Chapter 4: Flowing Fluids & Pressure Variation (part 1) Qualitative description of flow Types of visualizations Types of flows (part 1) Frames of reference (part 1) Euler’s equation of motion

Understanding Flow - Measurements To measure / understand flow, we often use a number of visualization techniques Experiments Computations Theoretical predictions Types (defined in a moment) Pathline Streakline Streamline Timeline (less common)

Flow visualization – pathline vs. streakline

Flow visualization - streamline Streamline – vectors tangent to the direction of flow at every point (not exactly shown here)

Why do we care?? Understanding, measuring flow Two different distinctions of flow (for now) Steady vs. unsteady a question of time Uniform vs. non-uniform a question of space Two different “frames of reference” Eulerian Lagrangian

Steady flow: uniform vs. non-uniform Steady flow: at every point in space, the velocity is unchanging, independent of time Flow could be steady if qin = qout In the non-uniform flow, the fluid acceleration is not equal to zero qin qout

Un-steady flow: uniform vs. non-uniform unsteady flow: the velocity is changing, independent of time Flow could be unsteady if qin ≠ qout In the non-uniform flow, the fluid acceleration is not equal to zero qin qout

Summary

Some comments about (un) steady (non-) uniform flows and visualization Steady uniform flow: Streamline, streakline, pathline, are all the same, always (the visualization example was unsteady & uniform) (the computational example was steady & non-uniform) Steady flow: Uniform flow:

Eulerian vs. Lagrangian frame of reference Quick summary for now

Which best represents Lagrangian frame of reference? Eulerian frame of reference? (a) Streamline (b) Streakline (c) Pathline

Euler’s equation Valid for inviscid, incompressible flow only!

Euler’s equation Consider the fluid-filled accelerating truck. Where is the pressure greatest? How can we calculate the pressure of B relative to that of A?

Euler derivation, continued Now… what about the pressure difference between B and C? Which is greater? How can we calculate the pressure of C relative to that of B? Relative to that of A?

Euler derivation, continued Now, what do we do when g is not perpendicular to acceleration direction?