-
Hindawi Publishing CorporationInternational Journal of Aerospace
EngineeringVolume 2009, Article ID 793647, 12
pagesdoi:10.1155/2009/793647
Research Article
Combined Magnetohydrodynamic and Geometric Optimizationof a
Hypersonic Inlet
Kamesh Subbarao and Jennifer D. Goss
Department of Mechanical and Aerospace Engineering, The
University of Texas, Arlington, TX 76019, USA
Correspondence should be addressed to Kamesh Subbarao,
[email protected]
Received 3 June 2009; Revised 24 September 2009; Accepted 30
October 2009
Recommended by Anwar Ahmed
This paper considers the numerical optimization of a double ramp
scramjet inlet using magnetohydrodynamic (MHD) effectstogether with
inlet ramp angle changes. The parameter being optimized is the mass
capture at the throat of the inlet, such thatspillage effects for
less than design Mach numbers are reduced. The control parameters
for the optimization include the MHDeffects in conjunction with
ramp angle changes. To enhance the MHD effects different ionization
scenarios depending upon thealignment of the magnetic field are
considered. The flow solution is based on the Advection Upstream
Splitting Method (AUSM)that accounts for the MHD source terms as
well. A numerical Broyden-Flecher-Goldfarb-Shanno- (BFGS-) based
procedure isutilized to optimize the inlet mass capture. Numerical
validation results compared to published results in the literature
as well asthe outcome of the optimization procedure are summarized
to illustrate the efficacy of the approach.
Copyright © 2009 K. Subbarao and J. D. Goss. This is an open
access article distributed under the Creative Commons
AttributionLicense, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is
properlycited.
1. Introduction
Scramjet engine inlet flow is subject to many
engineeringtribulations, mainly due to the fact that the base
geometryof the engine is suited to a very narrow range of
flightconditions. In order to have the engine operate
efficientlyacross a broader range of flight conditions the inlet
flowmust be adjusted. One obvious method to tune the inlet flowis a
mechanically actuated, variable geometry inlet that canadjust its
shape in flight to achieve optimal inlet conditions[1–3]. Since
there are pressure losses across the inlet, theshape of the inlet
needs to be optimized to minimize theselosses. Another novel method
of inlet flow optimizationinvolves the use of magnetohydrodynamics
(MHD) to con-trol the incoming flow instead of the mechanically
actuatedsurfaces.
The scramjet engine is designed for hypersonic flightwith
supersonic combustion and is flown at speeds rangingapproximately
from Mach 5 to 15. At these speeds the airahead of the vehicle can
become ionized thereby makinga case for magnetohydrodynamic flow
control. Since theionization levels in this type of flow are quite
low, it isconsidered in the MHD community as a “cold flow.”
Several
methods have been explored in increasing the ionizationof this
type of “cold” flow including seeding and electronbeams. Work done
by Macheret and Miles suggests that themost efficient method is the
electron beam [4–6]. In thelimits of increasing conductivity in the
flow, increasing themagnetic field strength is of course also very
restricted.
It is to be mentioned that initial studies of optimizingduct
flows for MHD power generators [7, 8] later evolvedinto the “AJAX”
concept proposed by Russian scientists asa means to provide heat
protection for hypersonic flightvehicles [9]. The resulting
enthalpy reduction due to anMHD generator in the inlet of the
engine could theoreticallyallow the operation of a conventional
turbojet or ramjetengine in hypersonic flight instead of a scramjet
engine. If theenthalpy extraction was not sufficient to avoid the
need for ascramjet engine, the MHD power generator could still
lowerinlet temperatures to tolerable levels. As well the addition
ofa bypass system to further accelerate the exhaust gases
couldincrease the specific thrust of the engine during
off-designconditions and flow control [10–14].
Previous studies on the effects of MHD control on inletflow
optimization [15] were conducted on a symmetricaldouble ramp inlet
as depicted in Figure 1. These studies
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2 International Journal of Aerospace Engineering
Flow
α2
α1
Figure 1: Symmetric double ramp full inlet.
involved two cases, first with the magnetic field oriented
z-plane, or into the page, and secondly with the magnetic
fieldimplemented in the x-y plane. A number of simulationswere run
with varying conductivity and magnetic field angleand in all cases
no improvement was found in the pressurerecovery. The application
of a magnetic field to the chargedflow always results in an
increase in the pressure loss. This isconsistent with the theory in
that energy is not being addedto the flow and the resultant Joule
heating is only a detrimentto the pressure recovery.
