fluid temperature T may be defined at each point in the flow field. In such a case the governing equation for the intensity 1,.is Eq. (5.1.6), and for the optically thick case a solution may be obtained by a Taylor series expansion of 1,. about 37(T). The expressions for '5h and p8 as obtained by Zhigulev (18), Goulard (20), Scala and Sampson (31), and Pai (38), are pa 3 #3032901", (5.2.5) v.3n . - 54x13", optically thin 3... (5.2.4) 3 3R - 16/3(é%§-)VT, optically thick gas, (5.2.5) where XP is the Planck mean absorption coefficient defined 379 0 w a Q KP . 3(T)-{jrkvny(r)d , B(T) - 37(T)dvu £§§Lw (5.2.6) o and In is the Rosseland mean absorption coefficient 37 defined by, @ dd” . E 3" 7? 1/K3 8 fdfly V . (5.2.7) The Rosseland aean absorption coefficient KR as given by Scala and Saapson (31) for air as a function of tenperature and pressure is, K8,: (4.52 x 10"7)p1‘31 cxp(5.18 x 1o“r-7.13 x 10’9r2), (5.2.8) 1, the pressure p in 'where Kn is expressed in on' atmospheres, and the temperature T in °K. The Planck mean absorption coefficient KP for air was also given as, XP 8 8.3KB. (5.2.9) 5.3. Ionization and Electric Conductivity One of the acst important transport properties in magnetcgasdynamics is the electric conductivity of the gas which in part depends on the nunber of free electrons present or the degree of ionisation of the gas. The ionization occuring in high temperature gases, such as that surrounding the space vehicle, is referred to as thernal ionisation which is a general tern applied to the 38 ionizing action of aolecular collisions, radiation, and electron collisions. To determine the degree of ionization we consider a gas nixture of neutral particles, positive ions, and electrons which produce partial pressures and are related to the total gas pressure by, p a pn + p, + p° - (5.3.1) The pressure p is related to the temperature T by, p . nor. N/n? (5.3.2) where n is the nunber of nolecules per unit volume and k0 is the gas constant per aclecule or the Boltzmann constant. If we define the degree of ionization as 5 3'5 is. x: . , ' (5.3.3) where n m n1 are the number of electrons and ions e per unit volune, and n m n.n + no, then the relation developed by Saha is (#8) x2 -7 15/2 T? . (3.158 x 10 )7?pr 5:1). (5.3.h) where, pa? total pressure in atnospheres, q - ionization energy in Joules, T - temperature in °K, k0: Boltznann constant in Jouls/ ° . 39 Substituting pa 8 p/(1.013 x 105) into Eq. (5.3.t) 2 fir s (.032‘1‘5/2p'1)e1p(- :2!) I “7.9). (5.5.5) 2 where p is in Newtons per n . Solving Eq.(5.3.5) for the degree of ionization x 'we get, x 8 2'2 . (%)* 0 (50306) Substituting Eq.(5.3.2) for n into (5.3.6) we get the electron number density as a function of temperature and pressure of the gas, I r, ( P) )g - n (W (5.3.7) The nunber of neutral particles nay be obtained from uh I! n - n. . (Se’ee) Using Eqs. (5.3.2) and (5.3.7) we get the neutral particles as a function of temperature and pressure of the gas, “( 9P) )*. 85-m( 1 - (--g11r-7' (5.3.9) we note that in the limit as the temperature I becones large the quantity containing X(T,p) in Eq.