Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Welcome everyone! Today, we’ll dive deeper into the nature of shear stress, especially focusing on turbulent flow. Can anyone remind me what distinguishes laminar from turbulent flow?
Isn't laminar flow smooth while turbulent flow is chaotic?
Exactly! In laminar flow, shear stress is primarily from viscosity. However, in turbulent flow, we need to consider additional shear stress caused by turbulence. This brings us to Boussinesq’s model which introduces the concept of eddy viscosity.
What’s eddy viscosity, though?
Great question! Eddy viscosity quantifies the turbulent component of shear stress. It’s not just about fluid viscosity anymore, but how well the particles mix. And as turbulence increases, this effect gets more prominent.
Does that mean turbulent shear stress is always higher than laminar shear stress?
That's correct! In turbulent flows, shear stress is usually greater due to the additional turbulence factor. Remember, shear stress in turbulent flow can be defined as τ_turbulent = -ρ u' v'.
Oh, does that mean we're depending on u' and v' for calculations?
Yes, but these are fluctuating velocities which can be tricky to measure directly, leading us to the mixing length theory. Let’s summarize our key takeaways: we differentiate turbulent and laminar flow by their shear stress contributions and understand eddy viscosity plays a crucial role.
Excellent discussions team! Remember the acronym ET for 'Eddy Turbulence' to recall the additional shear from turbulence in your studies.
Now, let’s shift gears. What is the mixing length, and why is it essential in our discussions?
It seems to be the distance for particles to transfer momentum, right?
Exactly! The mixing length (lm) is the vertical distance between fluid layers that allows for mixing. Prandtl proposed that this length should be directly influenced by how far you are from a wall.
How does this help with turbulent shear stress?
By relating the fluctuating velocities, u', to mixing length, we can express shear stress in terms of a measurable quantity. Specifically, Prandtl defined u' as lm * (du_bar/dy).
And then what about the kappa constant?
Good catch! Kappa, or the von Karman constant, approximated at 0.4, allows us to linearize the mixing length, making it a function of distance from the wall. This helps simplify our calculations significantly!
Can we summarize what we've learned so far?
Certainly! We’ve established that mixing length is crucial for understanding turbulent flow and calculating shear stress. Remember, lm = kappa * y, where y is the distance from the wall. Keep this in mind as you work through calculations. What's our memory aid again?
ET for 'Eddy Turbulence' and K for 'kappa'!
Perfect! Let’s transition into how these concepts apply to turbulent flow in pipes.
To finish our discussion, let’s look at how this theory applies practically, especially concerning turbulent flows in pipes. What do you think the primary considerations are?
Is it about calculating the total shear stress?
Exactly! In turbulent flows within pipes, the viscous shear stress only exists near the boundary. We mainly consider turbulent shear stress there.
So, can we say that turbulent shear stress is dominant?
Yes, that’s correct! We can simplify our expression for total shear stress as τ = ρ lm² (du_bar / dy)². We primarily focus on the turbulent contributions in the core of the flow.
What's the takeaway about measuring shear stress in practical applications?
The importance lies in understanding averages and how they relate to fluctuating components. Once we comprehend this, we substantially ease our calculations.
Can we summarize what we've learned in this session?
Absolutely! Key points include understanding mixing length importance for shear stress calculation, the dominance of turbulent shear stress over viscous stress, and the reminder that turbulent flows primarily contribute to our calculations. Use the acronym T for 'Turbulent Dominance' to hold this concept in mind. Remember, practice is key!
The force per unit area exerted by fluid layers, crucial in determining flow behavior.
Represents turbulence's effect on shear stress calculations.
A model that simplifies turbulent shear stress calculations through the concept of a mixing length.
In a turbulent pipe flow, the mixing length can be used to calculate how much the velocity varies across different layers of the fluid, impacting shear stress predictions.
When analyzing natural rivers, incorporating the mixing-length model greatly improves predictions on sediment transport due to turbulent shear.
When fluid layers meet and mix, momentum's preserved, that's the trick.
Imagine a bustling kitchen where chefs are hustling. Each has their own area but needs to mix ingredients. The distance they have to stir and blend the flavors represents the mixing length in fluid dynamics.
KUD for 'Kappa, Upward, distance' helps recall mixing length in terms of distance from the wall.
MIX helps to remember: Mixing length, influencing shear stress, and 'mixing' brings velocities together.
Term
Eddy Viscosity
Click to flip
Term
Mixing Length (lm)
Click to flip
Term
Reynolds Shear Stress
Click to flip
A coefficient that quantifies the turbulent shear stress in a fluid.
The stress due to turbulent fluctuations, calculated as minus rho u' v'.
The vertical distance between fluid layers necessary for mixing to occur.
The von Karman constant, approximately 0.4, used in mixing length theory.
Prandtl’s Mixing Length Theory provides a framework for understanding shear stress in turbulent flows by relating fluctuating velocities to a defined mixing length.
This section explores Prandtl's Mixing Length Theory, which describes how shear stress in turbulent flow can be expressed in terms of a mixing length defined by the distance between fluid layers. The relationship between fluctuating velocities and shear stress is outlined, as well as the assumptions made to simplify calculations.
Prandtl's Mixing Length Theory, introduced in 1925, revolutionized the understanding of turbulent shear stress in fluid dynamics. It asserts that turbulent shear stress can be computed if the product of fluctuating velocities, represented as minus rho u' v' , is established. The challenge is determining u' v', as it is not easily measurable.
Prandtl introduced the concept of a mixing length (lm), which is defined as the vertical distance necessary for fluid particles from one layer to mix into another layer while preserving momentum. This relationship is crucial as it helps express shear stress in terms of a measurable quantity.
Additionally, Prandtl theorized that the mixing length is a linear function of the distance from a solid boundary (y), expressed as lm = kappa * y, where kappa is the von Karman constant (~0.4). This leads to a simplification where the turbulent shear stress can be calculated using the average velocity gradients, significantly aiding the analysis of turbulent flows.