Modeling of Heavy-Gas Effects on Airfoil Flows

May 11, 1992 - Roshko write equation (2.13) in more general form as. Pc ..... The results presented in this thesis were based on a quadratic fit for _bfrom ...
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Final

Modeling

Technical

to: Monitor:

NASA

Ames

Raymond

Research

Grant:

Submitted

by:

Group

NAG2-708

Department

Cambridge, Investigator:

Effects

Center

Aerodynamics

of Aeronautics

Massachusetts

Principal

! l

Hicks

Computational NASA

for:

of Heavy-Gas Airfoil Flows

on

Submitted

Report

Mark

MA

Astronautics

of Technology

02139

Drela

Associate MIT

Institute

and

Professor

Aeronautics

and

11 May

Astronautics

92

Nq2-2755 (NASA-C_-190_7) EFFECTS _N

&TRFO[L

MODELING FLO_S

_F _inJ1

H_AVY-GAS Report

(MII}

G_/uz

Uric| as 0091531

1

Summary

A non-ideal element

gas model

airfoil

has

program.

tunnels

employing

(SF6),

although

been

The

heavy most

developed

specific

gases. heavy

and

applications

The particular gases

retro-fitted

could

into

targeted

the

are

gas modeled

viscous/inviscid

compressible

in this

be implemented

MSES

work

airfoil

has

if adequate

been

state

flows

sulfur

and

multiin wind

hexMtuoride

caloric

data

were

available. Numerical airfoil

predictions

behavior

tunnels.

in transonic

The

dominant

in supersonic smaller)

zones

effect

somewhat

with

flows, effect

are

is that

more

MSES

especially

is that

lower,

and

for a given

resistant

indicate

that

the

at the

higher

for a given

shocks

edge

are

Mach

to an adverse

non-ideality total

freestream

pressures Mach

correspondingly

number,

pressure

of SF6

envisioned

number,

weakened.

a boundary

gradient

due

significantly

layer

for pressurized

local

Mach

Another

(but

in a heavy

to reduced

influences

gas

adiabatic

numbers apparently

is theoretically

heating

near

the

is valid

for

wall. As

pointed

non-ideal

out

gases.

by

Wagner

Similarity

and

Schrnidt

between

two

[1], transonic

flows

small-disturbance

can be obtained

if the

theory

transonic

similarity

parameter

1 - M 2

K-

[M2(7'+1)]2/3 is matched,

and

if the

pressure

coefficients

are

scaled

by

the

factor

M 2 A so that

the

above

are

quantity deft_ned

B for the

flows.

rameter the

for and

addition

of the

viscous Re

number

on

boundary

layer

that

the

parameters

M,

flows. and

the

of %, is not

effect

1 atm,

and

for

of a fixed

M.

Figure

and

SF6

Re)

Cp vs z/c at

7',

curves

good

for the

for air).

2 makes the

Clearly,

the

comparison matching

2822

airfoil

comparison at M*

a fixed

an

A

therefore

into

air

and

[2] at M

= 0.735

K

and

or K is more

M*

ACL

appropriate

scales viscous

flows. similarity

for air, = 0.765

It is rule

K,

ACL,

heavy-gas

parameters.

a fixed

pa-

and

matching

three

at

tran-

affects

one

con-

additional

number,

that

between other

in the

in compressible

indicate

correspondence

RAE

and

for viscous

that

all be combined

as the

K

in Appendix

is rigorous

Mach It

gradients

experiments

as significant

local

thickness.

7_ can

numerical

Figure makes

and

flows

C shows and

to pressure

is derived

be formulated

Appendix

properties

parameters

in MSES.

cannot

displacement

Re,

gives

nearly

3 atm.

3 in turn

CL = 0.743

still

the

gas

response

Fortunately,

M °, CL,

1 compares

= 0.735

Mach

on the

The

7', which

transonic

rule

Re,

flows.

heats

inviscid

number

depends

two

employed

a similarity

Reynolds

and

Figure

M

the

the

of specific

non-ideal

theory,

This

local

transonic (or

ratio

and

effects

unlikely

Apparently,

at

to

between

formulation

ideal

small-disturbance

In

displacement highly

between

(7'+1)-1__2

be matched

of an "equivalent"

7_ is introduced.

effect

also

small-disturbance

similarity

of transonic

sonic

must

in terms

second-order

Although text

ACL

=

flows.

for

SF_

instead

(corresponding for evaluating

to

RAE

....

Figure

1:

atm).

Cp distributions

CL

= 0.743,

transonic

flow

and

(3 atm)

SF6

comparisons, the K

the

ACL,

effects

the

unscaled.

However,

the

total

good

that match

slatted

Figure number reasonable.

= Voo/a_

incompressible

type

clear

drag

matching

a fairly

of M.

how

5.200.106

ALFA

:

2.q83

CL

=

0.7q30

CD

:

0.01367

CM

=

-0.0872

L/D

=

5q.36

="

9:00

= 0.735

......

for

air,

and

be scaled

ACL

(1 atm),

SF6

and

SF6

(3

described

strong

shock

It should

be

where

by

on

stressed

any

rise

much

A,

while

these

drag

the

are

results

drag

for air,

As

smaller not

coefficient friction

SF6

expected

were

CO

drag

components

theory

flows.

(1 atm),

from if M*

the

over

should

at

this

Cp

is used

performed

as

fixed

sweep.

In

be

left

perhaps

in an experiment,

is obviously

matching

Figure

6 shows

in reference

[3].

the

a somewhat

slat on

to M again

C/; gives

5.

sweep

profile

(or alternatively

Cj, distributions

unscaled

the

4 and

drag

Mach

behavior

since

only

survey.

heavy-gas

airfoil

drag-divergence

The

to scale

a wake

(corresponding

flows

M

on transonic

to separate

= 0.3257

and

:

in Figures

small-disturbance

air and

the

M*

of gas

should

from

K

between

at

further,

and

in lieu

configurations,

7 compares M*

M

it is impossible

two-element

air produces

of the

is obtained

For high-lift indicate

versus

it is not

pressure

drag

0.735

RE

NCRIY

airfoil

To illustrate

parameter

since

2822

=

= 6.2 million.

is shown

compressibility and

RAE

characteristics.

principle,

M

Re

for

2822

MACH

the

slat

= 0.30

that

a very

and

poor

gas non-ideality

M*

and

the

inviscid

A freestream

for the

for air)

simply

invalid,

three

and

matching

match

in all

CL)

the

usual except

gives

number

gas cases = 2.85.

cases,

still

shock

CL

numerical

studies

a reasonably

Cp distributions

Mach

weaker

but

on at

The

of M the

main

a fixed

of course

= 0.30

Mach

a in

element.

sonic

comparison

freestream

over

Mach is quite number

in effectively

is irrelevant.

3

RAE - 2.0 I"sE_

sF63 _T.-_ I "l'M

5F6

v 2.1

1,5 Cp

A IR -_7 L

_'_I

'

:

0.735

.E ALFA

:

_2.q83 2o0,_o'

CL

:

0.7q30

co --0.01367 _. :-oo872 _,o . s. 35

111 III

-1.0

2822

MACH

,

T = 9.00

-O.S

0.0

O.S i

1.0

j

f

Figure

2:

arm).

CL

Cp distributions = 0.743,

-2.0

Re

for RAE

2822

airfoil

at

M*

= 0.765

for

air,

SF6

(1 atm),

and

SF6

(3

= 6.2 million.

RAE WSE5

-1.5

Cp -1.0

2822

MACH

:

0.735

RE

=

6.200"!06

ALFA

:

2.483

CL

=

O, 7q30

C0

=

0.01367

CM

=

-0.

0872

L/O

=

5q.

36

Nc_Ir

=

9.00

-0.5

0°0

¸

0.5

1.0 f

Figure atm).

3: Cp distributions ACL

= 2.095,

Re

for ttAE

2822

airfoil

at

K

= 0.3867

for air,

SFe

(1 arm),

and

SFe

(3

= 6.2 million.

#

0.030

.TOTRL I

RIRFOIL /

R2822

RIR eI

R2822

SF6

1 RTM

R2822

SF6

3

RTM

/

0.020

,VlSC II

_D

.

0.010

WRVE

0.000 O. 55

O. 70

0.80

F1 Figure

4: RAE

C/: = 0.743,

2822 drag-divergence

behavior

versus

M for air, SF8 (I atm),

and

SF6

(3 atm).

Re = 6.2 million.

0.030 RIRFOIL '

R2822

/

RIR

0.020

CI]

0.010

0.000 O. 55

O. 70

O. 75

O. 80

H* Figure

5: RAE

CL = 0.743,

2822 drag-divergence

Re = 6.2 million.

behavior

versus

M ° for air, SF8 (1 atm),

and

SFs

(3 atm).

-]0.0

NASA A

MSES V 2.11

-9.0 -8.0

=

ALFA EL

= [9.q67 = 2.8500

CO

= 0.00582

0.300

= 0.1098 =q89.49

CM L/D

-7.0

SLAT 2

MACH

-6.0

C ,o

i

-q.O

I

J

-3.0

I i

-2.0

I I I

-1.0 0.0 1.0

Figure

6: Cp distributions

SF6

3

5FS

i

for slatted

airfoil

in air.

¢ITM _TM

#]R

NASA

-IO.OTwsES

_9.0j v2" -8,01

1 -7.0i -S.O

'

i

:

i

::i

:

::::::i:i!_z

SLAT =

ALF_

=

]9._66

CL

=

0.8601

CD CM

I

A

MACH

0

2

{EL

I)

0.300

= 0.00000 = 0.2355 :=:O;OO

:

I

-4.0 -3.0 -2.0 -i .0 0.0:

i

1.0

Figure

7: Cp distributions

(3 atm).

over slat at M* = 0.3257

j and CL = 2.85 for air, SF6 (1 atm),

and SF6

The bulk of the heavy-gas model development and applicationto transonic, inviscidflows is documented in the SM

Thesis of Marc Schafer,which isattached as Appendix A. As mentioned

previously,Appendix B derivesthe farfield behaviorof a non-idealairfoil flow. This was required for implementation of new outer boundary conditionsforthe MSES

code. Appendix C derives

the shape parameter compressibility correctionforan adiabaticboundary layerin non-idealflow. This was requiredto implement new heavy-gas correlations forthe MSES

integralhoundary layer

formulation.

