1 1 Director, Modeling & Simulation Research Center, Dept of Aeronautics, HQ USAFA/DFAN, AIAA

Member 2 Visiting Researcher, 2354 Fairchild Drive, USAFA, CO 80840

3 Research Engineer, AIAA Member

Computational Investigation of the Upsweep Flow Field

for a Simplified C-130 Shape

Keith Bergeron

1 and Jean-Francois Cassez

2

United States Air Force Academy, Modeling and Simulation Research Center

2410 Faculty Drive, Suite 108

USAF Academy, CO 80840-6400, USA.

Yannick Bury3

Institut Supérieur de l’Aéronautique et de l’Espace

Département Aérodynamique, Energétique et Propulsion

BP 54032, 31055 Toulouse Cedex 4, FRANCE Delayed Detached-Eddy Simulation (DDES) has been applied to various configurations of a

simplified C-130 wind tunnel model. This research represents a continuation of work to determine

the flow-field characteristics of the C-130 Hercules aircraft as part of the NATO supported Airflow

Influence on Airdrop (AIA) project. The models include both closed and open cargo door

configurations, and both the simulations and experiments use the same Computer-aided design

(CAD) geometries. A combined grid-resolution and time-step sensitivity study was performed for

each geometry and serves to ensure the numerical simulations are accurate for the separated flows.

Excellent agreement is found between experiments and DDES simulations for vortex location,

strength and orientation, and the DDES results are shown to compare more favorably than previous

Detached-Eddy Simulation (DES) results for these configurations.

Nomenclature

= free-stream pressure

St = Strouhal Number

= free-stream temperature

= free-stream velocity y+ = dimensionless wall distance

= aircraft angle of attack

β = aircraft sideslip angle = Reynolds number based on fuselage length, L

I. Introduction

This work supports the AIA project under The Four Power Air Senior National Representative

Cooperative Long term Technology Projects program. The AIA project began in July 2002 with the

aim of making recommendations to enhance airdrop capabilities. Program outcomes are to include

increased safety (for flight crew personnel, parachutists, and loads), improved airdrop delivery

accuracy, and efficient use of resources. A detailed overview of the AIA project and its relationship

with the NATO Joint Precision Airdrop Capabilities Working Group is given in Seeger, et. al.1

In particular, the upsweep region of cargo aircraft causes a region of detached flow which evolves

with aircraft configuration (e.g., angle-of-attack, airspeed, and cargo door open/closed). The strong

vortices generated not only affect airdrop operations but also have a significant effect on drag2.

Therefore, in addition to characterizing the effects of an aircraft's flow-field and developing validation

methods to predict these flow-field effects, the project also provides a mechanism to determine

potential aircraft modifications.

The main focus of the research presented in this paper is to computationally extend the aft flow-

field evaluation for a simplified C-130 configuration used in tests conducted at the S4 low-speed wind

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-90

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

2

tunnel facility, Institut Supériuer de l’Aéronautique et de Espace (ISAE). In conjunction with

simulations performed by the German Federal Office for Defense Technology and Procurement

(IABG) and flight tests conducted by the United Kingdom’s Joint Air Transport Evaluation Unit

(JATEU), this subtask addresses rigorous validation methods for the prediction of the aft body airflow

influencing airdrop missions. The close coordination among the experimental, computational, and

flight test teams has resulted in significant insight into the physics of the airflow and has provided a

well-documented database for the next phase of the project which includes recommendations and

evaluation of alternatives for future airdrop operations and aircraft.

II. Background

A. 1:48th

Scale Baseline

Initial CFD and experimental work by Claus, et al.3 and Morton, et al.

4 focused on the cargo door

closed configuration with an emphasis on following a systematic validation campaign. Experiments

used a 1:48th scale modified model which produced data based on loads, five-hole probe, and single

hotwire probe measurements. Combined with the DES CFD simulations, the results captured the

large contributions made to the flow by the upsweep vortices and the relatively small influence of the

nose and vertical stabilizer. Due to the sparseness of the grid, however, the 1:48 scale model was

found to be too small for proper visualization of the upsweep vortices in the aft fuselage region.

Therefore, a decision was made to conduct the next round of tests using a larger, simplified model.

B. 1:16th

Scale Results

At 1:16th scale, however, the width of the wind tunnel necessitated the removal of the main wings.

