and Fred R. DeJarnette. NORTH. CAROLINA. STATE. UNIVERSITY ...... than tile VSL algorithm. The flare at the end of the shock shape of Ref. [29] reflects.
NASA
Contractor
Report
AN APPROXIMATE TECHNIQUE REACTING
"i_ 1-
: I iJ i,r;i
v.
{ Y
.
,,'
................
i'
:-
", A_ Ti"d(_
Cheatwood
CAROLINA
Raleigh,
North
Cooperative
National Space
Agreement
1991
Aeronaulics
Hamplon,
Research Virginia
i--iYi>-i 20.
approximately is ide_ltical
approach
25 flow over
shapes
sharp
From
Mach
are p,_ = 0.00794
are generally
this difference
of the
identical
the
results
conditions
the
R,_o,, = 1.5 in.
conditions
method
Solutions
the shock
for the
freestream
current
(see Section
reached
region.
heating
downstream
that
flow has
cone
performed
conditions
wall condition
overexpansion
the
are
the VSL [14] and
9.92 shows
once
surface
calculations
freestream
fully-catalytic
able, the
for 20 ° sphere-cone,
= 1.5 in) with the same
Comparisons
this
comparison
2
As
and
shape
results
is
region of the
9.3.
NONEQUILIBRIUM
121
vsL
jj
3
2 I"
__
p_ : 1.14 psf
L./_
T.:486°R
'l/f
M =2s
0
5
10
Z Figure
9.93.
Shock
shape
comparison
for 20 ° sphere-cone,
R,_o,_ = 1.5 in (overexpansion
two methods lutions and
for both
are in good
show
only a ten
Since
catalytic
agreement percent
the freestream
conditions
identical
to Figures
they are
9.80
visibly
here.
different
the
through
it are nearly
are not shown
between
the
included
are the same noncatalytic 9.85.
The
two so-
five percent),
wall catalysis.
as the
previous
wall profiles
from
9.97. within
case,
those
arid mass
and
wall
of Figures
geome-
fractions
line are
solution,
the
9.80 through (Figures
counterparts,
although
within
five to ten
percent.
The
mass
in Figure
9.99,
since
trace
amounts
are
pro-
9.82,
9.98 and
noncatalytic-wal[ only
the
for the stagnation
For tile fully-catalytic
temperature
their
to the
in Figure (generally
is still
two methods
of NO + is not
due
indiscernible
The
from
are presented
wall conditions
difference
try is still spherically-capped, files of p, v, and
conditions for both
region).
so
9.99)
agreement
fraction present
profile for this
case.
Again,
profiles
(sb _ 250) affected profiles
near
are presented.
by the
wall
of p, v, and
minimum end
station,
the
body-normal shows
pressure Because
catalytic
the
profiles profiles
velocity these
minimum the
only
Figures
are withi_ arc
nearly
components
components
(sb _
pressure
condition,
_5 are presented.
the pressure
the
ure 9.103
the
and
velocity
and
to fifteen
percent
to be in excellent
percent
agreement
of the
axe not
body
greatly
wall
results
that
at the pressure
of each
At the
by twenty
end
profiles
9.101 show
iudistinguishable. differ
at the
tile fully-catalytic
9.100 ten
3) and
other
pressure
for the while
at
minimum,
(Figure
9.102).
Fig-
(within
five percent)
122
9.
10 "1
I_ESULTS
VSL
p_ =.1.14
Present
T = 486 °R M_=25
AND
DISCUSSION
psf
T w = 22 60 °R 10 .2
qw 10 .3
i 10"4
noncatalytic ....
0
i .... 50
, .... 100
I 150
, , , I .... 200
J 250
Sb Figure
9.94.
Heat
transfer
comparison
fin' 20 ° sphere-cone,
R,_o,_ -
0.05 VSL ..,..
0.04
" Present %%"_%.._%
0.03 qw 0.02
0.01
0.00 0.00
0.25
0.50
0.75
1.00
Sb Figure
9.95.
Heat R,ose
transfer
compariso,_
= 1.5 in (stagnation
for 20 ° sphere-cone, region).
1.5 in.
9.3.
NONEQUILIBRIUM
123
10° [
VSL Present
p.. = 1.14psf T. = 486 R
|
Pw 10
-1
M.=25
_
_T* = 2260_
10 .2 0
50
100
150
200
250
Sb
Figure
9.96.
Body
pressure
comparison
for 20 ° sphere-cone,
P_os_ = 1.5 in.
10 "1
......
VSL
p. = 1.14
psf
Present
T. = 486 °R M.=25 T w = 2260
Cf
°R
10 "2
noncatalytic
10 .3
.... 0
J .... 50
_ ....
.' ,
100
150
,
,
,
I
....
200
1 250
Sb
Figure
9.97.
Skin
friction
comparison
for 20 ° sphere-cone,
Rnos_ = 1.5 in.
124
9.
RESULTS
AND
DISCUSSION
1.00 p. = 1.14 psf T = 486 °R M_=25 T w = 2260
0.75
°R
fully catalytic 11n 0.50
VSL
0.25
0.00 10 .2
'
J
'
_-_--'
Present
''''
.....
J 10 0
10 "1 T
Figure
9.98.
Temperature
profile
comparison
R,,os_ = 1.5 in (stagnation
0.8 I"
Cl
p. = 1.14 psf
k
T-
I\
"- =2s
0.6 _-
0.4 i
for 20 ° sphere-cone, line).
k
= 466 °R
Tw = 2260
_fully
N 2/
vsL °R
/
Preserit J
catalytic
0.2 0,0 0.00
0.25
0.50
0.75
1.00
11n
Figure
9.99.
Mass
fraction
profile
comparison
Rnosc = 1.5 in (stagnation
for 20 ° sphere-cone, line).
9.3.
NONEQUILIBRIUM
125
1.00
p®= T.
1.14 psf
= 486 °R
M.=25 T w = 2260
0.75
°R
fully catalytic _n
0.50
......