References

[11B. Wagner and W. Schmidt. Theoreticalinvestigations of realgas effectsin cryogenic wind tunnels.

AIAA

[2] P. H. Cook,

Journal,

16(6),

M. A. McDonald,

and boundary

Jun 1978. and M. C. P. Firmin.

layer and wake measurements.

Assessment,

AR-138.

AGARD,

Aerofoil

In Ezperimental

RAE

2822 pressure

distributions

Data Base for Computer

1979.

[3]E. Omar, T. Zierten,and A. Mahal. Two-dimensional wind tunneltestsof a NASA airfoil with varioushigh-lift systems.Contractor Report 2214, NASA, [4] M. Drela.

Two-Dimensional

Equations. Report

PhD thesis,

Aerodynamic

Design

Dec 1985. Also, MIT Gas Turbine

and

supercritical

Apr 1973. Analysis

& Plasma

Using

Dynamics

the

Euler

Laboratory

No. 187, Feb 1986.

[5] J.D. Cole and L.P. Cook. Mathematics [6] M. B. Giles Journal,

Transonic

MIT,

Program

Aerodynamics,

and Mechanics.

New-Holland,

and

Two-dimensional

M. Drela.

25(9),

Sep 1987.

[7] D. L. Whitfield.

Analytical

AIAA-78-1158,

Transonic

description

volume

Amsterdam,

New York,

transonic

of the complete

30 of New-Holland

in Applied

1986.

aerodynamic

turbulent

Series

boundary

design

method.

layer velocity

AIAA

profile.

1978.

6

N

!

Appendix

A

Modeling

of Heavy

Gas

Effects

on

Airfoil

Flows

by Marc

Submitted

Alan

to the Department

Schafer

of Aeronautics and Astronautics

on May

3, 1992

in partial fulfillmentof the requirements for the degree of Master

Thermodynamic

of Science in Aeronautics and Astronautics

models

were constructed for a caloricallyimperfect gas and

for

a non-ideal gas. These were incorporated into a quasi one dimensional flow solver to develop an understanding the perfect gas model.

of the differencesin flow behavior between the new models and

The

models

were also incorporated into a two dimensional flow

solver to investigate their effectson transonic airfoilflows. Specifically,the calculations simulated results but those

airfoil indicated

that

matching

testing that

in a proposed the

high Reynolds

non-idealities

caused

of an appropriate

number

significant

non-dimensional

heavy-gas differences

parameter

led

test

in the to flows

in air.

Thesis

Supervisor:

Mark

Drels,

Associate

Professor

of Aeronautics

and

facility.

Astronautics

flow

The field,

similar

to

4

Acknowledgments

I would liketo expressmy thanks to allthose who made thisthesispossible.First, to Mark Drela whose brilliance and ingenuityhave served as an inspirationin allof my studies.Also, to Harold 'Guppy' Youngren whose leadershipduring the Daedalus projecthelped me to reMize what itreallymeans to be an engineer.

I would also liketo thank my parents and the rest of my family. Without your support,I never would have made itas faras I have.

My

appreciationalso goes to the NASA

Ames

research centerand the NDSEG

fellowshipprogram without whose financial support thisthesiswould never have happened.

PRECED:NG

PAGE

BLANK

5

NOT

FILMED

Contents

Abstract

3

Acknowledgments

1

Introduction

10

2

Real

11

3

Gases

2.1

Calorically

2.2

Non-ldeal Gases

Solving

the

Imperfect

Euler

Gases

...............................

Equations

3.1 Calorically Imperfect Gas

3.2

4

Non-Ideal

.........................

11

12

15

..........................

Gas .................................

16

17

20

Results

4,1

One Dimensional

Duct Flow .........................

21

4,2

Two Dimensional

Res_flts

23

5

Conclusions

A

Curve

pRECAE_.r2.!NGPAGE

Fit

..........................

28

For SFs

State

2T

Equation

..... FILMED _:,_;._.._ _w:3T ,.

7

11

B

MSES

Bibliography

Subroutine

for Non-Ideal

Gas

Model

28

43

jJ7

List

4.1

Stagnation and

Pressure

SF8 at latin

of Figures

Ratio(Strength)

vs.

Upstream

Mach

No.

for Air

and 3atm

.........................

2O

Duct Flow

.........................

21

4.2

One dimensional

4.3

Shock Strength

and

Location

vs. _ .....................

22

4.4

Shock

and

Location

vs. Z0

22

4.5

Comparison

of Air and SF6 at Fixed M and

4.6

Comparison

of SFs at latm

and

3atm to Air, M* = .740, CL = .9

4.7

Comparison

of SFs at latin

and

3stm to Air, M" = .732, CL = .75

4.8

Comparison

of SFe at latm

and 3atm to Air, _¢ = .439, ACL

Strength

9

....................

Cr,

.............

24

= 2.18

. . .

24

. .

25

. .

25

Chapter

I

Introduction

In the past few decades, the design and development of large transport aircrafthas reLiedon wind tunnel data taken at significantly lower Reynolds numbers than those found in operation. The drawbacks of thissubscaledata become apparent when one considersphenomena

such as attachment linetransition or similaraspectsof boundary

layerbehavior at high Reynolds numbers.

The need for accurate wind tunnel data clearlymandates the constructionof a suitablehigh Reynolds number testfacility. However, the cost of buildinga largeatmospheric tunnel and largetunnelmodels isprohibitive. Higher Reynolds numbers are oftenachievedby pressurizing tunnelsto effectively increasethe densityof the air.This alternative ispracticalonly up to a point.

A potentialsolutionfollowingthe same basicidea reliesupon the use of gases with significantly higher molecular weights than air. Candidate gases include Freon-12 or SuLfur Hexaflouride(SFs), but the use of non-breatlutblegases clearlycauses some problems. These problems willlikelybe insignificant to the cost and operationaladvantagesof such a facility. Combining heavy gases with pressurization would allow test Reynolds numbers comparable to those on largetransportsin flight[1].

One complicationisthat Freon and SFs have si_plificantly different thermodynamic propertiesthan air,especial]y at elevatedpressures.Heavy gases do not followthe ideal equationof stateP - pRT nearlyas wellas airdoes,nor do they maintain a constant ratioof specific heats _ -- c_/c_ over any significant temperature range. The following discussionwillattempt to quant_y the potentialimportance of these effects through a computational study.

Chapter

Real

The thermodynamic

relations

2

Gases

specifically

subject

to real gas effects are the state

equa-

tion

p : p_T

(2._)

and the caloric equation, h = /cpdT

(_.2)

= cpT

theseparticularforms only being validfora perfectgas. Real gas effects may be divided into two cate$ories:

I. Calorically imperfectgases forwhich cp depends on temperature, but which still satisfyequation (2.I). 2. Non-ideal gases for which cp depends on both pressure and temperature, and equation (2.1)no longerholds.

The firsteffectresultsfrom the introductionof multiplevibrationalmodes for polyatomic molecules which become more important at higher temperatures. The second effectdepends on intermolecular fozceswhich become strongeras a gas moves towards liquefaction, ie.higher pressuresand lower temperatures.

2.1

Calorically

Imperfect

Gases

The only difference between a perfectand an imperfectgas stems from the dependence of c_ on temperature

in the imperfect

case.

A cursory _T_mln_tion

for $F6 shows that,

in the range of temperatures

11

of experimental

data

likely to be found in a wind tunnel

test,thisdependence islinearin temperature.

(2.3)

%(T) = a + bT Therefore,

equation

(2.2) becomes bT 2

(2.4)

h(T) = _T + -7which may be easily inverted

to find T(h).

- -_+

2.2

Non-Ideal

(2.5)

+T

Gases

The stateequationfora perfectgas (2.1)derivesfrom a kineticmodel of gas molecules which assumes that themoleculesarepointmasses and that they do not exertany forces on one another except instantaneouslyduring collisions.Clearly these assumptions become lessaccurateas the molecular weight of the gas increases.Van der Wsals's equation

(p + p2 )I1 contains molecular

two correction attraction,

to equation and _ corrects

(2.1):

= pRT

a corrects

for the volume

the pressure of the molecules

/26) to account

for inter-

themselves.

Using a non-idealstateequationlikeVan der Waals's causes many seriouscomplicationsas enthalpy,%, % etc.now depend on pressureas wellas temperature. Despite these complir2ttions, enthalpyand entropy must remain stateva_'iables regardlessof the form of the state equation. That is,localentropy and enthalpy must depend only on the localpressureand temperature and not on the upstream conditions(ie.the gas

history).

Liepmann and Roshko [2] equatethisconditionwith therequirementthat a canonical equationof statemust have one of thesefour forms:

e = e(_,p)

(2.T)

h

g Here e -- h - p/p

is the usual

internal

=

h(s,p)

(2.8)

=

f(T,p)

(2.9)

=

g(T,p)

(2.10)

energy,

f

-- e - Ts

is the free energy,

and

g _- h - Ts is the free enthalpy.

For a conventional specifying readily

flow solver,

the enthalpy

defintion

(2.8) appears

the state in this specific form is not convenient

available

to the flow solver.

Liepmann

because

and Rosh.ko propose

best; however,

the entropy

s is not

a more suitable

_LP = z(p,r)

(2.11)

pRT

which requires

T(p, h) to have a form which makes

For a Van der Waals's

form

h a state

variable.

gas

Z =

1 1 - _p

ap fit

(2.12)

which clearlyapproaches the ideal stateequation fora, _9 _

O. For typically

small

valuesof a and/3

Z

_

1 + p

where the second approximation Roshko

write equation

(2.13)

-

-_ 1 +

_-

(2.13)

is made to make Z = Z(p, T) explicitly.

