Mechanisms of
Hypersonic Boundary Layer Transition
on Two Generic Vehicle Geometries
Steven P. Schneider
Purdue University, School of AAE
AFOSR Contractors Meeting
12 September 2003
San Destin, Florida
School of Aeronautics and Astronautics
Acknowledgements
•Boeing/AFOSR/BMDO/Purdue/Sandia/Langley development
of Mach-6 Quiet Ludwieg Tube, based on previous LaRC
work. $1m, 1995-2001.
•Cooperative funding by Sandia and Northrop-Grumman
(ballistic RV’s/blunt cones, flight data review), NASA Langley
(scramjet forebodies)
•Graduate students Craig Skoch (AFOSR), Erick Swanson
(NG/AFOSR), Shann Rufer (Sandia), Shin Matsumura
(Langley)
•advice from Jim Kendall, Scott Berry, etc.
School of Aeronautics and Astronautics
Shadowgraph of Transition on Sharp Cone at Mach 4
turb. spot
wave from spot
turb. spot shock
laminar
noise radiated from turbulent boundary layer
5-deg. half-angle cone in NOL Ballistics Range at Mach 4.31. Shot 6728, Dan
Reda, AIAA Journal v. 17, number 8, pp. 803-810, 1979. Re=2.66E6/inch,
cone length is 9.144 inches. From Ken Stetson. Cropped
Aeroheating Rises By a Factor of 3-8 at Transition
13-foot Beryllium Cone at Mach 20 in Reentry
CFD predicts heating well --ONLY IF-transition location picked to match flight
qw, Btu/ft -sec
400
2
300
Transition Uncertainty 300%
Laminar Uncertainty 15%
Turbulent Uncertainty 20%
200
100
0
0
0.25
Hamilton, Re-Entry F, NASA-TP-3271.
0.5
z/L
0.75
1
Existing Correlations Have a Large Uncertainty
Kuntz, Sandia SWERVE maneuvering flight vehicle
Empirical Correlations Typically Scatter by a Factor 3 in Re,
or a factor 10 in Rex, for fairly general datasets
3
Re
10
102
2
Re = 150 M e (NASP)
Laminar Local Conditions
Turbulent Local Conditions
4
From Schneider, JSR, Jan. 99.
6
Me
8
10
12
Conventional Noise Corrupts Transition Expts:
Quiet Tunnels Needed
1.
2.
3.
4.
5.
6.
7.
8.
9.
Conventional fluctuation level typically 1%
Major Source: Acoustic radiation from the nozzle walls
Causes early transition
Can change trends in transition
Transition in Conventional Facilities is NOT a reliable
predictor for flight!
Quiet tunnels require laminar nozzle-wall boundary layers
Quiet tunnel development is an expensive and risky exercise
in laminar flow control
Langley developed quiet tunnels in 1970-1990, but only
Mach 3.5 is operational
Purdue effort leads hypersonic q.t. dev. Langley may restart
S.P. Schneider, Purdue AAE
Need Measurements of the Mechanisms of Transition
• Transition data by itself is ambiguous. What caused the
transition? Roughness? Crossflow? 1st mode? All 3?
Tunnel noise? stray roughness? AOA errors?
• Need detailed measurements of the transition mechanisms
(rare field measurements of small fluctuations,
preferably with controlled disturbances).
• Detailed measurements and computations of the
mechanisms can provide physical understanding.
• Can improve scaling from wind-tunnel to flight conditions
• Such measurements are difficult; development of the
capability requires a sustained effort. Purdue presently has
the only lab making hypersonic hot-wire measurements
S.P. Schneider, Purdue AAE
Reliable Predictions Must Be Based on Mechanisms
• Instabilities that lead to transition can be computed (now or
soon) (1st & 2nd mode, crossflow, Gortler, algebraic, etc.)
• Seek semi-empirical mechanism-based methods similar to
e**N, where N=ln(A/A0) is the integrated growth of the
most-amplified instability, incorporates mean-flow effects
on wave growth
• Computations must be developed and validated based on
detailed measurements in ground facilities
• Computations must be compared to flight data
• Dominant Mechanisms on RLV’s, airbreather forebodies,
and RV’s remain to be determined; little or no data at
present
• Bridge gap between users and researchers
S.P. Schneider, Purdue AAE
Three Primary AFOSR-Funded Tasks
1. Obtain quiet flow at high Reynolds number in the
Mach-6 Ludwieg tube. Craig Skoch and now also Matt
Borg. Joint with NASA Langley and Sandia.
