I'

NASA-CR-173254 19840008789

A Reproduced Copy OF

Reproduced for NASA

by the

I'IASA Scientific and Technical Information facility

111111111111111111111111111111111111111111111 NF01452

FFNo 6 72 AU~J 65

SEP 1 1981

LANGLEY 1\·~cSfAf?CH GEN I ER

1.18RArlY, NASA

tlAJYIf':ION, VlltGII':II"

https://ntrs.nasa.gov/search.jsp?R=19840008789 2018-05-27T15:35:31+00:00Z

, ,I

.-' ~ .

'. t

r,.' ,

THREE-DIHENSIONAL UNSTEADY EULER EQUATION

SOLUTIONS USING FLUX VECTOR SPLITTING

(NISI-CB-173254) TUDEE-DIMENSIONAL EULER EQUATION SOLUTIONS USING FL~X SPLITTING1Mississippi State Unlv.) He A04/MF AO 1

uect::mber 1983

Ui~STIHWY VECTOR

58 p cseL t2A

N84- J 605"1

Unclns 00555

.,/

, . .;

.. .. ,

. ' ~:. -. , '

.. ,.,," .. " ., . ' ..

. .

, iii'"

FOREWARD

Thls research was sponsored by the NASA Langley Resenn:h Center

under G~ant ~A~A NAG-1-226 with Dr. Frank Thames as TecJmifal Honitor,

and the United States Air Force, Eglin Air Force Base. Florida under

Grant F08635-82-K-0409 wi th Dr. Larry Lij m.rski as Tt'chnical:. Officer.

The support of NASA and the Air Force, as "Jell as tedmical. interactions

with Drs. Thames and Lije\"s1d. in gratefully acknotvledged~

. \

". , " , . ' .. . . ,./

" ,.\ " I " t '.: . . . ~' ,

,

"

" , ....

ABSTRACT

A method for numerically,.soJving the three-d:f.mensional unsteady

Euler equations using flu;cvt!"etuj: split.ti.ng is developed. The equations

are cast in curvilinear coordinates and a finite volume discretization

is used. ru. explici.t upwlnd second-order predictor-corrector scheme

:L8 used to solve the discreti:r.cd equations. The scheme is st[tble· for a

GFL num~er of 2 and local time stepping is used to accelerate convergence ,,",,; ..

for steady-state problems. Characteristic variabl~'boundary conditions

lIre developed and used in the far field and at surfaces. No additional

dissipation terms are included in the scheme. Numerical results are

compared with results from an existing three-dimensional1!:uler code

and experimental data, ... ..;

.. ,

.\

I. INTRODUCTION

In support of NASA's interest in the use of prop-fans as a propul-

sion device, acomput'll1,t'l:aliitl mert,Jlod was recently developed for numerically

solving the flowfield about a swept. tapered, supercriti.cal '''ing with a

propeller producing thrust and s\~irl (Ref. 1). The equations solved

were essentially Euler equations with body force terms included to

simulate the propeller. The Euler solver used was an extension of the

method of Jameson. et. aL (Refs. 2 and 3) xcferred to as FL057. Although

good agreement was obtained with exped.mental data (Ref. l). some

difficulty in convergence, particularly the energy equation, was encountered

using the FL057 central-difference scheme. Recently. convergence

difficulties Here also reported by Swafford (Ref. 4) in solving a sct

of hyperbolic equations \'Jith source terms (ltlh:tch can be consj.dered

similar to the force terms included in the Euler equations in Ref. 1)

using a similar central-difference scheme. An upwind,. scheme eU.m:l.nated

the convergence problems in Rcf. 4. Horeover, any additional smoothing

added to the upwind scheme was found to be detrimentaJ. with regard to

convergence •. (Additional smoot.hing was found to ahmrys be necessary

in usi,ng the central-difference scheme in Ref. 4,) Tlt~e):efore. it

seemed appr9priate to consi.der an upwind scheme for Sl:!lv:!.ng the thrce-

dimensional Euler equations.

The flux-vector-split form of the equations used is developed in

the following section. The motivation and bacl,ground' of using splitting

is given in the literature {see, for example, Steger and \varming (Ref. 5»

and is not repeated here. Horeover. many of the ma.tr:kes needed are

given in the literature (see Refs. 6 and 7); hot-lever. all equations

~ .

"

and matrices needed are developed and included in Section II fo·r clarity

and completeness. Formulation of t~e equations for numerical solution

a.nd the algorithm used to solve the discretized equations are discussed

in Section In,' 'i1!f~~;; 'Ciu.:ract"~'· ~ siic variable boundary conditions used

:l.n the far field and at surfaces are developed in Section IV. NUmerical

results, including compa:dsons with FL057 solutions and experimental

data, are presented and discussed in Section V.

"r'

., .,

II. SPLITTING

The conservation law vector form of the I:U))~$.!·Jr equations in

Cartesian coordina:t.#asx .• y. and z are

(2.1)

l~here

- [p, elT

q '" pu, pv, P\'1,

f :: Cpu, 2 + p, u(e -7i' p),lT pu puv, puw,

[pv, 2 . T

g '" puv, pv + p, pvw, v(e +. p}}

h '" [pt., • 2

+ p. wee +. p.~, J T puw, pvw • P'd

(y - 1) [e 222'

P = - JliP(u + v + w )1

Using curvilinear coordinates defined as

E:; :: E:;(x,y,z)

n '" n(x,y.z)

I; '" l;(x,y,z)

T '" t

it is straightforward to transform Eq. (2.1) t.'o

(2.2)

,,,here

[ el T Q" J p, pu, P\', pw,

"

'.. .

..

. 1

,. " "

.. , . "

T F" J[pU, puU + l;xP , pvU+E:yP, pwU + ~zP. U(e + p)]

T' G = J[pV. puV + nxp. pvv + nl' pwv + nzp. V(e '+ p)]

H" J[pW, puW + C;)~p, pvW + C;yP, pwW ... l;zP, W(e + p)}T

, J = xE;(ynzr; - znYI;) -YE;(~nzz; - znxr.) + z~(xnYr. - Ynx~)

nx ..

1; .. x

E; '" Y

n '" y

/; '" Y

E;, = z

-1 J (zf;Yl;

-1 ' J (Y E;zn

-1 J (z x

n /;

-1 J (Xt;Z I;;

-1 J (Zt;X

n

-1 J (x Yr.

n •

z y ) n r;

- y :r. ) t: I;;

zE;Yn)

- x z ) n r.

