NASA Technical Reports Server (NTRS) 20040105525: Numerical Simulation of Turbulent MHD Flows Using an Iterative PNS Algorithm

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by  NASA Technical Reports Server (NTRS)

AIAA 2003-0326
Numerical Simulation of Turbulent MHD
Flows Using an Iterative PNS Algorithm
Hiromasa Kat0 and John C. Tannehill,
Iowa State University, Ames, IA 5001 1
and
Unmeel B. Mehta
NASA Ames Research Center, Moffett Field, CA
94035
41s t AIAA Aerospace Sciences
Meeting and Exhibit
6-9 January 2003 / Reno, NV
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Numerical Simulation of Turbulent MHD Flows
Using an Iterative PNS Algorithm
Hiromasa Kato* and John C. Tannehillt
Iowa State University, Ames, IA 50011
and
Unmeel B. Mehtat
NASA Ames Research Center, Moflett Field, CA 94035
Abstract Stokes (PNS) equations in place of the N-S equa-
tions. The PNS equations can be used to compute
A new parabolized Navier-Stokes (PNS) algorithm three-dimensional, supersonic viscous flowfields in a
has been developed to efficiently compute magne- very efficient manner [ll].T his efficiency is achieved
tohydrodynamic (MHD) flows in the low magnetic because the equations can be solved using a space-
Reynolds number regime. In this regime, the electri- marching technique as opposed to the time-marching
cal conductivity is low and the induced magnetic field technique that is normally employed for the complete
is negligible compared to the applied magnetic field. N-S equations.
The MHD effects are modeled by introducing source
Recently, the present authors have developed a
terms into the PNS equation which can then be solved PNS code to' solve supersonic MHD flowfields in
in a very efficient manner. To account for upstream
the high magnetic Reynolds number regime [12].
(elliptic) effects, the flowfields are computed using
This code is based on NASA's upwind PNS (UPS)
multiple streamwise sweeps with an iterated PNS al-
code which was originally developed by Lawrence et
gorithm. Turbulence has been included by modify-
al. [13]. The UPS code solves the PNS equations u s
ing the Baldwin-Lomax turbulence model to account
ing a fully conservative, finite-volume approach in a
for MHD effects. The new algorithm has been used
general nonorthogonal coordinate system. The UPS
to compute both laminar and turbulent, supersonic,
code has been extended to permit the computation of
MHD flows over flat plates and supersonic viscous
flowfields with strong upstream influences. In regions
flows in a rectangular MHD accelerator. The present
where strong upstream influences are present, the
results are in excellent agreement with previous com-
governing equations are solved using multiple sweeps.
plete Navier-Stokes calculations.
As a result of this approach, a complete flowfield can
be computed more efficiently (in terms of computer
time and storage) than with a standard N-S solver
Introduction
which marches the entire solution in time. Three it-
erative PNS algorithms (IPNS, TIPNS, and FBIPNS)
Flowfields involving MHD effects have typically been
have been developed. The iterated PNS (IPNS) al-
computed [l-10) by solving the complete Navier-
gorithm [14] can be applied to flows with moder-
Stokes (N-S) equations for fluid flow in conjunction
ate upstream influences and small streamwise sepa-
with Maxwell's equations of electromagnetodynam-
rated regions. The time iterated PNS (TIPNS) algo-
ics. When chemistry and turbulence effects are also
rithm [15] can be used to compute flows with strong
included, the computational effort required to solve
upstream influences including large streamwise sep
the resulting coupled system of partial differential
arated regions. The forward-backward sweeping it-
equations is extremely formidable. One possible rem-
erative PNS (FBIPNS) algorithm [16] was recently
edy to this problem is to use the parabolized Navier-
developed to reduce the number of sweeps required
for convergence.
*Graduate Research Assistant, Student Member AIAA
t Manager, Computational Fluid Dynamics Center, and The majority of MHD codes that have been de-
Professor, Dept. of AEEM. Fellow AIAA veloped combine the electromagnetodynamic equa-
:Division Scientist, Associate Fellow, AIAA
tions with the full Navier-Stokes equations result-
Copyright 02003 by the American Institute of Aeronau-
tics and Astronautics, Inc., all rights reserved. ing in a complex system of eight scalar equations.
