Champneys, Alan

Gambaruto, Alberto

Webster, Mikaela

English

This work involves applying director theory to the modelling of fluid flow in straight and curvedpipes. Director theory assumes that the velocity of the fluid can be approximated by a summation of weighting functions multiplied by vectors that are dependent on the coordinate following the centre-line of the pipe. The aim is to investigate whether this could be a useful approach for modelling bloodflow in the human cardiovascular system. The approach allows for a more realistic geometric modelthan classical 1D approaches, but is computationally cheaper than full 3D simulations

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                          Webster, M., Gambaruto, A., & Champneys, A. (2020). ApplyingDirector Theory to the Modelling of Fluid Flow in Straight and CurvedPipes. In 6th International Conference on Computational andMathematical Biomedical Engineering - CMBE2019 (pp. 185 - 188).(CMBE Proceedings). CMBE Proceedings.https://www.compbiomed.net/2019/cmbe-proceedings.htmPeer reviewed versionLink to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia CMBE Proceedings at https://www.compbiomed.net/2019/cmbe-proceedings.htm . Please refer to anyapplicable terms of use of the publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/6th International Conference on Computational and Mathematical Biomedical Engineering - CMBE201910–12 June 2019, JapanP. Nithiarasu, M. Ohta, M. Oshima (Eds.)APPLYING DIRECTOR THEORY TO THE MODELLING OF FLUIDFLOW IN STRAIGHT AND CURVED PIPESMikaela J. Webster1, Alberto M. Gambaruto1, and Alan R. Champneys21Department of Mechanical Engineering, University of Bristol, Senate House, Tyndall Avenue, Bristol BS81TH, UK {m.webster,alberto.gambaruto}@bristol.ac.uk2Department of Engineering Maths, University of Bristol, Senate House, Tyndall Avenue, Bristol BS8 1TH,UK a.r.champneys@bristol.ac.ukSUMMARYThis work involves applying director theory to the modelling of fluid flow in straight and curvedpipes. Director theory assumes that the velocity of the fluid can be approximated by a summation ofweighting functions multiplied by vectors that are dependent on the coordinate following the centre-line of the pipe. The aim is to investigate whether this could be a useful approach for modelling bloodflow in the human cardiovascular system. The approach allows for a more realistic geometric modelthan classical 1D approaches, but is computationally cheaper than full 3D simulations.Key words: director theory, fluid, modelling, cardiovascular1 INTRODUCTIONThis work is about applying director theory (also known as Cosserat theory) to the modelling of fluidflow in pipes. The motivation behind this project is to explore if director theory could help the mod-elling of blood flow in the human cardiovascular system become a viable clinical tool. While it ispossible to reconstruct 3D computational models of individual patients using non-invasive medicalimaging techniques, only a section of this can be used for 3D CFD simulations, otherwise the compu-tational effort becomes too great. This can be coupled with 1D modelling for the arterial branching.The aim is to find a good balance between computational cost and accuracy. The director theoryapproach is computationally cheaper than full 3D modelling but retains more information about thegeometry of the pipes than classical 1D models.2 METHODOLOGYThe idea behind director theory is to assume that the velocity of the fluid flow can be approximatedby a series expansion of shape functions, which are generally polynomials of the cross-sectionalcoordinates, multiplied by vectors (called director velocities) that depend only on the coordinatefollowing the centreline of the pipe and on time, as shown by Eq. (1). Integrated versions of theconservation of momentum and ‘director momentum’ equations are imposed.v(ζ1, ζ2, ζ3) =K∑N=1λiN (ζ1, ζ2)wiN (ζ3, t) (1)There are a number of advantages to the director theory approach. The system of equations is closed ateach order, meaning that no assumptions need to be made about the form of the nonlinear and viscousterms. It is a hierarchical theory, so the accuracy can be improved, at the cost of the simplicity ofthe equations. The theory allows for the description of curvature and torsion, as well as non-circularcross-sections. This results in a more accurate solution of the flow field as compared to classical 1Dmodels which are