2. Measure mechanisms of transition on a generic RV:
a blunt cone including angle of attack. Shann Rufer and
Erick Swanson. Joint with Sandia and Northrop-Grumman.
3. Measure mechanisms of transition on a generic scramjet
forebody. Shin Matsumura, M.S., but work now
suspended. AIAA 2003-3592, 2003-4583. Joint with
NASA Langley.
School of Aeronautics and Astronautics
Boeing/AFOSR Mach-6 Quiet Tunnel
So Far, Quiet Only at Low Reynolds Number
Pressure Drops 30-40% During 10-sec. Run
School of Aeronautics and Astronautics
Schematic of Mach-6 Quiet Nozzle
School of Aeronautics and Astronautics
Tunnel Quiet Only at Low Reynolds No. – Why?
1. Nozzle length twice Langley Mach-6 quiet nozzle (was
quiet to 145 psia). We drop quiet at 8 psia in downstream
half of nozzle. Bypass!?
2. Fluctuations generated at bleed-slot lip? (Tried Case 7)
3. 0.001-0.002-in. step at aft end of electroform? Lack of
polish on downstream sections? (Polished downstream)
4. Leaks which we have not found yet?
5. Upstream effect of diffuser fluctuations? (current focus)
LaRC quiet tunnels all open jet
6. Vibrations of tunnel & bleed lip?? M4 had no lip. But
these damp with time, no time dependence observed
7. Residual noise in driver? Plan hot-wire measurements
8. Something else?
S.P. Schneider, Purdue AAE
Pitot Measurements on Nozzle Center, Near Exit
and 40 Inches Upstream. Mach Varies Slowly
6
z=45.0 in/Bleeds Open
z=84.5 in/Bleeds Open
Mach Number
5.8
nozzle end
at z=102
in.
5.6
throat at
z=0
5.4
Empty
tunnel
5.2
5
5
6
7
8
9
10
11
12
13
14
15
AIAA 2003-3450
Driver Tube Pressure (psi)
School of Aeronautics and Astronautics
Pitot Fluctuations Drop at Nearly Same Pressure
Near the Nozzle Exit and Halfway Up the Nozzle
5
% RMS Fluctuations
z=45.0 in/Bleeds Open
z=84.5 in/Bleeds Open
BYPASS
TRANSITION!
4
3
noz. end at
z=102 in.
throat at
z=0.
empty
tunnel
2
1
T0=160C
centerline
0
5
AIAA 2003-3450
6
7
8
9
10
11
12
13
14
15
Driver Tube Pressure (psi)
School of Aeronautics and Astronautics
pressure (psia)
Diffuser-Entrance Static Pressure Fluctuations are
50% when Bleeds are Open: Due to Bleed-Air Jets
in Diffuser? Fed Upstream? Trip BL?
0.016
Bleeds Open
0.014
0.012
0.01
0.008
0.006
5
5.02
5.04
5.06
5.08
5.1
pressure (psia)
time (sec)
0.016
Bleeds Closed
0.014
0.012
z=104.85
inches
0.01
0.008
0.006
5
AIAA 2003-3450
5.02
5.04
5.06
5.08
5.1
time (sec)
School of Aeronautics and Astronautics
Bleeds Plumbed Direct to Vacuum Tank: Mean
Mach number higher!
6
T0=160C
Probe on CL
Mach Number
5.8
Reduced B.L.
thickness?
First results
5.6
5.4
5.2
5
z=75.3 in/New Sting Mount
z=75.3 in/New Bleed System
4
6
8
10
12
14
Driver Tube Pressure (psi)
School of Aeronautics and Astronautics
Bleeds Plumbed Direct to Vacuum Tank:
Unchanged!
Noise
5
% RMS Fluctuations
z=75.3 in/New Sting Mount
z=75.3 in/New Bleed System
T0=160C
Probe on CL
4
New results,
preliminary
3
2
1
0
4
6
8
10
12
14
Driver Tube Pressure (psi)
School of Aeronautics and Astronautics
Crossflow Vortices on a Cone at Angle of Attack at Mach 6
Re = 3 x 106/ft, 7-deg. half-angle sharp cone, 6-deg. AOA, TD=160C, AIAA 2003-3450
Oil applied in smooth layer by brushing crosswise. Streaks develop ala china clay
School of Aeronautics and Astronautics
Hot-Wire Spectra on Sharp Cone Showing Waves
1st-mode
instab. waves?