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*. ..
These codes can theoreticdly be used for any mag-
netic Reynolds number which is defined as Re,,, =
u,peV, L where 0; is the electrical conductivity, pe
is the magnetic permeability, V, is the freestream Ohm's law
velocity, and L is the reference length. However, +
J = ue (E V x B) (4)
it has been shown that as the magnetic Reynolds
number is reduced, numerical difficulties are often where V is the velocity vector, B is the magnetic
encountered 241. For many aerospace applications field vector, E is the electric field vector, and J is the
the electrical conductivity of the fluid is low and conduction current density. The flow is assumed to
hence the magnetic Reynolds number is small. In be either in chemical equilibrium or in a frozen state.
these cases, it makes sense to use the low magnetic The curve fits of Srinivasan et al. [23,24] are used
Reynolds number assumption and reduce the com- for the thermodynamic and transport properties of
plexity of the governing equations. In this case, the equilibrium air.
MHD effects can be modeled with the introduction The governing equations are nondimensionalized
of source terms into the fluid flow equations. Several using the following reference variables.
investigators [4,8,17-191 have developed N-S codes
for the low magnetic Reynolds number regime where x*,y*,z = -X,Y,Z , u*,v*,w* = -U,V,W , t =-u mt
the induced magnetic field is negligible compared to L urn L
the applied magnetic field.
In the present study, a new PNS code (based on p * =-P , T=-T , pa=--- P
E
the UPS code) has been developed to compute MHD Pm T, Pm
flows in the low magnetic Reynolds number regime.
The MHD effects are modeled by introducing the a p ef = -et , =r*= - 5L , /A*=- Ir (5)
propriate source terms into the PNS equations. U p @a P aU CQ Pm
stream elliptic effects can be accounted for by u s
ing multiple streamwise sweeps with either the IPNS,
TIPNS, or FBIPNS algorithms. Turbulence has been
included by modifying the Baldwin-Lomax turbu-
lence model [20] to account for MHD effects using
the approach of Lykoudis [21]. The new code has
*
where the superscript refers to the nondimensional
been tested by computing both laminar and turbu-
quantities. In subsequent sections, the asterisks are
lent, supersonic MHD flows over a flat plate. Com-
dropped.
parisons have been made with the previous complete
N-S computations of Dietiker and Hohann [MI. In If the flow variables are assumed to vary in only two
dimensions (2, y) while the velocity, magnetic, and
addition, the new code has been used to compute the
electric fields have components in three dimensions
supersonic viscous flow inside a rectangular channel
designed for MHD experiments [22]. (2, y, z), the governing equations can be written in
the following flux vector form:
Governing Equations aU aEi aFi aEu dFu
-a+t -+-a=x - ay az +-+aSy MHD (6)
The governing equations for a viscous MHD flow with
where U is the vector of dependent variables and E;
a small magnetic Reynolds number are given by [18]:
and Fi are the inviscid flux vectors, and E,, and Fu
Continuity equation
are the viscous flux vectors. The source term SMHD
aP =o contains all of the MHD effects. The flux vectors are
-+V.(pV)
at given by
Momentum equation
Energy equation
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AMERICANIN STITUTE OF AERONAUTICS AND ASTRONAUTICS
The prime in the preceding equation indicates that
the streamwise viscous flow terms have been dropped.
For turbulent flows, the two-layer Baldwin-Lomax
turbulence model [20] has been modified to account
for MHD effects. Only the expression for turbulent
viscosity in the inner layer is changed. This modifi-
I cation for MHD flows is due to Lykoudis [9,21].