- z x ) t; c;

- x Z ) t; n

- Y n,cr;)

-1 n = J (Y t;'\ - x,y 1;) , Z

-1 - Y x ) /;z .. J (xE;,Y

n t;, n

U .. t; u + x t: v + Y

t: v.' z

V'" n u + n v + n \oJ x Y ~

W .. r; u + r; v + '(,zw x y

ORiGiNl~t. rt'~c!S r~ OF POOR QUALITY

The strong conservation law of the Euler equati.ons i.n curvilinear

coordinates (Eq. (2.2» can be wr.:ltten in the quasilinear form

(2.3)

.' ., ~

, .. . "

j. , -I' . .. "

" . ~ ~ .' I

, "',. i \ .. " .

where the matrices At B, and C are given by

A .. aF (lQ

B .. aG aQ

C = aB ClQ

•. I '.'<\. ~ ......... r"~' ORlb\'~k .. r' ~ ,1.;',," [0",

Of POOR QUf,Ul'V

Canying out these operaUons. one obtains a matr:[J~o it

0 k x

"fI-

kx</l - uOk kx (2 .·y)u + Sk

i~ = lcy4> - vak k v - k (y -x y

l)u

k <p - t"e 1. k

It \-1 - Ie (y -x z l)u

(24) - ..Y!:) a k (.~ - 4» - (y .. l)uG p k J( P k

k 0 z

k u z - k (y - 1)\., x

k (y - 1) x

k v - k (y - l)w k (y - 1) z y Y

k (2 z - y)w + e k

kz{y - 1)

k (E _ </I) •• (y - 1)\\10 z p . k Yak

where , v-I ( 2 + 2 + 2) </I .. ...1-_ U V w

2 .

a .. ku+kv+k\oT 'k x Y z

k Y'

Ie u - k (y - l)v y x

k {2 - y)v + Ok y 0

Ie w - k (y - l)v y Z

k (1.~ .~ 4» - (y -1)vO y p 1<

(2.4)

and 0 matrices A, B, and C are given by the matrix K depending on \olhether

'.

'0

, , , o! .

k in Eq. (2.4) is ~. n. or 1,;. That is

ic .. A for k .. ~

i( := B for k = Tl (2.5)

'.~ t.:" ", R", c for k '" l;

The ",lgensystem of the matri.ces A, B. and C: is important to

achieve :.he intented 3plil:ting. However, there are few zero elements

in Eq. ,I. .4) and the eigenvalues of A, B, and C are difficult to

deterIn1.!e us:i.ng Eq. (2. tl) directly. Hence. consider the nonconservative

vector form of the Euler equations in curvilinear coordinates

'l-lhere

T q '" J[p, u. v, t". p]

Note that Eq. (2.3) can be written as

(2.6)

(2.7)

Ivhere H is the matriit .~~. Multiply Eq. (2.7) on the left by M-1

to·obtain

'where I is the identity ma.trix. Then from Eqs. (2.6) and (2.7)

a to J:.C1AM

b '" M-1BH (2.8)

c .. H-1CM

,

"

, . ~j

;r-' I.

'.

. :

f;rHGn~,~\::. ~:; .... ;"';:~: :~~ OF POOf~ QU;\Lrr'1.

Therefore, the matrix a is similar to A. b is simHar to B, and c is

similar 'to C. Because similar matrices have the same eigenvalues

(Ref. 8). the eigenvalues of A, B. and Care KnO\>1t\ by determining the

eigenvuluesof a~>'i.·;·anJ >::. The matrices a, b, and c are more simple

to handle than matri~es A, B, and C because they contain several zero

elements as shown below.

The matrices a, b, and c are now determined. The matrix H gi' ~n

b lQ. Y aq :i.s

1 0 0 0 0 -I

u p 0 0 0

HI:: V 0 P 0 0 (2.9)

W 0 0 0

-.P_ I y-l

pu pv (lW y-l

The inverse of this matrix is ..., 1 0 0 0 0

u 1 0 0 0 - -p P

-1 v 0 1 0 0 (2.10) M :: - -p p

w 0 0 1 0 p p

~ -u(y-l) -v(y-·l) -w(y-l) (y-l)

rv·· ...

ORIGUISAL PAGE IS OF POOR QUALI1"V

Then by Eqs. (2.8) the matrices a, b, and c are given by

r :; l Ok pk pk pk x Y z

0 '\ 0 0

k K .. 0 0 Ok . 0 .J..

p (2.11)

0 0 O· Ok itz p

0 k pc 2 k pc 2

It pc 2

Ok x y z ..J

where K '" a for k '" t;

K '" b for k = n (2.12)

K "" C for k '" ~

2 The c in the pc 1:e~'lils:tn Eq. (2.11) is the speed of sot.md and is not:

to be confused with the matriJ~ c. The </> appearing in Eqs. (2.9)

and (2.10) is aga~n

just as in Eqs. (2. 4).

From Eq. (7..11) the eigenvalues of matrices. a. b. and care

easily determined. 'rhe eigenvalues are

Al '" A 2 ~, >.3 '" k u + It v + k W '" Ok k 1-. k x Y z

(2.13)

5 I A ., 0 I, - C I 17k It. !~

/

ORlGfN.4L P,l~Cf! ffJ OF POOI~ QUALiTY

1 2 345 ,,,here A

k, A

k, A

k, A

k, and Ak are the eigenvalues of a for k = E;, b for

k = 11. and c for k ::: r;. Correspondingly, the eigenvec tors are

where

Xl [k;x' 0, kz •

-k , y

O]T

x2 :: [It • ·-k

y z'

x = [It , k , 3 z y

1 [l~ x4 .. Ii e

1 P x5 = -- ['(! •

12

_ k x

kx = TViZr =

k

ky =~T

O.

-k • x

Kx'

-k x'

kx ' O)T

O. ('IT

-k • y kz •

-k • -k y

k x

(2.14)

pc] T

z' pc] 'I."

(2.15)

Sufficient equations have now been d,~veloped to carry out the

splitt:l.ng. The integral conservation 18\11 form of the Euler equations

will be formulated and discretized for numerical solution :i.n the

following section. This formulation requires evaluation of the

vectors F. G, and H at faces of the f:Lni.te volumes. By splitting

the vectors F, G. and Ii into thE! sum of separate vectors. each having

an eigenvalue as a coefficient. the evaluation of each separate vector

by extrapolating from the .!lppropdate direction indicated by the sign

of its eigenvalue can be carried out.

