Floquet stability analysis of pulsatile flow in toroidal pipes

The linear temporal stability of the fully developed pulsatile flow in a torus with high curvature is investigated using Floquet theory. The baseflow is computed via a Newton-Raphson iteration in frequency space to obtain basic states at supercritical Reynolds numbers in the steady case for two curvatures, $\delta =0.1$ and 0.3, exhibiting structurally different linear instabilities for the steady flow. The addition of a pulsatile component is found to be overall stabilizing over a wide range of pulsation amplitudes, in particular for high values of curvature. The pulsatile flows are found to be at most transiently stable with large intracyclic growth rate variations even at small pulsation amplitudes. While these growth rates are likely insufficient to trigger an abrupt transition at the parameters in this work, the trends indicate that this is indeed likely for higher pulsation amplitudes, similar to pulsatile flow in straight pipes. At the edge of the considered parameter range, subharmonic eigenvalue orbits in the local spectrum of the time-periodic operator, recently found in pulsating channel flow, have been confirmed also for pulsatile flow in toroidal pipes, underlining the generality of this phenomenon.

Figure 1

Computational meshes for the pipe cross-section for each considered curvature. $n_r$ is the number of radial Chebyshev grid points, and $n_{\theta }$ is the number of equispaced azimuthal grid points. Note that, due to the assumed symmetry, the computations are performed on half the grid.

Figure 2

Steady baseflows, dominant eigenvector, and spectra at $\text{Re}=1700$ at $\delta =0.1$ (a)–(c) and $\delta =0.3$ (d)–(f), with the real part of the eigenvector corresponding to the most unstable eigenvalue for $\alpha =1.7$ and 2.1, respectively. The left half of each flow field shows $u_s$ and the right half shows the in-plane velocity $u_p$. Left: Steady baseflow. The left half of the figure is overlaid with contours of the stream function. Center: Leading eigenvector. Right: Spectrum of the linear operator with symmetric (sym) and antisymmetric (asym) modes, red and blue, respectively. The line indicates the smooth variation of the dominant eigenvalues for variations of $\alpha$ in the range of roughly $\pm 20\%$. The circles are realizable values ($\alpha /\delta \in \mathbb{Z}$).

Figure 3

Parameter map for the pulsating baseflow for $\delta =0.1$ (left) and $\delta =0.3$ (right). The circles correspond to computed cases. Open symbols indicate that the truncated baseflow solution is not accurate to the full tolerance of the Newton solver (see Appendix pp2-s1 for details). The red crosses indicate where the Newton solver diverged (no attempts at higher $Q$ were made).

Figure 4

Examples of the baseflow profiles at $\text{Re}=1700$, $\delta =0.1$, and $Q=0.01$ at three representative Womersley numbers. (a) $\text{Wo}=5$. (b) $\text{Wo}=25$. (c) $\text{Wo}=60$. From left to right, the steady component ($n=0$) and the real (top) and imaginary (bottom) parts of the unsteady components $n\in [1,2,3]$ in order. The left half of each figure shows $u_s$ and the right half shows the in-plane velocity $u_p$. For the steady component, the left half of the figure is overlaid with contours of the stream function. Note the different color bars for each component.

Figure 5

Variation of the initial curvature ($Q\to 0$) with respect to the pulsating amplitude of the real temporal growth rate computed using least squares from data for $Q\le 0.003$. (a) Overview and full range of data. (b) Closeup of the values close to zero [box in (a)].

Figure 6

Required number of retained Fourier modes $N_f$ for convergence of the leading Floquet eigenvalue as a function of $Q$ and $\text{Wo}$. The area shaded in red indicates where the baseflow is not computed, and the red crosses indicate cases that cannot be converged with the respective maximum $N_f$.

Figure 7

Variation of the growth rate of the leading Floquet eigenvalue as a function of $\text{Wo}$ and $Q$ at $\text{Re}=1700$ for $\delta =0.1$ (top row) and $\delta =0.3$ (bottom row). In the frequency range where strong variations occur, the amplitude range is traversed in smaller steps. A variation in the line weight is used for improved readability. Left: Real part of the Floquet eigenvalues for $Q\ge 0.005$ compared to steady growth rate (${\omega }_0$, red cross). The full (dashed) lines and filled (open) symbols indicate that the effective accuracy of the baseflow is ${10}^{-9}$ ($\le {10}^{-6}$); see Appendix pp2-s1 for details. Right: A closeup of the area in the black box. The line indicates the extrapolation ${\omega }_q$.