VSL
0.25
Present
0.00 10"2
" " " ' 0"1
........
' 100
P Figure
9.100.
Pressure
profile
comparison
for 20 ° sphere-cone,
/_o,_ = 1.5 in (sb "_ 3).
1.00
p
= _1.14 pM
T. = 486 °R M.=25 T w = 2260
0.75
°R
fully catalytic _n
0.50
.......
VSL
0.25
Present
0.00 10 "2
.........
l 10 "1
•
•
,
,
,
•
•
I
10o
P Figure
9.101.
Pressure
prolile
comparison
R,,os_ = 1.5 i', (sb _ 250).
for 20 ° sphere-cone,
126
9.
Table
9.4.
Run-times"
RESULTS
for 20 ° cone,
stations time
587
157
7961
525
4 3.0
15 2.1
11
33
grid pts/sec shock iterations grid
pts/sec/shock
a - Sun Sparcstation
at
the
end
(Figure
station.
9.104)
other
in the
these
profiles
The
are
body-tangential
in agreement
interior
of the
axe virtually
the
body
layer.
identical.
profiles
and
Figure
1]
1+
velocity
near
shock
at the
shock,
9.105
Temperature
but
shows
profiles
pressure
deviate
that
of profiles
the
end
are
of the
Mass
fraction
pressure
the As
present,
a final
between
concentrated
so those
at the
Table
for this
from
9.4 shows
the overall
for the VSL results the
previous
case are
is the
one
end
an-
station
two stations
given
wall, and in Figminimum, both
another.
The
are
shown
in Figures
9.110
and
9.113
profiles
at
case,
the
electron
Figure
times
total
observed
station there.
again
here.
Only
at, the
to generate for three
trace
profiles
9.115
show
and fully-catalytic
profiles
required
and
wall.
five percent amounts
figures.
density
9.114
required
time
The
9.111
are indistinguishable,
region.
t¥om
for the noncatalytic
run
approximately
near-wall
are excluded
and
for the fully-catalytic
at the end of the body
are considered.
the methods
of one
are within
in the
profiles
respectively. The wiggles in the end probably due to poor grid resolution shown
9.112
sets of profiles
in the profiles
effects
above
two stations
in Figures
both
comparison
discussed
agreement
and
Differences
chemistry
of NO + are stations
wall,
five percent
from
identical.
at these
minimum,
of one another. with
approximately
are practically
profiles
for the noncatalytic At the
within
body
minimum
at the
above are shown in Figures 9.106 and 9.107 for the noncatalytic ures 9.108 and 9.109 for the fully-catalytic wall. At the pressure sets
DrlscussION
t?.,_o_e = 1.5 in. Present
CPU
AND
for the
good
wall solutions,
boundary-layer this solution. shock
three
generally
iterations.
edge The
are
value Trends
9.3.
127
NONEQUILIBRIUM
1.00 p. = 1.14 psf
'_
T., = 486 OR
_
M= = 25 0.75
,/
l,y'"
T w = 2260
°R
..2' r
I
_n 0.50 soso
ssss
//"""
0.25
i
VSL
Present 0.00 0.00
,
•
•
•
I
....
I
0.01
....
L
0.02
....
I
0.03
0.04
V Figure
9.102.
Normal
velocity R,o,_
profile
comparison
for 20 ° sphere-cone,
= 1.5 in (.% ,._ 3).
1.00 P= = 1.14 T
psf
/
= 486 °R
2/
M_=25 0.75
,_
Tw = 2260:R
_"
_n 0.50
VSL
0.25
Present
0.00 0.00
,
L
,
*
I
L .
J.
0.01
,
_
l
0.02
....
I
0.03
,
,
•
•
I
0.04
V Figure
9.103.
Normal
velocity R,o,_
profile
comparison
= 1.5 in ( Sb ,_ 250).
for 20 ° sphere-cone,
128
9.
RESULTS
AND
DISCUSSION
1.00 p=== 1.14 T. 0.75
psf
!
= 486 °R
M.=25 T w = 2260
] / _f
°R
fully catalytic
/y
Tlr, 0.50 ......
i I
VSL
/ "/
Present
,/_//
0.25 o S s S
0.00 0.00
•
0.25
0.50
l
0.75
,
.
|
|
1.00
U Figure
9.104.
Tangential
velocity
profile
comparison
for 20 ° sphere-cone,
R,,o,e = 1.5 iu (,_ _ 3).
1.00
p=.= 1.14 psf To. = 486 °R M_=25
0.75
T w = 2260
°R
fully catalytic ]In
0.50 VSL Present 0.25
0.0o 0.Oq
0.25
0.50
0.75
1.00
u Figure
9.105.
Tangential
velocity R,,o,e
profile
comparison
= 1.5 in (sb _ 250).
for 20 ° sphere-cone,
9.3.
NONEQUILIBRIUM
129
1.00
p. = 1.14 psf T.
= 486 °R
M_=25 0.75
T w = 2260
°R
noncatalytic 11n 0.50 VSL Present 0.25
0.oo , 10"2
,
'
i
i
i
i
|
|
i
i
|
|
10-I T
Figure
9.106.
Temperature
profile
R,,o_
1.00
p.=
comparison
for 20 ° sphere-cone,
= 1.5 in (sb _ 3).
1.14 psf
T® = 486 °R M.=25 T w = 2260
0.75
°R
noncatalytic
0.50 VSL Present 0.25
0.00 10 "2
......
,
_
,
i 10 "1
T Figure
9.107.
Temperature R,,o_
profile
comparison
= 1.5 in (sb _ 250).
for 20 ° sphere-cone,
130
9.
1.00
p
RESULTS
AND
DISCUSSION
= 1.14 psf
T. = 486 °R
%%%
M.=25 T w = 2260
0.75
°R
fully catalytic
0.50 VSL Present 0.25
•
0.00
,
•
|
10 -I
10 °2
T Figure
9.108.
Temperature R.o_.
