*2.1. Governing Equations*

We consider a viscous fluid with constant properties confined in a straight smooth rigid pipe of circular cross-section and diameter *D*. The fluid flow is driven through the pipe with a time-dependent pressure gradient, and is considered to be incompressible and governed by the Navier–Stokes equations (NSE)

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla p + \frac{1}{Re} \nabla^2 \mathbf{u} + \mathbf{F}\_d(t) + \mathbf{F}\_p(r, \theta, z, t) \text{and} \\ \nabla \cdot \mathbf{u} = 0. \tag{2}$$

Here, *u* and *<sup>p</sup>* denote the fluid velocity and pressure. The driving force *Fd*(*t*) represents a mean pressure gradient, which is adapted in a way such that the flow rate (*ub*) given in Equation (1) is maintained. The additional body force term **F***p*(*r*, θ, *z*, *t*) is used to model geometric imperfections in the pipe geometry, and thus to perturb the flow locally (see Section 2.4). Unless otherwise stated, all quantities are rendered dimensionless using the pipe diameter *D*, the statistically steady part of the bulk velocity *us* = *ub<sup>t</sup>* (see Equation (1)), and the fluid's density (ρ).
