1.7. Prandtl's Mixing Length Theory

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. The shear stress due to turbulence can be written as, minus rho u prime v prime whole bar. This can actually be derived, which we will do at some point in this hydraulic engineering course.

Reynolds shear stress is a critical concept in fluid mechanics that quantifies the effects of turbulence on fluid motion. It is expressed as the product of the density of the fluid (rho) and the average of the product of fluctuating velocity components in different directions (u' and v'). This mathematical expression helps in understanding how turbulence contributes to shear stress in flowing fluids. Reynolds formulated this relationship in the late 19th century, providing a foundation for modern turbulent flow analysis.

Think of a busy highway at rush hour where cars are moving at different speeds and changing lanes unpredictably. The average speed of cars in each lane represents a component of velocity, while the fluctuations in speed as cars weave in and out can be likened to the fluctuating velocity components in Reynolds shear stress. Just as in traffic flow, where the interactions between vehicles cause additional forces (stress), turbulence in a fluid generates complex shear stresses.

Now, there is a concept of Prandtl's mixing length theory. In 1925, Prandtl introduced the concept of mixing length to express shear stress in a measurable way. The mixing length, l_m, is the distance between two vertical fluid layers such that fluid particles can mix together while preserving their momentum along the direction of flow.

Prandtl proposed the mixing length theory to bridge the gap in measuring shear stress due to turbulence. According to this theory, mixing length (l_m) describes how far fluid particles can travel vertically before they mix and distribute their momentum. It serves as a conceptual tool to quantify the contribution of turbulent motion to shear stress. This intuitive approach allows engineers to calculate turbulent shear stress using measurable variables, which is essential in practical applications such as pipe flow.

Imagine a blender mixing ingredients for a smoothie. The distance the blades beat through the mixture can be thought of as the mixing length. Just like in fluid dynamics, where particles mix together to maintain momentum while being blended, the mixing length in a fluid describes how layers of fluid interact and transfer momentum in turbulent flow.

Prandtl related u prime to mixing length lm stating that u prime can be written as mixing length lm multiplied by the gradient of the average velocity: u prime = l_m * (du bar/dy). By substituting this into Reynolds' stress model, it becomes tau turbulence = rho l_m^2 * (du bar/dy)^2.

In this relationship, Prandtl formulated an equation that connects the fluctuating velocity (u') with the mixing length and the average velocity gradient. By substituting this relationship into the expression for turbulent shear stress, it simplifies the problem to a form that emphasizes the importance of the mixing length in calculating shear stress in turbulent flow. The derived expression shows how shear stress grows with the square of the mixing length and the velocity gradient, highlighting the influence of these factors in turbulent conditions.

Consider a basketball player dribbling while running. The rate at which they bounce the ball (rate of change of average velocity) represents the gradient of their motion, while the distance they travel while dribbling could be likened to the mixing length. Just as a player maintains control and transfers energy to the ball within a certain distance while running, the mixing length describes how fluid particles interact within a defined space, affecting the overall flow's shear stress.

Prandtl assumed that the mixing length lm is a linear function of distance y from the wall or any solid boundary: l_m = kappa * y. Here, kappa is known as the von Karman constant and is approximately equal to 0.4.

In this model, Prandtl simplifies the relationship between mixing length and distance from a boundary. By treating mixing length as a linear function of distance y, where kappa is a constant, it allows for easier calculation of mixing length across various flows. This assumption helps in practical applications, where predicting shear stress becomes more manageable because it relies on measurable quantities, significantly aiding in the design of systems involving turbulent flow.

Think of a tree growing taller as it moves away from a fence. The height of the tree (mixing length) increases linearly as the distance from the fence (the boundary) increases. Just like this relationship helps us visualize the tree's growth dynamics, Prandtl's model allows engineers to understand how mixing length varies with distance from the wall in fluid dynamics.

In turbulent flow, viscous shear stresses exist only near the boundary, while most regions are dominated by turbulent shear stress. Thus, the total shear stress can be approximated, neglecting viscous shear stress.

During turbulent flow, the effect of viscosity is primarily felt near the solid boundaries where the fluid can adhere and slow down. However, in the majority of the flowing body, turbulence dominates, leading to a different equation for shear stress that emphasizes the role of turbulence. This simplification is crucial in engineering scenarios where turbulent flow conditions are prevalent, making analysis and predictions more feasible.

Imagine a river flowing swiftly past a rocky riverbank. Near the bank, the water's surface is slower due to friction (viscous shear stress), whereas in the middle of the river, the flow is rapid and chaotic (turbulent shear stress). Just as most of the river's movement is dictated by turbulence away from the banks, most turbulent flows in engineering contexts are dominated by turbulent shear stress rather than viscous effects.

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