In this section I will summarise the simplest model for the behaviour of an air-jet instrument like the recorder or flute. It is known as the “jet-drive model”, and my account will closely follow the descriptions by Auvray et al. [1] and Fabre [2]. We can define the geometry of the problem via an idealised recorder mouthpiece, sketched in Fig. 1. Air is blown from a slot of height $h$. The open “mouth” of the recorder has length $w$, after which there is a sharp edge, the “labium”. The edge is a distance $y_0$ below the centre-line of the slot.
The mouth is open to the atmosphere on its upper surface, while its lower surface opens into the tube of the instrument. The pressure just inside the mouth is $p(t)$, and there will be a net volume flow rate into the tube that we will call $v(t)$ as we did in the case of the clarinet mouthpiece back in section 11.3.1. This pressure and volume flow rate are linked via the linear acoustics of the tube, characterised by its input admittance $Y(\omega)=V(\omega)/P(\omega)$ where $P$ and $V$ are the Fourier transforms of $p$ and $v$. (Recall that for an open-ended tube like a recorder, the resonances are marked by peaks of admittance. This contrasts with a closed-ended tube like the mouthpiece end of a clarinet, where resonances appeared as peaks of impedance as we saw in earlier sections.)
Key ingredients of the model are illustrated in cartoon form in Fig. 2. The air emerging from the slot forms a jet, outlined in orange. This jet is assumed to remain laminar over the distance to the labium. The jet is perturbed by the air flow associated with the acoustic response of the pipe, creating a sinuous disturbance that is convected along the jet, growing as it travels. We will denote the downward displacement of the centre-line of the jet (shown as a dotted line) by $\eta(x,t)$ where $x$ is distance measured horizontally from the end of the slot.
When the jet reaches the labium, the sharp edge acts as a “splitter” so that part of the jet flow is diverted into the tube while the rest is deflected outside. This splitting action exerts a force on the fluid, and that force acts as a dipole sound source (as explained in section 11.7.1), exciting the acoustic response of the tube. So the feedback loop is closed: sound pressure creates volume flow via the tube admittance, the flow triggers perturbations on the jet, and those perturbations interact with the labium to create the sound pressure. We need to look carefully at each of these stages, to approximate the physics by a set of equations that can be combined to give a complete simulation model. This is not a simple exercise: each stage involves teasing through tricky physics to find simple approximations that capture enough of the essence to give simulations that are at least qualitatively plausible.
The first step is to understand that if the air had no viscosity, the flow pattern would be completely different. As was explained back in section 11.2.1, any flow without viscosity will be an example of “potential flow”, which among other things means that the streamline pattern is reversible: if the flow went the other way, the streamlines would be exactly the same (with the arrows reversed). Figure 3 gives an indication of how the streamlines would behave as the air comes out of the slot. They fan out in all directions at the end of the slot, a pattern that is easier to imagine if you think of reversing it. If instead of blowing, you sucked air from the mouthpiece, flow would be gathered from all directions into the slot (as happens at the nozzle of your vacuum cleaner).
The biggest difference between Figs. 2 and 3 occurs at the sharp corners at the end of the slot. In the potential flow, there is a singularity at any sharp edge like this: the fluid acceleration and velocity would tend to infinity. Near such a point, even very low viscosity will have a strong influence on flow. The result is boundary layer separation, indicated by the two circles in Fig. 2. This is how the air jet forms. The resulting flow is definitely not reversible: you cannot make a jet by sucking, only by blowing. (The reversed “vacuum cleaner” flow would still involve boundary layer separation at the sharp corners, but this would affect the flow inside the slot rather than outside it.)
There is another effect of viscosity on the flow, illustrated in Fig. 4. This shows how the profile of flow speed across the width of the jet evolves as the jet travels. At the entry to the slot, the flow is driven by uniform pressure in the player’s mouth, so the flow speed is the same at all positions in the jet. This gives a “top hat” profile as illustrated on the left-hand side of the figure.
As the jet travels through the slot, viscous drag on the walls generates a boundary layer on each side of the jet. The boundary layer thickness increases with distance through the slot, as indicated by the dashed lines. By the time the jet reaches the exit of the slot, the speed profile has been rounded off on both sides as shown. If the slot had been longer, or the flow had been slower, the boundary layers would have increased further in thickness, to the extent that they might meet in the middle. The flow profile would then tend towards a parabolic shape known as “Poiseuille flow”.
Once the jet passes the sharp corners so that boundary layer separation occurs, the jet profile continues to evolve in a slightly different way because it is no longer in contact with the hard walls. The influence of viscosity means that it “entrains” some air flow from the surrounding space as indicated in the sketch. By the time the jet reaches the labium (provided it remains laminar) the profile might have turned into a smooth humped shape like the sketch on the right-hand side of the figure. In order to obtain simple model equations, we will shortly use a convenient mathematical function to describe a humped profile of this general form.
