Compressible magnetohydrodynamic boundary layer swirling nozzle and diffuser flows with a magnetic field

M!O-7!!~/79/1101-IImo?

00/o

COMPRESSIBLE MAGNETOHYDRODYNAMIC BOUNDARY LAYER SWIRLING NOZZLE AND DTFFUSER FLOWS WITH A MAGNETIC FIELD

M. KUMARI Department

of Plpplied

Mathematics.

Indian

(Communicated

and G. NATH ln\titute, by S.-l.

of Science,

Bangalore

260 012. India

PAI)

Abstract-The flow, heat and mass transfer problem for boundary layer swirling flow of a laminar steady compressible electrically conducting gas with variable properties through a conical nozzle and a diffuser with an applied magnetic field has been studied. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme after they have been transformed into dimensionless form using the modified Lees transformation. The results indicate that the skin friction and heat transfer strongly depend on the magnetic field. mass transfer and variation of the density-viscosity product across the boundary Payer. However. the effect of the variation of rhe density-viscosity product is more pronounced in the case of a nozzle than in the case of a diffuser. It has been found that large swirl is required to produce strong effect on the skin friction and heat transfer. Separationless flow along the entire length of the diffuser can be obtained by applying appropriate amount of suction. The results are found to be in good agreement with those of the local nonsimil~~rity method. but they differ quite significantly from those of the local similarity method.

1. INTRODUCTION

compressible electrically conducting fluid through a duct of variable cross section in the presence of an applied magnetic field is of considerable importance due to its application in power generation and space propulsion. So far, no studies have been reported in the open literature of this general problem. However, some particular cases such as non-swirling steady laminar compressible boundary-fayer flow of an electrically conducting fluid through a duct of constant cross section with an applied magnetic field have been studied in the past [l-S]. In the absence of a magnetic field, Back [6] has investigated the steady laminar compressible boundary layer swirling flow through a conical nozzle with constant fluid properties (i.e. the density-viscosity product pp is constant) using the local similarity method. Recently, Nath et ~11.[7,8] have studied both the nozzle and diffuser flow problems with variable gas properties (i.e. p,u varies across the boundary fayer). They have solved the partial differential equations governing the flow numericalfy using an implicit finite-difference scheme. In this paper, we have studied the flow, heat and mass transfer problem for boundary layer swirling flow of a steady compressible electrically conducting gas with variable properties through a conical nozzle and a diffuser with an applied magnetic field. In order to reduce the equations in dimensionless form the modified Lees transformation has been used. This transformation transforms the equations to a form where the derivative of the density-viscosity product pp does not occur unlike in the Lees transformation (where it occurs) used in Refs.f6-81. The present study has shown that the modified Lees transformation is more suited to problems with variable fluid properties and with highly cooled walls where some difficulty is encountered when the Lees transformation is used. This difficulty is caused by strong distortion in the velocity and total enthalpy profiles. Hence, in this case, a very small step size is required to calculate the skin friction and heat transfer on the wall. However, no such difficulty is encountered when the modified Lees transformation is used. The resulting partial differential equations have been solved numerically using an implicit finite-difference scheme[9, lo]. The results have been compared with those of the local similarity and local nonsimilarity methodsfll, 121.

