22.7. Velocity Distribution and Shear Stress

Now let us come back to very interesting experiments what it happened in 1930s... energy losses when you have a pipe length of L with the informations about the roughness, informations about flow Reynolds numbers.

In the 1930s, Professor Nikuradse conducted significant experiments on pipe flow, which focused on how flow characteristics change due to pipe roughness. In these experiments, he introduced roughness to the inner surface of pipes using materials like sand grains. This roughness influences the velocity distribution of the fluid as it flows and leads to energy losses in the system. Nikuradse's work established fundamental relationships between roughness, flow type (laminar or turbulent), Reynolds number, wall shear stress, and friction factors, which are crucial for understanding and calculating fluid flow in pipes.

Imagine walking on a smooth surface versus a rough surface. On a smooth surface, you can glide easily, while on a rough surface, you have to put in more energy to move forward due to friction. This analogy helps understand how pipe roughness impacts fluid flow – just like the roughness affects your movement, it disrupts the fluid's flow, causing energy losses.

Now let us come back to the noncircular conduit... hydraulic diameter as equal to the geometric diameters.

In noncircular conduits, finding the hydraulic diameter is essential for analyzing fluid flow. The hydraulic diameter is calculated using the formula: diameter = 4 × (Area / Wetted Perimeter). The wet perimeter includes only the portion of the perimeter that is in contact with the fluid. For circular pipes fully submerged in fluid, the hydraulic diameter equals the physical diameter of the pipe. For pipes that are not fully submerged or that have a different shape (like rectangular or triangular), the hydraulic diameter gives a single representative value to use in calculations, simplifying the analysis of flow behavior.

Think of a garden hose versus a square tube filled with water. The hydraulic diameter helps us understand how water flows through both shapes, allowing engineers to determine how much water can flow through them effectively, just like knowing how much water can flow through a narrow vs. a wide hose.

Now if you look it that if you have the turbulent flow... will have uniform, not the maximum at the midpoint as you got in case of laminar flow.

When examining turbulent flow through noncircular pipes, the wall shear stress distribution does not follow the same pattern as it does in laminar flow. In turbulent flow, the maximum wall shear stress occurs along the sides of the pipe edges rather than at the midpoint, as in laminar flow. This change is due to the chaotic nature of turbulent flow, which mixes fluid layers more efficiently and allows for a more uniform distribution of velocity at the wall. Understanding these differences is crucial for correctly calculating the forces and energy losses in different pipe shapes.

Imagine mixing a fruit smoothie; when you blend different fruits, they mix chaotically, creating a uniform taste throughout the drink. Similarly, turbulent flow mixes the fluid more thoroughly, resulting in more even shear stress around the pipe wall compared to the orderly flow of a laminar smoothie where the fruits stay in distinct layers.

But when you look it what could be the shear stress that is what is the wall shear stress... giving the what could be the wall stress for a circular pipe.

Wall shear stress is an important characteristic in fluid mechanics, especially in turbulent flow. It is the stress exerted by the fluid on the wall of the pipe. This stress can be quantified using empirical relationships derived from experimental data, considering factors such as average velocity and hydraulic diameter. The governing equations highlight the dependence of wall shear stress on variables like flow velocity and kinematic viscosity. Understanding these relationships allows engineers to predict how much force is exerted on the pipe walls, which is critical for design and material selection in piping systems.

Consider rubbing your hand against a flowing river; the harder you try to keep your hand flat against the flow, the more force you feel pushing against it. Similarly, the wall shear stress is like that force. Engineers need to calculate this to ensure pipes can withstand the pressures and forces exerted by the flowing fluid without failure.

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