$Q = A v$ — Q equals A v — is the most-used relation in the whole of applied fluids. $Q$ is the volumetric flow rate in m³/s, $A$ is the cross-sectional area the flow passes through, in m², and $v$ is the average velocity across that section, in m/s. Average is the operative word: the fluid at the wall is not moving at all and the fluid at the centre is moving fastest, and $v$ is the single figure that carries the same volume as the real profile.

Pipes are round, and a round pipe's area is $A = \dfrac{\pi D^{2}}{4}$, where $D$ is the inside diameter in metres. Substitute it and you get the working form, $v = \dfrac{4Q}{\pi D^{2}}$ — v equals four Q over pi D squared. The 4 is not a fudge factor; it is what is left when you write the area in terms of the diameter instead of the radius. The diameter is squared, and forgetting that is the single most common arithmetic error in this subject. A pipe one size up does not carry a little more — it carries a great deal more, because the bore grows with the square.

Now put two sections of the same closed pipe side by side. Nothing is created, nothing escapes, and water does not compress in any way a plant room would notice — so every second, the same volume must cross both. That is continuity: $A_1 v_1 = A_2 v_2$. The subscripts are a convention worth fixing now and never revisiting: 1 is upstream, 2 is downstream. Solve it for the downstream velocity and the areas turn into diameters, $v_2 = v_1\left(\dfrac{D_1}{D_2}\right)^{2}$ — the π and the 4 cancel clean off both sides. Halve the diameter and the water runs four times as fast. Your thumb on a garden hose has been demonstrating this for years.