(5.5.9) approaches unity so that nn -9- O, and we have a fully ionized gas, and as T heceaes small the quantity approa- ches zero and we have a neutral gas. 40 An equation for the electrical conductivity of a partially ionized gas which was found to agree very well with experiaent is (hi) 2) n.(e ’ nohs n.7(nnfion .- n15“) 11 (5.3.10) ‘3 - Where, '8 m electron rest lass, kg, e n electron charge, coulomb, ?' a neon thermal velocity of an electron, m/sec, '6“ :- electron-ato- mean collision cross section, .2, §;1 - electron-ion mean collision cross section, la, The nean electron thermal velocity V'ie given as a function of temperature by, 8k? )* . ’ (5.3.11) 5 3' (fi)*( n. ) . (5.3.12) From equation (5.3.8) we get by using (5.3.6), :2). %-’- 1 a (1 + K.1)* -1. (5.3.13) 0 C From Eq.(5.3.5) we have, k T [-1 I Law-'2 e (5e3elk) .032 T 41 Substituting Eq.(5.3.1t) into (5.3.13) we get, an poxp(q/koT) a: - (1 4 W) - 1. b (Se3e15) Since in our case noun1 we get from Eq.(5.3.12) by dividing top and botton of the last term by ne and “.1“; EQe(5e3e13) k a . (WWW . x") .. 1Y6“ . “r1. (5.3.16) The nean electron-ion collision cross section ‘5.1 is (41) 1. it 10'1° 1.251 10“ 12 2 6.1 I —J-—-(T——l ln((n.T) (2) ). I (5.3.1.7) .5816 T froa qu.(5.3.7) and (5.3.16) we have (n. in_ {/cn3) (P/ko)2)§ -3 (“.')* ' 1° ( 1 + x' . (5.3.18) Substituting (5.3.18) into (5.3.17) we get the collision cross section as a function of tenperature and pressure, -10 6 T2 6 a 2' 10 1 (8.8 10 . (503-19) ei "22£§"'-l n (p/k¢))*) (1+ K'I Equation (5.3.16) together with (5.3.19) gives the elec- trical conductivity of a partially ionized gas as a function of teaperature and pressure of the gas. (§;ns constant). Q2 6. SPECIAL CASE or v(r), w(z), ONLY In this section we consider a solution to the equations along streanlines as obtained in section t.t., by choosing the fern ef the streanlines so that v(r) only and w(z) only. 6.1. E nation of the Streanlines Introducing a streanfuncticn such that, v m 31%, -w a: 19% , (6.1.1) and the equation of continuity (&.3.7) is automatically satisfied. If we let 7 8 hr, V 8 -2b‘, (6e1e2) where b is a costant, we have by Eq.(6.i.i), 5,331;- . br, fig . 2bz. (6.1.3) Prcn this last result we find that 7)’ . 131-22, (6.1.5) which is the required streanfuncticn. It is readily verified that Eq.(6.1.t) satisfies the Laplace equation fi_r3_lz+_g.2_¥.. 0, (6.1.5) #3 From equation (6.1.11) we find that for 'V- 0, either r u 0, or z e 0, so that the z-axis is the stagnation etreanline and at z-O, we have the r-plane through that point. For 7” s 7f!- constant we get frcn sq.(6.1.h) ‘ ' 36$")! , (6.1.6) which is the equation of the streanlines and represents flow against a disk. To obtain a particular set of strean- lines it is necessary to evaluate the constant “b' in Eq. (6.1.6). For this purpose we use the definitien of the Stokes streanfunction; i.e., 211‘?’ is equal to the volune flow rate between any two streanlines for constant density flow. Thus, at any point z upstream of the wall the volune flow rate between the stagnatien etreanline and any otherstreamline r distance away fron it is given by my. (fl’r2)V, , (6.1.7) where V is the fluid velocity of the onconing strean. Thus, by Eq.