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"REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503. 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 1997 Final Report 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Problem of Supersonic Flow Declaration by Magnetic Field F6170896W0297 6. AUTHOR(S) Dr. Alexander Vatazhin 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER Central Institute of Aviation Motors Aviamotornaya St. 2 N/A Moscow 111250 Russia 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING AGENCY REPORT NUMBER EOARD PSC 802 BOX 14 SPC 96-4091 FPO 09499-0200 11. SUPPLEMENTARY NOTES 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for public release; distribution is unlimited. A 13. ABSTRACT (Maximum 200 words) This report results from a contract tasking Central Institute of Aviation Motors as follows: The contractor will investigate the problem of hypersonic flow control by magnetic field in channels with minimal losses. 19980203 029 14. SUBJECT TERMS 15. NUMBER OF PAGES 32 Plasma Physics, Magnetohydrodynamic (MHD), Hypersonic Flow Control 16. PRICE CODE N/A 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19, SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF REPORT OF THIS PAGE OF ABSTRACT UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. 239-18 298-102 PROBLEM OF SUPERSONIC FLOW DECELERATION BY MAGNETIC FIELD (The final report on Contract F61708-96-W0297) A.B.Vatazhin, V.I.Kopchenov, O.V.Gouskov, V.A.Likhter, E.K.Kholshchevnikova MOSCOW 1997 K CONTENTS Introduction ..................................................................................................... 3 1. Two-dimensional flow in magnetic field normal to flow plane .......................... 4 1.1. G asdynam ic equations ................................................................................... 4 1.2. Two-dim ensional electrodynam ic model ......................................................... 6 1.3. Quasi one-dim ensional electrodynam ic model ............................................... 8 1.4. D im ensionless param eters .............................................................................. 9 2. The problem s form ulation .............................................................................. 10 3. Results ................................................................................................................ 13 3.1. One exam ple of m odel testing ...................................................................... 13 3.2. Results for model I (2-D flow and quasi 1-D electrodynamics) ...................... 14 3.3. Results for model II (2-D flow and 2-D electrodynamics) .............................. 18 3.4. Results for supersonic flow deceleration due to electric power input without m agnetic field .............................................................................................. 19 3.5. Results for supersonic flow deceleration in the end zones of magnetic field ....... 20 Conclusion ............................................................................................................. 21 Resum e .................................................................................................................. 22 The possible future work developm ent ............................................................... 24 References .............................................................................................................. 26 INTRODUCTION Supersonic flow deceleration by magnetic field and estimation of this process efficiency is the main object of the investigations performed within this contract. In the previous reports [1, 2], supersonic flow deceleration in the axisymmetric duct by the axisymmetric magnetic field generated by solenoid or current loop was studied. The mathemati- cal, physical models and computer code were developed to simulate such flows. Inviscid (Euler equations), viscous laminar and turbulent flows (full Navier-Stokes equations) were analyzed. It was shown that complex interaction of gasdynamic, MHD effects and boundary layer are responsible for the flow structure generation. Moreover, due to this interaction, some unusual effects concerned with MHD interaction parameters S influence on the deceleration process efficiency are observed. The explanation of these effects was presented in [1, 2]. In this report, another scheme of MHD flow control is investigated. This scheme provides the supersonic flow deceleration in the 2-D channel when magnetic field is perpendicular to the flow plane and, depending on wall electric properties, electric energy may be extracted from the duct. This scheme was widely investigated in 1- D approach for MHD generators [3] and, in the case of moderate supersonic Mach numbers, in 2-D formulation [4]. The substantial peculiarity of this report is the analysis of supersonic flow (M = 5) deceleration and estimation of irreversible losses in such flows. The 2-D Euler and 2-D full Navier-Stokes equations are used for the gasdynamic flow description. Two models are used for the electrodynamic field description: the simplified quasi 1-D model [5] and full 2-D model [4]. It is interesting to note that scheme of