$($b$)$ Using dimensional analysis, show that the time and velocity scale of smaller turbulent eddies $(1$ and $9)$ respectively are: t $=($nu$/$epsilon$)^(1/2)$ and vartheta $=($epsilon$*$nu$)^(1/4)$

where, & is the rate of dissipation of turbulent kinetic energy and v is the kinematic viscosity.

Further show that the ratios of time and velocity scales of smaller and larger eddies are:

I$/$tau $=$ R $*$ e$\_\{$j$\}^(-1/2)$

$9/$u $=$ R $*$ e$\_\{$i$\}^(1/4)$

where r is the time scale of larger eddies, R$*$e$\_\{1\}$ is the Reynolds number based on the length scale

$(1)$ and velocity scale

$($u$)$ of the larger eddies