9.38 oC3 - NTRS - NASA

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Rno_ = 1.5 in (stagnation region) . ..... Distance measured along the shock wave. Time. Temperature. Reference temperature ... air. Body value. Boundary layer edge. Chemical equilibrium value .... flows in Earth, ...... kw. 7Lb. _ w. Since this may be written as or dh = CpdT. _'2 k,,, Oh _, qto. -- ...... a - Sun Sparcstation. 1+.

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

by shock-shape

An investigation

with

a lnarching

portion

phenomenon

of the

amount

faster

it is unclear

or to other

It would

the full VSL equations

outer

is significantly flows,

relation

cfficiency.

employs

the

technique

approximate

by a large

scenario,

present

present

for nonequilibrium

computational

be increased

region

the

especially

to Maslen's

In this

system. The

show

noticeable

streamin overall in the

138

REFERENCES

References [1] J. D. Anderson, ics".

[2] O.

AIAA

Jr.

Paper

M. McWherter,

Reentry

Layer

Vehicles".

[3] P. A. Gnoffo. cous,

[4] W.

"An

S. Helliwell,

Geometries 197, [5] P. A.

L. Lawrence,

Paper

[8] J.

AIAA

N. Moss.

Diffusion

[9] A.

and

L. Murray

N. Gupta,

Flowfield pages

Jill

Journal,

[12] J.

Mass

Solutions

Relaxation

NASA

TP

S. C. Lubard. AIAA

of In-

for Design

of

1984.

2953,

Algorithm June

"Viscous Journal,

Oww

Biconics

Stokes

Code".

AIAA

and

for Vis-

1990.

Flow

Volume

and

Solution

Viscous

over

Arbitrary

19: pages

191-

5):

Bodies".

and

TR

83-1666,

July

"Application

pages

1983.

of an Upwind Equations".

"ltypersonic AIAA

Shock May

June

with

AIAA

Equa-

1974.

Volume

Viscous

16: pages

MacCormack. Method".

Layer 1970.

Multicomponent

Three-Dimensional

Journal,

R. W.

Viscous 843-851,

Solutions

R-411,

by an Explicit-hnplicit

and

of Radiation

a Variable-Effective-

Navier-Stokes

Shock-Layer NASA

P. A. Gnoffo, May

Paper

of th4_ llypersonic 8(Number

C. It. Lewis. Blunt

Using

J. C. Tannehill. Parabolized

Volume

Injection".

Analysis

"Viscous AIAA

Journal,

Shock

1279-1286,

Shock-Layer Volume

23:

1985. J.

Gas

N. Moss. Mixtures

"Viscous-Shock-Layer in Chemcai

Solution

Equilibrium".

NASA

for Turbulent TM-X

72764,

1975.

V. Rakich,

ment

"Evaluation

1987.

"Reacting

723-732,

August

June

"Numerical

E. C. Anderson Flow

January

Flows

D. S. Chaussee,

Layer Flows over December 1978.

[10] R.

Navier

87-1112-CP,

tions".

and

to the Three-Dimensional

[7] R. T. Davis.

L. Oberkampf.

Point-Implicit

of Attack".

"Hypersonic

Parabolized

Algorithm

Aerodynam-

1981.

Gnoffo.

Gamma,

in Hypersonic

Navier-Stokes

84-0486,

Flows".

R. P. Dickson, Angles

W.

Parabolized

Paper

Perfect-Gas

at High

February

and

Upwind-Biased

Compressible

Research

1984.

Noack,

and

AIAA

of Modern

June

R. W.

viscid/Boundary

[6] S.

"A Survey 84-1578,

D. A. Stewart,

on the Catalytic

82-0944,

June

1982.

and

Efficiency

M. J. l_anfi'azwo. of the Space

Shuttle

"Results Heat

of a Flight Shield".

AIAA

ExperiPaper

REFERENCES'

[13]

E. W.

Miner

Flows

[14]

139

Over

Nonanalytic

"Viscous

Chemistry". 1969. and

Equations A1AA

Slip".

[18] S.

and

W. L. Grose. PhD

[20] W.

Layer

C. tl.

about

It. 1971.

[22] E.

V. Zoby Gas

of Attack

[23]

S. K.

[24]

Attack 4):

[25]

and

pages

NonequiAIAA

Paper

Symmetric

Institute,

Flow

Applications,".