Liepmann

and

in more general form as

Pc with Pc and Tc being the critical pressureand temperature of the gas,and _bevidently being a universalfunctionwhich they tabulatefor gases other than airbut with approximatdy

the same molecular

curve to experimental

data

weight.

For heavier

gases such as SFs it is best to fit a

as explained

in Appendix

A. For SFs, s good curve fit takes

the form

= c= It is now necessary function

to determine

h(p, T) can be obtained.

+

the specific heat c_pa_ity Liepmann

13

and

(2.15)

+

Roshko

c_(p, r) *o that the enthalpy combine

two forms of the

equation

of state

h(p, T) and

s(p, T) into the fundamental

reciprocity

relation

between

h(p,r) _d p(p, r) Oh Op which

is valid for any gas.

Oh Since Oh/Op = 5(r) the pressure

_

Combining

TO(1/p) OT

I p

this with the state

RT 2 (OZ)p

only depends

(2.16) equation

RT¢ _,(_)

on the temperature,

_ both

(2.11)

gives

9v(r )

(2.17)

h and cp must

be linear

in

as follows.

h(p,r)

=

/c-p(r)dT

+ pF(r)

(2.18)

Oh

cp(p,r) =-- aT =

(2.10)

d_" c-p(T) + p_-_

= cp(r) - R _p

As in the case of the calorically

imperfect

6p(T)

Substituting

(2.20) T_ _b" (_)

(2.21)

gas, c-p(r) has the form

= a + bT

(2.22)

thisinto the enthalpy equationgives

bT _ + pRTe h(p,r) = aT + -V --Pc ¢'( T/ ) It is also possible

to determine

the caloric

equation

(_) as e(p, r) [3].

14

by expressing

(2.23) the internal

energy

Chapter

Solving

These

gas models

the integral

may

the

be readily

form of the steady

3

Euler

integrated

Eater

into an existing

(pff . h ff + p_) dA ho -= h +

These equations

p to the enthalpy

used to capture

conditions

where () denotes

0

(3.1)

=

0

(3.2)

constant

(3.3)

lu_---_-2 = 2

of shock

the local Mach number

the equations,

quantitiy

and (),el

p =

_/_._

Y

_'/Y,d

=

h =

and

the upwinding

while the boundary

losses require the local stagnation

to nondimensionalize the dimensional

with a state equation

h and the density p. In addition,

the shocks requires

and evaluation

It is desirable

=

are exact for any fluid flow, but must be supplied

to relate the pressure scheme

flow solver which solves

equations:

j_._dA f

Equations

conditions.

the following

denotes

a reference

scheme

is used

quantity:

h _--_-P_

Furthermore, nondimensional

%, _,

and R are nondimensionalized

using

R resulting

in several

new

parameters.

a

=

a/R bT,,_ 2a

15

]q

For the resultspresented here, the referenceconditionsare chosen to be stagnation conditions.

3.1

Calorically

Imperfect

Gas

The nondimensional form of the caloricequation which governs the behavior of the imperfectgas is:

h(r)

=

/ %dT

(3.4)

=

aT + a_T 2

(3.5)

which may be invertedto give T as a functionof h. T(h)

With

T obtained

specified speed

from

value of p.

=

h, p may

-I

+ y_l +4nh/a 2j9

be determined

The local Mach

number

(3.6)

using the ideal

comes from

gas law (2.1)

the familiar

defintion

and a of the

of sound:

0p a2 =

_.

=7T

(3.7)

The localvalue of 7 may be found from equation (2.3). 7 =

The last remaining pressure,

density,

conditions



difficulty

a + 2a_T 1 - a - 2a_T

is the determination

and temperature.

from flow conditions.

_oo =

%-- =

These relations The familiar

of the isentropic are necessary

perfect

relations

to calculate

between

stagnation

gas relations _A_

-1

1+

(3.8)

M 2

do not hold for a calorically

imperfect

The proper forms are obtained

gas.

from the formal statement, dh = T d# + dp P

16

(3.9)

_'_

and for an isentropic

process

ds = O:

dh = -_

(3.zo)

P From the definition of enthalpydh = cpdT, and foran idealgas p/p = T, so equation (3.10)becomes cp(T)dT

= dp --

T

(3.1_)

p

Integratingthisequationgives

po = exp(-_logT + 2_(1 - r))

(3.12)

P and the isentropicdensityrelationthen followsdirectlyfrom the stateequation. p T(ho) Po T(h)

p = Po

(3.13)

Strictlyspeaking,solutionof the Euler equationsrequiresnothing else.However, if a Newton-IL_phson techniqueisused, allof the necessaryequationsmust be linearized for the Jacobian matrix. In the case of the calorically imperfectgas, the equations axe slightlymore complicated than for a perfectgas, but they may still allbe written explicitly. Therefore the linearizations axe easilydone by differentiating the relevant equations.

3.2

Non-Ideal

Gas

The nondimensional equationsdescribingthe non-idealgas are the stateequation

p

1 + p,_(,-_)

pT

(3.14)

Zo

and the caloric equation.

=

.q,,.

z ,l 1

(3.15)

Z0 isanother parameter which may be describedin terms of I-and r.

= 1 + _(

zo = po- o

1

)

(3.16)

17 ,,4"

The non-idealgas presentssome difficulty as the enthalpydepends on the temperature and the pressure.Therefore,from equations (3.14)and (3.15),p and T may be found using a Newton-Raphson

system to drivethe followingresidualsto zero. p

I + _(_)

R1(p,r)= pT

(3.17)

Zo

(3.1s)

_r + _T2 + p ¢'(_.r )

R2(p,r)= h-

The localMach number depends on the speed of sound which must be found from the definition: a2 :

_p,

(3.19)

This iscalculatedas follows:

dp but dh = dp/p

for an isentropic

=

-_P Op h dp +

process,

°2_

OP IIp -_

dh

(3.20)

and hence

(3.21)

_

The local7 reallyhas no me_nlng and need not be calculated.

The extra complexity of the non-idealgas appears in the calculationof the sensitivities. Since p and T axe found by an iterative processthey must be found by perturbing the Jacobian matrix of the converged Newton-Raphson and p is related remain

to a perturbation

system. A perturbationin h

in p and T by the condition

that the R(p, r, h, p) must

zero. 6RI

Numerically

--

inverting

-"

_

this system

oa

Jr

gives the required

derivatives.

(3.23)

The second derivatives are found in a similarfashionstartinginsteadwith !_ a_ the residuals.Using a subscriptnotationforthe derivatives (_

Jr

18

and -_p

_=Ph):

(3.24)

_

A similar system with Rzp invertinggives _

and R_p as residualsisalsoformed. As above, numerically

: -_, _

: -_-_p _ etc. These manipulationsare implemented in

the sottrce code in Appendix B.

The lastremaining task is calculationof the stagnationconditionsand, again,it is not possibleto find an analyticexpression. Another Newton-Raphson

system is

constructedwhere the first residualcomes from equation(3.15):

RI : ho- h(p,T)

(3.25)

The second residualisderivedby rearrangingequation(3.9) dh

ds

Integrating

T

+

dp

pT

(3.26)

=

_dT

+ d(p_) 7"

-

_dT

+ d(lx/rl_

dpz p e) -

(3.27) wd(p_)

dp P

(3.28)

gives:

-

(3.29)

The second residualmay then be formed

R2 = -_1.- siP, T)

(3.30)

where sz is the entropy of the staticconditions.

Driving these two residualsto zero gives the stagnation conditionsPo, To. The dezivatives-_p,_z, etc,needed forthe Newton-Rsphson

solvermay then be found by

perturbing the converged Jscobian matrix and relatingthe resultingderivatives to the staticconditionsthxongh the chain rulesad eqtmtions (3.15)sad (3.29).This process is identical to the one used above to findp sad T and theirderivatives.

19

Chapter

4

Results

After developingthe models forthe calorically imperfectand non-idealgases,the next step was to evaluatethe differences these changes caused in inviscidflows.The primary quantities of interest are thelocationof shocks and theirstrengthwhich isdefinedas the ratioof of stagnationpressuresacrossthe shock. For a perfectgas,the shock strength may be expressedas a functionof the upstream Mach number M1.

po_.__: = [1+ -y+ 27 1_''' _,,_I)]-I/('-I)[ m_ However,

for the non-ideal

(7+I)M_ [(#--T)M-_'_ -7 - 2'],/(,-I)

gas, this relation

must be calculated

1.000

""-'.._ _-.._

0.9511

,..

_

P er£ect

......

Po: Po:

",

%" ",

0.9049

N.

"',

%'% ",.

%'%" %.

•, "',

,

1._

Istm 3arm

"_%

-.

1.2oo

numerically.

-, """

Strength

0.8S0

(4.1)

,

,

%%"

1._

,

M1

Figure

4.1: Stagnation

SFs at latm

Pressure

Ratio(Strength)

and 3atm

20

vs. Upstream

Mach

No. for Air and

2.00 __......

Perfect rmperfect Non-Idead

Math 1.20

I; t_

0.40

!

I

I

o.2s

0.00

O.SO X

'

0.TS

'

1.o0

Figure 4.2: One dimensionalDuct Flow

4.1

One

The first

Dimensional

comparison

ing/diverging

Duct

of the different

Flow

gas models

was a study

of the flow in a converg-

nozzleusing a quasi one dimensionalEuler solver.This flowischaracter-

izedby sonicflow at the throat with a shock downstream to match the specifiedexit pressureas shown in figuxe(4.2).

As a basis dimensional

for comparison

reference

of the

enthalpy

(hopo/Po)

ho =

With

ho held constant,

also held tittle

constant.

