Sistemul Euler Pt Curgeri Compresibile

2

Sistemul Euler pentru Curgeri Compresibile

Cuprins Capitolul 1 Introducere. Sistemul Euler. ............................................................................... 3

1.1 Sistemul Euler. ............................................................................................................ 3 1.2 Adimensionalizarea ecuaiilor Euler. .......................................................................... 5

Capitolul 2 Proprietile matematice ale sistemului Euler. ................................................... 7 2.1 Formularea n variabile primitive a sistemului Euler. ................................................. 7 2.2 Valorile i vectorii proprii ai sistemului Euler n variabile primitive.......................... 7 2.3 Formularea diferenial conservativ. ....................................................................... 13 2.4 Legtura dintre formulrile n variabile primitive i conservative, ........................... 14 2.5 Vectorii proprii ai sistemului Euler n variabile conservative. .................................. 15 2.6 Relaiile de compatibilitate. Variabilele caracteristice. ............................................. 16

Bibliografie .......................................................................................................................... 18

3

Capitolul 1 Introducere. Sistemul Euler.

1.1 Sistemul Euler.

n cadrul Dinamicii Fluidelor modelul ecuaiilor Euler reprezint cel mai complet model al

micrii unui fluid fr vscozitate i fr conducie termic, din punct de vedere matematic poate

fi privit ca o aproximare a ecuaiilor Navier-Stokes pentru micri la numere Reynolds mari.

Sistemul Euler reprezint expresia matematic a principiilor generale de conservare n

care se neglijeaz influena teremenilor vscoi i de conducie.

Forma integral a sistemului Euler este constituit din ecuaiile de conservare a masei,

impulsului i energiei totale pe unitatea de volum:

Conservarea masei:

0

+ =

d dt

V n (1.1)

Conservarea impulsului:

( )

+ + =

d p d dt e

V V V n f (1.2)

Conservarea energiei:

+ =

E d H d dt e

V n f V (1.3)

Sistemul format din (1.1), (1.2) i (1.3) poate fi scris compact:

d d dt

+ =

U F n Q

(1.4)

unde:

U vectorul variabilelor conservative;

F

vectorul flux conservativ;

n

versorul normalei la frontiera , , domeniului ;

Q - vectorul surs.

4

[ ]Tu v w E =U (1.5)

= + + x y zF F i F j F k

(1.6)

( )0 Tex ey ez ex ey ezf f f f u f v f w = + + Q (1.7) componentele vectorului flux conservativ se scriu explicit:

2

2

2

T

T

T

u u p u v u w u H

v u v v p v w v H

w u w v w w p w H

= +

= +

= +

x

y

z

F

F

F

(1.8)

Pentru ca sistemul (1.4) s fie complet determinat trebuie completat cu o ecuaie de stare

care s defineasc proprietile termodinamice ale fluidului, vom folosi o ecuaie de forma:

( ) ( ), sau ,p p T e e T = = (1.9) Pentru gazul perfect (1.9) se poate scrie explicit:

ve c T

p R T

=

= (1.10)

Unde e energia intern a fluidului i T temperatura, vom folosi deasemeni relaiile valabile

pentru un fluid perfect:

1 1

, p vR Rc c

= =

(1.11)

Entalpia specific este definit:

pph e c T

= + = (1.12)

iar entalpia total:

2

2V pH h E

= + = + (1.13)

unde 2 2 2V u v= + .

5

Pentru aer n condiii standard avem constanta 2

2287mRs K

= , iar raportul cldurilor

specifice la presiune constant, pc , i la volum constant, vc , este 1 4.p

v

cc

= = .

Pe baza ecuaiilor de stare putem explicita presiunea n sistemul Euler:

2

12

( ) ( )Vp E = (1.14)

Observaie: uzual, n aplicaiile de aerodinamic vectorul surs se neglijeaz i sistemul

(1.4) se poate scrie:

0d dt

+ = U F n

(1.15)

1.2 Adimensionalizarea ecuaiilor Euler.

n vederea obinerii unei soluii numerice (aproximative) a sistemului (1.15) este util

rescrierea acestuia n form adimensional, acest lucru duce la o normalizare a variabilelor

sistemului care iau astfel valori ntre limite prescrise.

