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.

to be achieved by first-order and high-resolution

upwind schemes. Since 1959, first-order and highresolution 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-resolution
algorithms 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. This
interface 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-

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 high
subsonic 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 of
the aerodynamic parameters important to the design
of 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 of
strong and sharp shocks, becomes an important goal
Received: 19 Feb. 2007; Revised: 3 Apr. 2007; Accepted: 5 Apr. 2007
238

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

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, based
on the work of Mavriplis (1990) and Azevedo
(1992), are implemented. On the other hand, the
Liou 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 the
steady 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 calculation
domain 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.

u

Q = v ,
w

e

u
u 2 + p

Ee = uv ,
uw

(e + p)u

2. NAVIER-STOKES EQUATIONS
The Euler equations are obtained from the NavierStokes equations by ignoring the viscous flux
vectors. Hence, this section presents the formulation
of the Navier-Stokes equations. These equations
can be written, according to a finite volume
formulation, in integral conservative form and on an
unstructured spatial discretization context as:

QdV + [(Ee Ev )nx + (Fe Fv )n y + (Ge Gv )nz ] dS = 0

t V
S

(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 nz are 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 of
conserved variables; E e , Fe and Ge represent the
convective flux vectors in Cartesian coordinates;
and E v , Fv and Gv represent the viscous flux
vectors, also in Cartesian coordinates:

v
w
uv
uw

Fe = v + p , Ge = vw
vw
w2 + p

(e + p)v
(e + p)w

0
0

yx
xx
zx

1
1
1
xy
zy
yy
Ev =
, Fv =

and Gv =
Re
Re
Re

xz
zz
yz

zxu + zyv + zzw qz

xxu + xyv + xzw qx
yxu + yyv + yzw qy

(2)

(3)

The components of the conductive heat flux vector are defined as follows:
q x = ( Prd )ei x ,

q y = ( Prd )ei y

and qz = ( Prd )ei z

(4)

In these equations, the components of the viscous stress tensor are defined as:
xx = 2 u x 2 3 ( u x + v y + w z) , x y = ( u y + v x) , x z = ( u z + w x)

(5)

y y = 2 v y 2 3 ( u x + v y + w z) , z z = 2 w z 2 3 ( u x + v y + w z)

(6)

239

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

y z = ( v z + w y)

(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 of
nonlinear stabilities and odd-even uncoupled
solutions. Two artificial dissipation models were
implemented 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:

(7)

The quantities in the equations above are described

in Maciel (2007). The molecular viscosity is
estimated by Sutherlands empirical formula (Fox
and McDonald, 1988).
The
Navier-Stokes
equations
were
nondimensionalized in relation to the freestream
properties, as defined in Maciel (2007). To solve
the matrix system of five equations with five
unknowns described by Eq. (1), the state equation
of perfect gases is used:

p = ( 1) e 0.5(u 2 + v 2 + w 2 )

d (Vi Qi ) dt + [C(Qi ) D(Qi )] = 0

(8)

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:

3. JAMESON AND MAVRIPLIS (1986)

ALGORITHM
The spatial discretization proposed by Jameson and
Mavriplis (1986) expresses Eq. (1) in the threedimensional space as:
d (Vi Qi ) dt + C(Qi ) = 0

Qi(0) = Qi(n)

(9)

Qi(n +1)

( ) ( )n x i,k
(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,k was evaluated as:
Qi,k = 0.5(Qi + Qk )

) ( )]

(13)

Qi(k)

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).

nff
C Qi = E e Qi, k E Qi, k
v
k = 1

+ Fe Qi , k Fv Qi , k n y i, k

+ Ge Qi , k Gv Qi , k n z i, k S i , k

[(

Qi(k) = Qi(0) k t i Vi C Qi(k 1) D Qi(m)

with

( )

(12)

3.1

Artificial dissipation operator

The artificial dissipation operator implemented in

the Jameson and Mavriplis (1986) scheme to threedimensional simulations has the following
structure:
D (Qi ) = d (2) (Qi ) d (4) (Qi )

(14)

where

(11)

nff

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

(2)

(Qi ) = i,(2)k 0.5( Ai + Ak )(Qk Qi )

(15)

k =1

known as undivided Laplacian operator, is included

to establish numerical stability for the scheme in the
presence of shock waves; and

240

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

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 a = p .

nff

(4)

(Qi ) = i,(4)k 0.5( Ai + Ak )( 2 Qk 2Qi )

(16)

k =1

named bi-harmonic operator, is added to obtain

background stability. The term

nff
2
Q i = Q k Qi
k =1

(b) Azevedo (1992) model

(4)

is called Laplacian of Qi. In the d operator, every

time that 2 Qi is related to a special boundary cell,
recognized in the literature as ghost cell, its value
is extrapolated from its real neighbor value. The s
terms are defined as follows:

In the Azevedo (1992) dissipation model, the Ai

terms are defined as:
Ai = Vi ti

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.

