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Today, we'll delve into Reynolds shear stress, an essential component of turbulent flow. Can anyone tell me what shear stress signifies?
Isn’t it the force per unit area acting parallel to a surface?
Great! That's correct. In turbulent flow, we have to consider not only the viscous shear stress but also the additional shear stress due to turbulence. This is where Boussinesq's eddy viscosity concept comes into play.
What do you mean by eddy viscosity?
Eddy viscosity measures the turbulence in the flow. It enhances total shear stress significantly beyond what is seen in laminar flow.
So, does that mean the total shear stress in turbulent flow is always larger?
Exactly! It usually is, due to the turbulent contributions.
Reynolds introduced a formula for calculating shear stress in turbulent flow back in 1886. Can anyone recall how it's expressed?
It's noted as -ρ⟨u'v'⟩, right?
Exactly! This expression illustrates that the shear stress is based on the average of fluctuating velocities. Why might this average be a negative value?
Maybe because of the correlation between the flow directions?
You're on point! The negative correlation leads to positive shear stress once evaluated.
Now, let’s transition to Prandtl's mixing length theory, which is crucial in determining turbulent shear stress. Who can summarize what mixing length is?
It’s the distance between two fluid layers that can interact and exchange momentum?
Correct! It’s essential for quantifying shear stress. Prandtl simplified it by stating that mixing length is proportional to the distance from the wall, described mathematically as lm = k * y. What’s the constant k known as?
It's called the von Karman constant, and it’s approximately 0.4.
Exactly! This relationship helps us link shear stress with measurable quantities.
Finally, we must address how viscous shear stress operates within turbulent flow. Where does viscous shear stress largely apply?
It mainly exists near the wall, right?
Exactly! Most turbulent shear stress dominates away from the wall. So, how can we express the total shear stress in turbulent conditions?
It would be mixed with turbulent shear stress, significantly outweighing viscous stress?
That's right! As you can see, the turbulence in flow changes our approach significantly.
An example of Reynolds shear stress calculation in a turbulent pipe flow scenario, showing how fluctuating velocities influence the total stress experienced.
In a river, the turbulent flow of water over rocks can result in increased shear stress due to both turbulence and viscosity combined, illustrating the concepts discussed.
In turbulent flow, stresses do grow, shear and eddy, steal the show.
Once in a fluid realm, turbulence danced wildly. It swirled so much that it confused the walls. That's when eddy viscosity came in to quantify the chaos!
Remember 'REM' - Reynolds, Eddy viscosity, Mixing length for easy recall.
S.E.R.V.E - Shear stress, Eddy viscosity, Reynolds shear stress, Viscous flow, and Effect of turbulence.
Term
What is Reynolds shear stress?
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Term
Define mixing length.
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The force acting parallel to the surface per unit area.
A value representing turbulent flow characteristics affecting shear stress.
The average shear stress resulting from fluctuations in velocity in turbulent flow.
The vertical distance over which fluid particles can mix between layers.
A model that relates turbulent shear stress to measurable quantities through mixing length.
This section explores the concept of Reynolds shear stress in turbulent flow and its relationship with the flow's eddy viscosity and mixing length.
Reynolds shear stress is introduced as a critical parameter in turbulent flow theory, distinguished by the unique role of turbulent viscosity. The section discusses the essential concepts of Boussinesq's model, Prandtl's mixing length theory, and how shear stress components in turbulent flow differ from those in laminar flow.
In turbulent flow, shear stress is notably influenced by turbulence on top of the viscous effects already present in laminar flow. Boussinesq's model introduces the concept of eddy viscosity, which contributes significantly to total shear stress, leading to much larger values than those found in purely laminar conditions.
Reynolds shear stress, articulated in 1886, captures this turbulence effect and can be mathematically expressed as -ρ⟨u'v'⟩, which is a negative correlation indicative of the interaction between fluctuating velocity components (u' and v').
Prandtl's mixing length theory posits that the turbulent shear stress can be calculated using a mixing length (lm) that represents the vertical distance over which fluid layers can interact efficiently. This mixing length is defined as proportional to the distance from the wall, leading to the conclusion that turbulent shear stress can largely be attributed to the non-viscous components dominating the flow away from the wall.