THE

SWIRLING flow of a viscous

hl. K(llM4KI ,IllJ (; \.AlI’H

11x4

2. GOVERNING

EQUATIONS

We consider the steady laminar compressible boundary layer swirling flow of an electrically conducting gas with variable properties (p 5. T- ‘, p 72‘I’“, n x T”. Pr = 0.7, where p, T, g and Pr are, respectivety, the density, temperatL~re, viscosity and Prandti number, and w and n are the indexes in the power-law variation of the density-viscosity product and electrical conductivity across the boundary layer, respectively) through an axisymmetric porous insulated surface of variable cross section (see Fig. If with an applied magnetic field aligned in the normal direction. We assume that there is a free vortex at the edge of the boundary layer and also a mass transfer on the surface. The magnetic Prandtl number is assumed to be small (in most applications this is true). Therefore, the effect of the induced magnetic field on the how can be neglected in comparison with the applied magnetic field. Under these circumstances the partial differential equations governing the flow expressed in dimensionless form by using the modified Lees transformation are [6--g, 131 F”+fF’+ N/3(p,/p - F’)+ Na(p,/p - s’) - MN/%(p,/p)[(&~)F - 11= 2X(NFFx s”+fs’-

~~~,(p~/p)[(~/~~)s

Pr-‘g”+fg’+{(l

- I] = 2X(NFsx

- Pr “)(u:/H,)[FF’+

- Mua,(tr,/rr)(p,/p)[(rr/rr,)s - ll(u:/H,f The boundary

conditions

-fxF’)

-fxs’)

(la) (Lb)

(v,/u,)“ss’]} = 2X[NFgx

-fxg’].

(ICI

are

FIX, 0) =

sex.0) = 0, g(X,

0) = gw

F(X, r) = s(X, x) = g(X, x) = 1. The modified Lees transformation?

is expressed

@a)

6%)

as

(3) Here pu =

r--‘(a$/,lag). pw = - r-.‘(aI)/ag) F = f’/N = H/U,

3/ = (2X)“% s = v/v,,

g = HIK,

fw = -(2X)-“’

f =IL NFdZ+f,

(pw),r d[

Fig. 1. Co-ordinate system and velocity components for axisymmetric flow through a surface with arbitrary cross section f( is the longitudinal distance along surface, 9 is the tangential (circumferential) distance around surface, and b is the distance normal to surface).

tin the Lees transformation

2 = (psu,rf(ZX)~ ‘!’

(4a)

‘(p/p?)di and X is the same as in eqn (3)

(4b)

Boundary

layer swirling

M =

nozzle

CT&y/f&.,

and diffuser

field

1185

LY= (2Xir)f-drldX)(v,lu,)”

P = (2X/u,)(du,ldX), P&P= T/T,*

flows with a magnetic

p, = 2X/(pju5r2L2)

N = ~~/~~~~= (T/T,)“- ‘*

(4c)

crlffe= (T/T,)

T/TF = {g - (u32HeNF2+ (u,/u,wl~/u - (u512K)U + tv,/4)211.

(44

In the present case, .$ q and g are, respectively, the longitudinal, tangential and normal directions; u, u and w are, respectively, the velocity components in the 5, q, and 6 directions; X and 2 are transformed coordinates; II, and f are the dimensional and dimensionless stream functions, respectively; F(or f’) and s are the dimensionless longitudinal and tangential (or swirl) velocities, respectively; H is the total enthalpy; g is the dimensionless total enthalpy, BQ is the applied magnetic field; M is the Hartmann number; L is the characteristic length (length of the nozzle or diffuser); N is the ratio of the density-viscosity product across the boundary layer; cy and /3 are the dimensionless swirl (tangential~ and longitudinal acceleration parameters, respectively; uz/H< and vz/H, are the dimensionless dissipation parameters; f,,. is the mass transfer parameter at the surface; g, is the total enthalpy at the wall; r is the surface radius; PI is a dimensionless parameter; the subscripts e and w denote conditions at the edge of the boundary layer and on the surface, respectively; the subscript X denotes derivative with respect to X; and the prime denotes differentiation with respect to Z. The skin-friction coefficients in the longitudinal and tangential directions are given by]6-81

