fluid dynamics - Intuitive meaning of a special case of the Bernoulli equation

In the Bernoulli equation, if $h$ equals zero, it reduces to $$P_1+\frac12\rho v_1^2 = P_2+\frac12\rho v_2^2$$

The equation does not have an intuitive meaning other than the fact that it is a bare mathematical truth.

I have tried to interpret it in the following manner: The sum of kinetic energy per unit volume and pressure exerted by flowing fluid in a horizontal pipe is constant in the direction of flow. Yet, I still can't understand how (pressure) + (kinetic energy per unit volume) is a conserved quantity.

Please give your views on the same.

Answer

The fact that the Bernoulli equation contains the kinetic and potential energy density (energies per unit volume) terms should strongly suggest to you that the conserved quantity is actually energy. Seeing the pressure as an energy term is a little bit trickier, but really is easiest to see from your units: $$ \left[\,{p}\,\right]=\frac{\rm N}{\rm A}=\frac{\rm N\cdot m}{\rm A\cdot m}=\frac{\rm J}{\rm V} $$ which is the same unit as energy density.
NB: Torque and energy have the same units, but these are definitely not the same thing, so you cannot necessarily equate things based on their units. In this particular case, the two items (pressure & kinetic energy density) are equated, but not due to units (instead the physics), meaning that pressure can be viewed as an energy density.

A few instances that show pressure as proportional to an energy density are:

And probably a few other places that I am neglecting at the moment.

Further, in the Eulerian framework, the energy conservation equation takes the form $$ \frac{\partial E}{\partial t}+\nabla\cdot\left(\left[E+p\right]\mathbf u\right)=0 $$ where $E=\rho e+\frac12\rho\left(\mathbf u\cdot\mathbf u\right)$ is the total energy per unit volume (total energy density), so the pressure must be an energy term to be added to the energy density (as one cannot simply add unlike quantities).