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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=fρvvg¼ð=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

ðωDiÞξDi(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 kωSST (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/

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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 [Nm]

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

o¼min1,maxM2,M2

∞2½0;1(13)

where M

∞

is the free-stream Mach number.

(iv) Find the scaling function value

fo¼Moð2MoÞ2½0;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

max1σM2,0pRpL

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

ð5ÞðMRÞðρLþρRÞfaa1=2ðuLuRÞ

(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