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DLR LK6E2 Missile Airframe Validation Study Using
Wall-Modeled Large Eddy Simulations
Jordan B. Angel∗, Aditya S. Ghate, Gaetan K. W. Kenway, Man Long Wong, and Cetin C. Kiris
Volcano Platforms Incorporated, Palo Alto, CA 94304, USA
This paper presents numerical simulation results for a realistic generic missile in transonic
flow conditions using a recently developed Cartesian octree immersed boundary Wall-Modelled
Large Eddy Simulation (WMLES) solver, Volcano ScaLES. The missile used in this paper
is the subject of NATO Science & Technology Organization (STO) efforts AVT-316 (“Vortex
Interaction Effects Relevant to Military Air Vehicle Performance”, previously) and AVT-390
(“Vortex Flow Predictions for Stability and Control of Missile Airframes”, currently) which
study the prediction of an experimentally-demonstrated break in rolling moment for total
angles of incidence
𝝈
beyond
16◦
. Predicting the wind tunnel results from an angle of attack
sweep is considered as a validation case for the solver and the study is taken on in two parts.
First, results of a consistent grid refinement study are presented for three angles of attack.
Satisfactory accuracy is achieved with a medium resolution “best practice” grid that is suitable
for engineering analysis, and subsequently this grid is used to simulate three additional angles
of attack to complete an angle sweep. The LES solutions accurately capture the break in rolling
moment which eluded many Reynolds-Averaged Navier–Stokes (RANS) solvers in the results
from AVT-316. Finally, as a measure WMLES readiness for routine industrial design use,
computational resources and timings are also detailed e.g., the solver is capable of completing an
aerodynamics simulation using the 435M cell grid in two days using a commodity workstation
with only
2×
NVIDIA RTX 4090s while simulations using the best practice mesh at 761M cells
are completed in roughly twenty-four hours using a single server with
8×
NVIDIA L40S cards.
I. Introduction
Computational fluid dynamics (CFD), particularly steady-state Reynolds-Averaged Navier–Stokes (RANS) methods,
have been used widely in industrial aircraft development programs to reduce design costs or to accelerate testing cycles;
while new development programs are likely to expand their reliance on CFD, as solvers and hardware capabilities
continue to improve [
1
]. The applicability of numerical simulation to the design and testing of aircraft is limited to
flow conditions for which the solver has been validated, typically through a representative case study where data is
available to confirm the numerical predictions. The maneuverability of agile tactical aircraft and missiles requires
testing aerodynamic performance at flight conditions that are sometimes beyond the capability of most current methods
of analysis. Aerodynamic flows that occur during critical flight conditions (e.g. take-off, landing, and high-speed
maneuvers) are often highly unsteady and complex nonlinear dynamics such as shock buffeting, massive flow separation,
and complex vortex interactions are typical [
2
–
4
]. An insightful perspective on these limitations is given in a paper by
Smith [
1
] which details the requirements and the opportunities for expanded use of CFD in a development program of
a new fighter aircraft. In that work, a classification is defined where test conditions fit into either an ‘A’, ‘B’, or ‘C’
category with ‘A’ being the most routine and least demanding conditions (cruise, low alpha, no vortex separation) and
‘C’ representing the conditions which require scale-resolving simulations or wind tunnel testing owing to the several
complex nonlinear flow phenomena mentioned already. With some estimates from the F-35 development program,
estimates demonstrate that, for a fully digital development campaign, the conditions in group ‘C’ would dominate the
other two categories, consuming more than 80% of the total required core hours. The greatest reduction in program
cost through CFD would come from cost-effective methods for solving problems in the ‘C’ class. Two capabilities
are highlighted that could significantly reduce the computational costs of development: mature and well-validated
Wall-Modelled Large Eddy Simulation (WMLES) and GPU accelerated solvers. This paper may be seen as an effort
towards demonstrating these capabilities.
There are no universal criteria such that one can claim that a numerical solver has been “validated”; validation can
only be established for particular problems where reliable data already exists [
5
]. As an example, the NATO Science &
∗Corresponding author.
