Power Balance in Aerodynamic Flows
Mark Drela
MIT Aeronautics & Astronautics
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
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.
R =
CL
W
ln 0
TSFC CD
We
V
?
T
T =?
D
P
D =?
⇒ Will sidestep ambiguity by relating Power to flowfield directly
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.
Assumed Governing Equations
Mass:
~
∇ · ρV
Momentum:
Forming
(
!
= 0
~ · ∇V
~ = −∇p + ∇ · τ̄¯
ρV
~
momentum
· V~ gives Kinetic Energy equation:
)
~ · ∇ 1 V 2 = −∇p · V
~ + (∇ · τ̄¯ ) · V~
ρV
2
Control Volume
n
V
V
Vn
Side Cylinder
SC
n
dSO
n
TP
dSO
SB
dSO
d
dSB
V
V
u
v, w
n
V
z SB
x
y
SO
Trefftz Plane
• Outer boundary SO has . . .
– Trefftz Plane ⊥ to freestream
– Side Cylinder k to freestream
• Inner body boundary SB can . . .
– cover propulsor blading to include shaft power
– cover internal ducting to include flow losses, pump power
Integral Kinetic Energy Equation
ZZZ



~ + (∇ · τ̄¯ ) · V
~
~ · ∇ 1 V 2 = − ∇p · V
ρV
2
..



dV
PS + PK + PV = W ḣ + Ėa + Ėv + Ėp + Ėw + Φ
• Exact decomposition into physically-clear components
• Primary use:
⇒ Obtain total power (LHS) by evaluating all RHS terms
Integral KE Equation — Energy inflow or production
PS + PK + PV = W ḣ + Ėa + Ėv + Ėp + Ėw + Φ
Shaft power: PS =
ZZ
(−pn̂ + ~τ ) · V~ dSB
K.E. inflow: PK =
ZZ
P dV power: PV =
ZZZ
− p − p∞ +
Potential energy rate:
~ · n̂ dSB
V −V∞ V
2
~ dV
(p − p∞) ∇ · V
W ḣ
.
Wh
PV
1
2ρ
PS
PS
PK
PV
2
Integral KE Equation — Energy flow out of CV
PS + PK + PV = W ḣ + Ėa + Ėv + Ėp + Ėw + Φ
Axial K.E. outflow:
Ėa =
1
2
ZZ
ρ u2 (V∞ +u) dSOTP
2
2
Vortex K.E. outflow: Ėv =
ZZ
1
2ρ
Ėp =
ZZ
( p − p∞) u dSOTP
Ėw =
ZZ Pressure work:
Wave outflow:
u, v, w
v +w (V∞ +u) dSOTP
p − p∞ +
.
Ew
1
2ρ
2
2
u +v +w
u
2
Vn dSOSC
.
Ea
.
Ev
v, w
Integral KE Equation — Energy lost inside CV
PS + PK + PV = W ḣ + Ėa + Ėv + Ėp + Ėw + Φ
Viscous dissipation:
Φ =
ZZZ
Φprop
Φshock
Φduct
~ dV
(τ̄¯ ·∇) · V
Φjet
Φsurface
Φjet
Φwake
Outflow/Dissipation term balance
DiV
.
P−Wh
v, w
.
Ev ~ v 2 + w 2
.
E a ~ u2
u ∼− 0
u ∼− 0
u
v, w
.
Ev
v, w ∼− 0
Φvortex
Φ
Φ
Φsurface
Φ
x TP
LHS
Φwake + Φ jet
x TP
x TP
x
RHS
• Same lost power (LHS) is obtained for any chosen RHS
Trefftz Plane location
• Ė outflow terms account for dissipation outside of CV
Loss Calculation and Estimation
Ė, Φ components often available from other theories/methods
Ėv = Φvortex = Di V∞
Φprop
L2
= 1 2 2 V∞
2 ρV∞ π b e
L2
sin Λ
V∞
Ėw = Dw V∞ ≥ r
1 − M⊥2 12 ρV∞2 π b2
√
Tc + 1 − 1
Φjet = PS (1−ηfroude) = PS √
Tc + 1 + 1


1
−
(c
/c
)
tan
φ


d ℓ

= PS (1−ηprofile) = PS 1 −

1 + (cd/cℓ)/ tan φ
Dissipation in Boundary Layers and Wakes
y
ue
w
u
x, u
z, w
Φsurface,wake =
ZZZ
≃
ZZZ
≃
ZZZ
(τ̄¯ · ∇) · V~ dV
∂w 
∂u
+ τzy
(for 3D shear layer)
 dx dy dz
∂y
∂y
∂u
τxy
dx dy dz
(for 2D shear layer)
∂y