E. V. Zoby.

"Approximate

Like Bodies

at Large

TM-X

2843,

angles

bodies.

Flow

AIAA

in the Invis-

Inviscid

Program

December

University

of Spacecraft

CR

2133,

Bodies

D-6529,

September

the

at an Angle

1973.

Flow

of a Perfect

of Colorado, Flowfield

and

TN

Hyper-

for Calculating

Axisymmetric

Hgp_:rsonic

"Engineering

Atmosphere.

NASA

NASA

"A Computer

NASA

Journal

symmetric

January

1969.

Flow".

Blunt

July-August

91-0469,

Atmospheres".

About

A. L. Simmonds.

smooth

Paper

Nonequilibrium

for Nonequilibrium,

Field

thesis,

of the

Multicomponent

in the

AIAA

June

ttypersonic

PhD

and

Fligi_t in an Arbitrary

Venusian

Viscous-Inviscid

Solutions

1964.

Solution

Jr.

2281-2288,

to the Nonequilibrium

Polytechnic

Bodies.

398-404,

June

in Hypersonic

R. A. Graves,

The

Flows".

,Solution

and

Nonequilib-

pages

1986.

flow past

1055--1061,

with

12):

Low-Density

Improvements

Blunt-Body

Martian,

Inviscid

and

1975.

Point

7(Number

Nonequilibrium

June

"Recent

hypersonic

of 0 Degrees".

E. V. Zoby

Chemical

Lewis.

"Asymmetric

Jackson.

Smooth

May

Shock-Layer

Geometries".

at the Stagnation

"tlypersonic

"A Thin-Shock-Layer

Maslen.

Perfect

Viscous

"Three-Dimensional

Complex

a Vehicle

sonic Flows in Earth, December 1971.

[21] S.

Air

CR 2550,

Volume

86-13,t9,

Approximate

Virginia

L. Grose.

Layer

with

5: pages

An

thesis,

Over

Journal,

Paper

Inviscid

Volume

cid Shock

Shock

for Hypersonic

H. Maslen.

Jou_tal,

NASA

C. If. Lewis.

A. L. Simmonds.

Surface

B. A. Bhutta

and Flows

AIAA

Navier-Stokes

VSL Scheme 1991.

Bodies".

Ionizing

1983.

F. G. Blottner.

[16] R. N. Gupta

[19]

Blunt

Shock-Layer

January

rium Air December

"tlypersonic

M. D. Kim,

Viscous

83-0212,

[17]

C. H. Lewis.

S. Swaminathan, librium

[15]

and

August

Method

Rockets,

Gas

with

Volume

over

1966. Angle

22(Number

1985.

Heating

Analysis

of Attack".

for the Windward

In T. E. Horton,

editor,

Plane

of Shuttle-

Thermophysics

of

140

REFERENCES

of Atmospheric 82, ])ages

[26]

F.

Entry;

229-2,17.

C. Grantz.

|I.

Flow 1989.

C. Grantz,

cous Shock

[29]

Shock-Layer

York,

Hamilton

II, K. J.

Weilmuenster,

Shock

Viscous

Blunt-Nosed

Method

June

Volume

Layer PhD

the

and

F. M. Cheat-

in Aerodynamic

Method

thesis,

for

Calculating

North

"An

tlypersonic

Heating

the

Carolina

State

Approximate

Vis-

Flow over

Blunt-Nosed

1989.

and

R. A. Thompson.

for Hypersonic

Rockets,

Used

R. A. Thompson.

for Calculating

F. R. DeJarnette, Method

Bodies.

and

89-1695,

volume

1982.

Approximate

Paper

of Spacecraft and December 1990.

and Acro_mutics,

Methods 1985.

17. R. DeJarnette,

AIAA

A. C. Grantz,

New

It.

over

Layer

Bodies".

in Astro_'tautics

of Some Approximate Paper 85-0906, June

An

Hypersonic University,

[28] A.

AIAA,

R. DeJarnette,

wood. "A Review Analyses". AIAA

[27] A.

Progress

Flow

Volume

over

27(Number

"Approximate

Bhmt-Nosed 6):

Viscous

Bodies".

pages

597-605,

Journal November-

[3O] F.

M. Cheatwood and F. R. DeJarnette. "An Approximate Viscous Shock Layer Approach to Calculating Hypersonic Flows about Blunt-Nosed Bodies". AIAA Paper 91-1348, June 1991.

