NASA Technical Reports Server (NTRS) 20040105528: Simulation of 3-D Nonequilibrium Seeded Air Flow in the NASA-Ames MHD Channel

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AIAA 2005=0000
~
Simulation of 3-D Nonequilibrium
Seeded Air Flow in the NASA-Ames
MHD Channel
Sumeet Gupta, John C. Tannehill,
Iowa State University, Ames, IA 5001 1
and
Unmeel B. Mehta
NASA Ames Research Center, Moffett Field, CA
94035
43rd AIAA Aerospace Sciences
Meeting and Exhibit
10-13 January 2005 / Reno, NV
For permissiont o copy or republish, contact the copyright owner named on the first page. For AIAA-held copyright,
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Simulation of 3-D Nonequilibrium Seeded Air Flow
in the NASA-Ames MHD Channel
Surneet Gupta' , John C. Tannehillt
Iowa State University, Ames, IA 5001 1
and
Unmeel B. MehtaJ
NASA Ames Research Center, Moffett Field, CA 94035
Abstract cilities, launch assist, and power generation. One of
the critical technologies associated with these appli-
The 3-D nonequilibrium seeded air flow in the NASA- cations is MHD acceleration. In order to study MHD
Ames experimental MHD channel has been numeri- acceleration, an experimental MHD channel has been
cally simulated. The channel contains a nozzle sec- built at NASA Ames Research Center by D. W. Bog-
tion, a center section, and an accelerator section danoff, C. Park, and U. B. Mehta [1,2]. The channel
where magnetic and electric fields can be imposed is about a half meter long and contains a nozzle sec-
on the flow. In recent tests, velocity increases of up tion, a center section, and an accelerator section. The
to 40% have been achieved in the accelerator section. channel has a uniform width of 2.03 cm. Magnetic
The flow in the channel is numerically computed us and electric fields can be imposed upon the flow in
ing a 3-D parabolized Navier-Stokes (PNS) algorithm the accelerator section. A cross section of the MHD
that has been developed to efficiently compute MHD channel is shown in Fig. 1.
flows in the low magnetic Reynolds number regime: In the present study, the flow in the experimen-
The MHD effects are modeled by introducing source tal MHD channel is numerically simulated. Flow-
terms into the PNS equations which can then be fields involving MHD effects have typically been
solved in a very efficient mauner. The algorithm has
computed [3-151 by solving the complete Navier-
been extended in the present study to account for Stokes (N-S) equations for fluid flow in conjunction
nonequilbrium seeded air flows. The electrical con- with Maxwell's equations of electromagnetodynam-
ductivity of the flow is determined using the progam ics. When chemistry and turbulence effects are also
of Park. The new algorithm has been used to com-
included, the computational effort required to solve
pute two test cases that match the experimental con- the resulting coupled system of partial dif€erential
ditions. In both cases, magnetic and electric fields equations is extremely formidable. One possible rem-
are applied to the seeded flow. The computed results edy to this problem is to use the parabolized Navier-
are in good agreement with the experimental data. Stokes (PNS) equations in place of the N-S equa-
tions. The PNS equations can be used to compute
three-dimensional, supersonic viscous flowfields in a
Introduction
very efficient manuer [16]. This efficiency is achieved
because the equations can be solved using a space-
Magnetohyrodynamics (MHD) can be utilized to im-
marching technique as opposed to the timemarching
prove performance and extend the operational range
technique that is normally employed for the complete
of many systems. Potential applications include hy-
N-S equations.
personic cruise, advand Earth-to-orbit propulsion,
Recently, the present authors have developed PNS
chemid and nudear space propulsion, regenerative *
codes to solve 2-D and 3-D supersonic MHD flowfields
aerobraking, onboard flow control systems, test fa-
in both the high and low magnetic Reynolds number
'Graduate Research Assistant, Student Member AIAA regimes [17-191. The magnetic Reynolds number is
+Manager,C omputational Fluid Dynamics Center,& Pr* defined as Re,,, = a,peV,L where 0, is the electrical
fessor, Dept. of Aerospace Engineering. Fellow AIAA conductivity, pe is the magnetic permeability, V, is
$Division Scientist, Associate Fellow, AIAA
the freestream velocity, and L is the reference length.
