Chapter 4: Flowing Fluids & Pressure Variation (part 1)

Qualitative description of flowTypes of visualizationsTypes 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: 0z/v,y/v,x/v

0t/v

• 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?