Entry Aerodynamics - University Of Marylandspacecraft.ssl.umd.edu/academics/791S14/791S14L07.aerodynamics.pdfEntry Aerodynamics ENAE 791 - Launch and Entry Vehicle Design U N I V E

Entry AerodynamicsENAE 791 - Launch and Entry Vehicle Design

U N I V E R S I T Y O FMARYLAND

Entry Aerodynamics• Atmospheric Regimes on Entry• Basic fluid parameters• Definition of Mean Free Path• Rarified gas Newtonian flow• Continuum Newtonian flow (hypersonics)

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© 2014 David L. Akin - All rights reservedhttp://spacecraft.ssl.umd.edu



Basic Fluids Parameters

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M ⌘ Mach Number =v

a

a ⌘ speed of sound =

p�RT

ordered energy

random energy

=

12mv2

12mv2g

=

v2

3RT=

�

3

v2

a2=

�

3

M2

✓R =

<m

◆

Re ⌘ Reynold’s number =

inertial force

viscous force

Re =mv

⌧A=

⇢Av2

µ vLA

=⇢vL

µ



Random vs. Ordered Energy

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More Fluid Parameters

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K =

number of collisions with body

number of collisions with other molecules

K ⌘ Knudsen number

K =�

L

� ⌘ mean free path

L ⌘ vehicle characteristic length



Estimating Mean Free PathAssume:• All molecules are perfect rigid spheres• Each has diameter σ, mass m, and velocity• Consider a cube with side length L containing N

molecules• N/6 molecules are traveling in each direction

– ±X– ±Y– ±Z

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v



Consider Collisions in +Z Direction

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v

v2v

()

Area = ⇡�2

number of potential +Z collisions

n(+Z) =1

6N

⇡�2L

L3=

1

6N

⇡�2

L2

frequency of +Z collisions

f(+Z) =n(+Z)

�t=

⇡6N

�2

L2

L2v

f(+Z) =⇡⇢�2v

3m



Consider Collisions in +X Direction

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v

v

2v

()frequency of +X collisions

f(�X) = f(+Y ) = f(�Y ) = f(+X) f(�Z) = 0

Total frequency of collisions

f =⇡

3(1 + 2

p2)

⇢�2v

m

f(+X) =n(+X)

�t=

p2⇡⇢�2v

6m



Mean Free Path

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� =v

f=

m/�2

⇡3 (1 + 2

p2)⇢

✓/ 1

⇢

◆

at sea level: � = 6.7⇥ 10�8 m

at 100 km: � = 0.3 m ⇠ 1 ft



Five Basic Flow Regimes• Free molecular regime• Near-free molecular regime• Transition regime• Viscous merged boundary layer• Continuous regime

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Entry Flow Regimes

ref: Frank J. Regan, Reentry Vehicle Dynamics AIAA Education Series, NY, NY 1984

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Flow Regime Definitions

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K =�

RN

Mean free path after collision

Knudsen number in rarified flow

�c =4

p⇡�

✓Tw

T1

◆ 12 �1M1

If Tw ⇠ T1

�c⇠= 1.9

�1M1



Free Molecular Regime• Orbital flight• • Molecule encountering a boundary (e.g., surface of

vehicle) attains the state of the boundary aer a single collision

•

12

� � `

Kc � 10 or K1 > 5.24M1



Newtonian Flow• Mean free path of

particles much larger than spacecra --> no appreciable interaction of air molecules

• Model vehicle/ atmosphere interactions as independent perfectly elastic collisions

α

αV

V

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Newtonian Analysis

α

AA sin(α)

ρ

Vmass flux = (density)(swept area)(velocity)

dm

dt= (!)(A sin")(V )

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Momentum Transfer

• Momentum perpendicular to wall is reversed at impact

• “Bounce” momentum is transferred to vehicle

• Momentum parallel to wall is unchanged

Vsin(α)

V

F

V

F =dm

dt!V = !V A sin"(2V sin") = 2!V 2A sin2 "

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Lift and Drag

F

VD

L

α

cD =D

1

2!V 2A

= 4 sin3 "

cL =L

1

2!V 2A

= 4 sin2 " cos "

D = F sin! = 2"V 2A sin3 !

L = F cos ! = 2"V 2A sin2 ! cos !

L

D=

cos !

sin!= cot !

