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Engineering Applications of Computational Fluid Mechanics Vol. 1, No.4, pp. 238252 (2007)

SOLUTIONS OF THE EULER AND THE NAVIER-STOKES EQUATIONS

USING THE JAMESON AND MAVRIPLIS AND THE LIOU STEFFEN

UNSTRUCTURED ALGORITHMS IN THREE-DIMENSIONS

Edisson Svio de Ges Maciel

CNPq Researcher, Rua Demcrito Cavalcanti, 152, Afogados, Recife, Pernambuco, Brazil, 50750-080

E-Mail: edissonsavio@yahoo.com.br

ABSTRACT: In the present work, the Jameson and Mavriplis and the Liou and Steffen unstructured algorithms are

applied to solve the Euler and the Navier-Stokes equations in three-dimensions. The governing equations in conservative

form are solved, employing a finite volume formulation and an unstructured spatial discretization. The Jameson and

Mavriplis algorithm is a symmetrical second-order one, while the Liou and Steffen algorithm is a flux vector splitting

first-order upwind one. Both schemes use a second-order Runge-Kutta method to perform time integration. The steady

state problems of the supersonic flow along a ramp and of the cold gas hypersonic flow along a diffuser are studied.

The results have demonstrated that both schemes predict appropriately the shock angles at the ramp and at the lower and

upper walls of the diffuser, in the inviscid case. In the viscous study, only the Liou and Steffen scheme yielded

converged results, obtaining good ramp shock angles.

Keywords: Euler equations, Navier-Stokes equations, Jameson and Mavriplis algorithm, Liou and Steffen algorithm,

symmetrical and upwind schemes.

1. INTRODUCTION

The development of aeronautical and aerospace

projects requires hours of wind tunnel testing. It is

necessary to minimize such wind tunnel type of test

because of the growing cost of such tests. In Brazil,

there is a lack of wind tunnels that have the capacity

for generating supersonic flows or even highsubsonic flows. Therefore, Computational Fluid

Dynamics (CFD) techniques are receiving

considerable attention in the aeronautical industry.

Analogous to wind tunnel tests, the numerical

methods determine physical properties in discrete

points of the spatial domain. Hence, the

aerodynamic coefficients of lift, drag and

momentum can be calculated.

The Jameson and Mavriplis (1986) scheme is a

symmetrical scheme that had been widely used in

the CFD community during the 60s to 80s. It

provided a method for the numerical calculation ofthe aerodynamic parameters important to the designof airplanes and other aerospace vehicles.

Comments about this scheme can be found in

Maciel(2007).

The necessity to construct more elaborated and

more robust schemes, which allows the capture ofstrong and sharp shocks, becomes an important goal

to be achieved by first-order and high-resolution

upwind schemes. Since 1959, first-order and high-

resolution upwind schemes, which combined the

characteristics of robustness, good shock capture

properties and good shock quality, have been

developed to provide efficient tools to predict

accurately the main features of a flow field. Several

studies involving first-order and high-resolutionalgorithms were reported in the literature.

Liou and Steffen(1993) proposed a new flux vector

splitting scheme. They claimed that their scheme

was simple and its accuracy was equivalent to, and

in some cases better than, that of the Roe (1981)

scheme in solving the Euler and the Navier-Stokes

equations. The scheme was robust and converged

solutions were obtained as fast as when using the

Roe (1981) scheme. The authors proposed an

approximated definition of an advection Mach

number at the cell face, using its neighbor cell

values via associated characteristic velocities. Thisinterface Mach number was used to determine the

upwind extrapolation of the convective quantities.

In the present work, the Jameson and Mavriplis

(1986) and the Liou and Steffen(1993) schemes are

implemented in a finite volume context using an

unstructured spatial discretization to solve the Euler

and the Navier-Stokes equations in the three-

Received: 19 Feb. 2007; Revised: 3 Apr. 2007; Accepted: 5 Apr. 2007

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dimensional space applied to the steady state

physical problems of supersonic flow along a ramp

and the cold gas hypersonic flow along a diffuser.

The Jameson and Mavriplis (1986) scheme is a

symmetrical one with second-order accuracy and

therefore, an artificial dissipation operator is

required for numerical stability. Two models, basedon the work of Mavriplis (1990) and Azevedo

(1992), are implemented. On the other hand, theLiou and Steffen (1993) scheme is a flux vector

splitting one with first-order accuracy and hence

more robustness are expected. The time integration

uses a Runge-Kutta method and is second-order

accurate. Both algorithms are accelerated to thesteady state solution using a spatially variable time

step. This technique has introduced excellent gains

in terms of convergence ratio as reported in Maciel

(2005a).

