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Carlo A. Borghi*, Mario R. Carraro¶ and Andrea Cristofolini§

Department of Electrical Engineering of the University of Bologna, Viale Risorgimento, 2, I 40135, Bologna, Italy

ABSTRACT

A model for the analysis of the magneto-plasmadynamic regime, devoted to MHD systems in a hypersonic blunt body, is presented. In the model the discrete formulation of fluid dynamics and electrodynamics are coupled. The Navier-Stokes equations are discretized by means of a finite volume formulation. The electrodynamics is discretized by means of a finite element method. The model has been utilized for the analysis of the MHD interaction in the boundary layer of a test body in a wind tunnel. The activity is aimed to the design of the experimental activity in the frame of an ASI research project on the MHD interaction in hypersonic flows. Calculations has shown that MHD interaction can be strongly weakened by the effect of the Hall current.

INTRODUCTION

The concept of hypersonic flight has recently received an increasing interest. Hypersonic flight has been usually linked to the re-entering of space vehicles in the atmosphere. A new wide range of applications of hypersonic flight is related to the development of a single stage to orbit (SSTO) vehicle, that is now one of the main prerequisite for the commercial exploitation of space flight. An SSTO vehicle should be powered by air breathing engines: a turbojet for low speed operations, such as take off and landing, a ramjet for vehicle Mach numbers between 4 and 7, and finally a scramjet that could accelerate the vehicle up to Mach 20 in the upper layers of the earth atmosphere. Many applications of the MHD interaction in hypersonic vehicle has been proposed and discussed1. An MHD system has been proposed to control the fluid dynamics at the inlet of the scramjet; the Russian AJAX project, MHD techniques are utilized to bypass kinetic energy of the working fluid from the supersonic diffuser to the nozzle, reducing the flow velocity in the combustion chamber to acceptable value, even for high vehicle Mach numbers. Another interesting opportunity is to control boundary layer phenomena, shock waves and

heat fluxes by means of MHD interaction. In Fig. 1 the basic concept of the MHD interaction for the control of flow regime in the boundary layer of an hypersonic vehicle is shown. The current I flowing in the conductor located under the leading edge of the body generates a magnetic flux density which, interacting with the gas flowing with velocity X, produces an electromotive force Xu%. If the gas presents some ionization degree, the electromotive force generates a current density -which, interacting with the magnetic flux density, originates a body force -u% on the gas, opposite to the flow direction. Ionization methods vary according to the hypersonic regime. For moderate Mach numbers (M<7), a suitable ionization degree can be reached by injecting in the gas an alkaline element, K or Cs, with a low ionization potential. For intermediate regime(7<M<12), a strong MHD interaction can be obtained by applying RF or electron beam to excite the gas molecules. For highly hypersonic regimes (M>12), a suitable ionization degree is obtained solely relying on the fluid dynamic conditions. In this paper, a two dimensional model time dependent numerical model is utilized for the preliminary calculation of the magnetoplasmadynamic regime around a blunt body. The aim of the analysis is to get an insight on the effect of the electrical configuration of the blunt body on the MHD interaction. The result obtained will constitute an essential knowledge for the design and the optimization of the experimental activities to be performed in the frame of the ASI research project ‘MHD interaction in hypersonic flows’

PHYSICAL MODEL

The fluid-dynamic equation are given by the continuity equation for mass, momentum and energy, and by the state equations of gas as follows:

4)8 =⋅∇+∂∂W ,

(1)

),();,( 7HH7SS LL ρρ == (2)

where ρ is the specific mass and ei is the specific internal energy of the fluid. The quantities 8, ), 4 are defined as:

_*¶

§

34th AIAA Plasmadynamics and Lasers Conference23-26 June 2003, Orlando, Florida

AIAA 2003-3761

Copyright © 2003

________________ Full professor, AIAA member (ca.borghi@mail.ing.unibo.it) Ph.D. student (mario.carraro@mail.ing.unibo.it ), Associate professor (andrea.cristofolini@mail.ing.unibo.it)

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by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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σρ

ρρρ

2

0

,,

-7NHS

H LL

%-4XX

,XXX

)X8 ×=∇−⋅−

−+== .

