The flat-plate magnetohydrodynamic boundary layer in a transverse magnetic field

J

THE FLAT-PLATE MAGNETOHYDRODYNAMIC BOUNDARY LAYER IN A TRANSVERSE MAGNETIC FIELD

JANUARY 1967

by

f

,

I

J. K. Dukowicz

UTIAS TECHNICAL NOTE NO. 68

,"

•

THE FLAT-PLATE MAGNETOHYDRODYNAMIC BOUNDARY LAYER IN A TRANSVERSE MAGNETIC FIELD

by

J. K. Dukowicz

Manuscript received Jan. 1966

ACKNOWLEDGEMENTS

The author wishes to express his gratitude to Dr. G. N. Patterson for the opportunity to carry on this research at the Institute for Aerospace Studie s.

Special thanks are due to Dr. J. H. deLeeuw, the supervisor of this research, for many valuable discussions and suggestions.

The use of the computational facilities at the Institute of Computer Science, University of Toronto, is gratefully acknowledged.

This work was financially supported by the Air Force Office of Scientific Re searcp.

SUMMARY

The equations of motion for a flat plate boundary layer flow of an incompressible, electrically conducting fluid in the presence of a transverse magnetic field moving with the main stream velocity have been

solved numerically for the case of negligible induced magnetic field. The equations of motion have been transformed into a universal form with no characteristic parameters present. The velocity profiles have been calculated as they range .. from the Blasius profile to the asymptotic exponential profile. The asymptotic profile is reached in a distance xl ~

pU

_oo / 0 - B 2 from the leading edge. The error due to the finite-difference solution has also been calculated.

1. 2. 3. 4. 5. 6. 7. 8. TABI,E OR.CONTENTS NOTATION INTRODUCT ION

THE EQUATIONS OF MOTION

THE ASYMPTOTIC VELOCITY PROFILE OTHER SOLUTIONS

THE TRANSFORMATION OF THE EQUATIONS OF MOTION FOR TfIE PURPOSES OF NUMERICAL COMPUTATION THE NUMERICAL SOL UTION

RICHARDSON'S EXTRAPOLATION AND ERROR ESTIMATES DISCUSSION Page v 1 2 4 5 6 7 11 12 REFERENCES 14

APPENDIX: The Compressible Asymptotic Boundary Layer 15 TABLES

j k m n u v

,

u

,

v x y B B L T

,

NOTATION

G coordinate grid point location

thermal conductivity

magnetic parameter,

'l:.

coordinate grid point location

x component of veloçity

y component of velocity

nondimensional

S

component of velocity, u/uro nondimensional

VL

component of velo city,

v/'mv

coordinate parallel to the plate

coordinate perpendicular to the plate

magnetic induction transverse to the plate, usually constant equal to free stream value Bo

component of B due to induced currents

Specific heat at constant pressure

Hartmann number, BL

~

;'\5

characteristic length

characteristic length in

~

characteristic length in

XL

magnetic Reynolds number,

cr

f-

U roL

2

transformed u for purposes of numerical solution; temperature (appendix)

freestream temperature wall temperature

free stream velocity

Greek Symbols

p

magnetic parameter, B

~f'A ~

Ua:>

parameter

\S"f'-l

V •

ratio of diffusivities

nondimensional coordinate perpendicular to the plate,

~~

lj

magnetic permeability; coefficient of viscosity (appendix only) kinematic viscosity

nondimensional coordinate parallel to the plate.

~

-:x..

density

electrical conductivity; transformed coordinate perpendicular to the plate~

Z

transformed coordinate parallel to the plate.

~

1. INTRODUCTION

Following the original work on tne oscillating plate and the

Rayleigll problem in magnetohydrodynamics tnere nave recently appeared

extensions of the theory to the more practical case of the flat plate boundary

layer, originally by Rossow (Ref. 1), and then by Radlow and Ericson (Ref. 2)

and Dix (Ref. 3). These analyses all consider the case of incompressible

flow and constant electrical conductivity. The natural extension of tne

Rayleigh problem assumes no current flow in the free stream and this implies

an electric field ~ = - ~Cû X ~ far upstream of the plate. Such an electric

field may be approximated in practice by insulated walls wilich restrict the

flow of current in the main stream. This is the case called "magnetic field

fixed relative to the fluid" by Rossow.

Dix shows that the characteristic parameters of the general

,problem

are~=~}Av

, the ratio of the diffusivities, and

~2

=

1/Pf.J.f:[3/U~2,

the square of the ratio of the Alfve'n wave speed to the free stream velocity,

wnere essentially the parameter E:. determines the strength of the induced

magnetic field and the parameterE.ol2.determines the strength of the inter-action with the velocity field.

2. Rossow (Ref. 1) solves equations which are valid in the limit

ofE: ~ 0, ~ 0(

«

1 but only for ((SB 2 /

p)

x

<:.

<

1, that is, a series solution is obtained of first or second order in this parameter. Radlow and Ericson (Ref. 2) as well as Dix (Ref. 3) linearize the equations using an

Oseen-type linearization and solve them approximately using the Weiner-Hopf technique. In addition, Dix numerically solves the exact equations for a few selected cases but under rather restrictive boundary conditions.

The linearized analyses indicate that the induced magnetic

induction BI is of order -È,1/2 B. For most conducting fluids €::. is

extremely small, for example, some values are Fluid

Liquid metals

Ionized air (30000 - 50000K)

Air at 34000K seeded with

O. 10/0 potassium

Sears, W. R. Astronautica Acta, Vol. VII, 1961 Fasc. 2 -3, p. 223 Therefore it appears that for most conducting fluids it is a good approximation to neglect the induced magnetic field in the same way as is done by Rossow, however, this automatically eliminates such

interesting aspects of the problem as disturbances ahead of the leading edge

of the plate (C)(. 2

>

1) and

Alfv~n

waves sweeping back from the leading edge

( 0\

2

<.

1) as discussed by Dix.

The purpose of this note is to extend Rossowls calculation, since this is a case of much practical interest, to all values of the parameter

2. THE EQUATIONS OF MOTION

The equations of motion for a flat plate boundary layer flow of an

incompressible, electrically conducting. fluid in the presence of a transverse

magnetic field moving with the main stream velocity are given by Rossow (Ref. 1).

with the boundary conditions

·

L.ll"X)o)

=

v(X)O)=

0

U (

:X)ó:::J

J

=

U

co

(2. 1)

(2. 2)

(2. 3)

These equations have been simplified by applying the boundary

layer assumption, that is, the characteristic.length across the boundary layer

(boundary layer thickness) is much smaller than the characteristic length

along..the. boundary layer.

It is assumed that the magnetic Reynolds number

(2. 4)

is so small the B can be considered to be the applied magnetic field. It is

now further assumed that the electrical conductivity trand the magnetic induction B are constant.

Tpe coordinates are transformed by

::x...

=

lÁ._VVl oo

S

~

~VL

\...,lc>o~

(2. 5) t..Á.

-U

-

-

~YY\V

u'

where

VVI

crB2..