(2)
(2)
i,k = K MAX (i , k ) and

(4)

(4)
(2)
i,k = MAX 0, K
i,k

(17)

with

4. LIOU AND STEFFEN (1993) ALGORITHM

nff

i =

nff

p k pi

k =1

(p

+ pi )

The approximation of the integral equation (1) to a

tetrahedron finite volume yields a system of
ordinary differential equations with respect to time:

(18)

k =1

representing a pressure sensor, responsible for the

identification of high gradient regions. The K(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 k = i .
Two models of artificial dissipation are
implemented in the Jameson and Mavriplis (1986)
scheme. They are characterized by the different
ways in defining the Ai terms. They are described as
follows:

Vi dQi dt = Ci

Ci = F1 + F2 + F3 + F4

u
k =1

i,k S x i,k

+ vi,k S y i,k + wi,k S z i,k + ai,k S i,k

(22)

with F1 = F1e F1v , where e is related to the flow

convective contribution and v is related to the
flow viscous contribution at l = 1 interface.
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 i and its neighbor
respectively, multiplied by the interface Mach
number, and a scalar dissipative term. Hence, to the
l interface:

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

(21)

with Ci representing the net flux (residue) of

conservation of mass, linear momentum and energy
in the Vi volume. The residue is calculated as:

(a) Mavriplis (1990) model

Ai =

(20)

(19)

a
a
a 0
a

au S p

au

au
au
x

1
1
Fl = S l M l av + av l av av + S y p
2

2 aw
aw
aw
aw S z p

aH 0
aH
aH
aH

L
R
R
L
l

241

(23)

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

where Sl = [S x S y S z ] Tl defines the normal area

5. SPATIALLY VARIABLE TIME STEP

AND INITIAL AND BOUNDARY
CONDITIONS

vector to the l surface. Ml defines the advection

Mach number at the l face of the i cell, which is
calculated according to Liou and Steffen (1993) as:
Ml =

M Lp

M Rm

The spatial variable time step procedure is

described briefly in Maciel (2007) and in detail in
Maciel (2002, 2005b and 2006). Comments in these
works are valid for the initial and boundary
conditions.

(24)

where the separated Mach numbers M p / m are

defined by Van Leer (1982):
if M 1 ;
M,
2

M = 0.25(M + 1) , if M < 1 ;
0,
if M 1 ;
M

6. RESULTS

and

if M 1 ;
0,
2

= - 0.25(M 1) , if M < 1 ;
M,
if M 1 ;

Tests were performed by using a microcomputer

with the processor AMD ATHLON XP 2600+
running at 1.91 GHz and 512 Mbytes of RAM
memory. Converged results occurred at four orders
of reduction in the maximum residue value. The
downstream and longitudinal plane angles were set
to 0.0. In the numerical experiments performed in
this work, the Azevedo (1992) model was used.
The ramp and diffuser configurations in the xy
plane are described in Fig. 1 and Fig. 2. The
spanwise length of the ramp is 0.25 m, while that of
the diffuser is 0.10 m.

(25)

ML and MR represent the Mach number associated

with the left and right states respectively. The
advection Mach number is defined by:

M = S xu + S y v + S z w

) ( S a)

(26)

The pressure at the l face of the i cell is

calculated in a similar way:
pl = pLp + p Rm

(27)

with p p / m denoting the pressure separation defined

according to Van Leer (1982):
if M 1 ;
p,

2
P = 0.25p (M + 1) (2 M ) , if M < 1 ; and
0,
if M 1 ;
p

if M 1 ;
0,

2
P = 0.25p (M 1) (2 + M ) , if M < 1 ;
p,
if M 1 ;
m

(28)

The definition of the dissipative term determines

the particular formulation of the convective fluxes.
According to Radespiel and Kroll (1995), the choice
below corresponds to the Liou and Steffen (1993)
scheme:
l = lLS , with lLS = M l

Fig. 1

(29)

The time integration employs the Runge-Kutta

method described by Eq. (13). The viscous terms
are implemented in the same way as in the Jameson
and Mavriplis (1986) scheme. The Liou and Steffen
(1993) scheme presented in this work is first-order
accurate in space.
242

Ramp configuration in the xy plane.