The main purpose of this study is to increase themass capture of
a double ramp cowl style scramjet inlet.The performance metric is
optimized using a combinationof changing geometry (ramp angles) and
including MHDeffects. The discretized Euler’s equations are
augmentedwith MHD source terms and the flow solution is obtainedvia
the AUSM method. The flow solution methodology isvalidated against
test cases drawn from literature availablein the public domain. The
optimization methodology is alsovalidated together with the flow
solver on the traditionaldouble ramp inlet problem with and without
MHD terms.Finally, the optimization results for the combined MHD
+inlet geometry are presented.
2. The Cowl Style Scramjet Inlet
The double ramp scramjet inlet studied in this paper is acowl
style inlet. Such an inlet represents a forebody
externalcompression region followed by a small region of
internalcompression. This mixed compression configuration
canbalance the problems of high external drag in the case of
fullexternal compression and excessive viscous effects during
fullinternal compression. The optimal cowl inlet configuration
isthe well-known “shock-on-lip” condition shown in Figure 2.
During off-nominal flight conditions when the flowMach number is
less than the design value, the shocks willmove ahead of the cowl
lip and some of the compressed airwill escape the inlet resulting
in “spillage” and a decrease in
Flow
Cow1
Figure 2: Optimal cowl configuration with shocks converging
onthe cowl lip.
the mass capture. In flight conditions where the flow Machnumber
is greater than the design value, the shocks moveinto the inlet
causing multiple reflected shocks, loss of totalpressure, possible
boundary layer separation, and engineunstart [16]. It is proposed
that the optimal “shock-on-lip”configuration can be recovered via
flow control employingmagnetohydrodynamic source terms as in [16].
As such, theflow control methodology in this paper compares two
stylesof ionization. The first is inspired by the work from
Shneideret al. [16] who focused on a “virtual cowl” scenario in
whichlocalized off-body energy addition was used to increase
themass capture and pressure recovery at Mach numbers lessthan the
design value. In their work the virtual cowl is aheated region
upstream of and slightly below the cowl lip.This heated region acts
as a scoop by deflecting the incomingflow and increasing mass
capture [17].
The second ionization scenario was presented by Shnei-der et al.
[16, 18] as well as Sheikin and Kuranov [19] whoimplemented a
uniform magnetic field in which they assumean ionized region
enclosed by lines that are parallel to themagnetic field lines to
improve scramjet inlet performance atoff-design conditions. Figure
3 demonstrates these two sce-narios. The first scenario is
characterized by the stationary-ionized region upstream of and
centered on the cowl lip.The magnetic field is applied to this
charged region withvarious angles to determine the capabilities of
influencingthe inlet mass capture. The second scenario is
characterizedby a moving ionization region which is coincident with
themagnetic field implementation. In this case the magneticfield
and ionized regions will be applied together at variousangles in an
attempt to influence the flow.
Also, it has been independently shown in works such asin [1, 2]
that geometric optimization of inlet shapes (namely,the ramp angles
in this case) could lead to optimal pressurerecovery as well as
mass capture. This paper investigates thepossibility of
simultaneously optimizing the geometry of theinlet (as was done in
[1, 2] and the ionization beam angle[16, 18, 19]). The geometric
optimization portion of thework will adjust the two ramp angles, α1
and α2.
3. Euler Equations with MHD Effects
The equations of motion that govern inviscid, compressiblefluid
flow in a region are given by Euler’s equations [20] and
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International Journal of Aerospace Engineering 3
Flow
Cow1
B
θ
Lonizedzone
(a) Moving magnetic field and stationary ionized zone
Flow
Cow1
B
θ
Lonized zone
e-beam
(b) Moving magnetic field and coincident ionizing e-beam
Figure 3: Application of magnetic field and ionized region.
