Three-dimensional steady and oscillatory flow in a double bifurcation airway model

We investigate the steady expiratory and the oscillatory flow in a planar double bifurcation model with geometric proportions relevant to the respiratory human airways. Expanding on a previous study focused on steady inspiration [Jalal et al., Exp. Fluids 57, 148 (2016)], we use magnetic resonance velocimetry to characterize the three-dimensional velocity field for a range of Reynolds (Re) and Womersley (Wo) numbers. During expiration the velocity profiles are flatter than in inspiration, due to stronger secondary motions. The latter are characterized by counter-rotating streamwise vortices induced by curvature at the branch junctions. With increasing Re, the vortices gain strength, and for $\mathrm{Re}\ge 1000$ they propagate through successive branching generations, profoundly changing the secondary flow pattern. Under oscillatory conditions, as long as the ventilation frequency is in the normal respiration range, the flow topology for both inhalation and exhalation phases is similar to the corresponding steady cases over most of the breathing cycle. On the other hand, in the high-frequency ventilation regime ($\mathrm{Wo}=12$), the acceleration part of both inhalation and exhalation phases show signature features of oscillatory flows, with high-momentum regions located close to the walls. The phenomenon of counterflow is found to be prominent at $\mathrm{Wo}\ge 6$, with reverse flow pockets marking the velocity field especially during the inhalation-exhalation inversion. With increasing oscillation frequency, the secondary motions become more intense during the inhalation phase but are attenuated during the exhalation phase of the cycle. The cycle-averaged drift is found to be significant at low Wo but decreases with increasing ventilation frequency, suggesting that steady streaming is not the main transport mechanism during high-frequency ventilation.

Figure 1

(a) 3D rendering of the double bifurcation, with cross sections and streamwise abscissas indicated, A-A′, B-B′, C-C′, and D-D′ are at $S/d_0=0$, 3.75, 5.3, and 5.3, respectively. (b) CAD model of the bifurcation model (in yellow), with the extension tubes connected at $G_2$ (in blue), and the upper piece connected to $G_0$ (in white) fitted into the coil. (c) Azimuthally averaged velocity profile at section A-A′ during peak inhalation for $\mathrm{Wo}=1.2$ and 12 at peak $\mathrm{Re}=2000$. The velocity profiles are a somewhat flatter than the theoretical ones for the same Re and Wo due to the very short developmental length.

Figure 2

Position of the considered oscillatory cases in the flow regime diagram proposed by Jan et al. [9].

Figure 3

Measured flow rate at each phase for all Womersley numbers plotted against the theoretical waveform.

Figure 4

Contours of velocity magnitude along the bifurcation plane and streamwise velocity along cross sections A-A′ and B-B′, for steady exhalation at $\mathrm{Re}=250$ (a) and $\mathrm{Re}=2000$ (b). Secondary flow vectors are shown in the cross sections. A reference velocity vector is shown at the bottom left.

Figure 5

Streamwise velocity profiles under steady exhalation for all considered Reynolds numbers, at stations (a) B-B′ and (b) A-A′.

Figure 6

Momentum distortion parameter D [Eq. (2)] under steady exhalation for all considered Reynolds numbers. The flow direction is towards smaller values of the abscissa.

Figure 7

Isosurfaces of positive (light color) and negative (dark color) ${\mathrm{\Gamma }}_2$ [Eq. (3)] identifying vortical structures, and contour maps of ${\mathrm{\Gamma }}_2$ with secondary flow vectors at various cross sections for steady exhalation at $\mathrm{Re}=250$ (a) and $\mathrm{Re}=2000$ (b). The isosurfaces are only shown in the part of the domain boxed in the cross sections.

Figure 8

Normalized circulation Γ [Eq. (4)] of the streamwise vortices in steady exhalation at (a) $\mathrm{Re}=250$ and (b) $\mathrm{Re}=2000$. The flow direction is towards smaller values of the abscissa.

Figure 9

Contours of velocity magnitude along the bifurcation plane at $\mathrm{Wo}=1.2$ and peak $\mathrm{Re}=2000$ during the breathing cycle. The corresponding phase within the oscillatory cycle is labeled above the corresponding contour plots and by the flow rate diagram on the bottom right.

Figure 10

As Fig. 9, but for the case $\mathrm{Wo}=12$.

Figure 11

Velocity isosurfaces for successive phases of the inhalation and exhalation cycle at $\mathrm{Wo}=12$. Light and dark colors indicate flow moving in the inspiratory and expiratory direction, respectively.

Figure 12

Velocity isosurfaces at (a) exhalation and inhalation inversion and (b) inhalation and exhalation inversion for the case $\mathrm{Wo}=6$. Light and dark colors indicate flow moving in inspiratory and expiratory direction, respectively.

Figure 13

${\mathrm{\Gamma }}_2$ isosurfaces and cross section contours with in-plane velocity vectors, identifying vortical structures $\mathrm{Wo}=1.2$ (a) and 12 (b), at peak inhalation and peak exhalation.

Figure 14

Time-varying secondary flow strength averaged along $G_1$, for all considered Wo; the steady inhalation and exhalation levels are also shown for comparison.

Figure 15

Isosurfaces and cross section contours of cycle-averaged drift velocity for all considered oscillatory cases $\mathrm{Wo}=1.2$ (a), $\mathrm{Wo}=3$ (b), $\mathrm{Wo}=6$ (c), $\mathrm{Wo}=12$ (d). Light and dark colors indicate inspiratory and expiratory drift. The purple lines indicate zero drift.

Figure 16

Streamwise velocity profiles along the bifurcation plane at B-B′ for (a) $\mathrm{Wo}=1.2$ and (b) $\mathrm{Wo}=12$, for the different phases in the cycle.