The heat-transfer coefficient in the form of Stanton number can be expressed as]631 St = q/[(R - HH’)peUrl=

where

Pr = j.~~~lk,

J.Llr(2X)-"2C:,

G;v = Pr--‘(I -gw)-‘g:,.

(64 (6b)

Here T&and C, are the shear stress and skin-friction coefficient on the surface in the longitudinal direction, respectively; 7” and C, are the analogous expressions in the tangential direction; F: and s:. are the skin-friction parameters in the longitudinal and tangential directions, respectively; St is the Stanton number: q is the heat-transfer rate at the wall; g:. and Gk are, respectively, the total enthalpy gradient and heat transfer parameter; k is the thermal conductivity; and c, is the specific heat at a constant pressure. For a conical convergent nozzle with straight generators (see Fig. 2), we have[6,7]

For a truncated conical diffuser with straight generators (see Fig. 2), the above expressions become 181 u, = b/r’,

v, = r/r,

a = -2($bt)ll+

rlL=bl[l+(iE/b~)sinA],

b, = al/L

fSjbd sin h] sin ~(~~~u~)~

P =-4($bl)[l +($/br)sinA]-“sinA /?I = hzb:c[l +(.flb,) sin A]‘, AT = 2pUpeb)

(W

II86

M. KUMARI and G. NATH

’ (a) Conical

(b) Truncated

nozzle

with straight

conicat diffuser generators

generators

with straight

Fig. 2. Conical nozzle and diffuser geometry.

US/H,= [l + ($b,) sinA]-4(U:/H,)i v:/H, = [l+ (c/b,) sin A]-*(vQH,)i (G/U,)* = ]l + (abr) sin Al*(v,/~,)~

fw = A(.$/bl)“*[l +(.$/2b,) sin A]

(8b)

A = -(pw),a

(SC)

:‘*/(2p,~eb)“*

where I is the circulation; A,, A?, and b, are dimensionless constants; L and A are the length and semi-vertical angle of the nozzle or diffuser; al and b are dimensional constants; and the subscript i denotes the inlet conditions. The parameter A 20 according to whether there is suction or injection. In the potential flow region (i.e. outside the boundary-layer region) we have used the appropriate approximation corresponding to the low-speed flow. For a low-speed flow, the density-viscosity product pepP can be considered as a constant along the longitudinal distance < (i.e. p,~~ is independent of 5) without any appreciable error. For a high-speed flow also, as a first approximation, pepP can be taken as a constant along 5 (i.e. independent of 5) without introducing significant error in the analysis. Consequently, the mass transfer parameter A will become a constant if the surface mass transfer velocity (PW)~ is taken as a constant (i.e. (pw),, does not vary with t), because L, A, b, al and pepP are all constant. Hence f,,. will vary according to the eqns (7~) or (8~). It is evident from these equations that fw = 0 at $ = 0 whatever may be the value of A. We have taken the mass transfer parameter on the surface to be small so that the potential flow outside the boundary layer is not affected by it. Hence the velocity components at the edge of the boundary layer u, and v, given in eqns (7a) or (8a) which have been derived on the assumption that there is no mass transfer on the surface will be valid even in the presence of mass transfer provided the mass transfer on the surface is small. It may be remarked that for M = 0 (i.e. when there is no magnetic field) eqns (1) reduce to those of swirling flows (without magnetic field). Further, for M = (Y = 0, (VJU,)i = 0, they reduce

Boundary

layer swirling

nozzle

and diffuser

flows with a magnetic

field

1187

