The free-stream capturing technique for both the finite-volume (FV) and finite-difference (FD) framework is summarized. For an arbitrary motion of the grid, the FV analysis shows that volumes swept by all six surfaces of the cell have to be computed correctly. This means that the free-stream capturing time-metric terms should be calculated not only from a surface vector of a cell at a single time level, but also from a volume swept by the cell surface in space and time. The FV free-stream capturing formulation is applicable to the FD formulation by proper translation from an FV cell to an FD mesh

Obayashi, Shigeru

English

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NASA Contractor Report 177572
Free-Stream Capturing in Fluid
Conservation Law for Moving
Coordinates in Three Dimensions
Shigeru Obayashi
(NA_A-CR-!7757Z) FREF STreAM CAPTURING iN
FLU[0 CONStPVATION LAW FOR MUVING
C_OQDINATES IN THREE nIMLNSTONS (MCAT
[ns_:.) 15 p CSCL 01A
c3/o?
N91-I_I_9
uncl _
032_330
Contra_ Number NCC2-605
Janua_ 1991
National Aeronautics and
Space Administration
https://ntrs.nasa.gov/search.jsp?R=19910005826 2020-03-19T19:18:06+00:00Z

NASA Contractor Report 177572
Free-Stream Capturing in Fluid
Conservation Law for Moving
Coordinates in Three Dimensions
Shigeru Obayashi
MCAT Institute
San Jose, California
Prepared for
Ames Research Center
Contract Number NCC2-605
January 1991
RI/LSA
National Aeronautics and
Space Administration
Ames Research Center
Moffett Field, California 94035-1000

Free-Stream Capturing for Moving Coordinates
in Three Dimensions
Shigeru Obayashi*
NASA Ames Research Center, Moffett Field, California
Introduction
Body-conforming coordinates transformation of a fluid conservation-law form is
generally used in computational fluid dynamics. The metrics associated with the coordi-
nates transformation are required to satisfy certain geometric identities to maintain the free
stream. 1 These metrics are called free-stream capturing (or preserving) metrics. So far, nu-
merical techniques are known to capture the free-stream on stationary grids. 2-4 However,
the extension of the free-stream capturing metrics to moving grids is not straightforward.
The error introduced by the time metrics has been overlooked because it is negligible in
most cases, but it can be significant in certain applications such as helicopter rotor flow
fields, s
Rigorous formulations to avoid this error were suggested in Ref. 1, and demon-
strated, for example, in Ref. 6. Based on the work in Ref. 1, the present study describes
detailed formulas for constructing the free-stream capturing metrics in space and time on
both the finite-volume (FV) and finite-difference (FD) framework. The error introduced
by the inconsistent time-metric term is also evaluated.
*Research Scientist, Applied Computational Fluids Branch, MCAT Institute, San
Jose, California 95127.
Finite-Volume Formulation
Geometric Identities and Free-Stream Capturing
Following Ref. 1, the integral form of a conservation law for a given cell can be
written as
Iv QdV - QdV + n. FdS dt = 0 (1)(,_) (t_) (*)
where V(t) is the cell volume and ndS(t) is a vector element of surface area with outward
normal n. Considering the Euler equations, Q is a vector of conservative variables, viz.,
density, momentum and energy, and F is the flux tensor of Q. The flux F can be de-
composed into the flux in the stationary frame, Fst, and the contribution due to surface
element velocity, v as follows
F = Fat - vQ (2)
Let r, r0(t), v0(t), and fl(t) be the position vectors of a point in space, the origin, velocity
and angular velocity of the non-inertial frame relative to the inertial frame, respectively.
Then,
v = vr + vc (3)
where
v,. = v0(t) + f_(t) × [r- r0(t)] (4)
and vc(t) is the surface element velocity relative to the non-inertial frame, but expressed
in the inertial frame.
