Chapter 4 FLOWING FLUIDS AND PRESSURE VARIATION Fluid Mechanics Source:

Chapter 4

FLOWING FLUIDS AND PRESSURE VARIATION

Fluid Mechanics

Source:

http://www.geofys.uu.se/files/teacher/

Rotation and Vorticity

Rotation of a fluid element in a rotating tank of fluid(solid body rotation).

Rotation of fluid element in flow between moving andstationary parallel plates

o You can think of the “plus signs” as small paddle wheels that are free to rotate about their center.

o If the paddle wheel rotates, the flow is rotational at that point.

Rotation: the average rotation of two initially mutually perpendicular faces of a fluid element.

The angle between the bisect line and the horizontal axis is the rotation, θ.

As

And similarly

The net rate of rotation of the bisector about z-axis is

The rotation rate we just found was that about the z-axis; hence, we may call it

and similarly

The rate-of-rotation vector is

Irrotational flow requires (i.e., for all 3 components)

Applicable to ideal flow theory

The property more frequently used is the vorticity, which is a vector equal to twice the rate of rotation vector

Vortices

A vortex is the motion of many fluid particles around a common center. The streamlines are concentric circles.

Choose coordinates such that z is perpendicular to flow.In polar coordinates, the vorticity is (see p. 104 for details)

(V is function of r, only)

Solid body rotation (forced vortex):

or

Vortex with irrotational flow (free vortex):

A paddle wheel does not rotate in a free vortex!

In a cyclonic storm:Forced vortex (interior) andfree vortex (outside):

Good approximation to naturally occurring vortices such astornadoes.

Euler’s equation for any vortex:

We can find the pressure variation in different vortices(let’s assume constant height z):

In general:

1) Solid body rotation:

2) Free vortex (irrotational):

Application to forced vortex (solid body rotation):

with

Pressure as function ofz and r

p = 0 gives free surface