0
10
driver pressure:
turb.?
-2
10
45 psia
75 psia
95 psia
115 psia
135 psia
x=11.7 in.
7-deg. halfangle cone
O.H. 1.8
-4
10
l/d=140
225kHz
T0=160C
-6
10
0
0.5
1
1.5
frequency (Hz)
2
2.5
5
x 10
School of Aeronautics and Astronautics
TSP IMAGES ON HYPER-2000 SCRAMJET FOREBODY (VORTEX BREAKDOWN)
20
-2
15
-1
0
y (in.)
y (in.)
-2
0
1
3
0
5
0
2
-5
0
-1
0
y (in.)
-1
2
4
Streamwise Vortices
Streamwise Vortices
5
4
6
x (in.)
6
x (in.)
8
10
8
10
-5
q/qref
Breakdown and Spreading
10
15
Breakdown and Spreading
20
AIAA Papers
2003-3592, 2003-4583
0
1
-15
thick. : 4 mils
spacing : 5/8 inch
0
1
5
P0
: 120 psia
unit Re : 2.57 million/ft
10
2
2
q/qref
6
-10
7
-10
6
-5
7
8x (in.) 9
-5
0
8 10 911
x (in.)
5
0
10
q/qref
10
5
11
q/qref
10
Vortices generated at LE
grow, break down and spread
across the corner. Followed
by onset of turbulence?
Summary
• Tunnel runs quiet, now without separation at 8 psia, Mach
5.7. Model support/second throat had excessive blockage. A
new streamlined support avoids blockage problems.
• Bypass causes nozzle-wall transition at 8 psia
• Prime suspect is still fluctuations from downstream
• Work toward quiet flow continues
• Bypassing bleed-slot air direct to vacuum tank through fast
valve does not affect noise much.
• Oil flow shows crossflow vortices on cone at AOA
• Hot wire appears to show instability waves on sharp cone
• Hot-wire frequency response needs improvement for
measurements of second-mode waves
• TSP and oil-flow measurements on scramjet forebody
S.P. Schneider, Purdue AAE
What Next?
• Are Disturbances Fed from Downstream Tripping the
Upstream Boundary Layer? Centerbody DID cause
separation. Plumbed bleed air direct to vac. tank, but this
shows limited effect so far
• Modify diffuser further? How?
• Measure in diffuser and with hot-wire on nozzle wall
• Measure flow in contraction entrance, confirm low noise
• Check for leaks with helium sniffer
• Touch up throat polish, bleed-lip bump
• Need computation of flow in Bleed Slots,
more measurements
• Hot-wire, oil-flow, temp. paints on blunt cones incl. AOA
S.P. Schneider, Purdue AAE
Backup Slides
School of Aeronautics and Astronautics
General 3D Tunnel Data Scatter Over Rex = 105 to 107
Depending on
Noise,
Configuration,
Roughness, etc.
From “A Survey of NASA Langley
Studies on High-Speed Transition
and the Quiet Tunnel”, NASA TMX-2566, Beckwith and Bertram, as
reproduced in Bertin, “Hypersonic
Aerothermodynamics”, AIAA,
1994, p.379.
Context for Blunt Cone Research
1. Interest in gliding RV’s from Air Force (CAV) and DARPA
(Falcon). Transition is critical to aeroheating during
extended glide. A cone at AOA is one such glider.
2. Interest in HyFly and similar airbreathing missiles
3. Flight data review (incl. classified) provides:
a) suggestions regarding research directions
b) test cases for development/validation of mechanismbased computational prediction methods
3. Connections to user community at Sandia, NorthropGrumman, etc.
4. Sandia work now joint with Candler at Minnesota
School of Aeronautics and Astronautics
Transition Critical to Scramjet Cruise Vehicles
1. Multistage airbreathing to orbit will still be similar to NASP
– a large scramjet-powered vehicle
2. NASP review by DSB, 1988: “Estimates [of transition]
range from 20% to 80%”, affects GTOW by factor 2
3. NASP review by DSB, 1992: Two most critical tech. areas
are scramjet engines and boundary layer transition
4. Propulsion has made great progress under Hyper-X,
HyTECH, related programs. Engine works in direct
connect tests.
