Problem 4: Approximate Solutions of the Navier–Stokes Equations

(1) On a hot day (T = 30°C), a truck moves along the highway at 29.1 m/s. The flat side of the truck is treated as a simple, smooth flat–plate boundary layer, to first approximation. Estimate the \(x\)-location along the plate where the boundary layer begins to transition to turbulence. How far downstream from the beginning of the plate do you expect the boundary layer to become fully turbulent? Give both answers to one significant digit.

(2) A boat moves through water (T = 5°C), at 42.0 km/h. A flat portion of the boat hull is 0.73 m long, and is treated as a simple smooth flat plate boundary layer, to first approximation. Is the boundary layer on this flat part of the hull laminar, transitional, or turbulent? Discuss.

(3) What criteria can you use to determine whether an approximation of the Navier–Stokes equation is appropriate or not? Explain.

(4) In the nondimensionalized incompressible Navier–Stokes equation, there are four nondimensional parameters. Name each one, explain its physical significance (e.g., the ratio of pressure forces to viscous forces), and discuss what it means physically when the parameter is very small or very large.


1 This problem set contains 6 problems, and your final score will be based on the 4 problems on which you earned the highest scores.