are summarized as follows:
∂ρ
∂t+∂ρu
∂x+∂ρv
∂y+∂ρw
∂z= 0,
∂
∂t
(ρu)
+∂ρu2
∂x+∂ρuv
∂y+∂ρuw
∂z+
∂
∂xp = ρ fx,
∂
∂t
(ρv)
+∂ρuv
∂x+∂ρv2
∂y+∂ρvw
∂z+
∂
∂yp = ρ fy ,
∂
∂t
(ρw)
+∂ρuw
∂x+∂ρvw
∂y+∂ρw2
∂z+
∂
∂zp = ρ fz,
∂
∂t
(ρE)
+∂ρuE
∂x+∂ρvE
∂y+∂ρwE
∂z+∂pu
∂x+∂pv
∂y+∂pw
∂z
= ρq̇ + ρf ·V.(1)
These equations can be recast into the following form inorder to
facilitate the implementation of a flux splitting flowsolver method
used in this study. The equations are limitedto 2D and are
consistent with the formulations employed forthe study of
inlets:
∂U
∂t+∂F
∂x+∂G
∂y= S, (2)
where U is called the flow solution vector, F and G are knownas
the flux vectors, and S represents the source terms:
U =
⎡
⎢⎢⎢⎢⎢⎢⎣
ρ
ρu
ρv
ρE
⎤
⎥⎥⎥⎥⎥⎥⎦
, F =
⎡
⎢⎢⎢⎢⎢⎢⎣
ρu
ρu2 + p
ρuv
u(ρE + p
)
⎤
⎥⎥⎥⎥⎥⎥⎦
,
G =
⎡
⎢⎢⎢⎢⎢⎢⎣
ρv
ρuv
ρv2 + p
v(ρE + p
)
⎤
⎥⎥⎥⎥⎥⎥⎦
, S =
⎡
⎢⎢⎢⎢⎢⎢⎣
0
ρ fx
ρ fy
ρq̇ + ρf ·V
⎤
⎥⎥⎥⎥⎥⎥⎦
.
(3)
In addition, the following perfect gas relations are assumedto
hold
E = pγ − 1 +
12ρ(u2 + v2
), p = ρRT , a =
√γp
ρ=√γRT.
(4)
In modeling the source terms we consider the effectsof a charged
flow through the inlet with an appliedelectromagnetic field. This
can be accomplished with theaddition of appropriate electrodynamic
terms [21] to Euler’sequations. It is to be mentioned that the MHD
flows throughthe inlet are characterized as having a very low
(≈0.0001� 1) magnetic Reynolds number, Rm. (Rm = μ0σuL.
Highmagnetic Reynolds numbers (>1) are characteristic of
fusionresearch and astrophysical phenomena.)
This suggests that the conductivity in MHD flows is verylow and
therefore the current and hence the induced electricfield are also
very small. This allows us to assume B to beconstant; therefore
∇× E = −∂B∂t≈ 0. (5)
As a consequence of the above we may introduce a scalarelectric
potential ϕ such that E = −∇ϕ and ∇2ϕ = const.The current density
is then calculated using Ohm’s Law:
J = σ(−∇ϕ + V× B). (6)Thus we can then calculate the electric
potential for a givenmagnetic field and flow conductivity as
follows [22–24]:
∑
faces
σ(ϕN − ϕP
d
)Δs =
∑
faces
σ(V× B)Δs. (7)
Therefore,
ϕP = −(∑
faces σ(V× B)Δs−∑
faces σ(ϕN/d
)Δs
∑faces(σ/d)Δs
)
, (8)
where Δs is the elemental area. This allows us to thencalculate
the current density directly from (6), which willbecome necessary
as we implement the source terms into theequations. Note that we
neglect the effects of ion slip in thismodel.
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4 International Journal of Aerospace Engineering
With the application of a magnetic field to a chargedflow the
body forces and volumetric heating effects are nolonger negligible.
The body force term known as the Lorentzforce is given by the
vector J× B, and the volumetric heatingknown as Joule heating is
given as J2/σ . As seen earlier inthe development of the Euler
equations the contribution tothe momentum equation is strictly due
to the Lorentz forceswhere the contribution to the energy equation
is the total rateof energy addition, J · E = J2/σ + V · ( j × B),
due to bothvolumetric heating and work done by the Lorentz
forces.
In (2) the source term, S, is now given as
S =
⎡
⎢⎢⎢⎢⎢⎢⎣
0
(J× B)x(J× B)y(J · E)
⎤
⎥⎥⎥⎥⎥⎥⎦
. (9)
The left-hand side of (2) is unchanged and is hyperbolicfor Mach
numbers greater than one. The right-hand sidehowever is
unconditionally elliptic for smooth variationsof material
properties; we therefore need to implement aPoisson solver to
obtain the source terms. If however weimplement a magnetic field in
the x-y plane and consideran ideally sectioned Faraday MHD
generator such that theHall effect is neutralized, the resultant
current density is onlyin the z-direction. This greatly simplifies
the computationsas well as the implementation and is used in this
studyto investigate the feasibility of simultaneous MHD
andgeometric optimization:
V = (u, v, 0), B =(Bx,By , 0
), J = (0, 0, J),
J = σ(−∇ϕ + V× B).(10)
Therefore,⎡
⎢⎢⎢⎣
Jx
Jy
Jz
⎤
⎥⎥⎥⎦= σ
⎡
⎢⎢⎢⎣
0
0
uBy − vBx − ϕz
⎤
⎥⎥⎥⎦. (11)
For this study we set ϕz to zero which corresponds to a
shortcircuit of the electric field.