5. Will transition technology be ready when the engine is?
DTIC citations AD-A201124, AD-A274530
School of Aeronautics and Astronautics
Conventional Wind and Shock Tunnels are Noisy!
1. Fluctuation level typically 1%: > 10 times higher than flight
2. Major Source: Acoustic radiation from turbulent boundary layers on
the nozzle walls.
3. Causes early transition: perhaps 3-10 times earlier than in flight.
4. Can change trends in transition:
a) Sharp cone transition data in conventional tunnels scales with noise
parameters alone, independent of Mach number.
b) ReT, CONE = 2 ReT, PLATE in conv. tunnel, but ReT, CONE = 0.7 ReT, PLATE in
quiet tunnel and e**N analysis. Flat Plate is later, NOT cone!
c) Bluntness, crossflow, and roughness effects all differ in quiet and
noisy conditions.
d) Transitional extent typ. 2-4 times longer in conv. tunnel than in flight
or quiet tunnel.
5. Transition in Conventional Facilities is NOT a reliable predictor for
flight! Except for certain limiting cases, such as transition that occurs at a
roughness element.
S.P. Schneider, Purdue AAE
Quiet Tunnels Have Been Under Development Since
the 1960’s to Address the Noise Problem
1. Must solve the Acoustic Radiation Problem
2. Must Control Laminar-Turbulent Transition on the nozzle walls!
3. Quiet Tunnels also require low-noise core flows.
4. Laminar Nozzle-Wall Boundary Layers requires mirror-finish nozzle
walls, specially designed nozzles, particle-free flow
5. Accurate Fabrication of the Nozzle with tight tolerances and a mirror
finish is expensive and risky.
6. NASA Langley built a dozen nozzles between 1970 and 1990, and
worked out many of the problems: Mach 3.5 since 1982, Mach 6
from 1990-97 (presently boxed)
7. No High Reynolds Number Hypersonic Quiet Tunnel presently in
operation anywhere. Purdue effort leads. Langley Mach-6 may be
reinstalled ca. 2004.
S.P. Schneider, Purdue AAE
Hypersonic Transition is Critical to Large
Scramjet Accelerator Vehicles
• Multistage Airbreathing to Orbit will still be similar to NASP -- a large
hypersonic scramjet-powered vehicle
• National Aerospace Plane Review by Defense Science Board, 1988:
Estimates [of transition] range from 20% to 80% along the body …
The estimate made for the point of transition can affect the design
vehicle gross take off weight by a factor of two or more.
• National Aerospace Plane Review by Defense Science Board, 1992:
The two most critical [technology areas] are scramjet engine
performance and boundary layer transition… Further design
development and increased confidence in these two technical areas
must be of paramount importance to the NASP program.
• The propulsion problems are being worked under various programs.
However, transition research is reduced to a shell. Will transition
technology be ready when the combustor is?
AD-A201124, Report of the DSB Task Force on the NASP Program, Sept. 1988
AD-A274530, Report of the DSB Task Force on the NASP Program, Nov. 1992
Transition is Critical to RLV Reentry Aeroheating
• Aeroheating affects TPS weight, type, and operability –
a low-maintenance metallic TPS may not be possible if
transition occurs early
• Reentry trajectory is iterated to achieve acceptable
aeroheating, and therefore depends on transition
• Crossrange is critically dependent on aeroheating
• TPS selection affects roughness and surface temperature
and therefore boundary-layer transition
• Uncertainty in transition drives TPS temperature margin,
200+F for shuttle (per Stan Bouslog)
• A metallic TPS may have a more repeatable and smaller
roughness which might permit delaying transition
Simple Conventional Transition Measurements
Often Don’t Give “Correct” Trends
H
M
Q
N
Krogmann M= 5, c= 5 deg.
Stetson & Rushton M= 5.5, c= 8 deg.
Stetson 1981 M= 5.9, c= 8 deg.
Muir NOL, M= 6, c= 8 deg.
Holden M= 13.3, c= 6 deg.
McCauley M= 10, c= 6 deg.
King M=3.5, c= 5 deg., quiet
King M=3.5, c= 5 deg., noisy
N
Sharp Cone at AOA.
