Pointwise Fluid Motion

Objectives

Before we can try to find out how complete flow fields look, we first need to understand how small regions of fluid behave. In particular, how their motion consists of both rotation and deformation processes.

We also want to understand quantities such as particle paths and streamlines that are frequently used to examine experimental or computed flow fields.

Performance Criteria

• Ability to define the vocabulary
• Can find streamlines and particle paths
• Can find vorticity in any standard coordinate system.
• Can find moving time derivatives.
• Can identify the strain-rate tensor.
• Can find circulation by direct integration.

Vocabulary

• Lagrangian
• Eulerian
• Ideal stagnation point flow.
• Particle path.
• Streamline.
• Steady flow
• Lagrangian derivative.
• Vorticity.
• Rate-of-strain tensor.

Notes

For the linear shear flow of section 4.4,

$$A = \left(\begin{array} {ccc} \frac{\partial u}{\partial x... ...left(\begin{array} {ccc} 0 & U/h \ 0 & 0 \end{array}\right)$$

Hence, averaging A and its transpose, the strain rate matrix is

$$S = \left(\begin{array} {ccc} 0 & U/2h \ U/2h & 0 \end{array}\right) \qquad$$

with eigenvalues (principal strain rates) $\pm U/2h$ and corresponding eigenvectors $(\cos 45,\sin 45)$ and $(\cos 135,\sin 135)$. So the principal axes are under 45 degrees with the plates and flow direction.

The only nonzero vorticity component is in the z-direction:

$$\vec \omega= \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \hat k = - U \hat k /h$$