1.3. Shear Stress in Turbulent Flow

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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