numerical study of compressible viscous mhd equations with a bi-temperature model for supersonic...

b (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

flA A,

AIAA 2000-0449

NUMERICAL STUDY OF COMPRESSIBLE VISCOUS MHD EQUATIONS W ITH A BI-TEMPERATURE MODEL FOR SUPERSONIC BLUNT BODY FLOWS

P. Deb and R. Agarwal Nationa l Institute for Aviation Research (NIAR) W ichita State University W ichita, Kansas 67260-0093

38th Aerospace Sciences Meeting & Exhibit

IO-13 .Januaty 2006 / Reno, NV For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

Numerical Study of Compressible Viscous MHD Equations with a Bi-Temperature Model for Supersonic Blunt Body Flows

Prasanta Deb* * and Ramesh K. Agarwal* National Institute for Aviation Research

Wichita State University Wichita, KS 67260-0093

ABSTRACT

In recent years, the possibility of supersonic drag reduction by imposing a magnetic field on a slightly ionized plasma has received a great deal of attention. Some of the results reported in the Russian literature indicate that the shock structure in a slightly ionized gas (plasma) is significantly weaker than that in nonionized gases at the same temperature. For example, the shock standoff distance of a sphere moving at supersonic speed in a slightly ionized air heated to plasma temperature is considerably larger than in a nonionized air heated to the same temperature. Furthermore, the shock front is highly diffused, sometimes to the point of being scarcely visible besides lacking sharp boundary normally observed in photographs obtained under such conditions in nonionized gases. These concepts and others for shock wave modification/dissipation in supersonic flow are currently being investigated by the U.S. Air Force under the AJAX program. One of the concepts of interest is the effect of large magnetic field (approximately 2 Tesla or more) on supersonic tlowfield about blunt bodies to evaluate the possibility of shock wave dissipation/elimination. This paper evaluates this concept by numerical simulation.

In this paper, the effect of magnetic field on the weakly ionized flow is studied for a blunt body moving at hypersonic speeds using the compressible viscous MHD equations with a bi-temperature model. Two-dimensional MHD equations in generalized coordinates with and without a bi-temperature model are solved using a modified Runge-Kutta time integration scheme with second-order accurate spatial discretization. A symmetric Davis-Yee Total Variation Diminishing (TVD) flux limiter is employed to damp the oscillations in the shock regions. Numerical results indicate the feasibility of

* Bloomfield Distinguished Professor and Executive Director, Fellow AIAA

** Postdoctoral Research Associate 0 by the authors

supersonic drag reduction by application of a strong magnetic field on the surface of a body moving at supersonic speed in a weakly ionized gas.

Q E F x Y P 24 V

Ix

BY

P Pi Pe Yf Ye x? et G VOX

NOMENCLATURE

field vector flux vector component in the x-direction flux vector component in the y-direction X Cartesian coordinate Y Cartesian coordinate density velocity component in the x-direction velocity component in the y-direction velocity component in the z-direction magnetic field component in the x- direction magnetic field component in they- direction magnetic field component in the z- direction full pressure hydrodynamical ionic pressure electronic pressure specific heat ratio of ionic gas specific heat ratio of electron gas electron entropy total energy per unit mass speed of sound AlIven wave velocity in the x- direction Alfven wave velocity in the y-direction eigenvalue generalized coordinate parameter generalized coordinate parameter Jacobian of transformation fast wave velocity in the 5 -direction fast wave velocity in the q -direction slow wave velocity in the 5 -direction slow wave velocity in the ?j -direction inverse right eigenvector associated

with Jacobian matrix 2

American Institute of Aeronautics and Astronautics 1

- (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

right eigenvector associated with Jacobian

matrix A diagonal eigenvalue matrix associated with

Jacobian matrix 2 inverse right eigenvector associated with Jacobian matrix i? right eigenvector associated with Jacobian matrix B diagonal eigenvalue matrix associated with Jacobian matrix B flux limiter function vector associated with

flux limiter function vector associated with

flux limiter function vector component

entropy correction variable limiters magnetic permeability of the fluid electrical conductivity free-stream Mach number Reynolds number Magnetic Reynolds number Prandtl number free-stream velocity kinematic viscosity Hartmann number

