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ARO 404-2 High Speed Aerodynamics
Review
June 11, 2013
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Shock-Expansion Theory--- A review of Gas dynamics/oblique shock/isentropic flows
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Problem: A Pitot tube is inserted in the aft of a double wedge and its reading is 2.596 atm.
The local pressure on the backface point a is measured as 0.1 atm. Find free stream M1
a
Assume M1= 3.5, then =29.2, Mn1=3.5sin=1.71. Then from Table A.2 Mn2=0.638Then from E.1, we have 2.6*sin(29.2-15)= 0.638 (thus it checks)! M1=3.5
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4
2
0,
2
0,
2
0,
0,
1
1
1
21
121222
M
cc
M
cc
M
CC
CC
V
uC
VxVxVV
uC
m
m
l
l
P
P
PP
P
P
Linearized velocity potential equation analyses results
Insert transformation results into linearized CP
Prandtl-Glauert rule: If we know theincompressible pressure distribution over anairfoil, the compressible pressure distributionover the same airfoil may be obtained
(subsonic flowonly)
Lift and moment coefficients are integrals ofpressure distribution (inviscid flows only)
Perturbation velocity potential for incompressible
flow in transformed space
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]2
)()()[(
1)(
4
1)(
4
222
2
2
leue
ee
e
e
ee
MCd
MC
Ackeret Supersonic Linearized Theory
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cX
C
xM
C
CP
cm
mx
5.0
0
)02
1(
1
4
5.0
20
PROBLEM: Find the aero coefficient on the double wedge airfoil with Ackeret Theory
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3D wing aerodynamics
In subsonic flow
First find the 3D wing in incompressible flow (e.g., Prandtl lifting line
theory)
Then applying Prandtl-Glauert rule
In supersonic flow
If it is a rectangular wing, use Bertin/Cumming book Chapter 11
Table 1
--- Applicable for 3 types of airfoil shapes (double wedge,
modified double wedge and bi-convex shapes)--- Aspect ratio effects
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This Table gives a rectangular wing aerodynamics in supersonic flow including effects
of Mach number, aspect ratio, gas properties (), thickness/chord ratio, airfoil configuration,
and angle of attack, .
1)( 2 M
BertinsBook Chapter 11 Table 11.1
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Example: Use conical method to find the lift andDrag on a rectangular flat plate wing (1storder)
]~
[sin2
,
~'tan;
1
1
1tan
tan'tansin2
,
wingtipthebyinfluencedregiontheIn
,2
1
2,
,2
1
2,
1
2
2
1
2
22
22
x
y
Cp
CpThus
x
y
M
CpCp
sideleewardM
Cp
sidewindwardM
Cp
d
d
d
d
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Example: A rectangular wing of aspect ratio 2.5 is subject to a uniform stream of Mach 1.4.
the wing is constructed from a double-wedge airfoil of thickness ratio=0.04. Use Table 11.1
of Bertin to give the following aerodynamic characteristics of the wing first symbolically
and then numerically for arbitrary angle of attack,.
(a) Lift coefficient
(b) wave drag coefficient(c) Moment coefficient about the leading edge
(d) Location of the center of pressure
CPXandCMCDCLFind 0,,:
airfoilwedgedoubleaisgeometryAirfoiAR
MGiven
5.2
04.0
4.1:
297.3
)]04.0*2
1
)14.1(2
)24.1(4.1*4.11(
14.1*5.2*2
11[
14.1
4
)1(2
)2(
1)(;21'
)]'*1()(2
11[
4)(:
2/32
224
22
2/32
224
3
2
3
L
L
L
C
C
M
MMc
MA
ACAR
CaSolution
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22
2
2
1
2
2
1
97.3006532.097.314.1
)04.0(44
*1
*)(
CD
airfoilwedgedoubleafork
CM
kCb LD
cccAcARARAcARX
pressureofcenteroflocationd
CM
ARAcARAR
CMc
CP 5.0~43.0]'*1*
)1*'*(*3/2*[
)(
417.1
)]114.15.2)(04.0(2
1*
)14.1(2
)24.1(4.1*4.1
3
214.1*5.2[
)14.1(*5.2
2
)]1*'*(*3
2*[
*
2)(
3
3
0
2
2/32
2242
2
320
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Swept-back wing of infinite span
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Geometrical Description of Wing Sweep
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Equivalent 2-D Flow on Swept Wing
Freestream Mach number resolved into 3 components
i) vertical to wing
ii) in plane of wing, but tangent to leading edge
iii) in plane of wing, but normal to leading edge
sinM
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i)Mvert Msin
ii)M|| Mcossin ii)M Mcoscos
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Equivalent Mach Number normal to leading edge
Meq M 2 Mvert2 Msin 2
Mcoscos 2
M 1 cos2 cos2 1 sin2 M 1 sin2 cos2
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Equivalent angle of attack normal to leading edge
tan eq MvertM
Msin
Mcoscos
tan cos
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Equivalent chord and span Chord is shortened
Span is lengthenedceqc cos beq
b
cos
cos
1)()(
)(cos
dx
dz
dx
dz
xx
eq
eq
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Equivalent 2-D Lift Coefficient
CL eq L2pMeq
2c cos bcos
L2pMeq
2cb
L
2pM
2cb 1 sin2 cos2
CL
1 sin2 cos2
)(cos,cossin1
1cos,1
222
eqLLLL
Leq CCorCCC
when
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Equivalent 2-D
Drag Coefficient
CD eq D/ cos
2
pMeq2c cos b
cos
D/ cos
2
pMeq2cb
D/ cos
2pM
2cb 1 sin2 cos2
CD / cos
1 sin2 cos2
3cos,1 DeqD CCAs
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Summary on Swept-back Wing (
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Effects of swept on lift-to-drag ratio
Double wedge airfoil at Mach 2
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Example: Show that the section lift coefficient for a swept airfoil
with a supersonic leading edge is given by:
smallare
AOAandratiothicknessthatsassumptionthewith
MC
,,
1cos)(
cos4
22
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Critical Mach Number
MCRcan be estimated from(1) Prandtl-Glauert rule
(2) Karman-Tsien
(3) Laitone
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Every where is subsonic flow on the airfoiol
Critical Mach number
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28
IMPROVED COMPRESSIBILITY CORRECTIONS
0,2
22
2
0,
0,
2
22
0,
2
0,
12
2
11
1
2111
1
P
P
P
P
PP
P
P
CM
MM
M
CC
C
M
MM
CC
M
CC
Prandtl-Glauret
Shortest expression
Tends to under-predictexperimental results
Account for some of nonlinear
aspects of flow field
Two other formulas which showexcellent agreement
1. Karman-Tsien
Most widely used
2. Laitone
Most recent
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Isentropic relation between theFree stream and the point on the
airfoil
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Cp
Prandtl-Glauert rule
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Determination of Critical Mach Number
Isentropic relation
2)
11(1
min
2
22 MIN
MINp
Cp
M
MM
CpC
ruleTsienKar
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M Cp
0.7 -0.4469
0.725 -0.4672
0.75 -0.4913
0.775 -0.5202
0.8 -0.5556
0.825 -0.6
0.85 -0.6582
M Cp
0.7 -0.7791
0.725 -0.681
0.75 -0.5910.775 -0.5095
0.8 -0.4346
0.825 -0.3057
0.85 -0.302
Problem: Use the Karman-Tsien rule to calculate the critical Mach number for an airfoil
Whose Cp-min =-0.3 at low speed for a given altitude. Give the key equations, generate a
Table and make a full page graph on engineering paper to determine critical Mach number
To three significant figures. Hint: the values is between 0.70 and 0.85.
2)
11(1
min
2
22 MIN
MINp
Cp
M
MM
CpC
ruleTsienKar
Karmin-Tsien Cp-crit
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 0.2 0.4 0.6 0.8 1
Cp-CRKarman_tsien
Mach
Cp
Two curves intersects at M-cr=0.75, Cp =-0.49
3.0, n
micrit cpsettingCpCptwotheseSetting
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PS 8 Problem 1: Starting with a rectangular flat plate wing at an arbitrary supersonic
cruise Mach number. Modify the wing in such a way that it would experience only
Two-dimensional flow. Then proceed to calculate the wing total lift, drag andmoment the leading edge.
PS 8 Prob. 1
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M
Isentropic compression in the lower part of airfoil
Isentropic expansion in the upper part of airfoil
There exists trailing shock at the rear end of airfoil
But on the airfoil, it is shockless
Problem 2: 2D air foi l with zero thickness in supersonic f li ght. F ind an air foil , i t would
Produce no shock waves (except possibly at its trai l ing edge)
In-coming flow is tangent to the airfoil at the leading edge.
Trailing shock
Expansion waves
Shock-less airfoil
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Example: Find Lift and drag coefficient on this airfoil
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Example: Find Lift and drag coefficient on this airfoil
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Prob. 3 :Shock wave and boundary layer interactions usually has a detrimental effects on the
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Adverse pressure gradient
Aerodynamics of an airfoil. Conceive a modification to the airfoil in the vicinity of the
Interaction that would significantly reduced the adverse pressure gradient effects and prevent
the Boundary layer separation.
Applying suction here
Applying wall suction here
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Reduction of viscous skin friction or delay the boundary layer separation
using suctions at the wall at the place that has adverse pressure gradient.
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Subsonic flow over a thin airfoil --- Kutta conditions at trailing edge
Streamline patterns are dictated by the Kutta trailing edge conditions
PS 8 Problem 4: Consider the flow past a flat plate airfoil (zero thickness) at an AOA.
For both supersonic and subsonic free stream, sketch the streamlines and the wall pressure
distributions
b i l fl l hi i f il
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Subsonic flow
Suction force
(V>>1) at
leading edge
PL
PU
CD = 0 DAmbertparadox
CL >0
Subsonic Flows over a flat plate or thin airfoil
The real flow patterns may have leading edge
Flow separation
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PS 8
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Compare to Flat Plate in subsonic flows
CL
pl
p
pu
p
2M
2
cos CD= 0 (2D flow)
FpLift
Drag
Supersonic flows over a flat plate
streamlines
In 3D flow, CD is no longer zero
Prandtlslifting line theory indicates
Downwash would produce induced drag
CD > 0
FpLift
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pLift
Drag
Subsonic flow
Streamlines around a NACA 0012 airfoil at moderate angle of attack
Supersonic flowStreamline pattern
http://localhost/var/www/apps/conversion/tmp/scratch_9//upload.wikimedia.org/wikipedia/commons/b/b3/Streamlines_around_a_NACA_0012.svghttp://localhost/var/www/apps/conversion/tmp/scratch_9//upload.wikimedia.org/wikipedia/commons/b/b3/Streamlines_around_a_NACA_0012.svghttp://localhost/var/www/apps/conversion/tmp/scratch_9//upload.wikimedia.org/wikipedia/commons/b/b3/Streamlines_around_a_NACA_0012.svg -
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Supersonic flows over a thin flat plate
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