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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 effi-
ciency is the main object of the investigations performed within this contract. In the previ-
ous reports [1, 2], supersonic flow deceleration in the axisymmetric duct by the axisym-
metric 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 bound-
ary layer are responsible for the flow structure generation. Moreover, due to this interac-
tion, 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 per-
pendicular 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 for-
mulation [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 simpli-
fied 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 con-
sidered 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, includ-
ing 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 coef-
ficient. This coefficient is supposed to be the function of temperature (Sutherland's viscos-
ity law). The transfer of heat according to Fourier's law of heat conduction may be pre-
sented 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 den-
sity. 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 de-
fined 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 mag-
netic 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 re-
port.
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 distribu-
tions 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 solu-
tion of equations (1.14)-(1. 15) exists. All variables in this solution (may be except for elec-
tric 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 applica-
tions 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 mag-
netic 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 pa-
rameters 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.