MHD control considered in this report at- tracts particular interest because of proposals of its application in the hypersonic propul- sion system of aerospace plane [6, 7]. In this connection, the results obtained in this report concerning viscosity influence on the supersonic MHD flow become very important. All gasdynamic and electric characteristics of this scheme are presented for all considered models. Additionally two limiting cases are also investigated. One of them is concerned with deceleration process in 2-D duct due to electric energy input to supersonic flow without magnetic field. The second one is the supersonic flow deceleration in the 2-D duct with isolator walls in the end regions of magnetic field. The resume of the work performed within this contract and some possible directions of future investigations are presented also. 1. TWO-DIMENSIONAL FLOW IN MAGNETIC FIELD NORMAL TO FLOW PLANE In this report, the hypersonic flow deceleration in 2-D channel by magnetic field that is perpendicular to flow plane is investigated. The channel walls are nonperforated (for fluid) and may be electrodes, sectioned electrodes or isolators (for electric current). The generator regime is considered essentially. But some examples when electric power is not extracted from channel are discussed too. 1.1. Gasdynamic equations. The governing equations for the present analysis are the unsteady compressible 2-D Euler and full Navier-Stokes equations. The time dependency of the governing equations permits the solution to progress from an arbitrary initial guess to the steady state. One can write the full Navier-Stokes equations in 2-D case aU aF 8G + +--= Q, F = F(U), G = G(U) (1.1) at ax ay Here U is the vector of conservative variables, F and G - vectors of fluxes, including viscous stresses P PU pu 2 ++ p -t xx PV puV - t'Y peJ Lpuh* - u xx- vXY + q(12 G= puv- u(cid:127) , Q= fX (1.2) pv2 +Pp _- Y y pvh * - u'cx - vIC u + qhy)KI The total enthalpy and energy are defined by h = h + (u 2 + v 2) e (u 2 + v 2( 2 2 where e and h are specific internal energy and enthalpy. The components of viscous stresses tensor are given by TX = 2g au- 2--g(a-+-j ax 3 ax a, Sgav+ (1.4) ay 3 ay trxy = Tyx = __L- X Here p, p, and v are density, pressure, and velocity vector, p. is the dynamic viscosity coefficient. This coefficient is supposed to be the function of temperature (Sutherland's viscosity law). The transfer of heat according to Fourier's law of heat conduction may be presented by qhx- ah (cid:127)ah (1.5) Pr ax' qh=_ Pr7ay where Prandtl number Pr = ptCp/X is supposed to be a constant (in laminar case for air Pr=0.72). The system (1.1) - (1.5) is closed by the relationships for perfect gas with constant heat capacities h= p = p91T (1.6) Here y is the specific heat ratio and 91 is gas constant. In relations (1.2) , fy and q are correspondingly volume densities of MHD force and electric power supplied to the gas. The expressions for f and q are following f=jxB, q=jE (1.7) Here B, E and j are vectors of magnetic field, electric field and electric current density. In (1.7) the volume electric charge density is neglected. In turbulent case, the averaged full Navier-Stokes equations are used. The Bussinesq hypothesis is used to express the turbulent tenses through tensor of deformation rates defined by averaged velocity components with changing of dynamic viscosity pt on the effi- cient transport coefficients p.e'.t+PVt in equations (1.4). Here vt is turbulent viscosity. In equation (1.5), (p/Pr+pvt/Prt) are used instead of t/Pr. Here Prt is turbulent Prandtl number. The one equation differential model "vt-91" [8] for turbulent viscosity vt is used. This equation can be presented in the following form: a+ a ____ + _____ +_ __a__ V --aF + Vt 1 Fj(t +V aV1 at ax = ax axJ a [L t t +ay +2PVG+c 3Vt(U+P a 2 G 2 C5V 2 + C6V tV +C2 ++3- V -C4PVt a2 ax ay) + 11.2vtv + 12.8v2 C 2 2' ll.2vtv + 64v2 (1.8) C CD U\2 2 2 2 G =2 +2 + + +2---U (,a a!, a ax ay el C2' C3 C4 05 C6 2 0.2 0.7 0.5 3 50 Here a is sound velocity, S is minimal distance to the wall and v is the kinematic viscosity. 1.2 Two-dimensional electrodynamic model When induced magnetic field is neglected in comparison with the applied external magnetic field, the equations for E, j and B deter- mination are following [3, 1] j= a(-V(p+vxB)- a(jxB), divj=O (a =B (1) JB (1.9) rotB=0, divB=O, E=-Vcp Here (p is electric potential, a is electric conductivity, 03 is the Hall parameter, cc and Y are known functions of thermodynamic parameters. In these equations B is applied magnetic field. The equation rot B = 0 is satisfied in the regions where external electric cir- cuits are absent. It is suggested that transport coefficients (viscosity, conductivity) are not dependent upon the magnetic field B. Let's assume that applied magnetic field has the following structure B = (Bx(x,z), 0, B2(x,z)) 1.10) Such form of magnetic field is realized in many practical situation when condition Bx = 0, B = B = const is fulfilled in the main magnetic field zone and distribution 2 0 (1.10) is fulfilled in the end zones of magnetic field. Let's suggest that flow structure and medium properties are following: v = (u(x,y), v(x,y), 0), a = O(x,y), a = a(x,y) (1.11) Procedure of z-averaging of equations (1.9) at above mentioned assumptions was produced in [3]. According [3], the magnetic field is given by B = (0, 0, B(x)), B(x) =< B,(x,z) >, (1.12) where B(x) is the known function and < >z designates the averaging in z - direction. The solution of (1.9) is sought in the form j = (jx(x,y), j,(x,y), 0), (p = P(x,y) (1.13) As a result, the following 2-D electrodynamic equations may be formulated: ix =( To[- + vB + P + uB a x a( 1.14 ) CauBp+P Fap - +v yy = CrP I' )y (x O-x- + O-y- =0 (% -1 3 3aB) (.5 ax ay 1+P1 From (1.14) - (1.15) the following elliptical equation for (p is obtained Pa(p~ +~o L1nOpap )' ap( afrny p ahi(IOCT)D ay aay( - -if 0(1.16) Sax Goo~f={ ~[Ba(v + 13u)] - P[a p(U - P3V)]} 1 6 The forcesff,, fy and local electric power q supplied to gas are defined as fx =jB, f, =-ixB, q =-j-x Jy (.17) ax ay The solution of equation (1.16) requires the formulation of the boundary conditions on the duct walls and in the entry and exit duct cross-sections. The formulation of these boundary conditions depends upon the physical problem. The following wall boundary conditions will be considered in typical cases of this report. Walls are ideal electrodes. Then the potential is constant on each wall. Thus = (p- = const, on the lower wall P= 9 = const on the upper wall (1.18) 2 Walls are ideally sectioned electrodes. In this case the electric potential distributions along the upper and lower walls are supposed to be known. In some cases these distributions may be taken from experiments (see [4]). Thus 9 = (p- (s) along the lower wall (p = 9 + (s) along the upper wall (1.19) Here s is coordinate along the wall. Walls contain electrodes and isolators. Then the electric potentials on the electrodes are constant values. For isolators, the normal component of electric current density on the wall is equal to zero. Thus p = const on each electrode J= 0 on the isolators (1.20) It is necessary to note that gasdynamic equations (1. 1)-(1.6) along with equation (1.8) for turbulent viscosity can be solved using (1.17). In turn, electrodynamic equations (1.14) -(1.15) can be solved if velocity components are known. Therefore the solution of the whole gasdynamic - electrodynamic problem can be obtained using iterative procedure when the electrodynamic subsystem is solved using known gasdynamic parameters and gasdynamic subsystem is solved taking into account known characteristics of electromag- netic field. 1.3. Quasi one-dimensional electrodynamnic model. Simplified electrodynamic models are widely used in technical applications instead of 2-D models. These models are based on the following line of reasoning. When gasdynamic velocities u and v, values of a and cc are dependent only on transverse coordinate y, and magnetic field B, channel height h and electric boundary conditions do not vary with longitudinal coordinate x, the exact solution of equations (1.14)-(1. 15) exists. All variables in this solution (may be except for electric potential) are depend only on coordinate y. Let's assume now that { = (i, v, C, aC, h, B) and boundary conditions vary slightly with x. This assumption allows to use above mentioned solution with { = (cid:127)( x, y) as an approximate solution. This procedure was discussed firstly in [3]. Some examples of such model applications are presented in [5]. The general formulations of main equations for different types of boundary conditions was presented in previous report [2]. In this report, the one version of quasi one-dimensional models will be considered that corresponds to the real situation when electric potential distributions on the upper and on the lower walls are given. In practice, electrodes are sectioned and potential varies along a wall. In numerical modeling, it is convenient to use the wall potential distributions from experiments [4]. The above mentioned quasi one-dimensional approximation for this case is described by equations Ev= _J + B(u - O~v) (1.21) B(u - fv) - (&p/ h) _ Jv =- (C-FO 1 ) /) ' = avB - f3j, (1.22) h 4 fx= jB, = -jxB, q = jYEY, (< ý >= h-1.J (x,y)dy ) (1.23) 0 Here 8p = 9- q-p,h and B are given functions of x. They are the control functions. For f,, f4, q we have, according to (1.23), the relationships using only local parameters. 1.4. Dimensionlessp arameters. The system of equations is used in dimensionless form. The main values used for reference are following: the duct height Do at the entry, the free stream velocity Vo, density po, gas constant ý1, dynamic viscosity po, characteristic magnetic field B., the reference difference 59, = (p-- 9+o of electric potential on the duct walls, electric conductivity 0o and the value of ao. Then all coordinates and length are re- ferred to Do, velocity components to Vo, pressure to poV02, energy and enthalpy to Vo2, temperature to Vo2/91, viscosity to [to and turbulent viscosity to po/po, magnetic field inten- sity to B., electric conductivity and a to o0 and ac0. Then the following dimensionless parameters appear in the system of equations Re = poVoDolto , S = B,2 Do/poVo, Ko V°B*D°, Pr, Prt, y, 003 = a0oB where Re is Reynolds number, S is MHD interaction parameter, K0 is the load parameter and 13o is the Hall parameter.