1.6. Turbulent Flow

Turbulent motion is an irregular motion which is associated with a random fluctuation of swirling regions of fluid called eddies. These fluctuations are caused due to disturbances like a roughness of solid surface.

Turbulent flow is characterized by chaotic and irregular fluid motion. In this state, the fluid does not flow in a smooth manner but instead exhibits fluctuations known as eddies. These eddies occur because the flow encounters obstacles, like rough surfaces. The turbulence leads to more energy dissipation compared to smooth laminar flows.

Imagine a calm lake (laminar flow), where you can see straight reflections on the surface. Now picture throwing a stone into it. The ripples that form and disrupt the calm surface represent turbulent flow; the patterns become random and chaotic, similar to eddies in a turbulent fluid.

Important to know is, when the Reynolds number of the flow is less than Reynolds number, which is a critical Reynolds number, for if you remember, in the pipe flow it was 2300, the kinetic energy of the flow cannot sustain the random fluctuations and then this is called the laminar flow. When Reynolds number is more than the critical Reynolds number for example, in pipe flow it was 4000 the kinetic energy of the flow supports the growth of the fluctuation.

The Reynolds number is a dimensionless quantity used to predict flow patterns in different fluid flow situations. Specifically, when the Reynolds number is below a certain critical value (e.g., 2300 for flow in a pipe), the flow is characterized as laminar, meaning it flows in smooth layers. When the Reynolds number exceeds the critical threshold (e.g., 4000), the flow becomes turbulent as the kinetic energy in the system allows for chaotic fluctuations.

Think of a smooth, calm river flowing gently (laminar) and a fast-moving stream that creates rapids and whirlpools (turbulent). The swift stream's ability to throw you around represents the high Reynolds number where liquid can swirl and interact with itself chaotically.

One of the important components of the turbulent flow is the Reynolds experiment. Osborne Reynolds verified the existence of laminar and turbulent flow regimes by injecting dye streaks into the flow in a glass pipe.

Osborne Reynolds conducted an experiment to visually demonstrate the difference between laminar and turbulent flow. By injecting dye into a flowing fluid, he observed how the streaks of dye behaved at various flow speeds. In laminar flow, the dye remained as a smooth line, while in turbulent flow, the dye spread out and became chaotic, showing the mixing and fluctuating nature of turbulent flow.

Imagine stirring cream into coffee. At first, the cream swirls smoothly into the coffee (laminar). As you stir faster, it becomes harder to see distinct lines of cream—it blends and disperses (turbulent). This visual change reflects Reynolds' experiment and illustrates how flow characteristics shift with speed.

One of the important components of the turbulent flow is the Reynolds decomposition. What is that? It is the decomposition of an instantaneous value of a hydrodynamic quantity into time-averaged value and its fluctuation.

Reynolds decomposition separates the instantaneous values of velocity, pressure, and other fluid characteristics into two components: an average value and a fluctuation around that average. This approach helps in analyzing turbulent flow by distinguishing between mean behavior and the random variations that occur over time.

Consider a student's grades over the semester. The average grade represents their usual performance, while individual test scores are the fluctuations around that average. In turbulent flow, the average velocity is like the student's average grade, and the fluctuations are like the ups and downs in their test scores.

If f’ and g’ are any two general fluctuating parameters, we have seen u prime, v prime, w prime and p prime then the following result holds true. So, if we take the average of the fluctuating component it will be 0. If we multiply both fluctuating component and then take the average it will never be 0.

Reynolds conditions state that when averaging fluctuating components of a turbulent flow, the mean of the fluctuations will equal zero. However, when considering the product of two fluctuating components or their derivatives, they generate non-zero values due to their interaction, which leads to additional complexities in turbulent flow equations.

Think of a busy restaurant where customers frequently come and go (the fluctuations). On average, the number of customers sitting down at the same time might be constant, but on a busy night, some tables may have many patrons while others are empty. The averages and interactions (similar to products of fluctuations) need to be understood for managing service efficiently.

--