effectively straight rods. Preliminary discussion and comparison of director theoryto classical 1D models are outlined in Robertson and Sequeira [1].We began following the work of Caulk and Naghdi [2], taking a director approach to modelling fluidflow in a straight cylindrical pipe, where the radius of the pipe could vary along the coaxial direc-tion. Satisfying boundary, incompressibility and integrated (over the cross-section) versions of themomentum equations, leads to a system of six partial differential equations dependent on the coaxialcoordinate and time. These equations can be simplified for particular geometries or conditions, whichin some cases leaves a system which can be solved analytically.Then we began applying the theory to curved pipes, starting with a toroidally curved pipe, initiallyfollowing the work of Green et al [3], later diverging from their approach in how the appropriateequations of motions are derived. We set up a local coordinate system that follows the curve of thepipe. The shape functions are assumed to be of polynomial form. To determine their exact form, thevelocity expansion, for a chosen order, is substituted into incompressibility and boundary conditionequations. The shape functions for N = 1 to N = 6 are shown in Fig. 2. As can be seen, cross-sectional flow does not enter until the sixth order. Once the shape functions have been determined,the director velocities are determined through the satisfaction of the integrated momentum equations.We employ Powell’s hybrid method to find solutions to these equations.Figure 1: The shape functions for order N = 1 to N = 6 in each of the ζi directions.3 RESULTS AND CONCLUSIONSIn the most simple case of a straight cylindrical pipe of constant cross-section, following the workof Caulk and Naghdi, we were able to recover the Poiseuille solution through the director theoryapproach. We were also able to recover a decaying swirling solution in the straight pipe as seen inFig. 2. This shows the advantage of the director theory approach over classical 1D modelling, whereit is not possible to capture flow in the cross-section.Figure 2: Decaying swirling solution in a straight pipe.In the toroidally curved pipe, the flow field from our solutions, which can be seen in Fig. 3, matchesthe flow field from a 3D simulation created using STAR-CCM+. This is with a curvature ratio of 0.01and a Reynolds number of approximately 2.Figure 3: Solution for a toroidally curved pipe of curvature ratio 0.01.Figures 4 - 6 show the contours of the velocity in the ζ3, ζ1 and ζ2 directions respectively in a cross-section of the toroidal pipe. The left images are from a 3D simulation created using STAR-CCM+.The right images are the solutions obtained through the director theory approach.Figure 4: Coaxial velocity contours in the cross-section of the pipe. Left: STAR-CCM+ simulation. Right:director theory approach.Figure 5: Vertical velocity contours in the cross-section of the pipe. Left: STAR-CCM+ simulation. Right:director theory approach.Figure 6: Radial velocity contours in the cross-section of the pipe. Left: STAR-CCM+ simulation. Right:director theory approach.This provides a promising start to applying this approach to curved pipes. We can now begin to lookat increasing the complexity of the curvature and including torsion into this theory, with the hope ofbuilding up to realistic geometries for blood vessels.4 ACKNOWLEDGEMENTSThe author gratefully acknowledges support from EPSRC Dortoral Training Partnership, with schol-arship reference 1823818.REFERENCES[1] Robertson AM, Sequeria A. A director theory approach for modeling blood flow in the arterialsystem: An alternative to classical 1D models. Mathematical Models and Methods in AppliedSciences. Vol 15, No.6 871-906, 2005[2] Caulk DA, Naghdi PM. Axisymmetric motion of a viscous fluid inside a slender surface ofrevolution. Journal of Applied Mechanics. Vol 54, 190-196, 1993.[3] Green AE, Naghdi PM, Stallard MJ. A direct theory of viscous flow in pipes ii. Flow of in-compressible viscous fluid in curved pipes. Philosophical Transactions of the Royal Society ofLondon A: Mathematical, Physical and Engineering Sciences. 1993.

Applying Director Theory to the Modelling of Fluid Flow in Straight and Curved Pipes

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Applying Director Theory to the Modelling of Fluid Flow in Straight and Curved Pipes

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