Development of a Pre-processor for Grid-Free Solvers
K. Anandhanarayanan, R. Krishnamurthy and Debasis Chakraborty
Scientists, Defence Research and Development Laboratory, Hyderabad
Abstract
A versatile pre-processor has been developed to generate data-structure for grid-free
solvers. The grid-free solver requires a distribution of points in the computational domain
and a set of neighbours around each point. The pre-processor uses overlapped unstructured
grids to obtain the distribution of points. The points that lie inside the body are removed
using hole-cutting methods and the neighbours around each point are obtained using the grid
information. The neighbours of the point in the overlapping region are obtained using donor
search method. Approximate inverse map and alternating digital tree methods are used to
accelerate the donor search. The implicit hole cutting method is used to reduce the
overlapping region, by which the computational time can be reduced and as well the solution
accuracy can be improved. The pre-processor has been applied to generate connectivity for
multi-body problems such as wing-store, aircraft-store and launch vehicle with fairings and
grid-free solver is applied to solve subsonic flows around the above configurations.
1. Introduction
The Grid-free Euler and Navier Stokes Solver (GEANS)1 has been developed to
operate on an arbitrary point distribution. The solver requires a distribution of points and a set
of neighbours or connectivity for each point. The governing partial differential equations
(PDE) for fluid flow are solved at each point using the least squares method to approximate
the spatial derivatives of fluxes in the PDE using the flow field variables available at the
neighbouring points. The solver has been validated for geometrically simple configurations
using cloud of points that are obtained using structured or unstructured grids and the
neighbours are obtained using the grid connectivity information. With the objective of
extending the application of the solver to multi-body problem, a pre-processor has been
developed to generate a distribution of points and a set of neighbours around each point. The
purpose of the pre-processor is to utilize a set of overlapping grids, known as overset or
chimera grids2,3, as shown in Figs. 1 and 2, to get the distribution of points. The complex
geometry can be considered as blending of various simple components. The component grids
are generated around portions of the geometry without regard for other portions of the
geometry, which greatly simplifies the grid generation process. It has been widely used to
simplify the grid generation requirements for complex geometries. The use of an overset grid
system enables for the simulation of bodies in relative motion, such as a store separation and
rotorcraft4, 5. Due to overlapping of multiple grids, some points may lie inside the body of
other component, such points are referred to as hole points and such points should be
excluded from the computations. There could be some points in the overlapped region that
may be outside the body, but to be excluded in favour other component grid points. The
points adjacent to the holes become inter-grid boundary points, termed as “fringe" points. The
marking of holes and the surrounding fringe boundaries forms the first phase of an overset
grid assembly process. In the grid based solvers, the boundary values required by the flow
field solution at the fringe points are obtained by interpolating the solution from appropriate
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donor elements using information from other grids that overlap the region. Whereas, in the
grid-free solver, the fringe points are also considered as active points but they include points
from other overlapping grids6. Therefore, there is no interpolation involved in the grid-free
solver. The fringe points, essentially active points in the grid-free method, couple the
solutions between component grids using points from all the overlapping grids as
neighbours7. The donor cell of a point is a cell in the overlapping grid that contains the point.
The pre-processor identifies the hole points and fringe points, known as hole cutting. Then,
the connectivity of fringe points is enhanced by identifying the donor cell using donor search
method from overlapping grids. The following sections give details about the various
algorithms to realize the above tasks and in addition, implementation of those algorithms.
2. Overset Hole Cutting
The hole-cutting procedure is a critical element of the overset grid assembly process.
It is desirable for the hole-cutting process to be fast, efficient, and require a minimal amount
of user input and control. The surface meshes of component grids are used to represent the
geometry for hole-cutting. The unstructured surface meshes are composed of triangles and a
triangular surface face is a planar geometric simplex with well known geometric properties,
such as the face normal. The intersection tests and other operations can be easily performed
for a triangular face, since analytical solutions are available for planar elements. The
following methodologies have been implemented in the pre-processor for hole-cutting.
