1.6. Reynolds Shear Stress

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.

Reynolds shear stress refers to the stress that occurs in a turbulent flow between two different fluid layers. In 1886, scientist Osborne Reynolds introduced a mathematical expression describing this shear stress. It is formulated as the negative product of the fluid density (rho) and the average product of the fluctuating velocity components in two different directions (u' and v'). This means that Reynolds found a way to quantify the chaos of turbulent flow mathematically.

Imagine two layers of honey being stirred in a pot. The top layer moves quickly, while the bottom layer is more stationary. The interaction between these layers creates stresses that can be felt as resistance while stirring. Reynolds' expression helps us understand the nature and the magnitude of that resistance in a turbulent flow scenario.

So, you have to understand, Reynolds shear stress is given by and it does not have only one component, it has minus u dash w dash, it will have minus v dash w dash. So, there are different, there are some normal shear stress, but this is one of the shear stress components.

Reynolds shear stress consists not only of the interaction between fluctuating velocities in the x and y directions (u' and v') but also includes combinations with another fluctuating component (w'). This means that Reynolds shear stress is multi-faceted; various directional forces contribute to it. The various components can influence how turbulent flows engage with one another as well as their surrounding structures.

Think about a dance floor where dancers are moving in different directions. Each dancer's movements affect others around them, creating a complex interaction of forces. Similarly, the fluctuation of different velocity components can create a complex interplay in turbulent flows, contributing to the overall shear stress experienced by the fluid.

Experiments show that u prime v prime is usually a negative quantity. Therefore, the tau turbulence or minus rho u prime v prime whole bar is total positive quantity, it has negative correlation that we will see.

In turbulent flow, the product of the fluctuating velocity components (u' and v') generally yields a negative value. Despite this negative nature, when combined with the negative density factor in Reynolds' equation, the final Reynolds shear stress becomes positive. Understanding this is crucial as it indicates that turbulence contributes positively to overall shear stresses, even if the individual fluctuating components may have a negative correlation.

Consider a football game where each player tries to evade opponents while maintaining possession of the ball. While a player's action (like moving away) might create a negative impact on how close they can get to the opponent, it ultimately leads to positive outcomes, like scoring. In flow dynamics, seemingly negative interactions can also lead to positive results in terms of turbulence and shear stress.

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. Because as we see, the tau turbulence by Reynolds was given by minus rho u prime v prime whole bar. So, what a nice thing it would be if we can calculate u prime v prime whole bar because that is unknown until now.

Prandtl's mixing length theory provides a framework for calculating turbulent shear stress. The theory posits that if we can determine the average of the product of the fluctuating velocities (u' and v'), we can then precisely quantify shear stress in a turbulent flow. The goal here is to bridge the gap between the challenging task of measuring the shear stress directly and finding an elegant way to calculate it using measurable parameters.

Imagine measuring the flow of traffic on a busy highway. Instead of counting every car, you might take periodic snapshots to gauge average speed and behavior. Similarly, in fluid dynamics, Prandtl's theory allows scientists to indirectly measure turbulent stresses through averages rather than direct observations.

So, mixing length l m, he said it can be described in terms of mixing length l m. He said mixing length l m is the distance between 2 fluid layers in the vertical direction, in the y direction, such that, the bundles of fluid particles from one layer could reach the other layer and mix in the new layer in such a way that the momentum of the particle along the flow direction is the same.

The mixing length (l m) is defined as the vertical distance between two fluid layers in which fluid particles can mix while maintaining the same momentum in the flow direction. This definition is fundamental as it allows us to relate the turbulent flow and momentum transfer between different layers. The mixing length essentially reflects how turbulence aids in distributing energy within a fluid.

Think of two layers of colored water in a glass. If you stir them, the mixing length would be akin to the space through which the color particles transition smoothly from one layer to another, effectively blending without losing their individual momentum. This mixing is key to understanding how momentum behaves in turbulent flow.

Prandtl related u prime to mixing length lm. He said that this 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. He related u prime to the mixing length lm as, he said this u prime can be written as, mixing length lm multiplied by the gradient of the average velocity.

Prandtl established a proportional relationship between the fluctuating velocity component (u') and the mixing length (l m) combined with the velocity gradient (/changed velocity). This means that the intensity of turbulence can be quantified and directly related to how steeply the average velocity changes across a distance. Understanding this relationship is crucial for predicting shear stresses in turbulent flows.

Imagine measuring how steep a hill is while biking. The steeper the hill (the greater the gradient), the harder it is to pedal. Similarly, in fluid dynamics, a steeper velocity gradient means more turbulence and, correspondingly, greater momentum transfer, impacting overall shear stress.

And if you substitute equation 11 and equation 12 in Reynolds stress model, which was tau turbulent is equal to minus u prime v prime whole bar you get, so, minus and minus will become positive it will become tau turbulence is .

By substituting the relationships derived from mixing length theory into Reynolds' shear stress equation, we can simplify and express turbulent shear stress in terms of known quantities like the average velocity gradient and mixing length. This process illustrates how to calculate turbulent shear stress with more manageable variables, which is essential for engineers and fluid dynamicists.

Consider a chef refining a recipe. By systematically adjusting ingredients (substituting known factors), the chef can create a delicious new dish. Similarly, fluid dynamicists can refine complex turbulent flow equations into practical formulas for predicting shear stress in real-world applications.

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