Without the main wing on the model, there are two changes in the flow-field which must be addressed

for the current simulations. First, the main wing provides a downwash on the empennage and rear

section of the fuselage. To alleviate this affect during airdrop operations, the aircraft is flown between

an α = 2º-8º depending on the mission. Therefore, the baseline wind tunnel and CFD tests were

conducted at α = 0º since the positive α was no longer needed to simulate low α airdrop flight

conditions. The second flow-field change occurs when investigating aircraft configurations with the

cargo door open. The wingless flow now allows an increased adverse pressure gradient, relative to

configurations with the main wing, to be present in the “cavity” created above the cargo-bay ramp and

below the upward swept body. No configuration changes were made to account for this flow-field

change, but it was noted in order to make more accurate and careful interpretations of the validation

results.

The experimental data associated with the second tests was collected via Particle Image

Velocimetry (PIV) flow visualization versus five-hole probe and hotwire measurements. The PIV

data allows for better investigation of the physics of the flow and was easily compared with the

Computational Fluid Dynamics (CFD) simulations. The CFD simulations were carefully done with an

extensive grid convergence study and turbulence model comparison. The final results used Reynolds

Averaged Navier-Stokes (RANS) methods for steady-state simulations and Detached-Eddy

Simulations (DES) for unsteady simulations. This work demonstrated very good agreement between

experiments and simulations for both loads and off-body vorticity. Figure 1, from the CFD study,

shows the main features associated with the flow-field for the closed door geometry. The upsweep

vortices are the result of increasing pockets of vorticity formed along the upsweep region, and the

numerical simulations predict the vortices detaching just before the leading edge of the tail. This flow

behavior is slightly modified in the experimental data which shows an earlier separation. As the

upsweep vortex interacts with the horizontal tail the flow stalls and generates a counter rotating vortex

labeled in as the “Detached vortex”, in the figure. In addition, with respect to vortex location and

shape, the DES results correlated well (better than the RANS models) with PIV cross-plane data,

especially in the region aft of the aircraft tail. The DES data indicated only slightly different intensity

profiles and vortex orientations as compared with the PIV data in this region.

Recent experimental work by Bury, et al.5 have refined the precision of the PIV measurements,

introduced a systematic method for vortex core identification and parameter definition, and extended

3

the database for the open cargo door configuration at α = 0º. Figure 2 illustrates the data acquisition

cross plane orientation used for both experiments and computations.

C. Objectives

This computational research makes use of recent advances in CFD methods in the form of DDES

6

turbulence modeling to attain high resolution characterization of the flow-field behind the cargo door

of the C-130 for various configurations. The results reported in this effort also detail the initial efforts

to couple grid and time-step studies based on characteristic features of the flow. This portion of the

effort follows the guidelines set forth by Cummings, et al.7

III. Computational Methodology

A. Flow Solver—Cobalt8 and DDES with SARC Turbulence Model

CFD simulations were made with the commercially available flow solver Cobalt v4.2 which

solves the unsteady, three-dimensional, compressible RANS equations. As a cell-centered, finite

volume code it is applicable to arbitrary cell topologies including hexahedra, prisms, and tetrahedra.

Second-order accuracy in space is achieved using the exact Riemann solver of Gottleib and Groth10

and least squares gradient calculations using QR-factorization. To advance the discretized system a

point-implicit method using analytic first-order inviscid and viscous Jacobians is used. A Newton

sub-iteration method is used to improve time accuracy of the point-implicit method. The method is

second-order accurate in time.

Cobalt v4.2 uses DDES turbulence modeling for unsteady calculations. Like DES, DDES models

the turbulence inside the boundary with Reynolds Average Navier-Stokes (RANS) and outside the

boundary layer using Large Eddy Simulation (LES). In this manner, a compromise is achieved with

respect to the computational costs associated with LES methods and the inability of RANS to

accurately model highly separated flow regions outside the boundary layer. However, DES can give

incorrect behavior in thick boundary layers. Specifically, this phenomenon is seen when the grid

spacing parallel to the boundary is less than the boundary-layer thickness, and sometimes is a

consequence of grid refinement. Therefore, the difference between DES and DDES is the mechanism

used to switch between RANS and LES. Whereas DES implements a switch based on the underlying

grid spacing, DDES includes the physics of the flow, eddy-viscosity field, to determine the

appropriate transition between RANS and LES. As a result modeled-stress depletion (MSD) is

mitigated, and the switch happens more quickly in DDES following flow separation.