Thus, the source terms are then trivially found as
followswithout the need for a Poisson solver:
S =
⎡
⎢⎢⎢⎢⎢⎢⎣
0
σBy(vBx − uBy
)
σBx(uBy − vBx
)
J · E
⎤
⎥⎥⎥⎥⎥⎥⎦
. (12)
4. Flow Solution
4.1. Numerical Approach. We employ the flux splittingmethod
developed by Liou and Steffen [25], known as theadvection upstream
splitting method (AUSM). It is a firstorder scheme that is
relatively simple to implement and yet
still has the ability to resolve shock structures. It requires
onlyO(n) operations in contrast to O(n2) operations needed fora Roe
splitting scheme (where n is the number of equations).The choice of
the scheme is motivated by the simplicity andease of implementation
for a feasibility type study. Of coursemore accurate solutions can
be obtained using higher-orderschemes. The scheme is developed as
follows: the flux vectoris split into two components, convective
and pressure terms:
F =
⎡
⎢⎢⎢⎢⎢⎢⎣
ρ
ρu
ρv(ρE + p
)
⎤
⎥⎥⎥⎥⎥⎥⎦
u +
⎡
⎢⎢⎢⎢⎢⎢⎣
0
p
0
0
⎤
⎥⎥⎥⎥⎥⎥⎦
= F(c) +
⎡
⎢⎢⎢⎢⎢⎢⎣
0
p
0
0
⎤
⎥⎥⎥⎥⎥⎥⎦
. (13)
The convective terms are propagated at the cell interfaces byan
appropriately defined velocity u and the pressure term ispropagated
at acoustic wave speeds. This leads to the twoterms being
discretized separately. For the convective termsat an interface L
< 1/2 < R,
F(c)1/2 = u1/2
⎡
⎢⎢⎢⎢⎢⎢⎣
ρ
ρu
ρv
(ρE + p)
⎤
⎥⎥⎥⎥⎥⎥⎦
L/R
=M1/2
⎡
⎢⎢⎢⎢⎢⎢⎣
ρa
ρau
ρav
a(ρE + p)
⎤
⎥⎥⎥⎥⎥⎥⎦
L/R
, (14)
where
(�)L/R =⎧⎨
⎩
(�)L if M1/2 ≥ 0,(�)R otherwise,
M1/2 =M+L + M−R .(15)
The Mach number splitting method for the left and rightstates
utilizes Van Leers definitions as follows:
M± =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
± 14(M ± 1)2 if |M| ≤ 1,
12(M ± |M|) otherwise.
(16)
Similarly for the pressure terms,
p1/2 = p+L + p−R , (17)where the pressure splitting is weighted
using the second-order polynomial of the characteristic speeds (M ±
1)2 as
p± =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
1
2p(M ± 1)2(2∓M) if |M| ≤ 1,1
2p(M ± |M|)/M otherwise.(18)
This splitting of the advection and pressure terms allows forthe
complete definition of the inviscid flux vector.
4.2. Grid Generation. The grid chosen for this study isa simple
algebraic style grid with a Thomas Middlecoffcontrol function
applied for smoothing. The algebraic grid
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International Journal of Aerospace Engineering 5
utilizes uniformly spaced grid points in the x-direction and
atransformation in the y-direction which allows for clusteringof
the grid points near the wall boundary [26]:
x = x,
y = 1− ln[{β + 1− (y/h)}/{β − 1 + (y/h)}]
ln{(β + 1
)/(β − 1)} 1 < β
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6 International Journal of Aerospace Engineering
Flow
α2
α1
1
2
3
4
(a) Initial inlet configuration with shock canceling
Inlet flow mach number contours
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Y(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
X (m)
α1 = 2.882 degα2 = 9.342 degU =Mach 14
6
7
8
9
10
11
12
13
(b) Mach contours for the shock canceled configuration
Figure 4: Full inlet demonstrating initial
conditions—Shock-canceled case.
