1.3 - Shear Stress in Turbulent Flow
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Introduction to Shear Stress in Turbulent Flow
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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.
Understanding Eddy Viscosity
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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.
Reynolds Shear Stress
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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.
Prandtl's Mixing Length Theory
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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.
Application in Turbulent Flow in Pipes
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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.
Introduction & Overview
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Quick Overview
Standard
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.
Detailed
Detailed Summary
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.
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Introduction to Shear Stress in Turbulent Flow
Chapter 1 of 7
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Chapter Content
So, shear stress in turbulent flow. We are going to talk about a model that is called Boussinesq’s model, where the total shear stress, in case of laminar flow it was due to the viscosity. But in a turbulent flow, there is an additional component of shear stress that happens because of the turbulence in the flow.
Detailed Explanation
This chunk introduces the concept of shear stress in turbulent flow. In laminar flow, shear stress arises solely due to viscous forces, which is the internal friction within the fluid. However, in turbulent flow, the situation becomes more complex as an additional component emerges due to turbulence effects. Turbulence mixes the fluid flow, leading to increased shear stress. The Boussinesq model provides a theoretical framework to understand these dynamics.
Examples & Analogies
Think of laminar flow like a smooth, straight river where the water flows steadily without much mixing. In contrast, turbulent flow is like a wild river with rapids, where water swirls and churns. In the smooth river, water sticks together and moves uniformly, leading to lower shear stress, while in the wild river, the chaos of turbulence increases friction between layers of water, resulting in higher shear stress.
Eddy Viscosity and Shear Stress Calculation
Chapter 2 of 7
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Chapter Content
Therefore, the shear stress in total is much larger than the viscous flow. At least it is definitely larger than the viscous flow because there is shear stress that is associated with turbulence too. So, Boussinesq’s says as for laminar flow, the shear stress due to the turbulence component is that eddy viscosity du / dy.
Detailed Explanation
In this chunk, the concept of eddy viscosity is introduced. Unlike laminar flow, where viscosity ( ) determines shear stress, turbulent flow involves eddy viscosity, which is a measure of turbulence's contribution to shear stress. The overall shear stress in turbulent flow is represented as the sum of viscous shear and turbulent shear, indicating that in turbulent conditions, the shear stress dramatically increases due to turbulence-related movements in the fluid layers.
Examples & Analogies
Imagine stirring a thick soup. The soup flows smoothly due to viscosity when left calm, but once stirred, the mix creates swirls and eddies. The stirring action (similar to turbulence) introduces additional friction, resulting in the soup resisting motion more than it would if still, exemplifying how turbulence increases shear stress through the action of eddy viscosity.
Reynolds Shear Stress
Chapter 3 of 7
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Chapter Content
Now, coming to what is Reynolds shear stress. So, Reynolds in 1886 gave expressions for turbulent shear stress between two fluid layers separated by a small distance. And he said that the shear stress due to turbulence can be written as, minus rho u prime v prime whole bar.
Detailed Explanation
This chunk discusses Reynolds shear stress, which is a critical concept in turbulent flow. The Reynolds shear stress formula captures the interaction between fluctuations in velocity within the turbulent flow. Each component u' and v' represents fluctuations in fluid velocity. The negative sign in the equation indicates a relation between the average momentum transfer across fluid layers, which generally results in a positive value for shear stress even though the raw calculation may yield a negative product.
Examples & Analogies
Think of a crowded dance floor where people (fluid particles) are moving around. Some are dancing smoothly (average velocity), while others are weaving through the crowd (fluctuations). When calculating how hard dancers push against each other, we might need to consider both their average swirling motion and the chaotic movements (u' and v'), just like how Reynolds shear stress accounts for turbulent velocity fluctuations and their impact on overall flow.
Prandtl's Mixing Length Theory
Chapter 4 of 7
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Chapter Content
Now, there is a concept of Prandtl's mixing length theory. So, turbulence shear stress can be calculated if this thing is known, u prime v prime whole bar is known. In 1925 Prandtl introduced the concept of mixing length, which can be utilized to express the shear stress here, in terms of some measurable quantity.
Detailed Explanation
In this chunk, we explore the Mixing Length Theory proposed by Prandtl. This theory helps simplify the process of measuring turbulent shear stress by relating it to a measurable quantity, the mixing length (l_m). This length describes the average distance that fluid particles travel within their layer before crossing over to a neighboring layer. Understanding how to express shear stress in terms of mixing length provides valuable insights into turbulent dynamics.
