Boundary Layer and Turbulent Flow
The boundary layer is a thin layer of fluid that forms on the surface of an object in contact with a fluid. The flow of the fluid in the boundary layer is affected by the viscosity of the fluid. When the velocity of the fluid is low, the flow in the boundary layer is laminar, i.e., the fluid flows smoothly in parallel layers. However, when the velocity of the fluid is high, the flow in the boundary layer becomes turbulent, i.e., the fluid flows in a chaotic and unpredictable manner.

Reynolds Number and Turbulent Flow
The transition from laminar to turbulent flow in the boundary layer is determined by the Reynolds number. The Reynolds number is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in the fluid. When the Reynolds number is below a certain critical value, the flow in the boundary layer is laminar. However, when the Reynolds number exceeds the critical value, the flow becomes turbulent.

Determination of Turbulent Length
To determine the length at which the boundary layer becomes turbulent, we need to calculate the Reynolds number. The Reynolds number can be calculated using the formula:

Re = (ρ * v * L) / μ

where ρ is the density of the fluid, v is the velocity of the fluid, L is the length of the plate, and μ is the viscosity of the fluid.

Substituting the given values, we get:

Re = (1200 * 0.3 * L) / (2 * 10^-3)

Simplifying the equation, we get:

Re = 180000L

Critical Reynolds Number
The critical Reynolds number for the transition from laminar to turbulent flow in a flat plate is approximately 5 × 10^5. Therefore, the length at which the boundary layer becomes turbulent can be calculated by equating the Reynolds number to the critical Reynolds number:

180000L = 5 × 10^5

Solving the equation for L, we get:

L = 2.78 meters

Therefore, the length of the plate at which the boundary layer becomes turbulent is approximately 2.78 meters.

2.78