1.9. Turbulent Flow in Pipes

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. Therefore, the shear stress in total is much larger than the viscous flow because there is shear stress that is associated with turbulence too. So, Boussinesq’s says, the shear stress due to turbulence is given by eta du/dy, where eta is a new coefficient of viscosity called eddy viscosity.

In turbulent flow, the shear stress is influenced by both traditional viscous forces and additional turbulent forces. This is captured in Boussinesq’s model, which introduces a new parameter known as 'eddy viscosity'. Eddy viscosity accounts for the chaotic and irregular movements within the fluid that lead to increased momentum transfer. As a result, the total shear stress is significantly higher in turbulent flows compared to laminar flows, where shear stress is solely a factor of viscosity.

Imagine a crowded dance floor where dancers (fluid particles) are moving smoothly in pairs (laminar flow). When the music tempo increases and everyone starts dancing in a chaotic way, they collide with each other frequently, leading to faster movements across the dance floor. This change in behavior is similar to how turbulence increases shear stress in fluids.

Unlike the dynamic viscosity mu and kinematic viscosity nu, eta and epsilon are not fluid properties. The values of eta and epsilon are dependent on the flow conditions. So, epsilon decreases towards the wall becoming 0 at the wall.

Eddy viscosity (eta) and kinematic eddy viscosity (epsilon) differ from traditional viscosity measures, as they are not constant properties of the fluid; rather, they vary with the flow conditions. Specifically, kinematic eddy viscosity approaches zero as you move closer to a surface (like a pipe wall) because the turbulent fluctuations diminish in those layers due to frictional effects. This implies that the turbulent behavior is most pronounced away from solid boundaries.

Think of a river flowing past a rocky riverbed. Near the rocks, the water doesn't flow as vigorously due to friction (zero turbulence), but in the middle of the river, the water swirls and moves faster. Just like the turbulent velocity increases in the middle of the river, the eddy viscosity is also higher away from the walls of a pipe.

Reynolds in 1886 gave expressions for turbulent shear stress between two fluid layers separated by a small distance. Reynolds shear stress is given by minus rho u' v' whole bar, where u' is the fluctuating velocity component in x direction, and v' is the fluctuating component of velocity in y direction.

Reynolds shear stress represents the effect of turbulence on the momentum transfer between fluid layers. The formula includes fluctuating components of velocity (u' and v') to account for the disordered movements that occur in turbulent flows. This results in a shear stress that is influenced not only by average velocity but also by the variations in velocity. An important characteristic is that experiments show the product of these fluctuations is often negative, suggesting a correlation between the two directions.

Imagine a busy highway where some cars are speeding while others are slow. The 'fluctuating velocities' mimic how traffic behaves: some cars move fast while others slow down or stop. The shear stress here represents the pressure exerted among cars, akin to how Reynolds shear stress is derived from the differences in velocities.

In 1925, Prandtl introduced the concept of mixing length, which can be utilized to express the shear stress. Prandtl assumed that the mixing length lm is a linear function of distance y from the wall.

Prandtl's mixing length theory simplifies the calculation of turbulent shear stress by introducing a variable called mixing length (lm). Mixing length is defined as the average distance over which fluid particles can interact with one another and transfer momentum. Prandtl proposed that this mixing length increases linearly with the distance from the wall, suggesting a predictable relationship between the flow near the wall and the turbulent flow in the bulk of the fluid.

Think of a classroom full of students where they move around to collaborate. Near the walls, students are constrained and cannot move (similar to low mixing length), while in the center, they can move freely and organize (higher mixing length). The concept of mixing length reflects how far students can influence one another's movement based on their position in the room.

If we substitute the expressions related to the mixing length into the Reynolds stress model, we arrive at an equation relating shear stress to average velocity gradients and mixing length.

By substituting Prandtl's expressions for u' related to the mixing length into the Reynolds shear stress equation, we can derive a clear relationship between turbulent shear stress, fluid density, and the velocity gradient of the flow. This relationship allows engineers to calculate shear stress in turbulent flow scenarios more readily, using measurable parameters.

Think of an air conditioner in a room that affects air flow throughout the space. Just as the placement of the air conditioning unit dictates the movement and mixing of air in the room, understanding mixing length helps determine how forces are distributed in turbulent flows.

In turbulent flow, the viscous shear stresses exist only near the boundary, and most of the region is dominated by turbulence. Therefore, we can neglect the viscous shear stress for practical calculations, allowing us to focus on the turbulent component.

In turbulent flow situations, the impact of viscous shear stress is minimal except near boundaries, meaning that the turbulent shear stress is the dominant factor affecting fluid behavior. This allows for simplifications in calculations, where engineers primarily consider turbulent effects in their analyses rather than the relatively minor viscous stresses.

Picture moving through a busy crowd at a concert. The individuals near the edges (viscous stresses) might be barely moving, while the chaotic center (turbulent stresses) is where the majority of the action occurs. For engineers, it's easier to focus on this turbulent middle rather than the still edges.

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