A new strategy for solving store separation problems using OpenFOAM

Abstract

The ability of OpenFOAM to solve the problem of a store separating from an air vehicle (store separation problem) has been evaluated using a dynamic mesh (Overset/Chimera) technique for an industry-class (transonic and generic) benchmark test case. The major limitations of the standard libraries have been determined. To tackle these challenges, a new strategy has been proposed and implemented using only open-source libraries and tools. The strategy combines porting, modifying, and adapting an overset library from the OpenFOAM fork platform (foam-extend) to the standard OpenFOAM platform (ESI). Furthermore, in order to overcome the well-known weakness of the standard OpenFOAM compressible solvers, the newly adapted overset library was integrated with an open-source, density-based, and coupled solver ( HiSA), which uses the OpenFOAM technology. Additionally, a force restrained model was developed to consider the externally applied forces on the store by the store ejectors. The accuracy of the developed strategy has been compared with wind tunnel tests and the solutions of two well-known commercial codes, showing good agreements with them. While the study has focused on simulations with inviscid Euler equations (typical of the test case considered here), the viscosity effect on the solution has also been studied with Navier–Stokes equations and compared with other results in the literature, showing minor differences. To the best of the authors’ knowledge, this is the first work which studies and validates the store separation problem in transonic regime with OpenFOAM.

Full text

Original Article
Proc IMechE Part G:
J Aerospace Engineering
2022, Vol. 0(0) 1–15
© IMechE 2022
Article reuse guidelines:
sagepub.com/journals-permissions
DOI: 10.1177/09544100221080771
journals.sagepub.com/home/pig
A new strategy for solving store separation
problems using OpenFOAM
Saleh Abuhanieh
1,2
, Hasan U. Akay
1
and Barıs¸ Bicer
2
Abstract
The ability of OpenFOAM to solve the problem of a store separating from an air vehicle (store separation problem) has
been evaluated using a dynamic mesh (Overset/Chimera) technique for an industry-class (transonic and generic)
benchmark test case. The major limitations of the standard libraries have been determined. To tackle these challenges,
a new strategy has been proposed and implemented using only open-source libraries and tools. The strategy combines
porting, modifying, and adapting an overset library from the OpenFOAM fork platform (foam-extend) to the standard
OpenFOAM platform (ESI). Furthermore, in order to overcome the well-known weakness of the standard OpenFOAM
compressible solvers, the newly adapted overset library was integrated with an open-source, density-based, and coupled
solver (HiSA), which uses the OpenFOAM technology. Additionally, a force restrained model was developed to consider
the externally applied forces on the store by the store ejectors. The accuracy of the developed strategy has been compared
with wind tunnel tests and the solutions of two well-known commercial codes, showing good agreements with them. While
the study has focused on simulations with inviscid Euler equations (typical of the test case considered here), the viscosity
effect on the solution has also been studied with Navier–Stokes equations and compared with other results in the literature,
showing minor differences. To the best of the authors’knowledge, this is the first work which studies and validates the
store separation problem in transonic regime with OpenFOAM.
Keywords
trajectory prediction, large mesh movement, overset method, compressible flow, parallel computing, open-source tools
Date received: 12 July 2021; revised: 11 December 2021; accepted: 25 January 2022
Introduction
Safe separation of stores is a crucial mission for many air
vehicles. Predicting the trajectory of the store (to decide
the safe separation envelop) can be done using wind tunnel
testings
1
and flight tests.
2
However, both methods are
costly and raise some safety issues. The recent develop-
ments in computer hardware as well as parallel Compu-
tational Fluid Dynamics (CFD) algorithms have made
computer simulations possible and more feasible. This
reduced the need for these methods.
There are two CFD approaches to solve this problem,
which are inherited from the wind tunnel testing.
3
The first
approach is the off-line (or grid survey)
4
and the second
approach is the on-line (or captive trajectory simulation).
5
In the off-line approach, an aerodynamic database of
forces and moments are obtained by solving the flow fields
(as a steady-state case) over a static mesh for different
scenarios. The scenarios (or grid test matrix) are mixture
of different Mach numbers, altitudes, angle of attacks, side
angles, store positions, and orientations. Finding an op-
timum set of scenarios is normally obtained using the
Modern Design of Experiments (MDOE) method.
3
After
creating the aerodynamic database, a six degrees of
freedom (6-DOF) solver is used to obtain the linear dis-
placements, linear velocities, angular displacements, and
angular velocities. For the on-line approach, a transient
simulation which utilizes a dynamic mesh technique is re-
quired. Since the resulting deformations are normally large,
the suitable dynamic mesh techniques are either based on
a deformable mesh (with re-meshing capabilities)
6,7
or an
overset/chimera
8–11
technique. At each time step, the flow
fields are computed, the forces and moments are calculated,
and using a 6-DOF solver, the displacements and velocities
are calculated. Finally, the mesh is moved (or deformed/
re-meshed) according to these displacements.
In general, if the required number of cases to decide the
safe separation envelop is enormous for the same ge-
ometry, using the first approach by generating large da-
tabase will be efficient. However, if the number of cases
1
Department of Mechanical Engineering, Atilim University, Turkey
2
Turkish Aerospace, OpenSource CFD Group, Turkey
Corresponding author:
Saleh Abuhanieh, Turkish Aerospace, OpenSource CFD Group, Ankara,
Turkey.
Email: salehkhairisaleh.abuhanieh@tai.com.tr

