CombinedMagnetohydrodynamicandGeometricOptimization ...ﬂow control methodology in this paper compares two styles of ionization. The ﬁrst is inspired by the work from Shneider et

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

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

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.


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


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


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



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.


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


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


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


B = 0 teslaB = 0.25B = 0.5


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


B = 0 teslaB = 0.25B = 0.5



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


B = 0 TeslaB = 0.25B = 0.5


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


B = 0 TeslaB = 0.25B = 0.5



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


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


B = 0 teslaB = 0.25B = 0.5


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


B = 0 teslaB = 0.25B = 0.5



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


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


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


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.

References

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