7 therefore

Under

different

these

was made

models

in a duct

equal for all three

flow,

the

non-

cases.

3' 7-1

(4.2)

=

a(1 + ]3)

(4.3)

=

a(1 + a) + x_'(_)

(4.4)

zo

depends

on a, _, _r, and _'. The exit presure ratio is

conditions,

or no effect on shock strength

gas

the slope of the c_ versus T curve 09) had

or position

a e(4.s).

21

relative

to the perfect

gas as shown

in

0.9440

__ ___

0.860

Peffect Imperfect

__ Perfect ___/mperfe_

0.84G

0.9360 Location

Strength

0.82C

0.9280

0.9200

I

I

0.00

I

!

1.00

0.80_

!

2.00

o.oo

3.00

1;0

Figure

For

the

combined

non-ideal into

functions

Zo.

of Z0 and

stagnation

enthalpy

the

effects

of the

gas.

The

difference

becomes

less

'

2;o

'

and

may

3_00

Beta

Beta

gas,

4.3:

Shock

_ and

Figure(4.4)

r are not shows

the corresponding as above.

the

plots

by changing

in shock

strength

and

and

really

Location

vs.

independent

variation

perfect

These

non-ideality

Strength

in shock

gas results clearly

show

7 as in the position

parameters strength

with

7 adjusted

that case

becomes

and

position

to preserve

it is not possible

be as the

to mimic

of the

calorically

imperfect

larger

and

as the gas

larger

ideal.

0.9400

__ ___

0.860

Perfect Non-Ide,d

__ ___

0.S40 /

Strength

ss

Location

/

.s J

0.9300 ////_'

0.820

// // //

0.9200 0.60

0._)

'

O.SO00.60

1.00

'

Figure

4.4:

Shock

Strength

22

and

Location

0.80

Zo

ZO vs.

Zo

'

1.00

Perfect Non-Ide_

The Last test conducted effects

of the various

numerical

scheme.

with the one dhneusional

gas models

flow model

on the upwindlng

The flow solver

drives

scheme

the momentum

was to determine

needed

equation

for stability residual

(4.5)

2

speed

is defined

qi

and p_ is non-zero

as

:

qi

only if M_ is greater

_(M,(q,))

Initially,

the exact

earizations converged

and used in the upwinding

value of 7 even though perturbations.

_'/(qi

(4.6)

-- q_-l)

than Me.

1-

(4.?)

M?J

at each node along scheme.

analysis

with all the necessary

Under these conditions,

the upwindlng

the stability

Using a constant

--

= 7

7 was calculated

with Mc ___1. However,

of the

to zero,

R1 - p_qiA_(¢_ - q;_) + p_A_ - p_-IA___ +/_ + P_-I (A_ - A__I) where the upwinded

the

is relatively

used to derive

value of 7 had absolutely

l_-

the flow solver

insensitive

to the exact

equation(4.7)

ignored

7

no effect on the viable range

for Mc or the rate of convergence.

4.2

Two

Dimensional

Results

The subroutine

which appears

element

of the two dimensional

version

Numerical

experiments

clearly demonstrate Mach distributions

in Appendix

transonic

Airfoils

tests

airfoil design/mmlysis

the effect of the new gas model.

inviscid

Figure(4.5)

freestream

and 3atm and

in heavy

a stagnation

the airfoil

code ISES [4]. cases to more

conditions

Mach number and lift coefficient. The SFs is characterized temperature

performance

and Note

by stagnation

of 310K.

gases will be much more worthwile

be found so that the tests reflect

the multi-

shows am overlay of the

for a test airfoil run in 5Fs at two different stagnation

that they are not at the same angle of attack. of latin

into MSES,

carried out were limited to single-element

in air. All three cases axe at matched

pressures

B was incorporated

if some relationship

in air.

may

The only parameters

-2.0

RRE

_.us ,Y

2822

IEL

1)

i_,l

J L

-I.Sl

-1,0

//_/SPII

(161m)

" t

......

-11.51 I

0.0! I

0.51 l.n

Figure

which

4.5: Comparison

may be adjusted

ditions,

and angle

M and

C_ constant:

in a wind tunnel

of atttack

experimentatatJon,

of Air and SF6 at Fixed

or CL.

clearly,

test are the Math

Fignre(4.5)

shows

the best match

sonic conditions.

Figure(4.6)

(latin

at the same M*.

3atm)

-2.0

number,

an attempted

this is not an eft'ective technique. was achieved

After

velocity

stagnation match

_RE

2022

con-

keeping

a good

deal of

gases at the

to the speed of sound at

shows the case in air from fig_ze(4.5)

mill

C_

by running the different

same M* which is defined as the ratio of freestream

and

M and

tEL

compared

with SF6

l)

V_T,I

-t.5 _-%///--

C_.

-r_ l (l,.m)

s_'e(3,o--)

-[.0

-0.5

o.o: O.S

1.0

Figure

4.6: Comparison

of SFe at latin

24

and 3atm to Air, M* = .740, Cr_ = .9

2822

Rfi[

((I.

tl

vl.I

-1.51

/_ •_z

,s,lJr

_-

SPe (3.*=)

I ......

-0.5

l

0.0_

11.5[ 1.0

Figure

4.7: Comparison

of SFs at latin

and 3arm to Air, Mr* = .732,

A case with s weaker shock, figure(4.7) The match

is slightly

was used to further

worse, but this is to be expected

more sensitive

to small changes

M*, Anderson

[5] proposes

in M than a strong

mateh/ng

because

CL -- .75

verify this relationship. a weak shock

one. As an alternative

the small disturbance

similarity

is much

to matching

parameter

_ and

ACL where

(,M_(7, A

+ 1))2/s

(4.8)

' + z) z-

(4.9) _g[

211:P1 ((/

I!

Vi.I

-l.S

Cp -l.O

___--

s_

illllt,lte_lll

13.m)

istTl_VlVlllllelllgl_lllll

-0.5

0.0

O.S

1.0

Figure 4.8: Comparison

of 5Fs at latm

and 3arm to Air, x = .439, ACe

25

= 2.18

Chapter

5

Conclusions

The models derivedabove adequately describethe thermodynamic behaviorof non-ideal and calorically imperfectgases.Despite some minor complicationsin l£nearizing these models, they were implemented in routinessuitableforincorporationintoexistingflow solversbased on Newton's method. First,a quasi one-dimensionalflow solverwas used to examine the in_uence of the variousnon-dlmensionalparameters which govern the behavior of the different gases.

Transonicairfoil testcasesfor air and SFs were then used to study the influenceof parameters which may be controlledin a wind tunnelexperiment: stagnationpressure, freestreamMach number, and angle of attack. The goal of thisstudy was determine the conditionsunder which a wind tunnel testin a heavy gas would produce results comparable to those found in air. Matching M*

and CL or _ and ACL

were both

effective for the test cases presented here. Further study is necessary to determine which isbest formulti-elementcases.

The resultsare encouragingin that they definitely hint at the possibility of directly relating mentally SOWS.

heavy

gas test data to performance

the mode]

for SFs,

and to invest_ate

in air. It is first necessary

to verify experi-

the effects of non-ideal

gases on viscous

Appendix

Curve

A curve

Fit

For

A

SF6

fit may be found for the function

State

$ (_)

Equation

for any gas given experimental

state

data. With the density(p) measured at a number of different pressures(p) and temperatures(T), a vectoris definedcontainingthe difference between the realgas and a perfectgas at each data point. at-_t

- 1

Z=

(A.1)

_-I Defaxing0 = -_, the matrix A containsthe stateinformation.

1



:

.oo

:

:

:

(A.2)

/_.e!

The goal isto find a stateequationagreeingcloselywith the experimental data in but of the simple form: e,,

Z(p,T)=l+

p

f | C,_ L

_-1 (A.3)

C._I ... Co] 1

Therefore --_ A_ and _ is found

by the technique

of linear

regression:

= (ATA)-ZATZ The results presented data for SF6.

in this thesis were based on a quadratic

The required

(A.4)

data may be found in [6].

27

(A.S) fit for _bfrom approximate

Appendix

MSES

Subroutine

Jubroutine

for

B

hgpare(alfl,btal,

Non-Ideal

Gas

ccO,ccl,cc2,

taul,

Model

hO)

C---

non-ideal derived

Initializes Formulation

Input: alfl

Constants

gas routines. Schafer SM thesis.

in

for

Cp(T)

in

caloric

equation:

Cp

= a(1

+ bT)

betl

taul

Constant

ccO ccl

Constants

cc2

in

phi(T)

in

defining

phi

= cO

non-ideality

phi(T)

+ cl(tau/T)

factor

in

polyuonial

+

c2(tau/T)**2

Z(p,T)

fore:

Ousput: hO

Enthalpy

Tnternal

at

reference

conditions

pO,

TO

output:

zO

Non-ideality

factor

Z(pO,TO)

at

reference

conditions

C .......

real*4

implicit common

/nongss/

coamon

all, /nonfit/

k k

c2,

(a-h,m,o-z)

bta, cl,

pi,

tau,

zO

cO

c pu$

input

all

=

bta

= btal

t_



paraaetera

into

comaon

blocks

all1

taul

cO =ccO cl

:ccl

c2

= cc2

pi

:

c 1.0

c calculate zO

=

hO :

reference

1.0

+ pie(c2/tause2

(airs(1.