1
Technology Organization (STO) efforts AVT-316 and AVT-390 [
6
–
9
] were organized to assess the capability of CFD
solvers to predict an experimentally-demonstrated break in rolling moment for total angles of incidence
𝜎
beyond
16◦
.
The transonic regime and high-angle of incidence create conditions for several complex and unsteady phenomena:
shock-induced separation at the nose, shock boundary layer interactions over the wings, shock-vortex interactions from
the leading-edge vortices, and largely separated regions over the wings at higher angles. The geometry of the missile
is depicted in a rolled configuration in Figure 1 along with a visualization if Q-criterion that shows some of the flow
features described.
Wing 3
Wing 2
Wing 1
Wing 4
Nose Shock
Wing Separation
Fig. 1 Left: The missile frame considered in this paper shown with a perspective view. Right: Iso-surface of
Q-criterion colored by the streamwise velocity component for a typical solution.
It has been noted that the break in the rolling moment is due to an asymmetric breakdown of the leading edge vortices
causing a sudden change in the lift behavior for increasing angle [
9
]. This problem is viewed to be a representative case
2
that may arise during the industrial design of a missile. Predicting these kinds of critical phenomena is essential for
enabling the practical usage of LES for industrial design and so, the problem is taken up here as a meaningful validation
case for the GPU native Volcano ScaLES solver. Other validation studies using Volcano ScaLES include: aerodynamics
of the F-16XL at high angles of attack [
10
], aerodynamics of iced straight wings [
11
], automotive aerodynamics [
12
],
subsonic jets [13], vibroacoustics of a launch vehicle [14], and commercial aircraft high-lift aerodynamics [15].
The next section describes the solver with some discussion of designing simulation software for modern hardware.
The simulation results of Section III are divided into two parts. The first half presents results from a systematic grid
refinement study performed using three mesh resolutions at three different angles of incidence. For these three angles,
results are compared that show sharply diminishing differences as the mesh spacing is refined. Based on the results of
this study, a satisfactory grid resolution is chosen and used to perform three additional simulations for a total angle
sweep using six angles. The results of the sweep are presented in the second half of Section III where details of the flow
are examined. Section IV will provide an overview of the results and some discussion of future work planned.
II. Volcano ScaLES
Use of general-purpose programmable GPUs has been mainstream in the scientific supercomputing community for
more than a decade at this time. Recent breakthroughs in generative artificial intelligence and deep learning technologies
that rely heavily on this hardware are driving a surge in GPU production and it appears likely they will continue. The
increased production and proliferation of GPUs have made them easily accessible outside of dedicated computing
centers. Engineering companies that use computationally aided design are now building on-premises GPU clusters.
These devices potentially offer an enormous boost in computational throughput, but the degree to which this potential is
realized depends entirely on the design of the simulation software which will look very different compared to CPU
based predecessors. The GPU hardware is of a fundamentally different design than the CPUs which developers have
programmed for decades and effectively using the hardware requires different design patterns and algorithms. In this
section we briefly describe the computational software used for the simulations.
Volcano ScaLES is a compressible Navier–Stokes solver utilizing a nominally fourth-order spatially accurate
finite difference discretization with favorable kinetic energy and entropy consistency properties making it suitable for
Large Eddy Simulations (LES) over a vast range of flow regimes. Motivated by the need to fully exploit all available
resources on modern GPUs, Volcano ScaLES is written exclusively in C++17 and CUDA. The solver is designed for
Cartesian octree meshes which are well-suited for large-scale simulations on GPUs. The octree meshes are generated
very rapidly which significantly lowers the barrier for rapid experimentation by the user. Grids with over a billion
cells are generated in a couple of minutes. Changing geometries is often as simple as a one-line change in the input
file. The majority of computations are compute bound, and high performance is achieved even on graphics-oriented
devices, bandwidth-restricted devices which are far less expensive compared to higher memory bandwidth hardware
like the H100. In simulations involving non-canonical geometries, such as those considered in this paper, we achieve
approximately 30% of the peak FLOPs on compute-oriented devices along with nearly perfect (>95%) weak scaling
efficiency from 1 to 8 GPUs. The combination of Cartesian methods and an explicit time-stepping scheme yields
extremely low memory requirements: ScaLES can simulate between 8.5 and 10 million cells per gigabyte of GPU
memory, depending on the complexity of the configuration. Simulations containing 400 to 500 million cells are routinely
performed on desktop computers containing just two RTX 4090 GPUs with 24 GB of memory each. The ability to fully
exploit lower-specification hardware significantly lowers the cost of LES simulations to a practical level for use in an
industrial setting.