(6.1.7) and (6.1.t) we htv. 'y’. inz c bran, (6.1.8) and b I *V/z. (6e1e9) Now if the velocity V is known at sone point z 1- z1 upstrean fron the wall; i.e., at z 3 z1, V - V1, and we have by Eq.(6.1.9) b I iv1/I1 . (6.1.10) Ml Substituting the result .1 Eq.(6.1.10) 1hto (6.1.4) we 83t9 )V-=(1V, 21)r21 . (6.1.11) Solving (6.1.11) for r we get, ’9’ r .(W)*, (V70. (6.1.12) It. may now obtain explicit expressions for the pressure and temperature distribution along the streanlines given by Eq.(6.i.12). Iron Bq.(6.i.2) we have v2 + w2 a b2(r2 4- hzz). (6.1.13) Substituting Eqs.(6.1.13) and (5.2.3) 1hto (6.1.25) and noting that V2 a v2 + w we get, 8% 2 1y 2 tagg‘ B2 me1+§F-+}?Vf-}fb(fi+§£)--F-§F, (5.1.15) where we have also used (6.1.12) to elininate r. We next obtain the expression for the temperature distribution along the streanlines fron 811.01.11.28) by using qu.(5.2.§), (6.1.2), (6.1.12) and (6.1.13), _ 2b2 1128 1'" gg-l. Tv-é-g-fi—(tg— + 1122) + fiz- . (6.1.15) 1” acoust. ‘15 Equations (6.1.it) and (6.1.15) are two equation for the two unknowns p and T along the streanlines given by (6.1.12). We now consider the nagnetic field of the following form 8 1.2 B I -:"T1' e (6e1e16) Introducing a g (Vb) (6e1e17) and using Eq.(6.i.i2) to elininate r in (6.1.16) we get 2 81r1z a . (6.1.18) B: Substituting (6.1.18) into (6.1.1&) and (6.1.15) we get, 32r8'2 p . p1 . p_ . 1fo - 1982(1/1. . 6.3) - pa - 2,1"; , (6.1.19) g . 1R1" z - A'(i/a + 333-), (6.1.20) Y aconst. ‘ where 225K 3') ( 3ft: ) ‘3 I W , and A" I 73T— . (6.1.21) 7 46 6.2. Temperature and Pressure Distribution Along Streanlines. The pressure distribution is given by Eq.(6.i.i9) which can be evaluated once the temperature distribution 1. known. The tenperature distribution 1. given by (6.1.20) which is a first order non-linear ordinary differential equation of the following general form, §§|8 f(z,T), (6.2.1) with the condition of T s T , at z 3 z1. 1 We propose a solution of Eq.(6.1.20) by a nethod of successive approximation. A.proper development of this method is given by Coddington (57). The successive approximate solutions to Eq.(6.2.i) are defined to be the functions T1, T2, T3, ”“‘, given recursively by the fornulae, T1(z1) a T1 (initial condition) z T2(z) a: T1 +ff(z,T1)dz, '1 z T3(z) m T1 +jf(z, T1(z))dz, 21 z T.*1(z)- T1 edjrf(z, Th(z))dz, (6.2.2) z 1 where n - 1,2,3,°"’°°°'. ‘17 It nay be noted that the ncre nearly correct a particular approximation Tn(z) is, the better will be its successor Tn*1(z). In our case we will obtain a good first approxi- nation by integrating Eq.(6.1.20) with the nagnetic term neglected. Thus, by neglecting the magnetic term in Eq. (6.1.20) and integrating by separation of variables we get for our first approxination, T2 3 (c1 - 3Aaln8)-1/3’ (6.2.3) where c1 is obtained fron the initial condition 01 I T;3 ‘f 34811121. ‘6e2eb’ To obtain the second approxination we substitute Eq.(6.2.3) into (6.1.20) and (6.2.2), which gives z z ‘11 ‘l'3 - T1 1» '32“1 - 3ARlnz)'&/3dz - 3— (i + l1z3/a)dal.