Copyright @ZOOS by the American Institute of Aeronau-
tics and Astronautics, Inc., all rights resewed. The new MHD PNS codes are based on NASA's up-
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A-UERICAN INSTITUTE OF AERONAUTICASN D ASTRONAUTICS
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wind PNS (UPS) code which was originally developed The governing magnetogasdynamic equations are
by Lawrence et al. [20]. The UPS code solves the PNS nondimensionalized using the following reference
equations using Roe's scheme in a fully conservative, variables:
hite-volume approach in general nonorthogonal co-
ordinates.
For many aerospace applications, including the
present experimental MHD channel, the electrical
conductivity of the fluid is low and hence the mag-
netic Reynolds number is small. In these cases, it
makes sense to use the low magnetic Reynolds num-
ber assumption and reduce the complexiQ of the gov-
erning equations. The MHD effects are modeled with
the introduction of source terms into the PNS eqy*
tions.
pi=---P-e -1, a,=- ue
Previously [19], the present authors used the low 8
Pem 'Jew
magnetic Reynolds PNS code to compute both Z
*
where the superscript refers to the nondimen-
D and 3-D flows in the NASA-Ames MHD channel.
These perfect gas (7 = 1.25) calculations assumed sional quantities. For convenience, the asterisks are
dropped in the following equations.
that the magnetic and electric fields, as well as the
The governing equations written in vector form in
electrical conductivity, were constant in the accelera-
a 3-D Cartesian coordinate system become
tor section. In the present study, the 3-D simulations
have been extended to include both equilibrium air aU aEi BFi aG aE, aFu aG,
flows as well as nonequilibrium seeded air flows. For -a+t -+-a%+2 ay a2 =-+-a+x-+ sMH%D az
the latter case, the electrical conductivity is variable
(6)
and is computed using the program of Park [21]. where U is the vector of dependent variables, E,, Fi
and Gi are the inviscid flux vectors, and E,, Fua nd
G, are the viscous flux vectors. The source term
Governing Equations
S Mcon~tain s all of €he MHD effects. The flux vec-
tors are given by
Magnetogasdynamic Equations
The governing equations for a viscous MHD flow u=[P , P, pv, pw, pet ] = (7)
with a small magnetic Reynolds number are given
by [14]:
Continuity equation
-aaPt + v . (pV) = 0 (1)
Momentum equation
Gi =
[
a(pv)+V. p W + p I ] =V.?+JxB (2)
at
I
Energy equation
+ E, =
a(pet)+V.[(pet p)V]= V-(V-i)-V-U+E-J( 3)
at
Ohm's law
J =ae(E+V x B) (4)
where V is the velocity vector, B is the magnetic
field vector, E is the electric field vector, and J is the
conduction current density.
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AMERICAN INSTEVIX OF AERONAUTICS AND ASTRONAUTICS
where
SMHD = &n
The primes in the preceding equations indicate that
the streamwise viscous flow terms have been dropped.
For turbulent bws, the twPlayer Baldwin-Lomax
turbulence model [22] has been modified to account
for MHD effects. Only the expression for turbulent
where viscosity in the inner layer is changed. This modi&
cation for MHD flows is due to Lykoudis [23].
1 : +
pet = p e+ - (uz V’ +u2) In order to “close” the preceding system of PNS
equations, relations between the thermodynamic
variables are required along with expressions for the
and the nondimensional shear stresses and heat fluxes
transport properties /I and k. For a perfect gas, the
are defied in the usual manner [lS].
pressure is computed from the relation
The governing equations are transformed into com-
putational space and written in a generalized coordi- p=(?-l)pe (18)
nate system (& 7,C ) as where 7 = +ym,a nd the transport properties are com-
puted using Sutherland’s formulas [16]. For equilib-
rium air computations, ;V and all other thermody-
namic and transport properties are obtained fkom the
simpxed curve fits of Srinivasan et al. [24,25]. For
where
nonequilibrium computations, the thermodynamic
($) ($) and transport properties are determined using the
E = (E, - E,) + (F, - F,) procedures described in the next section.
(5)
+ (Gi - G,) Nonequilibrium Flow Equations
(5) (?) For nonequilibrium flows, the species continuity equa-
F = (E, - E,) + (Fi - F,) tions must be solved in addition to the magnetogas
(5)
dynamic equations given previously. The magnet-
+ (Gi - G,)( s> gasdynamic equations remain the same except for the
($) additional term in the energy equation, which is due
G = (E, - E,) + (F, - F,) to the diffusion of the species. The nondimensional
species continuity equations, expressed in 2-D trans-
formed coordinates for a steady flow, are given by
and J is the Jacobian of the transformation.