are not so many, and for different geometries, the second
approach can be more efficient. Moreover, the second
approach is time-accurate. In this study, the second ap-
proach is used.
Although OpenFOAM
1
is a very popular and suc-
cessful open-source platform for CFD, it has been rarely
used for solving the store separation problem. The work of
Wadibhasme
12
can be mentioned, where the Mesquite
dynamic mesh library of Menon
13
was used, which was
available under OpenFOAM in earlier releases. However,
Wadibhasme solved only a sample projectile case using an
incompressible solver (pimpleDyMFoam) for demon-
stration. The reasons that OpenFOAM is not used widely
in store separation problems are twofold. Firstly, store
separation analysis is normally required in transonic and
supersonic regimes, whereas in OpenFOAM, there is
a well-known limitation in its standard compressible
solvers, such that there are no density-based coupled
compressible solvers, which are normally used in these
regimes. Secondly, the OpenFOAM dynamic mesh li-
braries have limited capabilities in this area.
Because of these limitations, OpenFOAM was not used
in the past for store separation simulations, neither was
compared before to commercial codes which are used
extensively for this kind of problems. Therefore, this work
carried out to address this gap in the existing literature.
Accordingly, the main objects of this study are as follows:
(a) To develop effective and accurate strategy in
OpenFOAM for solving the store separation problem.
(b) To compare the developed strategy with the standard
OpenFOAM.
(c) To compare the developed strategy with commercial
codes.
(d) To analyze the viscosity effect on store separation at
the transonic regime.
The Eglin
1
test case has been used for code validation
and comparisons due to the availability of the wind tunnel
results.
Problem formulation
Governing equations. The problem can be described as
solving a transient, compressible, and viscous flow over
a moving body. The governing equations can be written as
∂ρ½
∂tþ=ρvðÞ¼0 (1)
∂ðρvÞ
∂tþ=fρvvg¼ð=pÞþ=fτg(2)
∂½ρe
∂tþ=ðρveÞ¼=qþfσg:f=vg(3)
where ρis the density, tis the time, vis the velocity vector,
pis the pressure, τis the viscous stress tensor, eis the
specific internal energy, qis the heat flux vector, and σis
the mechanical stress tensor.
Additionally, a dynamic mesh model driven by the
6-DOF equations of motion, described below, is to be
integrated with the CFD model
XF¼d
dt mVðÞ (4)
XM¼d
dt HðÞ (5)
where Fand Mare the applied force and moment vectors
computed at the center of gravity of the store from the
CFD analysis, mVis the linear momentum vector, and His
the angular momentum vector. The linear and angular
displacement vectors of the store are then computed by
integrating equations (4) and (5), respectively, at each time
step.
The overset/chimera method
Since the first time it was proposed to the aerospace
community four decades ago,
14
the overset method proved
to be a very useful technique in CFD. Its basic idea is to
assemble the computational domain using separate
meshes (sub-domains). It has been mainly used to solve
two problems. Firstly, to simplify the meshing of complex
geometries. In this case, each mesh can be prepared alone,
and that allows for generation of high-quality meshes
much easier including the block-structured meshes. Sec-
ondly, to simulate the cases where a solid body is moving
inside a computational domain. The latter capability has
been used widely to solve the store separation problems.
An example is shown in Figure 1, where a pitching airfoil
case is presented. The airfoil mesh (the overset mesh is in
red) is prepared separately and merged/overlapped with
a background mesh (in blue). This configuration allows
the airfoil to pitch or to move significantly during the
simulation without any need for re-meshing.
The main task in the overset method is to connect the
multiple meshes in a single computational domain where
the discretized governing equations are solved. This
process has been commonly termed in the literature as the
Overset Grid Assembly (OGA).
15–18
In OpenFOAM, the
same has been called as the cell-to-cell mapping.
2
The OGA process can be divided into the following steps:
Figure 1. The pitching airfoil case. The background mesh is
in blue and the overset mesh is in red.
2Proc IMechE Part G: J Aerospace Engineering 0(0)

(i) Hole identification: Finding the cells which rep-
resent the solid bodies or the cells which are located
outside the computational domain. These cells are
excluded from the calculation (hole cells
2
in
OpenFOAM).
(ii) Fringe construction: Deciding the cells which shall
receive the information from the other mesh(s).
These cells are called the fringe, receptors, acceptors,
or in OpenFOAM, they are called the interpolated
cells
2
.
(iii) Donors search: Identifying the cells which deliver
the information to the interpolated cells (the donor
cells
2
in OpenFOAM). Normally, those are the cells
which their centers are the closest to the interpolated
cells centers; however, other criteria can be used too.
The cells which are not hole cells and not interpolated
are considered as calculated cells.
2
For these cells, the
governing equations is solved. Thus, the donor cells are
under the calculated cells category.
The output information from this process is usually
called the Domain Connectivity Information (DCI).
15,18
In
the OpenFOAM code and literature, the names cellTypes
and cell classifications
19
have been used as well.
The DCI for the pitching airfoil case is presented in
Figure 2. For the overset mesh (Figure 2(a)), all the cells
are calculated (in blue) except the cells at the outer
boundary (overset patch). These cells are interpolated (in
yellow) from the background mesh. For this case, there are
no hole cells in the overset mesh.
For the background mesh (Figure 2(b)), the hole cells
(in red), which are the projection of the airfoil, are ex-
cluded from the computation. The neighbor cells of the
hole cells are considered as interpolated cells. The re-
maining cells are the calculated cells.
Once the DCI are computed, the donor cells (e.g.,
D
0
D
4
in Figure 3) for each interpolated cell (e.g., I
0
in
Figure 3) are known. This is often called the interpolation
stencil.
The next step, which is normally at the linear system
solver level, is to use the interpolation stencil to evaluate
the required fields values at each interpolated cell. Dif-
ferent interpolation schemes can be used, for instance, the
inverse distance is expressed as:
ξI0¼X
ND
i¼1
ðωDiÞξDi(6)
where ξI0is the value of the field variable ξat the in-
terpolated cell I
0
,N
D
is the total number of donor cells for
cell I
0
,ωDiis weight contribution by the donor cell D
i
, and
finally, ξDiis the value of ξat the donor cell D
i
. The weight
is defined as:
ωDi¼
1
dDj

P
ND
j¼1
1
dDj

(7)
where d
Di
is the distance between the donor D
i
cell center
to the interpolated cell center, the denominator is the sum
of all the donors’inverse distances.
The test case
Eglin
1
is by far the most dominating test case for code
validations for the store separation problem. That can be
attributed to the availability of the geometry, the experi-
mental test reports, and many published numerical results.
Basically, it is a wind tunnel test which was conducted in
Arnold Engineering Development Center (AEDC) in 1990.
The test object includes wing, pylon, and finned store
(Figure 4). The test results are available at two free-stream
Mach number (Ma): 0.95 and 1.2, while the angle of attack
was fixed to 0
o
for the two tests. In this work, the transonic
(Ma = 0.95) test case has been used where the store tra-
jectory data is available from time = 0.0 s to time = 0.33 s.
Geometry
The full-scale geometry as per the test report
1
has been
created using OpenVSP
3
(for the airfoils) and Salome.
4
Both codes are free and open-source. The combined wing,
pylon, and store geometry are shown in Figure 4.
Figure 2. The overset DCI for the pitching airfoil case. The
overset/store mesh (a) and the background mesh (b). The
calculated cells are in blue, the interpolated cells are in yellow,
and the hole cells are in red.
Figure 3. The interpolation stencil of cell I
0
consists of the
donors D
0
D
4
.
Abuhanieh et al. 3

