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A short class with Sarah and her students. The other ways are under More.
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Let's start by discussing shear stress in turbulent flow. Can anyone tell me what shear stress at the wall is typically assumed to be?
Isn't it constant at the wall, represented as tau_naught?
Exactly! We assume it’s constant, especially for small distances from the wall, or 'y'. This assumption helps us simplify our equations.
How does that relate to the velocity profiles we study?
Great question! It’s part of deriving our velocity profiles, especially the logarithmic ones we use for turbulent flows. This is a crucial aspect - remember that!
In fact, let's remember 'tau_naught at the wall, stability for the fluid's call.' This rhyme helps you keep in mind that shear stress is stable at the wall.
Now, let's discuss velocity profiles. Student_3, can you describe how a laminar flow profile appears compared to a turbulent flow profile?
Uh, I think laminar flow has a parabolic profile, while turbulent flow has a sort of logarithmic shape?
Precisely! The laminar flow is smooth and parabolic, whereas turbulent flow creates a fuller velocity profile. It reflects the chaotic nature due to eddies.
What do we call the difference between the maximum velocity and actual velocity?
That's called the velocity defect! Great catch, Student_4. Remember, 'Velocity defect: the flow’s suspect.' It's a crucial point when analyzing turbulent flows.
Let's dive into Prandtl's mixing length theory. How does it help us?
Isn't it supposed to help derive those logarithmic profiles we just talked about?
Exactly! By considering factors like turbulent intensity, we can derive key velocity equations from it.
What about the boundary conditions used in the equations? How important are they?
Boundary conditions are critical! They help us set our equations properly to derive relationships such as shear stress from turbulence theory. 'Boundary here, simplify near!'
Now, let's apply what we've learned to calculate shear stress. Can someone remind us how to calculate wall shear stress?
We use tau_naught equals rho times u_star squared, right?
Correct! And let’s recall that u_star is the frictional velocity. Are there any other parameters we need to consider?
The density of the fluid too, right? Like for water, it’s around 1000 kg/m³.
Very good! Remember to consider fluid density. 'Shear stress, density's a must!' Let’s try a calculation together next.
Shear Stress at Wall:
It is often assumed to be constant and equal to in small values of y (distance from the wall).
Velocity Profiles:
The flow velocity is described by a logarithmic profile in contrast to the parabolic profile observed in laminar flow.
Integrating derived equations leads to the establishment of the famous velocity defect law.
Boundary Conditions:
Boundary conditions are crucial for integrating fluid motion equations and determining shear stress.
Examples include setting velocity conditions to derive constants and thus solving for turbulent properties.
Characteristics of Turbulent Flow:
Turbulent flow consists of multiple regions with varying characteristics, including the viscous sublayer and turbulent layer.
This section provides a comprehensive framework for understanding how mathematical equations can model turbulent flow and predict key parameters such as shear stress at the pipe wall.
If water has a velocity of 4 m/s at the pipe center and 3.5 m/s at 2 cm from the center, the wall shear stress can be calculated using derived equations.
During the discussion of turbulent flow, if one considers a pipe with a rough surface, the wall shear stress would reflect greater turbulent effects compared to a smooth pipe.
When shear appears, at the wall it steers; constant it may stay, guiding the flow’s way.
Imagine a pipe where water flows freely, at the surface it's calm, but inside it swirls chaotically—a contrast between the outside and turbulent currents.
PVT: Prandtl’s Velocity Theory helps remember that turbulent flows have varied velocity behavior.
FLOW: F = Friction, L = Logarithmic shapes, O = Overlap in layers, W = Wall shear stress.
Term
What is shear stress?
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Term
Define frictional velocity (u*)
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Term
What is the velocity defect law?
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The stress that acts parallel to the surface of a material, particularly in a fluid, often assumed constant at the wall in turbulent flow.
A measure of the velocity used to characterize the effect of turbulence near the wall in flow.
A law stating the difference between maximum velocity and actual velocity in turbulent flow.
A representation of velocity distribution in turbulent flow which shows how velocity increases logarithmically with distance from the wall.
This section focuses on the application of equations in turbulent flow analysis, highlighting shear stress, velocity profiles, and boundary conditions.
In this section, we explore the mathematical relationships governing turbulent flow, particularly the shear stress at the pipe wall and the velocity profiles derived from Prandtl's mixing length theory. Key equations are manipulated to reveal insights into flow behavior and shear stress calculations.
This section delves into the fundamental equations used to analyze turbulent flow, focusing primarily on shear stress and velocity profiles derived from fluid mechanics principles. Equations such as equation 14 and equation 15 are referenced to illustrate how shear stress () behaves under turbulent conditions.
Shear Stress at Wall:
Velocity Profiles:
Boundary Conditions:
Characteristics of Turbulent Flow:
This section provides a comprehensive framework for understanding how mathematical equations can model turbulent flow and predict key parameters such as shear stress at the pipe wall.