The governing equations are parabolized by drop
ping the time derivative term and the streamwise
direction (0v iscous flow terms in the flux vectors. (3 = 1,2, ...,?a) (19)
Equation (14) can then be rewritten as
where cs is the mass fraction of species s, Ljs is the
nondimensional production term, D is the noendimen.-
sional binary diffusion coefficient, and ,& =
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AMERICAN INSTITUTE OF AERONAUTICASN D ASTRONAUTICS
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The chemical model used in the present calcula- for many applications. For cases where upstream
tions is similar to the clean-air model of Blottner et (elliptic) effects are important, the flowfield can be
al. [26] and Prabhu et al. [27]. It consists of molecular computed using multiple streamwise sweeps with ei-
oxygen (02), atomic oxygen (0),m olecular nitrogen ther the IPNS [28], TIPNS [29], or FBIPNS [30] algo-
(Nz), atomic nitrogen (N), nitric oxide (NO), ni- rithms. This iterative process is continued until the
tric oxide ion (NO+) and electrons (e-). The follow- solution is converged.
ing reactions are considered between the constituent For the iterative PNS (IPNS) method, the E vec-
species. tor is split using the Vigneron parameter (w) [31].
This parameter does not need to be changed for the
(1) 02+M1+2O+Ml
present low magnetic Reynolds number formulation.
(2) Nz+M2+2iV+Mz In the previous high magnetic Reynolds number code
(3) N2+N+2N+N [17] it was necessary to modify the Vigneron param-
(4) NO+kf3FfN+O+& eter to account for MHD effects. After splitting, the
E vector can be written as:
(5) NO+Ot:Oz+N
(6) Nz+O+NO+N
E=E*+EP (23)
+
(7) NO Trt NO+ e- (20)
where MI,M 2, M3, are catalytic third bodies. The where
clean-air chemical model has 7 species (n = 7) and
Seven reactions (m= 7). In order to simulate the
seeded air flow in the MHD channel, the potassium
seeding reaction has been added to the above chem-
istry model. Thjs reaction is the ionization of atomic
potassium (K)a nd is given by the following equation:
(8) K+e-+K’+e-+e- (21)
Using the law of mass action, the mndimensional
mass production rate of species s is
m Rt
Ws = ~ s x ( v : , s- vi.,s) [ ~ j . b (n~~)a yrl’*~
k=l t=l
fi (22) +-
GI
-Kb,k (T) [pr+ 14..] J
r= 1
where Tr is the nondimensional molemass ratio of
the reactants, Ms is the molecular weight of species
I
s, and are the stoichiometric coefficients and 0
nt is the number of reactants. Mher details on +Y 0
the reaction rates and the thermodynamic and trans-
port properties can be found in Ref. [as]. The electri-
cal conductivity is determined from the species mole
fractions, along with the temperature, density, and The streamwise derivative of E is then Berenced
pressure of the gas, using the program of Park [21]. using a backward difference for E* and a forward dif-
ference for the “elliptic” portion (W):
Numerical Method
Solution of PNS Equations
The governing PNS equations with MHD source (25)
+
terms have been incorporated into NASA’s upwind where the subscript (i 1) denotes the spatial index
PNS (UPS) code [20]. These equations can be solved (in the E direction) where the solution is currently
very efficiently using a single sweep of the flowfield being computed. The vectors E;+l and E:+, are then
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kUEFUCAN INSTITUTE OF AERONAUTICS AND ASTJ~ONAUTICS
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linearized in the following manner: Solution of Species Continuity Equations
+ (m aE' )i For chemical nonequilibrium, the species continuity
E;+l = E; (Ui+-l U,)
equations, Eq. (19), must be solved in addition to
the magnetogasdynamic equations. The equations
E:+l = E: + (aEu )(U ;+i -Vi) (26) have been integrated using the loosely-coupled ap-
proach of Tannehill et al. [32]. In this approach,
the species continuity equations and magnetogasdy-
The Jacobians can be represented by
namic equations are solved separately. The coupling
between the two sets of equations is then obtained
in an approximate manner. The species continuity
equations are modeled using a second-order-accurate,
(27)
upwind-based TVD scheme for the convective terms
and second-order-accurate central aerences for the
After substituting the above linearizations into
diffusiont erms. The assumption of zero net charge of
Eq. (25), the expression for the streamwise gradient
the gas is used to eliminate the electron mass conser-
of E becomes
vation equation. In addition, the species continuity
equation for the nth species is eliminated by using
the requirement that the mass fractions must sum
to unity. The term representing the rate of produc-
]
+ m+2 - E:> (28) tion of species, ws,is treated as a source term, and is
lagged to the previous marching level.