1.00
p. = 1.14
profile
comparison
for 20 ° sphere-cone,
= 1.5 in (._ _ 3).
psf
T.. = 486 °R M.=25 0.75
Tw = 2260
°R
fully catalytic Tin 0.50 VSL Present 0.25
0.00
t
I
I
.
!
|
I
10 -1
10 .2
T Figure
9.109.
Temperature profile comparison R,,o_e = 1.5 in (_ _ 250).
for 20 ° sphere-cone,
9.3.
NONEQUILIBRIUM
131
0.8
VSL
Prese___ 0.6
'''''_
_,,_,/"
p, = 1.14
__,_,'"
CI
N2 psf
T, = 486 °R M =25
_"-
T w = 2260
0.4 N
°R
noncatalytic
.2
_
0.0 0.00
0.25
0.50
0.75
1.00
T_rl
Figure
9.110.
Mass
fraction
R,,o,
profile
comparison
for 20 ° sphere-cone,
= 1.5 in (sb "_ 3).
0.8 J
N2
0.6 Cl
VSL
po.= 1.14 psf T. = 486 °R
Present
M.=25 T w = 2260
0.4
noncatalytic
0.2
02
0.0 0.00
Figure
9.111.
°R
0.25
Mass
fraction
0.50
profile
0.75
comparison
Rno,e = 1.5 in (sb _ 250).
1.00
for 20 ° sphere-cone,
132
9.
RESULTS
AND
DISCUSSION
0.8 N2 0.6 p==
1.14
psf
T= = 486 °R
cI
...... 0.4
VSL
M_=25
Present
Tw = 2260
°R
fully catalytic
_,
0.2
0.0 0,00
0.25
0.50
0.75
1.00
11n Figure
9.112.
Mass
fraction
profile
comparison
for 20 ° sphere-cone,
R.o_ = 1.5 in (sb _ 3).
0.8 _ 0.6 cl
N2 p= = 1.14 psl =
I
--
VSL
T, = 486 °R
_
--
Present
Moo= 25
L
Tw = 2260
0.4 _
°R
fully catalytic
O 0.2
02
0.0 0.00
0.25
0.50
0.75
1.00
11n Figure
9.113.
Mass
fraction
profile
comparison
R,,o,_ = 1.5 iu (_b _ 250).
for 20 ° sphere-cone,
9.3.
NONEQUILIBRIUM
133
0.75
p. = 1.14 psf 1". = 486 °R
_
1In 0.50
_.
Tw : 2260
_
noncatalytic
_, _,
°R
1
t
J 0.25
_
0.00
..... _ 10 5
.........
...... "_ ....... _". ..... .J ...... .J __.-i
10 6
10 7
10 8
10 9
Ne Figure
9.114.
VSL
Comparison
1010
I
..... ..J .,e.........
1011
1012
.J¢. .... I
1013
10 TM 10 ls
[electrons/cm3]
of electron concentration R,_o_ = 1.5 in.
profiles
for 200 sphere-cone,
,00 0 0 7s [-
\
°'25
10 s 10 e 107
P_"= 1.14 psf
.........
"_\
VSL
10 e 10 e 10 lo 1011 1012
1013
10 TM
10 TM
H e [electrons/eras] Figure
9.115.
Comparison
of electron R,,o_
concentration = 1.5 {n.
profiles
for 20 ° sphere-cone,
134
9.
RESULTS
AND
DISCUSSION
10 .2
p_ = 0.40 psf T_ = 455 OR M.=
15.7
T w variable
fully catalytic noncatalytic
0
STS-2
10 "4
0
10
20
Zb Figure
9.116.
Case
tleat
transfer
for 42.75 ° hyperboloid,
R,_os_ = 4.489
ft.
3
In tile past, of the Shuttle method
hyperboloids at angle
is used
approximates 42 °. The
the
to calculate windward
nose radius T_
= 455°R,
have
of attack
is R,o,e
of "%e,,d = 25. The freestream psi and
calculations
been
used
to model
the windward
(see Ref. [74], for example). the
Mach
symmetry
15.7 plane
= 4.489 fl and conditions
with a variable
flow over of the
the solution
for all altitude
wall temperature
symmetry
As a final case,
the present
a 42.75 ° hyperboloid,
Shuttle
at an angle
is computed of 60.56 input.
plane which
of attack
for a body
of
length
krn are p_, = 0.00276
Solutions
are calculated
for noncatalytic and fully-catalytic wall conditioas. Figure 9.116 compares the two heating rate distributions with Shuttle windward center]ine measurements from STS2 [31]. As expected these two calculations bracket the measured values. Note that the flight also show
data is closest the flight data
for a hyperboloid of Ref. [8].
geometry
to the fully-catalytic result. Results presented in Ref. falls closer to the fully-catalytic solution. Those results as well, and
the solution
is calculated
from
VSL
[74] are
method
135
Conclusions
10
A new approximate metric
blunt
hyperboloids, profiles,
has
over
when
perfect
VSL approach
bodies
been
a freestream
compared
mentum
This
erally
within
component
of velocity
of the
from
to be smooth over the in the
in lieu of numerically
region.
five percent
seen
pressure
As a result,
VSL values.
the continuity
velocity
of the approximate
is minimal
Turning
except
to heat
normal
in the
transfer
calculations
the
approximate
_pproach_,s
overexpansion/recoml_ression
rcgi(m,
comparable
with
results
sharp-cone
the VSL solution,
to be due the VSL
results
the
The
Reynolds
surprising
approaches well.
are
limited
the
method
of Ref.
gas
and
CPU interval,
present [29] or the
time comparisons reacting
requirements. the
present
flows
show
technique
velocity
flows, the regions
approach
approach
their
from the
having
agreement
here,
stagnation
an effect
skin
present
heating
in Ref.
with friction,
and
results
VSL agree
can be obtained [29], with
the surface values
on
nonequi-
agreement
to the the
appears
and
equations
is used used
downstream,
good
good
the
methods
This deviation
rate
where
form of the governing
of the formulation
Further
exhibit
results
Within
present
equilibrium
heating
agreement
region.
above)
both
consistently relates
excellent
only
properties
more
smoothly
VSL solution.
between
Further
gen-
normal
tangential
shows
[29] are higher. (mentioned
A dilferent
the
with
Run
line. result
than
technique
skin friction
a limiting
are the
the approximate
[29] and
VSL solution.
in those
As a direct
calculated those
the
mo-
pressure
pressures
in the nose
Grantz
the
of llef.
approach
flows,
present
reacting
analogy
agreement
stagnation
success.
