Now Reynolds number for a system is defined as follows
$$Re=\frac{\rho*u*L}{\mu}$$
where $L$ is the characteristic length. I am wondering how this $L$ is decided (in case of flat plate it is $x$ where $x$ is the length of fluid traveled, but in a arbitrary system how do we decide it). In general how do we calculate this characteristic length, ( this doubt arises in Nusselt number as well).
Nondimensional groups such as the Reynolds number do not generally characterise the flow as a whole, but a feature that you choose in the flow. If the flow considered is not an academic problem, you will have several such features, which have different lengths, velocities... So there may be several different relevant Reynolds and Nusselt numbers in your system. See e.g. How do I calculate the Reynolds number in multiphase flows? for an example of such a flow.
If you want a more precise answer, then you should give more detail on the system you are modelling.