F, =
Numerical Met hod
1 0 The governing PNS equations with MHD source
&(Ey + wBz - uB,) terms have been incorporated into NASA’s upwind
PNS (UPS)c ode 1131. These equations can be solved
very efficiently using a single sweep of the flowfield
for many applications. For cases where upstream
(elliptic) effects are important, the flowfield can be
-&(Ey + wBz - uB,) computed using multiple streamwise sweeps with ei-
Ez(Ez + vB, - wB,) ther the IPNS [14], TIPNS [15], or FBIPNS [16] algo-
+Ey(Ey ++ wB, - uB,) sriotlhumtiso.n Tish cios nivteerragteivde. process is continued until the
+Ez(E, uBY - vB,)
For the iterative PNS (IPNS) method, the E vec-
where
tor is split using the Vigneron parameter (w) [25].
Pet = Z1 P (u2 + v2 + WZ) + -7-P1 Tprheisse npta rlaomw emtearg dnoeetisc nRoet ynneoeldd st onu bme bcehra fnogremdu floart itohne.
and 7 can be determined from the curve fits of Srini- In the previous high magnetic Reynolds number code
[12] it was necessary to modify the Vigneron param-
vasan et al. [23] for an equilibrium air flow or is equal
eter to account for MHD effects. After splitting, the
to a constant (7)fo r a frozen or perfect gas flow. The
E vector can be written as:
nondimensional shear stresses and heat fluxes are de-
fined in the usual manner [ll].
E=E’+W (17)
The governing equations are transformed into com-
where
putational space and written in a generalized coordi- 1
nate system (t,~as)
-
Pu2 +WP
where
+-
FY
J
and J is the Jacobian of the transformation.
The governing equations are parabolized by drop
ping the time derivative term and the streamwise
direction (E) viscous flow terms in the flux vectors. Ep
Equation (13) can then be rewritten as
The streamwise derivative of E is then differenced
using a forward difference for the “elliptic” portion
where
(W:
(9) (9)
+
= (Ei -E:) (Fi -FL) (16)
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.I
+
where the subscript (i 1) denotes the spatial index Test Case 1: Supersonic laminar and
(in the direction) where the solution is currently turbulent flows over a flat plate with
being computed. The vectors ET+l and are then applied magnetic field
linearized in the following manner:
In this test case, the supersonic, laminar and turbu-
lent flow over a flat plate with an applied magnetic
field is computed. This case corresponds to the flat
plate case computed previously by Dietiker and Hoff-
mann [18] using the full N-S equations. A strong
magnetic field is applied normal to the flow as shown
The Jacobians can be represented by in Fig. 1. The dimensional flow parameters for this
test case are:
M, = 2.0
pa, = 1.076 x lo5 N/m2
T, = 300K
After substituting the above linearizations into Re, = 3.75 x lo6
Eq. (19), the expression for the streamwise gradient 7 = 1.4
of E becomes
L = 0.08 m
u, = 800mho/m
The plate is assumed to be an adiabatic wall and a
(22) perfect gas flow is assumed. The magnetic Reynolds
number (based on the length of the plate) is 0.056 and
The final discretized form of the fluid flow equa- can be considered negligible when compared to one.
tions with MHD source terms is obtained by substi- The normal magnetic field component (By)ra nges
tuting Eq. (22) into J3q. (15) along with the linearized in value from 0.0 to 1.2 T. The magnitude of the
expression for the flux in the cross flow plane. The magnetic field can be represented by the parameter
final expression becomes: m which is defined [I81 by
ueB 2
[&(A; m = Y (24)
Po0 urn
and has units of (l/m). For By = 1.2 T, m is equal
= RHS (23) to 1.33.
where A highly stretched grid consisting of 50 points in
the normal direction was used to compute this case.