+ bta)

non-ideality

factor

+ cl/tau

+

+ pi/tausphid(1./tau))

28

and

enthalpy

cO) /

zO

_/

subroutine

nideal(hO,r,q,

p

t

,p_r

,p_q,

msq,msq_r,msq_q)

C ....

c

Calculates

pressure

c

stagnation

enthalpy,

and

Mach

density,

number and

for speed.

c c

Input

c

hO

: stagnat

c

r

density

c

q

speed

ion

enthalpy

c c

:

Output

c

p

pressure

c

p_r

dp/dr

c

p_q

dp/dq

c

msq

square

c

msq_r

dM" 2/d.r

msq_q

d]4" 2/dq

c C ....

of

Math

number

N'2

"

implicit

set

static

h

= hO

h_q

=

subroutine

real*4

(a-h,n,o-z)

enthalpy -

0.Seqe*2 -q

ngasp$(h,r,p,p_r,p_h,p_rr,p_hh,p_rh,

29

specified



_,t_r,__h,t_rr,t

C

Calculates

C

specified

pressure

and

static

temperature

enthalpy

and

hh,t_rh)

for

density.

C C

Input:

C

h

enthalpy

C

r

density

C C

Output

:

C

p

preHure

C

p_r

dp/dr

C

p_h

dp/dh

C

p_rr

d'2p/dr*2

C

p_hh

d'2p/dh'2

C

p_rh

d*2p/drdh

C

t t_r

t emp er aSur • dr/dr ... etc.

C

C ........................................................

implicit

real*4

(a-h,m,o-z)

dimension

a(2,2),

b(2,2),

bh(2,2),

couon

ai(2,2),

aih(2,2),

air(2,2),

br(2,2)

/nongu/ all,

bta,

pi,

tau,

zO

C

c ....

Newton data

convergence eps

tolerance

/5.0E-6/

C

c ....

initial

guess

from

if(bta.eq.O.O) t

imperfect

ideal

gas

then

= h/all

else t

=

(-1.0

+

sqrt(1.O

+ 4.0ebtaeh/alf))

to

converge

on

/

endif p = ret

Newton itcon do

=

100

set

loop 16 iter=l,

and

correct

p,t

itcon

lineazize

non-ideality

ttc

=

ttc_t

= -l./(taue_ee2)

factor

1./(taue_)

z = I. z_p = z_t =

÷ p_pi*phi(tt¢) pi*phi(ttc) p_pi*phld(t¢c)*ttc_t

C

residual

I:

eta_e

p/(r*t)

equation

reel

=

rl_p

=

l./(r*_)

-

- z

/zO

rl_t

:

-p/(rstse2)

- z_t/zO

z_p/zO

3o

Z(p,t)

(2.0ebta)

tml

=

tnl_p

= O.

(alf*t

+

tml_t

=

tm2

= p*pi/tau*phid(ttc)

/

zO

tm2_p

=

/

zO

tm2_t

= p*pi/tau*ph/dd(ttc)*ttc

/

zO

(all

all*bract**2)

+ 2.*alf*bta*t

)

2: caloric

zO

equation

= h - (tml

+ tm2)

r2_p

=

- (tml_p

+ tm2_p)

r2_t

=

- (tml_t

+ tm2_t)

Jacobian

matrix

a(1,1)

= rl_t

a(1,2)

= rl_p

a(2,1)

= r2_t

a(2,2)

= r2_p Jacobianmatrix

find

inverse

dstinv

=

1.0

ai(1,1)

=

a(2,2).detinv

ai(2,2)

=

ai(1,2)

=

-a(1,2)*dstinv

ai(2,1)

=

-a(2,1)edstinv

Newton

/

-

(a(1,1)*a(2,2)

a(1,2)*a(2,1))

a(1,1)*dstlnv

changes

dt

: -(ai(l,l)*reml

+ ai(l,2)*re82)

dp

= -(ai(2,1)*resl

+

rlx

/

t

res2

sot

zO

pi/tau*phid(ttc)

residual

set

/

al(2,2)sras2)

= 1.0

if(rlx*dp

.gt.

2.Sap)

rlx

= 2.5*p/dp

if(rlxsdp

.It.

-.8*p)

rlx

= -.8*p/dp

if(rlxsdt

.gt.

2.5*t)

rlx

= 2.Set/dr

if(rlxsdt

.it.

-.Set)

rlx

= -.8*_/dt

updat o varlablo8 t = t + rlxsdt p

= p

+ rlx*dp

convorgenca if

check

(abs(dplp)

.le.

spa

.and.

abe(dr/t)

.le.

epm)

C

IO0

coal_ians

C

grtte(s,*)

'IGISPT:

rrite(e,e)

'dp

write(.,*)

'p

Convergence

dT T h

r:',

p,

failed.'

dt

:', dp, t,

h,

r

C

3

continue

C

set

residual

derivatives

r1_r

= -p/(ree2et)

rl_h

= O.

er_

input

z,h

31

variables

So:o

S

r2_r=O. r2_h=l.

b(l,l)

= rl_r

b(l,2)

= rl_h

b(2,1)

= r2_r

b(2,2)

= r2_h

C C ....

set t_r

p,t derivatives wrt r,h = -(ai(1,1)*b(1,1) + ai(1,2)sb(2,1))

t_h

= -(ai(1,1)*b(1,2)

+ ai(1,2)*b(2,2))

p_r

= -(ai(2,1)*b(1,1)

+ ai(2,2)sb(2,1))

p_h

= -(ai(2,1)*b(1,2)

+ ai(2,2)mb(2,2))

set

second

C C C ....

residual

derivatives

tic

=

ttc_t

= -1./(tau*t**2)

tic_it

=

wrt

r,h

i,l(tau*t) 2./(tau*t**3)

z

= 1.

z_p

=

+ p*pi*phi(ttc) pi*phi

z_pt

=

pi*phid(ttc)*ttc_t

z_pp

= O.

z_t

=

pspiSphid(ttc)*ttc_t

z_tt

=

p*pi*

rl

=

rl_p

=

rl_pt

=

rl_pp

=

rl_t

=

rl_tt

='2.*p/(ret**3)

(tic)

(phidd(ttc)*ttc_t**2

p/(r*t)

-

z

/zO

l./(r*t)

-

z_p

/zO

-l./(r*t**2)

-

z_pt/zO

-

z_pp/zO

-

z_t

-

z_tt/zO

-p/(r*t**2)

r1_r

=

rl_h

= O.

rl_hp

= O.

rl_ht

= O.

rl_rp

=

r1_rt

=

r1_rr

=

+ phid(ttc)

/zO

-p/(r.*2et)

-l./(r*s2st) p/(rss2st*s2) 2.*p/(r**S*t)

tal

=

(alf*t

+

tal_t

=

(all

+

/

zO

2.*alf*bta*_

alfebtaet**2) )

/

tal_tt

=

(

zO

2.*alf*bta

)

/

Cml_p¢

= O.

zO

Cml_p

= O.

tml_pp

= O.

tm2

= pspl/tau*phid(ttc)

/

zO

ta2_p

=

pl/tauSphid(ttc)

/

zO

ta2_p¢

=

pl/tau*ph/dd(ttc)sttc__

/

zO

ta2_pp

=

tm2_t

= p*pi/tau*

tB2_tt

= pspl/taue(ph/ddd(t¢c)*ttc_t**2

O. ph/dd(ttc)*ttc_t

/ +

32

zO

*tic_it)

&

ph/dd(ttc)*t_c_tt) r2

= h

-

(tml

+ tm2)

r2_p

=

-

(tml_p

+ tm2_p)

r2_t

=

-

(tm1_t

+ tm2_t)

r2h

=

/

zO

I.

c c c ....

set

and

ph

= p_h

linearize

th

= t_h

new

+ rl_t

*th

rlh

rlh

= rl_p

rlh_ph

= r1_p

rlh_th

=

rlh_p

= r1_pp*ph

+ rl_pt*th

+ rl_hp

rlh_t

= rl_pt*ph

+ rl_tt*_h

+ rl_ht

rlh_h

= O.

rlh_r

*ph

residuals:

= drl/dh

= O,

r2h

= dr2/dh

= 0

+ rl_h

rl_t

= -ph/(r**2*t)

-

+ th*p/(r**2*t**2)

r2h

= I.

r2h_ph

=

tml_t*th

r2h_th

=

-

tal__

r2h_p

=

-

tml_pt*th

-

r2h_t

=

-

tml_tt*th

-

r2h_h

= O.

r2h_r

= O.

a(1,1)

= rlh_th

a(1,2)

= rlh_ph

a(2,1) a(2,2)

= rlh_th = r2h_ph

detinv

:

1.0

aih(l,l)

:

a(2,2)edetinv

/

aih(2,2)

=

aih(l,2)

=

-a(l,2)*dotinv

aih(2,1)

=

-a(2,1)*do_Inv

-

tml_p*ph

-

tml_p

-

tm2_t*th

-

tm2_p*ph

-

tm2_p

-

t12_t

tm1_pp*ph

-

ti2_pt*th

-

tm2_pp*ph

tml_pt_ph

-

tm2__*th

-

tm2_p_eph

(a(l,l)ea(2,2)

-

a(l,2)*a(2,1))

a(l,l)edetinv

dth

= -(a:Lh(1,1)*rlh

+ aih(1,2)*r2h)

dph

= -(aih(2,1)*rth

+ alh(2,2)*r2h)

c C

_

l

c

_h

= th

_l_

+

I_h

+ dth

c c c ....

sot

end

pr

= p_r

lt.no_tzÙ

tr

= t_r

rlr

= rl_p

rlr_pr

= rl_p

rlr__r

:

nor

*pr

roetduL].s:

+ rl_t

*tr

rlr

+ rl_r

rl_t

33

= drl/dr

= O,

r2r

= dr2/dr

= 0

rl=_p rlr__

= zl_ppspr

+ rl_pt*_r

+ r1_rp

= r1_pt*pr

+ rl_tt*tr

+ rl_rZ

rlr_r

= r1_rp*pr

+ r1_rt*_r

+ r1_rr

rlr_h

=

r2r

= -

r2r_pr

=

r2r_tr

= -

tml.t

r2r_p

= -

tml_pt*tr

-

r2r_t

= -

tml_tt*tz

-

r2r_h

--

O.

r2r_r

=

O.