OH!GiN;\~. i:;:.,~l~ ~r~

OF POOl1 QUt>.U'f'l

The ve..:tors F, G, and H are homogeneous func.t:h:vns (I.e degree one

in Q; therefore, by E41er's Theorem (Ref. 9)

K = KQ ;2.H)

where K == F and K == A for k .= E;. K = G and K 'C B for It == TJ. and

K = Hand K '" C for k = 1;. The matrices K can be· t;rrlt.ten

(2.17)

where Ak is the diagonal w.atrb: whose diagonal elem.€nt·"I are the eigen-

values of K (cr, also, K) given by Eq. (2.13)

by Eq. (2.12) can be Hrittcn

K == P II p-1 k k k

The matri.ces K given

(2.18)

where .t·he columns of Pk

are the (;.~gel1\ eCLot!:; of K corresponding to the

respective eigenvalues. From Eqs. (2.8)

- -1 K '" MKM

Using Eq. (2.18) in Eq. (2.19)

Th,en from Eqs. (2.17) and (2.20)

··1 -1 P M k

(2.19)

(2.20)

(2.21a)

(2.7.1b)

The matrices M and M-1 are given by Eqs. (2.9) and (2.10). The

matrix Pk

is

'\

. ,

..... :- .;

.' ,\ , , \

-~ ~,

\

:~

\ ;. ! !

'-..... ~'" i

\' 1' • ..., L I "

,,,/~' I. , /

> ,.

It'

ORIGINAL PI\[~2 ;r:J OF POOR QUAtnv

f i k 1, a ~

I )C y z

.. k k

0 -k k -2S. __ !S

Z Y Ii Ii

k It P

k .. k 0 -k _'L __ .l:'

(2.22) z x If .,If

k k -k k 0

Z & ---Y x If 1'2

0 0 0 2 2 ac ac

where

... ] Rather than inVel"t P

k cUrcctly. the mntrix P

k' can bu dt\t(~rl\linerl casily

from the left: eigenvectors of Ko '."lli;! left (~igmlV(lctors of K are the rO\,6

-1 of Pk

• Le.

k x

.. k Y

k z

o

o

0

-k z

k Y

k _l£.

If

k x - ----

/2

k -k z y

.. 0 It x

-k 0 (2.23) x

k k -..Y-_ z

fi Ii

k k .. _'1- z

12 1£

" .. . ' ~ .

I"~ 4 ..

-:. :, ". ,

.!/. . \

'.

\-Ihere 1

P "" .-=---Ii ~;i~

OmG~NAl ~AGI! V~ Of' pom~ Q"AU1'Y

The mat.rices Tk and T;l can now be determined using Eqs. (2.21).

These matrices are

k k k It Y z

uk uk pkz

uk + pk x y z y

T "" vI< + pit vk vk -P~ It x z y z

wk pk \<lk + pk wkz x y Y It

....L k + p (vl~ - wk ) y-I x z y

uk) J.....1

k + p (uk - vk ) z y- z y x

u(u + ckx) a(u - ckx)

~

a(v + ck ) a.(V - ck 'I (Z.24a) y y'

. a(w + ck) a.(\J - ck )

,f!

,tt 2 ~ j>+c2

1"C Cllk

) ,.( -- + eEl ) et( -- -'V-I k ),-1

ll~ ~ ,

k u(y-I!. x 2

c

-k 1:. + k u(v-H, zp y 2 c

-1 ~ A.' 1 ~ • T

1: .. k (1- ..:r._) -+ ~-(vk -uk ) ~ z 2 p x Y c

where

k 1: + k \~.(X-l) xc), y c2

k~L z 2

c . 13[ckz - w(y-l)]

. -e[ckz + w{y-ll]

fHckx - u(y-I) 1

~

-13[ckx

+ u(y-l) I

-k y-1 x 2

c

. rl -k

Y 2 c

f3(y-1)

13(y-1)

.r:.! 2 2 2 • = 2 (u + v + w )

-0) =ku+kv+k\J

~ x y z

(l= .-L lie

1 !3 .. ----Ii pc

Using Eqs. (2.16) and (2.17)

• 1 • v( '-1) -It -- + k ~'i.._

xp 3 2 c

-B[ek - v(y-I)]

y

-Plcky + ,,(y-1)]

(2.24b)

(1.25)

" "

. • •

" "

•• ( , .

ORIGINAl PAC';' '.'" • ~I. t..J> The diagonal matrix OF POOR QUAUn

,,1 ,1:.-

0 0 0 0

0 . 2

>-k 0 0, fJ

II .. It

0 0 ;.,3 It 0 0 (2.26)

0 0 0 I~·

>-, .t. Q

0 0 0 or . •. ~~ •. i ;

,can be writt:en

,,1 I ).4 '.l II c: + 1,+ + ;It ,r I., k k 1,2,3 k 'fu. ,) ..

(2.27)

1iJhere 11 2 3 is a matrix \vhich has unity lAS t:he. £:h:st three diagonal , . elCluents and Clll other elc:mcnt$ zero, 14 has !)A1ity as the fourth diagonal

.~lement and all other ele;nents zero, and IS i\13:'", unity as the fifth

diagonal element and all other elements zerO'. lirO!t.e that only three

terms are needed on. the right hand side of Eq: •. (2 • .27) instead of five

because the first three eigenvalues are I'he same <w shown by Eqs. (2.13).

Using Eqs. (2.25) and (2.27). the split form. Qf K is

or

K=K +K +K 123

(2.28)

.... ..

where 1 -1

.~;~ Tk 11 2 3 Tk Q . , K2 '"

4 Ak Tk 14

-1 Tk Q

! " 1<3 .. ' -1

Ak TIt 15 Tk Q

Everything is now available to carry out these operations and determine

the split form of the vector K. The operations describ(!d by Eq.

(2.28) give

(2.29)

where

K = F for k." E;

K = G for k '" n

K = H for k = ~

p

pu

pv

pw

,

~ · ., , • ·

.. ;~

., . . ·

., '\.

"

ki k '" -.--

i l'Vk/

ORiGINAL i'il.cm fS OF OOR QUAL!"

pv + pck y

p\>, + pck . z

.P

pu - pck :It

pv - pek y

pw .• pck z

6 akxu+kv+kw k y Z

Th:la if} the Dame form of the equations a6 presented by Reklis and

Thomas (Ref. 10).

, ',J."

.,,'

, , . 11 '.+ , '.

"'f".

'. \ .

/

/'

.~ .

