AIAA 2009-4223
40th AIAA Plasmadynamics and Lasers Conference
22 - 25 June 2009, San Antonio, Texas
Numerical Simulation of Aero-Optical Distortions by a
Turbulent Boundary Layer and Separated Shear Layer
Kan Wang∗ and Meng Wang†
University of Notre Dame, Notre Dame, IN 46556
Compressible large-eddy simulations are carried out to study the aero-optical distortions
caused by a Mach 0.5 flat-plate turbulent boundary layer and a separated shear layer over
a cylindrical turret with a flat window. The fluctuations of refractive index are calculated
from the density field and ray tracing is employed to compute the optical path differences
(OPD). In the flat-plate boundary layer, the log layer and wake region contribute most
to the optical wavefront distortions. Consistent with previous experimental findings, the
distortion magnitude is dependent on the propagation direction due to anisotropy of the
boundary-layer vortical structures. In the case of separated shear layer over a cylindrical
turret, wavefronts are tilted toward downstream due to the angle between the flat window
and separated shear layer, and the low-density region therein. The root-mean-square of
OPD (OPDrms ) caused by the separated shear layer is found to be five times as large as that
caused by the attached boundary layer upstream of the turret. The predicted OPDrms for
both flows are in good agreement with experimental values despite much reduced Reynolds
numbers in the simulations, suggesting relative insensitivity to the Reynolds number.
Nomenclature
KGD
OP L
M
OP D
R
ReR
Reθ
c∞
k
n
u, v, w
x, y, z
x′ , y ′ , z ′
∆x, ∆y, ∆z
δ
δ∗
λ
ρ
φ
θ
Subscript
min
rms
∗ Graduate
† Associate
Gladstone-Dale constant
Optical path length
Free-stream Mach number
Optical path difference
Radius of the cylindrical turret
Reynolds number based on the radius of the cylindrical turret
Reynolds number based on momentum thickness
Free-stream sound speed
Optical wavenumber
Refractive index
Streamwise, wall-normal, and spanwise components of velocity
Coordinates in streamwise, wall-normal, and spanwise directions for flow simulation
Coordinates for optical calculation
Grid spacing
Boundary-layer thickness at inlet
Displacement thickness
Optical wavelength
Density
Phase distortion
Momentum thickness
Minimum value
Root-mean-square
Student, Department of Aerospace and Mechanical Engineering, Student Member AIAA.
Professor, Department of Aerospace and Mechanical Engineering, Member AIAA.
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Institute
ofofAeronautics
andAstronautics,
Astronautics
Copyright © 2009 by Kan Wang and Meng Wang. Published American
by the American
Institute
Aeronautics and
Inc., with permission.
I.
Introduction
When a collimated optical beam passes through a compressible turbulent flow region, its wavefront
becomes distorted due to fluctuations of the refractive index, which is related to density variations. This aerooptical effect is detrimental to airborne communication, targeting and imaging systems. Many experimental,
analytical and numerical investigations have been conducted in order to understand and mitigate aero-optical
effects.1
In recent years, large-eddy simulation (LES) has emerged as a promising tool for studying aero-optics
because of its ability to capture a broad range of turbulence scales relevant to optical aberrations. Childs2
was among the first to apply LES to study aero-optics. He simulated a compressible turbulent mixing layer
and applied ray tracing through it to determine the wavefront distortions. Jones and Bender3 calculated
optical aberrations for a fuselage/turret configuration using LES. Tromeur et al.4, 5 carried out LES to
study the aero-optical distortions of subsonic and supersonic flat-plate boundary layers and evaluate the
applicability of Sutton’s statistical model6 relating optical phase distortions to turbulence statistics. Visbal
and Rizzetta7 used LES to simulate a turbulent free shear layer and examine optical distortions caused by
the shear layer.
Mani et al.8, 9 simulated flow over a circular cylinder with a sixth-order, non-dissipative LES code and
investigated the aero-optical effects of separated shear layers and turbulent wakes at Reynolds numbers of
3900 and 10,000. They analyzed the far-field optical statistics and their dependence on optical wavelength,
aperture size, and beam position in the flow field. They also examined the effect of different flow scales on
aero-optics and established a grid-resolution criterion for accurately capturing the aero-optical effects.10 The
basic conclusion is that an adequately resolved LES can capture the aero-optics of highly aberrating flows
without requiring additional subgrid scale modeling for the optics.
