A smooth two-dimensional flat plate is exposed to a wind velocity of 100 km/h. If laminar boundary layer exists up to a value or Rex = 3 x 105 , find the maximum distance up to which the laminar boundary layer persists and find its maximum thicknes
You have been given the reynolds number and the velocity of the flow
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To determine the maximum distance up to which the laminar boundary layer persists and its maximum thickness, we need to use some equations related to boundary layer theory and fluid mechanics.
Firstly, we need to calculate the characteristic length scale of the flat plate, which is typically taken as the distance along the plate from the leading edge to the point where the laminar boundary layer transitions to a turbulent boundary layer. This is known as the transition point, and for a smooth flat plate, it occurs at Rex = 5 x 10^5.
Since the given Reynolds number Rex = 3 x 10^5 is less than the transition Reynolds number, we can assume that the entire boundary layer is laminar.
Next, we can use the Blasius solution to estimate the maximum distance up to which the laminar boundary layer persists (known as the boundary layer thickness) and its maximum thickness. According to the Blasius solution, the boundary layer thickness (δ) can be calculated using the following equation:
δ = 5.0x / √(Re_x)
where x is the distance along the flat plate from the leading edge and √(Re_x) is the square root of the Reynolds number at that point.
To find the maximum distance up to which the laminar boundary layer persists, we need to find the point at which the boundary layer thickness becomes comparable to the thickness of the flat plate (i.e., δ ≈ thickness of the flat plate). Assuming that the thickness of the flat plate is negligible compared to the length of the plate, we can use the following approximation:
δ ≈ x
Substituting the given values into the equation for δ, we get:
δ = 5.0x / √(Re_x) δ = 5.0x / √(3 x 10^5)
Setting δ ≈ x, we can solve for x:
x = δ ≈ 5.0x / √(3 x 10^5) x ≈ 0.014 m
Therefore, the maximum distance up to which the laminar boundary layer persists is approximately 0.014 m.
To find the maximum thickness of the boundary layer, we can substitute the value of x into the equation for δ:
δ = 5.0x / √(Re_x) δ = 5.0 x 0.014 / √(3 x 10^5) δ ≈ 0.00084 m
Therefore, the maximum thickness of the laminar boundary layer is approximately 0.00084 m.
To solve this problem, it is necessary to use the Reynolds and boundary layer numbers, as well as formulas describing their relationship.
The Reynolds number determines the mode of flow movement, and is calculated by the formula:
Re = ρvL/μ,
where ρ is the air density, v is the flow velocity, L is the characteristic size of the plate (in our case the plate width), and μ is the viscosity of the air.
To determine the maximum distance up to which the laminar boundary layer persists, we need to find the characteristic size at which the Reynolds number reaches Rex = 3 x 105. We know the wind speed (100 km/h), so we need to determine the air density and its viscosity.
The air density under normal conditions (temperature 20 °C, pressure 1 atm) is about 1.2 kg/m3. The viscosity of air at the same temperature is about 1.8 x 10^-5 Pa*s.
Now we can calculate the characteristic size of the plate L:
L = Rex μ / (ρv) = 3 x 10^5 x 1.8 x 10^-5 / (1.2 x 1000 / 3.6) = 1.35 m.
Thus, the maximum distance up to which the laminar boundary layer is maintained is 1.35 m.
The maximum laminar boundary layer thickness can be calculated using the formula:
δ = 5L / (Rex)^0.5,
where L is the characteristic size of the plate, and Rex is the Reynolds number at which the laminar boundary layer becomes turbulent.
Substituting the values, we get:
δ = 5 x 1.35 / (3 x 10^5)^0.5 = 0.014 m = 1.4 cm.
Thus, the maximum thickness of the laminar boundary layer is 1.4 cm.
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