Figure 8

Difference between the exact Floquet eigenvalue $\omega$ and the quadratic extrapolation ${\omega }_q$ relative to the steady eigenvalue. Note that only $Q\ge 0.005$ is shown since the data for smaller $Q$ were used to compute the quadratic dependence.

Figure 9

Evolution of the eigenvalue orbit of the dominant eigenvalue ${\lambda }_1(t,Q)$ as a function of $t\in [0,T]$ and $Q$ for $\text{Re}=1700$, $\delta =0.1$, $\text{Wo}=35$. The black line indicates the variation of the leading eigenvalue with $Q$ at $t=T$ from which the tracking is initialized. The crosses correspond to the symmetric (s) and antisymmetric (a) part of the spectrum ${\mathrm{\Lambda }}_0$ of the corresponding steady problem for reference. The dominant eigenvalue moves along the orbit in the anticlockwise direction during the period.

Figure 10

Intracyclic variation of the dominant Floquet eigenfunction compared to the unstable mode in the steady case ($Q=0$) for $\delta =0.1$ (left) and $\delta =0.3$ (right) for intermediate Womersley numbers. Top row: Instantaneous growth rate $\gamma \left(t\right)$ over the pulsation period $T$. Symbols = computed, lines = interpolated. Black line: Baseflow pulsation (not to scale). Center row: Integrated perturbation energy over time. Bottom row: Absolute value of the maximum (full) and minimum (dashed) variation of the instantaneous intracyclic energy growth rate $\mathrm{\Delta }\gamma =\gamma -2\text{Re}\left({\omega }_0\right)$ as a function of $Q$. The vertical dashed red line indicates the parameter combinations for the data in (a)–(d).

Figure 11

Growth rate of the dominant Floquet eigenfunction over the pulsation period and the corresponding spatial structure at instants marked by dots. The contour lines on the cross-sections are isolines of the instantaneous streamwise baseflow velocity. (a) Instantaneous growth rate. (b) Symmetric perturbation fields corresponding to the blue curve. $\text{Re}=1700$, $\delta =0.1$, $\text{Wo}=25$, $Q=0.05$. (c) Antisymmetric perturbation fields corresponding to the red curve. $\text{Re}=1700$, $\delta =0.3$, $\text{Wo}=40$, $Q=0.01$.

Figure 12

Spectra for the steady flow in the torus at different azimuthal resolutions. The red (blue) symbols correspond to symmetric (antisymmetric) modes.

Figure 13

Energy norm of the pointwise difference of the computed eigenmodes at each resolution with respect to the solution on the reference grid for each Fourier component at different radial (left) and azimuthal (right) resolutions. In each plot, the kinetic energy distribution of the reference solution is shown in black for comparison, and the resolution marked by a star is the resolution chosen for the study. Top: $\delta =0.1$. Bottom: $\delta =0.3$.

Figure 14

Variation of the relative flow rate $Q^{\left(n\right)}/Q$ of the higher harmonics ($n\in [2,3]$) at selected pulsation amplitudes (where available). Both plots use the same symbols.

Figure 15

The least stable part of the Floquet spectrum for symmetric modes computed with $N_f=11$ Fourier components for $\text{Re}=1700$, $\delta =0.1$, $\text{Wo}=25$, $Q=0.02$, and $\alpha =1.7$ (circles) compared to the corresponding symmetric steady spectrum (red crosses). (a) Floquet spectrum (circles) with the converged eigenvalues marked with filled symbols. (b) Physically relevant spectrum showing only converged eigenvalues, colored by their growth rate. (c) Temporal Fourier components of the kinetic energy of each converged eigenpair normalized by their respective sum. The color code is the same as in (b), and the spectrum of the dominant eigenvector is emphasized with a thick line. The dashed line indicates the chosen convergence threshold.

Figure 16

Difference in the growth rates computed using a reduced-order model ${\omega }_i$ ($i=1$: full lines, $i=2$: dashed lines) compared to the accurate data $\omega$. For ${\omega }_1$, the Floquet problem was solved retaining only one unsteady baseflow Fourier component from the solution computed at $K_b=3$. For ${\omega }_2$, the approximate baseflow solution computed at $K_b=1$ was used. The thin dashed horizontal line indicates the threshold for the eigenvalue computation.