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

and Mavriplis (1986) scheme presents good

symmetry properties. The Liou and Steffen (1993)
solution presents good symmetry characteristics
without non-homogeneity problems. It is a better
solution when compared to the Jameson and
Mavriplis (1986) scheme, mainly because of its
characteristics of shock capturing, typical of flux
vector splitting and flux difference splitting
schemes.
Fig. 2

Diffuser configuration in the xy plane.

The meshes employed in this work were generated

on a structured context, obtaining cells of
hexahedra. Through the generation of connectivity,
neighboring and node-coordinate tables, such
meshes were transformed into meshes of tetrahedra.
Although this procedure of mesh generation does
not produce meshes with the best spatial
discretization, meshes with reasonable quality were
obtained for the present study. The ramp and
diffuser meshes were generated in the xy plane and
projected in the z direction, in xy planes parallel
to the original xy plane. The computational data of
the generated meshes are presented in Table 1.

Fig. 3

Density contours (JM).

Fig. 4

Density contours (LS).

Table 1 Data of the unstructured meshes.

Inviscid

Viscous

Ramp

Diffuser

Ramp

Diffuser

61x50x10

61x41x10

61x60x10

61x51x10

Cells

158,760

129,600

191,160

162,000

Nodes

30,500

25,010

36,600

31,110

6.1

Inviscid simulations

The first problem to be analyzed is the ramp

problem. The freestream Mach number adopted as
initial condition to this simulation was 5.0,
characterizing a supersonic flow.
Figures 3 and 4 exhibit the density contours
obtained by the Jameson and Mavriplis (1986) and
the Liou and Steffen (1993) schemes respectively.
The shock is well captured by both schemes. In
Fig. 3, a loss of homogeneity in the density
contours is noted, which affects the solution
symmetry in the parallel planes. However, this
phenomenon is typical of unstructured solutions and
does not suggest errors in the algorithm
implementation and in the algorithm solution. Even
with this characteristic, the solution of the Jameson

Figures 5 and 6 show the pressure contours

obtained by the Jameson and Mavriplis (1986) and
the Liou and Steffen (1993) schemes respectively.
The loss of homogeneity is again observed in the
Jameson and Mavriplis (1986) scheme although
better symmetry properties are demonstrated. The
Liou and Steffen (1993) scheme presents again
good homogeneity and symmetry properties. The
pressure field generated by the Jameson and
243

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

Mavriplis (1986) scheme is more severe than that

generated by the Liou and Steffen (1993) scheme,
indicating that the former is a more conservative
algorithm.

Fig. 5

Fig. 6

Fig. 7

Mach contours (JM).

Fig. 8

Mach contours (LS).

Fig. 9

Cp distributions at ramp (JM and LS).

Pressure contours (JM).

Pressure contours (LS).

Figures 7 and 8 exhibit the Mach number contours

generated by both schemes. The Jameson and
Mavriplis (1986) scheme presents a post-shock
oscillation at the ramp end. This scheme is secondorder accurate in space and solution oscillations can
occur immediately before or after the shock. The
symmetry properties are excellent, being kept
constant the Mach number contours at the
z = constant planes and at the ramp, without loss of
homogeneity. The Liou and Steffen (1993) scheme
presents again good symmetry and homogeneity
properties, as expected. It is free of pre- or postshock oscillations. The Mach number field
generated by the Jameson and Mavriplis (1986)
scheme is more intense than the one generated by
the Liou and Steffen (1993) scheme.

