The Reynolds number is defined as the ratio of ´inertial´ forces to viscous forces.
$$ Re = \frac{\text{Inertial Forces}}{\text{Viscous Forces}}$$
Now, viscous forces make sense to me. They are frictional shear forces that come about due to the relative motion of the different layers in a flowing fluid, resulting in different amount of friction, hence, different viscosity values.
However, I am not really sure how to think about the 'inertial force'. This, to me, is somewhat of a dynamic effect since large Re numbers indicate turbulence in most cases, where there is a lot of motion, vortices and eddies. But what exactly is the inertial force and how can it be explained physically?
Answer
Inertial force, as the name implies is the force due to the momentum of the fluid. This is usually expressed in the momentum equation by the term $(\rho v)v$. So, the denser a fluid is, and the higher its velocity, the more momentum (inertia) it has. As in classical mechanics, a force that can counteract or counterbalance this inertial force is the force of friction (shear stress). In the case of fluid flow, this is represented by Newtons law, $\tau_x = \mu \frac{dv}{dy}$. This is only dependent on the viscosity and gradient of velocity. Then, $Re = \frac{\rho v L}{\mu}$, is a measure of which force dominates for a particular flow condition.
The inertial forces are what gives rise to the dynamic pressure. Another way to look at the Reynolds Number is by the ratio of dynamic pressure $\rho u^2$ and shearing stress $μ v/ L$ and can be expressed as $$Re =\frac{\rho u^2} {μ v/ L} = \frac{ u L} {\nu} $$
At very high Reynolds numbers, the motion of the fluid causes eddies to form and give rise to the phenomena of turbulence.
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