standoff distance of the bow shock from the frontal surface of the sphere on the null streamline. Figure 1 shows the ratio of the experimentally determined standoff distance z? = Ap /R (A is the standoff distance and R is the radius of the sphere) of the bow shock from a sphere in a plasma to the corresponding calculated value of zr in air at the plasma temperature as a function of the velocity of the model in supersonic range from 1350 to 2300 m/s. It is evident from this figure that the standoff distance of the bow shock from a sphere with a velocity of 1400 m/s in a plasma is more than 2.2 times the classical “thermal” value for the given conditions of motion. The ratio & I& decreases as the velocity increases. The investigations of flow around a sphere in a glow- discharge plasma in argon and xenon also revealed a pattern qualitatively similar to the pattern observed in air plasma. The Toepler shadowgraph of the flow around the sphere in glow-discharge plasma showed the shock front highly diffused, sometimes to the point of being scarcely visible besides lacking the sharp boundary as normally observed in photographs obtained under such conditions in nonionized gases. It is also important to note that the experiments using duralumin, nickel, copper, teflon, and polyethylene sphere models indicated that the material from which the model is made does not affect the flow pattern around bodies in a plasma.

1.0 INTRODUCTION

In recent years, the possibility of supersonic drag reduction by imposing a magnetic field on a slightly ionized plasma has received a great deal of attention. The feasibility of supersonic drag reduction is being investigated through computation and experiment by US Air force in collaboration with NASA. The interest in this concept has originated from results reported in the Russian literature which indicate that the shock structure in a weakly ionized gas at high temperature (plasma) is significantly weaker than that in nonionized gases at the same temperature. Most often quoted are the experiments of Mishin and his co-workers at the A.F. Ioffe Physicotechnical Institute of the Russian Academy of Sciences. The results of these experiments are documented in Refs. 1-3. The details of the experiments are not given here but some of the relevant results are presented. In one of the experiments, studies of flow around a sphere were conducted in a slightly ionized gas- discharge plasma in various gases (air, argon and xenon). The measurements were made for the

Since the relative standoff distance of the bow shock from the body is a function of the Mach number and the specific heat ratio, the Mach number of a sphere in a glow-discharge plasma in air can be determined from the values measured in air plasma. This allows the determination of the “effective” sound velocity in the air plasma from known velocities of the model. The effective sound velocities obtained by this procedure are plotted as a function of the model velocity in Figure 2. Under the given experimental conditions, the acoustic wave velocity corresponding to the plasma temperature should be 717 m/s, but it is evident from Figure 2 that the average sound velocity determined from the data for flow around a sphere in a plasma is equal to 1084 m/s, that is the effective sound velocity in the plasma is higher than the thermal velocity by a factor of approximately 1.5. This disparity between the sound velocities cannot be attributed to error of measurement of the plasma temperature, because the occurrence of such a velocity of weak disturbances in heated air requires a temperature of - 3000 “K which is impossible in a glow discharge.

Several alternative hypotheses have been advanced to account for the above observed non- trivial dynamic properties of a thermal-non-



equilibrium gas discharge plasma. These hypotheses can be divided into two groups. The first group is based on the assumption that the weak disturbances propagate in the plasma at a velocity greater than the thermal sound velocity. The second group (the “energy hypothesis”) rests on the assumption that the energy is released in the shock layer after the shock front (either the release of energy is concentrated in the molecular degrees of freedom or it is a consequence of current flowing in the shock wave). The lack of substantive experimental data does not allow definitive conclusions as to the mechanism of observed phenomenon. It should be mentioned at this point however that in 1996, a series of “quick and dirty” shock tube experiments were performed using the shock tube facility at Wright Air Force Laboratories in Dayton, Ohio. The experiments measured the shock propagation velocity and shock overpressures in a glow discharge plasma. The results showed that as the plasma intensity was increased, the shock velocity also increased and the overpressure decreased, both of which indicate decreased shock strength with increased plasma intensity. When the plasma intensity reached a high enough level, the overpressure ceased to be measurable, indicating that the shock had been effectively eliminated. The degradation of shock strength, as indicated by the pressure trace, was also accompanied by a thickening of the shock front. All these observations are similar to those obtained at Ioffe Institute by Mishin et al. described earlier.

As a result of the experimental observations at Ioffe Institute in Russia and at Wright Air Force Laboratories, the Air Force has launched a combined experimental/computational program “AJAX to investigate the possibility of supersonic drag reduction by dissipation/elimination of shock waves using the plasma/MJXD flow control/modification concept. The goal of this paper is to evaluate this concept by numerical simulation.

Many experimental and analytical studies 14-71 were performed in the late 1950s and 1960s to study the effects of MHD on hypersonic flow fields. For the present study, especially noteworthy is the paper by Kantrowitz [S] wherein he proposed the use of MHD to control a weakly ionized plasma created by strong shock waves. More recently, Mnatsakanian et al. [9], Bityurin et al. [lo] and Cole et al. [l l] have studied the MHD effects on aerodynamic and propulsion system flow fields. MHD equations along with Maxwell equations essentially characterize the flow of a conducting fluid in presence of magnetic and electric fields.

