ARO Final Review Session

8/12/2019 ARO Final Review Session

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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

Documents

ARO Final Review Session