Regime change in the recorder: Using Navier–Stokes modeling to design a better instrument

Conceptually the notes produced by a recorder are simple to understand. Gently blowing a recorder produces a musical tone whose fundamental frequency corresponds to a wavelength that is (apart from end corrections) twice the length of the instrument; i.e., a standing pressure wave inside the instrument with nodes at both ends. Increasing the blowing speed eventually causes the tone to abruptly increase (i.e., “jump”) by an octave corresponding to a wavelength equal to the length of the instrument. This jump is often referred to as regime change. While it is straightforward to explain why notes that correspond to these standing waves are preferred, it is not straightforward to give a quantitative explanation of regime change and to predict the blowing speed at which it occurs, other than to note that nonlinearities likely play a role.

The importance of regime change in the recorder was brought to our attention in discussions with recorder players. They noted that regime change seems to occur more readily, i.e., at a lower blowing speed, in bass recorders than in soprano or sopranino instruments.^1,2 From the player's perspective, the fact that the note being played jumps an octave at a low blowing speed could reduce the dynamic range available with the lowest notes and thus limit the musical possibilities with such instruments. In this paper we study regime change in the recorder using a combination of Navier–Stokes modeling and experiments with real recorders. We first establish, through modeling, a connection between certain physical dimensions of a recorder and the blowing speed at which regime change occurs, showing that a simplified model for a bass recorder does indeed exhibit regime change at a lower blowing speed as compared to a sopranino instrument. Modeling results for the air flow pattern near the labium are used to suggest how changes in the geometry in that region might affect the behavior. Those suggested changes are then explored through modeling, and it is found that they do indeed cause the regime change threshold to shift back to higher blowing speeds. To validate these results, we have used 3D printing to make recorders with the geometries explored in the modeling studies and find behavior that agrees well with the modeling predictions. Specifically, our modified bass-style recorders have increased blowing speed thresholds for regime change in quantitative agreement with the modeling. This increase in the blowing speed threshold extends the range of blowing speeds and hence the dynamic range accessible to the player for the lowest notes of these instruments, suggesting also that our new recorder design may be worth pursuing by instrument makers.

The Navier–Stokes equations in the limit that applies to the air flow in a recorder (a viscous, compressible fluid at low Mach number, taking the ideal gas equation of state and assuming adiabatic conditions) can be written as^3,4

Here, we denote the components of the velocity along x, y, and z as v_x, v_y, and v_z, respectively, the density as ρ, the speed of sound by c, and the kinematic viscosity by ν. Variations in pressure from the background value (denoted $p=P−P0$⁠, where P₀ is the pressure when the blowing speed is zero) are related to the corresponding variations in the density (⁠$ρ−ρ0$⁠, where ρ₀ is the density when the blowing speed is zero) by

We use the MacCormack method,⁵ an explicit, finite-difference time-domain algorithm, to solve the NS equations on a nonuniform Cartesian grid (like that described in, e.g., Ref. 6) The grid has it is smallest spacings (typically 0.1 mm) inside and near the surface of the instrument, as needed to capture the detailed motion of the air jet near the labium. The grid spacings then increase smoothly as one moves away from the instrument, where the important length scales increase and approach the wavelength of sound. Using such a grid reduces the total number of grid elements in the calculation, thus reducing the computational time (by typically a factor of 2 or 3 in our work) as compared to what would be required if a uniform grid were used. More details of our approach to using the Navier–Stokes equations to model the recorder have been given previously.^7,8

The geometry of one of the models studied in detail previously⁸ will be central to our work on regime change and is shown in Fig. 1(a). This model is shorter than any of the instruments in the real family of recorders; it has a resonator tube of length $L=121 mm$ with a cross section that is square with $h=w=10.0 mm$⁠, where h is the height and w is the width of the resonator in Fig. 1(a). This rectangular cross section makes the numerics with a Cartesian grid simpler and also allows us to study how changes in h and w separately affect the behavior (although we focus on variations in h in the present paper). This smaller resonator length was chosen to reduce the computational time required for the simulations. We will refer to this as a sopranino-style recorder even though it is approximately half the length of a true sopranino recorder, to emphasize that it produces notes at the high end of the musical range for the recorder family. In the modeling calculations the instrument is “blown” by imposing flow at a specified speed in the entry half of the channel (on the left in Fig. 1), with a spatial dependence that is constant along the x direction and follows Poiseuille's law in the transverse directions. The flow speed on the central axis of the channel and along the x direction in Fig. 1 is denoted by u.

