Table Of Content1:_t i 7 +L
AIAA 96-0764
High Resolution Euler Solvers
Based on the Space-Time Conservation
Element and Solution Element Method
_X(..y. W&l'i_g, C.Y. Chow
University of Color.do
Boulder, CO
and
S.C. Chang
NASA Lewis
Cleveland, OH
34th Aerospace Sciences
Meeting & Exhibit
January 15-18, 1996 / Reno, NV
'1 ' I II IIIn _ I ill I '
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
370 L'Enfant Promenade, S.W.,-Washington, D.C. 20024
tlIGtt RESOLUTION EULER SOLVERS BASED ON THE SPACE-TIME
CONSERVATION ELEMENT AND SOLUTION ELEMENT METHOD
Xiao-Yen Wang,* Chuen-Yen Chowt
Department of Aerospace Engineering, University of Colorado, Boulder, CO 80309
Sin-Chung Changl
NASA Lewis Research Center, Cleveland, OH 44135
Abstract variables and their spatial derivatives are consid-
ered as individual unknowns to be solved simultane-
The I-D, quasi 1-D and 2-D Euler solvers based
ously at each grid point. Third, no approximation
on the method of space-time conservation element
techniques other than Taylor's series expansion, no
and solution element are used to simulate various
monotonous constraints, and no characteristic-based
flow phenomena including shock waves, Mach stem,
techniques are used in the design of the scheme. De-
contact surface, expansion waves, and their inter-
tailed description of this method and the accompa-
sections and reflections. Seven test problems are
nied analysis are referred to Refs. 1and 2.
solved to demonstrate the capability of this method
Various efficient numerical schemes based on the
for handling unsteady compressible flows in various
CE/SE method have been developed for solving dif-
configurations. Numerical results so obtained are
ferent flow problems, especially the problems in the
compared with exact solutions and/or numerical so-
presence of shock waves with discontinuous flow
lutions obtained by schemes based on other estab-
properties. Of those schemes, the one- and two-
lished computational techniques. Comparisons show
dimensional time marching Euler solvers are em-
that the present Euler solvers can generate highly
ployed here to solve problems involving flow phe-
accurate numerical solutions to complex flow prob-
nomena that are more complex than those shown in
lems in a straightforward manner without using any
Refs. 1and 2. In addition, a quasi one-dimensional
ad hoc techniques in the scheme.
Euler solver is constructed in this work which is
aimed at dealing with problems in axisymmetric con-
1. Introduction
figurations. Comparisons of the computed results
The method of space-time conservation element with published data are made to demonstrate the
and solution element (to be abbreviated as the simplicity and accuracy of the present Euler solvers.
CE/SE method) is a new numerical method devel- Detailed description of the governing equations
oped by Chang for solving conservation laws. 1-4 It and CE/SE l-D, quasi l-D, 2-D Euler solvers used
is different in both concept and methodology from in the following numerical tests is referred to Ref. 3,
the well-established traditional methods such as the while numerical results and discussions are described
finite difference, finite volume, finite element and in details as follows.
spectral methods. It is designed from a physicist's
perspective to overcome several key limitations of 2. Numerical Results
the traditional methods.
2.1) 1-D shock-tube problems
Simplicity, generality and accuracy are weighted in
the development of the present method while satis- In the 1-D weighted-average Euler a-e scheme used
fying the fundamental computational requirements. in the numerical tests for the following four shock-
Its salient properties are summarized briefly as fol- tube problcms, the parameter e is set as 0.5, whilc
lows. First, the concepts of conservation clement c_is l excel)t in the first problem where c_is 4.
and solution element are introduced to enforce both
local and global flux conservation in space and time a) The Lax problem
instead of in space only. Second, all the dependent The initial conditions in the region [-8, 6] on the
x axis are defined as
*Graduate Research Assistant.
tProfessor, AIAA Associate Fellow.
p1=0.445, u1=0.6989, pt =3.5277 x<0 (1)
!Senior Scientist.
