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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: [email protected]
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
QDQCVtQQ
=
=
=
+
(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):
( )