O.

tml_t

a(1,1)

= rlr_tr

a(1,2)

= rlr_pr

a(2,1)

= r2r_tr

a(2,2)

= r2r_pr

*tr

-

tml_p

-

tml_p

*pr

-

t=2_t

*tr

-

tm2_p

-

tm2_p

*pr

-

tm2_t

tml_pp*pt

-

tm2_pt*_r

-

tm2_pp*pr

tml_pt*pr

-

tm2_tt*tr

-

tm2_pt*pr

-

a(1,2)*a(2,1))

C

detinv

=

1.0

air(l,1)

=

a(2,2)*detinv

/

(a(1,1)*a(2,2)

air(2,2)

=

a(1,1)*detinv

air(l,2)

= -a(1,2)*detinv

air(2,1)

=

-a(2,1)*detinv

dtr

=

-(air(1,1)*rlr

+ air(1,2)*r2r)

dpr

=

-(air(2,1)*rlr

+ air(2,2)*r2r)

C

c

pr

c

tr =

= pr

+

dpr

tr + dtr

C C

c ....

calculate

responses

in

dZ/dh

and

dxlh

= rlh_h

+ rlh_p*ph

+ rlh_¢*¢h

dr2h

= r2h_h

+ r2h_p*ph

+ r2h_t*¢h

dxlr

= rlr_h

+ rlr_psph

+ rlr__s_h

dr2r

= r2r_h

+ r2r_psph

+ r2r_tsth

dth

=

-(alh(1,1)*drlh

unit

h

pertuxba_ion

_o

unig

r perturba¢ion

+ alh(2,2)*dr2h)

*.hh

= dth

d_h



-(alr(1,1)*drlr

+ alz(1,2)*dr2r)

dph

= -(alr(2,1)*drlz

+ alr(2,2)*dz2r)

_hr

= dth

phr

= dph

responses

in

dt/dh

and

dzlh

ffi rlh_r

+ rlh_p*pr

+ rlh_tstr

dr2h

=

+

+ r2h_t*tr

r2h_r

to

+ aih(1,2)*dr2h)

dl_ = -(alh(2,1)*d=_h

calculate

dp/dh

r2h_p*pr

dp/dh

34

drlz

= rlr_r

÷ rlz_pepz

+ rlz_t*tr

dr2z

= r2r_r

+ r2r_pepr

+ r2r_t*tr

dth

= -(ath(1,1)*dxlh

+ aih(1,2)*dx2h)

dpb

= -(aih(2,1)*dxlh

+ aih(2,2)*dx2h)

trh

= dth

prh

= dph

d_h

= -(air(1,1)sdxlr

+ air(1,2)*dx2r)

dph tx_r

= -(air(2,1)*dxlr = d_h

+ air(2,2)*dx2r)

prr

= dph

set

final

first

p_r

= pr

t_r

=

p_h

= ph

t_h

=

th

p_hh

=

phh

t_hh

= thh

p_rr __rr

= prr = trr

p_rh

=

.5*(prh+phz)

t_rh

=

.S*(trh+thr)

and

second

derivatives

wrt

(r,h)

tr

return end

subroutine

nonstag(hO,rho,q,

pO,pO_r,pO_q,



rO,rO_r,rO_q

C

Calculates

C

specified

s_agna_ton

presses

and

:_a_attenauthalpy,

C

Input:

C

hO

stagnat

C

rho

denJlty

C

q

.pe_

£on

enChalpy

C C

OUtlm_:

¢

pO

station

C C

po_r pO_q

dpO/dr epO/dq

C

rO

8ta6na¢

C

rO_r

drO/dz

C

rO_q

drO/dq

lapllclt dimension reals4

reals4 a(2,2),

pzeesure

ton

density

(a-h,a,o-z) aI(2,2),

d_usi_y

denslSy,

C

b(2,2)

h_p,h_t

35

)

for and

speed.

co--on

/non,

u/ all,

comaon

bta,

pi,

tau,

zO

/non.fit/ c2,

cl,

cO

C

data

eps

/S.OE-6/

C

CCC

z(pp,tt)

=

z_p(pp,tt)

=

z_t(pp,tt)

=

h

= hO -

h_q

=

h_hO

=

1.

+ pp*pi*phi

(1./(tau*tt))

pi*phi

(1./(tau*tt))

pp*pi*phid(l./(taustt))

/

(-tau*tt**2)

.5.q*.2

-

q

1.0

C

r

= rho

set

input

call

preliure

and

temperature

and

derivatives

ngaspt(h,r,p,p_r,p_h,p_rz,p_hh,p_rh, t,t_r,t_h,t_rr,t_hh,t_rh)

C

set ttc

entropy =

• and 1./(tau*t)

ttc_t

= -l./(tau*t**2)

tic_it

=

derivatives

wrt

p,t

2./(tau*tee3)

ph

= phi(tic)

phd

= phid(ttc)

phdd

= phidd(ttc)

phddd

= phiddd(ttc)

C

ph_t

= phd



ttc_t

phd_t

= phdd

*

ttc_t

phdd_t

= phddd

* ttc_t



= alfelog(t) -

i_p • _t

+ 2.0*alfebtaet

p'p•*(

_*phd

= - pi*( = alf/t

t*phd

-

p*pi*(

*ttc_t *ttc_t + 2.0*alfebta

phd



*ttc_t

+ ph

)

-

log(p)

÷ ph

)

-

1.0/p

+ ph_t

+ t_hd_tettc_t + t*phd

)

*ttc_tt

c c ....

initial

gue|s

cc cc

if(bta.eq.O.O) tO = hO/al£

cc

elle

¢c cc

tO audif

cc

pO

= = p

(-1.0

for

pO,tO

from

imperfect

+

•qrt(1.O

• exp(-alfelog(t)

+ 4.0*btaehO/alf)) + alf*2.0ebtae(1.O-t))

C

tO

gas

then

= t

pO = p

36

/

(2.0*bta)

lerton itcon

C ....

do

=

100

loop IS

to

converge

iter=l,

correcZ

pO,tO

i$con

tic

=

tic_tO

= -l./(tau*tO**2)

tic_fro

=

ph

on

l./(tau*tO)

2./(tau*tO**3)

= phi(tic)

phd

= phid(ttc)

phdd

= phidd(ttc)

phddd

= phiddd(ttc)

ph_tO

= phd

* ttc_tO

phd_tO

= phdd

* ttc_tO

phdd_tO

= phddd

* tic_tO

enthalpy

residual

reJl

=

(alf*(tO

rl_pO

=

(

rlotO

=

(alf*(l.O+

+ b_a*tO**2)

+ pO*pi/tau*phd

)/zO

pi/tau*phd bta*tO*2.)

)/zO

+ pO*pi/tau*phd_tO)/zO

C

residual

entropy reg2

= all*log(tO)

&

+ 2.0*alf*bta*tO

pO*pi*(

r2_pO

=

-

r2_tO

=

all/tO

k

pie(

tO*phd

*tic_tO

+ ph

)

-

log(pO)

tO*phd

*ttc_tO

+ ph

)

-

l.O/pO

+

pO*pi*(

k

2.0*alf*bta

phd

*ttc_tO

+ ph_tO

+ tOSphd_tOsttc_tO + tOaphd

_ttc_ttO

)

C

getup

and

invert

a(1,1)

= rl_tO

a(1,2)

= rl_pO

a(2,1)

: r2_tO

a(2,2)

= r2_pO

Jacobian

detlnv

=

1.0

ai(1,1)

=

a(2,2)edetlnv

/

ai(2,2)

=

a(1,1)edatlnv

ai(1,2)

= -m(1,2)sd.tlnv

ai(2,1)

= -a(2,1)edetlnv

matrix

(aCi,1)em(2,2)

-

a(1,2)ea(2,1))

C

sa_

Ionon

vtrimbles

dt



d_

= -(_L(2,1)e=esl

r].z

-(ai(1,1)eresl

:

+ 8i(1,2)eres2) + ai(2,2)eres2)

1.0

if(r].xsdt:

.IF.

2.S,pO)

rlx

: 2.S*pO/dp

if(rlx*_p

.it.

-.8,I_0)

rlx

:

£f(r].xedt if(rlxedt

._t. .it.

2.5stO) o.SStO)

rlx rlx

= 2.5etO/d_ = -.8.tO/dr

update

Q.e_pOl_p

variables

37

hO

pO = pO + rlx*dp _0 = tO + rlx*dt

convergence

check

if(abm(dp/pO)

.le.

eps

.and.

abs(dt/tO)

.le.

apt)

go to

C

lO0

continue

C

write(*,*)

'NONSTiG:

Convergence

write(*,*)

'dp

dT

:',dp,

write(*,*)

'po

To

h r:',pO,tO,h,r

failure.'

dt

C

2

continue

C C ....

set rl

residual s

=

derivatives

wrt

(s,hO)

O.

r2_s

= -I.0

rl_h

=

r2_h

= O.

-1.0

b(1,1)

= rl_s

b(1,2)

= rl_h

b(2,1)

= r2_s

b(2,2)

= r2_h

C C ....