Copyright (_)1996 American Institute of Aeronautics and As-
tronautics, Inc. All rights reserved. p,.--0.5, u_=0.0, p_ =0.571 x>0 (2)
Thenumericaslolutionat
I = 100At obtained by Pr = 1+ 0.2sin 5x, u_ = 0, p_ = 1 otherwise (6)
using the present scheme, under the same compu-
This problem does not have an exact solution.
tational conditions (CFL=0.95 and Ax = 0.1) as
Several upwind schemes have been used to solve this
those used in Ref. 5, is shown with the exact solu-
problem to compare their abilities in resolving the
tion represented by solid lines in Fig. 1. Compar-
peaks appeared in the solution. 9 The CE/SE solu-
isons were made by Harten 5to appraise four numeri-
tion at t=l.8 obtained by using 800 grid points with
cal schemes, namely, the ROE, LW (Lax-Wendroff),
ULT1, and ULTIC schemes, which were used to At = 0.0015 (CFL=0.582) is shown in Fig. 3. The
present solution is comparable to those obtained in
solve the same problem. The last two are the TVNI
Ref. 9 by using the TVD1 and TVD2 schemes with
finite-difference schemes designed by Harten. The
the same number of grid points.
numerical results plotted in Figs. 2(a)-(d) of Ref. 5
are not reproduced here. Among the four solutions, d) The Woodward-Colella problem
the LW solution is the worst showing serious oscil- This problem, concerning the interaction of two
lations near the shock discontinuity. In the ROE blast waves in a close-ended tube, was proposed by
solution, the oscillations disappear but up to four Woodward and Colella without an exact solution.I°
grid points are needed to resolve a shock wave, while The initial conditions are
a significant smear is found either near the contact
surface or within the expansion fan. Much improve- Pt = 1.0, ut=0, Pl= 1000 x<0.1 (7)
ment is revealed in the ULT1 solution, which re-
solves a shock discontinuity with only 2 grid points Pm = 1.0, u,,,=O, p,,=0.01 0.1 <x<0.9 (8)
and shows an excellent agreement with the exact so-
p,.= 1.0, u,. =0, p,. = 100 0.9<x< 1.0 (9)
lution except in the diffusive region near the contact
surfade. The ULT1 solution closely resembles the The two clads are at. x = 0 and x = 1where the re-
solution plotted in Fig. 1 obtained by use of the flecting boundary conditions are imposed. Detailed
present Euler solver. Iiarten showed that a further reflecting boundary conditions used in the present
refinement can be obtained by using the higher order scheme are referred to Ref. 4. The CE/SE so-
accurate ULT1C scheme. lution at t=0.038 based on 800 grid points with
At = 1.25 x 10-5(CFL=0.3524) is shown in Fig. 4.
b) The Sjogreen problem The flowfield at. t=0.0a8 contains three contact sur-
This problem is taken from Ref. 6, whose initial
faces and two shock waves. It can be seen that the
conditions are
contact surfaces are much smeared than the shock
discontinuities. A comparison with the numerical
Pt= 1.0, u_=-2, pl=0.4 0<x<0.5 (3)
solutions obtained by using AUSM +, Roe, Van leer,
pr = 1.0, ur ---- 2, Pr = 0.4 0.5 < X < 1.0 (4) AUSMDV and AUSM+-w splitting schemes for the
same problem shown in Fig. 5of Ref. 11reveals that
The initial velocity discontinuity causes two rar-
the CE/SE solution is at least of the same accuracy.
efaction waves to propagate in opposite directions,
It has been demonstrated that the present 1-D Eu-
leaving in between a region of high vacuum. It
let solver can generate highly accurate solutions to
was mentioned 7that several Godunov-type schemes
shock-tube problems involving various discontinu-
failed in this problem due to the extremely low pres-
ities, even though it does not need the implement
sure in the middle region. The CE/SE solution at
of monotonous restraints, TVNI, and entropy con-
t = 50At based on 100 grid points and At = 0.002
is shown in Fig. 2, in which the exact solution is ditions as did in Ref. 5. This simple scheme can
represented by solid lines. It can be seen that the be used without difficulty to solve any 1-D problems
governed by Euler equations.