RHS = --1 - The first point off the wall was located at 2 x
A< m. Initially, the flow was assumed laminar and sev-
eral values of Byr anging from 0.0 (no magnetic field)
to 1.2 T were used. The velocity and temperature
profiles at z = 0.06 m are shown in Figs. 2 and 3 for
and the superscript k+l denotes the current iteration By = 0.0 T, 1.0 T, and 1.2 T. The velocity profiles
(Le. sweep) level. are compared to the N-S results of Dietiker and Hoff-
mann in Fig. 2 and show excellent agreement. The
magnetic field generates a Lorentz force which acts in
Numerical Results a direction opposite to the flow. Thus, the flow is de-
celerated as the magnetic field is increased as seen in
In order to investigate the utility and accuracy of Fig. 2. For By = 1.2 T the flow is slightly separated.
the present PNS approach of solving MHD flowfields The temperature profiles cannot be compared at this
at low magnetic Reynolds numbers, a few basic test time since no temperature data is given in Ref. 1181.
cases were computed. The supersonic viscous flow in The turbulent flow over the flat plate was then
these cases is altered by the presence of the magnetic computed using the modified Baldwin-Lomax turbu-
and electric fields which are applied to the flow. lence model that accounts for MHD effects. The flow
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was assumed laminar prior to the point (z = 0.04 m) To = 7500 K
where transition from laminar to turbulent flow was
triggered. Again, several values of By ranging from The laminar flow was assumed to be in chemical equi-
librium. The computed flowfield at the end of the
0.0 to 1.2 T were used in the computations. The tur-
bulent velocity and temperature profiles at z = 0.06 center section was then used as the starting solu-
tion for the flow calculation of the accelerator section.
m are shown in Figs. 4 and 5 for By = 0.0, 1.0 T,
The MHD parameters used in the accelerator section
and 1.2 T. The turbulent velocity profiles in Fig. 4
were:
are in good agreement with the results of Ref. [18].
The variation of skin friction coefficient is shown in
u, = 50 mho/m
Fig. 6. The present laminar/turbulent skin friction
By = 1.5 T
variations are compared with the results of Ref. [18]
and show good agreement. The difference in results E, = -Ku,By
near the transition point may be due to the coarse Re,,, = 0.05
grid and smoothing used in Ref. [l8].
where the load factor (K)r anged in values from 0.0 to
All of the present laminar computations were per-
formed using a single sweep of the flowfield except for 1.4, and the centerline velocity (ue)a t the beginning
the separated flow case (By= 1.2 T). For this case of the accelerator section had a value of 3162 m/s.
The velocity profiles at the end of the accelerator
as well as for all the turbulent cases, multiple sweeps
section are shown in Fig. 8 for different load factors.
were used to account for upstream effects.
The velocity profile with no electric or magnetic fields
is denoted by K = 0. The increase in the centerline
Test Case 2: Supersonic viscous flow in velocity with distance (z) for various load factors is
a rectangular MHD accelerator shown in Fig. 9. The centerline velocity increases
by about 30% with a load factor of 1.4. It should
In this test case, the supersonic flow in an experi-
be noted that the flow decelerates because of friction
mental MHD channel is simulated. This facility is
when no electric or magnetic fields are applied.
currently being built at NASA Ames Research Center
by D. W. Bogdanoff, C. Park, and U. B. Mehta [22]
to study critical technologies related to MHD bypass Concluding Remarks
scramjet engines. The channel is about a half meter
long and contains a nozzle section, a center section, In this study, a new parabolized Navier-Stokes al-
and an accelerator section. The channel has a uni- gorithm has been developed to efficiently compute
form width of 2.03 cm. Magnetic and electric fields MHD flows in the low magnetic Reynolds number
can be imposed upon the flow in the accelerator sec- regime. The new algorithm has been used to com-
tion. A schematic of the MHD accelerator section is pute both laminar and turbulent, supersonic, MHD
shown in Fig. 7. flows over flat plates and in a rectangular accelera-
This test case was previously computed by tor section. Although only limited results have been
R. W. MacCormack [lo] using the full N-S equations obtained thus far, it can be seen that the present
coupled with the electromagnetodynamic equations. approach is quite promising. Computations of other
The electrical conductivity in his calculations was set test cases are currently underway in order to validate
at 1.0 x lo5 mho/m resulting in avery large magnetic the current method.
