The primary interest of the authors is in the area of grid generation, in particular, optimal domain decomposition about realistic configurations. A grid generation procedure with optimal blocking strategies has been developed to generate multi-block grids for a circular-to-rectangular transition duct. The focus of this study is the domain decomposition which optimizes solution algorithm/block compatibility based on geometrical complexities as well as the physical characteristics of flow field. The progress realized in this study is summarized in this paper

Soni, Bharat K.

Yoon, Yonghyun

English

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I # <¢
OPTIMAL DOMAIN DECOMPOSITION STRATEGIES
Yonghyun Yoon*
Department of Aerospace Engineering
Korea Air Force Academy
Cheong-Ju, Korea
and
Bharat K. Soni**
NSF Engineering Research Center for
Computational Field Simulation
Mississippi State University
Mississippi State, Mississippi
N95- 28759
ABSTRACT
The primary interest of the authors is in the area of grid
generation, in paticular, optimal domain decomposion about realistic
configurations. A grid generation proced_ with optimal blocking
strategies has been developed to generate multi-block grids for a
circular-to-rectangular transition duct. The focus of this study is the
domain decomposition which optimizes solution algorithm/block
compatibility based on geometrical complexities as well as the physical
characteristics of flow field. The progress realized in this study is
summarized in this paper.
INTRODUCTION
Most solution algorithms for irregular configurations presently solve
a discrete form of the fluid equations of motion. The discretized
equations must be solved on a discretized computational region, that is,
on the grid nodes. A discretization process of grid generation is to
establish a relationship between the physical and computational domain,
thus allow the solution algorithm to be performed in the transformed
domain.
In principle it is possible to make a correspondence between any
physical region and a single computational region. However, for general
complicated three-dimensional configurations the resulting grid is likely
to exhibit excessive skewness and coarseness. Despite the power and
sophistication of present grid generation capabilities, it remains difficult
to generate a reasonable, single-block grid about geometrically complex
flow field comfigtn'atiorL
* Assistant Professor of Korea Air Force Academy
**Professor of Aerospace Engineering, Mississippi State University
597
https://ntrs.nasa.gov/search.jsp?R=19950022338 2020-06-16T07:36:51+00:00Z
A better approach with complicated physical boundaries is to
decomposethe physical domains,each boundedby six curved sides and
each of which transforms to a rectangular block in the computational
domain(ref.1). This domain decomposition has the merits of flow solver
efficiency, grid smoothness and orthogonality, and flexibility of griddi_
for complex geomeixies by choosing suitable block topologies.
There are _,nany approaches and philosophies which can be included
under the domain decomposition strategies. Structured grid analysis of
flow around complex geometries in widespread use is blocked
decomposition, in which the solution domain )_s divided into regions
with common internal boundaries. The grid lines at adjoining blocks
can be set up to match in a variety of ways, with various levels of
slope continuity. There is no theory which governs the way in which a
flow field should be partitioned. A limited number of papers in the
literature(ref.2-5) address basic flow-field-decomposition criteria and
guidelines. Eiseman has developed an automatic block decomposition
algorithm(ref.6). This algorithm is based on the geometrical
complexities. However, no comprehensive, systematic studies have been
done to determine the effect on the computed solution of using different
blocks for the same geometry. Currently, the major bottleneck is in the
design and implementation of the blocking plane(ref.7)
To remedy this problem, the focus of this study is the domain
decomposition which optimizes algorithm/block compatibility based on
geometrical complexities as well as the physical characteristics of the
flow field.
Generally, the use of analytical shapes is not enough to satisfy
the unusual geometrical requirement. Hence, sculptured curve/surface
definitions such as Non-Uniform Rational B-Splines(NURBS), and quick
elliptic grid refinement algorithms are developed. The application of
these algorithms to grid adaption and domain decomposition is
demonstrated. Grid generation associated with the circular-to-rect
angular transition duct has been accomplished by applying these
techniques. After careful consideration of the various alternatives, the
structured muli-block approach must be chosen as the most suitable,
from the point of view both grid generation and flow analysis.
