Chapter 6 Conduits and channels

Tubes and pipes carrying fluid often deform away from their straight shape when the flow rate through them exceeds a critical value. Straight pipes can spontaneously curve or start oscillating because the straight shape is unstable. In cases, such as the whipping of a firehose, the instability is driven by the reaction force on the hose as the fluid leaves it. Such a force can easily be calculated using control volume analysis, such as the kind you learned in IB. The possibility is not merely academic, but is a serious engineering concern, as these videos and numerous others on the internet demonstrate (see video 1, video 2).

The hose is also unstable to perturbation away from an exit. To understand the principles behind this instability, consider an infinite conduit under tension $T$ carrying an inviscid fluid of density $\rho$ at speed $U$, as shown in fig. 6.1. The cross section of the conduit is uniform along its length with area $A$, and the conduit has mass $\mu$ per unit length. The unperturbed shape of the conduit is straight, say along the $x$ direction, and the fluid flows parallel to the local centreline of the conduit. The fluid dynamics is identical to the setup described in section 3.2.1, whereas the structural dynamics derives from section 2.4.

The steady centerline of the conduit is $𝑿(s,t)=s{\widehat{𝒆}}_x$ and let us perturb it perpendicular to its length as$𝑿(s,t)=s{\widehat{𝒆}}_x+y(s,t){\widehat{𝒆}}_y$, where $y(s,t)$ is an infinitesimal transverse displacement. The equation governing the perturbation is

$\mu {\displaystyle \frac{{\partial }^{2}y}{\partial t^2}}=T{\displaystyle \frac{{\partial }^{2}y}{\partial s^2}}-\rho A{\left({\displaystyle \frac{\partial }{\partial t}}+U{\displaystyle \frac{\partial }{\partial s}}\right)}^{2}y.$

(6.1)

(Note that we have circumvented the details of the steps writing the equation governing a finite perturbation and then linearizing it for small magnitude. Also, we have dispensed with the prime notation for perturbations.)

We now seek modes in the form $y(s,t)=\widehat{y}(s)e^{i\omega t}$, where $\widehat{y}$ satisfies

$-\mu {\omega }^2\widehat{y}=T{\displaystyle \frac{{\text{d}}^2\widehat{y}}{\text{d}{s}^2}}-\rho A{\left(i\omega +U{\displaystyle \frac{\text{d}}{\text{d}s}}\right)}^2\widehat{y}⟹(T-\rho A{U}^2){\displaystyle \frac{{\text{d}}^2\widehat{y}}{\text{d}{s}^2}}-2i\omega \rho AU{\displaystyle \frac{\text{d}\widehat{y}}{\text{d}s}}+{\omega }^2(\mu +\rho A)\widehat{y}=0,$

(6.2)

This is a constant coefficient ordinary differential equation, in an infinite domain . The solution to this equation is clearly of the form $e^{rs}$ for some $r$, but the perturbation can neither grow as $s$ approaches either $\mathrm{\infty }$ or $-\mathrm{\infty }$. Therefore, the only possibility is that $r$ is purely imaginary, and the solution oscillates. It is customary to take $r=ik$, where $k$ is called the wavenumber (of course, the symbol $k$ is of no consequence, and cam be replaced by another convenient symbol). This observation turns eq. 6.2 into the dispersion relation

${\omega }^2(\mu +\rho A)+2\rho UA\omega k-(T-\rho A{U}^2)k^2=0.$

(6.3)

Observe that in the absence of the fluid ($\rho =0$) the dispersion relation reduces to that of a stretched string, $\omega =\pm k\sqrt{T/\mu }$. In the presence of the fluid but in the absence of the flow ($\rho \ne 0$ but $U=0$), the mass per unit length increases to $\mu +\rho A$. In both these cases, $\omega$ is purely real, implying that any perturbation travels as a wave which neither grows not decays. Since eq. 6.3 is a quadratic for $\omega$ with real coefficients, and the solutions are real when $U=0$, the solutions are bound to remain real for small values of $U$. Instability is possible only if $\omega$ has a negative imaginary part, i.e. when $\omega$ is complex. Complex roots for $\omega$ are possible when $U$ exceeds a threshold, which can be determined by setting the discriminant to negative as

