Turbulent Flow in Pipes

Welcome, everyone! Today, we will continue our discussion on shear stresses in turbulent flow. Did anyone notice how turbulent flow differs from laminar flow?

Yes, I think turbulent flow has more chaotic movements?

Exactly! In turbulent flow, there's an added component of shear stress from turbulence itself, unlike laminar flow, which is purely viscous.

So, how do we quantify that?

Great question! We use a model called Boussinesq’s model which introduces eddy viscosity. Recall, eddy viscosity is crucial for turbulent flow and varies based on flow conditions.

How does eddy viscosity relate to regular viscosity?

Eddy viscosity is not a fixed property; it changes with flow conditions, particularly decreasing toward the wall and becoming zero at the wall. Remember this: it’s frequently denoted by eta (η).

So, do we still consider traditional viscosity?

Yes, but in turbulent flow, the focus shifts more to the turbulence-induced shear stress.

To recapitulate, the key components we're looking at are shear stress due to eddy viscosity and the necessity to account for both turbulence and viscosity in analysis.

Next, let's talk about Reynolds shear stress, developed by Osborne Reynolds in 1886. Can anyone summarize what this is?

Isn’t it the average turbulence component that affects shear stress between layers?

Exactly! It’s defined as τ = -ρ (u'v')̅. Here, u' and v' are fluctuating velocity components. Why do you think Reynolds stressed the negative value?

Because it’s conceptualized between different fluid layers that are moving in different velocities?

Correct! There’s a negative correlation between fluctuating velocities, which makes total shear positive despite the negative signs in the equation.

So how do we derive that completely?

That involves more advanced derivations we’ll cover later, but for now, keep in mind the importance of this concept in calculating turbulent shear stress.

Summing up, Reynolds shear stress is crucial for understanding how shear behaves in turbulent flows and sets the stage for further analysis.

Continuing our discussion, let’s dive into Prandtl’s mixing length theory introduced in 1925. How do you think this theory makes analyzing turbulence easier?

By allowing us to relate shear stress to measurable quantities?

Exactly! The theory describes mixing length (lm) as the distance that fluid particles can travel from their layer allowing for mixing.

How is that practically used in fluid dynamics?

We express shear stress in terms of lm and the velocity gradient. For instance, we can write τ = ρ lm² (du/dy)², where 'du/dy' is the velocity gradient.

And how does one determine the mixing length?

Prandtl assumed that lm is linear with respect to the distance from the wall, so lm = κy, where κ is von Karman’s constant, roughly 0.4.

This simplifies the whole turbulent shear stress equation!

Precisely, this theory allows engineers to predict shear stress effectively. In summary, Prandtl’s mixing length provides a key link in analyzing turbulent flows.

Finally, let’s summarize what we've learned about shear stress in turbulent flow. Why is this different from laminar flow?

The dominance of turbulent shear stress rather than viscous shear stress, especially away from the wall.

Right! Most shear stress in turbulent flow is attributable to turbulence itself, allowing for approximations in calculations.

So we can focus primarily on Reynolds shear stress and shortcuts from Prandtl’s theory?

Exactly! Thus, in practice, you’ll often neglect viscous shear stress when analyzing turbulent flow far from walls.

And that helps simplify fluid dynamics considerably?

Absolutely! To wrap up, we focused on shear stress in turbulent flow, understanding key models and theories that simplify these analyses.