.1

1

,J I"

"

"

III • ALGORI THM

ORIGII".i\L Htm~ it] OF POOR QUALrnr

The discretized integral form of Eq. (2.2) in computational

space for a cell with center denoted as i,j,k is (see, for example,

Ref. 11)

~qT M';lInlil; ..... (Pi +, - Fi 1 )lInllr,; + (Gj+1, - G. 1 )lJ.~l'lt;

... . 1l -Yg '2 J -11

(3.1)

or

(3.2)

The central differenceqperatornotatlon in Eq. (3.2) :i.ndicates the

flux vectors are evaluated at. cell faces in this ffnite v()lt\li1e f()t'mu-,

lation as Ulust:nited in Fig. 1.

The. numerical scheme used is a finite volume version of the

second-order upwind scheme of Warming and Beam (Ref. 12), The

present scheme is an extension.of that used by Deese (Ref. 13) for two-

dimensional flow. To illustrate this scheme consider the simple lI1ode1

equation

(3.3)

'where. F(u) <: au; and a is a constant taken as greater than zer.o

for illustration. The sec.ond-order upwind schmne of '.Janning and Beam

is

(3.4a)

..

" . , . " ," " .. ~ . ~ ., , ..... '. " : . ',~,

/

~ .. ,­IbA'ij., .

where

and

ORIGINAL Pf.\GC ,9 OF POOR QUALITY

V2Fn VFn+1 n+l n -n+l 8t 1 At. i

u i '" ~(ui + u 1 ) - "2 ---,;:;- - "2 -iX

The finite volume forlD of Eq. (3.3) corrf!spollding to Eq~ (3.2) is

lIu oF1 -+--"0 lit lIx (3.5)

By using the one- point upwind extrapolations. ui

for u1

+); and ui

_1

for ui

~. the predictor step for Eq. (3.5) is -'2

(3.6)

"'hieh is the same as Eq. (3.4a). By using the t\vo-point upwind

extrapolations. 2ui

- ui

_1 for ui

+% and 2u1_1 - u1

_2 for u1_% • the

finite volume corrector step is

or

Using Eq. (3Jw) , Eq. (3.7) eCln be written

2 n n+l f (n ·-n+1 1J.t: V F i lit VFn+l

u i .. ~ u i + III ) - -2- -t:x- - 2"" ~~

(3.7)

(3.8)

'" '. .. .. . .,!<.

~

. .' ,

.,'

/

/

/ .'

j

ORIGINAL Pf,G,'f. itt OF POOH QUAlll"Y

which is the same form as \olarming and Beam's corrector step given by

Eq. (3.4b). The stability, dissipation, and dispersion analyses of

Ref •. ,12, ;t1:.~;~:t';re, arc.rpplicable to the present finite volume scheme.

In addition, it is not difficult to show that this scheme is also.

consistent.

The scheme for the complete three-dimensional unsteady equation

is:

Predictor:

(3.9)

where the subscript t corresponds 'to one part 0f the split vector in

Eq. (2.29); and

1 4 5 . if the corresponding Xr,. A~. or A, eigenvalue

<in CI Qn i+~,j,k i+l,j,k

~n

and Similarly for Qi , . k' . -!2.J.

~n 1 4 Qi j f k except A • A • and

• -12. I] n ~n

Similarly, for Qi j 1+' and • , (. ':I

evaluated at :t+!z,j ,ft is > a

din ,1, II ,5 i' 1 if the corl'espon ' g "'E.; ,,, f;' or "E; e genva ue

evaluated at i+l2,j,k is < a

~n

The same is done for Q and i,j+~>/, k

5 interrogated; and A eigenvalues arc n

~n Al 4 and AS Qi,j.k-~ except

1;' >'/;;' r,;

eigenvalues are interrogated.

i.

" "

..

OR/(~/N I..\./ r)f' /.,,: .... .. " .. r .t.:d .... :.-3

Corrector: . OF POOR QUALITY

(3.10)

'::n+l where the Qi+~,j,k' etc, variables are determined: in the same way

~n ) as the Qi+'ii,j ,k' etc, varia.bles as described belo\Yi Eq. (3.9 except

- n Qi.l,k' etc, are used in place of Qi,j,k' etc.

"n The Qi+lti.j.k' etc, are

determined by

or

"n n Qi I j k" 2Qi+l +~, ,

n Ql-1,j,k

,1 4 \ 5 if the corr:iEsponding A~ • A~, or At;

eigenvalue evaluated at i+~,j.k

is :;. 0

1 4 5 if the conr.e:sponding A~, I.E;' or I.E;

eigenvalue evaluated ~t i+~,j,k

is < 0

"n and sim:l.1arly for Q, I. j k'

l.-'~. ,

... n Again, the same is oY:r.ne for Qi j+l_ k

, -"'2,

n and Qi,j ,k±.lli by interrogaUng the appropriate nor. t eigenvalues.

Although the algorithm given by Eqs. (3.9) and< (3.10) is an

extensi.on to thr.ee dimcnsiona of that used in Ref. 13 for two dimensions,

the interrogation of ei.genvalues differs significD.l'1.t1y • Three methods

were tried in Ref. 13 to handle the computation of flow variables at

cell faces Hhen e:lgenvalues were of different sign on either- side of

'.)

.. ~

ORIG1rJP.L FJ\G;;." ~'.; OF POOR QUALn:y

the cell face. The approach taken in the present sdH~me is to compute

e:Lgenvalues at cell faces rather than at cell centers as illustrated in

F:lg. 2 on a z; .. constlmt':~lan"', The difficulty ass.<il'.ciated with eigenvalues

changing sign is thus eliminated. This see.ms.a nat'Ural approach because

the use of Eq. (2.29) in the finite volume discret±2led equations (Ecl' (3.2».

r,equires that eigenvalues be known at cell faces. ']lhe td.genvalues are

computed by averaging the information on either sj.de of a cell face that

is necessary for their computation according to Eq. (2.13). Then,

depending on the sign of the eigenvalues, the information necesoary to

compute the remaining terms in the split vectors given by Eq. (2.29) for

use in the algoritlun givE!o by Eqs. (3.9) and (3 ',10), is determined by

e,xtrapolation from the apPJ:opriate direction,

The time step l1t i ,j,k is determined from

(3.11)

,,,,here

(3.12)

for k '" ~, n. and 1';,. Beca.use the eigenva lues are' CCHl1pU ted at cell faces

.e . rather than cell centers, Ak is the average of the e.igenvalues on cell

faces in the k .. , constant computational planes (Wh(~r.f" k = 1;" fI, or 1;;),

Bnd .e is eigenvalue 4 or 5 because one of these will always have the

maximum absolute value. To accelerate convergenc(': for steady-state

solutions tho maximum allowable time step in each volume is used \vhere

CFL < 2.