(6.2.5) 1 ‘1 Integrating and using (6.2.5) for c1 we get from (6.2.5) T3 3 (T‘;‘3--3ARln(z/z1))"1/3 + #(zrz) .- 3%“: + z"). (6.2.6) We note that for z 1. z1, T3 1. T1 as required by the initial condition, and as as ->- 0, T3 -> (‘H’i/a + AM‘i/az). #8 A higher approxination may be obtained by re-sub- stituting Eq.(6.2.6) into (6.2.2) which gives, ~IT1+2(81-2)+-‘§(8:-2§)+ Q TL... [01-3A31n2)-1/3 g ¥(z1-g) + A”(21-4" dz. (6.2.7) 8 Prom Eq.(6.2.7) it is apparent that the formal integration process becones more and more complicated for higher approximations so that a numerical process would have to be used sooner or later in order to obtain the nth order of approximation. Therefore, we propose a piecewise application of our second order approxination (6.2.6) over a nunber of smaller intervals by dividing the range of integration into a finite number of snaller intervals. Thus, dropping the subscript 3 in Eq.(6.2.6) which denoted the 2nd approxination , we any use Eq. (6.2.6) to conpute the tenporature in the range zzszsz1 where z2 nay be taken as close to z1 as desired to obtain the necessary accuracy. After computing the tenporature at z2 we nay consider this point as our initial condition and apply Eq.(6.2.6) over the next interval z3£ z fizz with z2 playing the role of s1. We nay continue in this nanner until the entire range of interest is covered. ‘19 In general we nay write Eq.(6.2.6) in the following fern, 11.1 m ('l"1"3--3Anln(z/z£))"1/3 .';!(.1-.) +13%(z:-zk), (6.2.8) where T1+1 is the tenporature at any point in the interval z1 Sz521 and T1 is the tenporature at +1 the point zi; i 3 1,2,3,""”', represents the nunber of intervals under consideration. Thus, we consider equation (6.2.8) as the solution to the tenporature distribution over the entire range of interest. 1,e., 0< s25 :11. PART II: VISCOUS RADIATION MAGNETOHYDRODYNAMICS 7. GOVERNING EQUATIONS OP VISCOUS FLOW 7.1. Fundanental Equations We consider a viscous, heat-conducting, steady flow of an ionized gas in an electro-nagnetic field with thermal radiation. The governing equations for the present case may be obtained by nodifying the systen of equations derived in section 2.3. The nodifioation consists of adding the viscous stress terns to the equations of nonentun (2.3.2), and the viscous dissipation term to the equation of energy (2.3.3). The heat flux vector '5 is also nodified to account for the heat conductivity of the gas. The viscous stress term is given by, (56) IEEJ 9 (7.1.1) 1) where 7“” are the conponents of the stress tensor given by 1‘" _ F933 , 3L3) - $5“ . (7.1.2) The viscous dissipation function is obtained as, F1 . gnflJ—g-S. (7.1.3) 50 51 The heat flux vectors now becone fron (5.2.t) and (5.2.5) 6 . (11t + jig—we, optically thick gas, v.3 .V-(KtVT) - hKPJBT", optically thin gas, The system of equations now become, I...ntu, fVJ-g—S- + 313-33» - raj-351- - {ES-J- . 0. energy, 181—8 18:8 » pNJHfi-§§}8- VJ-ggsi+ A§E;'+'—+. nagnetic field Eq. VJ 3H1 . 