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i

Total No. Pages iv and 55 Copy No. ( 2 r o f (74

TECHNICAL REPORT NO. 331

AXISYMMETRIC LAMINAR AND TURBULENT

JETS O F HYDROGEN WITH SIMPLE CHEMISTRY

By Harold Rosenbaum

P r e p a r e d for

George C. Marshal l Space Flight Center Huntsville , Alabama

Under Contract No. NAS8-2686

P repa red by Resea rch Deaprtment

Merr ick and Stewart Avenues Westbury, L . I . , New York

General Applied Science Laboratories , Inc.

The author wishes to express h is thanks L U M A . A CAUL A.. U L " W J - A I - u----- - - - - - - ~ for their many helpful suggestions.

A- n- n,...l A T 4Lh-r 9 - A r . A s l i a K l e i n n t e i n

January 4, 1963

P r e.s i dk n t

t

ii

TABLE O F CONTENTS

SECTION TITLE

I

I1

LIST O F SYMBOLS

SUMMARY

INTRODUCTION

ANALYSIS

CONC LU SI0 NS

iii

1

2

4

17

Reference s 18

... 111

LIST O F SYMBOLS -

radius of je t a

Ai constants appearing in Equation ( 29)

constants appearing in Equation ( 2 0 )

stagnation enthalpy

s ta t ic enthalpy

Pi C

H

h

hi

K

s ta t ic enthalpy of species i

p for laminar flow;

constant = 0.025

offset c i rcu lar probability function

pfor turbulent flow

n

P

rad ia l coordinate r

T s ta t ic temperature

T r

U

constant appearing in Equation ( 20)

velocity ra t io u /ue

velocity component in x-direction U

velocity component in r -direction

volumetric r a t e of production of ith species

mean molecular weight of mixture

V

axial coordinate X

xi mole fraction of i th species

Y i

'i

mass fraction of ith species

element mass fraction of ith element

constants appearing in Equation (20)

eddy viscosity for compressible flow €

- f eddy viscosity fo r incompressible flow

constants appearing in Equation ( 2 9 )

density of mixture

wi P

iv

. Subscripts

f radius at flame sheet

inc incompressible values

I laminar values

t turbulent values

e external flow

AXISYMMETRIC LAMINAR AND TURBULENT

JETS O F HYDROGEN WITH SIMPLE CHEMISTRY

By Harold Rosenbaum

SUMMARY

32088 The problem of a uniform axisymmetr ic j e t of pure hydrogen issuing into

a paral le l f r ee s t r e a m of air is discussed. There a r e considered both laminar

and turbulent flows under the assumption that all Prandt l and Lewis numbers a r e

equal to unity, and that the limiting chemical behaviors of frozen and equilibrium

flow prevail; the flame sheet approximation is applied to the equilibrium model.

Calculations a r e performed along a point on a typical t ra jectory for miss i le

launch.

2

f. INTRODUCTION

During the launch of multistage vehicles it i s sometimes necessary to discharge

overboard hydrogen, o r other fuels, f rom the upper stages. Since combustion of

these gases with the surrounding a i r s t r e a m may occur in the neighborhood of the

vehicle, localized convective heating and alterations of the forces and momenta

on the vehicle may resul t . A s e r i e s of studies of the phenomena involved in the

ejection, mixing and combustion of gaseous fuels is being c a r r i e d out.

The problem considered h e r e is that of a uniform axisymmetr ic je t of pure

hydrogen issuing into a uniform paral le l f ree s t ream. Both turbulent and laminar

flow models with unity Prandt l and Lewis numbers a r e treated.

shown schematically in Fig. 1.

The flow field i s

The analysis presented he re represents an extension of work performed by

Libby (Ref. 1) and Kleinstein (Ref. 2). In Ref. 2 a l aminar axisymmetr ic je t

of pure hydrogen issuing into and reacting with a moving air s t r e a m was analyzed.

