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Linearized Compressible Potential Flow Governing Equation

Recall the 2-D full potential eqn is:

02

)(1

1)(1

12

2

2

2

2

=

+

xy y x yy y xx xaaa

Where [ ]2222 )()(1 y xoaa +=

As you saw, for small perturbations to a uniform flow, the linearized form of theequation was:

Where

y

x J y j x

i

v

uuuuu

=

=+=+=

vv

v

v

vv

This equation is valid for both 1 M . Note, though, it is not correctfor 1 M or for M large, say greater than about 2 or so.

So what happens to the linearized potential equation for :1> M

It also turns out that 0)1( 2 =+ yy xx M is much easier to solve when 1> M .

Define0

12

2

=

=

yy xx

M

Then,)(),( = y x where y x =

is a solution to the linearized potential. To see this:

0)1( 2 =+ yy xx M = perturbation

potential

perturbationvelocity

Subsonic Flow

01

12 >

M

V 1

.

2 =

===

M

const y x

const

1

12

= M dx

dy

nv

dy

dx

= V v

= V u

)

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Linearized Compressible Potential Flow Governing Equation

16.100 2002 3

This is very useful because the linearized pressure coefficient is:

==V

u

V

p pC p

2

21 2

on boundary!

0> pC for 0> (i.e. a compression)0< pC for 0

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Linearized Compressible Potential Flow Governing Equation

16.100 2002 4

M 0

d C (linearpotentialtheory)

0