(:::>

to obtain and

l.J"\

"du.\

+

d'l

\ \

~u.;

-+

dl)

=

0

-aS

'OYL

LL ( S)

0 )

=

Lr \

(~~

0') =- 0

L-è

(S/X))

= \

(2. 6) (2. 7) (2. 8)

This transformation puts the equations into non-dimensional form with no characteristic parameters being present. However, it introduces certain scaling distances and velocities which may be considered as the characteristic quantities of the problem.

LlelO

(;:)

lloo

L\

=

VVL

=

c::J g~

;

A distance from the leading edge when

the inertia force s become approximately equal to the electromagnetic forces.

\rS:L _

~y.e~

L

₂

=

VYVl -

cr'S2.

;

A boundary layer thickness when the

viscous forces become approximately equal to the electromagnetic forces, it is related to the reciprocal of the Hartmann number in that they are numeri-cally equal when the characteristic length used in the Hartmann number is equal to unity.

\}2

=

~Y'(\~= ~

\),

L,

The free stream velocity.

; Not an independent quantity.

The

ratiL \ _

~

f'

\..l':b

=

R.2..

L

2. -

cr-

B'v

\-\0...

occurs in the calculation of the stability of the Hartmann boundary layer (Ref. 4). The magnetic Reynolds number can now be expressed as

2-Ra

M

=

cr-

fA-

U~ 1E.~

==

CJtt

~

::.

E:.

(2.9)

The magnitude of this parameter has been shown to be generally very small and this implies that the induced magnetic field is small.

The boundary layer assumption requires that

L::2..

<:<

L \

That is,

V~t[

=

~e:ct2...:::.<..

\

Hence the equations to be solved represent the limiting case of:

E:..

~

0 ·

)

é

ct'-<:::..< \

Since the associated equation for the magneticfield is neglected,

it is not necessary to consider the electric boundary conditions on the plate, that is, it is not important whether the plate is electrically conducting or not.

It is not difficult to show that there is no similarity solution

possible for equations (2. 6, 2. 7) and it becomes necessary to solve the partial differential equations numerically.

3. THE ASYMPTOTIC VELOCITY PROFILE

Equations (2. 6, 2. 7) have a solution that is independent of That is, assuming

one obtains and with

LJ\=

0

ç\3-

l..C. ::.

l,.Á' - \

Ó.

yt:4

~(O)=O) ~(OO)=\

This equation has the solution

Lol\

= \

e

't.