Despite previous computational efforts, accurate predictions of flow-induced optical aberrations remain a
significant challenge, particularly for realistic complex geometries and at high Reynolds numbers. Furthermore, there is a general lack of quantitative experimental validation of simulation results, even for simple
geometries.
Gordeyev et al.14 (see also Refs.11–13) conducted a series of flat-plate boundary layer experiments, in
which they measured aero-optical distortions using Malley probes. They examined statistical properties of
optical distortions and found a scaling law for the root-mean-square (rms) of optical path difference (OPD),
which is shown to be proportional to the boundary-layer displacement thickness, free-stream density and
square of free-stream Mach number. In another investigation, Gordeyev et al.15 measured optical distortions
caused by a separated shear layer over a flat window on a cylindrical turret mounted on two flat surfaces
of different altitude. They investigated the use of different passive control devices placed on the upstream
surface for mitigating the optical effects.
In the present study, LES is used to predict optical aberrations caused by a turbulent boundary layer
and separated shear layer from a cylindrical turret. The experiments of Gordeyev et al.14, 15 are used
as references for comparison, although the Reynolds numbers are reduced in the simulations due to high
computational expenses. The boundary layer considered has the same characteristics as that upstream of
the cylindrical turret. It is used for a detailed analysis on several issues of fundamental interest such as
the contributions from different flow regions to wavefront distortions, the effect of propagation angle and its
relation to flow structures, and correlation length scales of the refractive index field and wavefront distortions.
The simulation of separated shear layer over a cylindrical turret is used to demonstrate the capability of
LES to predict more complex and realistic flows and study the optical distortions by a separated turbulent
shear layer. In addition, this simulation will provide a baseline case for comparison in future studies of the
effects of passive control devices for suppressing flow separation and mitigating optical distortions.
II.
Numerical Approach
The aero-optical study is performed in two steps: first, a large-eddy simulation is conducted to provide a
detailed description of the turbulent flow field, especially the fluctuating density field; second, the refractive
index is obtained from the Gladstone-Dale relation16 and a ray tracing calculation is performed to compute
the optical wavefront distortions.
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II.A.
Flow Simulation
The flow-field simulation is performed using an unstructured-mesh compressible LES code developed at the
Center for Turbulence Research, Stanford University.17 The code solves the spatially filtered compressible
Navier-Stokes equations in conservative form with the state equation using low-dissipative, robust numerical
algorithms. It employs a second-order finite volume scheme for spatial discretization and an explicit thirdorder Runge-Kutta method for time advancement. The effect of subgrid-scale motions is modeled using the
dynamic Smagorinsky model. This code is parallelized with MPI and achieves very high parallel efficiency.
The flow configurations and boundary conditions for a spatially developing boundary layer and a separated turbulent shear layer over a cylindrical turret are discussed along with simulation results in sections
III and IV, respectively. In both cases, it is necessary to prescribe a time-dependent turbulent inflow at the
inlet of the computational domain. The turbulent inflow data are generated by an auxiliary simulation of
a flat-plate boundary layer adopting an extension of the rescale and recycle technique of Lund et al.19 to
compressible flows.20, 21 The basic idea, based on self-similarity of a turbulent boundary layer, is to rescale
the density, velocities, and temperature at a downstream station and re-introduce them into the inlet, thus
allowing a sufficient time for turbulence structures to develop inside the computational domain. The computational domain size for inflow generation is 12δ, 4δ and 2.4δ in the streamwise, wall normal and spanwise
directions, respectively, where δ is the boundary-layer thickness. The plane at x = 9δ from the inlet is
selected as the recycling station, and the inflow data are collected at x = 7.3δ at each time step.
II.B.
Optical Computation
The refractive index is related to density by the Gladstone-Dale law16
n − 1 = KGD ρ,
where KGD is the Gladstone-Dale constant. It can be approximated by9
ρ
n − 1 = 2.8 × 10−4
,
ρ∞
(1)
(2)
for air under standard condition and optical wavelengths between 1 µm and 10 µm. The optical path length
(OPL) of a beam is defined as1, 6
Z z1′
n(x′ , y ′ , z ′ , t)dz ′ ,
(3)
OP L(x′ , y ′ , t) =
z0′
where x′ , y ′ , and z ′ are the coordinates for optical calculations, with z ′ in the direction of propagation and
x′ and y ′ in the plane normal to the beam. z0′ and z1′ are boundaries of the turbulence region along the
beam path. Wavefront distortions are described by the beam’s optical path difference (OPD), which is the
variation of OPL about its mean,
OP D(x′ , y ′ , t) = OP L(x′ , y ′ , t) − OP L(x′ , y ′ , t),
(4)
where the overbar denotes averaging in the x′ -y ′ plane. The phase distortion for a single frequency optical
beam with wavelength λ can be calculated by
φ(x, y, t) =
2πOP D
.
