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Chapter 9: Problem 25

An airplane flies at a speed of \(400 \mathrm{mph}\) at an altitude of \(10,000 \mathrm{ft}\). If the boundary layers on the wing surfaces behave as those on a flat plate, estimate the extent of laminar boundary layer flow along the wing. Assume a transitional Reynelds number of \(\mathrm{Re}_{\mathrm{xcr}}=5 \times 10^{5} .\) If the airplane maintains its 400 -nph speed but descends to sea- level elevation, will the portion of the wing covered by a laminar boundary layer increase or decrease compared with its value at \(10,000 \mathrm{ft}\) ? Explain.

Short Answer

Expert verified
The laminar flow extends to about 42 cm from the leading edge at an altitude of 10,000 ft. If the plane descends to sea level while maintaining the same speed, the extent of the laminar flow will decrease.

Step by step solution

01

Calculate the Reynold's number

The Reynold's number \(Re_x\) at any point along the wing can be expressed as \(Re_x = \frac{ρVx}{μ}\) where \(V\) is the velocity of the aircraft, \(ρ\) and \(μ\) are the air density and viscosity at the given altitude and \(x\) is the distance along the plate from the leading edge. This can be rearranged to calculate the distance \(x\) as \(x = \frac{Re_x μ}{ρV}\).
02

Use given Reynolds number for transition

Substituting the given transitional Reynold's number \(Re_{xcr} = 5 \times 10^5\), the viscosity for air at 10,000 ft (about \(μ = 1.8 \times 10^{-5} kg/(m.s)\)) and estimated air density at 10,000 ft (about \(ρ = 1.056 kg/m^3\)), and the speed in SI units \(V = 400 mph = 178.2 m/s\), gives the limit of the laminar flow as \(x_{laminar} = \frac{Re_{xcr} μ}{ρV} ≈ 0.42 m \). Thus the laminar flow extends to about 42 cm from the leading edge.
03

Estimate the effect of change of altitude

If the aircraft descends to sea level while maintaining the same speed, the air density increases (to about \(ρ= 1.225 kg/m^3\) at sea level) while the viscosity remains relatively unchanged. Using these values in the equation for \(x_{laminar}\) shows that the extent of laminar flow will decrease compared to its value at 10,000 ft. This is because the increase in \(ρ\) makes \(Re_x\) reach the transitional value at a lower \(x\), i.e., closer to the leading edge of the wing.

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