Aula Teórica 6

Pressure distribution in “Rigid Body” type flows.

What is a rigid body flow?

• A flow is of the “Rigid body” if the adjacent elementary volumes maintain their relative position, i.e. if there is no shear.

• In this case forces are exclusively resulting from:– Pressure or – Gravity

Force resultant

dxdydzgWeight

ii

i gx

p

dt

du

Examples

• Uniform linear acceleration:

ii

i gx

p

dt

du

x1

x3

33

33

3

0 gx

p

gx

p

dt

du

1

1

11

1

x

p

dt

du

gx

p

dt

du

If there is a free surface

• The pressure is hydrostatic and consequently there is inclination….

• If acceleration is constant, the horizontal pressure gradient is constant and the free surface is a plane:

11

11

1

1

1

1

Cxg

a

ax

g

x

p

gp

ax

p

x

x

x

001 x

x1

x3

1

11

1

000

xg

a

Cx

x

mzx

xg

021.07.0

7

Rotating flow

• Rotation implies acceleration. i

i

i gx

p

dt

du

• Cylindrical coordinates are the most adequate

• i.e. the parabola vertices is at z=0 and r=0.

• The line along which p=p1 is:

• At free surface p=p0 and thus:

g

rzsurf 2

22

• i.e. the free surface is a parabola.

Free surface formation• The volume above the resting

level is equal to the volume below that level.

• The free surface can cut the bottom.

Curvilinear stream lines

• Can we have any curvature?

• We are limited by the pressure gradient.

Summary

• If there is acceleration in a free surface flow, free surface inclination will appear.

• If acceleration is linear and constant, the free surface is plan

• If there is rotation at constant angular velocity, the free surface is a parabola.