1.8. Mixing length (lm)

In 1925, Prandtl introduced the concept of mixing length, which can be utilized to express the shear stress in turbulent flow in terms of some measurable quantity.

In 1925, the engineer Prandtl proposed the mixing length concept to understand turbulent flow better. He aimed to find a way to express turbulent shear stress in measurable terms. This approach emerged because calculating turbulent shear stress directly is complex and difficult.

Think of mixing in a kitchen. Just as you might mix ingredients like flour and water to make dough, fluid particles in turbulent flow mix together. Mixing length is like measuring how far these ingredients have blended — it helps us understand how energy or momentum is transferred in the flow.

Mixing length, lm, is the distance between two fluid layers in the vertical direction, so that the bundles of fluid particles from one layer could reach the other layer and mix in the new layer.

The mixing length (lm) is defined as the vertical distance that allows fluid particles from one layer to reach and mix into an adjacent layer. Prandtl illustrated this by assuming there's a specific distance where the mixing happens, crucial for calculating momentum transfer in turbulent flows.

Imagine two layers of ingredients in a salad bowl, one above the other. Mixing length represents how far you need to stir the salad to ensure the dressing reaches and blends all the ingredients below — similar principles apply to fluid layers in turbulent flow!

Prandtl related u' to mixing length lm, stating that u' can be expressed as mixing length lm multiplied by the gradient of the average velocity du/dy.

Prandtl established a mathematical relationship connecting the fluctuating velocity (u') to mixing length (lm) and the velocity gradient (du/dy). This equation helps us quantify the turbulent flow's characteristics in relation to mixing length, making theoretical predictions possible.

Consider a busy street where cars have to merge. The speed of merging cars (u') can be thought of as a reflection of how well the two lanes are mixing. The mixing length (how far back the first lane extends) would influence how quickly cars can enter and mix with the faster-moving lanes.

Substituting findings into Reynolds’ turbulence model yields tau turbulent = rho lm² (du bar/dy).

By substituting the relationships developed into Reynolds' turbulence model, we derive a new equation for turbulent shear stress. This equation expresses the shear stress in terms of known quantities, such as density (rho) and average velocity gradient (du/dy), incorporating mixing length (lm) as an essential factor.

Imagine you are trying to gauge how much force is needed to mix paint. The paint thickness and the distance you can stir before it becomes homogeneous (mixing length) are crucial. This derived equation helps engineers predict the force needed to mix fluids effectively in engineering processes.

Prandtl assumed that the mixing length lm is a linear function of distance y from the wall, where lm can be written as lm = kappa * y.

To simplify the calculation further, Prandtl assumed that mixing length varies linearly with the distance from a boundary. This assumption means that as you move away from a wall, the mixing length increases proportionately, represented mathematically by lm = kappa * y, where kappa is a constant.

Think of how deep water gets stirred in a swimming pool. Near the pool walls, it mixes less than in the middle. The distance from the wall (y) might give us a way to estimate how much energy needs to be applied to achieve good mixing across varying depths.

In turbulent flow, viscous shear stresses exist only near the boundary, while the region is dominated by turbulence, allowing us to approximate total shear stress using derived equations.

In turbulent flow, majority shear stress comes from turbulence rather than viscosity, especially away from the boundary. This means engineers can primarily rely on the derived equations related to turbulent shear stress for most flow conditions, simplifying their analyses.

It's like cooking pasta. When the water is bubbling with turbulence, it's easy for moving water to mix the pasta uniformly. In this case, most of the stirring happens in the middle far away from the pot's edges, allowing us to focus on how the pasta is moving rather than the subtle effects near the edges of the pot.

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