MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Compressible Flow Over Airfoils:Linearized Subsonic Flow

Mechanical and Aerospace Engineering DepartmentFlorida 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

VVt

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

00

0

yv

yv

xu

xuV

VVV

VVt

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

yv

yv

xu

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

0211112

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 Equation2. Momentum Equation3. Energy Equation

• May be solved to obtain for fluid flow field around any two-dimensional shape, subject to boundary conditions at:1. Infinity2. Along surface of body (flow tangency)

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

1. Obtain 2. Calculate u and v3. Calculate a4. Calculate M5. Calculate T, p, and from isentropic relations

yv

xu

1000

20

22

2220

2

211

21

TT

pp

MTT

avu

aVM

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

ˆˆ

xV

vvuVu

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

ˆˆ12121

0ˆˆˆ2

ˆˆˆˆ

0ˆˆˆ

2ˆˆˆˆ

2222222

2222

2

2

22

22

22

2

vuVaVaVa

yuvuV

yvva

xuuVa

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

xv

yu

Vu

VvM

yv

Vu

Vv

VuM

xu

Vv

Vu

VuM

yv

xuM

ˆˆˆ1

ˆ

ˆˆ2

1ˆ2

1ˆ1

ˆˆ2

1ˆ2

1ˆ1

ˆˆ1

2

2

2

2

22

2

2

2

22

2

Line

arN

on-L

inea

r

14

HOW TO LINEARIZE

• Limit considerations to small perturbations:– Slender body– Small angle of attack

xv

yu

Vu

VvM

yv

Vu

Vv

VuM

xu

Vv

Vu

VuM

yv

xuM

ˆˆˆ1

ˆ

ˆˆ2

1ˆ2

1ˆ1

ˆˆ2

1ˆ2

1ˆ1

ˆˆ1

2

2

2

2

22

2

2

2

22

2

1ˆ

,ˆ

1ˆ

,ˆ

2

2

2

2

Vv

Vu

Vv

Vu

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

xv

yu

Vu

VvM

yv

Vu

Vv

VuM

xu

Vv

Vu

VuM

yv

xuM

ˆˆˆ1

ˆ

ˆˆ2

1ˆ2

1ˆ1

ˆˆ2

1ˆ2

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

yv

xuM

17

BOUNDARY CONDITIONS

constantˆ0ˆ

ˆ

constantˆ0ˆ

ˆ

0ˆˆ

yv

xu

vu

tanˆ

tanˆ

ˆˆ

ˆtan

Vy

VvVv

uVv

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

VuC

Vvu

VuC

Vvu

VuM

pp

TT

pp

cVT

cVT

pp

MC

Vpp

qppC

P

P

pp

P

P

ˆ2

ˆˆˆ2

ˆˆˆ22

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 equationLinearized 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

yx

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

yx

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

VuC

VxVxVVuC

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,

122

111

2111

1

P

PP

P

PP

PP

CM

MMM

CC

C

MMM

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