J.b.. _ \

e-

~~~i: ~

\....loo-

-or

s·

(3.1) (3. 2) (3. 3) (3. 4) (3. 5) (3. 6) It can be seen that this solution must occur far downstream from the leading .. edge oL the. flat plate where the viscous shear force is balanced by the magnetic force. The actual solution of equations (2. 6, 2:7) must tend asymptotically to the velocity profile (3. 5). An interesting result of the solution will be the value of

5

at which the velocity profile has essentially become like (3. 5). Assuming that this value is ~:, then the distance from the leading edge will be

::c,::'

= ~:, ~ •

The velocity profile (3. 5) is the same as that of the Hartmann boundary layer which occurs in channels at high Hartmann numbers. >!<*

4. OTHER SOL UT ION S

The approach that has been used by Rossow (Ref. 1) is to assume that the stream function

'{I

can be expanded in a power series in terms of a suitable magnetic parameter

~

=

~

U,.;V:l(

l

f

0

oT

'(y)':x:,

~

'2. -I-

(VV1':l()

2..ç.ij.

+ - -

J

(4. 1)

The first term of the expansion is the B\~function. The

functionsf_o ~-Ç'2.~~,

---

are all functions of the variable~~~ which

appears in the Blasiu:s. transformation. Equation (4. 1) is used together with equations (2. 1, 2. 2) fo obtain a series of simultaneous ordinary differential equations forfo

;Ç2,

)-Ç4r' etc.

It can be shown that

~

-

u..,V

_V:x...

LA ...

Lf·

I

f\\

\)'2..

\I

1

(4.2) 0

+

VVl

"X.. .:l.

+ (

VVl:x.

~

4+

-0'8

where \

o-~'2.. ~

VY\:x.

-

-f:J

UCQ

,

Applying transformation (2. 5) yields

'OU-

\

[ -(' +

S.ç:

-I-

s2f~\

-I-

--1

-0

V{.

-

_\Js

(4. 3)

At

1

= 0, Rossow's results give

~~l

=

~

[O ..

~~'2.0(o

-+

\.141

S -

\."3S~~

2.+_

-1

(4.4)

Ae

.

~isadvantages

of this approach are that the question of convergence is left open and that the higher order approximations become progressively more difficult to calculate.

>~>!< The question of whether a similar asymptotic boundary layer is possible in compressible flow is examined in the appendix.

The linearized solutions of RadIowand Ericson (Ref. 2) and Dix

(Ref. 3) are obtained by sol:v:ip.g .. the. Oseen-linearized equations for velocity and magnetic field to first order in ~ by the Weiner-Hopf technique. For

the case ê..:::::....<...I. E:::.

Cl...

2

c::..<...

1; the results for the slope of the velocity

profile of the plate is given in both cases by

" :;

oL0 \

= \

e

151,,-

r l ....

J

I

~/\/

\

\/2

(4.5)

,

C

YL

0

~\\

s/"-

-+

~

r-,- \...

,... )

where ~,'is an arbitrary constant chosen by RadIowand Erickson equal to

O. 3464 5'0' that this function would agree with the Blasius solution in the limit

as

cr ...

~~.~:

. ,It should be noted that this result is exactly that obtained from the

corresp011ding Rayleigh problem provided time t is replaced by x/uro

(except fo,r the constant K).

Dix also calculates several cases numerically. He considers the

case of a ty,ro-dimensional cascade of plates and establishes a downstream

boundary with a prescribed velocity distribution. These boundary conditions

introduce effects such as "channeling" which do not exist in the unbounded

case. The cases of interest calculated by Dix are

e:

=

0,

E:

ö...

2

=

10- 3 and

10-2 ; and each one for two different positions of the transverse and downstream

boundaries. ' These cases are classified as follows:

Case Position of Trans- Position of

Down-verse boundary,

'1..,

stream boundary.

St)

-3

1 - 1 10 11. 0 O. 65

2 - 1 10'':'2 11. 0 3.0

3 - 1 10- 3 _{18. 8} 0.65

4 - 1 10-2. 11.0 1. 90

The calculations of Rossow. Dix, and RadIowand Ericson are

presented in Table 13 and in Fig. 2 for comparison with the present

calculation.

5. THE TRANSFORMATlON OF THE EQUATlONS OF MOTlON FOR THE

PURPOSES OF NUMERICAL COMPUT ATlON

The presence of an asymptotic velocity profile and the lack of a

similarity ~olution are similar to the case of the flat plate boundary layer

with uniform suction treated by R. 19lisch (Ref. 5). This section follows closely

the method of this reference.

The equations of mot ion (2. 6, 2. 7) are first transformed using

the von Mis.es .transf.ormation

Introdueing the stream function automatieally satisfies the

eontinuity equation_J and the momentum equation beeomes

Letting

N ow the transformation \ ~

~=-~~ ~-=

\Jrt-Z(;;)~) =T(~,~J

puts equation (5. 3) in the form

with the boundary eonditions

T(~.)OO)::r

\

T('t~o~"O

(5. 2) (5. 3) (5. 4) (5. 5) (5. 7)

This final transformation puts the equation in a eonvenient form for eomputation and eliminates a troublesome diseontinuity in the boundary eonditions at the leading edge or the point ~- ~=O as is seen in (5.4). A major advantage of this transformation is that the boundary layer thiekness decreases only slightly from the leading edge onwards. This simplifies the applieation of the boundary eondition at infinity and it means that a uniform grid ean be used in eomputation throughout the flow field. This is not possible in the untransformed equations sinee the boundary layer is very thin near the leading edge and an extremely fine grid would be neeessary to avoid loss of accuraey.

In addition to the boundary conditions (5. 7), (at

1::=0)

is neeessary for the solution of equation (5. 6). (5. 6) beeomes

i I i

"37.'0

4

(cr--

'G '\

ëf\~

-

0

~:a

ocr.2.

~'i) ~~

-where

T~(~)= \J\~(O)=O

\o(crj

= \"' (

OJ

\J)

7 an initial eondition At ~.

0,

equation (5. 8)

It is not difficult to show that equation (5. 8) and its boundary conditions are equivalent to the Blasius equation. Thus the Blasius solution for the incompressible flat plate boundary layer forms the initial condition for the present problem. as has been postulated by Rossow (sec. 4). It has been necessary to recompute the Blasius solution for the purposes of this problem to an accuracy of eight significant figures.

6. THE NUMERICAL SOLUTION

Equation (5. 6) is a nonlinear parabolic partial differential equation. Applying the usual criterion for the stability of the finite difference approxi-mation as given by Richtmyer (ReL 6) shows that for small values of

1:.

