Introduction to Compressible Fluid Flow, Second Edition, 2nd Edition

461An Introduction to Two-Dimensional Compressible Flow

Now consider the boundary conditions at the surface as discussed in the derivation of

Equation 14.47. Let

y = f(x)

describe the shape of the actual body and let

η = F(ζ)

describe the shape of the body in the transformed plane.

The boundary condition in the actual ow is

∂

∂

=

∞

Φ

p

s

y

u

df

dx

(14.60)

whereas in the transformed plane the boundary condition is

∂

∂

=

∞

Φ

p

s

u

dF

d

ηζ

(14.61)

However,

∂

∂

=

∂

∂

∂

∂













=

∂

∂













ΦΦ Φ

p

s

p

s

p

s

y

y

yη

β

η

β

β

() ()

1

i.e.,

∂

∂

=

∂

∂

ΦΦ

p

s

p

s

yη

(14.62)

Equation 14.62 therefore shows that the lef-hand side of Equations 14.60 and 14.61 are

equal, so

df

dx

dF

d

=

ζ

(14.63)

This means that the shape of the body is the same in real and the transformed planes,

i.e., the function that relates y to x on the surface of the body in the real plane is identi-

cal to the function that relates η to ζ on the surface of the body in the transformed plane.

Since Φ

p

is determined by Equation 14.59, which is identical to the equation that applies

in incompressible ow, and since the shape of the body is the same in the real and trans-

formed planes, it follows that

Φ

p

is the same as the linearized velocity potential function

that would exist in incompressible ow over the body being considered. Hence, if

Φ

p

is

462 Introduction to Compressible Fluid Flow

determined from the solution for incompressible ow over the body, the actual linearized

velocity potential and x and y being given by

ΦΦ Φ

pp p

Mx

yM

== −=== −// //βζηβ η

11

22

,,

Now, Equation 14.54 gives

C

udxud

p

pp

=−

∂

=−

∂

∞∞

22

ΦΦ()

/β

ζ

i.e.,

C

ud

p

p

=−

∂

















∞

12

βζ

Φ

(14.64)

However, the variation of

Φ

p

with ζ is the same as in incompressible ow over the body

shape being considered, i.e., by virtue of Equation 14.54, Equation 14.64 gives

CC

C

M

pp

p

==

−

∞

1

1

0

0

2

β

(14.65)

where C

p0

is the value of the pressure coefcient that would exist in incompressible ow

over the body being considered. This means that if the pressure coefcient distribution is

determined for incompressible irrotational ow, the pressure coefcient distribution is com-

pressible ow at Mach number M

∞

can be found by applying a “compressibility correction

factor” equal to

11

2

/ −

∞

M

. This only applies in subsonic ows, i.e., to ows in which M

∞

< 1.

Consider the lift force on a body, i.e., the force normal to the upstream ow, resulting

from the variation in the pressure over the surface of the body. If L is the lift per unit span,

it will be seen from Figure 14.11 that

Lp

pd

s=−

∞

∫

()cosθ

the integral being carried out over the surface of the body.

Lift

p

θ

θ

u

∞

p

∞

ds

FIGURE 14.11

Calculation of lift from pressure distribution.

463An Introduction to Two-Dimensional Compressible Flow

This equation can be written as

C

L

uc

Cd

s

c

Lp

=





















=













∞∞

∫

1

2

2

ρ

θcos (14.66)

where C

L

is the lift coefcient and c is the wing chord. From Equation 14.65, it follows that

C

C

M

L

L

=

−

∞

0

2

1

(14.67)

C

L0

being the lift coefcient that would exist in incompressible ow. Since

11

2

−<

∞

M

, the

above equations indicate that compressibility increases the coefcient of lift.

Now for smaller angles of attack, α, this being dened in Figure 14.12, C

L

= aα in incom-

pressible ow. In compressible ow, then

Ca

M

L

=−

∞

α/1

2

. This is shown in Figure 14.13.

Lift

Drag

Angle of

attack

u

∞

α

FIGURE 14.12

Denition of airfoil angle of attack.

Compressible

flow

Angle of attack, α

i.e., incompressible

flow

a (= slope of line)

M

∞

<< 1

Coefficient of lift, C

L

a/√M

2

–1

∞

FIGURE 14.13

Effect of compressibility on lift coefcient variation.

464 Introduction to Compressible Fluid Flow

Linearized Supersonic Flow

In supersonic ow disturbances are, as discussed in Chapter 2, propagated along Mach

lines as indicated in Figure 14.14.

The ow upstream of the Mach line is undisturbed by the presence of the wave. It is to

be expected therefore that the perturbation velocity potential, Φ

p

, will, in supersonic ow,

be constant along a Mach line. Now, along a Mach line

y

x

M

=

−

∞

1

1

2

i.e.,

xM y

=−

∞

2

1

where it has again been noted that because sin α = 1/M

∞

,

tan/α= −

∞

11

2

M

.

Hence, it is to be expected that

ϕ

p

= f(x − λy) = f(η) (14.68)

where

λ= −

∞

M

2

1 (14.69)

and

η = x − λy (14.70)

Small disturbance

Mach wave

y

x

M

∞

α = sin

–1

(1/M

∞

)

FIGURE 14.14

Mach wave.