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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
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