[31]

D. A. Throckmorton.

"Benchmark

Determination

dynamic Heat-Transfer Data". pages 219-224, May-June 1983. [32]

F. M. White. 1974.

[33] E.

W.

Viscou.s

Miner,

E.

Flows". 1971.

[34]

Reacting

dynamic

[35] C.

F. Hansen.

ties of High

[36]

C. H. Lewis. Arbitrary 1983.

and

Formulas

of Equilibrium Equations".

Air".

"VSLNOSE:

Nosetips".

and

Tables

TN

for the NASA

Thermodynamic TR

Report,

R-50,

Volume

Boundary

for Layer

Institute,

and

February

York,

Equilibrium

Some

Air for Use in Solutions

D-19.t,

A Three-Dimensional

Technical

Turbulent

New

20:

Program and

Polytechnical

of Density

Dissociating

NASA

"Approximations Temperature

and/or

Gas

Aero-

Volume

Company,

Perfect

Virginia

Entry

Rockets,

"A Computer

Nonreacting

Transitional,

Orbiter

and

Book

C. II. Lewis.

VPI-E-71-8,

"Correlation

Properties

Boundary-Layer

Laminar, Report

of Spacecraft

McGraw-llill

aJld Axisymmetric

Technical

N. B. Cohen.

Flow.

C. Anderson,

Two-Dimensional Chemically

l'Yuid

Journal

of Shuttle

May

Thermoof the

1960. and

Transport

Proper-

1959. Viscous

Shock-Layer

1: Engineering

Analysis,

Code

for

January

t

REFERENCES

[37] W.

141

E. Moeckel

and

of Air in Chemical

[3S] W.

L. Bade.

Dissociated

[39] S.

K. C. Weston.

"Composition

Equilibrium".

NACA

"Simple Air".

Srinivasan,

Analytical

ARS

[4O]S.

and

Properties

[41]

E. C. Anderson Shock-Layer

[42]

Virginia

R. R. Thareja,

K. Y. Szema,

Rockets,

Volume

R. N. Gupta, personic Reynolds ber 2):

[45] T.

pages

Cebeci

ible

175 184,

Boundary

T. Cebeci.

"Behavior

into

N. Moss, over

and

Long

Laminar

and Report

Laminar

Journal

or

of Spacecraft

1983. R. A. Thompson.

Slender and

of a PerFebruary

Technical

Equilibrium

Flows".

"Hy-

Bodies-Part

Rockets,

I: High

Volume

27(Num-

1990.

"A Finite-Difference

Layer

of Turbulent Conference,

J.

Flows

1975.

"Chemical

of Spacecraft

March-April

Transfer

March

Viscous-

D-7865,

September-October

Solutions Journal

TN

of Attack".

Shock-Layer

454-460,

attd A. M. O. Smith.

Turbulent

Computation IFP-Stanford

Equations

Boundary Stanford

Solution

by an Eddy

Layer+', Volume University, 1968.

of Turbulent,

Flow

near

to the

Viscosity

1, pages

a Porous

Wall

IncompressConcept".

346-355.

with

In

AFOSR-

Pressure

Gra-

t

dient".

[47] T.

AIAA

Cebeci

gineering,

Journal,

and

Compressible

[48]

Flows".

of Mass

all Angle

Transport

Hypersonic

Turbulent

NASA

Institute,

E. V. Zoby,

Shock-Layer

Number

"Effects

for the

Fits

August

1987.

of the

and

Bodies".

Cones

Viscous

K. P. Lee,

Viscous

Transitional,

of

Curve

RP 1181,

December

Solution

and C. 1I. Lewis.

20: pages

"Simplified

Fits

CR 178411,

of State

1959.

Curve

Polytechnical

Three-Dimensional

4), April

"Simplified

C. H. Lewis. over

Equation

NASA

Symmetric

Layers

VPI-AERO-031,

and

[46]

, and

Boundary

to the

29(Number

"Num_,rical

Properties

1958.

Air".

for Laminar, Axially

4265,

K. d. Weilmuenster.

NASA

J. N. Moss.

Blunt

M. C. Frieders,

Turbulent

[44]

Air".

Thermodynamic

of Equilibrium

C. Tannehill.

Equatious

Turbulent

[43]

J.

and

Over

and

Properties

of Equilibrium

fect Gas 1975.

