Shear stress in turbulent flow in a pipe

In turbulent flow, the fluid particles move in a random manner, resulting in chaotic motion and mixing. This flow regime is characterized by the presence of eddies and vortices, which cause rapid fluctuations in velocity and pressure within the fluid. Understanding the distribution of shear stress in turbulent flow is important in various engineering applications, such as pipe flow.

Shear stress distribution

The distribution of shear stress in turbulent flow in a pipe depends on the velocity profile across the pipe cross-section. The velocity profile is the variation of velocity with respect to the radial distance from the center of the pipe. In turbulent flow, the velocity profile is typically parabolic in shape, with maximum velocity at the center and decreasing towards the pipe walls.

Maximum shear stress

The shear stress is defined as the force per unit area acting tangentially to the flow direction. In turbulent flow, the maximum shear stress occurs at the boundary, which is the pipe wall. This is because the fluid particles in contact with the wall experience a higher velocity gradient compared to those near the center. The velocity gradient is the change in velocity per unit distance, and it is responsible for the generation of shear stress.

Decrease in shear stress towards the center

As we move from the wall towards the center of the pipe, the velocity gradient decreases. This is due to the parabolic velocity profile, where the velocity near the center is lower compared to the wall. Consequently, the shear stress also decreases linearly from the wall towards the center. This linear decrease in shear stress is a result of the variation in velocity profile and is a characteristic of turbulent flow in a pipe.

Logarithmic decrease in shear stress

Option c) suggests that the shear stress decreases logarithmically towards the center. However, this is not the case in turbulent flow. In the logarithmic velocity profile, the velocity varies logarithmically with respect to the distance from the wall, but the shear stress does not follow the same trend. The shear stress in turbulent flow decreases linearly towards the center, as explained earlier.

Therefore, the correct answer is option b) - the shear stress is maximum at the boundary (wall) and decreases linearly to zero value at the center.