2.1. Implicit hole cutting
The implicit hole cutting8 is the simplest hole-cutting approach that uses only a
standard donor search procedure (which will be discussed in the later section) to find the
potential candidate donor elements. It is basically selection of the quality cells in the
overlapped grid for computation while other cells are blanked. In the present work, wall
distance, distance to the closest boundary, is used for selection of cells. The wall distances
are computed at all the points in a grid from the wall boundaries within the grid. In the next
step, the donor cell is obtained for each point and the wall distance of a point is compared
with the wall distances of the vertices of the donor cell. The point is marked as inactive if it is
closer to the other boundary than to its own boundary. The effect is to locate the fringe
boundary equidistant between the bodies and minimize the overlap. In the grid-based method,
since the fringe points do not take part in computation, the fringe points isolate solid
boundary and therefore, the name is implicit hole-cutting which does not require separate
hole cutting procedures. In the grid-free method, there is no interpolation involved and
therefore, the points inside the body should be blanked. In the present work, implicit hole
cutting is used only for optimising the overlap region and one of the following hole cutting
procedure is used for blanking solid cells. This method works well when the grid has
homogeneous spacing in the overlap region but requires a donor search for all points in the
mesh. If there is insufficient overlap in the grid system, the approach may fail with much of
the grid system erroneously marked as holes.
2.2. Analytic Shapes
The simplest, and typically fastest, hole cutting method is the use of analytic shapes.
Simple analytical shapes such as cubes, spheres, cones, cylinder, parallelepiped as holecutting geometry9. Testing of a point, that is inside the body or not, can be easily done against
analytical shapes. However, this approach can be time consuming as the set of analytic
shapes must be chosen and positioned to approximate the actual geometry. Sometimes, it may
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not be possible to define such shapes for closely positioned multi-bodies. This method is
implemented for cases when the bodies are positioned at farther distances.
2.3 Point-Normal Comparison
The computational surface grids are used as hole-cutting surfaces. The dot product
between the surface normal vector and a vector drawn from the surface to a point in question
is used to decide if the point lies inside the surface or not10. An example of showing the
point-surface normal test is shown in Fig. 3. The cost of the approach is proportional to the
number of component grid points being tested and the number of component used to define
the cutting surface. The cost can be reduced by considering only the points that are within the
bounding box of cutting surface. The hole cutting surface should be convex for successful
application of this method. To overcome this difficulty, a closest point on the cutting surface
is used to apply this test.
2.4 Cartesian Grid Approximations to the Geometry:
The Cartesian hole-map offers an efficient means of hole point determination. A
closed surface S can be enclosed within a uniform Cartesian grid. Points in the grid can be
marked as being inside or outside of S very easily. As a pre-processing step, the bounding
box of surface faces are compared with the cells of Cartesian grid and the intersecting cells
are marked as cut-cells. Then, a cell far away from the boundary is marked as outside and
traversing through neighbours recursively, all the unmarked neighbour cells are set as outside
cells. After marking all the outside cells, the rest of cells are marked as inside cells. A
uniform Cartesian grid so marked becomes an approximate representation of S and is referred
to as a hole-map11 (Fig. 4). The proximity of any point P with respect to surface S can be
determined by checking the hole-map of S.
Given the coordinates of point P, the corresponding bounding hole-map element can
be computed directly as
( x p  xmin )
( y p  y min )
( z p  z min )
(1)
i
,j
,k 
dx
dy
dz
where xmin, ymin, zmin are the coordinates of the hole-map origin, and dx, dy, dz are the holemap spacings in three co-ordinate directions. If the eight vertices of hole-map are all marked
as a hole, then P is inside the hole-cutting surface. If the eight vertices are all unmarked, P is
outside the surface. However, if the eight vertices are of mixed type (marked and unmarked),
P is near a hole-cutting plane and a vector intersection test can be used to determine the
actual status of P. The test of a point status relative to the hole-map can be made completely
robust and very fast. But, the memory required to store hole-maps of cutting surfaces can be a
serious drawback to this approach for some applications. Hole-maps require 3-D arrays that
can be very large when precise hole-cuts are needed.
