[freixas]

Please read The Compressibility of Air and Mass Flow, Speed, and Pressure before proceeding.

I introduced some basic fluid dynamics concepts in prior posts. We can apply these in any situation where we have a single air inlet and outlet, such as when we play a single note on the melodica. If we want to understand what happens when more than one note is played, we need to study the physics of branching flows.

Unfortunately, the physics of branching flows gets very complicated, very quickly. I’ll cover what I currently understand now and may expand this topic in the future.

Let’s diagram a very simple branched flow:

Here a single flow is split into two equal flows of half the size. Without thinking too hard, it’s easy to figure out that each branch will get half of the mass flow and that the air’s velocity will be unchanged.

Let’s return to the basic principle that was goes in, comes out. For a branched flow like this we would say: $$\begin{align*}V_1 &= V_2 + V_3\\A_1 v_1 t &= A_2 v_2 t + A_3 v_3 t\\A_1 v_1 &= A_2 v_2 + A_3 v_3\end{align*}$$

What’s interesting is that, even if A₂ and A₃ are half the size of A₁ and we know v₁, it still doesn’t tell us much about v₂ or v₃: $$\begin{align*}A_1 v_1 &= A_2 v_2 + A_3 v_3\\A_1 v_1 &= \frac {1}{2}A_1 v_2 + \frac {1}{2}A_1 v_3\\A_1 v_1 &= \frac {1}{2}A_1 ( v_2 + v_3)\\v_1 &= \frac {1}{2} ( v_2 + v_3)\end{align*}$$

If $v_1 = v_2 = v_3$, the equation would be true, but it would be equally true if $v_2 = 0$ and $v_3 = 2v_1$ and for many other values (v₂ would be 0 if, for example, I capped the end of the A₂ branch).

Let’s look at a slightly more complex branch.

Again, we would guess that, barring any downstream blockages, that while the mass flows would be divided unequally, all flow velocities would remain the same.

Here’s another simple case:

It looks like the mass flows are divided evenly, but the velocities at A₂ and A₃ will be slower (but equal) because the branches have increased the cross-sectional area of the system.

Now that we’re nice and over-confident, let’s try this branch:

Branch A₃ narrows, but only after A₁ splits into A₂ and A₃. Will this affect what happens at the branch? Yes. The degree to which the narrowing will affect the flow at the branch will depend on the distance from the split to where the one branch narrows. And determining the mass flows requires a lot of mathematics.

Why do we care? As I noted earlier, we may want to understand what happens when two notes are played. Here is a rough diagram of a melodica (not to scale) showing this:

This is still a branched flow. The branching is not clear cut, Branch A₂ is closer to the air source than branch A₃. Branch A₂ is also bigger than A₃. There is also no guarantee that air that heads toward A₃ won’t turn around and head back to A₂. Further analysis would get us out of the realm of a background article.

There are two things you should retain from this post:

• The air that goes into a system of branched flows is still exactly the same as the air that goes out, just as when we work with a single tube or pipe.
• Simplistic ways of deciding how the mass flow is divided at a branch will probably not work except for the most basic cases.

[Author]

Posts