Figure 9 shows the Cp distributions obtained by

the Jameson and Mavriplis (1986) and the Liou and
244

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

Steffen (1993) schemes at z = 0. They are compared

with the results of the oblique shock wave theory
and Prandtl-Meyer expansion theory. As observed,
both schemes present smaller width of the pressure
plateau (shock plateau) in comparison with the
expansion theory. The Jameson and Mavriplis
(1986) scheme presents oscillations at the plateau,
typical of second-order schemes, while the Liou and
Steffen (1993) scheme is free of oscillations and
presents a pressure plateau closer to the theory. The
expansion pressure obtained by the Liou and
Steffen (1993) scheme is closer to the theory results
than the Jameson and Mavriplis (1986) one, which
presents oscillations.
To the cold gas hypersonic inviscid flow along
the diffuser, the freestream Mach number adopted
as initial condition was 10.0.
Figures 10 and 11 exhibit the density contours
obtained by the Jameson and Mavriplis (1986)
scheme and the Liou and Steffen (1993) scheme
respectively. Good homogeneity and symmetry

properties are observed in the Jameson and

Mavriplis (1986) solution. The interference between
the upper and lower shock waves is well captured
by this scheme. The Liou and Steffen (1993)
scheme also presents good homogeneity and
symmetry properties. It detects appropriately the
shock interference as well.
Figures 12 and 13 show the pressure contours
obtained by the Jameson and Mavriplis (1986) and
the Liou and Steffen (1993) schemes respectively.
In the Jameson and Mavriplis (1986) solution, the
symmetry of the pressure contours is well
characterized at the z = constant planes, presenting
a small discrepancy at z = 0. The shock interference
is well highlighted. In the Liou and Steffen (1993)
scheme, homogeneity and symmetry properties are
well highlighted and so is the shock interference.
The pressure field generated by the Jameson and
Mavriplis (1986) scheme is again more severe than
the one generated by the Liou and Steffen (1993)
scheme.

Fig. 10 Density contours (JM).

Fig. 11 Density contours (LS).

Fig. 12 Pressure contours (JM).

Fig. 13 Pressure contours (LS).

245

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

wall. They are again compared to the results of the

oblique shock wave theory and Prandtl-Meyer
expansion theory. As observed, both schemes
present smaller width of the pressure plateau (shock
plateau) in comparison with the theory. Both
schemes present oscillations at the plateau although
their solutions are closer to the theory. The
expansion pressure obtained by the Liou and
Steffen (1993) scheme is closer to the theory than
that obtained by the Jameson and Mavriplis (1986).

Fig. 14 Mach contours (JM).

Fig. 16 Cp distributions at diffuser (JM and LS).

One way to verify quantitatively whether the

solutions generated by each scheme are satisfactory
is to determine the shock angle of the oblique shock
wave, , measured in relation to the initial direction
of the flow field. Anderson (1984) (pages 352 and
353) presents a diagram with values of the shock
angle, , to oblique shock waves. The value of this
angle is determined as a function of the freestream
Mach number and of the deflection angle of the
flow after the shock wave, . In regard to the ramp
problem, where = 20 (ramp inclination angle)
and the freestream Mach number is 5.0, a value of
=30.0 is noted from the diagram. As for the
diffuser problem, where = 20 (diffuser
inclination angle) and the freestream Mach number
is 10.0, noted from this diagram is a value of
=26.0. Using a transfer in Fig. 5 and Fig. 6, to the
ramp problem, and in Fig. 12 and Fig. 13, to the
diffuser problem, considering the xy plane, it is
possible to obtain the values of and the respective
error percentages for each scheme in each case, as
shown in Table 2. The findings demonstrate that the
schemes present practically equivalent accuracy.

Fig. 15 Mach contours (LS).

Figures 14 and 15 exhibit the Mach number

contours obtained by the Jameson and Mavriplis
(1986) and the Liou and Steffen (1993) schemes
respectively. In the Jameson and Mavriplis (1986)
solution, there is no pre- or post-shock oscillations,
as observed in the ramp problem. The homogeneity
and symmetry properties observed in Fig. 14 are not
as good as those observed in Fig. 10 and Fig. 12;
nevertheless, they are still conspicuous. On the
other hand, the Liou and Steffen (1993) scheme
presents excellent homogeneity and symmetry
characteristics. The Mach number field generated
by the Jameson and Mavriplis (1986) scheme is
again more intense.
Figure 16 shows the Cp distributions obtained by
the Jameson and Mavriplis (1986) and the Liou and
Steffen (1993) schemes at z = 0 at the lower diffuser
246

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

Table 2 Shock angles and percentage errors.