2.5 Object X-ray
Object x-ray12 method is similar to Cartesian hole-map, but approximates the
geometry in two dimensions and rigorously evaluates the geometry in the third dimension. In
a pre-process step, a 2-D uniform Cartesian mesh is defined for a plane located at an
elevation below an object to be used for hole-cutting purposes. From each vertex in the 2-D
plane, a ray is defined normal to the plane and is of sufficient magnitude to reach above the
object. Intersections (i.e., pierce-points) between the rays and the object are then computed
and saved as shown in Fig. 5. The blank status of a point P can then be determined as a
function of the pre-computed ray casts in the vicinity of the projection of P onto the 2-D
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plane. In the present implementation of this method, the 2-D plane, or "image plane," is
always aligned with the Cartesian X-Y plane, and the rays are aligned with the Cartesian Z
direction. Accordingly, the complete definition of an object can be efficiently represented
with a single plane of data, analogous to how common x-ray images for medical purposes can
be stored on the photograph paper. Given this construct, the location of P on the image plane
is simply defined by xp, yp. The corresponding bounding element i,j in the image plane can
be found from Eqn. 1 , where xmin, ymin are the coordinates of the image plane origin and dx,
dy are the resolution of the object x-ray. Hence, the robustness of the ray casting method is
coupled with the efficiency derivable from the structure of uniform Cartesian grids. Further,
the memory limitations of the hole-map method discussed earlier do not exist, since only
planar data (as opposed to volumetric data) is required here. Use of object x-rays to cut
chimera holes is simple. Object x-rays can be used any number of times, applied to any
combination of grid components, and used to cut any size of hole about the respective
objects. The hole cutting operations can be carried out in a fully automatic way. A crosssection of the geometry and object x-rays that intersecting the geometry are shown in Fig. 6.
Consider a point P that P is associated with one of the grid components to be cut by the
object. Further, any point within a small distance of the surface defined by the object x-ray is
to be part of the hole. Their indices of the x-ray element that bounds P can be computed using
Eqn. (1). Determination of the blank status requires comparison of the Z coordinate of P with
that of pierce-points on the four rays that surround P in the image plane. The hole generated
by the object ray is called the "minimum hole" because it is a cut from the actual object x-ray.
Given the minimum hole, however, it is easy to expand the hole any distance away from the
object. The x-ray method reduces the memory required while providing a better
approximation of the geometry.
2.6 Direct cut approach
A direct cut approach13 uses the surface grid of a geometry to remove the cells inside
the component and is applied in two distinct processes. The first process will identify the grid
elements that intersect the cutting faces. The second process is required to identify the
remaining elements or points that are inside the geometry and should also be cut. In the first
process, the direct cut method identifies the elements cut by the geometry. This is typically
performed by finding the intersection of the cutting surfaces with the target grid and breaking
the links in the grid at these intersections. It is the most accurate method, since it does not use
an auxiliary approximation to the geometry. Once the cutting phase is completed, a fill
operation is required to mark the points that lie interior to the bodies. The fill operation is
initiated from some point that is known to be inside the geometry and iteratively sets
neighbours to be inside the geometry. Neighbours across the links broken by the cutting
process are not set as inside the geometry and iteration halts in that direction. If the cut is not
complete and water tight, this fill operation can leak out of the actual geometry and
erroneously mark the entire grid as inside the geometry. The “hole” in the geometry where
the leak initiates can be difficult to locate. A robust fill procedure is critical to the success of
the direct cut method. An efficient search process is required to limit rigorous tests to only
reasonable cells. The present approach uses Alternating Digital Tree (ADT)14 for this purpose
as it is a very efficient spatial decomposition data structure. The ADT optimizes the search
operation to O(Log2 N) as opposed to O(N). A search tree is built for the cells of each grid
with the root covers the complete domain. The tree is used to identify the possible
overlapping cells of a grid with each cutting face of other grids. A list of candidate cells that
overlap with the bounding box of a specific cutting surface face are found using ADT tree.