The instrument geometry in Fig. 1(a) is quite simple but could certainly be made and played (as we have confirmed; see below). For our study of regime change we need for comparison a model of a bass recorder. A straightforward way to design such a model would be to copy the geometry of an actual bass recorder. However, straightforward modeling of this larger instrument would lead to a much larger number of computational grid elements and hence require a much greater computational time, and would not be feasible with computers available to us at present. Fortunately, we were able to devise an alternate model of the bass recorder that appears to exhibit similar regime change behavior.

That alternative model was motivated by the work of Blanc et al.,⁹ who have given a very useful summary of how the dimensions of various baroque recorders vary in spanning from the sopranino to the bass instruments. A key finding from the results of Blanc et al. is that most of the physical dimensions vary approximately in proportion to the length L of the recorder. Since this length is inversely proportional to the fundamental frequency of the lowest tone produced by the instrument, this means that the cross-sectional area of the resonator for a recorder would, if this scaling were followed, increase by a factor of 4 when the range is extended an octave lower. Interestingly, Blanc et al. observed that an exception to this scaling “rule” is found for the bass recorders; for those instruments the cross section is much smaller than would be expected from the scaling observed for the other physical dimensions. In other words, dimensions such as h and w in Fig. 1 are not as large as would have been expected.

Another way to express the dimensional scaling observed by Blanc et al. is that the ratio h/L is approximately constant, except in the bass recorders for which this ratio is smaller than for the other recorders. For this reason, in previous work one of us¹⁰ studied the recorder geometry in Fig. 1(b); the physical dimensions are similar to the sopranino-style recorder in part (a) of the figure except that the height h_b of the resonator is reduced by a factor of 2. The idea was that by keeping the length L fixed, this reduction in the height of the recorder would make the ratio $hb/L$ smaller and therefore make such a model a good proxy for a true bass recorder. Of course, it is still necessary for us to demonstrate that such a recorder model, with the same length L and a smaller height h_b, exhibits regime change behavior like that of a true bass recorder. The results from previous work¹⁰ along with the results presented below do indeed confirm this. The recorder geometry in Fig. 1(b) will therefore be referred to as a “bass-style” recorder and in our modeling it has dimensions $L=121 mm, w=10.0 mm$⁠, and $hb=5.0 mm$⁠.

Figure 1(c) shows the third instrument for which modeling results will be shown below. It is very similar to the bass-style recorder with the same values of L, w, and h_b shown in part (b) of the figure, but with a cut-out (“trench”) region in the bottom of the resonator beneath the labium. The motivation for this of the modification will be discussed below. The recorder in Fig. 1(c) will be referred to as a “modified bass-style” recorder.

The behavior observed for the sopranino-style model was similar to that found previously.⁸ Results for the sound pressure outside the instrument (i.e., at the location of a hypothetical listener) are given in Fig. 2 which shows the steady state behavior. These results are typical for low to moderate blowing speeds. The sound pressure is seen to be approximately sinusoidal; this is confirmed by the power spectrum [Fig. 3(a)], with a dominant spectral component having a frequency of $f1≈1200 Hz$⁠. This value of f₁ is consistent with the length L of the instrument along with the expected end corrections, and is also the value found in our experiments as will be seen below. The expected position of the second harmonic f₂ is indicated; it is seen that the power at f₂ is only slightly above the noise level, and nearly negligible on this scale.

Figure 3(b) shows the spectrum for the tone produced by the bass-style recorder model at the same blowing speed as used to obtain the results with the sopranino-style instrument in part (a) of the figure. The dominant spectral component again has a frequency near 1200 Hz, which is not surprising since this model has the same length as the sopranino-style model. However, the second harmonic now has a power that is only slightly smaller than the first harmonic. To understand the implications of these two spectra, we first recall that as the blowing speed is increased starting from some low value, the relative power in the second harmonic will increase smoothly until it is equal to that of the fundamental at the regime change threshold (see, e.g., Ref. 8) Hence, the fact that the fundamental and second harmonic are close in magnitude in Fig. 3(b) shows that this case is very close to the regime change threshold, while the sopranino model [Fig. 3(a)] is far below that threshold.

