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Welcome, everyone! Today, we are talking about shear stress in turbulent flow. In contrast to laminar flow, where shear stress arises only from viscosity, turbulent flow introduces additional complexities.
Can you explain why turbulence adds more shear stress?
Great question! In turbulent flow, the chaotic and irregular movements of fluid particles increase the shear stress beyond what is caused just by fluid viscosity. We can understand this using Boussinesq's model.
What is Boussinesq’s model exactly?
Boussinesq’s model states that the total shear stress in turbulent flow comprises two components: one from viscosity and another from the turbulence, represented by eddy viscosity.
Now let's dive deeper into eddy viscosity. Eddy viscosity varies with flow conditions and is crucial for calculating shear stress in turbulent flows.
So, it's different from the regular viscosity we talked about, right?
Exactly! Unlike dynamic or kinematic viscosity, which are inherent properties of fluids, eddy viscosity is dependent on the turbulence of the flow. It's a concept that helps us understand real-world fluid dynamics.
Does eddy viscosity change near the walls of a pipe?
Yes! Eddy viscosity decreases as we approach the wall, and it's zero at the wall boundary where the fluid is stationary.
Let's now discuss Reynolds shear stress, introduced by Reynolds himself in 1886, which helps evaluate turbulence's effect on shear.
How is Reynolds shear stress calculated?
It’s computed as the negative product of the fluctuation in velocity components in two directions. More formally, it’s expressed as minus density times the fluctuating velocity components, which highlights the relationship between turbulent fluctuations and shear stress.
Does this have any practical applications?
Absolutely! Understanding Reynolds shear stress is essential for designing efficient hydraulic systems and predicting flow behaviors in various engineering applications.
Next, let’s explore Prandtl’s mixing length theory. This theory provides a framework to estimate turbulent shear stress based on observable quantities.
So how does mixing length relate to shear stress?
Prandtl identified the mixing length as the distance over which two layers of fluid mix, and he described it as a linear function of distance from the wall. It connects directly to the velocity gradients.
And what about the von Karman constant?
The von Karman constant is a proportionality factor in these relationships, typically accepted as approximately 0.4, tying the mixing length to the distance from the wall.
Finally, let's discuss how we apply our understanding of turbulent shear stress in pipes. Most shear stress in turbulent flow comes from turbulent shear stress, with viscous shear stress having a negligible effect away from the boundary.
So, we can simplify our calculations?
Yes! For practical purposes, we can neglect viscous shear and focus on the turbulent component, allowing us to write total shear stress in terms of mean velocity and mixing length.
This seems crucial for designing effective engineering solutions!
Exactly! Understanding these dynamics is key in many fields, including hydraulic engineering and environmental science.
Describes shear stress in turbulent flow combining viscous and turbulence components.
A unique viscosity parameter relevant to turbulent flows, dependent on the flow character.
Quantifies the shear stress induced by turbulence through velocity fluctuations.
Prandtl's concept linking shear stress to fluid layer interactions, expressed in measurable terms.
A crucial constant in mixing length theory, used to relate mixing length to distance from a wall.
Shear stress at play, in turbulent sway, adds turbulence's way to viscosity’s say.
Imagine a calm river (laminar flow) turns into a wild, swirling stream (turbulent flow), where not just the water (viscosity) but the chaos of currents (turbulence) shape its flow – both cause shear stress!
SHEAR: Stress from Heat, Eddy And Resistance.
B.E.R.M: Boussinesq's Equation Relating Mixing.
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What is shear stress?
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What is Boussinesq's model?
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What does Reynolds shear stress quantify?
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What is the relationship of mixing length to distance from a wall?
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The stress component that acts parallel to the surface.
A coefficient that describes the viscosity due to turbulence.
Shear stress due to turbulence, calculated from fluctuating velocity components.
Distance over which fluid particles interact or mix, essential in estimating turbulent shear stress.
A proportionality constant in mixing length theory, approximately 0.4.
This section discusses shear stress in turbulent flow, emphasizing Boussinesq’s model, Reynolds shear stress, and Prandtl’s mixing length theory.
Understanding shear stress in turbulent flow is crucial for hydraulic engineering. This section elaborates on Boussinesq’s model where shear stress is impacted by turbulence, introduces Reynolds shear stress, and explains Prandtl's mixing length theory to estimate turbulent shear stress based on measurable quantities.
In turbulent flow, unlike laminar flow where shear stress arises solely from viscosity, a significant additional component is attributed to turbulence. This concept is captured by Boussinesq’s model, which defines total shear stress in turbulent flow using eddy viscosity. The shear stress related to turbulence is expressed as product of eddy viscosity and velocity gradient. Importantly, eddy viscosity is not a constant fluid property but varies with flow conditions.
The concept of Reynolds shear stress, introduced by Reynolds in 1886, helps quantify turbulent shear stress through the fluctuation of velocity components. This leads to the formulation of shear stress as a product of density and fluctuating velocity components.
Prandtl further advanced these theories by introducing the mixing length, a measurable quantity related to turbulence. He defined the mixing length as the distance over which fluid layers interact and mix, providing a means to express turbulence in terms of measurable variables. By establishing a proportional relationship between the mixing length and vertical distance from a boundary, Prandtl paved the way for understanding shear stress dynamics in turbulent flow, simplifying the analysis with the introduction of the von Karman constant.