Turbulent Pipe Flow and Roughness Coefficient

Turbulent Pipe Flow:
In fluid mechanics, pipe flow can be classified into two categories: laminar flow and turbulent flow. Laminar flow occurs when the fluid flows smoothly in parallel layers, while turbulent flow is characterized by chaotic and irregular motion of the fluid particles. Turbulent flow is more common in practical situations and is characterized by high velocities and mixing.

Roughness Coefficient:
The roughness coefficient (k) is a parameter used to quantify the roughness of the inner surface of a pipe. It represents the average height of the irregularities on the pipe surface and is typically measured in millimeters. A rougher surface will have a higher roughness coefficient, while a smoother surface will have a lower roughness coefficient.

Laminar Sublayer Thickness:
The laminar sublayer thickness (ẟ) is a characteristic length scale that represents the thickness of the laminar boundary layer near the pipe surface. The laminar sublayer is a region of the fluid flow where the velocity profile is approximately linear, and the flow is predominantly laminar. As we move away from the pipe surface, the flow becomes increasingly turbulent.

Ratio of Roughness Coefficient to Laminar Sublayer Thickness:
The ratio of the roughness coefficient (k) to the laminar sublayer thickness (ẟ) is given by k/ẟ. This ratio is used to determine the relative importance of the pipe roughness on the flow behavior.

Explanation:
The correct answer is option 'C' - smooth. This means that for a turbulent pipe flow, if the ratio of roughness coefficient to laminar sublayer thickness (k/ẟ) is less than one, the pipe is considered smooth.

When the ratio k/ẟ is less than one, it indicates that the roughness of the pipe surface is relatively small compared to the thickness of the laminar sublayer. In such cases, the influence of pipe roughness on the flow behavior is negligible, and the flow can be considered as smooth.

On the other hand, if the ratio k/ẟ is greater than one, it indicates that the roughness of the pipe surface is significant compared to the thickness of the laminar sublayer. In such cases, the flow becomes more affected by the roughness, and the flow behavior transitions from smooth to rough.

Therefore, for a turbulent pipe flow, the ratio of roughness coefficient to laminar sublayer thickness (k/ẟ) is used to determine the roughness category of the pipe. If the ratio is less than one, the pipe is considered smooth.

For smooth pipes with turbulent flow, (k/ẟ) < 0.25.="" />