INS3D(ref.8), a three-dimensional multi-block incompressible Naver-
Stokes code is used for this study.
598
DOMAIN DECOMPOSION PROCEDURE
The work focuses on effects for optimizing domain decomposition
strategies associated with circlar-to-rectangular transition duct_ The
_ansition duct is designed to connect a typical circular engine exhaust
to a high aspect ratio rectangular nozzle. The application presented is
of considerable engineering importance in internal fluid flow designs.
To take full advantage of the flexibility of multi-block stl"uctured
techniques, one has to decide upon a suitable blocking topology to yield
an optimal block arrangement for a given flow solver.
[ CAD
f
Geometry
definition
Analysis of solver
: scheme
: block capability
: flow properties
Compter resources
: CPU time
: amount of memory
Decision of block topologyJ
I
Transformation of physical]to computational domain
I
I Generation of surface grid]
I
[ Generation of field grid ]
I
[ Set B.C. and I.C. [Solve the flow equations
I
--NO--[Anaysis : postprocessor I
Figure 1. Basic procedure for domain decomposition
599
This step is an art, requiting knowledge-based techniques and
trial-and-error. Typically, a domain decomposition strategy is performed
and then surfaces and volume grids are created for each individual
block. Figure 1 shows the basic procedure. The desired characteristics
of multi-block grid are strongly dependent on the needs of the flow
solver and computer.
GEOMETRY DEFINITION FOR TRANSITION DUCT
The circle-to-rectangular transition duct can be designed by the
equation of a superellipse.
--
Where a and b are the semi-major and semi-minor axes, respectively,
of the superellipse. The x axis is coincident with the streamwise
direction and the y and z axes are parallel to the major and minor axes
of the duct. The cross-sectional area at a given streamwise location is
given by
r(1) z 2
An: q ( -_- )(4ab) (2)
r(_)
Where n refers to the gamma fuction defined as
l'(_)= f :(e -t t'-t)dt (_>0) (3)
The design procedure for the transition duct is to specify the
sWearnwise variation of the semi-major and semi-minor axes, and
superellipse exponent(_) defined by fifth order polynomial functions.
Special cases of the superellipse include a circle( a=b, __-oo), an ellipse(
a/ b, _=_).
Steady, incompressible, turbulent, swirl-flee flow through a
circle-to-rectangular transition duct has been studied experimentally
(ref.9). For comparison, the same geometry has been simulated. Figure
2 shows the lower half of the duct. The transition duct has an inlet
diameter of 20.43cm, a length-to-diameter ratio of 1, and an exit plane
60O
aspect ratio of three. The transition region length-to-diameter ratio is
1.5. The cross-sectional area remains the same at the exit as at the
inlet, but varies through the transition section to a maximum value
approximately 15% above the inlet value.
SURFACE REDISTRIBUTION AND REMAPPING
The principal requirement for generating a grid about a complex
geometry is the ability to efficiently redistribute points on the surface
while maintaining the integrity of the geometry. Because the regions
where high gradients are expected in the flow solution require high
density of grid points. The redistributed surface grid is accomplished
by evaluating the NURBS surface at th_ respective parameter
associated with the desired distribution space. The NURBS is used for
the standard surface description. The convex hull, local support, and
variation disrninishing properties of B-spline functions contribute to the
generation of the well-distributed smooth grid.
Let r:[xl(L_), xz(L4), xa(¢,)] denote the parametric representation
of the surface with coordinate(x1, x_ x3) and parameter(_, n). A
control point form of the NURBS surface(in 3-D) is defined as a tensor
product formula in 4-D(ref.10).
NI N
,'_oyoo (4)
P(_, n): m N M
_B, (_)B_(n)U¢i
i=o./=o
Where B_(_):theM th degree B-spline basis functions in i direction.
BM(n):theK _ degree B-spline basis functions in j direction.