The geometric identities given in Ref. 1 are as follows. A mathematical expression
of a closed cell is given by
sndS = 0 (5)
The relative rigid motion of two frames of reference is given by
n. vrdS = 0 (6)
2
Using Eqs. (4) and (6), one obtains
_ r × ndS= 0 (7)
The conservation of volume for a time-varying cell is given by
V(t2) - V(tl) = n. vcdS dt (8)
(t)
The free-stream preservation due to these geometric identities can be demonstrated by
substituting Qoo, F,too and Eqs. (2) to (4) to Eq. (1):
J" /sQoo[v(t2) - g(t,)] = - [f,too. ( ndS) - Ooo(v0 - fl × r0)-( ndS)
1 (9)
P 1"
(_r× ndS)- Q_o(_s n. vcdS)]dt
The geometric identity, Eq. (5), suffices to capture the free stream in a fixed coordinate
system, where most of steady-state computations are carried out, and in a moving coor-
dinate system without rotation (_ = 0). When the grid is moving with rotation (fl # 0),
the second geometric identity, Eq. (7), is to be satisfied. For the general motion of grid
with changing cell volume, the third geometric identity, Eq. (8), is also required.
The geometric identity, Eq. (5), preserves the free stream at any instance t when
v = 0. Thus, the time-differential form of Eq. (1) is often used in the FV formulation.
However, if the grid moves, the geometric identities have to be satisfied correctly in the
integral form.
Free-Stream Capturing in the Inertial Frame
To preserve the free-stream perfectly with a moving grid, Ref. 1 suggests to consider
the rigorous FV formulations in space and time. One of the rigorous FV formulations with
a grid velocity expressed in the inertial frame is shown here in detail.
Assuming that v is given and then by substituting Q_o and F,too to Eq. (1), one
obtains
i,"i i,"i Qo_[V(t,) - V(t,)] = -F,t_o • ndSdt + Q_ n. vdSdt (10)
t 1
3
The first term in the right-hand side of the aboveequation yields Eq. (5). Then the rest
of the equation becomessimilar to Eq. (8).
Figure 1 showsa regular hexahedral cell. We assumethat all edgesare straight
lines. Reference1 describesthe formulas for the surfacevector S (note that S is taken
in the positive coordinate direction here) so as to satisfy the geometric identities on the
hexahedron:
1
-- - × -
$1562 = (r56 - ri2) x (rls - r26)
1 S
V12345678 -- 5( 1485 -11- S1234 J¢" S1562) • (rT - rl)
(11)
(12)
(13)
where r56 = ½(r5 + re), and so on. Note that Eqs. (11) and (12) result in the same
expression. In fact, the surface vector is defined uniquely as long as the edges are straight
lines. Also note that there are other consistent ways to compute the cell volume instead
of Eq. (13). However, Eq. (13) is the simplest form. Let $1562 = S15s2n. With either
Eq. (11) or (12), one obtains
Thus, Eq. (5) is satisfied.
ESn=O (14)
cell
It is essential to compute the second term in the rlght-hand side of Eq. (10). It can
be rewritten as
"(ESn.v)dt=E(ft2Sn.vdt) (15)
i cell cell Jtl
Let Vs be a volume swept by a surface S between the time interval [el, t2]:
Vs = Sn. vdt (16)
Let S(t_) = $1562 and S(t2) = S_,5,6,2, (see Fig. 2). The volume Vs can be computed
similarly to Eq. (13) as
VS 1234 -'V122,1,566'5'
(17)1
==($11'5'5 + S122'1' + SISS2)" (re, - ri)
3-
4
where Sll,s,s and S122,1,are the surfacevectors in spaceand time domain. Note that this
formula requiresonly the differenceof the positions of grid points betweentl and t2, not
the grid velocity v itself. Then, instead of Eqs. (6) and (8), one obtains
AV = E V s (18)
cell
This identity does not mean to satisfy Eq. (7) but does satisfy Eq. (6) in the time-
integral form. Therefore, Eq. (18) leads to the perfect free-stream capturing with the
use of Eq. (11).
Free-Stream Capturing in the Non-Inertial Frame
It is convenient to use the non-inertial frame for certain applications. Thus, the
FV formulation with a grid velocity expressed in the non-inertial frame is shown next.
The analysis also provides a deep insight for the free-stream capturing because it considers
three types of motion given in Eqs. (3) and (4) separately. The discretized forms of the
geometric identities in the FV method can be expressed as
ESn=O (19)
cell
ESrxn=0
cell
(20)
cell
where Vsc is obtained from Eq. (16) by replacing v with vc (see Ref. 1 for more details).