The proposed strategy
The preliminary results of this work, which have been
presented in Ref. 20, indicate clearly that the standard
OpenFOAM implementation has two main limitations:
(i) The flow solver: More accurate and stable com-
pressible solver is required.
(ii) The OGA algorithm: A more efficient, faster and
robust algorithm is required that can classify the cells
accurately for complex cases. Having an accurate
classification/DCI would improve the accuracy of the
results as well.
More details on the observed limitations may be seen in
Appendix II.
The flow solver. HiSA
5
is an external (non-standard
OpenFOAM) open-source and free solver which utilizes
the OpenFOAM libraries. It is a density-based and cou-
pled solver which can solve both transient and steady-state
cases. It has been developed at the Council for Scientific
and Industrial Research, South Africa (CSIR).
6
HiSA
solver has been verified and used here together with the
overset method. The code is also parallelizable through
OpenFOAM parallel library.
Similar to most density-based coupled solvers, it solves
the following coupled vector form of conservation of
mass, momentum, and energy equations:
∂W
∂tþ=FWðÞ¼QWðÞ (8)
where Wis the conservative variables vector, F(W) is the
flux vector, and Q(W) is the source terms vector. More
information about the theory, code, and many validation
cases can be found in Refs. 21 and 22.
The adapted OGA algorithm. Under the OpenFOAM
platform, there is an alternative open-source overset li-
brary in the foam-extend fork.
7
Its algorithm is more
robust and faster than the standard OpenFOAM overset
library. The first step was to test the capability of this
library. After that, the library was ported to OpenFOAM,
since many data structures were not compatible between
the two forks of OpenFOAM. The third step was to im-
plement this library as a new “cellCellStencil”inside the
openFOAM overset library as shown in the block diagram
in Figure 5. For this purpose, the interface class “foa-
mExtendStencil”has been developed.
The implementation in this way allows the selection of
the adapted OGA at run time, without changing any line of
code in the standard OpenFOAM. The developed interface
class contains “oversetMesh”object from the ported
overset library as shown in Listing 1. After reading all
necessary data from the case folder (e.g., fringes selecting
settings, hole patches, and the cells in each overlapped
mesh), the object is initialized (Listing 2). Any Open-
FOAM solver, which supports the dynamic meshes, calls
the “update()”function at each time step; thus, the OGA-
related part is written inside the inherited “update()”
function. The “oversetMesh”object provides the required
DCI information for each mesh (Listing 3). Finally, the
interpolation stencil depicted in Figure 3 and the inter-
polation weights from each donor (according to the se-
lected overset interpolation scheme) are returned to the
flow solver (Listing 4).
Listing 1: Declaration of the adapted overset object
(foamExtendStencil.H).
Listing 2: Initialization of the adapted overset object
(foamExtendStencil.C).
Listing 3: Sample code for obtaining the DCI infor-
mation from the adapted overset object
(foamExtendStencil.C).
Figure 4. Eglin test case full-scale geometry.
Figure 5. Block diagram showing the adapted foam-extend
overset library as a new “cellCellStencil”inside OpenFOAM in
blue color.
4Proc IMechE Part G: J Aerospace Engineering 0(0)

Listing 4: Sample code for transferring the interpolation
weights to the flow solver (foamExtendStencil.C).
Listing 5: Sample code for using the “globalCellCells”
function (foamExtendStencil.C).
A shortcoming has been discovered during the testing
is that foam-extend library was not able to collect all the
donors from all the processors in case of parallel run (step
(iii) in The overset/chimera method section). As a conse-
quence, the results were changing according to the number
of the used cores.
This problem has been resolved by getting only the
main donors (D
0
in Figure 3) from the “oversetMesh”
object. Then, for collecting the remaining/extended do-
nors (D
1
D
4
in Figure 3) for each main donor, the
“globalCellCells”function from the main class “cell-
CellStencil”have been utilized properly (Listing 5). This
function uses the faces of the cell to find its neighbors
(face-walk). Furthermore, it implements all the necessary
synchronization tasks between the processors in case of
parallel executions.
The final step was to integrate the new library with the
HiSA solver. During the early stages of the HiSA code
investigation, it was observed by the authors that the solver
in principle shall be able to run any dynamic mesh library
from the standard OpenFOAM platform. Since the standard
OpenFOAM overset class (“dynamicOversetFvMesh”)is
derived from the dynamic mesh class, and it uses the
“cellCellStencil”objects for performing the OGA op-
erations only, a proper implementation for the new OGA
was sufficient. Thus, the HiSA solver works with new
library without any modification on its source code. That
was an advantage of using the standard OpenFOAM as
the common development platform.
The differences between the adapted OGA library
which was developed in this work and the original foam-
extend library can be summarized as follows:
(i) The functionality for finding the extended donors
cells is different as explained previously.
(ii) In order to maintain the compatibility with the
standard OpenFOAM, the overset interpolation
functionality was implemented in a way that the
function is called for each acceptor. Thus, the foam-
extend interpolation methodology, where one func-
tion call calculates the weights for all acceptors, was
not used.
(iii) In some cases, the overset is used only to assemble
the computational domain from different meshes,
andnomotionisinvolved.Inthiscase,runningthe
OGA algorithm every time step is unnecessary.
Thereisanoptioninthefoam-extendlibraryto
prevent the assembly again after the first time
(“cacheFringe”); however, it works only with
a specific fringe type (“overlap”). This option was
not used in this work; instead, this functionality was
implemented at a higher level inside the “foa-
mExtendStencil”class. In this way, the overall
OGA algorithm works only at the first time step,
while the interpolation (computationally much
cheaper than the OGA) is done at each time step as
usual. Furthermore, this functionality has been
extended by introducing a user-defined variable
(“oversetFrequency”). This extension can be used
to reduce the overhead for the transient cases too
by executing the OGA every N time steps instead
of each time step. Lijewski
23
compared the
overhead and the accuracy of assembling the OGA
every 2 and 10 time steps. According to his
conclusion, the reduction in accuracy was small.
Comparison with the standard OpenFOAM
After checking all the standard solvers and the dynamic
mesh libraries in OpenFOAM, the only available option
to solve the store separation problem has been by using
the overRhoPimpleDyMFoam solver. This solver is
a segregated, pressure-based compressible solver which
supports the overset/chimera dynamic mesh library.
Mesh. To create the mesh, different open-source tools
have been used, including Tetgen,
24
SUMO,
25
cfMesh,
8
and snappyHexMesh.
9
After several trials, snappyHexMesh
(unstructured, octree-based, and hexa-dominant) has
been selected for mesh generation. Compared to the other
meshing tools mentioned above, it retrieves the surface
mesh with good quality while keeping the minimum cell
volume relatively high and the number of cells relatively
low. The minimum cell size has a direct effect on the
computation time of the overRhoPimpleDyMFoam solver
since it is a segregated solver. Although it is an implicit
solver, the maximum Courant number, which can guarantee
a stable run is still limited because of the complexity of the
overset scheme. Increasing the minimum cell size reduces
the maximum Courant number, which guarantees a more
stable run. The Courant number defined in OpenFOAM is
expressed as:
Co ¼Δtλ;λ¼1
2VX
faces
χfacei
(9)
Abuhanieh et al. 5

