11.2 Fluid phenomena

PREVIOUS SECTION

In the following sections we will discuss models for the excitation of various kinds of wind instrument, and for that we will need a bit of knowledge about fluid flow. This section gives a qualitative overview of key fluid phenomena, and a side link will give some technical background about how the subject can be approached mathematically.

So far our only contact with the world of fluid dynamics has been through the very special case of linear acoustics. However, we will need to go beyond this to understand such things as how a clarinet or flute mouthpiece works. If you think about it for a moment, you already know that everyday fluid dynamics must involve nonlinear effects. Look at the small waterfall in Fig. 1. You see smooth water coming over the lip of the fall, but it turns into complicated turbulent flow at the bottom. The smooth initial flow is known as “laminar flow”.

Figure 1. A small waterfall illustrates the contrast between laminar flow (over the lip of the fall and during the descent) and turbulent flow (in the churning water at the bottom).

Figure 2 shows another example. This is a schlieren image of the plume of hot air rising from a candle flame. At the bottom the flow is laminar, but there is a rather abrupt transition to turbulent flow in the middle of the image, without anything obvious happening there to trigger the transition (nothing analogous to the waterfall edge, for example). This is a spontaneous transition to chaotic behaviour, in the sense discussed in section 8.4.

Figure 2. A schlieren image of the plume rising from a candle flame. The flow is laminar in the lower half of the image, but exhibits a spontaneous transition to turbulence higher up. Image: Gary Settles, CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0), via Wikimedia Commons.

Chaotic behaviour like this can only occur in nonlinear systems, although if you recall the example of the double pendulum from section 8.4 you will appreciate that it may only take a rather simple-looking nonlinear system to exhibit such behaviour. The origin of the main nonlinearity in the governing equation for fluid flow (known as the “Navier-Stokes equation”) is nothing more than a quadratic term, as shown in the next link, but this is enough to allow the full complexities of turbulence.

SEE MORE DETAIL

The possibility of turbulence is surely enough to tell us that we are not expecting easy mathematical solutions to the governing equations for fluid flow. It then makes sense to explore approximations of different kinds: these can simplify the mathematics under certain circumstances and allow us to make some progress with understanding the physics behind fluid behaviour.

We have already met one approximation: when we derived the linear wave equation back in section 4.1.1, we assumed that all relevant quantities like pressure and flow velocity were small in some appropriate sense, and that allowed us to ignore a lot of complicating factors. However, in order to treat sound waves we certainly had to allow our fluid (usually air) to be compressible. But in the waterfall seen in Fig. 1, the water behaves as if it were incompressible. This turns out to make a major simplification in the governing equations, and for many problems it is a useful approximation to make. The previous link gives some details.

You might think that the incompressible approximation would have nothing to do with the acoustics of wind instruments, since we are surely always talking about sound waves? But this would be misleading. Compressibility is indeed always important for the acoustic resonances of instrument tubes, and for the internal pressure waveform when an instrument is played. But when we are thinking about how a mouthpiece works, we are mainly concerned with a different aspect of fluid flow, associated with the air blown into the instrument by the player. We can usually get a rather good approximation to the behaviour by ignoring the compressibility of the air moving through the mouthpiece. The underlying reason for this is that the air-flow is very slow compared to the speed of sound (around 340 m/s), or in other words the Mach number is very small. This is the mathematical condition for compressibility effects to be unimportant.

We will make use of an important result which takes its simplest form in the case of incompressible flow. This is called Bernoulli’s principle. The mathematical details are given in the previous link, but in words Bernoulli’s principle says that if you follow a laminar air-flow along its streamlines, a region where the flow speeds up is automatically associated with the pressure going down, and vice versa. So if a jet of air is squeezed through a narrow gap, for example between the reed and the lay of a clarinet mouthpiece, the pressure will be lower there. This low pressure tends to make the reed close towards the lay. We will see in section 11.3 that this is an important ingredient of how a reed mouthpiece works.

There is another physical property of fluids we need to think about, called viscosity. Figure 3 shows a familiar sight, honey running slowly off a spoon. If you imagine doing the same thing with a spoonful of water, the behaviour would be quite different: almost all the water would fall off the spoon as soon as you tip it up. This contrasting behaviour happens because the viscosity of honey is far greater than the viscosity of water.

Figure 3. Honey running slowly off a spoon, to illustrate the concept of viscosity.

But you may have noticed that I said “almost all water would fall off the spoon”. Actually, a few drops of water continue to fall, after most of it has gone. All normal fluids, including water, have some viscosity. When you tip the spoon up, a thin layer of the water is reluctant to run off. It clings to the surface, and only runs down gradually — very much like the honey, except that with water it is only a very thin layer that shows the behaviour. The same thing happens when you wash up crockery: if you leave it to drain for a while, that gives the last thin layer of water time to drip off.

So what exactly is viscosity? It is the property of a fluid that resists shear deformation. Figure 4 shows a sketch of a layer of fluid between two metal plates. (In essence, this is a sketch of an apparatus you could use to measure viscosity.) The upper plate is forced to move to the right. This plate does not slide over the fluid, it carries it along with it: fluid in contact with a solid surface is “stuck” to it by molecular forces. So the fluid at the top of the layer moves rightwards with the plate, but the fluid at the bottom is anchored to the bottom plate. Provided the motion is slow enough, the fluid in between will show a linear velocity profile as sketched: this can be described as uniform shearing flow. Now the viscosity relates to the force necessary to move the plate, at a given speed and with a layer of given thickness and area.