We will present further results below that show the regime change behavior in more detail, but first we consider how one might explain the difference in the thresholds for regime change of the two models.

Figure 4 shows images of the flow patterns for the sopranino-style recorder model in part (a) of the figure, and for the bass-style model in part (b). For both models we show images at approximately equally spaced times during one oscillation cycle and for the same blowing speed (⁠$u=28 m/s$⁠) as considered in Fig. 3. For both models the air emerges from the channel as a jet and impinges on the labium. For the sopranino-style model [part (a)] the air jet oscillates above and below the labium, spending roughly equal time above and below, with one oscillation each period of the fundamental frequency. In addition, a vortex appears to form beneath the window and labium, and the flow speed directly beneath the channel exit appears to be strongest when the air jet is moving upward [most notably in panel V of Fig. 4(a)], suggesting that the flow associated with this vortex is the mechanism by which the sound field in the resonator interacts with the air jet, causing the motion of the jet to be synchronized with the resonant frequency of the resonator. This is the same behavior reported previously for this model geometry.⁸ 

The behaviors seen in Fig. 4 for the sopranino-style and bass-style models are distinctly different. While in both cases the air jet oscillates at the fundamental frequency of the resonator, the oscillation amplitude of the air jet is more robust, i.e., it extends much higher above the labium for the sopranino-style model. In particular, the air jet in the bass-style model never reaches a point above the height of the labium. In addition, while a vortex seems to form beneath the labium in both cases, the speed of the vortex beneath the channel exit is qualitatively stronger in the sopranino-style model. This suggests (but certainly does not prove) that the smaller resonator height acts to suppress the vortex which in turn changes the coupling between the sound field and the air jet in the bass-style model.

While this comparison of the flow patterns in Figs. 4(a) and 4(b) is admittedly very qualitative, it motivated us to investigate various design changes in the bass-style model. Specifically, we were curious if changes to the geometry near the channel and labium that allow a larger vortex to form beneath the labium could alter the coupling to the air jet and hence affect the regime change behavior. We therefore investigated a number of different model geometries in which the region beneath the labium was made deeper, as shown in Fig. 1(c). The extent of this extra depth along both the x and y directions in Fig. 1(c) was varied over a wide range and we did indeed find that this affects the behavior. For the many models studied, the largest effect on regime change was found with the geometry shown to scale in Fig. 1(c); we will refer to this as a “modified bass-style” recorder. Results for the flow pattern in the mouthpiece of the modified bass-style model are shown in Fig. 4(c), obtained again at a blowing speed of $u=28 m/s$⁠. Comparing to the results in Figs. 4(a) and 4(b) we see that the air jet reaches points well above the labium, in a manner very similar to that found for the sopranino-style model [part (a) of the figure], in sharp contrast to the behavior of the bass-style model [in part (b)]. Results for the power spectrum for the modified bass-style model, again at $u=28 m/s$⁠, are shown in Fig. 3(c). Here, the power at the fundamental frequency, which is again near 1200 Hz, is much larger than at the second harmonic, as was also found for the sopranino model at the same blowing speed [shown in part (a) of Fig. 3]. This shows that the regime change threshold is now at a much larger value of blowing speed for the modified bass-style model than for the bass-style instrument [part (b) of the figure], and confirms that the change in the mouthpiece geometry in Fig. 1(c) has indeed greatly affected the regime change behavior.

Figure 5 collects results for all three recorder geometries, showing the frequency of the dominant mode (i.e., the spectral component with the largest sound power) as a function of blowing speed u. For all three models we see an abrupt change in the dominant mode, with regime change occurring at approximately $u=38 m/s$ for the sopranino-style model, 28 m/s for the bass-style model, and 37 m/s for the modified bass-style model. This figure shows two important results that were already evident from the spectra of the three different models in Fig. 3 and from the flow images in Fig. 4. First, comparing the results for the sopranino-style and bass-style models shows that the bass-style model does indeed have a much lower threshold for regime change. Second, the modified bass-style design shifts the threshold back up to a value very close to that found in the sopranino-style model. We have thus succeeded in changing the design in a way that increases the threshold for regime change in our modified model for a bass recorder.