where Co is the Courant number, Δtis the time step size, λ
is the characteristic time scale, Vis the cell volume, and χ
is the volumetric flux at each face. The maximum Courant
number is the highest value of Co among all cells.
For comparison purposes with the developed strategy,
and considering the high computational time of the
overRhoPimpleDyMFoam solver, including the overset
overheads (see Appendix II), a very coarse mesh with
0.58 Mcells has been generated and tested. A slice from
this mesh is shown in Figure 6.
Case setup and the standard OGA algorithm. For the sake of
conciseness, the case setup and the details of the used
OGA algorithm are included in Appendix II. The case
parameters such as mass, center of mass, and moment of
inertia have been taken from the test report.
1
Results
Steady-state studies
First, the steady-state Euler solution has been obtained.
Since the overset in OpenFOAM can be used only with
transient cases, a transient simulation (without motion) is
driven to steady-state by running the case until the re-
siduals reach convergence. The residuals are plotted in
Figure 7, where the density-based HiSA solver used with
the developed strategy yields the density residuals, while
the pressure-based solver overRhoPimpleDyMFoam used
with the standard OpenFOAM yields the pressure
residuals.
The pressure coefficient (C
p
) at roll angle 5° is plotted
in Figure 8 for the standard OpenFOAM (over-
RhoPimpleDyMFoam plus the cellVolumeWeight OGA)
and for the developed strategy (HiSA plus the adapted
OGA) compared with the experiment. Although the mesh
is coarse, the developed strategy solution is able to match
the C
p
trend of the experiment. However, for the standard
OpenFOAM solution, there is a significant deviation
specially in the middle section. That can be attributed to
the inaccuracy of the OGA algorithm, since the 5° roll
angle resides in the small region between the store and the
pylon. Examining Figure 33(a) shows that this region was
not solved by the flow solver; the cells there are either
interpolated or holes. Thus, the value of the fields there
take the free-stream values. That explains the value of zero
for the (C
p
) at this region in Figure 8. This example il-
lustrates the direct effect of the OGA accuracy on the
results’accuracy.
Transient simulations
After initializing the field variables by the obtained steady-
state solution, the actual transient run started by applying
6-DOF motion solver. The Mach number at time = 0.17 s
over a plane is plotted in Figure 9 for the standard
OpenFOAM. A diffused solution can be observed in the
store mesh near the outer patch (the outer boundaries of
the red box in Figure 6). This can be attributed to the first-
order overset interpolation scheme which has been used in
this case.
Unlike the standard OpenFOAM results, with the
developed strategy shown in Figure 10, the outer patch of
the store cannot be distinguished, primarily due to the
Figure 6. A slice from the 0.58 Mcells mesh. The background
mesh is in blue and the overset mesh is in red.
Figure 7. The residuals convergence for the developed
strategy and the standard OpenFOAM.
Figure 8. Pressure coefficient versus X/L for store at time =
0.0 s and roll angle = 5
o
for the standard OpenFOAM and the
developed strategy using the 0.58 Mcells mesh.
Figure 9. Mach number plot at time = 0.17 s (plane: y normal
at the initial center of mass) for the standard OpenFOAM.
6Proc IMechE Part G: J Aerospace Engineering 0(0)

higher order interpolation scheme used (inverse distance).
Furthermore, the shocks are more observable, thanks to
the used second-order upwind scheme (AUSM
+
-up).
26
A
brief description of this scheme is presented in Appendix III.
The linear displacements of the store versus time are
plotted in Figures 11–13. Compared with the experiment,
both results are in good agreement. However, the results of
the developed version of the code are better in all three
directions.
The angular displacements of the store versus time are
plotted in Figures 14–16, where a general agreement with
the experiment can be observed for the two results. The
developed strategy results are better for pitch and yaw
angles. However, for roll angle, the standard OpenFOAM
shows a better agreement with the experiment. That may
be attributed to the smaller value of the moment of inertia
for rolling, compared to the pitching and yawing (27 kg.m
2
versus 488 kg.m
2
for the Eglin case store at full-scale
1
).
That makes the roll angle too sensitive to the errors in
aerodynamic force calculations. Furthermore, with the
absence of results with finer meshes for the standard
OpenFOAM, it is hard to evaluate the reliability of this
Figure 10. Mach number plot at time = 0.17 s (plane: y
normal at the initial center of mass) for the developed strategy.
Figure 11. Linear displacement in the x-direction for the
store with comparison to the experiment for the standard
OpenFOAM and the developed strategy.
Figure 12. Linear displacement in the y-direction for the
store with comparison to the experiment for the standard
OpenFOAM and the developed strategy.
Figure 13. Linear displacement in the z-direction for the
store with comparison to the experiment for the standard
OpenFOAM and the developed strategy.
Figure 14. Roll angle of the store with comparison to the
experiment for the standard OpenFOAM and the developed
strategy.
Figure 15. Pitch angle of the store with comparison to the
experiment for the standard OpenFOAM and the developed
strategy.
Figure 16. Yaw angle of the store with comparison to the
experiment for the standard OpenFOAM and the developed
strategy.
Abuhanieh et al. 7