For
present
method,
the
inaccuracies
to see that
in good
In the present use on
velocity profiles.
normal
computing
on the
gas
is seen the
from
the results
from
so it is not
for
while
enthalpy
solutions.
deviate
is approached,
to the normal
near-wall
librium,
which
limit
variable
more consistent with a full to the tangential velocity,
expression
for perfect
pressure as the
for [27],
region.
between yield
VSL and
near-wall
accurate
of the
of using
layer
of Grantz
the
body
instead
the
impact
and
outside
predicted
profiles which are is small relative
profiles
shock
results
Not surprisingly,
and
The
integrating
relation of Ref. [29], is shown to yield VSL solution. Because this component
axisym-
transformation
profiles
equation,
about
to sphere-cones
of 10 to 25.
to differences
to give accurate
overexpansion/recompression
range
are
flowfields
is applied
is an improvement
relation
is shown
hypersonic
method
uumber
attributed
pressure
equation
Mach
flows.
and can be at least partially for the normal direction. Maslen's
The
to VSIJ solutions,
gas and reacting
Using
to solving
developed.
the
typically analysis can
Ire'sent an shows
process
aud
order that
VSL [9, 14] approaches of magnitude for a given
two to three
times
reduction shock
shape
more
grid
for perfect in overall and points
time than
136
the
10.
VSL algorithm.
Based
on the
accuracy
of tile computed
surface
new approach could be useful in the preliminary design environment. it could be used to gcne,'ate an initial shock shape t'o1" more exact require starting solutions.
CONCL
USIONS
properties,
this
Alternately, methods which
137
11
Recommendations
For this new approach three-dimensional provide mate
the form
flows
desired certain
the
The
so tile
would
facets
inviscid
useful
is required.
accuracy,
of it) probably
In addition, cerning
to be most
in the design approximate
transverse
need
approach
momentum
to be included
of the ongoing
solution
environment,
research
in the
flows
to
of Ref. [29] does
not
equation
by Riley
of three-dimensional
its extension (or an approxi-
governing and
might
equation
DeJarnette
set.
[69] con-
be extended
to viscous
flows.
Run-time
comparisons
VSL approach.
However,
is due
primarily
herein
to increase
time
would
included.
the
method
where
the physically subsonic passes
correct
portion followed
Currently,
into the effect calculations
proximately
45°).
ASTV-type
vehicles.
would
require
As a final wise
step
pressure
cannot
The
comment,
be made prevents
shock
shape
before
such
a more
marching
requirements.
overexpansion/recompression
region
the
equation
for
would region.
blunt
yield
As approach,
approach bodies
(greater
subsonic-transonic be performed.
would
than over
most
apmany
region the
improvements be
global
is warranted.
the
significant
the
upstream.
of flowfields
for determining
in
a result,
through
could
procedure
improvements
iterations
to propagate
calculation
computations
sophisticated Such
for very
global
the VSL
in the current
run
equation
upstream
With
this
in a shock-normal
is supersonic.
information
the
developed
momentum
without
layer
whether
to see if the
be solved
propagating
allow
of this omission
limitation
modification
size in the
computational
This
shock
is neglected.
smoothing
normal
procedure
of the
layer
the
than
techniques
be interesting
would
of information
boundary
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An investigation
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amount
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It would
the full VSL equations
outer
is significantly flows,
relation
cfficiency.
employs
the
technique
approximate
by a large
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present
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the
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S* 8
n
R*_ose
Z*
7"*
--
I'
--
V
--
U*
v= p*
--
.
tt T4
/_.*
*
T-
l%_s
n* _=c---:-
c; c,
c_-
pc_
C*
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T*
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' "*
(Vi
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rr_
--
wi
_i,:,s
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v_
tl
q*
The
t_sR*uose
*
p----
It _4
q=--
=
=
h*
h-
.
P_4
r
Ks
'V*
_
P--
following
--
R_.ose
?2 =
to the
7l*
--
Z
according
h"_4 =
C*
poo
=
qre]
• V.3 floo
_:'Jand #*_4 is the coefficient In addition, the following
" c_
n*.o..
., Wre.f
PooP'* --
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J;;s- R:o,.
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rela-
146
A.
IE_E1
_ '_ QUANTITIES RENCE
Lez2i = p _Pl i,,,, k*
Units
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dimensional
/_o_e
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.
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q_# =
Lf t2-s ]
_./= LZ3H BTU C*
- p_
=
.slug'°R_
T;_ : [°R]
147
B
Fluid The
development
sources
within
peated
here.
of the
the
so in the Ref.
curvilinear
these
Navier-Stokes
literature,
In particular,
an orthogonal flow,
Equations
equations
written
interest
(p.u,h,a)
Os* S*
-
Navier-Stokes
in a variety
of
derivation
is not
re-
equations
written
in
ourselves
to axisymmetric
form.
(p*v*h,h*a)
On*
(B._)
0
i
momentum:
{ U*
p*
-
. Ou* u*v* Ohl _ lop* + v On---;+ ht On*] + hl Os*
OU*
----
or:. r:. - r* Oh;
1 Or;* hi Os* _*
0
+
be found this
Restricting
in dimensional
continuity: 0
the
system.
below
can
of brevity,
[75] presents
coordinate
are
cquations
an*
+
h,l,,_
2Oh, _Oh; )
_¢ O," +
/-_--On* +
1 h_On*] r,,,.