In both analyses a l inearization of a modified Oseen nature was employed together

was a s tandard von Mises transformation. The equation for the velocity field in

the t ransformed plan was then reduced to a heat flow equation whose solution is

known. The chemical behavior of the j e t flow and the dominating viscous phenomena

ei ther laminar o r turbulent, then determine the inverse transformation to the

physical plane ~

In this study the laminar reacting j e t is considered by coupling the chemical model

used in Ref. 1 with the fluid mechanical model discussed above. The turbulent

reacting je t is analyzed much as in Ref. 1 , but with the fo rm of the eddy viscosity

given in Ref . 1 modified according to the model of the eddy viscosity suggested in

.I 3

Reference 3; this model is presently believed to be m o r e c o r r e c t than that employed

in Reference 1.

Two limiting chemical behaviors a r e studied: frozen and equilibrium flow.

The equilibrium composition is approximated by a "flame sheet model" af ter

I

Reference 1. Assumption of unity Prandtl and Lewis numbers allows an application

of Crocco integrals to the solution of the energy and mass fraction conservation

equations.

4

11. ANALYSIS

The equations of motion describing laminar and turbulent je t s a r e taken

to be identical in form, with mean turbulent quantities replacing their laminar

counterpar ts , Assuming all laminar and turbulent Prandt l numbers and Lewis

numbers equal to unity the governing equations are:

Conservation of Momentum:

Conservation of Energy:

aH 8H i a Pu a, t p v a , =r - a r

Global Conservation of Mass:

Conservation of Species: -#

p' a, -t PV a, a y i l a [Kr '$1 a yi

= w i + - T a, where for laminar flow:

K = p. = laminar coefficient of viscosity

and for turbulent flow

( 4 )

i 5 j

K = E p ; E = eddy viscosity coefficient

. The initial and boundary conditions are:

x = 0 ; O S r < a

U = Uj ; v = 0; H = Hj ; Y , = Y3= Y4 E 0 ; Y z = 1

x = 0 r > a

u = ue v = 0

x r o

i i m ; H = H,; u = ue; v = 0 ; Y2= Y3= 0; Y l = Yl e ; Y4 = Yqe r + m

with regular i ty condition applied along the je t centerline, r = 0.

Introduce a s t r e a m function:

pur = peueGQr ( 7 )

- p v r = peueGGE,.

and apply the von Mises transformation, i, e. t ransform f rom x, r +x, % ;then

and equation ( 1) becomes without approximation

Laminar Flow

Following the modified Oseen approximation of Kleinstein (Ref . 2) for

laminar flow , as sume

6

* Employing this assumption and introducing the additional t ransformation,

Equation ( 9 ) becomes

where:

In the present report the approximation function f ( X ) in Equation ( 11) is along

the jet center l ine, then

( 1) and the inverse of equation ( 11) becomes,

Turbulent Flow

F o r turbulent flow Libby (Ref , 1) has shown:

and has approximated F( x) by,

Po €

P e U e F ( x ) = -

( 1) In Ref . ( 2 ) pc is taken to be a constant equal to p . At the suggestion of G. Kleinstein it was decided here to take pc as a I unction of x to be evaluated along the je t center line.

7

where po = some reference density

E = incompressible eddy viscosity

4

-

is the jet half radius in the incompressible plane. % inc and ( r

It has been shown, however, by F e r r i et a1 (Ref . 3 ) that a m o r e accu ra t e

representation of F( x) might be:

where now , the half radius in the compressible plane is defined by,

Employing the approximation ( 14) and the additional transformation:

Equation ( 9 ) t ransforms to:

The in~er-p trz .nsfnrm-st inn for the axial coordinate is

( 1 7 )

8

The equations describing the velocity field, equations (12) and ( 16) , for

laminar and turbulent flow respectively, are identical in f o r m and independent

of the density and viscosity. The initial and boundary conditions fo r both cases

corresponding to the flow field shown in Fig. 1 a r e as follows:

\k u (0,q = u. 3 = U j /Ue

with the regular i ty conditions applied along the je t center l ine, = 0. The

solution of either equation, 12 o r 16, subject to the above condition is well

known in the theory of heat conduction ( cf Ref. 1 and 2) ; it i s :

( 18)

where P i s the' o f fse t c i rcu lar probability function and has been tabulated by

Masters (Ref . 4 ) .