.,

" '.~

, ,

~' ~ . ,

...

,/

\ \

" \ 1-!

----~ !

/­/'

For the computation of the first points inside the cw,putational

domain, the scheme is only first order accurate and stable for a CFL

of 1. The time steps at these outside poinl:s are, therefore, decreased

by a factor of 2.

~.

:!.

'. < ':

..... , .-.

" ..

/

..;

-.

"

;'

CRl(;i~ ~ ~.;:. ~'>.;,!~' ~;3

~)F t!r('t~i~ '~~. ;f\:.!,,\,~

IV'. BOUNDARY CONDITIONS

To derive the characteristic variable boundary conditions. the Euler

equations need to be cast in characteristic variable form. Therefore.

the characteristic variable form of U.e equati.ons is derived first,

follo1t.'ed by the derivat.ion of the boundary conditions for the specific

cases of supersonic and fmbson:tc inflow and outflow, and impermeable

surfaces.

4.1 Character:lstic Variables

Consider the nonCOnf.wrvative form of the Euler equations given

by Eq. (2.6). Using Eq. (2.18) in Eq. (2.6). and multiplying Eq. (2.6)

-1 on the left by P k • gives

P- 1 .£g •. p.-1 P • 1'.-1 19. + p-l P II p-l .o..q k ot' 'r k k "k . k ol~ k m m m am

o (!~ .1)

,·,here k "" 1;" n. or ~. and m includes the remaining two curvilinear

coordinates out of the set 1;" n. or l; where I'll :f. k. Therefore, there

1.s a two-term Slll:''1lation on m in the third term ill Eq. (4.1). Define

the third term in Eq. (/1.1) as

(4.2)

l,;rith the t\olo-terrn summation on m understood. Equation (4.1) can be

-log -1 ~ c Pk -;:-. + Ak Pk "k -I- "k dt Q ,In

o (4.3)

-1 -1 Consider Pk to be a constant matrix denoted as Pk,o' Then

Eq. (4.3) can be written

~ .

•••• #, '~"'v •• ,,,'>-r"""'_·""".~~""""'~""'''_''' __ ~-''''·''_'~'' ___ ''''''-'''---_''--_'· - .• ~-----,.,---.,.... ... ,

Ci~IGli~f"tl P!:.C!-~ tS OF POOR QUALITY

a(p-l q) . h.,o

aT -I- It k

a(p-l q) k,o + s ., 0 . ak k,m (4.4)

Defining the characteristic vector, Wk

, as ,I' .

(4.5)

Eq, (4.4) becomes

(4.6)

The elements of the characteristic vecto.r, Hk

, are -::.alled characteristi!!

variables, and are denoted hy wk,:i.'

The charac-teristic variables, wk

., are determLed using Eq. (.f. 5). ,~

The vector q is

[ p]T

q = J p, U, v, w,

-1 and the matrix PI is given by Eqs, (2.23) ,.here the variables p and

( ,0

c (speed of sound) in Eq. (2.23) are denoted rand c to indicate o c

-1 a reference ~onditil)n. Using q and P in Eq. (4.5), the elements k,o

of the characteristic vector

(4.7)

are

J [kx(p - -1-) + k v k \~'l wk,l .,

IVkl z y c °

(4, Ra )

J [k (p - .J?i) - k u + l~ 'v] wk 2 == ---

Il1k/ y z x , c

0

(I •. 8b)

wk 3 =_J_. [k (p - _1.'_) + k u - k v)

IVkl z 2 y x , c °

(4.8e).

G, I.

"

.'

~, . \ .

"

· ,. · ,,, "

· ... '. .. ',.' .,

,,' '. " , ..

J 1 .--- (4. 3d), IVkl Ii

J 1 .. ----IVkl 12

(4.8e)

The chat'llcteristic variables correspond in o'l"der to the cis<mvalues

A1 .. ku+ kv+ k w'" ':1' (4.911) k x y 2 k

1..2 ... k Ok (I •• 9h)

3 "k r.: (lk (4.9c)

/f "" k Ok + clvkl (4. 9d)

AS k .. €lk - c!Vkl (4. 90)

4.2 Characteristic Variable Boundary Cond'it ions

The houndary t;.onuitions are derived below assmuing locally one-

dimensional flow. l'hi5 assumption is probably bettcr for fm:- field

boundary conditions than for boundary conditions applied on or nc3r

surfaces. Howe'll~r. numerical experimonts using zero pressUl:c gradi~nt.

extrapolation~ t~ormal pressure eradient, llnd locally one-dim:::nsioll.:ll

characLeristic variable. boundary conditions" indic<lte tlmt similar l"~~tllts

can be obtained using llny of these four Inethods for impcrnwuhle "'1l11

boundary conditions HS long as the grid is not extraordimn::U.y CO'H"S(h

For the CO\\\putations performed thus fat" ~ the locully ono-dimCllSioll<il

characteristic variable impermeable ,,,,111 boundary conditions are, to l~t~

preferred 0\'01' the other three methods; hence, this method is dc\'eh)pctl

"

ORIGINAL flAG!! r~ Of POOi:'t QUAllN

below. (Further investigations should be performed without the

locally one-dimensional assumption. ho,,'cvel.'.)

By ncglccting:th-c::r:directions in Eq. (4.6) one obtuins

(4.10)

This equation cnn be written

(4.11)

The eigenvalues, ther(\fore. indicate a direction in comput,ltional

'space. Accordtng to Eq. (4.11) one particular eigenvalut1 is ussociated

with one particular characteristic variable.

illdicat(;.~s thn dil:t:ction across the k '" constant computational surface

that information contained in the associnted characteristic variable.

Wk,i' proppgatt)s. This lresult is the basis for determining the

boundary conditiolls referred to here as characteristic variable bouudnry

cvnditiollR.

boundary conditions are now developt~d for the fo11lwiug five

specific CUHEIS

1. supe1sonic inflow

2. supersonic outflow

3. ~ubsonic inflow

4. subsonic outflow

5. impermeable surface

'"

. .. , .'~

I,

',' .~~

'. • .

Supersonic I~t~ow

This is a situation whcl.'c all eigenvalues have the same sigll,

Because flow is coming into~.Jhecomput ational domain, all flow \liH'tables ; ..

are specified.

SupeE.l!~nic Outflm"

This is another situation where all eigenvalues have the SI1l\l~ sign.

Because flow is leaving the computational domain nU flo,,, vllrhihltHl at

the boundary must be obtained from the solution in the comp\ltati\,)lHll

domain. All flow variables are extrapolated from inside the computntional

domain to the boundary.