an explicit scheme would always be unstable and it is therefore necessary to apply a stabIe implicit scheme.

Using RichtmyerYs notation. one defines a grid such that

"t:

=

VL

e=.

~

<S

=

~ ~\l""

..J

=

O~

'J

'2..J

-and

\~= \(V\~~)

jö'3'")

The finite difference equation corresponding to equation (

5.

~) is

. constructed using a six-point centered scheme.

TI/)

T

Vlot \ J+\ ~~\ ~V\ J - ,

'T

VI_{+ \} ~-\

The equation is taken to apply at a fictitious centered point

't:

.

'=

(VI +-

\/2.

)b

t )

<:J

=

~

b.r;s

\

~~

\ _ ,\Y\+ \

+

T"" _ ""'\

~ ...) + \ .j - \ ..\..;. \ \...,) - \

4ACj

(6. 1) (6. 2) (6. 3)

The equation is linearized using the approximation

(6. 4)

It is now written in fuU

A "\

~.,.\ --r\l\~

\

""

+ \

~\~+\-~,~ -+Cj\~_\

+\)==-0

(6.6)

and it is assumed that

~

.

--r-

v

_{\1-\ _}

_,V\-\-\

_-+

_\='"

_(6.7)

\.J -

E~ ~.\

~

It is easy to see that

Since

E~ -=

A

~

~.)-C~'E...r\

F

~

.::

')::)..1

~

Col

~.l

- \

\6+~

0

~~-c.~'E..~-\

Therefore Eo

=

F 0

=

0

The procedure is to evaluate

E

~) ~j

according to (6. 8) for increasing values of ~ until the condition

E~+ ~~==

\.0

(6. 8)

(6. 9)

required by the outer b~)Undary condition is satisfied to the desired degree of accuracy. This is the condition that determines the maximum value of j.

As pointed out in section 5, the boundary layer thickness decreases and there-fore this condition wiU continue to be satisfied at this value of j. Equation (6. 7)

is th~used to calculate

,_tl

~\, starting with the value 1. 0, for aU values of j.

This procedure is simple since it takes into account the special proI?erties of this set of simultaneous equations. Reference 6 also points out that the

quantities Ej and FJ wiU remain bounded and therefore contribute to numerical accuracy.

The grid size is determined by a rather severe loss of accuracy near the plate. At

'!":O

.

equation (5. 6) takes the form

~~jl î5L-~L~~"l~

-+

8~2.=

0

(6.10)

The quantities in the square brackets above are in an

indeterminate forma In particular. this means that \,)

~

~ ~d&

-~

~

are zero at the plate. Due to the limited number of significant figures

~

\vailable in the floating point numbers used in a computer. calculation of

~-a.-è ~

when it approaches zero involves loss of significant digits.

As a criterion, the error ïntroduced in this way should be no greater than the

truncation error of the finite dilference approximation used.

This criterion is now applied at a typical grid point next to the plate. Very roughly. for eight significant figures, the round off error in T is of the order of T x 10- 8. Therefore. the error in the expression above is of the order Of\K\Ö 6 / .6~:' . Since the truncation error is of the order of

~\S'2and

since T itself is of second order in

~

T)(\

a-a

IA

'&-!

~

A.

~

"2-~~3 -

o\S

-

~ '-J (6.11)

:2:>

-8

Thus ~ ~ \

0

and therefore A~ should be greater than about 10-2• Since the finite difference equations are stabie.

4Ö~

is

independent of

t:;:;.

\S""

but it should be such that the respective truncation errors should be of the same size. Since the truncation error with respect to

t

is of a lower order than that with respect to ~, due to the type of linearization used in equation (5. 8),

AAr::..

should be somewhat smaller than .o~. With this in mind. the grid size was chosen to be ~~

=

0.025 and ~~

=

0. 01.

The maximum value of j was taken to 111.

intervals of

The calculation is carried. out and the results are presented at

t:

=

O. 1. The calculated quantities are put in a suitable form by

VL -

~ ~

~ ~

t..aY\S+,

c:

2.

t

~a- ~o..(j

\.Á=

~,

0

rr

(6.12)

Simpson's rule is used to evaluate

vL

0 The intergrand is

indeterminate at ~:.O . and is evaluated with the help of L'Hospital's rule.

These results are presented in tables 1-12 and in Fig. 1.

The shear stress at the plate is proportional to the slope of the velocity profile <1tthe plate and thus

--aG0

_...L dT

is evaluated at the plate using L'Hospital's rule

~\\

_ \ ë:)'Z.T\

O~ ~.O

-

~~

aCS'"'Z.

"l"' ..

~

(6. 14)

The second derivative is calculated with a five point formula to give third order truncation error. The re sult is shown in table 13 and in Fig. 2.

7. RICHARDSON'S EXTRAPOLATION AND ERROR ESTIMATES Equation (2. 6) reduces to

'2. \

.

~ ~2.

= :- \

0+

0--=0

(7. 1)

This fact has been used to check on the accuracy of the solution together with the fact that

~\

at the plate tends to unity as

t

tends to infinity. These checks only~~tablish the fact that the finite difference solution correctly approximates the true solution but they do not provide a measure of the error.

A technique used in the numerical solution of ordinary differential equations to give an estimate of the error, known as the

Richardson extrapolation (Ref. 7) or the deferred approach to the limit, can be extended to apply to partial differential equations.

Assuming no round off error, the error at any point is a function of the step size. If f is some function of the true solution at some point and if

.ç

(v,J

Ó)

is the same function of the finite difference solution with step size

'n

and

<5

at that point, then

'2..-Ç-f

(Vl)&)

=

-Ç'

+

VllfV1.

+

ó

~

ot

~

Vt

2

~ V\~

1-

\16

?~~ó

+

.L

Ól.

'd2..ç

-T

0

r

Vt

~

Ó

~.,

2. d~2.

L..)

~

The error is given by

(7. 2)

e. -

-Ç(v"Ó)-t

=

'nll~ ~g5

1

-- J ~\-1-aÖ

(7. 3)

If the step size is small enough so that third order contributions to the error can be neglected, then six different finite diffeal'nce-eso(tions are nece"ä'if to calculate f and components of the error

~)

ö> ) ö"

~

)

~~\aJ

and

-a

~

.

This does not mean too much additional computation time since the step sizes are all chosen equal or larger than the basic ones. At the same time the larger grid size means that the effects of round-off are smaller.

If Vt:6~

6=,A't:

and f1

=

f(0.025, 0.01) f 2 ;: f(O. 025, O. 02) f3

=

f(O~ 05, 0.01) f4

=

..

f(O. 05, 0.02) (7.4) f5 = f(0.05, 0.0125) f6

=

f(O. 0375. O. 02) 11

then

-Ç'=

4f,+

2f~+

COf

3

+

ç-Ç'4-\?'~s-6.ç"

(7. 5)

Using an IBM 650 digital computer, the basic solution f 1 up to