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Flat Plate Newtonian Aerodynamics

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 20 40 60 80 100

Angle of Attack (deg)

Lift coeff. Drag coeff. L/D

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Example of Newtonian Flow Calculations

Consider a cylinder of length l, enteringatmosphere transverse to flow

dF

dDdLV

r!dm = !dA cos "V = !V cos "rd"d#

dF = dm!V = 2!V 2cos

2 "rd"d#

dA = rd!dl

dD = dF cos ! = 2"V 2cos

3 !rd!d#

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dL = dF sin ✓ = 2⇢V 2cos

2 ✓ sin ✓rd✓d`



Integration to Find Drag Coefficient

Integrate from

By definition, and, for a cylinder

D =

! + !

2

!

!

2

! !

0

dD = 2!V 2r

! + !

2

!

!

2

! !

0

cos3 "d"d#

D =1

2!V 2AcD A = 2r!

! = !

"

2"

"

2

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= 2⇢V 2r`

Z +⇡2

�⇡2

cos

3 ✓d✓ =

8

3

⇢V 2r`

⇢V 2r`cD =8

3⇢V 2r` =) cD =

8

3



Near Free Molecular Flow Regime• Also known as “slip region”• Gas molecule only attains state of moving boundary

aer several collisions• Molecules near the wall will have a different velocity

from the wall• Temperature will be nearly discontinuous function

of separation from wall•

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10 Kc 1

3

or 5.4M1 K1 0.175M1



Transition Region• Very difficult to treat analytically• For engineering purposes, usually treated as

interpolation between slip and viscous flow•

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0.175M1 K1 1



Viscous Merged Layer Regime• Viscous effects in forming shock and boundary layer

must be treated in a unified manner– Boundary layer on the wall alters the conditions for the

forming shock wave– Large pressure gradients across the shock wave

significantly alter the boundary layer• Neither shocks nor boundary layers can be treated as

discontinuities•

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1 K1 0.1

⇢s/⇢1



Continuous Regime• Classical fluid mechanics of high Reynolds number• Shock waves and boundary layer treated as

discontinuities• • Subdivided based on Mach number

– Incompressible (subsonic)– Transonic– Supersonic– Hypersonic

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(M ⇠ 0.8)

(⇠ 0.8 M ⇠ 1.3)

(⇠ 1.3 M ⇠ 5)

(⇠ 5 M)

K1 >0.1

⇢s/⇢1



Continuum Newtonian Flow (Hypersonics)

• Air molecules predominately interact with shock waves

• Effect of shock wave passage is to decelerate flow and turn it parallel to vehicle surface

αV

Shock wave

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Continuum Newtonian Flow (Hypersonics)

• Treat hypersonic aerodynamics in manner similar to previous Newtonian flow analysis

• All momentum perpendicular to wall is absorbed by the wall

αV

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Mass Flux (unchanged)

α

AA sin(α)

ρ

Vmass flux = (density)(swept area)(velocity)

dm

dt= (!)(A sin")(V )

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Momentum Transfer• Momentum

perpendicular to wall is absorbed at impact and transferred to vehicle

• Momentum parallel to wall is unchanged

Vsin(α)

V

F

Vcos(α)

F =dm

dt!V = !V A sin "(V sin") = !V 2A sin2 "

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Lift and Drag

F

VD

L

α

L

D=

cos !

sin!= cot !

L = F cos ! = "V 2A sin2 ! cos !

D = F sin! = "V 2A sin3 !

cL =L

1

2!V 2A

= 2 sin2 " cos "

cD =D

1

2!V 2A

= 2 sin3 "

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Modified Newtonian Flow• Coefficient of pressure in “classical” Newtonian flow

• Coefficient of pressure in modified Newtonian flow

• Cp(max) is the pressure coefficient behind a normal shock at flight conditions

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cp = cpmax

sin2 (↵)

cp = 2 sin2 (↵)

cp

max

=Pshock

� P112⇢1v21



Maximum Coefficient of Pressure

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cpmax

=2

�M21

((� + 1)2M2

14�M2

1 � 2(� � 1)

� �

��11� � + 2�M2

1� + 1

�� 1

)

cpmax

�!(� + 1)2

4�

� �

��1

4

� + 1

�

cpmax

�! 1.839 for � = 1.4

cpmax

�! 2 for � = 1

as M �! 1

Documents

Entry Aerodynamics - University Of Marylandspacecraft.ssl.umd.edu/academics/791S14/791S14L07.aerodynamics.pdfEntry Aerodynamics ENAE 791 - Launch and Entry Vehicle Design U N I V E