particular behavior. Unlike the linear displacements, the
accuracy of the angular displacements depends on the
accuracy of the calculated moments from CFD, which in
turn depends significantly on finding the accurate shock
location. That is why obtaining accurate results for angular
displacements is more challenging.
Code performance
The developed strategy improves the speed of solving the
store separation problems in two ways:
(i) The adapted OGA algorithm is faster. Table 1 shows
a comparison between the two algorithms (the
standard OpenFOAM cellVolumeWeight and the
adapted one) using the same node on TRUBA HPC
10
(mesh size 0.58 M) for one time step. It is clear that
the improved algorithm is much faster for this mesh
configuration (at least 7X). However, the effective
speedup with larger size meshes may need further
investigation.
(ii) The used flow solver is coupled and is able to solve the
problem with a higher Courant numbers, which allows
obtaining solutions with larger time steps.
Accuracy studies for the developed strategy. In this section,
the accuracy of the developed version of OpenFOAM and
its mesh independency have been studied by considering
different levels of mesh refinements and comparing the
results with the Eglin test case experiments. The results are
also compared with the solutions of two popular com-
mercial CFD codes, starCCM,
10
and Fluent,
11
that have
also used an overset algorithm for this test case.
Four different meshes (coarse with 1.04 Mcells, me-
dium with 1.62 Mcells, fine with 4.48 Mcells, and very-
fine with 6.20 Mcells) have been generated for this study
using snappyHexMesh. A slice from the fine mesh (at
center of mass and y normal) is shown in Figure 17.
Case setup and the OGA algorithm results
For the sake of conciseness, the case setup details and the
used OGA algorithm results are included in Appendix II.
Results
Steady-state simulations
First, the steady-state (Euler) solution has been obtained.
The pressure coefficient (C
p
) at roll angle 5° is plotted in
Figure 18, and compared with the experiment, the
starCCM code results in Ref. 10, and the Fluent code
results in Ref. 11.
It can be seen that the developed strategy results for C
p
agree quite well with the experimental results and with the
two referenced numerical solutions. Apparently, the two
shocks at X/L; 0.27 and at X/L; 0.67 have been captured
well. The three Euler numerical results (and many others,
for example, Refs. 6,8, and 9) show deviation in the small
region (X/L; 0.87 X/L; 0.94) or the last two measurement
points in the experiment report. This deviation can be
attributed to the viscosity effect (see The viscosity effect),
since in literature, for example, Refs. 6and 27, the Navier–
Stokes equations have been solved and there was an
improvement in that particular region for the same Mach
number and roll angle. This roll angle (f=5
o
) is of special
interest, since it is located in the tiny gap between the store
and pylon.
28
Unlike the viscosity effect, adding the sting
(shown in Figure 19), which was gaged to the store to
measure the pitching and yawing moments
1
during the
testing, apparently did not improve the results
appreciably.
8,9
There is a third shock which appears at the tail of the
store (at X/L; 0.98), which can be observed clearly in
Figure 20; however, there is no measurement point at this
location in the experimental data.
Table 1. Timing comparison between the standard
OpenFOAM cellVolumeWeight OGA algorithm and the adapted
foam-extend one.
The standard
OpenFOAM
The adapted
foam-extend Standard/adapted
time ratio
# Of cores Time [s) # Of cores Time [s) [-]
1 48.0 1 5.75 8.35
4 33.0 4 4.54 7.27
8 21.0 8 3.02 6.95
16 19.0 16 2.19 8.68
Figure 17. Slice from the fine mesh (4.48 Mcells), the
background mesh in blue and the overset mesh in red.
Figure 18. Pressure coefficient versus X/L for store at time =
0.0 s and roll angle = 5
o
for the developed strategy (fine mesh),
starCCM, and Fluent.
8Proc IMechE Part G: J Aerospace Engineering 0(0)

For the small oscillations observed in the present re-
sults, the authors have noticed similar oscillations in other
cases too while using snappyHexMesh for mesh gener-
ation. The octree concept used by snappyHexMesh does
not preserve the input surface mesh; instead, it snaps the
surface of the cells after cutting them. That can be the
reason for such oscillations which may be considered as
a minor post-processing issue.
In Figure 20, the Mach number contour for steady case
is plotted at the lower side of the wing showing the store as
well. It may be observed that multiple shocks are ap-
pearing there and they are not symmetrical around the
store. From this view (normal to z-direction), it can be
observed that the un-symmetry of the shocks around the
store is the main reason of yawing the store during
separation.
Transient simulations
The next step, is to start the transient simulation by
applying the 6-DOF motion solver. The effect of the
mesh refinement level on the solution has been studied
here. The angular displacements obtained by solving the
Euler equations for the four meshes (coarse, medium,
fine, and very-fine) are compared in Figures 21–23,
respectively. As may be observed, in general, the solution
is converging monotonically toward the experimental
resultsasthemeshisrefined. That indicates that the
discretization error is decreasing. Additionally, minor
differences between the solutions of the fine and very-
fine meshes indicate that the mesh convergence of the
solution is also reached.
The change of store location with time is shown in
Figure 24. The blue color represents the experimental
results and the Mach number contours represent the ob-
tained CFD results for the fine mesh. Minor deviations
Figure 20. Mach number contour at the lower side of the
wing and store at time = 0.0 s for the fine mesh.
Figure 21. Roll angle of the store. Comparison between the
Euler solutions for the four different meshes.
Figure 22. Pitch angle of the store. Comparison between the
Euler solutions for the four different meshes.
Figure 23. Yaw angle of the store. Comparison between the
Euler solutions for the four different meshes.
Figure 19. The used sting to measure the pitching and
yawing moments attached to the tail of the tested/metric store.
1
Dimensions are in inches (1/20 scale).
Figure 24. Store location with time. The Mach number
contours represent the fine mesh results and the experimental
results are in blue.
Abuhanieh et al. 9

from the experiment can be noticed, which reflects that
accurate displacements are obtained.
To highlight the importance of the applied external
forces to the store by the ejectors during separation, the
Center of Pressure (CoP) has been calculated at time =
0.05 s using the following relation:
CoP ¼Zxp
Zp
(10)
where CoP is the center of pressure coordinate vector, xis
the point coordinate vector, and pis the static pressure.
Figure 25 shows the location of the calculated CoP (the
red sphere). Since the center of mass for the store (the
black sphere) is located after the CoP and the lift force
direction is upward at this time step, the store is expected
to pitch nose-down. However, as per the results (e.g.,
Figure 22), the store is pitching nose-up. That is simply
due to the external forces, and the same can be verified
easily by calculating the total moments on the store with
and without the ejector forces. To incorporate the external
forces by the ejectors, a new force restrained model
(“ejectorExternalForce”) has been developed in this work.
This model applies a fixed force on a specific point which
moves with the body up to a certain length (the stroke
length specified in the experiment).
Since the angular displacements are normally con-
sidered the critical results for this kind of analysis, the
developed strategy results for the fine mesh are compared
to the experiment, the starCCM code results in Ref. 10 and
the Fluent code results in Ref. 11 as shown in Figures 26–28,
for roll, pitch and yaw angles, respectively.
It may be noticed that for the three angles, the de-
veloped strategy results are comparable to the results of
starCCM and Fluent. Both codes used the overset as
a dynamic mesh technique to obtain the results presented
here.
The viscosity effect. To evaluate the viscosity effect on this
particular test case, the Euler solution of the fine mesh
(4.48 M) has been compared to the Navier–Stokes solution.
The average y
+
(the non-dimensional wall distance) for
this mesh is around 200; thus, wall functions were used.
Furthermore, the kωSST (shear stress transport) tur-
bulence model
29
has been used.
Figure 25. The calculated center of pressure (the red sphere)
and the center of mass (the black sphere) at time = 0.05 s. The
contour plot shows the static pressure on the store.
Figure 26. Roll angle of the store with comparison to the
experiment and two commercial codes for the fine mesh.
Figure 27. Pitch angle of the store with comparison to the
experiment and two commercial codes for the fine mesh.
Figure 28. Yaw angle of the store with comparison to the
experiment and two commercial codes for the fine mesh.
Figure 29. Pressure coefficient versus X/L for store at time
= 0.0 s and roll angle = 5
o
for the fine mesh. Comparison
between the Euler and the Navier–Stokes solutions.
10 Proc IMechE Part G: J Aerospace Engineering 0(0)