Reference 1 introduces the area moment M = fs r x ndS so as to satisfy the
discretized geometric identity, Eq. (7), on the hexahedron:
M15_2 = r165 x $165 + r126 × S126 (22)
1 ½(r where rles = _(rl +r6 +rs), $16s = -rl) x (rs-rl), and so on. Note that
Mls62 # rls82 x S156_. The expression, r1562 x $1562, is not well-defined for computing
area moment. In contrast, Eq. (22) is well-defined. To see these, let r<>, rA, S o and Sz_
be r15_2, r16s, [S1s62]n and [$165]n, respectively. After simple algebraic manipulation, one
= (21)
obtains
in contrast,
r_, × S<> _ 0 (23)
cell
rA xSA----0 (24)
cell
Therefore, in addition to Eq. (21), the free-stream_ capturing can be shown as
[Fs,oo- (v0- f_ x r0)Qo_]" (_ S<>)- Q_f_. (_ M)--0 (25)
cell cell
The inconsistency of Eqs. (23) and (24) was pointed out in Ref. 1 and had been overlooked
for constructing the free-stream capturing metrics in space and time. In other words,
surface vectors and moments have to be computed to satisfy Eqs. (5) and (7), respectively.
The error introduced by the use of the inconsistent area moment has a unique
feature. Let the cell surface be a parallelogram (S¢, = S,_ + S_ = 2SA and r<> =
½(rzx + rv)). Now the difference between Eqs. (23) and (24) can be shown as
(r<>xS<>-rzxxS_-rvxSv) =_([(r_>-r_)+(r_-rv)]xSz_) =0 (26)
cell cell
Thus the error disappears, for example, on the Cartesian grid. For a hexahedron having
arbitrary quadrilateral surfaces, the error can be written as
1
- × : 0(1) (27)
cell
where -rzxl - O(h), ISAI= O(h and V = O(ha). This error may be ignored as long
as the effect of Coriolis force is negligible in the flow field.
Alternate Ways of Free-Stream Capturing
There are two other ideas to preserve the free stream. The simple way, especially
for the rigid motion of a grid, is to use the free-stream subtraction technique, _ when the
free stream is uniform. One can obtain the equation by replacing Q and F in Eq. (1) with
Q - Q_ and F - F_, respectively.
6
The other conceptually different way is to use Sa always; that is, to regard the
hexahedral cell as dodecahedronor to divide the hexahedral cell into tetrahedron. Then,
instead of Eq. (14), one will obtain
(2s)
cell
for either dodecahedron or tetrahedron in addition to Eq. (24). The resulting metric terms
will preserve the free stream. The use of the tetrahedral cell allows the most compact and
consistent metric formulation. Note, however, that the use of the tetrahedron results in
unstructured-grid formulations.
Finite-Difference Formulation
Geometric Identities in the Finite-Difference Formulation
The analysis of the FD method can be simplified with the aid of the above discussion
of the FV method. The FD formulation has to be derived from the integral form, Eq. (1).
Again from Ref. 1, the differential form for Eq. (1) can be written with a generalized
coordinate transfor;aation,
as follows:
r = r(_,_,(,r), t=r (29)
where subscripts indicate partial differentiation,
O=QV,
and where
S e = r_ × r¢,
(3O)
and
_=S_.F
V=re.r _ xr_ (32)
S _ = r_ × r e, S _ = r e x r, (31)
Thesemetrics are related with the usual FD notations as
J _ (_,_y,_z) T = S _ (33)
and
= -S_ • r,. (34)
J
where rr = v and J is the transformation Jacobian, J = 1/V. Analogous definitions can
be derived for the other directions. The differential forms of the geometric identities are
known as
(s_)_ + (s.). + (s¢)¢ = 0 (35)
and
Vr = (S _" r,)_ + (S '1. r,-)_ + (S ¢- r_)¢ (36)
Equation (36) is called the differential statement of the geometric conservation law (GCL). 7
Free-Stream Capturing Metrics in Space and Time
Following Ref. 1, let the edges of the hexahedron in Fig. 1 be redefined as a double-
sized cell in the FD grid (rl = r_-l,j-l,k-1, r2 = r_-l,/+l,k-1, r3 = ri-l,/+l,k+l, "" ",
r8 = ri+l,/-l,k+l). Also let the time level advance from tl to t2. Then, all the discussions
for the FV formulation in the previous section can be applied to the FD formulation for
the central differencing.