The final discretized form of the fluid flow equa- The coupling between the fluids and the chemistry
tions with MHD source terms is obtained by substi- is performed in an approximate manner. First, a fluid
tuting Eq. (28) into Eq. (16) along with the linearized step is taken fiom marching station i to i+l assuming
expressions for the tluxes in the cross flow plane. The frozen chemistry. Then the fluid density and velocity
final expression becomes: at i+l are used in the solution of the species continu-
ity equations to obtain species mass fractions at i+l.
Finally, the species mass fractions, molecular weight
of mixture, fluid density, and internal energy at i+l
are used to obtain the new temperature, pressure,
a ('Ga) u ]Isf1 (AU,)k+l= RHS (29) speciiic enthalpy, and frozen specific heats at the i+l
+q a mardung station.
The temperature is obtained by performing a
where
Newton-Raphson iteration of the following form:
-
(AU;)k+=l (U;+l Ua)'+l
where
(g)*" n
(yD)k+l
- +
a
n
and the superscript k+l denotes the current itera-
tion (i.e. sweep) level. In the preceding equation, the s=1
MHD source term, SMD, is treated explicitly since it and k is the index of iteration. The iterations are
is evaluated using the velocity at station i (Vi). For continued until
most cases, this will not degrade the accuracy of the
I T ~ + ~ - - T ~ I
solution since At is small and the velocity changes
slowly. If this is not the case, a predictor-corrector
procedure can be implemented whereby a predicted where E is asmall positive quantity. Once the temper-
velocity at station i+l (VG~)is first obtained using ature is determined, the pressure can be computed us-
Eq. (29). The solution at station i+l is then recom- ing Dalton's law of partial pressures. Further details
puted by evaluating SMHDw ith Vzl. of this procedure can be found in Re&. [32] and [33].
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A-MERICAN INSTITUTE OF AERONAUTICASN D ASTRONAUTICS
Numerical Results where the subscript o denotes total conditions at the
nozzle entrance and w denotes wall conditions.
The numerical calculation of the 3-D supersonic flow This case was computed using several Herent elec-
in the experimental h4HD channel is now discussed. tric field strengths in order to properly simulate the
The flow in the nozzle section was computed using a experiment. In the experiment, the voltage applied
combination of the OVERFLOW code [34] and the to the electrodes was approximately 134 V for this
present PNS code (without MHD effects). For the case, however, due to the sheath voltage drop, the
OVERFLOW nozzle calculation, a highly stretched actual voltage applied to the flow is smaller than the
grid consisting of 150 x 80 x 80 grid points was used. electrode voltage. The voltage drop was measured for
The normal grid spacing at the wall was 1.0 x the central inviscid core flow, and was approximately
m. For the PNS calculation of the flow in the re- 67 V [2]. Since the boundary layer is computed in the
mainder of the nozzle and the rest of the MHD c h a ~n umerical solution, the applied electric field must be
nel, a highly stretched grid consisting of 90 points in approximately the voltage drop across the electrodes
both the y and z directions was used and the nor- minus the sheath voltage drop. Unfortunately, it is
mal grid spacing at the wall was 2.0 x m. As not a trivial task to measure the sheath voltage drop.
a consequence of flow symmetry, only one-fourth of Therefore, several different electric field strengths
the channel cross section was computed in the 3-D were chosen in the numerical calculations so that
calculations. the corresponding voltage drop across the electrodes
The calculations were performed assuming turbu- would be between 67 V and 134 V. The voltage drop
lent flow throughout the MHD channel. The chan- of 67 V corresponds to E, = 3955 V/m and a voltage
nel wall temperature was assumed to be isothermal drop of 84.7 V corresponds to E, = 5OOO V/m
since quasi-steady flow conditions were maintained The computed streamwise variation of static pres-
in the experiment for only about 1.2 milliseconds. A sure for the nonequilibrium seeded-air calculations
schematic of the powered portion of the MHD &an- is shown in Fig. 3 for the different electric field
ne1 along with the directions of the applied magnetic strengths. The pressure variation with no electric
and electric fields is shown in Fig. 2. The values of the field or magnetic field is denoted by Ev = 0. The re-
magnetic field (E$), and the electric field (E,) were sults for E, = 5000 V/m are in excellent agreement
-
kept constant in the powered portion of the channel. with the experiment. The numerical results show an
Three Herent chemistry models were used in this increase in static pressure as the electric field strength
study to simulate the flow in the MHD chamel. is increased. The computed streamwise variation of
These were: (1) perfect gas (T = 1.25), (2) equi- static pressure for the *rent chemistry models is
librium air, and (3) nonequilibrium seeded-air chem- shown in Fig. 4 for E, = 5OOO V/m. The noneqoi-
istry. For the perfect gas and equilibrium air calm librium seeded-air model gives the closest agreement
lations, the electrical conductivity (a,)w as assumed with the experimental pressures.