Reynolds number. In the present study, the calcu-
lations are performed in the low magnetic Reynolds
Acknowledgments
number regime using a realistic value of electrical con-
ductivity. The flow is computed in two dimensions,
This work was supported by NASA Amea Fbsearch
but later will be extended to three dimensions. Be-
Center under Grants NCC2-53T9 and NCC2-5517
cause of flow symmetry, only half of the channel is
and by Iowa State UniversiQ. The Technical Mon-
computed in the 2-D calculations.
itor for the NASA grant is Dr. Unmeel B. Mehta
The flow in the nozzle section and the center sec-
tion was computed using a combination of the OVER-
References
FLOW code [26] and the present PNS code (without
MHD effects). The initial conditions for the nozzle
(flow at rest) were: [l] Gaitonde, D. V., “Development of a Solver for 3-
D Non-Ideal Magnetogasdynamics,” AIAA Pa-
PO = 8.0 x lo5 N/m2 per 99-3610, June 1999.
5
AMERICANIN STITUTE OF AERONAUTICAS ND ASTRONAUTICS
.. ..
Damevin, H. M., Dietiker, J.-F., and Hoffmann, [15] Tannehill, J. C., Miller, J. H., and Lawrence,
K. A., “Hypersonic Flow Computations with S. L., “Development of an Iterative PNS Code
Magnetic Field,” A M P aper 2000-0451, Jan. for Separated Flows,” AIAA Paper 99-3361,
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Hoffmann, K. A., Damevin, H. M., and Di- [16] Kato, H. and Tannehill, J. C., “Development
etiker, J.-F., “Numerical Simulation of Hyper- of a Forward-Backward Sweeping Parabolized
sonic Magnetohydrodynamic Flows,” AIAA Pa- Navier-Stokes Algorithm,” AIAA Paper 2002-
per 2000-2259, June 2000. 0735, Jan. 2002.
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MacCormack, R. W., ”Numerical Computa-
tion in Magnetofluid Dynamics,” Computational [18] Dietiker, J.-F. and Hoffmann, K. A., “Boundary
Fluid Dynamics for the 21st Century, Kyoto, Layer Control in MagnetohydrodynamicF lows,”
Japan, July 2000. AIAA Paper 2002-0130, Jan. 2002.
Deb, P. and Agarwal, R., “Numerical Study of [19] Cheng, F., Zhong, X.,G ogineni, S., andKimmel,
MHD-Bypass Scramjet Inlets with Finite-Rate R. L., “Effect of Applied Magnetic Field on the
Chemistry,” AIAA Paper 2001-0794, Jan. 2001. Instability of Mach 4.5 Boundary Layer over a
Flat Plate,” AIAA Paper 2002-0351, Jan. 2002.
MacCormack, R. W., “A Computational
Method for Magneto-Fluid Dynamics,” AIAA [20] Baldwin, B. S. and Lomax, H., Thin Layer
Paper 2001-2735, June 2001. Approximation and Algebraic Model for Sep
arated Turbulent Flows,” AIAA-Paper 78-257,
Gaitonde, D. V. and Poggie, J., “An Implicit
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Technique for 3-D Turbulent MGD with the
Generalized Ohm’s Law,” AIAA Paper 2001- [21] Lykoudis, P. S., “Magneto Fluid Mechanics
2736, June 2001. Channel Flow, I1 Theory,” The Physics of FZu-
ids, Vol. 10, No. 5, May 1967, pp- 1002-1007.
Dietiker, J.-F. and Hoffmann, K. A., “Numerical
Simulation of Turbulent Magnetohydrodynamic [22] Bogdanoff, D. W., Park, C., and Mehta, U. B.,
Flows,” AIAA Paper 2001-2737, June 2001. “Experimental Demonstration of Magnet*
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Hydrodynamic (MHD) Acceleration Facility
MacCormack, R. W., “Three Dimensional and Conductivity Measurements,” NASA TM-
Magneto-Fluid Dynamics Algorithm Develop 2001-210922, July 2001.
ment,” AIAA Paper 2002-0197, Jan. 2002.