Once the list of candidate cells has been obtained, each cell must be tested to determine if the
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candidate cell overlaps the specific cutter surface face. Any edge of the element that is
intersected by the face is marked as cut and the point of edge on the positive side of the face
normal is marked as exterior to the body while the opposite point of the edge is marked as
interior to the body. Intersection of an edge with a surface triangle and the points are marked
as interior (I) and exterior (E) based on surface normal (n) are shown in Fig. 7. There are
certain cases where a point can be marked as interior to a face and the point will be marked as
exterior to some other face. This can occur for the thin-cut case where an element is cut by
two faces that point in opposite directions such as an airfoil trailing edge. Therefore, both
points of the edge are set as exterior, but the edge is set as cut so that they do not form
neighbours during the connectivity generation. Determining the intersection between element
edges and the cutting face is a critical component of the current hole cut procedure. Since,
computing the intersection can be expensive, simpler tests are used to eliminate candidate
intersections. An intersection can only occur if the end points of the edge lie on both sides of
the face, i.e. one is above and other is below the face. Therefore, checking for an intersection
of the element edge with the cutting face is only performed if one of the element edge nodes
is above while the other is below the cutting face. The current process begins by testing the
edges of the element for intersection with the cutting face. All element edges must be
checked for intersections so that all appropriate element edges will be marked as cut. The
pyramid, prism or hexahedral elements will have quadrilateral faces. The bilinear
representation of the quadrilateral face is used to maintain consistency of representation of
the face between two cells. The intersection tests use a parametric representation of the line
segment, triangle, or quadrilateral with an intersection occurring when the parameters are
between 0 and 1. The initial phase of the direct cut approach will only mark those elements
intersected by the cutting faces as shown in Fig. 8. A second phase is required to mark the
remaining nodes or elements that are interior to the body. This is usually implemented as a
flood/fill operation that starts at a location know to be interior to the body. Neighbouring
points or elements that were not cut are iteratively also marked as interior to the body. The
procedure is terminated when no additional elements or points are marked. Once, the points
are identified as interior, they are set as hole points, the exterior points of cut edges are set as
fringe points and rest of the points are assigned as active points. The flow field information
across various overlapping grids are exchanged through fringe points. The grid-based solver
interpolates the flow field at the fringe point using the value in the cell, called as donor cell,
which contains the fringe point. In the grid-free solver, the fringe points are also active
points, but its connectivity include neighbours from all the overlapping grids. It is obvious
that the vertices of the donor cell are neighbours of the fringe point. Therefore, it is critical to
identify the donor cell for the fringe points from the overlapping grids. The process of
identifying the donor cell is known as donor search.
3. Donor Search
The grid-free method does not require donor cell for interpolation as in grid-based
method. But, identification of donor cell helps in identifying the closest neighbours from the
overlapping grids. The donor cell is identified using stencil jump algorithm15. The central
element of this algorithm is testing whether a point is inside a cell or not. The fringe point is
tested with a possible donor cell using the shape functions of the donor cell. The preprocessor has been developed to operate on hybrid grids whose elements are tetrahedron,
pyramid, prism and hexahedron. Therefore, the shape functions are defined for the above
element types and the local co-ordinates of the fringe point are iteratively solved. The local
co-ordinates indicate whether the fringe point lies inside the cell or not. If the point does not
lie inside the donor cell, then the adjacent cell in the direction of local co-ordinates of the
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fringe point is checked. The cells are traversed till the actual donor cell is identified. The
efficiency of the search depends on selection of the good initial condition for the donor
search. An approximate inverse map (AIM)16 is used to identify the possible donor cell. In
this method, as an initial step, virtual uniform Cartesian grid is defined over the domain. The
grid cell whose centre closest to the Cartesian cell centre is stored in the Cartesian cell.
During the donor search, using the co-ordinates of the fringe point, the Cartesian cell is
identified and in turn possible donor cell is identified. This strategy is very efficient but
typically results in a few off-body orphans, fringe points without donor cell. These orphans
are generated because the donor-search fails near partition boundaries. There is no way of
jumping across grid boundaries in this approach and therefore, the donor search fails. The
method is improved by storing all the intersecting boundary cells, whose one or more faces
are boundary faces, in the virtual Cartesian cell. During the donor search, if a boundary face
is reached, the virtual Cartesian cell that contains the centre of the boundary face is identified.
Starting from other boundary cells, that are stored in the corresponding virtual cell, the search
is continued till the donor cell is identified. The search, with the present modification, fails
only for the hole points and those points are set as hole points. This approach helps in hole
cutting also and no separate hole cutting procedure is required. The holes are generated as a
byproduct of the donor search. An Alternating Digital Tree (ADT) is also implemented for
the donor search. The ADT method provides the most accurate and robust donor search, i.e.,
no orphans are generated. However, it can be more than an order of magnitude slower than
the AIM method, which is unacceptable especially for dynamic mesh problems wherein
domain connectivity needs to be carried out frequently during the solution process.
