Power Balance in Aerodynamic Flows - MIT -...
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Power Balance in Aerodynamic Flows
Mark DrelaMIT Aeronautics & Astronautics
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Objectives
• Quantify Mechanical Energy sources and sinks in flowfield
• Use for aerodynamic analyses, in lieu of Thrust and Drag
• Framework for configuration comparisons and evaluations
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Motivation — I
• Thrust and Drag accounting is just a means to an end:Range, or Flight Power, or Fuel Consumption
• Thrust and Drag are ambiguous for integrated propulsion
• Standard performance relations become inapplicable, e.g.
PT
D
VTSFC
R = ln 0
e
C
CL
D
?
T = ?
D = ?
W
W
⇒ Will sidestep ambiguity by relating Power to flowfield directly
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Motivation — II
Want to identify . . .
• flowfield power sinks (what’s causing the loss/fuel burn?)
• power sink locations (where’s the loss?)
• opportunities of power savings (can I reduce the loss?)
and quantify . . .
• aero performance of disparate aircraft configurations and concepts
• effectiveness of BL ingestion and flow control systems
• parasite and interference drags in conventional force analysis
• etc.
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Assumed Governing Equations
Mass:∇ ·
(
ρ~V)
= 0
Momentum:ρ~V · ∇~V = −∇p + ∇ · ¯τ
Forming{
~momentum}
· ~V gives Kinetic Energy equation:
ρ~V · ∇(
12V 2
)
= −∇p · ~V + (∇ · ¯τ ) · ~V
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Control Volume
V
n
n
d
SB
dSB
SO
V
ndSO
x
z
y
dSOTP
Vu
SB
Trefftz Plane
Vn
dSO
V VnSide Cylinder
SC
v, w
• Outer boundary SO has . . .
– Trefftz Plane ⊥ to freestream
– Side Cylinder ‖ to freestream
• Inner body boundary SB can . . .
– cover propulsor blading to include shaft power
– cover internal ducting to include flow losses, pump power
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Integral Kinetic Energy Equation
∫∫∫
ρ~V · ∇(
12V 2
)
= −∇p · ~V + (∇ · ¯τ ) · ~V
dV...
PS + PK + PV = Wh + Ea + Ev + Ep + Ew + Φ
• Exact decomposition into physically-clear components
• Primary use:
⇒ Obtain total power (LHS) by evaluating all RHS terms
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Integral KE Equation — Energy inflow or production
PS + PK + PV = Wh + Ea + Ev + Ep + Ew + Φ
Shaft power: PS = ©∫∫
(−pn + ~τ ) · ~V dSB
K.E. inflow: PK = ©∫∫
−[
p − p∞ + 12ρ(
V 2−V 2∞)]
~V · n dSB
P dV power: PV =∫∫∫
(p − p∞)∇ · ~V dV
Potential energy rate: Wh.
hW
P
PS
K
PV
PV
PS
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Integral KE Equation — Energy flow out of CV
PS + PK + PV = Wh + Ea + Ev + Ep + Ew + Φ
Axial K.E. outflow: Ea =∫∫
12ρ u2 (V∞+u) dSTP
O
Vortex K.E. outflow: Ev =∫∫
12ρ(
v2+w2)
(V∞+u) dSTPO
Pressure work: Ep =∫∫
( p − p∞) u dSTPO
Wave outflow: Ew =∫∫ [
p − p∞ + 12ρ(
u2+v2+w2)]
Vn dSSCO.
E
.E
.E
a
v
w
u
v, w
u, v, w
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Integral KE Equation — Energy lost inside CV
PS + PK + PV = Wh + Ea + Ev + Ep + Ew + Φ
Viscous dissipation: Φ =∫∫∫
(¯τ ·∇) ·~V dV
Φshock
Φjet
Φwake
Φjet
Φsurface
Φprop
Φduct
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Outflow/Dissipation term balance
VDi
ΦΦ
Φ
x
u −u ∼ 0−u ∼ 0
Φsurface
Φvortex
Φwake + Φjet
.~ u2E
E.
~ 2+v2 wE.
−P.
Wh
−∼ 0v, w
a
v v
v, wv, w
xTP xTP xTP
LHS RHS
• Same lost power (LHS) is obtained for any chosen RHSTrefftz Plane location
• E outflow terms account for dissipation outside of CV
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Loss Calculation and Estimation
E, Φ components often available from other theories/methods
Ev = Φvortex = Di V∞ =L2
12ρV 2
∞ π b2 eV∞
Ew = Dw V∞ ≥ sin Λ√
1 − M 2⊥
L2
12ρV 2
∞ π b2V∞
Φjet = PS (1−ηfroude) = PS
√Tc + 1 − 1√Tc + 1 + 1
Φprop = PS (1−ηprofile) = PS
1 − 1 − (cd/cℓ) tan φ
1 + (cd/cℓ)/ tan φ
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Dissipation in Boundary Layers and Wakes
x, uz, w
yue
w
u
Φsurface,wake =∫∫∫
(¯τ · ∇) · ~V dV
≃∫∫∫
τxy
∂u
∂y+ τzy
∂w
∂y
dx dy dz (for 3D shear layer)
≃∫∫∫
τxy∂u
∂ydx dy dz (for 2D shear layer)
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Dissipation Coefficient
It’s convenient to work with a dissipation coefficient CD:
Φsurface =∫∫
[
∫ δ0τxy
∂u∂y dy
]
dx dz
=∫∫
ρeu3e CD dx dz
Analogous to skin friction coefficient Cf :
Df =∫∫
[ τxy ]y=0dx dz
=∫∫
12ρeu
2e Cf dx dz
• CD and Φ capture all drag-producing loss mechanismsCf and Df still leave out the pressure-drag contribution
• CD and Φ are scalars – no drag-force direction issues
• CD is strictly positive – no force-cancellation problems
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CD and Cf in Shear Layers – Laminar
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
2 2.5 3 3.5 4 4.5 5
Re
C
/2
,
R
e
C
fD
θθ
H
Re C
Re C /2f
D
θ
θ
CDCf
CD
Cf
Laminar CD is very insensitive to pressure gradient —
dissipation region just moves on/off the surface
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CD and Cf in Shear Layers – Turbulent
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
1 1.5 2 2.5 3 3.5 4
C
/2
,
C
f
D
H
Re = C /2
C
f
D
θ 50001000
200
CDCf
CD
Cf
Increased mixing increases CD in adverse pressure gradients
Note: A turbulent free shear layer has CD ≃ 0.02 (!)