Transformat ion to Physical Coordinates - General Remarks

To apply the solution given by equation ( 18) the inverse of the t r ans -

formations given by equation ( 8) and by either equation ( 13) o r equation ( 17)

must be employed. Therefore it i s necessary to consider tne cnemicai modei

to obtain the density ra t io p / p e and the center line propert ies needed to

per form these t ransformations. The chemical model employed here and

the resul t ing solutions for the density ratio follows that of Libby (Ref , 2)

the je t is assumed to be of pure gaseous hydrogen and the external s t r e a m

i. e .

to be air. Moreover, there is considered

9

a mixture of four species , denoted by the subscr ipts i = 1 , 2, 3 , 4 , respectively,

molecular oxygen, molecular hydrogen, water and nitrogen. The two limiting

chemical c a s e s of frozen and equilibrium flow a r e discussed,

“flame sheet model” is used to replace the equilibrium condition and nitrogen

Moreover, a

is considered to occur a s an iner t diluent only.

Inspection of equations ( 1 ) and ( 2 ) and the boundary and initial condi-

t ions leadato a Crocco integral relation for the stagnation enthalpy;

It is advantageous to employ an analytical representation for the static species

enthalpy h i = hi ( T ) . Assume therefore,

where b i , E p i , T r a r e constants selected to represent hi in the tempera ture

range des i red , Equations (19) and ( 20) then yield the tempera ture T .

10

F rozen Flow

F o r frozen flow hi and Y, E O ; thus injection of equations ( 1) and(4)lead

to Crocco integral relations of the form

Y , = ( 1 - U ) (1-Uj) I y, ..E 0

Eauilibrium Flow

Since the re can be no net production of a toms in a reacting gas i t is

advantageous to introduce the element m a s s fraction, ( c f ; e. g . , Ref. 1 )

N

,J

Y4 = Y4

Substitution of equation ( 22) into equation ( 4 ) yields:

i = 1 , 2 , 4

11

Once again consideration of these equations and of the initial and boundary

conditions lead to Crocco relations of the form:

Y, = Y,, ( U - U j ) ( 1 - U j ) t .4

Y , = (1 -U) ( l - u j )

( u-uj) ( 1-u . ) Y = Y4 = Y4,

I Tu

l J Iv

4

where the additional equation for the determination of the species Y. is obtai.ned 1

f rom the equilibrium condition. Since the equilibrium constant is la rge for the

tempera tures of interest here , Libby (Ref . 1 ) , suggests a flame sheet model

approximation. It is assumed that, for U . < 1 for example, Y, 0 when J

U j I U < Uf and Y, = 0 when Uf < U , where Uf is defined such that Y, = Y, E 0.

Therefor e:

I

Wl I

Once the concentrations have been determined, equations ( 19) and ( 20) m a y be

solved for the tempera ture by employing the definition;

4 - h = Yi hi

i = I L

hence:

-1 2 2

2 T-Tr = [ He (U-Uj ) + H j (1 -U) ] ( 1 - U j ) ' - Ue U - GYi tAj )

c Yi c P i

* .

U s e of the equation of state:

allows determination of the density ra t io;

Pe P

Determination of Center Line Viscosity

In o rde r to perform the t ransformation corresponding to equation ( 1 3 ) , the

viscosity along the jet center l ine, p c , of the gas mixture must be calculated.

Following Bromley (Ref . 5) write

n

where @i j is given by, I

w 'i and X . = mole fraction = -

wi 1

12

where Ai and ( L Y ~ a r e constants such that Equation ( 3 0 ) represents pi in the

t empera tu re range desired.

1 3

All of the dependent flow variables have now been determined as functions

of the t ransformed variables '&, f ,

either equation ( 1 3 ) o r equation ( 1 7 ) is now possible to re la te the dependent

variables to the physical coordinates x and r .