The surface vector evaluations, Eqs. (11) and (12), can be regarded as the evalua-
tions of the free-stream capturing metrics in the stationary grid for the FD method. For
example, Eq. (11) has been applied for the central-differencing part in Ref. 4. Equation (12)
can be rewritten as
sL,,_,_ = (r56- rl_) × (r,5 - r_6)
_ (rs-r,)÷(r6-r2) x (rl-r2)+(rs--r6) (37)
-- 2 2
1
-- ---_(_¢ri--l,j--l,k nu 6gri-l,/+1,k) x (6,ri-,,/,}-i + 6,1ri-l,j,k+l)
8
where6 denotes the difference operator. After scaling by one-fourth to adjust the area from
double-sized to regular cell, the first component of the above expression can be written as
_ = (#¢,5,_y)(#,lr¢z) - (#¢6,1z)(/_,_6(y) (38)
where /J denotes the arithmetic averaging operator. Thus, Eq. (12) is equivalent to the
consistently differenced metrics in Ref. 3 that are based on the averaging procedure so as
to satisfy the differential chain rules numerically.
The main discrepancy between the FV and FD formulations appears in the defini-
tion of cell volume. The cell volume defined by Eq. (13) is different from the one defined
by the discretized form of Eq. (32) because the FD method does not use the cell concept.
Nevertheless, Eq. (13) can be applied to the FD method with a scaling factor of one-eighth
instead of Eq. (32). Then, the FV space metrics on the double-sized cell become identical
to the FD ones.
The FD time-metric evaluation is also considered from the FV point of view. It is
easily found that the time-metric evaluation, Eq. (34), will not maintain the free stream
even with the use of the free-stream capturing metrics, Eqs. (11) or (12), in case of a
rotating frame because of Eq. (23). Also, it can be shown that such inconsistent time
metrics do not satisfy GCL. The discretized form of GCL can be written as
AV = Ar[6_(S ¢ • rr) + 6_(S 'l. rr) + 6¢(S ¢. r,.)] (39)
Let the grid move in the rigid rotation, that is, V_ = 0 and r,- - f_ × r. Then the left-hand
side of Eq. (39) is zero. But the right-hand side results in 6_(r ×S_)+6,_(r x S_)+_5¢(r × S ()
0 (Eq. (23) appears again). This indicates that the use of the GCL condition, Eq. (39),
for computing AV can be erroneous. In other words, the GCL condition, Eq. (39), is
necessary to preserve the free stream, but not sufficient to construct consistent metrics in
space and time.
It is easily found that the time integration of Eq. (39) from tl to t2 results in
Eq. (18) and thus both equations are equivalent. Therefore, the consistent time metrics
can be obtained, for examplein the ( direction, by replacing Se. r_ in the right-hand side
of Eq. (34) with the time average,Vs_/At, of Eq. (16) as,
-- = S_ • r_dt = -_ (40)
J At J,, At
Note that _t defined by Eq. (40) contains all information about the movement of a cell
surface, such as translation, rotation and deformation. In contrast, Eq. (34) is a sim-
ple product of surface area and velocity of cell centroid and thus can represent only a
translational motion.
The free-stream subtraction technique will be useful for the rigid motion of the
grid, because a rigorous evaluation of Eq. (40) is expensive computationally. Note that
the subtraction is required only for the time-metric terms with the use of the free-stream
capturing metrics in space, s
Concluding Remarks
This paper summarizes the free-stream capturing techniques for the finite-volume
(FV) and finite-difference (FD) formulations following a 1989 journal article by M. Vinokur.
For an arbitrary motion of the grid, the FV analysis shows that volumes swept by all
six surfaces of the cell have to be computed correctly. This means that the free-stream
capturing time-metric terms should be calculated not only from a surface vector of a cell
at a single time level, but also from a volume swept by the cell surface in space and time.
