# Is there a basis for naming these empirical fluid model coefficients in terms of flow regime?

Given any flow restrictive device (e.g. pipe, orifice, screen, etc.) one can measure data as the pressure drop across the device relative to the flow rate through the device. And from this data one can generally closely fit the polynomial

$$\Delta P = K_2Q^2+K_1Q$$

where $\Delta P$ is the pressure drop and $Q$ is the volumetric flow rate

Furthermore there have been people that (I believe) mis-appropriately attribute these factors to concepts involving Reynolds number, calling, for example $K_2$ the 'turbulent' flow factor and $K_1$ the 'laminar' flow factor.

While it is true that $K_1$ dominates the pressure drop at low flow, and $K_2$ dominates at high flow, I don't believe there is any basis to attribute these empirical factors to laminar and turbulent flow characteristics.

Am I maybe missing something? Is there anything in the analysis of fluid dynamics that might support such a naming convention or otherwise refute it?

Suppose geometry of your flow restrictive device is fixed, so that we can take any one geometric dimension of the device (which is relevant to the flow) as your length scale, call it $d$. Assuming fluid properties remain constant, changing flow $Q$ is equivalent to changing average speed $U$ (at any particular cross-section), which we shall take as our velocity scale. Now pressure difference $\Delta p$ required to create this flow depends on fluid properties ($\rho,\mu$), geometric parameters (characterized by $d$), and flow rate (characterized by $U$). Dimensional analysis then gives
$\frac{\Delta p}{\rho U^2}=f(Re)$, where $f$ is some function of Reynolds number.
In a turbulent flow it so happens that pressure drop is determined primarily by processes other than viscous dissipation (mixing of momentum by advection of fluid elements rather than by molecular diffusion), so to a good approximation viscosity is unimportant, in which case there cannot be dependence on $Re$ i.e. $\frac{\Delta p}{\rho U^2} \approx$ constant, which means for a turbulent flow, $\Delta p~\alpha~ U^2~\alpha~Q^2$. I think those who call $K_2$ turbulent flow factor are drawing upon this resemblance. But it is only a resemblence, may be it helps the person remember which factor is which.
If we write out $\Delta p$ as a polynomial series of $Q$, then for small enough $Q$, we may neglect higher powers of $Q$ and we thus have $\Delta p~\alpha~Q$. This is laminar flow regime, and calling $K_1$ laminar flow factor is again only by way of resemblance.