hypersonic 1
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HYPERSONIC FLOW THEORY
1.0 FASTER AND HIGHER1903: 35 mph @ Sea Level Wright Flyer
1969: 26,591 mph or 36,000 ft/s or M = 36 Apollo lunar capsule
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NASP and X - 30
Hypersonic Transport
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1.1 Design Considerations
Subsonic Flight M!! 1( )
Wing ! L Engines !T
Fuselage !"olume
Airplane components can be designed separately, independent of each other.
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Supersonic Flight 5 ! M!
! 1( )
Wing ! L Engines !T
Fuselage !"olume
Airplane components are slightly coupled through area ruling but still easily identifiablewhen looking at the airplane; can be designed separately.
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Hypersonic Flight M!! 5( )
! Entire undersurface of the vehicle ! L (wave rider) + contributes to T (compressesair before entering Scramjet)
! Wings: only small size is necessary! Fuel H
2!much larger volume requirement
! Components that generate L, T, and !olume are closely integrated.
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2.0 QUALITATIVE ASPECTS OF HYPERSONIC FLOW789:")(;&< =>(?@ M
!! 5
A(?:B:"C )(D: (= -A: 9A:;(D:;E DE8 F:G&; -( F:
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2.1 Thin Shock Layers
! Shock Layer = between shock wave and body
! For a given flow deflection angle !, as M! "#$% ! Consider M = 36 flow over a wedge with a half angle ! = 150" # = 180! shock
wave sits very close to the body surface ! shock layer is very thin.
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2.2 Entropy Layer
Consider a reentry vehicle with a blunt nose:
! In the nose region: shock wave is highly curved.! s increases across a shock wave.! The stronger the shock, the larger the !s .! A streamline passing through the strong, nearly normal portion of the curved shock
near the centerline of the flow will experience a larger !s than a neighboring
streamline, which passes through a weaker portion of the shock away from the
centerline ! strong s gradients exist in the nose region.! Entropy layer flows downstream, wetting the body for large distances from the nose.! BL grows inside this entropy layer and is affected by it.! Entropy layer is a region of strong vorticity.! Entropy layer and BL interaction is also called vorticity interaction.
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Viscous interaction effect: induced p on a sharp cone @
M!= 11, Re = 1.88
x10
5
per foot
! High surface p increases aerodynamic heating of the surface!
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2.4 High Temperature Effects
! High V, hypersonic flow! very large KE! Viscous dissipation: when flow is slowed by viscous effects inside the BL or behind a
strong normal shock wave! lost KE! e ! T""! vibrational E excitation,
dissociation, ionization! If an ablative heat shield is used !products of ablation are present within BL!
complex hydrocarbon chemical reactions.
! High T chemically reacting flows influence L, D, M and aero heating onhypersonic vehicles:
o Convective HTo Radiative HTo Apollo re-entry: radiative HT > 30% of total heatingo Jupiter space probe: radiative HT > 95% of total heating
! Consider reentry vehicle @ M = 36@ altitude =59km! Ts /T
" =252.9
@altitude =59km:T! =258K" Ts =65,248K !6x Tsurface of the sunMMMo NOP@ E&" &) ;( >(;G:" E ("&>8 9:"=:E8:" &; -A: ;(): ":G&(; (= -A: "::;-"8 F(Q8 &) E 9E"-&E>>8 &(;&]:Q9>E)DE :
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3.0 HYPERSONIC SHOCK WAVE RELATIONS! Oblique shock relations derived in AE164 are exact, hold for all M > 1, assuming
calorically perfect gas.
! In the limit as M! becomes very large!hypersonic shock relations.
Exact:p
2
p1
= 1+2!
! +1M
1
2sin
2 "# 1( ) as M1!" : p
2
p1
=2#
# +1M
1
2sin
2 $
Exact:!
2
!1
=
" +1( )M1
2sin
2 #
"$1( )M1
2sin
2 #+2 as M
1!" :
#2
#1
=$ +1
$ %1
T2
T1
=
p2 / p
1
!2 /!
1
as M1!" :
T2
T1
=
2# # $1( )
# +1( )2 M
1
2sin
2 %
Exact:u
2
V1
= 1!2 M
1
2sin
2 "! 1( )# +1( )M1
2 as M
1!" :
u2
V1
= 1#2sin
2 $
% +1
Exact:!
2
V1
= 1"2 M
1
2sin
2 #" 1( )cot#$ +1( )M1
2 as M
1!" :
#2
V1
=sin2$
% +1
NB: p2 / p
1, T
2 / T
1!" as M
1!"
But: !2 /!
1,u
2 / V
1,"
2 / V
1# finite values as M
1#$
The hypersonic pressure coefficient can be written as:
Cp! p
2" p
1
q1
=2
#M1
2
p2
p1
"1$
%&'
() =
4
# +1sin
2 *" 1
M1
2
$
%&'
()as M
1!" : C
p =
4
# +1sin
2 $
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4.0 A LOCAL SURFACE INCLINATION METHOD:NEWTONIAN THEORY
From linearized theory: Cp =2!
M"
2#1
The local cp depends only on the local surface inclination angle !
For hypersonic flow: local surface inclination method ! Newtonian Theory:
Cp = 2sin2!