The physical model of the electrodynamics has been obtained assuming the condition Rem <<1. The governing equations can be obtained from the Maxwell equations and the generalized Ohm’s law:

ϕ−∇=⇒=×∇ (( ,(1)

0=⋅∇ - , (2)

( ) ( )2%% LHH

%%-%-%X(- ××+×−×+= βββσ ,(3)

where βe and βi are the Hall parameter electrons for and ions respectively An explicit two dimensional model has been developed for the solution of the magnetohydrodynamic problem2.Referring to Fig. 1, an x-y plane is utilized as calculation domain. In the reference system fixed with the body, the gaseous medium flows with hypersonic velocity X lying on the x-y plane, while z-component of the velocity is neglected. The magnetic flux density %generating the MHD interaction lies on the x-y plane too. Thus the electromotive force Xu% is directed along the z-axis. The current density - has a component along the z-axis and an Hall component on the x-y plane. At each time step the space discretization is performed. A cell-centered finite volume formulation has been adopted for the spatial discretization of the fluid dynamics3. Diffusive terms are evaluated by means of a centered scheme based on a standard Galerkin approach. Convective terms are calculated utilizing a MUSCL4 approach, in order to get a second order accuracy by a combination of upwind and centered discretization of the gradient. Turbulent fluxes are calculated by means of a Nε method. Time integration is performed utilizing an explicit scheme, based on a fourth-order Runge-Kutta method. In the given two dimensional geometry, the generalized Ohm equations can be rewritten as:

)(- += , (4)

where the conductivity tensor has the following expression:

−−+

+

+=

1

1

1

12

2

2

[\

[\\[

\\[[

ββββββ

ββββ

βσ

where:

%% H[[

ββ = , %% H\\

ββ =

and the forcing term ) is:

( )

−−

+=

12 [

\

[\\[ %X%X ββ

βσσ)

Electrodynamics is discretized by mean of a finite element approach based on a variational formulation5.The finite element method utilized to discretize the electrodynamics is based on the following formulation of the Poynting theorem:

∫∫ ⋅−=⋅Φ96

G9G6 -(Q- ,(5)

where 9 and 6 are respectively the volume and the boundary surface of the calculation domain. In the two dimensional calculation domain, a variational formulation is obtained:

( ) ( ) 0=⋅Φ∇−Φ ∫∫ΩΓ

G6GO -- ∂∂ , (6)

where Ω and Γ are the two dimensional domain and the boundary of it respectively. The calculation domain is discretized and in each volume element Φ is expressed by means of the shape functions 1 :

( ) Φ=Φ 71\[, . (7)

Due to (21), from (1) and (3) the following matrix relations can be written:

[ ] Φ∇−= 1( ,

(8.a)

[ ][ ] )1- +Φ∇−= . (8.b)

Introducing (8.a) and (8.b) in (6), a matrix equation for each element of the domain is derived:

[ ] )- 33. +−=Φ . (9)

The matrix [ ]. is the conductivity matrix, defined as follows:

[ ] [ ] [ ][ ]G6. 11 7

∇∇= ∫ . (10)

The vector -3 is the contribution of the externally

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imposed current:

∫Γ

= GO-13 Q- , (11)

where -Q is the current density component normal to Γ.

The vector 3) represents the effect of the interaction

between the flow and the magnetic flux density:

[ ] G6)13 7

) ⋅= ∫Ω

.(12)

The solution6,7 of the steady state electrodynamic model at the nWK time step is utilized for the (n+1)WKtime step in the time dependent fluid dynamic equations.

APPLICATIONS

The model described has been utilized to analyze the magnetohydrodynamic field over a blunt body in hypersonic regime. The dimension of the considered body are shown in fig. 1, where the position of the conductors are shown as well. The body considered boards a magnetic circuit constituted by a ferromagnetic material. This allows to reduce the current needed to generate a given magnetic flux, and gives some degree of control on the distribution of the magnetic flux density lines. The conductor windings are deployed so that the total current flowing in the section numbered 2 in fig. 2 is equal to the sum of the currents flowing in the sections 1 and 3.

15 cm

10 cm

Ferromagnetic material

1

2 3

x

y

)LJ6FKHPDWLFRIWKHEOXQWERG\FRQVLGHUHG The distribution of the magnetic flux density produced by the described arrangement is shown in fig. 2. When the current I2 is equal to 52 KA, the magnetic flux density is about 1 T on the surface of the wedge, with a maximum of 2.29 T .

)LJ0DJQHWLFIOX[GHQVLW\DURXQGWKHEOXQWERG\

The wedge can be arranged in a differential configuration, shown in fig. 3. In this arrangement, the magnetic flux density, being confined in the ferromagnetic core, will be almost null in the lower boundary layer of the wedge. As a consequence, the forces caused by the MHD interaction will act only on the upper side of the wedge.

Ferromagnetic material

1

2

3

MHD interaction

No MHD interaction

)LJ6FKHPDWLFRIWKHEOXQWERG\FRQVLGHUHG In order to carry out the experimental activity, a wind tunnel facility, located at the Centrospazio laboratories, Pisa, Italy, capable to produce a hypersonic flow up to 7 Mach, will be utilized. Thus, the MHD regime around the blunt body subject to a Mach 7 air flow is analyzed. A free stream flow with a 76 Pa pressure and a 10-3 Kg m-3 density are is considered, in order to simulate the condition of a vehicle at a 50 Km altitude. The calculation of the flow condition around the blunt body has been performed on a 26,000 nodes unstructured triangular mesh. The calculated mach isolines, isobars and isothermals distribution when no MHD interaction is present are shown in fig. 4.a, 4.b, 4.c respectively.