3.1.3.3. + Eli-351- sql-g-lj-(g‘n (’31). and the equation of state, p a. fRT. continuity, (7.1 (7.1. (7.1 (7.1 (7.1. (7.1 (7.1 We note that the above system of equations are con- siderably nore complex than the classical NavierbStokes equations of classical fluid dynamics. .41) 5) .6) .7) 8) .9) .10) 52 7.2. Transfornation to Streanline Coordinates The transfornation of equations (7.1.6) and (7.1.9) was given in section 3. and will not be repeated here. By introducing the velocity and nagnetic field conponents fron (3.1.1), (3.1.2), into the equations of nonentun (7.1.7) and the equation of energy (7.1.8) we get, Pvflcgg-Lll . EL}? - thJ-g-g-‘fl - €51 . 0, (7.2.1) (VJ-35- .a'1(.”-“—Ufirfiox§“k a Fh - SHWJ‘F‘ a m“ ) + ,1V.311h1 gig-11 — 17.313" 35- + gn'ru-g-SLI. (7. 2. 2) Enanding (7.2.1) and using si-g—f :- -3-.- etc., and also x (3.1.3) “4 (3.1.11) we get fv-gl. . Pvzn + 313-33 - 23131 .- 0. (7.2.3) Expahdug equation (7.2.2) and 11.1111 (3.1.11) we get, fv_3!l- . _&-1(‘Jr air an ) _ $3.22 + '3; + gki’r‘kJ-g-b-v-‘h. (7.2.6) 53 7.3. Stream inc-Pressure variation Curvature and Torsion The variation of the total pressure Pt along the tangent, principal normal, and binornal directions of the streanlines nay be obtained by taking the scalar product .1 (7.2.3) with 31k.k k , ‘ikn , gikbk, respectively, P J 37'1 k 1 k 11 k d t 2.453 f7 s 51k. 4- szkn ‘ik’ + g 51k. a -g1ks a 0 (7.3.1) ..3.. . H.781.” -818 3838 (7.3.2) P J fv 3.1319" + fvzkh‘gnb" + 51-);1kbk-3-j- «“115-33— .0 (7.3.3) Making use of the orthogonal properties of s 1,n1,b1, and i Nah-58.113); 4.3— 43-;- .31...- we get frcn (7.3.1) to (7.3.3). P “'3'; * '31: ' 818L351 ' 0’ ”'3'” szt O '33" " Sikn ‘gg’ . 09 (70305) P '35; - sunk-23f ‘1 . 0, (7.3.6) where the total pressure Pt I (p + p-'+ pa). 54 Fro. Eq. (7.3.4) we find that the pressure variation along the streanlines depends on the nonentun change as well as on the viscous stresses. The sane holds true for the pressure variation in the nornal direction of the streanlines. Fron Eq. (7.3.6) we see that the pressure is no longer constant in the binornal direction of the streanlines for the viscous case under consideration here. Streanline Curvature An expression for the curvature k of the strean- lines nay be obtained in terns of the fluid density f , the velocity V, the nornal pressure gradient -3-:-t- , and the viscous stress torn by solving Eq.(7.3.5) tor k, P A k . (Fv9)'1(giknk_313'1 - .331). * (7.3.-n Torsion To develop an expression for the torsion of the streanlines as a function of the flow field paraneters we begin with the Frenet fornula 1 -'I'n1 - 3%. (7.3.8) The_unit binornal vector in Eq.(7.3.8) is by definition b1 . eijkgjpgkqspnq. (7.3.9) 55 An expression for the nornal vector 11‘1 any be obtained by solving Eq.(7.2.3) as, n‘1 s: (fvzk)’1(-?V-3l;sq - ‘11'_3_:_:, +-313;). (7.3.10) Substituting Eq.(7.3.10) into (7.3.8) and taking use of the following identities, eungpgkqsps‘l e o, gkngr e 6:, ngsp - VJV'1, we get the binornal vector as a function of the flow field para-eters, p b1 . (1"xr3lx)’1.13"(ngk fl - vJ-g-it). (7.3.11) '1 ax? differentiatinx (7.3.11) along a etreanline and sub- stituting into (7.3.8) we get the following eqression for the torsion of the streanlines, P -111 . .133-3;{(W3k)"vj(ekq-3§: - fig. (7.3.12) 56 7 . ll . Vorti 01 tz Substituting Eq.