NB: Incompressible flow:Cp,max = +1 @ a stagnation point
Hypersonic flow:
Cp,max = +2
@ a stagnation point
Modified Newtonian Theory
Cp =Cp,maxsin2!
! This eq. is more accurate for pressure calculations around blunt bodiesAnother way to arrive @ Newtonian Theory:
Exact oblique shock relation for cp: Cp =4
! +1sin2 "#
1
M$2%&' ()*
M!" ! : Cp =
4
# +1sin
2 $
!" 1: Cp " 2sin2 #
in the last eq. "is the wave angle, NOT the flow deflection angle.
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!L/ D !
monotonically as
!"
and
L / D!" as #! 0
However, this is
misleading:
! When skin friction is added:D! finite@" =0# L /D! 0@" =0
!L !max @" # 55
0
(54.7 deg to be precise, from Newtonial Theory). Thisresult is very realistic; the max lift coefficient for many hypersonic vehicles
occurs @ an aoa in this neighborhood.
! Lift curve is nonlinear @ low aoain contrast to subsonic and supersonic flows
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4K
! Newtonian Theory is more accurate for 3-D bodies.! Newtonian Theory accuracy improves as M increases.
4.1 Circular Cylinder of infinite span
cd =D
q!
S=
"D
q! 2R( )
S =2R b( )
cd =4
3from Newtonian theory
NB: this result does NOT depend on Mach; it simply assumes that M is high enough for
the flow to be hypersonic (Mach independence principle).
4.2 Sphere
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cd =D
q!
S
S = "R2
cd = 1 from Newtonian theory
NB: this result does NOT depend on Mach; it simply assumes that M is high enough for
the flow to be hypersonic (Mach independence principle).
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5.0 AERODYNAMIC HEATING! AH is the dominant design consideration for hypersonic vehicles.
o Blunt LE in hypersonic vehicles to dissipate as much Q as possible.o AD is of secondary importanceo SS Columbia accident, 01 February 2003: Thermal protection tiles damaged
during launch ! hot gases penetrated surface ! destroyed internal wing
structure!
5.1 The Connection between Hypersonic Flow and AH
CH !!qw
"eue haw #hw( )
Stan tonnumber
!qw : heat flux !heat transfer rate per unit area @ a given po inton the body W /m2( )
"e : local density@the edgeof the BL
ue : local velocity@the edgeof the BL
hw : enthalpy of thegas@thewall
haw : adiabatic wall enthalpy =enthalpyof thegas@thewall whenTw =Taw =
Tw is sohot$no moreE is conductedintothewall fromthegas
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Example:hypersonic flow over a flat plate @ aoa = 0-deg:
!e = !
"and u
e = V
" ignoring viscous interaction effects as described earlier.
For high M laminar flow over the plate: Taw
!0.88T0
i.e., the adiabatic wall T is ~ 12% less than the total T in the free stream.
Make the approximation: Taw ! T
0"h
aw ! h
0
where h0 = h
!+V!
2
2
@ hypersonic speeds: V!
is very large
@ high altitude: T!
is very cool ! h" =cpT" is relatively small
! high speeds: h0 !
V!
2
2
Ts(hot by normal standards) ! Tmelting or Tdecomposition of the surface !! T0 ! h0 ""hw !
haw !h
w " h
0 !h
w " h
0 "
V#
2
2
re-write Stanton number for a flat plate in hypersonic flow:
CH =
!qw
!"V" haw # hw( )
$
!qw
!"V" V
"
2/ 2( )
%
!qw $
1
2!"V"
3CH
PA: >E)- :`# )A(?) -AE- /7 "E-: &;(
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PA:": &) E; E;E>(G8 F:-?::; )'&; ="&:Q c: E;E>(G8# ^(" E >ED&;E"
=>(?C c: E;E>(G8 (? 9"(9:"-8X
^(" E&" H )-E;QE"Q 7Pc &;-( -A: B:A&: 7Pc E)@dQ
dt=
dQ
dV!
dV!
dt=
dQ
dV!"
1
2m#!V!
2SCD
$%&
'()
0`,E-&;G -A&) -( -A: :E">&:" :b9":))&(; =(" -A: -(-E> 7Pc@
dQ
dV!" 1
2m#!V!
2SCD$%&
'() =
1
4#!V!
3SCf*
dQ
dV!="
1
2mV!
Cf
CD*
dQ ="1
2mCf
CD
dV!2
2
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A simpler engineering formula for AH by Tauber & Meneses: !qw = !"NV"
MC
For the stagnation point: M =3, N =0.5, C = 1.83x10!8R!1/2 1!h
w
h0
"#$
%&'
Units:!qw W /cm
2
( ), V! m/ s( ), "! kg/m
3
( ), R m( )
So, for the stagnation point we have: !qw =
!"0.5V"
3
1.83x10#8R#1/2( ) 1#
hw
h0
$%&
'()
Again, we see that:
! AH ~ cube of the velocity! Stagnation point AH ~ 1 / R ! Stagnation point AH ~ !
"
The last result appears to be inconsistent with: !qw =
1
2!"V"
3
CH
which shows that Stagnation point AH ~ !"
However CH !1
Re!
1
"#
$ !qw ! "#CH ! "#