to those of non-swirling flows and eqn (lb) becomes redundant. For X = 0 ({ = 0) they, eqns (l), reduce to Blasius type of equation, because at X = 0 (I = 0), (Y= p = PI = 0. When the variation of the density-viscosity product pp is constant across the boundary layer (i.e. when w = N = I ), and M = 0 eqns (1) are same as those of Refs.[b8], but for variable pp flows (w Z I), the derivative of N does not occur in the equations unlike those of Refs.[6-81 where the Lees transformation was used. It may also be noted that w = 1 represents constant density-viscosity flows and w = 0.5 product simplification (N = I), w = 0.7 is appropriate for low-temperature may be considered as a limiting value for high temperature flows[l4]. 3. RESULTS

AND

DISCUSSION

Equations (1) under conditions (2) using reiations (7) or (8) have been solved numerically using an implicit finite-difference scheme. Since the complete description of the method is given in Refs.[9, IO], for the sake of brevity, its description is not presented here. The computations have been carried out on IBM 360/44 digital computer for various values of the parameters both for a highly cooled wall (g, = 0.05) and for a moderately cooled wall (gw,= 0.2). In the nozzle flow, /3. us/H, and vz/H, tend to infinity as
M=10,~-0.05,A=O,(~~~),=0.01,(v,/~,)~=10, 0.2

Pr-0.7,nP1

-f=O -.-FE

0.2

----j-o.4

L

Fig. 3. Longitudinal

tThe fThe

velocity

profiles

(Nozzle).

profiles for other values of the parameters are available with the authors. results for g, = 0.2 or n = 3 or A > 0 may be obtained from the authors.

0

M. KUMARI

M=l0,~-0.05,

arid G. NA’TH

&He)i” O.OI,(v&e

A-0,(

6 -10,

R-o.7 n-l.0

--g=O ----_?

mo.2

----ijao.4 0

I 1

I 2

Z

I 3

I 4

5

Fig. 4. Tangential (swirl) velocity profiles (Nozzle).

M*lO,~=0.05,

A=O,(“2,IHe)i’O,Ol,(v,lu,):

110,

-tJf-0 --a-+=0.2 ----f=o.4 0

1

2

2

3

Fig. 5. Total enthdpy profiles (Nozzle).

4

5

Boundary

layer swirling

nozzle

and diffuser

.‘.

\

flows with ;i magnetic

\

’ ‘\ Y\\ \e--.. . \ \‘\ \ ‘\ 1, \’ \ . \\ ’ \\ \ ‘\L,

\

I

I

l

/

a ”

1

9 5

1.

7

‘&\

N

field

l

-2

+\ \\ ‘U

O0

M. KtiMAKl

and G. N’ATH

1.5. g,B0.05,(uZ/He)i

-OOl,(~,l~e;f-1, A--0.5,Prc0,7,n.10

-w-1.0 ----w=O.6,,-----y

-ii

0.2

0

Fig. 8. Variation

01 0 Fig. 9. Variation

of longitudinal

skin friction

with t (Nozzle,

I

1

0.2

0.4

of longitudinal

? skin friction

0.6

0.4

r

with 5 (Nozzle,

(a,/~,)~

= I, A = -0.5).

5

(Y~/u,.): = IO, A = -0.5)

1191

Boundary layer swirling nozzle and diffuser flows with a magnetic field

~sO.O5,(&kie)i

b

Local

-O.Ol,(vel~~~=l,

nonsimilarity

(3-Equation

0

A-O,Pr=0.7,n-1.0

model)

I 0.6

I 0.4

I 0.2

0

? Fig. IO. Variation

of tangential

skin friction

with < (Nozzle. (c,/u,)~ = 1, A = 0)

4 ~-0.05,(u~/H,),-O.Ol,(~~lu~~f-10,~.0,Pr=0.7,

-w=l.O ---b

lIJ=o.5 Local

model

)

w=.l.O

3-

/-

/-

,~L+ZO

0.2

/,A

I

I

I

Oh

0

Finite difference

nonsimilarity

(3-Equation

nf

0.4

0.6

E

Fig

I I.

Variation

of tangential

skin friction

with i (Nozzle,

(v,/u,)f

= IO. A = 0).

1

M. KUMARI

and G. NA’I’D

1 ~=O.O~,(~~~H~)~=O.O~,(~~~~A=-O.~,P~=O.~,~~~.O -

w -1.0

----

w-o.5

,/-----\ \

/ /

/ /

1

‘1 \ \ i-1

3 -m

C

Fig. 12. Variation

of tangential

skin friction

with i (Nozzle.

(L.,/u, 1’ = I. A = -0.5).

1.5 g,,,~O.O5,(&He)i

-O.ol,(~,/~ef

-w=l.O ----WxO.5

510,A--0.5,Pr=0.7,n=l.0

,/--_ ‘b. ‘3, ‘.

// /

0

0.4

0.2

0.6

E Fig. 13. Variation

of tangential

skin friction

with [ (Nozzle,

(u,/u,)T

= IO, A = -0.5).

Finite difference

Fig. 14. Variation

(3-Equation

T

of heat transfer

“.L

model)

Local nonsimilarity

----w-O.5

-wwl.O

with [ (Nozzle,

(r,/u..)f

“.-+

.

= I. A = 0).

l

U.”

.

L

?i

l-

2-

3-

:-

6

l

Fig. 15. Variation

of heat transfer

“.L

uith

E

w=l.O

[(Nozzle.

n=l.O

(I.,/u, )f = IO. A = 0).

V’-

= O.Ol,( velue)~=10,A=0,Pr=0.7, Finite difference

i similarity

7

Local nonsimilarity i ( 3-Equation model ),

Local

-wwl.O ----wwIO.5

gw=0.05,(u$He),

V.”

1194

1.5

-____g~=O.05,(u~/H,)~~001,(~~~~,~~-~,A=-0.5,Pr=O.7,n=lO -oJwll.@ ----wsO.6