The objective of the present work is to extend the previous work of Agarwal and Augustinus by including a b&temperature model to simulate the

behaviour of a weakly ionized gas [ 121. Two- dimensional compressible viscous hJHD equations proposed by Powell [ 131 with a bi-temperature model proposed by Brassier and Gallice [14] are solved using a modified four-stage Runge-Kutta (R-K) time integration scheme with second-order accurate spatial discretization. A non-Monotonic Upwind Scheme for Conservation Laws (non-MUSCL) with Total Variation Diminishing (TVD) flux limiters is used to stabilize the modified R-K scheme. The governing equations are solved in the generalized coordinates. A set of eigenvectors is developed for the implementation of the numerical scheme.

The 2-D unsteady compressible viscous MHD code WSUMHD2D reported in Ref. [12] is extended to include the bi:temperature model in the presence of magnetic field. These calculations are compared with the Navier-Stokes calculations in the absence of magnetic field and the bi-temperature model. This set of calculations is then used to investigate the concept of supersonic drag reduction by applying a magnetic field to a weakly ionized plasma.

23..

18 ..

.t44 . . . . ; , . ; ; , , .- uoo %W woo ma 2zoo .ma vx t&Y.

fig. 1 Ratio of experimentally determined relative standoff distance of the bow shock from a sphere moving through a glowddischarge plasma to the corresponding calculated value of the relative standoff distance in air at the same temperature as the plasma (b, is the relative standoff distance of the bow shock in the plasma, hp is the relative standoff distance in air at T = 1350 K, and v is the velocity of the sphere), from Ref. 1.


” (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

Fig. 2 “Effective” (governing the flow pattern) sound velocity in a plasma in the range of velocities of a spherical model 350-13 m/s (a is the effective sound velocity, and v is the velocity of the sphere), from Ref. 1.

2.0 GOVERNING EQUATIONS

The 2-D viscous h4HD equations with a bi- temperature model in the Cartesian coordinates can be expressed as

aQ aE aF -+++-t=HM at h ~JJ

aE" aF, -z+- ?Y

where Q = [P PU PV PW 4 By 4 pet P&P

E=

uB,,-v& uB, - wBx

B;+B;+Bf B w+ 1

u-- uB,+vB,+wB, 47vf (

pus,

(2.2a)

,)

(2.2b)

F: =

pv2 +p+ B, -B; +B;

87~5

B$t pw - 4ltp, vB, - uB,,

0 vB, - wB,

Pe,+P+ B;+B;+B;

87~9 v-B'(uBx+vBy+wB;

4v, pvSe

H,= B, 2 I B B uvw UB, +vB, +wB, T

0 47v, 47vJ 47w, 4w, 1

(2.2d)

(2.2e)



F, = (233

P pe, =- +pu2+v2+w2 +B:+B:+B: ( )

Y-l 2 W-Q P. 39

(2.2h)

(2.2i)

(24)

(2.2k)

(2.21)

q*=-kaT ax

(2.2m)

qu =-kE ?Y

(2.2n)

y = yi (l+b)l(l+a)

a=Pe(yi -lYPi(Y, -1)

b = aye fY*

IJI Note that the Joulean dissipation - has been 0 neglected on the right side of the energy equation (equation for pe, ).

2.1 Nondimensionalization

The viscous MHD equations are nondimensionalized as follows:

The resulting nondimensional parameters are as follows: Reynolds number:

Prandtl number: pr=.Ei k

Magnetic Reynolds number: Rem, = cyp,L

Freestream Mach number: Mm =um l&icxw

The nondimensional viscous MHD equations in Cartesian coordinates can thus be expressed as (the “*” notation for non-dimensional quantities has been dropped)

where (2.3)

Q =b pu pv pw B, B,, B, pe, pSelT (2.4a)



E=

r

PU 1 pu2+p+

-B; +B; +B,2 8n

f& PUV- 4x

BB puw-2 4n

0 uBy -vBx

s pe, +p+

(2.4b)

PV *A pVU-- 4x

BP2 pvw-- 4x

vBx-uBy F=

I 0

I 4 - wB,,

pe,+p+ B,2+B;+B,2

v-- 8n 1

~~(uB,+vB,+wB,)

p&e

(2.4~)

uB, i- vB,, + wB, T 0

4n 1

E, =

(2.4d) ’

(2.4e)

-‘lr

(2.40

(2.5a)

(2Sb)

(2.5~)

Paw 2, =-- Re, &

(2Sd)

(2.5e)