4. Development of pre-processor
The connectivity generation requires various search algorithms and therefore, good
data structure makes this process efficient and cost effective. The data structure consists of
lists of cells, faces, edges, nodes and link them mutually with each other. Generalisation of
this process implies minimal user input. Further, problems like store separation dynamics of
missile from aircraft, require repeated connectivity generation by moving some component
grids relative to other component grids. Some of the data structure related to a particular
component grid does not change during relative displacements. Considering all these facts,
the connectivity generation process is split into two codes. The first code generates all linked
lists, surface and volume details which are fixed for a particular component grid. In this code,
the component grids can be rotated and translated so that the component grids can be
positioned relative to other grid or the component grids can be aligned to a particular axis for
convenience. The memory intensive and complex operations are restricted to this stage. The
wall distance which is used in the selection process of implicit hole cutting is also obtained
by the code. The wall distances can also be used in the viscous simulation in the wall
damping terms of turbulence model. The second code reads the above details and generates
the connectivity information. The component grids can be again translated, if required at this
stage. In this stage, the pre-processor carries out hole-cutting and donor search and generates
the data-structure needed for a grid-free solver. During dynamic simulation, the second code
is applied repeatedly to generate the data structure. The minimal inputs such as component
name, component grid file name, three translation distances and three rotation angles are only
needed to be given for both the codes. The pre-processor has been completely automated and
various options are given for hole cutting and donor search. The pre-processor first marks the
points as hole, fringe or active respectively using hole cutting information. Donor cell is
identified for all the fringe points. The connectivity is generated for active and fringe points
using grid information and edge cut status. Further, the vertices of the donor cell are added to
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the connectivity of the fringe point and flag it as an active point. The boundary flag is set to
the corresponding boundary points. The surface tangents and normal (pointing inward) at the
boundary points are estimated using the area averaged of those values of the surface faces
that share the points. The wall surface and volume information is written for force integration
and flow visualisation.
5. Applications
The pre-processor has been applied to three cases to verify the capability of the code.
The unstructured grids are generated around each component and the pre-processor is applied
to generate the data structure. The geometries are very complex and therefore, X-ray hole
cutting is used to identify the hole points. Then, the donor search is applied along with AIM
to obtain the donor cells for all the points except hole points. The ADT method is applied
only for the orphan points which could not find donor cell in the previous step. It was
observed that there was no orphan point present in all the cases. Then, the implicit hole
cutting method is applied to reduce the active region. After identification of active points and
fringe points, the connectivity is generated using respective grid information for those points
and neighbours of fringe points are updated using donor cells of overlapping grids. Then, the
surface outward normal for the boundary points are estimated. The results for three cases are
presented in the following sub-sections.
(i)Wing-Store
The combined wing-store geometry is shown in Fig. 9. In this case, unstructured grids
with size of 125 and 79 thousand cells are generated around wing and store respectively. A
typical cross section of overlapped grids is shown in Fig. 10. The red and green colour grids
are around wing and store respectively. After applying the pre-processor, the points are set as
active, fringe and hole points. The cells connecting hole points are shown in Fig. 11. It can
be observed that apart from the cells inside wing in the store grid, the cells above the wing
are also marked as hole cells. It can be recalled that the implicit hole cutting optimise the
overlap region by the cell selection process. Therefore, minimum numbers of cells are
marked as active and hence, the flow information exchange takes place across the component
grids at minimum number of points. This can clearly be seen in the Figs. 12 and 13 which
show the fringe cells and active cells respectively. The grid-free flow solver GEANS is
applied on this data-structure to simulate the inviscid flow past wing-store configuration. The
free-stream condition of Mach 0.8 and AOA 0o has been simulated. The pressure contours
and Mach contours are shown in Figs. 14 and 15. The contours are smooth and all the flow
features are captured.