I

0.2

0

Fig. 16. Variation

Fig. 17. Sariation

of heat tranbfer

of heat transfer

F with i (Nozzle.

with j (Nozzle,

0.4

(t,/u,)‘=

(v,/u,)f

0.6

1. A = -0.9.

= IO, A = -0.5).

Boundary layer swirling nozzle and diffuser flows with a magnetic field

1195

s$, and G:. and these parameters increase as N increases or w decreases. It is also observed that FL is strongly dependent on the parameter (oJU,)i whereas the dependence of s’,,. and G:, on it is rather weak. The effect of n or suction (A >O) is to increase F:., ,s: and G: whereas injection does the reverse. When A 2 0, FL. and s: increase as c increases, but G’,. increases till $ = 0.55 and then begins to decrease. When A < 0, FL., s:, and GL. first increase and then decrease or oscillate or continuously decrease as 4 increases depending on the values of M, (u,/u,)~ and o. We find that there is little difference between the values of FL. .FL,and G:, for the non-swirling flow (CY= (u,/u,)~ = 0) and for the swirling flow with (c,/M,), = I. Hence, they are not shown in the above figures. This shows that large swirl is required to produce strong effect. In order to test the accuracy of the present method, the results of the similarity solutions (obtained by putting X = 0 (or 4 = 0) in eqns (1)) are compared with those of the quasilinearization technique[6] and the results are found to agree up to 4 decimal places. In order to compare our results with those of the local similarity and local non-similarity methods, we have also solved eqns (1) using these methods[ll, 12). We find that our (finite difference) result5 are in good agreement with those of the local non-similarity method except for large values of c%but they differ appreciably with those of the local similarity method except near the inlet. It may he remarked that the local similarity solution of eqn (lb) reduces to that of the similarity solution. Hence .F;. = 0.4696 for all 4. Finally, we have also compared our results for M = A = 0,w = 1. gH.= 0.2with those of Ref.171 and found them in excellent agreement. Diffirser remits. in the case of a diffuser, the flow is against an adverse pressure gradient, hence it is likely to separate from the boundary even near the inlet. However, separation along the whole length of the diffuser can be prevented by applying suction. The amount of suction required to ensure separationless flow throughout the entire length of the diffuser depends on the values of M, (u,/u,)~, w and A. Less suction is required if M is large or (u,/u,);, h and w are small. For the values of the parameters used here we find that A = 2 is necessary to ensure separationless flow throughout the entire length of the diffuser. The results? (for g1 = 0.05, rt = 1) given in Figs. 18-23 show that M has strong effect on FC., s:, and G:, and they increase as M increases. The parameters F:,., s ‘,.and G I are comparatively weakly dependent on w and

r-

g -0.05,(u~/He~i-QOl,~v,/~)~-l,A-2,Pr-0.7,n-1,X-O.2,b~=0.5 W

?

15 .-Wx-1.0 ----_w=O.5

Finite difference

l

Local

b

Local nonsimilarity (3-Equation model

similarity

)]

10

-!3

5, .-

1 C)_

0

Fig. 18. Variation of longitudinal skin friction with < (Diffuser, (I,/u, I: 2 I) tThe

results

for ,g% = 0.2 and n = 3 are available

with the authors

M. KUMARI

11%

-w-1.0 ---__~=0.5

15

l

0

Local

and Ci. NATH

Finite

difference

i similarity

7

Local nonsimilarjty (d-Equation

w-1.0 /

1c -ii

5

0

i

I 0.25

OS0

I

I

1.00

0.75 e

Fig. 19. Variation of longitudinal skin friction with .$ (Diffuser, (Q/U,)” = 10).

( SEquation

0

0

0.25

model)

-0.50 5

1.00

0.75

Fig. 20. Variation of tangential skin friction with i (Diffuser, (v,/u,)i