te:

= 1. 0 took approximately 2 hrs. of computing time with the remaining solutions requiring a total of 4 1/2 hrs.

\

Both

r7

and

Ll.

are extrapolatedo Since there are errors in both, it is necessary to correct the extrapolated \.J,.' so that the u..'.)S can be compared at the same

VL

0 Since the error in

yt

is very small, only the

first two terms of the Taylor series are used to give sufficient accuracy. That

is, if

yt)

lJ..I are the calculated values and

'1e:<..

~ lt~are the extrapolated values,

th en , \

a \

\.Äo

=

u.~

+

(V(.-

VL~) ~~

, , 0 . \"Q:I(" (7. 6)

The error.

U

-Uo)is plotted in Fig. 3. The step size choice (7.4) means

that the error could be calculated at only seven points across the boundary layer. The error near ~

0

is not shown since here the round-off error plays a large role (co f. section 6).

d.f.

It is found that the largest contribution to the error comes from

6

~

and

~

~'4, ~.

The other components, if not zero, are

sufficiently small so thaf t1t:y vary randomly due to round-off. This indicates that there is no interaction between the two coordinates and that the truncation error with respect to ~ is essentially first order while that due to

cr-

is second order.

8. DISCUSSION

The solution as plotted in Fig. 1 shows the velocity profiles as they approach the asymptotic exponential profileo In order to determine when the asymptotic velocity profile has effectively been reached. it is necessary to consider when some characteristic property of the velocity profile such as the momentum thickness or the shear stress at the plate has reached some percentage of its asymptotic value. Since in general this will occur at different velocity profiles. such a determination is rather arbitrary and so it is also possible to choose a velocity profile simply by inspection.

It is convenient to choose ~* lil: \ • At this point the velocity profile is

visibly close tothe asymptotic profile and the shear stress at the plate is within one percent of the asymptotic value. as is seen in table 13 and Fig. 2.

One characteristic of the solution that may be noticed in Fig. 1

is that the velocity profiles approach the asymptotic profile much sooner near

the plate than out near the edge. Since the asymptotic profile represents a balance between the electromagnetic force and the viscous shear force. this

indicates that the balance is first achieved near the plate and then later out near the boundary. In fact. the equations of motion show that this balance

The error. that is, the difference between the solution given in

tables 1-12 and the true solution is plotted in Fig. 3. The error accumulates

as the calculation proceeds from the leadin~ edge.. This is not all due to the

finite step size in

1:::::.. •

The error due to

A\S'

also grows although at a

slower rate than that due to

At.

In order to reduce the error it is possib1e

to reduce.ce:.~ but it is not possible to reduce A~ without changing the scheme

. for the numerical solution or employing double-precision arithmetic to correct

the loss of accuracy near 0"0:.

0

.

_,

The error in the tabu1ated values of

i~

at

yt=

0

is not shown.

Since these values were calculated using the first five values of T near the

plate where round-off would produce a randomly fluctuating error in T, it is

likely that the five point formula would tend to average these out and produce

a maximum error

in~

of the order of 10 -4 - 10 - 5. However. if there were

no round-off error. the probable slopes of the error curves of Fig. 3 at

show that this error could be as high as 10-2 - 10- 3.

Figure 2 compares Rossow's second order calculation as well

as Radlow and Erickson 's linearized solution and Dix's finite difference

calculation of the slope of the velocity profile at the plate with the present

calculation. Rossow's solution agrees well up torg ~ O. 3 and it may be

expected that the velocity profiles will also agree to this point. The higher

order approximations become progressively more complex although they

would extend the agreement to higher values of

S .

Radlow and Ericson 's

value of

~\oiS

initially slightly higher but it crosses over and approaches

the asymptotic value sooner than the present calculation. In view of the

similarity of the result to the "Rayleigh solution" it is believed that the

difference is simply due to the Oseen-type linearization. The calculated

values of Dix alllie above the present calculation but it is easy to see that

the primary reason for this is the artificially placed downstream boundary

which acts as astrong constraint.

1. Rossow, V. J. 2. Radlow, J. Ericson, W. B. 3. Dix, D. M. 4. Lock, R. C. 5. Iglisch, R. 6. Richtmyer, R. D. 7. Richardson, L. F. Gaunt, J. A. REFERENCES

On the Flow of Electrically Conducting Fluids over a Flat Plate in the Presence of a Transverse Magnetic Field. NACA Report 1358, 1958.

Transverse Magnetohydrodynamic Flow Past a Semi-Infinite Plate. Phys. Fluids, Vol. 5, No. 11 Nov. 1962. p. 1428

The ,.l\I.IB.gnetohydrodynamic Flow Past a

Non-conducting Flat Plate in the Presence of a Transverse Magnetic Field.

Jour. Fluid Mech. Vol. 15, Part 3, March, 1963, p.449.

Stability of the Flow of an Electrically Conducting Fluid between Parallel Planes under a Transverse Magnetic Field. Proc. Roy. Soc. of London,

Ser. A. Vol. 233, No. 1192, 6 Dec. 1955, p.116. Exact Calculation of Laminar Boundary Layer in Longitudinal Flow over a Flat Plate with Homo-geneous Suction. NACA Tech. Memo. No. 1205, 1949.

Difference Methods for Initial- Value Problems. Interscience Publishers, Inc., New York, 1957. The Deferred Approach to the Limit. Trans. Roy. Soc. (London), 226A, PPD 299 -361, 1927.

APPENDIX

The Compressib1e Asymptotie Boundary Layer

It is interesting to consider the question of the existence of the eorresponding asymptotie "Hartmann" boundary 1ayer in compressib1e flow. The flat p1ate compressible boundary 1ayer equations rep1acing equations (2. 1, 2.2) are

We wish to obtain a solution such that u = u(y) T = T(y) Therefore again as in equation (3. 2)

\J"

=

0

