The Van Leer flux splitting is known to produce excessive numerical dissipation for Navier-Stokes calculations. Researchers attempt to remedy this deficiency by introducing a higher order polynomial expansion (HOPE) for the mass flux. In addition to Van Leer's splitting, a term is introduced so that the mass diffusion error vanishes at M = 0. Several splittings for pressure are proposed and examined. The effectiveness of the HOPE scheme is illustrated for 1-D hypersonic conical viscous flow and 2-D supersonic shock-wave boundary layer interactions

Liou, Meng-Sing

Steffen, Chris J., Jr.

English

NASA Technical Reports Server

IN92-23353
High-Order Polynomial Expanslons(HOPE) for Flux-Vector Splitting
Meng-Sing Liou_ and Chris J. Steffen,Jr.*
NASA Lewis Research Center
Cleveland,OH 44135,U.S.A.
t
Summary
The Van Leer fluxsplittingis known to produce excessivenumerical dissipationfor
Navier-Stokes cMculations.One example isthe incorrectpredictionof boundary-layer
profiles.We attempt in thispaper to remedy thisdeficiencyby introducinga higher-
order polynomial expansion(HOPl_ for short)for the mass flux. In addition to Van
Leer'ssplitting,a term isintroduced so that the mass diffusionerrorvanishesat M = 0.
Several splittingsfor pressure are proposed and examined. The effectivenessof the
HOPE scheme isillustratedfor 1-D hypersonic conicalviscousflow and 2-D supersonic
shock-wave/boundary-layer interactions.Also,we givethe weakness and suggest areas
for furtherinvestigationof the scheme.
Introduction
In the past decade, upwind differencingschemes have gained considerableattention
for theiraccuracy and robustnessin Euler flows with discontinuieties,hock waves in
particular.Naturally,significantresearcheffortinthe CFD community has been focused
on maxmlzing the accuracy and e_ciency, among other objectives.Four popular but
conceptuallydifferentfluxsplittingideashave been utilizedfornearly 10 years:Steger
and Warming, Van Leer, Roe, and Osher. However, each scheme has an associated
weakness when numerical accuracy and e_ciency are considered.
In thispaper,we dealspecificallywith the improvement of Van Leer'sfluxvectorsplit-
ting[l].Besidesitssimplicity,Van Leer'ssplittinghas the followingproperties:(I)itcan
be interpretedas a specialmember ofa familyof second-orderpolynomial expanslons[2],
and (2)the associatedfluxJacobian and eigenvaluesare continuous at the sonicpoints.
Van Leer'schoiceallowsone vanishing eigenvaluein the case of an idealgas,thereby
resultingin a crispshock representation.Furthermore, the continuous differentiability
ishelpfulfor convergence acceleration,e.g.,in multigridschemes.
However, failingto recognizethe contact discontinuity,the Van Leer splitting[i]pro-
duces excessivenumerical diffusionand thus requiresa huge number ofceilsto correctly
resolvethe boundary-layer flow.Some improvements have been demonstrated recently
by H_.nelet al[2]and Van Leer[3]for I-D conical,hypersonic viscousflow,but a pressure
glitcharises.A new scheme by the present authors[4]has been proposed that not only
correctsthispressuredi_iculty,but alsoisremarkably simple to implement. Neverthe-
less,the above schemes[2-4]have alreadydeparted from the ideasoffluxvectorsplitting
Senior Scientist, Internal Fluid Mechanics Division
* Aerospace Engineer, Computational Fluid Dynamics Branch
146
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and infactbecome more likethe fluxdifferencesplitting.Since the differentiabilityand
simplicityare desirableproperties,one would stillwish to search for a bettersplitting
scheme that isstrictlybased on the fluxvectorsplitting.
In this paper, we propose a family of higher-order polynomial expansions for the mass
flux that diminishes the diffusionerror as M --40. We give a detailedstudy of the