Volume

J. C. Tannehill,

Srinivazan

TN

Approximation

Journal,

for the Thermodynamic 1987.

and

Volume

A. M. O. Laminar

pages

523-535,

B. S. Baldwin

and

for Separated

Turbulent

8: pages

Smith.

and

September

Flows".

December

"A Finite-Difference

Turbulent

H. Lomax.

2152-2156,

"Thit,

Boundary

Layers".

1970.

Method

for

Journal

Calculating of Basic

En-

1970. Layer

A1AA

Approximation

Paper

78-0257,

and January

Algebraic 1978.

Model

142 [49]

REFERENCES

L. A. King with

and

D. A. Johnson.

a Nonequilibrium

"Separ_ted

Tnrbulence

Transonic

Model".

NASA

Airfoil TM 86830,

[5o] T.

J. Coakley. "Turbulence Modeling Methods for the Stokes Equations". AIAA Paper 83-1693, July 1983.

[51]

E. R. Van Sciences,

[52]

Driest.

"On

Volume

F. It. Clauser.

Turbulent

23(Number "The

Flow

Turbulent

P. S. Klebanoff. Pressure

[54] S.

Dhawan

and

Transition

l(Part

[56]

E. Harris.

"Numerical

F. G. Blottner.

T. Davis.

a Blunt

[6o]

AIAA

R. A. Thompson,

August

July

J. M. Yos, R. A. Thompson, and

Thermal

1990.

and

Transport

Nonequilibrium

during

with

"An

Paper

Zero

2):

Reacting

June

Laminar,

with Experimen-

of the pages

Aerother-

87-1475,

for Compressible Comparisons

of Solution

Boundary-Layer

193-206,

Binary

February

Mixture

Past

for the

Eval-

1970.

R. N. Gupta.

Properties,

Zero

Mechanics,

Layer

P. A. Gnoffo. AIAA

8(Number

Air Model".

and

and

Equations

Layers

70-805,

with

Flow

of Fluid

in a Boundary

of a Chemically

K. P. Lee, and

and Therinodynamic

for Chemical 1232,

Paper

of an ll-Species

R. N. Gupta,

of the

Volume

Layer

Aca-

1954.

Methods

Flow

of Thermodynamic

Constants

Rates

Journal,

"Hypersonic

Body".

uation

[61]

AIAA

1-15.

Layer

Journal

Vehicles".

Boundary 1971.

"Finite-Difference

Equations". 1970.

[59] R.

Solutions

and Turbulent NASA TR 368,

July

Conical

T. von

1958.

K. E. Wurster,

of Slender

and

pages

of Boundary

of Turbulence

E. V. Zoby,

of Aeronautical 1956.

in a Boundary

Motion".

3178,

Navier-

1955.

January

TN

Journal

Mechanics,

Properties

to Turbulent 418-436,

NACA

Study

Transitional tal Data".

"Some

Compressible

In H. L. Dryden

in Applied

1247,

Calculations

1985.

November

Layer".

of Turbulence

Report

"Characteristics

R. A. Thompson,

[57J J.

pages

Gradient".

modynamic 1987.

[58]

Laminar 4):

P. S. Klebanoff. Pressure

NACA

R. Narashirna.

from

Volume

[55]

"Characteristics

Gradient".

a Wall".

1007-1011,

Boundary

Karmen, editors, Vol. IV of Advances demic Press, New York, 1956.

[53]

Near

11): pages

Flow

"Computer

Transport NASA and

Codes

Properties, TM

102602,

K. P. Lee.

Properties Calculations

and February

"A Review

for an 11-Species to 30,000

K'.

Equilibrium 1990. of Reaction Air Model NASA

RP

REFERENCES

[62]

B. F. Armaly Gas

[63]

143

and

Mixtures".

E. A. Mason

and

361-369,

September-October

D. Van

[65] F.

Dyke

and

C. J. Riley.

"An

[68] C.

[69]

J. Riley

[71]

[72]

and

R. A. Thompson

and

Hypersonic

Heat

Transfer

27(Number

4):

J. W. Cleary. Distributions

5450,

October

[74] J.

Lay-

1979.

Problems.

Interscience

Inviscid

Flowfields

Flow".

Master's

Method

for

Approximate Flow

Fields".

Paper

and

Journal

TP

Over thesis,

Vehicle

Calculating

3018,

August

of Three-Dimen-

91-0701,

January

Parameter

of Spacecraft

July-August

1991.