Ho : weight (positive real values)
O_ : control points
The control points of the surface are determined using tensor product
formula associated with both ( and 4 parameters when a surface with
a set of data point is given. It is called the inverse problem.
The parametric space associated with NURBS is transformed as
the normalized arc length distribution mesh. The original surface which
is expressed into non-NURBS form can be converted to NURBS
representation. It is important to note that the redistributed surface grid
601
is obtainedby evaluating the NURBS surface at the desired distribution
point. However, the NURBS surface is smooth and has kept precisely
the fidelity of the original surface. A redistributed surface resulting
from the NURBS surface is demonstrated in Figure 3.
ELLIPTIC REFINEMENT
The grid should exhibit the desirable qualities of smoothness and
orthogonality especially near the solid surface. Some results of algebraic
grid methods such as transfinite interpolation .functions can not meet
the requirement of the grid qualities on certain geometries. Thus,
elliptic grid methods are used to make up the shortcoming of the
algebraic methods(ref.ll).
Derivation of the control function based on grid metrices begins
with analyzing the elliptic grid equation. Define the physical and
computational space with r: (xl, xa x3) and f_: (_1, _2, _3), respectively.
The covariant and contravariant vectors then appear as follows:
a, : covariant base vectors re, (i=l, 2, 3)
i
a : contravariant base vectors Az___ (i= 1, 2, 3)
g0 : ai. as = gJ_ (i=1, 2, 3), (j=l, 2, 3)
gO : a'' _ : g_ (i=l, 2, 3), (j:l, 2, 3)
g = detlgo[ = [ al • ( a2xaa)] 2
(gO) _ = derivative of go with respective to _k
- r¢,¢, reJ + l:e' res¢,, (i=l, 2, 3), (j=l, 2, 3), (k=l, 2, 3)
The elliptic grid generation system used in this study(ref.12) is
3 3 3
Z Zg ° 1: ¢,¢s+ kZ=tg_Pk 1: ¢, = 0 (5)
i:l/=t =
The determinations of the three control functions for the general
three-dimensional case can be summarized as follows. The three
components of the elliptic grid generation equation(5) provide a set of
three equations,
3 3 3
Zg _ (r¢,) l Pk = -i_-_lS_.lg° ( $2¢gJ)t, l=1, 2, 3 (6)
602
that can be solved simultaneouslyat each point for the three control
functions, Pk, k:l, 2, 3. The derivatives here are represented by
central differences. The smoothness is established by replacing the
control function at each point with the average of the four neighbors in
the two curvilinear directions other than that of the function. Thus,
p1(_1, _2, _3):1[pd¢1 ' ¢2+1 ' _3) + pl(¢l, ¢2_1, _3)
+ pl(_1, _2, _a+l) + p1(_I, _2,_3_i)' (7)
with analogous equations for P2 and P3. No smoothing is done in the
directionof the function because to do so would smooth the spacinq
distribution.The use of smoothed control functions evaluated from the
algebraic grid produces a smooth grid that retains essentially the
spacing of the algebraic grid(ref.13).
An application of these control functions results in a smooth-nearly
orthogonal grid in fewer iterations of the elliptic solver. These control
functions are applied in surface/volume grid refinement. It can be
observed that the elliptic grid provides smoothness and near
orthogonality in Figure 4.
BLOCK INTERFACES
There are three distinct configurations along the axial direction
which are a constant diameter circular entrance section, the transition
section, and a rectangular extension section. Because of the drastic
changes in the flow direction, a multi-block grid topology has been
adapted. This provides a smooth discretization of the entire volume
inside the transition duct_
Each block has its own curvilinear coordinate system irrespective
of that in the adjacent blocks. In order to keep complete continuity of
grid lines across the interfaces between blocks, the linkage among the
various blocks can be set by interpolation or fixing an actual boundary.
The grid points on an interface of one block are coincident in physical
space with those on another interface of the same or another block.
This facilitates the interface of block treatment without an application
of interpolation. This philosophy allows the flow solver to be run on
the multi-block grid system.