CL& zyr=-- Re, ?Y

P i3T 4x =-Re,Pr(y-l&f: %

i3T qy = - Re, Pr (y - 1)Mi -$

Gm

(2-W

(2.5h)

2.2 Generalized Coordinates

The nondimensional viscous MHD equations in generalized coordinates take the form:

@+aE+g+ at at ti - M

where (2.6)



(2.7b)

a,=&:i-$c; (2Sb)

a4 =tf +tt (2.Sd)

b, = q; + $7; (2.9b)

(2.9c)

(2.9d) (2.1Oa)

(2SOb) (2.1Oc) (2.1Od)

cs = 52-L + surly (2.1Oe) Defining the Jacobian matrices 2 andB as

shown below, the MHD equations can be written as

ai? -aQ -ap ai?, aF, dt+A-+B-=---+-,

3 % ag au (2.11)

where

A,dE+ ?!!5 aQ N aQ

B=E-, aB, aQ M aQ

(2.12a)

(2.12b)

(2.12d)

(2.12e)

3.0 NUMERICAL METHOD

3.1 Eigenvalues and Eigenvectors

The numerical scheme employed for solving the equation (2.1) requires the determination of eigenvalues and eigenvectors of the Jacobian matrices. In this section the eigenvalues and eigenvectors are provided. The eigenvalues of the Jacobian matrix 2 are %I< = 59 + tyv (3.la)

hl, =5x(Ufv,)+5r(v+%J (3.lb) A,, = &u + &v f vx (3.lc)

h,* = 59 + 5,v +_V& (3.ld) 1, = 5,u +5yv (3.le)

A,; =b +5yv (3.W (3.le)

where vi =+[a4(v,2 +cz)+.z4]

vi = f[a4(vz +cf)-z,] (3.2a)

(3.2b)

zI = &in (3.2~)

(3.2d)

(3.2e)

(3.20



(3 .%I

vf =v; +I$ +vi (3.2h)

The diagonal eigenvalue matrix, D, = LIzRR5 then becomes

h 0 it

0 0 0 0 00 0 h,, 0 0 0 0 0 0 0

0 0 a, 0 0 0 0 0 0 0 0 0 ii.,, 0 0 0 0 0

D<= 0 0 0 0 AA- 0 0 0 0 0 0 0 0 0 &+ 0 0 0 0 0 0 0 0 0 A+ 0 0 0 0 0 0 0 0 0 h, 0

00 0000 0 0 “AS

(3.3) where

Details of (3.4a) and (3.4b) are provided in Ref. 15. Similarly, the eigenvalues of the Jacobian matrix

F are

b, =TlP+~(yv (3.5a) h q*‘= rlx(fJ eJ+~y(.f’J (3.5b) h J$* =rlxu +rl,v+vfrl (3.5c) h sl)* =TP +rl,v*v, (3.5d)

kiq =r,*+rlyv (3.5e)

hq = rlxu +rl,v (3.59

where v; =+[b‘& +c,2)+zJ (3.6a)

vi =&‘&,2 +&J (3.6b)

z,, =&-I- (3.6~)

The diagonal eigenvalue matrix, D,, = I,,$$

then becomes

D, =

h a1

0 0 0 0 0 00 0 0 Lw 0 0 0 0 0 0 0 0 0 h+ 0 0 0 0 0 0 0 0 0 h, 0 0 0 0 0 0 0 0 A* 0 0 0 0 0 0 0 0 0 “- 0 0 0 0 0 0 0 0 0 O&O 0 0 0 0 0 0 0 0 h, 0

00 0000 0 0 h,

where (3.7)

R, = bo ‘+%, r-vm ‘+;Y,,, %, r+v,n tv,, ‘b,, %, 1

(3.8a)

(3.8b)

The right and left eigenvectors for the Jacobian matrix B are obtained by substituting q for 5 in relations (3.4a) and (3.4b).

3.2 The Modified Runge-Kutta (R-K) Time Integration Scheme

The numerical scheme employed for time integration is the modified R-K scheme. The R-K formulation for the 2-D viscous MKD equations in the generalized coordinate system is as shown below:

(3.9a)

(3.9b)

(3.9c)

(3.9d)

where

The following finite difference approximations are used for the viscous terms:

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- @i,j + Li-l,jxIwi,j+l +“i-l,j+l -“i,j-L -“i-l,j-l)l

(3.11c)

CM i+I,j+l + Mi+*,j -‘“i-l,j+l - Mi-l,j)

- (4, j + Li,j-l )@i+l, j + Mi+l,j-l - Mi-l,j - Mi-l,j-l)l

(3.lld) Central differencing is employed for the computation of metrics.