JM

LS

Diffuser
(Lower Wall)
JM
LS

()

30.0

29.6

26.4

26.0

26.0

26.0

Error (%)

0.00

1.33

1.54

0.00

0.00

0.00

Ramp

6.2

Diffuser
(Upper Wall)
JM
LS

Figures 17 and 18 exhibit the density contours

obtained by the Jameson and Mavriplis (1986) and
the Liou and Steffen (1993) schemes respectively.
Good homogeneity properties are observed far from
the ramp wall in the Jameson and Mavriplis (1986)
solution. At the ramp, a loss of homogeneity
characteristics is evident and the symmetry
properties in the z = constant planes are also
damaged. In the Liou and Steffen (1993) solution,
the homogeneity and symmetry properties are
reasonable. The shock is well defined and effects of
excessive dissipation due to viscous contributions,
which would tend to smooth the shock, are not
perceptible.
Figures 19 and 20 show the pressure contours
obtained by the Jameson and Mavriplis (1986) and
by the Liou and Steffen (1993) schemes
respectively. In the Jameson and Mavriplis (1986)
solution, the homogeneity and symmetry properties
are damaged. Moreover, the shock is not developed
at the ramp. The Liou and Steffen (1993) solution
does not present pre- or post-shock oscillations.
Good symmetry and homogeneity properties are
observed. The shock is fully developed.

Viscous simulations

For the viscous supersonic flow along the ramp, the

same freestream Mach number of the inviscid
simulation was adopted as initial condition. In
consideration of the ramp height, the characteristic
configuration length being equal to 0.0437 m and a
flight altitude of 20,000 m, the Reynolds number is
estimated to be 403,140, according to Eq. (8) and
the study of Fox and McDonald (1988).
The Jameson and Mavriplis (1986) scheme did not
present converged results to this and lower Mach
number values. The residue stops to converge after
5,000 iterations in the present case. Research
findings show a reduction of one order of
magnitude in the residue value.

Fig. 17 Density contours (JM).

Fig. 18 Density contours (LS).

Fig. 19 Pressure contours (JM).

Fig. 20 Pressure contours (LS).

247

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

Figures 21 and 22 exhibit the Mach number

contours obtained by both schemes. The Jameson
and Mavriplis (1986) solution, as shown in Fig. 21,
presents a smooth shock as a result of viscous
effects. There is no pre- or post-shock oscillations.
In the Liou and Steffen (1993) solution, a certain
loss of homogeneity and symmetry can be seen. As
observed, the Mach number contours at the ramp
surface do not present the tonality related to the
minimum field Mach number, which means that the
adherence condition is not fully satisfied. However,
the shock is well captured.
For the viscous cold gas hypersonic flow along
the diffuser, the same freestream Mach number of
the inviscid simulation was adopted as initial
condition.
Considering
the
characteristic
configuration length being equal to 0.14 m, the
diffuser entrance height, and a flight altitude of
40,000 m, the Reynolds number is estimated to be
110,880.

The Jameson and Mavriplis (1986) scheme did not

present again converged results to this and lower
Mach number values. The problem occurred in the
ramp scenario was also noted in this case. Findings
show a reduction of two orders of magnitude in the
residue value.
Figures 23 and 24 show the density contours
obtained by the Jameson and Mavriplis (1986) and
the Liou and Steffen (1993) schemes respectively.
The shock interference is well detected by the
Jameson and Mavriplis (1986) scheme. Good
homogeneity and symmetry properties are also
observed. The Liou and Steffen (1993) scheme also
presents good homogeneity and symmetry
properties. The shock interference is fully solved.

Fig. 23 Density contours (JM).

Fig. 21 Mach contours (JM).

Fig. 24 Density contours (LS).

Figures 25 and 26 exhibit the pressure contours

obtained by the Jameson and Mavriplis (1986) and
the Liou and Steffen (1993) schemes respectively.
The homogeneity and symmetry properties are not
observed in the z = constant and y = constant planes

Fig. 22 Mach contours (LS).

248

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

of the field, highlighting the fact that the adherence

condition is satisfied. The shock interference is well
detected by this scheme.

in the Jameson and Mavriplis (1986) solution. Even

so, the shock interference is captured. On the other
hand, the Liou and Steffen (1993) solution presents
good symmetry and homogeneity characteristics.
Moreover, the shock interference is well captured
and the solution and is free of pre- or post-shock
oscillations.