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a) a)

b) b)

c) c)

)LJ 0DFK LVROLQHV D LVREDUV E DQG LVRWKHUPDOVFZLWRXWK0+'LQWHUDFWLRQ

)LJ0DFK LVROLQHV D LVREDUV EDQG LVRWKHUPDOVFZLWK0+'LQWHUDFWLRQQHJOHFWLQJWKH+DOOSDUDPHWHU

In order to estimate the conductivity and Hall parameter, a 50 T-1 constant electron mobility has been assumed. This value corresponds roughly to the mobility in the boundary layer region. Assuming a ionization degree of 10-3, the higher value of the conductivity is 360 S m-1, in the denser region near the wedge walls. A first calculation has been performed in order to evaluate the effects of the MHD interaction when neglecting the hall current. The calculation results are shown in fig. 5.a, 5.b, 5.c, where mach isolines, isobars

and isothermals are plotted respectively. As can be easily noted, the evaluated MHD interaction strongly affect the flow around the blunt body. The pressure coefficient PC, defined as the ratio between the gas pressure and the free flow pressure Pin is reported along the body wall in the leading edge region, is reported in fig. 6.a, and compared with the pressure coefficient with no applied magnetic flux density. The leading edge vertex is the origin of the x axis. The reduction of the air speed caused by the specific force -u% generates a pressure rise along the

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wedge sides. The integrals of the forces along the direction normal to the wedge sides due to the wall pressure, increases by more than 100%. In Fig. 6.b, the friction coefficient FC, defined as the ratio between the viscous stress on the wedge wall τw and the free flow pressure is plotted along the body wall, with and without the applied magnetic flux density. The MHD interaction cause an decrease of the viscous stresses, as a consequence of the decreased flow velocity gradient. The integrals of the drag forces along the wedge sides is decreased by 36% due to the MHD interaction.

0

8

16

24

32

-7,50E-02 -4,50E-02 -1,50E-02 1,50E-02 4,50E-02 7,50E-02x[ m]

no B

applied B

a)

0

4

8

12

16

-7,50E-02 -4,50E-02 -1,50E-02 1,50E-02 4,50E-02 7,50E-02

[>P@

)&

no B

applied B

b) )LJ3UHVVXUHFRHIILFLHQWDDQGIULFWLRQFRHIILFLHQWE DUH FRPSDUHG ZLWK DQG ZLWKRXW WKH DSSOLHGPDJQHWLF IOX[ GHQVLW\ ZKHQ WKH +DOO SDUDPHWHU LVQHJOHFWHG The analysis has then be performed including the effects of the Hall parameter. Two electrical configurations has been studied. In the first configuration analysed, the current density normal to the wedge walls is set to zero. The second configuration, the wedge walls are set to the same electric potential. In fig. 7.a, 7.b, 7.c, mach isolines, isobars and isothermals are plotted respectively, in the case of insulating wedge walls. In this case, MHD effects are drastically reduced. In the pressure and friction coefficient plot, reported in figg. 8.a and 8.b, it can be noted that the pressure along the wedge side

weakly increases when the magnetic flux density is applied, while the friction coefficient remain largely unaffected. In this case, the variation of the integral values of normal pressure and drag force are +2% and +1% respectively. The same evidence are obtained for the shorted wedge walls. In Fig. 11 and 12, the pressure and the friction coefficient, are reported. The integrated pressure along the wedge wall increases of 4%, while the variation of the integral value of the drag force is less then +1%.

a)

b)

c) )LJ0DFKLVROLQHVDLVREDUVEDQGLVRWKHUPDOVFZLWK0+'LQWHUDFWLRQLQVXODWLQJZHGJHZDOOVFRQILJXUDWLRQ The forces per unit width due to pressure and friction stresses Ip and If can be obtained as follows:

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QQI

=

= ∫∫ GO3&33GO LQS ,

(13)

WWI

=

= ∫∫ GO)&3GO LQZI . (14)

0

4

8

12

16

20

-7,50E-02 -4,50E-02 -1,50E-02 1,50E-02 4,50E-02 7,50E-02

[>P@

3&

no B

applied B

a)