Subsonic Inflow

This situation is chal.'acterized by four eigenvalues of tho fbm\\~ sign

and one t)f differing sign~ For the subsonic inflow case o.ho\>111 in l~ig.

3a with the flow in '.i'! dir0ction of increaslng (~omputntl.onal cO(I:!.'dinHte

k, the first four eigenvalues ure posi.tivo nnd the fifth 113 ncgnth,"Iil.

For the subsonic inflow case 5hO\m in Fig. 3b ,·lith the flow in t.ht~

directioll of decreasing computational coordinate k, the first thl'.'tle nnd

fifth eigenvalues are negative. and the fourth eigenvalue is positivu.

For a totally general three··dimensional code eithfell' situation in rlg. 3

could occur. Each possiblity in !<'ig. 3 is ta~en into account in the'

foll(n~ing derivatioll. Using the characteristic ';im .. :t~lblcs given by

Eqs. (4.8) one obtains

[k (p P k v '. k \~) '" [k (p •.. 2_) k v - kywJ b

(I, .l~i.\) .. ---) + + x 2 z y a x 2 ;t; c c

0 0

[k (p _ ..R.) -ku+k\iJ .~ [k (p _ .,P-) - kzu + ~\o11b (11.Ub) y 2 z x a y 2

c c 0 0

• . "

,.

'" ~ . ,

.. . I ~

. .. 1

. .

/

lkz(p - ..Ii) + It u - k v) '" [k (p - ~) + k u - k v) y x a z y x b c c

(4.12c)

0 0

[J1Ylsl ± (It u + ltv +~~ ,~)y = [l1Y!J ± (k II + kv+ kzw> ]b PoCo x ':I ." z a PoCo . x Y

where the plus s:l.gll~ in Eq. (4. 12d) and the negat.ive signs in Eqs. (4.12(»

l:efer to the situation in F:i.g. 311, and th.e ?~her sigtl. option refers to

the situatIon in Fig. 3b. 1'ho subscript a refers to approaching the

boundary, subscipt b refers to the boundary, and subscript .e refers to

leaving the boundury (see ,Fig. 3). Note from the equations f<)l." "x' kyo

lind kz given holO\~ Eq. (2..2) that the products Jk • Jk • Dnd Jk aro x y' 2:

components of. lI\'cn veclt)rs, The computm: code actually uses those

components of cell sur-fnee 1:ll'~US. and because the boundary of interest.

j.n Eqs. (/1.12) is the cell face contaIning the boundary point h. the

Slrea vect.or (:omponcnttl of interest correspond to thj.s cell face. 'rhe

metrics at pointtl nand .e. nrc taken to be the same as those at point b,

"lnd the coefficients of the brack!?t terms in Eqs. (4.8) are not currIf!d

through the manipulution of the equations. The linearization point 0

ts taken to bo on tho boundary.

Equati.ons (' •• 12d) nod (/1,.12c) can ~.~ combined to obt.ain

~

pb

,"Ji/{Pa

+P,I!:J:Pc l k(l.I(l uo)+k(v -vo)+k(w -\~p»)} (4.1311) ~ () () X. ot: y a ot: z a -I,.

~,lhere kx

' k , flnd k ilK€< defined by Eqs. (2.15). Equat:ions (4. 12a), y Z

(4. 12b) , (4.12<:). nnd (A. 12d) can be solved for the four remu:Lning

1.:lUknm·m houndllry vulu0s. giv:tng

r·· ', .. > _. ~'" ,"

. ,

.' .. . '

\

...

:."

Ub

'" U ± l~ a x

± k vb .. v a y

± It Wb "" w a z

(/~.13b)

Pa - Ph ---- (4.13c)

Pa - Pb ---pc' o 0

(4.13d)

p - p 8 b -----Po c

0

(4.13e)

The plus sign option in Eqs. (4.13) refers to the computational coor-'

dinate and inflow situation depicted in Fig. 3a, and the negative sign

option refers to Fig. 3b. There is no sign option in Fig. (t., 13b) •

Note that these signs correspond to the s,ign of the first tht:ee e:i.gcllvalues,

and hen(~c this is a mealls of \tITiting the code for r,<mcY"lAl npplications

vlith al.'bHra~y orl.cntat:l.on of the computation coord:l,nl~t'"S' The point

a is outside the computat:tonal domain. point: h is on the 'complltatJonal

boundary, and point .e. is :inside the computational domain,.

,Subsonic Outflo\.]

This situation is also characterized by four eig,~mtalucs of the

same sign and one of opposite. sign. The development of subsonic out-

flow boundary conditions :ls sllllilar to t.hat: for subsonic inflow. Elnd

Fig. 3 can he used again for illustration. Hm.lever. for subson.i.c out-

flow only one characteristic variable is specified and four <\r(! (\cterm:i.ned

from information inside the computat ional domain. wheX;f·D.s. for ~mhsoni.c 1n-

flow four· characteristic variables \"ere speci.fiedand ont.1 ~ms determined from

information insi,de thc;!' computational domain. Usi,ng the: characteristic

variables given by Eqs. (14.8), and the signs of the eigunvDlues given

by Eqs. (4.9) for the situations in Fig. 3, one obtdins'8 Get of

.,. , . ,

'.

~ " ,j ., 'i

;

"~r

, .

-'f

~ ..

I.""" .,,-

equations identical to Eqs. (4.12). Note, hm'1(wer, t.hut although

the equations are identical, point a is now inside the computational

domain .andpoint .e .. i'S ouddde the computational domain; whereas, just

the oppo::::ite was true for'subsonic inflow.

Because the eq~.lations for subsonic outflo\~ ar.e the same as Eqs.

(4.12),. they have the same formal solution. However, hecause one

characterist:ic variable is specified, the resulti)j1g boundary conditions

differ somewhat from the Hubsonic inflow boundary c~ndltionG. Con~ider

l~q. (4.l2e). By ~'p(~dfying that the outflow is straight, then Pb .. P.e

according to Eqs. (4'.12('). The remaining four equations can be solved

for the remaining four variables giving

(4.14a)

(4.14b)

(4.14c)

(4 .14d)

(4 .14e)

The plus and minus 8 igns have the same meaning here as for Eqs. (4. 13) •

For external flo\. computations, P.e could be the ambient static pressure,

p • ex>

~~

"j;"

"" :'t If"" ~

t .

, . • \ . '.

