power balance in aerodynamic flows - mit -...

Power Balance in Aerodynamic Flows

Mark DrelaMIT 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.

PT

D

VTSFC

R = ln 0

e

C

CL

D

?

T = ?

D = ?

W

W

⇒ 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:∇ ·

(

ρ~V)

= 0

Momentum:ρ~V · ∇~V = −∇p + ∇ · ¯τ

Forming{

~momentum}

· ~V gives Kinetic Energy equation:

ρ~V · ∇(

12V 2

)

= −∇p · ~V + (∇ · ¯τ ) · ~V

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

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

Integral KE Equation — Energy inflow or production


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

Integral KE Equation — Energy flow out of CV


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

Integral KE Equation — Energy lost inside CV


Viscous dissipation: Φ =∫∫∫

(¯τ ·∇) ·~V dV

Φshock

Φjet

Φwake

Φjet

Φsurface

Φprop

Φduct

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

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

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)

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

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

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 (!)

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

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)

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)

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)

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

Drag Build-Up via Force Summation — Isolated Bodies

D

V

V = 1

= 1

V = 1 D

D

2

1

Dk

=

=

= =

1

100

101

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

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.

Drag Build-Up via Dissipation Summation — Isolated Bodies

D

V

ΦV

V = 1

= 1

V = 1 2

1

k

Φ =

Φ =

1

100

= =101

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.

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

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

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

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

power balance in aerodynamic flows - mit -...

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