The Spalart-Almaras with Rotation Correction (SARC) turbulence was chosen for the RANS

component of the simulation based on the results of Morton, et al4. Their recommendation was based

on the computational effort required for the two-equation Shear Stress Transport (SST) model which

did not justify the small additional increase in correlation with the wind tunnel data.

B. Setup

As with the geometry of the model, the computational domain (Figure 3) and simulation

parameters (Table 1) were chosen to match Reynolds, ReL=4.56 x 106 , and Mach, M = 0.1155,

numbers used for the ISAE wind tunnel experiments. The reference length, L, for the half-span model

is 1.70 m. Once the startup transients had dissipated (after approximately 1500 iterations) the

simulations were run second-order in time and space. To ensure the flow solution was converged at

every time step, three Newton sub-iterations were used. The no-slip adiabatic wall boundary

condition was used for the body surface, and the modified Riemann-invariant condition was imposed

at the far-field boundaries. A symmetry condition, which applies a tangency condition for the velocity

on the boundary, was implemented for the symmetry plane. On this plane, pressure and density are

calculated from the flow-field by assuming a zero gradient in the cell near the boundary.

Time-dependent calculations were performed on the Sun cluster, Midnight, at the Arctic Region

Supercomputing Center. Midnight, has 358 Sun Fire X2200 compute nodes with 16 GB of shared

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memory per node (4 GB per core), 2 dual core 2.6 GHz AMD Opteron processors per node, and one

4X Infiniband Network Card on PCI-Express Bus per node. Nodes are connected via a Voltaire

Infiniband Interconnect, and the system has 68 TB of storage.

C. Grids and Time Step Refinement

As noted earlier, a detailed grid convergence study was accomplished by Morton, et al.4 Grids

were based on the CAD files used to make the wind tunnel models and range in size from 3.05 x 106

to 13.59 x 106 cells for the investigated geometries. These grids contain 12 prismatic layers and an

average y+ value of approximately .1 which is required for good resolution of the velocity gradient in

the boundary layer. An associated grid-resolution study was made for the closed door grids, but a

study for the open door geometry was not addressed.

Cummings, et al7 have documented the high computational costs associated with making accurate

unsteady aerodynamic simulations. In particular, these simulations require a physical basis for

choosing an appropriate time step, and this time step is intimately related to the grid resolution used.

As a prototypical example the authors present an analysis of high angle of attack delta wing flows. By

identifying specific flow features with a Strouhal Number ( ), the computational

simulations may be optimized.

No such summary of flow features exists for the upsweep region of cargo aircraft. However, the

highly separated flows associated with this region, as seen in Figures 3 and 4, suggest the importance

for such a categorization, if one is to validate CFD results against experimental and flight test data.

Following the method of Cummings, et al.8 and based on the previous simulations, grids composed of

8.8 x 106 (fine grid = FG) and 10.5 x 10

6 (very fine grid = VFG) cells were used to conduct a Power

Spectrum Density (PSD) study for the open door configuration at various time steps, ranging from

.001s to .0001s. These time steps equate to a non-dimensional time step ( where l

is the characteristic length in this case equal to the fuselage length, L) between .0226 and .00226. This

range corresponds to values used by other researchers [11-13]. Balancing the grid resolution with this

range of time steps makes the use of DDES vs DES very important.

IV. Results and Discussion

A. Experiment vs DDES vs DES: Closed Door Configuration

The increased awareness of the link among physical phenomenon, grid refinement, and time-step

accuracy has accentuated the MSD associated with implementation of DES. The reduced eddy

viscosity leads to earlier switching between the RANS and LES models and also influences the

intensity, orientation, and geometry of the flow’s vortical structures. To determine the affect of using

a turbulence model more formally tied to the physics of the problem, a series of simulations were

completed using DDES for modeling the flow-field around the closed door configuration of the

simplified C-130 model. Using the end of the aircraft’s tail section as the origin, cross-planes of x-

vorticity (± 1000 rotations/s) were collected. Figures 5 and 6 show representative sequences of planes

for the PIV measurements and DDES simulations using 10.5 x 106 cells.

