Using coordinate transformation of Navier–Stokes equations to solve flow in multiple helical geometries

• Published in: Journal of computational and applied mathematics Vol. 234; no. 7; pp. 2069 - 2079
• Main Authors: Cookson, A.N., Doorly, D.J., Sherwin, S.J.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 01.08.2010; Elsevier
• Subjects: Approximations and expansions; Arterio-venous shunt; Bypass graft; Bypasses; Coordinate mapping; Coordinate transformations; Exact sciences and technology; Fourier analysis; Global analysis, analysis on manifolds; Grafts; Helical; Helical pipe; Mathematical analysis; Mathematical models; Mathematics; Mixing; Navier-Stokes equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Partial differential equations; Sciences and techniques of general use; Shunts; Spectral/hp; Spline; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Arterio-venous shunt; Mixing; Bypass graft; Coordinate mapping; Helical pipe; Spectral/hp; Spline; Grid pattern; Decomposition method; Fourier analysis; Spline approximation; Numerical approximation; Geometry; Numerical analysis; Navier Stokes equation; Applied mathematics
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2009.08.065

Recent research on small amplitude helical pipes for use as bypass grafts and arterio-venous shunts, suggests that mixing may help prevent occlusion by thrombosis. It is proposed here that joining together two helical geometries, of different helical radii, will enhance mixing, with only a small increase in pressure loss. To determine the velocity field, a coordinate transformation of the Navier–Stokes equations is used, which is then solved using a 2-D high-order mesh combined with a Fourier decomposition in the periodic direction. The results show that the velocity fields in each component geometry differ strongly from the corresponding solution for a single helical geometry. The results suggest that, although the mixing behaviour will be weaker than an idealised prediction indicates, it will be improved from that generated in a single helical geometry.