CCC

CCC

set

(tO,pO)

derivatives

_rt

(s,hO)

tO_s

= -(ai(1,1)eb(1,1)

+ ai(1,2)*b(2,1)) + al(l,2)*b(2,2))

tO_hO

:

pO_s

= -(ai(2,1)sb(1,1)

-(ai(1,1)*b(1,2)

+ ai(2,2)*b(2,1))

pO_hO

= -(at(2,1)*b(1,2)

+ ai(2,2)*b(2,2))

C C ....

conver_

derivatives

tO_t

= tO_s*s_t

tO_p

=

pO_t

= pO_seS_t

pO_p

= pO_s*s_p

(s,hO)

er_

to

er_

(p,t,hO)

tO_leS_p

C C Cw---

set

stqnation

zz

= z(pO,tO)

zz_p

= z_p(pO,tO)

zz__

= z_t(pO,_O)

rO

=

rO_z

= -zO/zze*2

zO/zz

rO

and

darivatlves



= rO_z*zz_p

+ zO/(zzezO)

= rO_z*zz_t

-

zO*pO/(zzetO*e2)

derivatives

from

rO_pOepO_p

+

rO_tOetO_p

= rO_pO_pO_t

+

rO_tOetO_t

= rO_pO*pO_hO

+

rOtOstO_hO

=

rO_t rO_hO

(pO,tO)

pO/tO

tO_tO

rO_p

_

* pO/tO

rO_pO

convert

CCC

density

m

(pO,tO)

C

38

to

wr$

(p,t,hO)

2

c ....

convor$ rO r = rO

derivatives rO_pep_r

q =

pO_r

=

pO_q

=

from grz + rO_t*t_r

(rO_pip_h

+

pO_p*p_r

rO_t*t

(p,t)

to

erz

(r,q,hO)

h)*h_q

+ pO_t*t_r

(pO_p*p_h

+ pO_t*t_h)*h_q

C

c¢¢

rO_hO

=

(rO_p*p_h

+ rO_t*t_h)*h_hO

+

rO_hO

¢cc

pO_hO

=

(pO_p*p_h

+ pO_t*t_h)*h_hO

+

pO_hO

C

return end

real*4

function

implicit

phi(ttc)

reale4(a-h,m,o-z)

C ........

c c

Returns function Z = 1 + pi*phi(ttc)

phi

used

in

non-ideality

C ........

common

/nonfit/

l

c2, phi

=

cl,

c2*ttc**2

cO + cl*ttc

+ cO

return end

reali4 inplicit common

function /nonfit/ c2,

i phld

phid(ttc)

reali4(a-h,m,o-z) cO

cl,

= 2.*c2ettc

+ cl

return end

reale4 intlli¢tt comion

functlonphldd(ttc) reali4(a-h,n,o-z) /noniit/ c2,

i plLtmkl

cl,

cO

= 2. ec2

retu.rn end

reil*4 iliplicii coumon •

functlonphlddd(ttc) rule4(a-h,n,o-z) /no.it/ c2,

cl,

cO

C

39

pazameter

phlddd

= O.

C

return end

subroutine

hgent(hO,r,q,

Returns

entropy

con:on

/nongas/

common

all, /nonfit/

t

s

= hO -

set

input

from

bta,

c2,

h

s)

input

pi,

cl,

vaziables

tau,

hO,r,q

zO

cO

.5.q*.2

C

c ....

call

pressure

and

temperature

and

derivatives

ngupt(h,r,p,p_r,p_h,p_rr,p_hh,p_rh, t,t_r,t_h,t_rr,t_hh,t_rh)

tic

=

1./(tau*t)

+tc_t

=

-1./(tau*t**2)

ph

= phi(tic)

phd

= phid(ttc)

s

= all*log(t)

+

2.0*alf*bta*t

- p*pi*(t*phd*ttc_t

+ ph)

-

log(p)

return end

subroutine

c

non_v(hO,r,q,

Returns

gam,gma_r,gal_q)

"equlvLlen_"

co-,,on

/nongu/

common

[nonfJ.$/

k

Llf,

k

ganma

bta,

c2,

cl,

pi,

for

tau,

BL density

prof£1o

zO

cO

C

c ....

see

e¢a¢ic

snChalpy

h

:

0.$*q**2

h_q

=

set

pressure

hO -

-q

C

c ....

call

and

ngaspt

(h,

temperatuzo

and

dorivm$ives

r,p,p_r,p_h,p_rr,p_hh,p_rh,

k

t, t_r,

t_h,

t_rr,

t_hh,

t_rh)

C

c ....

set asq

speed = p_r

of

sound

squared:

/

-

(I.

p_h/r)

a'2

-- alp/dr

(at

¢onmtant

s)

asq,r = p_rr

/

(I,

-

p_h/r)

-

/

(I.

-

p_h/r)**2

= p_rh

/

(I.

-

p_h/r)

+ p_r

/

(I.

-

p_h/r)**2

asq_h •

p_r

t_c

=

_Cc_t

= -l./(Cau*t**2)

tic_it

=

= phi(tic) = phld(t_c)

phdd

= phldd(_tc)

phddd

= phlddd(t_c) = =

z

t

cp

*p_hh/r

2,/(_au*t**3)

phd

z_p

- p_rh/r)

l./(tau*t)

ph

z

*(p_h/r**2

I.

+ p*pi*ph pi*ph

=

p*pi*phd*tZc_t

=

( alf*(1.0

phdds_tc_t

)

/

zO

phddst_c_t

)

/

zO

cp_p

=

(

cp__

=

( all*(



+ 2.0*bta*t)

+ p*pi/_au* pi/tau*

2.0*bta

)

+ pspi/taus(phddd*ttc_t*82

h/(cp*t)*(l.O

+

phddettc__t)

)

zet

=

- p*pi/(C*Cau)*phd/z)

* zO

zet_h

= l.O/(cp*C)*(1.0

-

p*pi/(t*_au)*phd/z)

* zO

zet_p

=

-

pi/(t*_au)*phd/z

h/(cp*_)*(

&

/

zO

-

p*pl/(t*Zau)*phd/zs(-z_p/z))

* zO

-

pepi/(t*$au)Sphd/z*(-z_t/z

-

l.O/t)

-

pmpi/(ts_au)*phdd*t$c_t/z

)

* zO

(ze_/cp)*cp_p zet_t

=

h/(cp*$)*(

& &

(zet/cp)*cp_t

-

_e/m

=

gsmr

=

gam_h

=

uq/(hszet)e(-zet_h/zet

gaa_p

=

aJq/(heze_)e(-zet_p/zet)

gu_t

= uq/(hezst)s(-ze__t/zet)

gaa.q

uq/(h*zet)

+

(ze_/t)

1.0 alq_r/(hezez) -

l.O/h)

gam_pep_h

+ gma_t*t_h

+ gam_h

gaa_p*p_r

+ gam_t*t_r

+ gam_r

= gaa_h*h_q

subrout4_e

sonAc(hO,pO,rO,

c

calculates

sonic

c

from

specified

q,p,r)

quan_i_ies sonic

quantities

q,p,r hO,pO,rO

41

+ aiq_h/(heze_)

C"

implicit

real

data

/ l.Oe-5

/

_ith

perfect

epm

initialize

(m)

gam

= rOshO

/ (rO*hO

gml

= gam -

1.0

gas

- pO)

q = sqr_(2.0*hO/(2.0/gml

:

trat p

1.0

+ 1.0))

+ O.5*gml

= pO*trat**(-gam/gml)

r =

rO*trat**(-1.O/gml)

converge do

on

non-ideal

I0 liars=l, call

values

by

forcing

N'2

nideal(hO,r,q,

p

,p_r

call

nonstag(hO,r,q,

resl

= msq

all

= msq_r

a12

= msq_q

rea2

= pstag

a21

= pstag_r

a22

= pstag_q

-

) )

1.0

-

pO

1.0/(all*a22

dr

=

-(resl*a22

dq

=

-(all

dp

= p_r*dr

rlz

and

pstag,patag_r,pstag_q, rstag,rstag_r,rstag_q

=

I,

,p_q,

msq,mJq_r,msq_q

detinv

=

15

*ros2

-

a12*a21)

-

a12

*res2)*detinv

-

reel*a21

)*detinv

+ p_q*dq

1.0

:

= 1.Set/dr

if(rlx*dz

.gt.

1.5*z)

zlx

if(rlx*dz

.i_.

-.6*r)

rlx

= -.6*r/dz

if(rlx*dq

.gt.

1.6.q)

rlx

=

1.6*q/dq

If(rlx*dq

.I_.

-.e,q)

rlx

=

-.e*q/dq

c r = r + rlxedr q

= q

+ rlxedq

p

-

+ rlz.dp

p

c dlax

= aaaxl(

abs(dz)/r

,

abs(dq)/q

)

c if(_

.it.

spa)

So

$o

11

c 10

continue 'sonic:

write(*,*) 11

conversence

fa/led.

continue

C

return end

!

son/c

42

,daax

patag

= pO

Bibliography

[1] J. H. McMasters, transport

airplane

[2] H. W. Liepmann

W. H. Roberts,

[4] M. Drela and number

and A. Roshko.

[5] W. Anderson. wind tunnels. [6] O. Coufal. 298.15

AIAA

AIAA

16(6),

of Gasdynamics.

air-freon

tests

of s

1988. Wiley, New York,

1957.

of real gas effects in cryogenic

June 1978.

25(10),

analysis

of transonic

and low Reynolds

Oct 1987.

study on the use of sulfur hexaflouride

AIAA-90-1421,

Thermodynamic

Recent

AIAA-88-2034,

investigations

Viscous-inviscid

Journal,

A numerical

Elements

Theoretical

Journal,

M. B. Giles.

airfoils.

F. M. Payee.

in high ].ift configurations.

[3] B. Wagner and W. Schinidt. wind tunnels.

and

as a test gas for

1990. properties

- 30,000 K and pressure

of sulphur hexaflouride

range .101325 - 2 MPa.

1986.

43

ACTA

in temperature Teehnica

range

CSAV, 31,

Appendix High-Order

Airfoil

for Ideal

The entropy

steady

flow around

and total

is an ideal

enthalpy,

or a non-ideal

the perturbation potential relations are obtained.

The

¢.

Boundary

Non-Ideal

away

and hence gas.

Farfield

and

an airfoil

B

from

Gas

flow can then

These

V_-_/

viscous

regions

properties

still be decribed

the freestream

4' =

Flows

shock wakes and

is also irrotational.

