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This assignment is devoted to the boundary layer developing when a uniform flow impinges on a half plane. In Workshop 5, we approximated the Navier-Stokes equation and divergence- free conditions by

ud_xu + wd_zu = d_zzu, d_xu + d_zw = 0, (1)

where we use scaled (non-dimensional) variables. We showed that the solution for x > 0 and z > 0 can be found in the form

For this assignment, you will write a code that solves (2) numerically and use it to:

1. Illustrate the behaviour of the solutions u(x, z) and w(x, z) as functions of x and z.

2. Obtain an explicit formula for the (non-dimensional) boundary layer thickness, defined

广8

8(x) = (1 — u(x,z)) dz.

70

3. Obtain an explicit a formula for the (non-dimensional) viscous stress at the wall

= dzu(x, z)|z=0.

(Each formula involves a constant that you can estimate from the numerical solution for f (n).)

Hints:

• The Python function scipy.integrate.solv^bvp solves boundary-value problems such as (3). You can use this or any other similar function.

• You will need to rewrite (3) as a system of first-order ODEs.

• For the numerical solution, the boundary condition lim₇T8 f (n) = 1 needs to be imposed at a large, finite 牛 I found that imposing f (10) = 1 works well.

• Boundary-value problems solvers are iterative and need an initial guess for the solution. I found that f (n) = f (n) = f (n) = e^—n is a suitable initial guess.

• Your solution for u = f^!(n) should look like the figure below.