Movie 1. Simulations based on time-integration of the linearized Navier--Stokes equations for flow of Newtonian fluid in a rigid straight tube with a smooth 75%-occlusion axisymmetric stenosis. Steady base flow and optimal transient disturbance for Re=400 and dimensionless peak-growth time-interval τ=4, azimuthal wavenumber k=1, corresponding to the case illustrated in figure 5. The animations cover 0<t<8.75, i.e. longer than the time interval at which maximum energy growth occurs for the case selected. In movie 1(a) the views are restricted to the meridional semiplane where the upper panel shows contours of vorticity in the base flow. The centre panel shows contours of axial velocity perturbation for the optimal disturbance. The lower panel also shows contours of axial perturbation in the optimal disturbance, but these are scaled such that the energy norm remains constant, in order to reveal structure at early as well as later times.

In movie 1(b) we see three-dimensional isocontours of normalized axial velocity perturbation field next to the meridional plane showing the base flow vorticity. The inset panel shows energy as a function of time. We observe that the optimal initial condition is located in the shear layer at the stenosis. The energy of this initial condition is then amplified by a factor of approximately 9×10⁴ as it propagates downstream with a maximal energy distribution centred around z=10. The wavelength of the perturbation is maintained as it propagates downstream, highlighting the convective nature of the instability. Because the timespan used here extends beyond that for maximum energy growth (in contrast to the other movies provided below), this case demonstrates that the energy growth is indeed transient: after t=4, corresponding to maximum growth, the perturbation dissipates and its energy reduces.

Movie 2. Pulsatile base flow and optimal transient disturbance for Re=400 and U_red=10, t₀=2.5, τ=8.75, azimuthal wavenumber k=1, corresponding to the case illustrated in figure 7(a–d). The animations cover 0<t<8.75, i.e. up to the time of maximum energy growth for the case selected, which has an initiation phase corresponding to peak systole in the base flow. As with movie 1(a) the upper panel in movie 2(a) shows contours of vorticity in the base flow, the centre panel shows axial velocity of the perturbation of the optimal disturbance while the lower panel also shows the axial velocity of the optimal perturbation but rescaled to maintain a constant energy norm over time.

Movie 2(b) again shows the three-dimensional normalized axial velocity perturbation isocontours next to the base flow vorticity. Owing to the length of the domain we have split the views of the first and second parts of the pipe. The inset plot in movie 2(b) shows the energy as a function of time (red line) and the pulsatile cycle (blue line). As for the case shown in movies 1(a, b) the optimal initial condition is located in the shear layer near its separation in the stenosis. The energy of the initial condition is amplified by a factor of approximately 1×10¹⁰ (while maintaining its wave-packet-type spatial form), achieving a maximum disturbance magnitude near z=23.

Movie 3. Pulsatile base flow and optimal transient disturbance for Re=400 and U_red=10, t₀=7.5, τ=13.75, azimuthal wavenumber k=1, corresponding to the case illustrated in figure 7(e–g). The animations cover 0<t<13.75, i.e. up to the time of maximum energy growth for the case selected. Similar to movies 1(a) and 2(a), in movie 3(a) the upper panel shows contours of vorticity in the base flow, the centre panel shows axial velocity of the perturbation of the optimal disturbance and the last panel also shows the axial velocity of the optimal perturbation but rescaled to maintain a constant energy norm over time.

Like movie 2(b), movie 3(b) shows the three-dimensional normalized axial velocity perturbation isocontours alongside the base flow vorticity. We recall that owing to the length of the domain we have split the views of the first and second parts of the pipe. The inset plot of movie 3(b) shows energy as a function of time in red and the pulsatile cycle in blue. In contrast to movies 2(a, b), the optimal initial condition for this value of t₀, corresponding to peak diastole in the base flow, is located upstream of the stenosis. The perturbation is then accelerated into the stenosis and this causes a shear layer perturbation similar to the conditions shown in movies 2(a, b). This perturbation is then amplified to achieve a maximum growth at a similar location to that for the case shown in movies 2(a, b), at z=23. Similar to the two other cases the spatial wavelength of the perturbation is maintained throughout its growth, indicative of a convective instability. The maximum amplification of energy in the initial condition is approximately 1×10⁸.