Geometries are represented using an immersed boundary method with appropriate asymptotic Reynolds number
properties using an equilibrium wall-model. The wall model that provides a shear stress constraint on the wall uses
solution information interpolated through probing at a fixed distance of
3Δ/2
in the fluid away from the walls, where
Δ
is the local grid spacing. The viscous flux discretization utilizes a mix of fourth-order and second-order operators
with high spectral bandwidth, thereby allowing further robustness for high Reynolds number flows without additional
numerical dissipation via the inviscid flux numerical operator in turbulence-resolving regions. Physics-based numerical
sensors that are functions of local velocity gradients, as well as pressure and density fluctuations, allow for localized use
of limiters needed to capture flows with discontinuities such as shocks and steep density gradients. A key convenience
afforded by the use of Cartesian grids with unit aspect ratio is the use of an explicit time-stepping scheme even in
flows involving highly complex geometries; the perfectly isotropic grid cells near the geometry do not introduce any
numerical stiffness associated with complex grid topologies. The simulations presented in this work use the dynamic
procedure Smagorinsky model [
16
,
17
] to model the subgrid-scale stresses. The Strong Stability Preserving (SSP)
3
variant of the classical third order Runge–Kutta scheme developed by Gottlieb and Shu [
18
] is utilized for all the work
presented in this paper.
III. Grid Refinement Study and Experimental Comparisons
This section provides a concise description of the test case and the computational setup for the grid refinement study.
Details of the test that are not immediately relevant are omitted. For a comprehensive overview, we recommend the
following papers, [6–9].
A. Initial and Boundary Conditions
The nominal conditions used to set up the problem are provided in Table 1 as a reference. The upper rows indicate
the quantities used to define the flow conditions with the symbols used in this paper and their units. All simulations are
initialized from the freestream conditions derived from Table 1. The reference length
𝑑ref
is the diameter of the missile.
Simulations are run in free-air with a domain
200 ×𝐿
where
𝐿=10𝑑ref
is the length of the body. The freestream Mach
number, speed of sound, and total angle of incidence, 𝜎, define the initial velocity vector field.
Table 1 Nominal freestream and reference conditions.
Static pressure Static temperature Roll angle Mach number Ref. length
𝑝∞[Pa] 𝑇∞[◦K] 𝜆[deg] 𝑀[-] 𝑑ref [m]
47 075 270 45 0.85 0.05
B. Mesh Considerations
Here we will describe the Cartesian octree mesh
∗
used for the grid refinement study. As already mentioned, the
domain is taken to be
200 ×𝐿
where
𝐿
is the length of the missile. The mesh is constructed using a bottom-up procedure.
A single spacing parameter
ℎ
is specified as input which represents the baseline spacing for all blocks intersecting
geometry. Other levels of the octree are filled on the basis of constraints given as input. Constraints can be volume or
surface refinement regions tagged so that their local spacing matches some refinement level relative to
ℎ
. That is, blocks
tagged with a local spacing corresponding to a relative level
ℓ
will have a spacing
ℎ/2ℓ
. For example, a base spacing
ℎ=1 mm
with a volume region specified to have a relative level
ℓ=−1
would refine all blocks in the region to a local
spacing of ℎ=2 mm. Surface regions can also specify the distance that the spacing should extend into the volume.