Numerical integration of equation ( 8) and of

Of particular interest here a r e the values of the dependent variables

along the jet center line and the length of the flame sheet; i. e . , the axial location

(denoted by xf, and q,, for laminar and turbulent flow respectively) of the in te r -

action of the flame sheet and the jet center line. This point is defined by the

condition

1

u ( 0 , J ) = Uf

14

Calculations

As an example of the foregoing analysis, calculations were performed for

conditions appropriate to those found to yield minimum react ion lengths and thus

to be c r i t i ca l on a typical launch t ra jec tory in (Ref . 6 ) . Following a r e the par-

t icular values of j e t and gas parameters considered:

He= 365.3 ca l /gm A2 = . 2 0 8 ~ 1 0 - ~ lb/f t s ec

u 2 / 2 - 370.5 ca l /gm A, = . 8 3 3 ~ 1 0 - ~ ' 1

6e= 1 , 29!jX1 0-5 lb/ft3 A4 = . 3 0 5 ~ 1 0 - ~ 1 1

Te= 275.75'K cc( 2 = 0.6

6. J = 7 . 9 2 6 ~ 1 0 lb/ft3 cL13 = 0.8

a = 1 inch 04 = 0. 65

C P1 . 26403 cal/gm°K W, = 32

-6

-

w, = 2

w, = 18

w4 = 28

.3654 I 1

C .56361 1 1

. 28382 I 1

- C

p2

P3 - - 534

AI = 185. 23 I t T, = 997.71'K

A 2 = 2687.9 1 1 YI e= 0.232

0.768

A4 = 200.05 I I n = 0.025

'4e = A, = 2881.3 1 1

15

For a jet temperature of T - 265"K, th ree values of the velocity ra t io j -

U j ( 0.33, 0.5, 0. 8) were chosen.

ra t io of Uj= 0 .5 was also considered.

obtained f rom the model for both laminar and turbulent equilibrium flow for

the flight and gas conditions cited above. F igures 6 - 25 show the distribution

of tempera ture , species concentrations and velocities along the jet center line

f rom the je t to the point of intersection of the flame sheet and the j e t center

line. F igures 26 - 37 show the variation of the tempera ture ra t io and the con-

centration of O,, H20 and H2 along the center line far f rom the je t ,

A jet t empera ture of T 60°K and velocity j =

F igures 2 - 5 show the flame sheets

It is of interest to note that for a given se t of flight conditions, there

appears to be a dimensionless flame length that is virtually independent of the

je t velocity ra t io U , within the range of calculations performed here .

l aminar flow this ra t io appears as:

F o r 1 j

Therefore , for laminar flow, the length of the flame sheet is proportional to

the square of the jet radius ( a,) and to the f i r s t power of the jet mass flow

For turbulent flow this ra t io appears a s :

16

Hence the turbulent flame length var ies as the first power of the jet radius ( a) and

the square root of the mass flow ratio ( "juj /&ue ) . It can be seen that the ratio of

laminar to turbulent f lame lengths is approximately:

a PjUj 2

Y 4 2.L 5 10 Xf, I X f , t ' - PjUj 1 2 c2

pe ue

for the range of conditions studied here.

CONCLUSIONS

17

The propert ies of a uniform axisymmetr ic je t of pure hydrogen issuing

into a uniform paral le l free s t r e a m has been presented for both laminar and

turbulent flow with simplified t ransport propert ies and simplified chemistry.

All Prandt l ar,d Lewis numbers have been assumed equal to unity and the two

limiting c a s e s of chemical behavior, frozen and equilibrium, a r e considered.

Calculations are presented for a particular se t of flight variables of

in te res t i n a n existing launch vehicle. It is found for laminar flow, that the

flame length var ies as the square of the je t radius and the first power of the

je t mass flux. Fo r turbulent flow the flame length is proportional to the f i r s t

Pj Uj

P eue power of the je t radius and to the 1 / 2 power of the mass flow ra t io

A comparison of the flame lengths encountered in laminar and turbulent flow,

for conditions considered here , indicate a rat io of

18

REFERENCES

1.

2.

3.

4.

5,

6.