As can be seen in Figures 5(a) and 6(a), the simulations are not capturing the exact location of the

flow separation. Indeed, for the x-position at -530 mm, PIV catches a separation of vorticity pockets

from the fuselage whereas Cobalt DDES does not. However, for positive x-positions the wind tunnel

and simulation results correlate extremely well in terms of spatial position and vorticity intensity.

This zone is the area of the highest energy eddies and therefore, is most problematic for airdrop

operations.

The next set of figures show a direct comparison between PIV, DES, and DDES results. Figure

7(c) reveals the better ability of DDES to predict flow behavior with the two characteristic “croissant”

shapes caught whereas DES calculates rougher shapes. In addition, Figures 8 (d) to (f) show that

DDES gives a better prediction of the orientation, width and the length of vorticity pockets, and

gradients. Figures 9 and 10 give a more explicit demonstration of the measures of correlation among the

different vortex data sets at two locations downstream of the model tail. The DDES modeled vortex at

x = 10mm in Figure 9 clearly shows a weaker distribution than the DES vortex. In addition, the

5

DDES “croissant” shape mirrors the geometry of the PIV vortex, while the DES vortex has much

smaller “tails”. These differences are maintained in the x = 200mm cross plane, and directly lead to

the orientation evolution depicted in Figure 10 by the orientation of the principle axis of the vortices.

To provide a measure of vortex core locations for the DES and DDES simulations 151 x 151 grid

of velocity taps whose data were reduced via a Matlab script. The grid was positioned, as shown in

Figure 11, to capture the flow-field for the port side of the aft region of the aircraft. Figures 12 and 13

depict the movement of the vortex cores for the up-sweep and detached vortices. The “error” bars

associated with the DES and DDES trajectories indicate the level of uncertainty based on the distance

between grid points. As can be seen the DDES results, on average, yielded a better correlation with

the wind tunnel date than the corresponding DES simulations. Of particular note is the divergence of

the DES and DDES data at or before the x = 0 cross-plane. As previously stated, neither DES nor

DDES flow the upsweep vortex behavior along the fuselage as the PIV data indicates that the flow has

already separated. With respect to the detached vortices for negative X positions, the Matlab routine

found the positions of the lower vorticity values, which don’t correspond to the detached vortices.

B. Experiment vs DDES: Open Door Configuration

The next phase of validation compared the results between the PIV measurements and the DDES

simulations. No comparisons were made with respect to earlier DES data/studies based on the better

agreement achieved by DDES described above. The methodology used for these comparisons

followed the outlined used for the closed door analyses. Figure 14 shows the PIV results and Figure

15 the DDES cross planes. Again, the computational results were not able to capture the separation

behavior of the wind tunnel vortices for negative x-locations. In addition, the simulations over-

predicted the vortex strength for these positions with the most significant difference at x = -430 mm.

This behavior reflects the anticipated difficulties associated with modeling the flow in this region, and

may be attributed to the sensitivity of the numerical methods to the relatively high longitudinal

pressure gradient present which in turn are due to the lack of downwash from the main wing. Another

potential explanation that was brought to the authors’ attention relates to the SARC turbulence model

used in the calculations14

.

It is well known that CFD turbulence models must include streamline curvature and system

rotation in order to accurately model turbulent shear flows. Indeed, the “rotation correction” for the

standard SA turbulence model was introduced to address this flow trait. However, once a grid is

resolved enough to resolve the relevant features of the flow, the rotation correction may actually

overcompensate as a model. This feature has been observed in other massively separated flows, and

serves as an additional indicator that accurate prediction of time-dependent flows must include a

detailed physical knowledge about the features of the simulated flow.

C. Characteristic instabilities and time step refinement for Open Door Configuration

A time step refinement study was initiated based on the open door results, and more refined

simulations were made for higher angles of attack with the door open. Figure 16 illustrates the

predictions of linear theory with respect to flows about an axisymmetric bluff body with upsweep

angle β. For the C-130 model used in this study β 25º. A plot of iso-surfaces of x-vorticity for an α

= 8º simulation on a 10.5 x 106 cell grid, shown in Figure 17, matches qualitatively the linear

prediction. Movies of the simulations also showed the unsteady phenomena of the upsweep vortices.