λ
(5)
To compute the optical quantities, a Cartesian beam grid is used to follow the optical path through the
turbulence region. For example, figure 1 illustrates the flow and optical grids used in the cylindrical turret
case. A linear interpolation scheme is employed to interpolate the density field from the unstructured LES
grid onto the beam grid. After the interpolation, the integration in Eq. (3) is carried out to obtain the OPL.
III.
III.A.
Flat-Plate Boundary Layer
Simulation Set-Up
A flat-plate boundary layer simulation with an inflow-outflow configuration is conducted to validate the
numerical approach and investigate optical distortions. The free-stream Mach number is M = 0.5, and
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Figure 1. Two grids in an x-y plane for the cylindrical-turret aero-optical simulations: an unstructured mesh (light
blue) for flow simulation and a Cartesian mesh (purple) for optical calculation.
the Reynolds number based on the momentum thickness at the inlet is Reθ = 1400. This simulation
employs a computational domain of 24δ, 5δ and 2.4δ in the streamwise, wall normal and spanwise directions,
respectively. A Cartesian mesh, uniform in the streamwise and spanwise directions and stretched in the
wall-normal direction, is employed. The grid spacings in viscous wall units are ∆x+ ≈ 50, ∆z + ≈ 17 and
+
∆ymin
≈ 1 based on inlet conditions. Turbulent inflow data generated from a separated simulation are fed
into the inlet at every time step. A sponge layer18 with thickness of δ is applied to the top and outlet
boundaries to damp out flow structures and acoustic waves. No-slip and adiabatic boundary conditions are
imposed at the bottom wall.
III.B.
Results
Instantaneous flow fields in terms of streamwise velocity (normalized by free-stream sound speed) and density
at a spanwise cut are plotted in figure 2. The mean velocity profile and rms of velocity fluctuations are
compared with those from the incompressible DNS of Spalart22 in figures 3 and 4, which show reasonable
agreement. The shape factor from the simulation H = δ ∗ /θ is approximately 1.38. The statistics are
calculated at x = 18.5δ from the inlet by averaging over a time period of approximately 200δ/U∞ and in the
spanwise direction.
Optical distortions are calculated for beams centered at x = 18.5δ, z = 1.2δ and normal to the wall. A
snapshot of wavefront distortions is shown in figure 5. The rms of OPD is computed at each time step and
then averaged over a time period of 100δ/U∞. The time-averaged OP Drms for a beam of size δ × δ is found
to be 7.46 × 10−7 δ. Based on analysis and experimental data, Gordeyev et al.14 proposed the following
scaling law for OP Drms :
ρ
OP Drms = (1.65 ± 0.30) × 10−5 δ ∗
M 2,
(6)
ρSL
ρ
where ρSL denotes the air density at the sea level. The ratio OP Drms /(δ ∗ ρSL
M 2 ) from the present simulation
−5
is approximately 1.69 × 10 , which agrees well with Eq. (6).
To investigate contributions to wavefront distortions from different flow regions in the boundary layer,
Eq. (3) is integrated from the wall to different y values, and the resulting OP Drms is shown in figure 6. As
can be expected, the viscous sublayer has a negligible contribution to wavefront distortions. The distortions
start to grow in the buffer layer, and most of the growth takes place in the log layer and wake region.
This is consistent with the distribution of density fluctuations, shown in figure 7, which shows that density
fluctuations peak at the lower edge of the log layer and remain significant throughout the log layer and
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Figure 2. Instantaneous streamwise velocity and density fields in a spanwise cut in the flat-plate boundary layer. Top:
streamwise velocity u/c∞ ; bottom: density ρ/ρ∞ .