6O3
APPLICATION
The two previously described methods, elliptic and algebraic, are
merged appropriately to obtain various multi-block grids on the
transition duct configuration. Moreover, by ufilizafing an approprite
blending of these methods allows a grid generation capability ranging
from cases with only a few blocks to cases with hundreds of blocks
for various shapes.
To demonstrate the capability of the present procedure, several
types of multi-block grids are designed such that the grid cell aspect
ratio and orthogonality are maintained with a reasonable range for all
the duct cross section. Figure 5-8 present several examples. For
comparative purposes, all types of domain decomposition are
implemented in the same flow solver code using equivalent boundary
conditions. Even the flow conditions for the computational analysis are
chosen to correspond to the experiments reperted in Ref.8. Inlet
conditions are as follows:
Reynolds number = 390,000(based on U and D)
Bulk velocity(U) = 29.95 m/s
Core velocity(UcAJ) = 1.083
Friction velocity(UtauAJ) = 0.04063
RESULTS AND DISCUSSION
The results of the flow calculation are shown in Figures 9-12.
Pressure contours of the axial flow component are plotted in Figure 9
(a)-(b) which are on the x-y plane and x-z plane, respectively. The
velocity contours on both planes are demonstrated in Figure 10.(a)-(b).
The transition segment produces saddle-shapod pressure distributions in
the y-z plane as shown in Figure 11. Within the boundary layer the
velocity is reduced as shown in Figure 12, but the cross-stream
pressure gradient imposed by the flow outside the boundary layer is
not reduced. This can result in significant flow turning in the boundary
layer and is refered to as skew-induced secondary flow or cross flow.
The rate of cross flow production increases as the amount of
streamline curvature increases(ref.13). Figure 13 shows the comparison
of sidewall normalized static pressure between the experimental data
and the computed solutions in the upper quarter of the duct only. The
results of computational simulation and the measured data of Ref.9.,
show a very good agreement.
604
CONCLUSIONS
Several blocking strategies have been considered in order to
analyze the transition duct flow. Optimal domain decomposition is
dependenton the method of flow solver and flow properties as well as
geometry concerns. This domain decompositiongives us the following
merits:
• A higher solution accuracy and faster convergence for the
computational fluid dynamics solver.
• Savings in the CPU time and the amount of memory.
• Maximization of the grid quality and optimization of the grid
distribution.
REFERENCES
1. Thompson, J. F., : A General Three-Dimensional Elliptic Grid
Generation System on A Composite Block Structure, Computer
Methods in Applied Mechanics and Engineering, p.377, 1987.
2. Allwright, S. E., : Techniques in Multi-Block Domain Decomposition
and Surface Grid Generation, Numerical Grid Generation in
Computational Fluid Mechanics '88,pp.559-568, Pineridge Press
Limited, Mumbles, Swansea, U.K., 1988.
3. Piperni, P. : Multiblock Grid Generation with CAD-Based Domain
Decomposition Techniques,:Numefical Grid Generation in
Computational Fluid Dynaimics and Related Fields, pp.109-121,
Pineridge Press Limited, Mumbles, Swansea, U.K., 1994
4. Rossow, C. C. and Ronzheimer, A., : Multi-Block Grid Generation
around Wing-Body-Engine-Pylon Configurations,Numerical Grid
Generation in Computational Fluid Dynamics and Related Fields,
North-Holland, New York, 1991.
5. Vogel, A. A., :A Knowledge-Based Approach to Automated Flow-
Field Zoning for Computational Fluid Dynamics, NASA TM101072,
1989.
6. Eiseman, P. R. etc, : Applications of Multi-Block Grid Generations
with Automatic Zoning,Numerical Grid Generation in Computational
Fluid Dynaimics and Related Fields, Pineridge Press Limited,
Mumbles, Swansea, pp.123-134. U.K., 1994
605
7. Dannehoffer, J. F., : A Block-Structuring Technique for General
Geometries, AIAA Paper 91-0145, Jan. 1991.