Libby, P. A . , Theoret ical Analysis of Turbulent Mixing of Reactive Gases With Application to Super sonic Combustion of Hydrogen. American Rocket Society Journal, Vol. 32, No. 3 , March 1962.

Kleinstein, G . , An Approximate Solution for the Axisymmetric J e t of a Laminar Compressible Fluid. ARL 51, Apri l 1961.

Polytechnic Institute of Brooklyn,

F e r r i , A., Libby, P. A . , Zakkay, V . , Theoretical and Experimental Investigation of Supersonic Combustion. Institute of Brooklyn, Sept. 1962.

ARL 62-4 67, Polytechnic

Master , J . I. , Some Applications in Physics of the P Function, Journal of Chemical Physics , October 1955, Vo. 23.

Bromley, L. A. ; and Wi-lke, C. R . , Viscosity Behavior of Gases , Industrial and Engineering Chemistry, Vole 43, July 1951.

Libby, P. A. of Flow Lengths Associated With The Combustion of Hydrogen-Air Mixtures l k r i c g a Launch Trajectory, GASL Tech. Report TR-330, December 1962.

Pergament , 13. S. , Taub, P., Engineering Es t imates

.. . L..

I

I 4 . , -- ,-

\_I

Tit

. '

r /a

2 .a

2

I

0 I 3

x /a

t a

-\

4

\ -\

i-

-I-

I % ii 5

FIG. 3 TURBULENY FLAME SHEETS T j=265"K

2 2

.

0 -

U

0 0 u 1:

G-

w 3 t- In

0 0 to

K a

W Z

K W c

W I I-

3 2

0 0

0

c (0

25

0

FIG. 7 TEMPERATURE DISTRIBUTION ALONG THE JET CENTER LINE TURBULENT

Tj 2 6 5 O K

26

.

0 \ X

t

In 0

3- W

0 (D

a W

W 0

I- W 7

W I c

c 3 m a - t v) -c 0

27

n

D

0 u)

0 \ X

I- W 7

c) I

N

-

0

I- Q

W a n z W I-

m

0 0 m

0 0 IC

0 0 Io

0 0 n

0 0 *

0 0 rr)

0 0 N

0 0

0

Y

v) (D cu n

0

G-

-.I K W I- z W u I-

+ w 7

W I I-

J a

z W 0 0 U 0 > I

LL 0

z 0 I- 3

- m a z - . r

0

1.0

.9

.0

.'I

.6

.s y2

.4

.3

.2

. I

0 0

FIG. II DlSTR

1 2 . 3 4 6

x /a

0

BUTION OF HYDROGEN ALONG THE JET CENTERLINE TURBULENT

Tj * 265 O K

30

0 Y) N N

0 0 0 N

0 0 n -

0 0 0

0 0 n

0 0

Y

0 (0

0

H

K

z I 4

a

a -

W z 4 K W I- z W 0

I- W 7

W 0 I \ I- X

z W Q 0 K 0 > I

LL 0

31

0 \ X

C

J a W I-

W I C

LL 0

z 0 b 3 m

0 O m

0 0 b

0 0 rD

0 0 n

0 0 *

0 0 n

0 0 (u

0 0

0 0

Y 0

tn (0 (v a

a a

i W 0

I- W 3

(3 z 0 4 a

L 0

t -

y3

.30

.2s

.20

. I 5

.IO

.os

0 I 2 3 4 5

t

x /o

FIG. 15 DlSTRl8UTlON OF H20 ALONG THE JET CENTERLINE TURBULENT

Tj 265 O K

0 r, el (u

0 0 N

0 0 2

B 1

3 3 n

3

I- W 7

0 \

Z 0 -I a

Ow I

LL 0

u,

n

\

Y

0 cc)

+- t

0

n

2

m a

w -J 3

3 b-

n w I

o b - \ X

Z 0 J t a

lL 0

3 3 0

3 2

3 3 V

3 3 n

3 3 4

3 3 9

3 0 N

0 0

0

I- W 7

w I I- O

\ X c3

Z w (3 0

k z a - IL 0

x /a

5 6

FIG. 19 DISTRIBUTION OF NITROGEN ALONG THE JET CENTERLINE TURBULENT

Tj = 265 O K

38

. 3 D s

3 3 !?