Fig. 27 Mach contours (JM).

Fig. 25 Pressure contours (JM).

Fig. 28 Mach contours (LS).

Figures 29 and 30 on page 250 exhibit the Cp

distributions generated by the Liou and Steffen
(1993) scheme at the ramp and at the diffuser lower
wall respectively. Findings show that only the Liou
and Steffen (1993) scheme gives converged
solutions. In Fig. 29, the pressure plateau is
practically characterized, with a small oscillation at
x = 0.19 m. In Fig. 30, the shock is detected, as well
as the expansion fan. However, the characteristic
pressure plateau is not formed. A general smoothing
as a result of viscous effects is observed in this Cp
distribution. The effects are more pronounced as the
Reynolds number decreases.

Fig. 26 Pressure contours (LS).

Figures 27 and 28 show the Mach number contours

obtained by both schemes. The Jameson and
Mavriplis (1986) solution (Fig. 27 refers) presents
good characteristics of homogeneity and symmetry.
At the diffuser entrance, an oscillation is observed.
The Mach number contours at the upper and lower
walls assume the minimum possible values of the
field, which means that the adherence condition is
satisfied. The Liou and Steffen (1993) solution
again presents good symmetry and homogeneity
properties. Moreover, the major regions of the walls
assume the minimum possible Mach number values
249

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

Table 4 exhibits the numerical data of the

simulations and the computational costs of the
schemes. The Jameson and Mavriplis (1986)
scheme is 61% cheaper than the Liou and Steffen
(1993) scheme in the inviscid case and 3.3% in the
viscous case.

Table 3 shows the shock angles to the ramp and the

diffuser in the viscous case. The data is obtained
from the Liou and Steffen (1993) scheme. It is
noted that as the viscous effects intensify, owing to
lower Reynolds numbers, the error percentages in
the shock wave geometry increases. An interesting
evaluative research would be to study shock wave
geometry in response to chemical reactions.

Fig. 29 Cp distribution at ramp (LS).

Fig. 30 Cp distribution at diffuser (LS).

Table 3 Shock angles and percentage errors with LS scheme.

Table 4 Numerical data of the simulations.

()

29.6

Diffuser
(Lower Wall)
21.0

Error (%)

1.33

19.23

Ramp

Diffuser
(Upper Wall)
21.0
19.23

Ramp

Diffuser

Inviscid

CFL

Iterations

CFL

Iterations

Cost*

JM

0.8

232

0.2

960

0.0000131

LS

0.3

575

0.3

720

0.0000211

Viscous

CFL

Iterations

CFL

Iterations

Cost*

JM

0.1

0.1

0.0000360

LS

0.2

1,037

0.1

4,981

0.0000372

* Given in seconds/per cell/per iteration.

vector splitting first-order scheme. The former

needs artificial dissipation to guarantee numerical
stability, while the later is an upwind scheme. Both
schemes use a second-order Runge-Kutta method of
five stages to perform time integration. The steady
state physical problems of the supersonic flow
along a ramp and of the cold gas hypersonic flow
along a diffuser are studied. A spatially variable
time step is employed to accelerate the convergence
to the steady state. The results have demonstrated
that both schemes predict appropriately the shock
angles at the ramp and the two shock angles at the

7. CONCLUSIONS
In the present work, the Jameson and Mavriplis
(1986) and the Liou and Steffen (1993) unstructured
algorithms are applied to solve the Euler and the
Navier-Stokes equations in three-dimensions. The
governing equations in conservative and integral
forms are solved, employing a finite volume
formulation
and
an
unstructured
spatial
discretization. The Jameson and Mavriplis (1986)
algorithm is a symmetrical second-order scheme,
while the Liou and Steffen (1993) scheme is a flux
250

