ME 634-001
HOMEWORK #2
Spring 2016
1. Demonstrate Galilean invariance of the 2-D momentum equations of the N.–S. system:
ut + uux + vuy = −px + ν∆u ,
vt + uvx + vvy = −py + ν∆v .
Carry this out with a general, but constant, coordinate velocity vector V ∈ R2 such
that V = (uG , v G )T .
(a) Write the form of the complete Galilean transformation, and inverse transformation, for this case.
(b) Give the form that must be taken by the transformed equations.
(c) Show that the original (untransformed) equations given above can be recovered
by applying the inverse Galilean transformation.
2. In Prob. 1, the N.–S. equations are formally time dependent. Now investigate validity
of Galilean invariance in the steady-state case. (Hint: keep in mind that, in general, whether a fluid flow is time dependent depends on the coordinate system chosen
to represent it. What are the implications of this with respect to transformed and
untransformed equations. Use this information to guide your analysis.)
3. It is often claimed that strain rate in a turbulent flow is irrotational. Investigate
the validity (or lack thereof, or conditions required for validity) of this assertion. Hint:
recall that for a velocity field U to be irrotational it must be curl free (i.e., ∇×U = 0).
Apply this to the definition of strain rate (possibly in usual matrix-vector notation
instead of Cartesian tensor notation) given in the lecture notes.
4. Derive the Galerkin approximation for solving Burgers’ equation,
ut + (u2 )x = νuxx .
Employ a basis set {ϕk (x)}∞
k0 having differentiation properties analogous to those of the
complex exponential, but without the imaginary i in the argument. Now restrict attention to a single wavenumber, and analytically solve the differential equation. Discuss
the mathematical properties of the solution, and, in particular, note to what extent
these do, or do not, mimic physics of the Navier–Stoke equations.