Ch9 Linearized Flow

9.1 Introduction up to the middle 1950s, before CFD comes

A uniform flow is changed, or perturbed, only slightly

Small-perturbation theories

1. Frequently (but not always) linear theory, e.g. acoustic theory in Sec. 7.5.

2. Highlighting some important physical aspects of the flow, explicitly identifying trends and governing parameters, providing practical formulas for the rapid estimation of aerodynamic forces and pressure distributions.

-A slender body immersed in a uniform flow

9.2 Linearized Velocity Potential Equation

x y zV V i V j V k ��������������������������������������������������������

'xV V u 'yV v'zV w

where , , denotes velocity perturbations from the uniform flow

'w'v'u

( ') ' 'V V u i v j w k ��������������������������������������������������������

total velocity potential

Define a new velocity potential － perturbation velocity potential

'ux

，'v

y

， 'wz

then

( , , ) ( , , )x y z V x x y z ，where

'xV V x u V xx x

'yV vy y

'zV wz z

Also， 2

2xx xxx

2

2yy yyy

2

2zz zzz

， ，substitute into

22 2

2 2 2

2 2 2

1 1 1

2 2 2 0

yx zxx yy zz

x y y zx zxy xz yz

a a a

a a a

22 22 2 2

2 2 22 2 2

2 2 2

2 2 2 0

a V a ax x y y z z

V Vx y x y x z x z y z y z

－ perturbation-velocity potential equation

or

22 2 2 2 2' ' '' ' '

' ' '2 ' ' 2 ' ' 2 ' ' 0

u v wa V u a v a w

x y z

v u vV u v V u w v w

y z z

－ (*)

0 .h const throughout the flow

2 2 22 2

0

' ' '

2 2 2

V u v wV Vh h h h

or 2 2 22 2 2 ' ' '

1 2 1 2

V u v wa V a

2 2 2 2 212 ' ' ' '

2a a u V u v w

－ (**)

Substitute (**) into (*), and algebraically rearranging

2 ' ' '1

u v wM

x y z

2 2 22

2 2

2 2 22

2 2

2 2 22

2 2

' 1 ' 1 ' ' '1

2 2

' 1 ' 1 ' ' '1

2 2

' 1 ' 1 ' ' '1

2 2

u u v w uM

V V V x

u v w u vM

V V V x

u w u v wM

V V V x

M

2 ' ' ' ' ' ' ' ' ' ' ' ' '1 1

v u u v w u u w u w w v v

V V y x V V z x V y z x

linear

nonlinear

－ an exact equation for irrotational, isentropic flow

Now specialize to the case of small perturbation, i.e. , , , are small compared to

'w'v'uV

'u

V ，

'v

V ，

'w

V1

2'u

V

，

2'v

V

，

2'w

V

1

1. 0≦M∞ 0.8 and ≦ M∞ 1.2 ≧ （ transonic flow (0.8 M≦ ∞ 1.2) is excluded ≦ ）

the magnitude of

2. M∞ 5 (approximately) ≦ （ hypersonic flow (M∞ 5 ) is excluded ≧ ）

2 2' ' '1 1u u u

M MV x x

2 2' ' '1 1u u u

M MV x x

2 ' ' '1u v v

MV y y

2 ' ' '1u w w

MV z z

2 ' ' ' '1 1 1v u u v

MV V y x

，

( 0)

2 ' ' '1 0

u v wM

x y z

or

2 2 22

2 2 21 0M

x y z

approximate equations are valid for subsonic ＆ supersonic flow only

Note ： The real physical problems associated with

1. transonic flow ： mixed subsonic –supersonic regions with possible shocks, and extreme sensitivity to geometry changes at sonic conditions.

2. hypersonic flow ： strong shock waves closed to the geometric boundaries, i.e., thin shock layers, as well as high enthalpy, and hence high-temperature conditions in the flow.

9.3 Linearized Pressure Coefficient

Pressure coefficient ：21

2

p

p pC

V

22 2 2

2

1 1

2 2 2 2

p VV V p p M

p a

2

1

2

p

pp

pC

p M

2

21p

pC

M p

22

2 2

VVh h

22

2 2p p

VVT T

c c

2 2 2 2

2 2 / 1p

V V V VT T

c R

22 2 22 2 2 2

2 2

' ' '1 1 11

2 2 2

V V u v wV V V VT

T RT a a

2 2 22

11 2 ' ' ' '

2

Tu V u v w

T a

1

,p T

isentropicp T

1

2 2 22

11 2 ' ' ' '

2

pu V u v w

p a

2 2 2 12

2

1 2 ' ' ' '1

2

u u v wM

V V

－ exact

Consider small perturbations ： '1

u

V ，

2

2

'u

V

，

2

2

'v

V

，

2

2

'1

w

V

11 11

p

p

2 2 22

2

2 ' ' ' '1

2

p u u v wM

p V V

2 2 22

2 2

2 2 ' ' ' '1 1

2p

u u v wC M

M V V

2 2 2

2

2 ' ' ' 'u u v w

V V

2 'p

uC

V － linearized pressure coefficient

valid for small perturbations

depends only on x-component of the perturbation velocity

9.4 Linearized Subsonic Flow

-Take incompressible results (theory or experiment) and modify them to take compressibility into account.