0

4

8

12

16

-7,50E-02 -4,50E-02 -1,50E-02 1,50E-02 4,50E-02 7,50E-02[>P@

)&

no B

applied B

b) )LJ3UHVVXUHFRHIILFLHQWDDQGIULFWLRQFRHIILFLHQWE DUH FRPSDUHG ZLWK DQG ZLWKRXW WKH DSSOLHGPDJQHWLF IOX[ GHQVLW\ LQ WKH LQVXODWLQJ ZHGJHFRQILJXUDWLRQ The integrals in (13) and (14) are carried on the walls of the blunt body. Utilizing the results obtained, the forces modules f’p and f’f acting on each wedge side are about 140 and 10 N m-1. The calculations carried out show that the expected variation of the forces modules ∆f’p

and ∆f’f due to the MHD interaction is in the order of some Newton and some tenth of Newton respectively. The differential configuration, shown in fig. 3, may be conveniently utilized to estimate ∆f’p. In this condition, neglecting the effect of the MHD interaction on the lower edge side, the total force per unit width fy acting on the blunt body in the \ direction is:

( ) ( ) ( )2cos’2sin’2cos’ ααα SIS\ IIII ∆−≈∆+∆−= ,

where α is the wedge angle. By measuring fy, ∆f’p. can be evaluated. The measurement of the variation of the total force in the x direction for the blunt body in the configuration shown in fig.1 can then be utilized to

evaluate ∆f’f , by means of the following expression:

( ) ( )2cos’2sin’ αα IS[ III ∆+∆=∆ .

0

4

8

12

16

20

-7,50E-02

-4,50E-02

-1,50E-02

1,50E-02

4,50E-02

7,50E-02

[>P@

3&

no B

applied B

a)

0

4

8

12

16

-7,50E-02 -4,50E-02 -1,50E-02 1,50E-02 4,50E-02 7,50E-02

[>P@

)&

no B

applied B

b) )LJ3UHVVXUHFRHIILFLHQWDDQGIULFWLRQFRHIILFLHQWE DUH FRPSDUHG ZLWK DQG ZLWKRXW WKH DSSOLHGPDJQHWLF IOX[ GHQVLW\ ZKHQ WKH +DOO SDUDPHWHU LVQHJOHFWHG

CONCLUSIONS

In this paper, a model for the analysis of MHD interaction in hypersonic regimes with low magnetic Reynolds numbers is presented. The model consists of the Navier-Stokes equations the Maxwell equations and the generalized Ohm’s law. Fluid dynamics is solved utilizing a MUSCL approach.. Electrodynamics are discretized by means of a finite element technique. An explicit scheme, based on a fourth-order Runge-Kutta method is utilized for the time integration process. The model has been applied for the analysis of the MHD interaction in the boundary layer of an hypersonic blunt body. Calculations show that MHD interaction can be significantly weakened by the Hall current phenomena.

ACKNOWLEDGEMENT

This work has been performed in the frame of the 0+',QWHUDFWLRQ LQ +\SHUVRQLF )ORZV research project, funded by ASI (Italian Space Agency).

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REFERENCES1.V.A. Bityurin, A.N. Botcharov, V.G. Potebnya, J.T. Lineberry, “MHD Effects in Hypersonic flows around Blunt Body”, 3URF RI WKH QG:RUNVKRS RQ0DJQHWR3ODVPD $HURG\Q ,Q $HURVSDFH $SSO Moscow, April 2000 2.C.A. Borghi, M. R. Carraro, A. Cristofolini and S. Domesi, “A Numerical Method for the Analysis of the MHD Interaction in the Boundary Layer of Hypersonic Vehicle”, 14th Int. Conf. On MHD Power Generation and High Temperature Technology, Maui, Hawaii, USA, Mai 2002. 3.L. Fezoui, B. Stoufflet, “A Class of Implicit schemes for Euler Simulations on Unstructured Meshes”, -RXUQRI&RPS3K\V84, pp. 174-206, 1989.

4.C. B. Laney, “Computational Gasdynamics”, Cambridge University Press, 1998. 5.C. A. Borghi, A. Cristofolini, G. Minak, "Numerical Methods for the Solution of the Electrodynamic Problem in Magnetohydrodynamic Flows.”, ,(((7UDQVDFWLRQRQ0DJQHWLFV, vol. 32, no. 3, May 1996. 6.C. A. Borghi, A. Cristofolini, P.L. Ribani, “Analysys of Magneto-Plasma Dynamic Transients in a Combustion gas Magnetohydrodynamic Generator”, 3K\VRI3ODVPD vol. 4, no. 8, August 1997. 7. C. A. Borghi , Andrea Cristofolini, "A Hybrid Implicit Numerical Method for the Analysis of the Magneto-Plasmadynamics in a Gas Discharge", ,(((7UDQVDFWLRQV RQ 0DJQHWLF 9RO pp. 3401-3404, September 2001.