For the steady-state solution, Figure 29 shows the
pressure coefficient (C
p
) comparison at roll angle 5°.
Compared to the experiment, for the Navier–Stokes so-
lution, the second shock at X/L; 0.67 is slightly shifted to
the right. That can be attributed to the high y
+
mesh used.
Afiner mesh in the viscous region can provide a better
result. With comparison to the Euler solution, the shock
near the tail of the store at X/L; 0.98 is detected more
sharply. However, since no measurement points are avail-
able at the test report in this region, improved compar-
isons of the Navier–Stokes solution with experiments
cannot be confirmed at this time.
Unlike the developed strategy results for both Euler and
Navier–Stokes solutions, the results obtained by Sunay
et al.
6
and Pandya et al.
27
matched better with the ex-
periment, for the last two measurement points. Further-
more, in their solutions, the second shock at X/Lx0.67
and the third shock at X/Lx0.98 are weaker, but still
respect the experimental results. Consequently, it is
expected that solving the Navier–Stokes equations with an
appropriate viscous mesh can provide better results for the
steady-state solution.
For the transient run, the angular displacements results
are shown in Figures 30–32 for roll, pitch, and yaw angles,
respectively. The work of Huang et al.
30
can be considered
as a comprehensive study which analyzed the effect of the
mesh (size and topology), the model scale, the wing
symmetry, and the turbulence models. Their best Navier–
Stokes result for angular displacements (using Spalart–
Allmaras turbulence model
31
) has been included along
the work of Sunay et al.
6
in these figures and compared
with the present results for both Euler and Navier–Stokes
simulations. Unlike the steady-state simulation, for the
trajectory prediction, it can be observed that solving the
viscous flow even with very well prepared meshes may not
necessarily provide better results than the Euler solution.
A similar observation can be made by reviewing the re-
sults of Figure 19 in Huang et al.
30
Conclusions
A new strategy which involves modifying and adapting
the foam-extend overset library, and integrating it with the
density-based coupled solver HiSA for solving the store
separation problem within the OpenFOAM platform has
been proposed and implemented. The new strategy is more
accurate and efficient than the standard OpenFOAM so-
lution for this problem; additionally, it is capable of
solving industry-scale cases/geometries. Furthermore, it is
faster at least by 7 times. A mesh refinement study has
been conducted, and an improvement in the results have
been observed with increasing the number of cells, which
is a good sign about the accuracy of the developed
strategy. The obtained results agree well with the exper-
imental results and are comparable with the results of the
two well-known commercial codes. A Navier–Stokes
solution has been compared to the Euler solution. It is
shown that for the transonic regime, at least for this case,
the Euler solution for predicting the trajectory can provide
satisfactory results, which in practice can save time and
effort. Similar observations have been noticed in the
literature.
Acknowledgments
The numerical calculations reported in this paper were partially
performed on the resources of the ULAKBIM High Performance
and Grid Computing Center of The Scientific and Technological
Research Council of Turkey. Wethank the Turkish Aerospace for
the support provided for this study.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
Figure 30. Roll angle of the store. Comparison between the
Euler and Navier–Stokes solutions for the fine mesh.
Figure 31. Pitch angle of the store. Comparison between the
Euler and Navier–Stokes solutions for the fine mesh.
Figure 32. Yaw angle of the store. Comparison between the
Euler and Navier–Stokes solutions for the fine mesh.
Abuhanieh et al. 11