and the equations (A. 1 - A. 3) become

~

(~~) ~ cr~~(

u.co-u..)

=

0

(A. 1) (A. 2)

(A. 3)

(A. 4) (A. 5) (A. 6) (A. 7)

~

(\e.~)

+

tA.(~Y-+ C-'S~Uoo-~t=

0

To solve these equations it is necessary to know k,

fAJ

and

0'"

as functions of the temperature, but it is possib1e to determine if such a solution exists without this detailed know1edge. Equations (A. 6, A. 7) together possess the

following integra1

cl

_

'R..

~

+

fA

l

\À -

~QC)

' )

~

-

c..

(A. 8)

Where

c..

is an integration constant that may be determined to be equal to zero sinee at the outer edge of the boundary 1ayer

6

'7d.~

)

0\

~d.:1

=

0

Dividing through by k, we have

2-ó.T

+

~

dlU-U-c) -

0

cg

.<i.J....

a:

:x..

-

(A. 9)

Under conditions where kj)--'. is constant (small degree of ionization in a pure gas) this ean be integrated, producing

\-T-

~

(LÀ-

\..ÀcO)-'

=

\CIQ

::2.;';'- (A. 10)

Where the integration constant is determined in the free stream. At the wall, equation (A. 10) takes the form

.,-

V2..

I l 2.. (A. 11)

T

_w

=

'00 -

~ '-"\~

Therefore, this result shows that an asymptotic solution is possib1e on1y if the wall temperature has a prescribed va1ue given byequation (A. 11). Under other conditions the boundary 1ayer will continue growing.

CALCULATED VELOCITY PROFILES

TABLE 1 TABLE 2 TABLE 3

~. 0.0100 ~- 0.0400

,,-

0 . 0 9 0 0 1\ lA' 0 .. ~OO U' O. ~O 0 0 U' 0.0000 0 . 0 0 0 0 0.0000 0.0000 0 . 0 121 0.0414 0 , 0231 ", 0 4 32 0 . 0 3 2 5 0.0459 0 . 0242 O.OR?? 0.0463 0.OA61 0.0653 0.0910 0.0363 0.1239 0.0695 O.1?86 0 . 0 9 8 4 0 . 1 3 5 5 0 . 0 4 8 4 O.1F.49 0.0929 0.1707 0.1317 0.1793 0 . 0 6 05 0.2058 0.1164 O.21?5 0.1653 O.22:?4 0.0727 0. 2 465 0 . 1 4 0 0 0.2539 0 . 1 9 9 2 0.2';49 0.0848 O.2R70 0.1637 O.ZQ"'8 0.2333 0.3067 0.0971 O.3?72 0.1875 0.3353 0.2677 0.3477 0.1093 0.3n69 0.2114 0.3753 0.3023 O.3AR1 0<. 1216 0 . 4 0 .... 3 0.2354 0 . 4 1 4 7 0 . 3 3 7 2 0 . 4 2 7 6 0 . 1 3 3 9 0.4451 O.2~96 0.4535 0.3725 0.4664 0.1463 a . d A 3 ) 0.2840 0.491.5 0.4080 0.5042 0.1587 0 .. 5? t} B 0.3085 0.528A 0,4439 0.5412 0.1713 0.5575 0.3331 u.!:)6~2 0 . 4 8 0 1 0.5771 0.1839 0.5932 0.3580 0.6005 0 . 5 1 6 6 0.6119 0.1966 O.6??9 0.3831 0. 6348 0.5536 0.6456 0.2093 0.6"15 0 . 4 0 8 ' 0.6679 0.5910 O.67RQ 0.2223 O.6~37 0.4340 0.6997 0.6288 0.7091 0.2353 0.7247 0.4599 0.7301 _g:~~~~ 0.7368 0.2485 0 . 7 5 4 1 0.4861 0.7591 0.7';70 0.2618 u.7819 0 . 5 1 2 6 0 .. 7FUi5 0.7454 0.7936 0.2753 u.Bloe1 0.5395 O.Rl:>'2 0 . 765"" 0.8186 0.2691 O."'):.!'" 0.5668 o. fI 362 0.8260 lJ.tJ"19 0.3030 0.8552 0.5945 O.85~. 0.867"" o. eli 3.

0.3171 O.A7"0 0.622.7 O.~7A7 0.9095 O.8A31

0.3315 O.E\!?49 0.6514 ( J . 850 0' 3 0.952.3 0.9010 0.3462 0.9120 0.6807 0.9140 0.9960 0.9171 0.3611 O.9?71 0.7105 O.9?eA 1.0""07 0.9315 0.3764 0.9405 0.7"" 1 0 O.'i419 1.0862 0 . 9 . 4 1 0 . 3 9 2 0 0.9521 0.7721 0.9532 1.1328 0.9550 0.4079 0.91i20 0.8039 O.QI\?9 1.1805 0.9"'41' 0 . 4 2 4 2 0.970"" 0.8365 0.9711 1.2293 0.9722 0.4409 0.9773 0.8698 O.Q77S 1.2793 0.9787 0.4580 0.9829 0.9040 0.9833 1.3305 0.9A40 0.4755 0.98741 0.9390 0.9A77 1.3829 0.9882 0 . 4 9 3 "" 0.9909 0.9749 0.9912 1.4367 0.9915 0.5118 0.9936 1.0117 0.993A 1.41919 0.9940 8:H86 0.9956 1.0""9"" O.99!=i7 1.5<484 0.9959 0.9971 1.0880 u.~971 1.606<4 0.9973 0.5698 0.9981 1.1276 0.99~1 1.6657 0.9982 0.5901 0.9968 1.1681 0.99~~ 1.7266 0.9989 0.6108 0.9993 1.2097 0.9993 1.7889 0.9993 0.6321 0.999" 1.2522 O.999ti 1.8527 0.9996 0.6539 O.999R 1.2957 O.99 Q€I 1.9179 0.9998 0.6761 0.9999 1.3402 0.9999 1.9847 0.9999

0.6989 0.9999 1.3857 0.9->99 2.0529 O.Y999

0.7221 1.0000 1.4322 1.0000 2.1227 1.0000

TABLE 4 TABLE 5 TABLE 6

~-0.1600 ~. 0.2500 ~- 0.3600 1

u'

O. ~O 00 \,\' 0 . 0 \ 0 0 UI 0.0000 0.0000 0.0000 0.0000 0.0404 O.u"!}l 0.0471 0.0526 0.0528 0.0561 0.0814 0.0970 0.0950 0.1035 0.1068 0.1102 0.1229 0.1439 0.1437 0.1530 0.1618 0.11i}!4 0.1648 O.lRQ? 0.1931 0.2011 0.2176 0.2130 0 . 2 0 72 0.2345 0.2432 0.2479 0.2745 o. 2t:; 18 0.2501 0.27A<4 0.2939 0.2933 0.3322 0.3089 0.2934 0.3212 0.3454 0.3374 0.3910 0.3544 0.3373 0.3631 0.3977 O.3A02 0.4508 0.3982 0.3816 0.4039 0.4507 0.4217 0.5116 0.4405 0.4264 0.4436 0.5044 0.4619 0.5734 O.4R12 0.4717 O.4R26 0.5589 0.5008 0.6363 0.5? 0 3 0.5176 0.5?03 0.6142 0.5385 0.7002 O.557Q 0.5640 u.~569 0.6703 0.5747 0.7653 0.5°39 0 . 6 1 1 0 0.592.3 0.7273 0.6097 0.8316 O.6aOili 0.6585 0.6265 0.78S2 0.6433 0.8990 0.6"13 0.7067 0.6594 0.6439 0.6754 0.9677 0.69 ?? 0.7556 0.6911 0.9036 0.7062 1.0376 O.7?26 0 . 8 0 52 0.7213 0.9643 0.73504 1.1089 O.750R 0.8554 U.75D1 1.0260 0.7632 1.1815 0.7775 0.9065 0.7773 1.0888 0.78904 1.2555 0.FJU?6

0.95 B iII O.t1lo30 1.1527 0 . 8 1 ] 9 1.3310 O.8?60

1.0111 0.8270 1.2178 0.8369 1.4081 o. A 4 ~I A 1.0648 0.80493 1.2842 O.8SRl 1.4867 0.8679 1.1195 0.81i99 1.3519 0.9777 1.5671 O.8R63 1 . 1 7 52 O.8RBA 1.4209 0.8956 1.6492 0.9031 1 . 2 3 2 0 0.9059 1.<49104 0.9118 1.7332 0.9183 1.2900 0.9213 1.5635 O.9?63 1.8191 0.9319 1 . 3 4 9 2 0.9350 1.6372 0.9392 1.9071 o. Q 4 :1Q 1.4098 0.9470 1.7126 0.9505 1.9972 0.9543 1.4718 0.957<4 1.7898 0.9602 2.0894 O.9Ft34 1.5352 0.9663 1.8689 0.9685 2.1840 0.9711 1.6001 O.9? 37 1.9499 0.9755 2.2810 0.9775 1.6666 0.9799 2.0329 0.9813 2.3805 0 . 9 0 2 9 1 . 7 3 . 6 0.9A49 2.1180 0.9860 2.4825 0.9872 1.8048 u.98a9 2.2053 O.9R97 2.5871 0.9906 1.8764 0.9920 2.29<49 0.9926 2.6945 0.9932 1.9499 0.99"4 2.3867 0.9948 2.8046 0.9953 2.0253 0.9961 2.<4809 0.9964 2.9176 O.996A 2.1025 0.9974 2 . 5 7 7 . 0.9976 3 . 0 3 3 · 0.9976 2.1817 0.9~83 2.6763 0.9985 3.1521 0.9986 2.2628 0.9989 2.1777 0.9990 3.2737 0.9991 2.3459 0.999"" 2.8815 0.9994 ' . 3 9 8 3 0.9995 2.4309 0.9996 2.9878 0.9996 3.5259 0.9997 2.5179 0.99Qe 3 .. 0966 0.9998 3.6564 0.9998 2.6070 0.9999 3.2079 0.9999 3.7899 0.9999 2.6980 O.9~!?9 3.3216 0.9999 3.9264 0.9999 2.7910 1.0000 3. <4 37 Q 1.00eo 4.0659 1.0000

CALCULATED VELOCITY PROFILES

TABLE 1 TABLE 8 TABLE 9