constant. For the nonequilibrium seeded-air calm For the noneqdibrium seeded-air computations,
lations, the electrical conductivity varied throughout
the electrical conductivity was not constant but. var-
the flowfield and was determined using the program ied throughout the flowfield. The average conduc-
of Park [Zl]. The seeding (as in the experiment) con- tivity (averaged over the channel cross section) at
sisted of 1% (by mass) of potassium. Two test cases the center of the powered portion of the channel
corresponding to Runs 15 and 16 of the NASA h e s (electrode pair 10) was found to be 130 mho/m for
experiments [2,35] were computed in this study and E, = 5000 V/m. This value of conductivity is within
are now dmxssed. the range determined in the experiments and is the
same constant value that was used for the perfect gas
Test Case 1: NASA Ames MHD Run 15 and equilibrium air computations.
(vc,.
= 320V) The computed streamwise variation of averaged ve-
locity for the noneqdibrium seedecE-air calculations
The dimensional flow parameters for this test case
is shown in Fig. 5. The velocities are averaged across
are:
the channel cross section and normalized using the
entrance velocity to be consistent with the experi-
po = 9.10I~O 5 N/m2
To = 5560 K ment. In the experiment, the velocities were obtained
T,,,= 300K by measuring the voltage generated by the flow at
ae = 130 mho/m (or variable) the last electrode pair (19) which is unpowered. This
B, = 0.0, 0.92 T procedure inherently involves an averaging of the ve-
locity profile. The numerical results indicate an in-
EV = 0, 3955, 5000Vfm
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AMERICANIN S- OF AERONAUTICAS ND ASTRONAUTICS
crease in the averaged velocity of about 2% with variation of averaged velocity for the different chem-
E, = 5000 V/m and this agrees exactly with the ex- istry models is shown in Fig. 11 for E, = 6000 V/m.
perimental value of 2%. Once again, the equilibrium air and the nonequilib
The computed streamwise variation of averaged ve- rims eeded-air models give similar results. The cen-
locity for the different chemistry models is shown in terline variation of static temperature for the equi-
Fig. 6 for E, = 5000 V/m. Both the equilibrium air librium model is shown in Fig. 12 for the different
and the nonequilibrium seeded-air chemistry mod- electric field strengths.
els give similar rdts. The centerline variation of
static temperature for the nonequilibrium seeded-air
Concluding Remarks
model is given in Fig. 7 for the different electric field
strengths. .
In this study, a new 3-D parabolized Navier-Stokes
algorithm with noneqdibrium seeded-air capability
has been developed to efficiently compute MHD flows
in the low magnetic Reynolds number regime. The
Test Case 2: NASA Ames MHD Run 16 new algorithm has been used to compute the flow
(Kq. = 38OV) in the NASA-Ames experimental MHD channel for
Runs 15 and 16. The numerical results are in good
The dimensional flow parameters for this test case
agreement with the experimental results.
are:
po = 9.92 x lo5 N/m2 Acknowledgments
To = 5560 K
Tu, = 300 K
This work was supported by NASA Am- Research
a, = 130mho/m (or variable)
Center under Grant NCC2-5517 and by Iowa State
B, = 0.0, 0.92 T
University. The Technical Monitor for this grant is
E, = 0, 4309, 5300, 6OOOV/m Dr. Unmeel B. Mehta The authors wish to thank
Dr. David W. Bogdanoff of NASA Ames for his help
This test case was also computed using several dif-
and comments regarding the MHD b e l ex peri-
ferent electric field strengths in order to properly
ments and Prof. Robert W. MacCormack of Stan-
simulate the experiment. The electric field strength
of 4309 V/m corresponds to the voltage drop mea- ford University for his assistance with the electrical
conductivity program of Dr. Chul Park
sured in the central inviscid core flow. The com-
puted streamwise variation of static pressure for the
nonequilibrium seeded-air calculations is shown in
References
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cal result with an electric field strength of 5300 V/m.
“Experimental Demonstration of Magnetc-
The computed streamwise variation of static pressure
Hydrodynamic (MHD) Acceleration - Facility
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