Introduction to 3D Ideal Flow

2

3D Steady Ideal Flow

x, i

y, j

z, k

3

Uniform Flow

x, i

y, j

z, k

V

4

Point Source/Sink

x, i

y, j

z, k

+

r

x, i

y, j

z, k

+r1

r-r1

5

Point Doublet

r, er

x, i

y, j

z, k

r

x, i

y, j

z, k

r1

r-r1

6

Flow Past a Sphere r, er

x, i

23 4.

2.

rrr

rr

eμeeμVVV

=-iUniform flow + opposing doublet

V=Vi

7

Flow Past a SphereUniform flow + opposing doublet

0 90 180 270 360

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Cp

SphereCylinder

8

Flow past sphere Re=300

Re=15000

Re=30000

ON

ER

A ph

otog

raph

s, W

erle

198

0

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Ideal Flow, AOE 5104

9

Lines/FilamentsSource, doublet or vortex distributed along a curved or straight line – source and doublet analysis analogous to panel

ds

r1

r

V

O

Filament path denoted by curve C

14 rr

qPoint source

10

Vortex FilamentMust obey Helmholtz’s Laws so filament strength must be constant, and filament must be looped or extend to infinity

ds

r1

r

V

O

Velocity field is determined from the Biot Savart Law:

The Biot Savart Law cannot be inferred from simple integration since there is no comparable point singularity. Instead it is determined from considering the general problem of determining a velocity field from a given vorticity field.

C

d3

1

11

||)()(

4)(

rrrsrrrV

Filament path denoted by curve C

11

Biot Savart Law1. How to invert

? VΩ

0. V2. For incompressible flow so we may write V as the curl of a vector potential AV

3. SoAAAAΩ 22).(

4. Which has the solution

spaceAll

d||

)()(41)(

1

11

rrrrrA

5. Which differentiates to

spaceAll

d3

1

111

||)()()(

41

rrrrrrAV

ds

r1

r

V

O

Curve C

6. Which, when applied to the singular vorticity field of a filament gives

C

d3

1

11

||)()(

4)(

rrrsrrrV

0. AChoose

12

Example: Velocity induced by a section of a straight filament

C

d3

1

11

||)()(

4)(

rrrsrrrV

r1

r

V

O

Curve C

dshs

2

1e

13

Example: Velocity induced by a section of a straight filament

rs

r

V

O

re

PanelsSource or doublet distributed over finite (often flat) sheet or ‘panel’

dS r1

r

V

O

Point doublet

31

1

||4).(

rrrrμ

n

Area SPerimeter C

ds

15

Constant Strength Doublet PanelSame flow as a vortex filament ring around the panel perimeter

r1 r

V

O

ds

C

d3

1

11

||)()(

4)(

rrrsrrrV

sw

nw

se

ne

=

Therefore for a quadrilateral panel,

),,(),,(

),,(),,()(

nwnefilnesefil

seswfilswnwfil

rrrfrrrf

rrrfrrrfrV

This makes doublet / vortex ring panels ideal for 3D panel methods, since their velocity fields are easy to compute. Since they contain a vortex element they can also be extended to situations (like wings) where vorticity is shed into the wake.