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Welcome back, everyone! Today, we continue discussing shear stresses in fluid flow, focusing on how they differ between laminar and turbulent flows. Can anyone remind me how shear stress works in laminar flow?
In laminar flow, shear stress is caused only by viscosity, right?
Exactly! In laminar flow, shear stress τ is directly proportional to viscosity μ and the velocity gradient. Now, what happens in turbulent flow?
There’s an additional component due to turbulence, making the total shear stress much larger.
Correct! This additional component is defined by eddy viscosity. It’s crucial to understand that while laminar flow has well-defined behavior, turbulent flow is influenced by chaotic fluctuations of the fluid particles.
Now, let’s talk about Boussinesq's model. Who can explain what eddy viscosity is and how it’s used?
Eddy viscosity refers to the additional viscosity in turbulent flow that accounts for energy transfer due to turbulence, right?
Exactly! We denote it as η. In turbulent flow, shear stress can be expressed as τ = η (du/dy). Can someone tell me how this differs mathematically from the laminar flow equation?
In laminar flow, it's τ = μ (du/dy). Here, we have a new coefficient, η, which varies with flow conditions.
Good point! Remember, η and the kinematic eddy viscosity ε are not fixed fluid properties; they depend on flow conditions. As we get closer to a wall, ε approaches zero.
Next, let’s explore Reynolds shear stress, introduced by Reynolds in 1886. Does anyone know how we express turbulent shear stress mathematically?
It's expressed as τ = -ρ u'v', where u' and v' are fluctuating velocity components.
Correct! The negative sign indicates a negative correlation. Do you remember what this implies about turbulent shear stress?
It implies that u' and v' are often negatively correlated, making τ a positive quantity overall.
Exactly! Understanding this relationship is pivotal for capturing the dynamics of turbulent flows.
Let’s delve into Prandtl's mixing length theory next. Can someone explain what mixing length (lm) entails?
Mixing length is the distance between fluid layers that allows for momentum transfer between them.
Yes! Prandtl linked mixing length to the velocity gradient. What equation represents this relationship?
Prandtl stated that u' is proportional to lm multiplied by the average velocity gradient.
Exactly! And do you remember how Prandtl defined lm concerning the distance from the wall?
He said lm is a linear function of distance from the wall, expressed as lm = k * y, where k is the von Karman constant.
Very well! With this understanding, we can derive formulations for turbulent shear stresses in practical scenarios.
Finally, how do these concepts apply to turbulent flow in pipes?
In turbulent flow within pipes, the main shear stress is dominated by turbulence, while viscous stresses are localized near boundaries.
Correct! So, we can simplify the shear stress equation significantly in these scenarios?
Yes, we can approximate the shear stress as τ ≈ ρ lm² (du/dy)².
Well done! Summarizing today's session, we covered the transition from laminar to turbulent flow, Boussinesq’s model, Reynolds shear stress, and how Prandtl’s theory helps us understand turbulent flows in pipes.
A type of flow where fluid moves in parallel layers with no disruption.
A type of flow characterized by chaotic changes in pressure and velocity.
A measure of the turbulent diffusivity of momentum due to eddies in a fluid.
A model to estimate shear stress in turbulent flow using a characteristic mixing length.
A dimensionless quantity used to predict flow patterns in different fluid flow situations.
In laminar flow, layers glide, viscosity is the only guide.
Imagine a group of children trying to mix colors in a jar—some are able to reach others, creating a beautiful swirl. This represents mixing length in ideal mixing.
Reynolds Shear Stress: Remember—R for Reynolds, S for Stress, T for Turbulence.
MEMS: Mixing, Eddy, Momentum, Shear to remember concepts related to turbulent flow.
Term
What is Boussinesq’s model?
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Term
What does Prandtl’s mixing length denote?
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The stress component that acts parallel to the surface of a material.
The additional viscosity in turbulent flow caused by turbulence.
Turbulent shear stress quantified as τ = -ρ u'v', considering fluctuating velocity components.
The distance between two fluid layers permitting mixing and momentum transfer.
A dimensionless constant in fluid dynamics used in mixing length theory, usually taken as approximately 0.4.
This section delves into the principles of shear stresses in turbulent flow, exploring concepts such as eddy viscosity, Reynolds shear stress, and the mixing length theory introduced by Prandtl.
The section focuses on the differences in shear stress between laminar and turbulent flows, emphasizing Boussinesq’s model which introduces the concept of eddy viscosity. It also discusses Reynolds shear stress and Prandtl’s mixing length theory, enhancing the understanding of turbulent shear stress in fluid mechanics.
This section provides insights into shear stresses in both laminar and turbulent flow, addressing key principles associated with hydraulic engineering. Initially, it clarifies that in laminar flow, shear stress is solely due to viscosity. However, in turbulent flow, a significant additional component arises from turbulence. This is encapsulated within Boussinesq’s model, highlighting how shear stress (τ) can be defined by including a new parameter known as eddy viscosity (η), leading to a revised equation: τ = η (du/dy).
The document then shifts to the concept of Reynolds shear stress as introduced by Reynolds in 1886, which describes turbulent shear stress in fluid layers. The relationship is explored mathematically but requires further derivation in future lessons. Subsequently, the section details Prandtl’s mixing length theory, which allows for the calculation of turbulent shear stress through a measurable quantity defined as mixing length (lm). Prandtl posited that this mixing length is the distance between fluid layers, contributing to momentum transfer.
Prandtl made a crucial assumption that mixing length varies linearly with the distance from a wall, leading to the formulation lm = k * y, where k (the von Karman constant) is approximately 0.4. The implications of these theories are important for understanding turbulent flow in hydraulic engineering, especially in applications like pipe flow, where turbulent shear stress dominates, and the effect of viscosity diminishes further from the boundary.