Assuming

Conditions

is aligned

hold whether

by the with

has

velocity the

z-axis,

constant the fluid

potential the

@ or

following

q_,(z + ¢)

(1)

=

q_[(1+¢_)i

+ CuJ]

(2)

q2 = iq-]2 =

q2 [(l+¢x)

2 + ¢2u]

(3)

1 _V(q 2) = qVq

2 qoo[ (¢x_+¢_

=

¢_+¢uCxu)i

+ (¢_u + ¢_ ¢_ + ¢_¢_)j] The governing

flow equation

(4)

is:

v.(pv_) v_

or

=

o

(5)

=

-V!.w

(6)

P In isentropic

flow (s = constant),

p = p(p), so

dp s Vp -_p

Vp = and

where

P a2 qVq

-

(7)

hence

a is the speed

to the speed:

of sound.

a = a(q).

W2_

=

1 -_qVq.

a 2V2dp

=

qVq.[(l+¢z)i

In isentropic,

For a perfect

and/or

adiabatic

(8) + _j]

(9)

flow, the speed of sound

is uniquely

related

gas, a(q) is given by

as while for an imperfect

[email protected]

=

non-ideal

2 a_

7 -1 2 (q2_q_)

gas it is necessary

(10)

here to linearize

a(q) about the freestream

conditions.

a2

,

d(a_)

_- a® + It is convenient

to define

an "equivalent"

7'

d(ql )

[q2-ql][

ratio of specific

=

d(a_) 2 d(q_)

1 8

heats

,

(il)

'_ 7' for the non-ideal

gas as

(12)

so that the a(q) relation for the non-ideal gas can be compactly

a2 For to

a perfect note

that

invariably

gas, 7'

this

can

reverts

easily

7 > 1 for perfect

Substituting

._

to the

be

less

2 a°°

_exact

than

written as

(13)

7'-1( 2 q2 _ q _) form

unity

(10)

since

for heavy

in this

gases

case

such

as

7' = 7. sulfur

It is interesting

hexafluoride,

while

gases.

for a s, qS, and

qVq

in equation

_ q_ (2¢. + _ + ¢

(9),

[¢= + ¢_]

we obtain

=

q_ [(1 + ¢_)(¢=

+ ¢_ ¢= + ¢_¢_)

+ ¢_ (¢_ + ¢_ ¢_ + ¢_¢_)1 F

l 7,-I 2 M_2 2¢=J [¢z=+¢_1

[1

'

+ ¢_

is the

freestream

- M®)¢== where

M=

= q=/a_

Equation non-ideal as

(16)

2D second-

compressible

derived

the

is the

potential

in references

"equivalent"

first-order

version

In terms

of the

:

M s [(7 +1)¢.¢_= Mach

(16)

P randtlIt has

Prandtl-Glauert

equation

same

that

the (12).

7' in lieu

form

usual

the

which

as the ratio

Wagner

and

+ 2_b_C=u]

+

()

CO ¢3

(16)

governs

equation

of specific Schmidt

small-perturbation

for a perfect heats [1] have

ideal

gas

7 is replaced

by

considered

the

of 7-

z//_

(17)

# = y

= _--k-_,

(15)

coordinates

=

where/3

O(¢3)

+ (7'-l)_bz¢_

Glauert

the

equation

using

+

number.

[5], except by

Cz=+2¢u¢=_l

'

order

7' defined

of equation

M2[¢-=+2¢=

flows.

[4] and

value

:

(14)

general

(is)

_s

:

_2 + _2

0

=

arctan_

solution

to equation

:

_r °

r_

+

D=cos0 2_r f

+

+

\

2_r )

(19) (20)

"V

(16)

is

D_sin8 27r

kl-'_'--ks

(21)

where

kl = _1(-/'+1 _--_- + Terms

of order

1/_ s and

above

have

(22)

._) been

discarded.

In a flow solver,the circulation r can be determined eitherdirectlyfrom the liftper unit span L p (Euleror Navier-Stokescode), Ll r-

p_q_

(23)

or indirectlyby specifyinga Kutta condition(potentialsolveror MSES). The source strengthZ can be determined from the totalprofile drag per unit span D _,or from the asymptotic mass defect behind the airfoil includingthe shock wake. £)i -

p®q_

(24)

In the case of a potentialsolver, D _ should not includethe wave drag sincethere isno shock wake (unlessan entropy correctionscheme is employed). Note that r and Z here have units of length since_bin (1) correspondsto a unitfreestreamspeed. Cole and Cook [5]give explicit expressionsforthe doublet coefficients Dz and D_ in terms of fieldintegralsover the domain. Unfortunately,these expressionsare unwieldy and for a non-ideal gas would be ratherexpensive.A simplerand economical approach isto iteratively update D_ and D u by minimizing the mismatch between V6

and the velocity_olution from the flow solveron the

outer boundary. The approach taken in reference[4], forexample, isto minimize the integral 1

I=_

/ IVq_× _olutionl _ dz

(25)

taken over the outermost streamlines.The doublet terms in the farfield expansion (21) decay fasterthan the others,and so can be neglectedforsufficiently distantouter boundaries.However, retainingthem greatlyreduces thesensitivity of the solutionto domain size,especially fortransonic

flows[6]. With itsterm coefficients defined,equation (21) gives a very accurate representationof the perturbationpotential_ away from the airfoil.The gradientof equation (21) accuratelygives the totalvelocity_ via relation(2).Either _b,_, or an appropriatederived quantitymay then be imposed at the outer domain boundary as a high-orderboundary condition. A potentialsolver would typicallyimpose _ or OctOn, whereas an Euler or Navier-Stokessolverwould typically impose the flow angle at the inflowand pressureat the outflow,both being determined from V6.

10

Appendix Shape for Ideal

The profile,

major which

influence alters

flows, this effect is mostly shape

parameter

V(y)

and

to pressure

and R(y)

by the correlation

are the velocity

of the

the velocity density

profile

profile

from its definition

U(y)

R(y)

(26).

near

Ha

is a non-uniform

In an integral shape

scheme

parameter

parameters

H, the kinematic are defined

as

(26)

profiles. R = p Pe

Ha are only weakly

the von-Karman

density

for adiabatic

f(1 - U) dy f(1 - U)Vdy

-

the wall due to adiabatic

In turn,

behavior

Me. The shape

and density

and hence

layer

between

V = u ue Since

Flows

gradients.

f(1 - RU) dy f(1 - U)RUdy

-

Gas

on boundary

response

captured

Relations

Non-Ideal

Ha, and the edge Mach number

H where

Parameter

of compressibility

the layer's

C

integral

(27)

affected

heating

by compressibility, will increase

momentum

reduction

H as can be seen

equation

dO dz shows that adverse ISES

an increase

pressure

C1 2

in H will increase

gradient.

[4]) employs

gas model

_

The

a correlation

(2 + gthe momentum

integral

boundary

of the form Hk(H,

in Appendix

A, the state

P

the corresponding dh the

the caloric

thickness

Me) for air.

(28) growth

layer formulation

equation

pT_T

where

_._-du....._ u, dz rate

in MSES

This is re-derived

d_/dz (and

for a given its precursor

for the non-ideal

as follows.

As developed

while

M_)

caloric equation

= _p(r) dT

approximation equation

across

the boundary

factor

Z for most

+ p._"(r)

dp __ 0 across

%(T-T,)

: 1= "_

The non-ideality

(29)

dT

= cp(p, T) dT a boundary

layer.

(30) Linearizing

layer we have

h-h,



that

as

form is

__ en(r) dT

on the basis

gas can be written

- Z(p, T)

in differential

+ d[pSr(T)]

is made

of a non-ideal

non-ideal

(31)

-1

c-_

-I

%,T_

(32)

gases has the form

Z(p, T) = 1 + P---¢(rc/r) Pc 11

(33)

where

Pc and Tc are the criticalpressure and temperature.

This can likewise be linearized about

the edge conditions as follows.

p_

-Z Combining

this

with

equation

(32),

the

equation

of state

(29),

= 1

-1

"-Z

(35)

%.T_

Z_ .

(36)

p_T,

we have

-_Using

T

=

1

the

- 1

Pc T_

density

profile

R-

p p_

is then

-

related

to the

T and

Z profiles

as

T_Zep T Z p_

T_ Z, with

the

the

density

usual

boundary

profile

layer

can

R

R

approximation

be written

- 1

--

1+

-_

1+(_-_-1)(

of the enthalpy

_

(_---_-1)

(37)

Z

p __ pe being

in terms

1+

=

T

1

Using

profile

P'Tc¢:_ peT, Z,/

relations

(32)

and

(36),

alone.

+

(38)

c ,T, z,J

-

T,

(1

h.

made.

O[(Lh,/h-

(39)

1 ,)] 2

(40)

where %,T,

CFor

turbulent

enthalpy

across

unity.

Since

in a gas the

the

the

the

boundary

layer,

turbulent

(convection

assumption

enthalpy,

adiabatic

by

velocity

and

this

diffusion

mechanisms

static

h.

true

enthalpy

numbers

h

for

are

are

then

ho - u_/2 ho _ U2/2

heat

typically With

related

a constant

a turbulent

and

is reasonable.

profiles

_

only

to assume

of momentum

Prandtl

enthalpy

(41)

Z,]

it is reasonable

is strictly

turbulent

stagnation

pcT,

flows,

although

eddies),

of constant

layer

1

h0

are

stagnation

Prandtl

number

essentially

the

close

to unity.

denoting

the

of same

Hence,

stagnation

by

1 - i_u2 U2 1 - _-_

(42)

u 2

h_ce_1

-

h

_

(U2-1) u2 U2 1_2_:_

(43)

and the density and velocity profilesare then related by u 2

R

=

1 +

_

1-2-- u2_ 12

(U 2U 2

1)C.

(44)

Since

u_/ho

and

( are both functions

implicitly

defines

Hk in terms

to assume

a small-defect

of the edge Mach

of H and

Me. To obtain

the density

this relation

Me,

the

in closed

density form,

profile

(44)

it is necessary

profile U = 1-e

so that

number

profile

;

can be approximated

e