present solution agrees very well with the exact so-
lution, without showing negative pressure values in
2.2) A quasi 1-D nozzle flow
the middle region. The solution displays an accu-
racy which is comparable to that obtained by Xu et An axisymmetric nozzle with cross-sectional area
al. r using a gas-kinetic scheme with 200 grid points. A(x) = 1.398 + 0.347tanh(0.Sx - 4) described in
Refs. 5 and 12 is reconsidered here. Numerical solu-
c) The Shu-Osher problem
Examined in this problem is the interaction of a tions obtained by use of the quasi I-D Euler solver
for CFL=0.9 with 20 and 32 grid points are shown
moving shock of M, = 3 with a sinusoidal density
wave. sThe initial conditions in the region [-5, 5] are in Figs. 5 and 6, respectively. The present solution
described as with 20 grid points is at least as good as those ob-
tained by ROE and ULT1 schemes shown in Fig. 6
pt=3.857, u1=2.629, pl= 10.333 x<-4 (5) of Ref. 5 with the same number of grid points, wlfile
thepresenstolutionwith32gridpoint.sis better
method with numerical techniques for controlling ar-
thantheULT1solutionwith50gridpoints.
tificial compressibility and dissipation. The flowfield
involves complicated phenomena including vortex,
2.3) 2-D supersonic flow past a step blast wave, rarefaction wave, normal shock and their
mutual interactions. The early time development of
Consider the supersonic channel flow past a step
vortex and shock diffraction and the subsequent flow
depicted in Fig. 7(a). The flow exhibits complicated
evolution were simulated in Ref. 15 up to 1.5 msec
phenomena which include Mach stem, slip surface,
to show a fair comparison with experimental data.
shock wave, expansion fan, and their interactions
This problem is solved here on cartesian coordinates
and refections. This is a standard benchmark prob-
using the CE/SE method to demonstrate its versa-
lem in the literature. It was used to test ttarten's
tility.
TVNI ULT1C scheme, s Giannakouros and Karni-
The two-dimensional shock tube configuration
adakis' spectral element-FCT method, lu and Van
adopted for numerical computation is depicted in
Leer's ultimate conservative difference scheme, a4 It
Fig. 9, in which the blank space is used to represent
was also adopted by Woodward and Colella 1° to
the solid tube wall above the plane of symmetry on
compare the accuracy of different numerical meth-
the x axis. Displayed schematically in the figure are
ods in handling a shock discontinuity. This problem
some representative grid points. A shock wave is
is solved again here to demonstrate the robustness
created by the sudden removal of a diaphragm at
of the CE/SE method.
the lip of the tube which separates a compressed
The 2-D weighted-average Euler a-c scheme with
fluid in region 2 inside the tube from the surround-
-- 0.5 and c_= 1is used in this numerical test. The
ing stagnant fluid in region I. The initial conditions
grid distribution shown in Fig. 7(b) indicates that
are described by
no grid point is placed at the upper corner of the
step to avoid the singular flow behavior there. The Pl = 1/1.4, Pl = 1.0, Ul =0.0, Vl =0.0 (10)
freestream condition is set at the inlet, while the
condition at the exit is extrapolated from the inte- P2=2.443, p2=2.28, u2=0.982, v2=0.0 (ll)
rior. The reflective boundary condition is imposed
The 2-D Euler solver with e = 0.5 and c_ = 1
on solid walls. Detailed description of boundary con-
ditions can be found in Ref. 3. is used to compute the flowfield with At = 0.0025
on a mesh of 49x97 grid points. The nonreflective
The density contours in the solutions obtained by
boundary condition is set at the inlet, outlet, and
the present Euler solver with 61x21, 121x41 and
upper boundary of the computational domain, while
241x81 grid points are shown in Fig. 8. Similar
the reflective boundary condition is set on the x axis
contour plots are displayed in Figs. 7(a)-(g) of Ref.
and at all tube walls.
