1.5. Eddy Viscosity

In 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 in viscous flow.

Eddy viscosity refers to the additional shear stress created by turbulence in a fluid flow, which is not present in laminar flow. In laminar flow, the shear stress is primarily due to the viscous properties of the fluid. However, in turbulent flow, due to chaotic fluctuations and mixing of fluid particles, there is an increase in shear stress. This leads to a higher total shear stress compared to that in laminar conditions, making the analysis of turbulent flows more complex.

Consider a calm river where the water flows smoothly (laminar flow) compared to a turbulent river section filled with rapids. In the calm section, the water flows steadily and evenly, resulting in lower shear stress. In contrast, in the rapids, the chaotic movement of water increases the shear stress due to turbulence, similar to how eddies form in a turbulent flow.

Eddy viscosity is represented by a new coefficient of viscosity, denoted as eta (η), which is different from the dynamic viscosity (μ) of fluids.

Eddy viscosity quantifies the effect of turbulence on the flow properties of a fluid. It is not a fixed property of the fluid; instead, it varies depending on flow conditions. Unlike the dynamic viscosity, which is a characteristic of the fluid itself, eddy viscosity changes with the flow's velocity and distance from the wall. This variability is crucial in understanding turbulent flows in engineering applications.

Think of a stirred pot of soup versus a pot left still. In the stirred pot, the motion creates eddies and turbulence, while the still soup simply flows without mixing. The dynamic viscosity of the soup remains unchanged whether stirred or still, but the eddies formed when stirring affect how the soup mixes, illustrating how eddy viscosity works in a real-world scenario.

In laminar flow, the value of eta is effectively zero, as it is related to turbulent viscosity, while the viscosity mu for laminar flow is typically around 10^(-3) Pascal seconds.

In laminar flow, the fluid particles move in parallel layers without disruption, making the contribution to shear stress solely dependent on classic viscosity (μ). In contrast, when considering turbulent flow, the turbulent effects become significant and require the introduction of eddy viscosity (η), which is not applicable in laminar scenarios. Thus, when flow transitions from laminar to turbulent, the factors affecting viscous flow calculations change drastically.

Consider pouring honey versus water. When you pour honey slowly, it moves smoothly (laminar flow), where only its viscosity plays a role. When the water is stirred rapidly, it starts to swirl and form eddies, and the internal forces change significantly (turbulent flow). In this way, the different viscosities give a practical understanding of laminar versus turbulent contexts.

Eddy viscosity values depend on flow conditions and decrease towards the wall, reaching zero at the wall.

Eddy viscosity is dynamic and depends heavily on how the fluid interacts with surfaces as well as the overall flow pattern. As you approach the wall of a channel or pipe, the turbulence lessens due to the no-slip condition (the fluid at the wall has zero velocity), leading to a decrease in eddy viscosity until it is effectively zero at the boundary. Understanding this profile can help in designing more efficient fluid systems and predicting flows in various engineering contexts.

Imagine how rougher surfaces like gravel or a smooth surface like a sheet of glass affect how air flows over them. Near a smooth surface, air moves with less turbulence and fewer eddies compared to when it interacts with a gravelly surface, illustrating how physical boundaries can impact fluid behavior.

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