(3.3.18) into (7.2.3) we set -PV(-3-l¥n1 + -3-E01) )+?V2kn1 + Vgir-g-I—r- + Sir-3'2}- 3 J30. (7.4.1) Adding and subtracting ivzgir-gé fro- Eq. (7.11.1) we get, x ‘Ffi'msiv'hi " "'31?“ * “um axr ” “1&3?- 421'2- .2 v.11. ()xr {- g ()1, 0. (7.11.2) Dividing (7.5.2) by P and transposing IOIO terns we get, -F sir-31$? + if’V2) + if 1V2¢1r-3-£;++-§L:—- xr (v21: — v-gh‘ - VHS-£9. (7.41.3) Introducing a function 5 defined by, F r sir-3:; - -P'1e"-g;;(l’t + 1}sz ) + iF‘stikg-E; +f'1-g-3n (7.1.4) equation (7.4.3) becones, 57 Big-E; . (v21: - v—S-Eh‘ - $39.1. (7.1.5) Multiplying equation (7.&.5) by 531' we get, 333;. . (v21: - V—ghfini - v-S‘Ygufi . (7.11.6) Taking the scalar product of Eq.(7.4.6) with (3.3.16) we get, ‘13-‘53: . o. (7.1..7) Taking the scalar product of Eq.(7.t.6) with sJ,we get .13.. ' .. I-g-Jv 0 (7&8) Thus, equations (7.h.7) and (7.t.8) inply that the surfaces ‘3'. cenetant contain both the streanlines and the vortex lines. 58 7 . 5 . General Cyl indri ca11__Coordinates The governing equations of viscous, radiation nagneto- hydrodynanics in cylindrical coordinates are as follows, ‘Sfifvg + £1.55 + F332=u .=._s..nsm .~-_HH enamel .5 .N 3e: sec 3.538 m CH ma om mm on mm o. 4 u u - Fl I H MD mfg . c a. .. a - w m_x Ni. . 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N 8:283. oz . . 1 1 . . mucuxooH 97 8335...; 3338.28 otooom oo oozooo 5:23. .255 to tote .83: too: .5 .~ one: not 3:339 2 o... a 1 on no . . o. o m S 1 q — — q - d _ lo. 117/ /// «2333.805 IIOII / 07/ . 1.. /or/ IO/l 23303.89 1|.Ol /o/ //Ol/ . / I / II N. u I o ’ ze\ /. 1:: ||.19I|11lo|11|.lol I|I9|1I1||0.111IIIOII1I. TI... - c. 8:23. oz 9. 1: Cf 01 )fi‘ )(1) S11 APPENDIX Typical Fortran Progran and Results This progran integrates the differential equation for the tenperature distribution by the nodified Bunge- Kutta nethod taking account of variable viscosity, heat conductivity, electron-ion collision cross-sections, and electric conductivity. At the sane tine the progran calculates the pressure, density, velocity, electron density, and electric conductivity distribution along the streanlines. Inasnuch as all flow variables are cal- culated as a function of s and Wyslong the streanlines we also obtain the coordinates r as a function of s snd‘?’ so that the final results say be interpreted as having the flow variables distribution given as a function of the two coordinates r and 2. Thus, by choosing as nany streanlines as desired it is possible to obtain the flow variables distribution throughout the flow field under consideration as a function of the field coordinates, r and s. 98 00000000000000 99 FORTRAN PROGRAM FOR IBM 1620 VISCOUS COMPRESSIBLE FLOW PROGRAM FOR THE CALCULATION OP TEE FLOW VARIABLES DISTRIBUTION NEAR A STACNATION POINT RS-STREANLINE ENTRANCE RADIUS AT E .40 CM ZsCOOBDINATE NORMAL To DISK RZ=COORDINATE PARALLEL TO DISK TbFLUID TEMPERATURE P-PLUID PRESSURE RO.PLUID DENSITY ECO-ELECTRON NUMBER DENSITY C.ELECTRIC CONDUCTIVITY OF THE PLUID VhFLUID VELOCITY _ QN-ELECTRON-ATOM COLLISION CROSS SECTION QI-ELECTRON-ION COLLISION CROSS SECTION RS..01 Do 200 1.1.5 PUNCH 2,RS PORMAT(AORSTREAMLINE ENTRANCE RADIUS IN METERS RS-EIA.8) T-2OOOO. Ros.ocooeé*515. 