10based on six selected numerical schemes. Accord-
Qualitative agreements between computed and
ing to Ref. 10, the ranking of these six methods in
measured positions of the vortex center (estimated
terms of accuracy is as follows: PPM(both PPMLR
and PPMDE), MUSCL, ETBFCT, BBC, MacCor- from Fig. 15 of Ref. 15) at six time levels are
mack's scheme, and Goduno_c's scheme. revealed in Fig. 10. A quantitative comparison is
Comparisons under the same computational con- not feasible due to the fact that the experiment was
performed in an axisymmetric configuration whereas
ditions (CFL=0.8, At = 0.0025 for 241x81 grid)
the computation was in 2-D. However, at an early
show that the present solution is as good as those
obtained by the accurate MUSCL and PPMDE stage the measured vortex positions are correctly
schemes and is much better than Godunov's solu- predicted by the 2-D code.
The velocity vector fields at six time levels are
tion. A direct comparison cannot be made with
other mentioned schemes because of their different plotted in Fig. ll to show the formation and move-
ment of both the vortex and blast wave. The vortex
time step sizes and CFL numbers. The Mach stem,
triple point, slip surface, expansion fan at the cor- can be recognized by its characteristic revolving flow
ner, and the interaction between the reflected shock pattern and the blast wave is represented by a shal-
low banded curve. The computed flow features can
with rarefaction waves are accurately simulated in
the present numerical solutions. be detected in photographs taken at different time
steps. 1_Our numerical results with 4753 grid points
uniformly distributed in the computational domain
2.4) A 2-D blast flowfield
are comparable to those reported in Ref. 15based on
Considered here is a blast flowfield generated by a finite volume method containing 7377 cells, with
an open-ended cylindrical shock tube, which was much smaller cells densely distributed in the neigh-
simulated in Ref. i5 using a TVD finite volume borhood of the tube exit..
Afterthistest,the2-Dblastflowfieldgenerated
without requiring any special treatments for flow dis-
bythesameinitialconditionisssimulatetdofurther
continuities, such as the inclusion of artificial viscos-
testtherobustnesosfthepresenEtulersolver.The
ity, blending of low- and high-order-accurate fluxes,
computationadlomainisenlargedto-1 < x < 3
the use of nonlinear solution to Riemann's prob-
and0 _<
y < 3, which is the same as that of the lem as suggested in Ref. 10, or the TVD property
axisymmetric flowfield simulated in Ref. 15. The used in Ref. 15. Its inherent features of simplicity,
boundary conditions are the same as those described generality and accuracy indicate that, with further
previously except the upper surface is now replaced improvements, the space-time conservation element
by a solid wall. The reflection of the blast wave from and solution element method may be developed to
the upper surface causes additional flow phenomena become a versatile tool for solving general fluid dy-
that were not observed in the experiment. namic problems.
To show the time history of flow development, nu-
merical solutions at eight time levels obtained us- Acknowledgment
ing 161x121 grid points are shown in Figs. 12 and
This work was supported by NASA Lewis Re-
13, in which the pressure and density contours rang-
search Center through Grant NAG3-1566.
ing from 0 to 5.88 with a constant interval of 0.049
are plotted. The sequential plots reveal that as the
References
blast wave initiated from the open end of the shock
tube propagates away, a vortex is developed at the
1Chang, S.C., "The Method of Space-Time Con-
lip of the tube wall, which moves downstream with servation Element and Solution Element A New
an ascending motion. When the blast wave reaches
Approach for Solving the Navier-Stokes and Eu]cr
the upper wall, it is reflected as shown in the plot
Equations," Journal of Compulalional Physics, Vol.
at t=l.0 msec. In the meantime on the tube axis a
191, 1995, pp. 295-324.
normal shock is formed ahead the vortex and is mov-
2Chang, S.C., _,Vang, X.Y., and Chow, C.Y., "New
ing slowly in the downstream direction, while a jet
Development.s in the Method of Space-Tilne Con-
shear layer is created at the lip of the shock tube. At
servation Element and Solution Element -- Appli-
t=l.5 msec, the portion of the blast wave that is re-
cations to Two-Dimensional Time-Marching Prob-
flected from the upper wall is shown to move toward
lems," NASA TM-106758, December, 1994.
the vortex. After passing the vortex, the blast wave
3Wang, X.Y., Chow, C.Y., and Chang, S.C.,
becomes curved and keeps moving forward to inter-
"High Resolution Euler Solvers Based on the Space-
act with the normal shock as shown in the plots at Time Conservation Element and Solution Element
t=l.7 msec. At t=l.9 msec, the flow pattern reveals
Method," to appear as a NASA TM.