Definim urmtoarele mrimi adimensionale:

/ , / , /x x L y y L z z L= = = (1.16)

/ , / , /u u V v v V w w V = = = (1.17)

( )2 2/ , / , / , /p p V T T T e e V = = = = (1.18) /t tV L= (1.19)

unde , , , TL V sunt valori de referin.

Sistemul (1.15) n variabile adimensionale, n care neglijm termenul surs, poate fi scris:

0t

+ =

U F

(1.20)

unde variabilele adimensionalizate sunt:

T

u v w E = U (1.21)

6

2

2

2

T

T

T

u u p uv uw uH

v uv v p vw vH

w uw vw w p wH

= +

= +

= +

x

y

z

F

F

F

(1.22)

Energia total i entalpia total adimensionalizate sunt:

2

2VE e= + (1.23)

i

2

2V pH h E

= + = + (1.24)

Ecuaia de stare adimensionalizat (1.14):

( )1p e = (1.25)

n cele ce urmeaz, pentru simplificarea scrierii am renunat la semnul pentru

adimensionalizare.

7

Capitolul 2 Proprietile matematice ale sistemului Euler.

2.1 Formularea n variabile primitive a sistemului Euler.

Exprimarea n variabile primitive a sistemului Euler uureaz calculul analitic a valorilor i

vectorilor proprii ale sistemului, i este folosit pentru dezvoltarea unor scheme numerice de

ordin superior de precizie. Variabilele primitive sunt mrimi direct msurabile i apar n mod

natural n impunerea condiiilor la limit.

Alegnd ca variabile primitive densitatea, componentele vitezei i presiunea, ecuaiile

sistemului Euler se vor scrie:

- ecuaia de continuitate:

0( )t + + =

V V

(2.1)

- ecuaiile de impuls:

0( ) pt

+ + =

V V V

(2.2)

- ecuaia presiunii:

2 0( ) ( )dp p cdt

+ + =V V

(2.3)

Sistemul format din ecuaiile de mai sus poate fi scris compact:

0( )t

+ =

q B q (2.4)

unde q reprezint vectorul variabilelor primitive:

[ ]Tu v w p=q (2.5) iar B este matricea iacobian:

= + + x y zB B i B j B k

(2.6)

2.2 Valorile i vectorii proprii ai sistemului Euler n variabile primitive.

innd seama de relaiile anterioare componentele matricii B sunt:

8

2 2

0 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 0 0

0 0 0 0 0 0

//

u vu v

u vu v

c u c v

= =

x yB ,B (2.7)

2

0 0 00 0 0 00 0 0 00 0 0 1

0 0 0

/

ww

ww

c w

=

zB

n vederea obinerii valorilor proprii ale sistemului Euler cutm o soluie de tip und

pentru forma cvasiliniar (2.4):

( ) i te = d xq q

(2.8)

n care q este amplitudinea complex a undei, x

este vectorul de poziie, d

este vectorul de

und, este pulsaia undei, t d x

reprezint faza undei care se propag n direcia d

cu

pulsaia . Introducnd (2.8) n (2.4) obinem :

( )

( )

( )

( )

i t

i tx

i ty

i tz

i et

d i ex

d i ey

d i ez

=

=

=

=

d x

d x

d x

d x

q q

q q

q q

q q

(2.9)

0

( ) ( )

( ) ( )

( ) ( )

( ) (

i t i tx

i t i ty z

i t i tx y

i t iz

i e d i e

d i e d i e

d i e d i e

d i e i e

+ +

+ + =

+ +

+ =

d x d xx

d x d xy z

d x d xx y

d x d xz

q B q

B q B q

B q B q

B q q

)

( )

t

x y y

x y y

d d d

d d d

+ + =

+ + = x y z

x y z

B q B q B q qB B B q q

(2.10)

Notnd matricea iacobian a sistemului cu:

x y zd d d= = + + p x y zK B d B B B

(2.11)

9

unde:

x y zd d d= + + d i j k

(2.12)

valorile i vectorii proprii ai sistemului Euler se obin din sistemul omogen:

unde am notat = =pK q q (2.13)

Impunnd condiia ca sistemul (2.13) s aib o soluie diferit de zero obinem ecuaia

caracteristic:

0det( ) =pK I (2.14)

sau explicit:

2 2 2

0

0 0 0

00 0 0

0 0 0

0

det( )

x y z

x

y

z

x y z

d d d

d

d

d

c d c d c d

= =

p

V d

V d

K I V d

V d

V d

(2.15)

din care obinem valorile proprii:

1 2 3 4 5, , d d dV V c d V c d = = = = + = (2.16)

unde am notat:

2 2 2, d x y zV d d d d= = + +V d

(2.17)

Definim vectorii proprii la stnga ai matricii iacobiene pK ca fiind soluii ale sistemului:

i = i p il K l (2.18)

Rezolvnd sistemul (2.18) pentru fiecare valoare proprie (2.16) putem construi matricea

-1L ale crei linii sunt formate de vectorii proprii il :

10

1 2 2 1 21 1 1 1

1 2 2 1 22 2 2 21 2 2 1 23 3 3 3

4 4 4 4

5 5 5 5

0

0

0

0

0

/ /

/ /( ) /( )

z y

z x

y x

x y z

x y z

d d c

d d c

d d c

d d d c

d d d c

=

-1L (2.19)

unde parametrii reprezint constantele de normalizare ale vectorilor proprii, care ramn a fi

specificate.

a) Presupunem c 0xd , atunci putem lua:

1 2 1 2 1 21 1 2 2 3 3 4 51 0 0 1 0 1 1 , , , , , , = = = = = = = = (2.20)

i din (2.19) obinem:

2

2 2

2 2

1 0 02 211 0 0 0

02 20 0 0

0 0 0 02 210

01 2 20

0 0 02 2

,

x xz y

z x

y yz z xy xy

x xx y z

y x z z zy

x xx y z

c cd dc d d

d dd dd d dd d d

d dd d d

d dc d d ddd dd d d

c c c

+ = = +

-1L L (2.21)

unde:

2 2 2 1x y zd d d+ + = (2.22)

b) Dac 0yd , atunci:

1 2 1 2 1 21 1 2 2 3 3 4 50 1 1 0 0 1 1 , , , , , , = = = = = = = = (2.23)


11

2 2

2

2 22

2

0 1 02 20 0 0

1 01 0 0 0 2 2

0 0 00

2 2100

1 2 20

0 0 02 2

,

z y

z yz x xx

y y

y yy xz x

x y zy x z z z

xy y

x y z

c cd dd dd d dd

d dcd dd d

d dd d d

d d d d dc dd dd d d

c c c

+ = = +

-1L L (2.24)

c) Dac 0zd , atunci putem alege:

1 2 1 2 1 21 1 2 2 3 3 4 50 1 0 1 1 0 1 , , , , , , = = = = = = = = (2.25)


2 2

2 2 2

0 0 12 20 0 0

0 0 0 02 211 0 0 0

01 2 20

01 2 20

0 0 02 2

,

z y

z yx x xz xy

y z

y yz x xy

x zx y z

z zy x

x y z

c cd dd dd d dd d d

d d

c d dd d ddd dd d d

c d dd dd d d

c c c

+ += =

-1L L (2.26)

Matricea vectorilor proprii la stnga (2.19) este inversabil rezult c vectorii proprii sunt

liniar independeni, aceasta mpreun cu faptul c valorile proprii (2.16) sunt reale duc la concluzia

c sistemul Euler nestaionar este de tip hiperbolic.

Observaie:

n cazul unidimensional valorile proprii (2.16) sunt reale i distincte, vectorii proprii (2.19)

sunt linear independeni i sistemul Euler nestaionar este strict hiperbolic.

Inversa matricii -1L notat L diagonalizeaz matricea iacobian pK :

12

=-1 pL K L (2.27)

unde este matricea diagonal a valorilor proprii.

Putem construi o matrice R a vectorilor proprii la dreapta (coloanele acesteia sunt

vectorii proprii), aceasta la rndul ei va diagonaliza matricea iacobian pK :

=-1 pR K R (2.28)

Din (2.27) i (2.28) obinem:

R L (2.29) adic vectorii proprii la dreapta sunt ortogonali pe vectorii proprii la stnga.

Din expresia lui L se pot obine expresiile matricelor care diagonalizeaz iacobienele

, ,x y zB B B .

Pentru 1 0x y zd , d d= = = (cazul a) obinem xL , care va diagonaliza xB :

1 0 0 2 20 0 0 0 5 0 50 0 1 0 00 1 0 0 00 0 0 2 2

/( ) /( ). .