25
20
U+
15
10
5
0 0
10
10
1
+
10
2
10
3
y
Figure 3. Mean velocity profile in the flat-plate boundary layer: ——–, LES; , DNS of Spalart;22 - - - -, U + =
2.44 ln y + + 5.2.
wake region. Since most distortions arise from the flow structure in the outer region of the boundary layer,
which are relatively insensitive to Reynolds number, the OP Drms is expected to be relatively insensitive to
Reynolds number. This explains why the scaling coefficients from the simulation and experiment agree well
despite the Reynolds number discrepancy. A quantitative evaluation of the Reynolds number dependence of
optical distortions will be carried out in future studies.
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+
u′ rms
3
2.5
1.5
+
v ′ rms
+
w′ rms
2
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
y/δ
′+
Figure 4. Root-mean-square of velocity fluctuations in the flat-plate boundary layer. ——–, u′ +
rms ; - - - -, v rms ; − · − · −,
′+
′+
′+
22
w ′+
rms : from present LES. , u rms ; , v rms ; △, w rms : from DNS of Spalart.
Figure 5. Wavefront distortions at a time instant caused by the turbulent boundary layer.
In practical applications it is of interest to shoot an optical beam through a turbulent boundary layer at
angles other than that normal to the wall. To investigate the directional dependence, wavefront distortions
of beams with different elevation angles β are examined. When β < 90◦ , the beam is directed towards
the upstream direction, while when β > 90◦ the beam is towards the downstream direction. All the beams
propagate to the same height above the wall. The values of OP Drms are plotted as a function of the elevation
angle in figure 8. Since OP Drms is a function of propagation distance inside the boundary layer which is
proportional to 1/ sin β, the function 1.6 × 10−5 / sin β is also plotted in the figure as a reference. It is found
that the optical distortions by a flat-plate boundary layer are dependent on the direction of propagation
and asymmetric with respect to the normal angle β = 90◦ . When the beam is tilted toward downstream, it
experiences more distortions than when it is tilted toward upstream at the same angle relative to the normal
direction. This phenomenon was observed earlier in the experiment of Gordeyev et al.14 The experimental
data and simulation results are in overall agreement (figure 8). As pointed by Gordeyev et al., the asymmetry
is due to the elongated oblique vortical structures in the flat-plate boundary layer which are tilted toward
the downstream direction.23, 24 From Sutton’s linking equation6
Z L
2
hφ2 i = 2KGD
k2
hρ′2 iΛ(z ′ )dz ′ ,
(7)
0
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7E-07
6E-07
OP Drms /δ
5E-07
4E-07
3E-07
2E-07
1E-07
10
0
10
1
10
y
2
10
3
+
Figure 6. Root-mean-square of OPD at different vertical distances in the flat-plate boundary layer.
0.007
0.006
ρ′rms /ρ∞
0.005
0.004
0.003
0.002
0.001
0 0
10
10
1
10
y
2
10
3
+
Figure 7. Root-mean-square of density fluctuations in the flat-plate boundary layer.
where k is the wavenumber of the optical beam and the angular brackets denote time averaging, wavefront
phase distortions are related to the correlation length Λ of the fluctuating density field along the optical path.
Although this equation is derived for isotropic turbulence, it can be used to explain the above phenomenon
qualitatively: in a flat-plate boundary layer, the correlation length is larger along an optical path tilted toward
the downstream direction than that toward the upstream direction, because the former is more aligned with
the coherent oblique vortical structures whereas the latter traverses those structures. To illustrate this, the
density correlations along the optical paths at three elevation angles are plotted in figure 9 as a function
of spatial separation along the path, at y = 0.05δ, y = 0.21δ and y = 0.70δ. It can be noted that at
the two angles β = 60◦ and 120◦ , which are symmetric relative to the wall-normal direction, the correlation
functions differ. The correlation lengths at 120◦ are larger than those at 60◦ at all y positions. Consequently,
optical distortions are more severe for the beam tilted toward the downstream direction than that toward
the upstream direction, even if they travel the same distance.
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ρ
M 2)
OP Drms /(δ ∗ ρSL
3E-05
2.5E-05
2E-05
1.5E-05
60
80
100
120
β
Figure 8. Root-mean-square of OPD as a function of beam elevation angle through the boundary layer. ——–, present
LES; - - - - -, 1.6 × 10−5 / sin β; with error bars, experiment of Gordeyev et al..14
0.8
0.8
0.8
0.6
0.6
0.6
Rρ
1
Rρ
1
Rρ
1
0.4
0.4
0.4
0.2
0.2
0.2
0
-0.5
0
0.5
∆z ′
0
(a) y = 0.05δ
-0.5
0
∆z ′
0.5
0
(b) y = 0.21δ
-0.5
0
∆z ′
0.5
(c) y = 0.70δ
Figure 9. Density correlations as a function of separation along optical paths at three elevation angles through the
boundary layer: ——–, 90◦ ; - - - - -, 60◦ ; − · − · −, 120◦ .