, .. . ' ., .' ·t· . ~ '1

~.: '. ,-' 'I.",'

\ \

ror a boundary across which there is no flOlli the first three

eigenvillues given by Eq,AA':'ff) are zero. the fouz:th is positive. and

'the f,ifth is negative. One' condition must, therefore,. be specified.

'rhe condition specified is that there is no flow across the boundary.

The following relations among the characteristic variables are used to

determine the boundary conditions

[~(p - -1) + kv - k w] = [k (p •. .....E.) +k.v - k w) z y b x 2 z y r c c

0 0

(4.158)

[k (p - ..Q_) k u + kxw]b [k (p •. _2...) - k. u + k w] y 2 z y 2 z x r c c

(4.15b)

0 0

._----. [k (p - .l?.) + k u - k v] g' 2 Y x b

Co

[k (p - L) +. 11 u - k v] z 2 Y x r c o

(4.15c)

/

..... ,' [k u + k v + k w]b = 0 x y z

kw) ) .. [.p I Vk I. -+ ~Jf.",~U + k v + It w») z b Pc '" Y z r o 0

(4.15c)

..

.(

The subscript r refers to a reference vahlc. which is selected as the

center of the first cell from the boundary. The'minus and plus signs

:In Eq. (4. 15e) corr~spond to the locntion of the point r. If the point

lC is in the positive k direction from the boundary then the minus

:;1gn is used in Eq. (l1.15e). and if it is in the In:bt.us direction then

the plus sign is used.

'.

.. .

., "f

-. , ~. .

" " , .1 .'

" "

, \

. '~

:I '

, ......

Finite volume codes only require the pressure at an impermeable

boundary and consequently Eqs. (4.15a). (4.l5b), and (4.l5c) are not

needed. However, to fad:;i.~;',;t:;lW.,"','Lhe handling of point~' near boundaries

and aid in code vectori.zllItion, phantom points are used in the present'

version of the code. The use of phantom points r('!quires information of

variables other than pressure to ensure, for eJtal1lple, l/!Cl'O flow across

an impermeable boundary. Such information can be obtained from Eqs.

(4.15) •

Equations (4.lSd) and (4.1Se) can be solved for Pb• Equations

(4.15a) - (4.1Sd) can then be solved for the remaining four variables.

The solution of Eqs. (4.15), is

~

Pb '" P r :j:.' p c (k u + kv o 0 x r y r

+ k w ) Z r (4.J.68)

Pb - P Pb '" Pr

;. r 2

(4.l6b) c

0

ub .. u - k (k u +kv + k " .. ) r x x r y r Z r (4 .16c)

v = v ,- k (k u +kv + k w ) b r y x r y 1'- Z r (4.16d)

wb '" wr .- k (k u +kv + k t" ) Z x r y r Z r (4.16e)

where the point r 1s the center of the first cell fr,?1n the boundary

and the minus sign in Eq. (4~16~ is used if r is in the positive k

direction from the boundary, and the plus sign is used if r is in the

negative k direction from the boundary.

. 4, '. '.

"

' . . -.

"

"

I

/ /

,/

./ .

. ,.\

\ . \

• I

. /

./

/

"

:>'~-~' .-

. i , I '.

......

. I /: I

4.3 Phantom Points

I I

Phantom points are denoted by the subscript p. The points are

obtained from the relations

",. :,.'"

(4.17a)

p :: 2p - p p b in (4.17b)

U :: 2u - u p b in

(4.17c)

(4.l7d)

(4.17e)

where th(,: subscript in refers to the center of the first cell inside

the computatj.onal domain and can be any of the points a, .t, 'or r used

in this section. For ~xample. the phantom points for an impermeable

. surface are ~

p =: Pr + 2p c (It u . + Ie v + k2\~r) (4.18a) p o 0 x r y r

2(Pb - Pr ) (4.1Ub) Pp .. p +-----

·r 2 c

0

u ., u 2k (k u + kv + k:e.\-lr> (4.18c) p r x x r y r

v .. v - 2k (k u +kv + k w ) (4.1Bd) p r y x r y r z r

- - -w .. w 2k (k u +kv +kw) (4.18e)

p r z x r y r z r

/."

The velocity vector components are the same as those used by Jacocks and

Kneile (R~!f. 14).

.. "

" . . ,

. ~ . ,',

1~ . . ' , .

;

I

.'

J I , I !

. I

/1 " I J , ,

i 1 I : .

I '\'

v

V. RESULTS

The computer program written to solve the three-dimensional un­

steady Euler equationn is r",:'.(:erred to as the SKOAL code. SKOAL is

an acronym for nothing.

To investigate the results of this code a numerical solution was

obtained for the ONERA H6 wing at Moo = 0,84 and (tt. on 3.06°. The pressure

distribution is shown in :Fig. 4. This solution is compared to a

solution from the FL057 code in Fig; 5. Identical 96x16x16 meshes

were used for both solutions in Fig. 5., The major difference between

the two solutiol1n occurs :I.n the outbuard region of the wing. More

detC'.i1e.d comparisons between the solutions, including comparisons

wHh experimental clflta (Ref. 15), are given ill Fig. 6. Comparisons of

spanwise distributions of lift and drag are given in Fig. 7. The

FL057 code gives a 4% higher value of lift to drag ratio than the

SKOAL code.

The number of supersonic points (NSUP) is sometimes used as a

crude indication of convergence. The FL057 solUiUon was obtained by

first using a 48x8x8 grid and then using this crude grid solution as

initial conditions for. the 96xl6x16 grid solution. Hence the NSUP

corresponding to nn impulsive start for the 96xl.6x16 grid \-lIlS not

available from the FL057 code for comparison wi1th the SKOAL code. In

order t.o compare the NS1JP history bet\<leen the two codes, a 48x8x8 grid

solution was obtainnd using the SKOAL code. These. results are presented

in Fig. 8. The FL057 code is stable for a eFL of 2.8 using a four-

stage Runge-Kutta Behome. whereas. the SKOAL code is stable fot' a eFL

of 2.0 using the predictor-corrector scheme. It if, interesting to note

(t~ '<J

.;

, . • t

,I

· .... ./ \ i

that the number of cycles required for the NSUP to become constant

using FL~57 is 246, wher~as, the number of cycles required for the

NSUP to become constant using SKOAL is 325. The ratio of these t\OlO

numbers is essentially the inverse x'aLio of CFL numbers. Another way

of looking at this is to note that FL057 passes through a flux balance

routine four times during each cycle, whereas SKOAL passes through I'l

flux balance routi.ne t.hn~e time dur:1.ng each cycle. A comparison of

the ratio of the NSUP to the final number of NSUP (denoted as NSUP/

(NSUP) ) as a 'fullcUon of flux balances is given :I.n Fig. 9. Bar:ed 011 c

the number of flux balances required to reach steady state for this

solution these methods are operating identically. However. one pass

through a dissipation routine (,.,hleh is similar i.n terms of computer

resources to an extra pass through the fhlJ( balance routine) is required

in the FL057 code \mich :l.s not required :tn the SKOAL code bec.aus.e no

additional dissipation terms were included. The :f.mportant comparisun.

however, :l.S tlio number of computer resource units required to reach

st-.::ady state. This depends on the coding, vectorization, storage, etc.,

of ea('h code on the same machine. Such a compnr:i.son c.annot presently

be made. As the codes now stand, the FL057 code is probably at least

twice as fast as the SKOAL code per cycle. Ho\vcver, bAsed on l~ig. 9,

it is anti.cipated that the SKOAL code CRn be improved.