Table 1: Shock-canceled inlet flow conditions.
Region Machnumber
Flowangle
Pti/Pt j Pt/Pt∞
1 14.000 0.0 1 1
2 12.107 2.882 0.927 0.927
3 8.953 9.342 0.669 0.620
4 6.385 0.0 0.626 0.388
Table 2: Summary of initial and final inlet conditions.
α1 (deg) α2 (deg)Pressurerecovery
Initial condition 2.882 9.342 0.35
Optimized value 4.129 7.976 0.617
Korte optimized values 4.263 7.621 0.625
one region to the other across the inlet. The
correspondingangles α1 = 2.882 degrees and α2 = 9.342 degrees and
atotal pressure recovery of 0.388 are thereby obtained.
Theresulting flow solution described here gives a maximum
totalpressure recovery of 0.35 for the above angles, as compared
tothe result of 0.372 from Munipalli [2] and 0.393 from Korte[1].
This is a reasonable result when considering the firstorder
accuracy of the flow solver and the much simplifiedgrid used in
this case.
We note that the results of this validation, given in Table 2are
consistent with those of Korte and Auslender. The initialand final
configurations are summarized in Table 2. Figure 5is a plot of the
Mach number contours for the final optimizedconfiguration.
Inlet flow match number contours
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Y(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
X (m)
α1 = 3.9041 degα2 = 7.265 degU =mach 14
7
8
9
10
11
12
13
Figure 5: Final optimized inlet configuration Mach
numbercontours.
6. Optimization Results for SimultaneousIonization Beam Angle
and GeometryChanges Applied to the Cowl Style Inlet
Buoyed by the confidence in the optimization procedureand the
flow solution after validation against publishedresults (see
earlier section), the procedure was applied tosimultaneous
optimization of the ionization beam angle and
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International Journal of Aerospace Engineering 7
Inlet flow mach number contours
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Y(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
X (m)
α1 = 2.2 degα2 = 8.9 degU =mach 14
6
7
8
9
10
11
12
13
Figure 6: Mach contours of optimized cowl inlet.
Inlet flow mach number contours
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Y(m
)
0 0.5 1 1.5 2 2.5 3 3.5 4
X (m)
α1 = 3 degα2 = 9 degU =mach 14
6
7
8
9
10
11
12
13
Figure 7: Mach contours for off nominal cowl inlet.
the ramp geometry. This section summarizes the resultsobtained
for the Cowl style inlet (the main focus of thestudy). The results
of these trials are shown in Figure 6 whereα1 = 2.2 deg, α2 = 8.9
deg, and the mass capture equal to6.1965 kg/s.
To simulate a less than design Mach number flow weadjust the
ramp angles to α1 = 3.0 deg and α2 = 9.0 degwhich results in a mass
capture of 5.8757 kg/s. See Figure 7for the Mach contours in this
case.
Given this off-nominal design condition, we investigatethe
ability of an applied magnetic field to direct theflow back to the
optimal mass capture configuration. Two
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
Mas
sca
ptu
re
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 teslaB = 0.25B = 0.5
(a) Coincident e-beam ionization
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2M
ass
capt
ure
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 teslaB = 0.25B = 0.5
(b) Stationary ionization
Figure 8: Mass capture for inlet with conductivity σ = 0.5
mho/m.Results are shown for (a) coincident ionization and (b)
stationaryionization.
different scenarios are considered: (1) a moving e-beamtype
ionization method [19] such that the magnetic fieldand ionization
region are movable and coincident and (2)a moving magnetic field
but stationary-ionized region [16](Figure 3 demonstrates each
scenario).
Figures 8, 9, 10, and 11 show the ability of an appliedmagnetic
field to direct the flow back to the optimal masscapture
configuration. It is clear from these results thatthe larger the
magnetic field strength and the larger theconductivity, the greater
the influence on the flow. It isinteresting to note that the
stationary-ionized zone has
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8 International Journal of Aerospace Engineering
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
Mas
sca
ptu
re
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 teslaB = 0.25B = 0.5
(a) Coincident e-beam ionization
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
Mas
sca
ptu
re
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 teslaB = 0.25B = 0.5
(b) Stationary ionization
Figure 9: Mass capture for inlet with conductivity σ = 1.0
mho/m.Results are shown for (a) coincident ionization and (b)
stationaryionization.
a much greater ability to manipulate the flow than
thecoincident-ionized region. For the largest values of B and
σ(line corresponding to B = 0.5 in Figure 11), we can see thatthe
mass capture is indeed approaching that of the
optimalsituation.