One way to describe the profile evolution shown in Fig. 4 is in terms of vorticity, making use of results from section 11.2.1. As soon as the flow enters the slot, it has regions that exhibit shear: differential velocity at different vertical heights in the jet. Shear motion automatically implies local rotation of the fluid, which is described by the vorticity. In the absence of viscosity, we learned in section 11.2.1 that vorticity is conserved along streamlines. But when viscosity is present, the vorticity diffuses, rather like heat. It is this diffusion of vorticity away from the walls of the slot that leads to progressive thickening of the boundary layers. Further diffusion in the free jet after separation leads to a progressively broader and smoother humped profile.
Vorticity is important for two ingredients of the model we are trying to build. The classical analysis of the instability of a single shear layer, or of a pair of mirror-image shear layers defining the sides of a jet, is phrased in terms of vorticity. (You can find an account of this analysis in section 10.3.2 of reference [2].) A small vorticity perturbation tends to travel along the shear layer or the jet, growing as it goes. We will need a description of this travel speed and growth rate.
But we also need to consider how the vorticity perturbation is applied to the jet in the first place. Remember that without viscosity, vorticity is conserved. So if we want to inject a vorticity perturbation associated with the acoustic flow (perpendicular to the jet), we are most likely to be able to achieve this at the point where viscosity has its most significant influence. We already know where this happens: at the sharp corners of the slot exit, where the boundary layer separation occurs.
Putting these two things together, and adding some semi-empirical estimates of functional forms and the values of certain constants, the first governing equations for our model describe the motion of the jet centre-line:
$$\eta(x,t)=e^{\alpha x} \eta_0(t-x/c_p) \tag{1}$$
where the growth rate
$$\alpha \approx \dfrac{0.3}{h}, \tag{2}$$
the propagation speed of disturbances on the jet is
$$c_p \approx 0.4 u_0 , \tag{3}$$
and
$$\eta_0(t)=\dfrac{h}{u_0} u(t) \tag{4}$$
where $u_0$ is the nominal speed of the injected air and $u(t)$ is the acoustic velocity, related to the volume flow rate by
$$v=u w d \tag{5}$$
where $d$ is the width of the mouth in the direction perpendicular to the diagrams in Figs. 1–4 so that $wd$ is the mouth area.
To complete the model, we need a relation between the jet motion, the acoustic velocity, and the pressure inside the mouthpiece. According to Auvray et al. [1] we can approximate this relation in terms of two components. First, there is the component arising from the acoustic source created by the jet interacting with the labium. The suggested relation involves the rate of change of the fraction of the jet volume flow that passes underneath the labium, into the tube. Assuming an idealised expression for the jet flow profile (known as a “Bickley profile”), they give the resulting contribution to the pressure jump across the mouth in the form
$$\Delta p_{source}=\dfrac{\rho_0 \delta_d u_0 b}{w} \dfrac{d}{dt} \left[ \tanh \left( \dfrac{\eta(w,t)-y_0}{b} \right) \right] \tag{6}$$
where
$$b \approx \dfrac{2h}{5} \tag{7}$$
describes the effective half-width of the jet profile, and
$$\delta_d \approx \dfrac{4}{\pi}\sqrt{2 h w} \tag{8}$$
describes the effective separation of a pair of acoustic sources above and below the labium: they combine to constitute the dipole source.
The second contribution to the internal pressure comes from interaction of the acoustic flow with the labium. Figure 5 shows in sketch form how the flow through the mouth would behave in the absence of viscosity. This is a potential flow pattern. The flow accelerates as it approaches the sharp edge, then decelerates symmetrically on the other side. As a result, Bernoulli tells us that the pressure decreases, then increases back to the original level.
Figure 6 shows, again in sketch form, what happens with non-zero viscosity. Just as we saw with the flow from the slot, the sharp edge creates a singularity in the potential flow, so that in reality boundary layer separation occurs at this point. A jet is formed on the downstream side, which then shrinks to a slightly narrower configuration due to the vena contracta effect. Now, the pressure falls as the flow approaches the edge, but it does not recover on the other side. The result is a pressure difference across the mouth, associated with a loss of energy further downstream as the jet degenerates into turbulence. We can estimate the pressure drop using Bernoulli’s principle, just as we did for the clarinet mouthpiece. The result is
$$\Delta p_{loss} = -\dfrac{1}{2} \rho_0 \dfrac{u^2 \mathrm{sign}(u)}{C^2} \tag{9}$$
where $C$ is a vena contracta coefficient with a value of the order of 0.6 as before.
