Answer / Explanation
To properly answer this question, we will be making few assumptions in respect to the question to make the learning process easier.
These assumptions are:
1) We assume that the flow is steady and in-compressible.
2) We assume that the critical Reynolds number is Recr = 5 × 10⁵
3) The surface of the plate is smooth.
We also assume according to the question that the properties, density and dynamic viscosity of water at 1 atm and 25°C are ρ = 997 kg/m3 and
μ = 0.891 × 10⁻³ kg/m⋅s
We move forward by representing the question in the diagram attached below:
On analyzing the diagram attached below, we see that the distance from the leading edge of the plate where the flow becomes turbulent is the distance xcr where the Reynolds number becomes equal to the critical Reynolds number,
Therefore:
Recalling the Reynolds equation, we have
Recr = ρ VXcr / μ
If we go ahead to make Xcr the subject of the formula, we have:
Xcr = μRecr / ρV
Now if we insert the values representing the parameter in the equation, we have:
Xcr = ( 0.891 x 10 ⁻³kg/m·s) (5 × 10⁵) / (997 kg/m3 ) (8m/s)
calculating the above further, we derive:0.056 meters
If we convert to centimeter, we then have 5.6 cm.
Thus,
The thickness of the boundary layer at that location is obtained by substituting this value of x into the laminar boundary layer thickness relation. Therefore, we have:
δcr = 5x / Reₓ⁰.⁵ = δcr = 5xcr / Reₓ⁰.⁵
= 5 (0.56m / 5 × 10⁵)⁰.⁵
= 0.00040 m = 0.4 mm
In respect to the answer gotten for the thickness of the boundary above, we can now say that the flow becomes turbulent after about 5 cm from the leading edge of the plate and the thickness of the boundary layer at that location is 0.4 mm. Also, from the calculation thus far, we can clearly state that when the flow becomes turbulent, the boundary layer thickness starts to increase, and the value of its thickness can be determined from the boundary layer thickness relation for turbulent flow.