Tunnel Noisy
Tunnel Quiet
What is the
"True" Trend?
xT/xT,=0
1.5
1 Q
N
H QH
M
M
Detailed Analysis
is Needed
HQ
M
H
N
N Q
NH
Q
M
0.5
windward
0
See JSR v. 38 n. 3,
May-June 2001, p. 328.
S.P. Schneider, Purdue AAE
H
-0.5
H
M
N
Q
leeward
0
/c
0.5
N
Q
Bridge the Gaps in Hypersonic BLT
• Designers using Empirical Methods/
Researchers doing PSE and DNS etc.
• Designers using CFD, often 3D/
But still crude models for BLT
• Improve BLT models at all levels:
Simple
for conceptual design to
Complex for
advanced design
• Improve coordination between
researchers and designers
• Improve coordination among researchers
S.P. Schneider, Purdue AAE
Summary of Purdue Effort, 1990-99
1.
2.
3.
4.
5.
6.
7.
8.
Development of Mach-4 Ludwieg Tube, Quiet to Re = 400,000, 1990-94.
Tests of Heated Driver Tube (Munro, 1996)
Development of Hot-Wire and Glow-Perturber Technique
Controlled Wave Growth of factor 2-3 on Cone at AOA under quiet
conditions (Ladoon Ph.D., 1998)
Development of Pulsed Laser-Perturber for Generating Local Perturbations
in Freestream for Receptivity Work (Schmisseur Ph.D., 1997)
Controlled Measurements of Damping in Forward-Facing Cavity,
Explained Low Heat Transfer in 1961 Flight Data (1997-99)
Developed of High-Sensitivity Laser Differential Interferometer ala
Smeets. Receptivity on Blunt Nose. (Salyer Ph.D., 2002 )
Development of High-Reynolds Number Mach-6 Quiet Ludwieg Tube
(1995-present)
S.P. Schneider, Purdue AAE
Summary of Purdue Effort, 1999-2002
1. Completion of Mach-6 Quiet-Flow Ludwieg Tube. Rufer, M.S.
2000, burst diaphragm tests. Skoch, M.S. 2001, heaters and initial
tests. Initial Operation, April 2001.
2. Development of Automated Vertical-Plane Traverse (probe profile
in single run). Swanson, M.S. Dec. 2002
3. Modifications to Bleed-Slot Throat Yield Initial Quiet Flow (but
only at low Reynolds number).
4. Hot-wires survive in Mach-6 flow, stable CTA operation, 2001-2002
(still not at full pressure).
5. Skoch/Rufer operate Ladoon’s glow perturber and hot wire
apparatus in Mach-4 tunnel, 2002. (New student education).
6. Matsumura/Swanson develop temperature-sensitive paints for
measuring stationary vortex growth, 2001-2002.
7. Matsumura measures streak/vortex growth on Hyper2000 with
controlled roughness perturbers.
8. Schneider surveys classified flight data, summer 2002
S.P. Schneider, Purdue AAE
Need National Plan for Hypersonic Transition
Research for Airbreathers and RLV’s
• Further development of existing mechanism-based
prediction methods
• Detailed measurements on generic geometries in quiet and
conventional tunnels to develop & validate the mechanismbased methods
• Comparisons of mechanism-based methods against
existing flight data
• Industry has long used mechanism-based methods for
transonic speeds – how long before they are available for
the more critical hypersonic problems?
S.P. Schneider, Purdue AAE
Deflected Control
Surfaces with
Compression-Corner
Separations:
-Transitional Heating
Can be 50% Larger than
Turbulent Heating
-Transition Occurs at
Low Reynolds Numbers
-Improved Predictions
Can Reduce Control
Surface TPS
Requirements
Horvath et al., AIAA 99-3558, Fig. 14. Mach
6, 40-deg. AOA, Re=2E6/ft., dBF=20 deg.