[23] Srinivasan, S., Tannehill, J. C., and Weilmuen-
Tannehill, J. C., Anderson, D. A., and Pletcher, ster, K. J., “Simplified Curve Fits for the
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[24] Srinivasan, S., Tannehill, J. C., and Weilmuen-
[12] Kato, H., Tannehill, J. C., Ramesh, M. D., and ster, K. J., “Simplified Curve Fits for the Trans
Mehta, U. B., “Computation of Magnetohydro- port Properties of Equilibrium Air,” NASA CR
dynamic Flows Using an Iterative PNS Algo- 178411, 1987.
rithm,” AIAA Paper 2002-0202, Jan. 2002.
[25] Vigneron, Y. C., Rakich, J. V., and Tan-
[13] Lawrence, S. L., Tannehill, J. C., and Chaussee, nehill, J. C., “Calculation of Supersonic Flow
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[26] Buning, P. G., Jespersen, D. C., and Pulliam,
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98-0226, Jan. 1998.
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AMERICANIN STITUTE OF AERONAUTICASN D ASTRONAUTICS
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YA
I
I
I
I
I
I
I
Ma,
I
- 1
I
I
Figure 1: Test Case 1
0.010 I
.'
I I I I I I i
l j
I j
Present result (Bo= 0.0")
0.008 .- - - - - Present result (Bo= 1.0") I !
I :
- - - Present result (Bo= 1.2") I .
00
0N -S result [ 181 (Bo= 0.0")
O. n I :
0 N-S result [ 181 (Bo= 1
0.006 1 ;
ON-S result [lS] (Bo=1 .2") I ;
4
k
0.004
0.002
0.000
.o
-0.2 0.0 0.2 0.4 0.6 0.8 1
n o 0
Figure 2: Laminar velocity profiles
7
AMERICANIN STITUTE OF AERONAUTICSA ND ASTRONAUTICS
.. .-
0.010
I ! I' ' I I I
I
I
I
Present result (Bo= 0.OT)
I
0.008
i I ---------- Present result (Bo= 1.OT)
I - - - Present result (Bo= 1.2T)
I
I
0.006 I
I
I
I
\
0.004
\
\
\
0.002
O.Oo0 I I I I \ I : I \ I I
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
Figure 3: Laminar temperature profiles
0.020 I I I I I 1 I I 1
I
I
Present result (Bo= 0.OT)
I
0.016 ____._.P _re sent result (Bo= 1 .OT)
- - - Present result (Bo= 1.2T) 0
I
0 N-S result [18] (Bo= 0.OT) cp
1 0 N-S result [18] (Bo= 1.OT) 4 ;
0.012
ON-Sr esult [ 181 (Bo= 1.2T)
i-
4
k
0
0.004
0.0°8
0.000 ~
.o
-0.2 0.0 0.2 0.4 0.6 0.8 1
n o 0
Figure 4: Turbulent velocity profiles
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AMERICAINNS T!,TUTE OF AERONAUTICASN D ASTRONAUTICS
0.0020
I : # II I I I
I
Present result (Bo= 0.OT)
1I -
0.0016 I __________ Present result (Bo= 1.OT).
- - -
I Present result (Bo= 1.2T)
I
I
0.0012 I
I
I
I
I
O.Ooo8 \
\
\
\
\
O.OOO4 \
\
\
O.oo00
0.8 1.0 1.2 I .4 1.6 1.8 2.0 2.2
Figure 5: Turbulent temperature profiles
I I I
Present result (Bo= 0.0T)
0.003
- - .- - - - - Present result (Bo= 1.OT)
- - - Present result (Bo= 1.2T)
0 N-S result [18] (Bo = 0.OT)
CI N-S result [18] (Bo= 1.OT)
0 N-S result [18] (Bo= 1.2T)
0.002
4 I
0.001
0.000
0.00 0.02 0.04 0.06 0.08
x, m
Figure 6: Laminar/turbulent skin friction coefficient
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AMERICANI NSTITUTE OF AERONAUTICASN D ASTRONAUTICS