(B.2)
momentum:
p_ (u*Ov*
,Ov"
u*2Oh,_
\EaT *+_ o..
1 Or* ar_,_____! , + rg, Oh_ * Iqh_ Os* + Os* + On**
_hi
Op*
1 Oh, Oh_ _ h--7_ ---7 On + hll On*,]
(
(B.3)
h, b-_/+ --v= On r*'_"
r;, I, Oh; h; On*
r_, aht hx On*
energy:
p. ( u* Oh* +v. _) Oh*) h-TOs--: [ 0
1
11-"
stress
flt;#'Oh*_
0
[
u* Op*
v* 03-[ =
h,o_*
,,,_Oh*\]
On*
_:.Ion,. .Oh,1 + -g, tos. + v g-_z..J
(B.4)
Or* % [_*ah; .Oh;1 r. [0_* .Oh,1 . Ou* + q. _ + _.*. O n--v + h_ t_ +" _n-;.*] h, [gTs, "bTq Sn
terms:
[
°
]
v*ah,] 2 _,* o (h;u*)+--(h,h;,,*) [lj7, 0s* --_+°"" h,--ff_n*J 3h,lt._ _ On*
.
ro, = 2#*
rn *n
--
2#*
0,* 2_,* o On* 3hit,; [o -67* (h>*)+ --z(h_h>*) On
. [ v* Oh_ r_ - 2#*[_ --_+ 0-;7,* hah_&*J
3h, h_
-_s* (h;u*)+_n
] " (h'h;v*)
(B.5)
148
B.
%* =
h-'_lOs --7
Using the definitions of Appendix in nondimensional form.
FL UID EQUATIONS
+ hl
A, Eqs.
(B.1)
through
(B.5)
are rewritten
below
continuity: 0
0
0-_ (pub3)
+ On (pvh,h3)
= 0
(B.6)
s - momentum:
10p P\(u__Ou h, o_ + _Ou + uvOhl) h--[o,---;.+ h, o_ ?
1 Or.,
Or,.
r_. - r_, Oh._
2 Oh,
Oh3_
(B.7)
h--io-V+-Y-#+ h,h_ o_ + h--,O,--7+ h_O.) *'"
n - momentum:
P (,_o.o _+vOnnoo c2{
._Oh,) +On= @ h_On
(B.S)
l O'rsn Tsn Oh3 l Oh3) h--( O---Y + _OTnn + h,h_ O_+ (lOb1 h7o---Z + h-_ o--Y_"" r¢¢ h_ Oh3 o.
h, o.
r_Ohl}
energy: ( u Oh
Oh_
u Op
+ _,, ._ + -bVj +_.. uOh,] h, [Ov _--_,,J stress
+_""_ + _°"N + _
(B.9)
h_o--2_ + o_ l
terms:
r'=2_[h,
Os + h, OnJ
3hih3
-_s(h3")+-_n
(h'h3v)
(B.10)
r¢_ = 2/1
where _2 --
re]
149 By assumingthat than
_, the
v and
standard
n are of tile or(h'r
viscous
shock-layer
tJf _ and
C¢luations
neglecting
terms
of higher
order
are obtained:
continuity: 0
8
0---_" (puh3)
+ _
(pvhlh3)
= 0
(B.11)
s - momentum: { u Ou
_ _
Ou
.v Ohz)
h, O,,11 +'
10p
_
0.--_+ --E#h_] _
h, _ )
(B.12)
n - momentum:
P (& \h, Ov Os +''onOv
u20ha) t,, 0. + _Op = 0
(B.13)
energy:
{ _ Oh
Oh'_
u Op
Op
tP_a.J + p,.o,, i,, o,-, + Ou Note: side
In order
Ohl'_
to keep the equations
of the n-momentum
equation
h--_ 0,----_
(10u parabolic
is ouly
of order
in nature, 1.
the right-hand
(B.14)
150
C.
C
Maslen's Maslen's
equation. others
pressure Briefly,
are assumed
expression
is an approximate
terms
present
are
to deviate
are given
observed
little
for p is a function
this derivation
from
of shock
below,
their
some
from
the continuity
equation,
From
to tile normal ii1 certain
values.
and
stream
function. results
momentum regions,
The resulting
of the intermediate
Eq. (B.II),
expression
is satisfied
works
by the
stream
while
closed-form The
details
are used
of
in the
chain
rule
(pvh]h3)
O_
0---_ = pvhlhz
On
system,
of differentiation
= 0
_,
0_
in a(_, q) coordinate
that
function
_=s The
shock
properties
since
0 o%(p, h3)+ N0
Maslen
solution
to be negligible
method.
Recall
This
METHOD
Method
relation
some
MASLEN'S
where
-
puh3
where
7/-
(c.1)
%
gives
0
0_ 0
07 0
0
O_ 0
071 0
Os
Os O_
Os Ort
On
On O_
On Oq
0,_ Os 07 On
] % 1 %
(C.2)
Eq. (C.1)
O( O_ 0( On For axisymmetric
OO Os 0o On
rI d_, _s
d_
flow, qj
2 = r_ 2
dq_ --=resinP_ ds
Thus,
0 Os
0
071 0
0
O_ + Os O-_
Or, -
puh3
0
_, 0,7
(c.3)
where OTI
1
Os
%
[pvhlh3
- qrs sin F,]
(C.4)
151
From
Eq. (B.13),
the
normal
momentum
equation
( u Ov Ov P I,h, Os + vo-nn Applying
the transformations
is
u 2ohm) Op h, On ] + -_n = 0
of Eq. (C.3)
to Eq.
(C.5),
(c.5)
substituting
Ohl On and
rearranging
gives
the
normal
momentmn
p_l{
Orl Apply
Eq. (C.3)
hlh3
u On --
hi O_ the
in this
coordinate
]
_)s [-_
system:
(C.6)
+ UtCs -
to n to get
V
Using
equation
+
rlr, sin F_
On
ph,h3
Orl
_
puh3
(c.7)
approximation n=
(r/-
1)
l
along
with
Eq.