accuracy of the scheme for 1-D conicalflow and 2-D 3hock wave/boundary-layer inter-
actions. The weakness of the scheme is alsopointed out and possibileimprovements
suggested.
Analysis
To exemplify the concept,letus considerthe quasitwo-dimensional system ofequations
for conicalflows:
OU OF
o--i-+ v,:s
where U T = (p,pu, pv, pE), F T = (pv,pvu,pv 2 + p, pvtt), E = e + 1/2(u 3 + v2), and
g = E + p/p. The flow considered consistsof a very thin shear layer at the wall
and a shock wave away from the wall. An algorithm must be capable of minimizing
the numerical smearing(diffusion)at the locationswhere an eigenvaluechanges sign or
approaches zero. For example, Van Leer'ssplitting[l]can representshock profilewell,
while greatlydiffusingthe boundary layer.The Van Leer splitmass fluxesare:
F1 = F + + F_'; F_ ='4-Pa14(M'4"l) 2 .
The net difference from the curve it approximates is largest at M = 0; its value equals
pa/2. This error, viz numerical diffusion, significantly broadens the boundary layer,
leading to incorrect velocity and temperature profiles. A simple way to remove the
diffusion at M = 0 is by adding an extra higher-order term that allows the split mass
fluxes to pass through the origin(Fig. 1), i.e.,
F_ = =hpa/4[(M .4- 1) _ + mI(M)(M 2 - 1)_1,
where the higher-order term has a coefficient m:, in general function of M. It should
have the following properties:
(1) rnl --+ -1 as M -+ O;
(2) ml(M) = ml(-M);
(3) ml --* 0 as M --* 4-1.
A formula satisfyingthose propertiesischosen as:
ml = (M 2-1)/(M 2+1) s,
where the exponent S is a free parameter; also shown in Fig. 1 is ml vs M with S = 4.
147
1.5
1.0
0.52
0.0
-0.5
-I .0
-I .5
(F, + +F;-)/pa /
r;/p_ /
..... F./p/
---'-V-7"- .... 7 _'---
II I 7
-1;0 0.0 1.0
Mactt No.
1.0
0.8
+_, 0.6
0.4
0.2
0.0
/"/
,/
-0.6 0.0 0.6
Mach No.
Fig. I. Mass Flux Splitting in HOPE. Fig. 2. Pressure Splitting.
In the conical flow calculations,the accuracy and convergence appear to be insensitive
to the specified values of S = 2,4,6. Now, regarding the flux as s sum of convective and
pressure terms, we can write the splittingformula for the flux vector:
I,) ()(0)0F2 = iovll = f i ll + P"Fs pvv + p v
F4 pv ]I 11 0
With the realization in [5] that the pressure splittingcould be considered separately in
the Van Leer formula[I], a whole host of freedoms for the pressure splitting becomes
possible. FoLlowing isthe llstof formulas tested:
and
(pl):
(p2):
(p3):
(p4) :
pi = :FI/4(M 4. 1)_(M :F2)p,
pi = (pl) :F314M(M' - 1)'p,
p:i:= (pl) ::k314m,M(M' - 1)'p,
p+ = 1/2(1 + 7M)p.
Figure 2 displays the distribution of the split pressure vs M. The first formula is that
used by Van Leer[l]. The second and third splits, (p2) and (p3), yield vanishing
slope at M = 0, thus corresponding to central dit_erencing. However, no instability
was encountered in the conical fiow problem with the (p2) or (p3) split used in an
implicit code. The fourth split(p4) is obtained from an approximate integration along
characterics. As will be seen later, the four formulas give essentially the same results
for the conical fiow calculated.
Resultl And Discussion
In this paper, two cases were tested to check the accuracy and convergence of the
HOPE scheme. The first case is the I-D self-similarconical fiow over a 10-degree
half cone at hypersonic speed, for which a detailed comparison study was conducted.
148
The flow conditions are: M= = 7.95, and Re= = 4.2 × 10 _. Since P, = 1.0, exact
solution gives adiabatic wall temperature, 13.64T.... The second case is the 2-D shock
wave/laminar boundary-layer interactions, for which experimental measurements were
available[6]. The conditions are: M= = 2.0, Re= = 2.96 × 10 a, and oblique shock
angle _ = 32.585 degrees. In both cases, the results from the Roe splitting are also