3.3 Total Variation Diminishing (TVD) Models

The modified R-K scheme described above becomes unstable for solving Riemamr-type problems because it employs central differencing of the convective terms, It is therefore necessary to add an artificial dissipation term to stabilize the scheme. The artificial dissipation term used in this research is based on the Total Variation Diminishing (TVD) model.

The TVD flux limiters are incorporated in the numerical scheme in a post-processor approach. The post-processor approach can be easily applied to many single and multi-step explicit schemes by adding an extra step to the original scheme. This approach becomes convenient because no modification is required for the existing numerical scheme. The addition of TVD model to equation (2.21) in the generalized coordinate system can be expressed as:

(3.12)

where Rc and R, are given in Section 3.1. The flux limiter functionvectors B’S and Q,, are:

(3.13b)

The vector elements of (2.25a) and (2.25b) are termed as the flux limiter functions and are given in the following section. Subscripts i+t, j and i -+, j denote that all other related variables are calculated based on the variables (p, U, v, w, B, , By, B, , P, 5, , 5,) r, and rly averaged at grid points (i + 1, j), ( i, j) and (i, j), ( i - 1, j) respectively.

3.4 Davis-Yee Second-Order Symmetric TVD Model ”

As an example of a TVD flux limiter, we describe here the Davis-Yee second-order symmetric TVD model. There are many models that have been proposed in the literature. WSUMHD2D has several TVD models, e.g., Harten-Yee model, Roe-Sweby model, Davis-Yee model, etc. The detailed description of these models is given in Ref. 16. Our calculations in 1-D and 2-D have shown the Davis- Yee TVD model to be most promising.

For the Davis-Yee symmetric TVD model, the lti symmetric flux limiter function is given by:

(3.14a)

where h’ ( > I i+i,j arId (h:, )i,j+l are the P eigenvalues 2

of the diagonal matrices D, and D,, . The function cp(.z) is an entropy correction to 2 with

(p(z) = i I4 I+-”

(z2 + S2)/26 )zI < s (3.15)

where 0 < 6 < 0.125 . However in the case of blunt- body steady-state calculations for M, > 2.5, slight modification is needed to the 6 term, otherwise



nonphysical solutions are likely to occur. The body problem, the minimum time step in the modification introduced is computational domain is obtained by :

(3.17a) (3.17b)

vfi and vfi are given in equations (3.2a) and (3.6a)

respectively and 0.5 < s < 0.7. The Davis-Yee symmetric limiters are:

Cd : i+$/ = n-&mod

ld

(3.18a) 1 rl i,j+t

= minmod

(3.18b)

Le =R;’ (3.19c) L, = q;’ (3.19d) minmoc(a,b,c ,... n)=S.maxtO,min(a,S.b,~.c ,... Jan)] S = sgn(a) . Relations (3.19c) and (3.19d) are given inRef. 15.

3.6 Local Time Step (LTS)

For a steady state solution it is advantageous to use the local time stepping to enhance the convergence. The main idea behind the LTS is to march the solution at different time steps at each individual grid point without violation of the stability condition, which is CFL 2 1 .O . Each time step is updated at every iteration. For the symmetric blunt

where h, and h, are the eigenvalues defined earlier in Section 3.1.

4.0 RESULTS AND DISCUSSION In this section, we present some 1-D and 2-D

calculations performed with WSUMHD2D to validate the code. Additional validation cases have been reported in Refs. 15, 16 and 18.

4.1 MHD Shock Tube Problem

WSUMHD2D code was employed to calculate the flow in a shock tube in the presence of magnetic field. Brio and Wu (Ref. 17) proposed this problem as a benchmark for studying the numerical schemes for the solution of inviscid compressible MHD equations. Over the years, this problem has been computed by several investigators using a variety of numerical methods, We have computed this problem for the parameters employed in Ref. 17 as shown in Figure 3. Computations have been performed for both the inviscid and viscous MHD equations using both the seven-wave and eight-wave models as reported in Refs. 16 and 18 respectively.

Flow conditions on L.H.S.: ~,p,v,B~,By,BJL =(I,1,0,0.75-&4&0)

Flow conditions on R.H.S.:

Computational parameters: At=0.2 and t =80.

y = 2.0 ) Ax = 1.0 )

Diaphram ruptured at t = 0 s.

High Pressure Side

Low Pressure Side

L=SOO

x=0 (i=l) x=800 (i=IM(801))

Figure 3: Magnetic shock-tube


v(c)2OOO American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

Figures 4-8 show the calculated values of the density, pressure, u-velocity, v-velocity and magnetic field along the shock tube at t = 80 sets. It should be noted that these results had excellent agreement with those reported in Refs. 16-18, and there was hardly any difference between the inviscid and viscous results. These calculations employed Davis-Yee TVD model.

