Chapter 7, Problem 31P

In an oscillating compressible flow field the volumetric strain rate is not zero, but varies with time following a fluid particle. In Cartesian coordinates we express this as

$\frac{1}{V}\frac{DV}{Dt}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$

Suppose the characteristic speed and characteristic length for a given flow field arc V and L, respectively. Also suppose that/is a characteristic frequency of the oscillation (Fig. P7-31). Define the following dimensionless variables,

$\begin{array}{l}t*=ft,V*=\frac{V}{L^3},x*=\frac{x}{L},y*=\frac{y}{L}\\ z*=\frac{z}{L},u*=\frac{u}{V},v*=\frac{v}{V},\text{and}w\text{*=}\frac{w}{V}\end{array}$

Noudimensionalize the equation and identify any established (named) dimensionless parameters that may appear.