It is also possible to derive the Grashof number by physical definition of the number as follows:
However, above expression, especially the final part at the right hand side, is slightly different from Grashof number appearing in literature. Following dimensionally correct scale in terms of dynamic viscosity can be used to have the final form.
Writing above scale in Gr gives;
Physical reasoning is helpful to grasp the meaning of the number. On the other hand, following velocity definition can be used as a characteristic velocity value for making certain velocities nondimensional.