The correct answer is option 'A': Logarithmic law.

The logarithmic law describes the velocity distribution in a turbulent boundary layer. It is an empirical relationship that relates the mean velocity of the fluid flow to the distance from the surface of the boundary layer.

The logarithmic law is derived from the concept of the law of the wall, which states that in a turbulent boundary layer, the velocity profile near the wall is linear. However, the velocity profile deviates from linearity as the distance from the wall increases.

Here is an explanation of the logarithmic law and how it describes the velocity distribution in a turbulent boundary layer:

1. Law of the wall:
The law of the wall states that in the inner region of the boundary layer, close to the wall, the velocity profile is linear. This means that the velocity increases linearly with the distance from the wall. This linear relationship holds until a certain distance from the wall, known as the viscous sublayer.

2. Logarithmic law:
Beyond the viscous sublayer, the velocity profile deviates from linearity. The logarithmic law describes this deviation and relates the mean velocity to the distance from the wall. According to the logarithmic law, the mean velocity (U) can be expressed as:
U = (1/k) * ln(y/y0) + B
where U is the mean velocity, k is the von Kármán constant (approximately 0.41), y is the distance from the wall, y0 is a reference distance, and B is a constant.

3. Interpretation:
The logarithmic law suggests that the velocity distribution in a turbulent boundary layer is logarithmic. As the distance from the wall increases, the velocity increases at a decreasing rate. This is because the turbulence in the flow causes the velocity to fluctuate, resulting in a logarithmic distribution.

4. Practical significance:
Understanding the velocity distribution in a turbulent boundary layer is important in many engineering applications. It helps in predicting the drag forces on objects, designing efficient pipelines, and optimizing the performance of aerodynamic surfaces.

In conclusion, the velocity distribution in a turbulent boundary layer follows the logarithmic law. This empirical relationship describes the deviation of the velocity profile from linearity and relates the mean velocity to the distance from the wall.