This type of phenomena appeared as a combination of a low frequency rotation combined with high

frequency vortices located on the outer edge of the main vortex. Once a particular phenomenon is

chosen, based on the observed flow topology, a variety of time steps are coupled with various grids to

determine if there is an appropriate convergence.

Three different time steps were tested on the open door configuration using the same 10.6 million

cells grid. These time steps are 0.001s, 0.0002s and 0.0001s. The study also included a simulation for

a 8.8 million cell grid with a time step of 0.0001s. For the spectral densities computed, if the average

forces for the entire model are used (i.e., using a single patch), it’s difficult to isolate the physical origin of each frequency peak. Therefore, the open door geometry of the C130 was been split into

several patches—fuselage, horizontal tail, and door. Power Spectral Density (PSD) data for the forces

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was collected for each patch and values for the fuselage are presented in Table 2. A Strouhal number

of for a time step of .0001s equates to a non-dimensional time step of .0026 which is in

agreement with the values found to be appropriate for high alpha delta wing studies. This value also

corresponds to the data which has been collected for numerical studies of axisymmetric bluff bodies

conducted at the USAF Academy16

. It was found that for bluff bodies that the aspect ratio AR = L/D,

as opposed to Reynolds Number, is the dominant physical factor governing changes in flow filed

phenomenology. For an aspect ratio, AR = 7, the researchers found a corresponding Strouhal number,

and helical-like vortical shedding. A similar calculation based on the assumption of the C-

130 model as a bluff body yields an . The analysis for the tail patch indicated a possible

correlation with the observed vortex shedding frequencies of the delta wing configuration at high

angles of attack. However, a more comprehensive study is needed to complete this subset of the

work.

V. Conclusion

DDES and DES computations were performed on a simplified C-130 model in closed and

open door configurations. As a result of these analyses, the DDES simulations demonstrated an

increased ability to predict the flow around geometric complicated configurations and thus

provide more accurate simulation capabilities for the AIA project. While DDES performed better

in fully separated flow, the requirement for more accurate numerical simulation in the region of

flow separation was noted. In addition, work is also needed for more precise tracking of vortices

and vortex characterization. The methods developed by one of the authors for experimentally

collected data holds much promise in this regard and will be implemented in future numerical

studies.

Unsteady simulation of the open door configuration indicates a possible sensitivity of the

SARC turbulence model to over correct in regions of strong longitudinal pressure gradients.

Future numerical studies will be conducted with different turbulence models to address this

observation as well as determine the different models’ ability to more accurately capture the flow

separation.

Based on an initial study of unsteady flow for the open door configuration, helical and

shedding instabilities were quantified for future validation with experimental data. These initial

numerical studies are in good agreement with work completed for simpler geometries. As such,

the methodology for using numerical simulations to accurately model time-dependent flows has

been extended to predict possible flow features of interest.

Acknowledgements

The authors would like to thank the Air Force Office of Scientific Research, the USAF

Academy Modeling and Simulation Research Center and the Natick Soldier Research

Engineering and Development Center for their generous financial support. Outstanding

computational support was provided by the DoD High Performance Computing Modernization

Program’s Allocated Distributed Center at the Arctic Region Supercomputing Center. In

addition, the authors are grateful to Scott Morton, Rich Charles, Dave McDaniel, Jürgen Seidel,

Stefan Siegel, and Russ Cummings for their insights and guidance.

References

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M., Wheeler, R., and Benney, R., "Four-Power Long Term Technology Projects: Airflow

Influence on Airdrop and 2nd

Precision Airdrop Improvements", AIAA 2005-1602 2Wooten, J.D. and Yechout, T. R., “Wind Tunnel Evaluation of C-130 Drag Reduction

Strakes and In-flight Load Loading Prediction”, AIAA 2008-348

7

3Claus, M.P., Morton, S.A., Cummings, R.M., and Bury, Y., “DES Turbulence Modeling

on the C-130 Comparison between Computational and Experimental Results”, AIAA 2005-884 4Morton, S.A., Vignes, B., Bury, Y, "Experimental and Computational Results of a

Simplified C-130 Shape Depicting Airflow Influence on Airdrop", RTO-MP-AVT-133

AC/323(AVT-133)TP/113, 2006 5Bury, Y., Morton, S.A., Charles, R., "Experimental Investigation of the Flowfield in the