The two-point spatial correlation of OPD in an x′ -y ′ plane can be computed from
ROP D (x′ , y ′ , ∆x′ , ∆y ′ ) =
hOP D(x′ , y ′ , t)OP D(x′ + ∆x′ , y ′ + ∆y ′ , t)i
.
OP Drms (x′ , y ′ )OP Drms (x′ + ∆x′ , y ′ + ∆y ′ )
(8)
Iso-contours of OPD correlations are shown in figure 10. The shapes of the correlation contours resemble
those of spatial correlations of wall-pressure fluctuations at moderate Reynolds numbers.25 It is noted that
the correlation length in the spanwise direction is comparable with that in the streamwise direction, which
differs from the observation of Gordeyev et al.14
IV.
IV.A.
Separated Shear Layer over a Cylindrical Turret
Simulation Set-Up
The flow configuration for the cylindrical turret simulation is shown schematically in figure 11. In the
experiment of Gordeyev et al.,15 the flat window and the cylinder can be rotated to position the laser beam
relative to the freestream at any angle between 90◦ and 150◦. This angle is fixed at 120◦ in the present
simulation. The computational domain size based on the radius R of the cylinder is 7.75R, 2.81R and 0.48R in
the streamwise, wall normal and spanwise directions, respectively. The free-stream Mach number is M = 0.5.
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Figure 10. Spatial correlation of OPD in the x′ -y ′ plane for wavefront distortions induced by the turbulent boundary
layer.
SPONGE
Rotating
Cylinder
TURBULENT
INFLOW DATA
Flat Window
OUTFLOW
WALL
Laser Beam
Figure 11. Computational set-up for flow over a cylindrical turret with a flat window.
The Reynolds number based on the freestream velocity and radius of the cylinder is ReR = 5.76 × 105 in
the experiment, which is not affordable for LES. To make the simulation feasible, the Reynolds number
is reduced to ReR = 7.2 × 104 , 1/8 of the real value. The turbulent boundary layer at the inlet has a
thickness of 0.2R, and the corresponding Reynolds number based on momentum thickness is Reθ = 1400.
The turbulent inflow data, provided from the auxiliary LES mentioned earlier, are interpolated onto the face
mesh at the inlet and then fed into the simulation. The boundary conditions for top and outflow boundaries
are dirichlet. A sponge layer with thickness of 0.5R is applied along these two boundaries to implement a
non-reflecting boundary condition. In the sponge region, the flow structures and acoustic waves are damped
out, and flow quantities are forced to prescribed reference values obtained from a RANS simulation. The
boundary conditions for the bottom wall are no-slip, no penetration and adiabatic.
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(a) vorticity magnitude ωR/c∞
(b) density ρ/ρ∞
Figure 12. Instantaneous vorticity magnitude and density in a spanwise cut in flow over a cylindrical turret.
IV.B.
Results
Instantaneous vorticity magnitude and density from the cylindrical turret simulation are shown in figure 12.
These figures show realistic turbulent boundary layer structures starting from the inlet. The boundary layer
separates at the upper corner of the flat window, forming a free shear layer behind the turret. Figure 12(a)
shows detailed vortical structures of a wide range of scales in the separated shear layer. These structures
are generated by the instability of the thin shear layer emanating from the top of the turret modulated by
the turbulent boundary-layer eddies. The corresponding density variations are shown in figure 12(b) , which
are the cause of optical distortions when an optical beam traverses this separated shear layer.
The aberrations of an optical beam of aperture size 0.4R × 0.2R through the center of the flat window
(as shown in figure 1) are calculated. A snapshot of the instantaneous wavefront distortions is shown in
figure 13(a). Due to the angle between the separated shear layer and the flat window, the wavefront is tilted
significantly in the streamwise direction. This tilt can be represented in terms of dOP L/dx′ .8, 9 The time
history of x′ -tilt is plotted in figure 14. It is noted that the x′ -tilt is always negative which indicates that the
wavefront is tilted toward the downstream direction. This is because the density in the separation bubble is
lower than that outside the bubble (figure 12(b)), and hence the downstream part of the beam propagates
through a longer low-density region than the upstream part, resulting in smaller OPL. The instantaneous
wavefront distortion after removing the tilt can be found in figure 13(b).