As mentioned before, based on the results from abroad parametric
study, the problem and the optimizationprocedure was set up. To
account for the sensitivity of theoptimization routine several
different initial conditions wereevaluated to see how well the
results converged. Finally,the initial conditions for the angle θBi
were chosen above
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
Mas
sca
ptu
re
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 TeslaB = 0.25B = 0.5
(a) Coincident e-beam ionization
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2M
ass
capt
ure
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 TeslaB = 0.25B = 0.5
(b) Stationary ionization
Figure 10: Mass capture for inlet with conductivity σ = 1.5
mho/m.Results are shown for (a) coincident ionization and (b)
stationaryionization.
Table 3: Table of optimized magnetic field angle for
coincidentionized zone.
θBi below optimal θBi above optimal
Initial condition (deg) 100 170
Optimized value (deg) 141 165
and below the approximate optimal values given by theparametric
study. Tables 3 and 4 summarize the resultsobtained from these
numerical experiments.
Comparing the results of the optimizer with those of
theparametric study for the stationary ionized zone we can see
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International Journal of Aerospace Engineering 9
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
Mas
sca
ptu
re
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 teslaB = 0.25B = 0.5
(a) Coincident e-beam ionization
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
Mas
sca
ptu
re
0 20 40 60 80 100 120 140 160 180
Magnetic field angle (deg)
B = 0 teslaB = 0.25B = 0.5
(b) Stationary ionization
Figure 11: Mass capture for inlet with conductivity σ = 2.0
mho/m.Results are shown for (a) coincident ionization and (b)
stationaryionization.
Table 4: Table of optimized magnetic field angle for
stationaryionized zone.
θBi below optimal θBi above optimal
Initial condition (deg) 80 140
Optimized value (deg) 108 111
a very nice correlation. The spread in the optimized resultsis
not large and is consistent with the parametric study.However in
comparing the optimizer results with those ofthe parametric study
in the case of the coincident ionization
5.75
5.8
5.85
5.9
5.95
6
6.05
6.1
Mas
sca
ptu
re
120 130 140 150 160 170 180
Magnetic field angle (deg)
B = 0.5 tesla
Flow field results for a constant conductivity
Figure 12: Zoom in of peak values for the moving ionization
zone.
zone we see a bit of discrepancy. There is a large spread inthe
values given by the optimizer depending on whether webegan the
optimization above or below the peak value shownin the parametric
study.
Additional Remarks. It is to be mentioned that a limitation
ofthe current study is that it ignores the specific effects of
powerdeposition due to the electron beam. The optimizationproblem
was solved with the intent of finding a solutionfor the most
optimistic scenario and as a result the powerdeposition was ignored
to keep the simulation complexitylow. To model the aspect of power
deposition [28], it wouldrequire one to include a more
sophisticated nonequilibriumtemperature model that includes
distribution of energy dueto particle collisions between the e-beam
particles and thegas molecules. Additionally, delays in the energy
distributionneed to be accounted for. It is also to be mentioned
thatthe power deposition due to the e-beam is a function of
thelocation where the e-beam is applied and the local
pressureconditions. As such, while the incorporation of the
powerdeposition definitely strengthens the overall analysis, it
doesnot take away from the fact that there is some
optimizationachieved within a very optimistic scenario of reduced
energylosses vis-a-vis the neglected energy losses due to the
powerdeposition.
In order to better understand this result we conductedanother
parametric study around the peak value. We limitedthe study to the
B = 0.5 Tesla and σ = 2 mho/m case andvaried the magnetic field
angle from 120 to 180 degrees. Ascan be seen in Figure 12 the
results of this study show that thecurve is not very smooth and
this in some measure explainsthe large range in values given by the
optimizer for thiscase.
