MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Compressible Flow Over Airfoils:

Linearized Subsonic Flow

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

2

WHAT ARE WE DOING NOW?• Goal: Examine and understand behavior of 2-D airfoils at Mach numbers in range

0.3 < M∞ < 1

• Think of study of Chapter 11 (compressible regime) as an extension of Chapter 4 (incompressible regime)

• Why do we care?

– Most airplanes fly in Mach 0.7 – 0.85 range

– Will continue to fly in this range for foreseeable future

– “Miscalculation of fuel future pricing of $0.01 can lead to $30M loss on bottom line revenue” – American Airlines

• Most useful answers / relations will be ‘compressibility corrections’:

2

0,

2

0,

2

0,

1

1

1

M

cc

M

cc

M

CC

mm

ll

pp

Example:1. Find incompressible cl,0 from data plot NACA 23012, = 8º, cl,0 ~ 0.82. Correct for flight Mach number

M∞ = 0.65cl = 1.05

Easy to do!

3

22

0,

1

5.0

1

MM

CC pp

For M∞ < 0.3, ~ constCp = Cp,0 = 0.5 = const

Effect of compressibility(M∞ > 0.3) is to increaseabsolute magnitude of Cp and M∞ increasesCalled: Prandtl-Glauert Rule

Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7

(Why not M∞ = 0.99?)

PREVIEW: COMPRESSIBILITY CORRECTIONEFFECT OF M∞ ON CP

SoundBarrier ?

M∞

4

OTHER IMPLICATIONSSubsonic Wing Sweep Area Rule

5

REVIEW

0

Vt

0

0

0

0

2

V

VVV

VV

t

Continuity Equation

True for all flows:Steady or Unsteady,Viscous or Inviscid,Rotational or Irrotational

2-D Incompressible Flows(Steady, Inviscid and Irrotational)

2-D Compressible Flows(Steady, Inviscid and Irrotational)

0

0

0

0

y

v

yv

x

u

xuV

VVV

VV

t

steady

irrotational

Laplace’s Equation(linear equation)

Does a similar expression exist for compressible flows?Yes, but it is non-linear

6

STEP 1: VELOCITY POTENTIAL → CONTINUITY

0

0

0

ˆˆ

2

2

2

2

2

2

2

2

yyxxyx

yyyxxx

y

v

yv

x

u

xuV

yv

xu

yxjviuV

Flow is irrotational

x-component

y-component

Continuity for 2-Dcompressible flow

Substitute velocityinto continuity equation

Grouping like termsExpressions for d?

7

STEP 2: MOMENTUM + ENERGY

2

22

2

2

2

2

2

22

2

2

22

222

2

2

22

yyyxxay

yxyxxax

yxd

ad

dadp

yxddp

vudVddp

VdVdp

Euler’s (Momentum) Equation

Substitute velocity potential

Flow is isentropic:Change in pressure, dp, is relatedto change in density, d, via a2

Substitute into momentum equation

Changes in x-direction

Changes in y-direction

8

RESULT

Velocity Potential Equation: Nonlinear EquationCompressible, Steady, Inviscid and Irrotational Flows

Note: This is one equation, with one unknown, a0 (as well as T0, P0, 0, h0) are known constants of the flow

021

11

12

22

22

22

22

2

yxyxayyaxxa

02

Velocity Potential Equation: Linear EquationIncompressible, Steady, Inviscid and Irrotational Flows

9

HOW DO WE USE THIS RESULTS?• Velocity potential equation is single PDE equation with one unknown, • Equation represents a combination of:

1. Continuity Equation

2. Momentum Equation

3. Energy Equation

• May be solved to obtain for fluid flow field around any two-dimensional shape, subject to boundary conditions at:

1. Infinity

2. Along surface of body (flow tangency)

• Solution procedure (a0, T0, P0, 0, h0 are known quantities)

1. Obtain 2. Calculate u and v

3. Calculate a

4. Calculate M

5. Calculate T, p, and from isentropic relations

yv

xu

1000

20

22

2220

2

2

11

2

1

T

T

p

p

MT

Ta

vu

a

VM

yxaa

10

WHAT DOES THIS MEAN, WHAT DO WE DO NOW?