-Applied for any 2-D shape, including

which satisfies the assumptions of small perturbations.

2-D flow over an airfoil

Flow over a bumpy or wavy body

Consider the compressible subsonic flow over a thin airfoil at small angle of attack (i.e. small perturbations)

－ inviscid flow boundary condition holds at the surface , i.e. at surface // to the surfaceV

��������������

'u V

'tan

'

df v

dx V u

if

'df v

dx V

tan

dfV

y dx

for subsonic compressible flow over a 2-D airfoil2 0xx yy ；

21 M

transformed to a familiar incompressible form by considering a transformed coordinate system

, ，

x

y

, ,x y

－ transformed perturbation velocity potential

1x

，0

y

， 0x

， y

1 1 1x x x x x

1xx

1 1y y y y y

yy

2 2 1xx yy

0 Laplace’s equation for incompressible flow

represents an incompressible flow in space, which is related to compressible flow in (x, y) space

,

The shape of airfoil is given by y=f(x) and in (x, y) and space, respectively. ,

q

1dfVdx y y

dqVd

df dq

dx d

The shape of the airfoil in (x, y) ＆ space is the same.

,

The compressible flow over an airfoil in (x, y) space transforms to the incompressible flow over the same airfoil in space. ,

2 ' 2 2 1 2 1p

uC

V V x V x V

Denoting the incompressible perturbation velocity in the direction by , where

u /u

1 2p

uC

V

0

21

pp

CC

M

0

2p

uC

V

Prandtl-Glauert rule

－ incompressible pressure coefficient in space ,

212

L

LC

V S

212

M

MC

V Sl

L ： Lift force is perpendicular to the V∞

S ： a reference area, for a wing, usually the platform area of the wing

： a reference length, for an airfoil, usually the chord length

l

0

21L

L

CC

M

0

21M

M

CC

M

，

－ Prandtl-Glauert rule

valid up to M∞≒0.7

An important effect of compressibility on subsonic flowfields

M

2

1 1'

1

u uu

x x M

'u ， － Compressibility strengthens the disturbance to the flow introduced by a solid body ！

c.f. incompressible flow

－ A perturbation of given strength reaches further away from the surface in compressible flow.

－ The spatial extent of the disturbed flow region is increased by compressibility.

－ The disturbance reaches out in all directions, both upstream and downstream.

In classical inviscid incompressible flow theory

d’Alembert’s paradox ： a 2-D closed body experiences no aerodynamic drag.

∵ No friction and its associated separated flow.

∴the pressure distributions over the forward and rearward portions of the body exactly cancel in the flow direction.

0

21

pp

CC

M

∴d’Alembert’s paradox can be generalized to include subsonic compressible flow as well as incompressible flow.

Similar results are obtained from nonlinear subsonic calculations (thick bodies at large angle of attack)

Ex 9.1

An uniform upstream M∞ flow over a wavy wall

Using the small perturbation theory, derive an equation for ＆

cos 2 /wy h x l

pC

sol ：Assume / 1h l

2 2

22 2

1 0d F d G

M G Fdx dy

2 2

2 22

1 1 10

1

d F d G

F dx G dyM

f(x) only f(y) only2

22

1 d Fk

F dx

2

222

1 1

1

d Gk

G dyM

2

2 22

1 0d G

k M Gdy

2 21 11 2( ) k M y k M yG y Ae A e

22

20

d Fk F

dx

1 2( ) sin cosF x B kx B kx

A1, A2, B1, B2 are determined by BCs.

1. y→∞ , V ( ) remains finite. →A2=0

→

2.

21

1 2 1, sin cos k M yx y B kx B kx Ae

' ' 1

'w w w

w w

dy v v

dx V u V V y

0

2 2sin

w y

xV h

y l l y

212

1 2 1sin cos 1 k M yB kx B kx A k M ey

21 1 2

0

1 sin cosy

Ak M B kx B kxy

2 2sin

xV h

l l

2

20,B k

l

2

1 1 1 1 2

21 ,

1

V hA B k M V h A B

l M

22 1 2

, exp sin1

MV h xx y y

l lM

＃

2

2

2 12 ' 4 2exp cos

1p

M yu h xC

V l l lM

2

2

2 12 2' cos exp

1

M yV h xu

x l l lM

2

2 ' 4 2cos

1pw

u h xC

V l lM

＃

－ the same cosine variation as the shape of the wall, but is 180˚ out of phase.

－ symmetrical distribution

∴ no net force in x-direction.

no drag.

2

2

2 11exp 0

1

M y

lM

0y as

2

1

1pwC

M

1

2

22

21

1

1

pw M

pw M

C M

MC

if incompressible M∞≈0

0

21

pw

pw

CC

M

9.5 Improved Compressibility Corrections

－ linearized solutions are influenced predominantly by free-stream conditions; they do not fully recognize change in local regions of the flow

nonlinear phenomena

Improved compressibility correction

－ Laitone0

21

pp

CC

M

local pressure coefficient

local Mach No., can be related to M∞

0

2 2 2 20

11 1 / 2 1

2

pp

p

CC

M M M M C

Cp0<<1 0

21

pp

CC

M

－ Karman and Tsien ： hodograph solution of the nonlinear equations of motion along with a simplified “tangent gas” equation of state.