τxy

Dissipation Coefficient
It’s convenient to work with a dissipation coefficient CD :
Φsurface =
=
δ
∂u
τ
xy
0
∂y dy dx dz
ρeu3e CD dx dz
ZZ "Z
ZZ
#
Analogous to skin friction coefficient Cf :
Df =
ZZ
=
ZZ
[ τxy ]y=0 dx dz
2
1
ρ
u
e
e Cf
2
dx dz
• CD and Φ capture all drag-producing loss mechanisms
Cf 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
CD and Cf in Shear Layers – Laminar
0.45
0.4
Reθ Cf /2 ,
Reθ CD
0.35
0.3
CD
CD
Cf
0.25
Cf
0.2
Reθ CD
0.15
0.1
Reθ Cf /2
0.05
0
2
2.5
3
3.5
H
4
4.5
5
Laminar CD is very insensitive to pressure gradient —
dissipation region just moves on/off the surface
CD and Cf in Shear Layers – Turbulent
0.008
0.007
CD
C f /2 ,
CD
0.006
0.005
CD
CD
Cf
0.004
Cf
0.003
0.002
200
1000
Re θ = 5000
0.001
0
1
1.5
2
Cf /2
2.5
H
3
3.5
4
Increased mixing increases CD in adverse pressure gradients
Note: A turbulent free shear layer has CD ≃ 0.02 (!)
Section Loss Quantification via BL and Wake Thicknesses
Dissipation via dissipation coefficient:
Z
dΦ =
x
3
0 ρe u e C D
Z
dx
Dissipation via K.E. thickness:
Z
dΦ =
3 ∗
1
ρ
u
e
eθ
2
=
ye 1
0 2
Z
u2e −u2
ρu dy
K.E. outflow via K.E. and momentum thickness:
Ėa =
3
1
ρ
u
e
e δK
2
2
y
= 0 e 12 (ue −u)2 ρu dy = 12 ρeu3e θ ∗ ∗ − 1
H
Z


Power Balance in Simple Cases – Propelled body
P
T
D
Φ surface
u >0
Φ jet
u <0
Φ wake
.
Ea noz~ u2
.
Ea TE ~ u2
Φ jet
Φ wake
Φ surface
Φ total = P
Isolated propulsor:
P
u ∼− 0
(no wake, no jet)
Φ surface
Φ surface
Φ total = P
P = Φtotal = Φsurface + ĖaTE + Ėanoz
Ingesting propulsor: P = Φtotal = Φsurface
Wake ingestion benefit is via elimination of wake and jet
dissipation (or equivalently, elimination of K.E. outflow)
Power Balance in Simple Cases – Laminar flat plate
K.E. defect
3 ∗
1
ρ
u
e
e θ (x)
2
=
x
0 dΦ
Z
along surface and wake
Potential wake-ingestion power savings is 21%
(possibly more from reduced Φjet)
Power Balance in Simple Cases – Transonic Airfoil
K.E. defect
3 ∗
1
ρ
u
e
e θ (x)
2
=
x
0 dΦ
Z
along surface and wake
Potential wake-ingestion power savings is 13%
(possibly more from reduced Φjet)
Dependence on Flow Velocity
Surface BL:
Φ ∼
Free shear layer: Φ ∼
Z
ρeu3e dx
Z
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 u2e
Drag Build-Up via Force Summation — Isolated Bodies
V= 1
D2 = 1
D=
V =1
V= 1
D1 = 100
Dk = 101
Drag Build-Up via Force Summation — Nearby Bodies
Assuming Dk ∼ V 2 . . .
V= 2
V =1
V= 1
D=
D2 = 4
D1 = 100
Dk = 104
Drag Build-Up via Force Summation — Nearby Bodies
Assuming Dk ∼ V 2 . . .
V =1
+∆p
D2 = 4
∆D 1 = 4
−∆p
D=
Dk = 104 108
D1 = 100
wrong !
Also present is buoyancy drag ∆D1 on large body,
due to small body’s source-disturbance pressure field ∆p.
Drag Build-Up via Dissipation Summation — Isolated Bodies
V= 1
Φ2 = 1
D=
V =1
V= 1
Φ1 = 100
Φk
V
= 101
Drag Build-Up via Dissipation Summation — Nearby Bodies
Assuming Φk ∼ V 3 . . .
V= 2
V =1
V= 1
Φ2 = 8
D=
Φk
V
= 108
Φ1 = 100
correct
Dissipation Build-Up captures “mystery interference drag”,
which cannot be estimated via simple Drag Build-Up.
Evaluation of Integrated-Propulsion Benefits
P
Pro
Before
∆Φ < 0
(Isolated propusion)
.
∆ Ea < 0
Changes
After
(Integrated propusion)
P +∆ P
Con
∆Φ > 0
Net propulsive power change is sum of all Pro,Con changes:
∆P =
X
∆Ė +
X
∆Φ
Motivation — I (again)
Usual Range Equation is ambiguous for integrated propulsion:
R =
CL
W
ln 0
TSFC CD
We
V
?
T
T =?
D
D =?
P
Motivation — I (resolved)
Power-balance form of Range Equation is unambiguous,
for any configuration:
R =
1
CL
W
ln 0
We
PSFC CE +CΦ
P
Φ
Ẇfuel
PSFC ≡ X
,
P
Φ
CE
.
Ea
P
Φ
.
Ea
Ė
≡ 1
,
3
2 ρ∞ V∞ S
X
CΦ ≡
.
Ea
Φ
1
3
2 ρ∞ V∞ S
X
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