3 > 2

> > n

Y 0

0 CD

II

G-

W z

i+r 7

(3 z 0 J a

z W (3 0

I- z a - IL 0

0 (u

Y

0 (D Y

G-

I- z W

K 3 I-

W z

I- W 7

W I I-

O

X z 0 J a

z W '3 0 K I- z - Ir. 0

0 0 W

8 I-

O 0 tD

0 0 n

0 0 4

0 0 rr)

0 0 N

0 0

0

Y 0

In (0 (u

N

G-

a a

a

z L -J

-

W Z J

W c z W 0

- a

I-

X W I I-

I-

0 u,

(u (u

1.0

.9

.e

.7

.6

.s

.4

.3

.2

. I

0

f x /a

5 6

FIG. 23 VE LOClTY DISTRIBUTION ALONG THE JET CENTERLINE TURBULENT Tj 8 265 O K

N

0 0 0 N

D D e

3 3 2

3 3 n

>

Y 0

0 (0

K a

J

w 0

0 c \

X e W I c

* I- 0

W >

- 3

43

Y 0

0 W

0 0 0 4

8 n n

0 0 ::

0 0 n N

0 0 ::

0 0 E

3 3 0

3 3 n

w

I- w 7

w 3 I- a

a

Y 0

In (D (u

H

+- I- z w -I 3 m a 3 I-

W

W a 3 I- a a w P I W I-

a t \ t

. 0 0 : 0 0 0 d -

8 0- p!

0 0 9 0

0

m 8

0 0 0 W

0 0 0 4

0 0 0 (u

0

Y

0 u)

0

I1

G-

W z A

W t- z w 0

a

c w 3

(3 : z 4

a X

> 0 w 0

a

W a 3 c a a w n z W t-

t

W z 5 a W t- z W 0

I- W 7

z 0 w 0

w e 3 c

W

a a a

0

. 0 0 0 4

0 0 n rr)

0 0 0 13

0 0 n N

0 0 0 N

0 0 n

0 0 0

0 0 n

0

z 0 I-

c z w 0 Z 0 0

z w (3 > X 0

LL 0

w v)

w 0

- a a

a a z 8

Y 0

5 t-

W 0

w

c U 0: I- z W

LL 0

W v)

W

u

a a z

'5 0

0 'u n

0 0 0" -

0 0 9 e!

0 0 9 s?

0 0

0 0 0 to

0 0 0 4

0 0 0 (u

0 0 Y)

9

a a z I a * J

-

J- w 3

W z 0 J a

z * X 0 0

J - 0 Q w a z O z 0 O

w G-

Y) t

0 *

Y) rr)

0 rr)

In (v

s

n

0

n

0

0 \ X

W z

i w 0

I- W 3

w I I-

c3 z

d o S F

; 6- *

a I- z W 0 z 0 u

z W 0 > X 0 LL 0

W m W K

r. -

M M

w - LL

1 0 0 0 *

0 0 Y) )r)

0 0 0 13

0 0 Y) N

0 0 0 N

0 5: -

0 0 0

0 5:

0 rr)

0 \ X

53

Y 0

Lo W N n

G-

I- Z w J 3 m a 3 I-

w 71 - a w I- 2 w 0

I-

%

LL 0

0 n 9

0 0 9 2

0 0 0- !!?

0 0 0- 0

0 0 0 OD

0 0 0 (0

0 0 0 *

0 0 0 (u

0 0

0 \ X

Y

0 W

0

n

6-

a a z I a -I

-

w z J

w I- Z W 0

I- W

- a

7

w I I-

(3 z 0 -I a

z 0 I-

[2: I- z w u z 0 0

- a

ON I

LL 0

> 0 w a

n

W M

c5 - LL

t

n P

0 P

n Io

0 Io

n N

0 N

E

0

n

0

55

Y

0 w

0

n

a 3 I-

w

. I - Z W 0

I- &I 7

W I I-

O A a z 0 I-

t z

- a a

w 0 z 0 0