3:287. P-BO*R*T 23.4 100 PUNCH 13,z,T,P,RO 33.01 2L-.01 0.10000. QTK-O. 5 E-O GO TO 100 10 AKi-FZ*H onurz z1-z Ti-T 2.21-E/2. TaTi-AKi/2. GO TO 100 15 AK2-FZ*H T-Ti-AKz/z. GO To 100 2c AK3=FZ*H 2.21-n TcTi-AK3 GO TO 100 25 AKh-FZ*H FZisFZ TbTi-(AK1+2.*AK2+2.*AK3+AK§)/6. ZaZi—H Rz=(RS*RS*zz/z)**.5 101 30 TK-.00199*T**.i/(i.e112./T) 31 QTY-TH*(Ez1-Ezo)/H PUNCH 13,z,T,P,RO 13 FORMAT (5H z.E1A.e,5H T.E1A.e,5H P-E1A.S,5H RO.E1A.8) ELECTRIC CONDUCTIVITY PROGRAM NEXT T2-T PE-P/A7.SS QNhl./(10.**19) UI=166000. x.(.o32*T2**2.5)/(47.88*P2*EXPP(UI/T2)) Y=(§7.88*P2*10.**23)/(i.38*T2) E=Y*SQRTP(x/(1.+x)) ENsY-E ECO:E/(10.**6) QI:(2.95/(T2**2*10.**10))*LOGF(8780.*T2**1.S/SQRTF(ECC)) C-(20.5/T2)**.5/(1O.**12) C-(G*E)/(EN*QN + E*QI) PUNCH 65,ECC,C,v,Rz 65 FORMAT (5H ECO-Ei&.8,58 C:E1A.8,5H vaE1A.8,5H R2.E1A.8) IE(2L-z)5,11o,11o 1OO R3287. CA-1.A STB-5.67/iO.**8 CL-iGo/(9o*10.**8) CB:STB*CL*T**3 v.2ooc. 102 223.4 SII=N*RS*RS/2. 818.2 31-.01 BS=Bi*Ri*Ri/RS**2 B2-BS*Z/Z2 PHS.(HS*HS*10.**7)/(8.*3.14) PVS-.5*R0*V*W PVh.5*R0*(H/(2.*ZZ))**2*(RS*RS*22/Z + h.*z*z) PB-B2*B2*10.**7/(8.*3.i§) Pi-25422h.6 P-Pi+PBS+PVS-PVAPB ROaR/(R*T) vs(w*w*z*z/(zz*zz) + 3!i*fl/(2.*Z2*Z))**.5 CVsR*T/(V*V) 80:.2 E-(HO/Io.**A)*E/(RS*RS*22) Px-.06 QRs-k.*PK*STB*T**§ zs.(1. + RS*RS*zz/(A.*z**3))**.5 VISCOUS PROGRAM NEXT 101 Nz.N/22 U-.000001&62*T**.5/(1.+1i2./T) UOs2.5*U*UZ*UZ STRES.U*NE/(RS*RS*zz/z + A.*z*z)**.5 QVb(QR¢UO+QTK)/V 103 102 DTN 323*(Cv;1.)*ov + ZS*C*V*B*B-ZS*CV*STBES DTDsCR + (P/T)*(1.+(aA/(CA.1.))*(Cv;1.)) FZs-DTN/DTD KuK+i GO TO (1 110 CONTINUE 200 RS:RS*2. END 0,15,20,25),K Typical progran printout for calculations along streanline‘?é; only internittent results are shown. STREAMLINE ENTRANCE RADIUS IN METERS RS: Zs.§0000000E+00 Za.39000000E+00 ECC=.7§1552863+18 23.350000003+00 3003.75311557E+18 Za.)00000008+00 ECO:.7&Ol6k60E+18 Tu.20000000E+05 TI.1986§6113+05 Ts.19287799E+05 Cs.12279§§0E+05 Cs.11665§163+05 Vh.19506569E+0h P-.27761739E+06 Vh.175081553+04 P-.30Q01610E+06 V5.15011105E+04 .200000003-01 R0-.#§2900003-01 303.4536686hE-01 RZ:.2025§787E-01 803.5015116AE-01 RZ:.21380899E-01 803.57165924E-01 32:.23094012E-01 OOCOOOOOOOOOOOOO Z:.250000003+00 ECO-.69753902E+18 Za.200000008+00 ECO:.63172579E+18 23.15OOOOOOEbOO ECC-.5§9576088+18 22.10000000E+00 ECCs.§538§093E+18 23.500000003-01 ECO-.338k7680E+18 22.100000008-01 ECC=.1962§103E+18 104 08.109722013+05 Tio1706§0963+05 CI.102198583405 Ts.163472663+05 C=.9§0577GOE+0§ Tb.15589313E+05 Ca.8§9059953+0§ TI.1§5678503+05 C=o733097363+0§ T-.13332793B+05 C:.5615007§E+0§ Vh.12515987E+0§ Pc.365632798+06 Vi.1002§968E+0§ P-o399375408+06 Vh.75§531138+03 Pa.§35&2k708+06 Vé.2872280§E+03 Pa.509§&799E+06 Vh.320156198+03 R0:.65326054E-01 823.25298223E-01 Roa.746576003-01 BZ:.2828§271E-01 303.85229311E-01 RZ-.32659862E-01 R0=.9731795§E-Oi BZ:.§0000001E-01 R0-.11231728E+00 RZ-.565685§&E-01 RO:.1329&087E+00 RZ=.126§91118+00 1. 2. 8. 10. 11. 12. 13. 105 BIBLIOGRAPHY Huber,P.w.; and Sins, T.E.: The Entry-Communications Problen. 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