1.4. Boussinesq’s Model

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, the shear stress is different from that in laminar flow. Unlike laminar flow where shear stress is solely due to viscosity, turbulent flow has additional shear stress components arising from velocity fluctuations within the fluid. This means that the total shear stress in turbulent flow is significantly higher than in laminar flow, highlighting the impact that turbulence has on fluid movement.

Think of laminar flow as a smooth, orderly dance where every dancer follows their path without deviation. In contrast, turbulent flow is like a lively, chaotic party where people are constantly bumping into one another, creating more excitement and energy — this chaos contributes to greater shear stresses.

Boussinesq’s says as for laminar flow, therefore, the shear stress due to the turbulence component is. 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. Therefore, if we want to write a kinematic eddy viscosity then we write it by epsilon, for example.

In the Boussinesq model, turbulent shear stress can be expressed using a new parameter called eddy viscosity (η). This is analogous to the dynamic viscosity (µ) used in laminar flow. Eddy viscosity accounts for the chaotic and fluctuating motion of turbulent flows and allows us to quantify the resulting shear stress. Additionally, kinematic eddy viscosity (ε) can be defined, which relates to the density of the fluid, further facilitating calculations in turbulent scenarios.

Consider eddy viscosity as how much warmer a room feels when people are moving around inside compared to a quiet, still room. Just like the mixing of warmer and cooler air creates a more dynamic and rich environment, the turbulent motions of fluid particles create a higher 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.

Reynolds shear stress is a fundamental concept in the study of turbulence. It quantifies the additional stress from fluctuations in velocity between different layers of fluid. The formula provided by Reynolds illustrates that this stress can be described as a product of average density (ρ) and the correlation of the fluctuating velocity components (u' and v'). This means that Reynolds shear stress incorporates the effects of turbulence on flow characteristics.

Imagine two layers of water in a lake, with the top layer being affected by wind and the bottom layer being calm. The interaction between these two layers can create stress at their boundary, similar to how Reynolds shear stress quantifies the turbulence between layers. Picture a tug-of-war where one side pulls more erratically — their combined strength has a significant impact on the overall outcome.

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 Prandtls introduced the concept of mixing length, which can be utilized to express the shear stress here, in terms of some measurable quantity.

Prandtl's mixing length theory provides a method for calculating turbulent shear stress by introducing the idea of mixing length (l_m). This mixing length represents the distance over which fluid particles can mix and exchange momentum. By determining this mixing length, we can derive a formula for turbulent shear stress that utilizes measurable quantities like average velocity gradients, thus making the calculations more practical.

Consider mixing two different colors of paint. The distance between the two colors where they blend creates a 'mixing length'. Similarly, in fluid dynamics, the mixing length is where the motion of one layer of fluid impacts another. The smoother the transition (just like a smooth blend of paint), the more accurately we can predict how the fluids will behave.

Prandtl 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, where kappa is known as von Karman constant and it has been found to be equal to 0.4.

Prandtl's assumption that mixing length (l_m) is linearly related to the distance from a boundary (y) simplifies the calculation of shear stress in turbulent flow. By expressing l_m as a product of a constant (kappa, approximately 0.4) and the distance from the wall, we can easily plug values into equations to find shear stress values based on measurable distances from a boundary.

Think of measuring the height of a tree. If we know how long the trunk is (the distance from the ground), we can estimate how high the branches reach by using a proportion or coefficient (like kappa). In turbulence theory, this relationship helps us predict how mixing length varies based on proximity to surfaces.

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