that as a result of the interaction, the blast wave is
4Wang, X.Y., Chow, C.Y., and Chang, S.C., "Ap-
broken into two parts while several new vortices are
plication of the Space-Time Conservation Element
created. More complex flow patterns are shown at
and Solution Element Method to Shock-Tube Prob-
t=2.1 and 2.3 msec, describing further reflection and
lem," NASA TM-106806, December, 1994.
interaction of shock waves and vortices.
5Harten, A., "High Resolution Schemes for Hyper-
Despite the difference between 2-D and axisym-
bolic Conservation Laws," Journal of Computational
metric configurations, the computed flow fields agree
Physics, Vol. 49, No. 3, 1983, pp. 357-393.
extremely well with those shown in the shadowgraph 6Einfeldt, B., Munz, C.D., Roe, P.L., and
pictures of Ref. 15 at early time steps of t=0.1996
SjSgreen, B., "On Godunov-Type Methods near Low
and 0.4937 msec. An axisymmetric Euler solver
Densities," Journal of Computational Physics, Vol.
based on the CE/SE method is being developed,
92, No. 2, 1991, pp. 273-295.
which will be used to simulate the realistic exper-
7Xu, K., Martine]li, L., and Jameson, A., "Gas-
imental flow conditions.
Kinetic Finite Volume Method, Flux-Vector Split-
ting and Artificial Diffusion," Journal of Computa-
3. Conclusions
tional Physics, Vol. 120, No. 1, 1995, pp. 48 65.
The l-D, quasi 1-D and 2-D Euler solvers have SShu, C.W., and Osher, S., "Efficient Implementa-
been validated using test problems. Numerical ex- tion of Essentially Non-Oscillatory Shock-Capturing
amples have been used to compare the CE/SE Euler Schemes, II," Journal of Compulalional Physics,
solvers with established schemes such as the TVNI Vol. 83, No. 1, 1989, pp. 32-78.
schemes designed by Harten, the upwind schemes °Huynh, II.T., "Accurate Upwind Schemes for the
used by Woodward and Colella, and others. It has Euler Equations," AIAA-95-1737-CP, June, 1995.
been demonstrated that tile present Euler solvers 1°Woo(lward, P., and Colella, P., "The Numeri-
can generate highly accurate numerical solutions cal Simulation of Two-Dimensional Fluid Flow with
Fig. 2 The CE/SE solution for SJogreen problem
Strong Shock," Journal o/Computational Physics, (At-0.00z, AX-O.OI, a-l, t-8OAt)
L6 __J--
Vol. 54, No. 1, 1984, pp. 115-173.
11Liou, M.S., "Progress Towards an Improved 1.8
Jii
CFD Method: AUSM+," AIAA-95-1701-CP, June,
1995.
12Shubin, G.R., Stephens, A.B., and Glaz, H.M.,
"Steady Shock 'rrncking and Newton's Method Ap-
plied to One-Dimensional Duct Flow," Journal o/
0.0 0.,11 0.4 0.8 0.8 1.0
Comp,fafional Physics, Vol. 39, No. 2, 1981, pp. X
364-374.
x3Giannakouros, J., and Karniadakis, G.E., "A
Spectral Element-FCT Method for the Compress-
ible Euler Equations," Journal o/ Computational 0._
Physics, Vol. 115, No. 1, 1994, pp. 65-85.
14Van Leer, B., "Towards the Ultimate Conserva- 0.10
tive Difference Scheme. V. A Second-Order Sequel
O.OO
to Godunov's Method," Journal o[ Computational -O.lOi
| . | . . , ! .
Physics, Vol. 32, 1979, pp. 101-136. 0.0 0.2 0.4 0.6 0.8 !.0
X
15Wang, J.C.T., and Widhopf, G.F., "Numerical
Simulation of Blast Flowfields Using a High Reso-
lution TVD Finite Volume Scheme," AIAA Paper
!
87-1320, June, 1987.