An unstructured discretization of the calculationdomain is usually recommended for complex

configurations because of the ease and efficiency

with which such domains can be discretized

(Mavriplis, 1990, and Pirzadeh, 1991). However,

the issue of unstructured mesh generation will not

be studied in this work.

2. NAVIER-STOKES EQUATIONS

The Euler equations are obtained from the Navier-

Stokes equations by ignoring the viscous fluxvectors. Hence, this section presents the formulation

of the Navier-Stokes equations. These equations

can be written, according to a finite volumeformulation, in integral conservative form and on an

unstructured spatial discretization context as:

[ ] 0dS)nG(G)nF(F)nE(EQdVt

S

zveyvexve

V

=+++

(1)

where V is the volume defined by the mesh

computational cell, which corresponds to a

tetrahedron in the three-dimensional space; nx, ny

and nzare the Cartesian components of the normal

vector pointing outward of the computational cell; S

is the area of a given flux face; Q is the vector ofconserved variables; and represent the

convective flux vectors in Cartesian coordinates;

and and represent the viscous flux

vectors, also in Cartesian coordinates:

ee FE , eG

vv FE , vG

, , , (2)

=

e

wv

u

Q

+

+

=

p)u(e

uwuv

pu

u

E

2

e

+

+=

p)v(e

vwpv

uv

v

F2

e

++

=

p)w(e

pwvw

uw

w

G2

e

++

=

xxzxyxx

xz

xy

xx

v

qwvu

0

Re

1E ,

++

=

yyzyyyx

yz

yy

yx

v

qwvu

0

Re

1F and

++

=

zzzzyzx

zz

zy

zx

v

qwvu

0

Re

1G (3)

The components of the conductive heat flux vector are defined as follows:

( ) xePrdq ix = , ( ) yePrdq iy = and ( ) zePrdq iz = (4)

In these equations, the components of the viscous stress tensor are defined as:

( )zwyvxu32xu2xx ++= , ( )xvyu yx += , ( xwzuzx ) += (5)

( )zwyvxu32yv2 yy ++= , ( )zwyvxu32zw2zz ++= (6)

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( ywzvzy ) += (7)

The quantities in the equations above are described

in Maciel (2007). The molecular viscosity isestimated by Sutherlands empirical formula (Fox

and McDonald, 1988).

The Navier-Stokes equations werenondimensionalized in relation to the freestream

properties, as defined in Maciel (2007). To solvethe matrix system of five equations with five

unknowns described by Eq. (1), the state equation

of perfect gases is used:

[ )()( 222 wvu0.5e1p ++= ] (8)

3. JAMESON AND MAVRIPLIS (1986)

ALGORITHM

The spatial discretization proposed by Jameson andMavriplis (1986) expresses Eq. (1) in the three-

dimensional space as:

( ) 0)C(QdtQVd iii =+ (9)

with

( ) ( ) ( )=

=

nff

1kki,x

nki,

Qv

Eki,

Qe

Ei

QC

ki,yn

k,iQ

vF

k,iQ

eF

+

k,iSki,znk,iQvGk,iQeG +

(10)

being the discrete approximation to the flux integral

of Eq. (1). In this sum, nff represents the total

number of flux faces of the computational cell. In

the present work, the computational cells were

adopted as being tetrahedra, resulting in nff = 4.

Details of the definition of a computational cell, the

calculation of its volume, the calculation of the flux

areas, as well as the calculation of the normal to the

flux faces can be found in Maciel (2002, 2005b and

2006). In this work, Qi,kwas evaluated as:

( kiki, QQ0.5Q += ) (11)

with i,k representing the respective tetrahedron

flux face, i being the computational cell under

study and k its neighbor.

The derivatives present in Eqs. (4) to (7) are

calculated using the procedure described in Maciel

(2007) in regard to the Jameson and Mavriplis

(1986) scheme. Details are documented in Maciel

(2002, 2005b and 2006).

As noted in the work of Maciel (2002, 2005b, 2006

and 2007), the Jameson and Mavriplis (1986)

scheme is a symmetrical scheme that needs to

include an artificial dissipation operator to

guarantee numerical stability in the presence ofnonlinear stabilities and odd-even uncoupled

solutions. Two artificial dissipation models wereimplemented in this scheme: the first is based on

the work of Mavriplis(1990) and the second on that

of Azevedo(1992). Equation (9) is rewritten as:

( ) [ ] 0)D(Q)C(QdtQVd iiii =+ (12)

The time integration is performed using a hybrid

explicit Runge-Kutta method of five stages, with

second-order accuracy, and can be represented in

general form as:

( ) ([ )](k)i

1)(ni

(m)i

1)(kiiik

(0)i

(k)i

(n)i

(0)i

QQ

QDQCVtQQ

QQ

=

=

=

+

(13)

where k= 1,...,5; m= 0 until 4; and 1= 1/4, 2=

1/6, 3 = 3/8, 4 = 1/2 and 5 = 1. Jameson and

Mavriplis (1986) suggested that the artificial

dissipation operator should be evaluated in the first

two stages when the Euler equations are solved

(m = 0, k = 1 and m = 1, k = 2). Swanson and

Radespiel (1991) suggested that the artificial

dissipation operator should be evaluated in odd

stages when the Navier-Stokes equations are solved

(m= 0, k= 1; m= 2, k= 3; and m= 4, k= 5).

3.1 Artificial dissipation operator

The artificial dissipation operator implemented in

the Jameson and Mavriplis(1986) scheme to three-

dimensional simulations has the following

structure:

( ) ( ) ( )i(4)

i(2)

i QdQdQD = (14)

where

( ) ( )( )=

+=nff

1k

ikki(2)ki,i

(2) QQAA0.5Qd (15)

known as undivided Laplacian operator, is included

to establish numerical stability for the scheme in the

presence of shock waves; and

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( ) ( )(=

+=nff

1k

i2

k2

ki(4)ki,i

(4) QQAA0.5Qd )

)

(16)

named bi-harmonic operator, is added to obtain

background stability. The term

is called Laplacian of Q

( )=

=nff

1kiQkQiQ

2

i. In the d(4)operator, every

time that is related to a special boundary cell,

recognized in the literature as ghost cell, its value

is extrapolated from its real neighbor value. The sterms are defined as follows:

iQ2

( ki(2)(2)

ki, ,MAXK = and

= (2)ki,

(4)(4)ki, K0,MAX (17)

with

( )==

+=nff

1k

ik

nff

1k

iki pppp (18)

representing a pressure sensor, responsible for the

identification of high gradient regions. TheK(2)and

K(4)constants have typical values of 1/4 and 3/256

respectively. Every time that a neighbor represents

a ghost cell, it is assumed that .ik =

Two models of artificial dissipation are

implemented in the Jameson and Mavriplis(1986)

scheme. They are characterized by the different

ways in defining theAiterms. They are described asfollows:

(a) Mavriplis(1990) model

In the Mavriplis (1990) dissipation model, the Ai

terms represent contributions of the maximum

eigenvalues of the Euler equations in the normal

direction to the flux face under study, integrated

along each cell face. These terms are defined as:

=

+++=

nff

1k

ki,ki,ki,zki,ki,yki,ki,xki,iSaSwSvSuA (19)

where ui,k, vi,k, wi,k and ai,k are obtained by

arithmetical average among their values at cell i

and at cell k. a represents the speed of sound,

defined as pa= .

(b) Azevedo (1992) model

In the Azevedo (1992) dissipation model, the Ai

terms are defined as:

iii tVA = (20)

which represents a scaling factor with the desired

behavior to an artificial dissipation term: (i) bigger

cells result in a bigger value to the dissipation term;

(ii) smaller values of time steps also result in a

scaling factor of bigger values.

4. LIOU AND STEFFEN(1993) ALGORITHM

The approximation of the integral equation (1) to a

tetrahedron finite volume yields a system of

ordinary differential equations with respect to time:

iii CdtdQV = (21)

with Ci representing the net flux (residue) of

conservation of mass, linear momentum and energy

in the Vivolume. The residue is calculated as:

4321i FFFFC +++= (22)

with , where e is related to the flow

convective contribution and v is related to the

flow viscous contribution at l= 1 interface.

v1

e11 FFF =

As shown in Liou and Steffen(1993), the discrete

convective flux calculated by the AUSM scheme

(Advection Upstream Splitting Method) can be

interpreted as a sum involving the arithmetical

average between the right (R) and the left (L) states

of the l cell face, related to cell iand its neighbor

respectively, multiplied by the interface Mach

number, and a scalar dissipative term. Hence, to the

l interface:

l

z

y

x

LR

l

RL

lll

0

pS

pS

pS

0

aH

aw

av

au

a

aH

aw

av

au

a

2

1

aH

aw

av

au

a

aH

aw

av

au

a

M2

1SF

+

+

= (23)

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where defines the normal area

vector to the l surface. M

[ Tlzyxl

SSSS = ]

l defines the advection

Mach number at the l face of the i cell, which is

calculated according to Liou and Steffen(1993) as:

m

R

p

Ll

MMM += (24)

where the separated Mach numbers Mp / m are

defined by Van Leer(1982):

( )