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Today, we're focusing on shear stress in turbulent flow. Recall that in laminar flow, shear stress is only due to viscosity. In turbulent flow, we have an additional component due to turbulence. Can anyone explain what this means?
Does this mean that turbulent flow has more shear stress compared to laminar flow?
Exactly, that's a great point. The total shear stress in turbulent flow is increased because of this turbulence-related shear. We use Boussinesq's model to define it. Can anyone tell me what eddy viscosity refers to?
Isn’t eddy viscosity a coefficient that represents the effect of turbulence on viscosity?
Correct! Eddy viscosity describes how turbulence increases momentum transfer, and it differs from the fluid's dynamic viscosity. Remember this distinction; it's vital for our further discussions.
Let's discuss Reynolds shear stress, established by Osborne Reynolds in 1886. Who can remind us how it's defined?
Reynolds shear stress relates to the fluctuating velocities between two fluid layers, right? Isn’t its expression negative due to how it accounts for turbulence?
Exactly! The expression includes the fluctuating velocities, u' and v'. The negative sign reflects the correlation between these components. What do we call the average value of this stress?
Is it tau turbulence?
Yes, good job! Remember this term because it ties back into how we calculate shear stress and influences many engineering applications.
Now, let's turn our attention to Prandtl's mixing length theory. Can someone explain what mixing length (l_m) is?
The mixing length is the distance over which fluid layers mix; it helps in understanding momentum transfer.
Right! Prandtl proposed that we could relate this mixing length to the average velocity gradient. How might we apply this in real-world scenarios?
So we can calculate shear stress using average velocity, which simplifies our engineering calculations a lot!
Excellent! And when we define l_m as a linear function of the distance from the wall, we get a common relation that uses the von Karman constant.
Finally, let’s discuss how these concepts apply to turbulent flow in pipes. What do we typically find regarding viscous and turbulent shear stress?
Viscous shear stress mostly occurs near the boundaries, while turbulent shear stress dominates in the majority of the flow.
Correct! That understanding lets us approximate total shear stress mainly through turbulent components. Any insights on how this affects real systems?
It helps engineers design more efficient pipes by understanding where to focus on turbulent flow dynamics.
Great summary! Always remember how fluid dynamics integrates fluid properties and flow regimes.
The stress parallel to fluid layers due to applied forces.
Additional viscosity due to turbulence impacting momentum.
Shear stress influenced by fluctuating velocities within turbulent flow.
Distance at which fluid layers interact and momentum is exchanged.
A proportionality factor in mixing length theory.
In laminar flows, the stress is clear, but turbulence adds to what we fear.
Imagine a busy river, layers rushing to mix; the turbulence creates chaos where the stillness used to fix.
Remember 'Eddy's Viscous Roller Coaster' – it represents how turbulence accelerates momentum transfer.
M.E.R.C – Mixing Length, Eddy Viscosity, Reynolds Stress, and Core Turbulence.
Term
What defines eddy viscosity?
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Term
What is the primary equation for Reynolds shear stress?
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The stress that arises from the force applied parallel to a surface or section.
An additional viscosity component in turbulent flow accounting for momentum transfer due to turbulence.
Turbulent shear stress expressed in terms of fluctuating velocities between fluid layers.
The average distance over which momentum mixing occurs between adjacent fluid layers.
A dimensionless constant (approximately 0.4) relating mixing length and distance from the wall.
This section discusses the concept of shear stress in turbulent flow and introduces Boussinesq's model, which includes the significance of turbulent viscosity and Reynolds shear stress.
The lecture continues from the previous discussion by explaining shear stresses in turbulent flow using Boussinesq’s model. It covers the concept of eddy viscosity, Reynolds shear stress, and Prandtl’s mixing length theory, highlighting their practical implications in hydraulic engineering.
In hydraulic engineering, the understanding of flow behavior is crucial. This section delves into shear stress in turbulent flow, contrasting it with laminar flow.
Boussinesq's Model: In this model, the total shear stress in turbulent flow incorporates not just viscous effects but also an additional component due to turbulence, represented by eddy viscosity. The shear stress due to turbulence is calculated as an equivalent of viscous shear but with a different coefficient.
Eddy Viscosity: Unlike conventional dynamic and kinematic viscosities, eddy viscosity (B7) and kinematic eddy viscosity (B5) are flow condition-dependent. B5 decreases towards the wall, reaching zero at the boundary.
Reynolds Shear Stress: Established by Osborne Reynolds, this quantity describes the turbulent shear stress, taking into account fluctuating velocity components (u’ and v’). It results in expressions that relate to the momentum transfer in fluid layers.
Prandtl’s Mixing Length Theory: In 1925, Prandtl proposed a method to quantify turbulence through a parameter known as mixing length (l_m), defined as the distance over which fluid bundles mix. This concept allows shear stress to be expressed in terms of measurable average velocity gradients. Through various equations, the mixing length is related to the distance from the wall, facilitating calculations in turbulent flow.
Application to Turbulent Flow in Pipes: The section recognizes that in turbulent flow, viscous shear stress dominates near the boundaries while turbulent stresses are more significant in the core regions of flow. Thus, shear stress can be generally approximated focusing on turbulent contributions.