The mesh is depicted in Figures 2 and 3. Figure 2 shows a center line slice of each mesh. This view is used to show
two volume refinement regions that are used in anticipation of the separated vortical flow. Since the same mesh is used
for all angles, the slope of the conical volume regions are chosen based on observations of coarse grid solutions at the
most separated angle. These are the only volume refinement regions we specify. Next, Figure 3 shows a constant
𝑥
cut
of the meshes near the leading edge of Wing 2. A surface refinement region can be seen on the suction side of Wing 2
where the refined region extends a fixed distance into the volume. As the mesh is refined, the interface will remain
roughly fixed in this location. Wing 4 has a similar refinement. These regions are tagged for refinement based on the
expectation that the flow will be largely separated will lift into a spiral vortex. A “well-balanced” property is maintained
such that adjacent blocks never differ in their relative levels by more than one. The geometry and refinement regions
define the topology for the mesh that will remain fixed and global refinement is carried out by reducing the baseline
spacing ℎ. A summary of three meshes with their spacing ℎand total number of grid points is given in Table 2.
∗We will use ‘mesh’ or ‘grid’ interchangeably.
4
𝐺1
𝐺2
𝐺3
Fig. 2 Constant 𝒚=0m slice of the computational mesh through the missile center line.
5
𝐺1
𝐺2
𝐺3
Fig. 3 Constant
𝒙=−0.22
m slice of the computational mesh looking downstream over the leading-edge of
Wing 2.
6
C. Grid Refinement Study
Table 2 provides details of the meshes used in the refinement study. We label the grids
𝐺1
,
𝐺2
, and
𝐺3
, numbering
the grids from coarsest to finest. The second and third columns give the baseline spacing
ℎ
and the total number of cells
in the grid. The mesh
𝐺1
has 435 million grid cells, but can be run on two NVIDIA RTX 4090 cards with 24 Gb of
memory each. Meshes
𝐺2
and
𝐺3
require more memory and are run on a server using
8×
NVIDIA L40S cards. The
third and fourth columns provide some performance metrics in terms of the wall clock time needed to integrate
50 ms
of
physical time and the number of millions of grid cells updated per second (MUPS), i.e.
7100
MUPS means that 7.1
billion cells are updated per second.
Table 2 Computational cost for the simulations presented in this section.
Label ℎNo. of cells Wall time / 50 ms MUPS Hardware
[mm] [106] [hr] [106cells/sec]
𝐺18.75 ×10−2435 42 1950 2×RTX 4090
𝐺27×10−2761 26 6800 8×L40S
𝐺35.5×10−21360 60 7100 8×L40S
Three angles of incidence are chosen, two from the pre-break regime, and one post-break angle:
14◦
,
16◦
, and
17.5◦
.
Each angle is simulated using all three grids and their roll moment time histories are plotted in Figure 5. The time
histories are plotted with solid lines and colored by their grids. Dashed lines are drawn to indicate the computed mean,
and a shaded band is also drawn to indicate a standard deviation about the mean.
Fig. 4 Rolling moment versus angle of incidence for the grid refinement study.
At
14◦
and
16◦
, refining the coarse grid significantly changes the solution, but
17.5◦
shows almost no difference. It
will be shown in the next section that at this angle, much of the coherent vortical flow has broken down into a large
separation bubble. With this in mind, it is not expected that grid refinement will change the rolling moment since
additional resolved content would not contribute to the moment anyway. The negligible differences in the rolling
7
moments are most likely due to the expected uncertainty inherent in any estimation of a mean and, with respect to rolling
moment, the solution is converged. At all angles, the solutions from the medium resolution
𝐺2
grid are essentially
the same as the fine grid. The surface pressure averaged over time is plotted in Figure 6. The differences between
grid resolutions are minor on this scale. Even the
𝐺1
solutions show an asymmetric vortex break down, although less
pronounced than the
𝐺2
and
𝐺3
solutions. Because of the roll angle, Wing 4 sees a slightly larger effective angle of
incidence compared to Wing 2 and as the angle of incidence increases, surface pressure at the trailing edge of Wing 4
decreases more than Wing 2, primarily in the outboard region. This difference in pressure gradients influences the
locations of the shocks and later it will be shown that the normal shock on Wing 4 vanishes at a slightly smaller
𝜎
than
the normal shock on Wing 2. See [8] for comparisons to the wind tunnel surface pressure.