(6.4)

Solving for the $U$ yields the threshold

$U>\sqrt{{\displaystyle \frac{T}{{\mu }^{\ast }}}}\text{where}{\displaystyle \frac{1}{{\mu }^{\ast }}}={\displaystyle \frac{1}{\mu }}+{\displaystyle \frac{1}{\rho A}}.$

(6.5)

At the critical $U$, the quadratic has two identical roots equal to

$\omega =-{\displaystyle \frac{kU}{\mu /{\mu }^{\ast }}},$

(6.6)

which defines the speed of the growing perturbation, $\omega /k=U({\mu }^{\ast }/\mu )$ at the threshold.

The physics of this instability is as follows. The fluid dynamical force on the conduit arises from the curvature of the streamlines, which arises from the curvature of the centreline. This curvature creates a pressure difference across the conduit, which exerts an unbalanced force on the conduit in a direction that increases the curvature (i.e. centrifugal). The magnitude of this force is proportional to $\rho U^2$, and it gets stronger with faster flow. When it exceeds the restoring force of tension, the perturbation spontaneously grows.

Inspired by the general model of a flow in a narrow space, such as through the vocal folds, consider the flow through a thin channel bounded by a stretched membrane, as shown in fig. 6.2. The elastic membrane is a model for a more general elastic structure. The membrane has mass $\mu$ per unit area and tension per unit length $T$, so its shape $h(x,t)$ satisfies

	$\mu {\displaystyle \frac{{\partial }^{2}h}{\partial t^2}}=T{\displaystyle \frac{{\partial }^{2}h}{\partial x^2}}+p(x,t),$			(6.7a)
where $p$ is the fluid pressure in the fluid pressure in the channel. The fluid in the channel has density $\rho$ and a fluid velocity $u(x,t)$ that is uniform across the channel height and width. The fluid conservation equations satisfy
	${\displaystyle \frac{\partial h}{\partial t}}+{\displaystyle \frac{\partial (uh)}{\partial x}}$	$=0,$		(6.7b)
	$\rho \left({\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}\right)+{\displaystyle \frac{\partial p}{\partial x}}$	$=0.$		(6.7c)

Equations 6.7b and 6.7c resemble eqs. 3.14 and 3.18, respectively, with $f=d(u,h)=0$. The purpose of this analysis is to examine the role of inertia of the fluid with the inertia and the elasticity of the surrounding structure. (A separate analysis can be carried out where the role of friction is examined in the absence of inertia.) Hence, as a deliberate choice with the intention of eliminating all extraneous physical effects, we accept $f$ and $d[u,h]$ to be zero.

Let us now interpret the result in question 6.2. Just as eq. 6.5 in section 6.1, ${\mu }^{\ast }$ is an effective mass per unit area, which combines the inertia of the membrane and the fluid in the channel. This ${\mu }^{\ast }$ depends on the wavenumber $k$, i.e. on the wavelength of the perturbation, and through it the threshold velocity $U_{\text{critical}}$ also depends on $k$. Of all the possible wavenumbers (i.e. wavelength), $k=0$ gives the lowest threshold $U_{\text{critical}}=\sqrt{T/\mu }$. Thus, we conclude that the uniform state $u=U$, $h=H$ will be impossible to maintain if $U$ exceeds the wave speed on the membrane $\sqrt{T/\mu }$. Based on this result, we suspect that for a finite channel and membrane, we expect the longest mode that fits in the domain to determine the threshold velocity. Of course, for a finite channel, boundary conditions will also play a role in determining the fate of any perturbation.

These are just two of the numerous examples possible of instability that arises when a structure interacts with a flowing fluid. The reader will benefit immensely in gaining a deeper understanding of the topic if they can formulate a modification of one of the problems in this document. For example, how could the singing of telephone wires hanging in the wind be described? Or the wild oscillations of iced electric cables in wind? How would the results of this chapter be modified in the firehose had a bending rigidity instead of being under tension? And for that matter, how could the oscillations of the vocal folds as we speak be described? Musical instruments, especially woodwinds, also have such instabilities underlyng their sound production mechanism. We leave the reader with these thoughts to ponder and with best wishes.