(ii) Fighter aircraft-store
The pre-processor is applied to a fighter aircraft with an axi-symmetric store. The
configuration is shown in Fig. 16. The unstructured grid sizes of 2.5 and 0.6 million cells are
generated around aircraft and store respectively. A typical cross-section of the overlapped
grids is shown in Fig. 17. The blue and red colour grids are around aircraft and store
respectively. The wall distances, that is used in the cell selection process, at a cross-section is
shown in Fig. 18. The hole cells, fringe cells and active cells are shown in Fig. 19, 20 and 21
respectively. As mentioned earlier, the minimum overlap of cells are marked as fringe cells.
This helps in stability by marking as many as bad cells as hole cells and hence, improves the
solution accuracy. The grid-free flow solver is applied on this cloud of points at a free stream
Mach number 0.8. The surface pressure contours are shown in Fig. 22 and it can be observed
all the flow features such as compression near nose and wing, tail leading edges and flow
expansion on the aircraft fuselage are captured accurately.
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(iii) Separation of a Fairing of a launch vehicle
Next case considered is a launch vehicle and its two separated fairings. The
unstructured grids are generated around vehicle and two fairings with size of 1.4, 0.17 and
0.19 million cells respectively. The geometry is shown in Fig. 23 and the overlapped grids are
shown in Fig. 24. The red, blue and green colour grids are around launch vehicle and two
fairings respectively. The pre-processor is applied to generate connectivity. The hole cells,
fringe cells and active cells are shown in Figs. 25, 65 and 27 respectively. The effect of wall
distance in the cell selection process can be clearly seen from the figures such that fringe cells
are far from the solid boundaries. The flow solver is applied at free stream Mach number 0.7.
The surface pressure contours are shown in Fig. 28. The stagnation pressure is observed near
the nose and vehicle attachment plates. The pressure gradients along the body is less except
near stagnation points.
6. Conclusions and further work
A pre-processor has been developed to operate on overlapped grids to generate
connectivity for grid-free solvers. Efficient algorithms have been implemented for hole
cutting and donor search. The pre-processor has been applied to three cases, viz., wing-store,
aircraft-store and launch vehicle with fairings to generate connectivity. The grid-free solver is
applied and proven the functioning of solver on the connectivity. As an immediate next goal
would be application of the pre-processor on hybrid grids to generate connectivity for viscous
solver and to be integrated with rigid body dynamics code for store separation studies.
Acknowledgements
The authors express their sincere gratitude to the Director, DRDL for his support and
encouragement throughout the work. They wish to thank Mr. Ankit Raj, Scientist, CFD
Division, DRDL for providing unstructured grids around various geometries.
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hole points
Fig. 1 Overlapped structured grids around the
flight vehicle
10
Fig. 2 Overlapped unstructured grids
around launch vehicle and fairings
P
nQ
Q
S
rP
rQ
Z
Y
X
Fig. 3 Surface normal test
Fig. 4 Hole map
x-rays
.
P
Image plane
Image plane
xmin,ymin
Fig. 5 Object x-ray and image plane
E
V1
Fig. 6 Hole point marking using object x-ray
n
grid1
V3
Cut-cells
I
V2
Fig. 7 Intersection of an edge with a triangle
Fig. 8 Cut cells of grid1 by surface triangles
11
Fig. 9 Wing-store geometry
Fig. 10 Overlapped meshes around wing-store
Fig. 11 Hole-cells of wing-store grids
Fig. 12 Fringe cells in wing-store grid
Fig. 13 Active cells in wing-store grid
Fig. 14 Surface Mach contours on wing-store
12
Fig. 15 Surface Pressure contours on wing-store
Fig. 16 Aircraft store geometry
6000
z
4000
2000
0
-2000
0
2000
4000
6000
8000
y
Fig. 17 Overlapped grids around aircraft-store
Fig. 19 Hole-cells of aircraft-store grids
Fig. 18 Wall distances around aircraft-store
Fig. 20 Fringe cells of aircraft-store grids
13
Fig. 21 Active-cells of aircraft-store grids
Fig. 22 Surface Mach contours on aircraft-store
Fig. 23 Geometry of launch vehicle with fairings
Fig. 25 Hole-cells of launch vehicle grids
Fig. 24 Overlapped grids around launch
vehicle
Fig. 26 Fringe cells of launch vehicle grids
14
Fig. 27 Active-cells of launch vehicle grids
Fig. 28 Surface pressure contours on
launch vehicle
15