/ /

c c

c c

=

xL (2.30)

Pentru 1 0y x zd , d d= = = (cazul b) obinem yL , care va diagonaliza yB :

0 1 0 2 20 0 1 0 00 0 0 0 5 0 51 0 0 0 0

0 0 0 2 2

/( ) /( )

. .

/ /

c c

c c

=

yL (2.31)

Pentru 1 0z x yd , d d= = = (cazul c) obinem zL , care va diagonaliza zB :

0 0 1 2 20 1 0 0 01 0 0 0 00 0 0 0 5 0 50 0 0 2 2

/( ) /( )

. ./ /

c c

c c

=

zL (2.32)

13

2.3 Formularea diferenial conservativ.

Aplicnd teorema divergenei n sistemul Euler scris n form integral (1.15)

0d dt

+ = U F n

(2.33)

se obine forma diferenial conservativ a acestuia:

0t

+ =

U F

(2.34)

Forma cvasiliniar a sistemului Euler se obine din (2.34) :

0t

+ =

U A U (2.35)

unde A este matricea iacobian:

= = + + x y zFA A i A j A kU

(2.36)

Forma analitic a matricilor iacobiene , ,x y zA A A este:

2 2

2 2 2

0 1 0 0 01 3 1 1 1

20 0

0 011 2 1 1

2

( ) ( ) ( )

[ ( ) ] ( ) ( ) ( )

u V u v w

uv v uuw w u

u E V E V u uv uw u

+ =

+

xA (2.37)

2 2

2 2 2

0 0 1 0 00 0 0

1 1 3 1 12

0 011 1 2 1

2

( ) ( ) ( )

[ ( ) ] ( ) ( ) ( )

uv v

v V u v w

vw w v

v E V uv E V v vw v

+ =

+

yA (2.38)

2 2

2 2 2

0 0 0 1 00 0

0 01 1 1 3 1

211 1 1 2

2

( ) ( ) ( )

[ ( ) ] ( ) ( ) ( )

uw w uvw w v

w V u v v

w E V uw vw E V w w

= + +

zA (2.39)

14

2.4 Legtura dintre formulrile n variabile primitive i conservative,

Pentru a obine o legtur ntre formulrile (2.4) i(2.35) considerm o transformare

neliniar de forma ( )=U U q , iacobiana transformrii este:

2

1 0 0 0 00 0 0

0 0 00 0 0

12 1

uvw

V u v w

= =

UMq

(2.40)

inversa lui M :

2

1 0 0 0 01 0 0 0

10 0 0

10 0 0

1 1 1 1 12

( ) ( ) ( ) ( )

u

v

w

V u v w

=

-1M (2.41)

Plecnd de la forma (2.4) vom folosi expresiile (2.40) i (2.41) pentru a obine explicit legturile

dintre cele dou formulri:

0( )t

+ =

q B q (2.42)

, , , t t x x y y z z

= = = =

q q U q q U q q U q q U

U U U U (2.43)

din (2.41) avem:

0( )t

+ =

-1 -1UM B M U (2.44)

nmulim la stnga cu M n (2.44) i avem:

0

0

( )

( )

t

t

+ =

+ =

-1 -1

-1

UM M M B M U

U M B M U (2.45)

15

comparnd (2.45) cu (2.35) identificm:

, = = -1 -1A M B M B M A M (2.46)

Notnd matricea iacobian a sistemului (2.35) cu:

x y zd d d= = + + x y zK A d A A A

(2.47)

i din relaiile (2.47) obinem:

= = = -1 -1pK A d M B d M M K M

(2.48)

adic matricile K i pK sunt similare i au aceleai valori proprii.

2.5 Vectorii proprii ai sistemului Euler n variabile conservative.

Notm cu matricea vectorilor proprii la dreapta ai matricii . Inversa matricii

notat diagonalizeaz matricea iacobian :

=-1P K P (2.49) unde este matricea diagonal a valorilor proprii ai lui .

Deoarece matricile i sunt similare i au aceleai valori proprii putem scrie:

= = -1 -1 pP K P L K L (2.50)

din (2.48) :

=

=

-1p

-1p

K M K M

M K M K (2.51)

i nlocuind n (2.50) vom avea:

= = -1 -1 -1P K P L M K M L (2.52) de unde:

=

= -1 -1 -1P M LP L M

(2.53)

Matricea vectorilor proprii la dreapta ai formulrii n variabile conservative se obine

nlocuind n (2.53) expresiile (2.41) i (2.21), (2.24), respectiv (2.26).