We should also emphasize that this increase in the threshold is not simply due to the increase in volume added by the trench beneath the labium. In studies of many other designs we found that the one considered here gave a higher regime change threshold than other designs with both smaller and larger added volumes. Our results (which will be presented elsewhere¹¹) suggest that the effect on the vortex beneath the labium is largest when the added volume has dimensions that reinforce an approximately symmetric and circular vortex flow. It would be interesting to devise some measure of the vortex “strength” that can be used to quantify the apparent connection of the vortex to the regime change behavior. Unfortunately, we have not yet been able to devise such a measure.

Another feature evident in Fig. 5 also deserves mention. The frequencies for the sopranino-style recorder model below and above the regime change threshold are slightly lower than the corresponding frequencies for the bass-style and modified bass-style models. This is because the resonator cross section for the sopranino-style model is larger resulting in smaller end corrections. The same differences in f₁ and f₂ with the same magnitudes were found in the experiments.¹¹ 

All of the results presented above were derived from modeling studies of the three different recorder designs in Fig. 1. To validate and complement the modeling, we have also carried out experimental studies of real recorders with these designs fabricated in our lab using 3D printing. The main focus of these experiments has been to compare with the modeling predictions and especially the results in Fig. 5.

The recorder designs in Fig. 1 were input to a CAD program¹² to produce instructions for a MakerBot Replicator+ printer.¹³ This produced recorders composed of the plastic PLA; an example is shown in Fig. 6.

Some trial and error was required to find the best orientation for depositing the PLA so as to obtain smooth sides (especially inside the resonator) and straight, well defined edges and corners. It was also important to avoid sagging, especially for the sharp edge of the labium, and to obtain a channel cross section (in the y–z plane) that is rectangular.¹¹ We found that the best results were obtained when the “growth axis” in the printing process was parallel to the long axis of the resonator, with the channel deposited first (i.e., with the final layer being at the open end of the resonator). Notably, this orientation of the instrument during deposition gave the best and most consistent definition of the labium edge. Further details on these aspects of the fabrication will be given elsewhere.¹¹ The nominal resolution of the printer was 0.1 mm, and we will see that this was an important factor in our analysis.

The experimental measurements were quite straightforward. A gas cylinder was connected through several metering valves to provide a controlled constant flow of $N2$ gas through a recorder.¹⁴ A pitot tube flow sensor¹⁵ placed just before the recorder measured the pressure at the inlet to the channel. The cross section and length of the channel were such that the pressure drop across the rest of the instrument was negligible compared to that across the channel. The sound was recorded using the microphone in a personal computer; this was satisfactory given the frequencies of interest and the fact that we did not attempt to measure the absolute sound pressure levels.

A key quantity needed for our analysis was the flow speed in the channel, which was deduced from the measured pressure drop across the channel, $Δp$⁠. The flow through the channel was expected to be laminar and follow Poiseuille's law, which leads to¹⁶ 

where ρ is the density, L_c is the length of the channel, ν the kinematic viscosity, and h_c the height of the channel. The result in Eq. (6) is expressed in terms of u, the flow speed along the x direction on the central axis of the channel. This is how the flow speed was specified in the modeling calculations presented earlier in this paper (and in previous work⁸) The uncertainty in the value of u deduced from Eq. (6) will turn out to be dominated by the uncertainty in the channel height h_c. As mentioned above, the dimensional precision specified for the MakerBot Replicator+ printer is $0.1 mm$⁠. We will see in Sec. VI that while the uniformity and accuracy limits on h_c are actually somewhat better than $0.1 mm$⁠, but will be the major source of uncertainty in our analysis of the experimental results.

We report results for two different recorders of each of the three types described in Fig. 1. Figure 7(a) shows experimental results for two sopranino-style recorders, where we plot the frequency of the dominant spectral component as a function of blowing speed. These two sopranino-style instruments were intended to be identical, but it is clear that the results are not exactly the same. The values of the frequencies at low blowing speeds are the same, as are the frequencies at high values of u, above the regime change threshold. However, the values of u at which regime change occurs are different. We believe that this difference is due to slight differences in the channel height. As explained in connection with Eq. (6), an error in the value of h_c for a given recorder will lead to a corresponding error in the value of u as derived from the measured pressure across the channel. A difference in the channel heights of the two recorders of only 5% is sufficient to cause the apparent difference in the regime change threshold observed for these two recorders. Since the channel height is $hc≈1.0 mm$⁠, this corresponds to a difference of only $0.05 mm=50 μm$⁠, which is smaller than the nominal resolution of the 3D printer used to make the recorders.