ORCID iD
Saleh Abuhanieh https://orcid.org/0000-0002-3620-8546
Notes
1. www.openfoam.com
2. www.openfoam.com/documentation/guides/latest/doc/
guide-overset.html
3. www.openvsp.org
4. www.salome-platform.org
5. hisa.gitlab.io
6. www.csir.co.za
7. www.sourceforge.net/p/foam-extend
8. www.cfmesh.com
9. www.openfoamwiki.net/index.php/SnappyHexMesh
10. https://www.truba.gov.tr/index.php/en/main-page/
References
1. Heim ER. CFD Wing/pylon/finned Store Mutual Interference
Wind Tunnel Experiment. ADB152669 Technical Report 1991.
2. Arnold RJ and Epstein CS. AGARD flight test techniques
series on store separation flight testing. AGAR Dograph
1986; 5(300).
3. Jamison KA. Grid-mode transonic store separation analysis
using modern design of experiments. In: 31st Congress of
the International Council of the Aeronautical Sciences.
Brazil: Belo Horizonte, 2018.
4. Davids S and Cenko A. Grid based approach to store
separation. In: 19th AIAA Applied Aerodynamics Confer-
ence. Anaheim, CA, USA, 2001.
5. Panagiotopoulos EE and Spyridon DK. CFD transonic store
separation trajectory predictions with comparison to wind
tunnel investigations. Int J Eng 2010; 3(6): 538–553.
6. Sunay YE, Gulay E and Akgul A. Numerical simulations of
store separation trajectories using the eglin test. Scientific
Tech Rev 2013; 63(1): 10–16.
7. Dehghan M, Davari AR and Manshadi MD. Numerical
investigation on the weight, speed, and installation location
effects on fuel tank separation trajectory. Proceedings of the
Institution of Mechanical Engineers, J Aerospace Eng 2017;
231(13): 2331–2344.
8. Lijewski L and Suhs N. Chimera-eagle store separation. In:
AIAA Atmospheric Flight Mechanics Conference. SC,
USA: Hilton Head Island, 1992.
9. Demir H, Alemdaroglu N and Ozveren V. External store
separation from fighter aircraft. In: RTO AVT Symposium on
Functional and Mechanical Integration of Weapons and
Land and Air Vehicles. Williamsburg, VA, USA: RTO-MP-
AVT-108, 2004.
10. MacLucasa D and Gledhillb I. Time-accurate transonic CFD
simulation of a generic store release case. R D J South Afr
Inst Mech Eng 2018; 34: 9–16.
11. Khaware A, Sivanandham A and Gupta VK. Numerical
simulation of store separation trajectory for Eglin test case
using overset mesh. In: AIAA SciTech Forum. FL, USA:
AIAA Aerospace Sciences MeetingKissimmee; 2018.
12. Wadibhasme R. Exploration and Implementation of Various
Dynamicmesh in OpenFOAM (Master thesis). Centre for
Modeling and Simulation, Savitribai Phule Pune University;
2016.
13. Menon S. A numerical study of droplet formation and be-
havior using interface tracking methods interface tracking
method (PhD thesis). Mechanical and Industrial Engi-
neering. University of Massachusetts Amhers; 2011.
14. Benek JA, Steger JL and Dougherty FC. A flexible grid
embedding technique with applications to the Euler equa-
tions. In: 6th Computational Fluid Dynamics Conference.
MA, USA: Danvers; 1983.
15. Martin JE, Noack RW and Carrica PM. Overset grid as-
sembly approach for scalable computational fluid dynamics
with body motions. J Comput Phys 2019; 390: 297–305.
16. Roget B and Sitaraman J. Robust and efficient overset grid
assembly for partitioned unstructured meshes. J Comput
Phys 2014; 260: 1–24.
17. Jingjing F and Chao Y. Enhancement and application of
overset grid assembly. Chin J Aeronautics 2010; 23(6):
631–638.
18. Noack RW, Boger DA, Kunz RF, etal.Suggar++:Animproved
general overset grid assembly capability. In: AIAA Computa-
tional Fluid Dynamics. San Antonio, TX, USA; 2009.
19. Chandar D. Development of a parallel overset grid frame-
work for moving body simulations in OpenFOAM. J Appl
Computer Sci Mathematics 2015; 9(2): 22–30.
20. Abuhanieh S, Akay HU and Bicer B. A new strategy for
solving store separation problems using OpenFOAM
(abstract), In: 32nd International Conference on Parallel
Computational Fluid Dynamics. Nice: France, 2021.
(virtual).
21. Heyns JA, Oxtoby OF and Steenkamp A. Modelling high-
speed flow using a matrix-free coupled solver. In: 9th
OpenFOAM Workshop. Zagreb, Croatia, 2014.
22. Abuhanieh S, Bicer B and Sahin M. A validation for the
OpenFOAM-Hisa solver for drag prediction (abstract), In:
32nd International Conference on Parallel Computational
Fluid Dynamics. Nice: France, 2021. (virtual).
23. Lijewski L. Comparison of transonic store separation tra-
jectory predictions using the Pegasus/DXEAGLE and
Beggar codes. In: 15th Applied Aerodynamics Conference.
Atlanta, GA, USA, 1997.
24. Si H. Tetgen, a delaunay-based quality tetrahedral mesh
generator. ACM Trans Math Softw 2015; 41(2): 1–36.
25. Tomac M and Eller D. Towards automated hybrid-prismatic
mesh generation. Proced Eng 2014; 82: 377–389.
26. Liou M. A sequel to AUSM, part ii: AUSM
+
-up for all
speeds. J Comput Phys 2006; 214(1): 137+170.
27. Pandya MJ, Frink NT and Noack RW. Progress toward
overset-grid moving body capability for USM3D un-
structured flow solver. In: 17th AIAA Computational Fluid
Dynamics Conference. Toronto, ON: Canada, 2005.
28. Snyder DO, Koutsavdis EK and Anttonen JSR. Transonic
store separation using unstructured CFD with dynamic
meshing. In: 33rd AIAA Fluid Dynamics Conference and
Exhibit. Orlando, FL, USA, 2003.
29. Menter F, Kuntz M and Langtry R. Ten years of industrial
experience with the sst turbulence model. In: Proceedings of
the fourth international symposium on turbulence, heat and
mass transfer. Turkey: Antalya, 2003.
30. Huang H, Blyth RH, Prior MA, et al. A comparison of
approaches to multi-body relative motion using the Kestrel
solver. In: AIAA Scitech 2019 Forum. San Diego, CA, USA,
2019.
31. Spalart P and Allmaras S. 30th Aerospace Sciences Meeting
and Exhibit. RenoUSA: NV, 1992.A one-equation turbu-
lence model for aerodynamic flows
32. Liou M and Steffen C. A new flux splitting scheme.
J Comput Phys 1993; 107: 23–39.
33. Liou M. A sequel to AUSM: AUSM
+
.J Comput Phys 1996;
129(2): 364–382.
12 Proc IMechE Part G: J Aerospace Engineering 0(0)

Appendix I
Notations
Co. Courant number [-]
CoP center of pressure coordinate [m]
ddistance vector magnitude [m]
D
i
donor cell i [-]
especific internal energy [J/kg]
Fapplied forces vector [N]
F(W)flux vector (convective flux and viscous flux)
Hangular momentum vector [kg m
2
/s]
I
i
interpolated cell i [-]
Mapplied moments vector [Nm]
mVlinear momentum vector [kg m/s]
N
D
number of donors [-]
pstatic pressure [Pa]
qheat flux vector [W/m
2
]
Q(W) source terms vector
ttime [s]
vfluid velocity vector [m/s]
Vcell volume [m
3
]
Wconservative fluid flow variables vector
xcoordinate vector [m]
y
+
non-dimensional wall distance [-]
θpitch angle (around y) [
o
]
λthe characteristic time scale [1/s]
ρdensity [kg/m
3
]
σmechanical stress tensor [N/m
2
]
τviscous stress tensor [N/m
2
]
froll angle (around x) [
o
]
χface volumetric flux [m
3
/s]
ψyaw angle (around z) [
o
]
ξ
i
field value at cell i
ω
i
interpolation weight of cell i [-]
Δttime step size [s].
Appendix II
overRhoPimpleDyMFoam case setup
The details of the setup for the case solved with over-
RhoPimpleDyMFoam are summarized in Tables 2–4.
The standard OGA
For this test case (Eglin 0.58 Mcells) only the first-order
cellVolumeWeight scheme worked with the over-
RhoPimpleDyMFoam solver. In the OpenFOAM im-
plementation, the cellVolumeWeight is not only an
interpolation scheme, to be used by the interpolated cell to
get the value from the donors cells, but it includes the
OGA process, which has been described in The overset/
chimera method section. A slice showing the overset DCI
for the overset/store mesh (Figure 33(a)) and the back-
ground mesh (Figure 33(b)) is shown in Figure 33.
In this work, studying the current/available capabilities
of the standard OpenFOAM to solve the store separation
problem reveals the following limitations:
(A) The solution obtained by the overRhoPimpleDyMFoam
is reasonable. However, the accuracy is not good
enough.
(B) A converged solution with a second-order upwind-
ing scheme was not obtained. This can be attributed
to the limited numerical stability of the used seg-
regated flow solver.
(C) The computational time is high (842 core-hours for
the Eglin case with 0.58Mcells), and more than the
half of each time step execution time is consumed
by the overset processes. Thus, running a fine mesh
may easily turn the case to be non-practical to be
solved.
(D) Only the first-order overset interpolation scheme was
used (cellVolumeWeight), since the other schemes
(inverseDistance and leastSquares) OGA algorithms
failed to classify the cells (either calculated, hole or
interpolated) correctly. A first-order overset in-
terpolation scheme is not enough to obtain accurate
results.
Table 2. Solver settings.
Solver overRhoPimpleDyMFoam
OpenFOAM version v2006
Flow Transient inviscid
Discretization schemes First-order upwinding in space
and first-order implicit in time
(backward euler)
Time step 1 × 10
4
s (maximum)
Maximum courant number 5.0
Pseudo time (inner loop)
settings
Maximum iterations = 30,
tolerance = 1 × 10
4
, there
is no dual time scheme option
Free-stream Mach number 0.95
Overset interpolation
scheme
cellVolumeWeight (first-order)
Table 3. Boundary conditions.
Patch/field Pressure Velocity Temperature
Farfield Free-stream
pressure
Free-stream
velocity
inletOutlet
wing_and_pylon zeroGradient Slip zeroGradient
Store zeroGradient movingWall
velocity
zeroGradient
oversetPatch Overset Overset Overset
Symm Symmetry Symmetry Symmetry
Table 4. dynamicMeshDict settings.
dynamicFvMesh dynamicOversetFvMesh
Solver sixDoFRigidBodyMotion
Abuhanieh et al. 13