Numerolls other results have been obtained using the SKOAL code

for var:lous two-and three-dimensional geometries. and for subson:tc,

transonic, and 8upersonic flow. The ONERA vri.ng. however, is the only

solution obtained thus far that can be con~ared to another Euler code

solution for wldch the same mesh \-las used.

(~ j

..

.,

., ,

:i .. .

,-j

I

. !

-, \ .: \.

, '.

I I,:

,-

I I

I I

, . / ,

II"':"

I

/

"

VI. CONCLUDING REMARKS

A metl>.od was presented for solving the three-dimensional unsteady

Euler equations" based on flux vector splitting. The equations were

cast in curvilinea.:coord5.nates and a finite volume discretization

1~as used for handling arbitrary geometries. The discretized equations

111ere solved using an explicit upwind second-order pred:l.ctor-corrector

:scheme that is stable for a CFL of 2. No addit:l.onal dissipation terms

111ere included in the scheme. Local time stepping was used to accelerate

~::onvergence for steady-state problems. . Characteristic variable boundary

I::onditions were developed and applied in the far field and at impermeable

imrfaces. Numerical results toJcre obtained and compared t<1ith resul1:8

ir.om· the FL057 code and experimental data for the ONERA 1-16 \11ng at

11", '" 0.84 and at .. 3.06".

, . '~

:"

, .'

J i

/

-,

......

REFERENCES

1. \yhitfield, D. L. and Jameson, A., "Three,,·Dimensional Eu] er Equation Simulation of"J)ropel1er-Hing Interaction in Transonic Flo:~," AIM

,Paper BJ-023f:; January 1983.

2. Jameson, A., Schmidt, 1-1., and Turkel, E •• "Numerical Solutions of the Euler Equatlons. by Finite Volume Methods Using Runge Kutta Time Stepping Schemes," AIM Paper No. 81-1259. June 1981.

3. Schmidt. ty., Jameson, A., and Whitfield. :0'. L., "Finite-Volume Solutions to 'the Euler Equations in Transonic Flo~]," Journal of Aircraft, Vol. 20. No.2, February 1983, pp. 127-133.

4. Swafford, T. H., "Three-Dimensional, 'Iime-·l}ependent, Compressible, Turbulent, Integral Boundary-Layer Equa.t:i0r!S in General Curvilinear Coordinates and Their Numerical Solution ,or, Ph.D. Dissertation, ' Mississippi State University, August, 19a.:,L

5. Steger, J. L. and Wa:::ming, R. F., "Flux Vector Splitting of The Inviscid Gasdynamic Equations .lith Ap.p1ication to Finite­Difference Nethods," Journal of Computational Physics, Vol. 40, 1981, pp. 263-293.

6. Har.ming, R. F., Beam, R. M., and Hyett, R. J., "Diagonalization and Simultaneous Symmetrization of the Gas,-,Dynamic Nutrices," MathcE!.at.ic of COm.P.utation, Vol. 20, No. 132, October. 1975, pp. 1037-1045.

7. Pulliam, T. H. and Chaussee, D. S., itA Diagonal Form of an Implid.t Approximate-Factorization Algorithm," .:!.~!:n1al of Computational PhYE.,ics, Vol. 39, 1981, pp. 347-363.

8. Nobel, B. and Daniel, J. H., AEplied I.inc::E __ Algebra. Second Edit:!.on, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1977.

9. Olmsted, J. M. n .• Real Variables. Appleton-Century-Crofts, Inc., New York, 1959.

10. Reklis, R. P. and Thomas, P. D. "Shock-Capturing Algorithm for the NHvier-Stokes Equations." AlAA Journal, Vol. 20, No.9, Septembp.r 1982, pp. 1212-1218.

11. Thomas, P. D. and Lombard, C. K., "Geometric Conservation Law and Its Application to F1ot.] Computations on HovIng Grids ," &~~~~.~ Vol. 17, No. 10, October 1979, pp. 1030-1031

12. Harming, l{. F. and Beam, K. M., "Upwind Second·-Order Difference Schemes .md Applications in Aerodynamic Flows," AIAA ,TournaI, Vol. 14. No.9, S€:ptember 1976, pp. 1241-1249. -------

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13. Deese, J. iL, "Numerical Experi1D:ents with tJ.e S;-,lit-Flux··Vector Form of the Euler Equations," AIM Paper No. 83-0122, January 1983.

14. Jacocks, J. L. and Kneile, K. R., "Computation of Three-Dimensional Time-Dependent Flow Using the Euler Equations," AEDC-TR-80-49, July, 1"91l1. "-')';.,

15. Schmitt, V. and C!1~rpin. F., "Pressure Distriubtions on the ONERA M6-Win& at Transonic l1ach Numbers," Experimental Data Baf<c for Computer Proar.wl Ass~ssment. AGARD AR-138, 1979.

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Fig. 5 Pressure Distribution Comparisons on the O!H:;RA 116 Wi,!lg for Mco '" 0.84 and a .., 3.060

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O;~ERA I-j6 HI NG O~iGff,!Al rAGE Vf£ ft, =: 0.84 OF POOR QUALITY

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----FLOS7 ---SKOAL

:-:-} C D (TOTAL.)

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ONEPJ\ N6 WING

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---- fl057 (48xS)<8)

-- SKOAL (48x8x8)

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.. _-- FL057 (48)(8xS)

b __ ~ SKOAL (iI8)(8x8)

,_...,JI _____ -.I~

GOO 800 1000 1200 FLUX BALMJCES

fig., 9 Numh(lr of Supers()u:tc Points (NS1.)P) as a Function (If Flu:lt Balances,

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End of Document