• Linearity: PDE’s are either linear or nonlinear– Linear PDE’s: The dependent variable, , and all its derivatives appear in a

linear fashion, for example they are not multiplied together or squared

• No general analytical solution of compressible flow velocity potential is known– Resort to finite-difference numerical techniques

• Can we explore this equation for a special set of circumstances where it may simplify to a linear behavior (easy to solve)?1. Slender bodies2. Small angles of attack– Both are relevant for many airfoil applications and provide qualitative and

quantitative physical insight into subsonic, compressible flow behavior

• Next steps:– Introduce perturbation theory (finite and small)– Linearize PDE subject to (1) and (2) and solve for , u, v, etc.

11

HOW TO LINEARIZE: PERTURBATIONS

ˆ

ˆ

ˆ

xV

vv

uVu

yxyx

yy

xx

ˆ

ˆ

ˆ

22

2

2

2

2

2

2

2

2

vy

ux

ˆˆ

ˆˆ

yy

xV

x

ˆ

ˆ

12

INTRODUCE PERTURBATION VELOCITIES

2

ˆˆ

12121

0ˆ

ˆˆ2ˆ

ˆˆ

ˆ

0ˆˆˆ

2ˆˆˆˆ

2222222

2222

2

2

22

22

22

2

vuVaVaVa

y

uvuV

y

vva

x

uuVa

yxyxV

yya

xxVa

Perturbation velocity potential: same equation, still nonlinear

Re-write equation in terms of perturbation velocities:

Substitution from energy equation (see Equation 8.32, §8.4):

Combine these results…

13

RESULT

• Equation is still exact for irrotational, isentropic flow

• Perturbations may be large or small in this representation

x

v

y

u

V

u

V

vM

y

v

V

u

V

v

V

uM

x

u

V

v

V

u

V

uM

y

v

x

uM

ˆˆˆ1

ˆ

ˆˆ

2

1ˆ

2

1ˆ1

ˆˆ

2

1ˆ

2

1ˆ1

ˆˆ1

2

2

2

2

22

2

2

2

22

2

Lin

ear

Non

-Lin

ear

14

HOW TO LINEARIZE

• Limit considerations to small perturbations:

– Slender body

– Small angle of attack

x

v

y

u

V

u

V

vM

y

v

V

u

V

v

V

uM

x

u

V

v

V

u

V

uM

y

v

x

uM

ˆˆˆ1

ˆ

ˆˆ

2

1ˆ

2

1ˆ1

ˆˆ

2

1ˆ

2

1ˆ1

ˆˆ1

2

2

2

2

22

2

2

2

22

2

1ˆ

,ˆ

1ˆ

,ˆ

2

2

2

2

V

v

V

u

V

v

V

u

15

HOW TO LINEARIZE• Compare terms (coefficients of

like derivatives) across equal sign

• Compare C and A:– If 0 ≤ M∞ ≤ 0.8 or M∞ ≥ 1.2– C << A– Neglect C

• Compare D and B:– If M∞ ≤ 5– D << B– Neglect D

• Examine E– E ~ 0– Neglect E

• Note that if M∞ > 5 (or so) terms C, D and E may be large even if perturbations are small

x

v

y

u

V

u

V

vM

y

v

V

u

V

v

V

uM

x

u

V

v

V

u

V

uM

y

v

x

uM

ˆˆˆ1

ˆ

ˆˆ

2

1ˆ

2

1ˆ1

ˆˆ

2

1ˆ

2

1ˆ1

ˆˆ1

2

2

2

2

22

2

2

2

22

2

A

B

C

D

E

16

RESULT• After order of magnitude analysis, we have

following results

• May also be written in terms of perturbation velocity potential

• Equation is a linear PDE and is rather easy to solve (see slides 19-22 for technique)

• Recall:

– Equation is no longer exact

– Valid for small perturbations:

• Slender bodies

• Small angles of attack

– Subsonic and Supersonic Mach numbers

– Keeping in mind these assumptions equation is good approximation

0ˆˆ

1

0ˆˆ

1

2

2

2

22

2

yxM

y

v

x

uM

17

BOUNDARY CONDITIONS

constantˆ0ˆ

ˆ

constantˆ0ˆ

ˆ

0ˆˆ

yv

xu

vu

tanˆ

tanˆ

ˆ

ˆ

ˆtan

Vy

Vv

V

v

uV

v

1. Perturbations go to zero at infinity

2. Flow tangency

Solution must satisfy same boundary conditions as in Chapter 4

18

IMPLICATION: PRESSURE COEFFICIENT, CP

V

uC

V

vu

V

uC

V

vu

V

uM

p

p

T

T

p

p

c

VT

c

VT

p

p

MC

V

pp

q

ppC

P

P

pp

P

P

ˆ2

ˆˆˆ2

ˆˆˆ2

2

11

,22

12

2

2

22

1

2

222

122

2

2

• Definition of pressure coefficient

• CP in terms of Mach number (more useful compressible form)

• Introduce energy equation (§7.5) and isentropic relations (§7.2.5)

• Write V in terms of perturbation velocities

• Substitute into expression for p/p∞ and insert into definition of CP

• Linearize equation

Linearized form of pressure coefficient, valid for small perturbations

19

HOW DO WE SOLVE EQUATION (§11.4)• Note behavior of sign of leading term for subsonic

and supersonic flows

• Equation is almost Laplace’s equation, if we could get rid of coefficient

• Strategy

– Coordinate transformation

– Transform into new space governed by and

• In transformed space, new velocity potential may be written

yx

y

x

yx

M

yxM

,ˆ,

0ˆˆ

1

0ˆˆ

1

2

2

2

22

22

2

2

2

22

20

TRANSFORMED VARIABLES (1/2)• Definition of new variables

(determining a useful transformation is done by trail and error, experience)

• Perform chain rule to express in terms of transformed variables

ˆˆ

1ˆˆ

,0 ,0 ,1

ˆˆˆ

ˆˆˆ

y

x

yxyx

yyy

xxx

y

x

yx,ˆ,

21

TRANSFORMED VARIABLES (2/2)• Differentiate with respect to x a second time

• Differentiate with respect to y a second time

• Substitute in results and arrive at a Laplace equation for transformed variables

• Recall that Laplace’s equation governs behavior of incompressible flows

0

ˆ

1ˆ

2

2

2

2

2

2

2

2

2

2

2

2

y

x

• Shape of airfoil is same in transformed space as in physical space

• Transformation relates compressible flow over an airfoil in (x, y) space to incompressible flow in (, ) space over same airfoil

22

2

0,

2

0,

2

0,

0,

1

1

1

21

1212ˆ2ˆ2

M

cc

M

cc

M

CC

CC

V

uC

VxVxVV

uC

mm

ll

PP

PP

P

P

FINAL RESULTS

• Insert transformation results into linearized CP

• Prandtl-Glauert rule: If we know the incompressible pressure distribution over an airfoil, the compressible pressure distribution over the same airfoil may be obtained

• Lift and moment coefficients are integrals of pressure distribution (inviscid flows only)

Perturbation velocity potential for incompressible flow in transformed space

23

OBTAINING LIFT COEFFICIENT FROM CP

2

0,

0

,,

1

1

M

cc

dxCCc

c

ll

c

upperplowerpl

24

IMPROVED COMPRESSIBILITY CORRECTIONS

0,2

22

2

0,

0,

2

22

0,

2

0,

12

21

11

2111

1

P

PP

P

PP

PP

CM

MMM

CC

C

M

MM

CC

M

CC

• Prandtl-Glauret

– Shortest expression

– Tends to under-predict experimental results

• Account for some of nonlinear aspects of flow field

• Two other formulas which show excellent agreement

1. Karman-Tsien

– Most widely used

2. Laitone

– Most recent