FILE. 1 The CE/SE soluUon for Lax problem
(CTL-O.96, AX-O.I. ,_-I, L-IOOAL)
1
4.0 ....... _ ' • '
0.0 02 0.4 0.8 0,8 1.0
s.o i X
F_i_. 3 The CI_/SE solution for Shu-Osher problem
(At-O.O0|5, _-0.0125. a-l, t-t.8)
t.6 i
I.O ! 44.6.oi!
, .,,
-8-?-4-6-4-$-2-1 0 ! 2 $ 4 6 8
X
1.715 1.8
1.0i
0.8, ..................
1.25 -4 -$ -2 -! 0 I 2
X
:p IlL6 • ,.,... ,. , ° , . .
10.8
-OJr6 l°IL8
-8-?-8-6-4-8-|-!01t_466
X
4.6
1.5
-...4 -8 -2 -1 8 I II
1.0 X
"B
-]
0.5
iLII
U
0.0 .............. J ......... |
-8-?-8-8-4-S-Z-! 0 I t S 4 6 iS lit
X l_.l
O.II
0.3
-0.2
-4 -:i -2 -I 0 I 2
X
5
Flg. 4 The CE/SE soluUon for blast wive )roblem Fig. 6 The CE/SE stead¥-|d_ste ImlutJon of nozzle probelm
(AL'O.O000I 9/t, /tX-O.O0125, a-I, t,,,O.O_8 ((:71,-0.11, IX-6/te, a-I)
0`8O
O.8O
O.VO
|
O.40
0`,qO
O./IO
0`0 0.2 0.4 0.8 0.8 1.0 0 I I 8 4 6 6 ? II I tO
Je _ " X " 1 " J 1_ ......... X
12
10 L4
- "°r t
0.0 02 0.4 U 0.8 1.0 0 I | S 4 5 | ? 8 | I0
X X
O.9O ,,,,.. ,L ° ,
O.8O
0.?O
j-
O.5O
180 0.80
0+40
IZO
O.qO
eO r O2O
0 ..... 0 . . , , . . 0.10
0`0 02 OA 0.4 0"8 1.0 0 l | 8 4 IS I T 8 | 10
X X
lq41. 5 The CE/SE steady-state smluUon of nozzle probalm
(C_-O.g, JX-O_, .-t)
Uo
_0`_
0`_
I | 8 4 6 i T i | io
x
|.8
1.8
t.4 1;'II. ?(s) Geometry or the duct with • step
j::
).
o.I
0`4 o.o - ["
0 t It 8 4 6 e ? 8 IJ JO 0.0 0_ O+l 0.I) 1.2 I..8 1.8 ILl 1.4 IL'/ 11.0
Ir
0.._ ., ,.+. , ,.-
O.8O
_|. ?(b) Grid dlztrlbuUou in the duct
0.'I'0
I-°
OJO
O.4O )-
0`,30 _- =
O.IO o 1
0.10 o s
1854||?|010 x
X
6
Fig. 8 Density contours for flow over a step at t=4
1.0' ,,. _ .... , .
0.8-
0.6
0.4
0.2
0.0
0.0 o.s 1.2 1.s 2.4 3.0
x (Slx21)
1.0
0.6
0.4
00..28 __
0.0
0.0 o.e 1.2 1.s 2.4 3.0
X (121x41)
Fig. 9 Grid distribution in the blastfield
1.2
0 • 0 • 0 • 0 • 0
• 0 • 0 • 0 • 0 • 0
0 • 0 • 0 • 0 • o
- • • 0 • 0
0 • 0 • 0
• 0 • 0 • 0
0 • 0 • o
.- : • 0 • 0 • 0
0 • 0 • 0 • 0 • 0
• 0 • 0 • 0 • 0 • 0
0 • 0 • 0 • 0 • 0
0.0
-0.25 0.35
x/_
Fig. 10 Locus of vortex center
0.00
0 CE/SE solution
0.80 Experimental data
0.70
O_
oR
0.60
¥
0.50
0.40
-0.25 -0.15 -0.05 0.05 0.15 0.25 0.35
X/D
Fig. 11 Velocity vector plots at early times
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