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Optimal inflow boundary condition perturbations in steady stenotic flow

  • X. Mao (a1), H. M. Blackburn (a1) and S. J. Sherwin (a2)

We determine optimal inflow boundary perturbations to steady flow through a straight inflexible tube with a smooth axisymmetric stenosis at a bulk-flow Reynolds number , for which the flow is asymptotically stable. The perturbations computed produce an optimal gain, i.e. kinetic energy in the domain at a given time horizon normalized by a measure of time-integrated energy on the inflow boundary segment. We demonstrate that similarly to the optimal initial condition problem, the gain can be interpreted as the leading singular value of the forward linearized operator that evolves the boundary conditions to the final state at a fixed time. In this investigation we restrict our attention to problems where the temporal profile of the perturbations examined is a product of a Gaussian bell and a sinusoid, whose frequency is selected to excite axial wavelengths similar to those of the optimal initial perturbations in the same geometry. Comparison of the final state induced by the optimal boundary perturbation with that induced by the optimal initial condition demonstrates a close agreement for the selected problem. Previous works dealing with optimal boundary perturbation considered a prescribed spatial structure and computed an optimal temporal variation of a wall-normal velocity component, whereas in this paper we consider the problem of a prescribed temporal structure and compute the optimal spatial variation of velocity boundary conditions over a one-dimensional inflow boundary segment. The methodology is capable of optimizing boundary perturbations in general non-parallel two- and three-dimensional flows.

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1. D. Barkley , H. M. Blackburn & S. J. Sherwin 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 14371458.

4. H. M. Blackburn & S. J. Sherwin 2004 Formulation of a Galerkin spectral element–Fourier method for three-dimensional incompressible flows in cylindrical geometries. J. Comput. Phys. 197 (2), 759778.

7. C. D. Cantwell , D. Barkley & H. M. Blackburn 2010 Transient growth analysis of flow through a sudden expansion in a circular pipe. Phys. Fluids 22, 034101–1–15.

8. J.-M. Chomaz 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.

9. P. Corbett & A. Bottaro 2001 Optimal control of nonmodal disturbance in boundary layers. Theor. Comput. Fluid Dyn. 15, 6581.

12. X. Mao , S. J. Sherwin & H. M. Blackburn 2011 Transient growth and bypass transition in stenotic flow with a physiological waveform. Theor. Comput. Fluid Dyn. 25 (1), 3142.

13. J. Nocedal & S. Wright 1999 Numerical Optimization. Springer.

14. P. J. Schmid 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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