The rms of OPD induced by this flow is 8.15 × 10−7 R. This is calculated by averaging over a time
period of approximately 34R/U∞ and in the spanwise direction. Based on the dimensions in the experiment,
R = 2 in, and the above OP Drms is approximately 0.041 µm, which agrees well with the experimental result
reported by Gordeyev et al.15 at the same Mach number. Although the Reynolds number in the simulaiton
is significantly lower than in the experiment, the large-scale coherent structures in the separated shear layer,
which contribute most to aero-optical distortions, are expected to be relatively insensitive to the Reynolds
number.
It is of interest to compare the magnitude of OP Drms induced by the separated shear layer with that
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(a) OPD with tilt
(b) OPD with tilt removed
Figure 13. Instantaneous wavefront distortions caused by the separated shear layer over the cylindrical turret.
-6E-06
dOP L
dx′
-8E-06
-1E-05
-1.2E-05
-1.4E-05
-1.6E-05
80
82
84
tc∞ /R
86
88
90
Figure 14. Time history of the streamwise tilt of the optical wavefront induced by the separated shear layer over the
cylindrical turret.
caused by the attached boundary layer upstream of the cylindrical turret. The boundary layer thickness is
0.2R at the inlet. By using the result from the flat-plate boundary layer simulation, described in section III,
OP Drms under the inlet condition is estimated to be 0.008 µm. Thus, the magnitude of wavefront distortion
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Figure 15. Spatial correlation of OPD in the x′ -y ′ plane at a streamwise position corresponding to the beam center,
for flow over the cylindrical turret.
caused by the separated shear layer is 5 times that of attached boundary layer.
The two-dimensional spatial correlation of OPD in the x′ -y ′ plane is plotted in figure 15. It is found that
the correlation length in the spanwise direction is significantly larger than that in the streamwise direction,
indicating that the large-scale coherent structures have 2D-like characteristics.
V.
Conclusions
In summary, large-eddy simulations of a compressible turbulent boundary layer and a separated shear
layer over a cylindrical turret have been preformed. The resulting density field is utilized to compute the aerooptical distortions. The predicted rms values of OPD from both simulations agree well with the experimental
results of Gordeyev et al.14, 15 It is found that in the turbulent boundary layer, optical wavefront distortions
are predominantly caused by the log layer and wake region. The anisotropic property of the turbulent
boundary layer renders the optical distortions dependent on the direction of propagation. An optical beam is
distorted more severely when its propagation path is tilted toward downstream than upstream, in agreement
with experimental findings.14 A correlation analysis of the density field confirms that the correlation length
is larger along downstream-tilted optical paths than upstream-tilted ones. For flow over a cylindrical turret,
wavefronts are tilted toward the downstream direction because of the angle between the flat window and
separated shear layer, and the low density region therein. The rms of OPD induced by the separated shear
layer is five times as large as that caused by the attached boundary layer upstream of the turret.
Although the Reynolds numbers have been much reduced in the simulations, the results of optical distortions from the simulations are very close to experimental values, suggesting that optical distortions may
be relatively insensitive to Reynolds number due to dominance by large-scale fluctuations of the index of
refraction in the boundary layer and separated shear layer. Futher investigation is needed to clarify this
important issue quantitatively. We will perform a systematic study of the Reynolds number effect as well
as the effects of Mach number and other important parameters. The simulation of flow over a cylindrical
turret has established a baseline case for studying the aero-optics of separated shear layers and evaluating
control effects. Future studies with this configuration will focus on simulating and analyzing the effect of
passive control devices for suppressing separation and mitigating the associated optical aberrations.
Acknowledgments
This work was sponsored through AFOSR Grant FA 9550-07-1-0504 by the High Energy Laser Joint
Technology Office (HEL-JTO). The U.S. Government is authorized to reproduce and distribute reprints
for government purposes notwithstanding any copyright notation thereon. We wish to thank Mohammad
Shoeybi of Stanford University for assistance with the LES code, and Stanislav Gordeyev and Eric Jumper
for helpful discussions.
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θ
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