P K P-1P K

K

K pK

16

2.6 Relaiile de compatibilitate. Variabilele caracteristice.

nmulind la stnga sistemul Euler (2.4) cu vectorul propriu pil acesta se transform ntr-

o ecuaie scalar n care apar derivate doar n lungul suprafeei caracteristice:

0( )t

+ =

pi piql l B q (2.54)

Relaia (2.54) se numete relaie de compatibilitate.

Relaiile de compatibilitate corespunztoare normalei la o suprafa de interes (frontiere

de intrare/ieire) sunt utile att n impunerea condiiilor la limit ct i n dezvoltarea anumitor

algoritmi numerici de integrare. Vitezele de propagare a undelor corespunztoare valorilor proprii

sunt:

( )

( )

c d

c d

= = =

= +

=

1 2 3 d

4 d

5 d

a a a V d i

a V d i

a V d i

(2.55)

Introducnd matricile L i -1L relaiile de compatibilitate se pot scrie:

0t + =

-1L B q (2.56)

sau n variabile conservative:

0At + =

-1 -1L M U (2.57)

Introducem variabilele auxiliare W , definite de relaia:

= -1W L q (2.58)

unde reprezint unul dintre operatorii de derivare spaial, operatorul de derivare temporal

sau o combinaie a acestora.

Componentele vectorului W se numesc variabilele caracteristice ale sistemului Euler. n cazul general componentele vectorului W nu se pot determina explicit.

Din (2.58) se observ c fiecare component a lui W reprezint o combinaie liniar a variabilelor primitive, putem scrie relaiile de compatibilitate:

17

0( )t

+ =

-1 W L B L W (2.59)

ecuaie care se numete ecuaia variabilelor caracteristice.

Putem exprima variabilele caracteristice i n funcie de variabilele conservative innand

cont de (2.40):

= =

-1W P UU P W

(2.60)

de unde, putem scrie :

j jj

w = U r (2.61)

unde jr sunt vectorii proprii la dreapta ai matricii iacobiene K .

Relaia (2.61) arat c variaia vectorului conservativ U poate fi descompus n baza vectorilor proprii la dreapta ai matricii K , coeficienii descompunerii fiind variaiile variabilelor caracteristice.

18

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Bucharest, 2005

20

33. R. C. Swanson, E. Turkel, Multistage schemes with multigrid for Euler and Navier-

Stokes equations, NASA TP 3631, 1997.

34. I. Gh. abac, Matematici speciale, E.D.P. Bucureti, 1981.

35. D. Vanderstraeten, A. Csik, D. Roose, An expert-system to control the CFL number

of implicit upwind methods, Report TW 304, 2000.

36. V. Venkatakrishnan, Convergence to Steady State Solutions of the Euler Equations

on Unstructured Grids with Limiters, Journal of Computational Physics 118, 120-130

(1995).

37. V. Venkatakrishnan, D. J. Mavriplis, Agglomeration multigrid for the three-

dimensional Euler equations, AIAA-94-0069.

38. Z. J. Wang, Y. Sun, A curvature based wall boundary condition for the Euler

equations on unstructured grids, AIAA-2002-0966.

39. J. F. Wendt (ed), Computational Fluid Dynamics, Springer-Verlag, 1992.

40. W. A. Wood, W. L. Kleb, Diffusion Characteristics of Upwind Scemes on

Unstructured Triangulations, AIAA Paper 98-2443.

41. K. Warendorf, U. Kster, R. Rhle, Multilevel methods for a highly-unstructured

Euler solver, ECCOMAS 98.

42. W. A. Wood, W. L. Kleb, Diffusion characteristics of upwind schemes on

unstructured triangulations, AIAA Paper 98-2443.

43. P. Woodward, P. Colella, The numerical simulation of two-dimensional fluid flow

with strong shoks, Journal of Computational Physics 54 (1984), 115-173.

Capitolul 1 Introducere. Sistemul Euler.1.1 Sistemul Euler.1.2 Adimensionalizarea ecuaiilor Euler.

Capitolul 2 Proprietile matematice ale sistemului Euler.2.1 Formularea n variabile primitive a sistemului Euler.2.2 Valorile i vectorii proprii ai sistemului Euler n variabile primitive.2.3 Formularea diferenial conservativ.2.4 Legtura dintre formulrile n variabile primitive i conservative,2.5 Vectorii proprii ai sistemului Euler n variabile conservative.2.6 Relaiile de compatibilitate. Variabilele caracteristice.