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Section Loss Quantification via BL and Wake Thicknesses
Dissipation via dissipation coefficient:∫
dΦ =∫ x0
ρeu3e CD dx
Dissipation via K.E. thickness:∫
dΦ = 12ρeu
3e θ∗ =
∫ ye
012
(
u2e−u2
)
ρu dy
K.E. outflow via K.E. and momentum thickness:
Ea = 12ρeu
3e δK =
∫ ye
012(ue−u)2 ρu dy = 1
2ρeu
3eθ
∗
2
H∗ − 1
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Power Balance in Simple Cases – Propelled body
Φsurface
−u ∼ 0
ΦsurfaceΦtotalΦsurface
T
D
u 0<
ΦwakeE.
TEa
Φtotal
Φwake
= P = P
~ u2
u 0> jet
.a ~ u2E
noz
Φ
Φjet
P
P
Φsurface
Φwake
(no wake, no jet)
Isolated propulsor: P = Φtotal = Φsurface + EaTE+ Eanoz
Ingesting propulsor: P = Φtotal = Φsurface
Wake ingestion benefit is via elimination of wake and jetdissipation (or equivalently, elimination of K.E. outflow)
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Power Balance in Simple Cases – Laminar flat plate
K.E. defect 12ρeu
3eθ
∗(x) =∫ x0
dΦ along surface and wake
Potential wake-ingestion power savings is 21%(possibly more from reduced Φjet)
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Power Balance in Simple Cases – Transonic Airfoil
K.E. defect 12ρeu
3eθ
∗(x) =∫ x0
dΦ along surface and wake
Potential wake-ingestion power savings is 13%(possibly more from reduced Φjet)
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Dependence on Flow Velocity
Surface BL: Φ ∼∫
ρeu3e dx
Free shear layer: Φ ∼∫
ρ(∆u)3 dx
• High-speed regions are very costly (Cp spikes, etc)
• Low-speed regions are innocuous (cove separation, etc)
• Local excrescense drag should be scaled with u3e, not u2
e
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Drag Build-Up via Force Summation — Isolated Bodies
D
V
V = 1
= 1
V = 1 D
D
2
1
Dk
=
=
= =
1
100
101
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Drag Build-Up via Force Summation — Nearby Bodies
Assuming Dk ∼ V 2 . . .
V
V = 1
= 1
V = 2 DD
D
2
1
Dk=
=
4
100
= =104
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Drag Build-Up via Force Summation — Nearby Bodies
Assuming Dk ∼ V 2 . . .
V = 1DD
D
2
1
Dk= =
= 100D1 = 4∆
p+∆p −∆104 108
wrong !
= 4
Also present is buoyancy drag ∆D1 on large body,due to small body’s source-disturbance pressure field ∆p.
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Drag Build-Up via Dissipation Summation — Isolated Bodies
D
V
ΦV
V = 1
= 1
V = 1 2
1
k
Φ =
Φ =
1
100
= =101
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Drag Build-Up via Dissipation Summation — Nearby Bodies
Assuming Φk ∼ V 3 . . .
1
2 D
V
ΦV
V = 1
= 1
V = 2 k
correct
Φ =
Φ = 100
8 = =108
Dissipation Build-Up captures “mystery interference drag”,which cannot be estimated via simple Drag Build-Up.
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Evaluation of Integrated-Propulsion Benefits
∆ < 0E.
∆Φ < 0
∆Φ > 0
(Isolated propusion)Before
After(Integrated propusion)
P P+∆
P
aChanges
Con
Pro
Net propulsive power change is sum of all Pro,Con changes:
∆P =∑
∆E +∑
∆Φ
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Motivation — I (again)
Usual Range Equation is ambiguous for integrated propulsion:
PT
D
VTSFC
R = ln 0
e
C
CL
D
?
T = ?
D = ?
W
W
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Motivation — I (resolved)
Power-balance form of Range Equation is unambiguous,for any configuration:
P
R = ln 0
e
CL
PSFC EC +CΦ
1
P
ΦΦEa.
Ea.
Ea.Φ
W
W
PSFC ≡ Wfuel∑
P, CE ≡
∑
E12ρ∞V 3
∞ S, CΦ ≡
∑
Φ12ρ∞V 3
∞ S
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Conclusions
• Flight power calculation without drag,thrust definition
• Quantities which determine flight power clearly identified
• Compatible with traditional force-based analyses
– can frequently use established drag estimation methods
– can use existing propulsive efficiency estimates
• Formulation is exact
– does not require isolation of wakes or plumes
– no small-disturbance or low-speed approximations are used
• Is more reliable for “interference drag” situations