1.2

1.0

0.8

2 0.0

0.4

02

0.0 j I , I , I I I 1

\ \ FR EC

\II al FRz Fastmode

SC: Slow mode

CD: Contact discontinuity SS: Slow shock

0 200 404 600 000 x

Figure 4: Computed density distribution.

Figure 5: Computed pressure distribution.

-I -0.3 ’

I I I 0 200 400 600

x

Figure 6: Computed u -velocity distribution.

0.2

0 200 400 600 x

Figure 7: Computed v -velocity distribution.

a; 0

000

\ a.0

3.0 \

-4.0 ’ , I , I 8 I I 0 200 4w 600

x

Figure 8: Computed By distribution.

10



This benchmark problem provides an excellent demonstration of the ability of-the numerical scheme to compute the local degeneracy considering the non- strict hyperbolicity of the MHD equations. As shown in Figs. 4-8, sometime after the diaphragm is ruptured, a slow mode compound wave (SC) is formed which consists of a slow mode shock and slow mode rarefaction wave which degenerate when B,, = 0 . The other discontinuities that are present are two fast mode rarefaction waves (FR), a contact discontinuity (CD), and a slow shock (SS). The discontinuity nomenclatures are displayed in the density plot in Figure 4.

4.2 Simulation of Hartmann-Poiseuille Flow

WSUMHD2D was employed to compute the fully developed incompressible viscous Hartmann- Poiseuille channel flow for which the analytical solution is available for comparison. The computational domain and the calculation parameters employed are shown in Figure 9. Computations were performed on a 195 x 195 uniform mesh.

Figure 9: Channel configuration

Inlet conditions p, =1.225kgme3 p, =101325.0 Pa T, = 288.203 IS v, = 0.0 m 5.’ w, = 0.0 ms-’

( 1 = 1.4494x lo-* T

(:)- =O.OT Frim Sutherland’s law,

Exit conditions p, =1.225kgm’j p,=101324.0 Pa T, = 288.200 K

y = 0.0 ms*’ w, = 0.0 m s-l

( 1 B y =1.4494x10-’ T e

(BEle = O.OT

Pm = 1.458~lodT;~

T, +llQ4 kgm’ So

Pe = 1.458x10dT,‘.’

T, +110.4 kg m“ s-l

)1~=47cxlO-~H me’ , o = 1000.0 (Q m)-’ ,

pG=Pm-Pe -025Pam” -- . s , L = 0.5 m .

Rem, = opp,L Rem, = opp,L The analytical solution is given by:

In Figures 10 and 11, a comparison of computed channel flow velocity u and the induced magnetic field B, respectively is shown with the analytical solutionatx=Ofor Ha,=54.17, M, =0.19, Re, = 2206381.65, and Rem, = 0.041.

I-

1 -0.60 Figure 10: u velocity along y -axis at x = 0.

0.5 0



0.wE+

-3.mE.

:I

-6.mE-

-9mE-

-0.60 -0.40 -0.20 o.m I 020 y(m)

Figure 11: Induced magnetic field B, at x=0.

0.40 0.60

along y -axis

Excellent agreement is obtained. It should be noted that the analytical solution is for incompressible flow while the computations were performed for low speed Mach number of 0.19 (WSUMHD2D is a compressible code).

4.3 Numerical Solutions for a Blunt Body

Calculations were performed for hypersonic blunt body flow in a weakly ionized gas to investigate the influence of magnetic field (generated by magnets on the body surface) on shock standoff distance and other flow variables.

As mentioned before, the simplest mathematical model that can simulate the weakly ionized plasma is the so-called bi-temperature model in which the gas is assumed to consist of electrons and ions each with its own translational temperature, In the calculations reported here, we have employed the model described in the paper by Brassier and Gallice [ 141.

Figure 12: Blunt body configuration.

The blunt body configuration employed in the calculation is shown in Figure 12. It is symmetric about the x-axis. The outer most boundary of the computational domain is an ellipse with its center at point B. The major and minor axes are 0.26 m and 0.1559 m respectively.

A 320x121 mesh was employed in the computational domain. Flow conditions are given in Table 1.