Finally, Figure 13 shows the outcome of a
simultaneousoptimization of the geometry and the ionization beam
angle.In reality, while this would require actuating the ramp
-
10 International Journal of Aerospace Engineering
60
70
80
90
100
110
120
130
140
150
160
170
An
gle
(deg
)
0 20 40 60 80 100 120 140 160 180
Iteration
3.5
3.55
3.6
3.65
3.7
3.75
3.8
3.85
3.9
3.95
4
Fob
j(s
cale
d)
α1α2
θEFobj
(a) Magnetic field angle history
0
1
2
3
4
5
6
7
8
9
10
An
gle
(deg
)
0 20 40 60 80 100 120 140 160 180
Iteration
3.5
3.55
3.6
3.65
3.7
3.75
3.8
3.85
3.9
3.95
4
Fob
j(s
cale
d)
α1α2
θEFobj
(b) Geometry angle history
Figure 13: Optimization design history for (a) the magnetic
fieldangle and (b) geometry angles.
surfaces as well as an ionization method and a way togenerate
the magnetic field, it is also represents the possibilityof fine
tuning the magnetic field. Again the off-nominal caseof α1 = 3.0
deg and α2 = 9.0 deg was used with the initialmagnetic field angle
of 70 deg. We limited this study to thestationary ionization zone
with a conductivity of 2 mho/mand a magnetic field strength of 0.5
Tesla.
Figure 13 shows the design history of the optimizationroutine at
each iteration of the inside loop in Figure 14.The outside loop
iterations or number of times that theHessian was updated is equal
to the total number of flowsolver iterations divided by the number
of design variablesplus 1 (iter/(i + 1)). From Figure 13 we can
tell that therewere approximately 170 iterations of the flow solver
and 3design parameters, giving us approximately 43 updates of
theHessian. The Fobj values plotted are scaled as per the
scalingfunction mentioned earlier.
Initial designvariables, xo
Grid generation
Flow solver
Calculate Fobj
Minimum?Yes
No
Return
Perturbvariable Δxi
For i = 1 → nn =number of design
variables
Calculate gradient ΔFobji
Determine search direction,α∗
Determine distance step size
Update design variables
Update hessian
Figure 14: Flowchart of optimization routine.
Table 5: Table of optimized geometry and magnetic field angles
forstationary ionized zone.
α1 (deg) α2 (deg) θB (deg)
Initial condition 3.0 9.0 70.0
Optimized value 2.14 9.11 110.5
The final configuration for the geometry is very close tothat of
the optimal mass capture with no MHD source termpresent (see Figure
6), and the final magnetic field angle isconsistent with that of
the previous case providing a highlevel of confidence in our
procedure and implementation.The results are also summarized in
Table 5.
7. Summary and Conclusions
This paper studied the numerical optimization of a doubleramp
cowl style scramjet inlet using magnetohydrodynamic(MHD) effects
together with inlet ramp angle changes. AnAUSM-based flow solver
was utilized to solve the 2D inviscid,compressible Euler equations
subject to MHD source terms.The objective function in the
optimization was the mass
-
International Journal of Aerospace Engineering 11
capture at the throat of a cowl style inlet, so that
spillageeffects for less than design Mach numbers are reduced.The
optimization procedure implemented in this study wasa numerical
Broyden-Flecher-Goldfarb-Shanno- (BFGS-)based procedure. Numerical
validation results compared topublished results in the literature
have been summarized atvarious stages that include flow solution
and the optimiza-tion procedure. It is shown that spillage
occurring from off-nominal geometries can be reduced by employing
MHDcontrol. We also demonstrate a more attractive case ofspillage
reduction employing simultaneous optimization ofthe ionization beam
angle and the ramp angles.
Nomenclature
α1 & α2: Ramp angles (degrees)B: Magnetic field strength
(Tesla)θ: Direction of the e-beam measured from x-axis.
(degrees)ρ: Fluid density (kg/m3)ρu, ρv, ρw: Fluid momenta in x,
y, z directions (kg/m2/s)p: pressure (N/m2)E: Energy (Joules)V:
Velocity (m/s)f : External force (N)U : Flow solution vectorF,G:
Flux vectorsS: Source termγ: Ratio of specific heatsT : Temperature
(◦K)R: Gas constant (Nm/kg/◦K)a: Speed of sound (m/s)Rm: Magnetic
Reynolds numberμ0: Permeability of free space (Wb/A/m)σ : Gas
conductivity (mho/m)L: Characteristic length (m)E: Electric field
intensity (N/C)ϕ: Electric potential (Nm/C)J: Current densityPti:
Total pressure at the ith stationFobj: Objective functionλ:
Lagrange multipliersC(x): Constraint function.
Acknowledgment
The authors gratefully acknowledge HyperComp Inc. (POC.Dr.
Ramakanth Munipalli) for the financial and technicalsupport for
this work.
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