The developed strategy case setup.
The details of the setup for the case solved with the
HiSA solver and the developed strategy are summarized in
Tables 5 and 6.
The adapted OGA results
For the adapted overset library, only one OGA algorithm is
available. However, for selecting the fringe (interpolated
cells) from each mesh, different methods can be selected
(manual, overlap, faceCells, etc.).
Unlike the standard OpenFOAM overset library, in this
implementation, the OGA algorithm is independent from the
interpolation scheme. In the present study, the interpolation
schemes which have been implemented in the overset library
are: inverseDistance and the leastSquares as shown in Figure 5
in blue under the interface class. The prefix“foamExtend”has
been added to distinguish them from the ones used by the
standard library to be able to select them during run time. A
slice showing the DCI for the overset/store mesh (Figure
34(a)) and the background mesh (Figure 34(b))isshownin
Figure 34. The overlap fringe has been used here.
Appendix III
AUSM
+
-up Scheme
AUSM
+
-up is a second-order upwinding scheme for shock
capturing, which is an extension to the AUSM (Advection
Upstream Splitting Method) family of schemes.
32,33
It has
been developed to improve the prediction of flows at low
Mach speeds, making the original AUSM scheme work ef-
ficiently at all speeds. The algorithm can be described as
follows
26
:
Table 5. Solver settings.
Solver HiSA (version 1.4.2)
OpenFOAM version v2006
Flow Transient inviscid
Discretization schemes Second-order upwinding in
space (AUSM
+
-up) and
first-order implicit in time
(backward euler)
Time step 1 × 10
4
s (maximum)
Maximum courant number 60.0
Pseudo time (inner loop)
settings
Maximum iterations = 100,
tolerance = 5 × 10
3
,
using dual time scheme
Free-stream Mach number 0.95
Overset interpolation scheme foamExtend_inverseDistance
(first-second-order)
dynamicMeshDict settings Same as the
overRhoPimpleDyMFoam
settings
Table 6. Boundary conditions.
Patch/field Pressure Velocity Temperature
Farfield Characteristic farfield pressure Characteristic farfield velocity Characteristic farfield temperature
wing_and_pylon Characteristic WallPressure Slip Characteristic WallTemperature
Store Characteristic WallPressure movingWall velocity Characteristic WallTemperature
oversetPatch Overset Overset Overset
Symm Symmetry Symmetry Symmetry
Figure 33. The overset DCI for the overset/store mesh (a) and the background mesh (b). The calculated cells are in blue, the
interpolated cells are in yellow and the hole cells are in red (plane: y normal at the initial center of mass). The observed limitations for the
standard OpenFOAM.
14 Proc IMechE Part G: J Aerospace Engineering 0(0)

(i) For each interface, calculate the Mach number’s left
and right states
ML=R¼uL=R
a1=2
(11)
where M
L
/R are the Mach number left and right states, u
L
/
R are the convective velocities (vn), and a
1/2
is the speed
of sound at the interface (can be obtained by averaging the
a
L
and a
R
).
(ii) Calculate the mean Mach number
M2¼u2
Lþu2
R
2a2
1=2
(12)
(iii) Calculate the reference Mach number
M2
o¼min1,maxM2,M2
∞2½0;1(13)
where M
∞
is the free-stream Mach number.
(iv) Find the scaling function value
fo¼Moð2MoÞ2½0;1(14)
(v) Evaluate the split Mach numbers (M
±
(4)
) for the left
Mþ
ð4ÞðMLÞand right M
ð4ÞðMRÞMach number states.
The used split Mach number is a fourth-order
polynomial function.
(vi) Calculate the interface Mach number
M1=2¼M
þ
ð4ÞðMLÞþM

ð4ÞðMRÞ
Kp
fa
max1σM2,0pRpL
ρ1=2a2
1=2
(15)
where K
p
and σare constants and p
R
and p
L
are the
pressure right and left states, respectively. ρ
1/2
is the av-
erage of the density left and right states.
(vii) Calculate the mass flux at the interface
_
m1=2¼a1=2M1=2
ρLif M1=2>0
ρRotherwise
(16)
(viii) Calculate the pressure flux
p1=2¼P
þ
ð5ÞðMLÞpLþP

ð5ÞðMRÞpR
kuPþ
ð5ÞðMLÞP
ð5ÞðMRÞðρLþρRÞfaa1=2ðuLuRÞ
(17)
where K
u
is constant. P±
ð5Þis a fifth-order polynomial.
(ix) Finally, evaluate the total flux at the interface
f1=2¼_
m1=2
ψLif _
m1=2>0,
ψRotherwise þp1=2
(18)
where ψis a vector quantity convected by _
m. In 1-D, ψ=
(1,u,H)
T
, where His the total enthalpy. pis the pressure
flux vector, which contains only one pressure term, p= (0,
p,0)
T
in 1-D.
Figure 34. The overset DCI for the overset/store mesh (a) and the background mesh (b) for the fine mesh. The calculated cells are in
blue, the interpolated cells are in yellow, and the hole cells are in red (plane: y normal at the initial center of mass).
Abuhanieh et al. 15