Figure 4. Illustration of shear deformation of a layer of fluid between two rigid plates

So now come back to our spoonful of water. What happens when you tip the spoon? The force of gravity acts in the same way on every particle of the water, so it all “wants” to move downhill at the same speed. But the water in contact with the spoon surface is stuck to it, and cannot flow away. The result is sketched in Fig. 5, which shows a magnified view of a small region near the spoon surface. Most of the water flows down at the same velocity, but a thin layer near the surface experiences shearing motion, which is resisted by the viscosity of the water.

Figure 5. Tipping water out of a spoon: sketch, and closeup showing the formation of a very thin boundary layer against the surface of the spoon. The thin black lines and red arrows show the velocity profile.

The result is a velocity profile of the kind sketched, featuring a boundary layer: a layer very close to the solid surface in which significant shear flow occurs. The thickness of this layer is determined by a balance between the force of gravity and the viscous force resisting the shear. The lower the viscosity, the thinner the layer. It is the lower part of this boundary layer that is left behind after the washing up, to drain away slowly in your dish-drainer.

A boundary layer related to what has just been described will form on the wall of a wind instrument when it is played. The acoustic field inside the tube involves cyclical longitudinal motion of the air. Very close to the tube wall, though, the “no-slip boundary condition” applies: the air is anchored to the wall. A viscous boundary layer is the inevitable result. Since shear motion resisted by viscosity always dissipates energy, this is one of the two main mechanisms for damping of the acoustic resonances of the pipe, as mentioned in section 11.1. The boundary layer will be at its thinnest if the internal wall of the tube is very smooth, but if that surface is rough then the boundary layer is likely to be thicker, and the damping increased.

Two effects we have described, nonlinearity and viscosity, are each described by one term in the governing Navier-Stokes equation (see the previous link). Fluid dynamicists use a quantity called the Reynolds Number to quantify the relative importance of these two terms (we described it briefly back in section 10.1.1). This number captures the ratio of strengths of the nonlinear term to the viscosity term. The formula for Reynolds Number turns out to involve the typical flow speed, multiplied by the typical length-scale, and divided by the viscosity.

In a flow with very low Reynolds number, nonlinear effects can be neglected. The flow will be dominated by viscosity effects, and it will be laminar because viscosity has a stabilising effect on the instabilities leading to turbulence. Examples would be our honey running off the spoon (because the viscosity is large), or the swimming of a micro-organism in water (because the length-scale is very small). A flow with very high Reynolds Number will have the opposite behaviour: viscosity can be neglected, nonlinear effects will be strong, and the flow is likely to be turbulent. Examples would be the air flow through a jet engine (because the flow speed is very high), or the swimming of a human being in water (because this time the length-scale is much bigger than for the micro-organism, and the viscosity of water is quite small).

The effective Reynolds Number will vary with depth through a structure like a boundary layer. There will always be a viscous-dominated layer very close to the wall, but it is perfectly possible (and indeed quite common) for the outer part of a boundary layer to become turbulent. See this video for a striking demonstration of a turbulent boundary layer.

Another phenomenon that can occur is that the shape of the solid object may create the conditions for the boundary layer to separate from the wall at some position, and then give rise to a turbulent wake. Figure 6 shows an example, in a wind-tunnel image of flow past a wing cross-section. The flow below the wing remains in contact with the surface, but above the wing it separates and produces a complicated wake structure.

Figure 6. Flow visualisation from a wind tunnel, showing an airfoil wing section. The boundary layer on the upper surface separates, and gives rise to a wake structure. Image: DLR, CC-BY 3.0, CC BY 3.0 DE https://creativecommons.org/licenses/by/3.0/de/deed.en, via Wikimedia Commons

Under some circumstances this separation process varies cyclically with time, giving rise to a strikingly beautiful “Kármán vortex street”, like the example in Fig. 7. The image is taken from this Wikipedia page, where you can find more detail. In the process of shedding vortices on alternate sides, an alternating force is exerted on the solid body, in the direction perpendicular to the flow. If the object is something like a stretched string or a power cable, this force can set it into vibration. This is the underlying mechanism of the Aeolian harp, for example.

Figure 7. A Kármán vortex street, generated by flow past a cylindrical body. The boundary layer separation point oscillates, shedding vortices in alternating directions. Image: Jürgen Wagner – Self-photographed, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=39746847

Another phenomenon involving vortices is more directly relevant to musical wind instruments. If a jet of air is blown out of a slot, as in a recorder mouthpiece for example, each side of this air jet will be a shear layer: a sudden jump in air flow speed. Well, there is a famous mathematical result in fluid dynamics proving that an ideal shear layer is unstable. If the interface between the two flow speeds is initially straight, but is then perturbed a little, the perturbation will grow.

With a narrow jet, the instability usually carries the whole jet up and down — we will say more about this in section 11.6 when we discuss air-jet instruments. For now, we can see one image for a rather wide air jet, in Fig. 8. In this schlieren image, we can see that the two shear layers on either side of the jet have generated elegant vortex shapes, in an alternating pattern. The image is taken from the doctoral thesis of Sylvie Dequand [1].

Figure 8. Schlieren image of a wide air-jet entering from a slot (the end of it is visible on the left), interacting with a sharp edge in the upper right of the picture. It shows unstable vortex growth of the shear layers on either side of the jet. Image copyright Sylvie Dequand, reproduced by permission.

[1] Sylvie Dequand, “Duct aeroacoustics: from technological applications to the flute”, Doctoral dissertation, Eindhoven University of Technology (2001), https://pure.tue.nl/ws/portalfiles/portal/3429613/445111.pdf