Table 1 Free stream density Free stream

) 4.3376 x low4 kg/m3 ( 36.6 N/m2

hydrodynamic pressure 1 Free stream electronic ( 36.6N/m2 pressure Ion gas specific heat ratio 1.4 Free-stream Mach number 10.6 Electron gas specific heat 5/3 ratio Free stream ionic velocity 3643.21 m/s component in x direction Free stream ionic velocity 0.0 m/s component in y direction Free stream ionic velocity 0.0 m/s component in z direction x-component of magnetic 2.163 x 1c6 T field y-component of magnetic 0.0 T field z-component of magnetic 0.0 T field Magnetic permeability 147~ x 10m7H/m



Boundary Conditions

There are four types of boundary conditions employed in this calculation.

Free stream boundary condition (i=JM) At this location both the flow and magnetic properties do not change, and therefore are equal to the free stream values.

Qwxu = Qc. (4.la)

ai JM+i = ’ (4.lb)

’ * Outflow boundary condition (i=l and i=IM) Assuming that i = 1 and i = IM are far downstream so that the flow is supersonic, simple extrapolation is employed. For simplicity, the zero&order extrapolation is used as follows:

CA, = Qz,, (4.2a)

Qw, = Qmm, (4.2b)

ati= (4.2~)

a -0 ,hd+?j - (4.2d)

Surface boun&rv condition (i=l) The no-slip, adiabatic and perfectly insulated wall conditions are implemented.

Pi.1 = Pi,2 (4.3a) UjJ = V,,l =wj,*= 0 (4.3b)

Figure 13: Surface boundary condition

(4.3c)

(4.3d)

(4.3e)

(4.30

Computational Test Cases

Computations are performed for the following three cases:

Case 1: This case represents the Navier-Stokes calculations without the magnetic field and bi- temperature model (BX)W y (BY), = (Bz)W = 0 T .

Case 2: This case represents the MHD Calculations without the bi-temperature model (I?,), = (B,), = 0 T and (BYjW=0.5T.

Case 3 : This case represents the MHD calculations with the bi-temperature model for the flow conditions given in Table 1.

Figures 14 and 15 show the contour plots of pressure and density for Cases 1 and 2 respectively. Comparing the location of the bow shock in Figure 14 and Figure 15, it is evident that the shock standoff distance is larger and the shock is more diffused in the presence of the magnetic field (applied as boundary condition on the surface of the body). These solutions are very encouraging and form the basis of further investigation of the concept of IvElD control of shock structure in hypersonic flows.

Figure 14: Contour plots for (a) pressure, and (b) density for Case 1.


-(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

/ Bow \

Figure 15: Contour plots for (a) pressure, and (b) density for Case 2.

Figures 16-21 show the distribution of various variables ( p, p, U, W, B,, , and B, ) along the centerline for Case 1 and Case 2. The variation in these quantities along the body centerline in the absence (Case 1) and presence (Case 2) of magnetic field further lends credence to the effectiveness of “MHD control of shock waves” concept articulated earlier. The most significant observation from Figure 16 is that there is a large reduction in pressure at the surface of the body for Case 2 as compared to Case 1 from 5195.8 Pa to 661.i Pa. This result clearly indicates the possibility of reducing supersonic drag by introducing the magnetic field.

Oooo ,a* 09 Figure 16: Computed density distributions at

y=Om.

Figure 17: Computed pressure‘distributions aty=O m.

Figure 18: Computed u distributions for at y = Om.

0.w 0.w x(m)

Figure 19: Computed w distributions at y = 0 m.


’ (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

405 -

4 IO I I I C-00 0.01 004 0.06 oo*

x(m) Figure 20: Computed B,, distributions at y = 0 m.

Figure 21: Computed B, distributions at y = 0 m.

Fig. 22 shows the density, ionic hydrodynamical pressure and electronic pressure contour plots with bi-temperature model for Case 3. It is easily observed from these plots that the shock standoff distance is larger and the shock is more diffixed in the presence of weakly ionized field.

Figure 23 shows the distribution of density, ionic hydrodynamical pressure, magnetic field and electronic pressure along the centerline of the body in the presence and absence of ionized field. It can be observed from the distribution of these variables from the body stagnation point to the shock along the body centerline, that these variables also indicate a larger shock standoff distance in the presence of a weakly ionized field compared to the standoff distance in the non-ionized gas. It should be noted that the ionized behaviour is simulated by a very simple bi- temperature model which may not be adequate to describe the behaviour of actual plasma. Nevertheless it serves a useful purpose to investigate the effect of ionization on shock standoff distance in the presence of a magnetic field.

(a) Density contours with bi- temperature model

(b) Hydrodynamic pressure contours with bi-temperature model

(c) Electronic pressure contours with bi-temperature mode!