(C.7)
gives I' 8
2p, u_ Differentiate
Maslen
with
neglects
respect
the
to { to get
last
term
0n _: By substituting variables
this expression
at the shock,
(l-.)
in this expression
to get
(' -,I) ,,_ [1 2 cosr_ t into Eq. (C.7)
the following
_.,'. ] _LJ
and evaluating
relation
Differentiate
with
respect
the remaining
dependent
for v is obtained
v. 1+--(_-1) cos F_J
v = v_ + '-5"
(c.9)
Ksrs
]
to q to get
O'v o7:-_v, [ _+
n_r_ ] _o-7-_.j
(C.10)
152
C.
Substitute this expression ables at the shock so that
where
the
expression
partial
derivative
to get
Maslen's
into
Eq. (C.6)
of v with
=p.
evaluate
respect
second-order
eft.,)
and
the
remaining
to _ has been
pressure
MASLEN'S
METHOD
dependent
neglected.
vari-
Integrate
this
equation:
+ p..(,-l)
(c.12)
+ p._
where _sT"s'll s PMz
--
2
and v, sin x,r, I-----_J ] 2 F, [ 1 + cos
PM2 -This result value:
is essentially
a truncated
p(_,r/)
=
po(_)
Taylor's
+
p,(()
series
[q --1]
expansion
+
of p about
P2(_)['/_! 112z
its shock
(C.13)
where 1)o =
P,
Pl
--
PM1
+ PM2
P2 =
PM2
i
The
streamwise
derivative
equations.
Differentiating
streamwise
derivative
Eq.
along
0p 0_
of p along (C.12)
lines of constant with
lines of constant
@s d_
v, sillP,
respect
r/,_ appears
to ( gives
in the governing
an expression
for the
)/:
dK,
&, ]
+v, ,l-(+ co_]_
d_
+ cosF,
d_
cosr,
(C.14) Recall
that dr__.__ d_ - sin F,
Make
these
substitutions
du, d--_- = _' sin F,
0 (sin [',) 0---_
= -t%
to obtain
O_l,7- d_ + --7
u,,,,si,,r', + ,,?,--_- + ,_],-,sinr',
cos F,
'
4
( _?-: +[1
Eq. (C.15) quantity that
is the partial
,s
+ cosF,]
derivative
is the partial
derivative
+_.si_,rs 1 cosr_, [sin F_ _--_
of p along of p along
Op _
Op]
153
'o_n, cos Ps
lines of constant lines of constant
(C.15) 7?. However,
r/,_. Recall
OqOp
Op _
the desired
from
Eq. (C.3)
puha Op
Tj
and
fi'om Eq.
Equate
Op ]
_ 71.-
Os
O_]..
op] op] o,7op ,I. - 1dn_o7, I.,, = o_1,+ o_o_ + nb d_ 07.
Op
expressions
that line.
the
nb
expression
By making
for OqlOs,
On/O_
is defined
line so this
in Eq. form
Eq. (C.4),
use of Eq. (C.7),
Os where
10p nb Orl,_
to get
&l
stagnation
Op
l dnb Op
On
out
stagnation
Op _
d_ 0_,,
these
It turns
(2.1.7)
(C.9).
is used
is not
this expression
puhanb
well-behaved may
be rewritten
puh3 On
_, Equation
everywhere.
near
the
as
(c.16)
O_ Eq.
Op
(C.16)
is well-behaved
near
the
154
D.
D
Shock With
of the
each
shock
iteration
layer
in Chapters
given
the
the
shock
solution the
The
to the
continuity
equation,
station. layer
dnb
0---_(nbp,#u,fih3) Integrate
across
the
- p,u, shock
these
This
continuity values
appendix
thickness.
a calculated
are
provides
Inherently,
value
equation.
with the geometric two
carl be rewritten
As
thickness
in agreement, the
expression
this expression
also
as
0
0
d_ O_l_
layer
THICKNESS
q..
Eq. (2.1.10),
0
is compared
next
q and
equations,
from the
When
shock
between
governing
be determined
equations.
this calculated relationship
of the
8, this thickness
body
LAYER
Thickness
solution
nb, may
7 and and
is advanced
for determining provides
in the
thickness,
explained by
Layer
SIIOCK
[(q.-
1)#(_hz]-psw---(#vhlh3)=0 uq,
(D.1)
to obtain dnb
(ps#vh,h3),
-
(ps#vh,h3)_
+ p,u,
1) #fih3]_ -
[(qn -
1) #fih3],,}
(D.2)
' _0 (nbpspus_h3)d,l_
= Recall
d---( {[(7In -
that sinF, Us
--
Uw
_
Uw
ths = 1
"-- 0
P_ Make
these
substitutions
into
Eq. (D.2)
with respect
nbp.us
= rs
to get
= "s sin Fs
o_ Integrate
h3s
(D.3)
to ( to obtain
[K_hadq.
d_ rs sin F_d_ = f0 _' rs sin rs_rdrs
=
(D.4)
Since drs
--
d_
this call
be rewritten
- sin P_
(D.5)
as
#fih.jdy_
=
rs
rsdrs
-
r_2 _ _ 2
(D.6)
155 Substitute for
h3 to get
nbp, usrs
This
yields
the
/0'
pf_drl,, -k nbpsu_nb
following
quadratic
cos F,
/0'
pfi {r/n -- 1) dr/,, -- _o
for rib, the normal
distance
from
the
(D.7)
shock
to the
body:
An
alternate
For a given
Similarly,
approach
station
for
(where
Eq. (2.1.7)
calculating
s is a constant),
shows
that drl dn
Equate
these
relations
the
-
shock-layer
utilizing
dr I
OrI
pu h 3
dn
On
d)_
along
a given
thickness
Eq. (C.3)
shock-normal
is given
below.
between
r/and
gives
line
O0
1 OrI
1 dq
On
nb Orl,_
nb drl,_
to get dT/ q)_-r-- = puh.:_n_ = p_[rusfth3nb arln
Integrate
to obtain _ s/
Substitute
r/=
is equivalent
7in. As a result, order
d71 = p,'u fl_b
fO
r_n
fifth3don
for h3 to get
At the shock,
which
r3
1 and
_1,, = 1 so that
to Eq. (D.8). it must
to find p for a given
be used value
Eq. (I).9) in conjunction of (_, _I,,)-
defines with
the
relationship
Maslen's
pressure
equation
in
156
E.