= I).

Boundar)

layer 4riing

nozzle

and diffuser

flow< with a magnetic

tield

15-

~~0.05,(~~i~~)~~O.O1,(~ei~~)~-lO,A=Z,

Pr=Q7,n=lf

x-02, b1’0.5

fig.

l

Local

‘1,

\‘arjation

skin friction

with i (Diffuser.

tt.,l&)~

= It’ll

slmilarlty

( ‘J-Equation

01 0

of tangential

modet)

I

0.25

Fig. 2.

Variation

I

o.50 r of heat transfer

I 1.00

I

with <(Diffuser.

0.75

(v,,/u,)i

= 1).

M.

KUMAKIand G. NA’fH

R=O.7,n=l,h-0.2,bl=0.5

a.

6

0 0

I

0.25

I

I

0.50

0.75

xl

T Fig. 23. Variation of heat transfer with i (Diffuser. (c,/u,.)z = 10).

they increase as o decreases. The effect of the swirl parameter (v,/u,), on FL,, s:. and G: is less than those of the nozzle flows. Further, FL., s: and CL. decrease as (U,/U,)i increases. The results for the non-swirling flow ((tlJU,)i = LY= 0) are found to be nearly same as those of the swirling flow with (v,/u,), = 1, hence they could not be shown in Figs. 18-23. As the velocity and total enthalpy profiles are qualitatively similar to those of nozzles, they are not given in figures for the sake of brevity. The results for M = 0, o = 1, A = 2, g, = 0.2 are found to be in excellent agreement with those of Ref. [8]. The results are also found to be in good agreement with those of the local nonsimilarity method except for large g but they differ considerably with those of the local similarity method except for small 1. This implies that the local similiarity method is not suitable for nozzle or diffuser flows. It may be remarked that the nonsimilar terms in eqns (1) (i.e. the right hand side terms) are not small for large ,$ and these terms are neglected in the local similarity method. Therefore, for large .$, the local similarity results differ considerably from those of the finite-difference results. 4. CONCLUSIONS

The skin friction and heat transfer are strongly dependent upon the magnetic field, the variation of the density-viscosity product across the boundary layer and the mass transfer. However, the effect of the variation of the density-viscosity product is more pronounced in the case of a nozzle than in the case of a diffuser. Large amount of swirl is required to produce strong effects on the skin friction and heat transfer. Separation along the entire length of the diffuser can be prevented by applying suction. The results for highly cooled walls can be obtained by using the modified Lees transformation. The results are found to be in good agreement with those of the local nonsimilarity method, but they differ significantly with those of the local similarity method. REFERENCES

[I] I. B. CHECKMAREV, J. Appl. Math. Mech. (U.S.S.R. English Translation) 26, 1194 (1962). [2] [3] [4] [S]

C. S. LIU and A. B. CAMBEL, Phys. Fluids 6,792 (1963). C. S. LIU and A. B. CAMBEL, Phys. Fluids 7, 564 (1964). F. J. HALE and J. L. KERREBROCK, AIAA J. 2,461 (1%4). U. P. HWANG, L. T. FAN and C. L. HWANG, AIAA J. 5, 2113 (1%7).

Boundary layer swirling nozzle and diffuser flows with a magnetic field

1199

161 L. H. BACK, AIAA J. 7, 1781 (1969). 171 G. NATH and M. MUTHANNA. Int. J. Heat Muss TrffnsfeT21. 1213 (1978). i8i G. NATH and B. K. MEENA, t&f Appl. Engng Sci. 5,341 (1977) 191J. C. MARVIN and Y. S. SHEAFFER, NASA TN D-5516 (1969). [lo] C. S. VIMALA and G. NATH, J. Nuid Mech. 70, 561 (1975). II I1 E. M. SPARROW, M. QUACK and C. J. BOERNER. AIAA .I. 8. 1936 (1970). il?j E. M. SPARROW and H. S. YU, J. Heat Transfer 93.328 (1971). f i3j W. S. KING and W. S. KING, In ~e~e~~)pmenf of ~~r~an~cs (Edited by S. Ostrach and R. H. Scanlen), Vol. 2, p. 107. Pergamon Press, Oxford (1965). [I41 A. WORTMAN, H. ZIEGLER and G. SOO-HOO, fnf. J. Heat Mass Transfer 14, 149 11971).

(&c&led 31 Janftar!: 1979)