included for comparison. An implicit Newton iteration procedure was used to achieve
steady-state solution with L= residual dropped by five orders of magnitude.
Figures 3 and 4 show the pressure and temperature distributions from the first- and
second-order solution on a 65-grid; little difference is seen• A monotone solution across
the shock is obtained with the first-order scheme while oscillation appears in the second-
order scheme, which can be eliminated by a TVD procedure. It is noted that the first-
order pressure is smooth at the edge of the boundary layer, unlike the ttoe solution which
shows a slight discontinuity(not shown here). Although the boundary layer exhibits a
steep temperature gradient, the HOPE scheme predicts the wall temperature correctly,
indicating removal of the numerical diffusion associated with the original Van Leer
splitting.
14.0
13.5
13.0
12.5
12.0
11.5
11.0
I0.5
10.0
0
oi
• O(a) HOPE
Q O(&Z) HOPE
M= 7.95
Pr=l.O
•o(A) /_oPz
o O(A z) HOPE
M = 7.95
Pr=l.O
14.0
13.5
13.0
12.5
_)
_ 12.0
 11.5
11.o
lO.5
10.0
5 0
I I I t i i t
I 2 3 4 4
p/p= T/T=
do = l=
I 1 I _ml
8 12T 6
fr_/r.9o...t
Fig. 3. Pressure Profile of Conic Flow. Fig. 4. Temperature Profile of Conic Flow.
Figure 5 displays the results using various pressure splittings; they are practically iden-
tical except the Van Leer pressure split (pl) shows some minor oscillation near the wall.
However, the pressure splittings show significant effect on the convergence rate. The
(p3) and (p4) splits are the best, comparable to the Roe splitting, while the other two
are roughly two to three times slower. These may indicate possible instability in a more
complex case.
Finally, for the 2-D case, the surface pressure and friction coefficient are plotted in
Figs. ? and 8. The first-order HOPE results compare fairly with R,oe's splitting and
experimental data. However, the second-order calculation experienced difficulty in con-
vergence in which the residual was reduced by only two orders of magnitude and the
result is not presented here.
149
14.C
13.5
13.0
12.5
,
_., 12.0
"u
11.5
11.0 t
10.5
lO.0
o
|
.!
. ROE Spt_t_g
= HOPE "with P_I
• HOPE "a_.ih t:,.1,
• HOPE "with, F_ a
o HOPE tuith P%
M = 7.9.,5
Pr = 1.0
10 t
|
I
I|
II|
i
i
!
!|
!
=
8
!
!
4
I 0 °
1 I t 10-5 I I"_ = t
1 2 3 5 0 2000 4000
p/p= zterat£orts
10-1
_ 10 .2
8
10-3
10-4
ROE Spltttin#
--- HOPE u,,iUt P*l
-- HOPE with. P*t
HOPE tui, ttt P*a
HOPE v_t& P*4
Fig. 5. Comparison of Pressure Splits. Fig. 6. Convergence History.
We suspect that a further investigation on other pressure splittings may lead to success
in stability and convergence. Nevertheless, a systematic study of the eigenv_lues of
the split fluxes and the complete discretized system will prove to be a useful endeavor.
Above all, the present research suggests that there are still possibilities in flux-vector
spirting after Van Leer's appeared nearly I0 years ago. The possibilities may very well
still lie in the mass-flux splitting.
2.0
1.5
12
0.8
0.4
O0
0.0
Roe SplittincJ
.... HOPE Splitting
o Ez'p. Re_,_t [6]
t | I /,
I .0 x 2.0
Fig. 7. Pressure at the Wall.
10
8
Roa Spl_ttino
6 .... HOPE Split_n 0
o Ex'p. Result [6]
¢_ • Separat_nt Regioni 4
× 2
-2
-4
0.0 1.0 z 2.0
Fig. 8. Friction Coefficient at the Wall.
Refel'@ncell
[1] B. van Leer, Lecture Notes in PhysieJ, Vol. 170, 1982.
[2] D. H_nel and R. Schwane, AIAA Paper 89-0274, 1989.
[3] B. van Leer, CFD Symposium on Aeropropulsion, NASA CP-10045, 1990.
[4] M.-S. Liou and C. J. Steffen, Jr., (in preparation).
[5] M.-S. Liou, B. van Leer, and 3. S. Shuen, J. Comput. Phllaica, VoL 87, 1990.
[6] R. J. Hakkinen, I. Greber, L. Trilling, and S. S. Abarbanel, NASA Memo 2-I8-
59W, 19,59.
150


High-Order Polynomial Expansions (HOPE) for flux-vector splitting

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