Table 3 Rolling moment coefficients by angle of incidence.
𝜎 𝐺1𝐺2𝐺3
Total angle (435 M) (761 M) (1.36 B)
14.0◦-0.529 -0.585 -0.595
16.0◦-0.609 -0.638 -0.644
17.5◦-0.558 -0.556 -0.569
Figure 4 and Table 3 show the mean and standard deviation respectively for all simulations. The results convincingly
show grid convergence for
14◦
and
16◦
. Based on the Grid Convergence Index (GCI) [
5
], the expected change from
further grid refinement is about 2%. As already discussed, due to the change in flow physics, increasing grid resolution
for
𝜎=17.5◦
is not expected to have a strong influence on the rolling moment so the answer is essentially converged for
our purpose. We conclude that the
𝐺2
grid is sufficiently accurate and provides the best trade-off between accuracy and
computation time.
8
Fig. 5 Time histories for rolling coefficient from the grid refinement study. Means are shown as dashed lines
and
±1
standard deviation is shown by a shaded region. The start of the shaded region indicates the start of
time-averaging.
9
𝜎=14.0◦𝜎=16.0◦𝜎=17.5◦
Coarse
MediumFine
Fig. 6 Surface 𝑪𝒑distributions for angles 𝝈=14.0◦,16.0◦, and 17.5◦.
10
D. Angle Sweep Results and Comparisons to Wind Tunnel Data
To fill out the sweep we add the three angles
𝜎=13◦
,
15◦
, and
19◦
. The rolling moment curve for the entire sweep
is shown in Figure 7 along with wind tunnel data collected from two different tunnels plotted in black with the data
spread shaded gray. The time histories for each of these simulations are plotted in Figure 8 with the time-average
value depicted as a dashed blue line. Figure 8 also shows the experimental data spread that is estimated for each angle
by using the shaded region in Figure 7. The simulations reproduce the wind tunnel results remarkably well and the
moment break beyond
16◦
is seen clearly. An individual simulation takes roughly twenty-four hours to complete using
8×
NVIDIA L40S GPUs. Two nodes with this configuration were used to run simulations simultaneously, completing
all six simulations in just over three days. This is a compelling indication that LES can be used, today, as a practical
design tool with demonstrated improvements at challenging flight conditions. Before examining the flow in detail we
remark that the angles chosen here are not clustered to sample the weaker break near
14◦
; future simulations are planned
for this regime.
Fig. 7 Complete angle sweep using the medium mesh. Mean values are shown with circles and connected by a
dashed line. Experimental values are shown in black with the spread shaded in gray [8].
It is important to note that wind tunnel experiments carried out at over a range of Mach numbers
0.3< 𝑀 < 0.85
did not show a roll moment break at
𝜎=16◦
for
𝑀 < 0.6
and the break near
14◦
was only seen at
𝑀=0.85
[
9
], so it is
reasoned that the shocks play an essential role in causing this behavior. Figure 9 shows density plotted on a constant
𝑧
slice above Wing 4 for
𝜎=13◦
as well as the core of the missile colored by the gradient of the average pressure
coefficient,
|∇𝐶𝑝|
. This figure highlights some of the well-known characteristics of shock-vortex interactions that are
also seen in the simulations. Near the leading-edge root of Wing 4, the vortex roll-up begins and a narrow vortex core
forms which can be traced downstream to the point where it passes through a normal shock which is evident in the
surface pressure. The adverse pressure gradient from the shock reduces the axial velocity of the vortex, changing the
balance between the centrifugal force and the core pressure gradient. The vortex begins to expand at an increased rate
compared to its pre-shock state which can be observed in the density. Eventually, all coherence is lost and the vortex is
completely broken down.