E
Geometric
form
appendix As_O,
which
provides
the
LIMITS
Limits
When the governing equations are applied forms (as ( ---, O) are required. These limiting terminant
GEOMETRIC
must
be evaluated
finite,
nonzero
oil the stagnation line, their expressions contain fractions
through
values
the
of these
use
of l'Hopital's
limiting in inde-
rule.
This
quantities.
drt b
uo = cos Fe _
0
r, _
0
d--'( --* 0
(E.1)
and du_
d cos r,
de
d_
dr,
- a, sinre -- mo
d_ -sinL
-* 1
(E.2)
Thus, ue
cos F_
rs
Applying
l'Itopital's
lim
from
re
0
rule,
_.--*0
Since
0
(}
cos I_______ = _-..o lim{d(cosPe)/d_} r e dr,/d_
= X_o
Eq. (2.1.4) ha=r,(l_n
c°sl'_) ?'s
/
as(_0, In addition first
appears
to these
fundamental
in Eq. (2.4.12),
v-,o
4
-_
quantities,
the limiting
ff_ ,_
form
(._,O l u,
two other
terms
of the streamwise
d_
merit
pressure
+_s0(r/-a)
Veo Laso _-,o(
L_--'" uZ--d-(J -%,_o Applying
l'tIopital's
rule:
lira { cos Fe-" 3" re':e } ,_--,'0 'U.s
3u_,
sin l s
attention.
Ke sin F, - rs d---( -
The
derivative:
(E.4)
157 so that lira
--
_-o
-
,_3
3,% 2
(E.5)
_, d_
T]lllS_
_-o 0--_,_lim1 u--_ 01,
= _-olim ,[_ d_
2
1)
(E.6)
1
A second term to be evaluated streamwise derivative of n:
Note
+ x, o (r/-
appears
in Eq. (2.4.16),
the
limiting
form
of the
that
lira _ cos F_, -- ,',_s _--.o L u_r,
}
k
rs
_-_0
Ua
so that (E.8) (--*0
rs
158
F.
F
Shock Boundary
conditions) ditions
along
are required the
the
for the
shock
appendix l)rovides rium flows.
vary
tile jump
In the shock-normal
body
and
solution
according
at the
co,lditions
the conservation
From
the
momentum
chemical
for l)crfect
the
energy
wave
(in the form
equations. nature
of the
gas, equilibrium,
The
of jump
jump
flowfield. and
conThis
nonequilib-
system tt_
_---/t s _
cOSFs
equation
equation P_ + po+V_ = Ps + PJ'_
From
shock
of the governing to the
voo = V£ sin I', From
PROPERTIES
Properties
conditions
across
SHOCK
= p, + p_2 k*
to the
energy
solution
term
diffusion
h3 On ] binary
Lewis
calculating
flowfields
conservation
derivation
with
equation
of this
for
equation
is
here:
04* o,,.. 4* (toh,+ _0h;) On* h;On* ] for species mass
number
(G.24)
species
excellent
+
of the
equation,
of the An
is repeated
.0_,x =_'. of production
(G.23)
n i _-;-c- ----
i=_ QV
mixture.
v@ -- =
u@
+"
-}- 2-' --
diffusivity,
to modifications
chemistry
chemical
presented
h i -_n
(G.22)
h, _]
i and
flux term
fli* is the diffusion is defined
by Fick's
(G.25) mass
flux
law to be
OCi
Jg = - p*'P';2On*
(G.26)
166 which
G.
can I,e written
k*
I lsing tile definitions sional tk)rn_:
(
_--
REACTING
FLOWS
;_s
J,* -
P
('.III,;MICALLY
(,
;'I
Oq . I.cr2
of Al)lwndix
0,_ + "_ )
--
On"
#*. Oq --l.,:t2--l'r On*
-
A, Eq. ((_.25)
{
is rewritten
(
';"- '_ --i)u -_J'
(G.27)
--b l O, +
below
, }
ha On )
ill nondimen-
(G.28)
wJlcrt'
k
J, -
1
Oq Lq,., ---
(.'pf
Substituting
2
Eq. ((L29)
O
kO__n
-Y_h'e_ i=1 Co, d,ining
Eq. (G.28)
+
into
F,q. ((1.23)
o,,
----+h3 h. o,,
L-_-I_ + J'
and
rearranging
-7_/+
\/j_--+Ou
(G.29)
J'tt
;,
.....h3 On '
and
Eq. ((1.30)
gives
,
u 07'
07"_
Pc'v (V, o_ +''_)
u Op
N,
07'
_3;C'"i
0,,
i=1
O'u
(G.30)
/q On
e2V, q..r _ OT z_..,,-,,_'pi On i=1
Op
;,,o,_ "o,,
+ tt ;,,,.[__J + _:;,,,;7, _{;;fa_,,] o,t,(, oO,,;`'+ ;,:. , o;,.:,'_ ---_1
-'
yields
N, i:1
2}
(0.31)
167
H
Reaction In nonequi]ibrium
production (when
terms
in terms
of these
Consider
where
flows,
partners
empirical
production
the
proceed
mixture) and
a finite
rate.
in the energy
species
continuity
The
equation equations.
below.
of diatomic
02 + M, --_
2 0 + M,
(or catalytic
third
a chemical
of formation
at
appear
in the
is presented
dissociation
do not undergo
rate
reactions
i in the
terms
the
partner
which
results,
chemical
of temperature)
as an example,
M_ is a collision
collision
the
(tb_ for each species
formulated
A discussion
Rates
oxygen:
(H.1)
body).
