Introduction:
The question asks us to determine the distance from the pipe wall at which the local velocity is equal to the average velocity for turbulent flow in a pipe. Turbulent flow is characterized by chaotic and irregular motion of fluid particles.

Explanation:
To solve this problem, we need to understand the concept of turbulent flow and its velocity profile in a pipe.

Turbulent Flow:
Turbulent flow is a type of fluid flow where the velocity of the fluid particles varies chaotically in both magnitude and direction. In a pipe, turbulent flow occurs at high Reynolds numbers (Re > 4000), which is a dimensionless parameter that characterizes the flow regime.

Velocity Profile in Turbulent Flow:
In turbulent flow, the velocity profile across the pipe is not a simple parabolic shape like in laminar flow. Instead, it has a more flattened profile with higher velocities near the center of the pipe and lower velocities near the pipe walls.

Distance from Pipe Wall:
The distance from the pipe wall at which the local velocity is equal to the average velocity can be determined using the concept of the velocity profile in turbulent flow.

The average velocity in a pipe can be calculated using the equation:

Average Velocity = (Maximum Velocity + Minimum Velocity) / 2

In turbulent flow, the maximum velocity occurs at the center of the pipe, while the minimum velocity occurs at the pipe wall. Therefore, the average velocity lies somewhere between the maximum and minimum velocities.

To find the distance from the pipe wall at which the local velocity is equal to the average velocity, we need to determine the location where the velocity profile intersects the average velocity.

Solution:
The correct answer is option 'D' (0.2228 R). This means that the distance from the pipe wall at which the local velocity is equal to the average velocity is approximately 0.2228 times the pipe radius (R).

This value is obtained by analyzing the velocity profile in turbulent flow and finding the location where the local velocity equals the average velocity. The specific calculations for this value are not provided in the question, but it can be determined using experimental data or computational fluid dynamics (CFD) simulations.

Summary:
In summary, the distance from the pipe wall at which the local velocity is equal to the average velocity in turbulent flow is approximately 0.2228 times the pipe radius. This value is obtained by analyzing the velocity profile in turbulent flow and finding the location where the local velocity equals the average velocity.