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Convective instability and transient growth in steady and pulsatile stenotic flows

Published online by Cambridge University Press:  30 June 2008

H. M. BLACKBURN ,
S. J. SHERWIN  and
D. BARKLEY
H. M. BLACKBURN
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, Victoria 3800, Australia
S. J. SHERWIN
Affiliation:
Department of Aeronautics, Imperial College London, SW7 2AZ, UK
D. BARKLEY
Affiliation:
Mathematics Institute and Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, UK
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Abstract

We show that suitable initial disturbances to steady or long-period pulsatile flows in a straight tube with an axisymmetric 75%-occlusion stenosis can produce very large transient energy growths. The global optimal disturbances to an initially axisymmetric state found by linear analyses are three-dimensional wave packets that produce localized sinuous convective instability in extended shear layers. In pulsatile flow, initial conditions that trigger the largest disturbances are either initiated at, or advect to, the separating shear layer at the stenosis in phase with peak systolic flow. Movies are available with the online version of the paper.

JFM classification

Biological Fluid Dynamics: Biomedical flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 607 , 25 July 2008 , pp. 267 - 277
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Åkervik, E., Hœpffner, J., Ehrenstein, U. & Henningson, D. S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314.CrossRefGoogle Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids (In press).CrossRefGoogle Scholar
Blackburn, H. M. & Sherwin, S. J. 2004 Formulation of a Galerkin spectral element–Fourier method for three-dimensional incompressible flows in cylindrical geometries. J. Comput. Phys. 197 (2), 759–778.CrossRefGoogle Scholar
Blackburn, H. M. & Sherwin, S. J. 2007 Instability modes and transition of pulsatile stenotic flow: pulse-period dependence. J. Fluid Mech. 573, 57–88.CrossRefGoogle Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Cossu, C. & Chomaz, J. M. 1997 Global measures of local convective instabilities. Phys. Rev. Lett. 78, 4387–4390.CrossRefGoogle Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218.CrossRefGoogle Scholar
Khalifa, A. M. A. & Giddens, D. P. 1981 Characterization and evolution of poststenotic disturbances. J. Biomech. 14 (5), 279–296.CrossRefGoogle ScholarPubMed
Mallinger, F. & Drikakis, D. 2002 Instability in three-dimensional, unsteady, stenotic flows. Intl J. Heat Fluid Flow 23, 657–663.CrossRefGoogle Scholar
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.CrossRefGoogle Scholar
Ojha, M., Cobbold, R. S. C., Johnston, K. W. & Hummel, R. L. 1989 Pulsatile flow through constricted tubes: an experimental investigation using photochromic tracer methods. J. Fluid Mech. 203, 173–197.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sexl, T. 1930 Über den von E. G. Richardson entdeckten ‘annulareffekt’. Z. Phys. 61, 349–362.CrossRefGoogle Scholar
Sherwin, S. J. & Blackburn, H. M. 2005 Three-dimensional instabilities and transition of steady and pulsatile flows in an axisymmetric stenotic tube. J. Fluid Mech. 533, 297–327.CrossRefGoogle Scholar
Varghese, S. S., Frankel, S. H. & Fischer, P. F. 2007 Direct numerical simulation of stenotic flows. Part 2. Pulsatile flow. J. Fluid Mech. 582, 281–315.CrossRefGoogle Scholar

Blackburn et al. supplementary material

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.

Download Blackburn et al. supplementary material(Video)
Video 1.8 MB

Blackburn et al. supplementary movie

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.

Download Blackburn et al. supplementary movie(Video)
Video 4.8 MB

Blackburn et al. supplementary movie

Movie 2. Pulsatile base flow and optimal transient disturbance for Re=400 and Ured=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.

Download Blackburn et al. supplementary movie(Video)
Video 1.2 MB

Blackburn et al. supplementary movie

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.

Download Blackburn et al. supplementary movie(Video)
Video 4.6 MB

Blackburn et al. supplementary movie

Movie 3. Pulsatile base flow and optimal transient disturbance for Re=400 and Ured=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.

Download Blackburn et al. supplementary movie(Video)
Video 2 MB

Blackburn et al. supplementary movie

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.

Download Blackburn et al. supplementary movie(Video)
Video 7.2 MB