Catalytic
change
during
of O can be written
bodies the
are
reaction.
those From
as
d dr* [O] = 2k} [02] [M_] where k} is the ibrward reaction rate coefficient each bracketed term represents the concentration The
reverse
(backward)
reaction
(H.2)
which is a function of that substance
of T* only, and (in moles/cm3).
is
02+M_Y--20+M_ so that
the
rate
of reduction
(H.3)
of 0 is d dt. [O] = -2k_
Combining
these
two elements
(H.4)
[O]2 [M.]
gives 02+M_=eO+Mr
so that
the
net
rate
of 0 formation
(H.5)
is
d dt-7 [0] = 2k} [02] [M,.] For chemical
the
more
species
for the overall
general and
change
case
of a multicomponent
N_ simultaneous from
reactants
chemical
Nt
_-'_,,,.,,.X* i=1
gas
reactions,
to products
(H.6)
2k_ [0] 2 [Mr] with the
N_ distinct
stoichiometric
reacting relations
are:
Nt
= _/3,.,_X* i=1
(H.7)
i
168
H.
where r = 1,2, •.., Nr and Nt is equal to the total and catalytic third bodies (No). The quantities coefficients
for reactants
concentration
and
of species
products,
third
body
, ci
RATES
number of the reacting species (N,) ai,r and fli,r are the stoichiometric
respectively•
i (or catalytic
REACTION
The
variable
X_
denotes
[moles]
(H.8)
x; = p, ,q, , t cm3] In the
above
example, ao2
= 1
_o_ = 0 For the
r-th
reaction
of Eq. (H.7), ( dX* _ k dr* 1,
while
the
backward
rate
the
\Mi]_ where obtained
net
rate
=\
p* is in gm/em by summing
aMr = 1
/30 = 2
3Mr = 1
the
forward
rate
of production
is
Nc I, II (T/p*)a'''
(H.9)
is
-(_i,r-oi,_)k*
of production
dr* ]
no=O
= (fll,r - eli,_) k* i=,
k dr* ], so that
the
i - N,):
r
(Eq.
(o*=p _
[78] is
= (Ni,r-_i,_)
a. The
(H.10)
i=,
k_
net
mass
H.11)
over
rate
i=1
br1-I(_ip*)_'"
(-rip*)_',_ -
k*
of production
all the
reactions
(H.11)
i=1
of the
i-th
species
may
be
N_:
= r=l _dt.] r = Mi r=l _,e,t.
r
'
(H.12)
or Nr
•*
w_,= e* Mi _
r=l
(_i,_- _i,_)(. R_,,_ - RL )
where Nc
R_,r
Nc
= l__r k*
p. i=1 II ('_/p*)°"_ = h_, *a_II ('_,)°"" i=1
k*
Nc
Nc
br " = --e* II ('y/)_"" R_,, lI ("1i#1_''_=/%p "* *'_i=1 i=l
N¢ =
_ i
Ot r
i=1
Oq, r --
1
(H.13)
169
Table
H.1.
(',[tenli('_tl l(ea('tions
Reaction
r
Mr
1
02+U,=
20+M,
N2, 02,
0, NO,
2
N.2+M_
2N+M2
0,
N2+
2N+
M3
N2, 02, N
N+O+M4
N2, 02,
O, NO,
3
M3_
4
NO+M4_--
5
NO + 0
6
N2+O
7
N+O
_ _
N
NO N
02 + N NO+N
_NO++e
-
N_ i=l
The
mole-mass
ratio
(or mole number),
_ti, is defined
as
for i = 1,2,...,N_ M_ ")'i =
(H.14)
N,
= .¥;
E n=l
where
Zi-N,.,,
modified
is the
Arrhenius
for i-
Zi-Ns,n'_'n
catalytic form,
p*
efficiency
the
reaction
of third r_tes
k*=AI'"T*_'_exI'(-TDs"/T*)I'_
T* is in third
are given rate
a seven-species
bodies
which
in Table
coefficients
Mi-N,
for specie
n.
Written
,
'
)
[ l(cm3_'_]_mole]
are
H.2 for the
are presented
model
considered.
'
various in Table
[15] for air is used. The
efficiencies
reactions. H.3.
in
(H.15)
\mole]
(H.16)
K.
In this study, and
body
are
k*b_= Ab,_T*B_"exp(--'ID_,_/ where
N_ + l,...,Nt
Finally,
Table of the the
H.1 lists the reactions catalytic forward
third and
bodies
backward
170
H.
Table
H.2.
Third
Body
Efficiencies
Relative
REACTION
RATES
to Argon
Efficiencies Catalytic Bodies
i - N,
M, M2 M4
1 2 3
e-
4
Table
N2
02
N
0
'NO
NO +
i=1
i=2
i=3
i=4
i=5
i=6
2
9
1
25
1
0
2.5
1
0
1
1
0
1
1
2O
20
20
0
0
0
0
0
0
1
H.3.
Chemical
Rate
(,10 effioents " •
]r
[,:,.:V,,_ol_-s_] or [c_/,,_ot__ -sec] 1
3.61×101ST-l"°exp(
2
1.92×1017T-°'Sexp(-1.131×lOS/T)
1.09×
1016T -o.s
3
4.15×1022T-l%xp(-1.131×lOS/T)
2.32×
1021T -l"s
4
3.97x102°T-l'Sexp(-7.56×104/T)
1.01 × 102°T -1"5
5
3.18x109
6
6.75×1013
9.63x1011T 1.50×1013
17
9.03×109
-5.94×
IO4/T)
3.01 × 101ST -°'s
T 1"° exp(-1.97×104/T) exp(-3.75×10a/T) T °'5 exp(-3.24x104/T)
1.80×
°'s exp(-3.6
1019T -1"°
x 10a/T)
171
I
Species
Thermodynamic
Properties For nonequilibrium
flows,
species
thermodynamic
properties
species present in the fluid. Ref. [60] has provided temperature range of 300K
