1.2. Laminar and Turbulent Flow (Contd.)

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 viscous. Sorry. Yeah, that was only due to the viscous. But in a turbulent flow, there is an additional component of shear stress that happens because of the turbulence in the flow. So, therefore, the shear stress in total is much, much larger than the viscous flow.

In turbulent flow, shear stress is influenced by two factors: the fluid's viscosity and the turbulence itself. In laminar flow, shear stress is caused solely by viscous forces, while in turbulent flow, turbulence adds an extra layer of complexity. This means the total shear stress encountered in turbulent flow is significantly greater than that in laminar flow. Boussinesq’s model helps explain this by incorporating both viscosity and an additional component due to turbulence.

Imagine water flowing smoothly through a pipe (laminar flow) versus water churning violently in a river (turbulent flow). In the smooth flow, the forces acting on the water are predictable and uniform. However, in turbulent flow, eddies and swirls create chaotic motion, adding extra force to the fluid's movement.

Here, you see, this is similar. So, instead of µ there is something called ɳ, a new coefficient of viscosity, and this is called eddy viscosity. The shear stress due to turbulence is eddy viscosity du / dy, very similar to the shear stress in the laminar flow.

Eddy viscosity is a key concept in understanding turbulent flow. It is a coefficient that represents the additional shear stress caused by turbulence in the fluid. This concept parallelly mimics laminar flow, where shear stress is a function of the fluid's viscosity. Eddy viscosity, however, varies based on flow conditions, unlike the standard dynamic viscosity, which is an intrinsic property of the fluid.

Think of a crowded dance floor where people move chaotically. They can bump into one another, creating a sort of 'friction' or resistance to movement. In this analogy, the 'crowding' corresponds to turbulence, and the influence of this crowding is similar to how eddy viscosity affects the flow of fluid, indicating how much additional resistance is present due to chaotic movement.

Now, coming to what is Reynolds shear stress. Reynolds in 1886 gave expressions for turbulent shear stress between two fluid layers separated by a small distance. Shear stress due to turbulence can be written as, minus rho u prime v prime whole bar.

Reynolds shear stress is a crucial concept for understanding how turbulence affects shear stress within a fluid. It describes the average shear stress resulting from velocity fluctuations (u prime and v prime) in the flow. This shear stress is often negative due to the nature of turbulence, indicating that the stresses act in opposing directions. Reynolds provided a mathematical expression for this shear stress, enabling engineers to analyze and predict flow behavior.

Imagine a group of people passing a ball among themselves while blindfolded. The unpredictability in passing the ball introduces random fluctuations in the flow of the game, similar to how turbulence introduces fluctuations in a fluid's velocity. The average effect of these chaotic interactions—like the cumulative stress on the ball during game play—can be understood through Reynolds' formula.

Now, there is a concept of Prandtls mixing length theory. So, turbulence shear stress can be calculated if this thing is known, u prime v prime whole bar is known. Consequently, in 1925 Prandlts introduced the concept of mixing length, which can be utilized to express the shear stress in terms of some measurable quantity.

Prandtl's mixing length theory simplifies the calculation of turbulent shear stress by introducing the idea of mixing length, denoted as lm. This length represents the distance over which fluid particles from one layer can interact with those in another layer, effectively mixing and conveying momentum. By relating shear stress to this mixing length, engineers can apply measurable quantities to compute complex turbulence phenomena.

Consider a classroom where students in the front row pass notes to those in the back. The 'mixing length' corresponds to the distance across the rows; if students in the front can convey a message effectively to those further away, there's a mixing dimension involved. Similarly, in fluid dynamics, mixing length allows for understanding how momentum is transferred across different layers of the fluid.

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 lm can be written as ky.

In his work, Prandtl established that the mixing length increases linearly with distance from a boundary, represented as lm = kappa * y. Here, kappa is a constant (approximately 0.4), known as the von Karman constant. This assumption greatly simplifies calculations for turbulent flows, especially near wall boundaries; the only unknowns become more manageable.

Imagine measuring how far you can throw a paper airplane—if you step further away from a wall and each step allows you to throw it farther, then the distance (y) you've moved directly affects your airplane's performance. In fluid dynamics, the further a particle is from a surface, the more room there is for momentum transfer (mixing).

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