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

7. Maciel ESG (2005a). Analysis of Convergence

Acceleration Techniques Used in Unstructured
Algorithms in the Solution of Aeronautical
Problems Part I. Proceedings of the XVIII
International
Congress
of
Mechanical
Engineering (XVIII COBEM), Ouro Preto, MG,
Brazil.
8. Maciel ESG (2005b). Relatrio ao Conselho
Nacional de Pesquisa e Desenvolvimento
Tecnolgico (CNPq) sobre as Atividades de
Pesquisa Desenvolvidas no Segundo Ano de
Vigncia da Bolsa de Estudos para Nvel DCRIF Referente ao Processo n304318/2003-5.
Report, National Council of Research and
Technological Development (CNPq). Brazil,
August, 54p.
9. Maciel ESG (2006). Relatrio ao Conselho
Nacional de Pesquisa e Desenvolvimento
Tecnolgico (CNPq) sobre as Atividades de
Pesquisa Desenvolvidas no Terceiro Ano de
Vigncia da Bolsa de Estudos para Nvel DCRIF Referente ao Processo n304318/2003-5.
Report, National Council of Research and
Technological Development (CNPq). Brazil,
August, 52p.
10. Maciel ESG (2007). Turbulent Flow
Simulations Using the MacCormack and the
Jameson and Mavriplis Algorithms Coupled
with the Cebeci and Smith and the Baldwin and
Lomax
Models
in
Three-Dimensions.
Engineering Applications of Computational
Fluid Mechanics 1(3):147163.
11. Mavriplis DJ (1990). Accurate Multigrid
Solution of the Euler Equations on
Unstructured and Adaptive Meshes. AIAA
Journal 28(2):213221.
12. Pirzadeh S (1991). Structured Background
Grids for Generation of Unstructured Grids by
Advancing Front Method. AIAA Paper
91-3233-CP.
13. Radespiel R and Kroll N (1995). Accurate Flux
Vector Splitting for Shocks and Shear Layers.
Journal
of
Computational
Physics
121(1):6678.
14. Roe PL (1981). Approximate Riemann Solvers,
Parameter Vectors, and Difference Schemes.
Journal
of
Computational
Physics
43(2):357372.
15. Swanson RC and Radespiel R (1991). Cell
Centered and Cell Vertex Multigrid Schemes
for the Navier-Stokes Equations. AIAA Journal
29(5):697703.

lower and upper walls of the diffuser, when

compared with the theory, in the inviscid case. The
pressure fields generated by the Jameson and
Mavriplis (1986) scheme, in both examples, are
more severe than those generated by the Liou and
Steffen (1993) scheme, although the later predicts
better pressure distributions along the wall in both
cases. In the viscous study, only the Liou and
Steffen (1993) scheme yielded converged results.
The shock angle at the ramp is well captured by this
scheme.
In conclusion, the Liou and Steffen (1993) scheme,
being a first-order accurate one, was the best
algorithm in this study. In the inviscid case, this
scheme provided solutions as accurate as those
obtained from the Jameson and Mavriplis (1986)
scheme, a second-order accurate one, in terms of
the shock angle of the oblique shock waves and in
terms of the wall pressure distributions in both
examples. In the viscous case, only the Liou and
Steffen (1993) scheme yielded converged
solutionsa contrast to the behavior observed with
the Jameson and Mavriplis (1986) schemeand
reasonable good values to the shock angles of the
oblique shock waves, as well as good pressure
distributions in both examples.
REFERENCES
1. Anderson JD (1984). Fundamentals of
Aerodynamics. McGraw-Hill, Inc., EUA, 563p.
2. Azevedo JLF (1992). On the Development of
Unstructured Grid Finite Volume Solvers for
High Speed Flows. Report NT-075-ASE-N,
IAE, CTA, So Jos dos Campos, SP, Brazil.
3. Fox RW and McDonald AT (1988). Introduo
Mecnica dos Fluidos. Ed. Guanabara
Koogan, Rio de Janeiro, RJ, Brazil, 632p.
4. Jameson A and Mavriplis DJ (1986). Finite
Volume Solution of the Two-Dimensional
Euler Equations on a Regular Triangular Mesh.
AIAA Journal 24(4):611618.
5. Liou M and Steffen Jr. CJ (1993). A New Flux
Splitting Scheme. Journal of Computational
Physics 107(1):2339.
6. Maciel ESG (2002). Simulao Numrica de
Escoamentos Supersnicos e Hipersnicos
Utilizando Tcnicas de Dinmica dos Fluidos
Computacional. Doctoral Thesis, ITA, CTA,
So Jos dos Campos, SP, Brazil, 258p.

251

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

16. Van Leer B (1982). Flux-Vector Splitting for

the Euler Equations. Proceedings of the 8th
International Conference on Numerical
Methods in Fluid Dynamics. E. Krause, Editor,
Lecture Notes in Physics, 70:507512,
Springer-Verlag, Berlin.

252