8. Rogers, S. E., Kwak, D. C., and Chang, J. L. C., : INS3D- An
Incompressible Navier-Stokes Code in Generalized
Three-Dimensional Coordinates, NASA TM100012, 1987.
9. Davis, D. O., : Experimental Investigation of Turbulent Flow
Through a Circle-To-Rectangular Transition Duct., NASA
TM105210, 1992.
10. Yoon, Y. H., : Enhancements and Extensions of EAGLE Grid
Generation System, Ph.D, Dissertation, Mississippi State University,
1991.
11. Thompson, J. F. Warsi, Z.U.A. and Mastin, C. W. : Numerical Grid
Generation, Foundations and Applications, bJ_rth-Holland , 1985.
12. Thompson, J. F. : Program EAGLE,Numerical Grid Generation
System User's Manual, Vols.II and III,AFATL-TR-87-15,1988.
13. Reichert, B. A., Hingst, W. R., and Okiish, : Circular-to-
Rectangular Transition Duct Flow without and with Inlet Swirl,
Journal of Propulsion and Power. Vol.10, No.l, Jan-Feb. 1994.
×
._._ z___ I_r STATION 6
,/_-_ImJ_ STATION 5
_'_ STATION 4
STATION 2
STATION 1
Figure 2. The lower half of the duct
606
Figure 3. A NURBS surface
Figure 4. Elliptic grid refinement
607
Figure 5. Two-block system along the flow direction
Figure 6. Two blocks, O-type grid system.
608
Figure 7. Two blocks, O-H type grid system.
Figure 8. 5 blocks H-type grid system.
609
(a)
-,,! /J
(b)
Figure 9. Pressure contours (a)x-y plane, (b)x-z plane.
(a)
:...... y:y:...
". r2 "-- ........ " (\| \ -- ,
Li • -
(b)
.....:>,....
Figure 10. Velocity contours (a)x-y plane, (b)x-z plane.
610
i', t : ".';
2
!
!! : . ...
Figure ii. Vorticity contuors.
......:- . . .
........-::::<!::::-:.iii?:::i:::i::-! ....
....-.-'::_:_::!_-Z"". " " " ":.......... _" i
•., -:.:.'." :. -... . •.... • . .
...... :.::-::?:::?.'- :::._.:..:..._.- " ...--=-----¢_- .
.... _:.."-- .... -._'-----7_ " " i ...
....- , ..... ..
::_:" : . - " -. " _ i-.- " .....
. ..
.. '"-
.-'" -''"
.., ."
• • :- -- :-" ........ 4.N*-
.
.....'°'"
.., .-'"
...... . ........... ..... _.
i..- ...... i
i : - "
: . .-'' !
°
k
Figure 12. Boundary layer effects.
611
cp
0.4
13eriperial wall along file upper quart_.
Figure 13. Comparison of i_.ripheral wall pressure
coefficient distribution between computed and measured
data. (symbol: experiment, simbol-line:caleulafion)
612
cp
04
03
02
O1
O0
-0.1
-0.2
0.0
I 1 I I
• x/R=2.2
• x/R=2.8
• x/R--4.0
• x/R=8.0
O----O x/R=2.2
t3 ...... o x/R=2.8
• <_ - --0 x/R--4.0
O'S.x) e _,- _ x/R--8.0 oo.O...o ....°- ....
• .1].- -_
"° • •
• . ..........._-'(___2..- .
• • .0 _ _O_AJ.
.........'-..........".-o.......................... j+_+ +-'_+_::!1
...........=.. _o. _- ]
.... ,_-- ,,--o-- • --"_ 1
0.2 0.4 0.6 0.8 1.0
periperial wall along the upper quarter.
Figure 13. Comparison of peripheral wall pressure
coefficient distribution between computed and measured
data_ (symbol: experiment, simbol-line:calculation)
613

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615



Optimal domain decomposition strategies

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