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Today, we’re discussing biophysical concepts, particularly the shear stress in turbulent flows. Can anyone tell me how laminar and turbulent shear stress differ?
I think laminar shear stress is only due to viscosity, right?
Exactly, Student_1! In laminar flow, shear stress indeed is only due to viscosity. However, in turbulent flow, there's an extra component due to turbulence. We can call it 'eddy viscosity'.
How does eddy viscosity differ from regular viscosity?
Eddy viscosity varies with flow conditions, unlike standard viscosity. It becomes zero at the wall! Remember: EDDY can help you remember 'Eddy viscosity varies'.
So, the shear stress in turbulent flow is higher than in laminar flow?
Correct! Now, what would be the mathematical form of turbulent shear stress?
Isn't it related to velocity difference?
Yes! It’s represented as τ = -ρ(u'v'). Great summary, everyone!
Moving on, let’s discuss Reynolds shear stress. Who remembers his expression for turbulent shear?
I remember it involves fluctuating velocity!
Exactly! It can be written as τ = -ρ(u'v'). This equation shows how turbulent shear stress is derived using velocity fluctuations. Would you like to know why it’s a negative quantity?
Yes, please!
Good! It’s negative because it represents opposing forces in different fluid layers. Remember: 'Negative reflects opposition!' Now, let's think of practical instances where this concept applies.
How about in design of hydraulic systems?
Yes! That's an excellent example. Keep thinking about these applications as we continue.
Now let’s delve into the mixing length theory proposed by Prandtl. Who can summarize what mixing length means?
Isn’t it the distance fluid particles travel to mix momentum?
Exactly! It's a critical concept for estimating turbulent shear stress. Could anyone write out how it relates to velocity?
It’s l_m = κy, right?
Great! And what does κ represent?
It's the von Karman constant, about 0.4!
Excellent! This relationship simplifies calculating turbulent shear stress. Remember: MIX for Mixing Length.
Let’s wrap up by applying these concepts. How might turbulent shear stress affect pipe design?
It could influence how we calculate pressure drops.
Yes! Understanding these stresses is crucial. Additionally, how does knowing ε help in flow predictions?
Eddy viscosity affects flow behavior close to walls!
Perfect! Now, let’s summarize what we learned today. What are three key aspects of Boussinesq's model?
Turbulent shear stress is more complex than laminar, influenced by eddy viscosity and mixing length theory.
Exactly! Keep those concepts in mind for future applications.
A model explaining shear stress in turbulent flow with the introduction of eddy viscosity.
A variable coefficient representing turbulence's contribution to shear stress.
Defined by the fluctuating components of velocity, representing turbulent shear stress.
A theory relating turbulent shear stress to measurable quantities through mixing length concepts.
To measure flow, the shear shows, An eddy here, as turbulence grows!
Imagine a river flowing smoothly, representing laminar flow. When it encounters rocks, it swirls and roils—this chaos shows turbulent flow, thus leading us to understand shear stresses caused by such movement.
Remember TREMBLE: Turbulent Reynolds Eddy Mixing Blends Laminar Energy for flow.
MIX for Mixing Length to recall how momentum mixes across layers.
Term
What is eddy viscosity?
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Term
Define Prandtl's Mixing Length Theory.
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A measure of how much force is applied parallel to a surface.
A coefficient that reflects the additional viscosity due to turbulence in a fluid.
The shear stress due to turbulent fluctuations in fluid velocity.
The distance over which momentum mixing occurs between fluid layers.
A dimensionless constant, approximately 0.4, used in the relationship between mixing length and distance from surfaces.
Boussinesq’s model describes the shear stress in turbulent flow, highlighting the concept of eddy viscosity and its significance in hydraulic engineering.
The focus of Boussinesq's model is on the differences in shear stress between laminar and turbulent flows, emphasizing the additional component of shear stress caused by turbulence. The model introduces eddy viscosity as a critical factor, which varies with flow conditions and becomes essential in calculating turbulent shear stress, leading to important insights in fluid mechanics.
Boussinesq’s model is a significant concept in hydraulic engineering, specifically when analyzing turbulent flow. Unlike laminar flow, where shear stress is due solely to viscosity, turbulent flow involves an additional shear stress component due to the chaotic nature of turbulence. This distinction leads to larger total shear stress values in turbulent flows compared to laminar ones.
In turbulent flow modeling, the shear stress contribution due to turbulence can be expressed similarly to that of laminar flow but utilizes a new coefficient known as ‘eddy viscosity’ (η). The relationship can be denoted as:
The kinematic eddy viscosity (ε) is defined as:
Where ρ is the fluid density. Unlike dynamic and kinematic viscosities in laminar flow (represented by μ and ν, respectively), eddy viscosity values depend on changing flow conditions, decreasing near the wall and becoming zero at the wall.
Developed by Reynolds in 1886, the formula for turbulent shear stress includes fluctuating velocity components in different directions (u' and v'). The Reynolds shear stress is defined as:
Prandtl introduced the mixing length concept in 1925 to quantify turbulent shear stress. He proposed that the mixing length (l_m) is defined as:
The correlations derived from mixing length theory provide insights into the relationship between turbulent shear stress and measurable parameters. Prandtl assumed that:
In turbulent flowing pipes, the shear stress predominantly arises from turbulent shear stress, whereas viscous shear stress is near the boundary only. Hence it can be simplified into a more manageable equation. This modeling insight is crucial for designing efficient hydraulic systems.