The error introduced by conventional inconsistent time metrics is also shown. The FV
analysis also gives a guideline to construct free-stream capturing metrics in space and time
for the FD formulation by regarding an FV cell as an FD mesh. The discretized geometric
conservation law is shown to be a necessary condition but not a sufficient one.
Acknowledgment
The author would like to thank Dr. Vinokur for his valuable discussions.
10
References
1Vinokur, M., "An Analysis of Finite-Difference and Finite-Volume Formulations
of Conservation Laws," J. Oomput Phys., Vol. 81, No. 1, Mar. 1989, pp. 1-52.
2Pulliam, T. H. and Steger, J. L., "On Implicit Finite-Difference Simulations of
Three-Dimensional Flow," AIAA Paper 78-10, Jan. 1978.
3Chaderjian, N. M., "Transonic Navier-Stokes Wing Solutions Using a Zonal Ap-
proach," NASA TM-88248, Apr. 1986.
4Brining, P. G., Chiu, I. T., Obayashi, S., Pdzk, Y. M., and Steger, J. L., "Numerical
Simulation of the Integrated Space Shuttle Vehicle in Ascent," AIAA Paper 88-4359-CP,
Aug. 1988.
SSrinivasan, G. R., Baeder, J. D., Obayashi, S., and McCroskey, W. J., "Flowfield of
a Lifting Hovering Rotor - A Navier-Stokes Simulation," Paper No. 1.3.5, 16th European
Rotorcraft Forum, Glasgow, Scotland, U.K., Sept. 18-21, 1990.
6Chen, C. L. and McCroskey, W. J., "Numerical Simulation of Helicopter Multi-
Bladed Rotor Flow," AIAA Paper 88-0046, Jan. 1988.
7Thomas, P. D. and Lombard, C. K., "Geometric Conservation Law and Its Appli-
cation to Flow Computations on Moving Grids," A/A/i J., Vol. 17, No. 10, Oct. 1979,
pp. 1030-1037.
11
/
S 1562
/
Fig. 1
/
i
S 4873
4
Geometry of a hexahedral cell.
61
5 2'
1
Fig. 2 Volume swept by a surface.
12
I IA. A Report Documentation Page
Na|onel Aermsu_s sr_l
_ce AdminislzaU_
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
NASA CR- 177572
4. Title and Subtitle
Free-Stream Capturing in Fluid Conservation Law for
Moving Coordinates in Three Dimensions
7. Author(s)
Shigeru Obayashi
9, Performing Organization Name and Address
MCAT Institute
San Jose, CA 95127
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, DC 20546-0001
5. Report Date
January 1991
6. Performing Organization Code
8. Performing Organization Report No.
A-91022
10. Work Unit No.
505-60
11. Contract or Grant No.
NCC2-605
13. Type of Report and Period Covered
Contractor Report
14. Sponsoring Agency Code
15. Supplementary Notes
Point of Contact: Peter M. Goorjian, Ames Research Center, MS 258-1 __
Moffett Field, CA 94035-1000 -*"
(415) 604-5547 orFTS 464-5547 , .-- __,_=---, _+[L t __
16. Abstract
This report summarizes the f r_stream capturing technique for both the finite-volume (FV) and finite-
difference (FD) framework.-For an arbitrary motion of the grid, the FV analysis shows that volumes swept
by all six surfaces of the cell have to be computed correctly. This means that the free-stream capturing time-
metric terms should be calculated not only from a surface vector of a cell at a single time level, but also
from a volume swept by the cell surface in space and time. The FV free-stream capturing formulation is
applicable to the FD formulation by proper translation from an FV cell to an FD mesh.
17. Key Words (Suggested by Author(s))
Conservation law, Moving coordinates, Finite-
volume method, Finite-difference method
18. Distribution Statement
Unclassified-Unlimited
Subject Category- 02
19. Security Classif. (of this report)
Unclassified
20. Security Classif. (of this page)
Unclassified
21. No. of Pages
16
22. Price
A02
_IASA FORM 1626 OCT86
For sale by the National Technical Information Service, Springfield, Virginia 22161



Free stream capturing in fluid conservation law for moving coordinates in three dimensions

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