Chapter 4 Flowing Fluids Pressure Variation part 1

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 • Pathline • 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?