Given the very important role that the value of h_c plays in our analysis, we used two different methods to independently measure the actual channel heights for each recorder. In one method we used a feeler gauge of the kind used to measure a narrow gap spacing for, e.g., a spark plug found in an old model automobile. The precision obtained in this measurement was estimated to be approximately $±0.07 mm$⁠. This measurement had the advantage that it gave some information on the value of h_c within the channel (i.e., along the y direction in Fig. 1). However, it had the disadvantage that it was hard to judge the magnitude of variations of h_c laterally across the channel (i.e., along the z direction in Fig. 1). The second method used to measure the channel cross section was to use direct observation with optical microscopy, which had somewhat better resolution than could be obtained with the feeler gauge. This optical method did not allow any measure of the channel height within the channel, but did reveal that h_c was quite uniform along the lateral (z) direction except for some small rounding at the sides of the channel at small and large z. This rounding was accounted for in estimating an average value of h_c, having only a 2% effect on the final estimate for the value of h_c used in Eq. (6). The two methods for estimating h_c gave consistent results. As a final observation, we note that we have no direct information on the uniformity of the channel height inside the channel, so slight variations of h_c inside the channel are certainly possible. In the analysis that follows we use the values from the optical method which had uncertainties in h_c of approximately $±0.05 mm$⁠. The value of h_c obtained in this way for each individual instrument was used to calculate the value of u for each instrument in Fig. 7(a), and for the bass-style and modified bass-style instruments considered in Figs. 7(b) and 7(c).

The horizontal error bar in Fig. 7(a) shows how the uncertainty in h_c contributes to the uncertainty in the blowing velocity. We see that, to within this estimated uncertainty, the results for the two physical recorders sopranino-style instruments agree with each other. These experimental results are also in agreement with the regime change threshold found from the simulations, which are indicated by the vertical arrow in the figure.

Figures 7(b) and 7(c) show experimental regime change results for the bass-style and modified bass-style recorders, respectively. The results for each pair of nominally identical recorders agree to within the estimated uncertainties with each other and also with the threshold values predicted by the modeling for the respective instruments. Just as importantly, the trends in the experimental regime change thresholds agree with the modeling; the threshold for the bass-style recorder is much lower than for the sopranino-style recorder, while the threshold for the modified bass-style is essentially as large as for the sopranino-style instrument.

In this study the Navier–Stokes equations were used to explore some rather detailed and specific behavior reported regarding regime change in recorders. We have found that simulation results for two recorder models exhibit regime change behavior similar to that found in real recorders. One of our recorder models, which we have labeled the “bass-style recorder” exhibits regime change at a much lower blowing speed than our “sopranino-style” model, just as found for real bass and sopranino recorders. This lower threshold for regime change can limit the range of blowing speeds, and hence limit the dynamic range available with a bass recorder. We then showed, again with modeling, that a simple change in the geometry of our recorder models can undo this change found in the regime change threshold of our bass-style recorder model. And last, in a set of experiments on real recorders we have verified the correctness of the modeling predictions.

We believe that the results of this project are of interest for several reasons. First, our design for the modified bass-style recorder could be used to increase the regime change thresholds for real bass and contrabass recorders. Our results indicate that this would extend the range of blowing speeds and hence also dynamic range available for the lowest notes of these instruments, suggesting a new recorder design that may be worth pursuing by instrument makers. Second, our results also suggest a way for instrument makers to tailor the regime change thresholds for all recorders. Third, the good quantitative agreement between our modeling and experimental results shows that Navier–Stokes based simulations can be used reliably to develop new instrument designs of value to instrument makers.

We thank Tom Rossing and Judy Cottingham for interesting discussions that inspired this project. We also thank Jeff Estep for essential guidance in working with the 3D printer, and Chris Easley for performing the microscopy measurements of the channel cross-sections. This work was supported by NSF Grants Nos. PHY-1513273 and PHY-1806231.