Close Wake of a Simplified C-130 Shape, A Model Approach of Airflow Influence on Airdrop",

AIAA 2008-6415 6Spalart, P.R., Deck, S., Shur, M.L., Squires, K.D., Strelets, M., Travin, A., "A new

version of detached eddy simulation, resistant to ambiguous grid densities", Theor. Comput. Fluid

Dyn. (2006) 20: 181-195 7Cummings, R, M., Morton, S.A., McDaniel, D.R., "Experiences in Accurately Predicting

Time-Dependant Flows", RTO-MP-AVT-147, 2007. 8Cobalt v4.2, www.cobaltcfd.com

9Gottleib, J.J. and Groth, C.P.T., “Assessment of Riemann Solvers for Unsteady

One-Dimensional Inviscid Flows of Perfect Gasses”, J. Comp. Phy., VOl 78, No 2, 1988, pp 437-458

10McLaughlin, T.E., Galloway, J.D., McClurg, J.P., and Brown, M., USAFA DFAN

Report 99-07, USAF Academy, September 1999 11

Strelets, M. “Detached eddy simulation of massively separated flows”, AIAA 2001-879 12

Görtz, S., “Detached-eddy simulations of a full-span delta wing at high incidence,”

AIAA 2003-4216 13

Schiavetta, L., Badcock, K., and Cummings, R., “Comparison of DES and URANS for

unsteady vertical flows over delta wings,” AIAA 2007-1085 14

Communication with Dr Cummings during M&SRC group meeting, August 2008 15

Christian Pujol, Aérodynamique, La théorie linéaire, Cours ENSICA, 2006 16

Seidel, J., Siegel, S., Cohen, K., and McLaughlin, T, “Simulations of Flow Control of

the Wake behind an Axisymmetric Bluff Body”, AIAA paper 2006-3490

Figure 1: Iso-Surfaces of X-Vorticity (±750 rotations/sec) colored by static pressure (dPa)

with identified phenomena (DDES simulation), α = 0º

8

Figure 2: X-vorticity cross-planes downstream of model.

Figure 3: Computational domain Table 1: Fixed Simulation Parameters

Figure 3: C-130 Fire retardant release

10 Figure 4: Unsteady Simulation of C-130

Flow Field, ReL = 4.56 x 106, α = 8º

Parameter Value

40 m/s

101325 Pa

298 K

β 0º

9

Figure 5: PIV

5 cross-planes of x-vorticity

(a): x=-530mm, (b): x=-75mm, (c): x=0mm, (d): x=100mm, (e): x=200mm, (f): x=400mm

Figure 6: DDES cross-planes of x-vorticity

(a): x=-530mm, (b): x=-75mm, (c): x=0mm, (d): x=100mm, (e): x=200mm, (f): x=400mm

10

Figure 7: Cross sections of x-vorticity, from top to bottom: PIV / DES / DDES

(a): x=-530mm, (b): x=-75mm, (c): x=0mm, same scale for x-vorticity

11

Figure 8: Cross sections of x-vorticity, from top to bottom: PIV / DES / DDES

(d): x=100mm, (e): x=200mm, (f): x=400mm, same scale for x-vorticity

Figure 9: Vortex geometry comparison

12

Figure 10: Vortex principal axis orientation comparison

Figure 11: Velocity tap discretization

13

Figure 12: y and z trajectories of the upsweep vortex cores

Figure 13: y and z trajectories of the detached vortex cores

Figure 14: PIV

5 cross-sections of x-vorticity for Open Door configuration

(a): x=-510mm, (b): X=-430mm, (c): X=-330mm, (d): X=-75mm, (e): X=0mm, (f): X=100mm

14

Figure 15: DDES cross-sections of x-vorticity for Open Door configuration

(a): x=-530mm, (b): X=-430mm, (c): X=-330mm, (d): X=-75mm, (e): X=0mm, (f): X=100mm

Figure 16: Upsweep angle β and its typical vortices

Figure 17: Iso-Surfaces of x-vorticity: -750 rotations/s (blue) & 750 rotations/s (red)

Time Step (s) 10.5 million cell grid 8.8 million cells

.0002 .35 N/A

.0001 .2 6 Table 2: Strouhal Number for Fuselage Patch