~. 0.4900 1- 0 . 6 4 0 0 '<j~ 0.8100 0.0\ 0 0 U' I{ lA' O. ~OO 0 lA' 0.0000 0 .. 00 0 0 0.0000 0.0000 0.0579 O.OSQ'; 0.0624 0.0~31 0.0666 O.Oti65 0.1172 0.11'~ 0.1265 O.1~34 _0.1351 0.1298 0 . 1 7 7 7 0.1719 0.1920 0.1 Rl 2 0.2052 0.1903 0 . 2 3 9 4 O.2?4R 0 . 2 5 9 0 0 .. 2., 6 '" 0.2769 0 . 2480 0.3022 0.2757 0 . 3 2 7 3 o. 'P~Q5 0 . 3 5 0 4 O.30?9 0.3663 O.)?,4'; 0.3971 0 . 3 4 0 1 0 . 4 2 5 5 0.3553 0.4316 0.3715 0.468~ O.3QRS 0.5024 0 . 4 0 5 0 0.4982 0.41";5 0.5413 0.4345 _0.5811 O.45?2 0.5661 O.45Q'; 0.6158 0.4785 0 . 6 6 1 6 0.4969 0 . 6 3 5 3 v .. ::)\..0<,) R 0.6918 0 .. 5? 0 2 0 .7441 0.5392 0 . 7 0 5 9 0.5402 0.7695 D.5SQ9 _0.8286 0.5792 0.7779 O.577~ 0 . 8 4 9 0 0.5976 0.9150 U.6170 0.8513 0 .. 613 ti 0.9302 0.6332 1.0036 o.há25 0.9262 0.6477 1 .. 01,::3 2 0.6669 1'.0944 0 . 6 858 1 . 0 0 2 6 O.6ROO 1 . 0 9 8 2 O.69AA _{1.1874 0.7171} 1.0e06 0 .. 7107 1.1850 D.72ft? _1.2827 O.74ti4 1.1602 0.7396 1.2738 0.7568 _{1 . 3 8 0 3} 0.7736 1.2415 0.71;69 1.3647 0.7R31 _{1.4805 0.7990} 1 . 3 2 4 6 0.7925 1.4578 O.~v76 _{1.5831 O.FI??5} 1.-40941: u.h164 1.5530 0.8304 1 . 6 884 0.8,,42 1 . 4 9 6 1 O.R386 1.6505 0.A514 _1.7964 o.M"'''1 1.5847 O.BSQ2 1.7503 0 . 8 7 0 8 _{1.9071 D.BR?)} 1.6753 0.6781 1.8526 0.8A86 2.0208 0.A989 1.7681 o. 8 ~ 5 41 1.9574 0.9047 _2.1375 0.913~ 1.8630 0.9111 2.0649 0.9192 2.3572 0.9~72 1.9602 0.9252 2.1750 0.9322 _2.3802 0.9392 2.0597 0.937A 2. 2118 0 0.9438 _{2.506" 0.94f}Q₇ 2.1617 0.9"'~A 2.41039 0.9539 _2.6361 0.95~9 2.2663 0.95A5 2.5229 0.96:?7 _{2.7692 0.9"68} 2.3736 0.91S6A 2.6"SO 0.9762 2.9061 0.9736 2.4836 0.9738 2.7703 0.9766 _{3.0466 0.9793} 2.596'" 0.9797 2.8989 0.9A19 _3.1910 0.9A40 2.7123 O.9R4S 3.0310 O.9~62 _{3.3393 0.9879} 2.8311 0.9RS4 3.1667 0.9897 _{3.4916 0.9910} 2.9~31 0.9915 3.3059 0.9925 _3.6481 0.9934 3.0782 0.9939 3.4488 0.9946 _{3.8087 0.9953} 3.2066 0.9957 3.5955 0.9962 _{3.9736 0.9967} 3.3383 u.yS71 3.7459 0.9974 _{4.1429 0.9978} 3.4734 0.9981 3.9003 0.9983 _{4.3164 0.9985} 3.6119 0.9987 4.0585 0.9989 _4.494. 0.9990 3.7538 0.9992 4.2206 0.9993 4.6768 0.9994 3.8991 0.9~9S "'.3867 0.9996 _{4.8636 0.9996} 4.0"'79 0.9997 4.5568 0.9998 _5.0549 0.9998 4.2002 o .99Q_f' 4.7308 0.9999 _{5.2507 0.9999} "'.3560 0.9999 4.9088 0.9999 _{5 .... 510 0.9999} 4.5152 1.0000 5.0908 1.0000 _{5.6557 1.0000} 4.6780 1.0000 5.2768 1.0000 _5.8650 1.0UQO

TABLE 10 TABLE 11 TABLE 12

1" 1.0000 _{~c 1.2100} "i..- 1.4400 0.1000 lAl O. ~O 00 lA' 'I. U' 0.0000 0.00 nO 0.0000 0 . 0 0 00 0.0704 O.O"~A _{0.0740 0.0730} 0.077" 0.0761 0.1430 0.13"0 0.150" O.14?O 0.1574 u. l .. 7A 0.2174 0.19°1 _0.2269 0.207" 0.2397 O.21S7 0.2937 0.2590 _{0.3094 O.?} A 97 0.32"'3 0.27~9 0.3716 1).3), S!..' _{0.3920 O • .3? R"} 0.4112 0.3405 0.4519 U. 36:J 9 _{0.4769 0.3 A '" 0} 0.5006 0.3!" 7 5 0.5341 0.4210 _{0.5640 0 . 4 ) 6 '} 0.5924 0.4510 0 . 6 1 8 3 O. " Ei 9:? 0.5534 0.4 n!'>,c:; 0.6868 0.501.2 0.7046 0.5147 _{0.7452 0.531R} 0.7840 0.54182 0.7931 0.5576 0.8396 0.5752 0.8839 0.5~?0 0.8840 0.5°79 _0.9365 ('1.615 A 0.9867 0.632A 0.9771 0.6357 _{1.0361 0.65 ),;} 1.0925 0.6706 1.0728 u . h ? l l i~~;g~ O.6R~8 1.2015 0.7057 1.1709 0.7041 O.7?15 1.3136 0.7)80 1.2716 0.7349 _{1.3519 0.751~} 1."'291 0.7';78 1.3750 o . ? " , ... _1.4632 0.7797 1.5'" 6 1 0.7951 1."'812 0.7R99 _{1.5777 v.t1v.!::t.4} 1.6707 0.8201 1.5903 0.8144 _{1 . 6 9 5 '} O.A?~O 1.7969 O.804?8 1.7023 0.R3J';9 _{1 . e} j 66 0.A506 1.9270 0.8~35

1.8173 O. FIo5~' 5 _{1.9412 0.A702} 2.0610 O.eA?l

1.9356 0.FIo7ti3 _{2.0';94 o.} ~ A "Iq 2.1991 0.89BA

2.0570 0.f:l9)4 _{2.2014 0.9039} 2.3414 0.~1)8 2.1818 0.90ef' 2.337 " O.Q1~) 2."'880 0.9?71 2.3101 0.92?7 _{2.4769 0.9310} 2.6392 0.9)B9 2.4420 0.9350 _2.6208 0.94? 3 2.7946 0.94°2 2.5776 0.9459 _{2.7668 0.95?2} 2.9553 0.95A2 2.7169 0.Q554 _{2.921? O. Q" 09} 3.1205 0.91559 2.8602 0.9637 _".0779 O.qA~) 3.2907 0.9725 3.0075 0.9709 _{3.2393 O.!l747} 3.4660 0.9782 . ".1589 0.9769 _{3 .... 053} O.9F\OO 3.6465 O.9B29 3.3146 0.9A?0 _{3.5761 O.9'i44} 3.B324 O.Q:J67 3.4746 0.9A61 _3.7517 (l.9~E"1 4,.0236 0.9899 3.6391 0.9895 _3.9323 0.9~10 4.2203 0.99?4 3.8082 0.99?2 _{4.1180 0.9!"34} ... 226 0.99 ... 3.9818 0.9944 4.308A 0.9~~2 4.6306 0.9960 "'.1602 0.9960 4.5049 o. 9 ~ A'; 4.8"'43 0.~972 4.3433 0.9972 4.70';2 O.Q~"ti 5.0639 0.9980 4.5312 0 . 9 9 ' , 4.9129 0.~98.4 5.2892 0.9987 .... 7240 0.99f'7 _{5.1249 n.Ygeo} 5.5205 0.9991 4.9217 0.9992 _{5.34a3 0.99°3} 5.7576 0.9994 5.1244 0.9995 _{5.5652 0.99°6} 6.0008 0.9997 5.3319 0.9997 _{5.7935 0.9997} 6.2498 0.9998 5.5445 0.999A _{6.0273 0.99°9} 6.5049 0.9999 5.7620 0.9999 6.2666 0.9999 6.7659 0.9999 5.9845 0.9999 6.5114 1.0000 7.0329 1.0000 6.2121 1.0UOO _{6.7616 1.0000} 7.3059 1.0000 6 •• • • 6 1.0000 7.0 j 7.4 1.0000 7.5849 1.0000

TABLE 13

THE SLOPE OF THE VELOCITY PROFILE AT THE PLATE

0.0 0.01 0.04 0.09 0.16 0.25 O. 36 0.49 0.64 O. 81 1. 00 1. 21 1. 44 0.0 .003 .011 .034 .0839 . 178 .253 .357 ".454 .552 . 65 1-1 14.26 8. 38 4. 17 2. 37 1. 61 1. 29 1. 21 1. 18 1. 13 1. 09 1. 10

o~\

_aK

0 present calc. 00 3.4339 1. 8789 1. 4206 1. 2250 1. 1278. 1. 0751 1. 0450 1. 0272 1. 0163 1. 0096 1. 0054 1. 0027

a~\

alt\.

0 Radlow (X) 3. 4511 1. 8616 1. 3874 1. 1871 1. 0915 1. 0439 1. 0203 1. 0088 1. 0035 1. 0012 1. 0003 1. 0000

d~1

a~ODix

Case 3-1 14. 07 8.25 4. 11 2. 31 1. 55 1. 23 1. 16 1. 10 1. 07 1. 01 1. 05

~

O. 0 .01 .03 .06 .11 · 19 · 34 · 551 .80 1. 05 1. 31 2-1 7.73 4.66 2.96 2. 18 1.7 1.4 1. 21 1. 13 1.1 1. 09 1. 09

9~\

'oYl

0 Rossow 00 3.4339 1.8786 1. 4136 1. 2004 1. 0646 .9427 4-1 7.73 4. 66 2.96 2.18 1.7 1.4 1. 21 1. 13 1. 09 1. 09 1. 09

p---~--~--~----~--~---.----~--~--~--~Q

~

o

---4----+_--_+----~--~--_4----+_--_+----~--4~

Q

~ ~

Q

N ~~--~----+----+----~--~--~Q

o

U) ~ ~ ~ ~ 0 ~ ~ ~ ~ ~ U 0 ~ ~

>

~

.

d ~ ~

1.1" ol a 7 ~ ~ A ? nr SI 7 ~ • A

•

? ,n, v.vv

o

.2

-•

! ••

. . . -

.

. .

I

-•

.4

.6

IC:

.8

1.0

§

;;;1;;',=_

§

11'

11

lil

1.2

1.4

unclassified

Security Classification

DOCUMENT CONTROL DATA· R&D

(Securlty ct.aalt/c.tlo.. ol tltt •. body ol .bat,.ct, .nd /ndel'ln, annotàtlon muat be ente,ed when the ove,." ,epo,t i. ~I.a.llled)

I. ORIGINA TIN G ACTlvl'!'Y (Corpo,.te .utho,) Ze. REPORT Sf;CURITY C L.ASSIFICATION Institute for Aerospace Studies,

University of Toronto, Zb, ~ROUP

Toronto 5 Ont-, Canada

J. REPORT TITLE

THE FLAT-PLATE MAGNETOHYDRODYNAMIC BOUNDARY LAYER IN A TRANSVERSE MAGNETIC FIELD

•. DESCRIPTIVE HOTES (TYI» ol report _d Inclu.l .... d.t •• )

Scientific Interim

5. AUTHOR(S) (L' •• t ... lI,.t n ... Inlll.I)

DUKOWICZ, John, K.

6. REPO RT DI. TE 7~. 'T'OTAL. NO. OF P"'~ES !7b. NO. O~ REFS