0

02 2 21 / 12

pp

p

CC

CM M H M

－ Karman-Tsien rule

9.6 Linearized Supersonic Flow

Linearized perturbation-velocity potential equation for 2-D flow2 0xx yy 2 0xx yy

for subsonic flow

for supersonic flow

21 M

2 1M

－ elliptic P.D.E.

－ hyperbolic P.D.E.

Consider the supersonic flow over a body or surface which introduces small changes in the flowfield, e.g. flow over a thin airfoil,

over a mildly wavy wall,

over a small hump in a surface.

2 0xx yy － wave equation

1

112

Mdx

dy

yxgyxf if g=0 , yxf － lines of =const. correspond to x-λy=const. (left-running Mach lines)

1/1arctan/1arcsin 2 MM

if f=0, yxg － lines of =const. correspond to x+λy=const. (right-running Mach lines)

1

112

Mdx

dy

－ disturbances propagate along Mach lines. the flowfield upstream of a disturbance does ∴

not feel the presence of the disturbance.

c.f.

M∞<1, disturbances propagate everywhere in the flowfield.( upstream ＆ downstream )

Letting g=0 , yxf

'

'tan

uV

v

dx

dy

'' fx

u

，'' f

yv

'

'v

u

B.C. on the surface

Vv' Vu'

Vu'

2'2

V

uC p

1

22

M

C p

pC

1

12

M

C p，

(local surface inclination w.r.t. V∞)

pCM ,

)(1

22

M

C ApA

，)(

1

22

M

C BpB

consistent with

a net pressure imbalance a drag (wave drag)

yxg For the upper surface,

1

22

MC p

Note ：

Although shock waves do not appear explicitly within the framework of linearized theory, their consequence in terms of wave drag are reflected in the linearized results.

d’Alembert’s paradox does not apply to supersonic flows ！！

For M∞=2, linearized theory yields reasonably accurate results for Cp whenθ<4°

Note ： CL ＆ CD are more accurate at large angle of attack then one would initially expect.

Ex. 9.2

An uniform supersonic flow over the same wavy wall in ex. 9.1

？ Cp ？

sol ：2 2

2 2 2

10

1x M y

yMxgyMxf 11 22

Let g=0

yMxf 12 2 2' 1 1f x M y My

B.C.

l

x

lhV

dx

dyV

yw 2

sin2

l

xhyw

2cos

w

ww

yVV

v

dx

dy

1'

02

sin2

'12

y

l

x

lhVxfM

l

x

lhV

Mxf

2sin

2

1

1'

2

.2

cos12

constl

x

M

hVxf

.12

cos1

1, 2

2

2 constyMxlM

hVyMxfyx

＃

yMxll

h

MxVV

uC p 1

2sin

1

42'2 2

2

l

x

l

h

MC p

2sin

1

42

＃

Note ：

1. The perturbations do not disappear

at y→∞ ↔ subsonic

2. ,

← characteristic lines

3. unsymmetrical streamlines

4. Cpw is a sine variation, whereas the wall is a cosine shape →a net force in the x-direction.(wave drag)

c.f.

..12 constCconstyMx p

.const

MMdx

dy 1sintan

1

1 1

2

9.7 Critical Mach NumberMc ≡M∞ at which sonic flow is first encountered in the airfoil

Assuming isentropic flow,

12

2

11

21

12

A

A

MP

P M

2

21p

PC

M P

12

22

112 2 1

11

2

Ap

A

MC

M M

MA=1M∞=Mcr

min

12

2

112 2 1

11

2

A A

cr

p p

cr

pcr

C C

MC

M

－ derived from fundamental gas dynamics, independent of the size or shape of the airfoil.

+ Prandtl-Glauert rule (or Laitone rule or Karman-Tsien rule)

Mcr for a given airfoil

Cp0

(measured or calculated) different for different airfoil

Thin airfoil

mild expansion over the top surface

Cp0 small ( curve B

is low )

Mcr large

c.f. Thick airfoil

stronger expansion

Cp0 larger ( curve B

is higher )

Mcr lower

∴ An airfoil designed for a high Mcr must have a thin airfoil.

1 > Drag-divergence Mach Number, MDD > Mcr

At MDD, the drag (CD) is massively increased.

“sonic barrier” before 1947

when M∞ > Mcr

The total pressure loss associated with the weak λ-shock will be small, however, the adverse pressure gradient induced by the shock tends to separate the boundary layer on the top surface, causing a large pressure drag. CD ↑ dramatically

Mcr ↑ MDD ↑ ( MDD ↑ more by “supercritical” airfoil)

ch 14

Two ways to increase Mcr ： (1) ↓ ( thinner airfoil )

t

c

0.09, 0.88DD

tM

c

0.04 0.06, 1DD

tM

c ﹟

(2) sweep the wing

The flow behaves as if the airfoil section is thinner. ,cr DDM M