The outside of the mouth is open to the atmosphere at ambient pressure, and as usual we neglect the small contribution from radiated sound and assume that the fluctuating component of pressure is zero there. So the pressure inside the mouthpiece is given by the sum of the two contributions:
$$p(t)=\Delta p_{source} + \Delta p_{loss} . \tag{10}$$
We can learn something interesting from these equations if we look at the threshold case when the pressure and volume flow signals are small and quasi-sinusoidal:
$$p \approx \bar{p}e^{i \omega t}, v \approx \bar{v}e^{i \omega t} . \tag{11}$$
The two complex amplitudes $\bar{p}$ and $\bar{v}$ are linked by
$$\dfrac{\bar{v}}{\bar{p}} = Y(\omega) . \tag{12}$$
We are only going to be interested in the phases of the various complex quantities in this calculation, so we will ignore the effect of the growth factor $\alpha$ and the labium offset $y_0$. The approximate version of equation (1) then says
$$\eta \approx \eta_0(t-\tau) \tag{13}$$
where
$$\eta_0 \approx \dfrac{h}{u_0 w d} \bar{v} e^{i \omega t} \tag{14}$$
and the delay
$$\tau=w/c_p, \tag{15}$$
so that
$$\eta \approx \dfrac{h}{u_0 w d} \bar{v} e^{i \omega (t-\tau)} . \tag{16}$$
Now we turn to equation (6). We are assuming that the amplitude of $\eta$ is small, so we can use the familiar approximation $\tanh \theta \approx \theta$ to simplify the equation to the form
$$\Delta p_{source} \approx \dfrac{\rho_0 \delta_d u_0 b}{w} i \omega \dfrac{\eta}{b} \approx \dfrac{\rho_0 \delta_d u_0 h}{w^2 u_0 d} i \omega \bar{v} e^{i \omega (t-\tau)} . \tag{17}$$
The term $\Delta p_{loss}$ is second order in $\bar{v}$ so we can neglect it and deduce that $p \approx \Delta p_{source}$. Substituting into equation (12) and cancelling the amplitude $\bar{v}$, we obtain something of the form
$$(\mathrm{constants})i Y(\omega) e^{i \omega (t-\tau)} \approx e^{i \omega t} \tag{18}$$
where $(\mathrm{constants})$ denotes a collection of real, positive quantities. In order for the phases of these complex quantities to balance, we must have
$$e^{[i \pi/2 +i \arg(Y) -i \omega \tau]} = 1 \tag{19}$$
where we have used the identity $i = e^{i \pi/2}$, so finally we require
$$\arg(Y) + \pi/2 – \omega \tau = 2m \pi \tag{20}$$
where $m$ could be any positive or negative integer (including zero). If we express this phase balance result in terms of the frequency $f=\omega/2 \pi$, we can deduce an expression for the delay $\tau$ as a fraction of the period length:
$$\tau f= \dfrac{\arg Y(\omega)}{2 \pi} + \dfrac{1}{4} -m . \tag{21}$$
This tells us something important. If our “recorder” is producing a note at a frequency close to one of the peaks of $Y$, then $\arg (Y)$ will be approximately zero: admittance is real and positive near each peak. So the player must adjust their jet speed (and the length $w$ in the case of a transverse flute) in order to achieve a delay which is approximately a quarter-period for $m=0$, or approximately 5/4, 9/4, 13/4,… periods for other values of $m$.
Section 11.8 showed some results of simulations using this model. In order to check the coding, the first case studied was an attempt to duplicate the results shown by Auvray et al. [1] in their Fig. 6(a) and (b). The parameter values were: $w=4 \mathrm{~mm}$, $h=1 \mathrm{~mm}$, $y_0=0.1 \mathrm{~mm}$ and $d=10 \mathrm{~mm}$. The jet speed $u_0$ was varied over the range 1–56 m/s. The input admittance was a three-mode approximation to measurements on a particular note of a recorder: the parameter values were given by Auvray et al., and a plot was shown in Fig. 12 of section 11.8.
In reference [1], the data was extracted from single long simulations in which the jet speed was slowly ramped up or down. Instead, the results shown here use a separate transient for each value of the jet speed, in steps of 0.1 m/s. Each transient was “seeded” with a small non-zero value of the IIR filter representing the lowest mode, in a similar way to results in earlier sections on reed and brass instruments. Each run gave a 1 s length of simulated waveforms at a sampling rate of 100 kHz. These waveforms were post-processed to extract amplitude and frequency information.
Figures 7 and 8 show the results, plotted in the same format as Fig. 6 of Auvray et al. [1]. The horizontal axis is a normalised version of the jet speed, the inverse of a standard dimensionless number known as the Strouhal Number which is often used in flow-induced vibration studies. This inverse Strouhal Number is equal to $u_0/\omega w$, where $\omega$ is the playing frequency (expressed in radians/s). Both figures are gratifyingly similar to the corresponding plots by Auvray et al. [1], although there are some differences of detail which probably arise from the different computational strategy adopted here.
In reference [1], the authors show a corresponding pair of plots computed using the discrete-vortex method. This alternative model shows qualitatively similar behaviour, but is different in a number of quantitative details. The authors suggest that the jet-drive model is most suitable for values of the inverse Strouhal Number above 5, whereas below that value the discrete-vortex model is better.
[1] Roman Auvray, Augustin Ernoult, Benoît Fabre and Pierre-Yves Lagrée, “Time-domain simulation of flute-like instruments: Comparison of jet-drive and discrete-vortex models”, Journal of the Acoustical Society of America 136, 389—400 (2014)
[2] Antoine Chaigne and Jean Kergomard; “Acoustics of musical instruments”, Springer/ASA press (2013): see Chapter 10, “Flute-like instruments”, by Benoît Fabre.