An explanation for the break in rolling moment beyond
𝜎=16◦
given in [
9
] states that around
𝜎=15◦
, the leading
normal shock on Wing 4 disappears. This changes the flow separation topology on Wing 4, leading to a large region of
separated flow on the outboard section of the leading edge. This is consistent with the present WMLES. Figure 10 plots
11
Fig. 8 Load histories for all six angles of the sweep. Experimental data band shown in gray and black.
constant
𝑦
slices of density for both Wing 2 and Wing 4 at
𝜎=13◦
,
14◦
, and
15◦
. Two normal shocks are seen, and
at
𝜎=13
the adverse pressure gradient due to the shock diverts the separated flow from the leading-edge outboard .
Looking at the shocks on Wing 4 for
𝜎=14◦
, the leading shock appears weaker and fluctuations behind the shock
suggest some rotational flow is starting to pass across the shock, but largely the separated flow is confined to the leading
edge. Increasing to
𝜎=15◦
the leading normal shock is almost diminished and the outboard separated flow begins to
cover more area of the wing. All the while, the shock structure on Wing 2 is relatively stable. The leading shock also
weakens on the Wing 2 side but not as severely. Importantly, due to the roll angle, the incoming flow on the Wing 2 side
tends to be glancing while Wing 4 tilts more towards the incoming flow which sees higher incoming momentum. The
same plots for the higher angles are shown in Figure 11. Once
𝜎=16◦
, the simulation results show the adverse pressure
gradient near the leading edge of Wing 4 has reduced to a smoother gradient although the presence of the shock has not
completely gone yet. Finally, beyond
16◦
, the pressure gradients over Wing 4 collapse, massive flow separation occurs,
and a weaker normal shock settles connecting Wing 4 and Wing 3. Figure 12 confirms this in the surface streamlines
12
Leading Edge
Roll Up
ShocksCore Formation
Shock/Vortex
Interaction
Vortex
Expansion
Wing 4
Fig. 9 Density is shown on a constant
𝒛
slice just above Wing 4. The fuselage is seen colored by
|∇𝑪𝒑|
. The
features of the shock-vortex interaction at
𝝈=13◦
are indicated with arrows. A perspective view of the slice is
shown for context.
computed from the instantaneous skin friction for all six incidence angles. The angles are plotted in increasing order
from left to right and from top to bottom. Starting from the top, the
13◦
and
14◦
simulations show a typical vortex
influence with some separation near the leading edges, more noticeably on Wing 4. As the angle increases to
15◦
and
16◦
, the separation region on Wing 4 moves more inboard and creates an identifiable separation zone. Beyond
16◦
, the
separation bubble has moved completely over Wing 4 and the leading edge vortex has broken down, while the leading
edge vortex from Wing 2 is largely intact. The Wing 2 leading-edge vortex remains mostly stable with some evidence of
vortex bursting seen in the streamlines towards the trailing edge.
Another way of demonstrating the loss of coherence for the Wing 4 vortex is shown in Figure 13 using density
on a plane at a constant
𝑧
coordinate that cuts the vortex core near the leading edge. The surface is also colored as in
previous figures to indicate the location of shocks. Shock-vortex interactions are seen clearly for the lower angles. At
𝜎=16◦
, the last evidence of coherent vorticity can be seen in the density fluctuations. Increasing the angle further
leads to a total loss of the vortex core and the flow over Wing 4 changes character to a large separated bubble with a free
shear layer originating at the wing edge.
13
𝜎=13.0◦
Wing 4 Wing 2
𝜎=14.0◦
Wing 4 Wing 2
𝜎=15.0◦
Wing 4 Wing 2
Fig. 10 Density viewed over Wing 4 and Wing 2 showing the normal shocks and regions of separated flow for
the lower angles. The surfaces are colored by skin friction with lighter regions indicting flow separation. Both
views are looking inboard so that the flow appears reversed.
14
𝜎=16.0◦
Wing 4 Wing 2
𝜎=17.5◦
Wing 4 Wing 2
𝜎=19.0◦
Wing 4 Wing 2
Fig. 11 Density viewed over Wing 4 and Wing 2 for the higher angles showing the normal shocks and regions of
separated flow. The surfaces are colored by skin friction with lighter regions indicting flow separation. Both
views are looking inboard so that the flow appears reversed.