Fig. 22 Contour plots for (a) density (b) ionic hydrodynamical pressure and (c) electronic pressure



;I;0

Y

p&g

t

I

ZOO- l--i,, , , ,

0 ----..-a.-.-_,A-

0 20 40 60 60 100. 120

E’ig. 23 Distribution of (a) density (b) ionic hydrodynamical pressure (c) magnetic field and (d) electronic pressure along the centerline of the body

Conclusions

The results reported in this paper for supersonic viscous MHD flow past a blunt body are very encouraging as they indicate the possibility of supersonic drag reduction by application of a magnetic field to a weakly ionized gas. This lends credence to the effectiveness of “MHD control of shock waves” concept.

References

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Mishin, G. I., Serov, Yu. L., and Yavor, I. P., “Flow Around a Sphere Moving Supersonically in a Gas-Discharge Plasma,” Sov. Tech. Phys. Lett., Vol. 17, No. 6, June 1991. Mishin, G. I., “Sonic and Shock Wave Propagation in Weakly Ionized Plasma,” in Gas Dynamics, Yu. I. Koptev Editor, Nova Science Publishers, 1992. Gordeev, V. P., Krasllinkov, A. V., Lagutin, v. I., and Otmennikov, V. N., “Experimental Study of the Possibility of Reducing Supersonic Drag by Employing Plasma Technology,” Fluid Dynamics, Vol. 31, No. 2, 1990, pp. 313-317 Joseph, W. C., “An Experimental and Theoretical Investigation of the Pressure Distribution and Flow Fields of Blunted AMS,” NASA Technical Report, No. D- 2969,1965. Ziemer, R., W. and Bush, W. B., “Magnetic Field Effects on Bow Shock Stand-off Distance,” Physical Review Letters, Vol. 1, 1958, p. 58. Resler, E. L., and Sears, W. R, “The Prospects of Magnetoaerodynamics,” J.of Aerospace Sciences, Vol. 25, 1958, p. 235. Levy, R. H., Gierasch, P. J., and Henderson, D. B., “Hypersonic Magnetohydrodynamics with or without a Blunt Body,” AIAA Journal, Vol. 2, 1964, p. 2091. Kantrowitz, A., R., “A Survey of Physical Phenomena Occurring in Flight at Extreme Speeds,” Proc. Conf. On High Speed Aeronaut., N.Y., 1955, p. 335. Mnatsakanian, A. Kh. and Naidis, G. V., “On MHD Effects on Spacecraft Motion in the Earth Atmosphere,” Research Report 92/11, IVTAN, 1991, p. 26. Bityurin, V. A. and Ivanov, V. A., “An Alternative Energy Source Utilization with an MI-ID Generator,” 331d Symp. On Eng.


* (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

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WI

Aspects of MHD, Tullahoma, TN, 1995, p. x-2.1. Cole, J., Campbell, J. and Robertson, A., “Rocket-Induced Magnetohydrodynamic Advanced Propulsion Concept,” AIAA Paper No. 4079, 1995. Agarwal R. K. and Augustinus J., “Numerical Simulation of Compressible Viscous MHD Flows for Reducing Supersonic Drag of Blunt Bodies,” AIAA Paper No. 0601, 1999. Powell, K. G., “An Approximate Riemann Solver for Magnetohydrodynamics,” ICASE Report No. 94-24, 1994. Brassier, S. and Gallice G., “A Roe Scheme for the Bi-Temperature Model of Magnetohydrodynamics,” Proc. Of Sixteenth Int. Conf. On Num. Methods in Fluid Dynamics, Arcachon, France, Bnmeau (ed), 1998, pp. 260-265. Augustinus, J., Hoffmann, K. A. and Harada, S., “Numerical Solutions of Ideal MHD Equations for a Symmetric Blunt Body at Hypersonic Speeds,” 2gfi Plasmadynamics Conference, AIAA Paper No. 1067, 1998. Augustinus, J., Harada, S., Agar-wal, R. K. and Hoffmann, K. A., “Numerical Solutions of the Eight-Wave Structure Ideal MHD Equations by Modified Runge-Kutta Scheme with TVD,” 28* Plasmadynamics and Laser Conference, AIAA Paper No. 2398, 1997. Brio, M. and Wu, C. C., “An Upwind Differencing Scheme for the Equations of Ideal Magnetohydrodynamics,” Journal of Computational Physics, Vol. 75, 1988, pp. 400-422. Harada, S., Augustinus, J., Hoffmann, K. A. and Agarwal, R. K., “Development of a Modified Runge-Kutta Scheme with TVD Limiters for Ideal 1-D MHD Equations,” AIAA Paper No. 97-2090, 1997.


numerical study of compressible viscous mhd equations with a bi-temperature model for supersonic...

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supersonic flow

supersonic speed

bitemperature model

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effect of magnetic field

shock structure