Influence

and Rockets,

on

Volume

1990.

of Attack Cone

NASA

Calculations

AIAA

"Flowfield

Drag".

of Angle

and

Bluntness

at a Mach

on Laminar

Number

J. N. Moss,

of 10.6".

Analysis

for the

Shuttle

bination

Rates".

AIAA

and

361-367,

HeatingNASA

TN

Plane

82-0842,

for a Flight

P. A. Gnoffo.

Surface

1982.

Exper-

"Aerothermody-

of Thermophysics

and

Heat

1989.

"Viscous-Shock-Layer

with June

"Comparison

1972.

October

A. L. Simmonds.

Windward Paper

2560,

Journal

pages

K. C. Wicker.

Measurements

TM-X

Vehicles".

4): and

NASA

K. E. Wurster,

Conical

3(Number

and

Heat-Transfer

F)".

E. V. Zoby,

of Slender

I,. B. Boney,

and

20 (Reentry

Volume

L. Shinn,

and

Predictions

R. A. Thompson, Transfer,

for Boundary

1988.

Fi(_lds".

C. B. Johnson,

at Mach

Study

hdtial-Value

"Engineering

Flow

on a 15 Degree

P. C. Stainback,

namic

of Blunt

1969.

of Theoretical

[73]

a Family

September

in Hypersonic

Inviscid

361-368,

"Effects

Rate

Past

Techniques

for Calculating

"An

E. V. Zoby.

pages

for

May

F. R. DeJarnette. llypersonic

Flow

79-0893,'

Bodies

F. R. DeJarnette.

C. J. Riley

iment

SAND

Method

University,

Three-1)imensional

lnviscid

Con-

5): pages

1957.

Approximate

Hypersonic 1990.

sional

[70]

and

Thermal

3(Number

1959.

Methods

York,

State

for the

Volume

"Supersonic

to Computational Release

Blunt-Nosed

Carolina

Formula

of Fluids,

rl'R R-l,

Difference

Three-Dimensional North

NASA

New

Ionized

1958.

"Introduction

Inc.,

Physics

Partially

1980.

"Al)proximate

The

Laboratories

R. D. Richtmyer.

of Multicomponent

July

H. D. Gordon.

Bodies".

Sandia

Publishers,

[67]

Mixtures".

G. Blottner.

ers".

80-1495,

S. C. Saxena.

of Gas

Axisymmetric

"Viscosity

Paper

ductivity

[64] M.

[66]

K. Sutton.

AIAA

Finite

Catalytic

Heating Recom-

144 [75]

REFERENCES

D.

A.

Anderson,

Mechanics 1984. [76]

J.

J.

and

IIeat

D. Anderson,

Jr.

McGraw-Hill

Book

[77] J. D. Anderson, Hill Book [78] W. John

Transfer.

Modern Company,

and

and Sons,

New

and

Compressible York,

York,

1965.

II. Pletcher.

Computational

Publishing

Flow:

Corporation,

with

Historical

Fluid

New

York,

Perspective.

1982.

and High

Temperature

Gas Dynamics.

McGraw-

1989.

C. H. Kruger, Inc.,

R.

Hemisphere

New

Hypersonic

Company,

G. Vincenti Wiley

Jr.

C. Tannehiil,

Jr.

Introduction

to Physical

Gas

Dynamics.

145

A

Reference

The tions:

governing

equations

Quantities are

nondimensionalized

S* 8

n

R*_ose

Z*

7"*

--

I'

--

V

--

U*

v= p*

--

.

tt T4

/_.*

*

T-

l%_s

n* _=c---:-

c; c,

c_-

pc_

C*

poo

T*

poo

' "*

(Vi

--

rr_

--

wi

_i,:,s

quantities

are

P .4 = P_o T* _4,

Pgo *

k=_ * q_4

reference

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'* --

R.nose

J;;s- R:o,.

of viscosity evaluated at T*re J" dimensionless parameters appear

in the

equations:

rela-

146

A.

IE_E1

_ '_ QUANTITIES RENCE

Lez2i = p _Pl i,,,, k*

Units

for the

dimensional

/_o_e

quantities

used

al)ovc

are

= [fl]

.

p,# =

h_j= L_._ J BTU

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.

[email protected] -- =

[email protected]

+"

-}- 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