15
𝜎=13.0◦
Wing 4
Wing 2
𝜎=14.0◦
Wing 4
Wing 2
𝜎=15.0◦
Wing 4
Wing 2
𝜎=16.0◦
Wing 4
Wing 2
𝜎=17.5◦
Wing 4
Wing 2
𝜎=19.0◦
Wing 4
Wing 2
Fig. 12 Instantaneous skin friction streamlines showing the asymmetric vortex breakdown. The separation on
Wing 4 moves inboard with increasing angle. Beyond some critical angle between
16.0◦
and
17.5◦
the leading-edge
vortex breaks down and the flow on Wing 4 shows a large separation bubble.
16
𝜎=13.0◦𝜎=14.0◦
𝜎=15.0◦𝜎=16.0◦
𝜎=17.5◦𝜎=19.0◦
Fig. 13 Density viewed over Wing 4 on a constant
𝒛
plane that cuts the vortex core near the leading edge. Loss
of coherent vorticity is evident with total break down beyond 𝝈=16◦.
17
IV. Conclusions
This work presented results of a validation study of an immersed boundary WMLES solver applied to a generic
missile in transonic flow. A consistent grid refinement study at three angles of incidence showed that a medium
resolution 756 million cell mesh provided essentially converged solutions. Additional simulations at other angles were
carried out to perform an angle sweep and results accurately predict a known rolling moment break that has challenged
state-of-the-art CFD codes. The sweep simulations were computed using only
8×
NVIDIA L40S cards taking about
twenty-four hours to complete per simulation. Besides the rapid turnaround time, a major enabler for industrial use is
the rapid and iterative grid generation procedure that is used by ScaLES. The 1.36 billion point grids used in this work
took fewer than 5 minutes to generate. This rapid workflow allows for systematic grid convergence studies even in
non-canonical real-world problems where flight or experimental data is not available. Finally, we note that explicit
time-stepping with structured grid blocks is highly amenable to GPU parallelism where fine granularity of parallelism
(down to the cell) is often key to achieve optimal performance.
Acknowledgments
The experimental data used in this paper was provided by the German Aerospace Center (DLR). The authors would
like to thank Kai Richter (DLR) and Christian Schnepf (DLR) for making the wind tunnel data available, and the NATO
STO AVT-390 project team for providing the geometry and conditions. We would like to acknowledge and thank
Nigel Taylor (MBDA UK), James DeSpirito (Army Research Labrartory), and Magnus Tormalm (Swedish Defense
Research Agency) and the rest of the AVT-390 team members for the generous invitation that initiated this work and for
discussions and suggestions that we found tremendously helpful when preparing this paper.
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Appendix: Surface Pressure Cuts
The diagram in Figure 14 shows the spatial location of several cuts where
𝐶𝑝
is compared to experimental data
for the simulations in Section III. Figures 15, 16, 17, 18, 19, and 20 plot the time-averaged surface data in solid blue
along with the wing tunnel data as black square markers. Some discrepancies can be seen but overall the agreement is
excellent.
Wing 3
Wing 2
Wing 4
Wing 1
𝑦′
𝑦′/𝐷=1.8
𝑦′/𝐷=1.2
𝑦′/𝐷=0.6
𝑥/𝐷=−4.4
𝑥/𝐷=−5.3𝑥/𝐷=−6.8
Fig. 14 Missile geometry at roll 𝝀=45◦. The coordinate 𝒚′extends along the span of Wing 4.
19
Fig. 15 Cuts for constant 𝒚stations at 𝝈=14◦.
20
Fig. 16 Cuts for constant 𝒙stations at 𝝈=14◦.
21
Fig. 17 Cuts for constant 𝒚stations at 𝝈=16◦.
22
Fig. 18 Cuts for constant 𝒙stations at 𝝈=16◦.
23
Fig. 19 Cuts for constant 𝒚stations at 𝝈=17.5◦.
24
Fig. 20 Cuts for constant 𝒙stations at 𝝈=17.5◦.
25