S.P. Schneider, Purdue AAE
Right body flap
Left body flap
Reattachment
2
Laminar approaching
boundary layer
1.5
LaRC
X-33
Expt.
hFR is
nose
Expansion fan
impingement
h/hFR 1
Turbulent approaching
boundary layer
0.5
0
0
0.5
1
Nondimensional flap chord length
Design of Seventh Bleed Slot Throat
• 1D streamtube analysis ala Beckwith, full 1D both sides of
bleed lip. See paper
• Increase from 30% to 38% suction
• Move stagnation point from 2/3 below top of hemicircle to
4/5 below top
S.P. Schneider, Purdue AAE
Crossflow Vortices on a Cone at Angle of Attack at Mach 6
Re = 2.6 x 106/ft, 7-deg. half-angle sharp cone, 6-deg. AOA, TD=160C,
AIAA 2003-3450. Oil applied in smooth layer by brushing crossways.
School of Aeronautics and Astronautics
Oil-Flow Image of Streamlines
on a Cone at Angle of Attack at Mach 6
Re = 3 x 106/ft, 7-deg. half-angle sharp cone, 6-deg. AOA, TD=160C, AIAA 2003-3450.
Oil Flow. In this case, oil was applied in dots, movement of dots shows streamlines.
School of Aeronautics and Astronautics
Near-Term Mechanism-Based Prediction Approach
• Compute approximate aeroheating and 1D heat
conduction, down the trajectory
• Compute accurate 3D mean flow (with chemistry) at
possible transition altitudes
• Compute 1st & 2nd mode instabilities on wind & lee planes
• Compute crossflow Reynolds number off centerplane.
Later compute crossflow instability growth
• Compute Gortler when relevant
• Compute Re_k, k/theta, etc. for roughness.
• Use linear instability, also PSE & nonlinear when needed
• Compare details to ground expts, results to flight & ground
S.P. Schneider, Purdue AAE
Diffuser Static Pressures with Throttled Bleeds
pressure (psia)
0.04
0.02
0
Bleeds Open
0.04
0.02
0
Bleeds 1/2 Open
0.04
0.02
0
Bleeds 3/8 Open
0.04
0.02
0
Bleeds 1/4 Open
0.04
0.02
0
Bleeds Closed
AIAA 2003-3450
0
2
4
time (sec)
6
8
School of Aeronautics and Astronautics
New Streamlined Sting Support, Installed
School of Aeronautics and Astronautics
New Streamlined Sting Support on Bench
School of Aeronautics and Astronautics
Mean Mach Number from Pitot on Centerline
New Streamlined Model Support Stops Separation
6
Mach Number
5.8
5.6
5.4
z=75.3 in/New Sting Mount
z=84.5 in/New Sting Mount
z=75.3 in/No Double Wedge
z=84.5 in/No Double Wedge
5.2
5
4
6
8
10
12
14
Driver Tube Pressure (psi)
School of Aeronautics and Astronautics
Pitot on Centerline: Noise Onset is Same for
Empty Tunnel and New Sting Support
z=75.3 in/New Sting Mount
z=84.5 in/New Sting Mount
z=75.3 in/No Double Wedge
z=84.5 in/No Double Wedge
% RMS Fluctuations
5
4
3
2
1
0
4
6
8
10
12
14
Driver Tube Pressure (psi)
School of Aeronautics and Astronautics
Single-Run Hot-Wire Profile on Sharp Cone at Pd=95 psia
1.6
0.08
mean
rms
x=11.7 in.
1.4
0.04
1.3
1.2
0.5
RMS voltage
0.06
mean voltage
1.5
7-deg. halfangle cone
O.H. 1.8
l/d=140
225kHz
0.02
1
1.5
0
2
height above cone (mm)
School of Aeronautics and Astronautics
HYPER-2000 MODEL GEOMETRY
Hyper-2000
Hyper-X
• Obtained from Hyper-X
program office
• Representative geometry
of current vehicles of this
class
• Approx. 1/5th scale of 12 ft.
full-scale vehicle
• Slightly longer first
compression ramp than
the Hyper-X
OIL-FLOW VISUALIZATION (“SMOOTH” MODEL)
y (in.)
-1
Appearance of streamwise
vortices immediately
downstream of first
compression corner
0
1
6
8
10
12
x (in.)
1.67 million/ft
1.93 million/ft
2.20 million/ft
2.46 million/ft
2.72 million/ft
RGB Counts (I'/Imean)
1
• Location and spacing of vortices
independent in this unit Re range.
0.5
• Effect of leading edge variation
0
• Suggests most amplified vortex
spacing of 0.19 inches (5.6
cycles/inch)
-0.5
-1
-0.5
0
y (in.)
0.5
1