Examples & Analogies
Consider baking a cake where you're mixing batter. The mixing length (l_m) represents the distance a spoonful of flour must travel before it completely blends with the rest of the batter. In flow dynamics, this distance translates to how fluid particle bundles from one layer reach and mix into another. The more effectively they mix (or the right mixing length), the smoother the batter turns out – just like better mixing in fluid leads to more predictable shear stress outcomes.
Calculating Velocity Fluctuations Using Mixing Length
Chapter 5 of 7
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Chapter Content
Prandtl related u prime to mixing length lm. He said that this u prime can be written as, mixing length lm multiplied by the gradient of the average velocity. So, he said u prime, as you can see in the figure here, he said proportional to u prime but I am going to write it in the next slide.
Detailed Explanation
In this chunk, we see how Prandtl's assumptions relate fluctuating velocity (u') to mixing length (l_m). By assuming that the turbulent velocity fluctuations are proportional to mixing length and the velocity gradient, it allows for calculating turbulent shear stress using known gradient values. This relationship lays the foundation for predicting and analyzing turbulent flow characteristics more effectively.
Examples & Analogies
Imagine driving a car – the speed you vary (u') is comparable to how bumps in the road change your speed patterns. The distance to the next bump can be thought of as mixing length (l_m). The faster you go, the more those bumps disrupt your smooth cruise. By relating speed changes to the frequency and severity of bumps (the velocity gradient), you can predict how turbulent your ride will be.
Linear Relationship of Mixing Length
Chapter 6 of 7
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Chapter Content
Prandtl also assumed that the mixing length lm is a linear function of distance y from the wall or any solid boundary. Therefore, he said l m can be written as ky.
Detailed Explanation
This chunk highlights an essential assumption made by Prandtl about the mixing length, establishing that it increases linearly with proximity to the wall or boundary. This linearity implies that as one moves away from the wall (y increases), the mixing length and therefore the associated shear stress also increase, which simplifies calculations and representation of turbulent behavior in boundary layer flows.
Examples & Analogies
Think of a trampoline—the closer you are to the edge (the wall), the more supportive it feels because there's less stretch. As you bounce higher (move outwards), the trampoline feels more flexible and responsive to your movements. Similarly, as fluid moves away from a boundary, mixing becomes more chaotic, directly impacting how swiftly layers mix and influence shear stress.
Total Shear Stress in Turbulent Flow
Chapter 7 of 7
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Chapter Content
So, in turbulent flow the viscous shear stresses exists only near the boundary and most of the region is dominated by the turbulence. So, we can neglect the viscous shear stress.
Detailed Explanation
In this conclusion chunk, the content emphasizes that in turbulent conditions, viscous shear stress predominantly affects only areas close to solid boundaries. In most regions away from those boundaries, turbulent shear generates the majority of the shear stress. As such, for practical calculations, it's often acceptable to ignore viscous shear, simplifying models and equations in hydraulic engineering.
Examples & Analogies
Think of a fast-flowing river. Near the riverbank, the water moves slower because it encounters resistance from the rocks and the bank (where viscous effects dominate). But further out where the flow is fast and wild, the interactions between layers (turbulent flow) hugely override these boundary effects. In engineering terms, we can simplify calculations by focusing on the driving turbulence in the middle of the flow rather than precisely measuring edge effects.
Key Concepts
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Boussinesq's Model: Describes shear stress in turbulent flow combining viscous and turbulence components.
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Eddy Viscosity: A unique viscosity parameter relevant to turbulent flows, dependent on the flow character.
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Reynolds Shear Stress: Quantifies the shear stress induced by turbulence through velocity fluctuations.
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Mixing Length Theory: Prandtl's concept linking shear stress to fluid layer interactions, expressed in measurable terms.
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Von Karman Constant: A crucial constant in mixing length theory, used to relate mixing length to distance from a wall.
Examples & Applications
In a turbulent pipe flow, significantly more shear stress occurs at higher velocities due to turbulence, impacting energy loss calculations.
Using Prandtl's mixing length, an engineer can predict how fluid layers mix within a pipe and use this to optimize pipeline design and efficiency.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Shear stress at play, in turbulent sway, adds turbulence's way to viscosity’s say.
Stories
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!
Memory Tools
SHEAR: Stress from Heat, Eddy And Resistance.
Acronyms
B.E.R.M
Boussinesq's Equation Relating Mixing.
Flash Cards
Glossary
- Shear Stress
The stress component that acts parallel to the surface.
- Eddy Viscosity
A coefficient that describes the viscosity due to turbulence.
- Reynolds Shear Stress
Shear stress due to turbulence, calculated from fluctuating velocity components.
- Mixing Length
Distance over which fluid particles interact or mix, essential in estimating turbulent shear stress.
- Von Karman Constant
A proportionality constant in mixing length theory, approximately 0.4.
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