January 1967 19

8 •. CONTRACT OR ~RANT NO. S •. ORIGINATOR'S REPORT .... Ut,lIlIER(S) AF-AFOSR-366-66

b. PROJEC TNO.

9783-02 UTIAS Technical Note No. 68

ç. 61445014

Ä

FHÜ~rir

N0(6(7

_o:·Öum

Ö"

ö t i Y

be . . . I.,.d d.681307

10. AVA IL ABILITY!LIMITATION NOTICES

l . Distribution of this document is unlimited.

" . SUPPL EMEN TARY NOTES 12. SPONSORING MILITARY ACTIVITY _(SREM)

Tech. Other Air Force Office of Scientific Research 1400 Wilson Boulevard

Arlington Virginia 2220Q

13· ABSTRAc,T

The equations of motion for a flat plate boundary layer flow of an incompressible, electrically conducting fluid in the presence of a transverse magnetic field mov-ing with the main. stream velocity have been solved numerically for the case of negligible induced magnetic field. The equations of motion have been tran~formed

into a universal form with no characteristic parameters present. The velocity profiles have been calculated as they range from the Blasius profile to the asymptotic exponential profile. The asymptotic profile is reached in a distance

xl~ ~ U /~B2 from the leading edge.

solution has also been calculated.

~, __ ~S~e~c~u~r~it~y~C~la~s~s~i~fi~c~a~ti~o~n~ ____________________________ ~---~---~---__ --~~

14, LINK A LINK B LINK C

KEY WORDS

MBD Boundary Layer

Flat Plate Boundary Layer Incompressible Boundary Layer

ROL..E WT ROL..E WT ROL..E WT

INSTRUCTIONS

1. ORIGINATlNG ACTIVITY: Enter the name and address

of the contractor, subcontractor, grantee, Department of

De-fense activity or other organization (corporate author) issuing

the report.

2a. REPORT SECUHTY CLASSIFICATION: Enter the ove~

all security classification of the report. Indicate whether

"Restricted Data" is inc1uded. Marking is to be in

accord-ance with appropriate security regulations.

2b. GROUP: Automatic downgrading is specified in 000 Di· rective 5200.10 and Armed Forces Industrial Manual. Entec

the group number. Also, woon applicable, show that optional

markings have been used for Group 3 and Group 4 as author·

ized.

3. REPORT TITLE: Enter the complete report title in all

capital letters. TiUes in all cases should be unclassified.

IC a meaningful title cannot be selected without

classifica-tiOD, show tiUe c1assification in all capitals in parenthesis

immediately following the title.

4. DESCRIPTIVE NOTES: If appropriate, enter t~e type' of

revort, e.g., interim, progress, summary, annual, or final.

Give the inclusive dates when a specific reporting period is

covered.

5. AUTHOR(S}: Enter the name(s} of author(s} as shown pn

or in the report. Ent et' last name, first name, middle initial.

If :r.ilitary, 'Show rank end branch of setvice. The name of

the principal anthor is an absolute minimum requirement. 6. REPORT DATE: Enter the date of the report as day,

month, year; or month, year. If more than one date appears on the report, use date of publication.

7a. TOTAL NUMBER OF PAGES: The tot al page count shou~d follow normal pagination procedures, i.e., enter the

number of pages containing informaHon.

7b. NUMBER OF REFERENCES: Enter the total number of

references cited in the report.

8a. CONTRACT OR GRANT NUMBER: IC appropriate, enter the applicable number of the cO!'\tract or grant under which

the report was written.

8b, Sc, & 8d. PROJECT NUMBER: Enter the appropriate

military department identification, such as project number,

aubproject number, system numbers, task number, etc.

9a. ORIGINAroR'S REPORT NUMBER(S}: Enter the offi·

cial report number by wbich the documet1t will be identified and controlled by the originating 'activity. Tbia number must

~e unique to this report.

9b. OTHER REPORT NUMBER(S): IC the report has been

assigned any other report numbers (elther by the originator

or by the sponsor), also enter tbis, number(s).

10. AVAILABILITY/LIMITATION NOTICES: Enter any lirn.

itations on further dissemination of the report, other than those

GPO 886-551

imposed by security c1assification, using standard 'statements

such as:

(1) "Qualifled requesters may obtain copies of this

report from DOe."

(2) "Foreign announcement and dissemination of tbis

report by DDC is not authorized."

(3) "u. S. Govemment agencies may' obtain copies of this report direcUy from DDC. Other qualified DDC users shatl request through

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(4) "U. S. military agencies may obtain copies of this

report directly from DOe. Other qualified users shall request through

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(5) "All distribution of this report is controUed. Quat·

ified DDC U8ers .hall request throu'gh

____ ... __ ... __ .... ____ ... _' ,'t

If the report has been furnished te the Office of Technical Services, Department of Commerce, for sale to the public, indi· cate tbis fact and enter the price, if known.

lL SUl'PLEMENTARY NOT ES: Use for additional

explana-tory notes.

12. SPONSORING MILITARY ACT!VITY: Enter the name of

the departmental project office or laboratory sponsoring (pa".

i(lg for) the research and developmet1t. Inc1ude address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, evet1 though

it mayalso appear elsewhere in the body of the technica 1 re·

port. IC additional space is required, a continuation sheet shaU

be attached.

It is highly desirabie that the abstract of c1assified reports

be unclassified. Each paragraph of the abstract shaU end with an indication of the military security c1assification of the in

-formation in the paragraph, represented as (TS)" (5), (C), ot (U),

There is no limitation cn the length of the abstract. How·

ever, the suggested length is from ISO to 225 words.

14. KEY WOROS: Key words are technically meaningful termll

or short phrases that characterize a repott and may be used

a.

index entries lor cataloging the report. Key words must be

selected so that no security classilication is required. Ident!·

fiers, such as equipment model designation, trade name, military

project code name, geographic location, riJay be used as key

words but will be followed by an indication of technica 1

con-text. The asslgnment of links, rule8, and weights is optional.