Bibliografie1. M. Aftosmis, D. Gaitonde, T. S. Tavares, Behavior of linear reconstruction techniques on unstructured meshes, AIAA Journal, Vol. 33, 1995.2. W. K. Anderson, Grid Generation and Flow Solution Method for Euler Equations on Unstructured Grids, NASA TM-4295.3. T. J. Barth, M. G. Larson, A posteriori error estimates for higher order Godunov finite volume methods on unstructured meshes, NASA Technical Report NAS-02-001.4. E. Carafoli, V. N. Constantinescu, Dinamica fluidelor compresibile, Ed. Academiei, Bucureti, 1984.5. E. Carafoli, Aerodinamica, Ed. Tehnica, 1951.6. E. Carafoli, Aerodinamica vitezelor mari, Ed. Academiei, 1957.7. S. Dnil, C. Berbente, Metode numerice n dinamica fluidelor, Ed. Academiei Romne, Bucureti, 2003.8. N. T. Frink, Upwind Scheme for Solving the Euler Equations on Unstructured Tetrahedral Meshes, AIAA Journal, Vol. 30, 1992.9. N. T. Frink, P. Parikh, S. 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Jameson, Essential elements of computational algorithms for aerodynamic analysis and design, ICASE 97-68.18. A. Jameson, W. Schmidt, E. Turkel, Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes. AIAA Paper 81-1259, June 1981.19. A. Jameson, Analysis and design of numerical schemes for gas dynamics 1 artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence, Journal of Computational Fluid Dynamics, 4, 1995, pp 171-218.20. A. Jameson, Analysis and design of numerical schemes for gas dynamics 2 artificial difusion and discrete shock structure, Journal of Computational Fluid Dynamics, 5, 1995, pp 1-38.21. M. J. Kermani, E. G. Plett, Modified entropy correction formula for the Roe scheme, AIAA 2001-0083.22. W. L. Kleb, Comments regarding two upwind methods for solving for two-dimensional external flows using unstructured grids, NASA TM 109078.23. L. Landau, E. Lifchitz, Mcanique des fluides, Ed. MIR 1971.24. R. J. LeVeque, Numerical methods for conservation laws, 1992.25. M. S. Liou, C.J. Steffen, A new flux splitting scheme, Journal of Computational Physics 17, (1993)26. M. S. Liou, A continuing search for a near perfect numerical flux scheme, NASA TM 106524 (1994).27. M. S. Liou, Ten years in the making AUSM family, NASA TM-2001-21097728. S. Lin, T. Wu, Y. Chin, Upwind finite-volume method with a triangular mesh for conservation laws, Journal of Computational Physics 107, 324-337 (1993).29. Xu-Dong Liu, A maximum principle satisfyng modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws, Dept. of Math., UCLA, LA, California, 9002430. D. Sidilkover, A genuinely multidimensional upwind scheme and an efficient multigrid for the compressibile Euler equations.31. S.P. Spekreijse, Multigrid solution of the steady Euler equations, CWI Tracts, 1980.32. M. Stoia-Djeska, A practical introduction to computational fluid dynamics, E.D.P., Bucharest, 200533. R. C. Swanson, E. Turkel, Multistage schemes with multigrid for Euler and Navier-Stokes equations, NASA TP 3631, 1997.34. I. Gh. abac, Matematici speciale, E.D.P. Bucureti, 1981.35. D. Vanderstraeten, A. Csik, D. Roose, An expert-system to control the CFL number of implicit upwind methods, Report TW 304, 2000.36. V. Venkatakrishnan, Convergence to Steady State Solutions of the Euler Equations on Unstructured Grids with Limiters, Journal of Computational Physics 118, 120-130 (1995).37. V. Venkatakrishnan, D. J. Mavriplis, Agglomeration multigrid for the three-dimensional Euler equations, AIAA-94-0069.38. Z. J. Wang, Y. Sun, A curvature based wall boundary condition for the Euler equations on unstructured grids, AIAA-2002-0966.39. J. F. Wendt (ed), Computational Fluid Dynamics, Springer-Verlag, 1992.40. W. A. Wood, W. L. 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